Co-operative response of chemically excitable membrane

I. theor. Biol. (1978) 71, 567-585

Co-operative Response of Chemically Excitable Membrane I. Formulation:

Unified Theory of Co-operativity

HIROMASA KIJIMA AND SHAKO KIJIMA Department of Biology, Faculty of Science, Kyushu University 33, Fukuoka 812, Japan (Received 23 March

1977, and in revised form 29 August 1977)

The lattice-model of Changeux, Thiery, Tung & Kittel(l966) was extended in order to examine the co-operative response of chemically excitable membrane and the exact mathematical correspondence to the Ising model was shown. In this model, two conformational states S and R with different affinities for the ligand are assumed to be accessible to each

protomer, which is interacting with the nearest-neighbor protomers. The model is applicable to any kind of symmetrically interacting system consisting of oligomers and lattices and is an extension of previously proposed

models of allosteric protein. It includes the model of Monod, Wyman, & Changeux (1965) and that of Koshland, Nkmethy & Filmer (1966) as the extreme cases of the oligomer. By assuming that a state-transition

from S to R in a protomer is accompanied by a unit increase in conductance, the characteristics of dose-response curves of chemically excitable membrane are examined. The Hill’s coefficient nH of dose-response curve, the measure of the co-operativity, is shown to be proportional to the square of the mean fluctuation of the state function, the fraction of protomers in R state.

1. Introduction There are two kinds of physiologically important excitable membranes: one is chemically excitable and the other is electrically excitable. The features of response are quite different from each other (Katz, 1966). Electrophysiological responses of chemically excitable membrane have been widely investigated. These are the endplate of frog muscle (Katz, 1966), innervated membrane of isolated electroplax from electric eel (Ruiz-Manresa & Grundfest, 1971), taste cells in mammalian taste buds (Beidler, 1971), insect bombicol receptor cell (Kaissling, 1974) and sugar receptor cell (Morita, 1972) etc. The response, the conductance change or potential change of the membrane, are more or less co-operative against the concentration of chemical stimulant or agoinst. That is, the concentration-response curves often deviate from the simple Langmur’s isotherm and become sigmoidal. 00.22-5193/78/0420-0567

$02.00/O

ID lY78AcademicPressInc. (London)Ltd.

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The strength of the co-operativity is, however, rather moderate in most cases. The Hill’s coefficients (izH) of most dose-response curves did not exceed the value two. But, there are some exceptions. Recent investigation on the sugar receptor in labellar marginal chemosensory hair of the blowfly showed existence of highly co-operative response (Shiraishi & Tanabe, 1974). One of the remarkable features of the chemically excitable membrane is that, in most cases, its excitation mechanisms seem to be independent of membrane potential, in contrast with the electrically excitable membrane whose functioning is essentially dependent on the membrane potential (Hodgkin, 1964). The change in ionic permeability or conductance of postsynaptic membrane of frog-muscle endplate produced by acetylcholine is independent of the membrane potential (Fatt & Katz, 1951; Takeuchi & Takeuchi, 1960). The Hill’s coefficient of dose conductance-change curve of innervated membrane of electroplax in the electric eel is also independent of membrane potential (Lester, Changeux & Sheridan, 1975). These facts show that co-operativity of response of chemically excitable membrane is not caused by the flow of ions through membrane which is drived by the nonequilibrium ion distributions across the membrane (cf. Blumenthal, Changeux & Lefever, 1970), but it may be caused by the mutual interactions between the receptor proteins (or between receptor-ionophore complexes) or between the subunits of them, like in the interactions of subunits of allosteric enzymes. In fact, the binding of isotope-labelled agonists such as acetylcholine and decamethonium in vitro to the membranebound acetylcholine receptor of electric organ from electric eel is co-operative (Weber & Changeux, 1974). But co-operativity of binding was completely lost when the receptor was solubilized by detergents from the membrane (Meunier, Sealock, Olsen & Changeux, 1974). There are two possible origins of co-operative response caused by the interaction of the constituents of the membrane. One is the co-operative interaction of oligomeric subunits (promoters) in a receptor or in a receptorionophore complex. Kirlin (1967) and Morita & Shiraishi (1968) analyzed co-operative response of electroplax and insect sugar receptor, respectively, based on the allosteric protein model proposed by Monod, Wyman & Changeux (1965). Another is the interactions in wide range between numerous promoters arranged repeatingly throughout the membrane, forming a two dimensional lattice. This was formulated by Changeux et al. (1966). Acetylcholine receptors in the innervated membrane of an electric organ of Torpedo was arranged in a close-packed lattice-like structure as shown by the electronmiscroscopy (Cartaud et al., 1973). Purified receptor was shown to consist of five or six subunits (Hucho & Changeux, 1973). There are, thus,

