Treatment of Hydration in Conformational Energy Calculations on

Chapter 24

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Treatment of Hydration in Conformational Energy Calculations on Polypeptides and Proteins Harold A . Scheraga Baker Laboratory of Chemistry, Cornell University, Ithaca, N Y 14853-1301

Calculations of the conformations of proteins in aqueous solution require a treatment of the influence of water on the dissolved solute molecule. Because present-day computers cannot accommodate a complete search of conformational space if the surrounding water molecules are treated explicitly, resort is had to solvent-shell or solvent-exposed surface-area models, with parameters for the component amino acids obtained primarily from experimental data on free energies of hydration but also from Monte Carlo or molecular dynamics simulations of solutions of small-molecule solutes. These models are presented here, and some of the thermodynamic parameters, viz. those for hydrophobic interactions, are rationalized in terms of theoretical treatments that yield results that have been checked by experiment. In order to understand the molecular mechanisms of chemical processes in aqueous solution (I), it is necessary to have potential functions for the solute molecule and for its interactions with the surrounding water molecules (2,3). Water has a considerable influence on the structure and reactivity of solute molecules. For example, in many protein structures, positively-charged arginine side chains are close to each other, and the strong electrostatic repulsion that would ensue in vacuum is modified by bridging water molecules, as demonstrated by semi-empirical quantum mechanical calculations on a model of two guanidinium ions surrounded by several water molecules (4). For an understanding of the role of hydration, one approach uses some type of quantum mechanical calculation, treating the solvent as a polarizable continuum (5), to obtain free energies of hydration. A recent example is the analysis of heterocyclic tautomerization in aqueous solution (6). Alternatively,

0097-6156/94/0568-0360$08.00/0 © 1994 American Chemical Society

In Structure and Reactivity in Aqueous Solution; Cramer, Christopher J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.


24.

SCHERAGA

Hydration in Conformational Energy Calculations

361

use is made of molecular mechanics, with the solvent molecules treated explicitly (7-70), or implicitly by either a solvent-shell model (11) or by a solvent-exposed surface-area model (12,13), especially for large-molecule solutes of arbitrary shape. Integral equation methods have also been applied to treat solvation; for example, use has been made of X R I S M (extended reference interaction-site model) methods to calculate pair correlation functions of protein atoms with surrounding water molecules, and thereby extract solvation free energies for each residue (14,15). In addition, for treating electrostatic interactions of such large molecules in solution, the solvent is considered to be a continuum dielectric, and the non-linear PoissonBoltzmann equation is solved numerically (16,17). For the particular application to aqueous solutions of macromolecules such as proteins, the quantum mechanical approach is not feasible, and molecular mechanics with an explicit treatment of the water molecules, as indicated schematically in Figure 1, is possible only if limited changes in the conformation of the solute are of interest. If, however, large changes in

Figure 1. Schematic representation water molecules.

of a macromolecule solute in a box of

protein conformation are allowed as, for example, in studying protein folding in aqueous solution, then the model of Figure 1 cannot be treated by presentday computers, and resort must be had to the solvent-shell model or the solvent-exposed surface-area model. For either approach, it is necessary to have solvation free energies for the components of the amino acids from


362

STRUCTURE AND REACTIVITY IN AQUEOUS SOLUTION


which the protein is constituted. While such thermodynamic quantities can be obtained for small-molecule solutes by quantum mechanical calculations (6,18,19),wç have used experimental free energies of hydration in accord with a similar use of experimental data to parameterize our potential function (20) for amino acid and polypeptide solutes. Therefore, in this paper, we will first describe the solvent-shell and solvent-exposed surface-area models, and then their parameterization. Finally, some applications to aqueous systems will be discussed. Solvent-Shell Model Our earliest use of the solvent-shell model (21) was made in 1967, and has been implemented more recently with improvements in the geometry of overlapping spheres (22-27) and with more recent experimental data on free energies of hydration (11,28-32). Figure 2 illustrates the free energy penalty involved due to the removal of water when the van der Waals sphere of a solute group overlaps the

