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Conformational effects on the activation free energy of diffusion through membranes as influenced by electric fields
J. theor. Biol. (1976) 61, 97-l 12
Conformational Effects on the Activation Free Energy of Diffusion Through Membranes as Influenced by Electric Fields FRANKLIN F. OFFNER AND SAN HYUNG KIM? Technological Institute, Northwestern University, Evanston, Illinois 60201, U.S.A. (Received 20 January 1975, and in revised form 6 February 1976) In biological membranes there are large differences in the permeabilities of K+ and Na+, the permeability of K+ being the higher in the polarized membrane. In excitable membranes both the absolute and relative permeabilities are strongly affected by the electric field. It is shown that these changes can be explained by simple electrostatically induced conformational changes at the mouths of pores, due to the deflection of ionized long-chain molecules. A conformational change requiring a low free energy for its production, may result in a large change in the free energy of activation, as a result of the relative magnitudes of elastic and inertial forces. The energy required to partially dehydrate the two ion species plays an essential role in the process. 1. Introduction The time and voltage dependence of the ionic permeability of the excitable membrane probably resides at the membrane-solution interfaces (Offner, 1970, 1972). Two factors have been considered: ionic adsorption at the mouths of the pores, assumed to be the avenue of ionic permeation; and electrodeformation of the membrane at the mouths. It has been shown (Offner, 1972) that these can account for a wide variety of the observed membrane phenomena: time-course of the differential Na+-Kf permeabilities, effect of Caf +, etc. Conformational effects have been previously discussed by Mullins (1959), Goldman (1964) and Hill & Chen (1972). While this paper will be illustrated with reference to a membrane model in which Kf and Na+ flow through the same channels, the principles would apply equally well to a model in which separate channels are assumed. The authors have adopted the single channel hypothesis at this time, since it appears that all experimental results can be demonstrated in such a model. The basic theory on which the model is based is that flow across the membrane is primarily limited by the activation energy required for ions 7 Dr Kim is now at Eastman Kodak Co., Rochester, N.Y. T.B. 97
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to cross the two membrane interfaces; even a moderate rise in activation energy can result in a large change in flow: a rise of 7.5 kJ/mol would, for example, result in a reduction in flow by a factor of 20, while a 15 kJ/M increase would decrease the flow 400-fold, assuming all other factors to be unchanged. It will be shown that large changes in activation energy can result from small deflections in atoms at the mouths of the pores. Such deflection is to be expected with changes in the interfacial electric field during activity, acting on ionized polar groups at the end of long-chain molecules, and involve only a small energy charge. A minor portion of this field change occurs instantaneously on the application of a voltage clamp step, but the major portion occurs thereafter, with a time course depending upon the diffusion of ions within the membrane, thus accounting for the time dependence of ionic flow. While the theory here presented would be applicable to any ionized longchain molecule, it appears likely that they are the phospholipids in the membrane. However, as has been well recognized, it would be unlikely that stable channels could be formed in a phospholipid bilayer, because of the fluidity of the hydrophobic central ends of the molecules. It therefore appears likely that the channels are formed by the lipid molecules directly surrounding the protein islets penetrating the membrane. A reason for favoring such a hypothesis is the effect of tetrodotoxin (TTX) : one molecule of TTX taken up by the protein could thus by allosteric effect influence a number of surrounding channels (Offner, 1972). This hypothesis has now received support from the experimental observation of the immobilizing effect of protein on adjacent lipid molecules (Jost, Griffith, Capaldi & Vanderkooi, 1973). In this paper, we will consider the electrodeformation effect; adsorption is considered in detail in a companion paper (Offner & Kim, 1976) particularly in terms of competitive adsorption between all species present in the bathing solutions.
2. Electrodeformation
and Activation Energies
The term “electrodeformation” is here used to mean the effect of the electric field on the membrane itself, resulting in a change in the free energy of activation required for ions to traverse the membrane-solution interfaces. This activation energy required for an ion to enter the membrane consists principally of the energy required to produce a void, or hole, in the pore, large enough to accept the ion with its hydration shell (partial or complete);
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and, depending upon the radius of the pore’s mouth, energy required to partially dehydrate the ion (Offner, 1972). In Eyring’s activation theory of diffusion, it is assumed that solutes diffuse through a solvent in steps of approximately one solvent molecule diameter (Glasstone, Laidler & Eyring, 1941). This is probably not the case, however, for ions crossing the solution-membrane interface. Since the dielectric constant of the lipid is considerably lower than that of water, even of the structured water close to the interface (approximately 2.5 versus 5 to 8), electrostatic forces will tend to extrude ions which have only partially entered themembrane.The resulting activationenergies required for a quadrihydrated univalent cation to enter the membrane, considering only the energy of hole formation, was therefore estimated to be of the order of 40 kJ/kJ/M (Offner, 1972). This energy must be the energy the ion possesses after crossing the interface; any energy loss resulting from the crossing of the interface itself will reduce the available energy, and thus increase the required activation energy. In particular, if the ion impinges on the membrane molecules, energy may be transferred to such molecules in the form of kinetic or potential energy. The factors affecting this energy loss is the subject of this paper. The activation free energy is in the form of kinetic energy of the entering ion ; it implies a high ionic velocity. A quadrihydrated K+ ion having a translational free energy of 40 kJ/M has a velocity of 850 m/s. It is this high velocity of the ions traversing the interfaces that plays an essential role in the electrostrictive process, since the velocity of an ion impinging on an interfering spring-loaded object (e.g. an ionized side-chain at the entrance to a pore) will determine the relative effect of the kinetic and potential energies imparted to the object. It will be shown that this in turn permits a relatively low energy electrostrictive process to result in a relatively large increase in activation energy. Consider the simple example of Fig. 1. Here, a spherical object (entering ion) of radius rl and mass m, impinges symmetrically on two interfering objects (side-chain atoms) each of radius I’~, mass m,, and separated by a center-to-center distance 2d. Each of the latter objects is held in position by springs of force constant kb. If, then, the initial velocity of the impinging object (ion) is L’il its velocity u/ after the collision, may be found by considering energy and momentum balance. If the spring forces are negligible, then M set’ O-1 Vf="iFiG2(l+t
where 0 = sin- ’ [d/(rl+r2)]
and M = m,/Zn1,. Thus the kinetic
(1)
energy
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FIG. 1. Spherical object (ion) moving with velocity v,, having collision with two other objects (side-chain atoms), each restrained by springs having bending coefficients kb. After
impact,the object(ion) hasa final velocity u,.
loss of the impinging
object is
AKE = ~ml(z$-v;)
= +ml$
(A4 set’ O- 1)2 1- (A4 sec2 0 + 1)’ *
1
(2)
Thus the loss in energy is a fraction of the initial energy of the impinging object, given by equation (2). The potential energy necessarily imparted to the interfering objects (side-chain atoms), in order for the impinging object to just pass, if only the stiffness of the springs is now considered, will be PE = 2&k,(dispiacement/L)2
= k,(r, f r2 - d)2/L2
(3)
where L is the bending radius. Thus the energy loss of the impinging object (ion) in the form of potential energy imparted to the interfering object is independent of the initial energy. The importance of these elementary considerations is in the effect of the mass ratio M on the relative distribution of energy loss on an impinging ion. If the mass of the interfering object (atom or group) is small, then M is large and AKE becomes small; the spring force is then the effective factor affecting the energy of the ion. But as the mass of the interfering object becomes progressively larger relative to the ion, the energy loss due to
imparted
KE becomes progressively
more important.