CO-OPERATIVE

RESPONSE

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569

two possibilities of occurrence of co-operative interactions in the cholinergic membrane. In this paper, we first extend and formulate the lattice model by Changeux ef al. (1966) and show that complete correspondence exists between this model and the Ising model. Its application to symmetric oligomers lead to a simple unified model of the allosteric protein, which includes both the model of Monod et al. (1965) and that of Koshland, Nemethy & Filmer (1966) as the extreme limits. Based on this model, we examine the characteristics of the co-operative concentration-response curves and Hill’s coefficients. In the subsequent papers the following will be presented: (:i) The difference between the co-operative dose-response curves of various systems (various kinds of oligomers, lattices with nearest neighbor interaction and lattices interacting with long range forces, etc.) will be examined based on the model proposed in this paper. When the curves with the same Hill’s coefficients are compared, there is a significant difference between the doseresponse curve of the oligomers with a few subunits and that of the lattice with long range interaction, and they will be experimentally distinguishable if ideal conditions of the model are maintained. (ii) If the ligand binds to the protomer not only in active state R but also inactive state S, the maximum response (r), and the Hill’s coefficient, ni become smaller than when it binds only to the R state protomer ((r),, = 1). Simple mathematical consideration will lead to the conclusion that the Hill’s coefficient cannot exceed the value two if the maximum response (r), for a ligand is smaller than two-thirds. This limit will not be overcome within the framework of two-state model. The extension to the three-state model will be proposed, in order to explain the co-operative responses of the insect sugar receptor ((r), < 3, & > 2). (iii) Correspondence to the experimental data will be showninvariousaspects 2. Mathematical

Formulation

Our model of chemically excitable membrane is based on the model of Changeux et al. (1966). But the associations of “protomers” defined by Changeux et al. is not limited to the two dimensional lattice-type but it can be oligomeric. The model has the following properties: (a) Two conformational states: R and S are reversibly accessible to the protomer. Transition of a protomcr from the state S to the state R causes unit increase g of the membrane conductance. (b) The protomer interacts with its nearest neighbor protomers. Thus, its conformation depends upon those of neighboring protomers. For simplicity,

570

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we assume that all the protomers has the same number of its nearest neighbors (z). Let us call oligomers in which all protomers have the same number of nearest neighbors as “symmetric oligomers”. (c) The protomer has at least one binding site for each type of specific ligands. The affinity of one or several binding sites towards the corresponding ligands is altered when a transition occurs from the state S to R or vice versa. (d) Interaction energies between protomers are independent of binding of ligand. First, we assume binding of only one ligand to a protomer. Let the total number of the protomers in the system (lattice or oligomer) be N, the numbers of the protomers in the state R and S be Na and Ns respectively and the numbers of the neighboring pairs RR, RS and SS be NRR, NRs and Ns, respectively. The following relations hold. NR + Ns = N, (1) 2NR, + NRs = zNR, (2) NRs +2N,, = zN,. (3) Because of these relations we can take NR and NRs as independent variables. The energy E of a state of the system with given NR and NRs is: E = NR~R + NSG + NRReRR + NRseRs + NSQS - (nRJR + nsJs) = (6s + hs)N + {(ER - ES)+ +(ERR - QJINR + &RR+% ~

NRS(~RJR+%Js)P (41 2 > where &R and ss are the energy of an isolated protomer in R and S state respectively and ERR, sSS, ESRare the interaction energies of the neighboring pairs of RR, SS and RS respectively. The binding energies of a ligand to the protomer in R and S state are denoted by JR and J, respectively, and the numbers of ligands bound to R and S protomers are nR and n,, respectively. The second row of equation (4) is obtained by eliminating N,, NRR and Nss making use of equations (l)-(3). The grand partition function 2 of the system is given, +

z = ,-~h++~essW .g, x { “EO nR!(~:nR)!

e-

e --B(u

+ f=essW

.

