摘要:

Small Parasitic Parameters and Chemical Oscillations BY €3. F. GRAY AND L. J. AARONS School of Chemistry University of Leeds Leeds LS2 9JT Received 23rd July 1974 In treating chemical oscillators it is common practice to use no more than two concentrations as dependent variables due to the mathematical difficulties with more than two. Other variables necessarily involved to make the system nontrivial-the "pool "chemicals-are treated as parameters which are independent of time. We investigate here under what conditions the "pool " chemicals can be treated as constants without qualitatively effecting the behaviour of the system. Mathematical methods developed by Tikhonov are used to study the effect of a small parameter on the roots of the characteristic equation.This small parameter may be chosen as the ratio of the initial concentrations of the reactants to the "pool " chemicals or for dilute solutions the reciprocal of the heat capacity. A number of interesting results are obtained in which the slow variation of the " pool " chemicals can either produce a limit cycle where there was none previously or place severe restrictions on the rate constants so as to exclude regions where interesting instabilities have been found in the two variable case. Multistability in the two variable system lends itself very well to the production of limit cycles of the relaxation type. Finally it is shown that it is possible to devise thermokinetic oscillators with very small temperature oscillations provided the energy equation is highly nonlinear.1. INTRODUCTION It has become common practise in designing models of chemical oscillators to limit the description to two dependent variables. The reasons for this are obvious being due to the formidable mathematical apparatus that exists for treating non-linear differential equations with only two variables. Several two dimensional kinetic schemes that exhibit undamped oscillations have been developed including the so-called "Brusselator ''.2* However the latter model employs a physically unrealistic termolecular step. In fact Hanusse has shown that it is impossible for any two variable kinetic scheme which has only unimolecular and bimolecular steps to exhibit undamped oscillations. The one exception to this the Lotka-Volterra ~cherne,~ is conservative and so is again physically unacceptable.It is implied in all these models that certain substances generally referred to as " pool " chemicals (such as fuel and end products) are held constant either by being in large excess or by flowing them in (out) as soon as they are used (produced). It is the purpose of this paper to investigate under what conditions the '' pool " chemicals can be treated as constant and what effect their variation may have. We will be concerned with a system of equations typically of the form where p is a small positive parameter. The solutions of these equations fall into two regions the region of slow motion and the region of rapid motion. In the first region a representative point moves comparatively slowly (ij bounded) within a and small neighbourhood (of order p) of the curve F(x;y) = 0.Outside this neighbour- hood the representative point moves in rapid jumps and in the limit as p+o the equations of rapid motion can be written 1 y = yo = constant X-= -F(x; yo). Lc s 9-5 129 CHEMICAL OSCILLATORS Furthermore for a point to stay within a small neighbourhood of F(x ;y) = 0 then this region must be one of stable equilibrium for the equations of rapid motion (2). This will be true if all s roots of the characteristic eqn (3) have negative real parts. The possibility of discontinuous oscillations then occurs if a representative point alternates between regions of slow and rapid motion. The points of transition between the regions of slow and rapid motion-the “jump ” points-are given by the inter- section of the curves F(x;y) = 0 and D(x;y) = 0 where D(x;y) is the Jacobian The remainder of this paper will be given over to four examples.First we consider the Lotka-Volterra scheme and show that if the fuel is allowed to vary the system can show non-conservative rather than conservative oscillations. In section 3 it is shown that when the “ pool ” chemicals are allowed to vary in the “ Brussel-ator” severe restrictions are placed on the rate constants so as to exclude certain regions where it was thought to oscillate. In section 4a kinetic scheme is devised in which slow variation of one of the “ pool ” chemicals gives rise to discontinuous oscillations of the type described above.Finally in section 5 a scheme originally proposed by Edelstein is modified and discontinuous oscillations are demonstrated when the temperature is allowed to vary slowly. 2. THE LOTKA-VOLTERRA SCHEME The set of differential equations originally proposed by Lotka in an ecological context can be reset in a chemical context via the following kinetic scheme kI A+X+2X k2 X+Y+2Y k3 A+Y+B. The rate equations for this scheme are dA -= -klAX-li,AY+k(A,-A) dt dX -= k,AX-k2XY dt where species A diffuses into the system from the outside where its value is assumed B. F. GRAY AND L. J. AARONS constant at Ao. It is convenient to change to new dimensionless variables A' = A/Ao X' = X/Xo Y' = Y/ Yowhere Ao Xo Yoare the initial values of A,X Y respectively and let Xo/Ao-Yo/Ao= 5 < 1.Eqn (6) become dA' __ = -k,XoA'X'-k,YoA'Y'+k(l -A') dt ; It can be seen that eqn (7) only reduce to the usual Lotka-Volterra scheme as 5+O provided k2Xo is of order 1/< while kl Xo and k3Yoare of order 1. Under these conditions eqn (7) possess a singularity at the point X' = Y' = A' = 3[-k/2+ (2k+k2/4)'] which is easily shown to be an unstable focus.8 These equations were programmed on a Solatron HS7 analogue computer and for a wide range of initial conditions the trajectories always wound onto a stable limit cycle. A typical run is shown in fig. 1. Thus by including the slow variation of the "pool " chemical A the system is made structurally stable. In the next section we will give an example of the opposite effect where the inclusion of a small parameter changes an oscillatory scheme to a nonoscillatory one.X' 0 1 2 Y' FIG.1.-Lotka-Volterra scheme. 3. THE BRUSSELATOR The following kinetic scheme was devised and has been exhaustively studied by the Brussels school 2* kl A+X k2 B+X+Y+D k3 2X+Y-+3X k4 X-+E CHEMICAL OSCILLATORS where in the original model A B D,E were assumed to be constant in time. Tyson later removed the necessity of the termolecular step by introducing another variable. The rate equations for the two dimensional model are dX -= k1A-k,BX+ksX2Y-kqX dt dY (9) -= k2BX-k,X2Y. dt The equations have a singularity for X = k,A/k4 Y = k2k4B/klk3A. It is easily deduced that the necessary condition for this point to be unstable is Lefever and Nicolis lo have shown that under certain conditions this stationary point is surrounded by a stable limit cycle.If A and B are allowed to vary subject to external fluxes we get the two additional equations Changing to dimensionless variables X' Y' A' B' as before letting Xo -Yo< Ao~Bo such that Xo/Ao = < and introducing a new time scale z = (l/c)t we get the four equations dA' -= <(-kl A' + J,/Ao) dz dY' -= k,XoB'X '-k,SXoYoX"Y'. dz It can be seen that eqn (12) only reduce to eqn (9) in the limit <+O provided k3 and k4are of order (115) and k and k are of order 1. Alternatively the flux terms have to become unrealistically large. Therefore under ordinary conditions this scheme will not oscillate in the region of parameters chosen by Lefever and Nicolis.lo This result shows clearly that one must be careful when maintaining various reactants at constant concentration.4. A CHEMICAL DISCONTINUOUS OSCILLATOR Consider the following kinetic scheme kl A+X-+2X k2 x+x-+x k3 X + B+products k4 B4products. B. F. GRAY AND L. J. AARONS The rate equations are dX --klAX-k2X2-ksXB dt dB = -k,XB-k,B+$B dt dA -7 -k,AX+$, dt where 4Aand 4Bare flow rates. Changing to dimensionless variables X' A' B' and letting Xo/Ao-Bo/Ao = 4 4 1 we obtain dA' -= -k,X,A'X'+&, dt where 4L = 4*/A0 and & = &/AO. If k2X0,k4,4Aand 4Bare all of order 1/5 and k,Xo is of order unity eqn (15) become dA' --A'X'+&k.dt They are then in the form of eqn (1). The equation of slow motion is given by F(x;y) = 0 which yields X' = 0,B' = 46 (17) and X'2+(1 -Af)Y+(4;-A') = 0 B' = $E;/(XI+f). (18) These solutions are sketched in fig. 2 in the X' A' plane. The branch X' = 0 is stable for A between 0 and P (A' = 4;) at which point it becomes unstable. The upper branch is stable down to the "jump " point R (X' = -1+ ,/#; ; A' = -1+ d&) and the middle branch between P and R is unstable. Thus for 41 < 1-2 J& + 46 a representative point will perform discontinuous oscillations as depicted in fig. 2. This limit cycle was verified for a wide range of initial conditions at the analogue computer. Discontinuous oscillators of this kind have been proposed by Lavenda Nicolis and Herschkowitz-Kaufman in connection with the Brusselator and by Rossler l2 for an autocatalytic scheme with feed back inhibition.However this scheme is perhaps simpler and is well suited as a model of a branched chain reaction of the type occurring in many combustion reactions,' I33 CHEMICAL OSCILLATORS 3 I 5 A’ FIG.2.-Equation ofslow motion and limit cycle for eqn (16) with 4’ = 5. 5. EDELSTEIN’S EQUATIONS A novel biochemical scheme that shows multiple steady states has been deviscd by Edelsteh6 The scheme is characterised by the following reactions ki A+X$2X k-1 kz X+E+C (1 9) k-2 k3 C+E+B k-3 where E is an enzyme and C is the enzyme-substrate complex.It is assumed that the total amount of enzyme is conserved i.e. E+C = constant (ET).It is trivial to produce a discontinuous oscillator of the type described in the last section merely by letting A vary slowly subject to a flux into the system. We have in fact succeeded in producing oscillations at the analogue computer this way. However a more inter- esting case arises if the first eqn in (19) is made exothermic and the reverse reaction has a strong temperature dependence. The rate equations for this system are (assum- ing we choose the rate constants such that A and B can be treated as constant) dT C -= C Ahi(RT -Ri)-L(T -To) dt i where Tis the reactant temperature Tois the ambient temperature Cthe heat capacity n.F. GRAY AND L. J. AARONS of the mixture Ahi is the enthalpy of the ith reaction R+ and R; are the rates of the forward and reverse ith reaction respectively and L is the heat loss proportionality constant.l4 Assume that k-l is the only significant temperature dependent rate constant and that it can be written as 2 exp( -E,/T). If the reactants are in dilute solution such that Cis large we can choose I /Cas our small parameter. The equation of slow motion is given by (with all rate constants other than k- put equal to unity) z exp( -E,/T)X3-[A-(2 +B)Z exp( -E,/T)]x~ + [ET-A(2+B)]X -BE = 0 (21) ZE, E = ______. X+B+2' For certain values of the parameters a hysteresis curve results such as the one shown in fig. 3. The variation of T is given by dT C __-= Ahl(AX-Z exp(-E,/T)X2)-L(T-T,).(22) dt Again for particular choices of the parameters a discontinuous limit cycle results typified by the one shown in fig. 3. 302 3 03 T FIG.3.-X T phase plane and limit cycle for Edelstein's equations (20). Parametric values chosen A = 10.2 B = 0.2 ET = 35 Ea = lo4 2 = 3~ lo" Ah1 = 1 1/C = To = 300,L = 4. One may be misled into thinking that an oscillator is basically isothermal merely on the grounds that the temperature oscillations are of small amplitude. As has been shown here that may not be the case and one suspects that oscillators of this type may be more common than suspected due to the highly nonlinear energy equation. 6. CONCLUSIONS By way of conclusion we wish to stress several points made earlier.Our main point is that one should be careful when setting as constant the concentrations of CHEMICAL OSCILLATORS various " pool "chemicals. It may be possible that the inclusion of a small parameter may change a stable equilibrium point into an unstable one (the reverse certainly cannot occur). More commonly if the rate constants are not treated correctly the scheme including the small parameter will not reduce to the degenerate scheme as the small parameter is reduced to zero. The results obtained are interesting and include the discovery of a number of counterexamples to Hanusse's conjecture (e.g. very small percentage variations in the pool chemicals are sufficient to allow sustained oscillations in the active chemicals).Multistability of the two variable system seems to lend itself very well to the pro- duction of limit cycles of the relaxation type and bistable systems are shown to oscil- late in this manner when the necessary variation of "pool " chemicals is taken into account. Biochemical models employing allosteric enzymes are particularly good candidates for discontinuous oscillators of this type. 5-Conditions are derived under which the " pool " chemicals can be treated as constants as a zero'th approximation and these conditions place severe restrictions on the rate constants and may exclude regions where interesting instabilities have been discussed in the two variable case ; e.g. the " Brusselator " appears to be unlikely to " Brusselate '' in the region where it is permissible to treat the two variable system.Finally it is shown that very small amplitude temperature oscillations in solution are not grounds for assuming that the oscillator is basically isothermal with secondary temperature effects. Since the energy equation is so highly nonlinear it should be far easier to devise thermokinetic oscillators than isothermal ones. Again if the two variable system shows multistability it is possibIe to obtain limit cycle behaviour by allowing small temperature variations. It is highly significant that we are able to treat systems of more than two variables exactly. This depends on being able to separate the system into regions of slow and rapid motion. It is surprising that little work has been done using this theory when the equations are analytically soluble.One of us (L. J. A) is grateful to I.C.I. for the support of a postdoctoral fellowship. A. A. Andronov A. A. Vitt and S. E. Khaikin Theory ofOscillators (Pergamon Press Oxford 1966). 'R. Lefever J. Chem. Phys. 1968,49,4977. P. Glansdorff and I. Prigogine Thermodynamic Theory of Structure Stability and Fluctuations (Wiley-Interscience London 1971). 4P. Hanusse Compt. Rend. 1972 274C 1245. R. Lefever G. Nicolis and I. Prigogine J. Chem. Phys. 1967 47 1045. B. B. Edelstein J. Theoret. Biol. 1970 29 57. A. J. Lotka J. Amer. Chem. Soc. 1920,42,1595. B. F. Gray and C. H. Yang Combustion and Flame 1969 13,20. J. J. Tyson J. Chem. Phys. 1973,58 3919. lo R. Lefever and G. Nicolis J. Theoret. Biol. 1971 30 267.B. Lavenda G. Nicolis and M. Herschkowitz-Kaufman,J. Theoret. Biol. 1971 32 283. l2 0.E. Rossler J. Theoret. Biol. 1972 36,413. l3 B. F. Gray Kinetics of Oscillatory Reactions (Specialist Periodical ed. P. G. Ashmore Reports The Chemical Society London 1974). l4 R. &is Efementary Chemical Reactor Analysis (Prentice-Hall New Jersey 1969). l5 H. R. Karfunkel and F. F. Seelig J. Theoret. Biol. 1972 36 237. l6 L. J. Aarons and B F. Gray to be published.

ISSN:0301-5696    

DOI:10.1039/FS9740900129

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

Kinetic Feedback Processes in Physico-Chemical Oscillatory Systems BY U. F. FRANCK Institute for Physical Chemistry Technical University of Aachen West Germany Received 29th July 1974 On the basis of a generalized feedback concept it is shown that the kinetic phenomena occurring in oscillatory systems can be understood as a result of two simultaneous counteracting feedback mechanisms. The positive feedback in particuIar brings forth excitability and propagation pheno- mena. The counteracting negative feedback brings forth recovery and overshoot phenomena. The cooperation of both feedback effects causes oscillation pulse formation and dissipative structures. 1. PHENOMENOLOGY Non-linear oscillations occurring in biological chemical electrochemical and other physico-chemical systems (fig.1) appear to be quite regular and uncomplicated. But on closer inspection the mechanisms bringing forth those rhythmic phenomena turn out to be unexpectedly intricate. In spite of the fact that they are of quite different physico-chemical nature the recordings of the oscillations look remarkably similar in their general shape suggesting that there may exist a common kinetic principle for their occurrence. As is well known the difficulties of elucidating oscillatory phenomena arise from the fact that the systems in question are essentially I E Ion Exchangebleembrane(Teor&f Iron in H2SO4+HCl5 !!Y.&!lLrnL t E FIG.1.-Oscillations of biological and non-biological systems. 137 KINETIC FEEDBACK PROCESSES ci 10 Nerve Membrane1 Heart Muscle 2 Cipid M~nhra~t? Ion ELC~R~CJF: N:mbram 0 lmrec 0 lam-0 lmin 0 Imm Iron in HNoJ0 Tunnel Dicde+NTC Hydrodynamic !3ystern7 Analog Computer Model7 0;hk 7V 0 1sac 0 1Ol.C 0 lQuc 0 tbru FIG.2.-Excitability (pulse formation) of biological and non-biological systems.I I Nerve Membrane iron Wire in HNO3 !=LlLLL 1-Ion Exchr Membrane !r==-=\ Belousov-Zhabotinskii Reaction FIG.3.-Propagation waves in oscillatory systems. The graphs represent wave profiles at two successive points of time (abscissa :length I in direction of propagation). U. F. FRANCK multi-variable systems with extremely non-linear kinetic relationships and complicated coupling mechanisms. By changing the environmental conditions slightly in such a way that oscillatory behaviour just ceases all these systems can be made to exhibit excitation phenomena and can be triggered by adequate stimuli obeying the all-or- none-law of excitation (fig.2). A superthreshold stimulus depending on its strength and duration releases an excitation pulse having the characteristic shape of the so-called action potential of nerve membranes. The third essential property of these systems is the phenomenon of decrement-free propagation signifying that a locally triggered excitation state or pulse propagates respectively along or through the entire system by means of eddy currents or fluxes (fig. 3). According to the phenomenology of nerve excitation refractoriness accommoda- tion adaptation etc.are also common features of all systeins capable of oscillation.’’ 2. THE PROBLEM OF OSCILLATING VARIABLES Systems exhibiting sustained oscillations are thermodynamically opm systems in which for the most part forces and fluxes behave rhythmically. Between the forces and fluxes there exist the well-known conjugation relationships and cross-effects. The kinetic variables appear in the mathematical description in form of temporal differential quotients combined in sets of simultaneous differential equations. force -flux flux -force conductance -resistance resistance conductance capacitance 1--3 inductance inductance capacitance wsitive feedm (non-monotonity) -negativefeedback circuitrv FIG.4.-The duality relationships between variables parameters feedback and circuitry of force dependent and flux dependent systems.KINETIC FEEDBACK PROCESSES From the kinetic point of view it must be borne in mind that there are two distinct classes of oscillatory systems ; those in which the fluxes are the result of driving forces and those in which the forces are the result of force generating fluxes. Accordingly classification of oscillatory systems with respect to force-dependent and flux-dependent interrelationships is essentially a matter of kinetic causality. Between the variables parameters and the "circuitries" of the system of both classes there exist defined duality-relationships as shown in fig. 4. Most of the known oscillations in particular those of chemical electrochemical and biological systems belong to the class of force-dependent oscillators.Here they Non-monotonic Systems high field strength high density of force -= I of flux t I force dependent systems t t I flux dependent systems t4'X- X- autoinhibitoryforce@ autocf~talyticforce autocatalytic autoinhibitoryflux FIG.5.-Non-monotonic flux-force-characteristics in force dependent and flux-dependent systems. Inductances" electric wlume flux inductance inductance -mercury inertia Lv L E*Lp ~-FW.6.4eneralized capacitances and inductances U. P. FRANCK are brought forth by force-depending non-monotonic flux-force-relationships (fig. 5a). Such kinetic characteristics arise frequently in systems containing structures where high field strength of forces can occur such as in membranes and at interfaces.On the other hand flux-dependent characteristics (fig. 5b) appear in systems in which high densities of fluxes occur. In biological and chemical systems such kinetic situations are extremely unlikely or even impossible. This obviously is the reason why flux-dependent oscillators are much less abundant than force-dependent oscillators. The time dependence of the oscillating variables arises from two kinds of intrinsic properties of the systems (a) STORAGE PROPERTIES l1 ' ;They may be of capacitance or inductance type whether the time dependence concerns the forces or fluxes :fig. 6 gives examples of "generalized "capacitances and inductances. They are defined by the well-known differential relationships capacitances inductances 1 = C(d X/dt) Xi = L(dZ/dt) I :flux into the capacitor (e.g.electric X :force of the inductor (e.g. electro- current molar flux heat flux motive force pressure) volume flux) C :generalized capacitance (e.g. elec- L :generalized inductance (e.g. electric tric capacitance volume of solvent inductance inertia of fluid) heat capacitance volume capa- citance) X force of the capacitor (e.g. electro- I flux of the inductor (e.g. electric motive force concentration tem- current volume flux) perature pressure) In the case of capacitances as illustrated in fig. 6a an influx of electric charges into a capacitor leads to a temporal increase of voltage an influx of matter into a volume of solvent leads to an increase of concentration an influx of heat into a heat capacitor leads to an increase of temperature and an influx of volume of fluid into a volume capacitor leads to an increase of pressure.In most cases the capacitances may be regarded as constants. They may however also depend on the forces but they are essentially positive. That follows from the well-known laws of conservation of charges matter volume of incompressible fluid and energy. (b) D I S S I P A T I V E PR 0P E RTI E S Whilst time dependence of variables effected by capacitances and inductances respectively concerns the storage properties of free energy dissipative properties such as conductivities and chemical reactions may also give rise to time dependences.That is in particular the case if such dissipative properties depend on their own driving forces or fluxes respectively. Evidently such behaviour is always a result of a feedback mechanism consisting of consecutive reactions and/or transportation pro- cesses which always cause a corresponding lag of time. This is the intrinsic reason why all phenomena effected by feedback mechanisms are time dependent and play a substantial role in all non-linear oscillatory systems. Strictly speaking any physico-chemical oscillator has as many variables as fluxes and reactions occur in the entire oscillatory process. Chemical oscillators in particular have as many chemical variables as chemical species or reactions take part KINETIC FEEDBACK PROCESSES in the oscillation mechanism.Biological and chemical oscillators therefore are usually systems of relatively many kinetic variables each of them giving rise to a differential equation of its own. For mere qualitative understanding of oscillatory behaviour the entirety of existing variables can mostly be reduced to only two “ essential variables ”. How-ever if the specific shape of a given oscillation has to be described and interpreted in detail more than two variables must usually be taken into consideration. 3. THE FEEDBACK CONCEPT OF NON-LINEAR OSCILLATIONS Coupling effects as well as autocatalytic and autoinhibitory mechanisms may lead as is well-known to positive or negative feedback in physico-chemical systems. In the case of oscillatory behaviour both kinds of feedback must obviously be present simultaneously.A direct consequence of strong positive feedback is the occurrence of non-monotonic flux-force-characteristics. As will be discussed later more in detail such characteristics cause the instability behaviour and excitation phenomena mentioned above. In contrast to positive feedback? negative feedback has a stabilizing effect and causes recovery and adaptation behaviour in oscillatory systems. Variables involved in negative feedback mechanisms are always free from instability situations. Obviously a general kinetic pattern l3 which is valid for all physico-chemical oscillatory systems can be designed according to fig. 7. Let X be a kinetic variable (e.g. voltage concentration pressure temperature etc.) which increases by a “formation process ” and decreases by a simultaneous “ consumption process ”.X becomes constant when both processes have equal turnover. As already mentioned oscillatory behaviour may arise if two antagonistic feedback mechanisms are effective inside the system simultaneously (a) a fast positive feedback in a “ favouring loop?’ (b)a slow negative feedback in a “ counteracting loop ”. \fast) / formation x consumption FIG.7.-General pattern of oscillatorysystems. Each loop as a rule includes several variables which may be arranged in direct stoichiometric reaction chains or in non-stoichiometric coupling mechanisms. Only the actual total effects and rates of both kinetic loops are essential here. It is obvious that for the occurrence of oscillations the lag of time in both feedback loops has to be sufficiently different? otherwise the system would gain a stable steady state.Favouring and counteracting effects can arise by autocatalytic or autoinhibitory action. It depends on whether the loop in question concerns the formation or the U. F. FRANCK consumption process. As a consequence the general pattern of fig. 7 can be realized in four different ways (fig. 8). Examples for all of these four types of oscillatory systems are known. For the elucidation of oscillatory systems it may be advantageous as a first step to find out to which of the four types the system in question belongs. 0 -X-u backward activation I forward inhibition c delayed delayed backward in hi bit ion backward inhibition I backward activation forward inhibition I delayed delayed I forward activation forward activation FIG.&-The four kinetic types of dissipative oscillatory systems (+ and -stands for autocatalytic or autoinhibitory effects respectively).4.EXCITABILITY PHENOMENA Instability as a result of positive feedback by backward activation or forward inhibition l4 is a necessary but not sufficient presupposition of oscillatory behaviour. However systems containing positive feedback only may already bring forth excita- bility and propagation phenomena. Fig. 9 illustrates how instability is brought forth by the action of a favouring loop. In both of the possible cases the result of auto-catalysis or autoinhibition is likewise the occurrence of a force-(i.e.concentration) dependent non-nionotonic kinetic characteristic.'O The stationary states of X(d X/dt = 0) are given by the intersections of the characteristics of formation and consumption of X. In the non-stationary states the difference of the reaction rates of formation and consumption corresponds directly to the time variation dX/dt according to the generalized capacitance equation rf-r = V(dX/dt) (rf,I', reaction rates of formation and consumption respectively Y:volume X concentration). KINETIC FEEDBACK PROCESSES formation x consumption rc ' r backward activation forward inhibition Ill x-II !X-FIG.9.-Instability and bistability of systems containing a favouring feedback loop.The resulting graph of dX/dt = (rf-r,)/ Vagainst Xis termed the " dynamic diagram " because it describes the dynamic behaviour of the system with respect to X. In this representation the stationary states are given by the intersections of the dynamic characteristic with the abscissa. The dynamic behaviour of the system in the immediate neighbourhood of the resulting three stationary states indicates that the outer states are stable and the inner state is unstable. In this way instability and bistability have the same causality. \ X' -I,/(-: t-t-t- bistabillty pulse formation scillation oQ FIG. 10.-Bistability pulse formation and oscillation as a result of feedback represented in the dynamic and the force-time diagrams.0. F. FRANCK Fig. 1Oaillustrates the effect of sub-threshold and super-threshold stimuli according to a concept which Bonhoeffer l6 introduced as early as 1943 into the theory of excitation kinetics. Some typical examples of systems in which non-monotonic force-dependent characteristics occur are shown in fig. 1 1. Nerve Membrane ,Icn Exchange Membrane .Tunncl Diode ,Ester Hydrolysis ,HydrodynamicSystem r E-IW14 P4 FIG. 11 .-Examples of systems exhibiting non-monotonic flux-force-characteristics by positive feedback. /local -‘\ eddy f luxes i f--.\ FIG.12.-Propagation as a result of bistability induced by positive feedback. Bistability exhibited by a favouring loop is the intrinsic cause of propagation phenomena (fig.12). The two stable states of the bistable system correspond to two different values of driving force. As a consequence at the boundary between ranges of different states transportation processes take place driven by the difference of the forces of both ranges. The local fluxes across the boundary act as stimuli for both KINETIC FEEDBACK PROCESSES ranges. The propagation of state conversion then proceeds in that direction deter- mined by the mutual stimulation process which succeeds first. By appropriate alteration of the environmental conditions it is possible to reverse the direction of propagation (fig. 13). Propagation. lower state upper state-lJ---4)c (activation wave) 1 I I 7r.e t-* 33cm Propagation lower state +upper state (passivation wave) IonExchange_ Membrane Propagation upper statedower state I 0,Bmin t-55cm .I lower statezupper state aemin 3 t-5,scm .+ FIG.13.-Examples of propagation reversal by changing of the environmental conditions.5. RECOVERY AND OSCILLATORY PHENOMENA Feedback systems differ in their responses to external stimuli in a very distinct and characteristic manner depending on whether the feedback is positive or negative. As shown in fig. 14a positive feedback exhibits transients which resemble qualitatively the temporal behaviour of capacitances. As already explained strong positive feedback giving rise to non-monotonic flux-force characteristic excitation transitions may occur in such systems. On the other hand systems containing negative feedback respond in the form of overshoot phenomena (fig.14b). In this respect the transients of negative feedback loops resemble inductances in their temporal behaviour. In other words negative feedback tends-with a certain temporal delay-to counteract or to compensate disturbances or alterations of the system. By combining a positive feedback with an appropriate negative feedback of sufficient long time delay then oscillations and excitation pulses are possible. In U. F. FRANCK this case the non-monotonic dynamic characteristic of the favouring loop also depends simultaneously on the state of the counteracting loop. Fig. lob c illustrate schema- tically how pulses and oscillations are brought forth by cooperation of both antagonisti- cally reacting feedback loops.1o Here for the sake of simplicity only two kinetic .positive feedback Lqative feedbad( monotonic non-monotonic t I mfTbof! 14.1 X- X- X- t-c FIG.14.-The time behaviour of dissipative systems exhibiting force-dependent conductance by positive and negative feedback. In the upper row of graphs the slope of the beams correspond to the conductance G(X) = I/X depending on the force X. In case of positive feedback (forward inhibition) the Conductance decreases and in case of negative feedback (forward activation) it increases with increasing force. As a result of the time-lag of the feedback sudden changes of flux bring about the characteristic transients of the lower row of graphs. variables are taken into consideration X being the “trigger variable ” characterizes the action of the favouring loop and Y,being the “ recovery variable ” characterizes the action of the counteracting loop.The primary excitation of the triggerable favouring loop recovers spontaneously by the delayed action of the counteracting loop in the course of which an unstable state may be attained too. In other words excitation and recovery similarly imply instability events. In the case of pulse generation excitation triggering is provoked by external disturbances (stimuli) and recovery is an internal spontaneous process. In case of oscillations however both trigger processes are internal events which occur spontaneously inside the system. Oscillatory behaviour considered from that point of view then consists of successive trigger events caused by the cooperation of both feedback loops.This idea that physico-chemical oscillators might imply two intrinsic trigger processes was suggested by Bonhoeffer l5 in 1948. In connection with the concept of antagonistic feedback loops in oscillatory systems it might be noticed here that the oscillating variables can be divided clearly into two classes depending on whether they belong to the positive or to the negative feedback loop. In most cases it is easy to decide experimentally to which loop a recorded oscillation variable belongs (fig. 15). Variables involved in the positive feedback as a rule exhibit characteristic breaks or steps in the course of their oscillation recordings.These breaks are the result of the unstable states occurring periodically KINETIC FEEDBACK PROCESSES in the kinetic mechanism of the positive feedback loop. Variables belonging to the negative feedback loop never have such breaks in their oscillograms because of the absence of instabilities in their kinetics. ion exchange iron electrode tunnel diode Belousov membrane in !NO3 NTC oscillator reaction I Lwtw:w,+w I c3e' I I I I I I I recoveryvariable ; I I I ld t+ t- t-t- FIG.15.-SimuItaneous oscillationsof trigger and recovery variables. 6. STRUCTURAL ASPECTS OF OSCILLATORY SYSTEMS In many cases the oscillations are bound to specific spatial structures such as membranes interfaces of electrodes or solid catalysts.In other cases oscillations clearly occur in continuous systems like the Belousov-Zhabotinskii-reaction and oscillating gas reactions. Concerning structural properties of oscillatory systems we have to keep in mind that all systems which oscillate in macroscopic spatial ranges necessarily imply macroscopic propagation processes. Otherwise only microscopic fluctuations or dissipative structures can occur. As mentioned before propagation generally is brought about by local transportation processes caused by local gradients of driving forces. These local transportation processes are decisive for the structural require-ments of oscillatory systems. For instance all systems in which electric local currents are involved as propagation-inducing transportation processes necessarily consist of interfacial structures since local currents require closed circuits which only can occur at interfaces.For this reason electrochemical and electrobiological oscillations are strictly bound to membranes or interfaces. On the other hand transportation processes inducing propagation in chemical systems (diffusion heat conduction) are independent of the existence of closed circuits. Here oscillatory behaviour may occur in continuous systems as well. 7. SUMMARY It has been shown that (1) The causality concerning the relationship between fluxes and forces in non-linear dissipative systems leads to two distinct classes of oscillators (a) formdependent oscillators (b)flux-dependent oscillators. (2) Time dependence in those systems arises from (a) storage properties of free energy (b)feedback properties of dissipative processes (chemical reactions and transportation processes).U. F. FRANCK I49 (3) Feedback properties are brought about by autocatalytic and autoinhibitory processes concerning chemical reactions and/or transportation processes. (4) Non-linear dissipative oscillators require the simultaneous cooperation of a strong (i.e. non-monotonic) positive feedback with a negative feedback being rela- tively delayed yet strong enough to recover excitation transitions of the positive feedback. (5) Non-linear dissipative oscillators can in principle be distinguished as four different types of systems depending on whether the formation process and the consumption process are influenced by autocatalytic or autoinhibitory effects.From the eight possible combinations four of them may induce oscillatory behaviour (a) forward inhibition-backward inhibition (b) forward inhibition-forward activation (c) backward activation-backward inhibition (d)backward activation-forward activation. (6) Positive feedback induces pseudo-capacitance behaviour (fig. 14a) and in the case of non-monotonity (a) excitability instability bistability all-or-none-behaviour (b) propagation phenomena. Negative feed back causes pseudo-inductance behaviour in particular (a) overshoot phenomena (b) adaptation behaviour accommodation refractoriness (c) recovery of excited states.Simultaneous cooperation of both kinds of feedback may lead to (a)oscillatory behaviour (b)pulse formation (c) dissipative structures. (7) All of the non-linear dissipative oscillatory systems are genuine " nerve models "exhibiting under appropriate environmental conditions the entire phenomen- ology of excitable nerve and muscle membranes. (8) Oscillations including electrical transportation processes are strictly bound to heterogeneous interfacial structures whereas chemical and thermal oscillations may also occur in continuous systems. I. Tasaki Handbook of Physiol. Neurophysiol. 1959 I 75. W. Trautwein Ergebn. Physiul. 1961 51 131. W. J. V. Osterhout J. Gen. Physiol. 1943 26 457. 4T. Teorell J. Gen. Physiol. 1959 42 831.U. F. Franck unpublished P. Muller and D. Rudin Nature 1968 217 713. 'U. F. Franck and F. Kettner to be published. U. F. Franck and F. Kettner Ber. Bunsenges. 1964 68 875. R. J. Field E. Koros and R. M. Noyes J. Amer. Chem. SOC., 1972,94,8649. lo U. F. Franck Biological and Biochemical Oscillators (Academic Press New York 1972) p. 7. U. F. Franck Chem. Ins. Techn. 1972 44 228. G. Viniegra-Gonzalez Biological and Biochemical Oscillafors (Academic Press New York 1972) p. 71. U. F. Franck Abstracts of contributed papers Congr. International Biophysics IUPAB Moscow (1 972). l4 J. Higgins Ind. Eng. Chem. 1967 59 19. l5 K. F. Bonhoeffer Z. Elektrochem. 1948,51 24. l6 K F. Bonhoeffer Naturwiss. 1943 31 270.

