Diffusion frequency factors in some simple examples of transition ...
Diffusion frequency factors in some simple examples of transition ...
Diffusion frequency factors in some simple examples of transition ...
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Proc. Nat. Acad. Sci. USA<br />
Vol 73, No. 3, pp. 679-683, March 1976<br />
Chemistry<br />
<strong>Diffusion</strong> <strong>frequency</strong> <strong>factors</strong> <strong>in</strong> <strong>some</strong> <strong>simple</strong> <strong>examples</strong> <strong>of</strong> <strong>transition</strong>state<br />
rate theory*<br />
(Eyr<strong>in</strong>g rate theory/dynamics versus diffusion)<br />
TERRELL L. HILL<br />
Laboratory <strong>of</strong> Molecular Biology, National Institute <strong>of</strong> Arthritis, Metabolism, and Digestive Diseases, National Institutes <strong>of</strong> Health, Bethesda, Maryland 20014<br />
Contributed by Terrell L. Hill, January 8, 1976<br />
ABSTRACT In the Eyr<strong>in</strong>g rate theory, the rate constant<br />
is expressed as a product <strong>of</strong> a <strong>frequency</strong> factor (kT/h) and a<br />
quotient <strong>of</strong> partition functions. Cont<strong>in</strong>u<strong>in</strong>g an earlier paper,<br />
it is shown here by means <strong>of</strong> <strong>simple</strong> <strong>examples</strong> that the Eyr<strong>in</strong>g<br />
formalism may be extended to <strong>in</strong>clude diffusion-controlled<br />
processes if a new <strong>frequency</strong> factor, D/RA, is substituted for<br />
kT/h, where D = diffusion coefficient, A = thermal de Broglie<br />
wavelength, and R = a characteristic distance that depends<br />
on the particular case. The Eyr<strong>in</strong>g formalism is also<br />
applicable <strong>in</strong> hybrid cases, <strong>in</strong>termediate between diffusion<br />
(D/RA) and dynamics (kT/h). Because these modified <strong>frequency</strong><br />
<strong>factors</strong> are not "universal" (as kT/h is), their ma<strong>in</strong><br />
use (other than conceptual) would appear to be <strong>in</strong> cases <strong>in</strong><br />
which one considers a <strong>simple</strong> model (with calculable <strong>frequency</strong><br />
factor) together with a related more complicated<br />
model. In the latter case, as an approximation, one would<br />
comb<strong>in</strong>e the "<strong>simple</strong>" <strong>frequency</strong> factor with the "complicated"<br />
quotient <strong>of</strong> partition unctions <strong>in</strong> order to obta<strong>in</strong> the desired<br />
rate constant. Examples are given.<br />
In the Eyr<strong>in</strong>g rate theory, the rate constant is expressed as a<br />
product <strong>of</strong> a <strong>frequency</strong> factor (f.f.) kT/h and a quotient <strong>of</strong><br />
partition functions (p.f.s), all relative to the same zero <strong>of</strong> energy<br />
(1). It was shown <strong>in</strong> an earlier paper (2) that the Eyr<strong>in</strong>g<br />
formalism can be extended to diffusion-controlled ligandmacromolecule<br />
association <strong>in</strong> solution by substitution <strong>of</strong> a<br />
new f.f., D/RA, <strong>in</strong> place <strong>of</strong> the usual dynamical factor<br />
kT/h. The primary object <strong>in</strong> this paper is to supplement this<br />
earlier discussion with additional elementary <strong>examples</strong> <strong>in</strong>volv<strong>in</strong>g<br />
diffusion. For completeness, we consider the correspond<strong>in</strong>g<br />
dynamical cases as well. Classical mechanics is<br />
used throughout. The paper <strong>of</strong> Chandrasekar (3), follow<strong>in</strong>g<br />
Kramers (4), provides the necessary background and much<br />
<strong>of</strong> the notation used.<br />
ONE-DIMENSIONAL EXAMPLES<br />
<strong>Diffusion</strong> Case. We consider the steady rate <strong>of</strong> <strong>transition</strong>s<br />
<strong>in</strong> the direction A -- B for the case shown <strong>in</strong> Fig. 1 (3, 4),<br />
where w = potential <strong>of</strong> mean force (1, 2). From the Smoluchowski<br />
