GENERALIZED BORN MODELS OF MACROMOLECULARSOLVATION EFFECTS

Transcription

1 Annu. Rev. Phys. Chem : Copyright c 2000 by Annual Reviews. All rights reserved GENERALIZED BORN MODELS OF MACROMOLECULARSOLVATION EFFECTS Donald Bashford and David A. Case Department of Molecular Biology, The Scripps Research Institute, La Jolla, California 92037; bashford@scripps.edu, case@scripps.edu Key Words solvation energy, continuum dielectrics Abstract It would often be useful in computer simulations to use a simple description of solvation effects, instead of explicitly representing the individual solvent molecules. Continuum dielectric models often work well in describing the thermodynamic aspects of aqueous solvation, and approximations to such models that avoid the need to solve the Poisson equation are attractive because of their computational efficiency. Here we give an overview of one such approximation, the generalized Born model, which is simple and fast enough to be used for molecular dynamics simulations of proteins and nucleic acids. We discuss its strengths and weaknesses, both for its fidelity to the underlying continuum model and for its ability to replace explicit consideration of solvent molecules in macromolecular simulations. We focus particularly on versions of the generalized Born model that have a pair-wise analytical form, and therefore fit most naturally into conventional molecular mechanics calculations. INTRODUCTION There are many circumstances in molecular modeling studies where a simplified description of solvent effects has advantages over the explicit modeling of each solvent molecule. One of the most popular models, especially for water, treats the solvent as a high-dielectric continuum, interacting with charges that are embedded in solute molecules of lower dielectric. The solute charge distribution, and its response to the reaction field of the solvent dielectric, can be modeled either by quantum mechanics or by partial atomic charges in a molecular mechanics description. In spite of the severity of the approximation, this model often gives a good account of equilibrium solvation energetics, and it is widely used to estimate pks, redox potentials, and the electrostatic contributions to molecular solvation energies (for recent reviews, see 1 6). For molecules of arbitrary shape, the Poisson-Boltzmann (PB) equations that describe electrostatic interactions in a multiple-dielectric environment are typically solved by finite-difference or boundary-element numerical methods (1, 7 12). These can be efficiently solved X/00/ $

2 130 BASHFORD CASE for small molecules but may become quite expensive for proteins or nucleic acids. For example, the DelphiII program, which is a popular program that computes a finite-difference solution, takes about 25 min on a 195-Mhz SGI processor to solve problems on a grid with 600 atoms. Obtaining derivatives with respect to atomic positions adds to the time and complexity of the calculation (13). Even though progress continues to be made in numerical solutions, and other approaches may be significantly faster, there is a clear interest in exploring more efficient, if approximate, approaches to this problem. One such simplification that has received considerable recent attention is the generalized Born (GB) approach (14, 15). In this model, which is derived below, the electrostatic contribution to the free energy of solvation is G pol = 1 2 ( 1 1 ε w ) i,j q i q j f GB, 1. where q i and q j are partial charges, ε w is the solvent dielectric constant, and f GB is a function that interpolates between an effective Born radius R i, when the distance r ij between atoms is short, and r ij itself at large distances (15). In the original model, values for R i were determined by a numerical integration procedure, but it has recently been shown that pair-wise approximations, in which R i is estimated from a sum over atom pairs, can be nearly as accurate and provide a simplified approach to energies and their derivatives (16 21). In the following sections, we present one derivation of this model and compare it to the underlying continuum dielectric model on which it is based. This is followed by a review of pair-wise parameterizations and of practical applications of GB and closely related approximations. Our emphasis is almost exclusively on the use of this approach to describe aqueous solvation of macromolecules. A comprehensive review of other applications of the GB model has recently appeared (6). GENERALIZED BORN AND RELATED APPROXIMATIONS The underlying physical picture on which the GB approximation is based is the two-dielectric model described above. To obtain the electrostatic potential φ in such a model, one should ideally solve the Poisson equation, [ε(r) φ(r)] = 4πρ(r), 2. where ρ is the charge distribution, and the dielectric constant ε takes on the solute molecular dielectric constant ε in in the solute interior and the exterior dielectric constant ε ex elsewhere. For gas phase conditions, ε ex = 1, whereas in solvent conditions, ε ex = ε w, the dielectric constant of the solvent (here, water); solving Equation 2 under these two conditions leads to potentials that can be denoted φ sol and φ vac, respectively. The difference between these potentials is the reaction field, φ reac = φ sol φ vac, and the electrostatic component of the solvation free energy is G pol = 1 φ reac (r)ρ(r) dv, 3. 2