ISSN:0301-5696    

DOI:10.1039/FS9740900137

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

GENERAL,DISCUSSION Dr. B. F. Gray (Leeds) said We have studied propane oxidation in a stirred flow reactor an apparatus in which questions of fuel consumption etc. do not arise because either perfectly steady states or limit cycles can be achieved exactly and their behaviour with respect to perturbations studied experimentally. Results which can only be guessed at from closed vessel studies (such as the existence of a stable focus) can be studied exactly in this case and in particular limit cycle oscillations can be made to persist indefinitely. Hysteresis and multistability are also shown in this system. In the closely related case of acetaldehyde oxidation the advantages of the flow system are relatively greater since the reaction parameters are such that in a closed system the behaviour is dominated first by the arbitrary initial conditions selected and then almost immediately by the monotonic decay towards equilibrium.The system never gets close to the kinetic stationary state (or sustained oscillation) and thus little information is obtained from experiments in closed vessels. On the other hand in the flow system the kinetic stationary state can be achieved exactly as can conditions of sustained oscillation. The system also exhibits hysteresis and multistability.2 Also as the kinetics of this system are reasonably well understood on a semi-quantitative basis computations have been performed including the energy conservation equation excellent agreement with experiment being obtained con-sidering the uncertainties in some of the experimental values of the rate constants used.This agreement leads one to feel that this thermokinetic oscillation is at least as well understood as the Belousov reaction following the detailed work of Noyes et al. but at the same time it displays a considerably richer phenomenology than the latter (which is not sufficiently nonlinear to show multistability or hysteresis). By analogy it would be interesting to pursue the possibility of obtaining spatial patterns in the acetaldehyde system. Dr. L. J. Kirsch (T/zornton)said At Thornton we have been working for some time on the mathematical modelling of hydrocarbon oxidation phenomena. These models starting with the acetaldehyde oxidation model of Halstead Prothero and Quinn have more recently been developed in a generalised form to describe alkane oxidation phenomena over a wide range of experimental conditions.They employ a degenerate branched chain mechanism with competing reactions of differing activation energies controlling the supply and consumption of branching agent and removal of radicals from the system. This competition leads to oscillation under certain con- ditions and the models are therefore examples of the thermokinetic type described by P. Gray et al. in their paper. In both analytical and numerical treatments of the model we have taken fuel consumption into account. The models give excellent simulations of all the patterns of behaviour illustrated in fig. 2 of the paper by Gray et al. I should like to mention some conclusions resulting from our experience of working with these models which are of relevance to the points raised by these authors.B. F. Gray and P. G. Felton Comb. Flame 1975. B. F. Gray and P. G. Felton Comb. Flame 1975. B. F. Gray P. G. Felton and N. Shank Comb. Flame 1975. M. P. Halstead A. Prothero and C. P. Quinn Proc. Roy. Soc. A 1971 322 377. 150 GENERAL DISCUSSION First with reference to the “ sustained ” oscillation illustrated in fig. 2(d),we have noted in our simulations that cool flame behaviour often terminates when a sizeable proportion of reactant remains unconsumed. There follows a lengthy period of slow combustion. Secondly we have observed during the course of a single simulation transitions froni the damped oscillations of fig.2(a) to the sustained oscillation of fig. 2(d). Both of these observations illustrate a property of the thermokinetic model we have considered-namely that the nature of the stationary point solutions are pro- perties not only of the initial conditions and heat transfer properties but also of reaction co-ordinate in as much as reactant consumption influences the appropriate rate coefficients. It would therefore be of interest to learn whether the statement that cool flame behaviour terminates abruptly when no fuel remains is based upon direct experimental evidence. Also whether Gray et al. have found any experimental evidence for a change in the nature of oscillatory behaviour during the course of a single experiment. This might particularly be anticipated in their experiments where a small quantity of acetaldehyde was added and clearly had a pronounced influence on the pattern of behaviour.Variation of the relative quantities of propane and acetaldehyde present may occur during the course of the reaction due to unequal reaction rates. Some rationalisation of the fact that our models predict changes in the nature of oscillation due to fuel consumption whereas this was not observed in the experiments of Gray et al. may lie in the orientation of the experimentally determined line separ- ating weakly and strongly damped oscillations (fig. 4). If the chemical mechanism does not involve any pressure dependent (termolecular) steps the effect of reactant consumption can to the first approximation be envisaged in terms of motion of the point describing the appropriate time-dependent reactant conditions parallel to the pressure axis as the reaction proceeds.The line separating weakly and strongly damped oscillations lies parallel to this direction of motion over much of its range. Thus transitions from one kind of oscillation to another will be unlikely in this system although an eventual transition into the region of slow combustion seems probable. Prof. P. Gray (Leeds) said The calculations which Kirsch has made attempt the most complete modelling of thermokinetic effects in hydrocarbon oxidations. They are based on the assumption of spatially uniform temperature and concentrations i.e. they represent a perfectly stirred system. The quantitative validation of the model therefore requires comparison with experiments done in well-stirred conditions.Unfortunately nearly all classical investigations of cool flames have not been made under such conditions they are all affected by natural convection and flame pro- pagation usually from a hot regi0n.l We believe that our experiments in a stirred reactor offer Kirsch experimental results with which he can compare his computer analysis ;we can also supply him with experimental values of heat transfer coefficients for this type of reactor. In answer to Kirsch’s point about cessation of reaction we find that temperature excesses and hence rates of heat release by reaction fall to zero almost discontinuously as soon as the oscillations have died away.(The time axis of our fig. 2 is too brief to display this.) The “piq d’arret ” is often exhibited ;Lucquin identifies this with the complete consumption of oxygen and the end of combustion. We have not measured J. F. Griffiths B. F. Gray and P. Gray Thirteenth International Symposium on Combustion (The Combustion Institute 1971) p. 239. L. R. Sochet J. P. Sawersyn and M. Lucquin Advances in Chemistry Series No. 76 Oxidation of Organic Conipounds-ZZ (The American Chemical Society 1968) p. 11 1. GENERAL DISCUSSION reactant concentrations by direct chemical analysis. Kirsch’s calculations that a large proportion of fuel remains and that combustion continues for a lengthy period even after the last undamped oscillation seem to show a real difference between experiment and the model.Alteration of the amount of acetaldehyde added has a pronounced effect on the reaction rate in the earliest stages. If concentrations of acetaldehyde are increased beyond 3 mol % not only are induction times to oscillations reduced but the ampli- tude of the first oscillation is also increased. Kirsch’s second point really has to do with the imperfections of trying to categorize chemical reactions in closed vessels in terms that are only properly appropriate to open systems. True limit cycles in these oxidations are possible only if fresh reactants can be constantly supplied. As experimentalists we can recognize in a closed system behaviour that is like a limit cycle in the phase plane when we see 5 or 6 successive undamped or weakly damped oscillations; the behaviour is more like a stable focus in the phase plane when we see strongly damped oscillations.Even these remarks do not apply to very many closed-vessel oxidations-propane is perhaps the most favourable case giving up to 11 pulses in a closed vessel. Acetal-dehyde is the opposite; no experimentalist seems to have observed more than one or two cool flames in a closed vessel. In an open system of course oscillations can persist for as long as we wish. Recent studies by B. F. Gray and P. G. Felton working in Leeds have revealed oscillatory oxidation supported indefinitely in a variety of conditions including acetaldehyde as well as propane. In our fig. 4 the broken line dividing the regions is correspondingly subjective and it ought perhaps to be drawn in its lower parts with a finite positive slope instead of as a nearly vertical line.(After all a single oscillation can hardly be called un- damped.) Even so we do not encounter either a train of (say 4)undamped oscillations followed by another train of (say 4) damped oscillations or the reverse sequence. It is as though Kirsch’s calculations assume that the development in time of the (real) closed system with a particular initial reactant concentration but suffering continual reactant consumption can be legitimately imitated by putting together a short sequence of open systems each without any reactant consumption at all but differing from one another by successively diminished initial reactant concentrations.So far as fig. 4 is concerned there is a similarity between a vertical traverse of the diagram and the course of events in time but to represent either fuel consumption (or reactant temperature) on it would require a third dimension. Prof. P. Gray (Leeds) (communicated) Among gaseous oxidations often said by reviewers of oscillatory reactions to be accompanied by oscillations is that of hydrogen sulphide. Our work shows it to involve thermal and kinetic feedback by self- heating and chain-branching but neither we nor any authors save Thompson have so far succeeded in producing oscillations. I think multiple ignitions reported may have arisen from their technique of admitting reactants successively to their relatively narrow cylindrical reaction vessel.At the pressures employed mixing could have been so imperfect that the first ignition would have been very incomplete and localized. When it was over time for mixing would be required to elapse before a second partial ignition could occur and so on. If this is correct then successive ignitions in H2S oxidation belong with old observations on phosphorus oxidation in unclosed vessels as originating in slow diffusive processes rather than in the chemistry. P. Gray and M.E. Sherrington,J.C.S. Furuduy I 1974,70,2338. * H. W. Thompson J. Chem. SOC.,1931 1809. GENERAL DISCUSSION Prof. P. Gray (Leeds) said In answer to Shashoua 1would remark that acetal- dehyde is added here principally to shorten the induction period before oscillatory cool flames set in-it is of course one of the many intermediate products of propane oxidation-and it also has some secondary effects on the oscillations observed.Rather than being a sink for radicals as in Shashoua’s polymerisations added acetal- dehyde in these oxidations is a potential radical source. I agree with Shashoua that it (and other added reactants) offer a means of varying conditions controlling be- haviour and investigating the chemical part of the tliermokinetic mechanism. We believe these would be worthwhile investigations though there are other fundamental aspects to clear up first. Dr. R. G. Gilbert and Mr. R. Ellis (University of Sydney) said Concerning Yang’s isothermal mechanism both he and P. Gray have mentioned the possibility of important non-isothermal effects.In a closed adiabatic system inclusion of temper- ature as an additional independent variable in numerical integration of the kinetic equations is fairly straightforward. We have written a general program to do this and can verify that such systems are very sensitive to this temperature variation. This program was written for our study of sound propagation in gas-phase reactions extending our previous work in this fie1d.l As a result of these studies we suggest that acoustic-kinetic interactions in systems which are capable of undergoing homo-geneous oscillations would lead to amplification and interference effects which could provide useful data for deriving kinetic information. Prof. C. H. Yang (Stonybrook) said In general the non-isothermal effects in the oxidation of CO is not as significant as in the case of hydrogen or hydrocarbon 3* oxidation.For sustained oscillations and long-lasting glows (minutes) in the CO system Dove showed that it is essentially isothermal. In the case of explosion or glow (seconds) for wet mixtures (greater than 1 % H20) the slow reaction rate is very low at sub-critical conditions. Limit pressures or temperatures are unlikely to be affected by the non-isothermal effect. Limits for drier mixture may be influenced by “ self-heating ”mildly. Only the duration of explosions will almost certainly be drastically shortened by thermokinetic considerations as the temperature difference between the reacting mixture and the bath will be high in such cases.However the duration of explosion has rarely been used as an important parameter in either theoretical or experimental studies. A unified thermokinetic approach to the problem probably would not add significantly to the understanding of the kinetic mechanism. Dr. L. J. Kirsch (Thornton)said :In the final paragraph of his paper Yangdiscusses the effect on his calculations of the rate constant k2 which describes the recombination rate between atomic oxygen and carbon monoxide O+CO+M -+ C02+M Yang uses the value recommended by Baulch et al. (1968) in his computations. Recently a number of studies of this reaction have been made many employing R. G. Gilbert H.-S. Hahn P. J. Ortoleva and J. Ross J. Chem. Phys. 1972 57 2672; R. G. Gilbert P. J. Ortoleva and J.Ross J. Chem. Phys. 1973 58 3625. K. K Foo and C. H. Yang Comb. Flame 1971,17,223. C. H. Yang and B. F. Gray J. Phys. Chem. 1969,73,3395. C. H. Yang J. Phys. Chem. 1969 73 3407. J. E. Dove Dissertation (Oxford 1956). I 54 GENERAL DISCUSSION direct monitoring of the atomic oxygen decay in a time-resolved This work has indicated that at room temperature the rate coefficient k2 is some orders of magnitude lower than that given by Baulch et al. Measurements over a range of temperature give a value of k = 2.4 x 10’’ exp( -4340(kcal/mol)/RT) cmG mok2 s-I . The reaction therefore appears to exhibit the sizeable positive activation energy that Yang states will improve the fit between calculated results and experimental data. At a temperature of about 700 K the overall value of the rate coefficient is in fact close to the temperature independent value employed by Yang.Perhaps the most striking fact to emerge from these recent studies of reaction (2) is its extreme sensitivity to traces of impurity presumed to be metal carbonyls. Thus the most stringent purification procedures must be followed even with supposedly high purity carbon monoxide in order that the true value of the rate constant be recorded. For this reason many earlier studies of the reaction must be disregarded. It follows that some doubt must be attached to any quantitative interpretation of more complex reactions in which reaction (2) plays a rate determining role unless similar precautions regarding purification of CO have been taken.Prof. C.H. Yang (Storzybrook) said A very interesting point has been raised by Kirsch namely that the rate constant of reaction (2) is extremely sensitive to traces of impurity according to recent experimental evidence. In fact the rate constant for reaction (4) may also be sensitive to impurities. While we have no direct evidence of this at the present time it appears to be true with some other excited species reported in the literature. Cvetanovic cited the data of Slanger and Welge indicating that the rate constants for the de-excitation reaction of the O(ls) atom with third bodies of N and H20 are 3.0 x lo7 and 2.1 x 1014~m-~ mol-’ s-l respectively. Water would clearly be a very sensitive impurity in this case. It is well known that the oscillation phenomenon in the CO system is difficult to reproduce under apparently identical experimental conditions.Often this is mysteriously attributed to the role played by the vessel surface. In view of our calculations,s* it is clear that oscillatory solutions may be completely inhibited by sizeable variations of either of these two rate constants. Perhaps purity of the reactants may be the most essential quality to strive for in kinetic experiments of this kind. Prof. R. M. Noyes (Oregon) said The behaviour of oxygen atoms in Yang’s mechanism is similar to that of the switched intermediate X in the Oregonator. Thus step (1) forms oxygen atoms at a rate independent of their concentration and the sequence of steps (5) and (6) accomplishes the same net reaction at a rate pro- portional to the concentration of OH radicals.Because the concentrations of OH and of 0 are positively coupled through step (7) the sequence of steps (5) and (6)generates oxygen atoms autocatalytically. The sequence (2) + (4) destroys oxygen atoms by a first order process and the sequence (2)+(3) essentially represents second order destruction of oxygen atoms. R. J. Donovan D. Husain and L. J. Kirsch Trans. Faraday SOC.,1971 67,375. F. Stuhl and H. Niki J. Chem. Phys. 1971 55 3943. T. G. Slanger B. J. Wood and G. Black J. Chem. Phys. 1972,57,233. R. Simonaitis and J. Heicklen J. Chem. Phys. 1972 56 2004. W. B. Demore J. Phys. Chem. 1973 76 3527. E. C. Y.Inn,J. Chem. Phys. 1973,59,5431. ’R.J. Cvetanovic Canad.J. Chetn. 1974 52 1452. C. H. Yang Comb. Flame 1974 23 97. C. H. Yang and A. L. Berlad J. C.S. Favaday 1 1974 70,1661. GENERAL DISCUSSION 155 As we have pointed out elsewhere,' systems with unstable steady states can not be modelled by elementary processes involving only two intermediates. Thus the state of the present system can not be described solely by the concentrations of CO; and 0 ; other species like H OH and H202undergo coupled variations with delays that are important for the instabilities exhibited. Although the calculations with this model show impressive similarities to experi- mental observations ; 1 am disturbed about the postulated COY intermediate. This excited species reacts with oxygen atoms very efficiently and without activation energy even though the net reaction involves breaking and forming chemical bonds each much stronger than 100 kcal/mol.However the same excited species must usually undergo over lo8 collisions before it loses its excitation energy by step (4). I am not aware of any precedent for an excited species that can so easily react with rearrangement of strong bonds or that is so inert to loss of excitation energy by collision; it would be particularly remarkable to find these unusual and seemingly contradictory tendencies in the same molecule. I am therefore unconvinced that the mechanism of this reaction is yet demonstrated. Prof. C. H. Yang (Stonybrook) said In reply to Noyes I would remark that Tyson and Light have shown that the type of oscillation prescribed by a limit cycle does not exist in a kinetic model which is constructed with two intermediates and involves only first and second order elementary process.This conclusion is of course invalid for a kinetic system with more than two intermediates. Under appropriate conditions a kinetic model of multiple components may be reduced to a binary system when the concentrations of some intermediates are eliminated by the steady state assumption. Again Tyson and Light's conclusion is inapplicable to such a reduced system. In our earlier work we replaced the concentrations of H OH and OE (where OE represents a vessel wall site which is occupied by an 0atom) by their steady state values. The kinetic model was reduced to a binary system containing only the concentrations of CO; and 0.A stable limit cycle was shown to exist in that system. In our present work on the other hand all intermediates in the proposed scheme have been con- sidered. Clearly the system is not solely prescribed by the intermediates CO,* and 0. It is useful to point out after comparing the present results with our earlier work that the assumption of steady states for some intermediates while greatly simplifying the mathematical complexities has not impaired the ability of the simpler model to predict all important kinetic features qualitatively. It is indeed essential to assume in our current work that the excited Cog molecule is quite stable (actually metastable) as far as the quenching reaction (4) is concerned.As indicated in the paper our sole criterion for selecting the rate constants for reactions (3) and (4) is based on a favourable fitting of the calculations to the oscillation and explosion limit data. There are no known values for these rate constants reported in the literature to the knowledge of the author. They probably remain to be in- dependently determined. However we find it difficult to accept the argument that the stability of the CO; molecule would imply a slow rate for reaction (3). Many bi- molecular reactions which involve a stable molecule and a radical are known to be fast. One example is the well known titration reaction NO+N -+ N 4-0 where a bond of I50 kcal is broken and a bond of 225 kcal is formed for which a rate constant of the value 3 x IOl3 cm3 mol-l s-l with nearly zero activation energy is a~cepted.~ R.M. Noyes and R. J. Field Ann. Rev. Phys. Chem. 1974 25 95. J. J. Tyson and J. C. Light J. Chem. Phys. 1973,59,4164. C. H. Yang Comb. Flame 1974,23,97. 4D. L. Baulch D. D. Drysdale D. G. Hoare and A. C. Lloyd High Temperature Reaction Rate Data (Leeds University 1969) no. 4. GENERAL DISCUSSION This value is almost three orders of magnitude greater than what we proposed for reaction (3). As noted before the rate constants for reactions (3) and (4) may be changed to 5.0 x 1013 and 1.25 x lo8 respectively if only one thousandth of the total excited CO; molecules produced in reaction (2) are effective in reaction (3). All computed results will be invariant for such a change.Reaction (5) is another example. At T = 1000 K the rate constant for reaction (5) is greater than or at least comparable to the one we suggested for reaction (3). Dr. J. R. Bond (Leeds) said Yang's theoretical paper on carbon monoxide oxidation is a carefully fitted complex of elementary reactions and quite narrow restrictions must be placed upon kinetic parameters (e.g. relative third body efficiencies of COz to other gases) to ensure reasonable agreement with experiment. It is also an isothermal model although it is for a highly exothermic reaction. To discover whether temperature changes in real systems are indeed negligible we have studied both " dry " and " wet " oxidations of carbon monoxide using exceedingly finethermo-couples as probes.In all but one case,2 oscillations observed by previous workers have occurred in dry mixtures and we too have observed many oscillations in such systems. However when we deliberately add small amounts of hydrogen to the initially '' dry " system we find that oscillations are still readily obtained over a range of temperatures and pressures. The number of cycles observed in a closed vessel is always far less than for the "dry " case but the reactant consumption in each cycle is much greater. In a mixture containing 0.1 mol % of hydrogen ten or more oscillations can be observed and this mixture is very " wet " indeed when compared with the low hydrogen content of mixtures used in the investigation of " dry " oxidation phenomena. The most striking feature of the oscillations is the size of the temperature pulse accompanying 0 10 20 30 40 50 time/s FIG.1 .-Temperature pulses in a " wet " CO+02 mixture at 480°C and 30 tom each cycle.Temperature peaks exceed 30 C for the first few cycles ; they reduce to 10°Cas reactant consumption nears completion at the end of the sequence. In " dry " mixtures amplitudes are smaller and the train of oscillations in a closed system is longer. Thus although thermal effects may be small or even negligible in " dry " systems it is probably necessary to allow for self-heating in "wet " systems and the isothermal model may be inappropriate. Moreover even the small temperature pulses may be more that the casual effects of reaction pulses-they may interact integrally with the kinetics.C. H. Yang and A. L. Berlad J.C.S. Farnday I 1974,70 1661. J. W. Linnett B. G. Reuben and T. F. Wheatley Comb. Flame 1968 12 325 These oscil-lations were observed during first or "glow " limit determination by the heating method. GENERAL DISCUSSION Prof. C. H. Yang (Stonybrook N. Y.)(communicated) Our calculated trajectories by oscillating 0 atom concentration also compared very well with the recent measure- ments of the successive emission peaks from the CO and O2 system by McCaffrey and Berlad. Their results will be published shortly. The destruction of hydrogen-containing intermediates on the wall will probably produce relatively more stable molecular species H2 and H20. At the present time the detailed mechanisms of these heterogeneous reactions are far from elucidated.We simply assumed that only H,O is produced from these reactions to avoid the complication of introducing many additional reactions into our calculation. Eqn (27) represents a conservation statement of the total oxygen in the vessel. In the early phase of the oxidation process the consumption of water is usually limited to less than a few percent of its initial concentration. Calculated limits of explosion glow and oscillations are not likely affected even if the products of the heterogeneous reactions contain a small fraction of H2. For the calculation of a long sustained oscillation or glow the overall oxidation rate will probably accelerate if H is slowly accumulated to reach a significant level as the rate of the reaction 0+H2 -+ OH +H is undoubtedly faster than the rate of reaction (7).The general kinetic behaviour however will remain unchanged. Dr. A. Perche (Lilfe)said Yang's paper concerning simulation of periodic carbon monoxide explosions would certainly have been much more informative if a direct @ 1 4 I i 4 6 t/min FIG.1.-Pressure variation (Ap) and luminous intensity (I) versus time. (a) Usual evohtiori- reaction vessel tap opened. (6) Tap closed. (c) Tap initially opened closed after the first ex-plosion. (d) Tap initially closed opened after 15 min. R. McCaffrcy personal communication. GENERAL DISCUSSION comparison with experiments had been performed. Thus in the case of high temper- ature oxidation of methane and carbon monoxide mixtures a similar periodicity was observed by me in Lucquin’s laboratory.As is shown in fig. 1 this oscillatory phenomena in our case mainly depends on the diffusion of initial reactants from the external dead volume into the reaction vessel. Another problem is that water for- mation from heterogeneous radical destruction (reactions 8,lO and 14) each producing +H20,does not seem very clear. Prof. P. Gray (Leeds) said Perche makes a valuable experimental point about the carbon monoxide oxidation “ lighthouse ” and speaking as someone with both theoretical and experimental interests in the system I have not the slightest doubt in asserting that however hard the computer calculations may be they are far less difficult in this system than the experiments.The open tap is clearly important in Perche’s study and the repetitive entry of fresh reactant through it may indeed con- tribute to repetitive reactions in his particular system. An extreme case is furnished by old Russian work.2 However in Ashmore’s (1 939) study there was no dead space at all and oscillatioiis were still foundn3 They persisted apparently unchanged when the vessel was reconnected to a ‘‘ dead space ”. Prof. R. M. Noyes (Oregon) (cornrntmicated) After further consideration I have concluded the argument of Section 3 of the paper by B. F. Gray and Aarons does nof invalidate the Brusselator as a model for many successive oscillations in a closed system. Such a model would require that only a small fraction OftheAandBreactants be consumed during a cycle in the concentrations of the intermediates X and Y.A necessary but not sufficient condition for applicability to closed systems is that k2B > k $ k,. (1) An additional restriction imposed by stability analysis of the steady state is that Although I have not yet proved it I am convinced the requirement of minimal fractional depletion of A and B during a cycle imposes a lower limit on k3A2just as eqn (2) above imposes an upper limit. I also believe that for sufficiently large values of k4/kl all the ilecessary restrictions can be satisfied simultaneously. It therefore appears the Brusselator is an acceptable model for oscillations in closed as well as in open systems although such a model requires rate constant ratios somewhat different from those usually used by the Brussels school.If rate constants are properly selected the Oregonator is also a satisfactory model for an oscillator that only very slowly depletes the major reactants. These ideas have been developed further in a manuscript submitted to J. Chenz. Phys. Dr. P. Hanusse (CNRS Talence) said In theoretical chemical models postulating that pool chemicals are held constants is generally a mere definition of pool chemicals rather than a conjecture. On the other hand it is perfectly justified to check the physical implications of such a definition as it is for instance to wonder about the existence of elementary autocatalytic steps. Now B. F. Gray shows that several well-known models may be structurally A.Perche Thtse de 3tme cyck (Lille. 1970). Tokarev and Nekrasov Russ. J. Phys. Chem. 1936 8 504. P. G. Ashmore Nnrure 1951 167 390. I<. J. Field and R. M. Noyes J. Chi. Phys. 1974 60 1877. GENERAL DISCUSSION unstable with respect to the introduction of some new steps. We think that this result depends on the perturbation introduced or the way it is desned. For instance one may replace a step of form A+X + . . . A constant by the two following steps A. + A diffusion at system limits A. constant A+X -+ . . . and A variable but other physically meaningful ways are possible for instance in a steady flow reactor this becomes A. + A influx constant rate A +. .. outflux A+X 4.. . We have already made experiments which illustrate the control of pool species as well as the effect of a small parameter namely temperature.In a steady flow reactor where influxes are controlled we have studied an oscillating reaction derived from Bray’s reaction as proposed by T. S. Briggs and W. C. Rauscher. The evolution of the system is continuously recorded by electrochemical potential spectrophotometric absorption and temperature measurements. Many interesting phenomena occur in this system oscillations are perfectly sustained very stable in magnitude and frequency-better than 0.5 %. Studies may be achieved in iso- thermal or adiabatic conditions. In both conditions temperature oscillation is observed. In some conditions a double frequency oscillation may be observed as those shown by Sarensen ;the high frequency part is due to purely chemical oscillation and the low frequency oscillation seems to be very drastically dependent on temper- ature.Several stationary states have been found and transitions between them have been studied. Hysteresis phenomena occur also when varying the constraints on the system namely mass flows. We think that the investigation of such completely sustained systems is the only way to avoid artefacts and they may lead to interesting new informations on oscillating reactions since the theoretical prerequisites are fulfilled. Dr. B. F. Gray (Leeds) said As Hanusse remarks our results depend on the perturbation of the system which we consider but our main point is that physically realistic improvements in the description of the system (such as trying to say what happens to the “ pool ” chemicals whose concentrations are not quite constant) can alter the behaviour of the intermediates is.remove or produce oscillation. We agree with the comment about completely sustained systems and are ourselves opera- ting such a reactor (see comment on paper by Gray Griffiths and Moule this Symposium). In response to Noyes our section 3 does not invalidate the Brusselator as a model for many successive oscillations in a closed system in general we simply said that it does not oscillate in the region of parameter space chosen by Lefever and Nicolis for their computations. Your condition (1) is stated in the text in Section 3 and its compatibility with the condition for an unstable singularity (your condition 2) has been discussed and shown to be unlikely.Briefly besides (1) one also needs P. Hanusse Compt. Rend. 1975. B. F.Gray Kinetics of Oscillntory Kmrrioiis (Specialist Periodical Reports 30 The Chemical Society London 1974). GENERAL DISCUSSION k3 >> k2. To be a little more precise if we take X = A. = 1 = Yo = Bo for example then the “ low consumption per cycle ” requirements become k3 k,/c k4 N kilt. Substitution of these into the instability conditions gives k < 0 i.e. a contradiction due to the leeway allowed in the -signs. In view of the limited physical interest of the Brusselator we decided not to pursue the matter numerically but the main con- clusion from our paper on this general aspect is that in future the onus is on any proposers of new oscillatory shemes to show that it is stable with respect to small parameter stability of this type.Dr. R. Lefever (Brussels)said 1. The question whether trimolecular steps as in the Brusselator are “physically unrealistic’’ should not be decided in a general a priori way. From the paper of Matsuzaki ef al. at this meeting it can be seen that sometimes they may appear as a good approximation to the behaviour of realistic systems. Often such a step can be regarded as the overall step of several realistic partial steps e.g. (i) of the isomerization process 2x -+ z Y+Z+ 3x (ii) of the enzymatic chain E+X + (EIX) (E,X)+X + (EiXX) (E,XX)+Y -+ (EIXXY) (EIXXY) -+ E” +3X. It can also easily be seen that for some values of the parameters the allosteric model of the glycolytic oscillator which Goldbeter Boiteux and Hess have men- tioned at this meeting would present a third order molecularity or even higher if more than two subunits are considered for the enzyme.2. The criticisms with respect to the effect of small parameters look exaggerated. (a) The Volterra-Lotka model is structurally unstable in any event. Thus we know that the smallest modifications will alter significantly even the qualitative behaviour sometimes in completely opposite ways. For example if instead of considering diffusion of the initial products one considers diffusion of the intermediates and in- vestigates the equations a2x ax- X-XY+& - at dr2 i?Y a2y- -= XY-Y+& at i?r2 then however small E no sustained oscillatory behaviour is possible.If on the other hand one considers the following slight change in the equation then their solution turns out to be a spiral which blows up in the first quadrant. I A. Roiteux A. Goldbeter aid B. Hess Proc. Not. Acnrl. Sci. 1975. GENERAL DISCUSSION (b) For the trimolecular model it is obvious that keeping A B constant is equivalent to infinitely large flows of A and B or perfect instantaneous stirring inside the system. The main test for the consistency of a model is to have it working in a certain limiting case. This is certainly so for the Brusselator. After all in physics the model of a perfect pendulum without frictioii is generally unrealistic but quite useful.Furthermore our own work in Brussels (see for example the recent papers by Nicolis and Auchmuty,l and Kaufman-Herschkowitz 2 has already shown that by modifying the original Brusselator one can obtain new types of behaviour such as localized structures. Dr. B. F. Gray (Leeds) (communicated) I. I agree with Lefever that a third order reaction may be a representation of a more complex system involving perhaps only first and second order steps but in making such an approxiination one is introducing yet another small parameter relating for example [EIXXY]to X and Y besides the original one relating X and Y to A and B. It may be possible to do this correctly but our main point is that this cannot be assumed. 2a. We do not agree that our criticism of the method of neglecting sinall parameters is exaggerated.The Lotka-Volterra model is well known to be structurally unstable (i.e. a characteristic root is zero) but we are not discussing this type of stability here we are discussing stability with respect to a small parameter and consequent increase of the degree of the characteristic equation producing new roots. If one or more of these is positive then it is possible for the system to be unstable even though the unperturbed system was structurally stable i.e. all its characteristic roots were <0. The other examples in our paper treat systems which are not structurally unstable and the structural instability of the Lotka-Volterra system is in this context incidental. 2b.The reference to a perfect pendulum without friction is completely misleading in this discussion since including friction does not increase the degree of the character- istic equation of the number of variables in the problem. This is a simple pertur- bation problem with no possibility of the type of instability we are discussing which is essentially characterised by non-uniform convergence in the neiglibourhood of E 3 0. On the contrary the convergence of a damped pendulum to an undamped one is completely uniform as the damping factor tends to zero i.e. the solution of the damped pendulum equation d2Y dY M-+E-+K~ =O dt2 dt which is y = exp( -Et/M} sin(K/M)ft is well approximated by the undamped solution yo = sin(K/M)*t provided t < M/E,and as E -+0 this interval gets larger and larger.The correct analogy within the realin of damped springs is where the mass of the spring tends to zero and the unperturbed differential equation is only first order dy/dt +K’ = 0; hence a spring with zero mass is not a sensible approximation to a spring with a small mass however small this may be. This is a singular perturbation problem as compared to Lefever’s example which is a secular perturbation problem. Dr. B. L. Clarke (AZberta)said The term “jump point ” used in Section 1 of the paper by Gray and Aarons can have two interpretations. The term is used in the G. Nicolis and A. Auchmuty Proc. Nat. Acad. Sci. 71 1974 2748. M. Kaufman-Herschkowitzand G.Nicolis J. Chem. Phys. 1972 56 5. S 9-6 162 GENERAL DISCUSSION paper to refer to the values of y for which the curve F(x ;y) = 0 has tangents which are perpendicular to the y axis.If eqn (2) moves x to a stable pseudosteady state on this curve x will essentially be determined by y. As y passes through the “jump point ” 5 becomes infinite briefly. However the trajectory (x ;y) through this point may still be a smooth curve. The sudden changes in the dynamical variables shown in the papers by Field and Noyes and by Franck are usually caused by a situation which is the second inter- pretation of &‘jump point ”. Since the stability of the steady state of eqn (2) depends on y the curve F(x;y) = 0 may have regions of stability and instability for pertur- bations Ax ofthe curve. The trajectory (x;y)will follow this curve in the regions of stability and jump off the curve at the points which separate the stable and unstable regions of the curve.The rapid evolution of x at essentially constant y which follows can move x into a limit cycle around the curve. If the solutions of F(x;y) = 0 arc multivalued for a given y the evolution may go to a second branch of the curve which is stable. The “jumping off” points of the curve are found by examining the Hur- witz determinants of the matrix (8F,/8xi). If the general equations of motion are used instead of eqn (I) the variables can often be treated as slowly evolving variables like y over part of their range and rapidly evolving variables like x over much of the remainder of their range. The variable q of the Oregonator behaves in this fashion.The argument of the preceding para- graph can still be used; however the stability of points on the possible curves must be determined for fluctuations in all the variables. Dr. A. Babloyantz (Brussels) said In connection with the paper of Gray and Aarons I would like to mention that structural stability formalism has already been applied to biological problems in the context of prebiotic evolution of informational macromolecules.1 It can be shown that a favourable mutation giving rise to a new macromolecule although in a small quantity may take over and destabilize the origin- ally stable system. Dr. B. L. Clarke (AZberta) said Regarding the paper by Gray and Aarons the model reaction systems which become unstable when the pool chemicals are allowed to fluctuate suggest a question.“ Which network features are necessary and sufficient to guarantee structural stability with respect to the inclusion of the pool chemicals? ” Several theorems given in a paper on the stability of topologically similar chemical networks give conditions which ensure that including the pool chemicals in the dynamics will not alter after the stability. If a pool chemical enters the network through a decomposition reaction of the form A + . . . permitting A to fluctuate is equivalent to adding a reactant X to the network as follows A 3X -+ . . .. I call the reactant X a “ type B flow through reactant ” (FTRB) and from Theorem 5 of my paper it follows that for large steady state concentrations of X the network is stable if and only if the network with X eliminated is stable.The only other way a pool chemical may enter a network is by reacting directly with other reactants as in A+Y + . . .. If A is allowed to fluctuate such a reaction is replaced by the following pair of reactions A -+ X and X + Y -+ . . .. I call the reactant X a ‘‘ type A flow through reactant ” (FTRA) and from Theorem 8 of my paper the following can be said if the original network has deficiency zero and if the reactant Y does not appear by itself on the left or right hand side of any reaction * I. Prigogine G. Nicolis and A. Babloyantz Physics Tuday November and December 1972. B. L. Clarke J. Chem. Phys. 1975 62 3726. GENERAL DISCUSSION of the extended network the extended network also has deficiency zero.This theorem is an extension of work by Horn Feinberg and Jackson and the significance of the deficiency is explained in the immediately following comment by Feinberg. Roughly speaking if the original network has deficiency zero it is stable and because the freeing of the pool species does not change the deficiency the extended network is also stable. From the Hurwitz determinants one can understand in general why pool chemicals which enter the network via reactions of the form A +Y -+ . . . sometimes destabilize the network when they are allowed to fluctuate. The extended network has a new positive feedback loop in which A and Y inhibit one another. Such loops will de- stabilize a network if they pass through reactants which have marginally strong autocatalysis such as Z+Y + 2Y and in addition certain other conditions are met.In all the examples of Gray and Aarons' paper the pool chemicals enter the network in this manner. Dr. M. Feinberg (Roclzester) said In the discussion regarding B. F. Gray's paper Clarke made reference to some theorems proved Horn Jackson and Feinberg which might bear upon the problem of how stability might be affected if a physico-chemical model is structurally perturbed. Horn and I have published a description of the most intriguing of these theorems.' Perhaps I should explain this theorem and some newer results (in terms more schematic than precise) and offer my views regarding their utility in answering questions of the type raised by Gray.According to a rather simple formalism described in the aforementioned article there can be assigned a non-negative integer (called the deficiency 6) to every chemical mechanism according to its algebraic structure. (Some of the reactions of a mech- anism might be " pseudo-reactions " incorporated to reflect special physico-chemical effects for a system under study). Thus mechanisms can be classified according to whether they are of deficiency zero one two etc. Mechanisms of deficiency zero are surprisingly common and it is for these that we have been able to prove what I think is an interesting theorem for open hoino- geneous reactors. For arbitrary positive rate constants (in the context of the usual mass-action kinetics) the existence of an equilibrium for which all species con- centrations are positive suffices for preclusion of pathological statics and dynamics e.g.sustained composition cycles. Moreover there exists such an equilibrium if aid only if the reaction " arrows " in the mechanism are directed such that the mechanism is what we call "weakly reversible ". The weak reversibility constraint is somewhat limiting since most models people have been considering these clays do not fall into this category. However since publication of the article cited above I have been able to prove the following. For mcchanisms of deficiency zero which are izot weakly reversible it is true that for arbitrary kiiretics (subjcct only to very weak constraints) the dynamical equations for open homogeneous systenis cannot give rise to sustained temporal composition cycles for which at some time all species concentrations are positive.(In fact this can be shown to hold for a class of inechanisms somewhat broader than those of deficiency zero.) Moreover for mechanisins of deficiency zero governed by mass-action kinetics with arbitrary positive rate constants sustained temporal composition cycles cannot be obtained at a11 whether or not the meclzarzism is weakly reversible. That is all mechanisms of deficiency zero taken with mass action kinetics are loosely speaking of stable character. ' M. Feinberg and F. Horn C/iet)t.Etiy. Sci. 1974 29 775. GENERAL DISCUSSION It can be shown that if one begins with a mechanism of deficiency 6 the addition of reactions to that mechanism results in a new mechanism of deficiency not less than 6.Similarly if one begins with a mechanism of deficiency S and one removes reactions the resulting mechanism has deficiency not greater than 6. In each case the deficiency may be unchanged. These ideas might in some instances help in deciding the effect of structural perturbations in a model (resulting in addition or removal or reactions) upon stability characteristics. If one begins with a mechanism of deficiency zero and one adds to it new reactions the resulting mechanism may also be of deficiency zero so that it falls within the realm of the theory. Thus the "perturbed " model will be of stable character. On the other hand if one begins with a mechanism of deficiency in excess of zero (perhaps exhibiting static and dynamic "exotica ") and one structurally perturbs the model by removing reactions then the resulting mechanism may have deficiency zero and the modified model will have essentially stable character.I might add that mechanism deficiency has a surprising bearing upon matters quite divorced from stability considerations (e.g. upon the determinability of com-plete sets of rate constants from certain classes of dynamic experiments). I have discussed these matters in a chapter of the forthcoming Wilhelm Memorial Volume on Chemical Reactor Theory several other chapters of which (particularly those by Luss and Bailey) might hold interest for participants in this Symposium. Dr.H. Tributsch (Berlin) said There have been many theoretical investigations on the Lotka Volterra model but there has been up to now a lack of experimental systems which permit a verification of conclusions derived from it. For this reason I would like to present a new and simple oscillating system which in my opinion corresponds to the first example which has been discussed in the paper by B F. Gray namely to that of a Lotka Volterra oscillator which has to be considered as being perturbed by small parasitic parameters. The system consists of a crystal of CuzS or Cu,FeS jn contact with an electrolyte that contains hydrogen peroxide. Oscil-lations appear during the reduction of H202at the sulphide electrode above a certain -0.9 -0.e -0.7 -0,s -0.5 -0.L -O,3 -0.2 -0.1 electrode potential USCE/V FIG.1.-Dynamical reduction curves for HzOz on CuSFeS4 electrode.GENERAL DlSCUSSlON 165 minimum potential. Fig. 1 shows two reduction curves which have been recorded at 2 mV/s in opposite potential-directions. For this system we could not only derive a reasonable kinetic scheme of the Lotka Volterra type but we were also able to find clear experimental evidence for the behaviour which is considered to be characteristic of this kind of oscillator especially an unstable frequency which is dependent on the initial conditions of the system as well as a complementary amplitude-frequency correlation which is a result of its conservative character and of a constant entropy production. (Consider the amplitude decrease which coincides with an accidental increase in the oscillation frequency in the central portion of one of the reduction curves (fig.1)). A closer investigation of the dynamics and shape of oscillations has shown that they are usually very periodic and well formed in the central region of the reduction curve but that they are often composed of compact spike groups of fast rising amplitude in the border regions where now and then oscillations also fail to occur. The available data are all consistent with the concept that the investigated system is basically of the Lotka Volterra type. Its instability is however so pronounced that it will-depending on the predominant chemical perturbation reactions-either slip into a limit cycle type of oscillation (case discussed in Gray’s paper) or result in a gradually rising or damped oscillation (cases discussed by Frank-Kamenetskii and Sal’nikov ’).I D. A Frank-Kamenetskii and I. E. Sal’nikov Zhur. Fiz. Khim. 1943 17 79.

