Dynamics of Macromolecular Interactions - American Chemical Society


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18 Dynamics of Macromolecular Interactions 1,3

Stuart A. Allison

1,

J. Andrew McCammon and Scott H . Northrup

2

1

Department of Chemistry, University of Houston, Houston, TX 77004 Department of Chemistry, Tennessee Technological University, Cookeville, TN 38505

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2

The rates of important processes in macromolecular solutions are often influenced or controlled by the binary diffusional encounter frequency of reactants. Two examples are the growth of polymer chains and the binding of ligands to receptors. Calculation of reaction rates in such systems generally requires consideration of such factors as anisotropic Coulombic and hydrodynamic interactions between reactants, and orientation dependent reactivity of the collision partners. A computer simulation approach has been derived that allows detailed bimolecular reaction rate constant calculations in the presence of these and other complicating factors. In this approach, diffusional trajectories of reactants are computed by a Brownian dynamics procedure; the rate constant is then obtained by a formal branching anaylsis that corrects for the truncation of certain long trajectories. The calculations also provide mechanistic information, e.g., on the steering of reactants into favorable configurations by electrostatic fields. The application of this approach to simple models of enzyme-substrate systems is described. The f r e q u e n c y w i t h which two r e a c t i v e s p e c i e s e n c o u n t e r one a n o t h e r i n s o l u t i o n represents an upper bound on t h e b i m o l e c u l a r reaction rate. When t h i s e n c o u n t e r f r e q u e n c y i s r a t e l i m i t i n g , t h e r e a c t i o n i s s a i d t o be d i f f u s i o n c o n t r o l l e d . Diffusion controlled reactions p l a y an important r o l e i n a number o f a r e a s of c h e m i s t r y , i n c l u d i n g n u c l e a t i o n , polymer and c o l l o i d growth, i o n i c and f r e e r a d i c a l r e a c t i o n s , DNA r e c o g n i t i o n and b i n d i n g , and enzyme c a t a l y s i s . Smoluchowski and Debye i n v e s t i g a t e d t h e problem o f d i f f u s i o n controlled reactions between u n i f o r m l y reactive spheres i n the absence (1_) and p r e s e n c e G O of c e n t r o s y m m e t r i c Coulombic f o r c e s . Since these pioneering works, t h e r e has been a p r o l i f e r a t i o n o f t h e o r e t i c a l s t u d i e s based on more r e f i n e d models. These have considered t h e i n c l u s i o n o f hydrodynamic i n t e r a c t i o n , O."^) s o l v e n t 3

Current address: Department of Chemistry, Georgia State University, Atlanta, GA 30303 0097-6156/86/ 0302-0216$06.00/ 0 © 1986 American Chemical Society

Eisenberg and Bailey; Coulombic Interactions in Macromolecular Systems ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

18. ALLISON ET AL.

Dynamics of Macromolecular Interactions

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c a g i n g e f f e c t s , 05) c o n c e n t r a t i o n e f f e c t s , (6^) o r i e n t a t i o n dependence of r e a c t i v i t y on one o r both s p e c i e s , (7-10) i n t e r n a l - c o n f i g u r a t i o n dependent r e a c t i v i t y , (11-13) and n o n c e n t r o s y m m e t r i c d i r e c t f o r c e s ( 14). A number o f e x c e l l e n t r e v i e w s d i s c u s s t h e s e and o t h e r f a c t o r s i n more d e t a i l ( 6 , 15-16). Perhaps t h e most advanced a n a l y t i c a l n u m e r i c a l methods a r e those based on t h e f o r m a l i s m o f W i l e m s k i and Fixman (17-18) and extended by o t h e r i n v e s t i g a t o r s , (19-21) and a l s o the n u m e r i c a l methods of Z i e n t r a , F r e e d , and coworkers ( 2 2 ) . These methods have been p a r t i c u l a r l y u s e f u l i n i n t r a m o l e c u l a r r e a c t i o n p r o c e s s e s such as r i n g c l o s u r e i n c h a i n m o l e c u l e s (20) and p r o t e i n domain c o a l e s c e n c e ( 2 2 ) . Here, a new method i s d e s c r i b e d i n which b i o m o l e c u l a r rate c o n s t a n t s a r e determined by a r e l a t i v e l y s i m p l e s i m u l a t i o n p r o c e d u r e (23). This method i s s u f f i c i e n t l y g e n e r a l t o model systems o f a r b i t r a r y configurâtional c o m p l e x i t y , a r b i t r a r y i n t e r - and i n t r a m o l e c u l a r f o r c e s , and a l l o w s f o r i n c l u s i o n of hydrodynamic i n t e r a c t i o n . When a v a r i e t y of i n t e r a c t i o n s a r e p r e s e n t between t h e r e a c t i v e s p e c i e s , t h e r e i s p r o b a b l y l i t t l e hope o f o b t a i n i n g a n a l y t i c a l rate c o n s t a n t s a t a d e t a i l e d l e v e l and r e c o u r s e t o s i m u l a t i o n methods becomes n e c e s s a r y . I n t h i s work, t h e r o l e o f l o c a l and l o n g range e l e c t r o s t a t i c f o r c e s on d i f f u s i o n c o n t r o l l e d r e a c t i o n s i s of p r i m a r y interest. Anisotropic r e a c t i v i t y and i n c l u s i o n o f hydrodynamic i n t e r a c t i o n a r e f a c t o r s t h a t a r e s t u d i e d as w e l l . I n t h e next s e c t i o n , we e x p l a i n how a r a t e c o n s t a n t c a n be d e r i v e d from t h e s i m u l a t i o n o f a l a r g e number o f t r a j e c t o r i e s and how a t r a j e c t o r y i s computed. I n t h e s e c t i o n on a p p l i c a t i o n s , t h e methodology i s a p p l i e d t o t h r e e p r o g r e s s i v e l y more complex model systems. In the f i r s t model (two r e a c t i v e s p h e r e s ) , t h e e f f e c t s o f centrosymmetrie coulomb i c f o r c e s , a n i s o t r o p i c r e a c t i v i t y , and hydrodynamic i n t e r a c t i o n s a r e considered. I n t h e second model ( a dimer r e a c t i n g w i t h a s p h e r e ) , i t i s shown t h a t non-centrosymmetrie Coulombic i n t e r a c t i o n s can a c t t o " s t e e r " t h e dimer i n t o a f a v o r a b l e o r i e n t a t i o n f o r r e a c t i o n w i t h t h e sphere. S i m i l a r behavior i s a l s o observed i n the t h i r d model, d e s i g n e d t o r e p r e s e n t t h e r e a c t i o n between t h e enzyme s u p e r o x i d e d i s m u t a s e and t h e s u b s t r a t e superoxide. I n t h e f i n a l s e c t i o n , we summarize t h e r e s u l t s o f t h e p r e c e e d i n g s e c t i o n and b r i e f l y discuss future applications. Methodology For diffusion-influenced bimolecular reactions, one i s o r d i n a r i l y most i n t e r e s t e d i n o b t a i n i n g a b i m o l e c u l a r r a t e c o n s t a n t k i n o r d e r t o make c o n t a c t w i t h e x p e r i m e n t a l s t u d i e s . To o b t a i n a r a t e c o n s t a n t by a s i m u l a t i o n p r o c e d u r e , one would, i n p r i n c i p l e , need t o s i m u l a t e a l a r g e ensemble o f r e a c t a n t p a i r s d i f f u s i n g from l a r g e s e p a r a t i o n t o the r e a c t i o n s u r f a c e . However, t h e need t o s i m u l a t e r e a c t a n t d i s p l a cements i n an i n f i n i t e domain was o b v i a t e d by a r e c e n t derivation connecting k t o a recombination p r o b a b i l i t y 8 f o r a p a i r of reactants d i f f u s i n g i n a f i n i t e domain. As d e p i c t e d i n F i g u r e 1, a h y p o t h e t i c a l sphere of r a d i u s b d i v i d e s t h e r e l a t i v e s e p a r a t i o n space r , i n t o an o u t e r r e g i o n ( r > b) and an i n n e r r e g i o n ( r < b ) . The r a d i u s b i s chosen s u f f i c i e n t l y l a r g e so t h a t i ) i n t e r p a r t i c l e d i r e c t and h y d r o dynamic f o r c e s a r e cent rosymme t r i e t o a good a p p r o x i m a t i o n a t r=b, and i i ) t h e ensemble r e a c t i v e f l u x through t h e r = b s u r f a c e i s

