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Towards a molecular theory of the nerve membrane: The sufficiency of a single-ion queue
J. theor. Biol. (1976) 58, 477-498
Towards a Molecular Theory of the Nerve Membrane: The Snmcieney of a Single-ion Queuet C. J. GILLESPIE
Medical Biophysics Branch, Whiteshell Nuclear Research Establishment; Atomic Energy of Canada Limited, Pinawa, Manitoba, Canada (Received 2 January 1974, and in revised form 11 October 1975) Some general properties of the nerve membrane are reviewed and the theory of absolute reaction rates is concluded to be an appropriate formalism for ionic currents through the membrane. The instantaneous current-voltage (I-V) relation from a model of the squid axon based on this approach is found to be quasi-linear in physiological conditions. Linearization results from single-fold queueing of K + ions. The independence principle of Hodgkin and Huxley can be derived from absolute reaction rate theory, and will be valid for this model in circumstances where neither queue occupation nor the number of conducting pathways is changed; this condition is violated for changes of internal K + concentration and to a lesser degree, external K + concentration. Examples are computed for peak transient and steady state I-V characteristics of Xenopus node which are compared with experimental data. The GoldmanHodgkin-Katz constant field equation is shown to approximate the I-V relation from the model due to compensatory changes in queue occupation. Such agreement does not imply that assumptions usually made in deriving the equation are valid. The range of potentials in which the approximation is predicted to hold is found to agree with recent studies on Myxicola for several concentrations of external K +. It is suggested that the significance of the variously applied assumptions--instantaneous linearity of the I-V relation, the independence principle, and a constant field electrodiffusion regime---should be re-examined since they may be misleading.
1. Introduction Constant instantaneous conductances for sodium and potassium currents were found by Hodgkin and Huxley (1952a) for the squid axon membrane and used as the basis for their mathematical description (Hodgkin & Huxley, 1952b). Although the technique involved in determining the "instantaneous" -~ This work is part IV of a series on this subject. Part of this work was presented at the 17th Annual Meeting of the Biophysical Society, 1 March 1973. T.II.
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current-voltage relation was complicated and not particularly accurate (Hodgkin & Huxley, 1952a), the "instantaneous" relation was certainly closer to linear than that found at times appreciably later than the stimulus. The subsequent success of the equations based on this foundation has confirmed the utility of this assumption (Cole, 1968). It is important, however, to distinguish between an assumption, or an empirical assertion, of approximate linearity on the one hand, and on the other a principle which requires linearity. Hodgkin & Huxley (1952c) observed this distinction with some care, and indeed their independence principle implies in general a quite nonlinear current-voltage relation, although they applied it only to current ratios at a given potential. In the general case for the squid axon (Noble, 1966) and for other tissues (e.g. Dodge & Frankenhaeuser, 1959) the instantaneous current-voltage relation is not necessarily linear. Thus it is desirable to consider intrinsically nonlinear ion transport mechanisms in the expectation that the appropriate mechanism will, for the squid axon in physiological conditions, be found to generate an approximately linear characteristic. In a previous work (Gillespie, 1973a) a Ifiodel was considered in which the current components were calculated using absolute reaction rate theory, and are thus exponentially dependent on the potential, but are limited by sin#e-fold queueing of ions. The model shows good agreement with voltage clamp data (Gillespie, 1973a) and with a more sensitive property, the temperature dependence of the quasithreshold amplification factor (Gillespie, 1973b), thereby demonstrating that an inherently linear transport mechanism is not required to explain this behaviour (cf., Cole, Guttman & Bezanilla, 1970). In the present work it will be shown that the same model gives rise to approximately linear instantaneous current-voltage characteristics in physiological conditions, and accounts for both the agreement and disagreement of experimental data with the constant field equation (Goldman, 1943; Hodgkin & K~tz, 1949) and the independence principle (Hodgkin & Huxley, 1952c). 2. Computational Methods (A) MODEL
Several modifications have been made to the model specified previously [Gillespie, 1973a, equations (1)-(19)]. This model assumes that ions traverse the membrane via low energy pathways provided by molecules fixed in the membrane. These molecules undergo transitions between conformational states sk (k = 0 to 5), which present different energy barriers 6Hik to the passive transport of ion species i (i = 0 for K +, i = 1 for Na+). The ion
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fluxes F~j k leaving side j (j = 0 outside, j = I inside) were given as Fijk
=
--
Rij SkA exp ( j F V - tSH~k)/RT
(1)
where Rij is a concentration-dependent frequency factor, A is the equivalent collision cross-section of the pathway, F and R have their usual meaning, T is the absolute temperature, and sk is the number of Sk per unit area of membrane. K + ions are assumed to be trapped by a fixed negative charge at the barrier peak .near the external surface, with time constant for K + desorption "rk. . The trap width is assumed to be small compared with the membrane thickness, so the appropriate expression for the inward K + fluxes is Fook = --s'd~k
(2)
where the prime indicates states with the trap occupied. This expression is used here in place of the corresponding terms of equation (1), although quantitative differences are small. For consistency, desorption of K+-con ducting states must be taken to occur in both directions
g'k = --2S;/Zk
(k = 3 to 5).
(3)
Since K + adsorption and desorption are much faster than other processes for states 2 to 5, the corresponding sk and s~ were computed from the equilibrium approximation to save integration time. The original estimate for the "gate-closing" time (z, = 100 nsee; see Gillespie, 1973a) was adjusted to the value determined from the quasithreshold properties (42 nsec; Gillespie, 1973b). The two latter changes are included for consistency with other recent calculations on the model, but do not significantly affect the results or conclusions of this work. All calculations are for a temperature T = 6.8°C unless otherwise stated. (B) SIMULATIONTECHNIQUE The equations were solved numerically using FORSIM, a FORT~N-oriented simulation language (Carver, 1973). Integrations were performed with Gear's variable-order, variable-step algorithm (Gear, 1971), which improves stability and reduces computation time for these extremely stiff equations. The maximum relative error per integration was set at 5 x 10 -7. Instantaneous changes of potential were not tolerated by this system, so potential changes were made exponentially with time constant 100 nsec and "instantaneous" responses were determined 2 gtsec after initiation of the potential step. By examining the time course of the response, the error introduced by this procedure was found to be less than 0.5 ~o of the current change, as compared to an ideally instantaneous step. In general the precision
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of the calculations is approximately an order of magnitude better than that of comparable experimental results (Hodgkin & Huxley, 1952a; Binstock & Goldman, 1971). 3. Choosing a Model
In this section the rationale underlying the choice of model is outlined in order to clarify its relationship with the properties of the instantaneous current-voltage characteristic. We are concerned here only with the basic conduction process; both the gate mechanism and transitions b&ween different conformational states (Gillespie, 1973a) are neglected. (A) BASIC CONSTRAINTS It is clear that the axon membrane presents a particularly difficultproblem for theoretical analysis. Agin (1967) has analysed the reasons for this difficulty in much more detail than required here. Some relevant points from his argument are summarized below. The first step in deriving a theory for a complex system, such as the axon membrane, is to choose a suitable approximation, i.e. one that includes the essential properties but eliminates excessive complication. In the squid axon (and probably all nerve membranes) the two obvious approximations are so inappropriate that they achieve neither of these objectives, a dilemma which can be described in several ways:
(I) We can assume neither that the membrane is thick enough to constitute a phase, nor that it is so thin we may treat it as an interface (with zero thickness). (2) The best estimate of the membrane thickness is of the same order as the best estimate of the Debye length in the membrane. (3) Ions traversing the membrane cannot reasonably be imagined to behave as a smooth charge distribution, nor to have no interaction in the membrane.
The obvious conclusion is that it is necessary to consider the individual interactions in greater detail. That this conclusion eliminates both the most elegant simplifying assumptions and the possibility of obtaining analytical solutions does not detract from its weight. (B) CHOICEOF APPROXIMATIONS
There are strong indications, particularly from studies using tetrodotoxin (e.g. Moore, Narahashi & Shaw, 1967), that transport occurs via a relatively small number of localized pathways; this permits some useful approximations. If the transport paths across the membrane are well separated,
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close to linear (in the geometrical sense) and not branching, the spatial variables are enormously simplified. A simple calculation (e.g. Cole, 1968, p. 534) indicates that such paths must present a significant energy barrier to ion flow, but of course provides no suggestion as to the barrier shape. A further calculation shows that with the expected number and target size of paths (Moore et al., 1967; Villegas, Bruzual & Villegas, 1968), the resting pathway must present a limiting energy barrier of order 5-10 kcal/mol. Extensive Monte CaTlo computations for a variety of multi-barrier configurations showed (unpublished calculations) that more than one ion in the membrane in one path is a negligibly infrequent event for all plausible situations examined. Thus a useful approximation for (like) charge-charge interactions in the membrane may be: any charge traversing a path prevents the entry of other charges (i.e. a queue of one). In view of the small physical dimensions involved, and the identification of ion energy in the membrane as the transport-controlling factor, the theory of absolute reaction rates (Glasstone, Laidler & Eyring, 1941; Parlin & Eyring, 1954) is an appropriate method for computing transport rates. That this method has not been widely utilized in assaults on the present problem may be attributed to the expectation that currents calculated in this manner would necessarily depart too far from linearity. If so, this is an unfortunate and, as will later be evident, erroneous assumption.
