Optimized Potential Energy Functions in Conformational Analysis of

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Chapter 11

Optimized Potential Energy Functions in Conformational Analysis of Saccharides Kjeld Rasmussen and Jesper Fabricius Chemistry Department A, Technical University of Denmark, DK-2800 Lyngby, Denmark

A short presentation of the Consistent Force Field is given, with emphasis on parametrization and optimization of energy function parameters. For best possible calculation of structure, potential energy functions with parameter values optimized on both structural and other properties must be used. Results from optimization with the Consistent Force Field on alkanes and ethers are applied to glucose, gentiobiose, maltose and cellobiose. Comparison is made with earlier and with parallel work. The meaning and use of conformational maps is discussed shortly. T h i s paper presents a few examples o f a p p l i c a t i o n s o f the program package c a l l e d the C o n s i s t e n t Force F i e l d (CFF). The program has been e x t e n s i v e l y d e s c r i b e d i n the l i t e r a t u r e ( 1 - 2 ) , as has the s t r a t e g y o f i t s use ( 3 4 ) , but a short overview may be p e r t i n e n t here. The CFF system The concepts. A l l i n t e r a t o m i c i n t e r a c t i o n s a r e modeled with a s e t o f mathematical f u n c t i o n s which, when summed over a l l i n t e r a c t i o n s , g i v e s the p o t e n t i a l energy o f a molecule. The p o t e n t i a l energy f u n c t i o n s , t h e PEFs, c o n t a i n a d j u s t a b l e parameters which, f o r a s t a r t , a r e taken from s i m i l a r work o r a r e merely guessed. 0097-6156/90/0430-0177$06.00A) © 1990 American Chemical Society

In Computer Modeling of Carbohydrate Molecules; French, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

178

COMPUTER MODELING OF CARBOHYDRATE MOLECULES


The way t o s t a r t a CFF p a r a m e t r i z a t i o n i s : S e l e c t a s e t o f PEFs, with a s s o c i a t e d parameters. Choose a s e t of molecules, c l o s e l y r e l a t e d t o t h e problem i n hand ( f o r carbohydrates: alkanes, c y c l o a l k a n e s , ethers, a l c o hols) ; t h e i r s t r u c t u r e s should be determined and t h e i r v i b r a t i o n a l s p e c t r a assigned t o a reasonable p r e c i s i o n . Put i n t h e i r s t r u c t u r e s by s p e c i f y i n g atomic c o o r d i n a t e s ; they need not be accurate. The methods. As shown i n Figure 1 , t h e p o t e n t i a l energy of each molecule i s minimized, g i v i n g t h e e q u i l i b r i u m conformations p e r t a i n i n g t o t h e chosen energy f u n c t i o n s with t h e a s s o c i a t e d i n i t i a l parameters. In p r a c t i c e , a l l energy g r a d i e n t s should be zero t o a h i g h p r e c i s i o n . The r e s u l t i s t h e s e t o f conformations a t e q u i l i b r i u m , and one can now c a l c u l a t e the second order d e r i v a t i v e s or f o r c e constants, i n d i v i d u a l values f o r each p a i r o f c o o r d i n a t e s , which w i l l g i v e t h e v i b r a t i o n a l s p e c t r a i n the form o f normal frequencies and normal c o o r d i n a t e s . From these s t a t i c and dynamic p r o p e r t i e s , thermo dynamic f u n c t i o n s and other p r o p e r t i e s may be c a l c u l a t e d . A l l c a l c u l a t e d values o f s t r u c t u r a l , v i b r a t i o n a l and other p r o p e r t i e s may then be compared with t h e c o r responding observed v a l u e s . The reason f o r the c l a i m t o c o n s i s t e n c y i s t h a t measured values o f observables such as bond lengths, angles, t o r s i o n s , frequencies, d i p o l e moments and, f o r c r y s t a l s , u n i t c e l l dimensions, a r e put i n . The program w i l l compare them with t h e corresponding c a l c u l a t e d v a l u e s , and w i l l optimize the energy f u n c t i o n parameters so t h a t , on t h e next run-through, a b e t t e r r e p r o d u c t i o n of t h e measured values i s obtained. When one i s s a t i s f i e d t o some c r i t e r i o n t h a t the model cannot do b e t t e r , one has a c o n s i s t e n t s e t o f parameters f o r t h e chosen s e t o f energy f u n c t i o n s ; see F i g u r e 1 . I t i s possible t o optimize on molecular s t r u c t u r e i n t h e gaseous phase and i n c r y s t a l s , and on molecular v i b r a t i o n a l frequencies and d i p o l e moments, i n one and t h e same calculation. Developing one's own PEFs i s much more time-consu ming than a p p l y i n g them, and some problems a r i s e . Here j u s t two p o i n t s a r e t o be emphasized. Molecular S t r u c t u r e . One i s the question o f which ex p e r i m e n t a l l y determined type o f molecular s t r u c t u r e t o use, as many s t r u c t u r e types a r e a v a i l a b l e i n t h e l i terature . They a r e d e r i v e d from x-ray and neutron d i f f r a c t i o n of c r y s t a l s , and from e l e c t r o n d i f f r a c t i o n and s p e c t r o s c o p i c measurements with microwave, i n f r a r e d and Raman techniques on t h e gaseous phase. F o r o p t i m i z a t i o n o f PEFs on small molecules, gas-phase s t r u c t u r e s a r e used. They a r e r a t h e r numerous, and they a r e a l l c a l c u l a t e d


