Energetics and Dynamics of Gated Reactions - Advances in Chemistry

6 Energetics and Dynamics of Gated Downloaded by UNIV OF SOUTHERN CALIFORNIA on June 18, 2016 | http://pubs.acs.org Publication Date: May 5, 1989 | doi: 10.1021/ba-1990-0226.ch006

Reactions Control of Observed Rates by Conformational Interconversion B r i a n M. Hoffman, M a r k A . Ratner, and Sten A . Wallin Department of Chemistry, Northwestern University, Evanston, IL 60208

Several unusual aspects of reactions in larger molecules, especially biological systems in the condensed phase, are discussed. In particular,we are concerned with processes involving "soft" (overdamped, low-frequency) degrees of freedom, such as might be associated with a conformational interconversion in a macromolecule. The role of subsidiary stable minima on the reaction surface is considered; subsidiary stable minima can provide an indirect transfer route whose activation barriers can be substantially smaller than the direct, concerted reaction path from precursor to successor. When these considerationsare applied to a cyclic electron-transfer scheme, "gating" behavior, in which conformational changes control the overall process, emerges naturally. This chapter also considers quite a different dynamic effect, where anisotropy in the diffusion of the system on the potential energy surface substantially changes the dynamic behaviorand can lead to transient decays that cannot be fit to a single exponential law.

CONVENTIONAL TRANSITION-STATE T H E O R Y MUST B E E X T E N D E D w h e n a

reacting system has certain k e y features associated w i t h its p o t e n t i a l - e n e r g y surface. T h e s e theoretical considerations w e r e i n i t i a t e d ( I , 2) to explore w h a t consequences m i g h t arise w h e n an electron-transfer system exhibits m u l t i p l e stable conformational states that can i n t e r c o n v e r t at rates c o m p e t i t i v e w i t h 0065-2393/90/0226-0125$06.50/0 © 1990 American Chemical Society

Johnson et al.; Electron Transfer in Biology and the Solid State Advances in Chemistry; American Chemical Society: Washington, DC, 1989.

126

E L E C T R O N TRANSFER IN BIOLOGY A N D T H E SOLID STATE

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e l e c t r o n transfer (3-5). A s n o t e d earlier (J), this situation is g e n e r a l a n d b y no means l i m i t e d to electron-transfer reactions (6-8). T h e f o l l o w i n g section describes a c y c l i c k i n e t i c s c h e m e appropriate for c o n v e n t i o n a l i n t r a m o l e c u l a r e l e c t r o n transfer ( E T ) , w h e r e reactant a n d p r o d uct each e x h i b i t a single stable conformation a n d conformational d y n a m i c s are not i m p o r t a n t . A f t e r that the role o f conformational i n t e r c o n v e r s i o n is discussed, a n d the k i n e t i c s c h e m e that applies to a system i n w h i c h e l e c t r o n transfer is c o u p l e d to the c o n v e r s i o n b e t w e e n two stable conformations is d e v e l o p e d (I). T h e n these equations are solved, a n d t h e i r most i n t e r e s t i n g l i m i t i n g forms are analyzed. F i n a l l y , the alternate situation (2) is d e s c r i b e d , i n w h i c h conformational d y n a m i c s q u a l i t a t i v e l y change the o b s e r v e d kinetics o f a reaction. T h e r e acting system that is c o n s i d e r e d undergoes anisotropic diffusion o n a t w o d i m e n s i o n a l e n e r g y surface, one d i m e n s i o n of w h i c h can b e c o n s i d e r e d a conformational coordinate. T h i s case corresponds to a generalization o f K r a m e r s ' d e s c r i p t i o n o f c h e m i c a l reactions i n the strongly c o u p l e d r e g i m e a n d leads to n o n e x p o n e n t i a l relaxations, e v e n for surfaces w i t h one stable c o n formation for each of the reactant a n d p r o d u c t .

Conventional Intramolecular Electron Transfer A general k i n e t i c s c h e m e for s i m p l e i n t r a m o l e c u l a r electron-transfer reac tions is g i v e n i n F i g u r e 1. A system consisting o f a n i n t r a m o l e c u l a r (linked) d o n o r - a c c e p t o r [d,a] p a i r i n its g r o u n d state, A , is e x c i t e d to a reactive state, A * . M o s t often this is d o n e b y flash p h o t o l y t i c excitation of d. E l e c t r o n

Figure 1. Simple kinetic scheme for electron transfer within a linked donoracceptor pair. State A represents the ground-state [d,a] pair. A * is the state formed at t = 0 by excitation to the donor excited state (d*,a). State A* can decay back to A (rate constant, k ) or can react to form the charge-transfer intermediate, I (rate constant, k ), composed of the (d ,a~~) pair. In turn, reverse charge transfer re-forms A from I (rate constant, kb). %

D

t

+


6.

127

Gated Reactions

H O F F M A N E T AL.

transfer, d * - » a (or d — * a*), t h e n produces t h e charge-separated i n t e r m e d i a t e , I = [d ,a~], at rate constant, k . T h i s i n t e r m e d i a t e can r e t u r n to +

t

the g r o u n d state b y t h e reverse electron-transfer process, a "

d

+

, with

rate constant, k . W i t h i n this scheme, the t i m e course o f t h e species A * a n d b

I are as follows (9): A*(t) = A*(0)e-**


I (t) = A*(0) (

y

(la)

f

k t

) Hp*** -

y

(lb)

e~ ) kbt

w h e r e A*(0) is t h e i n i t i a l concentration o f [d*,a]. A c c o r d i n g to e q l a , A * decays e x p o n e n t i a l l y w i t h the rate constant, k = k p

where k

D

D

(lc)

+ k

t

is t h e decay rate o f A * i n t h e absence o f E T . T h u s , k can b e t

d e t e r m i n e d from k b y i n d e p e n d e n t m e a s u r e m e n t o f k . p

D

E q u a t i o n l b represents t h e creation o f I from A * at a rate constant k

t

a n d its reaction to f o r m A at rate constant k . T h i s e q u a t i o n corresponds to b

an e x p o n e n t i a l rise a n d fall o f I, w i t h its concentration m a x i m u m at t i m e T: I(T) = A * ( 0 X * / * > ) exp(-fc/* - * a n d k< t

refer to t h e larger a n d s m a l l e r o f k a n d k . I f k < fc , t h e n k c a n b e h

t

b

b

p

d e t e r m i n e d from t h e slow decay o f i n t e r m e d i a t e I ; i f k > k b

py

t h e n k is b

o b t a i n e d b y m e a s u r i n g the r a p i d appearance o f I . T h e first p o r t i o n o f this scheme also applies to reactions w h e r e A * represents a state i n w h i c h a r a p i d b i m o l e c u l a r e v e n t has r e d u c e d d o r o x i d i z e d a. S u c h p r e p a r a t i o n c a n b e d o n e b y p u l s e radiolysis o r b y flash photolysis o f a sacrificial E T reagent (10); a subsequent

intramolecular

charge-transfer step produces I as t h e e n d p r o d u c t . T h e s e t w o cases differ from the flash p h o t o l y t i c excitation o f d (or a) i n that k

D

= 0.