-

Bten-ES+t:(enn - ess))NnX ep(p+JRq

*NgSQn(N~IN~~l*e _

(

8RS

* {“Z,

ns!(z:n,,!

eB(p+Js)“s)*

-tb'(Zms-ERR-ESSWRS

,-S(ER-CS+~Z(EHR-ESS))NR.{~

+eP~~~+J~)}N~.

li NR=O '(1

+eS'~+Js'}N~.

gSn(NR,

~~~).~-f8(2ens-en~-ess)N~s,

(5)

CO-OPERATIVE

RESPONSE

OF

MEMBRANE

571

where /I = l/kT, p is chemical potential of the ligand and Q(N,, NRs) denotes the number of configurations of the system to arrange R and S protomers in different ways for given values of NR and NRs. ‘We can calculate and rewrite 2 as : z = rNz’ (6) Z’ =

F

qNn c R(NR, NRS).eT2flUNfls

Nn=O =

.go

NRS rNR

N;s

QWR,

NRs)-ANRS”=

(7)

1 +eB(r’+JR) q =

e-fl(

eR-eS+fz(eRR-Ess))

x

1 +eS(~+Js)

where U is the measure whether the neighboring protomers favor homogeneous pairs (RR or SS) or heterogeneous pairs (RS) and is given by:

when U > 0, (< 0) the neighboring protomers favor homogeneous (heterogeneous) pair and the parameter of co-operative interaction A is defined as, A = exp (- 4/?zU).

(11) When we put, kR = e -BJR, k, = eBBrs and I = eSP, kR and ks are the dissociation constants between the ligand and the protomers in R and S state respectively and rZis absolute activity of the ligand, which is proportional to the concentration of the ligand in dilute solution. We further define according to Changeux et al. the relative concentration of ligand a and the constant c as, a = A/kR

(12)

c = k,lk,.

(13)

l-l-a ’ = L(1 +ca)’

(14)

The parameter ‘1 is expressed as:

where L is the equilibrium constant between a R protomer surrounded by R protomers and a S protomer surrounded by S protomers and is given? by: L = eS’“R-es +t~(.Qtn- ess)) (15) t The parameter L in this paper corresponds to LIIN in the work of Mondo, Wyman & Changeux (1965).

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The reduced grand partition function Z’ is equal to the reduced partition function of Ising model (cf. Domb, 1960) in the presence of magnetic field, if we identify the two states R and S to the two states (+, -) of the Ising spin and put, q = e2BH, (16) where H is the strength of magnetic field in (+) direction. The conductivity increase per lattice or oligomer (AG) caused by ligand is given, AG = WR)g = N
= (NR) ---=--=

1 din

2’

NRzo

NRqNIN;s

‘tNR7

NRS)‘ANRs’2r

. (18) N dln q NZ’ The curve of (r) vs. log q is essentially the same as the magnetigation curve of Ising model, since log YJis proportional to the strength of magnetic field H [equation (16)] in the Ising model. From the last expression,
5

N;qNR

NR=O

c !2(N,, -

d(r) -=

Z’ >

N E R

o NRV~~

N;s

WNR,

NRS)*““~‘“‘}~

Nr/Z”

drl

= F (r)

NRS)ANnSt2’

NRS

({r’)

- (r)2)

2 0,

is symmetric arround the point log rl = 0, (r)

=I-NR=O

(21)

= 4, because

5

(N - NR)qN - NR N;s R(N,, NRS)ANnS’2= N N N co qN-Nn c Q(N,, NRs)ANRS’2= R NRS

= 1-,. Here we used an equality a(NR, NRs) = IR(N-N,,

(22)

NRs).

CO-OPERATIVE

RESPONSE

OF MEMBRANE

573

The binding function (u) or fraction of the protomers bound by ligands is given as follows by differentiating the two expressions of equation (5) by jl p, (y)=~~R+n~)=ldlnZ=(r)--cl+(l_(r))~ N N din a

1 +ca’

1+ci

(23)

3. State Functions of Oligomers and Some Lattices (A) RING CHAINS

The grand partition function of the Ising model in the presence of magnetic field is easily obtained for symmetric oligomers and for one dimensional lattice. Thus the state function (r) is easily calculated for these systems.

(01

it)

^!

FOG. 1. Scheme of oligomers and lattice, (a) Ring chain (octamer); (b) B.W. octamer; (c) Bethe lattice, z = 4.

7Yhe grand partition function and the state function of ring chains with N protomers [Fig. l(a)] are given as follows (Thompson, 1968). Z’ = ;I; +A.;,

