I. tllevr. Biol. (1977) 66, 763 -773
Unified Theory of l/f and Conductance Noise in Nerve Membrane JOHN R. CLAY?
Department of Anatomy, Emory University,. Atlanta, GA 30322, U.S.,4. AND
MICHAEL. F. SHLESINGER~
The tnstitute ,fi>r Fundamental Studies, Department qf P/lyrics and Astronomy, The University qf Rochester, Rochester, NY 14627, U.S.A. (Received 21 May 1976, and i/l rel+sccl,fbrm 26 October 1976) A theory of l/f and conductance noise is given for ionic channels in nerve membrane. The theory is based on the assumption that the channels are in constant, stochastically independent, rotational motion within a fluid bilayer membrane. The resulting expression for the current noise power density S contains a conduction noise term consistent with Stevens (1972) and Hill & Chen (1972) and a l/j’ noise term consistent with Lundstrom & McQueen (1974) and Clay & Shlesinger (1976). The expression for S also contains a third term which is the spectrum of the product of the single channel conduction noise and l/f noise correlation functions. This term is independent of the number of channels in the membrane, R. Consequently, the expression for S effectively reduces to a sum of l/f and conduction noise for R .’ IO-100 which is in agreement with noise measurements on squid axon. The theory is applied in detail to potassium squid noise measurements of Conti, DeFelice & Wanke (1975) using the stochastic analysis of single file ion motion developed in our previous paper (Clay & Shlesinger. 1976).
1. Introduction We have developed a unified theory of flicker noise [power spectral densit) S(f) - l/.A .f = frequency] and conductance noise [S N c (c$+f2)-‘, i.e. a sum of Lorentzians] which are the two dominant types of noise observed in nerve membrane preparations for I-IO Hz < ,f < IO3 Hz (cf. Verveen & + NIH Research Fellow (HL05346-OIA! ). :: Supported by a grant from Physical Dynamics Inc., La Jolla, C‘A. 762
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DeFelice, 1974, for a recent review). The expression for the total power spectrum, equations (21) and (22) of this paper, is a synthesis of theoretical work on the conductance component (Hill & Chen, 1972; Stevens, 1972) and the flicker component (Lundstrom & McQueen, 1974; Clay & Shlesinger, 1976, hereafter referred to as CS). The theory is motivated by the finding that 70 mM TEA blocks both the potassium conductance noise as well a:\ part of the I/fnoise in the squid giant axon (Conti et ul.. 1975). [Fishman. Moore & Poussart, 1975, however, find very complex effects of IO mM TEA on squid axon potassium ion noise.] The total noise amplitude in the preparation of Conti et al. (1975) also appears to be a simple sum on the l/f and conductance noise components (S = S,/,+S,). It is not obvious a priori that a unified theory will give this simple decomposition, as the correlation function will nccessnrily contain a cross term proportional to the product of the I/f and conductance correlation functions. We find that this term has a complex spectral shape [equation (22)] with an amplitude which is about the same order of magnitude as SIII. or S,. for a membrane having a small number of K channels (R - 10-100). Since R i:; undoubtedly several orders of magnitude greater than IO-100 in typical membrane preparations, we find that the theoretical noise amplitude does in fact effectively reduce to a simple sum of the two noise components, because the cross term is roughly proportional to R-‘(S,,,+S,). The major assumption of this paper and of CS is that K ions translocate within the membrane by single file, non-dissipative, motion through narrow dynamic channels which are coupled to fluctuations in the fluid motion of the constituent membrane lipid molecules (cf. Singer & Nicolson, 1972; Edidin, 1974). In this paper we require that each channel have some number of discrete particles, all of which must be in a certain configuration for the channel to be open (Hill & Chen, 1972: Stevens. 1972). Random fluctuations in motion of the lipid molecules force the channel to rotate or move in some other way so as to block passageof ions even if the channel itself is in the conducting state. That is, we are hypothesizing two independent gating processes,one of which is carrsedby the internal state of the channel and the second of which is caused by lipid-channel interactions. We also assumethat the channel is in the proper orientation for only a short time, .Y, compared to the time the channel typically spendsin one of its conformational states. The details of the model are speculative, especially the prominent part played by lipid motion. However, recent evidence (Wolf et al., 1977)suggests that the diffusion constant of macromolecules in a lipid bilayer is quite large, D - I um’ s-I, for concentrations of the macromolecule - 10’ urn’, which is comparable to the density of K channels in squid axon (Conti
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et al., 1975). Consequently, bilayers and perhaps biological membranes may be strikingly dynamic systems. Moreover, the above formulation together with the model of passive channel kinetics of CS is consistent with: (1) lineal instantaneous I,-- I/ relation appropriate for squid and perhaps lobster (shown in this paper), (2) a simple decomposition of the total spectrum into I/f and conductance components (this paper), (3) a flicker spectrum of S = (a+bfi)/f observed in squid and lobster (CS), (4) the external potassium concentration dependence of a and b in lobster flicker noise data; a - [K], and b - independent of [K& (Poussart, 1971), (5) the temperature and voltage dependence of conductance noise in squid (Conti et al., 1975). and (6) the small temperature dependence of the l/f noise (Poussart, 1971). 2. The Model Channels have a Linear Instantaneous
I-V Relation
As described in CS, the single file motion of ions through a narrow channel having n ion sites, where n can be any positive integer (including II = I), is mathematically equivalent to a semi-Markovian random walk on an infinite one-dimensional lattice. Consequently, the mathematics of continuous-time random walks (cf. Montroll & &her, 1973) appropriate for this system may be employed. The average single channel current for univalent ions with electrical charge e is (CS) II = eN(5)I.F
= ei/(fL),
(Ii where W(Y) is the mean number of ions which pass through the channel during 3, i is the mean time between successful collisions (Hodgkin & Keynes, 1955; Macey & Oliver, 1967) of extra-membrane ions with the channel from either side of the membrane, fi is the mean distance the channel row moves with each of these collisions, and L is the distance between sites. The average in equation (1) implies that Y 9 f. That is, the typical lifetime of the channel in its conducting orientation must be significantly greater than the time for the n ion sites to be filled. We note that the local environment of each of the n ions need not be identical. For example, our description of channel kinetics is appropriate for the “funnel” model of the squid K channel proposed by Armstrong (1971). The essential assumptions of 0~11 theory are that the channel is always full between collisions, the ions do not slip past one another and the duration of the motion of the row after ;1 successful collision is small compared to i. If we further assume that onl~ one ion leaves the channel per collision, then f, = ei-‘[p(L)-p(
-L)],
(2) where p( L-L) is the (voltage-dependent) probability that the row moves one position toward the external (internal) fluid when a collision occurs.
