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J. Phys. Chem. 1980, 84, 2652-2662 11,473 (1977); (d) 0. Tapia and J. E. Sanhueza, Bbchem. Biophys. Res. Commun., 81, 336 (1976). C. Lamborelle and 0. Tapia, Chem. Phys., 42, 25 (1979). 0.Tapia, C. Lamborelle, and 0. Johannin, Chsm. Phys. Lett. submitted for publication. L. Onsager, J. Am. Chem. Soc., 58, 1486 (1936). (a) A. D. Buckingham, Proc. R . SOC. London, Ser. A , 248, 169 (1958); (b) Ibld., 255,32 and 39 (1969); (c) A. D. Buckingham, Trans. Faraday Soc., 56, 753 (1960); see also W. West and T. Edwards, J . Chem. Phys., 5, 14(1937); (d)E. Bauerand M. Magat, J. Phys. Radium, 9,319 (1938); (e) Ghost B. Chandre and S. Basu, J. Chlm. Phys. Phys.-Chlm. Blol., 66, 903 1903 (1969). D. Rinaldl and J. L. Rivail, Theor. Chlm. Acta, 32, 57 (1973); J. L. Rivail and D. Rinaldi, Chem. Phys., 18, 233 (1976); M. Barfieid and M. P. Johnston, Jr., Chem. Rev., 73, 53 (1973). L. Neel, Ed., “Non-Linear Behavlour of Molecules, Atoms, and Ions in Electric, Magnetic, and ElectromagneticField”, Elsevier, Amsterdam, 1979. R. C. Blngham, M. J. S. Dewar, and D. H. Lo J . Am. Chem. Soc., 97, 1285, 1294, 1302, and 1307 (1975). (a) M. J. S. Dewar, R. C. Haddon, and S. H. Suck, Chem. Commun., 612 (1974); (b) M. J. S. Dewar, A. Komamicki, W. Thiil, and A. Sweig, Chem. Phys. Lett., 31, 286 (1975); (c) M. J. S. Dewar and D. H. Lo, bkL, 33. 298 (1975): (d) M. J. S. Dewar, S. H. Suck, P. K. Weiner. and J. G. Bergmar, Jr.‘, /bib., 38, 226 (1976); (e) M. J. S. Dewar, S. H. Suck, and P. K. Weiner, M.,38, 228 (1976); (f) M. J. S. Dewar and 0. P. Ford, J. Am. Chem. SOC., 99, 1685 (1976); (9) M. J. S. Dewar and H. S. Rzepa, J. Mol. Stroct., 40, 145 (1977); (h) B. Silvi, J. Chlm. Phys. Phys.-Chlm. Blol., 76, 21 (1979); (I) K. Woiinski and A. T. Sadiej, J . Mol. Struct., 53, 287 (1979).
A. C. Huriey, Proc. R . SOC.London, Ser. A , 228, 179 (1954). (a) S. Bratos, Colloq. Int. CNRS, 82, 287 (1958); (b) P. Pulay, Mol. Phys., 17, 197 (1969); (c) K. Thomsen and P. Swanstrom, bid., 26, 735 (1973). (a) C. C. J. Roothaan, Rev. Mod. Phys., 23, 69 (1951); (b) J. A. Pople and D. Beveridge, “Approximate Molecular Orbital Theory”, McGraw-Hili, New York, 1970. QCPE Program no. 279; for methods used in the geometty optimbtbn step, see ais0 Murtagh and Sargent, Comput. J . , 13, 185 (1970); McIvers and A. Komanicki, Chem. Phys. Lett., 10, 303 (1971). B. Pedzlsz and R. Konopka, A&. Mol. Relaxation Interact. R-, 12, 129 (1978). “Handbook of Chemistry and Physics”, 46th ed., Chemlcal Rubber Publishing Co., Cleveland, Ohio, 1965. M. Losonczy,J. W. Moskowits,and F. H. Stiiiinger, J. Chem. Phys., 59, 3264 (1973); 81, 2438 (1974). S. Bratoz, J. Rios, and Y. Gulssanl, J. Chem. Phys., 52, 439 (1970); S. Bratos and J. P. Chestier, Phys. Rev. A , 9, 2136 (1974). (a) D. R. Clark, B. J. Cromarty, and A. Sgamellottl, Chem. Phys., 26, 179 (1977); (b) D. B. Bounds, A. Hinchliffe, and M. Barber, J. Mol. Struct., 37, 263 (1977); (c) K. Osapay and 0. Naray-Szabo, Adv. Mol. Relaxation Interact. Processes, 12, 97 (1978); (d) V. V. Lobanow, M. M. Aleksankin, and Yu. Krugiyak, Academy of Sciences of the Ukrainian SSR, Kiev, 1974. (22) A. E. Lutskii and S. N. Vragova, Opt. Spectrosc. (Engl. Trans/.), 31, 113 (1971). (23) T. J. Zieiinskl, D. L. Breen, and R. Rein, J. Am. Chem. Soc., 100, 6268 (1978). (24) G. Klopman, P. Andreozzi, A. J. Hopfinger, 0. Kikuchl, and M. J. S. Dewar, J . Am. Chem. Soc., 100, 6268 (1978).
Major Changes in Population Ratios Produced by Small Changes in a Potentlal Field. Relevance to Biomembranes Franklin F. Offner Blomedlcal Engineerlng Center, Northwestern Universlty, Evanston, Illinois 6020 1 (Received: December 15, 1978; In Final Form: April 30, 1980)
A nonequilibrium system is considered, whose members undergo a stochastic change of state. When a member of the system changes state, its local environment relaxes through the gradient of a nonequilibrium potential field in a sense to favor the current state. It is shown that if the relaxation time is long, as compared with the state transition time, a very large reversible change in the population ratio may be produced by a small perturbation applied to the system; cooperativity is not involved. It appears likely that the effect is of importance in electrically excitable biological membranes.