Eisenberg and Bailey; Coulombic Interactions in Macromolecular Systems ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

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COULOMBIC INTERACTIONS IN MACROMOLECULAR SYSTEMS

F i g . 1. Schematic I l l u s t r a t i o n o f t h e Method. T r a j e c t o r i e s are s t a r t e d a t b, which d e f i n e s the s e p a r a t i o n o f an a n i s o t r o p i c i n n e r r e g i o n (rb). Trajectories a r e t e r m i n a t e d upon r e a c t i o n o r when r>q (23).

Eisenberg and Bailey; Coulombic Interactions in Macromolecular Systems ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

18.

ALLISON ET AL.

Dynamics of Macromolecular interactions

219

isotropic. T h i s second c o n d i t i o n can be r e l a x e d t o y i e l d improved computational e f f i c i e n c y (42). Under steady s t a t e c o n d i t i o n s , k i s g i v e n by

k - kpOOp

(1)

where ρ i s t h e p r o b a b i l i t y t h a t t h e r e a c t a n t p a i r , s t a r t i n g a t i n i ­ t i a l s e p a r a t i o n r = b, w i l l r e a c t r a t h e r than d i f f u s e a p a r t and k ( b ) i s t h e f a m i l i a r Debye r a t e c o n s t a n t f o r p a i r s w i t h r i n i t i a l l y >D t o f i r s t a c h i e v e a s e p a r a t i o n r = b. Because o f t h e r e s t r i c t i o n s p l a c e d on b, k ( b ) c a n be d e t e r m i n e d a n a l y t i c a l l y and i s g i v e n by C5) D

D

exp k (b)

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n

[u(r)/k T] R

*

= (/ d r [ b

4irr

,

— ] )

(2)

D(r)

where u ( r ) i s t h e ( c e n t r o s y m m e t r i e ) p o t e n t i a l of mean f o r c e and D ( r ) i s t h e r e l a t i v e d i f f u s i o n c o n s t a n t d i s c u s s e d i n more d e t a i l l a t e r . To a v o i d t h e problem o f r e a c t a n t s d i f f u s i n g t o l a r g e d i s t a n c e s i n the d e t e r m i n a t i o n o f p, t r a j e c t o r i e s a r e t e r m i n a t e d i f r exceeds some c u t o f f d i s t a n c e q d e p i c t e d i n F i g u r e 1. What i s a c t u a l l y d e t e r m i n e d i n a s i m u l a t i o n over many t r a j e c t o r i e s i s a r e c o m b i n a t i o n probabi­ lity, 8. S i n c e i t i s p o s s i b l e t h a t a t r a j e c t o r y which reaches s e p a r a t i o n r > q would r e a c t i f not t e r m i n a t e d , ρ and β a r e n o t equal. U s i n g b r a n c h i n g arguments, however, i t i s p o s s i b l e t o c o r r e c t β t o account f o r t h i s d i s c r e p a n c y ( 2 3 ) . I n t h e s p e c i a l case where a l l r e a c t i v e surface c o l l i s i o n s lead to a reaction k (b)8 D

k

=

1-(ΐ-β)Ω

Ω - k (b)/k (q) D

D

(

3

)

(4)