Optimized Potential Energy Functions

RASMUSSEN AND FABRICIUS

TRIAL GEOMETRIES OF MANY MOLECULES


INITIAL PARAMETER SET

MEASURED OBSERVABLES

ENERGY

EQUILIBRIUM

MINIMISATION

CONFORMATIONS

VIBRATIONAL

NORMAL

ANALYSIS

VIBRATIONS

STi M I S T I C A L

THERMODYNAMIC

THER MCOYNAMCS

FUNCTIONS

OPTIMISATION

IMPROVED PARAMETER SET LA CONSISTENT SET

F i g u r e 1. The CFF C y c l e (Reproduced with p e r m i s s i o n from Ref. 4. Copyright 1989 Kluwer Academic P u b l i s h e r s . )


180


from molecular parameters f i t t e d t o reproduce experimen t a l data. Thorough p r e s e n t a t i o n s are not easy t o f i n d (5-6); a s h o r t summary i s given i n a monograph on the C o n s i s t e n t Force F i e l d (2.) . In p r i n c i p l e , the CFF should aim a t reproducing e q u i l i b r i u m s t r u c t u r e s r , but those are known only f o r a few small compounds. The r s t r u c t u r e s are very o f t e n presented, but they are temperature-dependent, being the thermal average values of the i n t e r - n u c l e a r d i s t a n c e s , and should t h e r e f o r e not be used. We use the r ° or the r s t r u c t u r e s which i n p r i n c i p l e are i d e n tical. They are d e r i v e d from s p e c t r o s c o p i c and e l e c t r o n d i f f r a c t i o n measurements and represent the d i s t a n c e s between average n u c l e a r p o s i t i o n s i n the v i b r a t i o n a l ground s t a t e at 0 Κ (5). They are temperature-indepen dent, and they are f a i r l y easy t o c a l c u l a t e from the most f r e q u e n t l y p u b l i s h e d s t r u c t u r e s r and r . Accor d i n g t o s t r u c t u r a l chemists (K. Kuchitsu, personal com munication) the r ° or r i s the molecular s t r u c t u r e which most c l o s e l y resemlbles low-temperature neutrond i f f r a c t i o n r e s u l t s . Therefore i t i s w e l l s u i t e d f o r d e r i v i n g a s e t of parameters t h a t should be r e l i a b l e f o r l a r g e r molecules whose s t r u c t u r e s , i f they are known, stem from low-temperature X-ray or p r e f e r a b l y neutron diffraction. 9