equation, Eq. 312 <strong>of</strong> ref. 3,<br />
dc = j c-D -+ dw\<br />
at ax' ax kT dx'<br />
we obta<strong>in</strong> for the steady A -- B flux j (per unit area), as <strong>in</strong><br />
Eq. 472 <strong>of</strong> ref. 3,<br />
= Dc 4e"- /IR [11<br />
where<br />
R fJ e-f(X/kTdX, [2]<br />
Abbreviations: p.f., partition function; f.f., <strong>frequency</strong> factor.<br />
*<br />
This paper is dedicated to Henry Eyr<strong>in</strong>g as a personal "supplement"<br />
to the recent volume [(1971) Advances <strong>in</strong> Chemical Physics,<br />
eds. Prigog<strong>in</strong>e, I. & Rice, S. (Wiley-Interscience, New York),<br />
Vol. 211 published <strong>in</strong> his honor on the occasion <strong>of</strong> his seventieth<br />
birthday.<br />
D = diffusion coefficient, CA = local concentration at x =<br />
XA, and f(X) - w w-w(X) <strong>in</strong> the neighborhood <strong>of</strong> x = xc.<br />
The total number <strong>of</strong> particles (per unit area) <strong>in</strong> state A,<br />
treated as at equilibrium, is nA = CAqAA, where qA = equilibrium<br />
one-dimensional p.f. and A = h/(27rmkT)1/2 = thermal<br />
deBroglie wavelength. This follows from the fact that<br />
qAA is the so-called configuration <strong>in</strong>tegral (1) for state A.<br />
For example (1, 3), qA = kT/hvA (classical harmonic oscillator).<br />
The (first-order) rate constant for A A B is then (compare<br />
Eq. 473 <strong>of</strong> ref. 3)<br />
a = jI/nA = (DIRA)(e-w*1kT/qA) [3]<br />
But s<strong>in</strong>ce, for this <strong>simple</strong> model, q* = e-w*/kT <strong>in</strong> the Eyr<strong>in</strong>g<br />
rate theory (1), we see aga<strong>in</strong> (2) that replacement <strong>of</strong> the f.f.<br />
kT/h by a new factor D/RA produces the proper rate constant<br />
<strong>in</strong> the diffusion case. Here, R is def<strong>in</strong>ed by Eq. 2. If we<br />
expand f(X) about X = 0 and reta<strong>in</strong> only the quadratic term<br />
Wc2mX2/2, which def<strong>in</strong>es wc, we f<strong>in</strong>d R = (2AkT/<br />
If the barrier <strong>in</strong> w(X) has<br />
Wc2m)1/2, or Wc2mR2/2 = 7rkT.<br />
the hypothetical shape shown <strong>in</strong> the <strong>in</strong>set <strong>of</strong> Fig. 1, then R =<br />
1. In either <strong>of</strong> these cases, for a wide barrier, i.e., R x, o we<br />
have a D/RA - - 0.<br />
As already po<strong>in</strong>ted out (2), D/RA is not a "universal" f.f.<br />
as kT/h is <strong>in</strong> the (generally <strong>some</strong>what approximate) Eyr<strong>in</strong>g<br />
theory. Rather, D/RA depends on the special case. Despite<br />
this, there are at least three advantages to rewrit<strong>in</strong>g diffusion<br />
rate constants <strong>in</strong> the modified Eyr<strong>in</strong>g form, i.e., (DI<br />
RA) X (p.f. quotient): (a) the considerable conceptual and<br />
<strong>in</strong>tuitive benefits deriv<strong>in</strong>g from the Eyr<strong>in</strong>g formalism are<br />
thereby extended to a new class <strong>of</strong> rate processes; (b) it may<br />
be possible to estimate R and hence to calculate the diffusion<br />
rate constant (with the aid <strong>of</strong> the appropriate p.f.s) <strong>in</strong> <strong>some</strong><br />
cases <strong>in</strong> which the ab <strong>in</strong>itio diffusion problem is too difficult<br />
to solve; and (c) R may be found for a diffusion problem <strong>in</strong><br />
a relatively small number <strong>of</strong> coord<strong>in</strong>ates (or the diffusion<br />
problem might be simplified <strong>in</strong> <strong>some</strong> other way) and then<br />
this R is reta<strong>in</strong>ed, as an approximation, when more coord<strong>in</strong>ates<br />
are added to the model via the p.f.s.<br />
An example <strong>of</strong> (c), above, has already been given (2):<br />
translation A translation + rotation <strong>in</strong> the b<strong>in</strong>d<strong>in</strong>g <strong>of</strong> spherical<br />
ligands. Another example is <strong>in</strong> the paragraph follow<strong>in</strong>g<br />