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3. Application
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to the Pore
Now consider the mouth of a pore, having ionized polar groups near the interface, at the end of long-chain molecules. This is shown in highly simplified form in Fig. 2; the ionized group is shown, for definiteness only, as PO,, and the molecular chain is shown straight, although the bond angles would actually be more nearly of the order of 100-120”; nor would the atoms lie in a plane. As a rough first-order approximation, however, the molecule terminating in the ionized group may be considered linear, with all the bonds
Pore
FIG. 2. Assumed configuration of membrane pore. The O- atom of an ionized PO1 group is deflected so that it may interfere with an entering ion. The interfacial electric field is distorted by the presence of the pore, producing a radial component of the field.
having the properties of the C-C bond: force constant kb = 0*54x 10-l’ dyn-cm/radian (Herzberg, 1945), and the bond length L = 1.54 A.? The deflection of such a chain, as shown in Appendix A, will increase approximately as the cube of the number of deflecting atoms in the chain. Thus a lo-atom chain would deflect l-7 x lo6 A/dyn, and a 20-atom, 1.3 x lo7 A/dyn. t Handbook
Ohio, 1969.
of Chemistry
and Physics,
50th edn., Chemical Rubber Co., Cleveland,
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Such a molecule will thus suffer a considerable deflection, if subjected to a component of electric field normal to its axis. Thus the 20-atom chain would deflect l-0 A due to a normal field component of 10’ V/cm acting on a single electron charge. The actual deflection that would occur is certainly less than this, due both to the bends in the chain, and to linkages between adjacent molecules, but these calculations show that substantial deformations are to be expected, under the influence of moderate electric fields. If now an entering ion impinges, for example, on the terminal O- of the molecule,t a complex dynamic transfer of energy to the whole molecule will occur. Rather than even attempt to calculate this, two simplified cases will be assumed, which will give an order-of-magnitude of the energy transfer to be expected. We first assume that the P atom is fixed rigidly in position, and only the O- atom deflects to allow the entering ion to pass; the actual energy transfer must be less than that so computed. To compute the PE, the bond constants for the P-O bond are used: kb = 2.0 x IO- ” dyn-cm radian (Herzberg, 1945) and L = I.56 A. The PE required to displace two 0 atoms (assumed symmetrically disposed across the mouth of the pore), as a function of the interference radius in Fig. 3. Also plotted in the figure is the KE transfer rO + j’ion -d, is plotted based on equation (2), for the case of quadrihydrated K+ as the impinging ion. The initial velocity for these KE calculations is such that the energy after collision is 40 kJ/M, i.e. the energy of hole formation. It will be seen that, except for a very small interfering radius, the PE is greater than the KE term. Thus the 0 atoms would have little residual KE after the collision and the energy absorbed would be essentially only PE. The same conclusion applies for Na+. A very different situation arises if it is assumed that the entire PO, group moves as a whole, with flexure at the O-C bond. For it, &, = 1.7 x 10-l’ dyn-cm/radian (Taylor & Vidale, 1957). The effective bending radius depends on the assumed steric configuration; a value of 5 A will be used. The result of these calculations is shown in Fig. 4; the system is inertialoaded, and the energy absorbed is in the form of KE. It is now seen that in the range of energy absorption of interest (less than 50 kJ/M) flexing of the P-Obond will result in less energy loss than motion of the whole PO; group; the actual dynamic effect will be energy transfer throughout the system. Thus, the actual energy loss would be somewhat less than shown in Fig. 3. t It should be emphasized that this discussion is only exemplary; the colliding atom need not be O-, nor even at the end of the lipid molecule.
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tll)
FIG. 3. Energy loss of a quadrihydrated K+ ion, after impinging upon an interfering O- atom, with the interference shown. The upper curve shows the loss in the form of potential energy, and the lower, in kinetic energy imparted to the O- atoms. The initial energy of the entering ion is that required to have 40 kJ/M after collision. For Na+, PE curve is identical, and KE differs only slightly.
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KE
Interference
FIG. 4.
Similar
to
(A)
Fig. 3, but for collision with POa group. Upper curve, KE; lower, PE,
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The essential point of these calculations is to show that a relatively large change in free energy of activation can result from the absorption of a much smaller energy by the membrane, in the form of ionized group displacement under the influence of an electric field. This difference is due to the timescale of the two processes: the electrical displacement can occur over a relatively long time (fraction of a millisecond), while the process affecting activation energy occurs in a time-scale of the order of 10-l’ s. It thus becomes clear that the activation energy for diffusion through pores should be highly sensitive to the boundary electric field, provided that the pore opening is of a suitable form, and carries charges at the interface. Otherwise, no special configuration is necessary. The terms entering into the activation energy necessary for an ion to cross the membrane interface have been previously discussed (Offner, 1972) but the results will be reviewed, with some modification and extension. The various terms are summarized in Table 1, for Na+ and K+ ions crossing in the di-, tri- and quadrihydrated states.
TABLE ___..~ Hydration number Dehydration energy Energy of hole formation Total energy Minimum opening, AU
1
Na+ 2 45 32 77 2.8 2.8
3 15 38.5 53.5 6.6 3.1
K+ 4 0 45 45 6.6 5.9
2 25 32 57 2.8 2.8
3 7 385 45.5 7.3 3.6
4 0 45 45 7.3 6.4
Energies are in kJ/mol. “Minimum opening” is the estimated minimum size rectangular opening, in ktgstrom units, which will pass a linear dihydrated ion, a tetrahedral hydrated ion with one water removed, and a quadrihydrated tetrahedral ion.
These terms include, for the first two states, the energy required to partiahy dehydrate the ion, from its normal quadrihydrated state. These energies have been obtained, as described in Offner (1972), from Dzidic & Kebarle (1970) who measured hydration energies in the vapor phase. We have attempted to correct their results to the liquid phase, by allowing for the difference in Born energy; which is estimated to be approximately 10 kJ/M for each water molecule. A theoretical calculation has also been made (Kim 8z Rubin, 1973), giving the same order of magnitude of values for these energies.