Introduction The equilibrium distribution of members of a system whose members can exist in two states p and q, differing in free energy AG, is given Boltzmann’s relation P / Q = exp(-AG/kT) (1)
where P and Q are respectively the populations of the p and q states of the system. Systems may react to a change in their environment. Such a change, which in biological systems is frequently termed a “stimulus”, affects the free energy difference AG between the states of membejs and thus results in a change in the population ratio P / Q . It is usually assumed that the change in AG resulting from such a stimulus is directly related to the magnitude of the stimulus. As an example, if the change in AG is due to a change in the electrical potential across a membrane acting on a molecule carrying an electric charge, then equilibrium theory indicates that the maximum change so produced in AG is the applied change in potential times the charge on the molecule. A similar result is obtained if the “stimulus” is the change in the chemical potential across the membrane of a reacting species. 0022-3654/80/2084-2652801 .OO/O
However, most biological systems are not in equilibrium, and thus no firm conclusions can be derived from the considerations of eq 1. In fact it will here be shown that when stochastic fluctuations along a nonequilibrium gradient are considered, the reversible rate of change of state with a stimulus may far exceed that which would be implied by eq 1. It will be shown that this may occur when the change of state of a member of the system results in a progressive change in the local environment of that individual member, in the sense to favor its current state. In such circumstances each member is able to “help itself” repeatedly to the energy provided by the stimulus and thus to move progressively towards a new state of greater free energy difference, It is thus found that stochastic fluctuations, such as the electrical “noise” observed in the current or voltage of the axonal membrane, is not merely an epiphenomenon but is essential to membrane function.* Reversible Changes of State in a Nonequilibrium System We consider a system composed of members, each of which can exist in two states, p and q, which differ in free energy by AG. As stated above, if the system were in 0 1980 American Chemical Society
The Journal of Physical Chemlsfry, Vol. 84, No. 20, 1980 2653
Changes in Population Ratios and a Potential Field a
Bath I
Bath
I
Bath
II
b
Bath
- IGQmV C a t i o n excessh N eu tra I
Figure 1. (a) Channel through membrane in unblocked state. Baths have equal concentration of predominantly 1: 1 electrolyte. Channel has negative fixtd charges such that, when electrically neutral, cation concentration whin the channel is equal to that in the baths. Uniform potential drop (constant electric field) through the channel results In uniform flow. Clhannel assumed to be Impermeable to anions and to have the same dielectric constant as membrane. b Channel as in (a) but with flow lblocked by adsorbed cation, e.g., CA+! Potentialdrop now occurs principally in vicinity of point of blocked flow; the electric field remote from this point Is necessarily substantially zero, as there Is no ionic flow. To obtain this potential distribution, charge separation is produced by 81 cation deficit within the channel in the vicinity of the point of blockage. This is almost balanced by the cation excess in the double layer in the adjacent bath.
equilibrium, the population ratio would be given by eq 1. We now consider what will occur if the environment of each member is affected by its present state, in such a manner that the probability of its change of state is affected by its present Btate. For example, when a member is in ita p state, we consider that the environment of the member progressively changes in a manner to facilitate the q p transition, while when the member is in its q state, this facilitation is progressively lost; it is important to note that the change in facilitation does not occur instantaneously, but progressively, and rather slowly compared to the frequency of the change of state of a member (mean p-q transition rate). The general performance of such a system will first be discussed in a qualitative manner, by considering the flow of ions through an isolated channel through a membrane, as shown in Figure 1. The flow through the channel will be intermittently blocked by some gating mechanism. We will identify tlhe unblocked channel with the p state, and the blocked with q. The channel may be 100 A long and 10 A in diameter. The solutions in the two baths are of equal ionic strength and are predominantly 1:l electrolytes. Fixed negative charges are distributed uniformly along the wall of the channel, with a density such that the ionic strength of neutralizing cations within the channel is equal to the cation strength of the two solutions. It will be assumed that only cations can diffuse through the channel. Finally, to furither simplify the model, it is assumed that the dielectric constant of the membrane is the same as that of the solution within the channel. We now consider the unblocked channel, as illustrated in Figure la, and assume that a voltage difference of 100 mV is established between the two baths, bath I1 being negative with respect to bath I. In the steady state the ionic flow along the channel must be constant; if the ionic mobility along the channel is also constant, constant flow
-
will be obtained with a uniform electric field (constant potential gradient) acting on a uniform ionic concentration. The constancy of the electric field is the result of the cation concentration neutralizing the fixed negative charge. With identical solutions at the two interfaces, the two double-layer potentials will be equal; if the Debye length at the interfaces is 1A, the double-layer potentials will be each about 1mV, the channel being negative with respect to bath I and positive to bath 11. The potential drop through the channel is thuu 98 mV, or an electric field of 0.98 mV/A. We now consider the case where the gate is closed, so that there is no flow through the channel. This, for example, may be due to adsorption of an impermeable cation species to the mouth of the channel: as shown in Figure lb. The impermeable cation may be Ca2+. The electrostatic binding of such an ion to a channel under the conditions of Figure l a would evidently be very weak-2 X eV or 0.04kT. If, however, the channel were blocked for a long time, a new steady state would exist: one consistent with zero flow through the channel, despite the 100-mV potential across it. To achieye zero flow in a region of uniform ionic concentration, the electric field in the channel must evidently be zero; this condition will be approximated in the region remote from the point of blockage of flow. Substantially all the 100-mV potential difference across the membrane must then occur in the region of blockage interface with bath I. Thus the potential difference in this double-layer region increases from -1 mV to a value approaching -100 mV. The electrical potential gradient resulting in this potential difference is equal and opposite to the chemical potential gradient, resulting in equilibrium and zero flow. The change in the electrical field through the channel is of course the result of charge separation (Poisson’s equation), so that the cation distribution through the channel can no longer be equal to the fixed charge distribution. It will rather be! approximately as shown in Figure lb. Primarily there is, major cation depletion in the region of the first interface. We now see that the two requirements of the system under investigation are met by a system composed of such channels. We have identified the open channel as the p state, and the closed as the q. If the Ca2+is bound long enough (q state), the electrostatic potential binding it will approach the membrane potential difference of -100 mV. On the other hand, when the channel is free of Ca2+ (p state),the potential will fall toward its minimum value (-1 mV). In each case, the interface potential tends toward the value favoring the cwreni; state: when in the q (bound) state, the binding energy increases, and conversely in the p (unbound) state. The interface potential does not change instantly but only as the cations redistribute toward their new steadystate distribution; the time required for this will depend upon the ionic mobility within the channels and across the interfaces. Denoting by -AG the free energy of the p q reaction, Le., the absorption of the Ca2+ion to the channel mouth, we see that AG will progressively increase while in the q state (diffusion blocked), approaching its maximum value, corresponding to the interface potential approximating -100 mV if the channel remains blocked for a prolonged period. Conversely AG would decrease to its minimum value if the channel were to remain in the p (open) state. The maximum decrease in AG will be represented by I’,
-
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The Journal of Physical Chemistty, Vol. 84, No.
