Geodesic
Monte Carlo methods for linear and non-linear Poisson-Boltzmann equation
Mireille Bossy1 ; Nicolas Champagnat2 ; Hélène Leman3 ; Sylvain Maire4 ; Laurent Violeau1 ; Mariette Yvinec5
1 INRIA Sophia Antipolis – Méditerranée, TOSCA project-team, 2004 route des Lucioles, BP. 93, 06902 Sophia Antipolis Cedex, France;
2 Université de Lorraine, Institut Elie Cartan de Lorraine, UMR 7502, Vandœuvre-lès-Nancy, F-54506, France; Nicolas.Champagnat@inria.fr — CNRS, Institut Elie Cartan de Lorraine, UMR 7502, Vandœuvre-lès-Nancy, F-54506, France — INRIA, TOSCA project-team, Villers-lès-Nancy, F-54600, France
3 CMAP, Ecole Polytechnique, CNRS, route de Saclay, 91128 Palaiseau Cedex-France
4 Aix-Marseille Université, CNRS, ENSAM,LSIS, UMR 7296, F-13397 Marseille
5 INRIA Sophia Antipolis – Méditerranée, GEOMETRICA project-team, 2004 route des Lucioles, BP. 93, 06902 Sophia Antipolis Cedex, France
ESAIM. Proceedings, Tome 48 (2015), pp. 420-446.

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The electrostatic potential in the neighborhood of a biomolecule can be computed thanks to the non-linear divergence-form elliptic Poisson-Boltzmann PDE. Dedicated Monte-Carlo methods have been developed to solve its linearized version (see e.g. [7], [27]). These algorithms combine walk on spheres techniques and appropriate replacements at the boundary of the molecule. In the first part of this article we compare recent replacement methods for this linearized equation on real size biomolecules, that also require efficient computational geometry algorithms. We compare our results with the deterministic solver APBS. In the second part, we prove a new probabilistic interpretation of the nonlinear Poisson-Boltzmann PDE. A Monte Carlo algorithm is also derived and tested on a simple test case.
DOI : 10.1051/proc/201448020

Mireille Bossy 1 ; Nicolas Champagnat 2 ; Hélène Leman 3 ; Sylvain Maire 4 ; Laurent Violeau 1 ; Mariette Yvinec 5

1 INRIA Sophia Antipolis – Méditerranée, TOSCA project-team, 2004 route des Lucioles, BP. 93, 06902 Sophia Antipolis Cedex, France;
2 Université de Lorraine, Institut Elie Cartan de Lorraine, UMR 7502, Vandœuvre-lès-Nancy, F-54506, France; Nicolas.Champagnat@inria.fr — CNRS, Institut Elie Cartan de Lorraine, UMR 7502, Vandœuvre-lès-Nancy, F-54506, France — INRIA, TOSCA project-team, Villers-lès-Nancy, F-54600, France
3 CMAP, Ecole Polytechnique, CNRS, route de Saclay, 91128 Palaiseau Cedex-France
4 Aix-Marseille Université, CNRS, ENSAM,LSIS, UMR 7296, F-13397 Marseille
5 INRIA Sophia Antipolis – Méditerranée, GEOMETRICA project-team, 2004 route des Lucioles, BP. 93, 06902 Sophia Antipolis Cedex, France 
@article{EP_2015_48_a20,
     author = {Mireille Bossy and Nicolas Champagnat and H\'el\`ene Leman and Sylvain Maire and Laurent Violeau and Mariette Yvinec},
     title = {Monte {Carlo} methods for linear and non-linear {Poisson-Boltzmann} equation},
     journal = {ESAIM. Proceedings},
     pages = {420--446},
     publisher = {mathdoc},
     volume = {48},
     year = {2015},
     doi = {10.1051/proc/201448020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/201448020/}
}
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%A Nicolas Champagnat
%A Hélène Leman
%A Sylvain Maire
%A Laurent Violeau
%A Mariette Yvinec
%T Monte Carlo methods for linear and non-linear Poisson-Boltzmann equation
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Mireille Bossy; Nicolas Champagnat; Hélène Leman; Sylvain Maire; Laurent Violeau; Mariette Yvinec. Monte Carlo methods for linear and non-linear Poisson-Boltzmann equation. ESAIM. Proceedings, Tome 48 (2015), pp. 420-446. doi : 10.1051/proc/201448020. http://geodesic.mathdoc.fr/articles/10.1051/proc/201448020/

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