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.