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Equation (2) may provide a model for leakage and “instantaneous” current. For this purpose we assume that the quantities p(L) and p( -1,) are the probabilities of flow of each of the n ions through a potential barrier of + P’/(n+ 1). If they arc related in the same way as passive influx and efflux. assuming independent movement of ions (Ussing, 1949), then
PC-U/p(L) with p(L)+p(-L)
= CWCW exp (eV/kJ) = exp Diet1’ - &YkJI,
(3)
= I. The average single channel current is then given by I, = et- ’ tanh [e( P’ - E,)j2k,T],
(4)
where e/kBT = F/RT has its usual meaning and E, = kBT log, ([K]J[K],). Although the independence principle may seem paradoxical in this context. it is appropriate for this model of channel kinetics for flux of ionic charge, as can be visualized using the simple analogy of H billard balls in a narrow tube. A collision of another billiard ball with one end of the tube will result in one ball leaving the other end regardless of the number of balls inside the tube. This does not necessarily imply that the ions in the channel are all in contact with one another. For example, water molecules may be situated between channel ions without modifying the effect of an extra-membrane ion so long as all of the molecular species move together. To modify the billiard ball analogy for ions in an electric field, we tilt the tube corresponding to the magnitude of transmembrane potential and we place indentations in the tube at each of the n sites so that the halls will not roll cut by the force of gravity alone. In this model a collision of an external ball with the lower end will not necessarily result in a ball leaving the higher end. Rather, it may cause the row to roll downard. We assume that when the analogous event occurs in a K channel, the row of ions drags along an ion from the other fluid so that the channel is always full between collisions. The “constant-field” approximation for this model is that each of the IZ ions plus the external ion which enters the channel moves through a potential barrier of k V/(n+ I) per collision, resulting in equations (3) and (4) for the channel current. That is, p(&t) is independent of n. This result does not hold true in general, although equations (1) and (2) are general results for single file diffusion current appropriate for any set of potential wells within the channel given that each of the II sites always contains an ion. Equation (4) with f independent of Y is approxinlately linear over the range of voltages of the noise measurements of Conti et al. (1975) (- 100 mV 5 V < -40 mV; EK 2: -60 mV). It is also sufficient to describe the lineal instantaneous I,- V for (V- EKj < 60 mV observed by Adelman, Palti & Senft (1973) with a voltage dependent increase of 7- ’ of 1.5. The frequency of successful collisions undoubtedly is a function of (V-&X,. although 1, - V
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should eventually saturate, as the collision frequency of all K ions with the membrane is finite. This aspect of the theory can be tested most fairly by “instantaneous” K current measurements under conditions of nearly equal external and internal K ion concentrations. This condition obtains in the experiment of Yeh et al. (1976). Their instantaneous I,- P’ data suggestan ultimate saturation of Ik, although values of 1V--&I > 100 mV will probably be required to observe the full saturation effect.
3. Leakage Current Noise Analysis If the model channels have only one conformational state, and if the probability density function for time intervals between successfulcollisions has an exponential form, the membrane current noise per unit area is given by (CS) S = [liBT/(4nydK,)][e2P/(f~)+
i’jf3]/J
Jmin< 1’< .fm:,x.
(5)
with 1 = eflv/(Z), where fl is the average density of channels in conducting orientation at any given time, d is the membrane thickness, K, is the elastic constant characterizing the fluid motion of membrane lipids, and y is the r.m.s. angular deviation of lipid moleculesfrom their equilibrium orientation normal to the membrane surface. (Note an error of a factor of 2 in equation (30) of CS.) Equation (5) was derived by neglecting fluctuations in single ion transit times which are presumably on a - 1 us time scale (Stevens, 1972). The minimum frequency cut-off is a result of the finite size of the fluid lipid patch surrounding each channel, and the maximum cut-off is a result of the non-validity of macroscopic hydrodynamic theory for distances less than lipid-lipid spacing. Our theory yields the j = 0 l/f’ term only if several criteria arc met involving the time scalesof the membrane processes,the signal averaging time, and the size of the experimental frequency window. As the signal averaging time approaches infinity, the frequency dependence of the f = 0 term disappears and it becomes a white noise (Nyquist) term. That is, the extent to which l/f noise is observed when f = 0 is a measure of the nonideality of the experimental conditions. This point will be discussedin detail elsewhere. Equation (5) is appropriate for analyzing noise from a membrane having a linear I-- 1/ relation such as the hyperpolarized squid axon in normal ionic environment. Flicker noise data from the preparation (Loligo dgaris) of Conti et al. (1975) is shown in Fig. 1. The TEA experiment indicates that the noise amplitude for - 100 mV < V 5 -60 mV is produced by leakage
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/
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-100
-90
.
-4ii
-80
30
mv FIG. I. Amplitude of flicker noise normalized to 1 cm” of membrane area from Conti et a/. (1975). The filled triangles are TEA data. The solid line through the (-100 mV < V < -60 mV) is a least squares fit of equation (7) to both TEA and non-TEA data in this range of membrane voltage. The dashed line is an extension of the fitted curve to V = -4OmV. The solid line through the non-TEA data for -6OmV N V< -40 mV represents the best fit to this part of the data obtained by Conti et 01. (1975) using cl, with c = 0.7 ?: 10-8cm2.
current, I,,. Using the linear term of a Taylor seriesexpansion of equation (4) to relate 1, to V, we find that I, - e’D,( V 4 60 mV)/(2k,Ti),
(6)
so that jS(.f,
= ~1L+b;~i'+60)1.
