- Email: [email protected]
499 bis—The surface compartment model—voltage clamp
Bioeleciiocfrernisrr and Bioene~etics, 9 (1982) 439-437 A s&t+ +. EL&ok&L Ckn+acd u~&tituting -Vo1.~141 il982)
439 -.
Depart&t of Physiology, C&ege of Physicians and Surgeons, Cohmzbia Unibersity, 630 W. 168th Street, N& Yorlr;-NY IO032 (U.S.+) W. @ATRICK
KAVANAUGH
Eiectmnic Associates Inc. Wat k.ang Branch, NJ 07764 (U.S.A.) GEtiEtiEVE
CERl=
Department of &ectricaI Engineering, Cohunbia University, New York, NY 1002S (USA) -(Manuscript &c&d
December 7th 1981)
SUM&Y
Solutions of the SCM equations under voltage clamp show that a transient inward sodium flux can oaxu at certain set&s of the parameters. If there is a non-specific increase in cation permeability upon depolarization. the positive sodium flux arises because the ion concentration changes in the electrical double layer compensate for the decrease in the ekctrical driving force+ while parallell changes in the potassium ion concentrations lead to a decreased (but still outward) p&&m flux. The sodium flux is quite sensitive to the magnitudes of the ionic petmeabilities and mobilities,, as well as the ion binding equilibrikn and release rate constants. The values of the parameters determined here suggest magnitudes for the properties of the membrane components associated with the pen&ability change.
I. THE SCM EQUATIONS DURlNG
VoLTAGE
CLAMP +*
The basic equations. of the Surface Compartment Model (SCM) have been developed [I] in terms of a model membrane system that consists of five discrete regions:___. .-. .: - an .outside bulk (ra.e&oir) phase, compktment 1, - an .outer skrfa=, compartment 2 with&pacitance CAPI, - a membrane having- surface charges X2 and X3 on the two faces, and a dielectric tipki+~~ CAP2, .- ad inner sllrface ‘&mlGrtm@t -3 with .capacitaixe CAP3, 1_. k inner$u& (reservoir) phase, compartment 4. ., : Resin+
..-
: ‘the 6& ~Inte&ta~onak Sy&p+&~& on Bio+ctrochemistry ,and Bioenergetics, Kiryat .. Auavim(Is~eI),‘June 281h-iuly 3id‘l98i. :. **. &ci+ of combu!er pr&am re&irem~ts~th~ sym&ls used in this paper.are not the usual tin&. Thel
$i&ols
ai
u&&&d
.--.
the cok&pokling~&&iti&~are _
,-. .. &2+98,82~OO&--ti,~~2aao/so2.75
_‘.
.-:c.
.
O-.1982 l&&r
$vm.+-ihe -.
‘et _a&iiiu Table. 1.
Sequoia SAA.
’
_-.
440
., .::
._’
.-
::,
We have derived a system of 15 first order non-linear -differential- .equations. to describe the changes that occur in the concentrations of ions and the eiectricaI potentials in the two surface compartments during the’ fIow .of current. (We, have introduced two additional equations for the surface charge to ahow us to tie into account the gating currents, but these have not altered the basic structure of the system.) Changes in the ion (N, K, A) concentrations are given by: ~2=fl/L2)*@Nl-JN-PN-&22)
0)
K2 = (l/L2)
* (JKl
(2)
Az=(l/L2)
*(JAI -JA-PA)
(3)
fi3=(I,‘L3)
* (JN+PN--JN3--33)
(4)
K3 = (l/L3)
* (JK + PK - JK3 - K33)
6)
A3 = (l/L3)
* (JA + PA - JA3)
- JK - PK - K.22)
(6) In the equations, L’s are compartment thicknesses, J’s are fhzxes given by the Nemst-Planck equation, P’s are pump fluxes and N, K, A are sodium, potassium and anions, respectivefy. Changes in the bound cations are given by: &22=BF*XZ*NZ-BR*N22
(7)
&22=BF*X2*K2-BR*K22
(8)
fi33=BF*X3*N3_BR’N33
(9)
K33=BF*X3*K3_BR*K33
(IO) where X2 and X3 are the total surface charge (fast f slow), and the ratio of the kinetic constants, BF/BR = KEQ, the binding equilibrium constant. Changes in the anionic surface charge on the membrane are given by: %2 = - (&22 + K22) - JXM
(II)
X3 = - (riI33 + K33) + JXM
02)
Equations 1I and 12 are split to aIlow for a fast moving fraclion of charge (labeled F) and a slow moving fraction (labeled S). This yields %2s = - (ti22 f k22) - .Ixs X3S = - (&I33 -t K33) + JXS X2F=
-JXF
X3F = JXF Finally, the currents in the surface comp~tments voltage clamp (I% = 0) are given by: I=FA*(JNl+JKl-JAI)-CAPl*&Z
@Ia) ..=::il _-.--,._~ _\ ,. ,xza,:.
‘-;
(Ilb)
(I2b) and in the membrane during -(I;) .’
.:
441 _.
.‘_
04) (J-k-+ JK .- JA - JkM + tiN + PK - PA) + CAP2 * (&
I = &i -
- k3)
(19
-where:thi.E’&- are.thi: ~le&%~oter&ls, ik. is the Faraday constant. and CAPl, CAP2 and :Ck3 ark fhe ca~~&nuS & the respective ,compartments. (We have retahiec-the.l% term -becauseit-is n&ssary ibr. the initial current tipon imposing the ~lamp~).Equations(13)-(U) &m be solved for derivatives of the E2 and E3 as well as -1. The 15 equtitions, (l)-(H), can be .solved for.. the 14 state variables and the current. :. / During &oltage &mp traqsient; E4 is set_&@ to a value that is different from its resting value, and-.the equations are solved for various values of the parameters. The symbols ‘used in the computer solution are shown in Table 1 where concentrations are C’s, bound ions are BC’s, fluxes are F’s and pump fluxes are P’S Where two indices qre u&d, @e first one is for the charged species (I= anion, 2 = sodium, 3 = slow charge, 4 z potassium and 5 = fast charge), and the second is for the phase in which the’charg&%pec+s is found (for concentrations) or from which it originate; (for fluxes). The relation between the symbols used in the equations and in the computer program is shown in Tables 2 and 3. TABLE
1
Diagram of the Surface Compartment Model illustrating the symbols used for the ions, compartment and fluxes in the FORTRAN program Outside surface
Inside
Compartment
Outside
ThiCbesS
Reservcir
SL (2)
Reservoir
Potential Capacitance
E (l)=O
E (9
E (4)
Anioli Passive. flux Pump flux
c 0,
sodium Bound sodium Passive flux
C(2,1)
Membrane
1
fzt~~
1
I
CAP (1)
1)
F(1, 1)
C (1. 2)
+
7
c (2, 2) BCCL 2)
-
-_,
F (2, 1) -
Pump flux Ch&e
Pota+sdm Bound potassium
PaGive- flux
C (3.2) :C (4. 1) F(4,
BC (492)
-_,
1) -
c 62) 3-.
-
-
(f&t)
curt-en;.‘
-
C (4.2)
‘.
Pumpfhk tih&&
-
-
I
I-
,
--F 62)
-
L
+*c
(5. 3)
FA
GM4 2) GM(I, 2)
GM@ 2)
P(4,2) P(l, 2)
N22 K22 x3 N33 K33 SL2 SL3 JN JK JA
moles/(cm2 s) moles/(cm2 s) cm/(s mV) D
WkmVa
5*00E-9 1.00 E-9 9.65 E4 cm/(s mV) a amps/mole
moles/cm’
5.09 E-11 1.11 E-II 5.00 EC10
E2 E3 N2 K2 A2 N3 K3 A3 x2
Equation
Symbol
mole.s/c~ moles/cm2 moles@? moles/cm’ mV mV moles/(cm2 s)
moles/cm’
Units
4.4 E-4 I.0 E-5 4.5 E-4 5.0 L5 400E-4 4.5 E-4 0 -65 -8.44 E-11
Magnitude Program E(2) E(3) C(2,2) (x4,2) C(l, 2) c(2, 3) c(493) (x1*3) {(x3* 2) CR 2) W2,2) BC(4,2) (c(3*3) c(5* 3) BC(2,3) BC(4,3) W2) W3) F(2,2) F(4,2) F(1,2)
Initial Conditions
2,47 E-12 1098E-l 1 3888 E-8 6,d9 E8 ‘aI4 E-11 -509 E-l 1 -l,ll Bll
2,B9 E-IO 6.56 El2 2095 E-I 1
-46 -78 2473 E-3 6021 &5 7,25 E-5 8,38 E-5 6,70 E4 2969 E-4 1006El0
Magnitude
’ Thcsc ‘inobility un$s arc consistent with *c flux equations but differ from the conventional units for mobility,
OK GA’ FA
PK PA GN
w* 1)
NI KI Al N4 K4 A4 El E4 PN
C(4, 1) c(k 1) c(2*4) (x4,4) w 4) 41) Y4) P(2,2)
Program
Equatiom
Symbol
Constants
Numerical data for SCM calculations
TABLE 2
moles/cm2 * moles/cm2 cm cm moks/(cm2 6): molcs/icm2 5) mo&s/(cm2 6)
moles/cm2 moles/cm2 moles/cm2
moles/cm’ moles/cm’ moles/cm’ moles/cm’ moles/cd moles/cm2
mV mV moles/cm3
Units
I, ’ ,’
:
443 TABLE3-
Paramktersin SCM &Iculatio&
Symbols f .KEQ,BEQ BRCAPI CAP2 CAP3 .Gx M
..