4. The 'Instantaneous' Current-Voltage Relation In the first paper of this series (Gillespie, 1973a) linearity of the instantaneous current-voltage relation was accorded no higher significance than other phenomena requiring explanation. We consider here, in terms of the model, the processes responsible for the form of the I - V relation, and whether it is sut~ciently close to linear. Hill & C h e n (1971) have examined the instantaneous I - V properties of a selection of somewhat similar models; all of these, however, differ significantly from that discussed here, particularly with respect to energy barrier shape and inclusion of non-selective conduction. As well, a more specific question is at issue. Is the intrinsic nonlinearity of the present model greater than the maximum nonlinearity consistent with experimental data? The system was subjected to a voltage clamp from rest ( - 75 mV) to 25 mV for 0.45 reset, chosen to give conveniently comparable magnitudes for the sodium and potassium "conductances". At this time the potential was damped to another potential V and the sodium and potassium currents were determined as described above. Several points can be made regarding
482 the
c.J. resulting
current-voltage
GILLESPIE relations,
shown
in
Fig.
l,
and
their
significance. (1) The currents are not as nonlinear as might be expected; with the exception of the sodium current at V > 40 mV, deviations from linearity are about 5 ~o of the current range ['i.e. considerably less than the experimental error of Hodgkin & Huxley (1952a)]. (2) The currents are linearized by K + blockage of transport pathways, or queueing, and by non-selective conduction at high potential ( > 50 mY). (3) The currents could be further linearized by simple modifications: e.g. optimizing the relative energy barriers for non-selective transport, or using a more complex barrier shape (Woodbury, 1969). There appears to be little purpose in pursuing such modifications at this stage. It is sufficient that linearity may be improved within the same basic framework. Although precise measurements are difficult, there are some
2
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I
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o
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o
_
f-
./ -
,,
zgf Z
-2 - I00
- 60
-
20
20
i
60
I
I00
v(mV)
FIG. 1. Instantaneous current-voltage relations computed from the model after damping at 25 mV for 0.45 msec with [K+]l = 0.5 M, [K+]o = 0.02 M, [Na+]t = 0.067 M and [Na+]o -----0.5 M. Filled circles show potassium current, open circles are sodium current and crosses show the total current (without added leakage current).
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indications (e.g. Chandler & Meves, 1965) that nonlinearity may be observed experimentally. While we thus have a definitive measure of neither the intrinsic theoretical nonlinearity, nor the upper limit determined empirically, it appears that the model is not inconsistent with the observation that the instantaneous current-voltage relation for squid axon in normal physiological conditions is approximately linear. It is in satisfying accord with experiments, both in different conditions and with other tissues, that the model suggests this near-linearity to be coincidental, rather than fundamentally significant. Certainly, until the degree of linearity required is further clarified, this criterion should be applied to any model with appropriate caution.
5. Relationship with the Independence Principle Segal (1969) has analysed in detail the applicability of the independence principle (Hodgkin & Huxley, 1952c). He showed that it does not have general validity and that its particular validity for the squid axon can only be discussed in terms of a microscopic model for the mechanism of ion transport. In terms of the present model we argue below that the principle is valid, with certain restrictions, for the squid axon. Hodgkin & Huxley [-1952c, equation (12)] express the independence principle as 1~ (C~/C3 exp ( E , - V ) F / R T - 1 -= (4) It exp (E i - V ) F / R T - 1 where I i and 1[ are the observed total currents (outward positive) due to univalent cation i when the external concentrations are C t and C~ respectively, and Et is the value of the membrane potential V (inside relative to outside) at which It is zero. This equation was derived by considering equilibrium conditions together with the postulate that the inward flux of any ion is proportional to its external concentration, and independent of other ion concentrations (although equivalent relations can be derived in terms of frictional motions of ions in the membrane, see Hodgkin & Huxley, 1952c). The same relation is readily derived, from the theory of absolute reaction rates, for a single thin barrier with its peak at any position. It follows that the model predicts sodium currents in accord with the independence principle provided the population of the Na ÷ selective state is not modified (e.g. by a change in internal K + concentration). On the other hand, from equations (I) and (2), the potassium current through potassium selective states $5 and S~ is given by
It~ = k.ssCt exp (VF/RT)-s~/rk
(5)
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C. J. G I L L E S P I E I
I
I
I
I
I
I
I
I
t
i
I
I
i
I
6 N
2
"6
g~ z
I
-I00
-60
-20
20
I
60
I00
v[mV)
Fro. 2. Change in queue occupation as a function of potential for the conditions of Fig. 1. Open circles are vacant K+-conductive sites (sD and filled circles are sites blocked by K+(sg); the sum s5 + s~ is almost constant since values were determined 2 Inset after the second potential step, before significant redistribution between different conformal states could occur.
where C~ is the internal concentration of K + (approximating the ion activity) and the constant/co includes the frequency factor. Trapping of K + ions is described by equation (3) and by ss = - kos5[Ci exp (VF]R T) + Co]
(6)
where Co is the external K + concentration, so that the equilibrium approximation [section 2(A), above] is ss/st = [1 -F½kaTk(Ci exp ( V F / R T ) + Co)] - i
(7)
with s, = ss + s's.
Thus, equation (5) becomes I K = ½k, s,[C, exp ( V F ] R T ) - Co]][I +½1¢o~k(C, exp ( V F / R T ) + Co)]
(8)
from which we find that, for a change to external concentration Co, the
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current ratio I~/l K has the form given by equation (4) multiplied by a factor s;[1 +½ko*k(C, exp (VF/RT) + Co)] m = s,[1 +½k,o'ck(Ci exp (VF/RT) + C')]
(9)
which represents the expected deviations from the independence principle. These arise from changes in both the density of potassium selective states and the fraction of thes.e blocked by K +. For conditions where these effects are small, such as moderate changes in external K + concentrations, both instantaneous and steady state K + currents should closely follow the independence principle.
0 E E
-75
-50
-25
0
25
v(mV)
FIG. 3. Steady state K + currents computed from the model for [K+]o = 2.5 m M (solid points and line), 12.5 mM, 62-5 m M and 114.5 m M (successively lower sets of points;
current scale offset) compared with the predictions obtained for the latter three concentrations (lines) by applying the independence principle to the values computed for [K+]o = 2.5 raM; cf., Frankenhaeuser (1962), Fig. 2.
T.B.
32
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GILLESPIE
The predicted deviations may be compared with measurements of steady state K + current by Frankenhaeuser (1962). Since these data were obtained from the myelinated fibre of Xenopus laevis the calculation was performed with an increased value of the "gate closing" time (~, = 400 nsec). This modification is not essential to demonstrate the point but shifts the HodgkinHuxley parameters to more negative potentials corresponding to the properties of the Xenopus node. This change may be shown to correspond with the reduced external [Ca ++ ] characteristic of this tissue compared with squid axon. Steady state K + currents were computed after 20 msec depolarizations from - 7 5 mV, for IK+]i = 120 mM with [K+]o = 2.5, 12.5, 62.5 and 114.5 mM (Fig. 3). The computed results for [K+]o = 2.5 mM were utilized to predict the current-voltage relation for the other three concentrations using the independence principle. As expected, changes in [K+]o produce small deviations from the independence principle. Similar deviations are found experimentally [Fig. 2 of Frankenhaeuser (1962)].
0.4
/ 0-2
/
/
~ V(mV)
E" E
.~
150
5O
o.o
E
,_.o /
k I
-0.2
-0.4
I
t -50
I 0
I 50
I I I00
V(mV)
FIG. 4. Peak transient Na + currents computed from the model for [Na+]o -~ 120-0 mM (lower line) and 39.9 mM (points) compared with the prediction obtained from the latter concentration by applying the independence principle to the values computed for the former case; cf., Dodge & Frankenhaeuser (1959), Fig. 5.
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By contrast, computed peak transient Na + currents (Fig. 4) with [Na+]o = 120 mM and 39.9 mM indicate that the results of changes in [Na+]o are predicted to high precision by the independence principle as expected, and in agreement with the corresponding experimental measurements of Dodge & Frankenhaeuser (1959, Fig. 5).
6. Relationship with the Constant Field Equation The constant field equation (Goldman, 1943; Hodgkin & Katz, 1949) may be derived from the properties of an electrodiffusion regime. That is, the assumptions inherent in it (other than the assumption of constant electric field) are precisely those which were earlier found unlikely to yield useful results for a system such as the squid axon membrane. The same assumptions have been questioned by Cole (1968) on the more empirical grounds that they apparently cannot generate some basic properties of the axon membrane and by others (see Schwartz, 1971) who find the constraints implausible for cell membranes. Nonetheless, the constant field equation was found to predict well the transmembrane potential, and less accurately the current-voltage relation in the squid axon (Goldman, 1943) and has been applied extensively to other systems, notably the frog node (Dodge & Frankenhaeuser, 1959) in which distinctly nonlinear instantaneous 1-V characteristics are inferred. The question arises: if we have reasoned correctly in showing that the assumptions used in deriving the equation do not apply to the nerve membrane, why does the equation predict successfully some of the observed properties ? We suggest below that the observed agreement may result from an approximation, valid over a limited" range. For this purpose, it is fruitful to examine the numerical relationship between the Goldman-Hodgkin-Katz equation and the rate theory expression. The former can be written (Hodgkin & Katz, 1949) for the potassium current (K subscript, eliminated for clarity in the concentration terms; the C superscript denotes the constant field expression) : ic = p F2V Co - C, exp (VF/RT) RT 1 - e x p (VF/RT)
(10)
where PK is a permeability coefficient for K +, Co and C i are the outside and inside K + concentrations, and the usual sign convention for current and potential has been used. From equation (8) we obtain the ratio IC/IK = kb V{1 +½ka'ck[Ci exp (VF/RT) + Co]}/[exp ( V F / R T ) - 1]
(11)
488
C.J.
