Bulletin ~?I Mathematical Bi~l~*g 3, Vol. 4tk pp. 637 649
O0117-4c/b5 78 tlgol 0637 ~,02.00 0
FClg.arl3Ol/ Pl~s~, Ltd. 197~. PHt/ted in Gloat Blitam t~ So~ct~ ic,t Mathematical Blolog)
ELECTRIC FIELD DISTRIBUTION, IONIC SELECTIVITY AND PERMEABILITY IN NERVE
• DOUGLAS K. MC1LROY and BRIAN D. HAHN Department of Applied Mathematics, University of the Witwatersrand, Jan Smuts Avenue, Johannesburg, 2001, South Africa
From a model of facilitated ionic transport across axonal membranes proposed by Mcllroy (1975), the equipotentials and electric field distributions in the vicinity of a potassium conducting pore and of a sodium conducting pore are computed and presented as two-dimensional mappings. The model is then extended to include the effect of impurity ions in the conducting pores viz. of potassium ions in a sodium pore and of sodium ions in a potassium pore. The ionic selectivities and permeabilities of the transported species are discussed in relation to the extended model. Bounds are deduced for the ionic selectivity coefficients for both the sodium and potassium current-carrying systems in squid giant axons and the electric-field distributions in the vicinities of the pores are computed for the extended model and compared with the impurity-free fields first calculated. Finally the permeability coefficients defined in terms of the extended model are shown to reconcile the results of attempts to measure permeability by means of radioactive tracer techniques, with the classical description of the resting nerve.
I. Introduction.
In p r e v i o u s w o r k ( M c I l r o y , 1975), the " i n s t a n t a n e o u s " c u r r e n t v o l t a g e r e l a t i o n s h i p s for a x o n a l m e m b r a n e s were a n a l y s e d in t e r m s of a m o d e l of facilitated ionic t r a n s p o r t across these m e m b r a n e s . It was a s s u m e d t h a t the dipole carriers of this m o d e l exhibit very s t r o n g selectivity in their i n t e r a c t i o n s with the c u r r e n t - c o n d u c t i n g species, a j p o r e being defined by the localization of carriers which, in the simplest f o r m of the m o d e l , i n t e r a c t exclusively with ionic species j (for t h a t discussion a n d here we t a k e j to m e a n either K + or N a + for simplicity t h o u g h the e x t e n s i o n to o t h e r c u r r e n t c o n d u c t i n g species is o b v i o u s ) ; if the a x o n a l m e m b r a n e is confined to the region 0 < x_< 6, with the inner surface of the m e m b r a n e l o c a t e d at x = 0 a n d the o u t e r surface at x = 6, a n d the c o n c e n t r a t i o n s in a j p o r e of the j t h species of 637
638
D O U G L A S K. MclLROY AND BR1AN D. HAHN
transported ion and its associated carrier are denoted by c + ( x ) and c f ( x ) respectively, we have the "electroneutrality" condition (Mcllroy, 1975, eqn 8) c+(x)=cf(x)=Q(x)
for
0_ PNa which implies ~ N~ > I k. In the classical description of the resting (excitable) nerve the Bernstein ( 1912) Hodgkin Huxley (1952) explanation that the membrane p.d. -~ OK because the permeability to K'- is much greater than to Na + therefore means in terms of the definition of (19) that DK>>DNa in the resting state. Now determinations of the permeability of squid giant axons by the use of radioactive isotopes o f N a + and K + seem to contradict the classical view. For example Tasaki (1963) states that "the Na permeability of the resting axonal membrane was found to be roughly equal to K permeability", Let us examine this conclusion in terms of the foregoing theory. Denoting radioactive tracer currents and concentrations by an asterisk we have from (2), (3) or (14l-(16) for efflux
(c* (,5 + )= 0) I* = e D a ¢ ) c * (0 - )((a/(oj- 1)/6,
and for i~flux
(20)
(c* ( o - ) = o ) I~ = - e D i ~ j c * (5 + )(O/cki - 1 )/6,
(21)
with Oj of course effectively independent of tracer concentration. Now Tasaki (1963) defines membrane permeability PJ as the radioactive tracer current density per unit original concentration of tracer which from (20) and (21) is Pir emux= eDj ¢, 'j (O/Oj - 1 )/6 = PJmnu~
648
DOUGLAS K. MclLROY AND BRIAN D. HAHN
as Tasaki indeed verifies. Furthermore since ~ K S f y , and P{a P~
DNa f N~ ~b/q~N~--1 DKf°K qS/~K--l'
(22)
we have for squid axons in a normal ionic environment T T PN,/Pk > 15DNa/DK for the resting nerve. Tasaki and others actually find for axons treated with metabolic poison (to remove ambiguities which would otherwise arise from currents due to active transport) that P{~/P~ ~ 1/3 to unity which would make, on our treatment of this section, the upper limit of DNa/D K in the range 1/45 to 1/15. It follows from (19) that PNa/PK lies in the same range of values. In other words, on our definition of the membrane permeability coefficients the radioactive tracer measurements support the classical view of the resting nerve. Thus we suggest that the results of the measurement of axonal membrane permeabilities by radioactive tracer techniques have been interpreted as being in conflict with the long-accepted description of the resting nerve because the definition of the membrane permeability coefficients used (e.g. P]) based as it is upon no detailed model of the membrane transport process, incorrectly includes a term depending upon membrane p.d. On the other hand, the definition of Pj of (19) which arises naturally from our detailed model of ionic transport across axonal membranes (McIlroy, 1975) includes no such terms; when the foregoing experiments using radioactive tracers are considered in terms of this definition they are seen to be in complete agreement with the classical description of the resting axon.