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which will be measured in units of kT. Evidently in the present example, the maximum value that r could have would be 8 (kT/2e = 12.5 mV); its actual value would be less than this, since the full interface potential would probably not be effective in binding the Ca2+ion2 At any instant AG will be something less than ita maximum value; y represents this decrease in AG again measured in units of kT. Thus depending upon the past p-q history of the channel, at any time y may vary between 0 and r, and the ratio of the closed time of the channel to ita open time will be proportional to e-, We will now examine how y may vary with time for a single channel. When the channel is in its p state (Ca2+ not adsorbed; free diffusion through the channel), the interface potential tends to decrease, i.e., y increases toward r; and conversely when in the q state (ea2+bound; diffusion blocked), y falls toward zero. This change between the p and q states will occur very rapidly as compared to the rate of change of y. Thus in the steady state of a channel which is most of the time in the q state (blocked), y might vary around a low value, as shown in Figure 2a. Here, each time the channel opens (goes to the p state), y increases, but, when it reverts to the q state, it remains in that state sufficiently long that y on an average varies around its low value. We now consider the effect of a small increase in the voltage across the membrane; this will increase I’. We consider the behavior of a channel, originally in the state of Figure 2a (in the region of low y), after the new voltage gradient has been established. The effect of this is illustrated in Figure 2b. When the channel opens, the ions within the channel will start redistribution toward a low interface potential and thus toward a higher value of y. On closing of the gate, the interface potential starts to increase; that is, it starts relaxing toward y = 0, but now the gate will open (undergo a p q transition) somewhat sooner because of a higher y (lower interface potential due to the higher value of I?). This alternate opening-closing (p-q) cycle will repeat, with an initially accelerating “walk” toward higher values of y, since each increase in y results in a longer fractional time open and thus a down (closed) step of shorter duration. However, the closer the interface potential approaches its maximum value, that is, the closer y approaches I?, the slower y will increase while the channel is open (p state), and the more rapidly it will relax toward y = 0 when closed (q state). There are thus two opposing factors involved in determining how far the biased random walk will proceed toward a higher value of y (and thus toward a greater fraction of time open) before reaching a new steady state: (1)the shorter dwell time in the closed state as y increases but (2) the more rapid decrease in y (Le., increase of the interface potential) and the further its state departs from y = 0. If these two factors nearly balance one another over a range-of y, then the probable state of the channel, Le., the P / Q ratio, will be very sensitive to small changes in r. The conditions for such a high sensitivity will now be considered. The problem will first be examined by considering stochastic p-q transitions for the limiting case of deterministic relaxation. Since the relaxation in the two senses will in general be unsymmetrical, the equation representing the time course of the system will not be of the usual Fokker-Planck type. We will then consider the limiting situation of the p-q transition rate so rapid that over any brief interval its average may be considered deterministic, as well as the relaxation processes. Finally, the case of
Offner a
rd Time b
r
Y
-
O L Time
Figure 2. Based random walk of the field y affecting a member, due to the stochastic changes In state of the member. When in the p state, y increases toward while In the q state, It relaxes toward zero. As the value of y increases, the mean dwell in the p state increases. (a) Path of the member with an applled fleld holding the member predominantly in the q state; y remains at a low value. (b) The applied field has been slightly altered In the direction to increase the dwell in the p state. The path of the walk is toward a higher value of y,and thus a greater dwell tlme in the p state, until a new steady state Is attained.
r;
rapid transitions, but with stochastic fluctuations in the relaxation process, will be considered. It will be found that the second (deterministic) case can give rise to metastable states, while the first and third result in continuous but rather sharp changes in population ratios in response to perturbations of the system. Kinetics of Change of State in a Nonequilibrium System The behavior of a system of the kind considered above will now be treated analytically. The activation free energy for the p q transition is G,, while that for the q p transition is G, - y. y relaxes toward its maximum value, r, when in the p state and toward its minimum value, taken as zero, when in the q state, according to some functional relationships; i.e. d r / d t =: fp(r- 7) (2a)
-
when a member is in the p state; and
+
The Journal of Physical Chemistry, Vol. 84, No. 20, 1980 2655
Changes In Poplulation Ratios and a Potential Field
dy/dt = -fq(y)
(2b)
when in the q1 state. The relaxation functions f ,and fq are monotonically increasing functions of their respective arguments, and are evidently zero when1 the argument is zero, but may otherwise take any functionid form. The simplest would be first-order kinetics (34 dy/dt = a,(r - y) d r / d t = -aqy
63b)
respectively. We now consider a system consisting of a large number of such members. Let P = P ( y )be thLe density of members of the system in state p, and having an energy shift y. That is, P dy is the fraction of the tlotal number of members in the p state and with an energy shift from y to y + dy; and similarly for Q. Then (4)
The system is thus characterized by two parameters at a given instant: (1) the value of y for each member and (2) its present state, either p or q. We will calculate the rate of change in density of members at each value of the y coordinate of the system, that is, the probability of a member having an energy shift y, arid the distribution of the members, at each value of y, between the p and q states. This csalculation will be made by considering the “flow” of the members in each state along their y coordinate, as well as their cross-flowfrom one state to another, i.e., the p-q transition. This will give a partial differential equation describing the time coursie of the state of the system following any perturbation applied to the system; and by setting time derivative equal to zero, the steadystate configuiration of the system will be obtained. Considering the members of the system in state p, at an energy shift ny, these members will “flow” toward y = I’ at a rate J , = wP(r- 7))
(54
and for memlbers in the q state Jq = -Q(fq(r))
(5b)
This flow will not be constant over yr so that there will be a change in local density of elements at coordinate y due to flow, given by = -aJ,/ay = -Paf,/ay - f,aP/ay (6a) (dQ/dt)lj = -aJq/dy = Qdfq/r3y + f,dQ/dy
(6b)
In addition, there will be a change in density of elements in state p at coordinate y, due to the t,ransitionsfrom state p to q and vice versa: (dP/dt)i, = v[Q exp(-G,
+ y) - P exp(-G,)]
(7a)
and similarly @ Q / a t ) , = Y[Pexp(-G,) - Q exp(-G,
+ y)]
(7b)
Here, and in what follows, energies are measured in units of kT. The frequency factor Y is, according to rate theory, of the order of k T / h , 6.3 x s-l. The total rate-of-change of state density at coordinate y is then given by adding eq 6 and 7: dP/at = -P[v exp(-G,) + afp/ar] t. QY exp(-G, + Y) - f p d P / a y (8a)
dQ/dt = PV exp(-G,)
-
Q[u exp(-Gq
+ 7) - df,/drl + fqaQ/ay (8b)