a

2

fl

a

P o t e n t i a l Energy Functions and Parameters. The second p o i n t i s the importance of non-bonded i n t e r a c t i o n s . The program was developed t o optimize a l s o on u n i t c e l l dimensions i n a d d i t i o n t o the usual conformational pro p e r t i e s , because t h i s g i v e s the p o s s i b i l i t y of o p t i m i z i n g on p r o p e r t i e s t h a t are very s e n s i t i v e t o non-bonded interactions. By f a r the most d i f f i c u l t i n t e r a c t i o n s t o model are the non-bonded, because of n e a r - c a n c e l l a t i o n of s t r o n g l y distance-dependent f o r c e s of opposite s i g n s . Only pro per handling of non-bonded i n t e r a c t i o n s w i l l g i v e sen s i b l e r e s u l t s i n the c a l c u l a t i o n of s t r u c t u r e s of mole c u l e s as f l e x i b l e as saccharides. Yet very few obser v a b l e s of small molecules depend s t r o n g l y on non-bonded i n t e r a c t i o n s (the -C-C- t o r s i o n i n η-butane i s an ex ception) , wherefore o p t i m i z a t i o n on c r y s t a l s i s needed as argued above. The terms used i n CFF are very simple; simpler than most other f u n c t i o n a l forms used by s i m i l a r programs; see F i g u r e 2. As we d e a l with the modeling of chemical systems, we d i v i d e the most important i n t e r a t o m i c i n t e r a c t i o n s i n t o two p a r t s : bonded and non-bonded. The bonded i n t e r a c t i o n s are almost always modeled with harmonic (parabolic) f u n c t i o n s which p r a c t i c e i s acceptable c l o s e t o e q u i l i b r i u m . For non-bonded i n t e r a c t i o n s , the van der Waals p a r t i s modeled with i n v e r s e power terms i n the i n t e r a t o m i c d i s t a n c e s , 12, or occa-



11.



s i o n a l l y 9, f o r the overlap r e p u l s i o n , 6 f o r the London attraction. In a d d i t i o n , e l e c t r o s t a t i c terms are essen t i a l when e f f e c t s of p o l a r groups are t o be considered. In these Coulomb terms, atomic monopoles are used; t h i s concept has h i t h e r t o proved t o be acceptable. Atomic charges are u s u a l l y taken from M u l l i k e n p o p u l a t i o n ana l y s i s of ab i n i t i o c a l c u l a t i o n s with b a s i s s e t s p r e f e r a b l y l a r g e r than minimal. They are reproduced i n CFF by a r a t h e r i n t r i c a t e algorithm from one charge parame t e r per atom type. In the energy c a l c u l a t i o n s a d i e l e c t r i c constant i s used. The b u i l t - i n v a l u e i s 2.0, but i t can be changed i n the input. The c h o i c e i s p u r e l y pragmatic: i f i t i s 1, s i m u l a t i n g vacuum, the e l e c t r o s t a t i c energy dominates e n t i r e l y ; i f i t i s 10, i t s e f f e c t can h a r d l y be seen. Values of 2.0 t o 3.5 are most p r a c t i c a l . (The value 1 i s used i n c a l c u l a t i o n s on purely i o n i c inorganic crystals.) Because of the simple f u n c t i o n s t h i s model i s too crude t o be of p r a c t i c a l use, and we must add a number of secondary terms; they depend e x p l i c i t l y on valence, t o r s i o n a l , and out-of-plane angles where a p p r o p r i a t e . F i g u r e 2 shows the terms r e l e v a n t t o the work r e p o r t e d here. The d e s i g n a t i o n s primary and secondary are con c e p t u a l l y s i g n i f i c a n t : the secondary terms are necessary because the present formulations of the primary terms i s not s u f f i c e n t l y accurate. In cases where c a r b o x y l , amido, imino and other groups occur, out-of-plane angles are u s u a l l y i n c l u d e d . Please note the meaning of the word "parameter". In the CFF context, K i s not a f o r c e constant of any bond i n any molecule, and 9 i s not the e q u i l i b r i u m v a l u e of any valence angle. They are energy f u n c t i o n parameters with u n i t s of f o r c e constant and angle. In the a c t u a l case, kJmol" À" and rad. b