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The next term is that required to form a hole in the liquid, of sufficient size to permit entry of the hydrated ion. The procedure for making these calculations is also given in Offner (1972). However, while this energy was previously taken as smaller for Na+ than for K+, due to the smaller size of the ion, they are now estimated to be about equal, due to the compensating effect of higher bonding of the Na+ ion to the secondary hydration shell. The value for trihydrated ions was obtained by interpolation. The total energy listed is the sum of the above two terms, and is that which would be required for an ion to cross the interface, provided the pore has an opening of a dimension large enough to pass an ion of the appropriate hydration number, and that the ion approaches the pore with an orientation such that it can pass through the opening unobstructed. The approximate dimension of the required opening, for an ion approaching normal to the interface, is shown in the last line of Table 1. Ions approaching from other than a normal angle will suffer a loss of energy, divided between deflection of the molecules forming the pore, as above discussed, and deformation of the hydration shell of the ion itself. This effect should be particularly important in the case of the unsymmetrical trihydrated ion. Considering the random direction of arrival of ions at the pore will thus add an entropic term to the activation energy, resulting in a progressively higher effective energy as the dimension of the opening is reduced. The above effects are far too complex to be calculated accurately for Na+ and Kf ions, but Fig. 5 shows the trend the activation energies would be expected to follow with deformation reducing the lesser dimension of the opening. This figure then illustrates the general order-of-magnitude of the variation in activation energy to be expected for the two ion species, with electrodeformation of the pore, as resulting from the boundary electric field. The gist is that the electrodeformation resulting from high electric fields will reduce the K+ permeability by a factor of about 100 (due to the rise of 11 to 12 kJ/M in the activation energy) and that the Na+ permeability will be reduced about a hundred times as much. Of especial interest is the effect of smaller deformations-under 1 &-on Kf and Naf activation energies. As a result of the substantial equality of the total of the activation energy terms for the tri- and quadrihydrated states in the case of K+, such smaller deformations should have little effect on K+ permeability, while that of Na” would be reduced by a factor of 25 to 50. It is suggested that the effect of tetrodotoxin (TTX) is to cause such a reduction in the maximum opening of the pore. This, as seen, would result in a large reduction in the Na+ permeability, without appreciable effect on that of K’.
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iX)
FIG. 5. Energy loss, and thus increase in free energy of activation, for KC and Na+ ions, as a function of reduction of the lesser dimension of the pore opening. The curves are based on an estimated combination of the energy loss due to molecular deflection, from Fig. 3, dehydration energy (Table 1). and the effect of random approach direction of the entering ion. Based on 3 A deformation reducing opening to diameter of dihydrated ion (approximately 2.8 A).
4. Discussion In these calculations, it has been assumed that energy is lost to a single process only. Thus, when the interaction energy is equal to the energy necessary for dehydration, it is assumed that dehydration will occur. This is an over-simplification; not only will, in general, some energy be lost to the membrane itself, but also the removed water molecules will take up energy, in the form of rotational and vibrational excitation, This will add to the total energy loss in dehydration. However, the estimate of total energy required to remove two water molecules from the ion is only very approximate in any case; what is more important is the difference in dehydration energies required for K+ and Na+. This estimate is believed to be much closer and would not be affected appreciably by these other factors.
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Another theoretical approximation that has been made is the assumption that all the activation energy is initially in the form of translational energy of the diffusing ion. In fact the energy would be partitioned between the various degrees of freedom involved. These would include the interfering side-chain groups, and the hydrogen bonds which must be broken for hole formation. This degeneracy introduces an entropy term, which in effect reduces the energy barrier. But the activation process herein considered is not truly adiabatic, in that the relative energy transferred to the side chains will depend upon the kinetic energy of the ion, and thus the activation energy barrier is a direct function of the initial translational energy on the ion. For example, if only half the activation energy is represented by kinetic energy of the ion, the energy absorbed by side-chain interference in the form of kinetic energy will be reduced by half. In the example calculated, the PO, group could then deflect as a whole with less total energy than the O-, for interferences of greater than about 0.15 A. To properly calculate the actual shape of the potential barrier it is thus necessary to compute the relative contributions of the various possible energy distributions between the degrees of freedom. We have not yet attempted the solution of this far more complex problem. Appendix A
Consider a linear chain of atoms, each separated by a distance L cm, and having bending force constant of kb dyn-cm/radian. Under a force f, the first atom will have an applied torque fL, resulting in a deflection fL?/kb. The second atom will have the force applied at a radius 2L, and a given angular deflection will result in twice the linear deflection, so the deflection due to the bending of its bond will be f(2L)‘/k,. The bending of the nth bond will result in a deflection f(nL)‘/k,. Thus the sum of all the bond deflections will result in a total deflection given by deflection = : i i2. bl
The sum of the series is +z3+jn2+n/6, for n = 20.
equal to 385 for n = 10, and 2870
Appendix B As a result of diffusion within the membrane, the change in potential at the external interface, AV,, eventually becomes of the order of the depolarizing voltage clamp step. If a simple Helmholtz double-layer is assumed,
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thickness X = 38 A (diameter of one water molecule in primary hydration shell, 2.8 A, plus ionic radius, approximately 1 A), the change in the interfacial electric field is AE, = -AVJX. For a 50 mV step, E0 = - 1.3 x lo6 V/cm. The lines of electric field are distorted in the vicinity of a pore (Fig. 2). If the effective angle of the field, relative to the axis of the pore, is, for example, 20”, the change in the radial component of the field is AEO sin 20” = -0.45 x lo6 V/cm.