Eq. 13 <strong>of</strong> ref. 2: R for the field-free b<strong>in</strong>d<strong>in</strong>g case was reta<strong>in</strong>ed,<br />
as an approximation, for the case with an external<br />
field (this approximation is not used below, <strong>in</strong> Eq. 24). As a<br />
third example <strong>of</strong> (c), let us extend, formally, the one-dimensional<br />
result <strong>in</strong> Eqs. 2 and 3 to three dimensions by allow<strong>in</strong>g<br />
yz-vibrational motion <strong>of</strong> the particle normal to the x-axis<br />
(the "reaction coord<strong>in</strong>ate"), both <strong>in</strong> the <strong>in</strong>itial state and <strong>in</strong><br />
the <strong>transition</strong> state. Then<br />
a= (DIRA)(q$ ,ze -u-*T/qAqA z)<br />
[4]<br />
where q*YZ and qAyz are the new vibrational p.f.s.<br />
679
680 Chemistry: Hill<br />
Proc. Nat. Acad. Sci. USA 73 (1976)<br />
w<br />
w=w* -f(X) s .w )) kT<br />
w = 0 * - - - - -,<br />
XA Xc XB<br />
0 X = x -Xc<br />
FIG. 1. Schematic. One-dimensional potential <strong>of</strong> mean force for study <strong>of</strong> <strong>transition</strong> A - B across barrier w* >> kT at xc. Inset shows<br />
hypothetical barrier with flat top and steep edges.<br />
General Relations. When (<br />
=<br />
kT/mD is not large compared<br />
to 2wc, we need a solution <strong>of</strong> the (steady-state) Kramers<br />
equation (Eq. 477, ref. 3)<br />
wa + Kow =<br />
OU aw + COW + qa0W<br />
ax au auau<br />
where W(X,u) = probability density <strong>in</strong> X and u (velocity), q<br />
= kTf3/m, and K = -m-1dw/dX. We write<br />
W(X,u) = CF(X,u)e-mu2/2kTee-w/kT [6]<br />
where F 1 at equilibrium. Then (Eq. 485, ref. 3)<br />
OF aF a2F aF<br />
Ua + K a= - AlU u [7]<br />
For the solution <strong>of</strong> Eq. 7 around the peak <strong>in</strong> w, the boundary<br />
conditions are F = 1 for X - -c and F = 0 for X<br />
+o. The required steady flux j across X = 0 (or, <strong>in</strong>deed,<br />
across any other value <strong>of</strong> X) is<br />
[5]<br />
j= f W(X=o,u)udu. [8]<br />
In the diffusion case 3>> 2wc and f = wc2mX2/2, we can<br />
obta<strong>in</strong> j from§ Eqs. 6 through 8 (3), compare the result with<br />
Eq. 1, and thus evaluate the normalization constant <strong>in</strong> W.<br />
One f<strong>in</strong>ds C = cA(m/27rkT)'/2. Thus, the normalization is<br />
such that, at equilibrium, fWdu = cAe-w/kT = local concentration.<br />
Incidentally, <strong>in</strong> the diffusion case, one can show<br />
from§ f Wdu at X = 0 (and verify from the Smoluchowski<br />
equation approach) that the local concentration at the <strong>transition</strong><br />
state <strong>in</strong> steady flow is one-half the equilibrium value,<br />
cAe/k<br />
c -w*IkT<br />
Dynamical Case. Here (3 = 0 and w = potential energy.<br />
Only the left-hand sides <strong>of</strong> Eqs. 5 and 7 rema<strong>in</strong> (Liouville's<br />
equation). The solution, for our boundary conditions, is F =<br />
1 or F = 0 with a step between these values along the curve<br />
u = [2f(X)/m]'/2 <strong>in</strong> the X,u-plane. If f = WC2mX2/2, this<br />
curve is the straight l<strong>in</strong>e u = wcX (Q = 0 <strong>in</strong> the notation <strong>of</strong><br />
ref. 3). On X = 0 (Eq. 8), where f = 0, F = 1 for u > 0 and<br />
§ For this purpose, it is helpful to use (at X = 0) F(u) = ('k) + (Wc2/<br />
27rfq)1/2u +.--.<br />
F = 0 for u 0 and F = 0, u 1.5.<br />
Thus, SYE and 5SD are merely two limit<strong>in</strong>g forms <strong>of</strong> f.f.<br />
(though no doubt the most important). Indeed, a further<br />
generalization <strong>of</strong> considerable physical <strong>in</strong>terest (which is<br />
be<strong>in</strong>g <strong>in</strong>vestigated) follows on use <strong>of</strong> (3 = (3(X) <strong>in</strong> Eq. 7 <strong>in</strong>stead<br />
<strong>of</strong> (3 = constant. For example, ( is a step function: (3 =<br />
(3- for X < 0 and (3 (3+ for X > 0 (see Discussion).<br />
Of course a generalized 5' (i.e., one other than 9rE or 9'D)<br />
can also be used <strong>in</strong> <strong>examples</strong> <strong>of</strong> type (c) above (see Eq. 29,<br />
below).