(24)

&=q+1+6,

(25)

A, = q+l-6,

(26)

where : 6 = (4&j

+(q- l)“>”

(27)

1 dln Z (t-1) = - ___ N din q 17[(rl+1+6)N-1{1+6-‘(2~+tl-l))+ +(~+1-S)N-1(1-lP(2JA+y-1)}] = (28) (?+1+s)N+()1+1-8)N The formula for one dimensional infinite lattice (infinite chain) is obtained

574

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when we take the limit N + 03, Z’ = q,

(N--+cQ)

(29) (30)

(B) OLIGOMERS

AND

LATTICE

INTERACTING

WITH

IN WHICH EACH

ALL

THE PROTOMERS

OTHER

ARE

(B. W. SYSTEM)

If all the protomers in the system are interacting with each other as nearest neighbors, the number of RS pairs (NRs) in the system is given as, NRs = N, x N, = N,(N - NR). The reduced grand partition function [equation (7)] is obtained as

(31)

N! -2/3UNn(N-Nn) (32) NR!(N- NJ! vNR e When the numbers of protomers in the system become infinite (N -+ co), the grand partition function is well approximated by the maximum term of the right hand side of equation (32). The state function (r) is obtained by differentiating the terms of the right hand side of equation (32) by NR and equating to zero, as :

=

2

=

1+,/-l

e2;“N(1-2(r))

=

The last formula of equation (33) is obtained number of nearest neighbor (z) is: z=N-l+ and, therefore the parameter A is, A

=

e-45zU

N, =

e-45NU:

1+,-1’,<.>-+’

(33)

by using the fact that the (34)

(35) Equation (33) is the same as the state function of lattices obtained by Changeux et at. (1966) with Bragg-William or molecular field approximation. It is clear that Bragg-William approximation (B.W. approximation) is the exact solution of this lattice, since B.W. approximation assumes equation (31) for all lattices. We denote hereafter the oligomers and lattices in which all the protomers are interacting as nearest-neighbors by Bragg-William (B.W.) oligomers and B.W. lattices, respectively, or simply as B.W. systems in common. B.W. system can be actually realized only in the symmetric oligomers containing no more than four protomers, i.e. dimer, trimer and tetramer. But if many of the promoters of the system interact with each other by long range

1

4

4

Ring tetramer (square)

B.W. tetramer (tetrahedral)

3

Ring

trimer

2

Dimer

N

3

2

2

1

z

I

;->I :- -,’

c

/

/\ 0 -

c

o--o

Fig.

1 -! 4Kq

1 i- 4Kq

1 + 3Kq

+ 6K”‘$

; 2K(K

-j 3K$

1 + 24 + q”

2’

/. 4Ktf

i- q4

i 2) t/S -j. 4Kq”

-t q’

.! ,I’

(Kq

:Kt/

(Kr/

-1 3K”3$

-i K(K

-i 2K$

WV + VW + $)/z’

$

-!- q’)/Z’

-I- 3K$

-t 3K$

+ 2) $



tf)/Z

The grand partition function Z’, and the state function of simple symmetric oligomers as the functions of 1 and A (K = Jii,

TABLE

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forces such as Coulomb interaction, dipole-dipole interaction, etc., B.W. systems may become to be good approximations for these systems. An example of B.W. oligomer (octamer) is shown in Fig. l(b). The grand partition functions and state functions of symmetric dimer, trimer and tetramers (ring tetramer and B.W. tetramer) are given in Table 1. 4. Two and Three Dimensional

Lattices

Exact grand partition functions of two or three dimensional lattices under the condition q + 1 (in the presence of magnetic field) have not yet been obtained, but many approximation methods have been devised (cf. Domb, 1960).

The simplest but rough approximation is B.W. approximation described above. More accurate, yet relatively simple method is Bethe’s approximation. In this approximation, the state function (r) is given as:

+lcZZ) = 1M(M +2MA”2’+M2’

(36)

where M is the root (real and positive) of the next equation: !f = (I”,;f$y.

(37)