The more g e n e r a l r e s u l t i n which o n l y a f r a c t i o n o f r e a c t i v e s u r f a c e c o l l i s i o n s l e a d t o r e a c t i o n i s g i v e n elsewhere ( 2 3 ) . In order to simulate the dynamical t r a j e c t o r i e s of a model system, t h e L a n g e v i n e q u a t i o n s o f motion a r e i n t e g r a t e d t a k i n g d i s c r e t e time s t e p s (24-29). Since i t i s the comparatively slow, l o n g - r a n g e r e l a t i v e motions o f r e a c t i n g s p e c i e s t h a t a r e o f primary i n t e r e s t here, h i g h l y damped L a n g e v i n o r Brownian dynamics i s t h e most relevant. A number o f Brownian dynamics algorithms are available, (24-29) but i n t h i s work, t h e a l g o r i t h m o f Ermak and McCammon i s used ( 2 4 ) . The i n t e r a c t i n g p a r t i c l e s a r e modelled as spheres or arrays of s p h e r i c a l s u b u n i t s . I f the i n i t i a l p o s i t i o n of s u b u n i t i i s r£ i n a space f i x e d r e f e r e n c e frame, i t s p o s i t i o n a f t e r a time s t e p o f d u r a t i o n A t i s

Eisenberg and Bailey; Coulombic Interactions in Macromolecular Systems ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

COULOMBIC INTERACTIONS IN MACROMOLECULAR SYSTEMS

220

r

- r? + Δ ϋ ( ^ Τ ) "

1

Σ D ?*F? +

(5)

S (At) i

where k i s Boltzmann's c o n s t a n t , Τ i s the a b s o l u t e temperature, F° i s the i n i t i a l f o r c e a c t i n g on s u b u n i t j e x c l u d i n g s t o c h a s t i c ( s o l v e n t ) f o r c e s and, i f p r e s e n t , f o r c e s of c o n s t r a i n t . i s a vec­ t o r of G a u s s i a n random numbers of z e r o mean and v a r i a n c e - c o v a r i a n c e

(6)

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- 2 D" At -i^J " i j

The components of represent s t o c h a s t i c displacements and are o b t a i n e d u s i n g the m u l t i v a r i a t e G a u s s i a n random number generator GGNSM from the IMSL s u b r o u t i n e l i b r a r y (30). g i s the initial hydrodynamic i n t e r a c t i o n t e n s o r between s u b u n i t s T a n d j« Although the e x a c t form o f D i s g e n e r a l l y unknown, i t i s approximated here u s i n g the Oseen tensox w i t h s l i p boundary c o n d i t i o n s . T h i s r e p r e s e n ­ t a t i o n has been shown t o p r o v i d e a r e a s o n a b l e and s i m p l e p o i n t f o r c e d e s c r i p t i o n of the r e l a t i v e d i f f u s i o n of f i n i t e spheres a t s m a l l separations (31). In t h i s case, one has 0

J

0

Β where

6^

i s

t

h

e

Kronecker

(7)

-I + (1

4TTna. « delta,

I i s the

identity

matrix,

J. ±

is

t h e Oseen t e n s o r 1

ij

3

i

+

a

r

j

ij

<

3

i

+

β

j

(9)

R = r

ij

ij

"

a

i

+

a

j

and a^ i s the r a d i u s of s u b u n i t i . I n some s i m u l a t i o n s , hydrodynamic i n t e r a c t i o n (HI) between the r e a c t i n g s p e c i e s i s i g n o r e d . In those c a s e s , the a p p r o x i m a t i o n i s made t h a t D = δ^. I where = k T/bi\r\a^ or i n o t h e r words, J . i s s e t _ e q u a l t o ^ e r o i n Eq. 7. S i n c e the n e g l e c t e d term f a l l s o i f as r . . , t h i s approximation i s e x p e c t e d t o work r e a s o n a b l y w e l l when i and j a r e v e r y f a r a p a r t . Subsequently, when we speak of the case of "no hydrodynamic i n t e r a c t i o n " (NHI) we s h a l l be r e f e r r i n g t o t h i s a p p r o x i m a t i o n . In the model of the monomer t a r g e t i n t e r a c t i n g w i t h a d i m e r i c l i g a n d , d i s c u s s e d i n s e c t i o n on sphere and dumbell dimer, i n t r a m o l e c u l a r HI between the s u b u n i t s of the dimer i s r e t a i n e d even though i n t e r m o l e c u l a r HI i s i g n o r e d i n p a r t i c u l a r NHI s i m u l a t i o n s . ±

If f o r c e s of c o n s t r a i n t a r e p r e s e n t , as i n the case of the monomer-dimer study where the d i s t a n c e between the dimer s u b u n i t s i s f i x e d , d i s p l a c e m e n t c o r r e c t i o n v e c t o r s must be added t o Eq. (5) i n

Eisenberg and Bailey; Coulombic Interactions in Macromolecular Systems ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

18.

ALLISON ET AL.

Dynamics of Macromolecular Interactions

221

order to enforce the c o n s t r a i n t s . Enforcing constraints i s a troublesome problem i n both m o l e c u l a r ( 3 2 ) and Brownian (26, 33) dynamics. N o n e t h e l e s s , they c a n be e n f o r c e d i n a r i g o r o u s manner. Where needed i n t h i s work t h e SHAKE - HI a l g o r i t h m d e s c r i b e d and implemented elsewhere, (33) i s used t o e n f o r c e c o n s t r a i n t s . When hydrodynamic i n t e r a c t i o n i s p r e s e n t , i t t u r n s out t h a t d i s p l a c e m e n t correction vectors must be added to unconstrained as w e l l as constrained subunits. I n t h e monomer dimer s t u d i e s w i t h HI, f o r example, a d i s p l a c e m e n t c o r r e c t i o n v e c t o r must be a p p l i e d t o t h e monomer when t h e c o n s t r a i n t between dimer s u b u n i t s i s e n f o r c e d .