Q

1

2

Saccharides The two main f i e l d s of a p p l i c a t i o n i n the CFF group i n Lyngby are saccharides and c o o r d i n a t i o n compounds. Here we s h a l l mention only the saccharide work. The f i r s t attempts t o c a l c u l a t e saccharide e q u i l i b r i u m s t r u c t u r e s were made by use of two PEFs developed by t r i a l - a n d e r r o r , PEF300 (7^8) without and PEF400 (9-10) with char ges. In s p i t e of t h i s , good r e s u l t s were obtained, both f o r s t r u c t u r e s of glucose and f o r the thermodynamic e q u i l i b r i u m between the anomers. In the present work we introduce two PEFs c o n t a i n i n g parameters optimized on s t r a i g h t - c h a i n and c y c l i c e t h e r s , some of them c o n t a i n i n g anomeric carbon atoms. In these o p t i m i z a t i o n s an anomeric carbon atom was g i v e n i t s own symbol and parameter a t t r i b u t e s . The f u n c t i o n s are named PEFAC1 and PEFAC2; the l a t t e r has Coulomb terms i n c l u d e d . T r i a l - a n d - e r r o r parameters from PEF400


181

182


M , = £ K ( b - b

e

)

s

bonds

primary

two-body interatomic

terms


interactions non-bonded interactions

angles secondary terms

correction

V.^Xd+coskO)

terms

single bonds

Vt o t a l = Vb ^+ Vn - b + vuv θ 0 Y

Y

Y

ν

τ

v

F i g u r e 2. P o t e n t i a l Energy Functions (Reproduced with permission from Ref. 4. Copyright 1989 Kluwer Academic P u b l i s h e r s . ) f o r hydroxo groups were appended f o r t h i s a p p l i c a t i o n . O p t i m i z a t i o n on a l c o h o l s was not done, as a v a i l a b l e experimental data are i n s u f f i c i e n t f o r our purpose. The work on PEFAC1 and FEFAC2 i s not y e t p u b l i s h e d . Table I shows t h e parameter v a l u e s i n PEFAC1 and PEFAC2. Glucose. The improvement i n the c a l c u l a t e d s t r u c t u r e which i s obtained by use o f t h e new PEFs i s i n d i c a t e d i n Table I I which i n c l u d e s comparison w i t h p r e v i o u s r e s u l t s (8,11). The measured v a l u e s p e r t a i n t o n e u t r o n - d i f f r a c t i o n data f o r α-glucose (.12) and x-ray d i f f r a c t i o n data f o r β-glucose (13). The separate treatment o f anomer carbon i m p l i e s t h a t the l a r g e s t d e v i a t i o n i n bond l e n g t h i s no longer found f o r t h e anomeric C-0. Two p a r t i c u l a r d e t a i l s a r e t h e oxygen r i n g angle and t h e angle a t the anomeric carbon. The comparatively open angles a r e b e t t e r reproduced than t h e more c l o s e d . At present, no e x p l a n a t i o n can be g i v e n . On the other hand, the e q u i l i b r i u m r a t i o o f the anomers has changed t o 0.48:0.52 r a t h e r than the value o f 0.36:0.64 which




Table I . Two p a r t i a l l y Optimized PEFs. U n i t s a r e chosen so as t o g i v e energy i n k c a l mol" 1

PEFAC1


K

O-H C-C K-C C-H K-H

o-c O-K

K-O-H C-O-H C-C-C K-C-C C-K-C 0-C-C O-C-K O-K-C C-C-H K-C-H C-K-H

c-o-c K-O-C K-O-K O-K-O O-C-H O-K-H H-C-H H-K-H

b

1070. 563.077 563.077 670.000 670.000 863.000 863.000 Κ

PEFAC2

b

0.955 1.5157 1.4824 1.0990 1.0990 1.4007 1.3945

0

80. 80. 142.447 142.447 142.447 143.837 143.837 143.900 93.500 93.500 93.500 143.353 143.897 143.900 143.882 93.498 93.498 74.800 74.800

0

θ

Ο

1.80 1.80 109. 109. 109. 109. 109. 109. 109. 109. 109. 1.8418 1.8816 1.8463 109. 109. 109. 109. 109. η