(7)
A least squares fit of equation (7) to Fig. I ( - 100 mV I V I -60 mV) gives uL = 3.9 x 10-”
amps’ cm-*;
6;- = 1.4~ lO-22 amps2mV-* cmw2. (8)
This result for a is about three orders of magnitude lessthan the corresponding result for lobster. A meaningful comparison of equation (7) to the data can be made using the ratio a,/bi
= Zik,T/(e’T
),
(9)
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since K, and y do not appear in this expression. The numerical results of equation (8) together with equation (9) give i/T w IO-‘, which is consistent with the assumption in CS that F < low4 s, as i is less than ion transit times across the membrane. Application of this result to HH channels may be inappropriate, but the simplest assumption to make as a first step toward applying the theory of CS to these channels is that lipid-channel interactions are independent of channel type.
4. Noise Analysis for Depolarized Axons As shown by the data of Fig. I, the Ilfnoise amplitude increases significantly for depolarizing voltages. A Lorentzian component is also observed for this voltage range. Both effects are eliminated by TEA, which suggests that they are produced by K ions passing through HH channels. The leakage current appears to be independent of K channel current, so we ignore it in our unified theory. The total noise amplitude may be obtained by simply adding the leakage current noise amplitude to our final result in equation (21). We calculate the power density for this case starting from the point of view of Hill and Chen (1972), who consider an ensemble of R equivalent and stochastically independent K channels each having x (x = I, 2,. . ., X; X < co) non-interacting, open-close particles. All particles must be open foi a channel to be conducting. The probability pj(f) that any single channel has .j (j = 0, I, 2,. . . , X) open particles is pi(r) = x!n(t)j[l - rt(t)]“-j/[.i!(s-,i)!], (10) whcrc /r(r) is the probability that any single particle is open at time f. This quantity is given by n(t)=
1~.+(n~--“*)exp(-1’r,,).
(11)
The quantities n,, II,, and t, are HH K channel parameters. The probability pi(l) may also be expressedas Pjtf)
=
i k=O
4jk(r)Pk(o).
(I?)
where d~,~~(f)is the probability that a channel has ,j particles open at time f given that k particles were open at t = 0. The average number of channel\ in state ,j, R,,(t), is also determined by the functions 4+(t). That is, Kj(t)
=
jJ
k=0
(bjk(f)Rk(O).
(13)
We model lipid-channel interactions in this theory by assuming:that the conformational state of any channel does not change during .F, since 7 < IO-‘s and T,, N 1O-2 s. Therefore, the average number of open
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channels al any time t. R*(t),
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is given by ,?(t)R,(t)
with
s(f) = s ,‘I@, t) dr/R
(14)
M
where r is a two dimensional vector in the plane of the membrane and M is the membrane area. The quantity s(t) is the fraction of channels in conducting orientation at time t, with equilibrium value, S, = fl,M/R. Lipid-channel interactions are assumed to arise from spatial and temporal fluctuations of P(r, 0. We stress that we are not hypothesizing interactions between channels, although the language of CS is not unequivocal on this point. A model of fluid motion in nerve membrane in which the lipids are in a single homogeneous fluid phase is oversimplified. A more likely picture of the state of membrane lipids is one of a mosaic of local domains, some of which are fluid, some solid, and some in between (Poste & Allison, 1973). We hypothesize each of the channels to be in a local domain of fluid lipid of a few to several hundred angstroms extent. In this revised model the range of validity of the prediction of l/f noise is not greater than 2-3 frequency decades, which is consistent with noise measurements to data. With the above formulation for membrane current the current noise correlation function is
ii3
C(t) = (~Z+&V2)([i?*(t)-~,*][R*(0)-~R,*])/.~2, where (6N2)*j2 is the r.m.s. deviation of the number through an open channel during ,F (given in CS), and R,* = S,p:R = S&R.
of ions which
pass
(16) That is, we ignore fluctuations in the transit times of individual ions which are probably on a 1 us time scale. These fluctuations will not produce significant low frequency noise. Moreover, a random channel open time will not alter the theory in a significant way, since F is also much smaller ---+ than the time scale of ms fluctuations. If the r.m.s. fluctuation 6F2 IS small compared to F, then the effect of a stochastic F will be to scale C(r) in -equation (15) by a multiplicative factor of (1 +SF2/F2). Noise on a ms time scale arises from the ms interstate kinetics of the channel protein molecule and the range of ms relaxation kinetics of the fluid motion of the lipids. This noise is represented by the bracketed term in equation (15) which is the correlation function of the number of open channels R*(t). Substituting equations (13) and (14) into this part of equation (15) gives .x c(f) = ($W.3) c 4xk(WWV,(O)) -$ i; 4x~(WV?C. (17) 0 0
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Using s(t) = $,+&s(t) and the expressions for (&(O)R,(O)) (k=0,1,2 )...) x) given by Hill & Chen (1972), we find that c(t) = nX,R[S,2 + (~%(t)6s(O))]{[n~
-i-(1 - 12,) exp (- t/z,)lx-
and &,(t)
n”,)j +
+ nz,xR2(&(?)&(0)).
(18)
Since the HH channels are assumed to be uncorrelated, the mean square fluctuation of open channels will be proportional to the total number of channels R. That is, (s/I(r, t)5/3(r, O)} N R. Therefore, c(t) = sfRc,(tj+n~:Rc,,f(t)+c,~f(f)c,(t):
R2
1.