.hiagnitudc
Units
‘comments
1000 SE5 l-E-4 .8 E-7 1E4 1E4 1 E-8 to 1 E-10
cm3/moIe s-1 F/Gil? F/cm2 F/cm2 1 s-’ mV-’ cm/@ mv)
fixed for maximum adsorptions corresponds to a characteristic time of 2 ps
II. OUTLINE OF COMPUTER
total series capkitance must give gating cur&t
-0.8
pF/cm2
characteritics
PROGRAM
equations have been solved on a PDP 11 minicomputer using a FORTRAN that en&s the constants and parameters of the system and then proceeds to evaluate the equations listed in the previous section. The algorithm the following: (I) The fluxes are evaluated from the C’s and Es. (2) The derivatives of the ionic concentrations, the bound ions and the surface charges are calculated from equations (l)-( 12). (3) Equations (13)-(15) are solved simultaneously to give the total current TI and the derivatives of the surface potentials E2, 6.3. (4) The above 14 derivatives are integrated with an initial step size of 10 ps, using a fourth order Runge-Kutta (with variable step size) program supplied by Digital Equipment Corp. (5) The integrated values are then used to reevaluate the fluxes and the new values of the derivatives for the next time step. Some segments of the program are given below. The
program
C C INITIALIZE c
VARIABLES-NOTE:
DO&_13
44)
= -20.0
.-
40 A(1, 1)~
IdDO A(l, 2) = CAP(I) A(I, 3)=O.ODO A(2,.2) =o.oDO AG3) = - cAp(3) A(3,2) ?-7C&P(2)-_ A(3,- 3).?.CAP(2)~ _ sL@j = x(2)/(+(1,2) swj
=~qyc-ca :
; -: +.c(2,2)+ c(4,2)) 3) + c(2,3) + c(lt, 3))
FOR VOLTAGE
CLAMP
444
DO 50 I = 2,4,2 DO50J=2,3 C(3, J)=(l.ODo-Pm) C(5, J) = PCT * X(J)
* X(J) _
50 BC(I, J) = BEQ * X(J) * C(I, J) C C POTENTIALS, FLUXES & PUMPS-INITIALLY
K = 0 FOR PUMPS
C DO2001=1,5 DO2OOJ=1,3 IF((C(1, J).LE.O.ODO).AND.(U(I, J).GT.O.ODO))F(1, J) = O.ODO
IF((C(1, J + I).LE.O.ODO).AND.(U(I, J).LT.O.ODO))F(1, J) = O.ODO IF((C(1, J).LE.O.ODO).OR.(C(I,J + l).LE.O.ODO))GO TO 200 U&J) = 5.8Dl * DLOGT(C(I,J)/C(I,J + 1)) + (- 1) ** I * (E(J) - E(J + I)) F(1, J) = GM(1, J) * U(I, J) * (C(1, J) + C(1, J + 1))/2.000 200 CONTINUE C C CALCULATE PUMPS IF K = 0 C IF(K. NE-O) GO TO 240 P(l, 2) = -F(l, 2) P(2, 2) = -F(2, 2) P(4, 2) = - F(4, 2) C CCALCULATE DERIVATIVES OF BOUND AND FREE CONCENTRAC TIONS 240 DO 250 1=2;4, 2 DO 250 J=2,3 DBC(E, J) = BR * (BEQ * X(J) * C(1, J) - BC(1, J)) .. 250 CONTINUE DO2601=1,5 DO 260 J=2, 3
DC(1, J) = (F(1, J - 1) - F(1, J) + P(1, J - 1) - P(1, J) - DBC(1, J))/SL(J) IF(I.EQ.3).0R.(I.EQ.5)) DC(I, J) = (- 1) ** (J - 1) * (F(1, J) + F(1, J - 1)) IF(I.EQ.3)DC(I, J) = DC(1, J) - DBC(2, J) - DBC(4, J) 260 CONTINUE C C CALCULATE DE’S AND TI C B(l)=FA * (-F(1, l)+F(2, l)+F(4, 1)) B(2) = FA * (-F( 1, 3) + F(2, 3) + F(4, 3)) - CAP(3) * DE(4) ‘. B(3)=FA * (-F(1, 2)+F(2, 2)+F(4,2):P(l, 2)+P(2, 2)+P(4, 2))$FA (- F(3, 2) - F(5, 2))
*
445
-_.
:--
%I .=-(B( 1j .* CAP(2)/CAP( I-) + B(2) * CAP(2)/CAP(3+ ‘- (A( 1;11) ,* CAq(2)/(CAP(.l) .+ .A(2, 1) :* CAP(2)/CAP(3) DE(Q=:(B(l)-TL)//CAy(I) : .: ‘y. ._. : .DE(3) + (TI-B(2))/CAP(3)
B(3));$_A(3;- -1)) :
The voltage ‘clamp is. imposed as a step with rapid exponential rise with a time ~constant of ‘l.kX 10’ S-’ so E4 changes rapi_dly and is within 1% of the clamp voltage at about 30 ius.:(The smooth variation ‘of E4 is necessary for the stability of the numerical integration algorithm.) The equations used are as. follows:
C-
:
C-CALCULATE VALUE OF INPUT VOLTAGE C IF(K.EQ.O)EINIT = E(4) TC = 3D0 * DLOGT(3DO)/STEP E(4) = EINIT - (ECLMP) f: (1 DO - DEXP( -TC * TIME)) DE(4) = -(EINITECLMP) * TC * DEXP(-TC * TIME) Under voltage clamp conditions (i.e. E4 is constant), all of the concentrations listed in the two reservoirs are fixed. Within the two surface compartments, the ionic concentrations are given by integrating equations (l)-(6) and the surface charge and bound ions by integrating equations (7)-(12). (SL2, SL3 and the pump fluxes required by the equations arc obtained, along with the values of the -other parameters.) The only other unknowns, E2, E3 and TI are obtained from the three equations for- the current, equations (13)-( 15). This enables a complete solution of the SCM. The parameters which must be initialized in order to make the calculations are: (1) Two of the three. binding parameters (KEQ, BF, BR). (2) Three ionic.mobilities in the surface compartments, Ml, M2, M3. (As a first approximation they will all be assumed to be equal to M.) (3) .Four mobilities in the membrane GN, GK, GA, GX. The first three are fixed by thenknown steady state fluxes, and GX is set so that the effect of the initial JXM does not exceed 0.3% of the total fixed charge flow [2]. (4) Three capacitances, CAPl, CAP&CAP3; Although it appears that twelve parameters must be initialized, a situation which ‘ought to enable :the description cf almost any kind of curve, upon closer examination most of these valuesare fixed.by experimental observations. From the following discussion it. yill become clear that the greatest uncertainty is associated with the choice of values fo? BR and M. II;. $‘ITIN~~O~ :
-...; 1. The
CONDITtON!i
:
Nl, _Kl, Al, N4, .K4.and _A4 are fixed at values normally the,gi+ $qtid axon 131, and specific values are g&en in .Table-2; El, as previously; is taken as- reference potential in’ the SCM and assumed to
bulk- coqFntrago&
f&rid-f& kention+ :
JNITL+ ., _
-’
:
.- :
446
have a value of zero, ‘while E4 is the normal resting potential; --The.fixed{negativesurface charge values for X2 and X3 and the surface values of goteitial FQ:ai~dE$ (which result from the surface charge distribution) are based onthe ~i&rlts~of Gilbe& [4]. Values of initial conditions for N2, K2, A2, N3, K3, A3 can be derived inthe following manner. In the steady state, the electrochemical p_ote@ials :for~the surface compartment fluxes are zero. Therefore, the el&ri&l field and chemical pot&&ls are equilibrated for each species in the internal and external surface compartments, yielding the following relations: 58 log(Nl/N2)
= -E2
06)
58 log(KljK2)
= --El
(17)
58 log(Al/A2)
= +EZ
(18)
58 log(N3/N4)
= E3 - E4
(19)
58 log(K3/K4)
= E3 - E4
(20)
58 log(A3/A4)
= E4 - E3
(21)
Since the bulk concentrations (NI, Kl, Al, N4, K4, A4) and E2, E3 are fixed, it follows that the state variables N2, K2, A2, N3, K3, A3 are determinable. SL2 and SL3 are calculated from the following relations: x2
SL2=N2+m-/Q
(22)
SL3=
(23)
A3 N3+K3-A3
From the surface compartment concentrations N2, K2, A2, N3, K3, A3 and .the surface potentials E2 and E3 in Table2, it can be seen that the electrochemical potentials across the membrane are not equal to zero at steady state. UN = 58 log(N2/N3)
+ E2 - E3 = 119.87 mV
(24)’
UK = 58 log(K2jK3)
+ E2 - E3 = -27.83
mV
(29
UA = 58 log(A2/A3)
+ E3 - E2 = -65.09
mV
(26)
The leak fluxes JN, JK and JA, which are offset by the pump fluxes, c-&i, be computed for steady state conditions in the following way: JN=1/2*(N2+N3)*GN*UN=-PN
(27)
JK=1/2*(K2+K3)*GK*UK=-PK
(28)
JA=1/2*@2+A3)*GA+UA=-PA
(29)
JN, JK and JA are known within certain limits [5], in which case GN, GK, and GA can be derived. Once set, there is no variation in GN, GK, and GA. HoweVer, ifthe electrochemical potentials change (i.e., the initial ion concentrations are changed), the corresponding pump fluxes (PN, PK, PA) must be recomputed to reStor+ a net initial flux of zero in the system. It should--be noted that ihere is a-f&r .amor;l;t of ..-
__.
.. ‘_~
:..