3 GENERALIZED BORN MODELS 131 or if the molecular charge distribution is approximated by a set of partial atomic point charges q i, G pol = 1 q i φ reac (r i ) i In the case of a simple ion of radius a and charge q, the potentials can be found analytically and the result is the well-known Born formula (22), ( G Born = q2 1 1 ). 5. 2a ε w If we imagine a molecule consisting of charges q 1 q N embedded in spheres of radii a 1 a N, and if the separation r ij between any two spheres is sufficiently large in comparison to the radii, then the solvation free energy can be given by a sum of individual Born terms, and pair-wise Coulombic terms: N q 2 ( ) i 1 G pol = N N q i q j 2a i ε w 2 r ij i i j i ( 1 ε w 1 ), 6. where the factor (1/ε w 1) appears in the pair-wise terms because the Coulombic interactions are rescaled by the change of dielectric constant on going from vacuum to solvent. The goal of GB theory can be thought of as an effort to find a relatively simple analytical formula, resembling Equation 6, which for real molecular geometries will capture, as much as possible, the physics of the Poisson equation. The linearity of the Poisson equation (or the linearized PB equation) assures that G pol will indeed be quadratic in the charges, as both Equations 1 and 6 assume. However, in calculations of G pol based on direct solution of the Poisson equation, the effect of the dielectric constant is not generally restricted to the form of a prefactor, (1/ε w 1), nor is it a general result that the interior dielectric constant, ε in, has no effect. With these caveats in mind, we seek a function f GB, to be used in Equation 1, such that in the self (i = j) terms, f GB acts as an effective Born radius, whereas in the pair-wise terms, f GB becomes an effective interaction distance. The most common form chosen (15) is f GB (r ij )= [ rij 2 + R ir j exp ( rij 2 /4R )] 1 ir 2 j, 7. in which the R i are the effective Born radii of the atoms, which generally depend not only on a i, the intrinsic atomic radii, but also on the radii and relative positions of all other atoms. Ideally, R i should be chosen so that if one were to solve the Poisson equation for a single charge q i placed at the position of atom i, and a dielectric boundary determined by all of the molecule s atoms and their radii, then the self-energy of charge i in its reaction field, q i φ reac (r i )/2, would be equal to (q 2 /2R i )(1 1/ε w ). Obviously, this procedure per se would have no practical advantage over a direct calculation of G pol using a numerical solution of the Poisson equation. To find a more rapidly calculable approximation for the effective Born radii, we turn to a formulation of electrostatics in terms of integration over energy density.

4 132 BASHFORD CASE Derivation in Terms of Energy Densities In the classical electrostatics of a linearly polarizable media (23), the work required to assemble a charge distribution can be formulated either in terms of a product of the charge distribution with the electric potential, as above, or in terms of the scalar product of the electric field E and the electric displacement D: W = 1 ρ(r)φ(r) dv 8. 2 = 1 E D dv. 9. 8π We now introduce the essential approximation used in most forms of GB theory: that the electric displacement is Coulombic in form, and remains so even as the exterior dielectric is altered from 1 to ε w in the solvation process. In other words, the displacement due to the charge of atom i (which for convenience is here presumed to lie on the origin) is D i q ir r This is called the Coulomb field approximation. In the spherically symmetric case (as in the Born formula), it is exact, but in more complex geometries, there may be substantial deviations, a point to which we turn presently. The work of placing a charge q i at the origin within a molecule whose interior dielectric constant is ε in, surrounded by a medium of dielectric constant ε ex and in which no other charges have yet been placed, is then W i = 1 8π (D/ε) D dv 1 8π in qi 2 dv + 1 qi 2 dv. 11. r 4 ε in 8π ex r 4 ε ex The electrostatic component of the solvation energy is found by taking the difference in W i when ε ex is changed from 1.0 to ε w, G poli = 1 ( 1 1 ) q i dv, 12. 8π ε w ex r 4 where the contribution due to the interior region has canceled in the subtraction. 1 Comparing Equation 12 to the Born Formula 5 or to Equations 1 or 6, we conclude that the effective Born radius should be R 1 i = 1 4π ex 1 dv. 13. r 4 1 It may be noticed that the interior integral contained a singularity at r 0. This a consequence of representing the charge distribution as a set of point charges, and similar singularities appear in treatments based on the electrostatic potential. The validity of canceling out such singularities can be demonstrated by replacement of these point charges by small charged spheres and consideration of the limit as the sphere radii shrink to zero.

5 GENERALIZED BORN MODELS 133 It is often convenient to rewrite this in terms of an integration over the interior region, excluding a radius a i around the origin, Ri 1 = ai dv, 14. 4π in,r>a i r 4 where we have used the fact that the integration of r 4 over all space outside radius a is simply 4πa 1. Note that in the case of a monatomic ion, where the molecular boundary is simply the sphere of radius a i, this equation becomes R i = a i and the Born formula is recovered exactly. The integrals in Equation 13 or 14 can be calculated numerically by constructing a set of concentric spherical shells around atom i and calculating the fractional area of these shells lying inside or outside the van der Waals volume of the other atoms, j i (15), or by using a cubic integration lattice (24). Ghosh et al (25) have proposed an alternative approach, in which the Coulomb field is still used in place of the correct field, and Green s theorem is used to convert the volume integral in Equation 14 to a surface integral. At this level, the S-GB (surface-gb) model is formally identical to the model outlined above. (There are potential computational advantages in the surface integral approach, especially for large systems and for evaluating gradients, but these have not yet been exploited.) In practice, empirical short-range and long-range corrections (discussed below) are added to improve agreement with numerical Poisson theory. Solute Dielectrics Other than Unity Strictly speaking, the Coulomb field approximation assures that the internal dielectric constant, ε in, does not appear in GB theory; the only dielectric constants that matter are those of the solvent and the vacuum. (See the passage from Equation 11 to Equation 12.) However, ε in can reappear in an indirect and somewhat deceptive way, in GB-based expressions for energy as a function of solute conformation or intermolecular interaction energies. In such cases, one would like to have a potential of mean force described on the hypersurface of the solute degrees of freedom. Its electrostatic component would be PMF elec = E elec,ref + G pol (ref sol), 15. where E elec,ref is the electrostatic energy of the solute in some reference environment that is chosen so that the calculation can be done simply, and G pol (ref sol) is the energy of transferring the system from this reference environment to solvent. If the solute is presumed to have an internal dielectric of 1, the obvious choice of reference medium is the vacuum, where E elec,ref can be calculated by Coulomb s law, and the usual GB expressions for G pol can be used unchanged for G pol (ref sol). However, if the internal dielectric has a value ε in that is different from 1, a more convenient choice is a reference medium of dielectric constant ε in, so that Coulomb s law can again be used. In this case, all occurrences