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Applications Two S p h e r e s . The s t e a d y s t a t e d i f f u s i o n c o n t r o l l e d r a t e c o n s t a n t f o r two u n i f o r m l y r e a c t i v e spheres interacting v i a a centrosymmetric p o t e n t i a l of mean f o r c e can be s o l v e d n u m e r i c a l l y and i n s p e c i a l cases a n a l y t i c a l l y as g i v e n by E q . 2. F o r a p o t e n t i a l o f mean f o r c e o f z e r o and no hydrodynamic i n t e r a c t i o n (NHI), E q . 2 reduces t o t h e Smoluchowski r e s u l t (1) k^b) = ^ D b (10) r e l

where b i s t h e c e n t e r - t o - c e n t e r d i s t a n c e a t which t h e spheres spon­ t a n e o u s l y r e a c t and D ^ - D. + where and a r e the t r a n s l a t i o n a l d i f f u s i o n c o n s t a n t s or t h e i n d i v i d u a l s p h e r e s . When HI i s i n c l u d e d , D ( r ) i s g i v e n by ( 6 ) 2k Τ D ( r ) = D r » [I - n ^ - T ] - r (11) rel D « rel w

where f i s t h e u n i t

relative

displacement

v e c t o r between t h e spheres

Τ i s approximated u s i n g Eq. 8. To t e s t t h e s i m u l a t i o n method, we f i r s t s t u d i e d u n i f o r m l y r e a c ­ t i v e spheres under c o n d i t i o n s such t h a t t h e s i m u l a t i o n s can be com­ pared d i r e c t l y t o a n a l y t i c r e s u l t s . C a l c u l a t i o n s were then c a r r i e d out f o r i n t e r a c t i n g spheres w i t h a n i s o t r o p i c r e a c t i v i t y . One o f t h e s p h e r e s was assumed t o be r e a c t i v e o n l y over h a l f i t s s u r f a c e whereas the r e m a i n i n g sphere was u n i f o r m l y r e a c t i v e . T h i s s h a l l be c a l l e d t h e hemisphere model. The r e s u l t s o f t h e s i m u l a t i o n s i n v o l v i n g two r e a c t i v e spheres a r e summarized i n T a b l e I . Sphere r a d i i o f a^ = a2 - 0.5 Â w i t h a " r e a c t i o n r a d i u s " o f 1 Â were used t h r o u g h o u t . In a d d i t i o n t o s t u d y i n g t h e e f f e c t o f hydrodynamic i n t e r a c t i o n , e f f e c t s of d i r e c t f o r c e s were a l s o c o n s i d e r e d u s i n g s i m p l e Coulomb and s c r e e n e d Coulomb i n t e r a c t i o n s . I n t h e case of t h e hemisphere model, s e v e r a l s i m u l a t i o n s were c a r r i e d out i n which t h e a n i s o t r o p i c sphere was allowed to rotate with a rotational diffusion constant of k T/8Trna F o r more d e t a i l s r e g a r d i n g t h e s e p r o t o t y p i c a l s t u d i e s , the r e a d e r i s r e f e r r e d t o r e f e r e n c e ( 2 3 ) . I t c a n be seen t h a t t h e s i m u l a t i o n s a r e i n e x c e l l e n t agreement w i t h a n a l y t i c r e s u l t s where the l a t t e r are a v a i l a b l e . Note t h e l a r g e i n c r e a s e i n k i n t h e p r e sence o f s c r e e n e d o r unscreened Coulombic a t t r a c t i o n a r i s i n g between two o p p o s i t e l y charged i o n s o f elementary charge magnetude i n a d i e l e c t r i c medium l i k e water (ε = 78). The i n c l u s i o n o f HI d e c r e a s e s k by 30% i n t h e n o - f o r c e case but o n l y by 6% w i t h a t t r a c t i v e f o r c e s present. I n t h e case of t h e hemisphere model, t h e i n c l u s i o n o f and

5

B

#

Eisenberg and Bailey; Coulombic Interactions in Macromolecular Systems ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Eisenberg and Bailey; Coulombic Interactions in Macromolecular Systems ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

D

none none none included included included none none none none none none none

0

none none included none included none included

-

Rotation

2

2

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κ

K r

τ

/er

2

2

i

Q

7

2 ^r

-rQ e

/er

2

- V ^ r 2 -rQ e /ετ none none none , -rQ,/ ^

f

-rQ e~ none

2

none

Forces^ )

1 1

Interparticle

Two R e a c t i v e

3 5 3 3 5 5 5 30 3 5 5 5 5

b(Â)

8 10 8 8 10 10 7 60 5 7 10 10 10

q(Â)

Spheres

1.03 7.18 4.81 0.72 7.08 4.66 .711 .700 .785 7.29 7.39 4.92 4.97

± + + + + + ± + ± + + ± ±

(iii) 0.07 0.14 0.12 0.06 0.14 0.14 0.032 0.041 0.012 0.14 0.16 0.15 0.18

K(simulation)

-

1.00 7.31 4.80 0.72 6.82 4.40 -.70(iv) -.70(iv) -.80(iv)

K(analytic)

β

3 χ

( i ) u n i f o r m - both s p h e r e s u n i f o r m l y r e a c t i v e ; hemis phere -- one s p h e r e u n i f o r m l y r e a c t i v e but o n l y h a l f the s u r f a c e of r e m a i n i n g s p h e r e r e a c t i v e . (i^.) Q - 1 e l e m e n t a r y c h a n g e u n i t , ε • 78 = s o l v e n t d i e l e c t r i c c o n s t a n t , κ = 0.1 Â ( c o r r e s p o n d i n g t o [Na ] 0.1 M). ( i i i ) Κ = k / k ° where k j . = 4 π ( + a )D . ( i v ) From F i g u r e s I I and I I I o f r e f e r e n c e 10.

uniform uniform uniform uniform uniform uniform hemisphere hemisphere hemisphere hemisphere hemisphere hemisphere hemisphere

Interaction

ReactivityHydrodynamic

Table I.