Appendix C
Readily deflectable molecules, of the type herein proposed, should suffer significant vibratory deformation due to Brownian movement. Such vibrations would result in a statistical variation in the activation energy for ions traversing the membrane interface and consequently a similar variation in the membrane current. Random fluctuations (“noise”) have been observed in current through excitable membranes by numerous workers, Verveen & Derksen, 1965; Poussart, 1971; Wanke, DeFelice & Conti, 1974; Fishman, 1973. The power spectrum of the variation of the membrane current has been described as varying as l/f, although the exact law appears to vary. Noise having a similar spectrum is also found both in non-excitable biological membranes, and in non-biological systems involving diffusion through small pores (DeFelice 8z Michalides, 1972; Dorset & Fishman, 1975). Therefore, the electrical noise observed in excitable membranes is undoubtedly largely due to phenomena not involved in the excitability process, and the noise contribution from Brownian motion of the electrodeformable molecules should represent only a small portion of the total noise. As a first approximation, the deformable molecule may be considered as a simple harmonic oscillator, with a resonant (angular) frequency o,=&jii (Cl) where s is the spring constant for bending of the molecule in its fundamental mode of vibration, and m is the “reduced” mass, considered concentrated at the end of the molecule. The power spectrum S,,(o) of the deflection of such a molecule due to Brownian motion has been shown (Papoulis, 1965; Uhlenbeck & Ornstein, 1930; Wang & Uhlenbeck, 1945) to be given by
Kx(~) = (oz-- 0;)’c1+ j?205
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where (C3) and (C4) D is the diffusion constant. Whereas w. will be of the order of 10” or higher, the electrical noise can in general only be measured as greater than thermal (Nyquist) noise for w < 104. This permits the simplification of equation (Cl) by setting (02 - u;)2 z 0:. Then by making appropriate substitutions,
The first term in the denominator will always be much greater than the second, for the range of w of interest: D is of the order of lo-’ cm2/s for large molecules in an aqueous medium, and s is about 8 d/cm for the very flexible 20-atom molecule considered. Thus the first term is greater than 10i4, permitting equation (C4) to be further simplified: 2
S,,(o) =;
7
. 09 ( > In accordance with the theory presented herein, the free energy of activation should be related to the displacement of the deformable molecules, in a manner such as illustrated in Fig. 5. The slope of the curve, in kJ/hi-cm, may be expressed in units of RT/F for univalent ions, giving a proportionality 3, of free energy of activation in volts, to displacement in cm. Then the power spectrum of the variation of the activation energy, measured in volts, is W7) S”“(O) = ~2Lb>. Through each pore there is a steady-state ion current i. of a (univalent) ion species (e.g. K+) due to a displacement V of the membrane voltage from its equilibrium value. The current flow across a potential barrier of height c( is proportional to exp (-a/kT-AVF/2kT), where AV is the electrical potential difference across the barrier (assumed symmetric) measured in the direction of current flow. Therefore a change in the activation energy is equivalent to a change in the membrane voltage; and to the extent that i, is proportional to V, we may write for the power spectrum of the statistical variation in i,, Sii(W) = ig S,,(O)/ V2. ((3)
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Equation (C7) thus gives the expected power spectrum of the current fluctuation across the interface, for a single pore. If there are n pores per cm2, then S,r(o), the power spectrum per cm2, will be n times this value. Alternatively, equation (C8) may be written in terms of IO, the current density per cm2. Then i,, = lo/n, so that Making
the appropriate
Sag = z; S,&D)/V2n. substitutions then gives S,,(o) =;
(C9)
T 2z;12,V2n. (C10) ( > From Fig. 5, the value of I for the foot of the curves, corresponding to substantial depolarization, is found to be approximately 4 x 10’ V/cm. The value of n is not known; although some workers have obtained evidence which they interpret as indicating there are relatively few channels (as few as I.5 x 107/cm2), kinetic considerations (C9) indicate that the number should be 100 to 1000 times as large. The value of D for large molecules in aqueous media is of the order of 10e6; the orientation of the water within the pores might be expected to raise its viscosity, and lower D to the order of lo-‘. Poussart (1971), equation (Cll) determined S,,(o) in lobster axon. Reducing his data to the basis of values per cm2, he found a low frequency value of S,,(W) of approximately 5 x IO- ” A2 s, for a steady-state current of ONl6 A/cm2 with V = O-1 V. The value of &(o) varied about as 0-l , falling to a value of approximately 5 x 10W2’ at high frequencies. Using parameters computed for 1 and s, based on the 20-atom chain, and the experimental conditions of Poussart (1971) equation (Cll) gives S,,(o) = 3.4 x lo-‘6/D& (Cl11 Using D = lo-’ and n = lOto, equation (ClO) gives Sr,(o) = 3.4 x 10-19, which is reasonably within the range of Poussart’s minimum value, considering the various approximations and uncertainties involved. Evidently the assumption of a less readily deflected molecule would result in a lower value of noise; a lo-atom chain would reduce S,,(o) to about 5.6 x 10v2’. It appears that in no case could the Brownian movement of the molecules controlling ion flow at the interface be responsible for the major portion of the current noise, because of its frequency spectrum. The above derivation indicates that the generated current fluctuation should be independent of frequency up to frequencies much higher than those measured. The current noise is, however, composed of two equal components: the current flowing across the interfacial barrier into the bathing solution, and that flowing from the second bathing solution, through the membrane, and across the barrier (Offner, 1972). The former component should appear practically
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undistorted in the external measuring circuit, while the second will be filtered by the diffusion time constant of the membrane. Thus the low frequency noise power due to Brownian motion should be four times as great as the high frequency limit. The fact that the current noise power spectrum varies by a factor of 100 or more with frequency indicates that the Brownian motion can be responsible for only a minor portion of the total noise. The remainder is almost certainly due to some process which also occurs in non-excitable membranes, biological or not. The computations in this section show, however, that even very readily deformable molecules may be responsible for the ionic control phenomenon, without contributing more than the observed statistical fluctuation to the passage of the membrane current. REFERENCES DEFJZLICE, L. & MICHALIDES, J. P. L. M. (1972). J. memb. Biol. 9,261. DORSET, D. L. & FISHMAN, H. M. (1975). J. memb. Biol. 21,291. DZIDIC, T. & KEBARLE, P. (1970). J. phys. Chem. 74, 1466. FISHMAN, H. (1973). Proc. natn. Acad. Sci. U.S.A. 70, 876. GLASSTONE, S., LAILILER, K. J. & EYRING, H. (1941). Theory of RateProcesses. New York: McGraw-Hill Book Co. GOLDMAN, D. (1964). Biophys. J. 4, 167. GOLDMAN, D. (1969). Handbook of Chemistry and Physics, 50th edn. Cleveland, Ohio: Chemical Rubber Co. HERZBERG, 0. (1945). Molecular Spectra and Molecular Structure, Vol. II. Princeton, New Jersey: D. Van Nostrand Co., Inc. HILL, T. L. & CHEN, Y. (1972). Proc. natn. Acad. Sci. U.S.A. 69, 1723. G. (1973). Proc. natn. Acad. JOLT, P. C., GRWFITH, 0. H., CAPALDI, R. A. & VANDERKOOI,
Sci. U.S.A. 70,480. KIM, S. H. & RUBIN, B. T. (1973). J. phys. Chem. 77, 1245. MULLINS, L. J. (1959). J. gen. Physiol. 42, 1013. OFFNER, F. F. (1970). .I. gen. Physiol. 56, 272. OFFNER, F. F. (1972). Biophys. J. 12, 1583. OFFNER, F. F. & KIM, S. H. (1976). J. theor. Biol. 61, 113. PAPOULIS, A. (1965). Probability, Random Variables and Stochastic Processes, Cb. New York: McGraw-Hill Book Co. POUSSART, D. J. M. (1971). Biophys. J. 11,211. TA~OR, R. C. & VIDALE, G. L. (1957). J. them. Phys. 26, 122. UHLENBECK, G. E. & ORNSTEIN, L. S. (1930). Phys. Rev. 36,823. VERVEEN, A. A. & DERKSEN, H. E. (1965). Kybernetick. 2, 152. WANG, M. C. & UHLENBECK, G. E. (1945). Reu. mod. Phys. 17,323. WANKE, E., DEFELICE, L. J. & CONTI, F. (1974). Pfliigers Arch. 347, 63.