6 134 BASHFORD CASE of (1 1/ε w ) in the GB theory expressions are replaced by (1/ε in 1/ε w ). The resulting expression for the electrostatic potential of mean force is PMF elec = 1 q i q j 1 ( 1 1 ) qi q j ε in r ij 2 ε in ε w f GB (r ij ) i j (compare with Equation 1). At long distances, where f GB goes to r ij, the ε in dependence disappears. It should be emphasized that ε in appears in these formulae not so much because it is the internal dielectric constant as because it is the external dielectric constant of the reference environment. In particular, GB theory, because of its Coulomb field approximation, and in contrast to Poisson-equation theory, cannot capture the tendency of solvation to increase the dipole moment of a dipolar solute, thus enhancing its solubility, through the use of an internal dielectric constant. Of course, such effects can be captured by methods that explicitly couple some other theory of solute polarizability to GB theory, e.g. though quantum mechanical descriptions of the solute (6). Incorporation of Salt Effects GB models have not traditionally considered salt effects, but the model can be extended to low salt concentrations at the Debye-Huckel level by the following arguments (21). The basic idea of the GB approach can be viewed as an interpolation formula between analytical solutions for a single sphere and for widely separated spheres. For the latter, the solvation contribution in the Poisson model becomes ( ) G pol = 1 e κr ij qi q j, 17. ε w r ij where κ is the Debye-Hückel screening parameter. The first term removes the gasphase interaction energy, and the second term replaces it with a screened Coulomb potential. For a single spherical ion, the result is (26, 27) G pol = 1 ( 1 1 ) q 2 2 ε w a q 2 κ 2ε w (1+κb), 18. where a is the radius of the sphere and b is the radial distance to which salt ions are excluded, so that b a is the ion-exclusion radius. To a close extent, these two limits can be obtained by the simple substitution ( 1 1 ε w ) i,j ( 1 e κ f GB ε w in Equation 1. This reduces directly to Equation 17 for large distances, and the salt-dependent terms become, as r ij goes to zero, ) 19. qi 2 ( κ 2ε w κ R ) 20. i

7 GENERALIZED BORN MODELS 135 through terms in κ 2. To terms linear in κ, Equations 20 and 18 agree, but the quadratic terms differ by the replacement of b with 1/2(R i ). In practice, Equation 19 gives salt effects that are slightly larger than those predicted by finite-difference linearized PB calculations, but which are strongly correlated with them. One likely reason is that the GB model outlined here does not have the concept of an ionexclusion radius and, hence, tends to overestimate salt effects (compared with the usual PB model) by allowing counterions to approach more closely to the solute than they should. A simple ad hoc modification that leads to acceptable results can be obtained by a simple scaling of κ by 0.73 in Equation 19 (21). Figure 1 compares linearized PB and GB estimates of the effect of monovalent added salt on the solvation energy of a 10-bp DNA duplex. The GB model is clearly capturing most of the behavior of linearized PB, especially at low salt concentrations. It is worth emphasizing that the linearized PB model itself is an imperfect model for salt effects (28), so that Equation 20 should only be viewed as a rough approximation; it does, nonetheless, introduce the exponential screening of longrange Coulomb interactions, which is one of the hallmarks of salt effects. Limitations and Variations of the GB Model The crux of the GB approximation for the self-energy terms and effective radii is the Coulomb field approximation, Equation 10, and this is also the main source of its deviation from solvation energies calculated using solutions of the Poisson Figure 1 Difference in the solvation energy at finite and zero added salt for a 10-bp DNA duplex, calculated by numerical solutions to the linearized PB model, and from Equation 38. (Data from Reference 21.)

8 136 BASHFORD CASE equation. In general, the electric displacement generated by a charge q i located at a position r i within the low dielectric cavity of the solute will consist of a Coulomb field and a reaction field component D reac, the latter being a consequence of the nonuniformity of the dielectric environment. The reaction field contains no Coulombic singularities within the molecular interior and is usually fairly smoothly varying within this region. If the dielectric boundary between solvent and solute is sharp, which is the usual assumption, then D reac can be thought of as arising from an induced surface charge density on the dielectric boundary. In spherically symmetric cases, such as the case analyzed in the original Born theory, D is given exactly by the Coulomb field (D reac is zero), and GB solvation goes over into Poisson-equation solvation. Schaefer & Froemmel (29) have analyzed deviations from the Coulomb-field approximation for the case of charges at arbitrary positions within a spherical dielectric boundary, a case for which analytical solutions of the Poisson equation are available (26). They found that the Coulomb field approximation leads to significant errors in both self-energies and in the screening of charge-charge interactions, and they proposed an image-charge approximation for D reac that is very successful in recovering the energetic behavior of the exact Poisson model. Some additional quantitative sense of the limitations of the Coulomb field approximation can be gained by considering the case of a charge near a planar dielectric boundary (Figure 2). This can be thought of as the infinite-radius limit of the situation where a charge is a distance d below the surface of a spherical macromolecule with a large radius (R d) and a dielectric constant ε in, and the macromolecule is transferred from an external medium of dielectric constant ε in to a medium of dielectric constant ε w. The electrostatic potential can be found exactly by the method of images (23). φ z>0 = φ z<0 = q ε in r 1 + q ε in r 2 q ε ex r 1, 21. where q = q ε ex ε in ε ex + ε in 2ε ex q = q. ε ex + ε in 22. The reaction field in the z > 0 region corresponding to a change of ε ex from ε in to ε w is ( ) φ reac = q εw ε in 1 = q, 23. ε in r 2 ε w + ε in ε in r 2

