Experimental and Theoretical Bases for Accurate Modeling - ACS

Chapter 2

Experimental and Theoretical Bases for Accurate Modeling An Experimentalist Looks at Modeling

Downloaded by EAST CAROLINA UNIV on March 12, 2017 | http://pubs.acs.org Publication Date: July 6, 1990 | doi: 10.1021/bk-1990-0430.ch002

G. A. Jeffrey Department of Crystallography, University of Pittsburgh, Pittsburgh, PA 15260

Canned Science has the great advantage that it allows the experimentalist to carry out theoretical calculations on molecules of his interest. Conversely the theoretician has a vast resource of structural data available in data bases or can carry out routine crystal structure analyses to test his predictions. Both adventures have pitfalls for the inexperienced. In the molecular modeling of carbohydrates, hydrogen-bonding is particularly troublesome because of the orientational freedom and the donor and acceptor property of the many -OH groups. This leads to a multiplicity of local minima of nearly equal energy. The hydrogen bonding in peptides and proteins, in comparison, is easier to model, since the predominant NH and C=O groups have neither orientational freedom nor donor and acceptor properties. Calculations on isolated molecules tend to emphasize intramolecular hydrogen bonding, which is frequently superceded by intermolecular bonding in solution or in the solid-state. Electrostatic interactions are difficult to parameterize due to polarization effects and the difficulty of defining the point-charge on an atom. To overcome these obstacles to successful modeling of oligo- and polysaccharides, the hydrogen-bond structure under investigation needs to be constrained to take into account existing data on such factors as (1) most probable bond lengths and bond angles, (2) three-center hydrogen bonding (about 25 percent), (3) cooperativity; favoring infinite or finite chains, homodromic loops, ribbons and nets, (4) the "Excluded Region", which places constraints on hydrogen-bond lengths versus hydrogen-bond angles. In the last decade, there has been an explosion of "Canned Science". By Canned Science, I refer to computer programs and data bases which can be bought and used successfully simply by following the "instructions on the can". It is no longer necessary to be a theoretical chemist to carry out computations at any of the three levels of molecular modeling; ab-initio, semi-empirical, or empirical force field (1-3). By the same token, it is not necessary to be a crystallographer to have ready access to 70,000 organic and organo-metallic crystal structures, of which about 2000 are carbohydrates (4). A minimal knowledge of crystallography is required to carry 0097-6156/90/0430-0020$06.00/0 © 1990 American Chemical Society

French and Brady; Computer Modeling of Carbohydrate Molecules ACS Symposium Series; American Chemical Society: Washington, DC, 1990.