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Dynamics of Macromolecular interactions

a t t r a c t i v e Coulombic f o r c e s has an even more d r a m a t i c e f f e c t on k, r e s t o r i n g i t t o t h e v a l u e o b s e r v e d when both spheres a r e u n i f o r m l y reactive. E v i d e n t l y , a t t r a c t i v e f o r c e s serve t o hold the spheres t o g e t h e r long enough f o r them t o a c h i e v e a f a v o r a b l e configuration for reaction. This i s likely t o be a f e a t u r e o f some enzymesubstrate i n t e r a c t i o n s , Since surface charges on enzymes a r e u b i quitous. The e f f e c t o f r o t a t i o n i s more modest, h a v i n g t h e l a r g e s t e f f e c t i n t h e absence o f d i r e c t a t t r a c t i v e f o r c e s . Since the r o t a t i o n a l d i f f u s i o n c o n s t a n t o f a p a r t i c l e v a r i e s r o u g h l y as a , the e f f e c t o f r o t a t i o n on r e a c t i o n r a t e i s e x p e c t e d t o be s m a l l when an a n i s o t r o p i c t a r g e t (enzyme) i s much l a r g e r t h a n t h e s u b s t r a t e .

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3

Sphere and Dumbell Dimer. Dumbell dimers r e a c t i n g w i t h a s p h e r i c a l target represent the simplest case of s t r u c t u r e d reactants. The model u s e d i s d e p i c t e d i n F i g u r e 2. The r a d i i o f t h e t a r g e t (subunit 1) and dimer ( s u b u n i t s 2 and 3) were 2.0 and 0.5 A, r e s p e c t i v e l y . The t a r g e t sphere and e i t h e r one o r both dimer s u b u n i t s were t a k e n t o be r e a c t i v e . The c r i t e r i a f o r a r e a c t i v e c o l l i s i o n were R < 3 Â , and ( f o r t h e cases w i t h o n l y one dimer s u b u n i t r e a c t i v e ) Θ < 9 0 ° . To study the e f f e c t s of d i r e c t forces on r e a c t i o n r a t e s , variable c h a r g e s (Q^) were p l a c e d a t t h e c e n t e r s o f t h e s u b u n i t s . To d e t e r m i n e a r a t e c o n s t a n t , k ^ b ) and t h e r e c o m b i n a t i o n p r o b a ­ b i l i t y 8 must be o b t a i n e d . For p a r t i c l e s i n t e r a c t i n g v i a a centrosymmetric p o t e n t i a l o f mean f o r c e , E q . 2 c a n be used t o o b t a i n k p ( b ) . F o r most o f t h e model s t u d i e s c o n s i d e r e d i n t h i s s e c t i o n , however, Eq. 2 i s n o t s t r i c t l y v a l i d s i n c e t h e p o t e n t i a l o f mean f o r c e has an a n g u l a r ( Θ ) dependence. Making t h e r e a s o n a b l e assumption t h a t t h e r e l a t i v e o r i e n t a t i o n o f t h e dimer f o l l o w s a Boltzmann d i s t r i b u t i o n a t R b, an e q u a t i o n s i m i l a r t o E q . 2 c a n be d e r i v e d (34)· To d e t e r ­ mine 8, d y n a m i c a l t r a j e c t o r i e s a r e computed using Eq. (5) o r e q u a t i o n s d e r i v e d t h e r e f r o m , ( 3 4 ) s t a r t i n g a t R - b = 8 Â and w i t h r e l a t i v e o r i e n t a t i o n s s e l e c t e d a t random from a Boltzmann d i s t r i b u tion. T r a j e c t o r i e s were t e r m i n a t e d a t R > q = 10 Â and 8 was d e t e r mined from t h e r e s u l t s o f 5,000 t o 10,000 s e p a r a t e t r a j e c t o r i e s . The i n t e r e s t e d r e a d e r i s r e f e r r e d t o r e f e r e n c e 34 f o r more d e t a i l s . The r a t e c o n s t a n t r e s u l t s a r e summarized i n T a b l e II· a

To e l u c i d a t e more d i r e c t l y t h e r o l e o f e l e c t r o s t a t i c and h y d r o d y namic f o r c e s i n " s t e e r i n g " t h e dimer toward p r o d u c t i v e collision g e o m e t r i e s , t h e d i s t r i b u t i o n o f r e l a t i v e o r i e n t a t i o n s f ( R , cos©) as a f u n c t i o n o f R was d e t e r m i n e d from t h e s i m u l a t i o n s . Specifically, f(R, cos©) represents the p r o b a b i l i t y that a reactive/unreactive dimer a t R has an o r i e n t a t i o n l y i n g between cos0 ± 0 . 1 . In the spec i a l c a s e o f an i s o t r o p i c d i s t r i b u t i o n , f ( R , cosG) - 0 . 1 s i n c e Θ space has been d i v i d e d i n t o t e n e q u i v a l e n t " b i n s " (-1 < cos© < +1). The observed d i s t r i b u t i o n i s nearly i d e n t i c a l t o t h e Boltzmann d i s t r i b u t i o n f o r R > 4.5 ( 3 4 ) . For R c l o s e t o the r e a c t i o n radius of 3 Â , however, t h i s i s n o t t h e case. T h r e e examples a r e shown i n Figures 3-5 where R = 3.05 ± 0.05Â. A c h a r a c t e r i s t i c feature of dimers which approach this close without eventually reacting ( u n f i l l e d symbols) i s t h a t they a r e i n an u n f a v o r a b l e o r i e n t a t i o n . The f a c t t h a t dimers i n r e a c t i v e as w e l l as u n r e a c t i v e t r a j e c t o r i e s have h i g h p r o b a b i l i t y o f b e i n g i n an u n f a v o r a b l e o r i e n t a t i o n i s i n t e r p r e t e d i n t h e f o l l o w i n g way. A dimer which approaches t h e target c l o s e l y i n a favorable o r i e n t a t i o n tends t o r e a c t quickly

Eisenberg and Bailey; Coulombic Interactions in Macromolecular Systems ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