9 GENERALIZED BORN MODELS 137 Figure 2 A point charge q near a dielectric interface at z = 0. The dielectric constant is ε in or ε ex in the positive or negative z regions, respectively. The potential on the +z side is a sum of the Coulomb potential of the real charge q at z = d, and an image charge q at z = d. The potential on the z side is the Coulomb potential of an image charge q at z = d. The distances of an arbitrary point r from the z = d and z = dcharge locations are denoted as r 1 and r 2, respectively. where the r 1 term has canceled in the subtraction of the potential in the ε ex = ε in case from the potential in the ε ex = ε w case. The electrostatic solvation energy can be found using Equation 4, ( G pol (exact) = q2 ε w ε in 1 4dε in (ε w + ε in ) = q2 1 ) d ε in ε w 1 + ε in /ε w The corresponding formula for the solvation energy according to the Coulombfield approximation and an integration of the energy density difference over the exterior (z < 0) region can be obtained using Equation 12: G pol (Coulomb) = ( 1 ε in 1 ε w ) 1 8π q 2 z<0 r2 4 = q2 8d ( 1 ε in 1 ε w ). 25. Note that in the usual case where ε w ε in, the magnitude of G pol is underestimated by a factor of almost 2 compared with the exact expression, Equation 24, although the form of the dielectric-constant dependence, a factor of (1/ε in 1/ε w ), is approximately correct. This suggests that for charges buried somewhat below the surface of large macromolecules, the W i of Equation 11 may be underestimated, and thus the effective Born radii overestimated because of the Coulomb-field approximation. Of course the methods of approximating density integrals (such as

10 138 BASHFORD CASE the pair-wise descreening approximation decribed below) will also affect the results. On the other hand, most of the solvation energy of a macromolecule will be due to charged or highly polar groups that protrude into solvent, and for these groups, GB theory may be expected to work nearly as well as for analogous groups in small molecules. Luo et al (30) have also examined errors arising from replacing the true field with a Coulomb approximation. Essentially, they assume that E E vac /d F, where E vac is the vacuum Coulomb field and d F is a screening parameter having different values in the interior region or exterior regions. In the centrosymmetric case, d F is identical to the dielectric constant, but for more general shapes, it is a parameter to be optimized to provide realistic solvation energies. This leads to G pol = 1 ε w 1 8π d F ex E 2 vac dv. 26. Noting that in the Coulomb field approximation D = E vac, this expression is similar to Equation 12 with (1 1/ε w ) replaced by (ε w 1)/d F. Its other difference from GB theory is that this expression is used for the entire solvation energy, implicitly including charge-charge interactions terms rather than using a formula such as Equation 7. On a number of test cases, it was found to give good agreement with Poisson calculations, but the numerical integration required had roughly the same computational cost as solving the Poisson equation numerically. The limitations of both the density integration methods and the Coulomb field approximation are also addressed in the electrostatic component of the SEED method for docking small molecular fragments to a macromolecular receptor (31). The desolvation of the receptor by the low dielectric of the ligand is taken to be the integral of D 2 /(8π) over the ligand volume, and D is assumed not to change on ligand binding, as in conventional GB theory. The integration is done numerically using a cubic lattice, and the user can choose whether to estimate D by the Coulomb field approximation, as in GB theory, or to calculate it by a finite difference solution of the Poisson equation for the ligand-free receptor. In its use of a D obtained by solving the Poisson equation, this method is similar in spirit to the SEDO approximation described below, except that the latter is based on solvation energy density rather than D 2. Solvation Energy Density In Equation 12, the solvation energy is expressed as the volume integral of an energy density that is zero within the solute volume, so that the integral need only run over the solvent volume. This could be thought of as a solvation energy density, but it is only by virtue of the Coulomb field approximation that it falls to zero in the solute region. It is possible to give a more rigorous definition of solvation energy density that does not depend on approximations (32). In classical continuum electrostatics, an important problem is to find the energy change caused by placing a dielectric

