Water in Polymers - American Chemical Society

2

S o l v a t i o n : A M o l e c u l a r D y n a m i c s S t u d y of a Dipeptide in Water MARTIN KARPLUS Department of Chemistry, Harvard University, Cambridge, MA 02138

Downloaded by UNIV OF SYDNEY on May 3, 2015 | http://pubs.acs.org Publication Date: August 19, 1980 | doi: 10.1021/bk-1980-0127.ch002

PETER J. ROSSKY Department of Chemistry, University of Texas, Austin, TX 78712 Water has an essential role in living systems and is u l t i mately involved in the structure and function of biological polymers such as proteins. However, in this contribution we shall focus primarily not on what the water does for the biopolymer but rather on the effects that the biopolymer has on the water that interacts with i t . Of interest are alterations in the structural, energetic, and dynamic properties of the water molecules. Studies of the rotational mobility of water molecules at protein surfaces have been interpreted by dividing the solvent molecules into three groups (1). The most rapidly reorienting group has a characteristic rotational reorientation time (T ) of not more than about 10 s. The next most rapid group exhibits a rotational reorientation time of about 10 s and has been tentatively identified as the water molecules that are strongly associated with ionic residues. The third group exhibits a T of about 10 s; these solvent molecules are considered to be essentially irrotationally bound to the macromolecules; an example might be the four waters in the interior of the bovine pancreatic inhibitor. The population exhibiting the fastest times is expected to include molecules which form hydrogen bonds to the peptide backbone and those which are influenced by the presence of nonpolar groups. It is this group which forms the major part of the solvation shell and, therefore, is likely to play the dominant role in the solvent effect on protein properties. Because of the difficulties involved in studies of protein solutions per se, it is of particular interest to investigate systems of small molecules that incorporate functional groups present in proteins. In this contribution we describe the results of a molecular dynamics simulation of such a molecule, the alanine dipeptide in aqueous solution. In such a simulation, one treats a sample of molecules with fixed volume and an energy and density corresponding to the system of interest. Given the internal and interaction potentials for the molecules in the box, and certain initial conditions for the coordinates and momenta of each particle, one solves the classical equations of motion for a l l of the particles r

-11

-9

-6

r

0-8412-0559-0/ 80/47-127-023505.00/ 0 © 1980 American Chemical Society In Water in Polymers; Rowland, S.; ACS Symposium Series; American Chemical Society: Washington, DC, 1980.


24

WATER IN POLYMERS

to o b t a i n the phase space t r a j e c t o r y of the e n t i r e system over a p e r i o d of time. An i n i t i a l i n t e g r a t i o n p e r i o d during which c e r t a i n p r o p e r t i e s (e.g., i n d i v i d u a l p a r t i c l e v e l o c i t i e s ) are ad j u s t e d i s used to o b t a i n a system that i s e q u i l i b r a t e d . A f t e r the e q u i l i b r a t i o n p e r i o d , i n t e g r a t i o n i s continued f o r a l e n g t h of time s u f f i c i e n t to y i e l d time averages that approximate e q u i l i b r i u m averages. In a d d i t i o n , the time e v o l u t i o n of the p a r t i c l e t r a j e c t o r i e s can be used to determine time-dependent proper ties. We examine s t r u c t u r a l and dynamic aspects of both the dipept i d e s o l u t e and the aqueous s o l v e n t . For the d i p e p t i d e , primary emphasis i s placed on the i n t e r n a l motions. The s i z e and dynami c a l character of f l u c t u a t i o n s r e l a t i v e to the average s t r u c t u r e are i n v e s t i g a t e d i n vacuum and i n the presence of s o l v e n t . The d i p e p t i d e v i b r a t i o n a l degrees of freedom have frequencies v a r y i n g from approximately 50 ( d i h e d r a l angle t o r s i o n s ) to 3500 cm (bond s t r e t c h i n g ) , corresponding to c h a r a c t e r i s t i c times i n the range of 7 χ ΙΟ"-^ to 1 χ 10~-^s. For such a range i n c h a r a c t e r i s t i c times, a s i g n i f i c a n t v a r i a t i o n i n s o l v e n t e f f e c t s (e.g., damping of f l u c t u a t i o n s ) i s expected. The s t r u c t u r a l and dynamic p r o p e r t i e s of the aqueous s o l v e n t i n the r e g i o n immediately surrounding the d i p e p t i d e s o l u t e are of s p e c i a l i n t e r e s t . The p r i n c i p a l questions which we address are: F i r s t , how i s the dynamic behavior of the s o l v e n t a l t e r e d by the p r o x i m i t y of the s o l u t e ? Second, what i s the range of i n f l u e n c e of the s o l u t e ; that i s , are the e f f e c t i v e s o l v e n t - s o l u t e i n t e r a c t i o n s of s u f f i c i e n t l y short range that i t i s reasonable to r e gard the water molecules i n contact w i t h the p o l a r (peptide) groups as q u a l i t a t i v e l y d i f f e r e n t from those i n contact w i t h the nonpolar (methyl) s u b s t i t u e n t s ? F i n a l l y , we i n v e s t i g a t e the s t r u c t u r a l o r i g i n s of observed d i f f e r e n c e s between the dynamic p r o p e r t i e s of the b u l k s o l v e n t and that i n contact w i t h the s o l ute. Model The d e t a i l s of the model used to simulate the d i p e p t i d e s o l u t i o n have been presented p r e v i o u s l y ( 2 ) ; a b r i e f review of the i n t e r a c t i o n s present i n the system and the methods used to c a r r y out the s i m u l a t i o n i s given here. The a l a n i n e " d i p e p t i d e " s o l u t e (CH C ONHCHCH C ONHCH ), shown i n F i g u r e 1, i s a n e u t r a l molecule terminated by methyl groups, r a t h e r than by the c a r b o x y l i c a c i d and amino groups of an amino a c i d . This i s an a p p r o p r i a t e choice to o b t a i n a system that models an amino a c i d as p a r t of a p o l y peptide c h a i n . The s t r u c t u r e shown i n F i g u r e 1 i s the e q u a t o r i a l C7 conformation (Cj ^) that i s the g l o b a l minimum i n the dipep t i d e p o t e n t i a l surface i n vacuum and i s b e l i e v e d to be the favored conformation i n both aqueous and nonaqueous s o l u t i o n s . The i n t e r n a l degrees of freedom of the d i p e p t i d e are governed by a molecular mechanics f o r c e f i e l d (2) that i n c l u d e s terms f