Introduction of Conformational Variation A n electron-transfer event is c o n t r o l l e d b y t h e F r a n c k - C o n d o n p r i n c i p l e , a n d thus is h i g h l y sensitive to t h e a c c o m p a n y i n g n u c l e a r rearrangements. I n t h e standard t h e o r y o f M a r c u s a n d H u s h ( I I , 12), t h e c o u p l e d n u c l e a r modes o f m o t i o n are d i v i d e d into i n n e r - a n d o u t e r - s p h e r e m o d e s . I n n e r sphere modes i n c l u d e b o n d - l e n g t h changes i n the redox centers t h e m s e l v e s , a n d c o n t r i b u t e a reorganization energy,

X ; t h e s u r r o u n d i n g m e d i u m is f

treated as a continuous d i e l e c t r i c , w i t h reorganization e n e r g y X , a n d t h e 0

total reorganization energy i s X = \

0

+ X . A potential-energy diagram l

d e p i c t i n g this situation is p r e s e n t e d i n F i g u r e 2. I n most cases, a n d p a r t i c u l a r l y for small inorganic ions reacting near r o o m t e m p e r a t u r e , i n n e r - a n d outer-sphere reorganizations are treated o n e q u a l footing.


128



potential

Inner Figure 2. Schematic potential-energy surface for ordinary electron transfer without conformational state change. The abscissa is the inner-sphere vibrational coordinate, the ordinate is either the outer-sphere coordinate or an internal low-frequency motion. The reaction proceeds over a saddle point located on a line joining precursor minimum (upper left) to successor minimum (lower right). (Reproduced from ref. 1. Copyright 1987 American Chemical Society.)

T h i s f o r m a l i s m assumes that, i n a n E T process s u c h as t h e f o r w a r d transfer ( A * —• I) a n d back-transfer (I —> A ) reactions, t h e r e is b u t a single stable conformational f o r m for each o f the p r e c u r s o r (P) a n d successor (S) electron-transfer states ( F i g u r e 2). H o w e v e r , there m u s t b e m a n y cases i n w h i c h a n " o u t e r - s p h e r e " coordinate is associated w i t h a conformational m o d e that has t w o (or more) m i n i m a c o r r e s p o n d i n g to alternative stable confor mations ( F i g u r e 3) a n d the substates of the P a n d S species display a d y n a m i c conformational e q u i l i b r i u m that c a n m o d u l a t e t h e E T rates. M o r e o v e r , major p r o t e i n conformational changes can o c c u r at rates that are c o m p e t i t i v e w i t h o b s e r v e d rates o f E T (13,14). S u c h " g a t i n g " m a y o c c u r i n t h e c o m p l e x b e t w e e n z i n c c y t o c h r o m e c peroxidase ( Z n C c P ) a n d c y t o c h r o m e c (3, 9,15-17) o r for c y t o c h r o m e c i t s e l f (18). M o r e g e n e r a l l y , i t has b e e n p r o p o s e d that a variety of other c h e m i c a l reactions are conformationally


6.


129

Gated Reactions


potential

Inner Figure 3. Schematic potential-energy surface for electron transfer with conformational equilibrium. The ordinate is a conformational coordinate, and the surface describes processes such as the A * —» I transfer described in the text and in Figure 4. Transfer from the conformational^ favored precursor state (upper left) to the stable successor conformer (lower right) occurs by a two-step process (either across and then down or down and then across), rather than by the combined upper left to lower right path, which has a higher barrier. (Reproduced from ref 1. Copyright 1987 American Chemical Society.) gated (19-21). A two-state case, w h i c h represents the s i m p l e s t m o d e l for any a n d a l l s u c h reactions, was p r e s e n t e d p r e v i o u s l y (I). T h i s c h a p t e r gives a m o r e general solution to the k i n e t i c equations that arise. C o n s i d e r a d o n o r - a c c e p t o r pair, [d,a], as part of a system that exhibits t w o conformations, (B) boat a n d (C) chair. F o r concreteness, w e m a y i m a g i n e d a n d a as attached 1,4 o n a cyclohexane r i n g ; the same f o r m a l situation w i l l b e r e a l i z e d i n far m o r e i n t e r e s t i n g ways w i t h p r o t e i n complexes. I n this case, each o f the three system states s h o w n i n F i g u r e 1, (A, A * , a n d I) is c o m p o s e d o f two conformational substates (A = A + A ; A * = A * + A * ; I = I + I ) . I n the illustration, the distance a n d the t h r o u g h - b o n d relationships that govern e l e c t r o n transfer b e t w e e n d a n d a w i l l differ i n the B a n d C conformations of the r i n g . B

c

B

c

B

c

F o r nonadiabatic e l e c t r o n transfer, this v a r i a t i o n w o u l d cause the m a t r i x


130


e l e m e n t that enters as a prefaetor i n t o the (nonadiabatic l i m i t ) rate expres sions to differ i n the B a n d C conformers (II,12). G e n e r a l l y , the energetics of charge creation o r a n n i h i l a t i o n w i l l b e c o u p l e d to conformation. I n the i l l u s t r a t i o n , this c o u p l i n g obviously occurs, for the c o u l o m b i c stabilization of a charge-separated state w i l l b e greater i n the boat conformer, w h e r e d and a " are close, than i n the chair. M o r e o v e r , the process of f o r m i n g t h e reactive state A * can change the conformational energetics. B y l i n k a g e r e lationships, this means that the e q u i l i b r i u m b e t w e e n conformers can b e different i n system states A , A * , a n d I. Downloaded by UNIV OF SOUTHERN CALIFORNIA on June 18, 2016 | http://pubs.acs.org Publication Date: May 5, 1989 | doi: 10.1021/ba-1990-0226.ch006

+

I n t r o d u c t i o n o f the conformational e q u i l i b r i a i m p l i c i t i n F i g u r e 3 ex pands the k i n e t i c scheme of F i g u r e 1 to that of F i g u r e 4. H e r e , the A , A * , a n d I states each e x h i b i t two conformations of the l i n k e d d o n o r - a c c e p t o r system (B a n d C ) , each p o t e n t i a l l y w i t h different i n t e r c o n v e r s i o n rates a n d energetics ( e q u i l i b r i u m ratio). Because o f these differences, the activation energy for E T is not the same for the two conformers. T h i s scheme i n c l u d e s E T rate constants o n l y for the d * —» a a n d a " —»

Figure 4. Kinetic scheme for electron transfer for a donor-acceptor couple within a system where the states A, A * , and I each undergo conversion between two stable conformers C and B. The symbols and rate constants are as shown in Figure I, with the following additions. The conformational substates and associated electron-transfer rate constants are labeled by conformer; the conformational conversions are defined by the equilibrium constant in A (1L ) and by rate constants in A* (k ,ka) and in I (k„i,kdi). eq

u


6.

