Proc. Nat. Acad. Sci. USA Vol. 72, No. 12, pp. 4859-4862, December 1975
Biochemistry
Spike-forming model of the neural membrane: Computer simulation of some perfused axon experiments (membrane conductance/gates and channels/gate conversion/action potentials/voltage clamps)
DEAN E. WOOLDRIDGE 4545 Via Esperanza, Santa Barbara, California 93110
Contributed by Dean E. Wooldridge, September 29,1975
ABSTRACT This paper is an attempt to account for the spanning the 50-100 A thickness of the membrane, guarded action potential spikes observed when the normal axoplasm at the two surfaces by complex molecules I have called of a squid giant neuron is replaced by a potassium-free, so"gates." The current/voltage relation governing the flow of dium-rich fluid, and the axon is immersed in a potassiuma particular type of ion through a pore is given by Eq. 1 of and sodium-free, calcium-rich bath. For this purpose a reref. 5. With a slightly different arrangement to display some cently described model of the neural membrane [Woolof the physical constants more explicitly, the relation is: dridge, D. E. (1975) Proc. Nat. Acad. Sci. USA 72, 3468-3471] is extended. Allowances are made for the flow of Ca++, Cl-, and Na+ ions through some of the membrane pore configuraI = nk,,(puaje"' - p0,,f) + (en' + a) [1] tions, as well as for some electron conductance, all of amounts too small to be significant in the normal neuron. It In this equation v is the membrane potential divided by 25 is also postulated that the nature of the perfusing fluid affects the rates of some of the reactions that change the conmV (the value of the Maxwell-Boltzmann coefficient at vertible gates from one configuration to another, as well as room temperature). n is the number of electron charges on the ionic permeabilities of the resulting configurations. The the ion, and can be + or -. v is positive when the interior of result of the modifications is a single membrane model that the neuron is positive with respect to the exterior fluid, accounts for the 3-sec, 15 gA/cm2 action potential spikes of while a positive value of I signifies an outward flow of posithe perfused axon as well as for the O.5-msec, 1 mA/cm2 tive ions. pi and po are the internal and external densities of spikes of the normal neuron. the ion. 9i and co are the capture cross-sections of the inner In certain experiments on the squid giant neuron, the norand outer gates, for ions that approach from the adjacent mal contents of a segment of the axon are replaced by a perfluid. a is the ratio of the average amount of time the ion fusate containing chemical ingredients selected by the exmust spend in the channel trapping site next to the outer perimenter, while the fluid in contact with the outer surface gate in order to escape into the fluid surrounding the neuron of the axon is also controlled in composition (1-3). It has to the time it must spend in the site next to the inner gate in been found that action potential spikes, of as high voltage as order to escape into the interior of the neuron. ko is a conthose of normal neurons but persisting for seconds instead of stant whose value depends on the total number of pores, the milliseconds, can be generated in such modified axons. This average ion velocity, and the charge on the electron. is despite the absence of the combination of sodium-rich exAs pointed out in ref. 5, for gates with complex molecular terior and potassium-rich interior to which spikes in the norconfigurations, a and a/olai can independently take on a mal neuron are generally attributed (4). wide range of values. The resulting variety in the shapes of Incompatibility of these findings with the sodium/potassithe current/voltage curves for different ions and different um counterflow scheme is prevented by the fact that the gate configurations has been found to play an important role membrane current density involved is only about 1% of that in the development of an explanation of the axon perfusion characteristic of normal action potential spikes. Thus the results, just as it did in connection with the properties of the perfusion phenomena would not be expected to have a signormal neuron. nificant effect on the electrical properties of a normal neuA second basic feature of the model is its use of two sets of ron, and could involve entirely different physical mechapores through the membrane: the "fixed" pores are relativenisms. Nevertheless, it would be an attractive simplification ly simple in gate configuration and unchanging in their if it should turn out that the two sets of spiking phenomena properties; the "convertible" pores have complex inner gates are merely the different responses of a single set of memwhich are subject to atomic rearrangement that markedly brane mechanisms to different chemical