Figure 2. A schematic drawing illustrating vartious stages of the approach of two solute groups 1 and 2. A , No overlaps. B , Overlap of the hydration spheres of groups 1 and 2. C, Overlap of the van der Waals sphere of group 2 with the hydration sphere of group 1. D , Overlap of three hydration spheres of solute groups 1, 2 and 3. (Reproduced with permission from reference 33. Copyright 1981 New York Academy of Sciences.) hydration shell of another solute group. In Figure 2B, the free energy of hydration of groups 1 and 2 is unaltered; in Figure 2C, however, while the hydration volume of group 2 is unaltered, that of group 1 is reduced by the densely shaded volume of overlap. Figure 2D illustrates a multiple-site interaction (33). This model has been used to compute the contribution of hydration in molecular mechanics calculations on polypeptides and proteins in water. As an example, we cite the role of water in the helix-coil transition of polyamino



24. SCHERAGA


363

acids (34), such as poly-L-valine, where the helix content increases as the temperature is raised because of the well-known behavior of the temperature dependence of hydrophobic interactions (in this case, between the nonpolar side chains of the helix). As another example, the experimental change in energy to convert the triple-helical collagen-like poly(Gly-Pro-Pro) to the single-chain statistical-coil form is 1.95 kcal/mol per Gly-Pro-Pro tripeptide unit (35); a theoretical value, making use of the shell model to treat hydration, is 2.4 kcal/mol (36), in fair agreement with the experimental value. If hydration is not included, the theoretical value is much higher, viz., 5.0 kcal/mol (36). As a third example, Han and Kang (37) have recendy used the hydration shell model to compute the ratio of cis-to-trans isomers of N-acetylN'-methylamides of Pro-X dipeptides, where X is a series of amino acids. These authors report good agreement with experimental data for this ratio and also for the theoretical propensity of the Pro-X dipeptides to adopt βbend conformations. Solvent-Exposed Surface-Area Model Alternatively, solvent-exposed surface area is used as a basis to assess the hydration contribution, using surface (instead of volume) free-energy density parameters (see Figure 3). Various models are used to compute the solventSolvent-

Convex

surfaces

Figure 3. The solvent-accessible surface (shown as a dashed line) for atoms i,j, and k with a probe radius of r . (Reproduced with permission from reference 13. Copyright 1992 John Wiley.) p



364


exposed surface area (13,38) in order to compute the free energy of hydration (39-41) o f a solute molecule. Vila et al (42) have evaluated a number of models and several sets o f hydration parameters. From a computational point of view, the model of Perrot et al (13) is much faster than that of Connolly (38), and Williams et al (43) have used it to show that computed structures of bovine pancreatic trypsin inhibitor, with hydration included with a solventexposed surface-area model, show better agreement with the X-ray structure than do those in which hydration is not taken into account. The increased speed of the algorithm of Perrot et al (13) makes it a practical one for inclusion (together with a molecular mechanics algorithm) in a procedure to minimize the sum of the energy and hydration free energy of a macromolecule. However, such minimization algorithms, which involve the computation of first derivatives, sometimes encounter discontinuities in the first derivatives of the accessible surface area and, hence, in the hydration free energy. Wawak et al (44) have recently shown that there are only two situations in which such discontinuities arise, and work is currently in progress to surmount these problems. Work is also in progress (32) to improve the hydration parameters for polypeptides by making use of a recent compilation of free energies of hydration of small molecules (31). By calculating the free energy of solvation for the residues of melittin with both the accessible surface-area approach and the X R I S M method, Kitao et al showed that the two methods agree qualitatively, but that the X R I S M approach consistently overestimates the solvation free energy (15). Rationalization of Some Hydration Parameters While our approach is based on the use of experimental free energies of hydration to parameterize either a solvent-shell or solvent-exposed surfacearea model, it is of interest to conclude this article with a theoretical rationalization (and corresponding experimental verification) of some of these parameters, viz.,as an example, those involving the hydrophobic interaction between nonpolar groups in water. Our earlier theoretical treatment of hydrophobic interactions (45,46) was based on a statistical mechanical theory of the thermodynamic properties of liquid water (47) and of aqueous solutions of hydrocarbon (48). The free energy of formation of a pair of nonpolar groups making maximum contact in water was expressed as AG°