GILLESPIE
which should be constant as a function of V if the two expressions are equivalent. The value of the constant kb is inconsequential, since there is an arbitrary constant (Pr,) involved in using the constant field equation in any case, but to evaluate the constancy of the ratio, the value of/ca is required. This can be conveniently obtained from equation (7) together with data of the type presented in Fig. 2 since
½k.z,[Cl exp (VF/RT) + Co] = 1 (12) where Vt is the p o t e n t i a l a t w h i c h ss/(s5 +s'5) = 0.5, o r - 1 4 m V f o r t h e c o n d i t i o n s o f Fig. 2. T h e resulting value, ko = 6.7 x 10 8 sec - t , c o u l d also b e d e r i v e d f r o m first principles, b u t t h e m e t h o d u s e d here is simple a n d clarifies the role o f t h e q u e u e i n g process. T h e r a t i o f r o m e q u a t i o n (11) is p l o t t e d in Fig. 5 as a f u n c t i o n o f p o t e n t i a l f o r T = I ° C a n d several c o m b i n a t i o n s o f i n t e r n a l a n d e x t e r n a l K + c o n c e n t r a t i o n . I n each case t h e r a t i o has a shallow m i n i m u m a t s o m e p o t e n t i a l
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o
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t
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I 0
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v(mV)
Fro. 5. The ratio of the constant field equation and model expressions for instantaneous K + current as a function of potential for 310 mM internal K + and three external K + concentrations: triangles, 10 raM; diamonds, 50 mM; and squares, 440 mM. In each case the arrows indicate the range of potential for which the ratio is within 5 ~ of the minimum value, and the circle on the connecting line shows the potential at which both currents are zero. The connecting line thus indicates the approximate range of potential within which the two expressions could be superimposed to better than experimental precision.
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close to the potential for zero current V o = (aT/F) In (Co~C,) (13) which is marked with a circle, together with arrows indicating the potential extremes where the ratio is 5~o greater than the minimum value. Thus, within the range defined by the arrows, a "permeability" PK can be defined so as to render the two expressions identical to better than experimental precision. This range covers approximately 40 mV in each case, but where Vo falls outside the range (as it does for physiological conditions) the requirement that both expressions pass through zero current at Vo effectively extends the region of apparent agreement. It is interesting to observe that the "queueing" term from equation (7) is intimately involved in determining the existence and positions of the minima in Fig. 5; that is, the change in queue occupation is necessary for approximate agreement with the constant field equation. The predicted range of agreement can be tested by comparing calculations from the present model with the constant field equation and with the experimental data of Binstock & Goldman (1971). Their data incorporate particularly careful correction for time dependent components of the "instantaneous"
1"6
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ooOQe o ° a ° °Qe°o oo
--
0.8
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0.0
0-8
I -I00 ! -50
I
0
I
I 50
I
I00
v(mV)
FIG.6. InstantaneousK + current-voltagerelationfor [K.+]= = 310 mM, [K+]o = 10 mM computed from the model (points) and fitted line from the constant field equation (5); cf., Binstock & Goldman (1971), Fig. 3B.
490
GILLESPIE
c.J.
current measured in Myxicola giant axons, a system whose electrical properties are similar to those of the squid giant axon (Binstock & Goldman, 1969). The comparison is made (Figs. 6-8) for [K+]~ = 0.31 M with [K+], = 0.02 M, 0.1 M and 0.44 M, which are the concentrations used by Binstoek & Goldman (1971). The constant field equation (lines) is in excellent agreement with the prediction of the present model (points) in the regions anticipated from Fig. 5 which cover the ranges of potential in which experi-
1"6
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T 0 VImV
I
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I 50
i I0 0
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FIG. 7. Instantaneous K + current-voltage relation for [K+]~ = 310 m M , [K+]o = 50 m M c o m p u t e d f r o m the model (points) a n d fitted line from equation (5); cf., Binstock & G o l d m a n (1971), Fig. 3B.
mental agreement was found (arrows, Figs. 6-8). The only deviation apparent in the data of Binstock & Goldman is matched in the model ( I / < - 5 0 mV, Fig. 7). As suggested by the upward curvature in Fig. 5, the current derived from the constant field equation is greater in magnitude than that calculated from the model at potentials both higher and lower than the range of agreement. Deviations of this kind from the constant field equation are thus a prediction of the model.
S U F F I C I E N C Y OF A S I N G L E - I O N 1-6
I
i
I
I
I
I
491
QUEUE
'//.
,
,
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E E 0-0
-0.8
gl O0000 o II'°'II qlql° /° ~ o, I I iel -I00
-50
I
0
,50
IgO
V(m V)
FIG. 8. Instantaneous K + current-voltage relation for [K+], = 310 raM, [K+]o = 440 mM computed from the model (points) and fitted line from equation (5); cf., Binstock & Goldman (1971), Fig. 3A. 7. Further Comments on Eiectrodiffusion Models
It would appear from the foregoing that there is no basis for interpreting the observed agreement with the constant field equation in a limited range of potential as indicating approximate validity of the constant field or electrodiffusion regime assumptions. It is, however, instructive to consider the "single ion queue" model as one extreme of a class of models, with electrodiffusion models at the other extreme. Any electrodiffusion model assumes that it is meaningful to consider the concentration of an ion as a continuous function of position in the membrane. If localized transport pathways are involved, this assumption implies at least that a large number of ions are associated with each path at any instant. This requirement may be relaxed, while still retaining the diffusion equations appropriate to a continuous charge distribution, to approximate the properties of paths with only a small number of binding sites, e.g. the diffusion model investigated by Roy (1971), but the validity of this treatment is questionable (Fishman & Volkenstein, 1973) on grounds which reduce to the dilemma discussed in section 3(A) above. The present
492
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model can be regarded as a limiting case in which each transport path can bind only one ion; the single binding site is either empty or saturated, and no basis for integrating the electrodiffusion equation can be found. Nonetheless, once the fluxes have been computed by other means, the essential role which changing queue occupation plays in generating constant-field behaviour (see previous section) may perhaps be regarded as a primitive analogue of the changing ion concentration profile in the membrane portrayed by eleetrodiffusion theory, though not a sufficiently accurate analogue for the latter to be a good approximation. In these terms we can summarize the conclusion of the previous section with a different perspective. Rectification of the constant-field type may be taken to imply interactions of mobile charges in the membrane; but quantitative agreement is obtained with the Myxicola data by assuming that two or more ions may not simultaneously occupy a single transport path. More complex assumptions are apparently not necessary, nor, following the general discussion in section 3, are they desirable (see also the Appendix for a discussion of the implications of flux ratio measurements). Since the concept of ion concentration in the membrane is no~ definable in such a milieu, it follows as a corollary that permeability coefficients derived from fitting the constant field equation to the data cannot be assigned their usual meaning (cf., Goldman & Binstock, 1969). It may be noted that the same arguments can be applied to sites whose availability follows the kinetics of the m, h and n parameters of the HodgkinHuxley axon, but whose ability to transport ions is calculated from absolute reaction rate theory with single-fold queueing.
8. Implications for the Transient and Steady-state Current- voltage Relations It is readily shown that the form of the Hodgkin-Huxley equations requires that the peak transient and steady-state current voltage relations be concave upwards for V > 0 (i.e. the gradient aI/dV is a monotonically non-decreasing function of V in this range). Experimental I-V characteristics, by contrast, often display decreasing gradients in the range V > 50 mV. Numerous examples have been published for a variety of tissues; those for the squid axon in near-physiological conditions include Narahashi, Anderson & Moore (1967), Armstrong & Binstock (1964), Taylor (1959), and Narahashi & Anderson (1967). Characteristics of this type may be simulated by a modified form of the HodgkinHuxley equations, but are quite compatible with the present model, which may show curvature in either direction, or both (Gillespie, 1973a).
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493
There has in the past been little motive to examine in detail minor nonlinearities of the I - V relation. Such data may, however, provide the most accessible source of information regarding energy barriers such as those postulated in the model.
9. Summary The proposed model has been shown to generate quasilinear instantaneous current-voltage characteristics under physiological conditions, despite the use of reaction rate theory which introduces into the expressions for the currents an exponential factor varying by more than three orders of magnitude in a 200 mV range. This linearization results from the queuing effect, together with nonselective conduction at large depolarizations. The independence principle is approximated by the present model in many circumstances. In general, it should apply accurately for changes of external Na + concentration, approximately for changes of external K ÷ concentration, and should be subject to severe errors for changes of internal K + concentration. Discrepancies arise from changes in the degree to which queueing blocks the transport path, and from redistribution of the state populations. Redistribution may give rise to large deviations from the independence principle; an extreme example is the effect on the transient current of reduced internal K ÷ (Chandler, Hodgkin & Meves, 1965). As found experimentally, instantaneous K + currents computed from the model are well-fitted by the constant field equation over a range of potential, which depends on concentration. In this case, the change in population due to queueing provides a correction factor which results in approximate numerical agreement between the constant field and model expressions. In other words, approximate agreement with the constant field equation results from the same process which restricts the validity of the independence principle. While some basis has been found for comparing this mechanism with electrodiffusion, it should be clear that conformity to the constant field equation due to such a tenuous relationship does not constitute a verification of any assumptions on which the equation is usually based (see, e.g. Barr, 1965). In particular, it does not establish that the axon membrane can be treated as an electrodiffusion regime. In overview, the model provides a conceptual framework within which the linearity of the instantaneous 1 - V relation and both the successes and limitations of the independence principle and the constant field equation in predicting axon membrane 1 - V relations can be understood. The model rests on assumptions which are different from those used in the above approaches and given apparent verification by their success. It is suggested here that the
494
c.J.