(19)
where c,(t) is the single channel conductance noise correlation function and cr,/(t) is the single channel (in the sensedescribed above) llf correlation function given by CS; Lundstrom & McQueen, 1974) c(t) = S,k,T/@~yK,cl)
J dq exp t -K,g’tlq)/q, ‘I,“,,>
(20)
where q is membrane viscosity, and y arises from a Fourier transform 01‘ /?(r, t), with qlni,, - R-’ and qmaX- /.‘-‘, where i, is the smallest linear dimension of the fluid domain and & is the lipid-lipid spacing. Equation (18) clearly shows that the cross term will be insignificant for very large R. A straightforward calculation of the power spectrum of the first two terms of equation (19) using the Wiener-Khintchine theorem (Wang & Uhlenbeck. 1954) gives s = 4(s,a + I#)?”
q
; 0
p’/jU + W~,,/j*)l- I + + k,7’[u + I;/(s’,R)]/(4nyK,clf),
(21)
and the spectrum of the cross term is (ii,,,, -+ ~0) S = k,T[cr/R +i;/(S,R2j]/(4nydK,f)
1 (;> p’
where
i, = en:,S,RV/(iL),
LI = e*S,Rn2,“/(W),
p = ( I - n,)jn,.
(23) The first term of equation (21) is identical to the result of Hill & Chen (1972) except that R-‘ii is replaced by R-‘Ii+S,a. The S,,, term is the same as in CS except for the HH parameter n”,“. In principle the non-TEA data (- 60 mV < V < - 40 mV) of Fig. 1 may be fitted to the Slif theoretical expression, as was done for the leakage current noise, to obtain an estimate of the ratio il.7 for K channels. However, the narrow voltage range as well 1.11. 51
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as the scatter of the data essentially invalidate the exercise. A meaningful fit to this data is possible using the approximation J?(f) = cl; which gives c = 0*7x Io-8 cm2 (Conti et al., 1975). Comparing this result to equation (21), we find that kBT/(4nydK,S,R)
= @7x lo-’
cm2
(24)
or (yK,S,R) = 0.6 dyn/cm’, using d = 75 A, and T = 280” K. Comparison of the above expression for S, to the conductance noise data [ignoring the (tY)-l term] gives R - 60 channels/pm2 (Conti et al., 1975) so that yK,s:, = lo-‘*
dyn.
(25)
The parameters y, K, and S, are unknown except that S, < I, and probably y < I, as well. For example, an r.m.s. deviation of 1” from equilibrium orientation of the lipids gives 17- 4 x IOm5. A typical value of K, for liquid crystals is IOe6 dyn (Stephan & Straley, 1974), so that equation (25) would predict S, = O-25, i.e. each channel in conducting orientation about 25% of the time. However, this calculation is not meant to imply anything more than the fact that apparently reasonable values of y, S, and K, are consistent with equation (25). The I/f aspect of our theory may be appropriate for membranes other than unmyelinated nerves in which significant I/fcurrent noise is observed and the channel density is low. The Paramecium membrane may be one other likely candidate (Moolenaar et al., 1976). Our theory is also consistent with membranes which do not exhibit l/.f current noise, since the passage of ions alone does not produce I/f noise in our model. An interaction between lipid motion and ion channels is the siue qua 17071 of our I/,f theory and lipid-channel interactions of the type described in this paper need not exist in all membrane types. The ultimate test of the model requires detailed understanding of the dynamic state of biological membrane and the nature of lipid-channel interactions. However, further noise measurementsover a wider range of membrane voltage, and noise data for 10 mM < [K& < 500 mM would give more constraints on the range of values of the various parameters of the model, as well as providing another test of its predictions. A report on this work was presented at the 21st annual meeting of the Biophysical Society, New Orleans,Louisiana, 1977. REFERENCES ADELMAN, W. J., PALTI, Y. & SENI;T, J. I’. (1973). J. wwh. ARMSTRONG, C. M. (1971). J. gen. Physiol. 58, 413. CLAY, J. R. StILESINGER, M. F. (1976). Bio&vs. J. Xi, 121.
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CONTI, F., DEFELICE, L. J. & WANKE, E. (1975). J. P/t.ssiol. 248, 45. E~IDIN, M. (1974). Ann. Rev. biophys. Bioeng. 3, 179. FISHMAN, H. M., MOORE, L. E. & POUSSART, D. J. M. (1975). J. tttettzb. Biol. 24, 305. HILL, T. L. & CHEN, Y. D. (1972). Bz’ophys. J. 12, 948. HODGKIN. A. L. & KEYNES. R. D. (1955). J. Phvsiol. 128. 61. LUNDSTR& I. & MCQUEEN, D. (1974). >. theo;. Biol. 45; 405. MACEY, R. I. & OLIVER, R. M. (1967). Biophys. J. 7, 545. MONTROLL, E. W. & SCHER, H. (1973). J. Stat. Phys. 9, 101. MOOLENAAR, W. H., DE GOEDE, J. & VERVEEN, A. A. (1976). Mtture 260, 344. POSTE, G. & ALLISON, A. C. (1973). Biochitrt. biophj,.I. .lcro 300, 121. POUSSART, D. J. M. (1971). Biophys. J. 11, 212. SINGER. S. J. & NICOLSON. G. L. (1972). Science. N. y. 175. 720. STEPHA;, M. J. & STRALE~, J. P. iI 974). Rev. Mod. Plr~~s.46, 617 STEVENS. C. F. (1972). Biophys. J. 12, 1028. USSINC,, H. H. (1949). Ph&ioI. Rev. 29, 127. VERVEEN. A. A. & DEFELICE. L. J. (1974). Prop. Bio~hm. trtolec. Biol. 28. 189. WANG, M. C. & UHLENBECK, G. E. (1954). In Sekrted Papers ott Noise ctttd S~oclxtstic Proce.sse.s.p. 113 (N. Wax, ed.). New York: Dover Publications. WOLF, D. E., SCHLESSINGER, J.. ELSON, E. L... WEBB, W. W.. BI.uMEI\‘TII-~[., K. & HINKAKI P. (1977). Biophys. J. 17, 133a. YEtI, J. Z.. OXFORI>. G. S., Wu, C. H. Rr NARZHASHI. T. (1976). Bioph,w. J. 16, 77.
Unified Theory of l/f and Conductance Noise in Nerve Membrane JOHN R. CLAY?