COULOMBIC INTERACTIONS IN MACROMOLECULAR SYSTEMS

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F i g , 2. Sphere - Dimer M o d e l . The t a r g e t sphere ( s u b u n i t 1) and one or both dimer s u b u n i t s (2 and 3 ) , a r e u n i f o r m l y r e a c t i v e . The t a r g e t sphere r a d i u s i s 2 Â and the t o u c h i n g s p h e r e s of the dimer each have a r a d i u s of 0.5 Â. Charges ( Q ^ Q , Q3) a r e p l a c e d a t the s u b u n i t c e n t e r s . 2

• 3

T

Δ

Ο

.21 Φ CO Ο

OA

υ

.11

-+-

-.9

-.7

-.5

-.3

ΟΛ

ÇL

+.3

+.5

-4-

-.1 cos

+.1

+.7

+.9

Θ

F i g . 3. O r i e n t a t i o n F a c t o r Near the R e a c t i v e S u r f a c e . R • 3.05 ± .05 A, Q = Q = Q = 0. Only s u b u n i t 2 o f the dimer i s r e a c t i v e . F i l l e d / e m p t y c i r c l e s (*/o) represent reactive/unreactive t r a j e c ­ tories where hydrodynamic interaction (HI) i s included, and f i l l e d / e m p t y t r i a n g l e s (Α/Δ) r e p r e s e n t r e a c t i v e / u n r e a c t i v e t r a j e c ­ t o r i e s where HI i s not i n c l u d e d . E r r o r bars a r e p l a c e d on o n l y c e r t a i n data p o i n t s . x

2

3

Eisenberg and Bailey; Coulombic Interactions in Macromolecular Systems ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

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ALLISON ET AL.

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Dynamics of Macromolecular interactions

.3^

ok

• 21 Φ

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ο ο

ι

Δ



.

·· I

o Δ

Δ

+ -.9

-.7 -.5

-.3 -.1

-+- —\— +.1 +.3

3

+.5

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?

+.7

*

+.9

cos θ Fig. Fig.

4. O r i e n t a t i o n F a c t o r Near t h e R e a c t i v e 3, but Q = e, Q = -e, Q - 0. x

-H

-.9

2

1

Surface.

Same as

3

1

-.7 -.5

1

1

1

1

1

1—

-.3 -.1

1

+.1

+.3

+.5

+.7

+.9

cos θ F i g . 5. O r i e n t a t i o n F a c t o r Near t h e R e a c t i v e S u r f a c e . Same as F i g s . 3 and 4, but Q = 2e, Q - -e, Q - +e. U n l i k e F i g . 3 (no c h a r g e s ) o r F i g . 4 ( n e t Coulomb charges on both r e a c t a n t s ) , t h e dimer i n t h i s case i s a pure e l e c t r i c d i p o l e . x

2

3

Eisenberg and Bailey; Coulombic Interactions in Macromolecular Systems ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

COULOMBIC INTERACTIONS IN MACROMOLECULAR SYSTEMS

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whereas a dimer i n an u n f a v o r a b l e o r i e n t a t i o n spends a c o m p a r a t i v e l y l o n g time i n c l o s e p r o x i m i t y t o the t a r g e t e v e n t u a l l y d i f f u s i n g away ( u n r e a c t i v e ) or r e o r i e n t i n g to a f a v o r a b l e c o n f i g u r a t i o n at which point i t quickly reacts. T h i s i n t e r p r e t a t i o n i s a l s o s u p p o r t e d by the r a t e c o n s t a n t s which do not show a s u b s t a n t i a l i n c r e a s e when both dimer s u b u n i t s a r e made r e a c t i v e . Net charges on b o t h t a r g e t and dimer and t o a l e s s e r e x t e n t hydrodynamic i n t e r a c t i o n have a s u b s t a n t i a l e f f e c t on o v e r a l l r a t e but a s u r p r i s i n g l y s m a l l e f f e c t on the dimer o r i e n t a t i o n s . E.g., F i g u r e 3 and 4 a r e n e a r l y s u p e r i m p o s a b l e d e s p i t e the > 2 - f o l d d i f f e r e n c e i n r a t e . Comparing F i g s . 3 and 4 t o F i g u r e 5, i t i s seen t h a t d i p o l a r f o r c e s have a s u b s t a n t i a l e f f e c t on o r i e n t i n g the dimer and, from T a b l e I I , a l s o a s i g n i f i c a n t e f f e c t on overall rate. Comparing the t h i r d and s i x t h l i n e s of T a b l e I I i t would appear t h a t a t t r a c t i o n of the d i p o l e by the inhomogeneous e l e c t r i c f i e l d of the t a r g e t makes a s m a l l e r c o n t r i b u t i o n t o the r a t e enhancement than does the orientâtional s t e e r i n g e f f e c t . From the sphere-dimer s t u d i e s , two major c o n c l u s i o n s emerge. The f i r s t i s t h a t the t r a j e c t o r y method can be extended t o s t r u c t u r e d r e a c t a n t s w i t h a n i s o t r o p i c r e a c t i v i t y and a n i s o t r o p i c d i r e c t f o r c e s and hydrodynamic i n t e r a c t i o n s . The second major c o n c l u s i o n i s t h a t c o m p l i c a t e d e l e c t r o s t a t i c i n t e r a c t i o n s between s p e c i e s w i t h a n i s o t r o pic r e a c t i v i t y can " s t e e r " the a p p r o a c h i n g p a r t i c l e s i n t o f a v o r a b l e o r i e n t a t i o n s and enhance the r e a c t i o n r a t e . F o r t h e s e model s t u d i e s , r a t e enhancements up t o 20% have been o b t a i n e d . The second c o n c l u s i o n i s l i k e l y t o be of c o n s i d e r a b l e r e l e v a n c e to m o l e c u l a r b i o l o g y . In the t h i r d and f i n a l s e r i e s of s i m u l a t i o n s , the Brownian dynamics t r a j e c t o r y method i s a p p l i e d t o a p a r t i c u l a r b i o l o g i c a l system. S u p e r o x i d e Dismutase and Superoxide. Electrostatic interactions i n f l u e n c e the r a t e s of many b i o m o l e c u l a r a s s o c i a t i o n s ( 1 5 ) . A part i c u l a r example of t h i s i s the d i f f u s i o n c o n t r o l l e d d i s m u t a t i o n of superoxide ( 0 ) catalyzed by the enzyme copper, z i n c s u p e r o x i d e d i s m u t a s e (SOD) (35-36). Although both s u b s t r a t e and enzyme a r e n e g a t i v e l y charged at p h y s i o l o g i c a l pH, the r e a c t i o n r a t e i s h i g h and increases with decreasing i o n i c strength at moderate s a l t concentrations (35). This i s opposite of the t r e n d e x p e c t e d on the b a s i s of the net charges and may be due to l o c a l e l e c t r o s t a t i c i n t e r a c t i o n s which may s e r v e to s t e e r 0« i n t o the a c t i v e s i t e o f SOD. 2