131

Gated Reactions


d electron-transfer processes, i n w h i c h the system conformation is c o n s e r v e d (the " h o r i z o n t a l " reactions i n F i g u r e 3), a n d conformational a n d E T steps o n l y occur sequentially. I n t u i t i v e l y , i t m i g h t be e x p e c t e d that a k i n e t i c scheme for E T i n the presence of an energetic c o u p l i n g b e t w e e n charge state a n d conformation m u s t i n c l u d e E T that is synchronous w i t h a confor m a t i o n a l change i n the m e d i u m coordinate, n a m e l y the " d i a g o n a l " processes +

i n F i g u r e 3, P - » S , a n d P —» S . T h i s i n t u r n w o u l d o p e n the issue o f " f r i c t i o n " along the c o u p l e d diagonal p a t h , p a r t i c u l a r l y i n p r o t e i n systems w h e r e large m o l e c u l a r masses are i n v o l v e d a n d at l o w t e m p e r a t u r e s . Downloaded by UNIV OF SOUTHERN CALIFORNIA on June 18, 2016 | http://pubs.acs.org Publication Date: May 5, 1989 | doi: 10.1021/ba-1990-0226.ch006

B

c

c

B

H o w e v e r , i n the cases of interest h e r e i t is not necessary to i n c l u d e the synchronous process (e.g., A * —> I ) for reasons that can b e d i s c e r n e d u p o n e x a m i n i n g F i g u r e 3. O n a potential-energy surface i n w h i c h the p r e c u r s o r , A * o r I , a n d also the successor, I or A , e x h i b i t two stable m i n i m a , the transition states o n the diagonal reaction paths (e.g., P to S ) l i e at a h i g h e r e n e r g y than a l l o t h e r transition states, i n c l u d i n g those for the isoconform a t i o n a l E T reactions ( P —» S a n d P —> S ) a n d those for the confor m a t i o n a l conversions ( P —> P a n d S —» S ) ( F i g u r e 3). T h e r e can b e n o c o m p e n s a t i o n for this increased activation e n e r g y b y the p r e e x p o n e n t i a l t e r m w h e n E T is adiabatic, n o r is effective c o m p e n s a t i o n l i k e l y i n the nonadiabatic, long-range E T processes o f interest h e r e . T h u s , i n this case w h e r e the c o n v e r s i o n is b e t w e e n stable conformers, the rates for the s e q u e n t i a l c o n formational a n d E T processes are greater t h a n those for the synchronous reactions, w h i c h can be o m i t t e d . c

B

c

c

c

c

B

B

c

B

B

B

I n some cases the conformational step w i l l b e slower t h a n the E T step, a n d these deserve to be c a l l e d gated reactions. T h i s p o i n t perhaps represents the most i n t e r e s t i n g feature o f o u r analysis a n d w i l l b e s h o w n later to have p r o f o u n d consequences for efforts to ascertain the d e g r e e o f conformational c o n t r o l i n electron-transfer reactions. O u r o r i g i n a l p u b l i c a t i o n (I) speculated that the synchronous r e a c t i o n always can be o m i t t e d . B r u n s c h w i g a n d S u t i n (22) c o n f i r m that this c o n d i t i o n is t r u e for the case of c o n c e r n i n o u r o r i g i n a l p a p e r a n d h e r e , w h e r e the m i n i m a are stable. T h e y f u r t h e r show that i t is not t r u e i n the l i m i t i n g case w h e r e the second m i n i m u m is barely stable; this l i m i t a t i o n is w h o l l y u n i m p o r t a n t i n the present context.

Electron- Transfer Kinetics with Conformational Dynamics W e w i l l discuss the k i n e t i c scheme s h o w n i n F i g u r e 4 i n terms o f the p h o t o i n i t i a t e d (d* —» a) electron-transfer a n d t h e r m a l ( d a") processes i n a d o n o r - a c c e p t o r p a i r c o u p l e d w i t h i n a m o l e c u l e o r c o m p l e x that exhibits two stable conformations. Reactions i n i t i a t e d b y a second-order r e d u c t i o n o f d are d e s c r i b e d b y the first h a l f o f this scheme. T h e p o p u l a t i o n o f A * i n the C conformation is d e f i n e d as ( A * ) , i n the B conformation as ( A * ) , a n d the total p o p u l a t i o n o f A * as (A*) = ( A * ) + ( A * ) . T h e p o p u l a t i o n o f the e l e c t r o n +

c

B

B

c


132


transfer i n t e r m e d i a t e , I , a n d its conformational substates are d e f i n e d a n a l ogously. I n the g r o u n d state, A , the conformational e q u i l i b r i u m is d e s c r i b e d b y the e q u i l i b r i u m constant, ( A / A ) B

c

= K

e q

. However, coupling between

e q

e l e c t r o n i c state a n d conformation m i g h t alter the conformational e q u i l i b r i u m in A * or I: ( A * / A * ) B

K

C

* = kjk ;

(I /I )

d

e q

B

c

e q

-

K

e q

' =

kjk . dl

A n i m p o r t a n t feature o f most e x p e r i m e n t s is that the d e t e c t i o n m e t h o d s e m p l o y e d d o not d i s t i n g u i s h b e t w e e n conformers. T h u s , w e r e q u i r e e q u a tions for the total p o p u l a t i o n s , A * a n d I , f o l l o w i n g the f o r m a t i o n o f A * at


t = 0 b y flash excitation o r s e c o n d - o r d e r r e d u c t i o n (oxidation) o f A . G e n e r a l l y , the d* — » a transfer process is e x p e r i m e n t a l l y c h a r a c t e r i z e d b y f o l l o w i n g the loss o f A * , the d

^ - a " process b y f o l l o w i n g t h e t i m e course o f

+

I. W e w i l l examine t h e two processes i n t u r n . The kinetic

R e a c t i o n o f Initial State: A * - » I: E x a c t Solution.

schemes i n F i g u r e s 3 a n d 4 c o r r e s p o n d to the f o l l o w i n g k i n e t i c equations for the A * populations: + c^e-»)

Cl

(3)

2

(4a)

lc

= F(c e-fl

+ c e-*)

1B

(4b)

w

with the definitions (f,g)

= l/2[2fe

D

+ k

d

+ k

u

+ k

tB

+

V(fc„ + k f + (k d

tB

k

tc

-

ktcXks - k

t c

+ 2k d

2k )] (5)

ci =


u

(6a)

6.


c

c

2

l c

= £ — = g - /

(6b)

-

(7a)

^ g

C2C =


133

Gated Reactions

c

1B

c

2B

/

g

(7b)

- /

= ^ g =

" /

/

g

(7c)

" /

(7d)

- /

where *« = k

D

*„c = k

D

^

= *

D

+ (1 -

+ fc„ + k

d

+ k

+

+

+ | F

tB

1 -

F ) * „ + Fktc

(8) (9a)

F

(9b)

