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.