ISSN:0301-5696    

DOI:10.1039/FS9740900150

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

C. Membranes Heterogeneous and Biological Systems Factors Controlling the Frequency and Amplitude of the Teorell Oscillator R.PAGEAND PATRICK BY KENNETH MEARES* Biophysical Chemistry Unit Department of Chemistrv The University of Aberdeen Old Aberdeen Scotlaqd Received 12th July I974 The properties of the Teorell oscillator are analysed by considering a membrane containing parallel cylindrical pores with a low surface charge density. Particular attention is given to alter- ations in the properties produced on varying the pore diameter and surface charge. Brief considel- ation is given to the relation between the behaviouc of the oscillator and of biological mechano- electric transducers. The Teorell oscillator is a device which makes use of the coupling of ion and water fluxes within a highly porous membrane of low internal charge.It was originally devised as an analogue for the study of biological excitability.' It is particularly interesting in relation to the problems of mechano-electric transduction i.e. the transformation of mechanical stimuli into electrical signals which is found in organs such as baroreceptors. In this paper some factors which affect the frequency and amplitude of the oscillations are examined because these may have a bearing on the study of natural mechano-electric transduction. The analysis relies on a theory developed by Meares and Page 2* which has been experimentally verified using Nuclepore filters supplied by General Electric. These filter membranes have an exceptionally well-defined and uniform parallel pore structure.In the Teorell oscillator the membrane separates two well-stirred electrolyte solutions of different concentrations. Each solution is contained in a compartment with a vertical capillary on the top. A net movement of solution through the mem- brane therefore generates a hydrostatic pressure opposing the flow. When as in the case examined here the membrane surface charge is negative an electrical potential across the membrane generates an electro-osmotic flow in the direction of the cathode. The solutions are arranged so that this flow takes place from the dilute to the concen- trated side. Sustained oscillations of the hydrostatic pressure AP and electric potential AY across the membrane may be produced when a constant electric current greater than a critical minimum is passed from one compartment to the other.The cycle of events can be visualised as follows. Initially Ap is low and the electro-osmotic flow causes dilute solution to enter the membrane pores. Thus the membrane conduct- ivity is low and the potential AY is large. As the hydrostatic pressure builds up the flow progressively reduces until a point is reached when the direction of flow is re- versed. The membrane then fills with the concentrated solution and its conductance rises. Hence the potential difference decreases and the reversed flow allows the pres- sure to fall until a point is reached when electro-osmosis once again takes control and a new cycle commences. 166 K.R. PAGE AND P. MEARES THEORY Attention is restricted to the case of a Nuclepore filter membrane separating two solutions of sodium chloride. It has been found empirically that the negative surface charge density on the membrane is directly proportional to the cube root of the salt concentration in contact with it. Owing to the uniform pore structure of the membrane the overall flow may be analysed in terms of that in a single pore. Flows from the dilute to the concentrated solution will be assigned positive values. The motion of the fluid is governed by a balance between four forces the hydro- static pressure P,,the electro-osmotic force PE,the viscous drag force Fqand an inertial force caused by accelerations of the fluid. It has already been shown that the inertial force may be neglected in the cases considered here.3 The viscous drag force Fq is obtained from the Navier-Stokes equation the result being Fq = -8qvl/a2 (1) where u is the volume flow per pore a the radius and I the length of the pore.q is the coefficient of viscosity of the fluid in the pores. P,is related to the difference in height between the menisci in the two vertical columns. The geometry of the cell permits the variation of P with time to be related to the volume flow by u = (l/yA)(dP,/dt). (2) Here y and 2 are constants of the system and include the total area of pores open to flow and the cross sectional area of each vertical tube. The electro-osmotic force also is a function of the volume flow because of convect- ive coupling between the ion fluxes and the flow of fluid.Either the dilute or the concentrated electrolyte solution is swept into the pores to an extent which depends on the magnitude and direction of the volume flow and the thickness of the electrical double layer. Provided the current is held constant the magnitude of the membrane potential varies accordingly with the concentration profile in the pore. Although this concentration profile cannot change instantaneously with changes in u provided dv/dt is sufficiently small the difference between the actual profile at any instant and the one appropriate to a stationary state will be negligibly small. When this holds it is possible to estimate PEby using the stationary-state equations given elsewhere.2 In order to do this the ion fluxes are expressed by the Nernst-Planck equations extended to include a term for convective flow and the Gouy-Chapman theory is employed to describe the electro-osmotic component of the volume flow.The local equations have to be averaged over the pore cross-section and integrated along its length. Allowance must also be made for the presence of unstirred layers of solution immed- iately adjacent to the membrane faces. The final result can be expressed in the form PE =f(Cm cji i V) (3) where c and cp are the concentrations on either side of the membrane and i is the current density in the pore. It is convenient to regard the forces P,and PEas being in opposition and neglecting the inertial force to express the force balance on the fluid by PE-pc = Fq.(4) When eqn (1) to (3) are taken into account eqn (4) can be expressed P = PE -Fq =f(dP,/dt). (5) The complete functional relationship in eqn (5)is complicated and the detailed formula- tion given by Meares and Page,3 shows that it can be related to the Van der Pol THE TEORELL OSCILLATOK equation. Without going into the details of this relationship its properties will be discussed here with the aid of numerical solutions. FIG.1 .-(Pc V,) limit cycle when I > I*. Plot calculated for membrane 2MA at a current density of 690 A m-2. Fig. 1 illustrates the relationship between P and the total volume flow in the membrane V, calculated for membrane 2MA at a total current density I greater than the critical minimum I* required to produce oscillations.V and I are related to the corresponding quantities in a single pore by eqn (6) and (7) V, = na2Nv (6) I = .na2Ni (7) where N is the number of pores per unit area of membrane. The curve in fig. 1 represents the solution of eqn (5) calculated by using the surface charge density pore density and hydrodynamic permeability of membrane 2MA. These quantities were determined in separate experiments. The system is in a stable state when on curves AB and CD. On curve AB the volume flow is negative and concentrated solution is being dragged into the pores. In this state the behaviour of the system is controlled chiefly by the hydrostatic pressure. Curve CD represents the opposite condition ; dilute solution is being dragged into the pores and the flow is dominated by electro-osmosis.On curve BD the system would be unstable and this region is inaccessible under constant current conditions. Points B and D mark transitions between the two stable states. Point E at which Vm would be 0 lies on DB and the system cannot therefore achieve a truly stationary state. Instead the state of the system will constantly progress around the closed path ABCD that represents the limit cycle of the oscillations. Provided the times taken to traverse BC and DA are small compared to those taken along AB and CD the period of the oscillations z is given by K. R. PAGE AND P. MEARES In fig. 2 the relationship between pressure and volume flowis shown for the same membrane at a current density less than the critical value I*.The point E at which V = 0 now lies on curve AB and hence represents an accessible state When lis less than I* the system will move around the curve BCDA until E is reached at which it will have attained a truly stationary state. If the curve were plotted for the case Z = I* the turning point B and the zero point E would coincide. 50 + L 0- 2 -5 0 FIG.2.-(Pc V,) limit cycle when Z c I*. Plot calculated for membrane 2MA at a current density of 250 A m-2. DISCUSSION The theoretical equations permit the calculation of the limit cycle and period of oscillation for a given set of membrane characteristics electrolyte soIutions and electric current density. They also indicate the value of Z*.The theory has been tested under a variety of conditions with several types of Nuclepore membrane^.^ Table 1 lists three of the results in order to indicate the extent of agreement between 1.-CHARACTERISTICS TABLE OF(Pc V,) LIMITCYCLES AND PERIODSFOR 0.1M-0.1 M NaCl IN THREE MEMBRANES [Units L -rnPa-ls-l; K=rnCm-'kg-l; a=prn; I=Am-2; P,=kPa; Vm = pm z = s. The letter bracketed after V denotes point on limit cycle shown in fig. 1.1 membrane 2MA L = 2.35~ lo-*; K= -9.84; a = 0.22; I= 690 Pc mnx Pcmia Vm(A) Vm(B) Vm(C) Vm(D) r observation 3.12 1.30 -78 -2.4 166 27 1194 prediction 4.21 1.06 -84 -3.9 119 33 1463 membrane 6MC L = 5.62~ lo-*; K= -5.21; u = 0.42; I= 1070 observation 1.15 0.396 -73 -3.5 130 50 576 prediction 1.28 0.340 -62 -4.6 91 26 519 membrane 2MD L = 36.2~ lo-* ;K = -4.35 ; a = 0.76;I = 1790 observation 0.262 0.058 -110 -6.1 180 42 90 .prediction 0.366 0.075 -120 -6.8 160 42 93 THE TEORELL OSCILLATOR prediction aiid observation.Each example refers to the same pair of electrolyte solutions 0.1 M and 0.01 M NaCl. The values of the hydrodynamic permeability L and the proportionality constant relating the surface charge density to the cube root of concentration K for the mem- branes were measured in separate experiments. The pore radii a were calculated from Lpand the pore density N which was obtained from photomicrographs of the membrane. The observed quantities which characterise the oscillations were taken from recorder traces such as fig. 3 which shows the pressure oscillations produced with membrane 6MC.The volume flows were calculated from the slopes of the oo 500 iooo 15CJO 71s FIG.3.-Oscillations of Pc observed using membrane 6MC. The positions of the turning points on the (Pc Vm) limit cycle are indicated ABCD. pressure traces. The predicted quantities were calculated by using the appropriate values of L, K c, ca and i. It can be seen from table 1 that the correspondence between prediction and observation is generally good the worst discrepancy being the overestimation of the maximum pressure Pc for membrane 2MA. A variety of work has shown that many biological mechano-electric transducers respond to strain rather than In organs such as the Pacinian corpuscle the sensory process appears to involve a two-step mechanism in which a sensory membrane produces a graded potential (generator potential).This in turn triggers off an oscil- latory discharge at an associated afferent nerve ending.4 It is possible however that some receptors operate by a single step process. In these the sensory membrane produces the oscillatory discharge directly. The baroreceptors in the carotis sinus and crustacean muscle stretch receptors may be examples of the single step mechan- ism.1 It is of interest therefore to examine what effects strain might have on the properties of the oscillator studied here. The tensile strength of the Nuclepore filters precludes any direct experimental study of strain in the present system. As indicated in table 1,however a series of Nuclepore filters with different pore radii and surface charges was studied.The consequences of these differences in properties were successfully described by the theory and it is possible therefore to investigate theoretically the effect of varying either the pore radius or the surface charge. Thus one may infer the effects of radial tension upon the behaviour of an extensible membrane which had otherwise exactly the same properties as a Nuclepore filter. As shown by Burt~n,~ a membrane of this type should be highly sensitive to strain provided its Poisson ratio was not too low. If such a membrane were stretched the pores would act as foci of stress and a small increase in the membrane diameter would cause a disproportionately greater increase in pore diameter.Fig. 4(i) shows the effect on the limit cycle of varying the pore radius at constant K. The points labelled A B C and D correspond exactly with those shown in fig. 1 for membrane 2MA at a current density of 690 A m-2. To clarify the presentation,- the pairs of points A-B and C-D are joined by straight lines. The lines labelled (a) K. R. PAGE AND P. MEARES PJkPa PJkPa FIG.4.-(i) Effect of altering pore radius on (Pc Vm) limit cycle. Lines AB and CD join turning points for membrane 2MA. Lines (a) and (6)correspond to AB and CD when the pore radius is 0.81 and 1.44 respectively times that of membrane 2MA. (ii) Effect of altering the surface charge constant on the (Pc V,) limit cycle. Lines AB and CD join turning points for membrane 2MA.Lines (a) and (b) correspond to AB and CD when the surface charge constant is 0.81 and 1.44 respectively times that of membrane 2MA. TABLE 2.-EFFECTS OF CHANGES IN PORE RADIUS a AND SURFACE CHARGE CONSTANT K UPON TIMES TAB TDC AND THE PRESSURE AMPLITUDE hp [Units a = pm; K = mC m-l kg-*; T~,TDC= s; AP = kPa. a' K' APL values appropriate to the experimental run on membrane 2MA.1 U K ala' K/K TAB TDC ~AB/~DC APc APGIAPc' 1.78 9.84 0.81 1.00 2316 1080 2.14 7.76 2.44 3.17 9.84 1.44 1.00 264 78 3.38 0.63 0.20 2.20 9.84 1.00 1.00 930 420 2.21 3.15 1.00 2.20 14.20 1.00 1.44 1068 552 1.93 5.37 1.69 2.20 7.96 1.00 0.81 912 348 2.62 2.24 0.72 correspond to AB and CD calculated for pores of radius 0.81 times those in membrane 2MA.The lines labelled (b) were similarly calculated for pores of radius 1.44 times those in 2MA. It may be seen that the extremum values of P,,ie. Pc(max) and P,(min) both increase as the radius decreases. The amplitude of the pressure oscillations APc also increases (see table 2) and the slopes of AB and CD decrease. Table 2 lists the times TAB and rCDtaken to traverse AB and CD respectively. Clearly the period increases as the radius decreases and TAB becomes smaller relative tozcD. As a result of becoming closer to unity the oscillation curves become more nearly symmetric with decreasing pore radius. THE TEORELL OSCILLATOR The changes caused by alterations in the surface charge constant Kat constant pore radius are shown in fig.4(ii). Here also points A B C and D correspond with those in fig.1. The lines labelled (a) were calculated for a charge density 0.8 1 times that of 2MA and those labelled (c) for a charge density 1.44 times that of 2MA. Although Pc(max) Pc(min) and AP increase as the charge density is increased the change is not as marked as for an equal proportionate change in the pore radius. This can be seen from the last column of table 2 which lists the relative changes in APc. The slopes of the lines AB and CD are scarcely changed when Kis varied. As a result the period of the oscillations changes less for a given change in K than for an equal change in a. This effect is illustrated in fig. 5 where the relative period is plotted as a function of relative pore radius and relative charge density.2.0 3-0i 0' I 1.0 1.5 (4ala' (b)KlK' FIG.5.-Relative period r/r' plotted against relative pore radius a/a' {line (a))and relative surface charge constant KIK' (line (b)}. Primed quantities denote properties of membrane 2MA. The great sensitivity of the system to changes in pore radius arises from the depend- ence of L on the fourth power of the pore radius whereas electro-osmosis is more nearly related to the first power of K. These considerations explain why dust proved to be a major hazard in early work with this system ; repeatable results were obtained only after carefully cleaning the apparatus and filtering all solutions. During a typical early experiment K was found to remain reasonably constant while L changed markedly owing to the progressive accumulation of small particles in the pores.Although it is unlikely that all pores became silted in the same uniform manner the results indicated the great sensitivity of the system to L,. Fig. 6 shows the trace 20 > \ % IG 4 0 0 1000 2030 71s FIG.6.-Oscillations of pressureP,line (a)and potential between probe electrodes A y5 line (6)observed using a membrane of pore radius 0.8 pm. At point A the hydrodynamic permeability was 32 x m Pa-' s-' and at point B 17x lo-* m Pa-' s-l K. R. PAGE AND P. MEARES obtained in an experiment during which L fell to 54 % of its original value. P,(max) and Pc(min) increased as also did AP, and the oscillation period lengthened as the run progressed. These features are all in agreement with the predictions made in the discussion above.This investigation has shown that a radial extension would increase the frequency and decrease the amplitude of the oscillations in a Teorell oscillator. When the pore walls are extended the surface charge density is likely either to remain constant or to decrease. If the latter occurred this would reinforce the changes produced by the increase in pore radius. As shown in fig. 4 an increase in pore radius and a decrease in Keach move point B closer to the line V = 0. If B reached this line oscillations would cease because the current density applied would then be less than the critical current density I* of the stretched membrane. Too great a stretch would therefore stop the system from oscillating and allow a stationary state to be reached a pheno- menon known as " overstretch ".Lack of space prevents a detailed discussion of the variations in membrane poten-tial but the frequency of its oscillations must match those of pressure. Experiment-ally the amplitude of the membrane potential oscillations was found to be less sensitive than APc to changes in L (see fig. 6) because most of the potential drop recorded occurred between the sensing electrodes in the solutions and the membrane surfaces. There are some similarities between the predicted properties of the membrane oscillator under stretch and the behaviour of natural mechano-electric transducers. An increase in frequency with increasing strain is observed in most such organs whilst some also show the " overstretch '' phenomenon.For example the oscilla- tions in cray-fish stretch receptors cease if the stretch is excessive.6 This behaviour is reversible and oscillations recommence when the strain is reduced. It is possible that some biological transducers rely on processes that phenomeno- logically resemble the mechanism analysed in this paper and this conclusion is inde- pendent of the nature of the processes occurring at the molecular level. Structurally the present system differs in a number of important ways from a biological tissue and a detailed correspondence with natural processes should not be sought. The present findings show only that a single step mechanism involving the deformation of a per- meable membrane may provide a useful working model with which to guide future investigations in this field.'T. Teorell Handbook of Sensory Physiology ed. W. R. Loewenstein (Springer Verlag Berlin 1971) vol. 1 chap. 10. P. Meares and K. R.Page Phil. Trans. A 1972,272,l. P. Meares and K. R. Page Proc. Roy. SOC.A 1974,339,513. W. R. Loewenstein Cold Spring Harbour Symp. Quaiit. Biol. 1965 30 29. A. C. Burton Permeability and Function of Biological Membranes ed. L. Bolis et al. (North Holland Publishing Co. Amsterdam 1970) p. 1. C. Terzuolo and Y. Washizu J. Neurophysiol. 1962 25 56.

