225 - The surface compartment model of the steady state excitable membrane

Revised

manuscript

received

May. roth

1978

,’

Snmmarv . Membrane bound .io& can- i&uen& the ki~~~ics..‘?f~-p~~f~l~~~ .: transport processes, and may be the’ source bf tinustial ,transieine:~idriiti’ flG..-.-es. To’ study. these effects we have used .th&-“Siirface, C&i&&&t. Model (SCM) approximation; which. divides’. a-~meni&& .‘.$+tkrn-!into discrete- regions : two. bulk reservoirs : (the: inner-:.-and. outer -‘solut&s); two surface layers (on the inner and outer face5 ‘of the-titirdbrane) .&id-. the ultra-thin membrane. Ion flux and conservati0.n equ&i&s-. cari-‘& written in terms of the various compmmerits;. asstiming: thati tie.: trans; port. processes are discontinuous and,-that the -flu& tie pro‘por&& to the differences in the electrclchemical potential,-:, : 1 : .:‘, :-,_-,:’:-:;,.: Using the SCM- it. can be shown that- the,. co_ncehti$ticmS gf. ions. in:“’ the surface regions of excitable membranesand the-&lcuiated @rmtibilities of the membranes to these ions are-quitedifferent, from: tho&:that.-: are generally accepted‘on the basis it appears that, small changes in.

bf bulk

concetitragqns.

:Fnr&&-nore;\;

the surface ‘charge &ti.~alter.the~ iotic concentrations at the membrane surfaces and.th&efore;,-.the ~io&:&&s~ This is especially true. if apprediable- numbers of ,ioris.,are ..b@d to :.themembrane surfaces and ion exchange processes occur aLa &.sult’~~6f:stidd&~ changes in ionic concentrations :o.f current flow. The. physiolb&al !+rG’ cesses (e.g., excitation and act&e. transport) .thA ‘ii.&tilve L~~u&~. idn‘: fluxes in terms of bulk concentrations and .pqtentials, say’ &+AI~-;.be occurring in the e.xpected direction% dgring, transient:- States.:’brought, about

by perfusion, &&-id

currents

or chemical ‘reactions_ .- ::-:; :: : ::, y; Y. .< .,I..-. ,,_,._ ..: ._..: .: .’ ,’

1

l

\Voods

’ Discussed ai the -_+td International Hole (Mass.). 2-S October J-gj’T_

0 Reader’s attention quirements, some symbols for the, given quantity.

is draw-n

Syngksitinj :

1

: ‘: :

to- the, fact ,that’,

used in this

paper tier

on &&c&&uistry,, &ca&_‘of

_;1 ~

.; . .

: ; ,_

.komput,ee,,re-

frb‘m, th&. 1 in:_~n,~~$ ~-6.

53’5

Blank

and Britten

Introduction

Escitable membranes show a great diversity of properties, but in all systems a specific sequence of ionic conductance changes provides the basis for our operational understanding of the process. Because the conductance changes do not appear to be in line with our expectations acaccording to the laws of physical chemistry, there has been a tendency to consider them in terms of membrane components with unusual properties, e.g_ channels with ion specificity that can achieve the known properties A more.conventional approach to the probin as yet unexplained ways. lem was suggested some time agoi in terms of processes that can occur at the surfaces of membranes and that would lead to transient changes in ion concentration and surface potential. If the surfaces of membranes bind ions (i-c-, act as sources and sinks of ions), changes in ionic concentrations due to transference, perfusion, chemical reactions and ion eschange processes in the surface regions can cause transient changes in membrane conductance. These processes appear to lead to the sequence of ionic conductance changes seen in excitable membranes,’ and may also provide the links between the chemical reactions at receptor sites and the For this reason we have decided to take a ensuing physical changes. more detailed look at the properties of a simple membrane system that behaves in this way. Ideally, it would be desirable to develop a continuous treatment of ion flow through a membrane, that is. to describe the ionic concentrations and the electrical potential as a function of a space coordinate across the entire system.’ This is a very difficult problem, even for a simple membrane system containing three ionic species_ However, it is possible to approximate the essential properties of a membrane with the aid of the Surface Compartment Model (SCM). The SCM divides the entire system into discrete regions : two bulk reservoirs (the inner and outer solutions), two surface layers (on the inner and outer faces of the membrane) and the ultra-thin membrane itself. The surface layers are regions where the ionic concentrations and the electrical potentials vary significantly with the space coordinates, but average values are chosen for these quantities and theJr are assumed to be uniform throughout the The flus and conservation (of mass and charge) equations can region. then be set up in terms of the various compartments assuming that the membrane is a boundary and not a compartment (I--e_. it is permeable to ions but does not dissolve any appreciabIe number)_ The fluxes are proportional to the differences in the electro-chemical potential according to XERSST-PL_-\SCK type espressions, and the derivatives of the fluxes are discontinuous across the interfaces between compartments_ Recent esperiments have shown that the rate of ion flow through monolayers 3 and bilayers4 varies with the surface charge density of the layer and the ionic strength of the aqueous solutions. The observed dependence of ion transport on the surface charge in the two systems can be explained by assuming that the ion concentration at an interface, as theory; determines the rate of ion flo~.~ given by el ec t rical double-layer

Blank and Britten

546

r/S of the membrane area. The fluses per unit area given in equations 7 and S must therefore be larger by a factor S in these regions, e.g. functional pores. If

‘UN

>

if

UN

If if

o,

JN

=

S(GN)

(Nz)

(UN)

< O.

