The specificity of molecular processes involved in neural transmission

J. Theoret. Biol. (1965) 9, 471-477

The Specificity of Molecular Processes involved in Neural Transmission R. WERMAN Indiana University

Medical Center, Indianapolis,

Indiana, U.S.A.

(Received 1 June 1965) Assuming constant field conditions, the membrane process allowing ionic flow is defined by the coefficients of a Goldman-like equation. If n ions are involved, it is demonstrated that two conditions utilize the same process when: (a) the equilibrium potential for each condition is the same; and, (b) after each of n-2 changes in different ion species concentration, each new pair of equilibrium potentials are also the same. Some implications for identifying neural transmitters are discussed. In the process of neural transmission a chemical mediator is released from the terminals of the presynaptic nerve. After a brief period, called the synaptic

delay, an increase in ionic conductance across the postsynaptic membrane is manifested. This synaptic delay is thought to represent the transit time of the mediator in diffusing across the less than 500 A synaptic cleft to interact with postsynaptic molecules. This interaction is highly specific, resulting in increased permeability for only certain ion species. The specificity results both from the nature of the transmitter, different transmitters producing different changes, and also from the nature of the postsynaptic molecule, the same transmitter producing different changes at different synapses. For example, acetylcholine produces increased Na+ and K+ movement in vertebrate skeletal muscle and only increased K+ movement in the heart. A pharmacological agent which produces transmitter-like action may involve either the same molecular mechanisms as does the transmitter, or other mechanisms that produce apparently similar changes. In attempting to identify transmitters it is important to distinguish between these possibilities. Increased success in this differentiation has led to a strengthening and greater utility of the criterion of identity of action, a criterion which states that a suspected transmitter must act in the same manner as the physiological agent (Werman, manuscript in preparation). The present paper establishes the minimum amount of information necessary to show whether a suspected transmitter and the physiological agent utilize the same postsynaptic molecular mechanisms. 471

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The equilibrium potential of an ionic process can be experimentally determined. Assuming conditions compatible with current understanding of membrane ionic processes, this potential is also given by an equation of this form (Goldman, 1943): E = RT -~ ln P,a,+P,b,+P,c, -__-___-F

...+P,-lr,~,+P,n,

P,a,+PZbz+P3~~...+P,_11~~2+P,n2

where: E is the equilibrium potential in volts, R is the Boltzman gas constant, F is the Faraday. a,, bi, cl,. . . , n, are the concentrations of all cationic species on the outside of the membrane and all anionic species on the inside of the membrane, a2, bZ, c2,. . . , nz are the concentrations of the same ion species a, b, c, . . . , n on the opposite side of the membrane, P1, P2, P3,. . ., P, are those fractions of the ionic current carried by ions a, b, c, . . . , n, respectively. The sum of these coefficients are taken to be equal to one. The coefficients in this equation are derived from those used by Hodgkin & Katz (1949) by dividing each of their coefficients by the sum of their coefficients. Since the following useful form of the constant Pm = I-P,--Pz-P3.. . -P,,-l, field equation can be written: E =

RT -gn

P,(W-n,)+P,(b,-n,)+P,(c,-n,)...+P,-,(m,-n,)+n, P1(~2-nJ+P2(~2-nJ+Pdc~-n~)...+P,-,(m,-n,)+n~

* (1)

The molecular process resulting in the complicated distribution of values p,,p,,p,,.. ., P,,- i may be thought of as a singular process which is defined by these coefficients. Any means of achieving these values must then be assumed to invoke the same molecular process. It will be shown that a suspected transmitter and the physiological agent operate by the same molecular membrane processes when n ions are involved if both the transmitter and the suspect have (1) the same equilibrium potential and (2) the same new equilibrium potentials after each of n-2 concentration, changes of different ion species on either side of the membrane. Some transmitter processes apparently involve only one ion. Thus, in the case of the crayfish neuromuscular inhibitory process, only the chloride ion is involved (Boistel & Fatt, 1958). In this case, equation (1) reduces to the Nernst equation :

ECl = Fin ccl-12

F [cl-II’ The molecular mechanism allows only chloride to permeate in appreciable amounts and is thus highly specific and restrictive. If a suspected transmitter replicated inhibition by allowing only chloride flow, the agent would be

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presumed to invoke the same mechanism as does the physiological transmitter (see Discussion). There are cases where two or more ionic species are involved. For example, the vertebrate neuromuscular excitatory transmitter involves only Na+ and K+ movements (Takeuchi & Takeuchi, 1960). In the two ion case, equation (1) reduces to: E = ~_TlnPl(al-bl)+h 1 F P,(a2--b,)+b,’ If our suspected transmitter produced an ion movement with the same equilibrium potential, E,, we write: E = p1

lnm~l-b,)+b’

F

P:(a,-b,)+b,

thus,

p:h-bl)+b,

P,(u,-bl)+bl

PXQ,-b,)+b,

=P,(a,-b,)+b,

and P,(a, b, -a, b,) = P:(a, b, -a, b*).