0.8<br />
0.6<br />
0.4<br />
0.2<br />
X<br />
Chemistry:<br />
\1/2s<br />
Hill<br />
\ \<br />
' \ ----<br />
I-s ".<br />
I...<br />
0 1 2 3 4<br />
s *l/2wc<br />
-<br />
FIG. 2. Solid curve: /9E as a function <strong>of</strong> s = :X2wc (Eq. 12).<br />
/E = 1 is the limit<strong>in</strong>g value (O = kT/h) at s = 0; the <strong>in</strong>itial slope<br />
is -1. For large s, 9/E- 1/2s (i.e., 9 -. D = D/RA).<br />
F<strong>in</strong>ally, we note that, <strong>in</strong> the <strong>in</strong>termediate case, we also<br />
have (M)cAe-w*/kT for the local concentration at X = 0,<br />
s<strong>in</strong>ce F(u) on X = 0 is equal to i plus an odd function <strong>of</strong> u.<br />
THREE-DIMENSIONAL EXAMPLES (WITH<br />
SPHERICAL SYMMETRY)<br />
Bimolecular Association <strong>in</strong> Solution. We have already<br />
considered (2) the case <strong>in</strong> which freely diffus<strong>in</strong>g po<strong>in</strong>t (ligand)<br />
particles may be bound at sites on relatively large adsorbent<br />
molecules once a ligand molecule crosses a hemispherical<br />
shell <strong>of</strong> radius rc (Fig. 3, dashed l<strong>in</strong>e) about any<br />
site. It was found that, <strong>in</strong> the f.f. D/RA for this case, R = rc<br />
(Eq. 9, ref. 2).<br />
The above model can be modified easily to treat a bimolecular<br />
reaction <strong>in</strong> which a molecule <strong>of</strong> species 1 (ml, Dl)<br />
comb<strong>in</strong>es with another <strong>of</strong> species 2 (m2, D2) to form a bimolecular<br />
complex whenever a mutual approach is made<br />
with<strong>in</strong> a critical distance rc. The second-order rate constant<br />
a' for this process is found to be 47r(DI + D2)rc (Eq. 447,<br />
ref. 3). The p.f.s <strong>in</strong>volved are (ref. 1, p. 155)<br />
q1 = V/A13, q2 = V/A23, q* = (V/A123)(27rkT/h2)4rrc2<br />
[14]<br />
where ml + M2 appears <strong>in</strong> A12 and 1A = Mlm2/(Ml + m2).<br />
Proc. Nat. Acad. Sci. USA 73 (1976) 681<br />
If, for this case, <strong>in</strong> view <strong>of</strong> the above paragraph, we write<br />
the f.f. <strong>in</strong> the form (D1 + D2)/rcX, that is,<br />
'= 47r(D, + D2)rc=<br />
[(D1 + D2)/rcA](qt/V)/(q1/V)(q2/V) [15]<br />
then we f<strong>in</strong>d, as one might expect, that X = h/(27rgukT)l/2.<br />
If, say, species 2 is very large (M2 >> ml, D2
6.82 Chemistry: Hill<br />
If w(r) has the simplified shape shown <strong>in</strong> <strong>in</strong>set (a) <strong>of</strong> Fig.<br />
3, Eq. 18 gives R = rc, as found for the reverse processl (2).<br />
If w(r) appears as <strong>in</strong> <strong>in</strong>set (b) <strong>of</strong> Fig. 3, we f<strong>in</strong>d R = rcl/(rc<br />
+ 1).<br />
A more realistic case: if w(r) = w* for r > rc (dashed l<strong>in</strong>e<br />
<strong>in</strong> Fig. 3) and f = wC2mX2/2 for r < rc, then Eq. 18 gives R<br />
rc + Ar, where Ar = (7rkT/2oc2m)'/2. The curvature<br />
<strong>of</strong> f(r) for r < rc must be such that WCc2mrc2/2 = O(wt*) >><br />
kT. Therefore, rc >> Ar and R rc [i.e., essentially as <strong>in</strong><br />
the <strong>in</strong>set (a) case].<br />
If, <strong>in</strong> Fig. 3, w* - WB >> kT and f = WC2mX2/2 on both<br />
sides <strong>of</strong> r = rc (X = 0), then we get R n 2Ar, as already<br />
found <strong>in</strong> the one-dimensional problem. Hence, the rate constant<br />
for escape is much larger here than <strong>in</strong> the preced<strong>in</strong>g<br />