To find the resulting steady state, we set the time derivatives equal to zero, giving dP/dy = {-P[Yexp(--G,) + df,/drl + Qv exp(-G, + y)l/fP ( 9 4 dQ/dr = {-Pvexp(-G,)
+ Qbexp(-G, + 7) - df,/drll/f,
(9b)
Integration of eq 9 over the region y = 0 to y = F and normalizing the result by application of eq 4 then gives the fractions P and 8 of all of the members of the system that are in the p and q states, respectively. If now the energy barrier between the states is shifted, by changing G, of G, (or both), so that AG is changed, then P and Q will change with time according to eq 8. This time variation will be the result of constrained random walks of each member of the system; the energy term y of each member will alternately “drift” toward I’ or 0, depending upon whether the member is in the p or q state. Equation 8 represents the time course of the system averages, and eq 9 their final value. Usually f , and fq will be everywhere small compared to v exp(-G,) and Y exp(-G 1; that is, there will in general be a large number of oscilfations of each element between p and q states before the system approaches a new steady state after any change in AG. The question of interest is how the ratio P / Q varies as AG is changed, e.g., by some externally applied force on the system. The answer to this question will first be obtained by making the simple assumption of first-order kinetics for the change in y, so that eq 3 is used for f, and f,. Equations 9a and 9b then become dP/dy = {-& exp(-G,) - a,] + QV exp(-G, + r)l/ap(r- 7) (loa) dQ/dr = {-Pvexp(-G,)
+ Q[Yexp(-Gq + 7) - aqll/a,y (lob)
These two equations have singularities at y = r and y = 0, respectively. They may be solved numerically as a boundary value problem, expanding P and Q in series at the boundary y = 0. P and Q are evidently both zero at each boundary. In Figure 3 is shown the result of such a calculation for a specific set of parameters. The exact value of the parameters selected is not critical to the result so long as a, and a are small compared with v exp(-G,) and v exp(-G,); the value of G, - G, at which the rapid transition of state occurs is dependent on the ratio ap/aq,but the steepness of the transition is not so dependent. However the magnitude of the shift in the state ratio P / Q is directly dependent on the value of r; this is to be expected, since the limiting change in AG/kT in eq 1 is equal to I’ plus the change in G, - G, resulting directly from the external applied force on the system. In the calculation illustrated, this latter term-is evidently very small compared to r. Thus the ratio P / Q changes by a ratio of 10 for a change of G - G, equal to O.OlkT; the effective change in AG is 2.3k!k The additional energy shift is provided reversibly from the energy stored in the force field. Analysis of the Equations of State It is instructive to examine eq 9, or their special case eq 10, in their limit as an aid to seeing why the large “magnification” of the applied driving force occurs and the
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The Journal of Physical Chemisfty, Vol. 84, No. 20, 1980
Qffner
IO
I .o
PI0
PI0
0.I
-6.I
3
6.5
G , - U n i t s of kT Flgure 3. Calculated change in P l d w i t h variation in G, using eq 10. Assumed parameters: ap = a, = 1/5000; = 6;G, = 3.
r
factors that may affect the extent of the magnification. Assume that everywhere f and fq are vanishingly small compared to v exp(-G ) ancfv exp(-G,) and have no singularities. Then dfp/& and df,/dy may be neglected in eq 9, giving dP/dy = [-Pv exp(-G,)
+ Qv exp(-G, + y)]/f,
d Q / d r = [-Pv exp(-G,)
+ Qv exp(-G, + r)l/f, (1lb)
(lla)
Because f p and f are vanishingly small, almost everywhere either P and Q must be vanishingly small or P exp(-G,) = Q exp(-G, + y) (12) that is
P / Q = exp(G,
- G,
+ 7)
(13)
Condition 13, when met where P and Q are not vanishingly small, corresponds to maxima of P and Q,which in the limit becomes delta functions. That is, the members of the system will cluster at one or more values of y.3 The position of these maxima cannot simply be obtained from eq 9 (or from their limiting form eq 11)but can be obtained by considering eq 5. Averaged over the infinitesimal width of a cluster, stationarity requires that Jp= -Jq,so that PlQ -fq/fp (14) Information as to the behavior of eq 11, that is, the limiting behavior of eq 9, is obtained by examining plots of eq 13 and 14, as shown in Figure 4 for the data used in calculating Figure 3. In the limit, the maxima and minima correspond to the intersections of the two curves; in the figure, for G, - G, = F/2 = 3, the maxima are at y = 0.5 and 5.5, corresponding to clusters of the population of the ensemble at these two points, where P/Q = 1/12 and 12 approximately. Each of these two points of intersection corresponds to conditions such as those illustrated in Figure 2, where the increase in y when in the p state on the average just equals the decrease in y when in the q state. When y is low, corresponding to the lower point of intersection, y falls at a low rate, but for the relatively long duration of the q
Flgure 4. Functlonal forms of eq 13 and 14. Solld line (eq 14) corresponds to f p and fq as in eq 1 0 dashed line is for quadratic variation in fGfqC= aq(y y2). Intersectlonsplace maxima of density of states, Pand Q(center intersectlon is minimum). Curves drawn for G, - G = r/2 = 3, and a, = aq.For very small values of a, Pand Qapproad! delta functions.
+
state. Thus, the mean height of the “up” steps equals that of the “down”. The converse situation exists at the upper point of intersection, where y has a high value: y decreases rapidly in the q state, but for a relatively short time, while y increases slowly in the p state. Again the height of the two steps is the same, corresponding to a second steady state. Such a condition of two possible steady states will exist whenever there are three solutions to eq 14, that is, when there are three points of intersection of the curves, as shown in Figure 4. With the symmetry existing under the conditions illustrated in Figure 4,the total population at each of the two points will be equal, so that P = Q. If G, is lowered in value (or G raised), the logarithmic line representing eq 13 is dispfaced toward the right, so that both intersections correspond to a higher ratio of P/Q, while the fraction of the total population corresponding to the upper intersection rapidly increases and the lower disappears. The converse occurs for an incretse in G,, resulting in the rapid change in the ratio of P / Q as illustrated in Figure 3. In the limiting case of vanishingly small fpand fq,if there are two points of intersection corresponding to maxima, as for example with first-order relaxations, the total population of the system will reside at only one point except for an infinitesimal region of the system parameters; variation of a parameter by any finite amount would then shift the total population to one maximum or the other. The sharpness of transitions is decreased as the p-q transition rates become slower as compared to the relaxation rates. Also, as will be discussed below, the transition region is broadened by stochastic fluctuations of the relation processes. This performance may be contrasted with that which would occur if the skew curve of fq/f were to be replaced by a horizontal line, i.e., if y remainec! unchanged between the p and q states. Then the abscissa of the intersection of the curves (lines) would only change as rapidly as the change in G, (or G,); there would be no amplification of the effect of the applied energy shift. In systems of the type herein considered, where the local field relaxes in a sense to favor the current state, the c w e s representing eq 13 and 14 will always shift more rapidly than the change in G, - G,; that is, there will always be an increase in the effective energy controlling the population ratio over that applied as a perturbation to the system.