15.
Conformational Effects on the Activation Free Energy of Diffusion Through Membranes as Influenced by Electric Fields FRANKLIN F. OFFNER AND SAN HYUNG KIM? Technological Institute, Northwestern University, Evanston, Illinois 60201, U.S.A. (Received 20 January 1975, and in revised form 6 February 1976) In biological membranes there are large differences in the permeabilities of K+ and Na+, the permeability of K+ being the higher in the polarized membrane. In excitable membranes both the absolute and relative permeabilities are strongly affected by the electric field. It is shown that these changes can be explained by simple electrostatically induced conformational changes at the mouths of pores, due to the deflection of ionized long-chain molecules. A conformational change requiring a low free energy for its production, may result in a large change in the free energy of activation, as a result of the relative magnitudes of elastic and inertial forces. The energy required to partially dehydrate the two ion species plays an essential role in the process. 1. Introduction The time and voltage dependence of the ionic permeability of the excitable membrane probably resides at the membrane-solution interfaces (Offner, 1970, 1972). Two factors have been considered: ionic adsorption at the mouths of the pores, assumed to be the avenue of ionic permeation; and electrodeformation of the membrane at the mouths. It has been shown (Offner, 1972) that these can account for a wide variety of the observed membrane phenomena: time-course of the differential Na+-Kf permeabilities, effect of Caf +, etc. Conformational effects have been previously discussed by Mullins (1959), Goldman (1964) and Hill & Chen (1972). While this paper will be illustrated with reference to a membrane model in which Kf and Na+ flow through the same channels, the principles would apply equally well to a model in which separate channels are assumed. The authors have adopted the single channel hypothesis at this time, since it appears that all experimental results can be demonstrated in such a model. The basic theory on which the model is based is that flow across the membrane is primarily limited by the activation energy required for ions 7 Dr Kim is now at Eastman Kodak Co., Rochester, N.Y. T.B. 97
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to cross the two membrane interfaces; even a moderate rise in activation energy can result in a large change in flow: a rise of 7.5 kJ/mol would, for example, result in a reduction in flow by a factor of 20, while a 15 kJ/M increase would decrease the flow 400-fold, assuming all other factors to be unchanged. It will be shown that large changes in activation energy can result from small deflections in atoms at the mouths of the pores. Such deflection is to be expected with changes in the interfacial electric field during activity, acting on ionized polar groups at the end of long-chain molecules, and involve only a small energy charge. A minor portion of this field change occurs instantaneously on the application of a voltage clamp step, but the major portion occurs thereafter, with a time course depending upon the diffusion of ions within the membrane, thus accounting for the time dependence of ionic flow. While the theory here presented would be applicable to any ionized longchain molecule, it appears likely that they are the phospholipids in the membrane. However, as has been well recognized, it would be unlikely that stable channels could be formed in a phospholipid bilayer, because of the fluidity of the hydrophobic central ends of the molecules. It therefore appears likely that the channels are formed by the lipid molecules directly surrounding the protein islets penetrating the membrane. A reason for favoring such a hypothesis is the effect of tetrodotoxin (TTX) : one molecule of TTX taken up by the protein could thus by allosteric effect influence a number of surrounding channels (Offner, 1972). This hypothesis has now received support from the experimental observation of the immobilizing effect of protein on adjacent lipid molecules (Jost, Griffith, Capaldi & Vanderkooi, 1973). In this paper, we will consider the electrodeformation effect; adsorption is considered in detail in a companion paper (Offner & Kim, 1976) particularly in terms of competitive adsorption between all species present in the bathing solutions.
2. Electrodeformation
and Activation Energies
The term “electrodeformation” is here used to mean the effect of the electric field on the membrane itself, resulting in a change in the free energy of activation required for ions to traverse the membrane-solution interfaces. This activation energy required for an ion to enter the membrane consists principally of the energy required to produce a void, or hole, in the pore, large enough to accept the ion with its hydration shell (partial or complete);
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and, depending upon the radius of the pore’s mouth, energy required to partially dehydrate the ion (Offner, 1972). In Eyring’s activation theory of diffusion, it is assumed that solutes diffuse through a solvent in steps of approximately one solvent molecule diameter (Glasstone, Laidler & Eyring, 1941). This is probably not the case, however, for ions crossing the solution-membrane interface. Since the dielectric constant of the lipid is considerably lower than that of water, even of the structured water close to the interface (approximately 2.5 versus 5 to 8), electrostatic forces will tend to extrude ions which have only partially entered themembrane.The resulting activationenergies required for a quadrihydrated univalent cation to enter the membrane, considering only the energy of hole formation, was therefore estimated to be of the order of 40 kJ/kJ/M (Offner, 1972). This energy must be the energy the ion possesses after crossing the interface; any energy loss resulting from the crossing of the interface itself will reduce the available energy, and thus increase the required activation energy. In particular, if the ion impinges on the membrane molecules, energy may be transferred to such molecules in the form of kinetic or potential energy. The factors affecting this energy loss is the subject of this paper. The activation free energy is in the form of kinetic energy of the entering ion ; it implies a high ionic velocity. A quadrihydrated K+ ion having a translational free energy of 40 kJ/M has a velocity of 850 m/s. It is this high velocity of the ions traversing the interfaces that plays an essential role in the electrostrictive process, since the velocity of an ion impinging on an interfering spring-loaded object (e.g. an ionized side-chain at the entrance to a pore) will determine the relative effect of the kinetic and potential energies imparted to the object. It will be shown that this in turn permits a relatively low energy electrostrictive process to result in a relatively large increase in activation energy. Consider the simple example of Fig. 1. Here, a spherical object (entering ion) of radius rl and mass m, impinges symmetrically on two interfering objects (side-chain atoms) each of radius I’~, mass m,, and separated by a center-to-center distance 2d. Each of the latter objects is held in position by springs of force constant kb. If, then, the initial velocity of the impinging object (ion) is L’il its velocity u/ after the collision, may be found by considering energy and momentum balance. If the spring forces are negligible, then M set’ O-1 Vf="iFiG2(l+t
where 0 = sin- ’ [d/(rl+r2)]
and M = m,/Zn1,. Thus the kinetic
(1)
energy
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FIG. 1. Spherical object (ion) moving with velocity v,, having collision with two other objects (side-chain atoms), each restrained by springs having bending coefficients kb. After
impact,the object(ion) hasa final velocity u,.
loss of the impinging
object is
AKE = ~ml(z$-v;)
= +ml$
(A4 set’ O- 1)2 1- (A4 sec2 0 + 1)’ *
1
(2)
Thus the loss in energy is a fraction of the initial energy of the impinging object, given by equation (2). The potential energy necessarily imparted to the interfering objects (side-chain atoms), in order for the impinging object to just pass, if only the stiffness of the springs is now considered, will be PE = 2&k,(dispiacement/L)2
= k,(r, f r2 - d)2/L2
(3)
where L is the bending radius. Thus the energy loss of the impinging object (ion) in the form of potential energy imparted to the interfering object is independent of the initial energy. The importance of these elementary considerations is in the effect of the mass ratio M on the relative distribution of energy loss on an impinging ion. If the mass of the interfering object (atom or group) is small, then M is large and AKE becomes small; the spring force is then the effective factor affecting the energy of the ion. But as the mass of the interfering object becomes progressively larger relative to the ion, the energy loss due to
imparted
KE becomes progressively
more important.