ISSN:0301-5696    

DOI:10.1039/FS9740900166

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

Electrical Oscillatory Phenomena in Protein Membranes BY VICTORE. SHASHOUA McLean Hospital Biological Research Laboratory and Department of Biological Chemistry Harvard Medical School Belmont Mass. 02178 U.S.A. Received 5th August 1974 Membranes were prepared by interacting polycations with polyanions at an interface to give a structured system in which a cationic phase was separated from an anionic phase by a neutral poly- ampholyte zone. Such a membrane system with a cation +anion junction exhibits electrical oscillations in an electric field. Measurements of the (current voltage) characteristics of the membranes under current clamp conditions shows a "negative resistance " region coincident with the polarization conditions required for producing electrical oscillations.Both proteins and poly- nucleic acids can be used as the polyelectrolyte components of the membrane. Biological membranes exhibit many types of electrical oscillatory properties. These include the relatively slow phenomena characteristic of plant cells and the fast events observed in nerve and muscle cells. Tasaki and Takenaka,I in an analysis of the electrical properties of the squid giant axon demonstrated that substantially all the electrical excitability and conduction properties of axons can be attributed to the cell membrane alone i.e. these properties remain intact even when practically all cytoplasmic components have been perfused out of the axons. Thus the axonal membrane consisting of a 70A thick layer of proteins coinplexed with lipids can generate the electrical properties of neurons.A number of model systems have been proposed for simulating various aspects of the excitability properties of axonal membranes. One of the first models was described by Lillie.2 He showed that an iron wire covered with glass tubing can propagate an electrical impulse in a manner suggestive of the characteristics of myelinated nerve fibers. In 1954 Teorell and subsequently Franck showed that glass membranes can generate slow oscillatory electrical signals. More recently lipid membrane systems based on the bilipid layer concept of Davson and Danielli have been the subject of many investigations.6-10 Mueller and Rudin showed that when lipid bilayers were modified by certain macromolecules they then could generate many of the electrical characteristics of neuronal membranes.This type of model suggests that specific macromolecules can convert a lipid bilayer which has the high electrical resistance characteristics of a good insulator (lo8 ohm/cm2) into a membrane with a low resistance (lo2 ohm/crn2) and a capacity to generate electrical oscillations. In addition "semiconductor like " properties are obtained for some protein-lipid interactions. Clearly no such properties can be predicted from an analysis of the bulk properties of either the lipid or the protein constituents of the membrane. Both classes of these molecules are insulators in the dry state. Lipids behave like detergents and proteins become polyelectrolytes in aqueous solutions. In an effort to find out if there are any fundamental characteristics of proteins or more generally polyelectrolytes which may be useful for defining the electrical properties of biological membranes we explored the possibility that specific changes might occur when polyelectrolyte membranes are organized into layered structures.* 174 V. E. SHASHOUA In this way we attempted to simulate the interaction of proteins with charged lipid monolayers organized in a smectic phase. The possibility exists that new properties may be obtained following such an interfacial interaction. ELECTRICALLY ACTIVE POLYELECTROLYTE MEMBRANES In the initial experiments aqueous solutions of polyacids were layered onto solutions of polybases.12 The membranes produced at the interface were found to be capable of generating random electrical oscillations in an applied electric field (see fig.1 A). In subsequent work,13 some membrane systems produced sustained electrical oscillations (see fig. 1B C,D). Essentially these membranes could simulate a transduction process with the properties of an electrical oscillator circuit to convert a d.c. potential into an a.c. output with " spike-like " characteristics. The amplitude (1-1OOmV) and duration (1-10ms) of the oscillations were of the same order of magnitude as those of the neuronal spikes. These properties were found to be directly attributable to the " sandwich-like " structure of the membrane in which a cation exclusion barrier (polycationic phase) was separated from an anion exclusion barrier (polyanionic phase) by a neutral polyampholyte zone.This type of structure may be called a polycation c-)polyanion (c f-) a) junction membrane. The double arrow in the c t)a symbol is used to designate the presence of a neutral polyampholyte layer between the two poIyelectrolyte phases of the membrane. FIG.2.-Experimental arrangement for study of electrical oscillatory properties of c ++a junction membranes :S Agar salt bridges with 0.15 N NaCl ;V and A are a voltmeter and ammeter ;output to oscilloscope is through a d.c. amplifier with high impedance ;insert shows diagram of the structure of a matrix supported c Ha membrane. In a typical experiment a c c-)a membrane separates two compartments containing 0.15 N NaCl (see fig. 2).The current is passed through the system via two agar salt bridges containing 0.15 N NaCl. These connect each compartment to a silver/ silver chloride electrode-immersed in 0.15 N NaC1. The potential across the membrane is detected by means of two Ag/AgCl electrodes connected to a d.c. differential amplifier with a very high input impedance (Metametrics Corp. Cambridge Mass.). This essentially draws no current from the system. The output of the amplifier is fed into an oscilloscope to display the pattern of the potential changes obtained. When a current is passed through the membrane to drive anions into the polycation phase and cations into the polyanion phase a sequence of electrical and 176 ELECTRICAL OSCILLATORY PHENOMENA IN PROTEIN MEMBRANES mechanical events takes place as a function of the applied voltage.At first the record- ing electrodes show that there is an instability region and that the output voltage recorded across the membrane becomes very sensitive to mechanical vibrations. Further increase in the applied voltage results in a loss of the mechanical instability followed by the generation of electrical transients with " spike-like " characteristics. Higher voltages produce a breakdown of the membrane. Thus there is a critical voltage range at which oscillations take place. MATRIX SUPPORTED c ++ a MEMBRANES The preparation of c c-) a membranes by the direct interaction of a polyacid with a polybase was found to be difficult to control. The membranes frequently had regions of imperfection and holes where short-circuiting occurred.In order to obtain a more experimentally feasible situation a matrix supported system was developed. This was achieved by using a matrix membrane to act as a neutral hydrophilic support polymer for the two polyelectrolyte phases. Fig. 2 shows a diagram of the cross- section of such a polymer matrix membrane cemented in place at the 2 mrn aperture of a cellulose nitrate tube separating the two electrode compartments. The membrane is generally about 1200A thick and contains a distribution of pores of 50-3000Ain diameter as shown by electron microscopy. This " sieve-like " structure acts as a support for the two polyelectrolyte components which are electrophoretically loaded into the films to produce a c w a membrane.The matrix membrane was prepared by evaporating thin layers of chloroforin solutions of the polyamide poly (sebacyl piperazine). (Pip-8) l4 onto a glass plate. The polymer and solvent were care-fully purified to eliminate contamination with dust particles water and traces of other organic matter. Pip-8 has a combination of properties which provide for a suitable matrix material. It is a neutral hydrophilic but water-insoluble polymer and has no peptide NH groups which could promote denaturation of any proteins that may be used as membrane additives. It generates no oscillatory activity in an electric field. The porous structure of Pip-8 is prepared by adding varying amounts of a water soluble impurity to g/ml) such as glycerol or polyethylene glycol to the 0.5 % Pip-8 solution.These are incompatible with Pip8 in the solid phase but remain in solution in CHC13. The porous Pip-8 structure is obtained because the added impurities separate into isolated regions as the film dries. These can then be extracted out with water. Fig. 3 shows a series of electron micrographs of the Pip-8 membranes showing the effect of additives to produce the porous types of films as well as the polyelectrolyte loaded systems. Experimentally the Pip-8 matrix is loaded simultaneously with a polyacid and polybase from opposite sides. The conditions of loading such as concentrations of the polyelectrolyte viscosity of the solutions and pH were adjusted so that the mobilities of the polyacid and polybase were the same and that they could interact within the pores to form films and thus the required barrier for a c c.)a structure.This type of experimental method was used to investigate a variety of synthetic polyelectrolytes proteins and polynucleic acids as membrane components (see table 1). EXPERIMENTAL PREPARATION OF MATRIX MEMBRANES The polymer for the matrix membrane was synthesized by the interfacial polycondensation method from sebacyl chloride and piperazine (Eastman Organic Chemicals Co.).l4 The polymer was rigorously purified by repeated extraction with 1 M sodium carbonate and water. The wet polymer was then dissolved in chloroform and precipitated by a mixture FIG. 1.-Oscilloscope traces of spikes generated by polyelectrolyte membranes in 0.15 N NaCl (A) dextran sulphate -polylysine membranes in 0.15 N NaCI (A) dextran sulphate + poly-L-sarcosine; scan = 50 ms amplitude = 20 mV per major division.(B) RNAse ;scan 20 ms and amplitude 5 mV per major division. (C) methylacrylate/acrylic acid -Ca2+ membrane upper record is aTd.c. trace at 100 mV/cm. (D) RNA -Ca2+ membrane scan = 0.1 s amplitude 20 mV per major division. [Toface page 176 FIG.3.-Electron micrographs of Pip-8 membranes upper left shows the nonporous matrix nieni- brane ; upper right and lower left show a matrix membrane with pores ; lower right shows a matrix membrane loaded with yeast RNA; the deeply stained areas are regions of RNA in the membrane. A B FIG.4.-Oscillatory patterns from a polylysine *DNA membrane A-initial output €3 stabilized output ; scales are 2 niV/cm and 0.5 s/cm for each major division on oscilloscope screen.FIG.8.-Current-clamp data for a polylysine -polvglutamic acid junction membrane ; the upper photographs shows an oscilloscope trace of the (I V) characteristics of a cwa membrane ; lower photograph is for the unloaded matrix membrane. The x and y axes correspond to the current and voltage values. Each major division corresponds to 1 V and 0.1 mA respectively. The zero voltage and current value is at the centre of the photograph. V. E. SHASHOUA of ethanol and water. This was repeated three times to remove low molecular weight components. The white fibrous polymer was then dried at 40°C for several days. The matrix membranes were obtained by casting a 0.5 % Pip8 solution in chloroform over a glass plate coated with a thin layer of a water soluble polymer such as polyethylene glycol (American Cyanamid) or Dextran sulphate (Pharmacia Inc.).The thickness of the chloro- form solution was adjusted by a knife edge supported at a distance of 0.025 cm from the glass. After drying for 20 min at room temperature the position of the Pip-8 film on the glass surface was visible only by tilting the glass to produce interference fringes. The film was then cut with a razor blade into 3 crn squares and distilled water was added at the cut edges. The water dissolves the water soluble polymer under the Pip-8 and dislodges the segments which then float onto the water surface. Next a cellulose nitrate tube with a 2 mrn aperture was used to pick up the matrix membrane.The Pip-8 film was sealed onto the plastic tube with a cement of Pip-8 (10 % in chloroform) at the edges to form a window as shown in fig. 2. The electrophoretic loading with polyelectrolytes was accomplished by placing the polyacid in the outside compartment and the polybase in the inner compartment. Each c-a membrane requires its own experimental parameters. In one example of a polylysine t-)DNA preparation the polylysine hydrobrornide (polycation phase) at a pH of 3 was electrophoresized against a DNA (polyanion phase) at pH 3. Initially the resistance of the system of salt bridges and electrolytes was 1.5x lo6ohm. A current of 0.02 mA at 1.5 V was passed through the system.After 15 min the current dropped to less than 0.005 mA. This was considered to be an indication of complete loading of the matrix membrane. Fig. 4 shows the type of electrical oscillations obtained with this membrane when the inner and outer compartment electrolytes were 0.15 N NaCl and when the system was polarized at a potential of 1.2 V. Trace A is the initial type of sustained oscillations obtained when the critical voltage of 1.2V was first applied. Trace B (fig. 4) shows the pattern after about 1 min of firing. This membrane continued to generate electrical oscilla- tions for a total of 10 min and then abruptly stopped due to the formation of a short circuit. MEASUREMENT OF (CURRENT VOLTAGE) CHARACTERISTICS Two methods were used for measuring the current-voltage characteristics of the c +-) a membranes.Fig. 5 shows the experimental arrangement for the measurements. The first method used Ag/AgCI electrodes to apply a potential across the membrane M. The current FIG. 5.-Experimental arrangement for measurement of current-voltage characteristics of c t*a membranes-S salt bridges ; Vm,voltage across membrane. flow in this circuit was detected by a milliammeter. The voltage across the membrane was detected by a voltmeter connected by the two salt bridges S to two Calomel electrodes as shown The results obtained in this type of measurements are shown in fig. 6. This method did not clamp the current or voltage during the measurement. The second method used a current clamp circuit to apply a given voltage Y across the membrane with a nanosecond time constant.Fig. 7 shows the circuit diagram with the operational amplifier in place. The current flowing was detected across a 10 k ohm resistance in series with the membrane. 178 ELECTRICAL OSCILLATORY PHENOMENA IN PROTEIN MEMBRANES Both the current and voltage measurements were carried out with Ag/AgCl electrodes connected with salt bridges across the membrane. A characteristic curve for the membrane was directly plotted onto an oscilloscope screen adjusted to display the voltage and current on the x and y axes respectively. All Ag/AgCl electrodes used were converted to chloride form immediately before each measurement. Fig. 8 shows a photograph of the oscilloscope screen for data of a polylysine -polyglutamic acid membrane system.RESULTS AND DISCUSSION Table 1 lists the various c ++ a junctions membranes studied to date and the polarizations applied to the membranes to initiate electrical oscillatory phenomena. Fig. 1 and 4 show the types of results obtained. In general random oscillatory patterns were more common. The constant frequency oscillations obtained ranged TABLE 1.-COMPOSITIONOF CATION f-) ANION JUNCTION MEMBRANES polycationic phase polyanionic phase polylysine HBr polyglutamic acid polydimethylaminoethyl acrylate polyacrylic acid Ca2+ yeast RNA BaZf dextran sulphate (m.w. 2x lo6) cytochrome c acrylic acid/acrylamide copolymer (50/50) Ca2+ acrylic acid/methylacryIate copolymer (50/50) poly L-sarcosine polyglutamic acid poly L-lysine HBr DNA During the oscillatory mode the positive and negative electrodes were connected to the polycations and polyanionic compartnients respectively of the c f-) a junctioii membranes.WLYGLUTAMIC-Ca" c 1234 .a2 E Lrolll) A B C FIG.6.-Current-voltage measurements of different types of c-a junction membranes ;electrolyte was 0.12 N NaCI + 1 mM CaCl for A and B and 0.12 N NaCI for C. Ec is the initial voltage at which electrical oscillations are observed. from a low range of 13 Hz for a RNA f-) Ca2+to about 100 Hz for a polyglutamic acid c+ Caz+ membrane. While it is not yet possible to specify the detailed procedures required for producing constant frequency patterns we believe that uniform loading and a narrow distribution of pore sizes in the matrix membrane may be among the critical factors.Fig. 6 shows the current-voltage characteristics of three c f-) a membranes measured under conditions of no current clamping. All three membranes show V. E. SHASHOUA characteristics of rectifiers with a ‘‘negative ” resistance region. The polyglutamic acid t)Ca++ membrane in fig. 6A has three regions of linear-current-voltage properties corresponding to resistance values of 888 ohm/cm2 45 ohm/cm2 and 110 ohm/cm2. The resistance of the unloaded matrix membrane was 12 ohm/cm2 considerably lower than for each of results for loaded membranes. Fig. 6C depicts ,111 10K 200 pf OP AMP +=-FIG.7.-Diagram of the current-clamp circuit used for obtaining (I V) curves for the co membranes.a more complicated (current voltage) curve obtained for an acrylic acid/acrylamide (50/50)c.)cytochrome c membrane. In all the graphs in fig. 6 the oscillatory pro-perties of the membranes are obtained when the membranes are polarized at the critical voltage E, i.e. at the negative resistance region of the membrane. The shift from non-oscillatory to oscillatory behavior can actually be seen on an oscilloscope. Experimentally the scope is set for a.c. recording then as the voltage is raised to a critical region there is a rapid transition and the oscilloscope trace shifts from one stable mode to another accompanied by the onset of “ spike ” generation. This transition in membrane properties occurs within a few seconds and is better illustrated in current-clamp type of measurements.Fig. 8 shows the current-clamp data for a polylysine t.)polyglutamic acid mem-brane. The photograph of the oscilloscope trace was obtained by a point by point setting of each voltage. It is seen that when a voltage of +3 V was applied across the membrane that a sudden decrease of voltage occurred at a constant current of -0.12 mA down to + 1.0 V. From then on the properties of the membrane had ohmic resistance characteristics. The only way to restore the membrane to its original state was to reverse the voltage and obtain (I V) characteristics as shown in the photograph. It is clear that the membrane behaves like a typical “ tunnel-diode ” with “ negative resistance ” characteristics. Katchalsky 15* l6 has proposed a molecular mechanism for the properties of these c -+ a membranes.It is based on the concept that polyelectrolytes undergo a phase transition at certain critical salt concentrations. Thus current flow through the membrane causes cations to move into the polyanionic phase but when they pass into the polyampholyte layer they suddenly encounter the polycation phase. This represents a very highly charged positive layer so the cations are repelled and they accumulate at the polyampholyte interface. Similarly anions arrive through the polycation phase. The net result is that the NaCl (or electrolyte) concentration builds up at the interface. At a critical concentration there is a sudden shrinkage of the polyelectrolyte and the membrane produces a “ breakdown ” region.This 180 ELECTRICAL OSCILLATORY PHENOMLNA IN PROTEIN MEMBRANES shrinkage is the result of the well known conformation change of polyelectrolytes at high ionic strength. The result is a conductance change and electrolyte rushes through to wash out the excess salt to regenerate the original membrane state. An electrical analogy to the properties of C-a junction membranes can be obtained from comparison of the c t)a structure to that of an n-p semiconductor. Fig. 9 shows a diagram of this type of consideration. Essentially the polycation and i-+lq JUNCTION --I--I--__I CATION INTERACTION ANION ZONE @ FIXED IONS -+ MOBILE IONS FIG.9.-Diagram of distribution of charges in a c-a junction membrane. The diagram indicates the presence of fixed charges provided by the polyelectrolyte and the mobile charges (current carrying) provided by the electrolyte.polyanion phases represent (in cross-section) regions of fixed ions. The poly- ampholyte zone is the interaction zone and the current carrying species are the mobile ions Na+ and C1-. Additional similarities are to be found from a consideration of the dynamic properties of c -a junction membranes. Katchalsky and Spangler l6 derived a theoretical equation for the frequency of the oscillations of c -a membranes. where v = frequency do = membrane thickness Lp = the filtration coefficient w = the salt permeability of the neutral zone and Cp = the concentration of the polymer in mol at the membrane surface.One remarkable aspect of this equation is that it predicts a square root relationship to concentrations of the polymer at the interface zone i.e. for the concentration of fixed ions at the interaction zone. The equation for the self-resonant frequency of a tunnel diode also has a square root relationship where R,,Cj and L are the junction resistance junction capacitance and series inductance of an equivalent circuit of a tunnel diode. At present we are not able suficiently to control the parameters for preparing the membranes SO as to test various aspects of the "tunnel diode " and the dynamic polyampholyte models. The general concept of a ce,a junction membrane and its analogy to semi- conductors can be applied to a number of membrane models. The requirements are that a two phase system with fixed ions should be present to act as a barrier for the current carrying mobile ions.In the simplest example (see table 1) both the cationic and anionic phases are fixed as polymeric components. It is also possible to have one fixed phase and one "pseudo " fixed phase. For example Ca2.+ ions can be used V. E. SHASHOUA as the polycationic phase. These ions are introduced into a matrix membrane already loaded with the polyanionic component to give a c c-)a membrane in which the Caz+ ions cross-link the polyanion to form a graded ‘‘ polycationic ” structure. Thus a preponderance of Ca2+ ions are present at the outer surface of the membrane and a “ neutral ’’ phase is developed in the centre of the polyanionic component.In such a system the polycationic phase is dynamic in nature and it is necessary to have extra Ca2+ ions present in the polycationic compartment so that any removal of the Caz+ by ion exchange can be replaced from the electrolyte. Examples of c -a membranes can also be prepared in which the polyanionic phase is derived from labile components. Thus polylysine can be used as the poly- cationic phase to provide the fixed cations and a bilayer of lipids (such as lecithin) can act as the anionic phase. If the lipid bilayer is stabilized as a smectic phase then the outer monolayer facing the electrolyte acts as the anionic (negatively charged) component while the inner monolayer interacts with the polycationic component to form the neutral zone of the c ++ a junction membrane The integrity of such a membrane can be maintained as long as the bilayer remains intact and extra lipid molecules are available to replace molecules lost by diffusion and electrophoresis.Such membrane systems have recently been prepared by Montal lo and earlier in some of the experiments of Mueller et a2.l7 using lipid bilayers with the addition of polylysine and protamine respectively. In both these examples the c t)a junction model can be used to provide a mechanism for the generation of the excitability and semiconductor properties of the membranes. This paper is dedicated to the memory of Aharon Katchalsky for his enthusiastic encouragement of the research. Thanks are due to Dr. K Kornacker for the circuit used in the current clamp experiments.I. Tasaki and T. Takenaka Pruc. Nat. Acad. Sci. USA 1964,52 804. ’R. S. Lillie J. Gen. Physiol. 1925 7 473. T. Teorell Exp. Cell Research Suppl. 1954,3 339. U. F. Franck Prug. Biuphys. 1956 6 171. ’H. Davson and J. F. Danielli In Permeubility of Natural Membranes (Cambridge University Press London 2nd Edition 1952). P. W. Mueller and D. 0. Rudin Nature 1968 217 713. ’C. Huang and T. E. Thompson J. Mul. Biol. 1965 13 183. A. M. Monnier J. Cell. Cump. Physiol. 1965 66 147. J. Del Castillo A. Rodriguez C. A. Romero and V. Sanchez Science 1966,153,185. lo M. Montal Biuchim. Biuphys. Acta 1973,298,750. l1 U. P. Strauss and P. L. Wineman J. Amer. Chem. Suc. 1953 75 3935 ;A. Katchalsky and I. R. Miller J. Polymer. Sci.1954 13 57. V. E. Shashoua Nature 1967 215 846. l3 V. E. Shashoua In The Molecular Basis of Membrane Function Ed. D. E. Tosteson (Prentice Hall N.Y. 1968) p. 147. l4 P. W. Morgan and S. L. Kwoleck J. Polymer Sci. 1962 62 33. l5 A. Katchalsky. In Neurusciencesv A Study Program Ed. G. Quarton T. Melnichuck and F. 0.Schmitt (Rockfeller University Press N.Y. 1967) p. 335. l6 A. Katchalsky and R. Spangler Quart. Rev. Biophys. 1968 1 127. l7 P. Mueller D. 0.Rudin H. T. Tien and W. C. Wescott Nature 1962,194,979.