The Journal of Physical Chemistry, Vol. 84, No. 20, 1980 2057
Changes in Population Ratios and a Potential Field Such an amplification will occur even though there is only one point of intersection of the curves. Such a case is illustrated by the dashed curve of Figure 4. A change in G, (or G,) will translate the line representing eq 13along the abscissa; that is, the P/Q ratio will be multiplied by some factor. The point of intersection of the curves will, however, be displaced by a greater m o u n t corresponding to a change in P / Q by a greater factor; that is, the effect of any perturbation to the systern is amplified. This change will he especially great if the two curves are nearly parallel in the region of intersectioin. Calculations of the system represented by the dashed curve confirm this conclusion. However, a system having only one intersection (one value of y corresponding to a steady state) will not, in cases approaching the theoretical limit of vanishingly small amd deterministic relaxation, result in the extreme steepriess of transition found with two intersections (two steady- state distributions). In Figure 4, the curve of eq 14 remains unchanged in form if the ratio fp/fq is changed but is translated along the P / Q axis. The same is true of the curve of eq 3 for a change of (G, - G,. Thus if the ratio fp/fq is multiplied by a factor ,8, an increase of log p in1 G, - G, will result in the same relative positions of the two curves but at P / Q values displaced by a factor p. Therefore, for the limiting case of small1 values of f, and f,, the form of the curve log ( P / Q )vs. G, - G, will be unchanged but shifted to higher or lower values of P / Q ; the basic phenomenon is unchanged. Stochastic Fluctuations of t h e Relaxation Processes Equation 13 may be written
P / Q = k,er
J, = SVpke/(ke (15)
Since under the assumed conditions where eq 13 or 15 is applicable Npand f q negligibly small compared to the p-q transition rates) the p and q states are effectively in local equilibrium, k, is the equilibrium constant for the p-q reaction whm y = 0. If we now write S=P+Q i.e., S dy is the fraction of the total number of elements in the region from y to y + dy, eq 15 may be rewritten to give the fraction of the total number of elements at y in each of the p and q states:
+ QJS = 1/(1 + keeY)
PIS = k,/(k,
(?-?)
(16a) (16b)
Equations 16 may alternatively be considered as giving the fraction of the time any element in the region y is in the respective state. As discussed in the previous section, the density of elements, that is, the value of S, will be other than infinitesimal only at the one or several values of y at the intersections of the curves of eq 13 (or eq 15) and 14 which correspond to maxima; the steady-state population ratio of the system, P/&, for these limiting conditions is obtained by integrating eq 9 as f p and f q become vanishingly small. The displacement of an element dong the y coordinate in time increment dt is the s u m of the displacements given by eq 2 for the fractional times given by eq 16; thus dr/clt = f , k , / ( k ,
+ e-7) - f,/(l + k,er)
If now the value of one or more parameters in eq 17 is altered-for example, if k, is changed, equivalent to changing G., as in Figure 3-dy/dt will no longer be zero, and the point of intersection will move to a new value of y with a velocity given by eq 17. At each value of y corresponding to the intersections giving the new steady state, the new values of P and Q will be given by eq 16, with the value of S at each point unchanged from its unshifted values. This is the result of the delta functions representing the total population at the poinb of intersection moving as a unit. While the P / & ratio of the system would then change, it would not change nearly as drastically as shown in Figure 3, for example. The difference in the steady states calculated by eq 17 from that using eq 9 (or eq 10) is the result of the idealized nature of the calculations, which permit the system to be trapped in a metastable state. Such a trapping in practice will only be partial, the reaction towards the true steady state proceeding due to two factors. First, the p-q transition rate is always finite; the approach to a new stable steady state is given by eq 8. However in many cases the transition rates are so much higher than the relaxation rates that the system could be expected to take a very long time to reach its new steady state, based on the theory as above presented. This is the result of the relaxation processes being taken as deterministic. The effect of stochastic fluctuations in relaxation will now be considered in an approximate manner for the limiting case of very rapid p-q transition rates. Multiplying eq 17 by S will give J,, the net flow of elements due to the (deterministic) relaxation processes:
(17)
The steady state requires that d*y/dt be zero at each point of intersection where S is other than infinitesimal.
+ eY)
- fq/U
+ k,eY)l
(18)
In fact the value of y for an element will not obey a deterministic law, as represented by eq 2, but will vary stochastically about its average value. This may be included in eq 18 by adding a diffusion term in s. With the very rapid p-q transition rate the relaxation of y may be considered to be a Markoff process. If then the fluctuations in y are independent of y, the diffusion term is of the Fick’s law form, -D(dS/dy), so that eq 18 becomes J, = S[fpke/(k,
+ e-9 - f,/U + keeY)l- D(dS/dr) (19)
Differentiating eq 19 with respect to y and noting that -dJ,/d y = dS/dt transforms eq 19 into the usual form of the Fokker--Planck e q ~ a t i o n . ~ The diffusion constant D represents the stochastic nature of the process; it may for example represent the Brownian motion of ions, resulting in fluctuations in the electric field. The value of D may be a function of y, but the present treatment will be restricted to a constant D. Consideration of cases where D is a function of y, as well as estimation of its magnitude, will be reserved for a future paper. Since S represents the density of all elements of the system, in the steady state J, must be zero, and the steady state is found by setting J, = 0 in eq 19. This is in contrast to the previous calculation of the steady state with a finite p-q transition rate, where the separate flows of the p and q elements needed to be considered. This could be included, but the addition of the Fick’s law term to eq 5 would result, on differentiation and addition of the cross-flow terms, in a rather complex equation. Calculations have been mad! of the effect of the value of D on the change of the PI& ratio of a system and on the P(y)and Q ( y )distribution functions. It is found that the effect of increasing D is essentially the same as lowering
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The Journal of Physical Chemistty, Vol. 84, No. 20, 1980
the p-q transition rates, except that, of course, y is no longer strictly limited to the 0-F interval because of the added component of “noise”. I t is this “noise” in the system which in many cases would be largely responsible for the steep change in population ratio actually observed with a small perturbation or signal; without such stochastic fluctuations in the system and with a very rapid rate of p-q transition, the response of the system would be so slow that for practical purposes it could appear to remain in one of several metastable configurations. The time course of the change of the system resulting from a perturbation is obtained from the Fokker-Planck equation, obtained as given above. Thus, a s / a t = -aJ,/ay =
a
--[S(f,ke/(k, dY
+ e-?) - fg/(l + k,eY))] + D(a2S/dy2)
(20) The population ratio may then be found, at any time, by first calculating S from eq 20 and then either P or Q from eq 16. P (or Q) is obtained by integrating P (or Q) over all values of y,which now theoretically ranges from --m to +a,but in practice the integration need cover only from a little less than zero to a little more than I?. Then Q=l-P.