environments. The changes their permeability to different types of ions. It was purpose of the study reported here was to determine whethin some of the chemical reactions that change the configuraer the membrane model shown in a previous paper (5) to tion of the inner gate that Ca++ ions were specified to play a have properties similar to those of the normal squid giant key role. Entering the membrane from the external fluid by neuron is also able to explain the perfused axon results. A way of the outer gate, these ions were specified to be able to preliminary reason for optimism was the report that calcium move through the channel, sometimes approaching the in the external medium (or strontium, which appears to be under side of the inner gate and attaching to suitable sites on equivalent to calcium) is essential to spike generation in perthe complex molecule, until dislodged by thermal agitation fused axons (ref. 2, p. 93), just as it is essential to the opera(electrical neutrality of the gate was assumed preserved by tion of the model. prompt change in the anion content of the upper, or fluid, side). Differing in the numbers of attached calcium ions, Review of the model principles three "states," or different molecular configurations, were In the model, ions pass between the external and internal postulated, among which interchange was said to occur with fluids by way of "pores." Each pore consists of a "channel" time constants shorter than the tenth millisecond intervals of 4859
4860
Biochemistry: Wooldridge
Proc. Nat. Acad. Sci. USA 72 (1975)
interest in normal neuron spiking. The so-called "K1" state was specified to have two calcium ions attached to the under side of the gate, the "A" state was assigned a single attached calcium, and the "B" state no attached calcium. The numbers of gates in the three states were thus in the proportions [2] B:A:K1 = l: Ye-2 'YAK~e-4v where 'yAe-2v is the ratio of the fixed AR-AB time constant to the voltage-dependent B-OA time constant, and YKle-2v is the corresponding ratio for the K 1-A and A -SK1 reactions. Three additional states were attributed to the inner gate of the model, called states "Na," "C," and "K2." Essentially one-way transitions were postulated, B-'Na----C-"K2. The chemical reactions causing these configurational changes were not attributed to ions in the channels, and thus were voltage-independent. Finally, the K2 configuration was held to be convertible back to an A configuration by attachment of a Ca++ ion to the channel side of the gate, with a reaction rate proportional to e-2v. The time constants of the B-o'Na-C---"K2- A reaction sequence were assigned values in the millisecond range. Adaptation of the model for simulation of perfusion experiments At the outset it is necessary to account for the fact that the sodium current through the membrane of a perfused axon, whether the sodium flows out from the perfusate or in from the external fluid, is a small fraction of the sodium currents observed in normal axons. A way to account for this is to expand the model specifications to include the requirement that potassium be incorporated into the molecular configuration of the inner gate in order for the gate in states Na, C, or K2 to be permeable-to sodium or to any other type of ion. This assumption is compatible with the observation that even a very small amount of potassium in the perfusate does indeed upset the current balance that permits spike formation (ref. 3, p. 339). As to the K1, A, and B states of the convertible pores, I have found it necessary to specify that a B gate is slightly permeable both to calcium and to sodium ions, and a K1 gate to chlorine ions. Also needed are small amounts of flow of calcium, sodium, and chlorine ions through the fixed pores (all in the iuA/cm2 range). Finally, I have added an electronic, ohmic, component to these ionic conductances. This is strongly suggested by a long straight tail on the low-v end of the current/voltage curve reported in ref. 1, p. 136. This may correspond to the normal electronic conductance of the membrane as a whole, it may be the electronic conductance of isolated inclusions or imperfections in the membrane, or it may even be partly due to leakage in the experimental measuring circuits. The resulting equation for the membrane current is I,,
= gv
+
kFNa(PiNaO'iFNae"
-
PoNaO'oFNaaFNa)
(ev + cXFNa) + kFCIPoCIaoFFCIaFCI + (e- + aFCI) + aFCa) - (1 - P) - 2kFCaPoCa1ToFCaOFCa + (e X [2kCCapocaiooCcaaCB a . (e~t + OaCBCa) - k('Na X (p,~aUWN,,e- PA.NfaO()(Naa(BN.) * (e + aCBNa) - k(( IYAYKle4 Pocio,,lcci CK1CI + (e + OcK1CI)] + (1 +
'yAe-2'
'YAYKe-4, ) [3] Here ge is the electron conductance; kFNa is the k0 of Eq. 1 for sodium passing through a fixed pore, kcca is for calcium traversing a convertible pore, etc. On all symbols subscripts +