Department of Anatomy, Emory University,. Atlanta, GA 30322, U.S.,4. AND
MICHAEL. F. SHLESINGER~
The tnstitute ,fi>r Fundamental Studies, Department qf P/lyrics and Astronomy, The University qf Rochester, Rochester, NY 14627, U.S.A. (Received 21 May 1976, and i/l rel+sccl,fbrm 26 October 1976) A theory of l/f and conductance noise is given for ionic channels in nerve membrane. The theory is based on the assumption that the channels are in constant, stochastically independent, rotational motion within a fluid bilayer membrane. The resulting expression for the current noise power density S contains a conduction noise term consistent with Stevens (1972) and Hill & Chen (1972) and a l/j’ noise term consistent with Lundstrom & McQueen (1974) and Clay & Shlesinger (1976). The expression for S also contains a third term which is the spectrum of the product of the single channel conduction noise and l/f noise correlation functions. This term is independent of the number of channels in the membrane, R. Consequently, the expression for S effectively reduces to a sum of l/f and conduction noise for R .’ IO-100 which is in agreement with noise measurements on squid axon. The theory is applied in detail to potassium squid noise measurements of Conti, DeFelice & Wanke (1975) using the stochastic analysis of single file ion motion developed in our previous paper (Clay & Shlesinger. 1976).
1. Introduction We have developed a unified theory of flicker noise [power spectral densit) S(f) - l/.A .f = frequency] and conductance noise [S N c (c$+f2)-‘, i.e. a sum of Lorentzians] which are the two dominant types of noise observed in nerve membrane preparations for I-IO Hz < ,f < IO3 Hz (cf. Verveen & + NIH Research Fellow (HL05346-OIA! ). :: Supported by a grant from Physical Dynamics Inc., La Jolla, C‘A. 762
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DeFelice, 1974, for a recent review). The expression for the total power spectrum, equations (21) and (22) of this paper, is a synthesis of theoretical work on the conductance component (Hill & Chen, 1972; Stevens, 1972) and the flicker component (Lundstrom & McQueen, 1974; Clay & Shlesinger, 1976, hereafter referred to as CS). The theory is motivated by the finding that 70 mM TEA blocks both the potassium conductance noise as well a:\ part of the I/fnoise in the squid giant axon (Conti et ul.. 1975). [Fishman. Moore & Poussart, 1975, however, find very complex effects of IO mM TEA on squid axon potassium ion noise.] The total noise amplitude in the preparation of Conti et al. (1975) also appears to be a simple sum on the l/f and conductance noise components (S = S,/,+S,). It is not obvious a priori that a unified theory will give this simple decomposition, as the correlation function will nccessnrily contain a cross term proportional to the product of the I/f and conductance correlation functions. We find that this term has a complex spectral shape [equation (22)] with an amplitude which is about the same order of magnitude as SIII. or S,. for a membrane having a small number of K channels (R - 10-100). Since R i:; undoubtedly several orders of magnitude greater than IO-100 in typical membrane preparations, we find that the theoretical noise amplitude does in fact effectively reduce to a simple sum of the two noise components, because the cross term is roughly proportional to R-‘(S,,,+S,). The major assumption of this paper and of CS is that K ions translocate within the membrane by single file, non-dissipative, motion through narrow dynamic channels which are coupled to fluctuations in the fluid motion of the constituent membrane lipid molecules (cf. Singer & Nicolson, 1972; Edidin, 1974). In this paper we require that each channel have some number of discrete particles, all of which must be in a certain configuration for the channel to be open (Hill & Chen, 1972: Stevens. 1972). Random fluctuations in motion of the lipid molecules force the channel to rotate or move in some other way so as to block passageof ions even if the channel itself is in the conducting state. That is, we are hypothesizing two independent gating processes,one of which is carrsedby the internal state of the channel and the second of which is caused by lipid-channel interactions. We also assumethat the channel is in the proper orientation for only a short time, .Y, compared to the time the channel typically spendsin one of its conformational states. The details of the model are speculative, especially the prominent part played by lipid motion. However, recent evidence (Wolf et al., 1977)suggests that the diffusion constant of macromolecules in a lipid bilayer is quite large, D - I um’ s-I, for concentrations of the macromolecule - 10’ urn’, which is comparable to the density of K channels in squid axon (Conti
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765
NOISE
et al., 1975). Consequently, bilayers and perhaps biological membranes may be strikingly dynamic systems. Moreover, the above formulation together with the model of passive channel kinetics of CS is consistent with: (1) lineal instantaneous I,-- I/ relation appropriate for squid and perhaps lobster (shown in this paper), (2) a simple decomposition of the total spectrum into I/f and conductance components (this paper), (3) a flicker spectrum of S = (a+bfi)/f observed in squid and lobster (CS), (4) the external potassium concentration dependence of a and b in lobster flicker noise data; a - [K], and b - independent of [K& (Poussart, 1971), (5) the temperature and voltage dependence of conductance noise in squid (Conti et al., 1975). and (6) the small temperature dependence of the l/f noise (Poussart, 1971). 2. The Model Channels have a Linear Instantaneous
I-V Relation
As described in CS, the single file motion of ions through a narrow channel having n ion sites, where n can be any positive integer (including II = I), is mathematically equivalent to a semi-Markovian random walk on an infinite one-dimensional lattice. Consequently, the mathematics of continuous-time random walks (cf. Montroll & &her, 1973) appropriate for this system may be employed. The average single channel current for univalent ions with electrical charge e is (CS) II = eN(5)I.F
= ei/(fL),
(Ii where W(Y) is the mean number of ions which pass through the channel during 3, i is the mean time between successful collisions (Hodgkin & Keynes, 1955; Macey & Oliver, 1967) of extra-membrane ions with the channel from either side of the membrane, fi is the mean distance the channel row moves with each of these collisions, and L is the distance between sites. The average in equation (1) implies that Y 9 f. That is, the typical lifetime of the channel in its conducting orientation must be significantly greater than the time for the n ion sites to be filled. We note that the local environment of each of the n ions need not be identical. For example, our description of channel kinetics is appropriate for the “funnel” model of the squid K channel proposed by Armstrong (1971). The essential assumptions of 0~11 theory are that the channel is always full between collisions, the ions do not slip past one another and the duration of the motion of the row after ;1 successful collision is small compared to i. If we further assume that onl~ one ion leaves the channel per collision, then f, = ei-‘[p(L)-p(
-L)],
(2) where p( L-L) is the (voltage-dependent) probability that the row moves one position toward the external (internal) fluid when a collision occurs.