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JEFFREY

Experimental & Theoretical Bases for Accurate Modeling

out a single crystal structure analysis using the modern computer-controlled X-ray diffraction equipment, which includes all the software necessary to measure the structure amplitudes, determine the structure phase angles and refine the atomic positional and thermal parameters. The results of advances in computer technology are especially valuable at a time when these aspects of science are becoming increasingly specialized. It provides the researcher with access to generations of expertise in fields other than that of his own. It permits the specialist once again to become a generalist, at least to some degree. The use of this computer software by investigators who are relatively inexperienced in the appropriate theory or experiment has its pitfalls, of course (5). There could be an increase in inaccurate data or misleading predictions. The experimentalist using a theoretical program will, if he is wise, use the general criterion; do the results make chemical sense? If possible, he will first apply the theory to a problem where he already knows the answer before extending it to a related problem where he desires an answer. Similarly, the theoretician will, or should,firsttest the theoretical methods against known answers. Most commonly available programs do, in fact, provide this 'evidence' of credibility. However, unlike experimental structure determination, the theoretical methods do not include an internal means of calculating standard deviations for the bond lengths, valence angles and torsion angles derived. There is no way of assessing their accuracy other than by comparison with experimental data, or with theoretical calculations carried out by other methods. It is not difficult to envision a next level of software that warns the user of possible erorrs, such as inconsistency with existing relevant data, unrealistic energies, unlikely molecular geometry, or unacceptable nonbonding distances. Editors of both experimental and theoretical journals should welcome these programs. They would both improve and make more standard the present refereeing procedures, which make relatively little systematic use of available computer data bases. Once computers leave the bureacracy of computing centers and become laboratory instruments, there is a burst of innovative and adventurous applications. The appearance of the departmental supercomputer, with its dedicated software for molecular modeling by ab-initio or molecular dynamics calculations, at a price comparable to that of the departmental NMR or X-ray diffraction equipment, suggests that this time is imminent. A conservative use of theory by the experimentalist is to correlate his observations and to obtain an insight into the electronic interpretation of what he observes. The early ab-initio calculations on methanediol, methoxymethanol, and dimethoxymethane in connection with the anomeric and exo-anomeric effect is such an example (6-$). In that work, ab-initio calculations were used to correlate a chemical observation, the anomeric effect (2), with the crystallographic observation of a bond-length shortening (10) and a preferred orientation of the glycosidic bond (Π). Another example is the use of a semi-empirical method, PCILO, to examine the cooperativity or non-additivity of cyclic systems of hydrogen bonds which were observed from the X-ray and neutron diffraction structure analyses of the α-cyclodextrin hexahydrate (12). The theoretician uses these programs to predict structure, either of single molecules or of assemblages of molecules, using X-ray or NMR data, when available, to test his predictions (13-15). It has been known for a long time that even the earlier molecular mechanics programs can predict the structures of certain types of molecules with excellent reliability. For the cyclic alkanes, an accuracy comparable to that of the best X-ray crystal structure analysis can be obtained. In fact, the method is more widely applicable since neither compound nor crystals are necessary (16). With monosaccharides, the structures of the relatively rigid pyranoses and methylpyranosides in the crystalline state can be accurately predicted if the hydroxyl groups are oriented as in the crystal, in the directions appropriate to form


21

22

COMPUTER MODELING OF CARBOHYDRATE MOLECULES

intermolecular hydrogen bonds (17). Oligosaccharides present a more difficult problem since some ad-hoc decisions must be made concerning the intramolecular hydrogen bonding. Intra-residue intramolecular bonds have to be prevented, while certain inter-residue intramolecular bonds have to be permitted. For cellobiose, for example, there are two quite different conformations, both with two inter-residue intramolecullar hydrogen bonds, shown in Figure la&b. In the crystal structure (18). only one is observed, as in Figure lc. It is difficult to predict the influence of these bonds on the conformational populations in solution without reliable solvation studies. With three linkage bonds as in the molecular mechanics study of the 1-6 disaccharide gentiobiose (12), the lowest 24 minima within 7 kcal mole which were reported did not include that observed in the crystal structure (2Q). Had the crystal structure conformation been one of the sets of starting parameters, it would have been included, but it is reasonable to assume that the conformational population in solution was also not fully represented. Another example of problems arisingfromhydrogen bonding is provided by an interesting comparison of semi-empirical MNDO, POLO, and molecular mechanics calculations on n-acetyl β-D-glucosamine (21). These three methods gave different energy sequences for the nineteen global minima considered. The lowest from PCILO was one of the highest from MNDO, and the lowest from a molecular mechanics method was not the lowest from the two semi-empirical methods. This could be a consequence of the different treatment of hydrogen bonding by the three methods considered. A previous paper in this symposium suggested that MNDO underestimates hydrogen bond energies, while PCILO may overestimate hydrogen bonding. The crystal structure of n-acetyl β-glucosamine has not been determined since it is the α-epimer that crystallizesfromaqueous solution. The crystal structure of the α-epimer suggests that all the hydrogen bonding will be intermolecular for the β-epimer in the solid state or in solution (22).