I n contrast to the exponential d e p e n d e n c e of A*(t) for the s i m p l e E T s c h e m e ( F i g u r e 2, e q l a ) , the general form of the o v e r a l l A*(t) p o p u l a t i o n , e q 3, is b i e x p o n e n t i a l , w i t h the two rate constants b e i n g the s u m (g) a n d difference (f) of two positive terms: 0 < g. T h e l o n g - t i m e b e h a v i o r of (A*) is g o v e r n e d b y the rate constant, / ; the short-time b e h a v i o r b y the larger rate constant, g. I n the f o l l o w i n g discussion, it w i l l frequently b e useful to c o n s i d e r s i t u ations w h e r e a single exponential t e r m controls the b e h a v i o r o f (A*(£)). T h e effective rate, fe , can be w r i t t e n as the s u m of the decay-rate constant, k , a n d a n effective electron-transfer t e r m , fc . D

obs

ET

fcobs =

k

D

+

(10)

k

ET

T h e r e are several i l l u m i n a t i n g l i m i t i n g cases of the g e n e r a l solution. A most i n t e r e s t i n g situation occurs w h e n < < 1 a n d thus the o n l y state i n i t i a l l y p r e p a r e d is A * . I f A * does not e x h i b i t r a p i d E T , n a m e l y k^ ~ 0, t h e n E T o n l y occurs subsequent to the A * —» A * conversion. T h i s is " g a t e d " E T i n the traditional sense that u n t i l the " g a t e " has b e e n o p e n e d (that is, u n t i l A * is f o r m e d from A * ) , there is no c o n v e r s i o n to p r o d u c t I. W h e n the state, A * , i n i t i a l l y p o p u l a t e d cannot u n d e r g o E T , t h e r e m u s t be a n c

c

c

B

B

c

c


134


i n i t i a l lag p e r i o d , w i t h k For decay of A *

= 0 (eq 10); the e a r l y - t i m e (t = 0) rate constant

ET

Kb* = h

(11a)

represents o n l y the A * —» A r e t u r n to g r o u n d state. S u b s e q u e n t l y , k increases w i t h t i m e as A * is f o r m e d a n d undergoes the A * —» I reaction. c

ohs

c


B

B

B

I n this gating situation, the l o n g - t i m e b e h a v i o r o f e q 3 exhibits t w o special cases of p h y s i c a l interest. W h e n the conformational e q u i l i b r i u m is fast c o m p a r e d to the E T rate, c o r r e s p o n d i n g to gated E T w i t h a fast gate, at l o n g t i m e , the A * A * p r e e q u i l i b r i u m is established, g i v i n g the steadystate r e s u l t , c

B

*obs ~

k

D

+ Ml -

+ [k (F) tB

(lib)

F)]

T h u s , the apparent electron-transfer rate k is a p o p u l a t i o n - w e i g h t e d av erage. T h e opposite extreme arises w h e n conformational c o n v e r s i o n A * —» A * is slow c o m p a r e d to the subsequent E T ; this is the t r a d i t i o n a l gated E T w i t h a slow gate. A t l o n g t i m e , the A * *" > A * conformational step is r a t e - l i m i t i n g . E a c h c o m p l e x attaining the h i g h - e n e r g y conformation o f the p r e c u r s o r crosses w i t h essentially u n i t efficiency to the p r o d u c t state I ; the o v e r a l l E T rate constant is j u s t k = k > and ET

c

B

c

ET

B

u

k

fcobs =

D

+

(11c)

k

u

R e a c t i o n o f E T Intermediate: I ~ » A (Scheme II): E x a c t S o l u tion. T h e E T i n t e r m e d i a t e state d e n o t e d I i n F i g u r e 4 is created from A * and simultaneously is b e i n g c o n v e r t e d to A b y E T . T h e t w o conformers of A * can u n d e r g o p a r a l l e l reactions to the c o r r e s p o n d i n g I conformers. I n this section the t e r m successor c o m p l e x refers to A , a n d p r e c u r s o r c o m p l e x is I. T h e k i n e t i c schemes i n F i g u r e 3 a n d F i g u r e 4 y i e l d the f o l l o w i n g k i n e t i c equations for the conformers of I: d(lc) dt

dt

= - ( * „ / + **c)(Ic) + U I B ) + M A c * )

(12a)

= Ulc) -

(12b)

(k

dI

+ k )(l ) bB

B

+ MA *) B

O n c e again, a n exact solution m a y be o b t a i n e d v i a the L a p l a c e - C a r s o n transformation. T h e solution to the general case for the t i m e course o f I(£) is a rather u n w i e l d y four-exponential function that does not l e n d itself to a discussion of its salient points. ( F o r the sake of completeness, this solution is g i v e n i n the A p p e n d i x . ) W e w i l l instead consider h e r e an i l l u m i n a t i n g


6.

135

Gated Reactions


subcase o f the general case, w h e r e I is f o r m e d o n l y t h r o u g h I k »

B

and where

k a n d k^ = 0. T h u s , the A * —> I process can b e d e s c r i b e d

k »

u

d

D

b y a single process, A * - » I , a n d b y a single e x p o n e n t i a l decay w i t h a rate B

&obs

= k

D

B

+ k . I n effect, w e ignore gating i n A * b y c o n s i d e r i n g o n l y state tB

A * i n o r d e r to see most clearly the effect o f I B

I o n the I —» A process.

B

c


T h e k i n e t i c equations for t h e conformers o f I o b t a i n e d for this case are

=

dt

+ k )(I ) + Uh)

-(K,

bc

(** + k )(I ) + * , ( A * )

= Ulc) "

dt

(13a)

c

bB

B

B

(13b)

B

B y s o l v i n g t h e c o u p l e d e q u a t i o n set (eqs 13a a n d 13b) for t h e b o u n d a r y conditions [ I ( * = 0)] = [l (t

= 0)] = 0 a n d b y u s i n g

B

c

= A*(0)g-

A *(t) t

(14)

feb!f

we find !(*) A*(0)

n -

m

L^obs

~ m

n -

J

k

ohs

(15) where

( ,) m n

= I [k

bB

+ k

bc

+ k

+ k T

dl

ul

Vfez + k f

+ (k

uI

bB

~ k )(k bc

bB

-

k

bc

+ 2*

d /

-

2fc )] M/

(16)

and *

z

= fcfcc + k

uI

+

(17)

T h e t i m e course o f (I) is d e t e r m i n e d b y a t h r e e - e x p o n e n t i a l f u n c t i o n i n v o l v i n g the A * —» I rate constant, fc , a n d t h e I - » A rate constants, m a n d n . T h e obs

rate constants m a n d n are composites o n l y of the f u n d a m e n t a l rate constants associated w i t h state I ; t h e f u n d a m e n t a l rate constants d e s c r i b i n g i n t e r c o n v e r s i o n a n d decay o f the A * substates d o n o t appear i n m a n d n . T o illustrate this exact d e s c r i p t i o n o f the I —» A process, w e e x a m i n e d the situation i n w h i c h gating w i t h i n I has t h e m a x i m a l effect. I is p o p u l a t e d o n l y t h r o u g h t h e A * - » I c h a n n e l , b u t t h e I conformer is u n r e a c t i v e , with k = 0. T h i s situation represents i d e a l i z e d conformational c o n t r o l o f B

B

B

bB


136


E T because t h e d

+

^- a

E T process c a n occur o n l y after a n I

B

-» I

c

c o n v e r s i o n t h r o u g h t h e sequential process, _ I where the I - » I B

c

kdi > I

B

he c

. >A

step is t h e "gate". T h e r e a r e t w o i n t e r e s t i n g subcases.