F and C refer to fixed and convertible pores; i and o refer to conditions at the inner and outer surfaces of the membrane; Na, Ca, and Cl specify the types of ion involved; B and K1 discriminate between the two conducting states of the convertible gates. P is a new variable to be discussed shortly, designating the fraction of the convertible gates that are "poisoned" and thus removed for a time from the K1/A/B reaction sequence that generates the convertible pore contribution to membrane current. In Eq. 3, 1/(1 + 'yAe-2v + YAYKle-4v) has been set in from Eq. 2 as the fraction of the non-poisoned convertible gates in state B, and YAYKle 4V/(l + YAe-2v + YAYKle-4v) has been set in as the fraction in state K1. Provision for two-way flow of ions has been made only for sodium, since all other types of ion consistently appear on the same side of the membrane in the perfusion experiments. It is now time to choose a specific type of neuron, for which enough data exist to suggest appropriate values for the constants of Eq. 3. A suitable choice appears to be an axon internally perfused with a 30 mM sodium phosphate solution, and surrounded by a 100 mM solution of calcium chloride. Both voltage clamping and spiking data are available for this axon (ref. 2, Figs. 5 and 6), which has therefore been used as the "reference neuron" for the numerical work of this study. For the reference neuron PoNa = PiCa. = 0. Values for the other constants of Eq. 3, which I have found to lead to model properties similar to those of the reference neuron, are as follows: ge = 1.3 ,A/cm2 OFNa > 1 aCKI C»
0kCCA9 /KcPml2occ
0.9,gA/CM2 2kFCaPoCaaoFCaaFCa 3.7 gA/cm2 YA 0.1; Ke 5 Insertion of these values in Eq. 3 gives the membrane current, in gA/cm2 as Im = 1.3v + 8.9 + ed/(1 + 0.5ev) - 3.7/(e2' + 0.4) - (1 - P)(61 - 23e" - 0.9e 4U)/(1 + 01e-2' + 0.5-4c) =
=
=
=
[4] Some comments on the value assignments are in order. First, it will be noted that the "sodium pump" current through the fixed pores is specified as 8.9 MA/cm2, compared with only 2.7 MA/cm2 in ref. 5 for presumably the same set of fixed pores and a roughly comparable density of internal sodium. The easiest explanation is that normal axoplasm contains some ingredient that decreases the sodium permeability of the fixed pore system. It is also evident that, when at rest, the model neuron as described does not balance to zero the net flow through the membrane of each type of ion. To be sure, such balance is not required in perfusion experiments, in which the desired composition of the internal and external fluids is maintained by steady replenishment. But for the normal neuron at rest, counterflow, amounting to a few MuA/cm2, due to the small amount of internal calcium and the substantial amount of
Biochemistry: Wooldridge
Proc. Nat. Acad. Sci. USA 72 (1975)
E +5
cli
0
-.
z
w
-
v
5
0 -10 w z
< -15 w
2 -20
FIG. 1. Computed membrane conductance curves, for different amounts of "poisoning" (P) of the convertible pores. The values of P are shown adjacent to the curves. Potential is in v units.
internal chlorine in the axoplasm, must be called upon to enforce ion flux balance. Finally, the values of 0.1 and 5 that are assigned to 'YA and 'YK1 must be reconciled with the values of 0.5 and 0.13 used previously. It would be expected that the new values would be about 10 times the old ones, because the 100 mM concentration of external calcium is about 10 times the calcium concentration in squid blood (ref. 2, p. 73). But this would lead to values for 'YA and YKI of 5 and 1.3, instead of 0.1 and 5. To account for the difference, I assume that replacement of potassium by sodium in the gate molecule, previously referred to, has an effect on the atom spacings and charge distributions of the K1, A, and B as well as the Na, C, and K2 configurations. Consequences are specified to include much looser binding of the single Ca++ to the under side of the gate in the A configuration and somewhat tighter binding of the second Ca++ in the K1 configuration. Eq. 4 is plotted in Fig. 1, for several values of P. It is easy to confirm that, when few of the convertible gates are poisoned, the current/voltage curve contains a type of negative conductance region that should contribute to spike formation. Thus, consider the P = 0 curve. It yields a resting potential of -1.62 (-40.5 mV), where stable equilibrium is provided by a combination of the conditions Im = 0 and dim/dV > O. If then an experimenter sends current Iext through the membrane by way of an internal probe and external electrodes, a new stable equilibrium will require that Im = Iext and dIm/dV > 0. For the P = 0 curve of Fig. 1, this must lead to a nearly discontinuous jump in membrane potential to approximately +0.8, when Iext is made slightly greater