3

f

3

3

BC

In Water in Polymers; Rowland, S.; ACS Symposium Series; American Chemical Society: Washington, DC, 1980.


2. KARPLUS AND ROSSKY

Solvation

25

Figure 1. Alanine dipeptide in the equa torial C conformation, (φ,φ) ~ (—60°, 60°). The structure is (left to right) CH CONHCHCH CONHCH ; the dashed line represents the internal hydrogen bond. 7

5

s

s


26

WATER IN POLYMERS


corresponding to harmonic bonds, anharmonic bond a n g l e s , d i h e d r a l angle t o r s i o n s , and nonbonded Lennard-Jones and 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 . No d i p e p t i d e degrees of freedom are c o n s t r a i n e d i n the s i m u l a t i o n . The water molecules are modeled by a m o d i f i c a t i o n of the ST2 model of S t i l l i n g e r and Rahman (_3) · The model c o n s i s t s of four p o i n t charges placed w i t h i n a s i n g l e LennardJones sphere centered a t the oxygen atom; two p o s i t i v e charges are l o c a t e d a t the hydrogen atom p o s i t i o n s , and two negative charges are l o c a t e d a t p o s i t i o n s r e p r e s e n t i n g the l o n e - p a i r o r b i t a l s . The only m o d i f i c a t i o n made i n the ST2 model i s to a l l o w i n t e r n a l f l e x i b i l i t y i n the water molecules. The i n t e r m o l e c u l a r i n t e r a c t i o n s among the waters are g i v e n by p a i r - w i s e p o t e n t i a l s . For the two molecules and W^, we have

w

J p t r - CAT

w

l 2

"

\

r

w

t

V°l°2/ *+

q . 1 q. 2

Σ

^—L^(r

. . , 1,3=1

)

r. .

(1)

12

13

where σ and ε are the parameters c h a r a c t e r i z i n g the Lennardw w w Jones i n t e r a c t i o n , q. i s the i t h charge i n water molecule W, TQ Q i s the i n t e r m o l e c u l a r oxygen-oxygen d i s t a n c e , S ( r ) i s a 1 2 s w i t c h i n g f u n c t i o n ( 3 ) . The i n t e r a c t i o n s between each water molecule and the d i p e p t i d e are g i v e n by a sum of LennardJones and e l e c t r o s t a t i c terms of the form -

, , - .12

6

/ - v \

4

w

-ι

atoms, λ wnere 5^ = ( a + 0χ)/2 and r g ^ i s the water oxygen-dipeptide atom d i s t a n c e . For the chosen v a l u e s (2) of dipeptide-atom LennardJones parameters, and ε·^, and charges, q-^, the water molecules a s s o c i a t e d w i t h the s o l u t e p e p t i d e groups have reasonable ener g i e s and geometries. I n p a r t i c u l a r , the o p t i m a l a s s o c i a t i o n ener g i e s (kcal/mol) f o r the four types of hydrogen bonds i n the s y s tem i n order of i n c r e a s i n g s t r e n g t h (the water HOH bond angle i s f i x e d a t the t e t r a h e d r a l angle) are NH ... H 0 (-6.0) < H 0 ... H 0 (-6.8) < C = 0 ... H 0 (-7.4) < N-H ... 0 = C (-8.1). The values of the a s s o c i a t i o n energies a r e adjusted to the waterwater i n t e r a c t i o n energy g i v e n by the ST2 model so that the hy drogen bond strengths are a l l s i m i l a r , i n accord w i t h a v a i l a b l e w

2

2

2


2

2.