H-C-C-•Η H-C-K-•Η H-O-C-•Η H-O-K-•Η

1.2809 1.2809 2.7849 1.6491 Α

C— Κ— 0— Η—

559.123 559.128 292.392 160.137

K

1070. 563.077 563.077 670.000 670.000 862.231 863.000 Κ

Β 18.865 18.818 12.599 7.746

0

80. 80. 142.447 142.447 142.447 143.838 143.838 143.900 93.500 93.500 93.500 143.336 143.897 143.900 143.882 92.477 92.477 74.800 74.800 κ

3. 3. 3. 3.

b

*

1.2809 1.2809 2.7575 1.6537 Α 559.123 559.128 292.394 160.139

b

o

0.955 1.5134 1.4853 1.0866 1.0866 1.3990 1.3948 θ

Ο

1.80 1.80 109. 109. 109. 109. 109. 109. 109. 109. 109. 1.8462 1.8807 1.8458 109. 109. 109. 109. 109. η 3. 3. 3. 3. Β 18.856 18.815 12.580 7.705

e CΚ. °· Η.

0.00001 -0.002 -0.108 0.140



184


agrees with the experimental value found i n aqueous solution. The rotamer r a t i o f o r the hydroxymethyl group i s s t i l l the same as c a l c u l a t e d before: g t t o gg = 0.78:0.22, whereas a compilation o f c r y s t a l s t r u c t u r e s gave a r a t i o o f 0.40:0.60 (14). T h i s discrepancy may be due t o the method: c a l c u l a t i o n on an i s o l a t e d molecule can not account f o r 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 i n c r y s t a l s . The i n c l u s i o n o f e l e c t r o s t a t i c terms i n PEF400 g i v e s an only m a r g i n a l l y d i f f e r e n t s i t u a t i o n , with no d i f f e r e n c e s w i t h i n the p r e c i s i o n o f the data given i n Table I I . Therefore only one column ( f o r PEFAC2) i s listed. Somewhat b e t t e r s t r u c t u r a l d e t a i l s were o b t a i ned f o r the monosaccharides by the use o f optimized PEFs such as PEFAC2, but the thermodynamic e q u i l i b r i a became less well f i t t e d . Disaccharides. The most f l e x i b l e d i s a c c h a r i d e c a l c u l a t e d before i s g e n t i o b i o s e (3., 9). The conformation as found i n the c r y s t a l (15) was minimized i n the new func t i o n PEFAC1 without e l e c t r o s t a t i c terms; the d i f f e r e n c e i n geometry i s n o t i c e a b l e e s s e n t i a l l y i n one t o r s i o n . I f charges are i n c l u d e d , PEFAC2, almost the same p i c t u r e obtains. Table I I . Glucose i n an Optimized P o t e n t i a l Energy Function PEFAC2 and Comparisons with Non-Optimized Functions (7-8, 10) α

C505C1 05C101

β

cale

meas

calc

meas

113.8 111.4

113.8 111.6

113. 8 109. 0

112.7 107.0

Deviations meas(12 -13) PEF300 Bonds/À max -0.039 rms 0.014 Angles/ max -4.5 rms 1.8 Torsions/ endocyclic max 4.5 rms 3.2 hybrid max 4.5 rms 3.2

-

calc

PEF400

PEFAC2

-0.034 0.014

0.029 0.004

-4.8 1.8

4.0 0.1

5.3 3.3

3.0 0.1

4.4 2.8

2.4 0.8


11.