GILLESPIE
verification should be carefully questioned, particularly in view of the influence these concepts may have had in restricting the choice of models. The universality of absolute reaction rate theory and its suitability for biological transport problems in general have been noted by Eyring & Urry (1965); the reasons for its selection as the most appropriate instrument for the particular problems of the axon membrane have been outlined above. These expectations are supported by the ability of the model to comprehend both the Hodgkin-Huxley axon and phenomena which lie outside its purview, and notably the three variously applied current-voltage relationships examined here, which have for too long given the appearance of pointing in other directions. The author is particularly indebted to Drs N. F. Clinch, T. L. Hill and G. Roy for helpful criticisms and suggestions.
APPENDIX A
Determination of Queue Properties from Tracer Flux Ratios Ussing (1949) has considered the dependence of the inward and outward ion flux components (M i and Me) on transmembrane potential V. For independent passive transport he derived an equation which Hodgkin & Keynes (1955) expressed for K ÷ ions as M i / M o = exp [-(V- EK)F/R T]
(A 1)
where EK is the equilibrium potential (Mi = Me) for K ÷. Thus the gradient of In (Mi/Mo) v. V should be F / R T ,~, 40 V- ~ if ions cross the membrane passively and independently. However, in squid axon poisoned to eliminate active transport, Hodgkin & Keynes (1955) found this value to be 100 V -~. They concluded that the inward and outward K + fluxes were not independent. Since exchange diffusion can only reduce the gradient, they suggested a competition mechanism with ions moving in single file through a narrow channel. From a simple calculation they showed that single file queueing should give rise to a gradient n-fold greater than that predicted by equation (A1), where n is the number of ions in the queue, and concluded that in squid axon the channels contain 2-3 K + ions. While the main thrust of this conclusion supports the arguments of section 3A, and certainly does not support an electrodiffusion formalism, it may be objected that the numerical discrepancy, between this estimate and the value assumed above for queue occupancy, is severe. The discrepancy is,
SUFFICIENCY OF A SINGLE-ION QUEUE
495
however, only apparent, since the two numbers do not represent the same property. To elucidate this problem we consider the simple system shown schematically in Fig. 9. Exchange transport between compartments at each end of the transmembrane transport path and the adjacent solutions is assumed to be
Inside
Membrene
Outside
1
i i
P
FIo " ~ ' ~ ~I
'
I I i i I
1 1 |
F2o
•
F3o
I i
Q
, , . t ~ l - - F3 i F2 i
_o.j
T
L___
I I
FIG. 9. Diagram of hypothetical transport path, with restricted exchange between terminal compartments (1 and O) and the adjacent solution. The inside and outside solutions contain isotopes x and y respectively. constant and additive with the net (transmembrane) fluxes. Within and outside the compartments the" solutions are well stirred, and the compartment volume is sufficient to contain several ions. The ratio n of the outward flux (Fao) to the inward flux (F2i) across the membrane barrier is assumed to obey the Ussing relation
F2o/F2~ = exp ( V - E e q ) F / R T =-. n
(A2)
which holds for the single-ion queue. Different isotopes (x and y) of the same ion in the inside and outside solutions contribute to the fluxes in proportion to their concentrations in the compartments. I f the internal and external exchange fluxes are Fez and Feo, we have
Fai = F2i+ F~o F3o = F2o + F~o Fii = F2~+Fei Flo = F 2 o + F,,i
(A3)
496
c . J . GILLESPIE
The fraction of y in 0 and x in I are then given by f and g respectively, where f = ( 1 - p)/(1-pq) q = (l -- q)/(1 -- pq)
(A4)
and p = F2o/(n + 1)F2z+ Feo q = F2z/(n + 1)F2z+ Fei
(A5)
The net isotopic flux ratio is then r =- F x / F , - q ( 1 - p ) F2o+Feo p(1 - q) F2i + Fez
(A6)
which in the case of zero exchange, Fee = Fez = O, reduces to r = n 3 = exp 3 ( V - E o q ) F / R T
(A7)
with gradient a(ln r ) / O V = 3 F / R T ~- 120 mV -I. In the general case the semilog plot is not linear except with isosmolar K +. However, with F2, and F2o computed from the model, and exchange fluxes of the order of the transmembrane fluxes at equilibrium, equation (A6) describes the data of Hodgkin & Keynes (1955, Fig. 7) at least as well as the straight line they proposed (Fig. 10). Derivation of a gradient greater than F [ R T depends here upon the assumption that a considerable fraction of the solution-compartment exchange is coupled to the transmembrane fluxes. This implies compartments which accommodate a fixed number of ions, or more rigorously, a large free energy for any increase or decrease in compartment occupancy. This can obtain only if the compartment is small, certainly smaller than the Frankenhaeuser-Hodgkin space (Frankenhaeuser & Hodgkin, 1956; Binstock & Goldman, 1971), otherwise a single ion would provide only a minor perturbation. It follows that the treatment above in terms of ion "concentration" in the compartments is not rigorous, but it may suffice to illustrate the point at issue. Of particular interest is the simple case with one ion in a potential well adjacent to the membrane at each end of the transport path, and a single-fold queue within the membrane. This corresponds closely to the suggestion of Hodgkin & Keynes (with tracer fluxes determined by a queue occupancy of three) but the electrical currents, which do not reveal the number of times a particular ion may traverse the membrane, are limited by the intramembrane single-fold queue. The above argument does not diminish the conclusion that inward and outward fluxes are coupled, though the coupling need not constitute a
SUFFICIENCY OF A SINGLE-ION QUEUE
497
x
V(mV) I
I
-~o
I
50
I
//
/Y -I
(
//~b~ /
-Z
FiG. I0. Semilog plot of flux ratio o. membrane potential relative to the equiJibrium potential: (a) computed from the model for [K+]t -----500 raM, [K+], = 20 m M and exchange fluxes Fa ----F,o = F2f at V = Eeq (see text); (b) line fitted to data by Hodgkin &
Keynes (1955). strict single-file queue. However, the coupling o f flows m a y occur at the interface(s), a non-trivial distinction in the present context since, in a configuration o f this kind, conclusions based on tracer studies are not necessarily pertinent to the electrical properties o f the membrane. REFERENCES Aoxr¢, D. (1967). Proe. hath. Acad. Sci. U.S.A. 57, 1232. ~ONG, C. M. • BI]qSTOCK,L. (1964). J. gen. Physiol. 48, 265. BAI~, L. (1965). J. theor. Biol. 9, 351. BINSa'OCK,L. & GOLDMAN,L. (1969). J. gen. Physiol. 54, 730. BINSTOCK,L. & GOLDMAN,L. (1971). J. Physiol. 217, 517. CARVER, M. B. (1973). Atomic Energy of Canada Limited, Report AECL-4311. ~LER, W. K., HOr)OrdN, A. L. & MEv~, H. (1965). J. Physiol. 180, 821. CrIArOLER, W. K. & MEre, H. (1965). J. Physiol. 180, 788. COLE, K. S. (I968). Membranes, Ions and Impulses. Los Angeles: University of California Press.
498
C.J.
GILLESPIE
COLE, K. S., GUTr~,N, R. & BEZANILLA,F. (1970). Proc. hath. Acad. ScL U.S.A. 65, 884. DODGE, F. A. & Fn-~',IKENH~USER,B. (1959). 3". PhysioL 148, 188. EYnn~o, H. & URRY, D. W. (1965). In Theoretical and Mathematical Biology (T. H. Waterman & H. J. Morowitz, eds) p. 55. New York: Blaisdell. b-'mHM~, S. N. & VOI.~ENSTr~, M. V. (1973). 3". Mem. Biol. 12, 189. ~ u s ~ , B. (1962). J. PhysioL 160, 40. ~ u s ~ , B. & HODGKn~, A. L. (1956). J. Physiol. 131, 341. GF.~n, C. W. (1971). Comm. A C M 14, 185. GnTF.~pm, C. J. (1973a). J. theor. BioL 40, 409. Ona.mPm, C. J. (1973b). J. theor. BioL 42, 519. GI~rONB, S., LAIDLER,K. J. & EYaI~G, H. (1941). The Theory of Rate Processes. New York: McGraw-Hill. G o t a ~ , D. E. (1943). J. gen. PhysioL 27, 37. GOLDMAN, L. & BrNsrocK, L. (1969). J. gen. PhysioL 54, 741. I-In.~, T. L. & C~N, Y. (1971). Biophys. I. 11, 685. HODGKn~, A. L. & Hox~Y, A. F. (1952a). J. PhysioL 116, 473. HODGK~, A. L. & HOXLBY, A. F. (1952b). J. Physiol. 117, 500. HODGK~, A. L. & I-Iux~y, A. F. (1952c). J. PhysioL 116, 449. HODGKn% A. L. & KATZ, B. (1949). J. PhysioL 108, 37. HOO~mN, A. L. & KsYI~s, R. D. (1955). J. PhysioL 128, 61. Moom~, J. W., N ~ m , T. & SHAW, T. I. (1967). J. PhysioL 188, 99. N~at~aAsm, T. & Ar~m~or~, N. C. (1957). Toxic. appL Pharmac. 10, 529. NAR~a.~HX, T., ANDVakSOr~,N. C. & MOORE, J. W. (1967). J. gen. PhysioL 50, 1413. NOBLE, D. (1966). Physiol. Rev. 46, 1. P~a.x~, R. B. & EvmNo, H. (1954). In Ion Transport Across Membranes. (H. T. Clarke, ed.) p. 103. New York: Academic Press. RoY, G. (1971). J. tool. Biol. 6, 329. SCHW~d~TZ, T. L. (1971). In Biophysics and Physiology o f Excitable Membranes. (W. J. Adelman, ¢d.) p. 47. New York: Van Nostrand Reinhold. SFx~.~, J. R. (1969). 3". theor. Biol. 24, 159. TA~LOR, R. E. (1959). Am. J. PhysioL 196, 1071. Ussn~G, H. H. (1949). Acta PhysioL Scand. 19, 43. VnJ~OAS, R., BRUZU~, I. B. & V n . x ~ , G. M. (1968). J. gen. PhysioL 51, 81. WOODntrsY, J. W. (1969). Biophys. J'. 9, abstracts A250.