ISSN:0301-5696    

DOI:10.1039/FS9740900174

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

On the Nature of Certain Oscillations on Bimolecular Lipid Membranes BY B&LAKARVALY Institute of Biophysics Biological Research Centre Hungarian Academy of Sciences H-6701 Szeged P.O.B. 521 Hungary Received 24th July 1974 The origins and nature of bioelectric oscillations are mostly still obscure. The sustained electro- mechanical oscillations observed by Pant and Rosenberg on bimolecular lipid membranes (BLMs) are therefore of great theoretical and practicalimportance and worthy of detailed investigations. It will be shown that lipid molecules arranged in a bimolecular structure are in a spontaneously excited state (called exciton state) and that these excitons are involved in the charge-transfer processes at the interfaces. The detailed mechanism of the interfacial electrode processes suggests that certain lipid complexes may serve as catalysts promoting interfacial charge transfer and gives further support for the view that a mechanism different from the electrohydraulic one (Teorell's oscillator) may be responsible for the oscillations mentioned above.1. INTRODUCTION Pulsating rhythmic and oscillatory phenomena at macroscopic microscopic and submicroscopic (molecular) dimensions alike are frequently observable in both inanimate nature and the living world. Marked attention has been paid to oscillatory processes in typically biological objects at organ cellular subcellular and molecular levels. We think first of all of rhythmic reactions in biochemical systems as well as of spontaneous and induced periodicities of bioelectric phenomena such as rhythmic changes in membrane potential during stimulus '; the propagation of excitation along nerve membranes ; the noise-like fluctuations of the membrane potential 39 ; the large voltage Auctuations on skeletal muscle membrane and other periodic processes in muscle,6 etc.The rhythmic phenomena in biology are comprehensively summarized in Sollen- berg's monograph. The thermodynamical bases of oscillatory systems have been laid down by Prigogine and his colleagues.8+ The general mathematical treatment of the kinetics of certain oscillations can be found in ref. (10) and (11). A detailed study of the dynamics and control of cellular reactions has been provided by Hig- gins l2; oscillatory properties in chemical systems have been reviewed by Nicolis and Portnow l3 ; those in biochemical systems by Hess and Boiteux l4 and the electrical oscillations on porous fixed charge membranes have been discussed by Teorel1,ls* l6 Franck l7 and Meares and Page.' Despite the intensive research work done in the field of bio-oscillations the mechanisms and the physical and physico-chemical background of these phenomena have not yet been cleared up.The present paper is not concerned with experimental details. It merely sum- marizes and adopts the most recent results obtained in our laboratory which furnish further evidence for the electronic nature of the Pant-Rosenberg oscillator. 2. COUPLED ELECTRO-MECHANICAL OSCILLATIONS ON BLMS Large-amplitude voltage and current oscillations accompanied by periodic change of the Plateau-Gibbs border of BLMs separating inorganic redox electrolytes have 182 B.KARVALY 183 been reported by Pant and Roseiiberg. ' These phenomena were found to be quite different from those observed on BLMs in the presence of proteins and proteinaceous substances,20*z1 and from spontaneous random fluctuations occurring on unmodified BLMs. Since the BLMs can be considered the best models of biomembranes the oscillations described by Pant and Rosenberg seein to be of great importance as regards the understanding of the rhythmicity of bioelectric phenomena and especially the membrane oscillations. The system to be discussed below is outlined in fig. 1. Comportment I Compartment 11 FIG.1.-The BLM oscillator.Let us recall the basic properties of this membrane oscillator 1. The necessary conditions for the appearance of oscillations are the acidity (pH 5 5) of the potassium ferricyanide solution and the alkalinity (pH 2 10) of the potassium iodide compartment. 2. The oscillations are sustained practically undamped of constant frequency and of constant amplitude. 3. The phenomenon is membrane-specific but it is independent of the nature of the membrane lipids. 4. The amplitude of oscillations is determined by the potassium iodide concen- tration but it is not affected by pH. 5. The frequency can be altered by changing the pH of the bathing solution but it cannot be influenced via the concentrations of the redox components. 6. The periodic electro-mechanical behaviour emerges only in a given range of the transmembrane potential i.e.the oscillatory process is voltage-controlled. 7. The voltage and current oscillations can be resolved into at least two sinusoidal or near-sinusoidal components. 8. The electric and mechanical oscillations are synchronized. 9. Periodic bulging of the membrane is not noticeable under the usual geometry of observation but periodic change of the Plateau-Gibbs border is detectable. 10. The system possesses rectifying character showing that the transport of negative charge-carriers is more likely from the potassium iodide compartment into the potassium ferricyanide compartment than in the opposite direction. Two basically different theories have been proposed to explain the experimental findings.Pant and Rosenberg presumed that a mechanism similar to that operating in the Teorell oscillator 5-1 or even the Teorell electrohydraulic mechanism itself gives rise to the electro-mechanical oscillations. It was recently suggested by the present author 22 that electrochemical electrode reactions may be involved in bringing about the prolonged periodic changes in the electrical properties of the BLM. Where-as the first interpretation tacitly assumes the existence of ionic conduction inthe BLM the latter considers rather the electronic properties and processes. OSCILLATIONS ON BiMOLECULAR LIPID MEMBRANE 3. CONDUCTIVITY MECHANISM OF LIPIDS AND BLMS The electrical properties of BLMs in the presence of the iodine/iodide redox couple have been extensively investigated in recent yea~s.~~'~~ Nevertheless both the nature of the charge-carriers and the mechanism of the conduction remain hotly- debated questions.Several scientists 25-27* 31 have arrived at the conclusion that the generation of the electromotive force and the dramatic resistance drop should be attributed to ionic processes rather than to electronic ones while others 28-30 have concluded that the iodine doped BLMs must be predominantly electronic semi- conductors the ionic processes being of secondary importance. According to the latter conception the BLM in the presence of iodine/iodide behaves like a semi-conductor electrode and electrochemical electrode processes are responsible for the emf.generation and charge-carrier injection while the increase in the conductivity can be attributed to the presence of lipid-iodine charge-transfer complexes. In spite of extensive investigations the observations could only be phenomenologically interpreted. In the following special attention is focused on the physical and physico- chemical aspects of conduction on which the electromechanical oscillatory proper- ties are based. 3.1 CONDUCTIVITY MECHANISM OF LIPIDS AND LIPID-IODINE COMPLEXES It has been pointed out in very recent investigations 32-37 that the conduction mechanism of wet oxidized bulk cholesterol samples and unmodified BLMs formed from oxidized cholesterol are very closely related to those of wet oxidized bulk cholesterol-iodine charge-transfer complexes and iodine-doped BLMs.Thus all the results concerning the conductivity properties of wet bulk samples give valuable information on the BLM conductivity too. Therefore from dielectric studies on oxidized bulk cholesterol and its iodine complexes,32* 36*37 as well as from conduc- tivity measurements on BLMs separating iodine/iodide solutions with different con- centrations the following could be concluded 1. The oxidized cholesterol (and probably all lipids) and its iodine complexes come into a very specific interaction with water thereby leading to the development of Maxwell-Wagner polarization zones. 2. The lipids and their complexes with iodine may become spontaneously excited especially in the presence of water and according to the reaction [L :I2 :W]+[L :I* W]* (3.1) in both the bulk samples and BLMs spontaneously excited states (molecular excitons) may be present in high density ([L :I2 W] denotes the lipid-water-iodine complexes in the ground state and [L :I2 :WJ*refers to the excited or exciton states).3. By means of exciton-exciton interaction these excitons are capable of producing free charge-carriers. As a result of the autoionization process [L :I~ :W]*+[L W]*+[L :I2:W]f,+e~ (3.2) there exist definite concentrations of mobile electronic charge-carriers and of less mobile (occasionally immobile) ionic species. 4. The BLMs (and probably the biomembranes and biomacromolecules too) are covered by an exciton coat the existence and properties of which are closely related to the stability dynamic structure and electric and transport characters of the membranes.Consequently it is practical to make a distinction between the molecular architecture and functional structure of membranes. The molecular architecture (the morpho- logical membrane) includes the lipid constituents proteins etc. and the substances B. KARVALY 185 dissolved in these components Cfig. 2a). The functional structure of membranes (the functional membrane) consists of two interfaces* and the bulk membrane phase (fig. 2b). 5. The injection of charge-carriers into the BLMs occurs according to the Poole-mechanism i.e. the injection process at the interfaces is governed by the trans-membrane potential Y and/or by the electric field strength E actually present in the interfacial region.In the event of a steady state the density of the electronic charge- carriers produced by field-enhanced electron/hole emission is proportional to exp(U) or exp(EE). (3.3) Here iiand 2 are constants and related to the potential distribution across the mem- brane (fig. 2c). 6. The electronic charge carriers produced move in the bulk samples and cross the membrane by hopping. bimolecular lipid layer \ eprotein coat a interfacial region b. A I1 C FIG.2.-The structure of membranes. a. The Danielli-Davson membrane model. b. Scheniatical representation of the functional structure of membranes. c. The potential distribution across membranes. * The interfaces are very special parts of the membrane-aqueous phase system which cannot be delimited on the basis of structural and geometricaI considerations.The interface means the space region at the membrane surfaces the processes taking place in which appear as membrane processes. Of course the dimensions of the interfaces depend upon the interaction in question and may vary from process to process. OSCILLATIONS ON BIMOLECULAR LIPID MEMBRANE 7. If ions are present in the bathing solutions exciton-ion interactions too may be involved in the generation of charge carriers. The results summarized above strongly support the idea that the iodine-doped BLMs are predominantly electronic rather than ionic conductors. It should be noted that the charge carrier generation by exciton-exciton and/or exciton-ion inter- actions is the explanation of the extremely high resistance-lowering ability of the iodine and water.So far as the hopping conduction is concerned this accounts for the apparent duality in the interpretation of conductivity data. In the case of such a conduction mechanism the ion-complexes formed as a by-product of autoionization convey their charges to the neighbouring neutral species this process being reflected as electrodiffusion of ionic charge-carriers. 3.2 MECHANISM OF IODINE EFFECT ON BLMS If only molecular iodine is present in the bathing solution two equilibria dominate (12) 421 (3.4) and L+[I,]+[L :121. (3.5) Here (I,) denotes the iodine in the aqueous phase [I,] that dissolved in the membrane phase and [L :12]the lipid-iodine charge-transfer complex (for technical reasons the water will not be indicated below).There exists however an equilibrium between the ground and excited states of the complexes as shown by eqn (3.1). Owing to the high hydrocarbon-solubility of iodine and to the strong affinity of lipids to form complexes with iodine the concentra- tions of lipid-iodine complexes on the two sides of the BLM will be practically the same. The partial reactions at the interfaces with positive polarity take place in the following sequence [L 12]*+ [L I?]* +[L :I,]'; + e+ (3.6) (autoionization process) 2[~ I,]; +[r,l=(~ :I2I2 :I);- (3.7) (chemical reaction) (12) =I1219 (3.8) (transfer reaction) ([L :1212 :I>;-+4[L :121+211-] (3.9) (chemical reaction) El4 (3.10) w9 (transfer reaction) [L :I ]+fL :IJ*.(3.1 1) (reexcitatory process) The above reaction set yields the overall reaction (Iz)+2(I-) + 2e+ (3.12) which is assumed to be enacted in the opposite direction at the interface with negative polarity (fig. 3). 3.3 THE MECHANISM OF IODIDE EFFECT ON BLMS For the sake of simplicity let the iodide content of the bathing solution be much higher than the iodine content. The equilibria given in eqn (3.4) and (3.5) are valid B. KARVALY 187 bulk brtcrfacr membrane interface FIG.3.-Overall interfacial reactions in the presence of iodine. of course and the iodide effect can be considered a result of exciton-ion interactions. The consecutive partial reactions are then (1-1 41-1 (3.13) (diffusion transport reaction) [L I2]*+[I-]+[L :I2]* [I-] (3.14) (chemical reaction) 2[L 12]* [I-]+([L 12] :[I])2 +2e- (3.15) (au toioniza tion) ([L 121 [1])2 +2[L :121+ [I,] (3.16) (chemical reaction) and finally the reexcitation described by eqn (3.1 1) follows.These partial reactions lead to the overall process (fig. 4) 2(I-) +(Iz) + 2e-. (3.17) This single electrode process generates a considerable part of the e.m.f. The basic difference between the effects of molecular iodine and iodide ions is that the interactions involved in liberating electronic charge-carriers are exciton-exciton interactions in the first case and exciton-ion interactions in the second case. As far as the propagation of the electronic charge-carriers is concerned they pass the membrane by hopping as shown in ref.(32) (36) and (37). 3.4 THE EFFECT OF K3[Fe(CN),] Unfortunately a detailed mechanism for potassium ferricyanide is not yet avail- able. However from preliminary studies it is clear that the effect of K3[Fe(CN),] is significantly different from that of either iodine or iodide. The ferricyanide ion appears to be capable of adsorption on the membrane surface but this adsorption process does not lead to drastic changes in the electrical properties of the BLM. Nevertheless this adsorbed state of the ferricyanide ion is a favourable one as regards participation in interfacial processes. The electrons supplied by the overall process (3.17) at the interface in iodide compartment may be picked up by the adsorbed ferricyanide ions producing ferrocyanide ions.4. SOME REMARKS ON THE OSCILLATIONS The general scheme of the Pant-Rosenberg oscillator has been outlined in an earlier paper.22 The further results summarized above confirm the suggestion that the interfaces can be considered thermodynamically open systems and that a signi- ficant portion of the oscillatory processes is electronic in nature. On the basis of J 88 OSCILLATIONS ON RIMOLECULAR LIPID MEMBRANE bulk intcrface memkane interface 21-Zfl_ 2 I' I I I I FIG.4.-Overall interhcial reactions in the presence of iodide. results presented in ref. (32) and section 3 two simplified versions of the oscillatory system can be given as shown in fig.5 and 6." The importance of the ionic processes arises primarily in the development of mechanical oscillations and in the voltage- controlled nature of the oscillatory behaviour. The appearance of the mechanical oscillation is thought to be due to the adsorption on the membrane of the products of the overall reaction and the excess surface charges result in the periodic change in the membrane surface at the expense of the Plateau-Gibbs border. Unfortunately we have no evidence as E FIG.5.-Overall system of interactions. to whether the iodide ion or the ferrocyanide ion or even both are involved in creating the mechanical oscillations. It is not clear either whether the iodide ions originate from the iodine dissolved in the membrane or from the lipid-iodine complexes.Although the latter possibility seems more probable this question still remains open and is under further examination. * The details of these branched reaction models will be described else~here.~' B. KARVALY 189 As can be seen from the reaction models in fig. 5 and 6 there are two systems capable of oscillating these being connected by an electronic charge-transfer process. Franck 39 has treated in detail the problems of coupling between two oscillating FIG.6.-Overall system of interactions. electrochemical metal electrodes. It was pointed out that in the low-frequency region such as ours the capacitive and inductive couplings play a subordinate role. Consider-ing that the bulk membrane phase has a relatively high resistance independently of the iodine dissolved in it and that the low-frequency capacity of iodine-doped BLMs is relatively high it is believed that the system in question may be replaced by two oscillating semiconductor electrodes connected by a parallel RC circuit (fig.7). Since the transmembrane charge transport is hopping the possible role of Iocal currents in coupling appears to be excluded. Finally some difficulties of a practical and theoretical nature should be noted. The reaction networks proposed in fig. 5 and 6 could be checked in principle by computer simulation for which however the equilibrium constants would be needed. Owing to the complicated structure of the system (diffusion layer interfacial phe- nomena charge-transfer complex forming etc.) these parameters are still unknown.CaJPLlNG RC-CIRCUIT FIG.7.-Equivalent circuit for the coupled oscillating interfaces. The theoretical difficulties originate from the fact that a considerable part of the trans- membrane potential drops in the interfacial region thereby causing an extremely high electric potential gradient of the order of lo6 V/cm or more (fig. 2c). The presence of such a high electric field in the interfacial region would make it difficult to restore the membrane system to the equilibrium state if it were perturbed. Conse-quently it seems plausible to assume that the membrane (i.e. the interfacial regions and the bulk membrane phase together) does not and can not exist in the equilibrium state of classical sense.Accordingly due to the high electrical potential gradient in 1YO OSCILLATIONS ON BIMOLECULAR LIPID MEMBRANE the interfacial region at the best a steady-state may prevail in the interfaces i.e. the natural state of the membrane is non-equilibrium.’ Another consequence of this enormously high potential gradient is that a thermodynamic treatment of such a system (e.g. the formulation of thermodynamic stability criteria for non-equilibrium states) cannot be performed for at present thermodynamical theories operate with space and time gradients of thermodynamical quantities these being assumed small. Despite many difficulties in explaining all the details of the oscillatory behaviour of the system discussed the basically electronic nature of the electronic oscillations can be considered to be proved.The physiological significance of this type of oscilla- tion is fairly obvious and these preliminary results demonstrate the suitability of BLMs for modelling certain bioelectric oscillatory processes as well. On the other hand the mechanism outlined here may give some more direct support for a nerve conduction mechanism similar to (or even the same as) that proposed by Lillie.40 A. L. Hodgkin and A. F. Huxley Nature 1939 144 710. A. L. Hodgkin The Conduction of the Nerve Impulse (Liverpool University Press Liverpool 1st ed. 1964 The Sherrington Lectures VIl). B. Katz The Release of Neural Transmitter Substances (Liverpool University Press Liverpool 1st ed. 1969 The Sherrington Lectures X). A.A. Verveen and H. E. Derksen Proc. Inst. Elec. Electron. Eng. 1968 56 906. E. Varga L. Kovacs and I. Gesztelyi Acla Physiol. Acad. Sci. Hung. 1972 41 81. ’E. Ernst Die Muskeltatigkeit (Verlag d. Ung. Akad. d. Wiss. Budapest 1958). ’A. Sollenberg Biological Rhythm Research (Elsevier Amsterdam 1965). * I. Prigogine and G. Nicolis Quart. Rev. Biophys. 1971 4 107. P. Glansdorff and I. Prigogine Thermodynamic Theory of Structure Stability and Fluctuations (Wiley New York 1971). lo J. I. Gmitro and L. E. Scriven in Intracellular Transport Symposia of the Int. SOC. for Cell Biology ed. K. B. Warren (Academic Press New York-London 1966) Vol. 5 p. 221. J. Higgins Ind. Eng. Chem. 1967 59 19. J. Higgins in Control ofEnergy Metabolism ed. B. Chance R. W. Estabrook and J.R. William-son (Academic Press New York-,1965) p. 13. l3 G. Nicolis and J. Portnow Chem. Rev. 1973 73 366. l4 B. Hess and A. Boiteux Ann. Reu. Biochcm. 1971 40,237. l5 T. Teorell Biophys. J. 1962 2 Suppl. 27. T. Teorell in Laboratory Techniques in Membrane Biophysics ed. H. Passow and H. Stampfli (Springer Verlag Berlin 1969) p. 130. l7 U. Franck Ber. Bunsenges. phys. Chem. 1963 67,657. P. Meares and K. R. Page Phil. Trans. A 1972 272,1. l9 B. Rosenberg and H. C. Pant Biocltim. Biophys. Acta 1971 225 379. 2o P. Mueller and D. 0. Rudin J. Theor. Biol. 1968 18 222. P. Mueller and D. 0. Rudin Nature 1968 217 713. ’’B. Karvaly Nature 1973 244 24. 23 P. Lauger W. Lesslauer E. Marti and J. Richter Biochirn. Biophys. Acta 1967 135 20. 24 P. Liiuger J.Richter and W. Lesslaucr Ber. Bunsenges.phys. Chem. 1968 71,906. ’’P.A. Peshayev and L. M. Tsofina Biophys. (Russ.) 1965 13,360. 26 A. Finkelstein and A. Cass J. Geii. Pliysiol. 1968 52,145. ’’G. L. Jendrasiak and H. E. Lyon 13th Annual Meeting of the American Biophysical Society Los Angeles 1969 Biophys. SOC.Absfr. p. 72a. 28 V. Ya. Vodyanoj 1. Ya. Vodyanoj and N. A. Fedorovich Fiz. Ttierd. Tela (Russ.) 1970 12 3321. (Societ Phys. Solid State). 29 L. I. Boguslavsky F. I. Bogolepova and A. V. Lebedev Chern. Phys. Lipids 1971 6,296. 30 13. Karvaly B. Rosenberg H. C. Pant and G. Kemeny Biophysik 1973 10,199. 31 G. Szabo G. Eisenmnn R. Laprade S. M. Ciani and S. Krasne in Membranes ed. G. Eisen- man (Marcel Dekker New York 1973) p. 179. 32 B. Karvaly Thesis for the Academy Degree Candidate of Sciences (Szeged 1974).* The biological macromolecules as a consequence of their structure relevant properties and localization aye presumably covered too by an cxciton coat and therefore their natural state is plausibly non-equilibrium in most cases for just this reason. This is outside the scope of this paper however and will be treated elsewhere. B. KARVALY 33 B. Karvaly Bioelectrochem. Bioenergetics 1975 2 124. 34 B. Karvaly I. Szundi and K. Nagy Bioelectrochem. Bioenergetics 1975 2. 35 I. Szundi and B. Karvaly Acta Biochim. Biophys. Acad. Sci. Hung. 1973 S (Suppl.) 200. 36 I. Szundi B. Karvaly and K. Nagy M.T.A. Biol. Oszt. kozl. 1974 17 in press. 37 B. Karvaly I. Szundi 2nd Conf. Condensed Matter Division of the European Physical Society Budapest 1974. 38 B. Karvaly (to be published). 39 U. Franck and L. Meunier 2. Nuturforsch. 1953 8b 396. 40 R. S. Lillie J. Gen. Physiol. 1920 3 107 129.

