Apr 29, 1974 - inspired by the ionic hypothesis of the action potential as described by Hodgkin (2) while the molecular basis of the model is largely derived ...
Proc. Nat. Acad. Sci. USA Vol. 71, No. 7, pp. 2858-2862, July 1974
A Molecular Model of Action Potentials* (proteins and bioelectricity/impedance variation cycle/acetylcholine/calcium)
D. M. DUBOISt AND E.
SCHOFFENIELSt§
t Institute of Mathematics, University of Libge 15, Avenue des Tilleuls, B-4000 Libge, Belgium; and and Comparative Biochemistry, University of Libge, Place Delcour 17, B-4000 Libge, Belgium
* Department of General
Communicated by David Nachmansohn, April 29, 1974 A quantitatively consistent model of nerve ABSTRACT activity is given in terms of two main biochemical cycles narrowly interlocked: an acetylcholine cycle and a calcium cycle. The activity of both cycles is controlled among other things by the electric field and various allosteric effectors. As shown by digital simulations our model accounts for the basic properties of an action potential as described by the electrophysiologists. Thus the shape, time course, and behavior under voltage clamping conditions of both sodium and potassium permeability variations are adequately reproduced.
ductance is transitory while the change in K conductance remains constant as long as the membrane is depolarized, thus suggesting a direct coupling between potassium permeability and membrane voltage. Moreover, we assume that the amount of calcium bound to the membrane is voltage-dependent. The more calcium bound to the membrane, the less permeable the membrane is to potassium ions. Together with electrophysiological evidence, pharmacological data are indicative of a spatial separation of both sodium and potassium channels, the gateways sometimes referred to as ionophores. In view of the peculiar evolution of the sodium conductance under voltage-clamping conditions, we assume that the sodium channel is controlled by an enzyme that hydrolyses a substrate released on proper stimulation. In our model we choose acetylcholine as the most probable candidate Whether or not the acetylcholine receptor (or part of it) is actually the gateway for sodium is so far irrelevant in our model. If an ionophore chemically distinct from the receptor is specifically responsible for the variable membrane permeability to sodium ions, one could then postulate that the conformational change of the receptor induced by acetylcholine is mediated to the sodium ionophore through an allosteric interaction between macromolecules. The same line of reasoning would apply to potassium ions, if future investigations reveal that the potassium ionophore is distinct from the protein that binds specifically Ca ions in the conducting membrane. In a recent publication (8), an attempt at an integral interpretation of nerve excitability has been presented. The authors discuss among other things, the various parameters that may affect the configuration of macromolecule. If, as we assume in the present paper, the changes in membrane permeability are related to the transconformation of specific macromolecules, the physico-chemical basis of such transitions as discussed in the above paper may well apply to our model. The fundamental assumptions being defined, let us describe the various parameters and the sequence of events that take place in our model.
We wish to report on a molecular model of the electrical activity of conducting membranes that takes into account the basic biophysical data well established in contemporary science. Since the Hodgkin-Huxley equations (1) for the squid axon membrane, many attempts at expressing electrical events in mathematical terms have been made. It should however be emphasized that unless a precise physical meaning is attributed to the various parameters or constants that inevitably appear in such formulations, their biological impact in terms of elementary molecular processes is rather weak. We believe that any model of nerve excitability should'take into account the following basic observations made over the years by generations of electrophysiologists: the threshold, the breakdown of membrane resistance, the graded response, the overshoot, the temporal dissociation of Na and K conductances, the spatial differentiation of the so-called Na and K channels, the different evolution of Na and K conductances under voltage-clamping conditions, the importance of Ca ions, and last but not least some fundamental concepts of general biochemistry and thermodynamics as applied to open systems far from equilibrium. Our phenomenological approach to the problem is basically inspired by the ionic hypothesis of the action potential as described by Hodgkin (2) while the molecular basis of the model is largely derived from the chemical theory of excitability proposed by Nachmansohn (3) more than two decades ago. An original addition is the description of what has been termed by one of us the enzymes of the impedance variation cycle (4-7). These are: acetylcholinesterase, choline acetylase, (Na + K)-ATPase, (Ca)-ATPase, an oxidoreductase. Our model takes into account the important observations that under voltage-clamping conditions, the change in Na con-
Release of acetylcholine (AcCh)
A proper stimulation, i.e., a depolarization of the membrane, induces a release of AcCh from a storage form. Whether or not we are dealing with a storage protein or nucleotide molecules such as CMP (9) has to be determined from direct experimental evidence. Nevertheless, our interpretation requires that the -storage molecule be field-sensitive, the binding conformation being favored at high membrane potential (positive outside). Neumann and Katchalsky (10) have provided ex-
Symbols used in this paper are defined on p. 2862. § To whom requests for reprints should be addressed.