11 GENERALIZED BORN MODELS 139 object within some volume, in the electric field of a fixed set of charges outside that volume. This is essentially the same as the present solvation problem: We seek the change of energy associated with changing the dielectric constant of the region outside the molecule V ex, from ε vac to ε w, in the presence of the atomic partial charges, which remain fixed in the molecular interior. Denoting the electric field and displacement before the dielectric alteration as E vac and D vac, and after the alteration as E sol and D sol, respectively, the energy change is G pol = 1 (E sol D sol E vac D vac ) dv, 27. 8π where the volume integration runs over all space. It can be shown (23) that Equation 27 can be transformed into G pol = 1 (E sol D vac D sol E vac ) dv π Using D = εe, it can be seen that the integrand falls to zero in the molecular interior because the dielectric constant there does not change (i.e. E sol ε in E vac ε in E sol E vac = 0). One can then write G pol = S(r) dv, 29. where the integral runs only over the exterior region, and S is the solvation energy density, defined by S = 1 8π (E sol D vac D sol E vac ). 30. No approximations have been introduced up until this point, and Equations 29 and 30 are fully equivalent to Equation 3. Of course, if one introduces the Coulomb field approximation, Equation 10, into the expression for S, the GB theory expression for self energies is obtained. A different approximation based on S and oriented toward estimating desolvation effects in intermolecular interactions has recently been proposed by Arora & Bashford (32). Suppose one would like to calculate the effect of the approach of a second molecule on the solvation energy of the first molecule. The effect of the low dielectric of the second molecule on the solvation energy density S of the first is twofold. First, S will go to zero in the interior of the second molecule, in other words a portion of S will be occluded. Second, in the solvent region near the second molecule, there will be some rearrangement of S. The approximation is to include the first effect but neglect the second. This means that one can precalculate S by solving the Poisson equation and differentiating the potential to obtain the field and displacement in both vacuum and solvent, for the first molecule alone. Then the desolvation effect of a second molecule can be obtained by G desol = S mol1 dv. 31. ex mol2

12 140 BASHFORD CASE This is termed the solvation energy density occlusion (SEDO) approximation. It is intended for use in docking calculations where the desolvation of one molecule by a second molecule must be calculated multiple times for multiple positions of the second molecule. The exact Poisson method would require resolving the Poisson equation, in both vacuum and solvent, for each new position, whereas the SEDO approximation requires only the Poisson equation solution for the first molecule alone and, thereafter, numerical integrations of S over the volume of the second molecule at its various locations. PAIRWISE GENERALIZED BORN MODELS Although expressions for the effective Born radii in GB models can be derived from continuum dielectric models and the Coulomb field approximation, as described above, in practice, additional approximations are often introduced to obtain faster execution times. These often involve adjustable parameters that can be fit either to experiment or (more commonly) to numerical continuum dielectric calculations. This dependence on new parameters may effectively make these new theories, whose behavior needs to be tested in each case, and whose strengths and weaknesses may differ from the pure GB models described above. This section gives a brief overview of some things that have been tried, with an emphasis on models that are computationally tractable for simulations of proteins and nucleic acids. For relatively small molecules, there has been considerable investigation of the original GB idea, where integrals over the solute dielectric regions are carried out numerically (30, 33). Of special interest for relatively small molecules are the SMx models of Cramer & Truhlar (6), which combine GB models for the electrostatic component of solvation with a surface-area dependent solvation term and various quantum mechanical treatments of the solute charge distribution. These sorts of models have generally not been fast enough to have received significant applications to macromolecules, and we do not consider them further, turning instead to the faster, pair-wise approaches. Overlapping Spheres Approach In the pair-wise versions of GB theory, the basic idea is to approximate the integrals in Equation 14 by a sum of contributions for each atom. If the molecule consisted of a set of nonoverlapping spheres of radius a j at positions r ij relative to atom i, then Equation 14 could be written as a sum of integrals over spherical volumes, R 1 i = a 1 i 1 4π j sphere j 1 dv. 32. r 4

13 GENERALIZED BORN MODELS 141 The integrals over spheres can then be calculated analytically (29), leading to R 1 i = a 1 i j a j 2 ( r 2 ij a2 j ) 1 4r ij log r ij a j r ij +a j. 33. Analytical expressions are also available for the case when atom j overlaps atom i (16, 29). A straightforward pair-wise summation using these ideas would overcount the solute region because neighboring atoms j themselves overlap with each other. Hawkins et al (16, 17) proposed scaling the neighboring values of R i as an empirical correction to compensate for this neglect of overlap. The expression for the GB radii then takes the form R 1 i = a 1 i j H(r ij,s j a j ), 34. where H is a fairly complex expression and the S j scaling factors are additional empirical parameters, fit either to experiment or to numerical Poisson results. Several groups have adopted this idea, using different training sets to determine how best to scale the neighboring radii (16, 17, 21, 34 36). The original work dealt primarily with small molecules, where solute charge distributions can be determined by quantum mechanical models (16, 17). Of more direct interest for macromolecules are approaches based on atomic charges derived from empirical force fields. Figure 3 shows one example, comparing solvation energies for a series of A- and B-form DNA duplexes computed by a pair-wise GB model with numerical solutions to the PB equations. The individual conformations were taken from explicit solvent simulations, as described elsewhere (37). As Scarsi & Caflisch note (38), a more stringent and useful test can be obtained by comparing the total electrostatic energies (including the gas-phase Coulomb term) because much of the solvation energy is simply a screening of the bare Coulomb interaction. Figure 3 also shows this gas-phase Coulomb interaction added to both the PB and GB solvation energies. Clearly, the range of total electrostatic energies is less than that of the solvation energies alone, but the agreement between GB and PB results is good in either case. Several studies (all using snapshots taken from explicit solvent molecular dynamics simulations) have shown that the conformational energy differences (for a variety of nucleic acid problems) using this pair-wise GB approximation are within 1 2 kcal/mol of numerical PB results (21, 37, 39, 40). The general agreement of solvation energies for different conformers, illustrated in Figure 3 (with similar behavior seen in many other GB studies), does not mean that individual interaction energies correctly model the corresponding PB results. There is generally a substantial cancellation of errors between atom pairs with the same and opposite charges (21); this can be particularly important for studies of pk a behavior. Furthermore, there is a general trend for more deeply buried atoms to have R i values that are smaller in this pair-wise GB approach than in comparable