447
.,
(.
‘.
varia~tin~.~~-thk &b&&d
vah& of t& ‘stkady-sta!e fh&, ‘.:.._ : 1 .. ~ I
_j&;be p-&i;,
so thai-thesevahws may
--:. CX is +t ~~~at.~=~~a~~dc of FA * J%M, i.e. the currentdue to the mobile r&&rn&&nti compbnent$ yill give -theproper mtignitudeand duration of the gating currents[2]__.: .:_.;.:: ::. ~..- -Y._. :.‘I!heinitial~&&itio~ for thestate v&abl&s dkkcribingthe bouud ‘ions, N22, K22, &I33and~K3~‘arc~aUdeterminablefrom the expressionkgiven in ~cquations (7)-( IO). ti setting$422 = &22.=33$= @33 =.O, we obtain;.
N22.=KE+N2-*X2
(30)
&2=I(EQfpX2_
x30
N33=KEQ*N3*%3
(32)
K33=KEQ*K.3*%3
(33)
This-completesthe descriptionof the set of initial conditions to be used. IV.VALU~
OF
THEPARAMETERS
The grious&&uts and initial conditionsof the SCM have been chosen in the above section, and vahks assumed for the parametersto be studied are shown in Table 3, These includethe capacitances,whichhave been discussedin general[ 11,the ion mobilitiesand the constants associatedwith ,ion binding. The capacitanceswill be disc+%+ in a separatesection. The ii&&s for the mobilifiesgiven in .Tible 3 are ~~WEECIto be equal for Na, M and typical dons A in solutions ch_arageristic of the exte_mal/intemalsolutionsof the squid axon. Their magnitudesare not lgmm and one can only assume that they -are,loycr than comparabb values in. aqueous solution. ‘J”hegate.~&L&I+ .BJ?a&d BR con&&s in equationsQ-( 10) are assumed to be thy same for Na and.5 ions, and are also assumedto be the same in the two surface compart~eqts. :The ratio qf BF at@ ,BR de&mines an equiIib.rium ,constant KEQ .which will have the same v&e for each of. the bound ions (N22, K22, N33 and k33). An u$perJimit for K;EQ c&t bc determinedfrom the following relationsand arguments: _. ~_
S2=X&N22+K22
: :
:
(34
S2 &sthe ,n&&+erof bkmin~ sites on the kternal surface of the membrane. For no +p&iable~ flow of chargeinthe membrtie(i.e. JXM-= 0) S2 is a constant. For ions bot$d.at all sites (i.e. X2 $=O):.: sgL~+&
_.T : : .. ‘.
_ .I, _‘,; ,. .
_’
;I.
,:
(W
From-~hj&%.l a& geometrical~~idera~~n~.~*~~g the size and structureof ‘th&$&$&uc m&terikJ &- es&& of ‘theupp&rlimit (S2MAX) on S2 carrbe mkde of..* .~~~i~~.~~.~ ‘m&w. ihat.each site ti~&~ & z&e+ ‘of 40 A2). Therefqre ~2n;rkx=~~~.~~~;;~wh~~‘=‘O~-: _.‘_- _‘I-.. ‘-..._.., ;- _ 1 _I -.I. (36) ._._ ’ _‘.a.:..- .. _..r : - .:, :z _. ,:,._
._
..
..-.
. :
~
44s
From equations (7) and (8) we have N22 KEQ=X2*N2
(37)
and K22
KEQ=m*m
On rearranging and adding equations (37) and (38), we get KEQ’X2*(N2+K2)=N22+K22 or (N22 f K22) KEQ = X2(N2
(39)
+ K2)
An upper limit for KEQ can be estimated by substituting SZMAX for (N22 + K22) and the known values of X2, N2 and K2 in equation (39). This yields a value of 1000 for the upper limit of KEQ. A lower limit of zero would mean that no binding was occurring and this is not of interest for the SCM. A similar argument could be made for S3, but since we have assumed the same rates we have the same KEQ’s. V. EFFECTS OF MEMBRANE CAPACITANCE
When a voltage clamp is applied so that E4 = -20 mV (i.e., a depolarization of 45 mV), the electrical potential across the membrane changes from the 32 mV in the resting state to a new potential, E23. The change is virtually complete in 30 &s, and depends upon the magnitudes of the capacitances chosen for the three regions defined in Table I. Since the three capacitances in series must have a combined-value of 0.8 pF/cm*, it is possible to obtain a range of E23 magnitudes, as shown in Fig. 1. If we assume CAP2 = 0.8 pF/cm* (a readily accepted value for the dielectric capacitance) and the two surface capacitances CAPI = CAP3 = 100 rF/cm* (a value close to measurements at mercurylwater interfaces), E23 = - 12 -mV. If we divide the capacitances more or less equally CAP2 = 3 pF/cm*, CAP1 = CAP3 = 2 pF/cm*, then E23 = +23 mV. The current transients in these two cases are shown in Fig.2, and despite the introduction of three capacitances, the early current across the membrane behaves as if there were a single time constant. The electrical double layer capacitances. are apparently undetectable, and one gets the same k&l response if the -total capacitance is the same. The different plateau currents at longer times are due to different values of E23. (See Fig. 3). When all the capacitances are approximately equal, the changes in E2 and E3 are approximately the same. When CAP2 is, much smaller than the double layer capacitances, the changes in,E3 and E4 are approximately the
same and E2 is virt@ly unchanged...These differences in the plateau values of E23.
two sets .of cqnditions
E23, the electrical driving force for the flow of cationsinto
lead to, large
the cell, decreases and
449 .-
Fig. 1. The plateau voltage across the membrane reached 1 millisecond after a voltage clamp is imposed, E23 (in mV) oersus the total capacitance of the three capacitances in series (in pF/cm2). The magnitude of the dielectric capacitance CAP2 (in pF/cm’) is indicated on the lines and CAPl=CAP3_ The parameters used were as in Table 3. except that M= 1 E-7.
reverses sign with an increase in the total capacitance. To increase the probability of a positive (inward) JN, it would appear to be advantageous to choose a low total capacitance along with a relatively high dielectric capacitance so that there is a significant electrical driving force for sodium ions into the cell. However, this is not so, as we shah see below. .The electrical drivmg force for sodium ion influx may decrease, but there are significant. compensatory changes in the chemical driving force that are due to ionic equilibria in the surface. This will be discussed in the next section. Vi. CQNDITIONS
FOR REVERSE
SODIUM
FLUX
On choosing different values for the parameters, solutions for the SCM system of . ‘equations give the-expected response of a passive system, I.e., a steadily decreasing negativecurrent :until *he establishment of. a new steady. state, with gradual and slight .changes in the variables-&f the system..However,. there are conditions under which the;SCM equations respond tith a. positive,. transient sodium ffux.~Fig. 4 shows one-such. case, Where .the permeability .of the membrane to sodium .and potaissium 3s increased : several orders -of m&guitude above resting values. The
_-__:--: ..
.-
-. _ :
20
40
60
100
used were the Fig. Z A semi-log plot of the total current in A/cm2 LWSUSthe time in gs. The parakers same as in Fi& 1. The two -es relate to approximately the same total capacitance, but different individual values. The 0.8 is due to 100.0.8.100 pF/cm2 and the 0.75 is due to 2.3.2 pF/cm2. for CAP], CAP2 and CAP3, respectively.
transient reversal of IN, i.e., an inwardly directed sodium fhk, is accompanied by increases in N2, the sodium concentration at the outer surface, and decreases inN3, the sodium at the inner surface, as shown in Fig.5 Apparently, the chemical potential difference more than compensates for the unfavorable electrical potential difference noted in Fig. 3. Solutions obtained for cases of favorable ekctrical potential differences, e.g. the second case shown in Fig. 3, give a lower positive INi The positive transient IN of Fig 4 is seen when the permeability of the membrane to both sodium and potassium is raised several orders of magnitude from..the resting values in Table2. When GN is increase&selectively (while GI&remainS~fhe same .a~.. in the resting membrane), there is no current reversal. .When. GK- ii -increaSed selectively, there is a slight reversal of IN that is three orders of magnitude smaller than in Fig. 4. It appears that in order to observe a positive IN of ap&oximately the : right magnitude,-both GN and-.GK must be iricrcascd.-(An .iricreasedGA- tcrids to . ,; ‘: -..z.. decrease the positive IN.) :/-..- _. The positive JN is quite sensitive to the magnitudes of the ionic +rr&abilities :
:
.’ LI_
‘.