than 1 MA/cm2. A similar effect can be elicited by a short current pulse, if the amount of charge it transfers to the membrane is enough to raise v above -0.8. The rapidity of the "nearly discontinuous jump" is determined by the membrane capacitance, but it takes place in a few milliseconds, compared with the several seconds that the membrane potential remains high before returning to a value close to the resting potential. Indeed, how v ever declines has yet to be provided for. In the model of the normal neuron, this was accomplished by means of differential time lags in the growth of the proportions of the Na and K2 states, with the suppressing effect of potassium outflow following the spiking effect of sodium inflow and thereby returning v to its resting value. Such a counterflow principle could conceivably apply to the perfused axons, but it would require substantial permeability of the convertible pores to each of the dozen or so multi-atomic internal positive ions that have been found to permit spiking. This seems unlikely. It is for this reason that I have postulated a "poisoning" process and introduced the factor P
4861
into the current/voltage equation. The process is conceived to be one which, over a period of several seconds, can change the configuration of the inner gates of most of the convertible pores so as to destroy their permeability. The easiest assumption is that this poisoning process is simply the B-Na transition (remembering that states Na, C, and K2 have been specified to be nonconducting for perfused axons). The time constant for this transition can be several seconds, despite its 0.5 msec value for the normal neuron; this can be another consequence of replacing potassium by sodium in the gate molecule, not fundamentally unlike the changes already attributed to such replacement. If the ensuing Na -C and C-"K2 transitions are rapid compared with the B-"Na transitions (although not necessarily in the millisecond range of the normal neuron), then the time constant for restoration to an unpoisoned condition can be essentially that of the K2-A transition. Proportional to e2v, this time constant can be large by comparison with that of B--Na at values of v attained near the peak of the spike, small by similar comparison at values near the resting potential. The effect on the spike of a P process with the features just specified may be seen by further reference to Fig. 1. Assuming no persistent Iext after the spike ascent described earlier, as P gradually increases from 0 to 0.2 to 0.4, and so on, the equilibrium membrane potential gradually decreases from 0.80 to 0.72 to 0.65, and so on. This gradual decline in v continues until P reaches 0.70, with v at 0.0, at which point there is a nearly discontinuous drop to v = -1.3. There then follows a much slower decrease to the resting potential of -1.62, as the K2-A reaction returns P toward zero. Although the model has now been equipped to generate spikes of several seconds' duration like those observed, it is not yet quite ready for quantitative comparison with experimental data. For certain voltage-clamping measurements imply the existence of a second pore-poisoning process, much more rapid than the one we have so far considered. The experimental observation is that there is a substantial decrease in current during the 50-100 msec immediately following imposition of a selected voltage across the membrane (ref. 1, Fig. 7). Thus the model must be equipped with a "fast" poisoning process as well as the "slow" one already described. To get good agreement with experiment I have found it best to assume that the reaction responsible for fast poisoning works only on the A and B gate configurations. The physical process could be one whereby, in these states, there is a site on the fluid side of the gate to which the positive ion can loosely attach, in addition to the site at which a similar ion enters rather permanently into the gate molecule. This second and more temporary attachment could change the configuration enough to freeze the gate in an inactive and nonconducting condition until thermal agitation releases the loosely bound ion. The interrelated reaction rate equations for the postulated fast and slow poisoning processes are: dPI/dt = (1 - Pf - PJ)(Cl + CGe-2) tf(1 +
Sy~e-2 +
yl-4ye)
-
Pf
+
tf
[5a]
dPjdt = (1 - P. - PJ + t5(1 + TyAe-' + YAY ,e ) - P, + kse [5b] Here the factor (1 - Pf - P8) + (1 + YAe-2v + YAYKle-4v) is the fraction of the convertible gates that are in the B state, while this factor multiplied by YAe-2v is the fraction in the A state. tf is the average time required to convert a Pf gate back to the A or B state (it doesn't matter which); tf*.+ C1 is
Biochemistry: Wooldridge
4862
Proc. Nat. Acad. Sci. USA 72 (1975)
+5 -3
-4I
-2I
E -U
Biochemistry