*
2858
Proc. Nat. Acad. Sci. USA 71
(1974)
Molecular Model of Action Potential
perimental evidence in favor of such a mechanism since they demonstrated that electric fields of about 20 mV/100 i, are able to induce conformational changes of biopolymers.
AcCh Receptor (Na ionophore)
AcChs
f
Storage
r(E, [Car] )
Release of calcium ions
The binding of calcium ions is field-dependent, i.e., the lower the membrane potential and thus, the less positive the outside, the more calcium ions are made free. The macromolecule to which calcium is attached has two main conformations, a relaxed one (Ca-free) and a tight one (Ca-bound). We also emphasize the idea that the relaxation time of the structural transition is such that the full effect of variation in sodium conductance, induced by acetylcholine, on the potassium flux is delayed. This is accounted for in our model by introducing an appropriate dipole moment for the K ionophore. Ca ions freed by the change in field strength are effective in controlling the rate of hydrolysis of acetylcholine by the esterase (4, 11, 12). We are dealing, in both instances, with regulatory mechanisms that have many features of allosteric control.
ltm
-E
([Cae])
t2 (E)
AcChp
t Choline acetylase
Ac
+
choline
AcCht
|
|
I1 3([CaE])
The acetylcholine cycle
According to our model the liberation of AcCh from its storage form (AcChS) is induced by a field-dependent transition of S(S S'). Thus [1] [E, (Cal)J AcChi + S' AcChS TI" This process is also sensitive to the presence of calcium ions. The free acetylcholine (AcChl) binds to the receptor molecule (R) that controls the sodium conductance. In the free state the receptor is in the configuration R and corresponds to a low conductance for Na ions. When acetylcholine forms a complex with R, it induces a transconformation R':
R'AcCh [2] R + AcCh that leads to the dramatic but transitory increase in sodium permeability. Indeed part of the free acetylcholine (AcChl) is subjected to hydrolysis by the esterase, a process also cal-
cium-dependent:
(AcChl) t,-1 (Cal) Ac + choline
[3] As a result, the complex R'AcCh is split due to the displacement of the bound AcCh allowing the receptor to return to its resting conformation R. AcCh R'i T2-1 (E) AcChi + R "_1 (Cal) o Ac + choline [4]
During the transition R
2859
-i
R', the sodium conductance
increases and the membrane potential V swings towards the equilibrium potential for sodium ions (VN.). This potential change, together with the applied stimulus, leads to an increase in the liberation of the calcium from its bound form. The AcCh cycle is summarized in Fig. 1.
FIG. 1. The acetylcholine cycle.
responding to experience of voltage clamping, the potassium conductance of the membrane remains constant contrary to what happens to the sodium conductance (transient increase). Ca ions can only be bound back to the receptor R2 provided that they are processed through the operation of a specific ATPase that is closely associated with the conducting membranes (ref. 13; J. -M. FQidart, unpublished results from this laboratory). The calcium cycle is given in Fig. 2. It is very sensitive to calcium ions and various substrates such as pyruvate as well as to the oxido-reduction state of some membranes components (4-7). In our model we postulate that the CaATPase is also field-sensitive, a hypothesis that should be accessible to experimental verification, or alternatively that calcium cannot bind to R2 as long as the change in field strength keeps it in the active configuration R2'. Thus under voltage-clamping conditions, the potassium conductance should remain constant. THE MATHEMATICAL INTERPRETATION OF THE MOLECULAR MODEL A. The membrane action potential
We define the total membrane current as follows: dV I = C + gK(V VK) + gN.(V-VN.) + gL(V-VL) [6] dt -
Ca, Receptor
The calcium cycle
(K onophore) (E)
The calcium is bound to the protein that controls the K permeability. The amount of complex R2Ca is field-dependent. Any lowering of the membrane potential from the resting value induces the transition
R2Ca 71(E) > R2' + Cai
[5] As long as the membrane potential is clamped to a value below the resting potential, the amount of R2' remains constant with the expected increase in potassium conductance. Thus, cor-
Ca-ATPase
[Caj
FiG. 2. The calcium
cycle.