14 142 BASHFORD CASE Figure 3 Comparison of GB and PB solvation (left) and total electrostatic (right) energies for 100 snapshots of double-helical DNA taken from an explicit solvent molecular dynamics simulation. (Data from Reference 36.) numerical Poisson calculations (21). Jayaram et al have also developed two sets of scaling factors for use with the AMBER molecular force field (35). The more successful one adds a modification to the functional form of f GB, which we discuss below. Asymptotic Approach The long-distance limit of Equation 32 is just V j r ij 4, where V j is the volume of the jth atom. The direct use of this functional form for pair-wise atom contributions at all distances has been considered (41), but it appears to lead to significant errors in the short-to-intermediate distance range (42). Qiu et al (18) have modified this idea, scaling each V j r ij 4 atomic contribution by a factor that depends on the number of covalent bonds between atom j and atom i, so that the sum over neighboring atoms becomes stretch j P 2 V j r 4 ij bend + j P 3 V j r 4 ij + nonbonded j P 4 V j C rij Here the P k are adjustable parameters and C is a close-contact function that adjusts radii for nonbonded atoms that are very close to the central atom i. The parameters were fit to R i values determined from numerical Poisson calculations for a diverse group of organic molecules, molecular complexes, oligopeptides, and a small protein. Overall, comparable solvation free energies were obtained from this modified GB expression or from numerical Poisson calculations; as with the approach of Hawkins et al (16, 17), described above, deeply buried atoms in larger molecules have GB values for R i that are smaller than those determined from numerical Poisson calculations, i.e. these buried atoms experience a higher local dielectric microenvironment in GB than in Poisson models.

15 GENERALIZED BORN MODELS 143 Edinger et al (19) compared this pair-wise GB model to results from numerical Poisson calculations for a database of peptide conformations. These showed good agreement in rank-ordering peptide conformations, but solvation energies for many of the oligopeptides were systematically about 5 kcal/mol less negative in the pair-wise GB theory than for the numerical Poisson results. In a similar vein, Dominy & Brooks (20) refit the adjustable parameters in a (slightly modified) version of Equation 35 to numerical Poisson results for a database of peptides, proteins, and nucleic acids, using charges and radii from the CHARMM empirical force field. These results show no overall bias toward higher or lower solvation energies for the GB and Poisson models. Other recent applications of this model include studies of nucleic acid base pairing and solvent-exposed salt bridges in proteins (43, 44). This approach has been available in the MacroModel program for several years, and seems generally to have given good results, even though there is little formal justification for the simple linear scaling of the V/r 4 terms in Equation 35. The Analytical Continuum Electrostatics Model Schaefer & Karplus (42) presented a general formalism for decomposing energy functions based on integration of the Coulomb-field energy density into pair-wise atomic terms. The integration over the solute interior in Equation 14 can be rewritten as an integral over all space with the integrand multiplied by a step function P(r) whose value is 1 in the molecular interior and zero elsewhere. This function can then be written as a sum of atomic terms, P(r) = j P j (r), 36. where, for example, the P j might be step functions corresponding to the Voronoi volumes of the atoms, but in principle, any set of P j s satisfying Equation 36 is admissible. In this context, the method of Hawkins et al (16, 17) amounts to approximating P j as spheres, excluding only the overlap with the central atom i. Its errors arise from the fact that in regions where two of the noncentral spheres overlap, P is approximated as 2, whereas in other regions that are inaccessible to solvent, but outside the van der Waals radii of any atoms, P is approximated as zero. In other words, P is approximated by a function that fluctuates significantly in the molecular interior about its correct value of 1. In order to smooth out such fluctuations, Schaefer & Karplus (42) proposed a Gaussian form for the P k, normalized according to the effective volume of each atom, which is a parameter that characterizes its contribution to the total solvent-inaccessible volume of the solute. Based on this idea, they developed an analytical, continuous, and differentiable pair-wise atomic expression for the electrostatic energy of the solute, called analytical treatment of continuum electrostatics (ACE). The model compares reasonably well with the results of numerical solution of the Poisson equation for a test

16 144 BASHFORD CASE set of small molecules and several proteins, and it is suitable for use in molecular mechanics calculations. In practice, the ACE model is similar to the pair-wise GB models discussed above. Equations 1 and 7 are used to compute charge-charge interactions, and the procedure for generating the effective Born radii (and hence the self-energies) is analogous to Equation 34, but with a different functional form for H. All the pair-wise models discussed here are analytical functions that can be readily differentiated and incorporated into effective force fields; this is discussed more fully below. It is not yet clear if there are intrinsic differences among the models that would systematically favor one over the others. The Surface-GB Model As mentioned above, the S-GB model adds a number of empirical corrections to the original GB idea of Equations 1 and 14. These are based on least-squares fits to numerical Poisson calculations on peptides and on ribonuclease A. The self-energy terms are corrected for short-range errors (involving atom-pairs whose spheres overlap) and longer-range corrections (based on an estimate of the amount of invagination of the molecular surface). In addition, interaction energies in the 2- to 6-Å range are modified in a way that depends on the number of nearest neighbors for each atom in the pair. This method is not strictly a pair-wise approach because it involves a global surface integral, but the empirical corrections, which appear to be important in practice, are of this form. This model outlined above has been applied to conformational profiles of peptides (25), and to ligand binding to enzymes (45). The electrostatic component of ligand-binding energies provides a useful substitute for explicit solvent simulations, in particular by ranking potential ligands in a similar way. As with all the GB models discussed here, many further studies will be required to see how well promising initial results hold up. RECENT APPLICATIONS Peptide Conformational Equilibria Hydrogen bonds are some of the most important interactions in proteins and nucleic acids, and the effects of solvent dipoles on the strength of these interactions is an important component of molecular modeling. Basically, water screens exposed hydrogen bonds, greatly reducing interaction energies compared with those expected in the gas phase; however, H-bonds that are partially buried should exhibit less solvent screening and be correspondingly stronger than surface H-bonds. Ösapay et al (46) investigated the ability of continuum solvent methods to model these effects, comparing numerical Poisson calculations for several H-bonding scenarios to results obtained from free-energy calculations in which explicit solvent simulations were used. Figure 4 shows results for the interaction of two formamide