451 -. . ...
._:
:
----------
...‘._
_.
:
---
-. 3
Fii 3. ‘he variations’of the potentials in the two cases shown in Fig. 2. Tbe_solid curves are for the 0.8 pF/&* and the dashed qmves for the 0.75 ~F/cm*. EA. the clamp voltage is the same in both cases, but U3 has opposite signs in the plateau regions.
:
452
and mobilities. The dependence .of the positive JN .on GN
I ., :.. ..
.
and ~GK~is~shownin..
Fig. 6. It is apparent that the membrane p~rmeabilitiqs must be at 1-t 5E&o’h$%e approximately the right magnitudes of. fluxes and times. From-Fig.‘7 we s+ that. :$e.
N2
K3 K2
N3
0
t
I
a2
I
,
0.4
I
t (ms) t
0.6
I
I
0.8
I
,
1.0
Fig. 5. The changes in the cation concentrations in the surface compartments are shown under the same conditions as in Fig. 4 and the solid lines in Fig. 3.
ionic mobility in the surface compartments must be at least an order of magnitude smaller then the GN end GK in order to sustain the concentration gradients (shown in Fig. 5) and give rise to a positive JN.
The binding equilibrium parameters also influence this flux markedly. In Fig. 8 we see that the equilibrium constant must be over 100 in order to have: an appreciable positive JN. The time of the JN peak is also infiuenced ‘-.by this parameter, but not as much as by the ion release rate constant shoF..in Fig, 9. The value of BR must be about SE4 or greater to show a peak at low values of time; The solutions of the SCM equations -under_vohage ch%p have been described for various values of the parameters. However, in order to approach &i&&~s~&at are
closer to the in oioo situation, it wouid be desirable to ik@Uce changes in permeability during the solution, but the me&a&m which could &e rise to Sudh changes is unknown, and the problem is also much more difficult; Serious distortions are introduced into the solutions because of the constant pernxabilities z&d .the high values of the resting (and ion pump) fluxes that must be incIuded when-the. membrane permeabilities are set to high values. These distortions probably:ac&nt for the peculiar responses : of the SC&f equations to- changes in jhree -:of. _t& parameters, the clamp potential, GX and-PCT. .. ..-I _. .I‘ .. Fig. 10 shows how the positive:JN depends upon. the ,magnitude of, .theT_c@inpi: potential. As one incrkses~the-magnitude of the depolarization, (i.e.$sthe~_ma@i~_.; :.
: ::
..
-- ; .> .,‘,”
_ .:..
.-_ .I
: .:-.
._
.‘.
.,
..
__
;
m.. _.
.j, 3- s p Y.
_, . -.
-:’
..
_’-.-
;
-3 -
Fig. 6.. The -variation of the sodium flux -with time as a function of GN and GK the membrane permeability to th+ two cations. The magnitudes of GN=GK in cm/(s mV) are shown on the curves.
.
Fig. 8. The variation of thesodiumfluxwithtimeas a function of KEQ, the bindingequilibrium constant. The maguitudcs of KEQ in c&/mole are shown on the cumes.
tude of the clamp potential becomes more positive), one expects that the positive JN would increase. This cleariy occurs up to about -40 mV and qualitatively up to
about -20
0-f
-1
mV. However, at higher values of the clamp voltages (e.g. i-20 mv)
0
-
: -2
-
Fig. 9. The variation on the sod&m fl& with time as a function of BR, the ion rekase rate constant The magnitudes of BR in s-’
are shown oil the c&es.
_
:. . ..
-.__L.
. .
:
...~
:
:_
:.
..,_-
455. -.
:
~.. _
is~&t&ji~ a .d&i&e. k:Mi &’ kxpecte& at the m&e, positive pqtentials; the. fi&ikn’fl&&~$t; to.&&&& ita++ but- t&5 peak fluxes tic. displ+ed to ,s+orter i-t&i&, tid :t$is’@nits’ the m&&itQde:of -the ri_se.~..From, Fig. 6 we- SF _.@a~ *hen the . _ifi&
:
i 2 cu_ -a-
'. .. '.
:_.
..'
.. __
..
.-
: ~
.- ,- ..
..
_g :
coo+..'sma'
0.5
t (ms)
1.0
-1 -
Fig. .10. ?he variation of the sodium flux yith time as a function of the magnitude of E, the clamp potent&l. The magnitudes of E in mV are shown on the curves.
permeabilities (GN, GK) tie high, the flux startseaflier, but also shuts-off earlier. If the permeability were to- start At a..low value and change gradually to the higher value and then back, one would sweep out a niuch smoother curve where .the flux : - : would not s$$off a%abruptly_ -. : ,The--paraqet& mated with thk movement of the rapid -charge, GX (the conduct&&‘-and PCT (the, f&ion, of ch&ge that is rapidly cmobile), have no infltienk on the‘ positive JN.‘Sixice .the.seI&meters do :not influence~the.ion flti kinetics;-it is probable @at. they are associated wi& the me&a&m that is-responsible-for the increase -in -membrane p&meability. _ .. ., -- j .. Fir&$ the cuqent levels dtig{a ckmp are much too &gh becauk of the high initial $erineabi#ty andthese would-be-b&x&t d&n to more reasonable values if a gradual .perqz@lity change ~&ch&nisni were -operative. This type of mechanism : YoUId alter the c&rent tqxkients significantly,~ b&cause -the curkent due to- the enhanced sodikn flux would then constitute a much &eater fact& of the totd. : : ‘-. .iTIi. IMPLICAiItiNS
FOR EXiXTATIilN
AhECHAiISM
.’
:-
.:
From t&&&s given in the&w0 papers it appears that a non-selectioe &r&se hi: c&on &&ji&b&t$ could .give ri& to. an apparently selective transient sodium .iqfh&.- Thk : s&l&lai; r&chan.$~_ &t- w&d. u&erQe :stich a ,Chtigc -& mor& in. :-ke$ng.witii the older ideas 0: &&tein i&] .&&with the~cti~ntl~ ptev&li& id&k .. :_ . . ~‘. : . ..
:
426 TABLE4
”
:.--
-.-
..
properties of channel pro& 1
Ionic mobility in the aqueous surface compartments is low M
2
Bindingof cations (Na+,
3
Cation release rate is fast Characteristic tirnc is l-10 ps
4
Upon depolarization there is a non-selective increase ih cation permeability to at least one. order of magnitude greater than that of aqueous medium
5
If formation of channel is by aggregation or disaggregation of oligomeric subunik, thiscould caused by:
Kf ) is substantial. Binding equilibrium constant is about 1 MS’.
a. Rapid shift of surface charge (gating current) b. Rapid changes in cation concentration (e.g. low Hf.
low Caz+)
.
be
at surface.
of a selective increase in membrane permeability to sodium.
The SCM has incorporated the selectivity shown by the membiane system in establishing the ion concentration and electrical gradients which characterize the resting state, but it has minimized the introduciion of selectivity beyond that. For example, the ion mobilities in the surface compartments and the cation-binding and release rate constants are assumed to be equal .for all ions on both sides of the membrane, an assumption which is almost certainly unjustified. Howtiver, it appears that the differences between the ions are relatively insignificant at this level of approximation. Bearing in mind that the model is approximate and that the results are far from complete, it appears that positive sodium fluxes can occur if there is a mechanism to increase the permeability as a result of depolarization. This raises interesting questions about possible mechanisms, but the results of our parameter study suggest ranges of properties of the channel protein that presumably give rise to the increase in membrane permeability. These are summarized in Table4, where approximate magnitudes of M, KEQ and BR are given. Since aggregation-disagtion mechanisms involve changes in surface charge and ambient ion concenirations, especially pH [7], these are probably reasonable possibilities for the basis. of a molecular mechanism. ACKNOWLEDGEMENTS
This investigation was supported by research grant PCM 78-09214 N.S.F. and NOOO14-SO-C-0027 from O.N.R.
from the
REFERENCES 1 M. Bl+ and W.P. Kavanaugh. Bioekkochem. Bioenerg.. 6 (1982) 427: : 2 R.D. Keynes and E Rojas. J. Physiol. (London), 239 (1974) j93..
‘. .: _ __ .: ” : .f .1.. :
..
I -:-_
.i-
:
:
:
..... :. . ‘~. -..
.. 457 .
. . .. . . .- --.. .3.:&L &d&in.Pr&~ k-&c., Ser. B: 148(1958)~‘1.~ 4 D++;~Gilb;e& i&Biophjsi& a& physiology of Excitable Membranes, _W_J_Ade!my. Jr_.(etor), Van I-:Ngtra~&NeW.~ork, 1971. pp_ 359-$78: : 5 ‘R.&&xii& id ~Bidphysics.and Physiolqy of Fkc&ble Membraks,. ii. Adelman, jr. (Editor), Van ‘. -,No.&a& N~~Yc&l971; pp. 96-124.. 16 J.-Bernstein; Pfbie&s kc< G-es. Physiol. Me&chin Tieren, 92 (1902) 521. .7- MC Blank; cdlloids Surf.,’ I.( 1980) 139:
._
._ --.
