Electrostatic solvation energy

Feig M, Onufriev A, Lee MS, Im W, Case DA, Brooks CL III (2004) Performance comparison of generalized bom and poisson methods in the calculation of electrostatic solvation energies for protein structures. J Comput Chem 25 265—284. 

The incorporation of the generalized Bom model into free energy calculation methods using FEP/TI and A-dynamics was carried out by Banba and Brooks.81 They define the electrostatic solvation energy for the hybrid system as follows... 

These definitions are interpreted as follows environment atoms recognize the average weighted states of the ligands each ligand recognizes only the environment atoms and itself with the full scale for estimation of its effective Bom radius. Alternate definitions of the electrostatic solvation energy and the effective Bom radius at the intermediate states are also possible.81... 

Fig. 2.2 The effect of solvent permittivity on the electrostatic solvation energy of an ion (curve 1) and that of a neutral dipolar molecule (curve 2). Curve 1 was obtained from Eq. (2.3) assuming r=0.2 nm. For curve 2, see4).

Figure 2-2. Analysis of electrostatic solvation energy error for an off-centered charge in a spherical cavity according to a monocentric multipole moment development. Left error as a function of the ratio r/a and inax- Right required /mM value as a function of the charge distance to the cavity center (r) assuming a = r +1.44 A...

Figure 2-3. Convergence of electrostatic solvation energy in the MPE method for a series of 18 polar molecules in cyclohexane...

The electrostatic solvation energy is only a part of the total solvation energy. Cavitation, dispersion and repulsion terms must be added. We show below that the MPE method leads to similar electrostatic energies than the polarizable continuum model (PCM) of Tomasi and co-workers [10], provided the same cavities are used. Therefore, non-electrostatic terms in these methods may be computed using the same computational strategies [15]. We emphasize the fact that accurate non-electrostatic contributions are often difficult to compute since they are based on parameterized formulae that cannot be directly compared to experiment. The obtained data must therefore be used with prudence, especially if they are expected to play a major role in the process under study. Fortunately, in many circumstances, non-electrostatic terms are small and/or vary little, so that they can be neglected. Tunon et al. [80] developed a parameterized expression for the MPE method using an expression of the type... 

Figure 2-4 Comparison of electrostatic solvation energies obtained with the PCM and MPE methods for 18 polar molecules in aqueous solution (see text). Left self-consistent reaction field calculations at the HF-6-31 G level. Right calculations using gas-phase CM2 net atomic charges...

The extremely general nature of the PCM technique makes it uniquely attractive, although the electrostatic solvation energies appear to be quite sensitive to choice of basis set.2 2,274,279-281,291... 

Substituting in equation 11 the known experimental parameters for phenol dissociation (AG, = 13.8 kcalmol" calculated from the ground-state equilibrium constant, pX, = 10.0), AGt((PhO ) — (PhOH)) of the phenolate/phenol system is about —76 kcalmoH, which is about 10% less than the accepted value for the electrostatic solvation energy of the chloride anion in water, AGe(Cr) = —85 kcalmol". These simple considerations imply that the AGt((PhO ) — (PhOH)) contribution to the overall free energy of solvation is largely electrostatic, and that relatively small differences in the gas-phase proton affinity of the base and in specific solvent-solute interactions of the photoacid and the base determine the relatively narrow (in free-energy units) acidity scale in aqueous solution. It... 

The second approach to the approximate description of the dynamic solvation effects is based on the semiempirical account for the time-dependent electrical polarization of the medium in the field of the solute molecule. In this case, the statistical averaging over the solute-solvent intennolecular distances and configurations is presumed before the solution of the SchrOdinger equation for the solute and correspondingly, the solvent is described as a polarizable dielectric continuum. The respective electrostatic solvation energy of a solute molecule is given by the following equation[13]... 

Figure 1. Dependence of the AMI SCa SCRF Calculated Electrostatic Solvation Energies (E), INDO/1 SCa Calculated Dispersion Energies (D) and SPT Spherical Cavity Formation Free Energies (C) on the Cavity Radius for Methanol (a) and Acetonitrile (b).

AMI SCRF Calculated Electrostatic Solvation Energies, Eei, INDO/1 Calculated Dispersion Energies, Edisp, SPT Cavity Formation Free Energies, AGcav, and Experimental Solvation Free Energies, AG(exp) (kcal/mol), [63] of 30 Organic Compounds in Aqueous Solution. 

This equation corresponds to the following three-step process. First, the cavity inside a solvent is created and the molecule is inserted into the cavity. Next, nonpolar interactions between the solute and the solvent are switched on. Finally, the electrostatic interactions between the solute and the solvent are switched on. Of the three components in Eq. (5), the electrostatic component of the solvation energy (AGeie) is by far the largest and is typically of an order of several thousands kcal/mol for an average protein. Consequently, it is convenient to start the examination of different approaches from the electrostatic solvation energy component. 

How can one then decide on the choice of the dielectric boundary One possibility is to benchmark PB calculations against explicit-solvent molecular dynamics (MD) simulations. Most of such efforts have been limited to small solute molecules [20-22]. However, it has been shown that the difference between MS and vdW results for electrostatic solvation energies depends on solute size [23]. Therefore parameterization on small solutes (either against explicit-solvent MD results or against experimental data) may not be reliable for calculating electrostatic contributions to protein-protein and protein-nucleic acid binding. 

Equation (11.20) is the primary PCM equation. It must be discretized for actual computation (see Section 11.2.2), but then given the solute s electrostatic potential evaluated at the surface discretization points, this equation can be solved for the induced surface charge at those points (i.e.,the discretized a). In an MM/PCM calculation, the electrostatic solvation energy is then immediately available via a discretized version of Eq. (11.3), although in QM applications the surface charge must be included in the next self-consistent field (SCF) iteration, and the SCF procedure is iterated until both the electron density and the surface charge have reached mutual self-consistency. 

Figure 11.3 Electrostatic solvation energy for classical histidine as a function of dielectric constant. The C-PCM approach is free of the matrix D and achieves the correct conductor limit as e -> oo. Reprinted from Ref [44] copyright 2011 Elsevier.

In order to obtain the matrix equations above, one must decide how to construct, and subsequently discretize, the cavity surface. The most widely used methods take the cavity to be a union of atom-centered spheres [77], as suggested in Fig. 11.1(a). The electrostatic solvation energy is quite sensitive to the radii of these spheres (it varies as in the Born ion model), and highly parameterized constructions that exploit information about the bonding topology [6] or the charge states of the atoms [31] are sometimes employed. The details of these parameterizations are beyond the scope of the present work, especially given that careful reconsideration of these parameters is probably necessary for classical biomolecular electrostatics calculations. 

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