F o r the i n i t i a l s t u d i e s d e s c r i b e d h e r e , the SOD dimer, which i s the a c t i v e form of the enzyme, was modeled as a sphere of 30 Â radius. Two r e a c t i v e patches corresponding to the active site r e g i o n s of the dimer were d e f i n e d by the s u r f a c e a r e a l y i n g w i t h i n 10° of an a x i s r u n n i n g t h r o u g h the c e n t e r of the sphere ( F i g u r e 6 ) . F i v e charges were embedded w i t h i n the s p h e r e t o r e p r o d u c e the monop o l e , d i p o l e , and q u a d r u p o l e terms a s s o c i a t e d w i t h the charged groups i n the 2 Â c r y s t a l l o g r a p h i c s t r u c t u r e of b o v i n e e r y t h r o c y t e SOD (37) a v a i l a b l e through the P r o t e i n Data Bank ( 3 8 ) . The net charge i s -4 i n u n i t s of the p r o t o n i c charge and the d i p o l e moment a p p r o x i m a t e l y v a n i s h e s due to the symmetry of the dimer. A d i e l e c t r i c c o n s t a n t of 78 was assumed throughout the system. On the b a s i s of p r e v i o u s e x p e r i m e n t a l (39) and c o m p u t a t i o n a l (36) s t u d i e s , t h i s s h o u l d p r o v i d e a r e a s o n a b l e as w e l l as s i m p l e d e s c r i p t i o n of the d i r e c t i o n and magnitude of e l e c t r o s t a t i c f o r c e s on 0 ~ . The θ" molecule was r e p r e s e n t e d by a s p h e r e of r a d i u s 1.5 Â w i t h a c e n t r a l charge of -1. 2

Eisenberg and Bailey; Coulombic Interactions in Macromolecular Systems ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

18.

ALLISON ET AL.

Table I I .

Dynamics of Macromolecular Interactions

R e l a t i v e Rate C o n s t a n t s

f o r Sphere - Dimer Models hydrodynamic interaction

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227

none included none none included none none included none included

k/k .906 .740 1.068 1.039 .802 069 080 873 51 21

(a)

.018 .008 (b) .017 0.18 C



0

1

9

0.016 .020 .017 .05 .05

C b ;

( a ) k = 4ïïD r where r = 3 Â and D i s t h e e f f e c t i v e r e l a t i v e d i f f u s i o n constant (see reference (b) Both s u b u n i t s o f dimer (2 and 3) r e a c t i v e . In a l l other simulat i o n s , only subunit 2 i s r e a c t i v e . Q

M

F i g . 6. Model o f SOD - S u p e r o x i d e . C r o s s e s (X) i n d i c a t e p o s i t i o n s of charges. A c t i v e s i t e s a r e i n d i c a t e d by t h e dark caps on t h e SOD sphere; Θ = 1 0 ° .

Eisenberg and Bailey; Coulombic Interactions in Macromolecular Systems ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

COULOMBIC INTERACTIONS IN MACROMOLECULAR SYSTEMS

228

Hydrodynamic i n t e r a c t i o n between SOD and 0 was i g n o r e d i n t h i s p r e liminary study. Between 25,000 and 100,000 t r a j e c t o r i e s were c a r r i e d out i n each s i m u l a t i o n w i t h t y p i c a l b and q v a l u e s of 300 and 500 Â respectively. F o r a d d i t i o n a l d e t a i l s , see R e f e r e n c e s 40 and 41. T a b l e I I I summarizes some c a l c u l a t i o n s c a r r i e d out t o e x p l o r e what e f f e c t s c o n t r i b u t e to the h i g h r e a c t i v i t y of SOD. For the n a t i v e - l i k e model w i t h a monopole charge of -4, i n c l u s i o n of the ( n o n - c e n t r o s y m m e t r i c ) q u a d r u p o l e i n c r e a s e s the r e a c t i o n r a t e by 40%. The q u a d r u p o l e e v i d e n t l y h e l p s to s t e e r 0 i n t o the a c t i v e s i t e . P a r a l l e l s i m u l a t i o n s were a l s o c a r r i e d out i n which monopole charges of 0 and +4 were u s e d . A l t h o u g h i n c r e a s i n g the monopole charge from -4 t o 0 t o +4 i n c r e a s e d the r a t e s by f a c t o r s of 2.5 and 5, r e s p e c t i v e l y , the s t e e r i n g e f f e c t i s p r e s e n t i n each c a s e . This suggests t h a t the enhancement i n r a t e due to s t e e r i n g by l o c a l e l e c t r o s t a t i c i n t e r a c t i o n s w i l l p e r s i s t i n the p r e s e n c e of added s a l t , which w i l l s u p p r e s s the e f f e c t s of the monopole f i e l d more s t r o n g l y than those of the s h o r t e r - r a n g e d q u a d r u p o l e f i e l d . The q u a l i t a t i v e e f f e c t of added s a l t has been examined d i r e c t l y by s i m u l a t i o n s u s i n g a s c r e e n e d p o t e n t i a l of the Debye-Huckel type and the r e s u l t s are shown i n F i g u r e 7. Above an i o n i c s t r e n g t h of about 3 χ 10 M, the r a t e d e c r e a s e s w i t h added s a l t as the s t e e r i n g f i e l d due to the q u a d r u p o l e i s s c r e e n e d . A qualitatively similar trend i s observed e x p e r i m e n t a l l y . In f u t u r e work, we p l a n to examine increasingly detailed models that include the irregular surface topography and f u l l charge d i s t r i b u t i o n of SOD, i n d i v i d u a l s a l t i o n s , l o c a l d i e l e c t r i c constant v a r i a t i o n s , e t c . 2

