Langevin dipoles solvent model

In the previous chapter we considered a rather simple solvent model, treating each solvent molecule as a Langevin-type dipole. Although this model represents the key solvent effects, it is important to examine more realistic models that include explicitly all the solvent atoms. In principle, we should adopt a model where both the solvent and the solute atoms are treated quantum mechanically. Such a model, however, is entirely impractical for studying large molecules in solution. Furthermore, we are interested here in the effect of the solvent on the solute potential surface and not in quantum mechanical effects of the pure solvent. Fortunately, the contributions to the Born-Oppenheimer potential surface that describe the solvent-solvent and solute-solvent interactions can be approximated by some type of analytical potential functions (rather than by the actual solution of the Schrodinger equation for the entire solute-solvent system). For example, the simplest way to describe the potential surface of a collection of water molecules is to represent it as a sum of two-body interactions (the interac-... 

The MFA [1] introduces the perturbation due to the solvent effect in an averaged way. Specifically, the quantity that is introduced into the solute molecular Hamiltonian is the averaged value of the potential generated by the solvent in the volume occupied by the solute. In the past, this approximation has mainly been used with very simplified descriptions of the solvent, such as those provided by the dielectric continuum [2] or Langevin dipole models [3], A more detailed description of the solvent has been used by Ten-no et al. [4], who describe the solvent through atom-atom radial distribution functions obtained via an extended version of the interaction site method. Less attention has been paid, however, to the use of the MFA in conjunction with simulation calculations of liquids, although its theoretical bases are well known [5]. In this respect, we would refer to the papers of Sese and co-workers [6], where the solvent radial distribution functions obtained from MD [7] calculations and its perturbation are introduced a posteriori into the molecular Hamiltonian. 

Treating the protein atoms as well as the surrounding solvent molecules explicitly constitutes the main alternative to continuum models. This obviously incurs a much heavier computational cost and simplified solvent models have been introduced to reduce it. The Langevin dipole (LD) model [299] is such an example here each molecule is represented by a polarisable point dipole, assumed to obey the Langevin polarisation law, located on a 3D grid with a cubic unit shell. This model has been progressively improved and recently used in free energy simulation calculations [370, 371]. 

Other recent microscopic approaches are based on the Langevin dipoles solvent model or on the all-atom solvent model, using a standard force field with van der Waals and electrostatic terms as well as intramolecular terms, and molecular dynamics simulations of the fluctuation of the solvent and the solute, incorporating the potential from the permanent and induced solvent dipoles in the solute Hamiltonian in a self-consistent way (Luzhkov and Warshel, 1991). 

The expressions of Vint which are now in use belong to two categories expressions based on a discrete distribution of the solvent, and expressions based on continuous distributions. The first approach leads to quite different methods. We quote here as examples the combined quantum me-chanics/molecular mechanics approach (QM/MM) which introduces in the quantum formulation computer simulation procedures for the solvent (see Gao, 1995, for a recent review), and the Langevin dipole model developed by Warshel (Warshel, 1991), which fits the gap between discrete and continuum approaches. We shall come back to the abundant literature on this subject later. 

The Langevin dipole model (LD) developed by Warshel (Warshel and Levitt, 1976) can be considered as an intermediate step between discrete and continuum models. Solvent (actually water) polarization is described by introducing a grid of polarizable point dipoles, responding to other electic fields according to Langevin s formula ... 

The most common boundary representation is periodic boundary conditions which assumes that the system consists of a periodic array (or a crystal ) of identical systems [1], Another common method, developed for the simulation of biomacromolecules, is the stochastic boundary approach, in which the influences of the atoms outside the boundary are replaced by a simple boundary force [78, 79, 80], Warshel uses a Langevin dipoles model in which the solvent is explicitly replaced by a grid of polarizable dipoles. The energy is calculated in a similar way to the polarization energy in a molecular mechanics force field (see above) [15]. 

An alternative approach to treating the dipole reorientation contribution that is particularly suited to the water surrounding the protein is the Langevin dipole model which describes the polarization due to a permanent solvent dipole of magnitude mLin terms of the applied field... 

Combining Equations (1) and (5) for the protein electronic polarizability and the solvent polarizability, respectively, with Eqn. (10) for the field Warshel and co-workers (Lee et al. 1993 Warshel and Aqvist 1991 Warshel and Russell 1984) developed the Protein Dipole Langevin Dipole (PDLD) model which was the first consistent model for treating protein/solvent polarizabilities in protein electrostatic applications. The electrostatic field distribution in this model is given by... 

Gibbs free energies of hydration, before and after ionization (AG y/l) and AGhyj(2)) were obtained by employing the Langevin dipole relaxation method (45-47) incorporated in the Polaris 3.2 program (46). Before and after ionization, solvent relaxation is modelled by evaluating the relaxation of discrete dipoles distributed on a lattice... 

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