Table III. Gentiobiose Conformations: Potential Energy Functions

PEF

PEF300

47. •177. •178. CO 114. C106C6 HI H 2.28 HIH 3.50

Φ


V

1

R

s

Crystal Conformation Minimized in Four

PEF400

PEFAC1

PEFAC2 C r y s t a l

85. -155. 152. 115. 2.66 3.56

60. -178. 173. 114. 2.42 3.07

63. -176. 173. 114. 2.43 3.10

63. -156. -178. 113. 2.40 3.12

Some conformational d e t a i l s are compared i n Table III. The very open COC angle i s w e l l reproduced, and so are the t h r e e t o r s i o n s along the g l y c o s i d i c l i n k a g e . Two Η Η d i s t a n c e s are of s p e c i a l i n t e r e s t because they can be estimated by NMR techniques and can t h e r e f o r e g i v e a c l u e t o the s o l u t i o n conformation; they are r e produced q u i t e w e l l . From a s i m i l a r comparison with non-optimized func t i o n s , PEF300 (8) without and PEF400 (10) with charges, we see t h a t the conformational d e t a i l s are not n e a r l y so w e l l reproduced. In the case of g e n t i o b i o s e the o p t i m i zed PEF has t h e r e f o r e made a r e a l improvement. F i r s t and foremost i t i s the o p t i m i z a t i o n of the non-bonded i n t e r a c t i o n s which has brought about the improvement. The c o n s i s t e n c y of the approach i s empha s i z e d by the f a c t t h a t the subset of parameters f o r C and Η were optimized on data f o r small alkanes, c y c l o a l kanes and alkane c r y s t a l s , and was used unchanged d u r i n g o p t i m i z a t i o n on e t h e r s ; a l c o h o l s were not y e t i n c l u d e d . These r e s u l t s prompted a reexamination of maltose (16) and c e l l o b i o s e (Γ7). French has r e c e n t l y presented comparisons of r i g i d and r e l a x e d conformational maps f o r c e l l o b i o s e and mal t o s e obtained with the MMP2(1985), which i n c l u d e s ano meric e f f e c t s . The f u l l y r e l a x e d maps show i n t e r e s t i n g details. The r e s u l t s f o r maltose and c e l l o b i o s e are shown i n F i g u r e s 3 and 4. They were obtained with the non-op t i m i z e d parameter s e t s PEF300 without charges (16-17), MMP2 which uses d i p o l e - d i p o l e i n t e r a c t i o n s i n s t e a d o f e x p l i c i t charges (18-19), PEF400 with charges (2,11), and the optimized s e t PEFAC1 without charges ( t h i s work). For maltose, F i g u r e 3 shows t h a t t h e r e are no s i g n i f i c a n t d i f f e r e n c e s between the geometric r e s u l t s found with the f o u r r a t h e r d i f f e r e n t p o t e n t i a l energy func t i o n s . The only discrepancy i s the absence of minimum 1 i n the map of French (18). Most c r y s t a l s t r u c t u r e data f a l l w i t h i n the v a l l e y j o i n i n g the t h r e e upper p o i n t s .


185

186

COMPUTER MODELING OF CARBOHYDRATE MOLECULES Τ

I

I

I

I

ι

ι

AO

80

Γ

80-

A0-

Ψ/

Ο χα 3

+

0 -

0

,

-A0-

°% 2


"Ô 1

-80-

-120-

-160°

160-

A

120-160

-120 - 8 0

-40

0

1 2 0 160

φ/ο

F i g u r e 3. Conformational Map o f Maltose. + PEF300, x PEF400, o PEFAC1, D MMP2 T h i s v a l l e y i s i n v e s t i g a t e d i n some d e t a i l i n the o r i g i n a l paper on maltose (10) where a l s o r e f e r e n c e s t o ex perimental work can be found. Some d e t a i l s are g i v e n i n Table IV; i t i s noteworthy t h a t i n PEFAC1 one conformer i s dominant, which i s i n agreement with the r e s u l t s o f PEF400 (3,10). Table IV. Conformer

1

Κ ν/ 01·0404/° HI Η4/Α AG/kJmol' 1

1

n

i

Maltose Conformers

-67.2 -56.2 117.5 3.149 0.000 0.924

2 -19.5 -35.9 116.3 2.366 6.336 0.072

i n PEFAC1 3 2.5 37.1 118.1 2.322 16.939 0.001

4 -40.0 173.3 118.6 3.642 13.714 0.003

For c e l l o b i o s e , the s i t u a t i o n i s s l i g h t l y d i f f e r e n t , as seen i n Table V and F i g u r e 4. The most obvious d i f f e r e n c e i s t h a t only f i v e minima are found with MMP2 and PEFAC1. I n PEFAC1, e s s e n t i a l l y two conformers are populated, and almost e q u a l l y so; they span the d i f f r a c t i o n r e s u l t s , as summarized by French (19).




1

1


1

•

• 1

1

60

A

Ψ/

0

C

0

w + Β

-60

-

-120 Ε

D

•

-

-180

-120

-60

1 0

60

ι 120

ι 180

ι -120

F i g u r e 4. Conformational Map of C e l l o b i o s e . + PEF300, x PEF400, o PEFAC1, D MMP2


188


Table V.