Towards a Molecular Theory of the Nerve Membrane: The Snmcieney of a Single-ion Queuet C. J. GILLESPIE
Medical Biophysics Branch, Whiteshell Nuclear Research Establishment; Atomic Energy of Canada Limited, Pinawa, Manitoba, Canada (Received 2 January 1974, and in revised form 11 October 1975) Some general properties of the nerve membrane are reviewed and the theory of absolute reaction rates is concluded to be an appropriate formalism for ionic currents through the membrane. The instantaneous current-voltage (I-V) relation from a model of the squid axon based on this approach is found to be quasi-linear in physiological conditions. Linearization results from single-fold queueing of K + ions. The independence principle of Hodgkin and Huxley can be derived from absolute reaction rate theory, and will be valid for this model in circumstances where neither queue occupation nor the number of conducting pathways is changed; this condition is violated for changes of internal K + concentration and to a lesser degree, external K + concentration. Examples are computed for peak transient and steady state I-V characteristics of Xenopus node which are compared with experimental data. The GoldmanHodgkin-Katz constant field equation is shown to approximate the I-V relation from the model due to compensatory changes in queue occupation. Such agreement does not imply that assumptions usually made in deriving the equation are valid. The range of potentials in which the approximation is predicted to hold is found to agree with recent studies on Myxicola for several concentrations of external K +. It is suggested that the significance of the variously applied assumptions--instantaneous linearity of the I-V relation, the independence principle, and a constant field electrodiffusion regime---should be re-examined since they may be misleading.
1. Introduction Constant instantaneous conductances for sodium and potassium currents were found by Hodgkin and Huxley (1952a) for the squid axon membrane and used as the basis for their mathematical description (Hodgkin & Huxley, 1952b). Although the technique involved in determining the "instantaneous" -~ This work is part IV of a series on this subject. Part of this work was presented at the 17th Annual Meeting of the Biophysical Society, 1 March 1973. T.II.
477
31
478
C. J. GILLESPIE
current-voltage relation was complicated and not particularly accurate (Hodgkin & Huxley, 1952a), the "instantaneous" relation was certainly closer to linear than that found at times appreciably later than the stimulus. The subsequent success of the equations based on this foundation has confirmed the utility of this assumption (Cole, 1968). It is important, however, to distinguish between an assumption, or an empirical assertion, of approximate linearity on the one hand, and on the other a principle which requires linearity. Hodgkin & Huxley (1952c) observed this distinction with some care, and indeed their independence principle implies in general a quite nonlinear current-voltage relation, although they applied it only to current ratios at a given potential. In the general case for the squid axon (Noble, 1966) and for other tissues (e.g. Dodge & Frankenhaeuser, 1959) the instantaneous current-voltage relation is not necessarily linear. Thus it is desirable to consider intrinsically nonlinear ion transport mechanisms in the expectation that the appropriate mechanism will, for the squid axon in physiological conditions, be found to generate an approximately linear characteristic. In a previous work (Gillespie, 1973a) a Ifiodel was considered in which the current components were calculated using absolute reaction rate theory, and are thus exponentially dependent on the potential, but are limited by sin#e-fold queueing of ions. The model shows good agreement with voltage clamp data (Gillespie, 1973a) and with a more sensitive property, the temperature dependence of the quasithreshold amplification factor (Gillespie, 1973b), thereby demonstrating that an inherently linear transport mechanism is not required to explain this behaviour (cf., Cole, Guttman & Bezanilla, 1970). In the present work it will be shown that the same model gives rise to approximately linear instantaneous current-voltage characteristics in physiological conditions, and accounts for both the agreement and disagreement of experimental data with the constant field equation (Goldman, 1943; Hodgkin & K~tz, 1949) and the independence principle (Hodgkin & Huxley, 1952c). 2. Computational Methods (A) MODEL
Several modifications have been made to the model specified previously [Gillespie, 1973a, equations (1)-(19)]. This model assumes that ions traverse the membrane via low energy pathways provided by molecules fixed in the membrane. These molecules undergo transitions between conformational states sk (k = 0 to 5), which present different energy barriers 6Hik to the passive transport of ion species i (i = 0 for K +, i = 1 for Na+). The ion
SUFFICIENCY
OF A S I N G L E - I O N
QUEUE
479
fluxes F~j k leaving side j (j = 0 outside, j = I inside) were given as Fijk
=
--
Rij SkA exp ( j F V - tSH~k)/RT
(1)
where Rij is a concentration-dependent frequency factor, A is the equivalent collision cross-section of the pathway, F and R have their usual meaning, T is the absolute temperature, and sk is the number of Sk per unit area of membrane. K + ions are assumed to be trapped by a fixed negative charge at the barrier peak .near the external surface, with time constant for K + desorption "rk. . The trap width is assumed to be small compared with the membrane thickness, so the appropriate expression for the inward K + fluxes is Fook = --s'd~k
(2)
where the prime indicates states with the trap occupied. This expression is used here in place of the corresponding terms of equation (1), although quantitative differences are small. For consistency, desorption of K+-con ducting states must be taken to occur in both directions
g'k = --2S;/Zk
(k = 3 to 5).
(3)
Since K + adsorption and desorption are much faster than other processes for states 2 to 5, the corresponding sk and s~ were computed from the equilibrium approximation to save integration time. The original estimate for the "gate-closing" time (z, = 100 nsee; see Gillespie, 1973a) was adjusted to the value determined from the quasithreshold properties (42 nsec; Gillespie, 1973b). The two latter changes are included for consistency with other recent calculations on the model, but do not significantly affect the results or conclusions of this work. All calculations are for a temperature T = 6.8°C unless otherwise stated. (B) SIMULATIONTECHNIQUE The equations were solved numerically using FORSIM, a FORT~N-oriented simulation language (Carver, 1973). Integrations were performed with Gear's variable-order, variable-step algorithm (Gear, 1971), which improves stability and reduces computation time for these extremely stiff equations. The maximum relative error per integration was set at 5 x 10 -7. Instantaneous changes of potential were not tolerated by this system, so potential changes were made exponentially with time constant 100 nsec and "instantaneous" responses were determined 2 gtsec after initiation of the potential step. By examining the time course of the response, the error introduced by this procedure was found to be less than 0.5 ~o of the current change, as compared to an ideally instantaneous step. In general the precision
480
c.J.
GILLESPIE
of the calculations is approximately an order of magnitude better than that of comparable experimental results (Hodgkin & Huxley, 1952a; Binstock & Goldman, 1971). 3. Choosing a Model
In this section the rationale underlying the choice of model is outlined in order to clarify its relationship with the properties of the instantaneous current-voltage characteristic. We are concerned here only with the basic conduction process; both the gate mechanism and transitions b&ween different conformational states (Gillespie, 1973a) are neglected. (A) BASIC CONSTRAINTS It is clear that the axon membrane presents a particularly difficultproblem for theoretical analysis. Agin (1967) has analysed the reasons for this difficulty in much more detail than required here. Some relevant points from his argument are summarized below. The first step in deriving a theory for a complex system, such as the axon membrane, is to choose a suitable approximation, i.e. one that includes the essential properties but eliminates excessive complication. In the squid axon (and probably all nerve membranes) the two obvious approximations are so inappropriate that they achieve neither of these objectives, a dilemma which can be described in several ways:
(I) We can assume neither that the membrane is thick enough to constitute a phase, nor that it is so thin we may treat it as an interface (with zero thickness). (2) The best estimate of the membrane thickness is of the same order as the best estimate of the Debye length in the membrane. (3) Ions traversing the membrane cannot reasonably be imagined to behave as a smooth charge distribution, nor to have no interaction in the membrane.