KARPLUS AND ROSSKY

Solvation

27

c a l c u l a t i o n s and data. The s i m u l a t i o n i s c a r r i e d out on a sample c o n s i s t i n g of one d i p e p t i d e and 195 water molecules i n a cubic box w i t h an edge l e n g t h of 18.2194 X; the d e n s i t y of 1.004 g/cnr* i s i n accord w i t h experiment. The d i p e p t i d e s o l u t e i s surrounded by approximately two molecular l a y e r s of water a t a l l p o i n t s . A f t e r the e q u i l i b r a t i o n , the s i m u l a t i o n analyzed i n the current work corresponds to 4000 time steps of 3.67 χ 1 0 - l ^ s , or 1.5 ps on a molecular time s c a l e . The mean solvent k i n e t i c temperature i s 303°K and that of the d i p e p t i d e i s 298°K.


Solute

Properties

During the s i m u l a t i o n , the s o l u t e remains i n the v i c i n i t y of the C f minimum. This i s not to be i n t e r p r e t e d as implying that the i s the most s t a b l e s o l u t i o n s t r u c t u r e , because there i s a very small p r o b a b i l i t y of observing a l a r g e conformational change i n such a short time. To determine the e f f e c t of the water on the d i p e p t i d e i n so l u t i o n , we d i d a corresponding s i m u l a t i o n of the d i p e p t i d e dynam i c s i n the absence of s o l v e n t . Neither the average s t r u c t u r e nor the magnitude of the l o c a l f l u c t u a t i o n s of the d i p e p t i d e i s s t r o n g l y a f f e c t e d by the solvent environment. The r e s u l t s are summarized i n Table I , where we g i v e r e s u l t s f o r t y p i c a l bonds, bond angles, and d i h e d r a l angles (see F i g u r e 1 ) . With the excep t i o n of the f l u c t u a t i o n s i n the d i h e d r a l angles ψ and χ the ob served d i f f e r e n c e s are w i t h i n the s t a t i s t i c a l e r r o r of the c a l c u lation. q

C o r r e l a t i o n Functions. We next consider dynamic c o r r e l a t i o n s of the s o l u t e f l u c t u a t i o n s . The time c o r r e l a t i o n f u n c t i o n f o r s o l u t e s t r u c t u r a l f l u c t u a t i o n s i s defined as

c (t) A

=

(3)

τ _ " 2

2

where A i s a p a r t i c u l a r s t r u c t u r a l parameter (e.g., bond l e n g t h , d i h e d r a l a n g l e ) , M ( t ) = A ( t ) - , = , and the brackets i n d i c a t e an average over the s i m u l a t i o n ; i n the nu merator of eq 3, the average i n c l u d e s a l l v a l u e s , τ, i n the simu lation. The unperturbed harmonic v i b r a t i o n of an i s o l a t e d bond of l e n g t h b would lead to a c o r r e l a t i o n f u n c t i o n , C , ( t ) , which o s c i l l a t e s without decay f o r a l l times. However, s h i f t s i n e i t h e r the frequency or phase of the o s c i l l a t i o n r e s u l t s i n an eventual de cay of C^(t) to zero a f t e r a time when the phase of Ab(t) i s , on the average, completely random w i t h respect to that of Ab(0). Even i n an i s o l a t e d molecule, such a decay can occur due to cou p l i n g of the motion of v a r i o u s degrees of freedom w i t h d i f f e r e n t f r e q u e n c i e s , although over long times the c o r r e l a t i o n f u n c t i o n cannot remain zero since there i s no d i s s i p a t i o n . The change i n 2

2


28

WATER IN POLYMERS

Table I Average Solute S t r u c t u r e

A

a

2

vacuum

vacuum

solution

Χ

solution

5

Bonds*


C

N

lT°L H

L- L

V R C

1.235

1.237

0.023

0.028

0.994

0.997

0.018

0.012

1.544

1.542

0.041

0.035

1.461

1.459

0.034

0.032

Bond Angles^

L- L-°L

122.31

121.89

3.30

3.23

W R

114.73

114.46

4.25

3.93

W B

107.71

108.15

3.51

3.85

120.97

120.68

3.97

4.29

C

C

C

C

C

R

-*R\

Dihedral A n g l e s

b , C

Φ

-67.21

-63.96

9.67

7.83

Ψ

63.45

59.33

11.53

22.57

-179.25

-179.67

8.74

9.67

-179.68

178.08

14.72

12.39

-59.10

-62.88

9.96

32.25

ω 1

ω 2

Χ

aA l l s t r u c t u r a l parameters, A, are d e f i n e d i n F i g u r e 1; = mean v a l u e , = . Bonds i n Xngstroms, bond angles and d i h e d r a l angles i n degrees. The vacuum minimum occurs at φ = 66.2°, ψ = -65.3°, ω = 179.2°, ω = 179.9° 2

2

/ 2

b

c

1

2



2.