= a + bT + c T

2

(1)

with a corresponding expression for the volume change, AV°. Near room temperature, AV° > 0, AG° < 0, AS° > 0 and ΔΗ° > 0. The unfavorable enthalpy of formation is more than counterbalanced by the positive entropy, resulting in a favorable free energy arising from changes in the surrounding water structure as the two nonpolar groups come into contact. As an illustration of the effect of temperature, the thermodynamic


24. SCHERAGA


365

parameters are given as a function of temperature in Table I for the leucineisoleucine hydrophobic interaction. These parameters were obtained from equation 1, with a = 7290, b = 47.8 and c = 0.0660 for AG° in cal/mol.


Table I. Thermodynamic Parameters for the Formation of a LeucineIsoleucine Hydrophobic Interaction of Maximum Bond Strength (45) t °c 0 10 20 25 30 40 50 60 70

AG" kcal/mol

ΔΗ° kcal/mol

AS" e.u.

2.3 2.0 1.6 1.4 1.2 0.8 0.4 -0.1 -0.6

11.6 10.4 9.2 8.5 7.8 6.5 5.1 3.6 2.1

-0.8 -0.9 -1.0 -1.0 -1.1 -1.2 -1.2 -1.3 -1.3

Source: Reprinted from reference 45. Copyright 1962 American Chemical Society. Table II illustrates the range of the theoretical thermodynamic parameters at 25 °C for several pairs of interacting nonpolar side chains of amino acids in maximum contact. Table II. Theoretical Thermodynamic Parameters for Hydrophobic Interaction at 25°C (45)

Side Chains Alanine-alanine Isoleucine-isoleucine Phenylalanine-leucine Phenylalanine-phenylalanine

âG° kcal/mol

ΔΗ° kcal/mol

AS° e.u.

-0.3 -1.5 -0.4 -1.4

0.4 1.8 0.9 0.8

2.1 11.1 4.7 7.5

Source: Reprinted from reference 45. Copyright 1962 American Chemical Society. A variety of experiments have been carried out to verify these thermodynamic parameters. One of these involves the dimerization of carboxylic acids in aqueous solution. From the increase in the observed dimerization constant with increasing chain length, it has been suggested (49) that, in aqueous solution, the dimers are side-by-side rather than cyclic as observed in the gas phase. Thus, dimerization involves single hydrogen bonds between the carboxyl groups, and hydrophobic interactions between the nonpolar portions. The experimental and theoretical values of AG° for the hydrophobic interaction are compared in Table III. In Structure and Reactivity in Aqueous Solution; Cramer, Christopher J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

366

STRUCTURE AND REACTIVITY IN AQUEOUS

SOLUTION

Table ΠΙ. Free Energy of Hydrophobic Interaction in the Dimerization of Carboxylic Acids at 25°C 0

A G (kcalVmole) Experimental


Side-Chain

Calculated

3

15

(1)

(2)

(3)

(4)

(5)

CH -

-0.79

-0.95

-0.80

-0.70

-0.70

CH3CH2"

-1.03

-1.06

-1.09

-0.90

-1.00

CH3CH2CH2*

-1.31

-1.47

-1.41

-1.15

-1.35

-1.57

-1.45

-1.63

3

C6H5CH2-

-

a

These three columns correspond to three different sets of experimental data (see Reference 49). b*rhese two columns correspond to two slight variations in the theory (see Reference 49). Similar binary complexes were examined by fluorescence quenching of phenols with carboxylates, from which the thermodynamic parameters for the hydrophobic interaction were extracted ( 5 0 . These are shown in Table IV together with the theoretical values. As a final example, we cite some experimental data for the volume decrease when nonpolar groups are added to water. The data in Table V were obtained from experiments on a homologous series (51) in which the data for the first member of the series were subtracted in order to obtain the contribution of the nonpolar group (52). The agreement between experimental and theoretical data is fairly good, especially at the higher temperatures. Of course, the volume change accompanying the hydrophobic interaction will have the opposite, i.e. positive, sign. Such increases in volume are observed in association reactions (53) where hydrophobic interactions may be involved. From the examples cited above, it appears that the theory of reference 45 provides a good rationalization of the origin of the thermodynamic parameters for hydrophobic interactions between nonpolar groups in water. The theory of reference 45 was based on a model in which partial clathratelike cages of water molecules surround the nonpolar group. This model has subsequently been validated by numerous Monte Carlo simulations of aqueous solutions of hydrocarbons (54,55); these include, e.g.,simulations of methane in water, in which C- Ό pair correlation functions (compared to 0 - 0 pair correlation functions in pure water) indicate the presence of clathrate-like structures.