]N

=

S(GN)

(N3)

(UN).

UK

>

JK

=

S(GK)

(Kz)

(UK)

UK

< 0,

JIG =

S(GK)

(K3)

(UK).

o,

Equation (II) must also be changed in the pores. j=

to correct

JN+JK+PN+PK-

+

;

1

(40) I ;

(41) the current

*Fz

+

c

per unit area

“c3

_

(42)

>

(Equation 42 assumes that the ion fluses occur through a fraction of the area, but that the surface charge and adsorbed ions are distributed There could also be a non-uniform distribution of charge uniformly. and bound ions.)

Extasion

of the SCM membrane to non-steady

state conditions

The behavior of the SCM equations during the flow of currents A depolarizing can be considered in a qualitative way at this stage. current, an outward flow of cations, across a cation permselective membrane, will cause a build up of ions in the outer surface layer and a deIf an ion exchange process can occur crease at the inner surface 1ayer.l with bound ions in the outer surface layer, the Na+ ion concentration gradient across the membrane will increase while the K f ion concentration gradient in the opposite direction will decrease. This would tend to cause an increased Naf influx and a decreased K+ efflus on the basis of chemical potentials. These changes are in the same direction as seen in (The electrical potential difference across the escitabIe membranes. membrane also changes because of the imposed current, but the magnitude of the depolarization, given by the solution of the equations, is not as easy to follow as the changes in chemical potential.) An important feature of the SCM approach is the limited number Therefore, an ion exchange process which reof sites for ion binding. leases Na+ in the outside layer cannot continue indefinitely, and the concentration gradient driving an enhanced Naf influx will dissipate and The inactivation mechanism is therefore an outgrowth be inactivated. of the activation mechanism and not a separate process. It should also be possible to interfere with the ion eschange mechanism in the surface layers (e-g_. by ion replacement) without changing any other steady state parameters_

Surface

Compartment

Mode1

547

SCM has been outlined in terms of Na+ and K+ ions and a If the concentration gradients, permeability selective membrane. coefficients and binding constants are of the proper magnitudes, other The

cation

Furthercations should also be able to cause unusual transient fluses. more, the mechanism need not be restricted to a cation selective membrane. From these qualitative comments it appears that the SCM offers the possibility of accounting for many of the properties of natural membranes on the basis of a physical chemical mechanism that involves no change in the membrane but rather a change in the ionic gradients imIon selectivity, which is characteristic mediately across the membrane. of many natural processes, could be the result of the normal asymmetry The unusual kinetics of the ionic concentrations across membranes. could arise from the transient changes in the ionic gradients between the

surface

compartments.

Acknowledgements the

This research was supported National Science Foundation.

by

Grant

NSF

PC31 76-11676

from

R&ren~ 1 9

,,’

B. BLAXK. J_ Colloid D.

GOLDSUS,

_~DELXIAN

in

Sci.

Biophysics

(Editor),

Van

20, 933 (1965) and Physiology

Nostrand

3 I 5

I.R. G.D. J.S.

6

D.L.

7

_-~DELMHXN (Editor), Van Nostrand in Biophysics and R.A. SJODIN. ADErhrAN (Editor), V+n Nostrand

Reinhold,

of

Ercifable

New

York

GILBERT,

in

Biofihysics

atld

Physiology Reinhold,

Physiology Reinhold,

Excitable

of

New

of

York

l3citabIe

New

York

>I.

9

DE SIbloNE, J. Theor. Biol. 68, 225 (-1977) &I. BLANK, IV Intemafio~zal Conference ox .%rrface Active don and Breach, Belfast (1967) vol. II, p_ 233 J-W. MOORE, T. NARAHASHI and T.I. SHAN-, J. Ph>lsioI.

11

p.

337

MILLER and hI. BL~XK, J_ Colloid Ifatsrfacs Sci. 26, 34 (1968) SWEENEY and hf. BLANK, J_ Colloid Interface Sci. 42, 210 (1973) Bioelcchocharn. Bioatzarg. 4, 109 (1977) BRITTEN and &I. BLANK,

8 10

A. J.

Membranes, (1971)

BLANK,

J.

Theor.

-BioI.

51,

127

A. J.

Membranes, (1971).

p. 359

Membranes, (1971)

A. J_

p. g6

(1975)

J.A.

(1@7)

Szlbstatzces, (Lotufon)

Gor-

iS?e, gg