The lattices for which Bethe approximation is exact are called Bethe lattices. The only realistic Bethe lattice is one dimensional infinite chain. An example of Bethe lattice with z = 4 is shown in Fig. l(c). For higher approximations, configuration of lattices (cubic, square, triangular, etc.) must be specified besides the number of nearest neighbors z. Among many higher approximations, Kikuchi’s approximation (1951) is simple yet relatively accurate and will be used in the subsequent paper. 5. Extension to Multi-Iigand

Binding

The theory can be generalized for cases in which two or more ligands bind to the same protomer by only modifying the expression of q [equation (14)]. We confine our discussion to the case of two ligands-binding. Let a1 and CI~ be the relative concentration (;l,/k,, and i2/kR2) of two ligands respectively and let us assume that the first ligand 1 is an agonist or a stimulant (cl = kRl/ksl c 1). The following two cases exist: (i) two ligands bind to the same site (mutually competitive binding), 1 +a, +rx, (38) v= L(1 +c,al +c2a2)’

CO-OPERATIVE

RESPONSE

OF

577

MEMBRANE

(ii) two ligands bind to distinct receptor sites (allosteric interactions).

(1 -l-%)(1+a,)

(39)

v = L(1 +crcQ)(1 +c,a,)’

If c2 is larger than unity, the second ligand is an antagonist or an inhibitor in both cases [a competitive inhibitor in (i) and allosteric inhibitor in (ii)], since the existence of the second ligand decreases the value of q and hence the value of(r) in both cases. 6. Hill’s

Coefficient

One of the most important indices of co-operative dose-response curve is Hill’s coefficient. Hill’s coefficient, n,, is defined (Changeux et al. 1966) as,

nR =dq;:;y::) =loge

dlog u 1

(r)-(r)o

1 +(r)m-(r)

d(r) .-*dlogr

dlog ? dloga’

(40)

where and (r).

= (r)(a

= 0).

In general, nH is dependent on the value of CI,except for the cases when (r) is described by Hill’s equation (A = 0, L-co and c=O; n,=N) or by Langmur’s isotherm (A = 1, L + 03 and c = 0; nH = 1) as described later, and the maximum value of n, (denoted by nHmax) is routinely used as the index of co-operativity. We adopt in this series of papers, nH at the middle point of response change (denoted by ni) as the index of co-operativity instead of namax. The former, n& is approximately equal to nHmax in most cases.

(41)

0 -2

h 3 O-5

I.0

-I

I.0

0

I

log a

Fm. 2.

2 3

4

5

CO-OPERATIVE

RESPONSE

OF

579

MEMBRANE

where (r)dd

=

(O-1,

+(Oo)/2.

From equation (21), we get: d(r) -=ln dlog 9

d(r) lO*rj----dtl = In 10*N((r2)

dlog q -=dloga

- (r)2)

1 l+ca

u --=---ca l+cr l+ca

(42)

1

(43)

l+a’

Thus nH is expressed as, 41~ N no = H

1 (r)-(r)o

1 ’ (r),-(r)

) CO-‘> - ‘) (Au

- i-&J

(4%

4N ((
-

,-(r)0

cr>= Wdd’

(45)

Equations (44) and (45) show that nH or ni is proportional to the square of the mean fluctuation of (r). The second row of equation (41) also shows that ni is proportional to the gradient of the curve of (r) vs log IX (dose-response curve) at (r) = (r),,,id [see Fig. 2(b)]. Next, the Hill’s coefficient of binding function (y) is described. FIG. 2. Relative concentration vs. relative response curves of ring tetramer. (a) with the definite A and various L and Lc. A = 0.00193 for all curves.

@IL =0*8 : g

1.25 1.0 ::;

Lc = 0

(r>o =

, =

n; = 1.27

log am,* = -0.48

0 00.79

0.325 X.677 0.325 0.101

:*OCX& 1.000 0.684 1.000

1.37 1.50 1.05 1.92

-0.40 -0.27 0.05 0.10

g @I @I g

,g loo0 loo0 :iz

0 0.1 O-5 0.79 1

O*OOO 0.000 0.005 0.111 0.000 ;:g

1.000 l*OOO 0.995 0.899 0.500 0.684

2.97 3.30 2.97 1.92 1.51 1.37

0.96 3.00 2.00 3.04 3.25 3.36 3.40

6

1000

1.2

0.000

0.354

1.29

3.39

amrd: thevalue of a giving O + m)/2 (b) with various A and definite L and c. L = 1000 and c = 0,
580