paragraph (R rc), the ratio be<strong>in</strong>g rc/2Ar.<br />
-<br />
S<strong>in</strong>ce we have now considered the diffusion-escape problem<br />
<strong>in</strong> both one (Eq. 3) and three (Eq. 20) dimensions, we<br />
can use the results to test, <strong>in</strong> a specific case, the approximation<br />
suggested <strong>in</strong> Eq. 4. If we had started with Eq. 3 (for<br />
present purposes, let us use RI to denote the R given by Eq.<br />
2) and then <strong>in</strong>troduced q yz = 27rrc2/A2 and qA- qAqAyz<br />
<strong>in</strong>to Eq. 4 <strong>in</strong> order to extend Eq. 3 to three dimensions, we<br />
would have obta<strong>in</strong>ed Eq. 20 but with R1 <strong>in</strong> place <strong>of</strong> the correct<br />
R3 (def<strong>in</strong>ed by Eq. 18). The approximation R1 - Rs is<br />
satisfactory to the extent that one is justified <strong>in</strong> plac<strong>in</strong>g r-2<br />
<strong>in</strong> Eq. 18 <strong>in</strong> front <strong>of</strong> the <strong>in</strong>tegral sign, as rc-2. Thus, the<br />
sharper the peak <strong>in</strong> w(r), the better the approximation R1 _<br />
R3. Indeed, we have already noted <strong>in</strong> the preced<strong>in</strong>g paragraph<br />
that, for the case f X2 (IXI > 0), R1 = 2Ar and R3<br />
2Ar.<br />
F<strong>in</strong>ally, the concentration at r = rc (<strong>transition</strong> state) deserves<br />
comment. If we take 1 = rA and 2 = rc <strong>in</strong> Eq. 16, we<br />
f<strong>in</strong>d<br />
= ce [1 - ( [22]<br />
where the <strong>in</strong>tegrands are the same as <strong>in</strong> Eq. 18. Of course, if<br />
there were equilibrium between the <strong>transition</strong> state and<br />
state A, [ ] 1. In the "dashed l<strong>in</strong>e" case (Fig. 3) above, the<br />
ratio f/f Ar/rc is small and [ ] 1. But <strong>in</strong> the next example<br />
(f - X2 for Xi > 0), cc is one-half <strong>of</strong> the equilibrium<br />
value cAe-w*/kT (as <strong>in</strong> the one-dimensional problem). For<br />
<strong>in</strong>sets (a) and (b) <strong>of</strong> Fig. 3, cc has the equilibrium value.<br />
We turn now to the <strong>in</strong>verse (b<strong>in</strong>d<strong>in</strong>g) process <strong>in</strong> the diffusion<br />
case. Fig. 3 and Eq. 16 still apply except that now c<br />
= 0 <strong>in</strong> state A and c = CB <strong>in</strong> state B (rB oo), If we choose<br />
1 = rA and 2 = rB <strong>in</strong> Eq. 16, we obta<strong>in</strong> for the second-order<br />
rate constant for b<strong>in</strong>d<strong>in</strong>g ligand to macromolecule (2),<br />
a' = -JICB = 2rDrc2e-(u- WB)/kT/R [23]<br />
where R is aga<strong>in</strong> given by Eq. 18. The special cases <strong>of</strong> Fig. 3,<br />
above, lead<strong>in</strong>g to various R values, apply here as well. For<br />
example, for <strong>in</strong>set (a), a' = 27rDrc as <strong>in</strong> ref. 2, Eq. 8.<br />
As already mentionedl, the f.f. D/RA <strong>in</strong> Eq. 20 must also<br />
be <strong>in</strong>volved here for the <strong>in</strong>verse process. But to verify this, a<br />
little more care is required with the partition functions (because<br />
the <strong>in</strong>verse process is bimolecular). Us<strong>in</strong>g qA and q2$<br />
as def<strong>in</strong>ed above, we now write (2) p.f.A = qA3qV, p.f.$ =<br />
q2tqV, qo = qV, and qB = (V/A3)e wB/kT, where qo is the<br />
p.f. <strong>of</strong> a macromolecule with an empty b<strong>in</strong>d<strong>in</strong>g site. Then<br />