17 GENERALIZED BORN MODELS 145 molecules that are constrained to form a linear H bond with a range of H-bond lengths. The gas-phase curve (computed using the CHARMM-19 empirical force field) has a well depth of 6.7 kcal/mol. The electrostatic component of solvation, however, is also distance dependent, with more effective solvation at larger distances. The figure shows both numerical Poisson and GB estimates of this, which are nearly parallel, with the GB estimate being slightly more negative than the Poisson estimate. In either case, the sum of the gas-phase and solvation energies shows a much-reduced minimum, about 0.8 kcal/mol. The latter value is consistent with explicit solvent simulations using the same force field (46). This illustrates the important effect of solvent screening on one of the key interactions in peptide conformational energetics. The extent to which hydrogen bond strengths are diminished is dependent on the environment (46), so that more buried H-bonds (such as in dimers of the alanine-dipeptide, a model for β-sheet formation) have considerably stronger net interactions than does the very solvent-exposed H-bond shown in Figure 4. These ideas play an important role in helix-coil and other folding/unfolding transitions in peptides. The generally good performance of Poisson models in describing peptide conformational transitions (47 53) may be expected to be carried over in large measure to GB models as well. Indeed, ACE calculations on peptide folding/unfolding transitions (54, 55) and on loop conformations (56) appear to be fully competitive with numerical Poisson calculations, and in overall accord with experiment. Figure 4 Energies for a linear formamide dimer as a function of hydrogen bond length. See text for explanation. (Gas-phase and Poisson results from Reference 46; GB results use parameters from Reference 21.)

18 146 BASHFORD CASE Nucleic Acid Helices Srinivasan et al (21, 37, 39) looked at the ability of numerical PB and GB models of solvation to provide a useful understanding of the relative energetics of A- and B-form helices for DNA and RNA. In general, the particular overlapping spheres GB model used here does an excellent job of reproducing the PB solvation energies for small molecules and for groups near the surface of larger molecules. There is a systematic tendency for this GB model to overestimate the effects of solvent screening (compared with PB) for pairs of buried atoms, but individual errors tend to cancel, and a good overall account of conformational energetics is obtained, in particular for the difference between A- and B-form helices. In many cases, it should be possible to replace PB calculations with much simpler GB models, but care needs to be taken for systems with extensive burial of charges or dipoles. pk a and Redox Problems Continuum solvent models allow a straightforward (if approximate) calculation of the free energy of the charge changes that are involved in protonation/deprotonation or redox events. For this reason, they have been widely applied to calculations of the pk a values of individual side chains in proteins, or to the corresponding calculation of redox potentials in metalloenzymes (4, 27, 57). A key challenge in these approaches is to carry out averaging over the large number of possible protonation states and over conformational states likely to be populated in solution. As the number of conformational states to be averaged increases, the benefits of using a model that is faster than numerical Poisson become more important. However, in spite of the generally good agreement between GB and Poisson theory demonstrated above for solvation energies, it is not clear that current GB models are sufficiently accurate to serve as a substitute for Poisson calculations in pk investigations. These difficulties arise from two principal limitations of current GB approaches. First, Poisson calculations of pk a behavior of side chains in proteins have generally used protein dielectric constants greater than unity. This allows for an (approximate) account of the conformational response of the protein to changes in charges on protonation/deprotonation events. In principle, one should be able to account for this effect through explicit averaging over protein conformational degrees of freedom, but this has seldom been achieved in practice, and currently, recourse to an effective dielectric constant appears to be required to obtain useful results. Second, pk shifts are sensitive to a relatively small number of electrostatic interactions, so that the overall behavior of solvation energies may be less relevant here. In particular, the differences between GB and Poisson theory discussed above tend to cancel for solvation energies (if a roughly equal number of positive and negative charges are excluded from solvent), but these differences can have a significant effect on predicted pk shifts.