_ .,-
439 -.
Depart&t of Physiology, C&ege of Physicians and Surgeons, Cohmzbia Unibersity, 630 W. 168th Street, N& Yorlr;-NY IO032 (U.S.+) W. @ATRICK
KAVANAUGH
Eiectmnic Associates Inc. Wat k.ang Branch, NJ 07764 (U.S.A.) GEtiEtiEVE
CERl=
Department of &ectricaI Engineering, Cohunbia University, New York, NY 1002S (USA) -(Manuscript &c&d
December 7th 1981)
SUM&Y
Solutions of the SCM equations under voltage clamp show that a transient inward sodium flux can oaxu at certain set&s of the parameters. If there is a non-specific increase in cation permeability upon depolarization. the positive sodium flux arises because the ion concentration changes in the electrical double layer compensate for the decrease in the ekctrical driving force+ while parallell changes in the potassium ion concentrations lead to a decreased (but still outward) p&&m flux. The sodium flux is quite sensitive to the magnitudes of the ionic petmeabilities and mobilities,, as well as the ion binding equilibrikn and release rate constants. The values of the parameters determined here suggest magnitudes for the properties of the membrane components associated with the pen&ability change.
I. THE SCM EQUATIONS DURlNG
VoLTAGE
CLAMP +*
The basic equations. of the Surface Compartment Model (SCM) have been developed [I] in terms of a model membrane system that consists of five discrete regions:___. .-. .: - an .outside bulk (ra.e&oir) phase, compktment 1, - an .outer skrfa=, compartment 2 with&pacitance CAPI, - a membrane having- surface charges X2 and X3 on the two faces, and a dielectric tipki+~~ CAP2, .- ad inner sllrface ‘&mlGrtm@t -3 with .capacitaixe CAP3, 1_. k inner$u& (reservoir) phase, compartment 4. ., : Resin+
..-
: ‘the 6& ~Inte&ta~onak Sy&p+&~& on Bio+ctrochemistry ,and Bioenergetics, Kiryat .. Auavim(Is~eI),‘June 281h-iuly 3id‘l98i. :. **. &ci+ of combu!er pr&am re&irem~ts~th~ sym&ls used in this paper.are not the usual tin&. Thel
$i&ols
ai
u&&&d
.--.
the cok&pokling~&&iti&~are _
,-. .. &2+98,82~OO&--ti,~~2aao/so2.75
_‘.
.-:c.
.
O-.1982 l&&r
$vm.+-ihe -.
‘et _a&iiiu Table. 1.
Sequoia SAA.
’
_-.
440
., .::
._’
.-
::,
We have derived a system of 15 first order non-linear -differential- .equations. to describe the changes that occur in the concentrations of ions and the eiectricaI potentials in the two surface compartments during the’ fIow .of current. (We, have introduced two additional equations for the surface charge to ahow us to tie into account the gating currents, but these have not altered the basic structure of the system.) Changes in the ion (N, K, A) concentrations are given by: ~2=fl/L2)*@Nl-JN-PN-&22)
0)
K2 = (l/L2)
* (JKl
(2)
Az=(l/L2)
*(JAI -JA-PA)
(3)
fi3=(I,‘L3)
* (JN+PN--JN3--33)
(4)
K3 = (l/L3)
* (JK + PK - JK3 - K33)
6)
A3 = (l/L3)
* (JA + PA - JA3)
- JK - PK - K.22)
(6) In the equations, L’s are compartment thicknesses, J’s are fhzxes given by the Nemst-Planck equation, P’s are pump fluxes and N, K, A are sodium, potassium and anions, respectivefy. Changes in the bound cations are given by: &22=BF*XZ*NZ-BR*N22
(7)
&22=BF*X2*K2-BR*K22
(8)
fi33=BF*X3*N3_BR’N33
(9)
K33=BF*X3*K3_BR*K33
(IO) where X2 and X3 are the total surface charge (fast f slow), and the ratio of the kinetic constants, BF/BR = KEQ, the binding equilibrium constant. Changes in the anionic surface charge on the membrane are given by: %2 = - (&22 + K22) - JXM
(II)
X3 = - (riI33 + K33) + JXM
02)
Equations 1I and 12 are split to aIlow for a fast moving fraclion of charge (labeled F) and a slow moving fraction (labeled S). This yields %2s = - (ti22 f k22) - .Ixs X3S = - (&I33 -t K33) + JXS X2F=
-JXF
X3F = JXF Finally, the currents in the surface comp~tments voltage clamp (I% = 0) are given by: I=FA*(JNl+JKl-JAI)-CAPl*&Z
@Ia) ..=::il _-.--,._~ _\ ,. ,xza,:.
‘-;
(Ilb)
(I2b) and in the membrane during -(I;) .’
.:
441 _.
.‘_
04) (J-k-+ JK .- JA - JkM + tiN + PK - PA) + CAP2 * (&
I = &i -
- k3)
(19
-where:thi.E’&- are.thi: ~le&%~oter&ls, ik. is the Faraday constant. and CAPl, CAP2 and :Ck3 ark fhe ca~~&nuS & the respective ,compartments. (We have retahiec-the.l% term -becauseit-is n&ssary ibr. the initial current tipon imposing the ~lamp~).Equations(13)-(U) &m be solved for derivatives of the E2 and E3 as well as -1. The 15 equtitions, (l)-(H), can be .solved for.. the 14 state variables and the current. :. / During &oltage &mp traqsient; E4 is set_&@ to a value that is different from its resting value, and-.the equations are solved for various values of the parameters. The symbols ‘used in the computer solution are shown in Table 1 where concentrations are C’s, bound ions are BC’s, fluxes are F’s and pump fluxes are P’S Where two indices qre u&d, @e first one is for the charged species (I= anion, 2 = sodium, 3 = slow charge, 4 z potassium and 5 = fast charge), and the second is for the phase in which the’charg&%pec+s is found (for concentrations) or from which it originate; (for fluxes). The relation between the symbols used in the equations and in the computer program is shown in Tables 2 and 3. TABLE
1
Diagram of the Surface Compartment Model illustrating the symbols used for the ions, compartment and fluxes in the FORTRAN program Outside surface
Inside
Compartment
Outside
ThiCbesS
Reservcir
SL (2)
Reservoir
Potential Capacitance
E (l)=O
E (9
E (4)
Anioli Passive. flux Pump flux
c 0,
sodium Bound sodium Passive flux
C(2,1)
Membrane
1
fzt~~
1
I
CAP (1)
1)
F(1, 1)
C (1. 2)
+
7
c (2, 2) BCCL 2)
-
-_,
F (2, 1) -
Pump flux Ch&e
Pota+sdm Bound potassium
PaGive- flux
C (3.2) :C (4. 1) F(4,
BC (492)
-_,
1) -
c 62) 3-.
-
-
(f&t)
curt-en;.‘
-
C (4.2)
‘.
Pumpfhk tih&&
-
-
I
I-
,
--F 62)
-
L
+*c
(5. 3)
FA
GM4 2) GM(I, 2)
GM@ 2)
P(4,2) P(l, 2)
N22 K22 x3 N33 K33 SL2 SL3 JN JK JA
moles/(cm2 s) moles/(cm2 s) cm/(s mV) D
WkmVa
5*00E-9 1.00 E-9 9.65 E4 cm/(s mV) a amps/mole
moles/cm’
5.09 E-11 1.11 E-II 5.00 EC10
E2 E3 N2 K2 A2 N3 K3 A3 x2
Equation
Symbol
mole.s/c~ moles/cm2 moles@? moles/cm’ mV mV moles/(cm2 s)
moles/cm’
Units
4.4 E-4 I.0 E-5 4.5 E-4 5.0 L5 400E-4 4.5 E-4 0 -65 -8.44 E-11
Magnitude Program E(2) E(3) C(2,2) (x4,2) C(l, 2) c(2, 3) c(493) (x1*3) {(x3* 2) CR 2) W2,2) BC(4,2) (c(3*3) c(5* 3) BC(2,3) BC(4,3) W2) W3) F(2,2) F(4,2) F(1,2)
Initial Conditions
2,47 E-12 1098E-l 1 3888 E-8 6,d9 E8 ‘aI4 E-11 -509 E-l 1 -l,ll Bll
2,B9 E-IO 6.56 El2 2095 E-I 1
-46 -78 2473 E-3 6021 &5 7,25 E-5 8,38 E-5 6,70 E4 2969 E-4 1006El0
Magnitude
’ Thcsc ‘inobility un$s arc consistent with *c flux equations but differ from the conventional units for mobility,
OK GA’ FA
PK PA GN
w* 1)
NI KI Al N4 K4 A4 El E4 PN
C(4, 1) c(k 1) c(2*4) (x4,4) w 4) 41) Y4) P(2,2)
Program
Equatiom
Symbol
Constants
Numerical data for SCM calculations
TABLE 2
moles/cm2 * moles/cm2 cm cm moks/(cm2 6): molcs/icm2 5) mo&s/(cm2 6)
moles/cm2 moles/cm2 moles/cm2
moles/cm’ moles/cm’ moles/cm’ moles/cm’ moles/cd moles/cm2
mV mV moles/cm3
Units
I, ’ ,’
:
443 TABLE3-
Paramktersin SCM &Iculatio&
Symbols f .KEQ,BEQ BRCAPI CAP2 CAP3 .Gx M
..