KARPLUS AND ROSSKY

29

Solvation

the c o r r e l a t i o n f u n c t i o n i n s o l u t i o n i s determined by the e f f e c t i v e n e s s of s o l v e n t - s o l u t e c o l l i s i o n s i n d i s s i p a t i n g the s o l u t e ' s dynamical i n f o r m a t i o n . In water, these " c o l l i s i o n s " can i n v o l v e the r e p u l s i v e f o r c e s c h a r a c t e r i s t i c of hard s p h e r e - l i k e systems and the strong hydrogen bonding f o r c e s ; weak a t t r a c t i v e van der Waals i n t e r a c t i o n s are expected to have only a s m a l l e f f e c t . C o l l i s i o n s w i t h solvent are more l i k e l y to a f f e c t the s o l u t e mo t i o n i f the l a t t e r i s a s s o c i a t e d w i t h a s m a l l c h a r a c t e r i s t i c f o r c e constant or i f the mass of the s o l u t e s t r u c t u r a l component i n v o l v e d i s s m a l l ; a l s o , damping i s g e n e r a l l y more e f f e c t i v e f o r motions i n v o l v i n g s t r u c t u r a l components of increased s p a t i a l dimensions. In the current study, a comparison of d i f f e r e n t d i peptide s t r u c t u r a l motions shows the expected q u a l i t a t i v e d i f f e r ences i n behavior. We i l l u s t r a t e the r e s u l t s by presenting the c o r r e l a t i o n f u n c t i o n s f o r (see Figure 1) a t y p i c a l bond ( ~ g ) Figure 2, a t y p i c a l bond angle ( ~ " ^ ) F i g u r e 3, and the d i h e d r a l angle χ i n Figure 4. In each f i g u r e , we show the r e s u l t obtained i n s o l u t i o n at the top and that obtained i n vacuum at the bottom. The s p e c t r a l d e n s i t y as a f u n c t i o n of frequency ω corresponding to C ( t ) i s c

c

i n

a

N

c

L

c

i n

a

A

£ () ω

= J °°dt cos(u>t)C (t) o

A

c

(4)

The l i m i t e d knowledge of ^ ( t ) f o r c e s us to truncate the time i n t e g r a l at t , r a t h e r than at i n f i n i t e time; the c a l c u l a t e d max s p e c t r a l d e n s i t y f u n c t i o n i s shown i n an i n s e t i n each case (the amplitudes are i n a r b i t r a r y u n i t s ) . Negative values of (^(ω) r e s u l t from the f i n i t e upper l i m i t on the i n t e g r a t i o n . On the picosecond time s c a l e considered, no s i g n i f i c a n t damping- i s seen i n the oscillatoryçorrelation f u n c t i o n s d e s c r i b ing the high-frequency (ω £ 300 cm" ) bond-length s t r e t c h i n g and bond-angle bending modes. I t i s c l e a r that f o r these h i g h frequency motions, the behavior of the time c o r r e l a t i o n f u n c t i o n s i s very s i m i l a r i n s o l u t i o n as compared to vacuum, and that i n both environments there i s no evidence f o r a s i g n i f i c a n t zerofrequency component i n the s p e c t r a l d e n s i t y . The l a t t e r i s ex pected i f the c o r r e l a t i o n f u n c t i o n contains a decaying component. Clear evidence of s o l v e n t damping i s found f o r the t o r s i o n a l angle χ ( F i g . 4) f o r which the vacuum motion can be seen to i n v o l v e p r i n c i p a l l y a s i n g l e frequency. During the current simula t i o n , the motion i n v o l v e s only l i b r a t i o n and not o v e r a l l r e o r i e n t a t i o n of the methyl group. By comparison w i t h the r e s u l t i n the absence of s o l v e n t , i t can be seen that the s o l v e n t i s e f f e c t i v e i n damping the o s c i l l a t o r y motion of the methyl group. The be havior i s manifest by the appearance of a low-frequency component i n the s p e c t r a l d e n s i t y , C^(o)). The short-time behavior of the s o l u t i o n c o r r e l a t i o n f u n c t i o n ( t

Water in Polymers - American Chemical Society

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