we f<strong>in</strong>d, as expected, that<br />
a' = (D/RA)(p-f.*/V)/(q0/V)(qB/V) [24]<br />
The f.f.s must necessarily be the same <strong>in</strong> the expressions for the<br />
two <strong>in</strong>verse rate constants, <strong>in</strong> view <strong>of</strong> the p.f. ratios <strong>in</strong>volved and<br />
their relation to the equilibrium constant (1).<br />
Proc. Nat. Acad. Sci. USA 73 (1976)<br />
agrees with Eq. 23. The equilibrium constant for b<strong>in</strong>d<strong>in</strong>g is<br />
(compare Eq. 5, ref. 2)<br />
K = a'/a = (pf-A/V)/(qO/V)(qB/V)<br />
=<br />
qA3A3ewRlkT. [25]<br />
To f<strong>in</strong>d cc, we choose 1 = rA and 2 = rc <strong>in</strong> Eq. 16 with<br />
the result<br />
Cc = cBe-(u- WB)/kT (f/ ) [26]<br />
In a certa<strong>in</strong> sense, this is the <strong>in</strong>verse <strong>of</strong> Eq. 22. For <strong>in</strong>sets (a)<br />
and (b), Fig. 3, cc = 0. In the "dashed l<strong>in</strong>e" case, cc n<br />
CBAr/rC, which is small. In the f X2 aIXI > 0) case, cc is<br />
one-half <strong>of</strong> the equilibrium value.<br />
_<br />
In the conventional treatment <strong>of</strong> this b<strong>in</strong>d<strong>in</strong>g problem,<br />
us<strong>in</strong>g a differential equation <strong>in</strong> c, the boundary condition c<br />
= 0 ("absorption") on the surface r = rc is used. The above<br />
paragraph shows that this is usually safe (the cotrect condition<br />
is CA = 0); however, <strong>in</strong> the f<br />
- X2 case, c £ 0 is justified<br />
only if w* - WB >> kT.<br />
Dynamical Case. We consider the rate <strong>of</strong> escape, <strong>in</strong> Fig.<br />
3, when 13 = 0. We are concerned with the three-dimensional<br />
Liouville equation with potential energy w a function <strong>of</strong> r<br />
only. But because <strong>of</strong> the condition rc >> Ar already mentioned<br />
<strong>in</strong> relation to this figure, the spherical surface can be<br />
treated as planar to a good approximation. Hence the discussion<br />
<strong>of</strong> the one-dimensional dynamical case, above, applies<br />
here with very little change. We obta<strong>in</strong> Eq. 9 for jr, as before,<br />
for any shape f(r). The total flux across the surface r =<br />
rc is J = 2lrrC2j,. The number <strong>of</strong> molecules <strong>in</strong> state A is nA<br />
= cAqA3A3. Thus we are led to<br />
a = J/nA = (kT/h)(q2t/qu), [27]<br />
<strong>in</strong> agreement with Eyr<strong>in</strong>g's formulation (q2* is given by Eq.<br />
21). If we use Eq. 19 for qA3,<br />
a = VA(rc /rA)e [28]<br />
Intermediate Case. For f = WC2mX2/2 and arbitrary f3/<br />
2wc, aga<strong>in</strong> because <strong>of</strong> the condition rc >> Ar we have essentially<br />
the one-dimensional differential equation, Eq. 5, to a<br />
good approximation. The one-dimensional solution (3, 4)<br />
can, therefore, be used but with J = 27rrC2j and nA =<br />
cAqA3A3. Hence<br />
al<br />
[29]<br />
= 3 (q2*/qA3)<br />
where 5f is given by Eqs. 12 and 13. Note that this follows<br />
from Eq. 11 simply by modify<strong>in</strong>g the p.f. quotient. The<br />
"exact" agreement <strong>in</strong> f.f. <strong>in</strong> the two cases is <strong>of</strong> course simply<br />
a consequence <strong>of</strong> the approximation we have made <strong>in</strong> handl<strong>in</strong>g<br />
the three-dimensional problem.<br />
DISCUSSION<br />