Application of the Theory As an example of the possible operation of the abovedescribed theory, we return to the diffusion of ions through a membrane and will consider the excitable axonal membrane. Hodgkin and Huxley5showed that for small changes of voltage applied across the axonal membrane the conductance of the membrane for K+ ions changes as exp(AV4e/kT), where A V is the applied change in voltage across the membrane and e is the magnitude of the electron charge, i.e., the charge on one K+ ion. If p and q represent channels with open and closed gates, respectively, and if control of K+ diffusion through the membrane is by such a gated diffusion process, then eq 1would imply that gating involves action of the full applied voltage on four electron charges. While this is not inconceivable, it appears unlikely, since initially at least the applied voltage appears across the whole membrane (of the order of 50 A), so that if four charges are involved, they would have to be displaced this full distance. An even more difficult phenomenon to explain has been the extreme sensitivity exhibited by the electric organ of certain fish, which respond to electrical fields as low as 0.1 pV/cma6 It appears highly unlikely that such fields (which would produce only a minute change in voltage across a single cell) could directly produce a response, detectable under the thermal fluctuation background. The present theory could provide the explanation of this problem. The most generally accepted view at present is that, during the activity phase of the axon, ions diffuse across the membrane through pores or channels and that the membrane permeability to the passage of ions is controlled principally at or near the external membrane-solution interface; that is, there is a change-of-state of the channel due to some “gating” mechanism in this region. Such gating may be of various types: one may be the adsorption and desorption of polyvalent cations (e.g., Ca2+)as discussed above, which can block the passage of ions through the membrane when ads~rbed;~,’?~ another form of gating will result from the deformation of ionized molecules.5~9 Figure 1may be used as a model of a channel through the axonal membrane as gated by adsorption of Ca2+ions.
Offner
R ISJ
Flgure 5. Highly simpllfled model of an axonal channel, showing a deflectable gate: and below, the approximate potential fall through the channel. The gate, and potential fall, are shown by solid llnes in the closed state of the gate and dashed lines in the open gate state.
Figure 5 shows a similarly simplified model of a channel as gated by the deflection of an ionized deformable molecule. The analysis of each model is essentially the same and is as has already been discussed. Both mechanisms are almost surely involved in actual biomembranes. When one recalls the earlier discussion of the model, if the gate is held closed (Ca2+adsorbed, or channel obstructed by the deflected molecule), substantially the full membrane potential drop will occur in the gating region, tending to hold the gate closed. When the gate opens, there will be no instantaneous change in the potential drop across the region, but it will only change as the ions redistribute themselves within the channel (and the associated double layers) toward their new steady-state distribution. If the gate stays open sufficiently long, the voltage across the gating will fall to a small fraction of its original value. In either of these examples the y field is the electrical potential difference across the gating region; the p state will correspond to the gate open, and the q to the gate closed. The gating mechanism, whether that of adsorption of conformational change, would be expected to occur with a rate of the order of at least lo7 s-l; the rate of change of the electric field may be estimated from the time course of the displacement currents following a voltage step across a membrane to be of the order of 10-5-104 s-l. Thus the conditions required for the present theory to apply appear to exist. We again consider -AG, the free energy of the p-q reaction, i.e., the opening of a channel’s gate. Recalling that r is the amount that AG would be reduced from its value for a gate held closed to that for a gate held open, it is seen that the value of r will depend upon the product of the voltage change which can occur across the gating region times the number of electron charges moving with the gate, In the axonal membrane the voltage across the membrane in the normal (resting) condition is of the order of -75 mV (inside of axon negative). This potential results from the membrane being semipermeable to Ktions in its resting state, Its voltage approaches the Nernst potential for K’, which has a high internal and low external concentration. During activity, the membrane permeability reverses, becoming more permeable to Na+ ions, which have a high external and low internal concentration. Thus during this phase the membrane potential reverses, the total change approaching as much as 150 mV. For adsorption of Ca2+ or any other gating process involving two electron charges, the value of I’ could theoretically approach a value as high as 1 2 (kT/2e = 12.5 mV). In practice, we have assumed a maximum value of the order of 6 for r. A decrease in the voltage across the membrane, essential to its excitation, directly reduces AG, that is, G,, since this voltage appears across the gating region when the gate is closed, and tends to hold the gate in the closed (9) state. But the reduction in the membrane voltage will also reduce
The Journal of Physical Chemistry, Vol. 84, No. 20, 1980 2659
Changes in Population Ratios and a Potential Fieid
I .o
PI0
0.I
61
6.5
6.3 G 4 - U n i t s o f kT
Flgure 6. Calciilation of change in Pldratio for tho same parameters equ,al to that of G, is used in Figure 3 but in which variation of assumed, each varying with the potential field.
r
the value of r. Thus in the calculation of the effect of a change in tlhe force field (e.g., thle voltage across the membrane), both r and G, must be changed in value. Figure 6 shows the result of such a calculation, based on the same parameters as used in calculating Figure 3, but including a simultaneous and equal variation in r and G . It is seen that a similar response is obtained but with a fess steep clhange in P / Q ,the change in G,, being partially compensated for by the change in r. I t must be emphasized that the models of the channel as shown in Figures 1 or 5 are intended oiily to illustrate the general principles involved and are highly idealized in order to clarify the presentation. In fact there is no reason to suppose that the channels are uniform as to ionic mobility, cross section, or negative fixed charge distribution. The theory does not depend upon these aseumptions, and calculations of very different (and possibly more realistic) models show similar changes in field distribution with opening and closing of gates.8J0
IO0
IO‘
102
Io’
Response Time of the Q-P Reaction. Effect of “Noise” The effect of the stochastic fluctuations in y on the response of a system to a perturbation may be found by the numerical integration of eq 20 and 16. Such calculations have been performed on the basis of parameters which could be appropriate to biomembranes. A relaxation rate constant ratio a p / a q= 10 is used, the relaxation of the membrane in the p (open) state being much more rapid than in the q (closed) state. A change in r is related to a change in k, because of the related change in G,, as discussed above, so that ke2/kel= exp(Fl - I’2). The results of the calculations are shown in Figure 7. The initial state is at I’ = 6 and k, = 0.0043. The curves show the time course of the value of P, the fraction of all the channels in the open state, at times following a change to I? = 5.9, for various values of D, corresponding to proportionally greater randomness of the y field. Times are in units of l/ap,which is probably of the order of 10 ps. The dramatic effect of D on the response time is evident, the time varying from seconds to milliseconds for the range of D from 0.003 to 0.02.11 Figure 8 illustrates the change in the distribution of the channel populations from the initial state with I‘ = 6 to the final state with r = 5.9 for D = 0.01. The effect of the value of D on the sharpness of transition is shown in Figure 9; other parameters are the same as those used in the calculation in Figure 7. The results illustrated in Figures 7-9 are typical of those calculated with relaxation functions such that two different steady states may exist, that is, where there are three solutions to eq 14. In such cases, where D = 0 (deterministic y), and at an infinitely rapid p-q transition rate, there is no possibility of any member of the system departing from its initial population group, either that at a low or that at a high value of y; an element undergoing an infinitesimal displacement always returns to its original steady-state position. The only effect of a parameter change, such as a change of I’ and a corresponding value of k,, _will be to shift the location of the steady state and thus P / Q . When the relaxation functions are such that there is only one solution to eq 14, that is, only one value of y corresponding to the steady state, it is similarly only the shift
TIME
1oq
I06
106
10’
Flgure 7. Time course of change of state of a system, showing effect of stochastic fluctuations of the y field: based on eq 2. Calculations employ parameter values which may roughly correspond to the axonal membrane: a,/a, = 10; = 6, k , = 0.0043 in the initial state, shifted to I’ = 5.9, k, = 0.00475 at zero time. Time scale in units of aP,which is probably of the order of 10 ps. Curves are for the values of parameter D in eq 20; increased D corresponds to increased randomness of y.