Conformer

2

3

4

φ 57.6 163.9 70.1 28.2 y 4.6 4.4 -165.0 -57.8 C1O4C4 114.0 115.5 116.5 115.5 HI H4 2.451 3.543 3.575 2.380 AG 0.359 13.845 3.994 0.000 nj 0.418 0.002 0.097 0.483 1


1

C e l l o b i o s e conformers i n PEFAC1. U n i t s as i n Table IV 5

6

merges 177.5 i n t o -150.7 conf. 121.2 3 3.924 54.788 0.000

Conformational Maps and Surfaces. The conformational maps i n F i g u r e s 3 and 4 are very small s e c t i o n s of the t r u e conformational s u r f a c e s of 136 dimensions (3 coor d i n a t e s per atom p l u s one f o r the energy). The s p e c i a l c h o i c e of φ and γ as the coordinates of the s e c t i o n i s the t r a d i t i o n a l one, which i s s e n s i b l e because the gross conformational f e a t u r e s are d e s c r i b e d w e l l by j u s t those two. One should j u s t not f o r g e t t h a t , on moving from one p o i n t t o another, many coordinates may change appre c i a b l e , w i t h i n a small energy i n t e r v a l . In consequence, a p o i n t i n the two-dimensional map r e p r e s e n t s an e n t i r e " f a m i l y " of p o i n t s i n 135—dimensional space. Only minima are shown, because they always i n t e r ested us most, as they correspond t o s t r u c t u r e s which i n p r i n c i p l e can e x i s t i n s o l u t i o n and i n c r y s t a l s . As argued above, a p o i n t i s not unique; a change i n , say, a CCOH t o r s i o n might cause a s l i g h t change i n φ or ψ or both. Therefore i t would be c o r r e c t t o say t h a t a m i n i mum i n conformational space i s represented by a small but u n s p e c i f i e d area around a p o i n t i n the conformatio n a l map, which was e a r l i e r ( 2 ) termed a manifold. The r e f o r e , a d i f f e r e n c e i n (φ,γ) of (10,10) i s r e a l l y no difference at a l l . Many people c a l c u l a t e v a r i o u s v a r i a n t s of " f u l l y r e l a x e d " conformational s u r f a c e s . A f u l l y r e l a x e d s u r face i s j u s t a s e t of p o i n t s , namely the conformations of minimum energy. One might ask the q u e s t i o n s : what i s the s i g n i f i c a n c e of those contour p l o t s ? -and what i s t h e i r use? I f the answer i s t h a t they may guide us i n modeling intermediate conformations which might be taken up i n c r y s t a l s , i n aqueous s o l u t i o n , o r near the a c t i v e s i t e of en enzyme, a more r a t i o n a l use of computer f a c i l i t i e s would probably be t o c h a r t v a l l e y s of the c o n f o r mational map (16-17). A b e t t e r approach than t h i s , though more c o s t l y , i s t o l e t the molecule deform along low-frequency normal c o o r d i n a t e s and f o l l o w the c o n f o r mational evolvement i n time with Molecular Dynamics. A procedure f o r s e l e c t i n g the normal c o o r d i n a t e s most


11. RASMUSSEN AND FABRICIUS


r e l e v a n t t o conformational interchange was worked out f o r t h e case o f a c o o r d i n a t i o n compound (20).