The obvious conclusion is that it is necessary to consider the individual interactions in greater detail. That this conclusion eliminates both the most elegant simplifying assumptions and the possibility of obtaining analytical solutions does not detract from its weight. (B) CHOICEOF APPROXIMATIONS
There are strong indications, particularly from studies using tetrodotoxin (e.g. Moore, Narahashi & Shaw, 1967), that transport occurs via a relatively small number of localized pathways; this permits some useful approximations. If the transport paths across the membrane are well separated,
SUFFICIENCY
OF A S I N G L E - I O N
QUEUE
481
close to linear (in the geometrical sense) and not branching, the spatial variables are enormously simplified. A simple calculation (e.g. Cole, 1968, p. 534) indicates that such paths must present a significant energy barrier to ion flow, but of course provides no suggestion as to the barrier shape. A further calculation shows that with the expected number and target size of paths (Moore et al., 1967; Villegas, Bruzual & Villegas, 1968), the resting pathway must present a limiting energy barrier of order 5-10 kcal/mol. Extensive Monte CaTlo computations for a variety of multi-barrier configurations showed (unpublished calculations) that more than one ion in the membrane in one path is a negligibly infrequent event for all plausible situations examined. Thus a useful approximation for (like) charge-charge interactions in the membrane may be: any charge traversing a path prevents the entry of other charges (i.e. a queue of one). In view of the small physical dimensions involved, and the identification of ion energy in the membrane as the transport-controlling factor, the theory of absolute reaction rates (Glasstone, Laidler & Eyring, 1941; Parlin & Eyring, 1954) is an appropriate method for computing transport rates. That this method has not been widely utilized in assaults on the present problem may be attributed to the expectation that currents calculated in this manner would necessarily depart too far from linearity. If so, this is an unfortunate and, as will later be evident, erroneous assumption.
4. The 'Instantaneous' Current-Voltage Relation In the first paper of this series (Gillespie, 1973a) linearity of the instantaneous current-voltage relation was accorded no higher significance than other phenomena requiring explanation. We consider here, in terms of the model, the processes responsible for the form of the I - V relation, and whether it is sut~ciently close to linear. Hill & C h e n (1971) have examined the instantaneous I - V properties of a selection of somewhat similar models; all of these, however, differ significantly from that discussed here, particularly with respect to energy barrier shape and inclusion of non-selective conduction. As well, a more specific question is at issue. Is the intrinsic nonlinearity of the present model greater than the maximum nonlinearity consistent with experimental data? The system was subjected to a voltage clamp from rest ( - 75 mV) to 25 mV for 0.45 reset, chosen to give conveniently comparable magnitudes for the sodium and potassium "conductances". At this time the potential was damped to another potential V and the sodium and potassium currents were determined as described above. Several points can be made regarding
482 the
c.J. resulting
current-voltage
GILLESPIE relations,
shown
in
Fig.
l,
and
their
significance. (1) The currents are not as nonlinear as might be expected; with the exception of the sodium current at V > 40 mV, deviations from linearity are about 5 ~o of the current range ['i.e. considerably less than the experimental error of Hodgkin & Huxley (1952a)]. (2) The currents are linearized by K + blockage of transport pathways, or queueing, and by non-selective conduction at high potential ( > 50 mY). (3) The currents could be further linearized by simple modifications: e.g. optimizing the relative energy barriers for non-selective transport, or using a more complex barrier shape (Woodbury, 1969). There appears to be little purpose in pursuing such modifications at this stage. It is sufficient that linearity may be improved within the same basic framework. Although precise measurements are difficult, there are some
2
I
I
I
I
I
I
I
I
I
I
-2
o
/
o
_
f-
./ -
,,
zgf Z
-2 - I00
- 60
-
20
20
i
60
I
I00
v(mV)
FIG. 1. Instantaneous current-voltage relations computed from the model after damping at 25 mV for 0.45 msec with [K+]l = 0.5 M, [K+]o = 0.02 M, [Na+]t = 0.067 M and [Na+]o -----0.5 M. Filled circles show potassium current, open circles are sodium current and crosses show the total current (without added leakage current).
SUFFICIENCY OF A SINGLE-ION QUEUE
483
indications (e.g. Chandler & Meves, 1965) that nonlinearity may be observed experimentally. While we thus have a definitive measure of neither the intrinsic theoretical nonlinearity, nor the upper limit determined empirically, it appears that the model is not inconsistent with the observation that the instantaneous current-voltage relation for squid axon in normal physiological conditions is approximately linear. It is in satisfying accord with experiments, both in different conditions and with other tissues, that the model suggests this near-linearity to be coincidental, rather than fundamentally significant. Certainly, until the degree of linearity required is further clarified, this criterion should be applied to any model with appropriate caution.
5. Relationship with the Independence Principle Segal (1969) has analysed in detail the applicability of the independence principle (Hodgkin & Huxley, 1952c). He showed that it does not have general validity and that its particular validity for the squid axon can only be discussed in terms of a microscopic model for the mechanism of ion transport. In terms of the present model we argue below that the principle is valid, with certain restrictions, for the squid axon. Hodgkin & Huxley [-1952c, equation (12)] express the independence principle as 1~ (C~/C3 exp ( E , - V ) F / R T - 1 -= (4) It exp (E i - V ) F / R T - 1 where I i and 1[ are the observed total currents (outward positive) due to univalent cation i when the external concentrations are C t and C~ respectively, and Et is the value of the membrane potential V (inside relative to outside) at which It is zero. This equation was derived by considering equilibrium conditions together with the postulate that the inward flux of any ion is proportional to its external concentration, and independent of other ion concentrations (although equivalent relations can be derived in terms of frictional motions of ions in the membrane, see Hodgkin & Huxley, 1952c). The same relation is readily derived, from the theory of absolute reaction rates, for a single thin barrier with its peak at any position. It follows that the model predicts sodium currents in accord with the independence principle provided the population of the Na ÷ selective state is not modified (e.g. by a change in internal K + concentration). On the other hand, from equations (I) and (2), the potassium current through potassium selective states $5 and S~ is given by
It~ = k.ssCt exp (VF/RT)-s~/rk
(5)
484
C. J. G I L L E S P I E I
I
I
I
I
I
I
I
I
t
i
I
I
i
I
6 N
2
"6
g~ z
I
-I00
-60
-20
20
I
60
I00
v[mV)
Fro. 2. Change in queue occupation as a function of potential for the conditions of Fig. 1. Open circles are vacant K+-conductive sites (sD and filled circles are sites blocked by K+(sg); the sum s5 + s~ is almost constant since values were determined 2 Inset after the second potential step, before significant redistribution between different conformal states could occur.
where C~ is the internal concentration of K + (approximating the ion activity) and the constant/co includes the frequency factor. Trapping of K + ions is described by equation (3) and by ss = - kos5[Ci exp (VF]R T) + Co]
(6)
where Co is the external K + concentration, so that the equilibrium approximation [section 2(A), above] is ss/st = [1 -F½kaTk(Ci exp ( V F / R T ) + Co)] - i
(7)
with s, = ss + s's.
Thus, equation (5) becomes I K = ½k, s,[C, exp ( V F ] R T ) - Co]][I +½1¢o~k(C, exp ( V F / R T ) + Co)]
(8)
from which we find that, for a change to external concentration Co, the
SUFFICIENCY OF A SINGLE-ION QUEUE
485
current ratio I~/l K has the form given by equation (4) multiplied by a factor s;[1 +½ko*k(C, exp (VF/RT) + Co)] m = s,[1 +½k,o'ck(Ci exp (VF/RT) + C')]
(9)
which represents the expected deviations from the independence principle. These arise from changes in both the density of potassium selective states and the fraction of thes.e blocked by K +. For conditions where these effects are small, such as moderate changes in external K + concentrations, both instantaneous and steady state K + currents should closely follow the independence principle.
0 E E
-75
-50
-25
0
25
v(mV)
FIG. 3. Steady state K + currents computed from the model for [K+]o = 2.5 m M (solid points and line), 12.5 mM, 62-5 m M and 114.5 m M (successively lower sets of points;
current scale offset) compared with the predictions obtained for the latter three concentrations (lines) by applying the independence principle to the values computed for [K+]o = 2.5 raM; cf., Frankenhaeuser (1962), Fig. 2.
T.B.
32
486
c.J.
GILLESPIE
The predicted deviations may be compared with measurements of steady state K + current by Frankenhaeuser (1962). Since these data were obtained from the myelinated fibre of Xenopus laevis the calculation was performed with an increased value of the "gate closing" time (~, = 400 nsec). This modification is not essential to demonstrate the point but shifts the HodgkinHuxley parameters to more negative potentials corresponding to the properties of the Xenopus node. This change may be shown to correspond with the reduced external [Ca ++ ] characteristic of this tissue compared with squid axon. Steady state K + currents were computed after 20 msec depolarizations from - 7 5 mV, for IK+]i = 120 mM with [K+]o = 2.5, 12.5, 62.5 and 114.5 mM (Fig. 3). The computed results for [K+]o = 2.5 mM were utilized to predict the current-voltage relation for the other three concentrations using the independence principle. As expected, changes in [K+]o produce small deviations from the independence principle. Similar deviations are found experimentally [Fig. 2 of Frankenhaeuser (1962)].
0.4
/ 0-2
/
/
~ V(mV)
E" E
.~
150
5O
o.o
E
,_.o /
k I
-0.2
-0.4
I
t -50
I 0
I 50
I I I00
V(mV)
FIG. 4. Peak transient Na + currents computed from the model for [Na+]o -~ 120-0 mM (lower line) and 39.9 mM (points) compared with the prediction obtained from the latter concentration by applying the independence principle to the values computed for the former case; cf., Dodge & Frankenhaeuser (1959), Fig. 5.
SUFFICIENCY OF A SINGLE=ION QUEUE
487
By contrast, computed peak transient Na + currents (Fig. 4) with [Na+]o = 120 mM and 39.9 mM indicate that the results of changes in [Na+]o are predicted to high precision by the independence principle as expected, and in agreement with the corresponding experimental measurements of Dodge & Frankenhaeuser (1959, Fig. 5).
