# Differential geometry based solvation model II: Lagrangian formulation

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DOI: 10.1007/s00285-011-0402-z

- Cite this article as:
- Chen, Z., Baker, N.A. & Wei, G.W. J. Math. Biol. (2011) 63: 1139. doi:10.1007/s00285-011-0402-z

## Abstract

Solvation is an elementary process in nature and is of paramount importance to more sophisticated chemical, biological and biomolecular processes. The understanding of solvation is an essential prerequisite for the quantitative description and analysis of biomolecular systems. This work presents a Lagrangian formulation of our differential geometry based solvation models. The Lagrangian representation of biomolecular surfaces has a few utilities/advantages. First, it provides an essential basis for biomolecular visualization, surface electrostatic potential map and visual perception of biomolecules. Additionally, it is consistent with the conventional setting of implicit solvent theories and thus, many existing theoretical algorithms and computational software packages can be directly employed. Finally, the Lagrangian representation does not need to resort to artificially enlarged van der Waals radii as often required by the Eulerian representation in solvation analysis. The main goal of the present work is to analyze the connection, similarity and difference between the Eulerian and Lagrangian formalisms of the solvation model. Such analysis is important to the understanding of the differential geometry based solvation model. The present model extends the scaled particle theory of nonpolar solvation model with a solvent–solute interaction potential. The nonpolar solvation model is completed with a Poisson–Boltzmann (PB) theory based polar solvation model. The differential geometry theory of surfaces is employed to provide a natural description of solvent–solute interfaces. The optimization of the total free energy functional, which encompasses the polar and nonpolar contributions, leads to coupled potential driven geometric flow and PB equations. Due to the development of singularities and nonsmooth manifolds in the Lagrangian representation, the resulting potential-driven geometric flow equation is embedded into the Eulerian representation for the purpose of computation, thanks to the equivalence of the Laplace–Beltrami operator in the two representations. The coupled partial differential equations (PDEs) are solved with an iterative procedure to reach a steady state, which delivers desired solvent–solute interface and electrostatic potential for problems of interest. These quantities are utilized to evaluate the solvation free energies and protein–protein binding affinities. A number of computational methods and algorithms are described for the interconversion of Lagrangian and Eulerian representations, and for the solution of the coupled PDE system. The proposed approaches have been extensively validated. We also verify that the mean curvature flow indeed gives rise to the minimal molecular surface and the proposed variational procedure indeed offers minimal total free energy. Solvation analysis and applications are considered for a set of 17 small compounds and a set of 23 proteins. The salt effect on protein–protein binding affinity is investigated with two protein complexes by using the present model. Numerical results are compared to the experimental measurements and to those obtained by using other theoretical methods in the literature.