Spike-forming model of the neural membrane: Computer simulation of some perfused axon experiments (membrane conductance/gates and channels/gate conversion/action potentials/voltage clamps)
DEAN E. WOOLDRIDGE 4545 Via Esperanza, Santa Barbara, California 93110
Contributed by Dean E. Wooldridge, September 29,1975
ABSTRACT This paper is an attempt to account for the spanning the 50-100 A thickness of the membrane, guarded action potential spikes observed when the normal axoplasm at the two surfaces by complex molecules I have called of a squid giant neuron is replaced by a potassium-free, so"gates." The current/voltage relation governing the flow of dium-rich fluid, and the axon is immersed in a potassiuma particular type of ion through a pore is given by Eq. 1 of and sodium-free, calcium-rich bath. For this purpose a reref. 5. With a slightly different arrangement to display some cently described model of the neural membrane [Woolof the physical constants more explicitly, the relation is: dridge, D. E. (1975) Proc. Nat. Acad. Sci. USA 72, 3468-3471] is extended. Allowances are made for the flow of Ca++, Cl-, and Na+ ions through some of the membrane pore configuraI = nk,,(puaje"' - p0,,f) + (en' + a) [1] tions, as well as for some electron conductance, all of amounts too small to be significant in the normal neuron. It In this equation v is the membrane potential divided by 25 is also postulated that the nature of the perfusing fluid affects the rates of some of the reactions that change the conmV (the value of the Maxwell-Boltzmann coefficient at vertible gates from one configuration to another, as well as room temperature). n is the number of electron charges on the ionic permeabilities of the resulting configurations. The the ion, and can be + or -. v is positive when the interior of result of the modifications is a single membrane model that the neuron is positive with respect to the exterior fluid, accounts for the 3-sec, 15 gA/cm2 action potential spikes of while a positive value of I signifies an outward flow of posithe perfused axon as well as for the O.5-msec, 1 mA/cm2 tive ions. pi and po are the internal and external densities of spikes of the normal neuron. the ion. 9i and co are the capture cross-sections of the inner In certain experiments on the squid giant neuron, the norand outer gates, for ions that approach from the adjacent mal contents of a segment of the axon are replaced by a perfluid. a is the ratio of the average amount of time the ion fusate containing chemical ingredients selected by the exmust spend in the channel trapping site next to the outer perimenter, while the fluid in contact with the outer surface gate in order to escape into the fluid surrounding the neuron of the axon is also controlled in composition (1-3). It has to the time it must spend in the site next to the inner gate in been found that action potential spikes, of as high voltage as order to escape into the interior of the neuron. ko is a conthose of normal neurons but persisting for seconds instead of stant whose value depends on the total number of pores, the milliseconds, can be generated in such modified axons. This average ion velocity, and the charge on the electron. is despite the absence of the combination of sodium-rich exAs pointed out in ref. 5, for gates with complex molecular terior and potassium-rich interior to which spikes in the norconfigurations, a and a/olai can independently take on a mal neuron are generally attributed (4). wide range of values. The resulting variety in the shapes of Incompatibility of these findings with the sodium/potassithe current/voltage curves for different ions and different um counterflow scheme is prevented by the fact that the gate configurations has been found to play an important role membrane current density involved is only about 1% of that in the development of an explanation of the axon perfusion characteristic of normal action potential spikes. Thus the results, just as it did in connection with the properties of the perfusion phenomena would not be expected to have a signormal neuron. nificant effect on the electrical properties of a normal neuA second basic feature of the model is its use of two sets of ron, and could involve entirely different physical mechapores through the membrane: the "fixed" pores are relativenisms. Nevertheless, it would be an attractive simplification ly simple in gate configuration and unchanging in their if it should turn out that the two sets of spiking phenomena properties; the "convertible" pores have complex inner gates are merely the different responses of a single set of memwhich are subject to atomic rearrangement that markedly brane mechanisms to different chemical environments. The changes their permeability to different types of ions. It was