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to the Pore
Now consider the mouth of a pore, having ionized polar groups near the interface, at the end of long-chain molecules. This is shown in highly simplified form in Fig. 2; the ionized group is shown, for definiteness only, as PO,, and the molecular chain is shown straight, although the bond angles would actually be more nearly of the order of 100-120”; nor would the atoms lie in a plane. As a rough first-order approximation, however, the molecule terminating in the ionized group may be considered linear, with all the bonds
Pore
FIG. 2. Assumed configuration of membrane pore. The O- atom of an ionized PO1 group is deflected so that it may interfere with an entering ion. The interfacial electric field is distorted by the presence of the pore, producing a radial component of the field.
having the properties of the C-C bond: force constant kb = 0*54x 10-l’ dyn-cm/radian (Herzberg, 1945), and the bond length L = 1.54 A.? The deflection of such a chain, as shown in Appendix A, will increase approximately as the cube of the number of deflecting atoms in the chain. Thus a lo-atom chain would deflect l-7 x lo6 A/dyn, and a 20-atom, 1.3 x lo7 A/dyn. t Handbook
Ohio, 1969.
of Chemistry
and Physics,
50th edn., Chemical Rubber Co., Cleveland,
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Such a molecule will thus suffer a considerable deflection, if subjected to a component of electric field normal to its axis. Thus the 20-atom chain would deflect l-0 A due to a normal field component of 10’ V/cm acting on a single electron charge. The actual deflection that would occur is certainly less than this, due both to the bends in the chain, and to linkages between adjacent molecules, but these calculations show that substantial deformations are to be expected, under the influence of moderate electric fields. If now an entering ion impinges, for example, on the terminal O- of the molecule,t a complex dynamic transfer of energy to the whole molecule will occur. Rather than even attempt to calculate this, two simplified cases will be assumed, which will give an order-of-magnitude of the energy transfer to be expected. We first assume that the P atom is fixed rigidly in position, and only the O- atom deflects to allow the entering ion to pass; the actual energy transfer must be less than that so computed. To compute the PE, the bond constants for the P-O bond are used: kb = 2.0 x IO- ” dyn-cm radian (Herzberg, 1945) and L = I.56 A. The PE required to displace two 0 atoms (assumed symmetrically disposed across the mouth of the pore), as a function of the interference radius in Fig. 3. Also plotted in the figure is the KE transfer rO + j’ion -d, is plotted based on equation (2), for the case of quadrihydrated K+ as the impinging ion. The initial velocity for these KE calculations is such that the energy after collision is 40 kJ/M, i.e. the energy of hole formation. It will be seen that, except for a very small interfering radius, the PE is greater than the KE term. Thus the 0 atoms would have little residual KE after the collision and the energy absorbed would be essentially only PE. The same conclusion applies for Na+. A very different situation arises if it is assumed that the entire PO, group moves as a whole, with flexure at the O-C bond. For it, &, = 1.7 x 10-l’ dyn-cm/radian (Taylor & Vidale, 1957). The effective bending radius depends on the assumed steric configuration; a value of 5 A will be used. The result of these calculations is shown in Fig. 4; the system is inertialoaded, and the energy absorbed is in the form of KE. It is now seen that in the range of energy absorption of interest (less than 50 kJ/M) flexing of the P-Obond will result in less energy loss than motion of the whole PO; group; the actual dynamic effect will be energy transfer throughout the system. Thus, the actual energy loss would be somewhat less than shown in Fig. 3. t It should be emphasized that this discussion is only exemplary; the colliding atom need not be O-, nor even at the end of the lipid molecule.
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tll)
FIG. 3. Energy loss of a quadrihydrated K+ ion, after impinging upon an interfering O- atom, with the interference shown. The upper curve shows the loss in the form of potential energy, and the lower, in kinetic energy imparted to the O- atoms. The initial energy of the entering ion is that required to have 40 kJ/M after collision. For Na+, PE curve is identical, and KE differs only slightly.
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KE
Interference
FIG. 4.
Similar
to
(A)
Fig. 3, but for collision with POa group. Upper curve, KE; lower, PE,
CONFORMATIONAL
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The essential point of these calculations is to show that a relatively large change in free energy of activation can result from the absorption of a much smaller energy by the membrane, in the form of ionized group displacement under the influence of an electric field. This difference is due to the timescale of the two processes: the electrical displacement can occur over a relatively long time (fraction of a millisecond), while the process affecting activation energy occurs in a time-scale of the order of 10-l’ s. It thus becomes clear that the activation energy for diffusion through pores should be highly sensitive to the boundary electric field, provided that the pore opening is of a suitable form, and carries charges at the interface. Otherwise, no special configuration is necessary. The terms entering into the activation energy necessary for an ion to cross the membrane interface have been previously discussed (Offner, 1972) but the results will be reviewed, with some modification and extension. The various terms are summarized in Table 1, for Na+ and K+ ions crossing in the di-, tri- and quadrihydrated states.
TABLE ___..~ Hydration number Dehydration energy Energy of hole formation Total energy Minimum opening, AU
1
Na+ 2 45 32 77 2.8 2.8
3 15 38.5 53.5 6.6 3.1
K+ 4 0 45 45 6.6 5.9
2 25 32 57 2.8 2.8
3 7 385 45.5 7.3 3.6
4 0 45 45 7.3 6.4
Energies are in kJ/mol. “Minimum opening” is the estimated minimum size rectangular opening, in ktgstrom units, which will pass a linear dihydrated ion, a tetrahedral hydrated ion with one water removed, and a quadrihydrated tetrahedral ion.
These terms include, for the first two states, the energy required to partiahy dehydrate the ion, from its normal quadrihydrated state. These energies have been obtained, as described in Offner (1972), from Dzidic & Kebarle (1970) who measured hydration energies in the vapor phase. We have attempted to correct their results to the liquid phase, by allowing for the difference in Born energy; which is estimated to be approximately 10 kJ/M for each water molecule. A theoretical calculation has also been made (Kim 8z Rubin, 1973), giving the same order of magnitude of values for these energies.