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2

Summary and

Conclusions

A p r i n c i p a l aim of t h i s work was to demonstrate the u t i l i t y and g e n e r a l i t y of a new s i m u l a t i o n method f o r d e t e r m i n a t i o n of the r a t e s and mechanisms of d i f f u s i o n c o n t r o l l e d r e a c t i o n s . W i t h r e g a r d t o the r o l e of e l e c t r o s t a t i c i n t e r a c t i o n s i n d i f f u s i o n c o n t r o l l e d r e a c t i o n s , several conclusions can be made on the b a s i s of the model s t u d i e s d i s c u s s e d i n the p r e v i o u s s e c t i o n . Net charges have a s i g n i f i c a n t and, i n some c a s e s , d r a m a t i c e f f e c t on o v e r a l l r a t e but may not p l a y an i m p o r t a n t r o l e i n s t e e r i n g the r e a c t i v e s p e c i e s i n t o o r i e n t a t i o n s favorable for reaction. Net charges do p l a y an i m p o r t a n t r o l e i n b r i n g i n g the s p e c i e s t o g e t h e r . L o c a l e l e c t r o s t a t i c f o r c e s can h e l p to steer species into productive o r i e n t a t i o n s . T h i s was m a n i f e s t i n r a t e enhancement of 20 t o 60% i n the above model s t u d i e s ; i t i s l i k e l y t h a t l a r g e r e f f e c t s o c c u r i n o t h e r systems. Such s t e e r i n g e f f e c t s are l i k e l y t o be i m p o r t a n t i n m o l e c u l a r b i o l o g y s i n c e v i r ­ t u a l l y a l l b i o m o l e c u l e s have complex charge d i s t r i b u t i o n s on t h e i r surfaces. F u r t h e r m o r e , the r e l a t i v e importance of l o c a l e l e c t r o s t a ­ tic interactions is expected to be greatest at moderate (physiological) s a l t concentrations. Hydrodynamic i n t e r a c t i o n s were o b s e r v e d to d e c r e a s e the r a t e by 5 t o 30% but had l i t t l e e f f e c t on "steering". I n f u t u r e work, the methods i l l u s t r a t e d i n t h i s paper w i l l be a p p l i e d to a v a r i e t y of problems i n m a c r o m o l e c u l a r k i n e t i c s . More d e t a i l e d s t u d i e s of s u b s t r a t e binding t o s u p e r o x i d e dismutase and a n t i g e n b i n d i n g to antibody molecules are i n p r o g r e s s . Other s t u d i e s t h a t are p l a n n e d or i n p r o g r e s s i n c l u d e the e x a m i n a t i o n of Coulombic c o n t r i b u t i o n s t o polymer growtn and to DNA-ligand i n t e r a c t i o n s .

Eisenberg and Bailey; Coulombic Interactions in Macromolecular Systems ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

18.

ALLISON ET AL.

Table I I I .

229

Dynamics of Macromolecular Interactions

R e l a t i v e Rate C o n s t a n t s f o r V a r i o u s SOD Models

SOD Charge

Model

k/k *

Monopole Monopole p l u s q u a d r u p o l e Monopole Monopole p l u s q u a d r u p o l e Monopole Monopole p l u s q u a d r u p o l e

0.056 0.079 0.12 0.19 0.26 0.41

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0

-4 -4 0 0 +4 +4

+ .003 + .004 ± .01 + .01 + .02 ± .03

A

T h e r a t e c o n s t a n t k i s n o r m a l i z e d by t h a t , k , e x p e c t e d f o r an SOD model w i t h no embedded charges and a u n i f o r m l y r e a c t i v e s u r f a c e . A t l e a s t 25,000 t r a j e c t o r i e s were computed f o r each model. Q

F i g . 7. Dependence o f R e l a t i v e Rate on I o n i c S t r e n g t h ("Salt"). S o l i d and d o t t e d l i n e s connect monopole + q u a d r u p o l e and monopole rates, respectively. The e l e c t r o s t a t i c p o t e n t i a l energy between t h e charge on 0« 'e Q i Q / where ε i s t h e d i e l e c t r i c c o n s t a n t (=78) and κ i s t h e Debye-Huckel p a r a m e t e r . a

a

r

e

2

£

r

2

Eisenberg and Bailey; Coulombic Interactions in Macromolecular Systems ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

COULOMBIC INTERACTIONS IN MACROMOLECULAR SYSTEMS

230

Acknowledgments T h i s work was s u p p o r t e d i n p a r t by g r a n t s from the Robert A. Welch Foundation and NIH ( H o u s t o n ) , the Research C o r p o r a t i o n and t h e P e t r o l e u m R e s e a r c h Fund as a d m i n i s t e r e d by ACS (Tennessee T e c h ) . SAA i s t h e r e c i p i e n t o f an NSF P r e s i d e n t i a l Young I n v e s t i g a t o r Award and a D r e y f u s Grant f o r Young F a c u l t y i n C h e m i s t r y . SHN i s t h e r e c i p i e n t of an NIH C a r e e r Development Award. JAM i s an A l f r e d P. Sloan F e l l o w and i s t h e r e c i p i e n t o f NIH C a r e e r Development and D r e y f u s T e a c h e r - S c h o l a r Awards.

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RECEIVED

June

10, 1985

Eisenberg and Bailey; Coulombic Interactions in Macromolecular Systems ACS Symposium Series; American Chemical Society: Washington, DC, 1986.