766
J.
R.
CLAY
AND
M.
F.
SHI.L‘SlNGER
Equation (2) may provide a model for leakage and “instantaneous” current. For this purpose we assume that the quantities p(L) and p( -1,) are the probabilities of flow of each of the n ions through a potential barrier of + P’/(n+ 1). If they arc related in the same way as passive influx and efflux. assuming independent movement of ions (Ussing, 1949), then
PC-U/p(L) with p(L)+p(-L)
= CWCW exp (eV/kJ) = exp Diet1’ - &YkJI,
(3)
= I. The average single channel current is then given by I, = et- ’ tanh [e( P’ - E,)j2k,T],
(4)
where e/kBT = F/RT has its usual meaning and E, = kBT log, ([K]J[K],). Although the independence principle may seem paradoxical in this context. it is appropriate for this model of channel kinetics for flux of ionic charge, as can be visualized using the simple analogy of H billard balls in a narrow tube. A collision of another billiard ball with one end of the tube will result in one ball leaving the other end regardless of the number of balls inside the tube. This does not necessarily imply that the ions in the channel are all in contact with one another. For example, water molecules may be situated between channel ions without modifying the effect of an extra-membrane ion so long as all of the molecular species move together. To modify the billiard ball analogy for ions in an electric field, we tilt the tube corresponding to the magnitude of transmembrane potential and we place indentations in the tube at each of the n sites so that the halls will not roll cut by the force of gravity alone. In this model a collision of an external ball with the lower end will not necessarily result in a ball leaving the higher end. Rather, it may cause the row to roll downard. We assume that when the analogous event occurs in a K channel, the row of ions drags along an ion from the other fluid so that the channel is always full between collisions. The “constant-field” approximation for this model is that each of the IZ ions plus the external ion which enters the channel moves through a potential barrier of k V/(n+ I) per collision, resulting in equations (3) and (4) for the channel current. That is, p(&t) is independent of n. This result does not hold true in general, although equations (1) and (2) are general results for single file diffusion current appropriate for any set of potential wells within the channel given that each of the II sites always contains an ion. Equation (4) with f independent of Y is approxinlately linear over the range of voltages of the noise measurements of Conti et al. (1975) (- 100 mV 5 V < -40 mV; EK 2: -60 mV). It is also sufficient to describe the lineal instantaneous I,- V for (V- EKj < 60 mV observed by Adelman, Palti & Senft (1973) with a voltage dependent increase of 7- ’ of 1.5. The frequency of successful collisions undoubtedly is a function of (V-&X,. although 1, - V
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767
should eventually saturate, as the collision frequency of all K ions with the membrane is finite. This aspect of the theory can be tested most fairly by “instantaneous” K current measurements under conditions of nearly equal external and internal K ion concentrations. This condition obtains in the experiment of Yeh et al. (1976). Their instantaneous I,- P’ data suggestan ultimate saturation of Ik, although values of 1V--&I > 100 mV will probably be required to observe the full saturation effect.
3. Leakage Current Noise Analysis If the model channels have only one conformational state, and if the probability density function for time intervals between successfulcollisions has an exponential form, the membrane current noise per unit area is given by (CS) S = [liBT/(4nydK,)][e2P/(f~)+
i’jf3]/J
Jmin< 1’< .fm:,x.
(5)
with 1 = eflv/(Z), where fl is the average density of channels in conducting orientation at any given time, d is the membrane thickness, K, is the elastic constant characterizing the fluid motion of membrane lipids, and y is the r.m.s. angular deviation of lipid moleculesfrom their equilibrium orientation normal to the membrane surface. (Note an error of a factor of 2 in equation (30) of CS.) Equation (5) was derived by neglecting fluctuations in single ion transit times which are presumably on a - 1 us time scale (Stevens, 1972). The minimum frequency cut-off is a result of the finite size of the fluid lipid patch surrounding each channel, and the maximum cut-off is a result of the non-validity of macroscopic hydrodynamic theory for distances less than lipid-lipid spacing. Our theory yields the j = 0 l/f’ term only if several criteria arc met involving the time scalesof the membrane processes,the signal averaging time, and the size of the experimental frequency window. As the signal averaging time approaches infinity, the frequency dependence of the f = 0 term disappears and it becomes a white noise (Nyquist) term. That is, the extent to which l/f noise is observed when f = 0 is a measure of the nonideality of the experimental conditions. This point will be discussedin detail elsewhere. Equation (5) is appropriate for analyzing noise from a membrane having a linear I-- 1/ relation such as the hyperpolarized squid axon in normal ionic environment. Flicker noise data from the preparation (Loligo dgaris) of Conti et al. (1975) is shown in Fig. 1. The TEA experiment indicates that the noise amplitude for - 100 mV < V 5 -60 mV is produced by leakage
768
J.
R.
CLAY
AND
M.
F.
SHLESINGER
I
/
A‘
OL-..-110
-100
-90
.
-4ii
-80
30
mv FIG. I. Amplitude of flicker noise normalized to 1 cm” of membrane area from Conti et a/. (1975). The filled triangles are TEA data. The solid line through the (-100 mV < V < -60 mV) is a least squares fit of equation (7) to both TEA and non-TEA data in this range of membrane voltage. The dashed line is an extension of the fitted curve to V = -4OmV. The solid line through the non-TEA data for -6OmV N V< -40 mV represents the best fit to this part of the data obtained by Conti et 01. (1975) using cl, with c = 0.7 ?: 10-8cm2.
current, I,,. Using the linear term of a Taylor seriesexpansion of equation (4) to relate 1, to V, we find that I, - e’D,( V 4 60 mV)/(2k,Ti),
(6)
so that jS(.f,
= ~1L+b;~i'+60)1.