2860
Biochemistry: Dubois and Schoffeniels
Proc. Nat. A cad. Sci. USA 71
where C is the capacity of the membrane, gK, gNa, and gL are the membrane ionic conductances ascribed to K, Na, and leakage; VK, VN., and VL are the potentials at which the corresponding current components are zero and are correlated ideally with the Nernst equilibrium electromotive forces for the ions, V is the membrane potential. Since the potassium conductance is related to the amount of calcium ions bound to the K ionophore we may write: [7a] gK = ggK[(Cai)'/[(Cai)4 + (Cal0)4]] where gK is the potassium conductance (mmho/cm2), go the potassium maximum conductance, (Cal) the concentration of calcium ions liberated by the potential change, (Cal,) the calcium concentration, which gives 50% of gK. Taking into account our hypothesis that the sodium conductance gN. is related to the amount of acetylcholine bound to its receptor molecule, we have gN5 = gNa [(AcChf) 4/ [(AcChf)4 + (AcChf0)4]] [7k] where g is the maximum sodium conductance, (AcChf) the amount of AcCh bound to receptor molecules, (AcChf') the amount of AcCh bound to receptor molecules which gives 50%W of gNa. B. The chemical kinetics
We write the following equation that relates the amount of calcium liberated from the potassium ionophore: d(Cal)/dt = -4-'(Ca,) + r-l(Caf) [8] where (Cal) is the concentration of free calcium, 74 a time constant related to the removal of free calcium under the influence of Ca-ATPase (Fig. 2), (Caf) the amount of Ca bound to the potassium ionophore, r a time constant related to the liberation of bound Ca. We may express the evolution of the amount of free AcCh, as follows (see Fig. 1): d(AcChl)/dt = --Ts'(AcChl) - d(AcChf)/dt + rr"'(AcChs) [9] where 73 and nr are time constants. The amount of AcCh in the storage form is then given by the following expression: d(AcChs)/dt = Tm-'(AcChp) - rl-'(AcChs) [10] while that bound to the receptor is: d(AcChf)/dt = -r2-'(AcChf) + rl-'(AcChl) [11] Moreover, the rate of liberation of (AcChs) and (Caf) from their bound form are field-dependent because of the existence of the nonvanishing dipole moment, characteristic of the conformation of the storage proteins. In other words, the time constants 7, 7r, and 72 are written: 7 = T' exp(-AKE/RT) [12] for calcium ions, [13] sT = rl' exp(-,usE/RT) for acetylcholine, and 72 = 72' exp(-MRE/RT) for the dissociation of AcCh from its receptor, where
lK
is the
(1974)
dipole moment (in Debye unit) for the K ionophore, us the dipole moment for the storage protein for AcCh, #R the dipole moment for the AcCh receptor, E the electric field. R and T have their usual meaning. Taking into account the effect of calcium ions on the rate of release of AcCh from its storage form and on the activation of the esterase, we write: 7,31 = Tg'-'[l + (Cai)2/[(Cal)2 + (Cal*)X]] [15] When (Cal) = (Cal*), in Eq. 15, the rate constant of release of AcCh increases about 50%. = ti'1[l - b[(AcChl)/(AcChj')]] X [1-r[(Cal*)-(Cai)11 [16] where the first term takes into account the fact that only a limited amount of AcCh may be released under the influence of a stimulus, r(X) is the Heaviside function which is zero for X -] , 0 [and equal to one for X e ]O, o [. When (Cal) > (Cal*) no more AcCh is liberated from its storage form (AcChs). We also assume that calcium ions control the fixation of AcCh on its receptor. Accordingly, we write for the rate constant of this process: 71-' = T-[1 - a[(Cai)/(Cal°)]] [17] Moreover, to take into account temperature effects on rate constants, all the r-1 are multiplied by the function 4 given by -m= 3(T-T')/1o [18] C. Digital simulation
The numerical simulation was performed on an I13M 370 computer. 1. Numerical Values. The numerical values are taken from Hodgkin and Huxley (1) or estimated from experimental data obtained in this laboratory. They are: C = 1 PF/cm2 gL = 0.3 mmho/cm2 gK = 35 mmho/cm' gN. = 35 mmho/cm2
VK = -12 mV VNa = 115 mV VL = 10.613 mV 1 = 72- = 1.5 msec-, these rate constants are chosen close to the duration of the variation of gNaw
= 13.6 msec-', this constant is related to the rate of hydrolysis of AcCh and is in relation with the time taken by gNa to go from its maximal to its resting value. = 1 msec ', this constant gives the rate of deT4 crease of gK that is 10 times slower than gNa. (Cal,) = 2 units of concentration (uc) (AcChf') = 10 uc I In view of the prescribed space limitation, the variations in the parameters that the equations tolerate will be discussed elsewhere.
