Probabilistic Interpretation for the Nonlinear Poisson-boltzmann Equation in Molecular Dynamics

The Poisson-Boltzmann (PB) equation describes the electrostatic potential of a biomolecular system composed by a molecule in a solvent. The electrostatic potential is involved in biomolecular models which are used in molecular simulation. In consequence, nding an e cient method to simulate the numerical solution of PB equation is very useful. As a rst step, we establish in this paper a probabilistic interpretation of the nonlinear PB equation with Backward Stochastic Di erential Equations (BSDEs). This interpretation requires an adaptation of existing results on BSDEs. Résumé. En dynamique moléculaire, l'équation de Poisson-Boltzmann (PB) permet de décrire le potentiel électrostatique d'un système moléculaire composé d'une molécule dans un solvant. Ce potentiel électrostatique intervient dans les modèles de simulation numérique permettant de comprendre la structure, la dynamique et le fonctionnement des protéines. La résolution numérique de l'équation de PB est donc une étape importante de ces simulations. Aussi, nous proposons dans un premier temps, une interprétation probabiliste de l'équation de PB non-linéaire à l'aide des Equations Di érentielles Stochastiques Rétrogrades (EDSR). Cette interprétation nécessite une adaptation des résultats d'existence et d'unicité des solutions d'EDSR. Introduction In this paper, we establish a probabilistic interpretation for the solution of the nonlinear Poisson-Boltzmann (PB) equation −∇. ( (x)∇u(x)) + κ(x) sinh {u(x)} = g(x), x ∈ R, where and κ are piecewise constant functions from R to (0,+∞), and g is a singular measure. Our probabilistic interpretation consists in the sum of a singular known function and the solution of a particular Backward Stochastic Di erential Equation (BSDE). We prove an existence and uniqueness result for this BSDE by extending to our situation the methodology developed in [BDH03] or [Par99]. The PB equation is used to calculate electrostatic energies and forces in molecular simulation. The numerical methods which are already used are Finite Element, Boundary Element or Finite Di erence (cf. [BBC06] for a discussion on these methods). The probabilistic interpretation we state in this work should provide theoretical foundations for the use of BSDEs simulation methods. In Section 1, we introduce the Poisson-Boltzmann equation in molecular dynamics and underline three di culties we have to face to get a probabilistic interpretation of the nonlinear Poisson-Boltzmann equation. In Section 2, we recall the probabilistic interpretation of divergence-form operators with discontinuous coefcients in term of SDE with weighted local time. 1 TOSCA project-team, INRIA Sophia Antipolis, France. [email protected] c © EDP Sciences, SMAI 2012 ha l-0 06 48 18 0, v er si on 3 1 M ar 2 01 2 2 ESAIM: PROCEEDINGS In Section 3, we very brie y introduce BSDEs theory, before giving a general result for the existence and uniqueness of the solution of BSDEs with time-dependent monotonicity (Theorem 3.3), which we apply in order to get a probabilistic interpretation theorem (Theorem 3.5) of the solution of the corresponding Partial Di erential Equation (PDE). In Section 4, we state our main result (Theorem 4.1). In order to prove it, we state an existence, uniqueness and regularity theorem (Theorem 4.2) for the solution v in H(R) to the regularised Poisson-Boltzmann equation. Then we conclude in Section 4.3 by giving the steps of proof for our main result. 1. The Poisson-Boltzmann equation in Molecular Dynamics The electrostatic potential of a biomolecular system in a medium of electric permittivity generated by a charge distribution ρc is given by the Poisson equation (cf. [BBC06]) −∇. ( (x)∇u(x)) = 4πρc(x), x ∈ R. We consider a molecule immersed in an aqueous solvent. The charge distribution ρc can be decomposed as a sum of the molecule contribution ρm and the solvent contribution ρs. The contribution of the molecule is modelled by a nite system of charges centred in the positions of the N atoms of the molecule