ISSN:0301-5696    

DOI:10.1039/FS9740900182

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

Current Oscillations in Iodine-doped Polyethylene Film BY G. T. JONES AND T. J. LEWIS School of Electronic Engineering Science University College of North Wales Dean Street Bangor Gwynedd North Wales Received 25th July I974 Very low frequency Hz) regular current oscillations may be induced in iodine-doped polyethylene films when subjected to electric fields in excess of about 3 x lo7 V m-'. The oscillations are similar whether iodine is introduced from aqueous KI electrodes or from the dry vapour. The frequency depends on film thickness and iodine concentration and has an activation energy of -1.2 eV. Most significantly it decreases with increasing field suggesting that space charge domains are propagating in the films encouraged by a negative differential charge carrier mobility-field char- acteristic.This is confirmed by direct measurement. The acceptor action of iodine probably generates mobile electron vacancies in the polymer chains the effective mass of which increases with field to give the negative characteristic. It has been demonstrated already that the electrical conductivity of thin poly- ethylene films will increase by several orders of magnitude when they absorb iodine by contact with aqueous sodium iodide solutions. It appears that neutral iodine rather than iodine ions diffuse into the polyethylene and the current growth at con- stant applied electric field follows a Fickian diffusion law.2 Neutral iodine is known to be preferentially absorbed in the amorphous regions of the polymer film and at the same time it appears that electron transfer from polymer molecules to vacant acceptor levels in the iodine system generates mobile "holes " in the polymer chain which leads to enhanced conduction2 Swan has also shown that when the field applied exceeded about 4 x lo7 V in-' regular slow current oscillations were superimposed on the steady background current the frequency of these depending on temperature iodine concentration and field strength.Swan considered the possibility that the oscillations might be due to the propagation of space charge domains across the film but concluded because the static (current voltage) characteristics did not appear to show negative differential character- istics that high field accumulation domains as observed in more conventional semi- conductors were not occurring.McCumber and Chynoweth have shown however that the absence of such characteristics is not in fact evidence against the existence of high field domain propagation in a solid. Our present studies of a system similar to that of Swan confirm the existence of oscillations with the same characteristics. We have also found that oscillations may be generated in a dry system in which iodine is diffused directly into the polyethylene from the vapour and also in a polyethylene+sulphuric acid system. Furthermore we have been able to make direct determinations of the charge carrier mobility in such systems and to show that this parameter exhibits a strong negative differential coeffi- cient with respect to the field such as would be required for the establishment of space charge instabilities in conventional semiconductors.The implications of this for curl.icr iiiotion in organic systems generally is briefly discussed. 192 G. T. JONES AND T. J. LEWIS EXPERIMENTAL The arrangement for most of the experiments was essentially similar to that used earlier.Z* Samples of low density polyethylene film without additives in the thickness range 50-350 pm were cleaned of surface grease by washing in methanol and sealed between p.t.f.e. cups each normally containing an aqueous 1 M solution of KI with iodine added and arranged so that a surface area of 0.32cm2 of the solution was in contact with the film on either side. Iodine added to the KI solutions in known concentrations was allowed to diffuse into the polymer thereby increasing the conductivity by orders of magnitude.Contact to the aqueous KI solution in each cup was made via tungsten leads and the whole cell could be housed in an oven of which the temperature could be controlled to 0.1"C. Dry samples were prepared by first outgassing the films for several hours at a pressure of about 10 N and then immersing in iodine vapour until saturated and excess iodine appeared on the surfaces. The excess was removed by gentle washing with methanol. The sample was then placed between flat copper electrodes one fitted with an insulated guard ring. This system was difficult to control since high field measurements had to be made under a vacuum in order to avoid surface leakage and consequently the iodine content of the sample gradually fell with time.Some improvement in stability was achieved by pre- iodizing the copper electrodes before making contact to the sample. This stabilised the contact and prevented an iodine deficiency adjacent to the electrodes. Gold electrodes were also tried but without success the surfaces deteriorating rapidly. A stabilised controlled voltage source was used to apply selected fields to the samples and currents (lo-" A to loe6 A) were measured by electrometer. RESULTS CURRENT OSCILLATIONS Provided the film thickness was greater than 50 pm and less than about 350 pm sustained slow current oscillations could be obtained for the "wet " system when the field was raised above a threshold value Et.A typical example is shown in fig. I where the threshold field lay between 3.9 and 4.3x lo7 V m-l. When the sample was thin (-50 pm) current oscillations could be induced only at high iodine concentra- tions (100 g/l. 1 M KI solution). At lower concentrations oscillations were inter- mittent and were rapidly damped out. The general chracteristics present in fig. 1 and time (intervals = 10 s) FIG.1.-Typical (current time) characteristic at the onset of oscillaticms. Polyethylene film 127 pm thick. Iodine concentration 10 g in 1 1. of 1 M KI solution at 30°C. At point A the field is changed from below threshold (3.9 x lo7 V m-l) to just above (4.3x lo7 V m-l). Note that the oscillations begin by a downward swing of current (B). present in all cases are as follows.On raising the field to the threshold value a fast capacitative transient is generated which does not appear to have any influence on the subsequent oscillations. Following the transient oscillations commence by a down-ward swing of current to a level below that existing before the increase in field. Following this the background steady current with uscillations superimposed rises to s 9-7 i94 OSCILLATIONS IN POLYETHYLENE a peak and then decays slightly to leave a regular oscillating pattern. The fact that the oscillations begin invariably by a downward swing of current is important for our later discussion of the mechanism. The amplitudes' of oscillations were not markedly dependent on iodine concen- tration but were always greatest after a field change and then (as in fig.1) decreased somewhat with time. Sometimes as Swan also found the oscillations were modu- lated in a slow way and sometimes there were simultaneous oscillations of slightly different frequency which distorted the sinusoidal form. Occasionally as in fig. 2 there would be a gradual transition from one frequency to a slightly different one reflecting presumably some slow change in conditions within the sample. tirne FIG,2.-Oscillations showing the slow transition from one frequency to another. Note also that there is initially some distortion of the wave due to the presence of other frequencies. Polyethylene film 127 pm thick. Iodine concentration 5 g in 1 1. of 1 M KI solution at 30T. Field 5.51 x lo7 V m-l.As fig. 3 shows the threshold field Et varied inversely as the sample thickness tl and also increased as the iodine concentration decreased. The dry system for which the iodine concentration was low but not precisely determined produced a result in agreement with the others. The characteristics if extrapolated converge to a limiting value Etz2.2x 10 'V m-1 for infinite thickness. It is interesting to note that if two films were put together as a single composite film of double thickness the oscillations were characteristic of a single film of double thickness for both wet and dry systems. 0 2 4 6 8 101214 d-l /m-'x lo3 Frc;. 3.-Variation of threshold field Et with reciprocal thickness and with iodine concentration at a temperature of 30°C.Concentration of iodinc (gin 1 1. of 1 M KI solution) U,25 ; 0, 10; A 5 ; a,1; x dry system. The frequency of oscillationfwas found to be a maximuin at E and to decrease as the field was raised above that value as Swan also found. Ultimately at a sufficiently high field oscillations become uncertain and periodicity was lost. By interpreting (arbitrarily at this stage)f-' as a transit time z of a disturbance across the film the mobility of the disturbance may be written as d/(sE)where E is the average applied field (E 3 &). The results for a range of fields and film thicknesses at ;Iconstant G. T. JONES AND T. J. LEWIS concentration lie on a universal curve (fig. 4) p decreasing with increasing E. Even the result for the double film (76f76) pm fits well on the curve.Arrhenius plots of the frequency against reciprocal temperature produced an activation energy of 1.2 eV in remarkable agreement with Swan.4 Oscillations could be obtained even when the * l6k \ I \ -I2 8- 4-'0 3 4 5 6 7 8 9 10 E/V m-I x lo7 FIG.4.-Mobility p determined from frequency of oscillations as a function of field E showing independence of thickness d. Iodine concentration 10 g/l. 1M KI at 30°C. x 178 pm 0(76+76) pm; 0 127pm; 76pm. temperature was low enough to freeze the electrodes. The period of oscillation decreased or the mobility p increased with increasing iodine concentration as shown in fig. 5 but the indication is that a limiting value might be reached ultimately. 30t I I I I 01 ' 0.I 1.0 10 100 iodine concentration/g 1.-l 1 M KI FIG.5.-Effective mobility 1-1 determined from f as a function of iodine concentration for a 127 pm sample at 30°C.0,4.7 x lo7V n1-l ; x 5.5 x lo' V m-' ; @ 7.9 x lo7V m-I ; .,dry system at 5.5 x 10' V m-'. The background static current has several interesting features which are related to the oscillations. Some (current field) characteristics are shown in fig. 6. The current first varies supralinearly with the field and then approaches a saturation value which for most samples begins to set in at a field just above 2x lo7 V m-l. Sub-sequently at the threshold field for oscillations E, there is a rapid increase in current and thereafter a variation with field which is less than linear.Again these results are very similar to those of Swan and illustrate how reproducible are the conductivity characteristics of iodine doped polyethylene in contrast to the poor reproducibility of such characteristics for natural polyethylene. The combination of oscillatory current the particular form of (current field) f 96 OSCILLATIONS IN POLYETHYLENE 4 3 m 2 X N E s2 x Y .m 3 a U s t: 21 L I I I I 1 0 2 4 6 8 applied field/\(' m-I x lo7 FIG. &-Static current-field characteristics showing the rapid increase of current above a quasi- saturation value as the threshold field Et is exceeded. Film thickness 127 pm temperature 30°C. Iodine concentration (g in 1 1. of 1 M KI solution) (a),I ; (B) 10 ; (c) 50..-. c1 .-x 0 I field E distance from injecting clcctrode (4 (6) V (4 FIG.7.-Characteristics associated with the onset of space charge domain propagation. (a)(Velocity field) characteristic with negative slope above the critical field Ec ; (6) field distribution in solid for various current ratios J/Jc (Jc = ywc where n is the carrier density) (c) general form of (current average field) characteristic deduced from (6). G. T. JONES AND T. J. LEWIS 197 characteristic and an apparent mobility which decreases with increasing field (fig. 4) supports a thesis that space charge domains are propagating in the films above the onset field Et. It is therefore worthwhile to review briefly the salient features of space charge domain propagation as developed for conventional semiconductor^.^* The essential requirement is that the mobile charge carrier concerned should have a (velocity field) characteristic with negative slope over the range of fields of interest (fig.7a). Assuming this velocity characteristic a uniform carrier concentration n in the bulk and an efficient carrier injecting electrode it has been shown that ' the field distribution will be as in fig. 7b. For current density J -cJ where J = qmc is the current corresponding to Ec and q is the charge of the carrier the field approaches a limit E < E which extends over most of the sample. For J > J, the field increases monotonically with distance from the injecting electrode. The (current field) characteristic may also be found and has the form shown in fig.7c the marked upturn in current coming when the ratio J/Jcis such that there is a rapid increase in field away from the carrier injecting electrode. In the negative slope regime (fig. 7a) space charge may accumulate locally and then propagate as a domain.5* When such an accumulation domain propagates the field ahead of it E, will tend to increase and that behind Eb to decrease (fig. 7a). If u < vb then accumulation continues but when zt > q,,any accumulation is dissipated. Thus depending on the field distribu- tion and notably on the operating mean bias field E so domains might propagate fully across the sample or first grow and then decline part way across or if the bias field were high enough (or low enough) so that operation moved outside the negative regime in fig.7a not be generated at all. This model thus suggests why there will be a critical field for the onset of oscilla-tions and why at much higher fields they will die out. It also provides a (current field) characteristic generally in agreement with experiment. MOBILITY MEASUREMENTS A central feature of the space charge domain theory is a negative differential (velocity field) or (mobility field) characteristic. Thus most convincing for our present argument is that we have been able to demonstrate directly the existence of a negative differential mobility coefficient by extending the method outlined by Davie~,~ Wintle and others to high fields. The experiment utilises essentially the same rime time time (4 (b) (4 FIG.8.-Typical results for Y and IdV/dtI as functions of time for a 76 pm thick sample containing 10 g iodinell.of 1 M KI solution at 23°C; -V,--IdV/dtI. (a)0.79 x lo7V m-l (b)3.9 x lo7 V m-' (c)7.9 x lo' V m-l. In (c) it should be noted how IdV/dtJ increases as Vdecreases with time which is indicative of the negative differential (mobility field) relationship. arrangement as for our other measurements reported above except that the constant voltage applied to the electrodes is replaced by a fixed charge placed on the high voltage electrode at an initial time and of sufficient magnitude to raise the field in the OSCILLATTONS IN POLYETHYLENE sample to the value required for the experiment. The electrode is then isolated so that charge can decay only through the sample and the electrode potential Y falls according to the mobility of carriers in the sample.The rate of fall is determined by monitoring the potential of the electrode using a rotating vane field-mill electrostatic voltmeter exposed to the field of the electrode and differentiating the output electronic- ally. t 000 0 2 4 6 002468 E/V m-I x lo7 EIVm-‘ x107 (4 (b) FIG.9.-Apparent mobility p’ from potential decay curves. (a)Iodine concentration 10 g per litre KI solution at 30°C. Film thickness in pm shown on curves. (b) Film thickness 76 pm,temperature 23°C. Concentration of iodine in g per litre KI solution shown on curves. It is possible to show that the apparent carrier mobility p’ is given by 2d Id V/dt 1 p‘ = (2a+l)p = v2 9 t=O where p is the true mobility Q = d2qn,/&V, q is the charge n is the concentration of thermally generated carriers and E is the permittivity.V and IdV/dtI are to be evaluated at time t = 0. In fig. 8 typical results for the decay of Y and Id V/dtI with time are shown at a low a critical and at a high field where the negative differential mobility-field characteristic holds good (see fig. 4). Determining values of Y and jdV/dtI at t = 0 from these it was possible to construct the apparent (mobility field) curves shown in fig. 9. It is seen that p’ reaches a maximum in the range 2-3 x lo7 V m-1 for all the con- ditions chosen and significantly exhibits a negative slope above this range.The onset of the latter regime coincides with the field range in which oscillations are found and for which there is also a negative slope of the mobility curve (fig. 4) derived from these oscillations. Our values of p’are in quite reasonable accord with those reported by Davies lo on iodine-doped polyethylene at low fields. The mobility p estimated from the period of oscillations (fig.4) is less than p’ (fig. 9) and it is very likely that the G. T. JONES AND T. J. LEWlS I99 difference is accounted for by the factor (2a+ 1) in eqn (1). From the daia of fig. 4 and 9a and the expression for 0 it is possible to determine n, assuming that the permittivity E of the film is known. We have taken E to be the value for undoped polyethylene which direct measurement of the permittivity of the filins in situ sup gested was appropriate.Values of n are shown in the table immediately below. ESTIMATES OF THERMALLY-GENERATED CARRIER DENSLTIES flt BASED ON MOBILITIES El AND p' (IODINE 10 g PER LITRE I M K1 SOLUTION AT 30°C) d=76pmE&'rn-':< 107 rlt/n1-3x 1020 E d = 127~111 nt TJ d= 178pm nt 5.25 2.8 4.5 I .4 3.5 0.7 6.0 2.7 5.O 1.7 4.0 0.9 7.0 3.0 6.0 1.9 5.0 I .3 7.0 1.8 6.0 1.3 8.0 1.8 7.0 1.3 The value of n remains sensibly constant for any given sample at a value of about lo2*m-3 in close agreement with trap densities estimated by Davies.lo Choosing a value of n, it is possible to calculate the contribution to the static current density from such carriers. For a 127 pm sample at a field of 4.5 x lo7V m-l and assuming a value of mobility from fig.4 we find a current density of 8 x A rr2. This is less than the measured value from fig. 6 (-1.5 x A m-2) but in view of the various possible inaccuracies is surprisingly close and suggests that a major part of the static current characteristic comes from thermally generated carriers. We should note that the onset of the peak in p' (fig. 9) coincides with the plateau region of the current-field curves (fig. 6) and also with the extrapolated threshold field Et (fig. 3). Although the magnitude of the peaks (fig. 9b) increases with iodine concentration the position of the peaks on the field axis is not so affected in agreement with the fact that the limiting Et in fig. 3 is also independent of concentration.The situation appears to be similar for the KI solution alone in fig. 9b but there will be some free iodine in this case also. Since the essential action of the iodine is likely to depend on its electronegativity and strong acceptor action it is important to try other similarly active dopants. In fact using sulphuric acid in water in a ratio 1 part in 5 in place of the iodine-KI solution produced similar although less repeatable oscillation phenomena and a (current field) curve with all the characteristic features of fig. 6 but at a lower level of current. The calculated mobility (p-10-13 m2 V-' s-l ) w as in agreement with that for the dry iodine samples (see fig. 5) i.e. those of lowest iodine concentration. DISCUSSION The direct evidence of a negative differential coefficient for mobility adds consider- able weight to the argument that the current oscillations are associated with domain f~rmation.~.There may also be significance in the fact that a second general condition for the onset of domain formation namely that the product of carrier density and specimen length should exceed a certain minimum,6 is also obeyed for the present system. There is more than one derivation of this condition but the general result for conventional semiconductors is that the product should be in the range 10l5 to 10l6 m-2. In our case if we choose values from the table we find the product to be practically constant at 2 x 1016m-2. We can see also from this condition why oscillations might not be possible in thin specimens (d = 50 pm) unless the iodine concentration was very large.OSCILLATIONS IN POLYETHYLENE It is now necessary to consider in more detail the action of iodine (or sulphuric acid) and the nature of the carrier whose motion could give rise to the behaviour illustrated in fig. 4 or 9. The iodination action in polyethylene may be twofold. First the strong electronegativity of iodine will create donor-acceptor complexes with the polymer and the evidence is that this occurs in association with the terminal double bond vinyl groups. Electron transfer to the iodine releases vacancies (“ holes ”) into the polymer chain which then become mobile in the crystalline region of the polymer structure since in the close-packed crystallite hole transfer between chains is also possible.* 2* Secondly neutral iodine as associated complexes 1 might be able to link polymer molecules across the amorphous gap between crystal- lites. Electron transfer along an I, chain could occur readily. Thus composite conduction pathways are opened up which unlike those in conventional semiconduct- ors may be localised and devious.2 Vacancies will have an effective mobility which takes account of both drift through the polymer crystallite lattice and trapping at iodine sites with an associated activation energy Wn2 Thus where y is the lattice mobility and 13 is a factor involving the trapping site density which will probably depend on the density of iodine at low concentrations and on the density of vinyl end groups at higher saturation concentrations of iodine.Eqn (2) is similar to that used by Davies * and W may be assigned the value 1.2eV estimated from the temperature dependence of the frequency of oscillations as well as the static current. The lattice mobility p1 can be represented by a relaxation time approxima- tion y1 = z/m:kwhere z is the relaxation time for collisions within the polymer crystal- lites themselves and m* is an effective mass for the carrier moving in an appropriate energy band ofthe crystallite structure. If as the field increases the carrier moment- um increases m* could increase markedly particularly if it is moving in a narrow band. Thus p1 could exhibit the necessary decrease with increasing field. At fields below the critical value Et we find according to fig.9 that p’,and therefore p increases rather than decreases with field. This might be explained by the counter influence of afield lowering of the activation barrier W. Much more work is required to confirm these ideas conclusively but it is certainly plausible that a decrease in effective mass of holes in the polymer chain is responsible for the mobility character- istic and thus for the current oscillations. The role of the electrodes has not been questioned so far but it has been implicit according to the model that the cathode and anode are ready hole and electron acceptors respectively. Diffusion experiments by Taylor and Lewis have indicated that it is certainly necessary to have iodine at the cathode interface for electron injection there but that under steady state conditions the current is bulk rather than injection limited.The experiments with double films either in the wet or the dry arrangement also lead to the conclusion that interfaces do not play a crucial role in the oscillatory process. It will be important to establish whether the phenomena observed with the poly- ethylene system may also be found with other long chain organic structures provided vacancies can be introduced into the chain. Negative differential mobility for carriers moving in narrow energy bands in organic crystals or even for movement along extended chain molecules might be a more common property than hitherto expected and overlooked in the past because of the need to operate at high electric fields and with low currents.Oscillatory currents have been observed in other organic polymers for example by Swaroop and Predecki l4 in dry polyethylene terephthalate and polystyrene as well G. T. JONES AND T. J. LEWIS 20 1 as polyethylene fhs withcut iodine doping. The likelihood in many such cases is simply that a relaxation breakdown phenomenon is being observed l5 especially since fields greater than 10' V m-l are required. Recently Toureille Reboul and Caillon have reported observations of oscillatory current phenomena in the form of regularly spaced peaks in the same three polymers. They find that oscillations coincide with the onset of a negative differential resistance which disappears along with the oscillations at higher fields.From the period of oscillations they estimate a carrier mobility of 10-13 m2 V-1 s-l for polyethylene which since their sample is undoped is much greater than ours would be in similar circumstances. The niodel they propose is that above a threshold field the injection capacity of an electrode increases markedly so that excess charge is momentarily injected into the film. While it propagates this charge serves to lower the field at the electrode and so interrupts the injection. Injection thereby becomes intermittent and the current fluctuates in a quasi-regular manner. Further work is required to confirm whether this phenomena differs fundamentally from that found for the iodine-doped system but it is encouraging that the subject is receiving continued attention.The authors thank Dr. D. M. Taylor and Mr. M. G. Jones for much help especially in earlier stages of this work. Many thanks are also due to Dr. F. J. Smith and the Monsanto Chemical Company for the supply of polyethylene samples and helpful discussions. One of us G. T. Jones is grateful for the award of a Science Research Council Research Studentship. D. W. Swan J. Appl. Phys. 1967,38 5051. T. J. Lewis and D. M. Taylor J. Phys. D Appl. Phys. 1972,5,1664. V. A. Marikhin A. I. Slutsker and A. A. Yastrebinskii Sou. Phys.-Solid State 1965 7 352. 4D.W. Swan J. Appl. Phys. 1967,38 5058. D. E. McCumber and A. G. Chynoweth Trans. I.E.E.E. Electron Deaices 1966 13 4. P. H. Butcher Rep. Prog. Phys. 1967 30 97. 'D. K. Davies Static Electrification I.P.P.S.Conference 1967 Ser. No. 4 29. 'H. J. Wintle J. Appl. Phys. 1970 41 4004. K. Keiji Kanazawa I. P. Batra and H. J. Wintle J. Appl. Phys. 1972 43 719. lo D. K. Davies J. Phys. D. Appl. Phys. 1972 5 162. I' D. K. Davies and P. J. Lock J. Electrocheiti. SOC. 1973 120 266. R. J. Fleming Trans.Furuduy SOC.,1970 66,3090. I3 E. A. Liberman and V. P. Topaly Biochim. Biophys. Actrr 1968 163 125. l4 N. Swaroop and P. Predecki J. Appl. Phys. 1971 42,863. N. Swaroop P. Predecki and J. P. Allen J. Appl. Phys. 1973 44 1943. l6 A. Toureille J. P. Reboul and P. Caillon Comnpt. Rend. 1974 278 849.