6. Relationship with the Constant Field Equation The constant field equation (Goldman, 1943; Hodgkin & Katz, 1949) may be derived from the properties of an electrodiffusion regime. That is, the assumptions inherent in it (other than the assumption of constant electric field) are precisely those which were earlier found unlikely to yield useful results for a system such as the squid axon membrane. The same assumptions have been questioned by Cole (1968) on the more empirical grounds that they apparently cannot generate some basic properties of the axon membrane and by others (see Schwartz, 1971) who find the constraints implausible for cell membranes. Nonetheless, the constant field equation was found to predict well the transmembrane potential, and less accurately the current-voltage relation in the squid axon (Goldman, 1943) and has been applied extensively to other systems, notably the frog node (Dodge & Frankenhaeuser, 1959) in which distinctly nonlinear instantaneous 1-V characteristics are inferred. The question arises: if we have reasoned correctly in showing that the assumptions used in deriving the equation do not apply to the nerve membrane, why does the equation predict successfully some of the observed properties ? We suggest below that the observed agreement may result from an approximation, valid over a limited" range. For this purpose, it is fruitful to examine the numerical relationship between the Goldman-Hodgkin-Katz equation and the rate theory expression. The former can be written (Hodgkin & Katz, 1949) for the potassium current (K subscript, eliminated for clarity in the concentration terms; the C superscript denotes the constant field expression) : ic = p F2V Co - C, exp (VF/RT) RT 1 - e x p (VF/RT)
(10)
where PK is a permeability coefficient for K +, Co and C i are the outside and inside K + concentrations, and the usual sign convention for current and potential has been used. From equation (8) we obtain the ratio IC/IK = kb V{1 +½ka'ck[Ci exp (VF/RT) + Co]}/[exp ( V F / R T ) - 1]
(11)
488
C.J.
GILLESPIE
which should be constant as a function of V if the two expressions are equivalent. The value of the constant kb is inconsequential, since there is an arbitrary constant (Pr,) involved in using the constant field equation in any case, but to evaluate the constancy of the ratio, the value of/ca is required. This can be conveniently obtained from equation (7) together with data of the type presented in Fig. 2 since
½k.z,[Cl exp (VF/RT) + Co] = 1 (12) where Vt is the p o t e n t i a l a t w h i c h ss/(s5 +s'5) = 0.5, o r - 1 4 m V f o r t h e c o n d i t i o n s o f Fig. 2. T h e resulting value, ko = 6.7 x 10 8 sec - t , c o u l d also b e d e r i v e d f r o m first principles, b u t t h e m e t h o d u s e d here is simple a n d clarifies the role o f t h e q u e u e i n g process. T h e r a t i o f r o m e q u a t i o n (11) is p l o t t e d in Fig. 5 as a f u n c t i o n o f p o t e n t i a l f o r T = I ° C a n d several c o m b i n a t i o n s o f i n t e r n a l a n d e x t e r n a l K + c o n c e n t r a t i o n . I n each case t h e r a t i o has a shallow m i n i m u m a t s o m e p o t e n t i a l
'
'
'
'
'\'
'Yl
t.0
c:
0-5
o
I -80
t
I
I -40
t
I
I 0
I
W 40
v(mV)
Fro. 5. The ratio of the constant field equation and model expressions for instantaneous K + current as a function of potential for 310 mM internal K + and three external K + concentrations: triangles, 10 raM; diamonds, 50 mM; and squares, 440 mM. In each case the arrows indicate the range of potential for which the ratio is within 5 ~ of the minimum value, and the circle on the connecting line shows the potential at which both currents are zero. The connecting line thus indicates the approximate range of potential within which the two expressions could be superimposed to better than experimental precision.
SUFFICIENCY
OF A S I N G L E - I O N
QUEUE
489
close to the potential for zero current V o = (aT/F) In (Co~C,) (13) which is marked with a circle, together with arrows indicating the potential extremes where the ratio is 5~o greater than the minimum value. Thus, within the range defined by the arrows, a "permeability" PK can be defined so as to render the two expressions identical to better than experimental precision. This range covers approximately 40 mV in each case, but where Vo falls outside the range (as it does for physiological conditions) the requirement that both expressions pass through zero current at Vo effectively extends the region of apparent agreement. It is interesting to observe that the "queueing" term from equation (7) is intimately involved in determining the existence and positions of the minima in Fig. 5; that is, the change in queue occupation is necessary for approximate agreement with the constant field equation. The predicted range of agreement can be tested by comparing calculations from the present model with the constant field equation and with the experimental data of Binstock & Goldman (1971). Their data incorporate particularly careful correction for time dependent components of the "instantaneous"
1"6
I
I
/
I
.~
ooOQe o ° a ° °Qe°o oo
--
0.8
Z
E E
!
0.0
0-8
I -I00 ! -50
I
0
I
I 50
I
I00
v(mV)
FIG.6. InstantaneousK + current-voltagerelationfor [K.+]= = 310 mM, [K+]o = 10 mM computed from the model (points) and fitted line from the constant field equation (5); cf., Binstock & Goldman (1971), Fig. 3B.
490
GILLESPIE
c.J.
current measured in Myxicola giant axons, a system whose electrical properties are similar to those of the squid giant axon (Binstock & Goldman, 1969). The comparison is made (Figs. 6-8) for [K+]~ = 0.31 M with [K+], = 0.02 M, 0.1 M and 0.44 M, which are the concentrations used by Binstoek & Goldman (1971). The constant field equation (lines) is in excellent agreement with the prediction of the present model (points) in the regions anticipated from Fig. 5 which cover the ranges of potential in which experi-
1"6
I
I
I
,
,/
/
,
..
0 D
00 QO
00 °
0.8-
E N
O0
-0-8
I - I00
I - 50
T 0 VImV
I
I
I
I
I 50
i I0 0
I
FIG. 7. Instantaneous K + current-voltage relation for [K+]~ = 310 m M , [K+]o = 50 m M c o m p u t e d f r o m the model (points) a n d fitted line from equation (5); cf., Binstock & G o l d m a n (1971), Fig. 3B.
mental agreement was found (arrows, Figs. 6-8). The only deviation apparent in the data of Binstock & Goldman is matched in the model ( I / < - 5 0 mV, Fig. 7). As suggested by the upward curvature in Fig. 5, the current derived from the constant field equation is greater in magnitude than that calculated from the model at potentials both higher and lower than the range of agreement. Deviations of this kind from the constant field equation are thus a prediction of the model.
S U F F I C I E N C Y OF A S I N G L E - I O N 1-6
I
i
I
I
I
I
491
QUEUE
'//.
,
,
I
I
I
I
I
0-8
E E 0-0
-0.8
gl O0000 o II'°'II qlql° /° ~ o, I I iel -I00
-50
I
0
,50
IgO
V(m V)
FIG. 8. Instantaneous K + current-voltage relation for [K+], = 310 raM, [K+]o = 440 mM computed from the model (points) and fitted line from equation (5); cf., Binstock & Goldman (1971), Fig. 3A. 7. Further Comments on Eiectrodiffusion Models
It would appear from the foregoing that there is no basis for interpreting the observed agreement with the constant field equation in a limited range of potential as indicating approximate validity of the constant field or electrodiffusion regime assumptions. It is, however, instructive to consider the "single ion queue" model as one extreme of a class of models, with electrodiffusion models at the other extreme. Any electrodiffusion model assumes that it is meaningful to consider the concentration of an ion as a continuous function of position in the membrane. If localized transport pathways are involved, this assumption implies at least that a large number of ions are associated with each path at any instant. This requirement may be relaxed, while still retaining the diffusion equations appropriate to a continuous charge distribution, to approximate the properties of paths with only a small number of binding sites, e.g. the diffusion model investigated by Roy (1971), but the validity of this treatment is questionable (Fishman & Volkenstein, 1973) on grounds which reduce to the dilemma discussed in section 3(A) above. The present
492
C. J. G I L L E S P I E
model can be regarded as a limiting case in which each transport path can bind only one ion; the single binding site is either empty or saturated, and no basis for integrating the electrodiffusion equation can be found. Nonetheless, once the fluxes have been computed by other means, the essential role which changing queue occupation plays in generating constant-field behaviour (see previous section) may perhaps be regarded as a primitive analogue of the changing ion concentration profile in the membrane portrayed by eleetrodiffusion theory, though not a sufficiently accurate analogue for the latter to be a good approximation. In these terms we can summarize the conclusion of the previous section with a different perspective. Rectification of the constant-field type may be taken to imply interactions of mobile charges in the membrane; but quantitative agreement is obtained with the Myxicola data by assuming that two or more ions may not simultaneously occupy a single transport path. More complex assumptions are apparently not necessary, nor, following the general discussion in section 3, are they desirable (see also the Appendix for a discussion of the implications of flux ratio measurements). Since the concept of ion concentration in the membrane is no~ definable in such a milieu, it follows as a corollary that permeability coefficients derived from fitting the constant field equation to the data cannot be assigned their usual meaning (cf., Goldman & Binstock, 1969). It may be noted that the same arguments can be applied to sites whose availability follows the kinetics of the m, h and n parameters of the HodgkinHuxley axon, but whose ability to transport ions is calculated from absolute reaction rate theory with single-fold queueing.