(a) The form D/RA for the diffusion f.f. is fairly obvious<br />
from an essentially dimensional po<strong>in</strong>t <strong>of</strong> view. In the first<br />
place, j D so we must have f.f. D. Then D = (r2)/6r or<br />
(x2)/2r, where r-1 is the <strong>frequency</strong> <strong>of</strong> steps <strong>in</strong> a three-dimensional<br />
random walk, and (r2) is the mean square displacement<br />
per step (3). S<strong>in</strong>ce f.f. and r-1 are both frequencies,<br />
D <strong>in</strong> f.f. must be divided by a quantity with dimensions<br />
(length)2. That one <strong>of</strong> these lengths should be A, leav<strong>in</strong>g the<br />
other (R) to characterize the <strong>transition</strong> state <strong>in</strong> <strong>some</strong> way,<br />
can be understood as follows. We are really forc<strong>in</strong>g a classi-
Chemistry:<br />
Hill<br />
Proc. Nat. Acad. Sci. USA 73 (1976) 683<br />
cal (diffusion) rate constant <strong>in</strong>to the form f.f. X p.f. quotient.<br />
S<strong>in</strong>ce any "classical" one-dimensional p.f. - h-1 (1), and<br />
s<strong>in</strong>ce, by def<strong>in</strong>ition, q* <strong>in</strong> the numerator <strong>of</strong> p.f. quotient always<br />
has one degree <strong>of</strong> freedom less than the denom<strong>in</strong>ator,<br />
it is necessary for f.f. - h <strong>in</strong> order that the classical rate<br />
constant itself be <strong>in</strong>dependent <strong>of</strong> h. An obvious length to appear<br />
<strong>in</strong> the denom<strong>in</strong>ator <strong>of</strong> f.f., with the required property<br />
length h, is A.<br />
Of - course this latter argument (f.f. h-1) applies just as<br />
well to 9E(=kT/h), or to any AT, s<strong>in</strong>ce classical cases (as <strong>in</strong><br />
this paper) must be encompassed by the rate theory. That is,<br />
use <strong>of</strong> p.f. quotient <strong>in</strong> the Eyr<strong>in</strong>g theory requires that 9E<br />
h-'. Note, <strong>in</strong>cidentally, that kT/h can also be written as<br />
v/4A, where iv = mean velocity. Thus, we have a (limited)<br />
physical <strong>in</strong>terpretation <strong>of</strong> (kT/h)-1: it is the time required<br />
(for a particle <strong>of</strong> mass m) to travel a distance 4A at velocity<br />
v. The analogous statement for diffusion is: (D/RA)'1 is the<br />
time required for a root mean square displacement <strong>in</strong> r <strong>of</strong><br />
(6RA)'/2 or <strong>in</strong> x <strong>of</strong> (2RA)1/2<br />
(b) When <strong>simple</strong> molecules react <strong>in</strong> essentially a vacuum,<br />
we have the pure dynamical case. This can be handled approximately<br />
by Eyr<strong>in</strong>g's theory or (<strong>in</strong> pr<strong>in</strong>ciple) exactly by<br />
quantum mechanical collision theory. When a rate process<br />
occurs <strong>in</strong> the presence <strong>of</strong> a surround<strong>in</strong>g medium, energy exchange<br />
with the medium via <strong>in</strong>termolecular forces becomes<br />
possible, and furthermore the medium may <strong>of</strong>fer velocitydependent<br />
resistance to the <strong>transition</strong>. For the relatively<br />
large molecular systems we are <strong>in</strong>terested <strong>in</strong> here, these effects<br />
are treated by Brownian motion theory. Although the<br />
medium is usually spoken <strong>of</strong> as a liquid (or sufficiently dense<br />