r
2880
The Journal of Physical Chemistry, Vol. 84, No. 20, 1980
Offner
a
Y b
Y Flgure 8. Distribution of channel population for system (membrane) with parameters employed in Figure 7, D = 0.01: (a) initial conditions = 6); (b) steady-state distribution (r = 5.9). Note logarithmic ordinates.
of its location that results in a change of PI& with a change in I?. However, as previously discussed, this change can be fairly sensitive to I?, depending upon the form of the relaxation functions. In such cases the performance of the system is much less influenced by the degree of randomness (noise) of the system. Such systems, which may be more relevant to the axonal membrane, will be discussed in a future publication. Hypothetical Chemical Application A result similar to that described above for the flow of ions through a channel, under the influence of an electrical potential difference, could occur in a purely chemical diffusion system, where the electrically activated “gates” are replaced by conformational changes due to ligand binding, and the electrical potential is replaced by the chemical potential of the ligand. Such a hypothetical channel is illustrated in Figure 10.
(r
A channel is shown through a membrane; the bath on the left is a solution having a low concentration Coof ligand R; on the right the ligand has a higher concentration Cp In, or forming, the wall of the channel at the left interface is a molecule M, capable of binding the ligand R. When the molecule is free of the ligand, the channel is open, as shown in Figure loa; but when the molecule binds R, it undergoes a conformational change, as shown in Figure lob. If bound a sufficiently long time, this would substantially block diffusion through the channel at the left interface. In this model, it is evident that if M has been ligand-free for a long period, the ligand concentration in the vicinity of M will be approximately Cot while if it has bound the ligand for a long period the solution in the channel will have assumed substantially the concentration Cp We now apply the theory developed above to this model, identifying the ligand-free molecule with the p state and
The Journal of Physical Chemistry, Vol. 84, No. 20, 1980 2001
Changes in Population Ratios and a Potential Field
site, each with Co taken as the standard state. We now see that in eq 21 [MR] = Q(7)and [MI = P ( y ) ,so that eq 21 may be written
“ O r - -
P / Q = k,eY (15) as before, where k, = kl/(k2Co).Since Co is assumed lower
E
0.I
c
;I
V
than Cr, the value of [R] will decrease in the p (unbound) state; that is, y will increase, and conversely in the q state. Thus y will vary with time according to eq 2. We thus obtain the same Fokker-Planck equations, eq 19 and 20, for this system, If we assume first-order kinetics for the relaxation process of y, the calculated responses would be similar to those already shown, for a concentration ratio across the channel was of the order of 4001 (T = 6). While this chemical model has interesting possibilities, it is not known whether it has any counterpart in nature.
k---
.-0
E
LL
6.0
r - Units
6.1
of kT
Flgure 9. Steady-state transition of a system with parameters as in Figure 7, for various values of 0;k, varies with k, = 0.0043 exp(6 - as explainled in text.
r),
Figure 10.
the bound molecule MR with the q state. We assume as before that the p-q reaction rate occurs very rapidly as compared with the rate of conformational change of M, so that local equilibrium may be assumed for the M + R = MR reaction: ki[MRI = ~z[Ml[lRl
(21)
kl and k2 being, respectively, the rate constants in the two directions. In eq 21 [MR] is the fraction of all the channels in which the concentration of ligand in the vicinity of the binding site on M is equal to [R] and which are in the bound (9) state, while [MI is the fraction of such channels which are unbound, that is, in the p state. We write 1- = In (Cr/Co)
(22)
7/ = -In ([R]/C0)i
(23)
the negative sign being due to the definition of y as the decrease in energy. Then X’(RT)is the chemical potential of R in the right-hand bath and -y(RT) is the chemical potential of R in the channel in the vicinity of the binding
Discussion In the preceeding treatment, it has been shown that a small perturbing force acting on an element can produce a change of state much greater than would be predicted from Boltzmann’s principle, if the energy involved were only the energy delivered to the element directly by the perturbing force. In fact, the force is applied to a much larger system, so that the total energy involved in the force field is not at all measured by the energy it delivers to the element itself but will be many times greater. Cooperative phenomenal2 have similarities to the described process, but there appear to be fundamental differences between the two. In cooperative phenomena energy is taken from the surroundings by a number of elements of the system. This energy is then shared among a much smaller number of degrees of freedom to produce a major change of state of the system. Such systems frequenty exhibit a large amount of hysteresis. In the process herein treated each element of the system acts completely independently of all other members. The total energy available to a degree of freedom (member) is thus obtained not by summing that of a number of elements but from summing that obtained from each of a large number of steps of the quasi-random walk of the element through the force field. This process exhibits no hysteresis but is continuously reversible along a single path. Cooperative systems can exhibit oscillatory behavior as a result of synchronization between elements; the present system cannot, although a stochastic fluctuation of state will exist. Some systems, such as the axonal membrane, may exhibit both independent and cooperative behavior and may thus result in oscillatory response. In the axon this can result from current flow through one channel affecting the local potential difference across the membrane seen by its neighbors, that is, the local value of r. Equation 2 contains an implicit approximation, since it implies that in each sense the force field y will always relax along the same pathway. This would not be stricqy true in a diffusion process, since the path of the quasi-random walk, as illustrated in Figure lb, will depend upon the succession of lengths of the steps, which would vary between members of the system. If, however, the individual steps are small, so that very many steps are taken along the path, as should be the case, the process will closely approach a constant functional form. Acknowledgment. I thank Dr. Erwin Bareiss for his assistance in formulating the numerical solution of the differential equations, and Drs. Irving Klotz, Arnold Siegert, George Schatz, and Mark Ratner for their helpful discussions. Dr. C. R. Hill was of great assistance in calculating the time-dependent behavior. This work was
2682
J. Phys. Chem. 1980, 84, 2662-2666
performed under NIH Grant No. NS 08137.