The authors thank Mr. T. Ling for his assistance with the numerical computations, which were carried out at the Computer Centre of the University of the Witwatersrand, and the C.S.I.R. who provided financial support under grant number 9/3/9-822. We also thank the reviewers for their comments.
LITERATURE Bernstein, J, 1912. Elektrobiologie. Braunschweig: Vieweg. Brinley Jr., F. J. 1965. Discussion report: "Conference on Newer Properties of Perfused Squid Axons." J. Gen. Physiol., 48, 41 47. Chandler, W. K. and H. Meves. 1964. "Sodium Inactivation in Internally Perfused Squid Giant Axons." Arch. Ges. Physiol.. 281, 25 26
ELECTRIC FIELD DISTRIBUTION
049
Cole, K. S. 1965. "'Electrodiffusion Models for the Membrane of Squid Giant Axon.'" Physiol. Rev., 45, 34(>379. Hodgkin, A. L. 1964. The Comluction of the Nert'ous hnpulse. Liverpool: Liverpool University Press. Hodgkin, A. L. 1975. "'The Optimum Density of Sodium Channels in an Unmyelinated Nerve". Phil. Trans. R. Soc. Lond. B., 270, 297 300. Hodgkin, A. L. and W. K. Chandler. 1965. "'Effects of Changes in Ionic Strength on Inactivation and Threshold in Perfused Nerve Fibres of Loligo.'" J. Gen. Physiol., 48, 27 30. Hodgkin, A. L. and A. F. Huxley. 1952. "'A Quantitative Description of Membrane Current and its Applications to Conduction and Excitation in Nerve". J. Physiol., 117, 500 554. Mcllroy, D. K. 1975. "Electric Field Distributions in Neuronal Membranes." Math. Biosci., 26, 191 206. Tasaki, I. 1963. "Permeability of Squid Giant Axon Membrane to Various Ions." J. Gen. Physiol., 46, 755 772. RECEIVED 9 - 2 1 - 7 6 REVISED 8-2-77
O0117-4c/b5 78 tlgol 0637 ~,02.00 0
FClg.arl3Ol/ Pl~s~, Ltd. 197~. PHt/ted in Gloat Blitam t~ So~ct~ ic,t Mathematical Blolog)
ELECTRIC FIELD DISTRIBUTION, IONIC SELECTIVITY AND PERMEABILITY IN NERVE
• DOUGLAS K. MC1LROY and BRIAN D. HAHN Department of Applied Mathematics, University of the Witwatersrand, Jan Smuts Avenue, Johannesburg, 2001, South Africa
From a model of facilitated ionic transport across axonal membranes proposed by Mcllroy (1975), the equipotentials and electric field distributions in the vicinity of a potassium conducting pore and of a sodium conducting pore are computed and presented as two-dimensional mappings. The model is then extended to include the effect of impurity ions in the conducting pores viz. of potassium ions in a sodium pore and of sodium ions in a potassium pore. The ionic selectivities and permeabilities of the transported species are discussed in relation to the extended model. Bounds are deduced for the ionic selectivity coefficients for both the sodium and potassium current-carrying systems in squid giant axons and the electric-field distributions in the vicinities of the pores are computed for the extended model and compared with the impurity-free fields first calculated. Finally the permeability coefficients defined in terms of the extended model are shown to reconcile the results of attempts to measure permeability by means of radioactive tracer techniques, with the classical description of the resting nerve.