-1

Why is Hydrogen Bonding a Difficult Problem in the Molecular Modeling of ÇarbQhydratçs? There are two obvious reasons. One is clearly the high level of hydrogen bond functionality that is present in carbohydrates, with four donors and six potential acceptors per monomer in an oligo- or polysaccharide. Each hydroxyl group can be both a hydrogen bond donor and acceptor, and eachringor glycoside oxygen may or may not be an acceptor. In contrast, the ^NH, -NH2, -SH3 functional groups, common in other biological molecules, are donors but rarely acceptors, while N and 0=C groups can only be acceptors. The second reason is that every - O H — Ο hydrogen bond involves knowledge of the -C-O-H torsion angle. This means that each hydrogen bond introduced into the molecular model involves at least one additional variable parameter and preferably more. The assumptions that hydrogen bonds are linear and that the hydrogen atom will lie on, or even close, to the Ο Ο line have been shown to be invalid by the many neutron diffraction studies of the hydrogen bonding in crystals (23). In the carbohydrate, nucleoside and nucleotide crystal structures, for example, about 25 percent of the hydrogen bonds are three-centered (bifurcated donor) bonds (24.25). s

For the ^NH N bonds that predominate in peptides, proteins and nucleic acids, there are no additional parameters added for each hydrogen bond, since the position of the hydrogen atom is defined by the adjacent non-hydrogen atoms. s


JEFFREY

Experimental & Theoretical Basesfor Accurate Modeling


2.

Figure 1. Conformations of cellobiose with inter-residue intramolecular hydrogen bonding. (a,b) conformations with two inter-residue bonds, (c) hydrogen bonding observed in the crystal structure (18).


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24



How is Hydrogen Bonding Treated in Modeling Methods? Ab-initio calculations include all hydrogen bond interactions. When these are isolated molecule calculations, the models are those with maximum intramolecular hydrogen bond energy. Even if the O-H—Ο angles are close to 90° and the Η — Ο distances are greater than 3.0 Â, there will still be an attractive coulombic term which favors an intramolecular hydrogen bond orientation of the OH groups. Such a calculation for glucose, for example, would orient the hydroxyls so as to form a cooperativeringof intramolecular hydrogen bonds around the perifery of the molecule (26). Such weak bonds are pre-empted by the geometrically more favorable intermolecular bonds in the crystal and to solvent molecules in solution. This makes ab-initio calculations inappropriate for most biological molecules which exist in environments where the hydrogen bonding is predominantly intermolecular. This can be avoided by fixing the orientation of the hydroxyl groups so that they cannot form intramolecular hydrogen bonds. Then the calculation ceases to be ab-initio. It contains some experimental content. The primary use of ab-initio calculations for such molecules is likely to be to provide source data for parameterizing molecular mechanics and dynamics programs. Hydrogen bonding is included in empirical force field calculations in two ways. In the M M series (27), bond dipoles are placed at the centers of the bonds and the Jeans equation is used: Σ Σ C μίμ^οοβΧ^ - 3cosocicosaj)D.rij . 3

This may be supplemented with a Morse potential (28) which involves more parameters and a knowledge of the equilibrium hydrogen bond lengths. It is not surprising therefore that force-fields directed at hydrogen-bonded molecules have favored the simpler charge-charge formula, qiq/eryorqiq/erij 2

if the dielectric constant is made distance-dependent. This term is used alone (29) or with a fine-tuning component of the form Σ Arij- + Bry- , 12

10

H-bond

which is parameterized for different groups of hydrogen bonds, as in AMBER QÛ, 21). The early version of the molecular dynamics program CHARMM (32) also included some rather complex X - H A angle-dependent terms, although there seems to be some uncertainty whether they are really advantageous. The coulombic term is an atom-pair atomic point charge model for the interaction between the electrostatic potentials within a molecule or between molecules. Considerable attention has been paid to deriving "appropriate" values for the qiqi parameters (33.34). One of the more sophisticated ways is to use ab-initio methods to calculate the molecular electrostatic potential and then least squares fit the atomic point charges to these potentials (35). This method is only applicable to relatively simple molecules, and the point charges derived are quite basis-set dependent, varying as much as 30% between STO-3G and 6-31G*. Unfortunately, atomic point charges are not well-defined physical properties. In the multipole method of experimental charge density analysis they correspond to the monopole populations (26). These populations depend upon the assumption made concerning the radial distribution functions. Within limits any point charge values can be derived by dividing up the electron density distribution in different ways between the atoms. Until there is a better way of distributing electron density between atoms in molecules