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AND

S.

KIJlMA

From equation (5), --1 d2Z = J- .dZZ = ((nR +ns)“) = N2(y2>. 2 d(ln a)2 b”Z dp2 By differentiating equation (23) by log LY,we obtain :

= N In 10~((y2)-
(46)

) 2( - (r>2) +

___ (l-(r))} (47) + (1 ia)2 <‘> + (1 -Zaj2 The last two terms of the third expression correspond to a(v)/alog a. Since (y), = 0 and (y), = 1 in all cases the Hill’s coefficient of binding function, ng, or nEb is obtained as:

&=loge*

-I_+ (y)

(-2)

nsb

c~)=+

Again, n’;i or nib is proportional

= ~N( - 8)=p

(48) (49)

to the square of the mean fluctuation of (y).

7. Characteristics of Dose-response Curve In this paper, dose-response curve of the membrane is expressed by the curve of (r) vs. log a. It deviates in general from the curve of (r) vs. log 1 since the relation between q and a is not simple as given by equation (14). The state function reduces to a simple form in the following cases: (i) when no co-operative interaction exists (V = 0, and hence A = 1), from equations (18) and (14)

(r)‘L=I+ 1+1

LIl. - cja I (1+L)2 l+L

1 + (1 +Lc)a’ l+L

(1 +Lc)a

(y)=-

l+L

l + (1 fLc)a’

l-IL

(51)

CO-OPERATIVE

RESPONSE

OF

581

MEMBRANE

We obtain from equations (40) and (48) : nx = n&= 1.

(52)

This is reasonable, since both (r) minus a constant (1 + L)-’ and (y) are described by simple Langmur’s isotherm. (ii) when co-operative interaction is infinitely large (U -+ co, A = 0):

(53)

Equation (53) is the same as the state function of the model by Monod,. Wyman & Changeux (1965). The properties of dose-response curve will be discussed in the following cases. Hereafter we confine our discussion to the case c 5 1, because the case c > 1 reduces to that of c < 1 if we exchange S and R. The dependence of the curves on various parameters is shown for ring tetramer in Fig. 2(a), (b) (a) L --t 03, Lc -+ 0: Yn the absence of ligands, S state is exclusively stable. Ligands bind to R state only. In this case, using equation (19) and equation (20), (r)O = 0, (r), = 1, ?o=O, v, =m, where q0 = ~(a = 0), qm = ~(a = co), and q is approximated a

i.

‘I = L = ht’

(54) as: (55)

in the region l/L < q 4 l/Lc. Thus, the curve of (r) vs. log a is essentially the same (only difference is a shift by IogL along the log c(axis) as the curve of (r) vs. log tf. For example when L = 1000 and Lc = 10e3, equation (50) substituted by q = a/L, is a very good approximation of exact equation in the region l/30 r q s 30 (relative error A{r>/(r) is less than 3 %) and (r) moves from 0.03 to O-97 in this region. Tn this case (a), our model for oligomers becomes the same as the model of Koshland et al. (1966), if we put L-kR = kD where kD is the apparent dissociation constant and the other parameters of them are redefined. .We get from equations (23) and (48) the relation: (Y> =
n; = nH.

(56) (57)

582

H.

KIJIMA

AND

S. KIJIMA

Equation (56) shows that the binding of ligand inevitably causes the change of state from S to R. The shape of the dose-response curve and the Hill’s coefficient &, i.e. the gradient of (r) vs. log a curve at (r)mid, is determined by the parameter of co-operativity A for a given system. The parameter L only shifts the curve along the log a axis. The examples of ring tetramer are shown in Fig. 2(b). If A is small (strong co-operativity), the curve is sharp and ni is large. When A > 1, 8: is smaller than unity showing negative co-operativity. When infinitely strong co-operative interaction exists (A = 0), we have from equation (53) (58) and from equation (40) nR = nk = N.

(59)

(b)Lc?fO,L-+co: ?fm = l/Lc