4.7 4.7

4.5 5.8

940 670

550 750 1050 780

-465 -735

-472 -580 -687 -786

4.8 ±1.3 5.8 ±1.1

3.6 ±1.7 5.2 ±1.7 6.1 ±2.3 4.2 ±1.6

860 ±280 940 ±240

470 ±380 880 ±390 980 ±590 350 ±400

-414

-649

-479

-552

-693

-761

-isobutyrate

-butyrate

Xylenol-acetate

-propionate

-isobutyrate

-butyrate

Probable error. The probable error in AG° is ± 100 cal/mol. Source: Reprinted from reference 50. Copyright 1968 American Chemical Society.

a

5.3

3.4

1.9 3.8

230 700

-349 -426

2.1 ± 1.8 2.6 ±1.4

Phenol-acetate 180 ±330

eu

AS°

-451

a

cal/mol

0

ΔΗ

-propionate

a

cal/mol

0

AG

100±450

eu

0

AS

-374^

0

ΔΗ cal/mol

0

Theoretical data

cal/mol

System

AG

Experimental data

Table I V . Thermodynaiiiie Parameters for Pairwise Hydrophobic Interaction between the Nonpolar Side Chains of Sodium Carboxylates and Phenolic Compounds at 25° (50)


368


Table V . Volume Decrease (in c.cJmole) Accompanying the Addition of Nonpolar Groups to Water (51) -AV 0°C

Exp.

Theor.

Exp.

Theor.

1.2

1.5

1.2

1.3

1.3

1.1

3.7

2.1

3.1

2.2

2.5

2.3

2.3

5.6

3.1

4.6

3.3

3.8

3.5

3.4

Theor.

Exp.

1.1

1.9

C2H5

1.9

C3H7

2.9

CH

3

50°C

40°C

Theor.

Exp.

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20°C

Some Applications of the Foregoing Theory of Hydrophobic Interactions Two applications of the foregoing theory (45) are presented here. The first one treats the entropy changes accompanying association reactions of proteins (56). If hydration is neglected, and the entropy change, A S , is attributed only to loss of translational and rotational freedom, then A S = -122 e.u. for a molecule the size of insulin. However, by taking hydration (and internal degees of freedom of the insulin dimer) into account, the computed value of AS is -10(±8) e.u.,in agreement with the experimental value (56). As a second application, nucleation (or chain-folding-initiation) sites were computed for initiating protein folding (57). The model was based on hydrophobic interactions, to remove the nonpolar groups from contact with water. The free energy change was calculated for the conversion of extended chain segments into hairpin-like structures with hydrophobic interactions between the side chains. By considering all segments along the whole chain, the most likely chain-folding-initiation sites were computed for several proteins. The computed sites for bovine pancreatic ribonuclease A (57) were subsequently verified by experiment and also by an alternative treatment based on triangular contact maps (58). a s s o c

a s s o c

a s s o c

Concluding Remarks While the data in Tables III-V provide a rationalization of the early theo retical treatment of the hydrophobic interaction, and the subsequent applica tion of simulation methods to aqueous systems, reliance is being placed, at least in the near future, on two alternative approaches to obtain the thermo dynamic parameters for the hydration of both polar and nonpolar groups. These are the use of experimental data, on the one hand, and Monte Carlo or molecular dynamics simulations on the other. These will improve the data for use in either a solvent-shell or a solvent-exposed surface-area model. As potential functions improve, so will the simulation results. Finally, if


24.

SCHERAGA


369

computer power increases by several orders of magnitude, it may then become possible to search the conformational space of a macromolecule solute by using explicit water molecules, as in Figure 1, instead of relying on these volume or surface models.


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14. 15. 16. 17. 18. 19. 20.

21. 22. 23. 24. 25.

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