.hiagnitudc
Units
‘comments
1000 SE5 l-E-4 .8 E-7 1E4 1E4 1 E-8 to 1 E-10
cm3/moIe s-1 F/Gil? F/cm2 F/cm2 1 s-’ mV-’ cm/@ mv)
fixed for maximum adsorptions corresponds to a characteristic time of 2 ps
II. OUTLINE OF COMPUTER
total series capkitance must give gating cur&t
-0.8
pF/cm2
characteritics
PROGRAM
equations have been solved on a PDP 11 minicomputer using a FORTRAN that en&s the constants and parameters of the system and then proceeds to evaluate the equations listed in the previous section. The algorithm the following: (I) The fluxes are evaluated from the C’s and Es. (2) The derivatives of the ionic concentrations, the bound ions and the surface charges are calculated from equations (l)-( 12). (3) Equations (13)-(15) are solved simultaneously to give the total current TI and the derivatives of the surface potentials E2, 6.3. (4) The above 14 derivatives are integrated with an initial step size of 10 ps, using a fourth order Runge-Kutta (with variable step size) program supplied by Digital Equipment Corp. (5) The integrated values are then used to reevaluate the fluxes and the new values of the derivatives for the next time step. Some segments of the program are given below. The
program
C C INITIALIZE c
VARIABLES-NOTE:
DO&_13
44)
= -20.0
.-
40 A(1, 1)~
IdDO A(l, 2) = CAP(I) A(I, 3)=O.ODO A(2,.2) =o.oDO AG3) = - cAp(3) A(3,2) ?-7C&P(2)-_ A(3,- 3).?.CAP(2)~ _ sL@j = x(2)/(+(1,2) swj
=~qyc-ca :
; -: +.c(2,2)+ c(4,2)) 3) + c(2,3) + c(lt, 3))
FOR VOLTAGE
CLAMP
444
DO 50 I = 2,4,2 DO50J=2,3 C(3, J)=(l.ODo-Pm) C(5, J) = PCT * X(J)
* X(J) _
50 BC(I, J) = BEQ * X(J) * C(I, J) C C POTENTIALS, FLUXES & PUMPS-INITIALLY
K = 0 FOR PUMPS
C DO2001=1,5 DO2OOJ=1,3 IF((C(1, J).LE.O.ODO).AND.(U(I, J).GT.O.ODO))F(1, J) = O.ODO
IF((C(1, J + I).LE.O.ODO).AND.(U(I, J).LT.O.ODO))F(1, J) = O.ODO IF((C(1, J).LE.O.ODO).OR.(C(I,J + l).LE.O.ODO))GO TO 200 U&J) = 5.8Dl * DLOGT(C(I,J)/C(I,J + 1)) + (- 1) ** I * (E(J) - E(J + I)) F(1, J) = GM(1, J) * U(I, J) * (C(1, J) + C(1, J + 1))/2.000 200 CONTINUE C C CALCULATE PUMPS IF K = 0 C IF(K. NE-O) GO TO 240 P(l, 2) = -F(l, 2) P(2, 2) = -F(2, 2) P(4, 2) = - F(4, 2) C CCALCULATE DERIVATIVES OF BOUND AND FREE CONCENTRAC TIONS 240 DO 250 1=2;4, 2 DO 250 J=2,3 DBC(E, J) = BR * (BEQ * X(J) * C(1, J) - BC(1, J)) .. 250 CONTINUE DO2601=1,5 DO 260 J=2, 3
DC(1, J) = (F(1, J - 1) - F(1, J) + P(1, J - 1) - P(1, J) - DBC(1, J))/SL(J) IF(I.EQ.3).0R.(I.EQ.5)) DC(I, J) = (- 1) ** (J - 1) * (F(1, J) + F(1, J - 1)) IF(I.EQ.3)DC(I, J) = DC(1, J) - DBC(2, J) - DBC(4, J) 260 CONTINUE C C CALCULATE DE’S AND TI C B(l)=FA * (-F(1, l)+F(2, l)+F(4, 1)) B(2) = FA * (-F( 1, 3) + F(2, 3) + F(4, 3)) - CAP(3) * DE(4) ‘. B(3)=FA * (-F(1, 2)+F(2, 2)+F(4,2):P(l, 2)+P(2, 2)+P(4, 2))$FA (- F(3, 2) - F(5, 2))
*
445
-_.
:--
%I .=-(B( 1j .* CAP(2)/CAP( I-) + B(2) * CAP(2)/CAP(3+ ‘- (A( 1;11) ,* CAq(2)/(CAP(.l) .+ .A(2, 1) :* CAP(2)/CAP(3) DE(Q=:(B(l)-TL)//CAy(I) : .: ‘y. ._. : .DE(3) + (TI-B(2))/CAP(3)
B(3));$_A(3;- -1)) :
The voltage ‘clamp is. imposed as a step with rapid exponential rise with a time ~constant of ‘l.kX 10’ S-’ so E4 changes rapi_dly and is within 1% of the clamp voltage at about 30 ius.:(The smooth variation ‘of E4 is necessary for the stability of the numerical integration algorithm.) The equations used are as. follows:
C-
:
C-CALCULATE VALUE OF INPUT VOLTAGE C IF(K.EQ.O)EINIT = E(4) TC = 3D0 * DLOGT(3DO)/STEP E(4) = EINIT - (ECLMP) f: (1 DO - DEXP( -TC * TIME)) DE(4) = -(EINITECLMP) * TC * DEXP(-TC * TIME) Under voltage clamp conditions (i.e. E4 is constant), all of the concentrations listed in the two reservoirs are fixed. Within the two surface compartments, the ionic concentrations are given by integrating equations (l)-(6) and the surface charge and bound ions by integrating equations (7)-(12). (SL2, SL3 and the pump fluxes required by the equations arc obtained, along with the values of the -other parameters.) The only other unknowns, E2, E3 and TI are obtained from the three equations for- the current, equations (13)-( 15). This enables a complete solution of the SCM. The parameters which must be initialized in order to make the calculations are: (1) Two of the three. binding parameters (KEQ, BF, BR). (2) Three ionic.mobilities in the surface compartments, Ml, M2, M3. (As a first approximation they will all be assumed to be equal to M.) (3) .Four mobilities in the membrane GN, GK, GA, GX. The first three are fixed by thenknown steady state fluxes, and GX is set so that the effect of the initial JXM does not exceed 0.3% of the total fixed charge flow [2]. (4) Three capacitances, CAPl, CAP&CAP3; Although it appears that twelve parameters must be initialized, a situation which ‘ought to enable :the description cf almost any kind of curve, upon closer examination most of these valuesare fixed.by experimental observations. From the following discussion it. yill become clear that the greatest uncertainty is associated with the choice of values fo? BR and M. II;. $‘ITIN~~O~ :
-...; 1. The
CONDITtON!i
:
Nl, _Kl, Al, N4, .K4.and _A4 are fixed at values normally the,gi+ $qtid axon 131, and specific values are g&en in .Table-2; El, as previously; is taken as- reference potential in’ the SCM and assumed to
bulk- coqFntrago&
f&rid-f& kention+ :
JNITL+ ., _
-’
:
.- :
446
have a value of zero, ‘while E4 is the normal resting potential; --The.fixed{negativesurface charge values for X2 and X3 and the surface values of goteitial FQ:ai~dE$ (which result from the surface charge distribution) are based onthe ~i&rlts~of Gilbe& [4]. Values of initial conditions for N2, K2, A2, N3, K3, A3 can be derived inthe following manner. In the steady state, the electrochemical p_ote@ials :for~the surface compartment fluxes are zero. Therefore, the el&ri&l field and chemical pot&&ls are equilibrated for each species in the internal and external surface compartments, yielding the following relations: 58 log(Nl/N2)
= -E2
06)
58 log(KljK2)
= --El
(17)
58 log(Al/A2)
= +EZ
(18)
58 log(N3/N4)
= E3 - E4
(19)
58 log(K3/K4)
= E3 - E4
(20)
58 log(A3/A4)
= E4 - E3
(21)
Since the bulk concentrations (NI, Kl, Al, N4, K4, A4) and E2, E3 are fixed, it follows that the state variables N2, K2, A2, N3, K3, A3 are determinable. SL2 and SL3 are calculated from the following relations: x2
SL2=N2+m-/Q
(22)
SL3=
(23)
A3 N3+K3-A3
From the surface compartment concentrations N2, K2, A2, N3, K3, A3 and .the surface potentials E2 and E3 in Table2, it can be seen that the electrochemical potentials across the membrane are not equal to zero at steady state. UN = 58 log(N2/N3)
+ E2 - E3 = 119.87 mV
(24)’
UK = 58 log(K2jK3)
+ E2 - E3 = -27.83
mV
(29
UA = 58 log(A2/A3)
+ E3 - E2 = -65.09
mV
(26)
The leak fluxes JN, JK and JA, which are offset by the pump fluxes, c-&i, be computed for steady state conditions in the following way: JN=1/2*(N2+N3)*GN*UN=-PN
(27)
JK=1/2*(K2+K3)*GK*UK=-PK
(28)
JA=1/2*@2+A3)*GA+UA=-PA
(29)
JN, JK and JA are known within certain limits [5], in which case GN, GK, and GA can be derived. Once set, there is no variation in GN, GK, and GA. HoweVer, ifthe electrochemical potentials change (i.e., the initial ion concentrations are changed), the corresponding pump fluxes (PN, PK, PA) must be recomputed to reStor+ a net initial flux of zero in the system. It should--be noted that ihere is a-f&r .amor;l;t of ..-
__.
.. ‘_~
:..