8. Implications for the Transient and Steady-state Current- voltage Relations It is readily shown that the form of the Hodgkin-Huxley equations requires that the peak transient and steady-state current voltage relations be concave upwards for V > 0 (i.e. the gradient aI/dV is a monotonically non-decreasing function of V in this range). Experimental I-V characteristics, by contrast, often display decreasing gradients in the range V > 50 mV. Numerous examples have been published for a variety of tissues; those for the squid axon in near-physiological conditions include Narahashi, Anderson & Moore (1967), Armstrong & Binstock (1964), Taylor (1959), and Narahashi & Anderson (1967). Characteristics of this type may be simulated by a modified form of the HodgkinHuxley equations, but are quite compatible with the present model, which may show curvature in either direction, or both (Gillespie, 1973a).
SUFFICIENCY OF A SINGLE-ION QUEUE
493
There has in the past been little motive to examine in detail minor nonlinearities of the I - V relation. Such data may, however, provide the most accessible source of information regarding energy barriers such as those postulated in the model.
9. Summary The proposed model has been shown to generate quasilinear instantaneous current-voltage characteristics under physiological conditions, despite the use of reaction rate theory which introduces into the expressions for the currents an exponential factor varying by more than three orders of magnitude in a 200 mV range. This linearization results from the queuing effect, together with nonselective conduction at large depolarizations. The independence principle is approximated by the present model in many circumstances. In general, it should apply accurately for changes of external Na + concentration, approximately for changes of external K ÷ concentration, and should be subject to severe errors for changes of internal K + concentration. Discrepancies arise from changes in the degree to which queueing blocks the transport path, and from redistribution of the state populations. Redistribution may give rise to large deviations from the independence principle; an extreme example is the effect on the transient current of reduced internal K ÷ (Chandler, Hodgkin & Meves, 1965). As found experimentally, instantaneous K + currents computed from the model are well-fitted by the constant field equation over a range of potential, which depends on concentration. In this case, the change in population due to queueing provides a correction factor which results in approximate numerical agreement between the constant field and model expressions. In other words, approximate agreement with the constant field equation results from the same process which restricts the validity of the independence principle. While some basis has been found for comparing this mechanism with electrodiffusion, it should be clear that conformity to the constant field equation due to such a tenuous relationship does not constitute a verification of any assumptions on which the equation is usually based (see, e.g. Barr, 1965). In particular, it does not establish that the axon membrane can be treated as an electrodiffusion regime. In overview, the model provides a conceptual framework within which the linearity of the instantaneous 1 - V relation and both the successes and limitations of the independence principle and the constant field equation in predicting axon membrane 1 - V relations can be understood. The model rests on assumptions which are different from those used in the above approaches and given apparent verification by their success. It is suggested here that the
494
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GILLESPIE
verification should be carefully questioned, particularly in view of the influence these concepts may have had in restricting the choice of models. The universality of absolute reaction rate theory and its suitability for biological transport problems in general have been noted by Eyring & Urry (1965); the reasons for its selection as the most appropriate instrument for the particular problems of the axon membrane have been outlined above. These expectations are supported by the ability of the model to comprehend both the Hodgkin-Huxley axon and phenomena which lie outside its purview, and notably the three variously applied current-voltage relationships examined here, which have for too long given the appearance of pointing in other directions. The author is particularly indebted to Drs N. F. Clinch, T. L. Hill and G. Roy for helpful criticisms and suggestions.
APPENDIX A
Determination of Queue Properties from Tracer Flux Ratios Ussing (1949) has considered the dependence of the inward and outward ion flux components (M i and Me) on transmembrane potential V. For independent passive transport he derived an equation which Hodgkin & Keynes (1955) expressed for K ÷ ions as M i / M o = exp [-(V- EK)F/R T]
(A 1)
where EK is the equilibrium potential (Mi = Me) for K ÷. Thus the gradient of In (Mi/Mo) v. V should be F / R T ,~, 40 V- ~ if ions cross the membrane passively and independently. However, in squid axon poisoned to eliminate active transport, Hodgkin & Keynes (1955) found this value to be 100 V -~. They concluded that the inward and outward K + fluxes were not independent. Since exchange diffusion can only reduce the gradient, they suggested a competition mechanism with ions moving in single file through a narrow channel. From a simple calculation they showed that single file queueing should give rise to a gradient n-fold greater than that predicted by equation (A1), where n is the number of ions in the queue, and concluded that in squid axon the channels contain 2-3 K + ions. While the main thrust of this conclusion supports the arguments of section 3A, and certainly does not support an electrodiffusion formalism, it may be objected that the numerical discrepancy, between this estimate and the value assumed above for queue occupancy, is severe. The discrepancy is,
SUFFICIENCY OF A SINGLE-ION QUEUE
495
however, only apparent, since the two numbers do not represent the same property. To elucidate this problem we consider the simple system shown schematically in Fig. 9. Exchange transport between compartments at each end of the transmembrane transport path and the adjacent solutions is assumed to be
Inside
Membrene
Outside
1
i i
P
FIo " ~ ' ~ ~I
'
I I i i I
1 1 |
F2o
•
F3o
I i
Q
, , . t ~ l - - F3 i F2 i
_o.j
T
L___
I I
FIG. 9. Diagram of hypothetical transport path, with restricted exchange between terminal compartments (1 and O) and the adjacent solution. The inside and outside solutions contain isotopes x and y respectively. constant and additive with the net (transmembrane) fluxes. Within and outside the compartments the" solutions are well stirred, and the compartment volume is sufficient to contain several ions. The ratio n of the outward flux (Fao) to the inward flux (F2i) across the membrane barrier is assumed to obey the Ussing relation
F2o/F2~ = exp ( V - E e q ) F / R T =-. n
(A2)
which holds for the single-ion queue. Different isotopes (x and y) of the same ion in the inside and outside solutions contribute to the fluxes in proportion to their concentrations in the compartments. I f the internal and external exchange fluxes are Fez and Feo, we have
Fai = F2i+ F~o F3o = F2o + F~o Fii = F2~+Fei Flo = F 2 o + F,,i
(A3)
496
c . J . GILLESPIE
The fraction of y in 0 and x in I are then given by f and g respectively, where f = ( 1 - p)/(1-pq) q = (l -- q)/(1 -- pq)
(A4)
and p = F2o/(n + 1)F2z+ Feo q = F2z/(n + 1)F2z+ Fei
(A5)
The net isotopic flux ratio is then r =- F x / F , - q ( 1 - p ) F2o+Feo p(1 - q) F2i + Fez
(A6)
which in the case of zero exchange, Fee = Fez = O, reduces to r = n 3 = exp 3 ( V - E o q ) F / R T
(A7)
with gradient a(ln r ) / O V = 3 F / R T ~- 120 mV -I. In the general case the semilog plot is not linear except with isosmolar K +. However, with F2, and F2o computed from the model, and exchange fluxes of the order of the transmembrane fluxes at equilibrium, equation (A6) describes the data of Hodgkin & Keynes (1955, Fig. 7) at least as well as the straight line they proposed (Fig. 10). Derivation of a gradient greater than F [ R T depends here upon the assumption that a considerable fraction of the solution-compartment exchange is coupled to the transmembrane fluxes. This implies compartments which accommodate a fixed number of ions, or more rigorously, a large free energy for any increase or decrease in compartment occupancy. This can obtain only if the compartment is small, certainly smaller than the Frankenhaeuser-Hodgkin space (Frankenhaeuser & Hodgkin, 1956; Binstock & Goldman, 1971), otherwise a single ion would provide only a minor perturbation. It follows that the treatment above in terms of ion "concentration" in the compartments is not rigorous, but it may suffice to illustrate the point at issue. Of particular interest is the simple case with one ion in a potential well adjacent to the membrane at each end of the transport path, and a single-fold queue within the membrane. This corresponds closely to the suggestion of Hodgkin & Keynes (with tracer fluxes determined by a queue occupancy of three) but the electrical currents, which do not reveal the number of times a particular ion may traverse the membrane, are limited by the intramembrane single-fold queue. The above argument does not diminish the conclusion that inward and outward fluxes are coupled, though the coupling need not constitute a
SUFFICIENCY OF A SINGLE-ION QUEUE
497
x
V(mV) I
I
-~o
I
50
I
//
/Y -I
(
//~b~ /
-Z
FiG. I0. Semilog plot of flux ratio o. membrane potential relative to the equiJibrium potential: (a) computed from the model for [K+]t -----500 raM, [K+], = 20 m M and exchange fluxes Fa ----F,o = F2f at V = Eeq (see text); (b) line fitted to data by Hodgkin &
Keynes (1955). strict single-file queue. However, the coupling o f flows m a y occur at the interface(s), a non-trivial distinction in the present context since, in a configuration o f this kind, conclusions based on tracer studies are not necessarily pertinent to the electrical properties o f the membrane. REFERENCES Aoxr¢, D. (1967). Proe. hath. Acad. Sci. U.S.A. 57, 1232. ~ONG, C. M. • BI]qSTOCK,L. (1964). J. gen. Physiol. 48, 265. BAI~, L. (1965). J. theor. Biol. 9, 351. BINSa'OCK,L. & GOLDMAN,L. (1969). J. gen. Physiol. 54, 730. BINSTOCK,L. & GOLDMAN,L. (1971). J. Physiol. 217, 517. CARVER, M. B. (1973). Atomic Energy of Canada Limited, Report AECL-4311. ~LER, W. K., HOr)OrdN, A. L. & MEv~, H. (1965). J. Physiol. 180, 821. CrIArOLER, W. K. & MEre, H. (1965). J. Physiol. 180, 788. COLE, K. S. (I968). Membranes, Ions and Impulses. Los Angeles: University of California Press.
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