ISSN:0301-5696    

DOI:10.1039/FS9740900192

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

Oscillations in Glycolysis Cellular Respiration and Communication BYARNOLD AND BENNO HESS * BOITEUX Max-Planck-lnstitut fiir Ernahrungsphysiologie Dortmund Received 3rd October 1974 In an attempt to understand oscillatory phenomena in chemical subcellular and cellular systems the mechanisms of oscillation as well as the general dynamics of glycolysis cellular respiration and cyclic-AMP-controlled oscillation in the slime mould Dictyostelirtm discoideum have been studied. The enzymic sources of the oscillations have been identified with glycolysis as well as for the slime mould and appropriate mathematical models have been developed. The primary reaction mecha- nism of mitochondria1 oscillations is not yet known in detail. However the studies show unequivo- cally that the conditions for the generation of oscillations can be summarized for all cases in four common statements (i) The "primary source " of the oscillations has a nonlinear kinetic characteristic (ii) The kinetic structure involves simple and multiple types of feed-back interactions.(iii) The system nioves on a limit cycle. (iv) The system operates far from equilibrium and is thermodynamically open. Recently oscillations in biochemical and biological systems have been observed (for summary see ref. (1)-(4)) recognized as results of thermodynamic conditions and defined as " dissipative structures " by Glansdorff and Prig~gine.~ Indeed theory and experiment demonstrate a number of dynamic states in nature which evolve in open systems operating with a critical degree of non-linearity distant from equilibrium.Three types of behaviour have been observed (i) The maintenance of multiple steady states with transitions from one to another. (ii) The maintenance of rotation on a limit cycle around an unstable singular point. (iii) The maintenance of sustained oscillations coupled to diffusion resulting in chemical waves. Recent analyses of biochemical and cellular oscillators provide a remarkable insight into the molecular source of non-linearity although detailed mechanisms might not be at hand in every case. In some cases promising progress has been made in biochemical studies of the dynamics observed and of the structures involved as well as in the establishment of quantitative mathematical models.This paper summarizes results obtained in our laboratory in the field of bioenergetic processes and of intercellular communication in partial collaboration with Dr. A. Goldbeter of the Weizmann Institute of Science Rehovot and Dr. G. Gerisch of the Friedrich- Miescher-Laboratorium der Max-Planck-Gesellschaft Tiibingen. OSCILLATIONS IN GLYCOLYSIS At the present time oscillating glycolysis is the best understood oscillating biochemical system. Since the original observation of NADH-cycles in yeast cells 202 A. BOITEUX AND B. FTESS initialed on the transition from aerobiosis to aiiaerobiosis (for review sec ref. (1 ,)) this phenomenon has received considerable attention. It has been observed in intact cells in cell ghosts in cell extracts and by a variety of analytical methods it can be reduced to its molecular mechanism and satisfactorily simulated by digital computer techniques.In a suspension of intact cells following the addition of glucose under anaerobic conditions a number of cyclic metabolic changes can be observed and directly demonstrated by recording the fluorescence of NADH located inside the cells and part of the metabolic process involved in the cyclic phenomena. A typical experiment is shown in the fluorometric record of fig. 1 which was obtained after a single addition of 100 pmol glucose leading to a series of cyclic changes of NADH-fluorescence FIG.1.-Oscillations in yeast cells grown aerobically for 32 h. The arrow indicates a single addition of glucose.(From ref. (7)). By means of an injection technique allowing the continuous addition of glucose at a steady rate undamped oscillations of NADH can be maintained as demonstrated in fig. 2. Whenever suitable glycolytic substrates-either glucose fructose sucrose maltose or trehalose-are injected into a suspension of yeast cells the cells continue to glycolyze their substrates in an oscillatory manner as long as the injection rate is maintained constant within an upper and lower limit.6* The period of the cycles varies with the experimental conditions between 15 and 75 s.' FIG.2.-Qscillatioas in yeast ceIls grown anaerobically for 32 h. Glucose is injected at a constant rate. (From ref. (7)). BIOCHEMICAL OSCILLATIONS IN TIME AND SPACE By a short treatment with toluene the cellular membrane of yeast can be made permeable to low molecular weight substances whereas the enzymic equipment of the cells is retained within the cellular volume.8 This technique allows a study of the functjon of the low molecular weight substances in the mechanism of glycolytic oscillation.It has been shown that as soon as a lower limit of NAD and adenine nucleotide concentrations is maintained substrate addition leads to glycolytic oscillations with a period between 30 s and a few minutes comparable to what is observed in intact cells. To a first approximation the NAD concentration determines the amplitude while the phosphate/nucleotide system strongly influences period and waveform of the oscillation of NADH.The complexity of the cellular system can be reduced further without losing essential properties. A cell-free extract obtained by treatment of cells with ultrasonic waves and high speed centrifugation still exhibits self-sustained oscillations when supplied with glucose or fructose at a limiting rate using the injection technique or a controlled enzymic generation of glucose from disaccharides (see ref. (1) and (7)). The period of the oscillations of NAD fluorescence in the cell-free extract depends primarily on the rate of substrate input and varies between several minutes and approximately 1 h.6 The prolongation of the period observed in cell-free extracts as compared with those observed in the intact suspended cells is mainly due to the dilution of the glycolytic enzymes during preparation of the extract.FIG.3.-Phase relationsof oscillating glycolytic intermediates. The amplitude of the concentrations is normalized. (From ref. (2)). A detailed analysis of the dynamics of glycolytic components during oscillation showed that in addition to the oscillation of NADH fluorescence the concentrations of all glycolytic intermediates oscillate in the range of 10-5-10-3M.6 Also pulsed production of protons and of carbon dioxide has been recorded by suitable tech- niques.2* The result of these studies is illustrated in the phase pattern given in fig. 3 where the normalized concentration changes per unit of time are summarized. The figure shows that the concentrations of metabolites oscillate with equal frequency A.BOITEUX AND B. HESS but different phase angles relative to each other. According to their time dependence they can be classified into two groups in which maxima and minima of the concentra- tions coincide in time. The two groups differ by the phase angle a which depends on the experimental conditions.6 Phase angle analysis of the concentration changes of glycolytic intermediates during oscillation allows location of the enzymic steps controlling the oscillatory state. The cross-over diagram given in fig. 4 is obtained by plotting the phase angles between adjacent glycolytic intermediates according to their reaction sequence along the glycolytic pathway. The phase shift of 180” between fructose-6-phosphate and fructose- 1,6-bisphosphate as well as between phosphoenolpyruvate and pyruvate indicates the enzymes phosphofructokinase and pyruvate kiiiase to be essential control points.FIG.4.-Phase angles of glycolytic intermediate concentrations during oscillations in a cross-over plot. (From ref. (2)). The question which of the kinases is the ‘‘primary oscillophore” could be answered by a demonstration that the substrate of phosphofructokinase fructose-6- phosphate does indeed induce glycolytic oscillation whereas an injection of fructose-I 6-bisphosphate or even phosphoenolpyruvate the substrate of pyruvate kinase does not. Furthermore the role of phosphofructokinase as the generator of glycolytic oscillations is stressed by the fact that its substrate fructose-6-phosphate clearly exhibits the relatively largest concentration amplitude of all oscillating glycolytic intermediates.6* The function of the control point attributed to pyruvate kinase can be described as control of pulse propagation along the chain of enzymically catalyzed reactions of glycolysis.Indeed it can be shown that ADP one of the products of the phospho- fructokinase reaction strongly interacts with pyruvate kinase whereas the other product fructose-l,6-bisphosphate does not influence the latter enzyme under conditions observed during glycolytic oscillation. Since ATP the product of the pyruvate kinase reaction reacts again with phosphofructokinase feed-forward (ADP) and feed-backward (ATP) components couple the primary oscillator phosphofructo- kinase with pyruvate kinase and propagate metabolic pulses via the nucleotide system.The function of the adenine nucleotides can be directly demonstrated by phase shifting experiments as shown in fig. 5 (see also ref. (10)). The addition of ADP at the NAD-minimum has no influence on the oscillation whereas addition of ADP at the NAD-maximum-corresponding to the lowest level of ADP-shifts the phase of the oscillation by 180” because of the rapid acceleration of the reactions catalyzed by phosphoglycerate kinase and pyruvate kinase both being susceptible to adenine nucleotide control. In addition to the feed-forward control by ADP the adenine nucleotide system imposes a strong feed-back activation on phosphofructokinase via AMP the allosteric activator of the latter.As should be expected ATP and BIOCHEMICAL OSCILLATlONS IN TlME AND SPACE AMP also shift the phase of glycolytic oscillation if added at appropriate time intervals. .-.. .. .. -. ._ . . .. . .I b15' -4 ... .-. -i-.. . . FIG.5.-Titration of glycolytic oscillations in yeast extract with ADP. The phase angle diagram of fig. 4 indicates a third control point located at the enzyme couple glyceraldehyde dehydrogenase/phosphoglycerate kinase. The first enzyme is part of the nicotinamide adenine dinucleotide loop. The latter one reacts with the adenine nucleotide pool. Both enzymes strongly cooperating exert a duplex control function on the glycolytic oscillation. This duplex control is responsible for the variation of the phase angle a by any change in the concentration of the nucleotide or dinucleotide systems compare ref.(6). The actual rates of the different enzymic reactions are coupled via a simple conservation equation to the time dependent concentration changes of the respective metabolites as follows eci,(t) = ~7~i)+(t)-~ci,-(t> indicating that the observed time course of a metabolite concentration reflects the imbalance between the source term (vci,+)and sink term (uci)-) for this metabolite. With given restrictions from the known input rates and the measured time course of the glycolytic intermediates the time dependences of the respective enzymatic reactions have been calculated.* The pattern obtained resembles a simplified version of fig. 3. The enzymic flux rates split into two groups with coinciding maxima and minima in each group.The maximum activity of enzymes in the ''upper " part of the glycolytic reaction sequence precede the maxima of the residual enzymes by the angle a. Since cc is in the range of 15"-60" it can be concluded that all enzymes operate almost synchronously which corroborates the experimentally observed temporal coincidence in the product ion of fructose- 1,6-bisphosphate and carbon dioxide. Again the amplitude of the phosphofructokinase activity which is the largest and a low amplitude of aldolase point to the generation of metabolic pulses by phosphofructokinase and to their propagation via the adenine nucleotide system (see also ref. (1 1)). In fact the oscillatory dynamics of glycolysis can be reduced to the operation of the " oscillophore " phosphofructokinase.An analysis of its structure and activity revealed complex properties. The molecular weight of the enzyme was found to be 720 000 the enzyme can be dissociated into 8 subunits of approximately 86 000 and 94 000 Daltons.7 The kinetics of the enzyme have been analyzed in the yeast extract under conditions observed during glycolytic oscillations as shown in fig. 6. The * in collaboration with Dr. H.-G. Busse. -f In collaboration with Dr. N. Tamaki. A. BOlTEUX AND B. HESS 207 Limit Conditions [mM] ATP ADP AMP A. 1 1.5 1.1 8 3 0,5 0,l 1 2 3 4 5 6 i [F-6-P]/103 M-’ FIG.6.-Activity of phosphofructokinase under conditions of oscillation. enzyme activity is strongly dependent on the state of the adenine nucleotide system as well as on the concentration of fructose-6-phosphate.In the oscillatory state the activity can change up to a factor of 80-90. The area on the left hand side of the diagram describes the range of activities which can be observed during oscillations. This kinetic behaviour fits an allosteric model with the homotropic effectors fructose-6-phosphate and ATP and the strong heterotropic activator AMP. ..... ’I 10”’-lo-’-K 22 IO-? 2 fl to-L-C .-..I 1o-s-&+Y ‘ 0 no oscillations (STABLE FOCUS OR NODE) 1d6-m oscillations (UNSTABLE FOCUS) osciilutions (UNSTABLE NODE) ano singularities K FIG.7.-Oscillatory domains of phosphofructokinase in the plan; of sourcz rate and sink con$tmt calculated for the osciilophore model with two dilferent allosteric constants (Lo).BIOCHEMICAL OSCILLATIONS IN TIME AND SPACE On the basis of the allosteric properties a mathematical model has been developed which is open and exhibits oscillatory self-excitation.12 The model displays a variety of dynamic domains which depend on the source rate the sink constant and the allosteric constant Lo (see fig. 7).* It is obvious that oscillations can only occur at conditions distant from equilibrium. The model describes in detail both period and amplitude of glycolytic oscillations as well as their change observed experimentally on variation of the injection rate of substrate. TABLE1.-INTERACTION OF GLYCOLYTIC OSCILLATOR (Tl)WITH MODULATED RATE OF SUBSTRATE INJECTION(T2).CALCULATED FROM EXPERIMENTS WITH YEAST EXTRACT relation of periods interaction entrainment on periodic (T2)pulsatory bursts of some cycles with the autonomous period TI 1.7~Ti >7'2 >1.2~Ti no synchronisation ;period and amplitude variable 2-2 N T,/n(n =1.2,.. .) entrainment on exactly IZ xT2 Recently entrainment of the glycolytic oscillophore to a periodically varied injection rate has been demonstrated experimentally. This behaviour is predicted from the common properties of non-linear oscillators. Whether and what type of entrainment is observed depends on the relation of the periods of the oscillophore TI and of the input rate T2 as shown in table 1. Given a stochastic variation of this rate the oscillophore settles at its autonomous frequency.Indeed a com-parison between the biochemical experiments and the properties of the model reveal a remarkable congruence. FIG.8.-Simultaneous record of the redox states of cytochrome b pyridine nucleotides and flavo- proteins in oscillating mitochondria. (Boiteux and Chance unpublished experiments.) *In collaboration with Dr. T. Plesser and V. Schwarzmann. A. BOITEUX AND €3. HESS In partial summary one now notes that analysis and simulation of the oscillating glycolysis reveal a number of basic principles which obviously are necessary condi- tions for this phenomenon (i) The "primary source " of the oscillations has a non-linear kinetic character- istic. (ii) The kinetic structure involves some type of feed-back interaction.(iii) The system moves on a limit cycle. (iv) The system operates far from equilibrium and is thermodynamically open. OSCILLATION IN MITOCHONDRIAL RESPIRATION The control system of cellular respiration possesses an inherent potential to break into self-sustained oscillations. In a number of laboratories this property has been observed during studies of intact cells as well as of isolated mitochondria.1 In a typical experiment fig. 8 demonstrates the dynamics of some components of the respiratory chain in suspensions of pigeon heart mitochondria supplied with substrate and oxygen. Since an integral part of the oscillatory system is the rate of ion transport across the mitochondrial membrane which must match the rate of electron flow through the respiratory chain the oscillation can be influenced by interaction with the mechanism of ion transport.Therefore in this experiment valinomycin an ionophoric antibiotic is used to activate the potassium transport. It is shown that the oscillations start right away as soon as potassium ions are added. The use of an ionophore however is not obligatory. Oscillations can be induced as well by other methods critically matching the rate of ion transport and electron flow. As shown in the figure the components of the respiratory chain cytochrome b pyridine nucleotides and flavoproteins are oxidized and reduced simultaneously with a period of about 70 s all three responding with identical kinetics to the change of oxygen potential which is maintained by vigorous stirring.The mitochondrial system is complex enough to show a variety of dynamic states. In a phase plane plot of the redox 0 40 1 0 10 20 30 cyt. b oxidation FIG.9.-Phase plane plot of the redox states of pyridine nucleotides andof cytochrome b in oscillating mitochondria. (Calculated from ref. (141.) states of pyridine nucleotides and cytochrome b as given in fig. 9 the transition from an unstable spiral to a stable limit cycle can be demonstrated on addition of potassium chloride to valinoinycin treated mitochondria. BIOCHEMICAL OSCILLATIONS IN TIME AND SPACE The oscillatory changes in the redox state of the respiratory chain are accompanied by periodic changes of the concentrations of protons and of potassium ions and by concomitant shrinking and swelling of the mitochondria1 matrix volume.Fig. 10 is a cutout obtained from a long train of virtually undamped cycles demonstrating the phase relations between the recorded parameters. Obviously the osciIlations of redox components ion fluxes and volume changes are not synchronous. First of all the directions of the fluxes of protons and potassium ions are opposite. This ion movement however is not a simple one-to-one exchange reaction. The amplitude of the ion fluxes is up to 15 p equiv. g-' of mitochondrial protein in the case of protons and up to ten times more for potassium ions depending on the experimental conditions. This electrical imbalance must be counteracted by a concomitant flux of ions pre- dominantly phosphate and substrates which in turn implies a simultaneous transport of water by osmotic forces.In fact the maximum of the potassium concentration inside the mitochondria slightly precedes maximum swelling indicating that the volume change is a secondary process. -1 NADH t fluorescence Reduction I g protein I K+ uptake T swelli ng 10% LS h -J FIG.IO. -Siniultaneous record of NADH fluorescence H uptake K+ uptake and swelling in oscillating mitochondria (A. Boiteux and B. Chance unpublished experiments.) By direct measurement of the oxygen potential at reduced pressure it has been shown that the mitochondrial swelling cycles correspond to an oscillating rate of oxygen uptake and the phase angle between swelling and respiration was calculated to be about 60" under similar experimental condition^.'^ This lag coincides well with an almost identical position of maximum proton influx as calculated from experiments of the type shown in fig.10. The same is true for minimum respiration which corresponds to the maximum rate of proton efflux from the mitochondrial matrix. The analysis of the phase relations clearly reveals a tight coupling of vectorial ion fluxes across the inner mitochondrial membrane to the respiratory activity of the mitochondrion. The volume cycle must be classified as a dependent process respond- ing to the metabolic changes with some time delay. Detailed analysis of mitochondrial oscillations have shown that the adenine nucleotide system is a control factor not only in the case of glycolytic oscillation but also for its mitochondrial co~nterpart.'~ First evidence for the involvement of adenosine triphosphate in the control mechanism was the observation that mito- chondrial oscillation can be suppressed by oligomycin,* an antibiotic inhibitor which prevents the interconversion of ADP and ATP at the mitochondrial phosphorylation sites The internal ATP-level in oscillating mitochondria clearly exhibits periodic * In collaboration with Dr.B. Chance. A. BOITEUX AND B. HESS 21 1 cycles as shown in fig. 11. The ATP-minimum succeeds maximum swelling with a lag of approximately 50° coinciding with the phase angle for maximum respiration rate. Accordingly the highest ATP-level corresponds to the lowest respiration rate.FIG.11.-Phase relations between swelling and endogenous level of ATP in oscillating mitochondria. (Boiteux and Chance unpublished experiments.) Final proof for adenine nucleotide control in mitochondrial oscillations comes from titration experiments. Addition of ADP at any phase angle of the oscillation results in a new synchronization of the mitochondrial oscillations at highest respiratory rate. Titrations with ATP lead to a synchronization at minimum respiration. Within a limited range of concentrations the synchronization by adenine nucleotides is completed within fractions of a second and the oscillation continues without time delay from the newly obtained position. The phase relations between volume change respiration rate redox state of cytochrome b and ion fluxes are given in the outer shells of fig.12. In addition the diagram shows in its centre sections the phase angles of the cycle obtained by titrations with ADP ATP potassium ions acids and bases respectively. Volume Ion Phase Angles obtained by Titration FIG. 12.-Phase relations in mitochondrial oscillations and shift of phase angles by titration. (From ref (14).) BIOCHEMICAL OSCILLATIONS IN TIME AND SPACE From these experiments it is obvious that the ADP/ATP-ratio in the mitochondria1 matrix is a control factor of the oscillations though not the only one. Experiments with valinomycin clearly demonstrate that the activity of the potassium carrier determines both amplitude and period of the o~cillation.'~ Thus the rate of ion transport across the mitochondrial membrane is another control factor in addition to the endogenous adenosine phosphate system.In fact membrane-bound carrier systems are the receptor sites for controlling and synchronizing signals in suspending medium and cytosol respectively. By titrations with acids and bases (cf. fig. 12) the external synchronizer of the mitochondrial population was shown to be a proton gradient l4 across the membrane. This gradient developing along with the proton flux serves as a very effective positive feed-back signal to the respiratory system promoting strong self-amplification. Fig. 12 shows that an addition of protons shifts the cycle to the onset of swelling and to further proton efflux from the matrix space.Addition of hydroxyl ions immediately shifts the mitochondrial system to the position where it starts shrinking and taking up protons from the medium. The study demonstrates that the mitochondrial energy production is tightly controlled by a multiplicity of signals from the cytosolic space the rate of ion fluxes across the inner membrane being a critical controlling parameter. The detection of the very effective feed-back action of a proton gradient raises the question of the rela- tion between the membrane charge and the structural and functional changes at critical control sites. Though the mechanism of generation of mitochondrial oscillations is not yet known in detail it is obvious that the statements from points (i)-(iv) of the previous section also hold for this case.(1) The involvement of vectorial transport processes for controlling reaction steps leads to non-linear kinetics. (2) The self-exciting proton gradient is the basis of the feed-back structure of the membrane system. (3) Fig. 9 shows that the system moves on a limit cycle. (4) Approximation towards the equilibrium state by substrate exhaustion or oxygen limitation leads to rapid damping of the oscillation which can be maintained only if a minimal flux is provided. The system is open for the source namely oxygen and for a number of sink reactions leading to the production of water carbon dioxide and heat. OSCILLATIONS IN CELLULAR COMMUNICATION The source and nature of the spatial organization of living systems is one of the fundamental problems of the biological sciences.More than 20years ago the question of stability of biological systems with respect to diffusion was analyzed by Turing and later by Glansdorff and Prigogine.' Nowadays it is known that rotation of chemical and biochemical systems around an unstable singular point coupled to diffusion does occur resulting in chemical waves. The generation of chemical waves has been observed in the case of the classical Belousov-Zhabotinsky reaction (see ref. (I) and (4))and it has been shown that this system is able to transmit information. Recently it has been found that the propagation of biochemical waves plays a significant role in the process of morphogenesis of the slime mould Dictyustelium discoideum.'' Within its life cycle the slime mould passes through a growth phase in which the organism exists in the state of single ameboid cells in a given life territory.After the end of the growth phase the single cells aggregate in response to chemotactic stimuli and finally form a multi-cellular body which differentiates to become the fruiting body which is the source of the next generation. This is an example of self-organization of spatial patterns by chemical communica- tion starting with a layer of randomly distributed identical cells. The aggregation territories are controlled by the centres which release chemical pulses of cyclic AMP A. BOITEUX AND B. HESS with a frequency of 0.2-0.3min-I. The pulses are propagated from cell to cell as excitation waves which spread in a uniform layer with a constant speed of 40-50pm mid.The waves are either concentric or spiral shaped. Formally the system can be treated as a set of diffusion coupled oscillators in which the DictyusteZium cells operate as the oscillating element by receiving amplifying and ejecting periodically concentration gradients of cyclic AMP. The periodic motion of the cells as well as the response of the cells to cyclic AMP can be recorded by light scattering indicating changes in cellular shape volume and degree of aggl~tination.'~~ A typical example is given in fig. 13 where periodic l8 spike formation and later on sinusoidal oscillations are recovered. The system is sensitive to cyclic AMP addition.l* A half maximal reaction is obtained with 1 x lo4 molecules of cyclic AMP per cell and the response is still detectable at a molarity of with a molecule to cell ratio of 3000.Cyclic AMP pulses interact also with the oscillating cellular system resulting in phase shift or suppression of spike formation. FIG.13.-Oscillations of light scattering in a suspension of Dictyostelium discoideum. The drift of the curve indicates shift towards larger aggregates (From ref. (18).) Biochemical analysis identified the function of a membrane-bound adenyl cyclase which is responsible for the production of cyclic AMP. The activity of a phospho-diesterase controls the upper threshold level of cyclic AMP at the membrane as well as in the intercellular space into which a solhble diesterase is secreted.The binding of cylic AMP to a receptor site is transitory which should be expected for an oscillat- ing binding function.' 7-20 It is interesting to note that the receptor system by which the slime mould aggregates in response to periodic cyclic AMP pulses is similar to the acetylcholine receptor/cholinesterase system by which nerve impulses are transmitted through the synaptic ~1eft.l~ It is beyond the scope of this paper to discuss the coupling mechanism by which waves of cyclic AMP are received and transduced resulting in an orientated chemotactic motion towards the aggregation centre. An analysis of the enzymic system involved in the production of cyclic AMP reveals an interesting network of interactions as demonstrated in fig.13 on the basis of recent experimental data. 19*2o The cyclase and pyrophosphohydrolase form a bienzyme system which is autocatalytically controlled by feedback. The hetero- tropic interactions of both enzymes with cyclic AMP and 5' AMP respectively are highly cooperative. Goldbeter recently analyzed the overall dynamics of the system applying the concerted transition theory of allosteric enzymes. A computer solution demonstrates that under suitable activation by 5' AMP the system readily settles on a limit cycle around a non-equilibrium unstable stationary state with a period independent of initial conditions. The model satisfactorily describes the experimental observations. The experimental results and model analysis demonstrate that intercellular communication transmitted by chemical waves plays a significant part in the morpho- genesis of highly complex species.Furthermore one of its essential elements namely BIOCHEMICAL OSCILLATIONS IN TIME AND SPACE the pulsewise production of the transmitting chemicals could be reduced and identified as the periodic function of the cyclic AMP producing enzyme in the cellular wall. Indeed nonlinear chemical and biocheinical oscillators are well suitable to serve as devices to measure time to memorize chemicals and to synchronize chemical activities. J( AT P Adenylcyclase Py rophos-phohydrolase Cyclic AMP G-AMP+ \ ? Phosphodiesterase1 FIG.14.-Enzyme network for the control of the cyclic AMP-level. (From ref. (17)). B. Hess and A. Boiteux Ann.Rev. Biochem. 1971 40,237. B. Chance E. K. we A. K. Ghosh and B. Hess Biological and Biochemical Oscillators (Academic Press New York and London 1973). B. Hess A. Boiteux H.-G. Busse and G. Gerisch Membranes Dissbatiue Structures in Evolution (Wiley-Interscience New York 1974) in press. G. Nicolis and J. Portnow Chem. Rev. 1973,73 365. P. Glansdorff and I. Prigogine Thermodynamic Theory of Structure Stability and Fluctuations (Wiley-Interscience N.Y. 1971). B. Hess A. Boiteux and 3. Kriiger Ah. Enzyme Regul. 1969 7 149. ’B. Bess and A. Boiteux Hoppe-Seyler’s 2.Physiol. Chew. 1968,349 1567. R. E. Reeves and A. Sols Biochem. Biophys. Res. Comm. 1973’50,459. B. Hess and A. Boiteux Biochim. Biophys. Acta Library 1968 11 148. lo B. Chance R. Schoner and S.Elsaesser Proc.Natl. Acad. Sci. USA 1964 52 335. l1 B. Hess Nova Acta Leopoldina 1968 33 195. l2 A. Goldbeter and R. Lefever Biophys. 1972 12 1302. l3 A. Boiteux and H. Degn Hoppe-Seyler’s 2.Physiol. Chem. 1972 353 696. l4 A. Boiteux Ergebn. exp. Medizin 9 (Volk und Gesundheit Berlin 1972) p. 347. l5 A. M. Turing,Phil. Trans. B 1952,237,37. l6 H.-G. Busse and B. Hess Nature 1973 244,203. l7 G. Gerisch D. Malchow and B. Hess Cell Communicatioiz and cyclic AMP Regulation during Aggregationof the Slime Mould Dictyostelium Discoideum (Biochemistry of Sensory Functions) (Springer Verlag Heidelberg Berlin New York) in press. l8 F. Gerisch and B. Hess Proc. Natl. Acad. Sci. USA 1974,71 2118. l9 E. F. Rossomando and M. Sussman Proc. Natl. had. Sci. USA 1973 70 1254. 2o D. Malchow and G. Gerisch Proc. Natl. Acad. Sci. USA 1974 71 2423. 21 A. Goldbeter Nature 1974.