Proc. Nat. Acad. Sci. USA 71
Molecular Model of Action Potential
(1974)
115
(Cal*) = 1.5 uc, it is the concentration of calcium ions that blocks the liberation of AcCh from its storage form. 0.08 cm mV-1, the dipole moment of the storage protein for AcCh. AK/RT = 0.02 cm mV-1, the dipole moment of the potassium ionophore. AR/RT = 0, the dipole moment of the sodium ionophore. a = 0; b = 1/3 T = 280 K, the temperature, (AcChs) n'-' = 25 uc msec-1 In first approximation the [rates of liberation of AcChs and Caf are considered as (Caf) r'-' = 0.64 uc msec-lj constants. In future calculation, their variation will be taken into account. 2. The Resting Potentil. At rest, the stationary state of the chemical reactions is given by the following equations: [19] d(Cal)/dt = 0 thus
As/RT
100
IAP
m IV
=
[20]
r,-'(Cal) = T'-(Caf),
d(AcChi)/dt
=
0
[21]
thus
ra-'(AcChl) = Ti'-l(AcChs), d(AcChs)/dt = 0
[23]
thus
Tl'-'(AcChs), and d(AcChf)/dt = 0
Tm-1(AcChp)
=
[24] [25]
thus Tr2-(AcChf) = T'-'(AcChl)
[26]
The conductances at rest are given by:
gNa,r = gN. [ [(AcChl) T2/T1 14/ [[(AcChi) fl/T1]4
2861
012
2
1
35
25 mmho/
3
4
5
1%
A
c~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
3
4
5
msec
g -
-
15
5, 0
1
2 msec
FIG. 3. Solution of Eq. 6 for initial depolarization of 30 mV. Evolution of V, gNa, and gx. Notice the interval from peak of potential to peak of gsw (0.1 msec), the position of the peak of go, the positive after potential and the shapes of g9N and gK with the characteristic patterns of onset and decay.
to a full strength action potential (Fig. 3) including the positive after potential. The time course and shape of this action potential is that observed in the case of the squid giant axon, The difference in shape, magnitude and time course of gNa and gM are also closely related to the experimental values. The threshold effect is well demonstrated by inspection of Fig. 4. From our equations, it is evident that under voltage clamping conditions, the evolution of gja is transitory while that of gK reaches a maximum value that remains constant as long as the membrane voltage is clamped. As observed experimentally (14), if a subthreshold preconditioning pulse of long duration is followed by a superthreshold stimulus, the latter gives rise to an action potential of about 50% the maximal value. This is what we observed with our model.
+ (AcChf0) 4]] [27] 11'
and gK,r = gK
[(Caf)7r4/r']4/[[(Caf)r4/i']4 +
(Cal )4]] [28]
In resting conditions the membrane potential is given by:
VR
=
(gN.,r VNa + gK,r VK + gL VL)/(gN.,r
+ gK,r + g) [29] The numerical values of gNa,r, gx.r and VR are: gK,, = 0.37 mmho/cm2 gNa,r = 0.01 mrnhO/Cm2
VR = 0.01 mV 3. The Subthreshold and Superthreshold Stimulation. In our stimulations, we deal essentially with the case of a spaceclamped membrane that gives an electrical response over a large surface area without propagation. Equation 6 is resolved numerically with I = 0 and the initial conditions that V equal the stimulus depolarization and gNa, gK, and jL have their resting value, when t = 0. A depolarization of 30 mV gives rise
msec
FIG. 4. Solutions of Eq. 6 for initial depolarizations of 10 (1), 20 (2), 40 (3), and 80 mV (4). Notice the slight decrease in potential from the initial depolarization value just before the onset of the action potential. In our molecular model this is explained by the time log of AcCh release, fixation on receptor, and hydrolysis by acetyloholinesterase. Compare with experimental tracings of membrane action potentials recorded by Hodgkin and Huxley (1) p. 525, and more particularly the parallelism of the falling phases of AP.