447
.,
(.
‘.
varia~tin~.~~-thk &b&&d
vah& of t& ‘stkady-sta!e fh&, ‘.:.._ : 1 .. ~ I
_j&;be p-&i;,
so thai-thesevahws may
--:. CX is +t ~~~at.~=~~a~~dc of FA * J%M, i.e. the currentdue to the mobile r&&rn&&nti compbnent$ yill give -theproper mtignitudeand duration of the gating currents[2]__.: .:_.;.:: ::. ~..- -Y._. :.‘I!heinitial~&&itio~ for thestate v&abl&s dkkcribingthe bouud ‘ions, N22, K22, &I33and~K3~‘arc~aUdeterminablefrom the expressionkgiven in ~cquations (7)-( IO). ti setting$422 = &22.=33$= @33 =.O, we obtain;.
N22.=KE+N2-*X2
(30)
&2=I(EQfpX2_
x30
N33=KEQ*N3*%3
(32)
K33=KEQ*K.3*%3
(33)
This-completesthe descriptionof the set of initial conditions to be used. IV.VALU~
OF
THEPARAMETERS
The grious&&uts and initial conditionsof the SCM have been chosen in the above section, and vahks assumed for the parametersto be studied are shown in Table 3, These includethe capacitances,whichhave been discussedin general[ 11,the ion mobilitiesand the constants associatedwith ,ion binding. The capacitanceswill be disc+%+ in a separatesection. The ii&&s for the mobilifiesgiven in .Tible 3 are ~~WEECIto be equal for Na, M and typical dons A in solutions ch_arageristic of the exte_mal/intemalsolutionsof the squid axon. Their magnitudesare not lgmm and one can only assume that they -are,loycr than comparabb values in. aqueous solution. ‘J”hegate.~&L&I+ .BJ?a&d BR con&&s in equationsQ-( 10) are assumed to be thy same for Na and.5 ions, and are also assumedto be the same in the two surface compart~eqts. :The ratio qf BF at@ ,BR de&mines an equiIib.rium ,constant KEQ .which will have the same v&e for each of. the bound ions (N22, K22, N33 and k33). An u$perJimit for K;EQ c&t bc determinedfrom the following relationsand arguments: _. ~_
S2=X&N22+K22
: :
:
(34
S2 &sthe ,n&&+erof bkmin~ sites on the kternal surface of the membrane. For no +p&iable~ flow of chargeinthe membrtie(i.e. JXM-= 0) S2 is a constant. For ions bot$d.at all sites (i.e. X2 $=O):.: sgL~+&
_.T : : .. ‘.
_ .I, _‘,; ,. .
_’
;I.
,:
(W
From-~hj&%.l a& geometrical~~idera~~n~.~*~~g the size and structureof ‘th&$&$&uc m&terikJ &- es&& of ‘theupp&rlimit (S2MAX) on S2 carrbe mkde of..* .~~~i~~.~~.~ ‘m&w. ihat.each site ti~&~ & z&e+ ‘of 40 A2). Therefqre ~2n;rkx=~~~.~~~;;~wh~~‘=‘O~-: _.‘_- _‘I-.. ‘-..._.., ;- _ 1 _I -.I. (36) ._._ ’ _‘.a.:..- .. _..r : - .:, :z _. ,:,._
._
..
..-.
. :
~
44s
From equations (7) and (8) we have N22 KEQ=X2*N2
(37)
and K22
KEQ=m*m
On rearranging and adding equations (37) and (38), we get KEQ’X2*(N2+K2)=N22+K22 or (N22 f K22) KEQ = X2(N2
(39)
+ K2)
An upper limit for KEQ can be estimated by substituting SZMAX for (N22 + K22) and the known values of X2, N2 and K2 in equation (39). This yields a value of 1000 for the upper limit of KEQ. A lower limit of zero would mean that no binding was occurring and this is not of interest for the SCM. A similar argument could be made for S3, but since we have assumed the same rates we have the same KEQ’s. V. EFFECTS OF MEMBRANE CAPACITANCE
When a voltage clamp is applied so that E4 = -20 mV (i.e., a depolarization of 45 mV), the electrical potential across the membrane changes from the 32 mV in the resting state to a new potential, E23. The change is virtually complete in 30 &s, and depends upon the magnitudes of the capacitances chosen for the three regions defined in Table I. Since the three capacitances in series must have a combined-value of 0.8 pF/cm*, it is possible to obtain a range of E23 magnitudes, as shown in Fig. 1. If we assume CAP2 = 0.8 pF/cm* (a readily accepted value for the dielectric capacitance) and the two surface capacitances CAPI = CAP3 = 100 rF/cm* (a value close to measurements at mercurylwater interfaces), E23 = - 12 -mV. If we divide the capacitances more or less equally CAP2 = 3 pF/cm*, CAP1 = CAP3 = 2 pF/cm*, then E23 = +23 mV. The current transients in these two cases are shown in Fig.2, and despite the introduction of three capacitances, the early current across the membrane behaves as if there were a single time constant. The electrical double layer capacitances. are apparently undetectable, and one gets the same k&l response if the -total capacitance is the same. The different plateau currents at longer times are due to different values of E23. (See Fig. 3). When all the capacitances are approximately equal, the changes in E2 and E3 are approximately the same. When CAP2 is, much smaller than the double layer capacitances, the changes in,E3 and E4 are approximately the
same and E2 is virt@ly unchanged...These differences in the plateau values of E23.
two sets .of cqnditions
E23, the electrical driving force for the flow of cationsinto
lead to, large
the cell, decreases and
449 .-
Fig. 1. The plateau voltage across the membrane reached 1 millisecond after a voltage clamp is imposed, E23 (in mV) oersus the total capacitance of the three capacitances in series (in pF/cm2). The magnitude of the dielectric capacitance CAP2 (in pF/cm’) is indicated on the lines and CAPl=CAP3_ The parameters used were as in Table 3. except that M= 1 E-7.
reverses sign with an increase in the total capacitance. To increase the probability of a positive (inward) JN, it would appear to be advantageous to choose a low total capacitance along with a relatively high dielectric capacitance so that there is a significant electrical driving force for sodium ions into the cell. However, this is not so, as we shah see below. .The electrical drivmg force for sodium ion influx may decrease, but there are significant. compensatory changes in the chemical driving force that are due to ionic equilibria in the surface. This will be discussed in the next section. Vi. CQNDITIONS
FOR REVERSE
SODIUM
FLUX
On choosing different values for the parameters, solutions for the SCM system of . ‘equations give the-expected response of a passive system, I.e., a steadily decreasing negativecurrent :until *he establishment of. a new steady. state, with gradual and slight .changes in the variables-&f the system..However,. there are conditions under which the;SCM equations respond tith a. positive,. transient sodium ffux.~Fig. 4 shows one-such. case, Where .the permeability .of the membrane to sodium .and potaissium 3s increased : several orders -of m&guitude above resting values. The
_-__:--: ..
.-
-. _ :
20
40
60
100
used were the Fig. Z A semi-log plot of the total current in A/cm2 LWSUSthe time in gs. The parakers same as in Fi& 1. The two -es relate to approximately the same total capacitance, but different individual values. The 0.8 is due to 100.0.8.100 pF/cm2 and the 0.75 is due to 2.3.2 pF/cm2. for CAP], CAP2 and CAP3, respectively.
transient reversal of IN, i.e., an inwardly directed sodium fhk, is accompanied by increases in N2, the sodium concentration at the outer surface, and decreases inN3, the sodium at the inner surface, as shown in Fig.5 Apparently, the chemical potential difference more than compensates for the unfavorable electrical potential difference noted in Fig. 3. Solutions obtained for cases of favorable ekctrical potential differences, e.g. the second case shown in Fig. 3, give a lower positive INi The positive transient IN of Fig 4 is seen when the permeability of the membrane to both sodium and potassium is raised several orders of magnitude from..the resting values in Table2. When GN is increase&selectively (while GI&remainS~fhe same .a~.. in the resting membrane), there is no current reversal. .When. GK- ii -increaSed selectively, there is a slight reversal of IN that is three orders of magnitude smaller than in Fig. 4. It appears that in order to observe a positive IN of ap&oximately the : right magnitude,-both GN and-.GK must be iricrcascd.-(An .iricreasedGA- tcrids to . ,; ‘: -..z.. decrease the positive IN.) :/-..- _. The positive JN is quite sensitive to the magnitudes of the ionic +rr&abilities :
:
.’ LI_
‘.