purpose of the study reported here was to determine whethin some of the chemical reactions that change the configuraer the membrane model shown in a previous paper (5) to tion of the inner gate that Ca++ ions were specified to play a have properties similar to those of the normal squid giant key role. Entering the membrane from the external fluid by neuron is also able to explain the perfused axon results. A way of the outer gate, these ions were specified to be able to preliminary reason for optimism was the report that calcium move through the channel, sometimes approaching the in the external medium (or strontium, which appears to be under side of the inner gate and attaching to suitable sites on equivalent to calcium) is essential to spike generation in perthe complex molecule, until dislodged by thermal agitation fused axons (ref. 2, p. 93), just as it is essential to the opera(electrical neutrality of the gate was assumed preserved by tion of the model. prompt change in the anion content of the upper, or fluid, side). Differing in the numbers of attached calcium ions, Review of the model principles three "states," or different molecular configurations, were In the model, ions pass between the external and internal postulated, among which interchange was said to occur with fluids by way of "pores." Each pore consists of a "channel" time constants shorter than the tenth millisecond intervals of 4859
4860
Biochemistry: Wooldridge
Proc. Nat. Acad. Sci. USA 72 (1975)
interest in normal neuron spiking. The so-called "K1" state was specified to have two calcium ions attached to the under side of the gate, the "A" state was assigned a single attached calcium, and the "B" state no attached calcium. The numbers of gates in the three states were thus in the proportions [2] B:A:K1 = l: Ye-2 'YAK~e-4v where 'yAe-2v is the ratio of the fixed AR-AB time constant to the voltage-dependent B-OA time constant, and YKle-2v is the corresponding ratio for the K 1-A and A -SK1 reactions. Three additional states were attributed to the inner gate of the model, called states "Na," "C," and "K2." Essentially one-way transitions were postulated, B-'Na----C-"K2. The chemical reactions causing these configurational changes were not attributed to ions in the channels, and thus were voltage-independent. Finally, the K2 configuration was held to be convertible back to an A configuration by attachment of a Ca++ ion to the channel side of the gate, with a reaction rate proportional to e-2v. The time constants of the B-o'Na-C---"K2- A reaction sequence were assigned values in the millisecond range. Adaptation of the model for simulation of perfusion experiments At the outset it is necessary to account for the fact that the sodium current through the membrane of a perfused axon, whether the sodium flows out from the perfusate or in from the external fluid, is a small fraction of the sodium currents observed in normal axons. A way to account for this is to expand the model specifications to include the requirement that potassium be incorporated into the molecular configuration of the inner gate in order for the gate in states Na, C, or K2 to be permeable-to sodium or to any other type of ion. This assumption is compatible with the observation that even a very small amount of potassium in the perfusate does indeed upset the current balance that permits spike formation (ref. 3, p. 339). As to the K1, A, and B states of the convertible pores, I have found it necessary to specify that a B gate is slightly permeable both to calcium and to sodium ions, and a K1 gate to chlorine ions. Also needed are small amounts of flow of calcium, sodium, and chlorine ions through the fixed pores (all in the iuA/cm2 range). Finally, I have added an electronic, ohmic, component to these ionic conductances. This is strongly suggested by a long straight tail on the low-v end of the current/voltage curve reported in ref. 1, p. 136. This may correspond to the normal electronic conductance of the membrane as a whole, it may be the electronic conductance of isolated inclusions or imperfections in the membrane, or it may even be partly due to leakage in the experimental measuring circuits. The resulting equation for the membrane current is I,,
= gv
+
kFNa(PiNaO'iFNae"
-
PoNaO'oFNaaFNa)
(ev + cXFNa) + kFCIPoCIaoFFCIaFCI + (e- + aFCI) + aFCa) - (1 - P) - 2kFCaPoCa1ToFCaOFCa + (e X [2kCCapocaiooCcaaCB a . (e~t + OaCBCa) - k('Na X (p,~aUWN,,e- PA.NfaO()(Naa(BN.) * (e + aCBNa) - k(( IYAYKle4 Pocio,,lcci CK1CI + (e + OcK1CI)] + (1 +
'yAe-2'
'YAYKe-4, ) [3] Here ge is the electron conductance; kFNa is the k0 of Eq. 1 for sodium passing through a fixed pore, kcca is for calcium traversing a convertible pore, etc. On all symbols subscripts +