W h e n t h e gating conformational change, I - » I , is slow c o m p a r e d to t h e B

c

rate o f E T w i t h i n t h e reactive B conformer (k

«


dl

A*(0)

k

dl

-

k ), bc

kobs

W h e n t h e gate is r a p i d c o m p a r e d to the subsequent E T step (k then

bc

l(t) A*(0)

KB

( -Ust _

~ fc

^

= ~

k ), dt

(19)

e

feeff

«

obs

where

he

r

+

k

(20)

f

ul

B o t h o f these subcases y i e l d a b i e x p o n e n t i a l functional f o r m i d e n t i c a l to that of the ungated k i n e t i c scheme, e q l b . O n e o f the rate constants is k , associated w i t h creation o f I ; t h e o t h e r describes t h e E T r e a c t i o n , I —> A . W h e n t h e gate is slow, t h e apparent rate constant o f the E T reaction is not that for t h e E T step, b u t rather is k , a conformational parameter. C o n v e r s e l y , w h e n t h e conformational change is fast, e q 19 says that t h e d ^ a " E T reaction w i t h i n t h e I c o n f o r m e r is t h e r a t e - d e t e r m i n i n g step for the I —> A process. ohs

dl

+

c

A n o t h e r i n t e r e s t i n g l i m i t c a n b e i l l u s t r a t e d b y again c o n s i d e r i n g t h e situation w h e n I is p o p u l a t e d o n l y t h r o u g h the A * - » I c h a n n e l , b u t n o w the I conformer is reactive a n d k > k ~ fc > k , k . I n this l i m i t , i f e q 15 can b e w r i t t e n i n the form B

bB

B

J£L

=

-k*

Cl€

dl

+

B

uI

obs

-mt

C2e

+

bc

-nt

( )

C3e

21

where -c

3

(22)

* c > c 2

t

T h i s result i m p l i e s that t h e i n t e r m e d i a t e appears w i t h rate constant n ~ k

bB

+ k

dI

a n d decays w i t h rate constant m ~ k

ul

+

k . bc


6.


137

Gated Reactions

Multimode Reaction Dynamics I n condensed-phase c h e m i c a l reactions, i n c l u d i n g b u t not r e s t r i c t e d to e l e c t r o n transfer, the relationship b e t w e e n the B o r n - O p p e n h e i m e r p o t e n t i a l e n e r g y surfaces (or free-energy surfaces) a n d the rate processes o c c u r r i n g o n those surfaces is n e a r l y always discussed i n t e r m s of the activation energies (or activation free energies) i n v o l v e d i n s u r m o u n t i n g barriers along a reaction p a t h l e a d i n g from a reactant to a p r o d u c t configuration ( I I , 12, 24). A t h i g h Downloaded by UNIV OF SOUTHERN CALIFORNIA on June 18, 2016 | http://pubs.acs.org Publication Date: May 5, 1989 | doi: 10.1021/ba-1990-0226.ch006

temperatures the rate constant w i l l t h e n b e g i v e n i n a g e n e r a l i z e d t r a n s i t i o n state theory f o r m k =

(23)

k Ke~

AG,RT

0

w h e r e k is the rate constant, k is an attempt frequency for crossing the b a r r i e r , K is a transmission coefficient d e s c r i b i n g recrossing of the b a r r i e r , a n d A G * is the activation free energy. T h e d o m i n a n t t e m p e r a t u r e d e p e n d ence arises from the exponential t e r m , a n d one generally expects a s i n g l e exponential time-course for the reactant concentration. T h e standard E T rate theory, i n the adiabatic l i m i t , is exactly of this f o r m , w i t h K & r e p l a c e d b y a u n i v e r s a l frequency near 1 0 s" , a n d A G * = (X 4- A ) / 4 X . F o r n o n adiabatic E T , the prefactor Kk becomes 0

0

1 3

1

2

0

i K k

_ ° "

2irV

P A

2

,

,

*(4irX*r)*

with V as the electron t u n n e l i n g m a t r i x e l e m e n t . F o r m u l a s 23 a n d 24 w o u l d b e appropriate for the i n d i v i d u a l electron-transfer rate constants (k , ktc, k , a n d k ) that appear i n the e a r l i e r discussion i n w h i c h the s i m p l e transition-state theory was e x t e n d e d to i n c l u d e m u l t i p l e stable m i n i m a . DA

tB

bB

bc

W e have o u t l i n e d the modifications that m u s t b e m a d e i n the standard nonadiabatic rate expressions of eqs 23 a n d 24 w h e n m u l t i p l e stable m i n i m a o c c u r o n the p o t e n t i a l surface. F o r other situations of real i m p o r t a n c e , d y namics o n the potential-energy surface, rather than the surface itself, r e q u i r e extension of these formulas. T h e first of these d y n a m i c extensions to receive extensive recent e x p e r i m e n t a l a n d theoretical w o r k arises f r o m d y n a m i c solvent relaxation effects (25-41); it is i m p o r t a n t for b o t h large- a n d s m a l l m o l e c u l e d y n a m i c s , especially at short t i m e s . Solvent relaxation is i m p o r t a n t i n several situations, most o b v i o u s l y w h e n it is r e q u i r e d to trap a successor c o m p l e x that m i g h t otherwise convert back to precursor. A s i m p l i f i e d p i c t u r e (33) of these d y n a m i c corrections replaces the standard f o r m for nonadiabatic transfer

kr E

=

k

NA


(25)

138


b y the c o r r e c t e d v e r s i o n

=

*ET

(26)

w h e r e w e have


h

_

?2E

P

4irV i

T* =

- T

ton\

-E /RT

V p A

A

2

(29)

d

w i t h r T , €oo, a n d e , respectively, the l o n g i t u d i n a l a n d D e b y e relaxation times a n d the o p t i c a l a n d static d i e l e c t r i c constants. F o r slow solvent relax ation, h

d

0

y

16irfc T B

2

exp

1 RT

K « K »

1 1

(30)

T h u s , i n this l i m i t , the rate is s i m p l y the p r o d u c t of the activation energy to attain the b a r r i e r top a n d the rate o f solvent relaxation r e q u i r e d to stabilize the p r o d u c t configuration. W e treated a second set of circumstances i n w h i c h the s i m p l e activated c o m p l e x p i c t u r e requires d y n a m i c a l extension. I n c o n d e n s e d phase, e x c i t e d translational or v i b r a t i o n a l states u n d e r g o frequent e n e r g y - r e l a x i n g collisions w i t h the solvent. U n d e r these conditions, the d y n a m i c b e h a v i o r of the system p o i n t m o v i n g o n the p o t e n t i a l surface w i l l generally b e diffusive, w i t h m e a n free p a t h short c o m p a r e d to characteristic distances a l o n g the p o t e n t i a l s u r face. K r a m e r s l o n g ago s t u d i e d the i m p l i c a t i o n s of this diffusive transport for c h e m i c a l rate processes (42). I n the past 15 years a large n u m b e r of studies, b o t h theoretical a n d e x p e r i m e n t a l , have u s e d g e n e r a l i z e d K r a m e r s models to analyze rates for processes as different as i s o m e r i z a t i o n , i o n i c c o n d u c t i o n , a n d e n z y m e - s u b s t r a t e b i n d i n g (43, 48). T h e o r i g i n a l K r a m e r s f o r m u l a t i o n , a n d the majority of its extensions, c o n s i d e r e d pictures i n w h i c h m o t i o n occurs along a single reaction c o o r d i nate, a n d thus the diffusive dynamics of b a r r i e r crossing occurs i n one d i m e n s i o n . Situations can frequently occur, h o w e v e r , i n w h i c h the o n e d i m e n s i o n a l p i c t u r e is inadequate. A g m o n a n d H o p f i e l d (49, 50) n o t e d that for reactions i n proteins, it m a y be necessary to treat i n t e r n a l m o t i o n of the


6.