References and Notes (1) F. F. Offner, Biophys. J . , 21, 84a (1978). (2) F. F. Offner and S. H. Kim, J . Theor. B/o/., 61, 113 (1976). (3) The reason for the clustering of the values of y for the members of the system over a narrow range may be seen from slmple statlsthl considerations. Consider a time interval At sufflclently short that f,At and f At are small compared to I?, but long compared to the mean p-qgransition time (assumed to be very short compared to the relaxation tlmes). Then, since the p-q transition is considered to be a stochastic process, the variation of y wiil also be stochastlc. I f there are Ncycles of transition of state and the root mean square change in y durlng a transition is Ay, then the variance in y will be
(4) (5) (6) (7) (8) (9) 10) 11) 12)
AylN'". For a very large value of N, corresponding to rapid p-q transltions, the variance in y approaches zero. M. C. Wang and G. E. Uhlenbeck, Rev. Mod. Phys., 17,323 (1945). A. L. Hcdgkin and A. F. Huxley, J. Physkl. ( L d n ) , 117,500 (1952). R. W. Mway, CWSprhg/f&wSynp. mnr. W., 30,233 (1965). B. Frankenhaeuser and A. L. Hodgkln, J . Physiol. (London), 137, 218 (1957). F. F. Offner, J . Gen. Physiol., 56, 272 (1970). F. F. Offner and S. H. Kim, J . Theor. Mol., 61, 97 (1976). F. F. Offner, Biophys. J., 12, 1583 (1972). Ow present estimates, based on h m a l agltatlon fluctuations (Nyquist noise) in the channels, give a value of Dof the order of 0.01, However, this problem requires further analysis. A revlew of the theory of cooperative systems far from equllibrium is given by H. Haken, Rev. Mod. Phys., 47, 67 (1975).
Ionic Associations of Amine Hydrochlorides Dissolved in Amines. Conductance Measurements and Consequences on Solvated Electron-Ammonium Ion Reactivity M. 0. Delcourt Laboratoire de Physico-Chimie des Rayonnements, Universit6 Paris-Sud, BEit 350, 9 1405 Orsay C&ex, France (Received: December 26, 1979; In Final Form: May 20, 1980)
Conductance measurements performed on solutions of ammonium chloride,RNH3+Cl-,in amino solvents, RNH2, give information about ionic associations in these media. Previous results obtained by a pulse radiolysis technique concerning recombination rate constants between solvated electrons e[ and RNH3+cations are discussed in terms of these associations. Three typical cases have been studied. In the first one (hydrazine,N2H4),ionic association is negligible so that the absolute rate constant for the recombination e; + RNH3' can be deduced from the experimental rate constants by the sole ionic strength effect (k$ = (6 f 1) X lo7 M-' s-l). In the second case (propylamine, n-PrNH2),pairing is so important that only the rate constant related to the pair can be obtained (k = (6.0 f 0.6) X lo8 M-' s-l). In the third case (ethylenediamine,EDA), conductance data show the presence of isolated ions, pairs, triple ions, and even quadrupoles, in the concentration range 10-4-1 M. The limiting molar conductance of the ammonium chloride, EDAH'Cl-, is found to be ho = 74.6 f 0.1 cm2f2-l mol-] at 25 "C. Equilibrium constants are deduced for the pairing (Kp = (6.35 f 0.10) X lo3M-l) and for the triple ion association (KT N 9 M-l). This knowledge enables us to obtain the value of the rate constant of the solvated electron with the isolated cation EDAH', Le., NH2C2H4NH3+:k$ = (2.0 f 0.2) X 1O'O M-ls-l.
Introduction Among the reactions which take place in the early stages of the radiolysis of liquids, recombination between solvated electrons e, and the parent cation C+ has been shown' to be the main phenomenon determining the survival probability of e;. The reaction efficiency, expressed as the ratio of the experimental rate constant to the diffusion limit, has been recently introduced in theoretical in order to describe the time-space evolution of the radiolytical species, thus opening a way toward a better understanding of primary processes in the radiolysis of liquids. It is then necessary to know experimental rate constants for this recombination in various media, particularly in solvents where e; is known to survive for a long time (i.e. microseconds) after the energy deposit. In amine solvents, RNH2,the countercation C+ is the stable acidic ammonium ion, RNH3+,so that the recombination reaction can be written as e,- + RNH3+ products (1) Pseudo-first-order rate constants kI, of reaction 1in the presence of an excess of cation (su% as ammonium chloride) have been measured with the pulse radiolysis technique by studying the decay of the IR absorbance of e,. They have already been published for various solvents, notably for hydrazine (N2H4),5n-propylamine (n-PrNH2)6a
-
0022-3654/80/2084-2662$0 1.OO/O
and ethylenediamine (EDA).6a The transient product of this elementary reaction is a long-lived tight pair with the stoichiometry RNH, which no longer exhibits an IR absorption up to X = 1400 nm.6b The variation of klQMvs. nominal acid concentration has been investigated in each case. However, because of possible ionic associations, the bimolecular rate constant for reaction 1 cannot be derived from kQ directly; conductance measurements allow us to solve this problem through the determination of the ionic composition of solutions as shown below.
Experimental Section Conductance and permittivity have been measured with a low-frequency (1592-332) Wayne Kerr bridge (B 642). The constant of the conductance cell, calibrated with KCl solutions, is 0.72 f 0.02 cm at 298 K. The permittivity cell constant is 12.30 f 0.02 pF. Viscosity has been measured with a falling ball apparatus (Haake-Karlsruhe). Solvents from Merck were pure grade reagents. Various techniques of purification have been used; specific conductances of solvents measured in each case were almost identical with those of commercial products. However, because of the hygroscopic properties of amines and mainly of EDA, the solvent flasks were opened in an inert atmosphere and all conductance experiments were performed in a glove bag flushed with nitrogen gas. 0 1980 American Chemlcal Society