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JEFFREY


(37), the coulombic qiqj values, like the ε dielectric constants, must be regarded as adjustable parameters, chosen so as to give the best fit between theory and experiment. Even then, multi-atom and cooperative effects affect hydrogen bond energies by the order of ten percent and need to be included in the calculations (38). This is particularly true for carbohydrates in the solid state and in solution, where finite and infinite chains and three-dimensional nets of hydrogen bonds are energetically favored. Both the dipole-dipole and charge-charge models alternate slowly with distance, as compared with the Lennard-Jones non-bonding interactions. They therefore tend to over-emphasize intramolecular bonding in the same way as abinitio calculations, when applied to isolated molecules.


Some Studies of Molecular Modeling for Hydration Biochemistry and chemistry takes place mostly in solution or in the presence of large quantities of solvent, as in enzymes. As the necessary super-computing becomes available, molecular dynamics must surely be the method of choice for modeling structure and for interpreting biological interactions. Several attempts have been made to test the capability of molecular dynamics to predict the known water structure in crystalline hydrates. In one of these, three amino acid hydrates were used; serine monohydrate, arginine dihydrate and homoproline monohydrate. The first two analyses were by neutron diffraction, and in the latter X-ray analysis was chosen because there were four molecules and four waters in the asymmetric unit. The results were partially successful, but the final comments of the authors were "this may imply that methods used currently to extract potential function parameters are insufficient to allow us to handle the molecular-level subtleties that are found in aqueous solutions" (39). In an oligonucleotide-drug hydrate complex, the appearance of a clathrate hydrate-like water structure prompted a molecular dynamics simulation (40). Again the results were only partially successful, prompting the statement, "The predictive value of simulation for use in analysis and interpretation of crystal hydrates remains to be established." However, recent molecular dynamics calculations have been more successful in simulating the water structure in the host lattice of a-cyclodextrin and β-cyclodextrin in the crystal structures of these hydrates (41.42). Some Suggestions for Modeling Hydrogen Bonding in Carbohydrates It seems clear that correct parameterization of the electrostatic terms is going to be the most difficult aspect of developing reliable predictions for carbohydrates in the solid state or in solution (43). What appears to be necessary to reduce the multiplicity of global minima are more constraints on the structure of the hydrogen bonding, similar to the constraints on bond lengths and valence angles used by protein crystallographers when refining their X-ray crystal structure analyses. Examples of such constraints are: (1) Use Η-bond length statistics to define the most probable bond lengths and angles as are used in parameterizing the covalent bond structure. A vast source of information on this subject is available in the Cambridge Crystallographic Data Base for all types of organic and organometallic molecules. Hydrogen bonds, like covalent bonds, can be expanded or compressed from the equilibrium value for a particular donor-acceptor pair. Since their force constants are about 15 times weaker, the range of values is much wider than for covalent bonds, of the order of 1 Â, as compared with 0.05 Â for C-C bonds. A similar spread of values is observed with the O-H- - -O angles (44). (2) Favor cooperative systems offiniteor infinite chains of O-H OH O H — or homodromic loops (45.46). as in Figures 2, 3 and 4.


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Figure 2. Schematic diagram of the hydrogen-bond structure in the crystal structure of gentiobiose (GENTOS01: REFCODE in Cambridge Crystallographic Data Base). The arrows indicate infinite chains. Distances are Η—Ο in Â, angles are O-H—Ο in degrees. The covalent O-H bond lengths have been normalized to 0.97 Â.