139

Gated Reactions

p r o t e i n as d e f i n i n g the " m e d i u m " coordinate ( F i g u r e 2). W h e n the p r o t e i n is h i g h l y conformationally m o b i l e , t h e r e are n o n e w effects. H o w e v e r , as m o t i o n i n this coordinate is s l o w e d , for e x a m p l e b y a decrease i n t e m p e r a t u r e , reorganization of the p r o t e i n b e c o m e s difficult a n d finally

i m p o s s i b l e ; the system no longer can reach the classical t r a n s i t i o n

state ( F i g u r e 2), a n d the reaction rate decreases. E x a m p l e s m i g h t i n c l u d e C O r e b i n d i n g (21, 51), the case o f E T w i t h stable i n t e r m e d i a t e states, or i s o m e r i z a t i o n about a d o u b l e b o n d , i n w h i c h extension of the b o n d v i b r a t i o n Downloaded by UNIV OF SOUTHERN CALIFORNIA on June 18, 2016 | http://pubs.acs.org Publication Date: May 5, 1989 | doi: 10.1021/ba-1990-0226.ch006

coordinate substantially changes the b a r r i e r height. U n d e r s u c h conditions, the K r a m e r s p i c t u r e s h o u l d be e x t e n d e d to two coordinates, a n d j u s t such an analysis of the i s o m e r i z a t i o n p r o b l e m has b e e n p r o v i d e d b y A g m o n a n d Kosloff(52). I n m a n y systems, h o w e v e r , p a r t i c u l a r l y i n proteins w i t h slow confor mational motions characterized b y v e r y large effective masses a n d s m a l l force constants, the characteristic times for the coordinates m a y b e v e r y different. A d d i t i o n a l l y , the different degrees of f r e e d o m m a y have totally different effective frictions, c o r r e s p o n d i n g to differing diffusion constants a l o n g the differing m o t i o n directions. A s a result, the e v o l u t i o n of the system o n the p o t e n t i a l surface can b e g o v e r n e d not o n l y b y the p o t e n t i a l surface itself (that is, b y the forces a r i s i n g f r o m the gradient of the potential), b u t also from n o n u n i f o r m frictional forces for differing coordinates. A s i m p l e example m i g h t h e l p to clarify this discussion. F i g u r e 5 A p r e sents a schematic p o t e n t i a l surface for C O r e b i n d i n g to m y o g l o b i n . T h e abscissa corresponds to translational m o t i o n of the C O away from the F e site, a n d the ordinate measures p r o t e i n conformational m o t i o n . T h e confor mational ("outer-sphere") motions w i l l o c c u r v e r y slowly; the small-species ("inner-sphere") motions along the abscissa are far faster. A g m o n a n d H o p field

(49, 50) therefore assumed that the effective reactive m o t i o n c o r r e

sponds to straight-line motions along the x coordinate, starting f r o m some d i s t r i b u t i o n of p r o b a b i l i t y (just a B o l t z m a n n - t y p e distribution) a l o n g the y axis. P h y s i c a l l y , this c o n d i t i o n w i l l i d e a l l y c o r r e s p o n d to a d i s t r i b u t i o n of b a r r i e r heights, c o r r e s p o n d i n g to crossing the r i d g e l i n e R at various cuts along the y axis. T h i s d i s t r i b u t i o n , i n t u r n , can m e a n that d y n a m i c s about the s i m p l e saddle p o i n t S of F i g u r e 5 A can b e c o m e i r r e l e v a n t for the o v e r a l l rate because b a r r i e r crossing w i l l o c c u r a l l along the ridge l i n e , not o n l y at the saddle p o i n t itself. T h e n the o v e r a l l rate w i l l be g i v e n b y a n average of the rate for each straight-line trajectory p a r a l l e l to x—that

is, at each c o n

formation, a rate of C O r e b i n d i n g can b e d e f i n e d , a n d the o v e r a l l o b s e r v e d rate w i l l be an average o v e r a l l accessible conformations. T h e appealing a n d p h y s i c a l l y reasonable p i c t u r e of A g m o n a n d H o p f i e l d r e p r e s e n t e d a n i m p o r t a n t step i n i n t e r p r e t i n g reactions o c c u r r i n g i n situa tions such as a p r o t e i n w h e r e the characteristic frictions a n d m o t i o n t i m e scales are substantially different a l o n g various coordinates. H o w e v e r , the straight-line trajectories p a r a l l e l to the x axis, c o r r e s p o n d i n g to fast-coordi-



140


Figure 5. (A) A schematic sketch of the potential surface for the ligand protein rebinding reaction. R denotes the ridge, T the separatrix, and S the saddle point. The stable reactant and product configurations are denoted by a and b, respectively. The shapes and orientations of R and T are schematic and other possibilities exist as discussed in the text. nate m o t i o n at fixed position along the slow coordinate, c l e a r l y represents a great o v e r - s i m p l i f i c a t i o n that cannot be g e n e r a l l y v a l i d . T h e issues r a i s e d , h o w e v e r , i n c l u d i n g the differences b e t w e e n reactions d e t e r m i n e d b y r i d g e l i n e crossing as o p p o s e d to saddle-point crossing, the effects of v e r y different frictions along the various motions, a n d possible n o n e x p o n e n t i a l rate b e havior a r i s i n g from short-time transients o n the p r o d u c t side of the r i d g e l i n e b u t not yet stable at p r o d u c t - t y p e geometries, m e r i t e x t e n d e d c o n s i d eration (53). R e c e n t l y , this p r o b l e m was c o n s i d e r e d b y u s i n g a S m o l u c h o w s k i e q u a tion approach (2). T h e equations of m o t i o n for the system are t h e n

1

w

x

(31a)

2

w

2

(31b)

x = — — + V2k T/my myi dx B

y =

-1

dV — + V2k T/my my dt/ B

2

w i t h yi a n d 72 the friction coefficients for x a n d y motions, r e s p e c t i v e l y ; a n d k , T, m, w and w are, respectively, B o l t z m a n n constant, t e m p e r a t u r e , mass, a n d two i n d e p e n d e n t w h i t e noise (random modulation) t e r m s . T h i s equation is e x p e c t e d to h o l d w h e n the m o t i o n is strongly d a m p e d , so that B

u

2


6.


141

Gated Reactions


B

Figure 5. (B) A plot of the potential corresponding to the lower eigenvalue of the 2x2 vibronic coupling problem. The ratio of the diffusion coefficients is 8 = 0.1. The ridge and separatrix shown in the figure have been calculated for this ratio. Unlike the situation in the schematic picture (A), R and T touch (at the saddle point), but do not cross each other. (Reproduced with permission from ref 2. Copyright 1989 American Institute of Physics.) y a n d y exceed a l l i m p o r t a n t frequencies; this s h o u l d b e t r u e i n p r o t e i n E T reactions. x

2

A n i s o t r o p y i n the m o t i o n along the p o t e n t i a l surface can have two causes. T h e first cause is the p o t e n t i a l V(x,y) itself, w h o s e structure can cause the d o m i n a n t e v o l u t i o n to skirt the saddle p o i n t , f o l l o w i n g instead a route t h r o u g h stable intermediates, as already discussed. T h e second cause is anisotropic diffusion or friction that arises because the effective d a m p i n g , or relaxation, is stronger along one d i r e c t i o n (say y) than along the other. T h e latter case introduces anisotropic b e h a v i o r o n the rhs of e q 3 1 , a n d results i n faster m o t i o n along one coordinate than along the other, just as i n the A g m o n - H o p f i e l d picture. The A g m o n - H o p f i e l d model, however, corre sponds to the l i m i t w h e r e there is no diffusive flow along the y coordinate (that is, to y hi w h i c h is not p h y s i c a l l y j u s t i f i e d . S o l u t i o n of e q 3 1 , 2