This shows that the curves of (r) vs. log a with the same A and Lc, i.e. (r),, = 0, (r), = constant, have the same shape with shift by log L along the axis log a for any value of L as long as L % 1. Hill’s coefficient ng given by equation (41) or (45) becomes smaller than the one for the curve of c = 0 with the same A. The precise analysis of the relation between n$ and (r),, and the reason why ng becomes small are rather complex and will be given in the subsequent paper with the analysis of a at (T),,,~~ (amid). An example of the decrease of ni with increasing c is shown in Fig. 2(a) for the ring tetramer. Chemically excitable membranes have the dose-response curves with various values of maximum responses for different ligands. The parameter c gives the freedom to set various values of (r),, but Hill’s coefficient inevitably decreases with the increase of c. (c) Lc + 0, L is not so large: VO = 1/L > 0, 0. The membrane is spontaneously excited in the absence of ligand. The doseresponse curve begins at the value (r). and gradually increases to the value 1

CO-OPERATIVE

RESPONSE

OF

MEMBRANE

583

as shown in the example of the ring tetramer in Fig. 2(a). Hill’s coefficient, ni, becomes smaller than the one for the curve of L 3 00 for the same reason as rzz becomes smaller when Lc + 0, L + co [case (b)]. Two (r) vs. log docurves with the same c and A in the same system, but wit.h different L’s (we denote L and L’, respectively) satisfying the relation LL’C = 1,

(61)

are symmetric with each other around the point log CI = -$log c, (r) = 3 (the proof is given in Appendix). If L -+ co, Lc + 0 (case b), L’ = l/Lc is not so large but L’c = l/L + 0 (case c). Therefore, spontaneously excited curve in this case is essentially the same as the inverted curve of the case b with (r), < 1. Since the Hill’s coefficients are the same in both curves, the spontaneously excited curve (for which (r’), + 0) has the same small Hill’s coefficient as that of the curve with (r:jm = 1 - (r’). in case (b). .If the spontaneous excitation exists, it is clear from equations (38) and (39) that specific antagonist or inhibitor (cl > 1) such as D-tubocuraline or flaxedil on cholinergic membrane, must cause appreciable decrease of conductance (decrease of (r)J or must cause hyperpolarization in the absence of agonist. At. present, such effects of specific antagonists are not known in any chemically exlcitable membrane. (d) Lc f 0, L is not so large: In this case (r). > 0 and (r), < 1. Hill’s coefficient decreases more than the case (b) or (c). An example is shown in Fig. 2(a). 8. Relations with Other Allosteric Models As shown above, this model becomes to be the same as the models of Koshland et al. (1966) at the limit L --) co and Lc --) 0. Thompson (1968) first formulated the model of allosteric protein based on the Ising model. He did not permit the discrepancy between the state function and binding function. His model is, thus, the same as the Koshland’s allosteric model. The lattice model by Changeux et al. is essentially the same as ours, but they confined their formulation to the Bragg-William approximation of the lattice. The first model of allosteric protein by Monod, Wyman and Changeux (MWC model) is identical to our models for oligomers when we put in our model the co-operative interaction to be infinite or A = 0 [equation (53)]. MWC model is insufficient for the description of the co-operative response of chemically excitable membrane known so far, since the co-operativity or Hill’s coefficient of the response is determined almost uniquely when the

584

H.

KIJIMA

AND

S. KIJIMA