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Figure 3. Schematic diagram of the hydrogen-bond structure of β-maltose monohydrate (MALTOS11). The arrrows indicate infinite chains. Distances and angles are from the neutron diffraction analysis.

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Figure 4. Schematic diagram of section of hydrogen-bond structure containing homodromic cycles in the crystal structure of cyclodextrin hexahydrate.

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JEFFREY


(3) Favor hydrogen bond structures in which three-center bonds cross-link their chains and loops into nets and thereby enhance cooperativity (47). (4) Use of the concept of the Excluded Region to limit the orientation of -OH groups and both orientation and translations of water molecules (48). Programming these constraints will require the same type of approach to the hydrogen bond aspect of molecular modeling as is already being applied to the molecular mechanics of covalent bonding. 1


Literature Cited 1. Hehre, W. J.; Radom, L.; von Schleyer, P. R; Pople, J. A. Ab-initio Molecular Orbital Theory: John Wiley & Sons: New York, 1986. 2. Malrieu, J. P. In Modern Theoretical Chemistry: Schaefer, H. F., Ed.; Plenum Press: New York, 1977; Vol. 7, Chapter 3, pp 69-104. 3. Burkert, U.; Allinger, N. L. Molecular Mechanics: American Chemical Society: Washington, DC, 1982; Monograph 177. 4. Allen, F. H.; Kennard, O.; Taylor, R. Accts. Chem. Res. 1983, 16, 146-53. 5. Jones, P. J. Chem. Soc. Rev. 1984, 13, 157-72. 6. Jeffrey, G. Α.; Pople, J. Α.; Radom, L. Carbohydr. Res. 1972, 25, 117-31. 7. Jeffrey, G. Α.; Pople, J. Α.; Radom, L. Carbohydr. Res. 1974, 38, 81-95. 8. Jeffrey, G. Α.; Pople, J. Α.; Binkley, J. S.; Vishveshwara, S. J. Amer. Chem. Soc. 1978, 100, 373-79. 9. Lemieux, R. U.; Chu, W. J. Abstr. Papers Am. Chem. Soc. 1958, 133, S1N. 10. Berman, H. M.; Chu, S. S. C.; Jeffrey, G. A. Nature. 1967, 157, 1576-77. 11. Jeffrey, G. A. In Anomeric Effect. Origin and Consequences: Szarek, W. Α.; Horton D., Eds.; Am. Chem. Soc. Symposium Series, 1979, No. 87. 12. Lesyng, B.; Saenger, W. Biochem. Biophys. Acta 1981, 678, 408-12. 13. Rees, D. Α.; Smith, P. J. C. J. Chem. Soc. Perkin II, 1975, 830-35. 14. Bock, K.; Meldal, M.; Bundle, D. R.; Iversen, T.; Garegg, P. J.; Norbert, T.; Lindberg, Α. Α.; Svenson, S. B. Carbohydr. Res. 1984, 130, 23-34. 15. Brady, J. W. J. Am. Chem. Soc. 1986,108, 8153-60. 16. Engler, E. M.; Andose, J. D.; von Schleyer, P. R. J. Am. Chem. Soc. 1973, 95, 8005-25. 17. Jeffrey, G. Α.; Taylor, R. J. Computat. Chem. 1980, 1, 99-109. 18. Chu, S. S. C.; Jeffrey, G. A. Acta Crystallogr., Sect. B (1968) 24, 830-838. 19. Melberg, S.; Rasmussen, K. Carbohydr. Res. 1980, 78, 215-24. 20. Rohrer, D. C.; Sarko, Α.; Bluhm, T. L.; Lee, Y. N. ActaCrystallogr.,Sect. B 1980, 36, 650-54. 21. Yadav, J. S.; Barnickel, G.; Bradaczek, H. J. Theor. Biol. 1982, 95, 151-66. 22. Mo, F.; Jensen, L. H. Acta Crystallogr. 1975, 31, 2867-73. 23. Jeffrey, G. Α.; Takagi, S. Accts. Chem. Res. 1978, 11, 264-70. 24. Ceccarelli, C.; Jeffrey, G. Α.; Taylor, R. J. Molec. Struct. 1981, 70, 255-71. 25. Jeffrey, G. Α.; Maluszynska, H.; Mitra, J. Int. J. Biol. Macromol. 1985, 7, 336-48. 26. Kroon-Batenburg, L. M. J.; Kanter, J. A. Acta Crystallogr.. Sect. B 1983, 39, 749-54. 27. Burkert, U.; Allinger, N . L . Molecular Mechanics: Am. Chem. Soc. Monograph, 1982, No. 177. 28. Taylor, R. J. Molec. Struct. 1981, 71, 311-325. 29. Lifson, S.; Hagler, A. T.; Dauber, P. J. Am. Chem. Soc. 1979, 101, 51115121. 30. Weiner, S. J.; Kollman, P. Α.; Case, D. Α.; Singh, U. C.; Ghio, C.; Alagona, G.; Propata, S. Jr.; Weiner, P. J. Am. Chem. Soc. 1984, 106, 765-84.