(7)
A least squares fit of equation (7) to Fig. I ( - 100 mV I V I -60 mV) gives uL = 3.9 x 10-”
amps’ cm-*;
6;- = 1.4~ lO-22 amps2mV-* cmw2. (8)
This result for a is about three orders of magnitude lessthan the corresponding result for lobster. A meaningful comparison of equation (7) to the data can be made using the ratio a,/bi
= Zik,T/(e’T
),
(9)
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769
since K, and y do not appear in this expression. The numerical results of equation (8) together with equation (9) give i/T w IO-‘, which is consistent with the assumption in CS that F < low4 s, as i is less than ion transit times across the membrane. Application of this result to HH channels may be inappropriate, but the simplest assumption to make as a first step toward applying the theory of CS to these channels is that lipid-channel interactions are independent of channel type.
4. Noise Analysis for Depolarized Axons As shown by the data of Fig. I, the Ilfnoise amplitude increases significantly for depolarizing voltages. A Lorentzian component is also observed for this voltage range. Both effects are eliminated by TEA, which suggests that they are produced by K ions passing through HH channels. The leakage current appears to be independent of K channel current, so we ignore it in our unified theory. The total noise amplitude may be obtained by simply adding the leakage current noise amplitude to our final result in equation (21). We calculate the power density for this case starting from the point of view of Hill and Chen (1972), who consider an ensemble of R equivalent and stochastically independent K channels each having x (x = I, 2,. . ., X; X < co) non-interacting, open-close particles. All particles must be open foi a channel to be conducting. The probability pj(f) that any single channel has .j (j = 0, I, 2,. . . , X) open particles is pi(r) = x!n(t)j[l - rt(t)]“-j/[.i!(s-,i)!], (10) whcrc /r(r) is the probability that any single particle is open at time f. This quantity is given by n(t)=
1~.+(n~--“*)exp(-1’r,,).
(11)
The quantities n,, II,, and t, are HH K channel parameters. The probability pi(l) may also be expressedas Pjtf)
=
i k=O
4jk(r)Pk(o).
(I?)
where d~,~~(f)is the probability that a channel has ,j particles open at time f given that k particles were open at t = 0. The average number of channel\ in state ,j, R,,(t), is also determined by the functions 4+(t). That is, Kj(t)
=
jJ
k=0
(bjk(f)Rk(O).
(13)
We model lipid-channel interactions in this theory by assuming:that the conformational state of any channel does not change during .F, since 7 < IO-‘s and T,, N 1O-2 s. Therefore, the average number of open
770
J.
R.
CLAY
AND
channels al any time t. R*(t),
M.
F.
SHLESINGER
is given by ,?(t)R,(t)
with
s(f) = s ,‘I@, t) dr/R
(14)
M
where r is a two dimensional vector in the plane of the membrane and M is the membrane area. The quantity s(t) is the fraction of channels in conducting orientation at time t, with equilibrium value, S, = fl,M/R. Lipid-channel interactions are assumed to arise from spatial and temporal fluctuations of P(r, 0. We stress that we are not hypothesizing interactions between channels, although the language of CS is not unequivocal on this point. A model of fluid motion in nerve membrane in which the lipids are in a single homogeneous fluid phase is oversimplified. A more likely picture of the state of membrane lipids is one of a mosaic of local domains, some of which are fluid, some solid, and some in between (Poste & Allison, 1973). We hypothesize each of the channels to be in a local domain of fluid lipid of a few to several hundred angstroms extent. In this revised model the range of validity of the prediction of l/f noise is not greater than 2-3 frequency decades, which is consistent with noise measurements to data. With the above formulation for membrane current the current noise correlation function is
ii3
C(t) = (~Z+&V2)([i?*(t)-~,*][R*(0)-~R,*])/.~2, where (6N2)*j2 is the r.m.s. deviation of the number through an open channel during ,F (given in CS), and R,* = S,p:R = S&R.
of ions which
pass
(16) That is, we ignore fluctuations in the transit times of individual ions which are probably on a 1 us time scale. These fluctuations will not produce significant low frequency noise. Moreover, a random channel open time will not alter the theory in a significant way, since F is also much smaller ---+ than the time scale of ms fluctuations. If the r.m.s. fluctuation 6F2 IS small compared to F, then the effect of a stochastic F will be to scale C(r) in -equation (15) by a multiplicative factor of (1 +SF2/F2). Noise on a ms time scale arises from the ms interstate kinetics of the channel protein molecule and the range of ms relaxation kinetics of the fluid motion of the lipids. This noise is represented by the bracketed term in equation (15) which is the correlation function of the number of open channels R*(t). Substituting equations (13) and (14) into this part of equation (15) gives .x c(f) = ($W.3) c 4xk(WWV,(O)) -$ i; 4x~(WV?C. (17) 0 0
UNIFIED
THEORY
01
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771
NOISE
Using s(t) = $,+&s(t) and the expressions for (&(O)R,(O)) (k=0,1,2 )...) x) given by Hill & Chen (1972), we find that c(t) = nX,R[S,2 + (~%(t)6s(O))]{[n~
-i-(1 - 12,) exp (- t/z,)lx-
and &,(t)
n”,)j +
+ nz,xR2(&(?)&(0)).
(18)
Since the HH channels are assumed to be uncorrelated, the mean square fluctuation of open channels will be proportional to the total number of channels R. That is, (s/I(r, t)5/3(r, O)} N R. Therefore, c(t) = sfRc,(tj+n~:Rc,,f(t)+c,~f(f)c,(t):
R2
1.