number of the protomers in the system is determined : i.e. n, = N for N-mers when (r), = 1 and other fixed value of $ with given (r),. The model of Koshland et al. (KNF model) is also insufficient for the description of co-operative response of the chemically excitable membrane. In this model, ligand binding inevitably causes the transition of state and (r), is always unity. Thus, to explain the variety of maximal responses with different kinds of agonists, it must be assumed that the value of g in equation (17) varies with different agonists. But this is unlikely at least in the nicotinic acetylcholine receptor. Reversal potential of the cholinergic membrane is the same for many agonists with different maximum responses (Rang, 1974), suggesting that the ion channel has the same ion-selectivity, so thatequalgfordifferent agonists. Our model includes, thus, all previously presented co-operative allosteric models of oligomers and lattices. We assumed the symmetry of the system or equivalence of the protomers in the system for the simplicity. For asymmetric systems as shown in Fig. 3, there must be more parameters because protomers have different numbers of nearest neighbors.

Symmetric

\

v

/

Asymmetric

FIG.

3. Symmetric and asymmetric tetramers

The authors sincerely thank Professor H. Morita for encouraging them throughout the work and for many useful discussions. They also thank Professor N. Go and Professor T. Mitsui for reading the manuscript and kind discussions and Mrs Fukagawa for preparing the manuscript. REFERENCES BRIDLER,L. M. (1971). In Handbook of Sensory Physiology Vol. 4-2, p. 200. Berlin: Springer Verlag. BLUIIIENTHAL, R., CHANGEUX, J. P. & LEFEVER, R. (1970). J. mem. Biol. 2,351. CARTAUD,

J., BENEDETTI,

FEBSLat. 33,109.

L.,

COHEN,

J. B., MEUNIER,

J. C. & CHANGEUX,

J. P. (1973).

CO-OPERATIVE Cwoeux,

J. P., THI~RY,

J., TUNG,

RESPONSE Y. & KITTEL,

OF

585

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C. (1966).

Proc.

nam.

Acad. Sci. U.S.A.

57, 335. DOMB, C. (1960). Adv. Phys. 9, 149. FATT, P. & KATZ, B. (1951). J. Physiof. 115,320. HODGKIN, A. L. (1964). The Conduction of the Nervous Impulses. Liverpool: Liverpool Univ. Press. HUCHO, F. & CHANOEUX, J. P. (1973). FEBS L&t. 38, 11. KAISSLING, K. E. (1974). In Handbook of Sensory Physiology, Vol. 4-1, p, 351. (L. M. Ikidler, ed.) Berlin: Springer Verlag. KATZ, B. (1966). Nerve, Muscle and Symqm, Ch. 8. New York: McGraw-Hill. KIKUCHI, R. (1951). Z’hys. Rev. 81, 988. &RLIN, A. (1967) J. theor. Biol. 16, 306. KOSHLAND, D. E., NEMETHY, G. & FILMER, D. (1966). Biochem. 5,365. LESTER, H. A., CHANGEUX, J. P. & SHERIDAN, R.E. (1975). J. gen. Physiol. 65, 797. MCINOD, J., WYMAN, 5. & CHANGEUX, J. P. (1965). J. mol. Biol. 12, 88. MORITA, H. (1972). Adv. Biophys. 3, 161. MORITA, H. & SHIRAISHI, A. (1968). J. gen. Physiof. 52, 559. MIXJNIER, J. C., SEALOCK, R., OLSEN, R. & CHANOEUX, J. P. (1974). Eur. J. Biochem. 45,371. RANG, H. P. (1974). Quart. Rev. Biophys. 7, 283. RUIZ-MANRESA, F. & GRUNDFEST, H. (1971). J. gen. Physiol. 57,71. SHIRAISHI, A. & TANABE, Y. (1974). J. camp. Physiol. 92,161. TAKEUCHI, A. & TAKEUCHI, N. (1960). J. Physiol. 154, 52. THOMPSON, C. J. (1968). Biopolymers 6, 1101. WI~BER, M. & CHANGEUX, J. P. (1974). Mol. Pharmacol. 10, 15.

APPENDIX

Let us consider two points: a point (log CC,(r)) on one curve and a point (log a’, (r)‘) on the other curve. If we assume the relation, logtx+loga’=-loge (Al) we obtain from equation (14), 1 +a’ L(l+ca) 1. W) 9’ = L’(1 +cal) = -~ 1 +a = -.PI’ Since (r) is dependent on a through q and the curve of (r) vs. log q is symmetric around the point log q = 0, (r) = 3, (r>’

= (r)(log

a’) =
$1 =
l/d

=
v)

= l-(r)(logq) = l-(r)(loga)= l-(r). (A3) From equations (Al) and (A3), two curves of (r) vs. log a are symmetric with each other around the point log a = -*log c, (r) = +.