I. Introduction.
In p r e v i o u s w o r k ( M c I l r o y , 1975), the " i n s t a n t a n e o u s " c u r r e n t v o l t a g e r e l a t i o n s h i p s for a x o n a l m e m b r a n e s were a n a l y s e d in t e r m s of a m o d e l of facilitated ionic t r a n s p o r t across these m e m b r a n e s . It was a s s u m e d t h a t the dipole carriers of this m o d e l exhibit very s t r o n g selectivity in their i n t e r a c t i o n s with the c u r r e n t - c o n d u c t i n g species, a j p o r e being defined by the localization of carriers which, in the simplest f o r m of the m o d e l , i n t e r a c t exclusively with ionic species j (for t h a t discussion a n d here we t a k e j to m e a n either K + or N a + for simplicity t h o u g h the e x t e n s i o n to o t h e r c u r r e n t c o n d u c t i n g species is o b v i o u s ) ; if the a x o n a l m e m b r a n e is confined to the region 0 < x_< 6, with the inner surface of the m e m b r a n e l o c a t e d at x = 0 a n d the o u t e r surface at x = 6, a n d the c o n c e n t r a t i o n s in a j p o r e of the j t h species of 637
638
D O U G L A S K. MclLROY AND BR1AN D. HAHN
transported ion and its associated carrier are denoted by c + ( x ) and c f ( x ) respectively, we have the "electroneutrality" condition (Mcllroy, 1975, eqn 8) c+(x)=cf(x)=Q(x)
for
0_ PNa which implies ~ N~ > I k. In the classical description of the resting (excitable) nerve the Bernstein ( 1912) Hodgkin Huxley (1952) explanation that the membrane p.d. -~ OK because the permeability to K'- is much greater than to Na + therefore means in terms of the definition of (19) that DK>>DNa in the resting state. Now determinations of the permeability of squid giant axons by the use of radioactive isotopes o f N a + and K + seem to contradict the classical view. For example Tasaki (1963) states that "the Na permeability of the resting axonal membrane was found to be roughly equal to K permeability", Let us examine this conclusion in terms of the foregoing theory. Denoting radioactive tracer currents and concentrations by an asterisk we have from (2), (3) or (14l-(16) for efflux
(c* (,5 + )= 0) I* = e D a ¢ ) c * (0 - )((a/(oj- 1)/6,
and for i~flux
(20)
(c* ( o - ) = o ) I~ = - e D i ~ j c * (5 + )(O/cki - 1 )/6,
(21)
with Oj of course effectively independent of tracer concentration. Now Tasaki (1963) defines membrane permeability PJ as the radioactive tracer current density per unit original concentration of tracer which from (20) and (21) is Pir emux= eDj ¢, 'j (O/Oj - 1 )/6 = PJmnu~
648
DOUGLAS K. MclLROY AND BRIAN D. HAHN
as Tasaki indeed verifies. Furthermore since ~ K S f y , and P{a P~
DNa f N~ ~b/q~N~--1 DKf°K qS/~K--l'
(22)
we have for squid axons in a normal ionic environment T T PN,/Pk > 15DNa/DK for the resting nerve. Tasaki and others actually find for axons treated with metabolic poison (to remove ambiguities which would otherwise arise from currents due to active transport) that P{~/P~ ~ 1/3 to unity which would make, on our treatment of this section, the upper limit of DNa/D K in the range 1/45 to 1/15. It follows from (19) that PNa/PK lies in the same range of values. In other words, on our definition of the membrane permeability coefficients the radioactive tracer measurements support the classical view of the resting nerve. Thus we suggest that the results of the measurement of axonal membrane permeabilities by radioactive tracer techniques have been interpreted as being in conflict with the long-accepted description of the resting nerve because the definition of the membrane permeability coefficients used (e.g. P]) based as it is upon no detailed model of the membrane transport process, incorrectly includes a term depending upon membrane p.d. On the other hand, the definition of Pj of (19) which arises naturally from our detailed model of ionic transport across axonal membranes (McIlroy, 1975) includes no such terms; when the foregoing experiments using radioactive tracers are considered in terms of this definition they are seen to be in complete agreement with the classical description of the resting axon.
The authors thank Mr. T. Ling for his assistance with the numerical computations, which were carried out at the Computer Centre of the University of the Witwatersrand, and the C.S.I.R. who provided financial support under grant number 9/3/9-822. We also thank the reviewers for their comments.
LITERATURE Bernstein, J, 1912. Elektrobiologie. Braunschweig: Vieweg. Brinley Jr., F. J. 1965. Discussion report: "Conference on Newer Properties of Perfused Squid Axons." J. Gen. Physiol., 48, 41 47. Chandler, W. K. and H. Meves. 1964. "Sodium Inactivation in Internally Perfused Squid Giant Axons." Arch. Ges. Physiol.. 281, 25 26
ELECTRIC FIELD DISTRIBUTION
049
Cole, K. S. 1965. "'Electrodiffusion Models for the Membrane of Squid Giant Axon.'" Physiol. Rev., 45, 34(>379. Hodgkin, A. L. 1964. The Comluction of the Nert'ous hnpulse. Liverpool: Liverpool University Press. Hodgkin, A. L. 1975. "'The Optimum Density of Sodium Channels in an Unmyelinated Nerve". Phil. Trans. R. Soc. Lond. B., 270, 297 300. Hodgkin, A. L. and W. K. Chandler. 1965. "'Effects of Changes in Ionic Strength on Inactivation and Threshold in Perfused Nerve Fibres of Loligo.'" J. Gen. Physiol., 48, 27 30. Hodgkin, A. L. and A. F. Huxley. 1952. "'A Quantitative Description of Membrane Current and its Applications to Conduction and Excitation in Nerve". J. Physiol., 117, 500 554. Mcllroy, D. K. 1975. "Electric Field Distributions in Neuronal Membranes." Math. Biosci., 26, 191 206. Tasaki, I. 1963. "Permeability of Squid Giant Axon Membrane to Various Ions." J. Gen. Physiol., 46, 755 772. RECEIVED 9 - 2 1 - 7 6 REVISED 8-2-77