(19)
where c,(t) is the single channel conductance noise correlation function and cr,/(t) is the single channel (in the sensedescribed above) llf correlation function given by CS; Lundstrom & McQueen, 1974) c(t) = S,k,T/@~yK,cl)
J dq exp t -K,g’tlq)/q, ‘I,“,,>
(20)
where q is membrane viscosity, and y arises from a Fourier transform 01‘ /?(r, t), with qlni,, - R-’ and qmaX- /.‘-‘, where i, is the smallest linear dimension of the fluid domain and & is the lipid-lipid spacing. Equation (18) clearly shows that the cross term will be insignificant for very large R. A straightforward calculation of the power spectrum of the first two terms of equation (19) using the Wiener-Khintchine theorem (Wang & Uhlenbeck. 1954) gives s = 4(s,a + I#)?”
q
; 0
p’/jU + W~,,/j*)l- I + + k,7’[u + I;/(s’,R)]/(4nyK,clf),
(21)
and the spectrum of the cross term is (ii,,,, -+ ~0) S = k,T[cr/R +i;/(S,R2j]/(4nydK,f)
1 (;> p’
where
i, = en:,S,RV/(iL),
LI = e*S,Rn2,“/(W),
p = ( I - n,)jn,.
(23) The first term of equation (21) is identical to the result of Hill & Chen (1972) except that R-‘ii is replaced by R-‘Ii+S,a. The S,,, term is the same as in CS except for the HH parameter n”,“. In principle the non-TEA data (- 60 mV < V < - 40 mV) of Fig. 1 may be fitted to the Slif theoretical expression, as was done for the leakage current noise, to obtain an estimate of the ratio il.7 for K channels. However, the narrow voltage range as well 1.11. 51
772
J.
R.
CLAY
AND
M.
F.
SHLESINGER
as the scatter of the data essentially invalidate the exercise. A meaningful fit to this data is possible using the approximation J?(f) = cl; which gives c = 0*7x Io-8 cm2 (Conti et al., 1975). Comparing this result to equation (21), we find that kBT/(4nydK,S,R)
= @7x lo-’
cm2
(24)
or (yK,S,R) = 0.6 dyn/cm’, using d = 75 A, and T = 280” K. Comparison of the above expression for S, to the conductance noise data [ignoring the (tY)-l term] gives R - 60 channels/pm2 (Conti et al., 1975) so that yK,s:, = lo-‘*
dyn.
(25)
The parameters y, K, and S, are unknown except that S, < I, and probably y < I, as well. For example, an r.m.s. deviation of 1” from equilibrium orientation of the lipids gives 17- 4 x IOm5. A typical value of K, for liquid crystals is IOe6 dyn (Stephan & Straley, 1974), so that equation (25) would predict S, = O-25, i.e. each channel in conducting orientation about 25% of the time. However, this calculation is not meant to imply anything more than the fact that apparently reasonable values of y, S, and K, are consistent with equation (25). The I/f aspect of our theory may be appropriate for membranes other than unmyelinated nerves in which significant I/fcurrent noise is observed and the channel density is low. The Paramecium membrane may be one other likely candidate (Moolenaar et al., 1976). Our theory is also consistent with membranes which do not exhibit l/.f current noise, since the passage of ions alone does not produce I/f noise in our model. An interaction between lipid motion and ion channels is the siue qua 17071 of our I/,f theory and lipid-channel interactions of the type described in this paper need not exist in all membrane types. The ultimate test of the model requires detailed understanding of the dynamic state of biological membrane and the nature of lipid-channel interactions. However, further noise measurementsover a wider range of membrane voltage, and noise data for 10 mM < [K& < 500 mM would give more constraints on the range of values of the various parameters of the model, as well as providing another test of its predictions. A report on this work was presented at the 21st annual meeting of the Biophysical Society, New Orleans,Louisiana, 1977. REFERENCES ADELMAN, W. J., PALTI, Y. & SENI;T, J. I’. (1973). J. wwh. ARMSTRONG, C. M. (1971). J. gen. Physiol. 58, 413. CLAY, J. R. StILESINGER, M. F. (1976). Bio&vs. J. Xi, 121.
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CONTI, F., DEFELICE, L. J. & WANKE, E. (1975). J. P/t.ssiol. 248, 45. E~IDIN, M. (1974). Ann. Rev. biophys. Bioeng. 3, 179. FISHMAN, H. M., MOORE, L. E. & POUSSART, D. J. M. (1975). J. tttettzb. Biol. 24, 305. HILL, T. L. & CHEN, Y. D. (1972). Bz’ophys. J. 12, 948. HODGKIN. A. L. & KEYNES. R. D. (1955). J. Phvsiol. 128. 61. LUNDSTR& I. & MCQUEEN, D. (1974). >. theo;. Biol. 45; 405. MACEY, R. I. & OLIVER, R. M. (1967). Biophys. J. 7, 545. MONTROLL, E. W. & SCHER, H. (1973). J. Stat. Phys. 9, 101. MOOLENAAR, W. H., DE GOEDE, J. & VERVEEN, A. A. (1976). Mtture 260, 344. POSTE, G. & ALLISON, A. C. (1973). Biochitrt. biophj,.I. .lcro 300, 121. POUSSART, D. J. M. (1971). Biophys. J. 11, 212. SINGER. S. J. & NICOLSON. G. L. (1972). Science. N. y. 175. 720. STEPHA;, M. J. & STRALE~, J. P. iI 974). Rev. Mod. Plr~~s.46, 617 STEVENS. C. F. (1972). Biophys. J. 12, 1028. USSINC,, H. H. (1949). Ph&ioI. Rev. 29, 127. VERVEEN. A. A. & DEFELICE. L. J. (1974). Prop. Bio~hm. trtolec. Biol. 28. 189. WANG, M. C. & UHLENBECK, G. E. (1954). In Sekrted Papers ott Noise ctttd S~oclxtstic Proce.sse.s.p. 113 (N. Wax, ed.). New York: Dover Publications. WOLF, D. E., SCHLESSINGER, J.. ELSON, E. L... WEBB, W. W.. BI.uMEI\‘TII-~[., K. & HINKAKI P. (1977). Biophys. J. 17, 133a. YEtI, J. Z.. OXFORI>. G. S., Wu, C. H. Rr NARZHASHI. T. (1976). Bioph,w. J. 16, 77.