gas), it could also be a quasi-solid (a prote<strong>in</strong> molecule, for example)-with<br />
quantitative differences, <strong>of</strong> course. Consider<br />
for example, the b<strong>in</strong>d<strong>in</strong>g (or escape) <strong>of</strong> a ligand to a site<br />
with<strong>in</strong> a prote<strong>in</strong> molecule that is <strong>in</strong> aqueous solution. In its<br />
approach to the b<strong>in</strong>d<strong>in</strong>g site, the ligand <strong>in</strong>teracts with the<br />
liquid medium (solvent) more or less up to the <strong>transition</strong><br />
state, but beyond this (i.e., <strong>in</strong> the site) the ligand <strong>in</strong>teracts<br />
with quasi-solid (i.e., with the prote<strong>in</strong> molecule plus possibly<br />
a few water molecules). This example might be characterized,<br />
<strong>in</strong> the above notation, by two 13 values, one for the<br />
quasi-solid (presumably relatively small) and one for the liquid,<br />
with more or less <strong>of</strong> a sharp step between the two values<br />
at the <strong>transition</strong> state. That is, 13 = 13(r) rather than 13 = constant.<br />
In an <strong>in</strong>ternal conformational change <strong>in</strong>volv<strong>in</strong>g a<br />
group or region with<strong>in</strong> a prote<strong>in</strong> molecule, a constant a<br />
might be appropriate but it would be a quasi-solid 1, presumably<br />
small, and not a liquid one.<br />
APPENDIX<br />
We sketch here a relatively <strong>simple</strong> derivation (compare refs.<br />
3 and 4) <strong>of</strong> the Smoluchowski equation from the Kramers<br />
equation<br />
aw/at + U'VrW + K-VUW = flVu (Wu) + qV.2W. [30]<br />
The local concentration and flux are<br />
c = J Wdu, i=f Wudu.<br />
co<br />
co<br />
[31]<br />
For t >> 13 (3, 4); the velocity distribution is practically<br />
Maxwellian:<br />
W _ c(m/27rkT)3"2 exp[-m(u"2 + u 2 + u22)/2kT]. [32]<br />
We multiply Eq. 30 by udu and <strong>in</strong>tegrate (as <strong>in</strong> Eqs. 31).<br />
To get first-order results, we need to use Eq. 32 explicitly <strong>in</strong><br />
the four <strong>in</strong>tegrals aris<strong>in</strong>g from aW/at and us VW, but not<br />
otherwise. The uaW/ax term, on <strong>in</strong>tegration, leads to (kT/<br />
m) ac/ax; the other three <strong>in</strong>tegrals give zero. The rema<strong>in</strong><strong>in</strong>g<br />
from Eq. 30, are <strong>in</strong>tegrated by parts us<strong>in</strong>g<br />
only the strong W 0 property as jUil -a c. The term<br />
KXaW/lau gives -Kxc; the term 13 a(Wu.)/au, results <strong>in</strong><br />
-1jx; the other seven <strong>in</strong>tegrals are equal to zero. In this way<br />
we f<strong>in</strong>d for the flux<br />
i -D[Vrc (mK/kT)cl = - [33]<br />
n<strong>in</strong>e <strong>in</strong>tegrals,<br />
The Smoluchowski equation then follows from the requirement<br />
<strong>of</strong> cont<strong>in</strong>uity, ac/at = -V. j. For a much more general<br />
treatment <strong>of</strong> this type, see Rice and Gray (5).<br />
1. Hill, T. L. (1960) Statistical Thermodynamics (Addison-Wesley,<br />
Read<strong>in</strong>g, Mass.).<br />
2. Hill, T. L. (1975) Proc. Nat. Acad. Sci. USA 72,4918-4922.<br />
3. Chandrasekhar, S. (1943) Rev. Mod. Phys. 15, 1-89.<br />
4. Kramers, H. A. (1940) Physica 7,284-304.<br />
5. Rice, S. A. & Gray, P. (1965) <strong>in</strong> Statistical Mechanics <strong>of</strong> Simple<br />
Liquids (Wiley, New York), pp. 249-253.