2862
Biochemistry: Dubois and Schoffeniels CONCLUSIONS
It is apparent from the results of the computer simulations presented above that our molecular model of the action potential reproduces faithfully the electrical events associated with the stimulation of a conducting membrane. The main features of the model are the control of the sodium permeability of the membrane by an enzymatic reaction and the association of the potassium permeability with the state of calcification of a specific ionophore. Thus, the well-established variations in sodium and potassium conductances that follows different time courses during an action potential are taken into account. Moreover the peculiar behavior of the sodium permeability under voltage-clamping conditions is welldescribed in our hypothesis of an enzyme-controlled step. Our two interlocked cycles for AcCh and calcium ions are electric-field sensitive and the sigmoid shape of the kinetics of liberation of both AcCh and Ca ions is explained in terms of allosteric interactions. It is certainly true that other effectors control in a subtle manner the operation of the two cycles as well as their interrelations. On the basis of experimental evidence, it has already been proposed that the cycle of impedance variation should include, besides the proteins involved in the processing of AcCh, a (Ca -Mg)-ATPase, the activity of which is among other things controlled by the oxido-reduction state of some membrane components (4-7). In this first analysis we have assumed that all the other conditions being unchanged, we should concentrate our attention on the molecular mechanisms that could underly the change in sodium and potassium conductances. It could be argued that AcCh is not associated with the control of the Na permeability in a conducting membrane contrary to the situation found at the postsynaptic membrane. It is sufficient to say that in our model any substrate that may undergo, with the proper turnover rate, an inactivation enzymatically controlled, could be a likely candidate. So far, as discussed many times by Nachmansohn (for a review, see ref. 3), AcCh remains the only chemical, the properties of which best explains at the molecular scale of dimensions the bioelectrogenesis. Symbols AcCh: acetylcholine C: membrane capacitance, per unit area
(sF/cm2) I: membrane current, per unit area (4A/
cm') t: time (msec) V: membrane potential (mV) E: electric field (mV/cm)
VR: resting membrane potential (mV) g: conductance, per unit area (mmho/ cm2)
Proc. Nat. Acad. Sci. USA 71
(1974)
IK, INa, IL: components of ionic membrane conduction currents as ascribed to K, Na, Leakage
VK, VNa, VL: potentials at which the corresponding
gK = IK/(V- VK) etc.:
gK, gNa, gL: r:
(Cal):
(Cal'): (Cal*): (AcChf):
(AcChf'):
current components are zero, and correlated with the Nernst equilibrium EMFs for the ions instantaneous membrane conductance component for K, etc. maximum conductance parameters, constants for conducting membrane time constants (msec), (r-1: are the rate constants) concentration of calcium released from the potassium ionophore concentration of calcium that gives a value of gK = gK/2 calcium concentration that inhibits the release of AcCh from its storage form amount of AcCh bound to the sodium ionophore (AcCh receptor) amount of AcCh bound to the sodium ionophore that gives a value of gNa = gNa/2
,As: dipole moment of the storage protein for AcCh AK: dipole moment of K ionophore AR: dipole moment of Na ionophore a, b: dimensionless constants This work is supported by Grant no. 790 from the Fonds de la Recherche Fondamentale Collective to E.S. 1. Hodgkin, A. L. & Huxley, A. F. (1952) J. Physiol. (London) 117, 500-544. 2. Hodgkin, A. L. (1958) Proc. Roy. Soc. Ser. B 148, 1-37. 3. Nachmansohn, D. (1973) in Structure and Function of Muscle (Academic Press, New York), 2nd ed., Vol. III, pp. 31-116. 4. Schoffeniels, E. (1970) Arch. Int. Physiol. Biochim. 78, 205-223. 5. Schoffeniels, E. (1971) Rev. Ferment. Ind. Aliment. 26, 5-14. 6. Schoffeniels, E. (1972) in Biomembranes, eds. Kreuzer, F. & Slegers, J. F. G. (Plenum Publ. Corp., New York), Vol. 3. pp. 499-513. 7. Schoffeniels, (1973) in Central Nervous System. Studies on Metabolic Regulation and Function, eds. Genazzani, E. & Herken, H. (Springer Verlag, Berlin), pp. 138-146. 8. Neumann, E., Nachmansohn, D. & Katchalsky, A. (1973) Proc. Nat. Acad. Sci. USA 70, 727-731. 9. Smythies, J. R., Benington, F., Bradley, R. J., Bridgers, W. F. & Morin, R. D. (1974) J. Theor. Biol. 43, 65-72. 10. Neumann, E. & Katchalsky, A. (1972) Proc. Nat. Acad. Sci. USA 69, 993-997. 11. Gridelet, J., Foidart, J.-M. & Wins, P. (1970) Arch. Int. Physiol. Biochim. 78, 259-264. 12. Wins, P., Schoffeniels, E. & Foidart, J. -M. (1970) Life Sci. 9, 259-267. 13. Glynn, I. M., Slayman, C. W., Eichberg, J. & Dawson, R. M. C. (1965) Biochem, J. 94, 692-699. 14. Hodgkin, A. L. & Huxley, A. F. (1952) J, Physiol. (London) 116, 497-506.