142


t h e n , extends the A g m o n - H o p f i e l d p i c t u r e to y i e l d the correct result for general anisotropic diffusion D^/D , w h e r e D ^ , the diffusion coefficient along ac, is g i v e n b y w


D„ « ^

(32)

T h e results o f m a t h e m a t i c a l analysis of the S m o l u c h o w s k i equations, 31a a n d 31b, are best u n d e r s t o o d i n terms o f F i g u r e s 5 A a n d 5 B , w h i c h s h o w cases of s m a l l a n d large anisotropy, respectively. F i g u r e 5 A is a schematic r e p resentation o f the C O - r e b i n d i n g p r o b l e m . F i g u r e 5 B is a p l o t o f a 2 X 2 v i b r o n i c c o u p l i n g ( J a h n - T e l l e r ) p r o b l e m w i t h large anisotropy ( D ^ / D ^ = 0.10). I n b o t h cases, the c u r v e l a b e l e d R denotes the ridge l i n e — t h a t is, the l i n e o n the p o t e n t i a l surface separating reactant a n d p r o d u c t regions. T h e c u r v e l a b e l e d T is c a l l e d the separatrix, a n d i t is the set o f points that separates d y n a m i c motions c o r r e s p o n d i n g to reactant a n d p r o d u c t . T h e ridge l i n e is d e t e r m i n e d solely b y the p o t e n t i a l surface, V(x y), a n d is the c u r v e that separates the p o t e n t i a l w e l l s o f the c h e m i c a l l y a n d p h y s i c a l l y d i s t i n c t reactant a n d p r o d u c t (for instance, p r e c u r s o r a n d successor states i n an E T process). T h e separatrix T is d y n a m i c a l l y d e t e r m i n e d as the " l i n e of no return". 9

Trajectories that pass o v e r T i r r e v e r s i b l y p r o c e e d to f o r m p r o d u c t , b u t a trajectory that has not r e a c h e d T m a y recross the r i d g e l i n e ( i n d e e d , w i l l recross i t i n l o n g - t i m e , anisotropic situations). I f = D ^ , the separatrix and r i d g e b e c o m e i d e n t i c a l a n d a n o r m a l , o n e - d i m e n s i o n a l reaction-path treatment is a p p l i c a b l e . H o w e v e r , t h e y can b e c o m e v e r y different i f the diffusion coefficient is h i g h l y anisotropic, as the figures i n d i c a t e , a n d i n this case q u i t e u n u s u a l kinetics are p r e d i c t e d . W h e n the r i d g e l i n e a n d separatrix d i v e r g e , i f t h e rate constant is meas u r e d b y appearance o f the system p o i n t near the b o t t o m o f the p r o d u c t w e l l , t h e n at v e r y l o n g times the reaction d y n a m i c s is exponential i n t i m e , w i t h the activation energy fixed b y the h e i g h t of the saddle b a r r i e r , E . B u t shortt i m e transients can appear, c o r r e s p o n d i n g to the system crossing back a n d forth o v e r the ridge l i n e i n t o the p r o d u c t r e g i o n before i r r e v e r s i b l y passing the separatrix T. A n actual m e a s u r e m e n t o f the reaction progress corresponds (as it n e a r l y always w i l l , g i v e n o u r d e f i n i t i o n o f p r e c u r s o r a n d successor i n terms of geometry) to an observable p h e n o m e n o n that changes w h e n the system crosses the ridge l i n e (e.g., C O b o u n d or u n b o u n d ; charges separated or not). T h u s , the kinetics o b s e r v e d w i l l , i n general, b e m u l t i e x p o n e n t i a l , because of the m u l t i p l e crossings a n d recrossings of the r i d g e . s

T h e S m o l u c h o w s k i m o d e l for anisotropic diffusive d i m e n s i o n a l p o t e n t i a l surface (eqs 31a a n d 31b) leads, al. s h o w (2), to a p r o p e r mathematical d e s c r i p t i o n of expressed b y A g m o n a n d H o p f i e l d (and b y G r o t e a n d

reactions o n a t w o as K l o s e k - D y g a s et the p h y s i c a l n o t i o n H y n e s (45) a n d v a n


6.


143

Gated Reactions

d e r Z w a n a n d H y n e s (46)). T h e actual t i m e d e p e n d e n c e of the p r e c u r s o r concentration thus contains not o n l y energetic information about the b a r r i e r at the saddle p o i n t (from the t e m p e r a t u r e d e p e n d e n c e of its l o n g - t i m e a s y m p totic value), b u t also i n f o r m a t i o n o n the anisotropy of the diffusion coefficient (from its s h o r t e r - t i m e dynamics). F r a u e n f e l d e r a n d coworkers have s t u d i e d extensively the n o n e x p o n e n t i a l transients i n the C O r e b i n d i n g situation (21, 51), b u t w e are aware of no such i n - d e p t h e x p e r i m e n t a l study of transient


n o n e x p o n e n t i a l b e h a v i o r i n E T reactions.

Discussion T h i s chapter examines two generalizations of the standard activated c o m p l e x rate theory that are n e e d e d to discuss d y n a m i c processes o c c u r r i n g o n p o tential surfaces o n w h i c h m u l t i p l e stable m i n i m a occur, o r o n w h i c h the system evolves b y anisotropic diffusion. W e a n a l y z e d the simplest k i n e t i c scheme for a system that displays m o r e than one stable conformational state a n d undergoes i n t r a m o l e c u l a r e l e c t r o n transfer. I n particular, w e focused o n the situation i n w h i c h the conforma tional c o n v e r s i o n rates a n d e q u i l i b r i u m can change u p o n excitation o r charge transfer. D e s p i t e this energetic c o u p l i n g , the synchronous processes (e.g., P ~ " * S ) can b e o m i t t e d from consideration because the activation free energy along the synchronous pathway is greater than that for e l e c t r o n transfer w i t h i n a single configuration. T h e fact that E T a n d conformational reactions are sequential, a n d not c o n c e r t e d , is a major factor i n efforts to disentangle conformational a n d electron-transfer influences. T h i s factor is of k e y i m p o r tance because standard detection methods m o n i t o r o n l y the E T event, a n d not conformational changes w i t h i n one electronic state. I n m a n y , i f not most, instances the m e a s u r e d t i m e course of a single gated E T reaction is l i k e l y to b e i n d i s t i n g u i s h a b l e from a reaction w i t h o u t gating. c