Ifa,b,

=albz, a1 -=a2

b, b,’

Under these conditions, it can be shown that the equation reduces to the one ion case (see above). If a, 6, # a, b2, then P, = Pi and the same molecular process is invoked by both conditions. In the three ion case, condition one (the transmitter) can be represented bv : E

1

= ~lnPl(al-cl)+P2(bl-~l)+cl

F

P,(%-c,)+P,(b,-c,)+c,’

and, if condition two (pharmacological agent) produces the same eqtilibrium potential, E =Eln P:(u,-c,)+P:(b,-c,)+c, 1 F P:(a,-c,)+P:(b,-c,)+c,’

Thus, P:(a,--cA+P;(b,-c,)+c,

Pl(Ul -c,)+P,(b,-c,)+c,

P&2 - cd + P,(b, - ~2)+ c; = P&z, - cJ +P;(b2 - CJ + c2’

(2,

Now we make one concentration change, u3 for a,, and, if both conditions give us the same new equilibriumSpotential E2, then, for condition one, E /Tin 2

F

Pl(a,-cl)+P2(bl-c,)+c, P,(~2-c,)+P,(b,-c,)+c,

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and, for condition two, E, = FFT ln

P:(a,-c,)+Pl(b,-c,)+c, -___ Y:(a,

-

c2) + Pi@,

- c2)

+ c2’

Thus, P,(n,-c,)+P,(b,-c,)+c, P1(a,-c,)+P,(b,-c,)+c,

P:(a,-c,)+P~(b,-c,)+c, = P:(a,-c,)+P&--c,)+c,’

(3?

Subtracting (2) from (3), we find:

PI P2@2- c2>+ cz p:= Jw2 - c2)+ c2’

(4)

Now, dividing (2) by (3) we find:

PI P,@,-c,)+c, p: = P;(b,-c,)+c,’

(5)

From (4) and (5) we get:

PXc,(b, - c2)- c,@,- 41 = ~,M~, - Q--c,(~, -cdl. If c,(b, - CJ- c,(b, -cr) = 0 Ct -.--

b,

c2

bz

which reduces to the two ion case (see above). If c,(b, - c2) - c,(b, - cl) # 0, P; = P,,

and from (5) Pi = P,.

Therefore, the same permeability constants are present in both conditions, and, presumably, the same molecular processes are involved. It will now be shown that the nth case can be reduced to the (n- l)th.$ase by one ionic substitution. In the nth case, equation (1) represents condrtron one and condition two can be represented by: El=?

RT

In

P&ynl)+P:(bl-nt)...P,‘-i(ml-ni)+nl P:(a,-n2)+P:(b2-n2)...P,‘_l(m2-n2)+n2’

Thus, P,(a,-n,)+... P&l,--nz)+...

Similarly,

P:(a,-n,)+... n, n2 = P:(u2-n,)+...n;

n,

(6)

by changing a, to a3 we get two new equations for E,, and P&--nl)+... nl P:(% --n,)+... P,(a,-n,)+...n,=P&-n2)+...n2’

nl

(7)

Subtracting (6) from (7), we get: P2(b2-n2)+... p: = P:(b2-n2)+...