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The next term is that required to form a hole in the liquid, of sufficient size to permit entry of the hydrated ion. The procedure for making these calculations is also given in Offner (1972). However, while this energy was previously taken as smaller for Na+ than for K+, due to the smaller size of the ion, they are now estimated to be about equal, due to the compensating effect of higher bonding of the Na+ ion to the secondary hydration shell. The value for trihydrated ions was obtained by interpolation. The total energy listed is the sum of the above two terms, and is that which would be required for an ion to cross the interface, provided the pore has an opening of a dimension large enough to pass an ion of the appropriate hydration number, and that the ion approaches the pore with an orientation such that it can pass through the opening unobstructed. The approximate dimension of the required opening, for an ion approaching normal to the interface, is shown in the last line of Table 1. Ions approaching from other than a normal angle will suffer a loss of energy, divided between deflection of the molecules forming the pore, as above discussed, and deformation of the hydration shell of the ion itself. This effect should be particularly important in the case of the unsymmetrical trihydrated ion. Considering the random direction of arrival of ions at the pore will thus add an entropic term to the activation energy, resulting in a progressively higher effective energy as the dimension of the opening is reduced. The above effects are far too complex to be calculated accurately for Na+ and Kf ions, but Fig. 5 shows the trend the activation energies would be expected to follow with deformation reducing the lesser dimension of the opening. This figure then illustrates the general order-of-magnitude of the variation in activation energy to be expected for the two ion species, with electrodeformation of the pore, as resulting from the boundary electric field. The gist is that the electrodeformation resulting from high electric fields will reduce the K+ permeability by a factor of about 100 (due to the rise of 11 to 12 kJ/M in the activation energy) and that the Na+ permeability will be reduced about a hundred times as much. Of especial interest is the effect of smaller deformations-under 1 &-on Kf and Naf activation energies. As a result of the substantial equality of the total of the activation energy terms for the tri- and quadrihydrated states in the case of K+, such smaller deformations should have little effect on K+ permeability, while that of Na” would be reduced by a factor of 25 to 50. It is suggested that the effect of tetrodotoxin (TTX) is to cause such a reduction in the maximum opening of the pore. This, as seen, would result in a large reduction in the Na+ permeability, without appreciable effect on that of K’.
CONFORMATIONAL
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of opening
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107
iX)
FIG. 5. Energy loss, and thus increase in free energy of activation, for KC and Na+ ions, as a function of reduction of the lesser dimension of the pore opening. The curves are based on an estimated combination of the energy loss due to molecular deflection, from Fig. 3, dehydration energy (Table 1). and the effect of random approach direction of the entering ion. Based on 3 A deformation reducing opening to diameter of dihydrated ion (approximately 2.8 A).
4. Discussion In these calculations, it has been assumed that energy is lost to a single process only. Thus, when the interaction energy is equal to the energy necessary for dehydration, it is assumed that dehydration will occur. This is an over-simplification; not only will, in general, some energy be lost to the membrane itself, but also the removed water molecules will take up energy, in the form of rotational and vibrational excitation, This will add to the total energy loss in dehydration. However, the estimate of total energy required to remove two water molecules from the ion is only very approximate in any case; what is more important is the difference in dehydration energies required for K+ and Na+. This estimate is believed to be much closer and would not be affected appreciably by these other factors.
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Another theoretical approximation that has been made is the assumption that all the activation energy is initially in the form of translational energy of the diffusing ion. In fact the energy would be partitioned between the various degrees of freedom involved. These would include the interfering side-chain groups, and the hydrogen bonds which must be broken for hole formation. This degeneracy introduces an entropy term, which in effect reduces the energy barrier. But the activation process herein considered is not truly adiabatic, in that the relative energy transferred to the side chains will depend upon the kinetic energy of the ion, and thus the activation energy barrier is a direct function of the initial translational energy on the ion. For example, if only half the activation energy is represented by kinetic energy of the ion, the energy absorbed by side-chain interference in the form of kinetic energy will be reduced by half. In the example calculated, the PO, group could then deflect as a whole with less total energy than the O-, for interferences of greater than about 0.15 A. To properly calculate the actual shape of the potential barrier it is thus necessary to compute the relative contributions of the various possible energy distributions between the degrees of freedom. We have not yet attempted the solution of this far more complex problem. Appendix A
Consider a linear chain of atoms, each separated by a distance L cm, and having bending force constant of kb dyn-cm/radian. Under a force f, the first atom will have an applied torque fL, resulting in a deflection fL?/kb. The second atom will have the force applied at a radius 2L, and a given angular deflection will result in twice the linear deflection, so the deflection due to the bending of its bond will be f(2L)‘/k,. The bending of the nth bond will result in a deflection f(nL)‘/k,. Thus the sum of all the bond deflections will result in a total deflection given by deflection = : i i2. bl
The sum of the series is +z3+jn2+n/6, for n = 20.
equal to 385 for n = 10, and 2870
Appendix B As a result of diffusion within the membrane, the change in potential at the external interface, AV,, eventually becomes of the order of the depolarizing voltage clamp step. If a simple Helmholtz double-layer is assumed,
CONFORMATIONAL
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thickness X = 38 A (diameter of one water molecule in primary hydration shell, 2.8 A, plus ionic radius, approximately 1 A), the change in the interfacial electric field is AE, = -AVJX. For a 50 mV step, E0 = - 1.3 x lo6 V/cm. The lines of electric field are distorted in the vicinity of a pore (Fig. 2). If the effective angle of the field, relative to the axis of the pore, is, for example, 20”, the change in the radial component of the field is AEO sin 20” = -0.45 x lo6 V/cm.