F and C refer to fixed and convertible pores; i and o refer to conditions at the inner and outer surfaces of the membrane; Na, Ca, and Cl specify the types of ion involved; B and K1 discriminate between the two conducting states of the convertible gates. P is a new variable to be discussed shortly, designating the fraction of the convertible gates that are "poisoned" and thus removed for a time from the K1/A/B reaction sequence that generates the convertible pore contribution to membrane current. In Eq. 3, 1/(1 + 'yAe-2v + YAYKle-4v) has been set in from Eq. 2 as the fraction of the non-poisoned convertible gates in state B, and YAYKle 4V/(l + YAe-2v + YAYKle-4v) has been set in as the fraction in state K1. Provision for two-way flow of ions has been made only for sodium, since all other types of ion consistently appear on the same side of the membrane in the perfusion experiments. It is now time to choose a specific type of neuron, for which enough data exist to suggest appropriate values for the constants of Eq. 3. A suitable choice appears to be an axon internally perfused with a 30 mM sodium phosphate solution, and surrounded by a 100 mM solution of calcium chloride. Both voltage clamping and spiking data are available for this axon (ref. 2, Figs. 5 and 6), which has therefore been used as the "reference neuron" for the numerical work of this study. For the reference neuron PoNa = PiCa. = 0. Values for the other constants of Eq. 3, which I have found to lead to model properties similar to those of the reference neuron, are as follows: ge = 1.3 ,A/cm2 OFNa > 1 aCKI C»
0kCCA9 /KcPml2occ
0.9,gA/CM2 2kFCaPoCaaoFCaaFCa 3.7 gA/cm2 YA 0.1; Ke 5 Insertion of these values in Eq. 3 gives the membrane current, in gA/cm2 as Im = 1.3v + 8.9 + ed/(1 + 0.5ev) - 3.7/(e2' + 0.4) - (1 - P)(61 - 23e" - 0.9e 4U)/(1 + 01e-2' + 0.5-4c) =
=
=
=
[4] Some comments on the value assignments are in order. First, it will be noted that the "sodium pump" current through the fixed pores is specified as 8.9 MA/cm2, compared with only 2.7 MA/cm2 in ref. 5 for presumably the same set of fixed pores and a roughly comparable density of internal sodium. The easiest explanation is that normal axoplasm contains some ingredient that decreases the sodium permeability of the fixed pore system. It is also evident that, when at rest, the model neuron as described does not balance to zero the net flow through the membrane of each type of ion. To be sure, such balance is not required in perfusion experiments, in which the desired composition of the internal and external fluids is maintained by steady replenishment. But for the normal neuron at rest, counterflow, amounting to a few MuA/cm2, due to the small amount of internal calcium and the substantial amount of
Biochemistry: Wooldridge
Proc. Nat. Acad. Sci. USA 72 (1975)
E +5
cli
0
-.
z
w
-
v
5
0 -10 w z
< -15 w
2 -20
FIG. 1. Computed membrane conductance curves, for different amounts of "poisoning" (P) of the convertible pores. The values of P are shown adjacent to the curves. Potential is in v units.
internal chlorine in the axoplasm, must be called upon to enforce ion flux balance. Finally, the values of 0.1 and 5 that are assigned to 'YA and 'YK1 must be reconciled with the values of 0.5 and 0.13 used previously. It would be expected that the new values would be about 10 times the old ones, because the 100 mM concentration of external calcium is about 10 times the calcium concentration in squid blood (ref. 2, p. 73). But this would lead to values for 'YA and YKI of 5 and 1.3, instead of 0.1 and 5. To account for the difference, I assume that replacement of potassium by sodium in the gate molecule, previously referred to, has an effect on the atom spacings and charge distributions of the K1, A, and B as well as the Na, C, and K2 configurations. Consequences are specified to include much looser binding of the single Ca++ to the under side of the gate in the A configuration and somewhat tighter binding of the second Ca++ in the K1 configuration. Eq. 4 is plotted in Fig. 1, for several values of P. It is easy to confirm that, when few of the convertible gates are poisoned, the current/voltage curve contains a type of negative conductance region that should contribute to spike formation. Thus, consider the P = 0 curve. It yields a resting potential of -1.62 (-40.5 mV), where stable equilibrium is provided by a combination of the conditions Im = 0 and dim/dV > O. If then an experimenter sends current Iext through the membrane by way of an internal probe and external electrodes, a new stable equilibrium will require that Im = Iext and dIm/dV > 0. For the P = 0 curve of Fig. 1, this must lead to a nearly discontinuous jump in membrane potential to approximately +0.8, when Iext is made slightly greater than 1 MA/cm2. A similar effect can be elicited by a short current pulse, if the