Pl

n2 n2’

(8)

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Now, dividing (6) by (7), gives us:

Pl

P,(b,-n,)+...n, p: =i’;(b,-n,)+...n,’

From (8) and (9): P,(bl-nl)+... P,(b,--nJ+...

n, P:(b,--n,)+... n2 = P:(b,-n,)+...

rzl nz’

But equation (10) is the relationship that is obtained in the case of n- 1 ion species when condition one and condition two both produce the same equilibrium potential. That is, it is the equivalent of equation (6) written for n-l species of ions. Thus, a coincidence of equilibrium potentials before and after one change in ionic concentration reduces the relationship between n ion species to the simpler one which relates a coincidence of equilibrium potentials in the case of n - 1 ion species. If n - 3 concentration changes are made, the equivalent relationship for three ionic species is obtained. This relationship has been shown to require one ionic change for its resolution. Therefore, n - 2 changes of ionic concentrations, one at a time, each of a different ionic species, both the original state and each change producing an equilibrium potential that is identical for both conditions one and two, establishes the identity of the coefficients for conditions one and two. Consideration of the concentration of an ion changing on only one side of the membrane was prompted by a specific model, i.e. altering the intracellular concentration of a neuron by iontophoresis. In this case, any accompanying changes in extracellular concentrations would be small because of the large extracellular volume. K. Krnjevic suggested that the treatment be generalized to the case of changing the extracellular ionic environment. The three ion case is given for such a situation. If in the three ion case, two conditions produce the same equilibrium potential, then P,a,+P,b,+c, P:a,+P:b,+c, PI a2 + P2 bl + c2 = P:a,+P:a,+c,

(11)

Thus, P:a,+P;bl+c,

= n(P,al+P2bl+c,)

P:a2+P;b2+c2

= n(P1a2+P2b2+cz),

and n#l

If the concentration of species c is now changed on both sides of the membrane so as to produce an increase cg on one side and c, on the other,

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and the two conditions produce new equilibrium those in equation (11) but still equal,

P,a,+P,b,+c,+c, P1a2+P,b,+c,+c, Cross-multiplication

potentials

n(P1a,+P,b,+c,)+c, = n(P,a,+P,b,+c,)+c,

different from (12)

and collection of terms leads to: ;l~l~;~~l~-! 12

(n-1) 22

= 2 (n-1)

2

Now if n # 1, the equilibrium potentials of equation (12) are the same as those of equation (1 I), a condition forbidden by definition. Therefore, n must be equal to one. It then follows that either P, = P: and P, = Pi or that al -=02 which reduces to the two ion case.

bl b,

Discussion In all studied cases of synaptic transmission, NaC, K+ and Cl- carry 95100% of the ionic current. Thus, measurement of equilibrium potentials after one or two ionic changes should be enough to specify the similarity of molecular mechanisms used by a suspected transmitter and the physiological agent to a high degree of probability. If two processes are different, changing the concentration of an ion species which carries little of the ionic current (one with a small P) may produce so little change in equilibrium potential that the differences in results produced by the two conditions might be small compared to measurement errors. Thus, in practice, it is important to change ionic concentrations of species with large P's. An appropriate choice of such species can be made by utilizing knowledge of the equilibrium potential of the individual ion species. It might be argued that the same constant field equation coefficients can be achieved by more than one molecular process. This argument gains greatest cogency in the few ion cases. Thus, in the case of crustacean neuromuscular inhibition it would seem that any process that let Cl- move would produce the same equilibrium potential as does inhibition. The argument for specificity can be made more persuasive by rewriting the constant field equation using terms for non-present ions that have hydrated radii, charges and chemical behavior similar to Cl-. Thus the equation for the one ion case might be written:

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This equation reduces to RT E inhibition =TIn---

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[Cl-],

ccl-11

because the concentration terms for I- and By are,physiologically zero. However, introducing these ion species into the medium or intracellularly by iontophoresis produces a more delicate discrimination, the three ion case. It is much less likely that two distinct ionic processes could discriminate among these three similar ionic species in exactly the same manner. The likelihood of any other ionic discrimination by a membrane process being duplicated by chance by a different membrane process can similarly be reduced, utilizing potentially permeant ion species not physiologically encountered. Although these formulations were prompted by problems in synaptic transmission, the conclusions are valid for any ionic processes governed by constant field considerations. Thus, the same molecular process is involved in each condition if the equilibrium potential is the same for both conditions and if each new pair of equilibrium potentials produced by n-2 changes of concentrations of different ionic species remain equal. The author was supported in part by grants NSF GB2692, USPHS NB 0440543 and United Cerebral Palsy Research and Education Foundation R174-64. REFERENCES J. & FAIT, P. (1958). J. Physiol., Lmd. 144, 176. GOLDMAN, D. E. (1943). J. gem Physiol. 27, 37. HODGKIN, A. L. & KATZ, B. (1949). J. Physiol., Land. 108, 37. TAKEUCHI, A. & TAKEUCHI, N. (1960). J. Physiol., Land. 154, 52. BOWEL,