ISSN:0301-5696    

DOI:10.1039/FS9740900202

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

GENERAL DISCUSSION Dr. B. L. Clarke (Alberta) said The cooperation between positive and negative feedback effects discussed by Franck shows up clearly in the linear steady state stability problem for chemical reaction networks. The necessary and sufficient conditions for the asymptotic stability of a linearized network are that the Hurwitz deter- minants are all positive. 1 have shown that each term in the Hurwizt polynomials can be interpreted as a product olfeedback loops. Negative terms are combinations of feed-back loops that cooperate to destabilize and positive terms are combinations which stabilize. The Hurwitz determinant can therefore be viewed as a mathematical algorithm for construct iiig all the combinations of feedback loops which are significant for stability and weighting them with coefficients and algebraic signs according to their importance and stabilizing or destabilizing character.Only one product of feedback loops is usually important at any one time. Franck has illustrated that the waveforms of oscillatory systems have portions which are dominated by various feedback loops. The reason for this can be under- stood from steady state stability analysis. Fast changing variables attain pseudo- steady states for given values of the slowly changing variables. The latter variables move slowly on a manifold which passes through regions of instability of the pseudo- steady state. In these regions the trajectory moves rapidly in a manner determined by the product of feedback loops which destabilize the pseudosteady state.Dr. R. Lefever (Brussels) said I must say that the necessity of two counteracting feedbacks positive and negative in order for a chemical system to produce an oscil- latory behaviour is not obvious to me or else that the definitions of negative and positive feedback are not so clear as it may seem. In the Brusselator there is indeed a step usually viewed as a positive (autocatalytic) feedback on the production of X but then no step can be associated with what would usually be called a negative feedback. Indeed what brings the concentration of X down during an oscillation is the fact that at the same time the autocatalytic step produces X and consumes Y. This consumption occurs at a rate proportional to X2 Y which cannot be matched by the linear Y production step (B +X -+ Y + D) if the concentration of X goes beyond a certain level.One thus sees that in a way the autocatalytic step is self inhibitory. On the other hand following the author’s rules since here both X and Y may have sharp breaks in the course of their oscillation both variables should be considered of the positive feedback type. As another illustration of the ambiguity between negative and positive feedback concepts I may perhaps mention the Yates-Pardee type of enzymatic chain regulation ; EO EI EZ En-1 A -+ X -+ X2 3 . . . -+ X B 3 Ot I In this chain of reaction the first enzyme of the chain Eo is inactivated by the end of chain product X,. This is generally considered as a purely negative feedback type of control.Nevertheless limit cycle behaviour can be observed in such systems. Dr. P.Rapp (Cambridge) said In his paper Franck notes the importance of time delays in allowing nonlinear feedback systems to oscillate. I would like to confirm B. L. Clarke J. Chern. Phys, 1974 60 1481. 215 21 6 GENERAL DISCUSSION the importance of this observation and also connect points raised by Auchmuty and Lefever in the discussion of this paper. Auchmuty has pointed out that if the dif- ferential equation contains terms with retarded arguments then it is possible for a system with a single variable to possess a stable periodic solution. As an example he cites the well known equation proposed by Hutchinson as a model for a fluctuating population dm --x(t)[l -Kx(t-T)].dt This equation has been investigated by Cunningham and Jones.3 1 would like to point out that the presence of time delays has an important effect on the stability properties of differential equations which describe phase shift oscillators mentioned by Lefever. Systems of this type consist of a sequence of reactions in which the first step is inhibited by the product of the last reaction. The most familiar class of differential equations describing these oscillations takes the following form (G~odwin,~ and Walter 5). = h(x,)-b,x ij= gj-lxj-l -bjxj ;j = 2 . . . n. For eqn (2) it can be shown that for an arbitrary nonlinear function h(x,) IZ is required for periodic solutions (Higgins and Rapp 7). Eqn (3) is the governing equation corresponding to eqn (2) where delays occur between each reaction.The constants Tj are positive real numbers. Pj = gj-1xj-1 -bj~j xj = yj(t-Tj) ; j = 2.. .IZ 1 Here it is possible to construct a one dimensional system which possesses a stable periodic solution. For example ; k dx --bx dt 1+ax(t-T)2 when k = 30 000 CI = 1 b = 1 T = 2 oscillates from xmin= 11.01 to xmax= 156.90 with period = 5.99. Clearly this is an extreme case which is valuable primarily as an object lesson. However such considerations are important when selecting between theoretical models of oscillatory processes. For example it can be shown * G. E. Hutchinson Ann N. Y. Acad. Sci. 1948,50,221. W. J. Cunningham Proc. Nat. Acad. Sci. 1954,40 708. G.S. Jones Interlinear Symposium on Nonlinear Differential Equationsand Nonlinear Mechanics ed. J. LaSalle and S. Lefschetz (Academic Press New York,1963). B. C. Goodwin Temporal Organization in Cells (Academic Press New York,1963). C. Walter Biochemical Regulatory Mechanisms in Eukaryotic Cells ed. E. Kun and S. Grisolia (Wiley Interscience New York 1971). J. Higgins R. Frenkel E. Hulme A. Lucas and G. Rangazas Biological and Biochemical Oscillators ed. B. Chance E. Pye A. Ghosh B. Hess (Academic Press New York 1973). 'P. Rapp FEBS (Federation European Biochemical Societies) General Meeting Budapest August 1974. GENERAL DISCUSSION that it is impossible to fiiid a set of physically realizable reactions Constants such that the following system oscillates i2 = gix1-62x2 (4) i3 = g2~2-b3~3 24 = 93~3-b4~4.However in the corresponding delayed system if c = Tl+T,+T3+T4 > 0 then it is possible to find reaction constants which give an oscillatory system. Dr. H. Tributsch (Bedin)said in his interesting review article Franck has demon-strated a remarkable similarity between artificial oscillating systems and a variety of non linear biological phenomena. In this connection it is interesting to note that there are apparently no simple model systems known which can simulate the property of biological receptors of modulating the frequency of excitable membranes according to the intensity of external physical and chemical stimuli. The transformation of continuous signals into spike patterns (pulse signals) is an important prerequisite for advanced data acquisition and processing methods.It would therefore be of considerable interest to develop technical detectors with digit a1 response . 10 N k8 light intensity 12%/ 100 'lo L 37 % 1 I I 1 I 1 0 I 0 5 10 15 20 25 30 35 40 45 time FIG.1 .-Influence of visible light on the oscillation behaviour of the system Cu2S-electrode/elec-trolyte+H202. Electrode potential -0.5 V against S.C.E. Our own experience with the development of a digital photodetector indicates that efforts in this direction might be promising. To achieve our aim it was necessary to find a suitable electrochemical oscillator with a light sensitive feedback mechanism. This condition was met by the new oscillating system Cu,S -electrode/electrode+HzOz.The light sensitive reactant (absorption between 400 and 600 nm) turned out to be an oxide bond on the electrode surface which is formed through the oxidizing action of hydrogen peroxide and which can be destroyed by a photo-electrochemical reaction. It is participating in an autocatalytical reaction which is crucial for the occurrence of the electrode current oscillations. Light can consequently be used to modulate the frequenc:y of these oscillations (fig. 1). Detectors of this type might become examples for a useful technical application of oscillating mechanism GENERAL DISCUSSION Dr. M. Blank (ONR London) said The ionic mechanism proposed by Shaslioua to account for the properties of a polycation w polyanion junction membrane can also explain the oscillations we have observed in a much thinner bilayer system.We formed the bilayers from decane solutions of cholesterol in contact with two identical aqueous phases containing NaCl and cetyltrimethylammonium bromide (CTAB). The resistance of these bilayers depended strongly upon the concentration of CTAB.l At low CTAB concentration the bilayer behaves as if it were negatively charged (probably as a result of adsorbcd anions) and as the CTAB concentration increases the bilayer becomes positively charged. The variation of the resistance with surface charge is in line with earlier experiments on mono1ayers.2 On passing current across bilayers formed in a range of CTAB concentrations around the point of zero surface charge we produced potential oscillations which we believe are due to the variations in CTA+ ion concentration that cause the bilayer to alternate between positive and negative surface charge.The concentration changes arise because of differences bet -ween the transport numbers of the aqueous phases and the bilayer. It should be possible to analyze the oscillatory behaviour of this bilayer system in terms of the physical properties of the components and the known behaviour of monolayer and interfacial systems. It may therefore serve as a model to elucidate the mechanism of oscillatory bioelectric phenomena involving ion flow in more complex biological systems. I would also like to point out that although oscillatory phenomena exist in bio- logical membrane systems excitation per se is not oscillatory.If we consider the prototype of an excitable cell a nerve axon the normal sequence is stimulus + de-polarization + potential reversal repolarization to complete the cycle. The cycle can be repeated but only with additional or continued stimuli. Oscillatory behaviour can be seen in the pacemaker cells of the cardiac muscle conduction system but these are specialized in as much as the membranes depolarize spontaneously with no externally applied stimulus. From an electrophysiological point of view these cells represent a special case where the membrane permeabilities to Naf and K+ ions do not stabilize on repolarization. Prof. P. Meares (Aberdeen) said On consideration of Shashoua’s paper it is not immediately obvious that the conformational “ shrinkage ” of a polyelectrolyte at high ionic strength would lead to an increase in the conductance or permeability of the membrane.The term shrinkage in this context refers to a decrease in the end-to- end vector of the polymer chains. Such a shrinkage does not require a decrease in the volume of the polymeric component such as would create pores. It is also un- certain whether the polyelectrolyte complex in the interaction zone of the c +a junction would respond to changes in ionic strength in the same way as a purely polycationic or purely polyanionic material. For these reasons I would like to offer for consideration an alternative mechanism of the periodic changes in conductance of the polyelectrolyte membranes.The material in the interaction zone is held together primarily by electrostatic attractions between polyanions and polycations (in this respect Ca2+ and Ba2+ can be regarded as polycations crosslinking polyanions by electrostatic forces). When during the passage of a current the salt concentration in the interaction zone is increased by the simultaneous arrival of Na+ and C1- from opposite sides the osmotic pressure in the interaction zone must increase. A point will be reached when this osmotic pressure is sufficient to overcome the electrostatic binding forces between the polyanions and G.W.Sweeney and M.Blank J. CulZoid Znterfuce Sci. 1973,42,410. I. R.Miller and M.Blank J. Colloid Interface Sci. 1968 26 34 GENERAL DISCUSSION polycations.These will then be forced apart adopting the Na-!-and Cl- ioiis respec-tively as their counterions by the incoming water i.e. the polyelectrolyte coinplex will be dissociated. The resulting conductance increase would cause the membrane to depolarize rapidly. Subsequent reformation of the polyelectrolyte complex by diffusion of NaCl away from the interaction zone would lead to a relatively slow recovery of polarization and the cycle of events would then be repeated. Dr. T. Keleti (Budapest)said 1 would like to make a comment coiicernirig one of the possible mechanisms of oscillations in biological systems. We are interested in the analysis of "three-body systems ",in which at least one substrate and two modifiers interact on the enzyme.These modifiers can be either one inhibitor and one liberator or two inhibitors. '* The liberator is a modifier which has no enect on the free enzyme but liberates the enzyme from the action of an inhibitor or a~tivator.~A peculiar effect can be shown in the case of certain types of inhibitions and liberations. One of these types is illustrated in fig. 1 as a function of liberator concentration its effect changes from partial liberation (i.e. inhibition) through complete liberation to activation. Competitive inhibitor non-competitive liberator If KSI KS,> 1 I --0s ILPIL'I -IL3=11'1 -lLl>Il!J FIG.1 .-Interaction of a non-competitive liberator with a competitive inhibitoi. Similar effects can be shown in the case of double inhibitions where different interactions between the substrate and/or inhibitors may appear (fig.2).4 One peculiar effect is the triple-faced enzyme-inhibitor relation. As a function of substrate concentration the antagonistic effect of the two inhibitors changes through the simple summation of their action to a synergetic effect (fig. 3).2 Moreover in the case of two partial inhibitors if the concentrations of the inhibitors and the substrate are higher than the "characteristic " one we obtain the inhibition paradox (fig. 4). In this case the initial rate of the enzyme in the presence of two inhibitors may be higher than in the absence of inhibit0rs.l. T. Keleti in Proc. 9th FEBS Meeting.Vol. 32. Symp. on Mechanism ofAcrion and Regulation of Enzymes (Akadtmiai Kiad6 Budapest and North-Holland Amsterdam 1975) ed.T. Keleti. Cs. Fajszi and T. Keleti in Mathematical Models of Metabolic Regulation ed. T. Krlei and S. Lakatos (Akademiai Kiado Budapest 1975). T. Keleti J. Theoret. Biol. 1967 16,337-355. T.Keleti and Cs. Fajszi Mathemat. Biosci, 1971,12,197. GENERAL DISCUSSION ESB I2 l-l(;. 3.-Schcnw of the general nicchanisni of double inhibitions. Triple-faced erlzyme inhibitor relation In the cuse of two purdy competltive lnhlbitors if f<oC<~ tihihrc of the two inhlYIIVI on the free enzyme n_ L Dc0:antagonism I/ K 0=IEIISI/ESl ~--IS]*lSoI -ISol=Ko (d-1) -ISI*ISo I-FIG.3.-The triple-faced enzyme-inhibitor relation. Assuming a linear chain of three enzymes where the substrate of the first enzyme is an inhibitor or liberator and the product of the third enzyme is the inhibitor of the second one we can obtain oscillations in the above-mentioned cases.This means that oscillations do not require the presence of an allosteric enzyme in the enzyme system. GENERAL DlSCUSSlON 22I Dr. H. Tributsch (Berlin) said As most researchers in the field are aware one must sometimes resist the temptation to interpret biological oscillations as true chemical oscillations. Periodic phenomena in complicated molecular structures such as sub- cellular and cellular systems may also be the consequence of a nonlinear energy con-version mechanism known as parametric energy coupling or variable parameter energy conversion. Parametric energy conversion proceeds through the periodical variation of an energy storing quantity (e.g.an electrical capacitance or an elastic constant) and is known to occur in all fields of physics both on the macroscopic and atomic levels (e.g. mechanism of a swing movements of a pendulum on a spring vibrating capacitor amplifiers or parametric amplifiers in electronics nonlinear (laser) optics the Raman effkct the parametric motor) (cf. N. Minorsky ref. (1)). An example of a parametric mechanism which could be operative in subcellular systems can easily be demonstrated with a physical model (fig. 1). If a pendulum is constructed with a piece of dielectric material and placed within the plates of a capacitor to which an alternating voltage is applied it will-under certain conditions- start oscillating and reach a stationary state of constant amplitude.A periodical FIG.1 .-Dielectric model of phenomenon of Bethenod. perturbation of a variable (energy storing) quantity (the capacitance) by a high fre- quency low aniplitude oscillation can thus lead to the generation of a low frequency high amplitude (capacity) oscillation. This type of parametric phenomenon which is difficult to understand in a purely intuitive way was first described by Bethenod with a model consisting of a coil carrying alternating current and a pendulum made of a piece of soft iron placed above it and has theoretically been investigated by Minorsky. By analogy with the phenomenon of Bethenod low amplitude relatively high frequency chemical oscillations (e.g.,of the type reported by Shashoua in protein membranes) could be the actual cause for the occurrence of pronounced low frequency (parametric) oscillations in subcellular (e.g.mitochondria) systems. It is suggested that a search for these faster oscillations of much lower amplitude should be made since parametric energy conversion has recently been proposed as playing a major role in membrane bound bioenergetical mechanisms. Dr. A. Boiteux (Dortmund)(communicated):Tributsch’s initial comment is perhaps based on a misunderstanding. We do not propose a sequence of simple chemical reactions as in glycolysis to cause oscillations in the mitochondria1 system. On the N. Minorsky in Nonlinear. Oscihtiortr (D. van Nostrand Inc. Princeton New York 1962) p.390 and p. 438. J. Bethenod Compr. Rend. 1338. 207. El. Tributsch J. Theor. bid. 1975 52 S47. GENERAL DISCUSSION contrary we have pointed out in our paper that the involvement of membrane per- meability and vectorial transport processes provide the necessary non-linearity for this parametrically controlled system. Compare also ref. (14) p. 354 cit. " The state of energy charge in the capacitor for the storage of chemical energy therefore is a controlling variable for the mitochondria1 oscillation. " Dr. A. Goldbeter (Rehovot)said Metabolic oscillations controlled by cyclic AMP in the slime mould Dictyostelium discoideum present a striking similarity to those observed in yeast glycolysis (see the communication of Boiteux and Hess at this Symposium for a detailed presentation of these oscillatory systems).In both cases models based on the molecular properties of the enzymes involved in the oscillatory mechanism indicate that periodic behaviour corresponds to a temporal dissipative structure i.e. to sustained oscillations of unique amplitude and frequency around a nonequilibrium unstable stationary state (see the communication of Nicolis and Pri- gogine at this Symposium). The models account for several experimental obser- vations in yeast and in the slime mould. In the following I briefly compare the predictions of the models with corresponding experiments and present evidence for a common molecular mechanism for sustained oscillations in the two systems. Glycolytic periodicities observed in yeast and muscle extracts as well as in single cells and cell populations of yeast originate from the activation of the allosteric enzyme phosphofructokinase by one of the reaction products.Recently Lefever and I analyzed a model for this reaction in the frame of the concerted transition theory of Moiiod Wyman and Changeux. The model considered is that of an open K-V system in which the product is a positive effector of the dimer enzyme. In the liomo- geneous case where diffusion is neglected this system is described by the following evolution equations for the substrate (a) and product (y) normalized concentrations (see ref. (3) and (6) for a definition of various parameters) da -= al-aM@ dt with The limit cycle behaviour of the model matches the oscillations observed in yeast extracts with a constant periodic or stochastic source of sub~trate.~ Qualitative and quantitative agreement is obtained in the model for the variation of period and ampli- tude in the oscillatory range of substrate injection rates which extends over one order of magnitude of parameter CT,,for the phase-shift exerted by the reaction product ADP and for the periodic change in enzyme acti~ity.~.' Thus in the middle of the G. Gerisch and B. Hess Proc. Nut. Acud. Sci. 1974 71,2118. B. Hess A. Boiteux and J. Kruger Adu. Enzyme Regul. 1969 7 149. A. Goldbeter and R. Lefever Bioplzys. J 1972 12 1302. J. Monod J. Wyman and J. P. Changeux J. Mol. Biol. 1965 12 88. .'I A. Boiteux A. Goldbeter and B.Eiess Proc. Nor. Acad. Sci. 1975 72 (IOj in thc press. A. Goldbeter and G. Nicolis Prugr. Tlreoret. Biol. 1975 4 in the press. GENERAL DISCUSSION unstable domain of o1 values the enzyme reaction rate v oscillates between 0.95 % and 73 % of the maximum rate V, with a mean value of 17.5 % VM(fig. 1). Experi-mentally Hess et al.' have reported a periodic variation of phosphofructokinase activity between 1 % and 80 % VM,with a mean value of 16 % V and an activation factor of 80 comparable to the factor 77 obtained theoretically. Further analysis of the model shows that the coupling between limit cycle behaviour and diffusion can give rise to propagating concentration waves at the supracellular level.2 t15 FIG.1.-Periodic variation of enzyme activity in the concerted allosteric model for the phospho- fructokinase reaction.The curve is obtained by integration of the evolution equations of a and y for u1 = 0.7/s k = O.l/s OM = 4/s L = lo6 c = E = c' = 8 = 1 (see text and ref. 6). The model for metabolic oscillations in Dictyosteliuin discoideum is based on the regulation of two membrane-bound enzymes involved in the synthesis of cyclic AMP namely ATP pyrophosphohydrolase and adenyl cyclase which transform ATP into S'AMP and CAMP,respectively (see ref. (8) and fig. 14 in the communication of Boiteux and Hess at this Symposium). The system is further coupled by a membrane-bound phosphodiesterase which transforms cyclic AMP into 5'AM P. Rossomando IIIIII 8 4E I '0 5 6 160-2 120-' ! I 1 I 1 02 0.3 0.4 05 0.6 0.7 UI Is-' FIG.2.-Variation of the period (7') and of the amplitude (A,) of noimalized cAMP concentration in the oscillatory domain of ATP injection rates ol in the model for the oscillatory synthesis of cAMP in D.discoideum (see text and ref.(8)). B. Hess. A. Boiteaux and J. Kruger Adv. Enzyme Regul. 1969 7 149. * A. Goldbeter Proc. Nat. Acad. Sci. 1973 70 3255. A. Goldbeter Nuture 1975 253 540. G. Gerisch and B. Hess Proc. Nut. Acud. Sci. 1974 71 2118. GENERAL DISCUSSION and Sussman have shown that cAMP activates ATP pyrophosphohydrolase whereas 5’AMP activates adenyl cyclase. In both cases regulatory interactions are highly cooperative pointing to the oligoineric structure of the enzymes.The variables considered in the model are the intracellular concentrations of ATP SAMP and CAMP. A stability analysis indicates that the cooperative and regul- atory properties of adenyl cyclase and ATP pyrophosphohydrolase can give rise to sustained oscillations in the synthesis of cyclic AMP around a nonequilibriuin unstable steady state.’ The oscillations have a unique amplitude and frequency and therefore correspond to a temporal dissipative structure. Simulations of the model as to the effect of continuous or discontinuous addition of CAMP match the observations of Gerisch and Hess in suspended Dictyosteliunz cells. It should be noted that sustained oscillations of intracellular cAMP may result through transport of this metabolite across the cell membrane in a periodic release of cyclic AMP into the extracellular medium.As cyclic AMP is the chemotactic factor in D.discoideum the mechanism for intracellular oscillations can also account for the periodic aggre- gation in this species of slime The pulsatory nature of cAMP oscillations in the model and the extreme stability of their period (fig. 2) correspond well to the observation that centre-founding cells in D.discoideunz aggregation release pulses of cyclic AMP with a period of 3-5 min.4 Both the pulsatory nature of the oscillations and the stability of the period with respect to a variation of enzyme parameters or environmental constraints (influx of substrate protein concentration etc.) result from the allosteric properties of the enzymes involved in the oscillatory mechanism as in the case of phosphofructokinase for glycolytic oscillations.A further similarity between these oscillatory systems becomes apparent when noting that in the absence of ATP pyrophosphohydrolase the mechanism of CAMP-controlled oscillations in the slime mould reduces to that of oscillating glycolysis of yeast and muscle.2 Thus the molecular basis of periodic behaviour is identical in the two metabolic systems both phosphofructokinase and adenyl cyclase are allosteric enzymes under positive feedback control. Dr. A. Winfree (Indiana) said Hanusse Noyes Tributsch and Goldbeter have independently used visible light or an injected reaction intermediate to test the limit- cycle stability of an oscillating reaction.In order to compare his analytic model for glycolysis with Pye’s corresponding experimental determinations Goldbeter went a step further he measured the inflicted phase shift after recovery to the limit cycle as a function of the original phase when ADP was injected. Is it too ambitious to suggest that consumable intermediates light thermal shock or even a differently-phased volume of the same oscillating reaction could be used as transient perturbations to measure new phase as a function of old phase? Particularly in the latter case this would allow us to approach chemical dynamics from a quite new perspective via some remarkably general theorems rccently proved by Guckenheimer in response to similar conjectures by Winfree. These theorems describe a variety of topologically-inescapable singularities and discontinuities which should be conspicuous in those measurements.The arrangement of these features reveals some thing about the reactions’ dynamical organization about the number of critical reactants and of stationary states about their stability properties etc. Such * E. F. Rossomando and M. Sussman Proc. Nat. Acad. Sci. 1973,70 1254. A. Goldbeter Nature 1975 253 540. G. Gerisch and B. Hess Proc. Nut. Acad. Sci. 1974 71 21 18. G. Gerisch Curr.. Top. Derd. Biol. 1968 3 157. .I. Guckenheitnct J. Mcrtli. Bid. 1975 I 259. GENERAL DISCUSSION measurements have begun to bear fruit in oscillating biological systems such as glycolysis the circadian “ clock ” * and the mitotic cell cycle.3 Why not in physical chemistry? Dr.Th. Plesser (Dortmund) said Goldbeter’s description of biochemical oscilla- tions by the concerted model of Monod Wyman and Changeux can be generalized very easily to the case of n promoters. The linear stability analysis gives the following results For all n there is only one stationary state in the positive quadrant of the phase plane. If the product is an allosteric inhibitor of the enzyme then the stationary state is always stable. If the product is an allosteric activator then the stationary state may be stable or unstable and give rise to self excited oscillations. The stationary state in the positive quadrant can never be a saddle point if the input rate of substrate is lower than the maximum velocity of the enzyme.If the number of protomers is odd there is only one stationary state. For even n there is a second one for negative and therefore unphysical con- centration. Numerical analysis shows that the oscillatory domain is not very sen-sitive to the number of protomers. A. T. Winfree Arch. Biochem. Biophys. 1972 149 388. ’A. T. Winfree Nature 1975 253 315. S. A. Kauffman private communication. S9-8

ISSN:0301-5696    

DOI:10.1039/FS9740900215

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

D. Theory of Excitable Media Excitability and Spatial Order in Membranes of Developing Systems BY B. C. GOODWIN School of Biological Sciences University of Sussex Brighton Sussex BNl 9QG Received 15th July 1974 A simple biological model of metabolic activities on membranes is explored with respect to excitability and spatial gradient formation. The model is shown to have the property of undergoing a phase transition from a non-excitable to an excitable condition. Once excitable it is shown that a spatially-stable gradient can be formed on the membrane. Periodic gradients can be formed by a slightly modified process. 1. INTRODUCTION The primary concern of this paper is the exploration of some temporal and spatial consequences of certain very elementary properties of biological membranes.The term " elementary " is employed in a biological not a physico-chemical sense. Thus it is taken as axiomatic that biological membranes are complex lipo-protein structures and that the proteins are as important as the lipids. This is the emphasis that has emerged in recent years as a result of the molecular biological approach to membrane structure and activity.l It is therefore assumed from the outset that membranes contain enzymes and proteins capable of reacting with specific ligands. Then on the basis of some very simple postulates about interactions between these proteins a model of metabolic excitability as a phase transition is presented ;ultimately it is shown how the postulates give rise to processes which result in either monotonic or periodic gradients of "morphogens " on the membrane.The reason for concentrating attention on biological membranes as the sites of spatial order in developing organisms is provided by experimental evidence which points very strongly in this direction. 2-6 Furthermore unicellular organisms show essentially the same morphogenetic capacities and behaviour as do the multicellulars so that a basic understanding of morphogenetic mechanisms must be arrived at independently of assumptions about cellular partitions. These points have been argued out more fully in a paper by Goodwin and McLaren in which the basic features of the model considered in this paper have been presented together with applications to particular developmental processes.The present goal is to look at membrane excitability from a particular perspective and to explore some further consequences of the basic model thus extending its range of applicability. 2. EXCITABILITY AS A PHASE TRANSITION The type of phenomenon now considered is the rather sudden appearance of spontaneous action potentials during the regeneration of the unicellular marine alga Acetabularia as reported by Novak and Bentrup.* Other developing systems show 226 B. C. GOODWIN similar types of state transition also but the electrophysiological behaviour of AcetabuZaria is particularly clearly defined. Whether this change of membrane state is correctly described as a phase transition of the type here described requires closer experimental study but the model draws attention to this possibility and hence to the eventuality that such transitions are an important aspect of development.OUTSIDE INSIDE FIG.].-Model of interactions between enzymes El and E2 metabolites U V and W,and the ionic species K+ which can result in propagating metabolic activity waves and the formation of a morphogenetic gradient on a membrane. For explanation see text. It is convenient first to describe the molecular organization within the membrane which characterizes the metabolically excitable condition underlying the model. This is shown in fig. 1. The circles labelled El lying within the membrane represent enzymes whose active sites face the interior of the cell. They catalyze the conversion of a metabolite U into another metabolite V.A second enzyme EZ,which is either a soluble enzyme or is loosely associated with the membrane converts V into W. It is assumed that an ionic species represented in the figure by K+,is in equilibrium between a membrane bound state and a free state and that V has the effect of displacing this equilibrium towzrds the free state. The ion is an activator of El so that a positive feed-back loop is created which introduces an instability into the kinetics. W on the other hand is an inhibitor of El so that the control circuit can be stabilized. A kinetic system of this type belongs to the same category of reaction system as the glycolytic oscillator extensively studied by Hess and his colleague^,^-^^ and to the general class of kinetic processes to be considered by Franck l1 in this symposium.Since there are many possible kinetic realizations of such a system there is no need to make particular assumptions ; one simply supposes that propagating waves of metabolic activity can occur providing certain constraints are satisfied. Several contributions to this Symposium are concerned with the definition of these constraints. It is evident that one condition which must be satisfied in order for there to be wave propagation in such a system is that the density of enzymes El within the membrane must be sufficiently large. The exact value will depend upon the stoichio- metries of the reactions the diffusion constants of the reacting species and of the activating ion the temperature etc.The density or the concentration of El in the membrane will depend upon the balance between the rate of its incorporation into the membrane and its rate of loss or dilution (if new membrane is being formed). All these factors will be represented by means of a biochemical potential p so that the concentration of enzyme is determined by this quantity. We may assume that El is effectively inactive until it takes the spatial configuration resulting from incorporation into the membrane so that the cytoplasm is not excitable. It is assumed that there are sites for enzyme El on the membrane and that their mean number per unit area is constant. The binary variable El is used to designate the state of site i on the membrane. This variable takes the value 0 or 1 accord-ing as the site is unoccupied or occupied respectively.The model now follows S 9-8* EXCITABILITY AND SPATIAL ORDER closely the lattice gas description of the process of condensation which is in turn based upon the Ising model of ferromagnetism using the mean field approximation.12* l3 This assumes that each unit in the system under consideration (in our case an enzyme) experiences a mean field of force due to the presence of other units and the essence of the procedure is to find an equation which expresses this field in terms of itself (the self-consistent field equation). The field under consideration in this model is assumed to arise from interactions between enzymes occupying membrane sites and those enter- ing the membrane the presence of an enzyme at a site facilitating the entry of an enzyme into the membrane at a neighbouring site.We designate this field as p = (Q the mean value of the occupancy averaged over all the sites. Let the total number of occupied sites be N so that Cci = N the sum being over all sites. And finally let the interaction between two neighbours be tii,. The appropriate function to use in determining average values in such a system is the grand partition function which allows for variations in both N the total number of active enzymes on the membrane and in the energy of the system resulting froin interactions. The Hamiltonian for such a system is then H = -+CU,~E~E~ -Np. (2.1) The self-consistent equation for p which is obtained from this expression via the grand partition function is (2P -I> = tanW?/2(pu(O)+PI (2.2) where u(0)is the Fourier transform of Cuijtaken at the point 0.The assumption here is that the interaction field is everywhere the same uij(R)being translation invariant. The parameter /? is l/kT. Thus it is assumed that the system is at a defined temper- ature. This temperature in fact implies that the relevant processes inolved are quasi-equilibrium ones in accordance with the use of a biochemical potential to represent the factors which determine the concentration of enzyme E in the mem- brane. Other processes such as those involved in the maintenance of the membrane and of a constant pool of precursor U are taken as constant parameters of the system. The consequences of this model are as follows.There is a region of parameter space in which eqn (2.2) has a single root and another region where it has three. The latter condition which is the interesting one for phase transitions requires that p = -u(0)/2 whence eqn (2.2) becomes (2p -1) = tanh[&(O)(2p -l)]. (‘2.3) For sufficiently large /? this equation has three roots in p one of which is always p = 3 and the other two are symmetric about this. The negative value of the potential in this region has no physical significance since this quantity is defined relative to an arbitrary constant. Plotting p as a function of T one gets a curve such as that shown in fig. 2. The region of triple roots lies below T, the critical temperature and the curve shown in this region is called the coexistence curve.Above T, no excitability is possible but below it a transition from the non-excitable to the excitable state can occur. Thus if the membrane is in the state shown by P and p is then increased (e.g. more enzyme is incorporated into the membrane) then the system can move to P’ an unstable point on the coexistence curve. A sudden state transition to Q’ can then occur so that the membrane becomes excitable. As the temperature is decreased this transition can occur at smaller values of p; i.e. thermal noise interferes less and less with the processes resulting in excitation. Now clearly in a biological system there will also be a lower temperature bound to the domain of excitability as well. Thus the coexistence curve should close on itself B.C. GOODWIN again. This could be modelled by making further assumptions about say the temperature-dependence of the quantities u(0) and p at lower values of T. However it is far from clear as yet that we are dealing with true phase transitions in excitability phenomena so a further analysis does not seem worthwhile. The point of this model is to demonstrate the possibility of such behaviour in membranes. The analysis could also be extended to a consideration of metastable regions of the coexistence domain corresponding to super-cooled and super-heated states. These would be conditions where very small fluctuations cause either excitation or the loss of excitation respect- ively. The former might describe the state of membranes which show latent excit- ability responding to small stimuli but are not recurrently active as in a pacemaker system.I I I T TC FIG.2.-Coexistence curve for non-excitable (1) and excitable (2) states on a membrane with molecu- lar interactions of the type shown in fig. 1. The region above Tccorresponds to a condition in which no transition to excitability can occur but below Tc such transitions are possible. The dotted line from P’ to Q’ describes such a transition which involves a rapid change in the variable p the fraction of the total number of sites which is occupied. 3. EXCITABLE MEMBRANES AND SPATIAL GRADIENTS Attention is now directed to certain spatial consequences of the model described in fig. 1. It is of interest to ask under what conditions activity waves can leave “ mem-ory ” traces on membranes in the form of gradients of a metabolite.Such a process would clearly be of significance in relation to both morphogenetic and neural activities. A simple mechanism for achieving this is to assume the existence in the membrane of proteins with binding sites for the metabolite V. These are shown as circIes marked B in fig. I. In order to get a graded distribution of bound V hereinafter referred to as morphogen we may assume that these sites are activated after the production of V by E Thus for example we could suppose that W is an activator of B as well as an inhibitor of El. This temporal asymmetry of the processes which constitute the propagating wave of activity on the membrane results in a graded distribution of rnorphogen with its maximum at the point of origin of the wave.’ To illustrate the principles of the model consider the following local representation.A pulse of metabolite of magnitude M is produced at the point x = p on a one-dimensional membrane as shown in fig. 3. The wave of activation of the binding sites is repre- sented by a pick-up function which starts at x = p and propagates with velocity v in the positive direction. We may then ask what the distribution of morphogen will be EXCITABILITY AND SPATIAL ORDER after the passage of the wave. The correct representation of the process involves the propagation of the pulse M as well as is done in another publication,’ but its essence is illustrated by the local model.X P FIG.3.-Local representation of the gradient-generating process on a membrane. The diffusible metabolite is produced in a pulse of magnitude Mat the point x = p on a one-dimensional membrane and the pick-up function (f)propagates from the point x = p at velocity D in the positive direction. The result is a gradient of bound metabolite represented by the function Y(x,a). The behaviour of the pulse of metabolite as a function of distance from the origin x and time t is given by the function M X(x t) = -== exp( -(p -~)~/4Dt) J4.nDt where D is the diffusion constant. The pick-up function is represented by f(s) = Kas exp( -as) s 2 0 =o s<o where s is the distance from the wave-front 01 is a parameter determining the distance from the wave-front to the maximum off($) which is Ke-l at s = l/a and K is the product of the concentration of binding sites and a rate constant for bound complex formation.The expression for the morphogen concentration Y(x,t),assuming that the bound complex is stable and that Y(x,0) = 0 is where the lower limit ro = 0 for x < p = x-p for -Y > p. Defining p(p-x>/z; and 7 = (p-x)~/~D, this becomes Y(x t) = C dr(z 4-fl)r-i cxp( -m(z +/I)-yir) 1:. where MavK c=--J4YD. For the case ro = 0 letting t+co this integral is j4 (3.4) B. C. GOODWIN 23 1 where 5 = p-x 2 0. For x > p the integral does not permit a simple closed form but it is easily shown that Y(x,co)decreases monotonically. The peak of morphogen occurs at a distance to= cr/cr(a+a) (where CT = a+ Jau/D) to the left of the origin of the pulse as shown in fig.3. Biologically plausible values of the parameters are v = 8 pm s-l o! = 0.025 pm-1 and D = 200 s-l giving j = 5.3 pm. Now if the temporal asymmetry is reversed so that the activation of the binding sites precedes the activation of enzyme and production of V then instead of (3.3) one obtains for Y the expression dx(Lir+p-x)t-* exp -a(vz+p-x)--402 * This represents a situation in which the pulse of metabolite occurs at the origin x = 0 while the pick-up function starts as before at x = p. Using the same procedure as previously one finds for the case T~ = 0 (x < p) This function increases monotonically up to p so that the maximum occurs for a value of x > p.Thus we see that the temporal relationships between metabolite production and pick-up determine which way the gradient of morphogen is formed. 4. PERIODIC DiSTRIBUTION OF MORPHOGEN In order to have spatially stable distributions of morphogen on membranes it is necessary to have well-defined self-stabilizing origins of wave initiation. This requires a further postulate in the model but this is easily provided by a consideration of how the morphogen is likely to act. It is reasonable to suppose that directly or indirectly it affects membrane properties as discussed more fully elsewhere. One such effect could be on the concentration of free K+itself. If the morphogen increases this concentration then the wave will tend to recur at the wave origin where the morphogen concentration is maximal.The kinetics of the system described in fig. 1 can then result in regularly recurring activity waves which will maintain dynamically a spatial gradient of morphogen. A spatially periodic distribution of morphogeii can result from the recurrent propagation of metabolic activation waves and binding-site activation waves if these propagate at different velocities from a common origin. For the waves will then pass one another at defined points on the membrane where their temporal relationships will be reversed. Thus the gradient will form in opposite directions on either side of these passing points. However in order for the waves to propagate independently of one another at different velocities it is necessary that the processes involved in metabolits production be different from those involved in binding site activation.Therefore the picture of fig. I must be modified so that each process has kinetics of the general type described for metabolite production. This could be achieved by assum- ing another pair of enzymes coupled as are El and E? with an ion activator distinct from K+. Then both these processes could be initiated by cz spatially-localized pacemaker region established by a gradient formed by a previous wave process. This complication of membrane-localized activities in no way overburdens this organelle with specific structure since a mitochondria1 membrane for example is very much more complex than this. The biological attraction of the above inodel is that it provides a plausible mech-anism for both aperiodic and periodic gradient formation in either unicellular or EXCITABILITY AND SPATIAL ORDER multi-cellular organisms.Furthermore having established a wave origin the spatially periodic process is a reliable one unlike the difficulties encountered by the Turing model in this respect. However although aperiodic gradients generated by the model regulate (are size-independent),' the periodic gradients do not ; so here the model fails biologically as does also the Turing process. This paper was written while the author held a Leverhuline Visiting Professorship at the Department of Theoretical Physics National University of Mexico and grateful acknowledgement is made to the Royal Society and to Professor M.Moshinsky Head of the Department for assistance and hospitality. C. Gitler Ann. Rev. Biophys. Bioeng. 1972 1 51. A. S. G. Curtis The Cell Surface (Academic Press London 1967). H. F. Stumpf Dev. Biol. 1967 16 144. D. L Nanney Science 1968 160,496. L. S. Sandakhchiev L. I. Puchkova and A. V. Pikalov Biology and Rodiobiofogy of Ai~nclcate Systems ed. S. Bonnotto 1972 p. 297. W. Herth and K. Sander Wilhelnt ROUX'Arch. Ent. Org. 1973 172 1. 'B. C. Goodwin and ID. I McLaren J. Theor. Biof. 1975. B. Novak and F. W. Bentrup Pfanta (Berl) 1972 108 227. B. Hess H. Kleinhaus and D. Kuschmitz Biological and Biochemical Oscillators ed. B. Chance and K. Pye (Academic Press New York 1973) p. 253. io A. Boiteux and B. Hess paper at this Symposium. * U. F. Franck paper at this Symposium. l2 R. Brout Phase Transitions (Benjamin New York 1965) Chap. 2 and 3. H. E. Stanley Introduction to Phase Transifions and Critical Phenomena (Clarendon Oxford 1971) chap. 6 and appendix A. I4 I. S. Gradsteyn and I. M. Ryzhik Tables of Integrals Series and Products (Academic Press New York 1965) p. 340. B. C. Goodwin Analytical Physiologj~ of Cells arid Detlefoping Organisins (Acadeiiiic Press 1976) chap. 5.

ISSN:0301-5696    

DOI:10.1039/FS9740900226

出版商:RSC    

年代:1974

数据来源: RSC