Dynamics of Macromolecular Interactions - ACS Symposium Series

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

Downloaded by UNIV OF PITTSBURGH on May 6, 2016 | http://pubs.acs.org Publication Date: March 28, 1986 | doi: 10.1021/bk-1986-0302.ch018

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

217


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



218

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).


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)


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


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)


- 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

Dynamics of Macromolecular Interactions - ACS Symposium Series

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