451 -. . ...
._:
:
----------
...‘._
_.
:
---
-. 3
Fii 3. ‘he variations’of the potentials in the two cases shown in Fig. 2. Tbe_solid curves are for the 0.8 pF/&* and the dashed qmves for the 0.75 ~F/cm*. EA. the clamp voltage is the same in both cases, but U3 has opposite signs in the plateau regions.
:
452
and mobilities. The dependence .of the positive JN .on GN
I ., :.. ..
.
and ~GK~is~shownin..
Fig. 6. It is apparent that the membrane p~rmeabilitiqs must be at 1-t 5E&o’h$%e approximately the right magnitudes of. fluxes and times. From-Fig.‘7 we s+ that. :$e.
N2
K3 K2
N3
0
t
I
a2
I
,
0.4
I
t (ms) t
0.6
I
I
0.8
I
,
1.0
Fig. 5. The changes in the cation concentrations in the surface compartments are shown under the same conditions as in Fig. 4 and the solid lines in Fig. 3.
ionic mobility in the surface compartments must be at least an order of magnitude smaller then the GN end GK in order to sustain the concentration gradients (shown in Fig. 5) and give rise to a positive JN.
The binding equilibrium parameters also influence this flux markedly. In Fig. 8 we see that the equilibrium constant must be over 100 in order to have: an appreciable positive JN. The time of the JN peak is also infiuenced ‘-.by this parameter, but not as much as by the ion release rate constant shoF..in Fig, 9. The value of BR must be about SE4 or greater to show a peak at low values of time; The solutions of the SCM equations -under_vohage ch%p have been described for various values of the parameters. However, in order to approach &i&&~s~&at are
closer to the in oioo situation, it wouid be desirable to ik@Uce changes in permeability during the solution, but the me&a&m which could &e rise to Sudh changes is unknown, and the problem is also much more difficult; Serious distortions are introduced into the solutions because of the constant pernxabilities z&d .the high values of the resting (and ion pump) fluxes that must be incIuded when-the. membrane permeabilities are set to high values. These distortions probably:ac&nt for the peculiar responses : of the SC&f equations to- changes in jhree -:of. _t& parameters, the clamp potential, GX and-PCT. .. ..-I _. .I‘ .. Fig. 10 shows how the positive:JN depends upon. the ,magnitude of, .theT_c@inpi: potential. As one incrkses~the-magnitude of the depolarization, (i.e.$sthe~_ma@i~_.; :.
: ::
..
-- ; .> .,‘,”
_ .:..
.-_ .I
: .:-.
._
.‘.
.,
..
__
;
m.. _.
.j, 3- s p Y.
_, . -.
-:’
..
_’-.-
;
-3 -
Fig. 6.. The -variation of the sodium flux -with time as a function of GN and GK the membrane permeability to th+ two cations. The magnitudes of GN=GK in cm/(s mV) are shown on the curves.
.
Fig. 8. The variation of thesodiumfluxwithtimeas a function of KEQ, the bindingequilibrium constant. The maguitudcs of KEQ in c&/mole are shown on the cumes.
tude of the clamp potential becomes more positive), one expects that the positive JN would increase. This cleariy occurs up to about -40 mV and qualitatively up to
about -20
0-f
-1
mV. However, at higher values of the clamp voltages (e.g. i-20 mv)
0
-
: -2
-
Fig. 9. The variation on the sod&m fl& with time as a function of BR, the ion rekase rate constant The magnitudes of BR in s-’
are shown oil the c&es.
_
:. . ..
-.__L.
. .
:
...~
:
:_
:.
..,_-
455. -.
:
~.. _
is~&t&ji~ a .d&i&e. k:Mi &’ kxpecte& at the m&e, positive pqtentials; the. fi&ikn’fl&&~$t; to.&&&& ita++ but- t&5 peak fluxes tic. displ+ed to ,s+orter i-t&i&, tid :t$is’@nits’ the m&&itQde:of -the ri_se.~..From, Fig. 6 we- SF _.@a~ *hen the . _ifi&
:
i 2 cu_ -a-
'. .. '.
:_.
..'
.. __
..
.-
: ~
.- ,- ..
..
_g :
coo+..'sma'
0.5
t (ms)
1.0
-1 -
Fig. .10. ?he variation of the sodium flux yith time as a function of the magnitude of E, the clamp potent&l. The magnitudes of E in mV are shown on the curves.
permeabilities (GN, GK) tie high, the flux startseaflier, but also shuts-off earlier. If the permeability were to- start At a..low value and change gradually to the higher value and then back, one would sweep out a niuch smoother curve where .the flux : - : would not s$$off a%abruptly_ -. : ,The--paraqet& mated with thk movement of the rapid -charge, GX (the conduct&&‘-and PCT (the, f&ion, of ch&ge that is rapidly cmobile), have no infltienk on the‘ positive JN.‘Sixice .the.seI&meters do :not influence~the.ion flti kinetics;-it is probable @at. they are associated wi& the me&a&m that is-responsible-for the increase -in -membrane p&meability. _ .. ., -- j .. Fir&$ the cuqent levels dtig{a ckmp are much too &gh becauk of the high initial $erineabi#ty andthese would-be-b&x&t d&n to more reasonable values if a gradual .perqz@lity change ~&ch&nisni were -operative. This type of mechanism : YoUId alter the c&rent tqxkients significantly,~ b&cause -the curkent due to- the enhanced sodikn flux would then constitute a much &eater fact& of the totd. : : ‘-. .iTIi. IMPLICAiItiNS
FOR EXiXTATIilN
AhECHAiISM
.’
:-
.:
From t&&&s given in the&w0 papers it appears that a non-selectioe &r&se hi: c&on &&ji&b&t$ could .give ri& to. an apparently selective transient sodium .iqfh&.- Thk : s&l&lai; r&chan.$~_ &t- w&d. u&erQe :stich a ,Chtigc -& mor& in. :-ke$ng.witii the older ideas 0: &&tein i&] .&&with the~cti~ntl~ ptev&li& id&k .. :_ . . ~‘. : . ..
:
426 TABLE4
”
:.--
-.-
..
properties of channel pro& 1
Ionic mobility in the aqueous surface compartments is low M
2
Bindingof cations (Na+,
3
Cation release rate is fast Characteristic tirnc is l-10 ps
4
Upon depolarization there is a non-selective increase ih cation permeability to at least one. order of magnitude greater than that of aqueous medium
5
If formation of channel is by aggregation or disaggregation of oligomeric subunik, thiscould caused by:
Kf ) is substantial. Binding equilibrium constant is about 1 MS’.
a. Rapid shift of surface charge (gating current) b. Rapid changes in cation concentration (e.g. low Hf.
low Caz+)
.
be
at surface.
of a selective increase in membrane permeability to sodium.
The SCM has incorporated the selectivity shown by the membiane system in establishing the ion concentration and electrical gradients which characterize the resting state, but it has minimized the introduciion of selectivity beyond that. For example, the ion mobilities in the surface compartments and the cation-binding and release rate constants are assumed to be equal .for all ions on both sides of the membrane, an assumption which is almost certainly unjustified. Howtiver, it appears that the differences between the ions are relatively insignificant at this level of approximation. Bearing in mind that the model is approximate and that the results are far from complete, it appears that positive sodium fluxes can occur if there is a mechanism to increase the permeability as a result of depolarization. This raises interesting questions about possible mechanisms, but the results of our parameter study suggest ranges of properties of the channel protein that presumably give rise to the increase in membrane permeability. These are summarized in Table4, where approximate magnitudes of M, KEQ and BR are given. Since aggregation-disagtion mechanisms involve changes in surface charge and ambient ion concenirations, especially pH [7], these are probably reasonable possibilities for the basis. of a molecular mechanism. ACKNOWLEDGEMENTS
This investigation was supported by research grant PCM 78-09214 N.S.F. and NOOO14-SO-C-0027 from O.N.R.
from the
REFERENCES 1 M. Bl+ and W.P. Kavanaugh. Bioekkochem. Bioenerg.. 6 (1982) 427: : 2 R.D. Keynes and E Rojas. J. Physiol. (London), 239 (1974) j93..
‘. .: _ __ .: ” : .f .1.. :
..
I -:-_
.i-
:
:
:
..... :. . ‘~. -..
.. 457 .
. . .. . . .- --.. .3.:&L &d&in.Pr&~ k-&c., Ser. B: 148(1958)~‘1.~ 4 D++;~Gilb;e& i&Biophjsi& a& physiology of Excitable Membranes, _W_J_Ade!my. Jr_.(etor), Van I-:Ngtra~&NeW.~ork, 1971. pp_ 359-$78: : 5 ‘R.&&xii& id ~Bidphysics.and Physiolqy of Fkc&ble Membraks,. ii. Adelman, jr. (Editor), Van ‘. -,No.&a& N~~Yc&l971; pp. 96-124.. 16 J.-Bernstein; Pfbie&s kc< G-es. Physiol. Me&chin Tieren, 92 (1902) 521. .7- MC Blank; cdlloids Surf.,’ I.( 1980) 139:
._
._ --.
_ .,-