B

F o r t u n a t e l y , the partial d e c o u p l i n g of E T a n d conformational processes afforded b y the absence of synchronous events i n p r i n c i p l e a n d i n practice allows for the identification of an o b s e r v e d decay rate constant. I f one c o n structs a series of homologous systems i n w h i c h the E T energetics (or elec t r o n i c coupling) is m o d i f i e d w i t h o u t change i n the conformational e q u i l i b r i u m , thus l e a v i n g the conformational rates u n c h a n g e d , t h e n the o b s e r v e d rate constants w i l l b e u n c h a n g e d i f the reaction is c o n t r o l l e d b y a confor mational rate, b u t w i l l vary i f this is not so. W i t h the aims of specificity a n d clarity, w e discussed conformational effects i n terms of i n t r a m o l e c u l a r electron-transfer p h e n o m e n a , focusing o n a d o n o r - a c c e p t o r c o m p l e x activated b y flash photolysis o r a second-order redox event. H o w e v e r , the general set of rate equations (eqs 2a a n d 9) a n d the k i n e t i c schemes of F i g u r e 4 are applicable to o t h e r types of rate process. F o r e x a m p l e , either proton-transfer or i s o m e r i z a t i o n reactions can b e c o n -


144



t r o l l e d b y g a t i n g — t h a t i s , b y conversion b e t w e e n t w o conformational geo metries that have different reactivities. T h e p r e s e n c e o f a d d i t i o n a l stable m i n i m a o n a t w o - d i m e n s i o n a l potential-energy surface that i n v o l v e s a c o n formational coordinate (the v e r t i c a l coordinate o f F i g u r e 4) distinguishes this class o f reactions. W e also briefly r e c o u n t e d t h e b e h a v i o r to b e e x p e c t e d w h e n t h e m o t i o n from p r e c u r s o r to successor involves n o secondary stable m i n i m a , b u t rather anisotropic diffusion, a r i s i n g from substantially different f r i c t i o n , o r d a m p i n g , along conformational a n d i n n e r - s p h e r e v i b r a t i o n a l modes. U n d e r such c i r cumstances, t h e system trajectories w i l l stray substantially from t h e steepestdescents pathway from reactant o v e r t h e saddle p o i n t to p r o d u c t geometries. I n this case t h e i n i t i a l p o p u l a t i o n o f reactant w i l l e x h i b i t a m u l t i e x p o n e n t i a l decay because t h e ridge l i n e , w h i c h separates reactant a n d p r o d u c t states o n t h e p o t e n t i a l surface, n o longer is c o i n c i d e n t w i t h t h e separatrix, w h i c h separates reactant a n d p r o d u c t regions o f reactive trajectories. S t u d y o f t h e transient decay b e h a v i o r c a n p r o v i d e i m p o r t a n t insights i n t o t h e differing frictions along the t w o modes, as w e l l as t h e shape o f t h e p o t e n t i a l surface.

Acknowledgments W e acknowledge s t i m u l a t i n g discussions o f E T processes w i t h A . N i t z a n , J . H u p p , a n d H . B . G r a y . T h i s w o r k has b e e n s u p p o r t e d b y t h e N a t i o n a l Institutes o f H e a l t h ( H L 1 3 5 3 1 , B M H ) a n d t h e N a t i o n a l Science F o u n d a t i o n (DBM86-06575, B M H and CHE8805585, M A R ) .

Appendix. Exact Solution to the General Case of I —> A T h e solution to t h e c o u p l e d equation set (12) was f o u n d b y u s i n g the L a p l a c e - C a r s o n transformation. T h e b o u n d a r y conditions are [l (t = 0)] = [I (t = 0)] = 0; A * ( f ) a n d A *(t) are g i v e n i n e q 4, a n d the total p o p u l a t i o n i n state I at t i m e t is c

B

B

c

/(*)

f

=

A*(0)

ap + bf+

l(g -f)(m am

2

(f -

c

-f)(n

_ -f)

6

+ hm + c

m)(g -

ag

2

fi

,

m)(n ~ m)

+ bg + c

(f-g)(m

+

-

g)(n -

g)

an + bn + c 2

(f -

n)(g -

n)(m -

n)

where a = kc + (k

tB

-b

= k [K tB

+ F(k

ul

+ k

dl

k, [k c

d

+ k

bc

+ (1 -

-

(A.2)

kc) F

+ kc + k )] + D

F)(k

ul

+ k

dI

+ k

bB

+ k

tB

+ k )] D


(A.3)

6.

H O F F M A N ET AL.

o = hsiKi + k

dl

Gated Reactions

145

+ fc )[fc„ + F(fc,c + k )] + ic

K(K

D

+ hi + he) lh + (1 -

F ) (ha + k )} D

(A.4)


/ a n d g are as d e f i n e d i n e q 5, a n d m a n d n are as d e f i n e d i n e q 16. A s i n the subcase e x a m i n e d earlier, m a n d n are composites o n l y o f the f u n d a m e n t a l rates o f state I .

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146 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53.

E L E C T R O N TRANSFER IN BIOLOGY A N D T H E

S O L I D STATE

Calef, D . R.; Wolynes, P. G . J. Phys. Chem. 1983, 87, 3387. Hynes, J. B . J. Phys. Chem. 1986, 90, 3701. Sumi, H . ; Marcus, R. A . J. Chem. Phys. 1987, 84, 4272. Nadler, W.; Marcus, R. A . J. Chem. Phys. 1987, 86, 3906. Sparpaglione, M.; Mukamel, S. J. Chem. Phys. 1988, 88, 3263. Rips, I.; Jortner, J . J. Chem. Phys. 1987, 87, 2090. Dakhnovskii, Yu. I.; Ovchinikov, A . A . Mol. Phys. 1986, 58, 237. Zusman, L . D . Chem. Phys. 1988, 51, 119. Murillo, M.; Cukier, R. I. J. Chem. Phys. 1988, 89, 6736. McManis, G . E.; Weaver, M. J . J. Chem. Phys. 1989, 90, 912. Gennett, T.; Milner, D . F . ; Weaver, M. J . J. Phys. Chem. 1985, 89, 2787. Simon, J . D.; Su, S. G . J. Chem. Phys. 1987, 87, 7016. Su, S. G.; Simon, J . D . J. Chem. Phys. 1988, 89, 908. Kahlow, M. A.; Jarzeba, W.; Kang, T. J.; Barbara, P. F. J. Chem. Phys. 1989, 90, 151. Kramers, H. A . Physica (Utrecht) 1940, 7, 284. Hynes, J. T. Annu. Rev. Phys. Chem. 1985, 36, 573-597. Skinner, J . L . ; Wolynes, P. G . J. Phys. Chem. 1980, 72, 4913-4919. Grote, R. F.; Hynes, J . T. J. Chem. Phys. 1980, 73, 2715-2732. van der Zwan, G . ; Hynes, J . T. J. Chem. Phys. 1983, 78, 4174-4185. Calef, D . F.; Wolynes, P. G . J. Chem. Phys. 1983, 78, 470-482. Nitzan, A . Adv. Chem. Phys. 1988, 47, 489. Agmon, N.; Hopfield, J . J . J. Chem. Phys. 1983, 78, 6947-6959. Agmon, N.; Hopfield, J. J . J. Chem. Phys. 1983, 79, 2042-2053. Frauenfelder, H . ; Young, R. D . Comments Mol. Cell. Biophys. 1986, 3, 347. Agmon, N.; Kosloff, R. J. Phys. Chem. 1987, 91, 1988. Onuchic, J . N . J. Chem. Phys. 1987, 86, 3925-3943.

R E C E I V E D for review June 6, 1989. A C C E P T E D revised manuscript October 13, 1989.


Energetics and Dynamics of Gated Reactions - Advances in Chemistry

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