amount of charge it transfers to the membrane is enough to raise v above -0.8. The rapidity of the "nearly discontinuous jump" is determined by the membrane capacitance, but it takes place in a few milliseconds, compared with the several seconds that the membrane potential remains high before returning to a value close to the resting potential. Indeed, how v ever declines has yet to be provided for. In the model of the normal neuron, this was accomplished by means of differential time lags in the growth of the proportions of the Na and K2 states, with the suppressing effect of potassium outflow following the spiking effect of sodium inflow and thereby returning v to its resting value. Such a counterflow principle could conceivably apply to the perfused axons, but it would require substantial permeability of the convertible pores to each of the dozen or so multi-atomic internal positive ions that have been found to permit spiking. This seems unlikely. It is for this reason that I have postulated a "poisoning" process and introduced the factor P
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into the current/voltage equation. The process is conceived to be one which, over a period of several seconds, can change the configuration of the inner gates of most of the convertible pores so as to destroy their permeability. The easiest assumption is that this poisoning process is simply the B-Na transition (remembering that states Na, C, and K2 have been specified to be nonconducting for perfused axons). The time constant for this transition can be several seconds, despite its 0.5 msec value for the normal neuron; this can be another consequence of replacing potassium by sodium in the gate molecule, not fundamentally unlike the changes already attributed to such replacement. If the ensuing Na -C and C-"K2 transitions are rapid compared with the B-"Na transitions (although not necessarily in the millisecond range of the normal neuron), then the time constant for restoration to an unpoisoned condition can be essentially that of the K2-A transition. Proportional to e2v, this time constant can be large by comparison with that of B--Na at values of v attained near the peak of the spike, small by similar comparison at values near the resting potential. The effect on the spike of a P process with the features just specified may be seen by further reference to Fig. 1. Assuming no persistent Iext after the spike ascent described earlier, as P gradually increases from 0 to 0.2 to 0.4, and so on, the equilibrium membrane potential gradually decreases from 0.80 to 0.72 to 0.65, and so on. This gradual decline in v continues until P reaches 0.70, with v at 0.0, at which point there is a nearly discontinuous drop to v = -1.3. There then follows a much slower decrease to the resting potential of -1.62, as the K2-A reaction returns P toward zero. Although the model has now been equipped to generate spikes of several seconds' duration like those observed, it is not yet quite ready for quantitative comparison with experimental data. For certain voltage-clamping measurements imply the existence of a second pore-poisoning process, much more rapid than the one we have so far considered. The experimental observation is that there is a substantial decrease in current during the 50-100 msec immediately following imposition of a selected voltage across the membrane (ref. 1, Fig. 7). Thus the model must be equipped with a "fast" poisoning process as well as the "slow" one already described. To get good agreement with experiment I have found it best to assume that the reaction responsible for fast poisoning works only on the A and B gate configurations. The physical process could be one whereby, in these states, there is a site on the fluid side of the gate to which the positive ion can loosely attach, in addition to the site at which a similar ion enters rather permanently into the gate molecule. This second and more temporary attachment could change the configuration enough to freeze the gate in an inactive and nonconducting condition until thermal agitation releases the loosely bound ion. The interrelated reaction rate equations for the postulated fast and slow poisoning processes are: dPI/dt = (1 - Pf - PJ)(Cl + CGe-2) tf(1 +
Sy~e-2 +
yl-4ye)
-
Pf
+
tf
[5a]
dPjdt = (1 - P. - PJ + t5(1 + TyAe-' + YAY ,e ) - P, + kse [5b] Here the factor (1 - Pf - P8) + (1 + YAe-2v + YAYKle-4v) is the fraction of the convertible gates that are in the B state, while this factor multiplied by YAe-2v is the fraction in the A state. tf is the average time required to convert a Pf gate back to the A or B state (it doesn't matter which); tf*.+ C1 is
Biochemistry: Wooldridge
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Proc. Nat. Acad. Sci. USA 72 (1975)
+5 -3
-4I
-2I
E -U