Appendix C
Readily deflectable molecules, of the type herein proposed, should suffer significant vibratory deformation due to Brownian movement. Such vibrations would result in a statistical variation in the activation energy for ions traversing the membrane interface and consequently a similar variation in the membrane current. Random fluctuations (“noise”) have been observed in current through excitable membranes by numerous workers, Verveen & Derksen, 1965; Poussart, 1971; Wanke, DeFelice & Conti, 1974; Fishman, 1973. The power spectrum of the variation of the membrane current has been described as varying as l/f, although the exact law appears to vary. Noise having a similar spectrum is also found both in non-excitable biological membranes, and in non-biological systems involving diffusion through small pores (DeFelice 8z Michalides, 1972; Dorset & Fishman, 1975). Therefore, the electrical noise observed in excitable membranes is undoubtedly largely due to phenomena not involved in the excitability process, and the noise contribution from Brownian motion of the electrodeformable molecules should represent only a small portion of the total noise. As a first approximation, the deformable molecule may be considered as a simple harmonic oscillator, with a resonant (angular) frequency o,=&jii (Cl) where s is the spring constant for bending of the molecule in its fundamental mode of vibration, and m is the “reduced” mass, considered concentrated at the end of the molecule. The power spectrum S,,(o) of the deflection of such a molecule due to Brownian motion has been shown (Papoulis, 1965; Uhlenbeck & Ornstein, 1930; Wang & Uhlenbeck, 1945) to be given by
Kx(~) = (oz-- 0;)’c1+ j?205
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where (C3) and (C4) D is the diffusion constant. Whereas w. will be of the order of 10” or higher, the electrical noise can in general only be measured as greater than thermal (Nyquist) noise for w < 104. This permits the simplification of equation (Cl) by setting (02 - u;)2 z 0:. Then by making appropriate substitutions,
The first term in the denominator will always be much greater than the second, for the range of w of interest: D is of the order of lo-’ cm2/s for large molecules in an aqueous medium, and s is about 8 d/cm for the very flexible 20-atom molecule considered. Thus the first term is greater than 10i4, permitting equation (C4) to be further simplified: 2
S,,(o) =;
7
. 09 ( > In accordance with the theory presented herein, the free energy of activation should be related to the displacement of the deformable molecules, in a manner such as illustrated in Fig. 5. The slope of the curve, in kJ/hi-cm, may be expressed in units of RT/F for univalent ions, giving a proportionality 3, of free energy of activation in volts, to displacement in cm. Then the power spectrum of the variation of the activation energy, measured in volts, is W7) S”“(O) = ~2Lb>. Through each pore there is a steady-state ion current i. of a (univalent) ion species (e.g. K+) due to a displacement V of the membrane voltage from its equilibrium value. The current flow across a potential barrier of height c( is proportional to exp (-a/kT-AVF/2kT), where AV is the electrical potential difference across the barrier (assumed symmetric) measured in the direction of current flow. Therefore a change in the activation energy is equivalent to a change in the membrane voltage; and to the extent that i, is proportional to V, we may write for the power spectrum of the statistical variation in i,, Sii(W) = ig S,,(O)/ V2. ((3)
CONFORMATIONAL
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Equation (C7) thus gives the expected power spectrum of the current fluctuation across the interface, for a single pore. If there are n pores per cm2, then S,r(o), the power spectrum per cm2, will be n times this value. Alternatively, equation (C8) may be written in terms of IO, the current density per cm2. Then i,, = lo/n, so that Making
the appropriate
Sag = z; S,&D)/V2n. substitutions then gives S,,(o) =;
(C9)
T 2z;12,V2n. (C10) ( > From Fig. 5, the value of I for the foot of the curves, corresponding to substantial depolarization, is found to be approximately 4 x 10’ V/cm. The value of n is not known; although some workers have obtained evidence which they interpret as indicating there are relatively few channels (as few as I.5 x 107/cm2), kinetic considerations (C9) indicate that the number should be 100 to 1000 times as large. The value of D for large molecules in aqueous media is of the order of 10e6; the orientation of the water within the pores might be expected to raise its viscosity, and lower D to the order of lo-‘. Poussart (1971), equation (Cll) determined S,,(o) in lobster axon. Reducing his data to the basis of values per cm2, he found a low frequency value of S,,(W) of approximately 5 x IO- ” A2 s, for a steady-state current of ONl6 A/cm2 with V = O-1 V. The value of &(o) varied about as 0-l , falling to a value of approximately 5 x 10W2’ at high frequencies. Using parameters computed for 1 and s, based on the 20-atom chain, and the experimental conditions of Poussart (1971) equation (Cll) gives S,,(o) = 3.4 x lo-‘6/D& (Cl11 Using D = lo-’ and n = lOto, equation (ClO) gives Sr,(o) = 3.4 x 10-19, which is reasonably within the range of Poussart’s minimum value, considering the various approximations and uncertainties involved. Evidently the assumption of a less readily deflected molecule would result in a lower value of noise; a lo-atom chain would reduce S,,(o) to about 5.6 x 10v2’. It appears that in no case could the Brownian movement of the molecules controlling ion flow at the interface be responsible for the major portion of the current noise, because of its frequency spectrum. The above derivation indicates that the generated current fluctuation should be independent of frequency up to frequencies much higher than those measured. The current noise is, however, composed of two equal components: the current flowing across the interfacial barrier into the bathing solution, and that flowing from the second bathing solution, through the membrane, and across the barrier (Offner, 1972). The former component should appear practically
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undistorted in the external measuring circuit, while the second will be filtered by the diffusion time constant of the membrane. Thus the low frequency noise power due to Brownian motion should be four times as great as the high frequency limit. The fact that the current noise power spectrum varies by a factor of 100 or more with frequency indicates that the Brownian motion can be responsible for only a minor portion of the total noise. The remainder is almost certainly due to some process which also occurs in non-excitable membranes, biological or not. The computations in this section show, however, that even very readily deformable molecules may be responsible for the ionic control phenomenon, without contributing more than the observed statistical fluctuation to the passage of the membrane current. REFERENCES DEFJZLICE, L. & MICHALIDES, J. P. L. M. (1972). J. memb. Biol. 9,261. DORSET, D. L. & FISHMAN, H. M. (1975). J. memb. Biol. 21,291. DZIDIC, T. & KEBARLE, P. (1970). J. phys. Chem. 74, 1466. FISHMAN, H. (1973). Proc. natn. Acad. Sci. U.S.A. 70, 876. GLASSTONE, S., LAILILER, K. J. & EYRING, H. (1941). Theory of RateProcesses. New York: McGraw-Hill Book Co. GOLDMAN, D. (1964). Biophys. J. 4, 167. GOLDMAN, D. (1969). Handbook of Chemistry and Physics, 50th edn. Cleveland, Ohio: Chemical Rubber Co. HERZBERG, 0. (1945). Molecular Spectra and Molecular Structure, Vol. II. Princeton, New Jersey: D. Van Nostrand Co., Inc. HILL, T. L. & CHEN, Y. (1972). Proc. natn. Acad. Sci. U.S.A. 69, 1723. G. (1973). Proc. natn. Acad. JOLT, P. C., GRWFITH, 0. H., CAPALDI, R. A. & VANDERKOOI,
Sci. U.S.A. 70,480. KIM, S. H. & RUBIN, B. T. (1973). J. phys. Chem. 77, 1245. MULLINS, L. J. (1959). J. gen. Physiol. 42, 1013. OFFNER, F. F. (1970). .I. gen. Physiol. 56, 272. OFFNER, F. F. (1972). Biophys. J. 12, 1583. OFFNER, F. F. & KIM, S. H. (1976). J. theor. Biol. 61, 113. PAPOULIS, A. (1965). Probability, Random Variables and Stochastic Processes, Cb. New York: McGraw-Hill Book Co. POUSSART, D. J. M. (1971). Biophys. J. 11,211. TA~OR, R. C. & VIDALE, G. L. (1957). J. them. Phys. 26, 122. UHLENBECK, G. E. & ORNSTEIN, L. S. (1930). Phys. Rev. 36,823. VERVEEN, A. A. & DERKSEN, H. E. (1965). Kybernetick. 2, 152. WANG, M. C. & UHLENBECK, G. E. (1945). Reu. mod. Phys. 17,323. WANKE, E., DEFELICE, L. J. & CONTI, F. (1974). Pfliigers Arch. 347, 63.
15.