Conclusion We can s t a t e , i n c o n c l u s i o n , t h a t o p t i m i z a t i o n o f t h e p o t e n t i a l energy f u n c t i o n parameters on experimental data o f small model compounds has l e d t o a parameter s e t t h a t g i v e s an o v e r a l l improvement o f t h e accuracy o f p o s t d i c t i o n and, by i m p l i c a t i o n , o f t h e v a l i d i t y o f p r e d i c t i o n . The improvement i s most marked i n t h e most f l e x i b l e substance. The phrase p o s t d i c t i o n i s used t o emphasize t h a t p r o p e r t i e s a r e c a l c u l a t e d which were not used i n t h e development o f the PEF and t h a t we a r e not d e a l i n g with j u s t reproduction. Postdiction i s therefore "predict i o n " o f known p r o p e r t i e s while p r e d i c t i o n d e a l s with so f a r unknown p r o p e r t i e s . T e c h n i c a l Matters The CFF program i s a v a i l a b l e , f r e e o f charge, and can most e a s i l y come on EARN o r BITNET, from KEAKJR a t VM.UNI-C.DK o r a t NEUVM1 o r from UNIJF a t VM.UNI-C.DK o r a t NEUVM1. No r e s p o n s i b i l i t y f o r problem s o l v i n g and t e c h n i c a l updatings can be accepted; the manpower a v a i l a b l e f o r b i d s s e r v i c e o f any k i n d . At the time o f w r i t i n g , d i s t r i b u t i o n through a software house i s under c o n s i deration. The CFF i s known t o run o r have run on CRAY XMP, Amdahl VP1100, many IBMs, Siemens, UNISYS, CDC, many VAXes, Ardent T i t a n . The program i s a patchwork p r e pared over 20 years, w r i t t e n i n IBM FORTRAN IV and l a t e r cleaned t o conform t o FORTRAN 77; new r o u t i n e s a r e w r i t t e n i n FORTRAN 77. Development i s now done on an Amdahl VP1100, and v e c t o r i z a t i o n i s used where a p p r o p r i a t e .

Literature Cited 1.

2.

3. 4.

Niketić, S. R.; Rasmussen, K. The Consistent Force Field: A Documentation; Lecture Notes in Chemistry, Vol. 3; Springer-Verlag: Berlin, Heidelberg, New York, 1977. Rasmussen, K. Potential Energy Functions in Conformational Analysis; Lecture Notes in Chemi stry, Vol. 37; Springer-Verlag: Berlin, Heidel berg, New York, Tokyo, 1985. Rasmussen, K. In Molecular Structure and Dynamics; Balaban, M., Ed.; Balaban: Jerusalem, 1980; pp 171-210. Rasmussen, K. In Strategies for Computer Chemi stry; Tosi, C . , Ed.; Kluwer: Dordrecht, 1989; pp 13-29.


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Kuchitsu, K.; In Phys, Chem. Ser. 1. Vol. 2, MTP Int. Rev. Sci; Butterworths: 1972; pp 203-239. 6. Kuchitsu, K.; Cyvin, S. J. In Molecular Struc tures and Vibrations; Cyvin, S. J., Ed.; Elsevi er: Amsterdam, 1972; Chapter 12; pp 183-211. 7. Kildeby, K.; Melberg, S.; Rasmussen, K. Acta Chem. Scand. 1977, A31. 1-13. 8. Melberg, S.; Rasmussen, K. Acta Chem. Scand. 1978, A32, 187-188. 9. Melberg, S.; Rasmussen, K. Carbohydr. Res. 1980, 78, 215-224. 10. Melberg, S.; Rasmussen, K. J . Mol. Struct. 1979, 57, 215-239. 11. Rasmussen, K. Acta Chem. Scand. 1982, A36. 323327. 12. Brown, G. M.; Levy, H. A. Science 1965, 147, 1038-1039. 13. Chu, S. S. C.; Jeffrey, G. A. Acta Cryst. 1968, B24. 830-838. 14. Lemieux, R. V.; Brewer, J . T. Adv. Chem. Ser. 1973, 117, 121-146. 15. Rohrer, D. C.; Sarko, Α.; Bluhm, T. L.; Lee, Y. N. Acta Cryst. 1980, B36, 650-654. 16. Melberg, S.; Rasmussen, K. Carbohydr. Res. 1979, 69, 27-38. 17. Melberg, S.; Rasmussen, K. Carbohydr. Res. 1979, 71, 25-34. 18. French, A. D. Carbohydr. Res. 1989, 188, 206-211. 19. French, A. D. In Cellulose and Wood - Chemistry and Technology; Schuerch, C., Ed.; Wiley: New York 1989; pp 103-118. 20. Niketic, S. R.; Rasmussen, K. Acta Chem. Scand. 1981, A35, 213-218. RECEIVED March 21, 1990


Optimized Potential Energy Functions in Conformational Analysis of

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