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31. Weiner, S. J.; Kollman, P. Α.; Nguyen, D. T.; Case, D. A. J. Computat. Chem. 1986, 7, 230-252. 32. Brooks, B. R.; Briccoleri, R. E.; Olafson, B. D.; States, D. J.; Swaminathan, S.; Karplus, M . J. Computat. Chem. 1983, 4, 187-217. 33. Singh, U. C.; Kollman, P. A. J. Computat. Chem. 1984, 5, 129-45. 34. Cox, S. R.; Williams, D. E. J. Computat. Chem. 1981, 2, 304-23. 35. Williams, D. E.; Jan, J. Adv. Atomic Mol. Phys. 1987, 23, 87-129. 36. Spackman, Μ. Α.; Stewart, R. F. In Methods and Applications in Crystallographic Computing: Hall, S. R.; Ashida, T., Eds.; Clarendon Press: Oxford, UK, 1984; pp. 302-20. 37. Bader, R. F. W. Accts. Chem. Res. 1985, 18, 9-15. 38. Barnes, P.; Finney, J. L.; Nicolas, J. D.; Quinn, J. E. Nature 1979, 282, 459-64. 39. Goodfellow, J. M.; Finney, J. L.; Barnes, P. Proc. Roy. Soc. London Β 1982, 214, 213-28. 40. Mezei, M.; Beveridge, D. L.; Berman, Η. M.; Goodfellow, J. M.; Finney, J. L.; Neidle, S. J. Biomol. Struct. Dynam. 1983, 1, 287-97. 41. Koehler, J. E. H.; Saenger, W.; van Gunsteren, W. F. Eur. Biophys. J. 1987, 15, 197-210. 42. Koehler, J. Ε. H.; Saenger, W.; van Gunsteren, W. F. Eur. Biophys. J. 1987, 15, 211-24. 43. Wilcox, G. L.; Quiocho, F. Α.; Levinthal, C.; Harvey, S. C.; Maggiora, G. M.; McCammon, J. A. J. Computer-Aided Molecular Design 1987, 1, 271-81. 44. Jeffrey, G. A. Landolt-Bornstein. New Series. Group VII, Vol. 1b; Saenger, W., Ed.; Springer-Verlag: Berlin, 1989; Sect. 2.7, pp. 277-348. 45. Jeffrey, G. Α.; Mitra, J. Acta Crystallogr., Sect. Β 1983, 39, 469-80. 46. Saenger, W. Nature 1979, 279, 343-4. 47. Koehler, J. Ε. H.; Saenger, W.; van Gunsteren, W. F. J. Biomolec. Struct. & Dnyamics 1988, 1, 181-198. 48. Savage, H.; Finney, J. Nature 1986, 322, 717-20. RECEIVED February 13,

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Experimental and Theoretical Bases for Accurate Modeling - ACS

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