19 GENERALIZED BORN MODELS 147 A few groups have looked at the ability of GB models to model pk a shifts in peptides and proteins. Jayaram et al (35) looked at a series of simple dicarboxylic acids, obtaining results that seemed to require a modification of the distance dependence in f GB to obtain good results. Other groups have seen encouraging results for small acids (58, 59) but poorer results (which suggests some different parameters or models) for the more buried titratable sites that often occur in proteins. Onufriev et al (59) have recently proposed a modification to compensate for the tendency of some models to underestimate the R i values for buried atoms, even though they work well for atoms near the surface (18, 21). If one assumes that the S j parameters used in Equation 34 do a reasonable job in correcting for sphere-sphere overlap, the dielectric shape represented still includes numerous small voids in macromolecular interiors. This is essentially the difference between defining the dielectric interface as the van der Waals surface, which is the assumption of conventional GB theory, or as the molecular surface (60), which is the usual definition in Poisson theory. For small molecules, the distinction makes little difference, but for macromolecules, use of the van der Waals surface leads to internal cavities of solvent dielectric that are unrealistic because they are too small for a solvent-sized probe to fit into. To compensate for this missed volume, Onufriev et al introduced a correction factor λ, which is conceived as the ratio of the correct solvent inaccessible volume (corresponding to the molecular surface definition) to the van der Waals volume, into Equation 34: R 1 i = a 1 i λ j H(r ij,s j a j ). 37. Based on cubic packing of spheres, one can estimate λ 1.3, and parameterization of λ to fit Poisson-equation results for proteins yields a similar value. Solvation energy contributions from atoms near the surface are not strongly affected by the introduction of λ because the second term on the right-hand side of Equation 37 is generally small compared with the first. This means that the parameterizations that have been developed for small molecules with considerable effort can be carried over into macromolecular calculations with little or no change. This model was used to estimate pk a values for ionizable groups in lysozyme, myoglobin, and bacteriorhodopsin. The predicted values agreed reasonably well with both experiment and calculations based upon solution of the Poisson equation. The agreement between the two models became even better when the GB approach was used to evaluate the difference in titration behavior associated with conformational change. For example, Table 1 compares GB and Poisson estimates of the difference in pk(1/2) values for the tetragonal and triclinic crystal forms of lysozyme. It has been known for some time that continuum models predict that these (static) structures would have significantly different titration behavior; results shown in the table suggest that a GB model could be profitably used to model this (and other) conformational dependencies.

20 148 BASHFORD CASE TABLE 1 Differences in pk 1 between 2 tetragonal and triclinic lysozyme a Residue Poisson GB LYS HIS GLU GLU ASP ASP ASP ASP ASP ASP ASP TYR TYR TYR LYS LYS LYS LYS LYS LYS LEU a GB, Generalized Born. From Reference 59. Generalized Born Models as Effective Force Fields There are obvious attractions to using continuum models as effective force fields for biomolecular simulations. In the first place, the number of atoms in the solute is typically only a small fraction of those required for an explicit solvent simulation. Furthermore, because the continuum model implicitly averages over the water and mobile counterion distributions, this averaging does not need to be done by the simulation itself; in particular, this can lead to a considerable simplification in calculation of thermodynamic parameters (37, 61). Although it is possible to use numerical PB methods in molecular dynamics simulations (62, 63), only the pair-wise models discussed above are (so far) efficient enough to be used in a nearly routine fashion as effective force fields in molecular dynamics simulations. Current implementations in CHARMM and Amber are about five to eight times slower than the corresponding gas-phase simulation.

21 GENERALIZED BORN MODELS 149 This in turn is significantly faster (although presumably less accurate) than explicit solvent simulations for the same solute; this latter ratio depends primarily on how much explicit solvent is included. Now that GB implementations are available in CHARMM (64), Amber (65), MacroModel (18), tinker (34), and NAB (66) (among others), one expects many simulations to begin testing the utility of this sort of approximation. Molecular dynamics simulations should be more stringent tests of the model than evaluation of snapshots of configurations generated by other methods, and they should help establish the strengths and weaknesses of this approach. Initial dynamics reports from several groups are encouraging, for both peptides and proteins (20, 54), and for nucleic acid helices and hairpins (36, 67). For example, molecular dynamics simulations of the A- and B-forms of a duplex DNA d(ccaacgttgg) 2, and the corresponding duplex RNA r(ccaacguugg) 2, resulted in good agreement with simulations using explicit water solvent in terms of both structure and energetics (36). In particular, the A B energy differences derived from GB trajectories for both DNA and RNA closely match those obtained earlier using explicit water simulations and finite-difference PB calculations. A GB simulation starting from A-form DNA converges to B-form DNA within 20 ps, more than 20 times faster than the transition from A- to B-form DNA in explicit solvent simulations. For B-form d(ccaacgttgg) 2, atomic fluctuations around the mean are highly correlated between GB and explicit water simulations, being slightly larger in the former, and the essential subspaces found from principal component analysis overlap to a high degree. Hence, for many purposes this parameterization offers an alternative to more expensive explicit water simulations for studies of nucleic acid energetics and structure. Frictional and other dynamic aspects of solvation are only included if some sort of Langevin model is used to replace Newton s equations. Most studies have concentrated on equilibrium averages (which are independent of solvent friction), and it is unclear whether a useful model of the dynamical behavior of macromolecules in solution can be constructed along these lines. CONCLUSIONS As we have outlined here, much of the current interest in GB models derives from repeated observations of usefully close agreement between GB and numerical PB results for the same system. These results strongly suggest that for many purposes, pair-wise GB simulations may serve as a useful (and fast) substitute for numerical PB calculations. The latter model has proved to be a useful, internally consistent approach for a variety of thermodynamic problems involving hydration effects in macromolecules (2, 4, 6, 37, 57, 61, 68, 69). Nevertheless, limitations of continuum models have been widely discussed and should not be downplayed (6, 70 72). In particular, these models often fail in the description of particular, short-range effects (6, 73), fail to predict temperature and pressure effects correctly, and fail to account for the frictional effects of water on solute dynamics.