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Variational Methods for Biomolecular Modeling

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Variational Methods in Molecular Modeling

Part of the book series: Molecular Modeling and Simulation ((MMAS))

Abstract

Structure, function and dynamics of many biomolecular systems can be characterized by the energetic variational principle and the corresponding systems of partial differential equations (PDEs). This principle allows us to focus on the identification of essential energetic components, the optimal parametrization of energies, and the efficient computational implementation of energy variation or minimization. Given the fact that complex biomolecular systems are structurally non-uniform and their interactions occur through contact interfaces, their free energies are associated with various interfaces as well, such as solute-solvent interface, molecular binding interface, lipid domain interface, and membrane surfaces. This fact motivates the inclusion of interface geometry, particular its curvatures, to the parametrization of free energies. Applications of such interface geometry based energetic variational principles are illustrated through three concrete topics: the multiscale modeling of biomolecular electrostatics and solvation that includes the curvature energy of the molecular surface, the formation of microdomains on lipid membrane due to the geometric and molecular mechanics at the lipid interface, and the mean curvature driven protein localization on membrane surfaces. By further implicitly representing the interface using a phase field function over the entire domain, one can simulate the dynamics of the interface and the corresponding energy variation by evolving the phase field function, achieving significant reduction of the number of degrees of freedom and computational complexity. Strategies for improving the efficiency of computational implementations and for extending applications to coarse-graining or multiscale molecular simulations are outlined.

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Notes

  1. 1.

    Water constitutes a large percentage of cellular mass and therefore biomolecules are mostly living in an aqueous environment where various types of ions such as sodium (Na\(^{+}\)), potassium (K\(^{+}\)), calcium (Ca\(^{2+}\)), and chloride (Cl\(^{-}\)) present at different concentrations.

  2. 2.

    In Sect. 2, we use S to denote the surface function, which is a domain indicator, and use \(\varPhi \) to denote the electrostatic potential following the traditional usage in the studies of biomolecular electrostatics. Here in Sects. 3 and 4 the models do not involve electrostatics, and we denote \(\phi \) the phase field function, while use S to denote the 2D surface embedded in \(\mathbb {R}^3\) when applicable. An interface in Sect. 2 refers to solvent-solute boundary region, whereas in Sects. 3 and 4, it refers a boundary curve on a given surface.

  3. 3.

    A protein unit consisting of several segments such as most ion channel proteins or G-protein-coupled receptors (GPCRs) is not taken as a distinct domain in this study. The whole unit is considered as a single protein instead.

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  1. Department of Mathematics, Michigan State University, East Lansing, Michigan, 48824, USA

    Guo-Wei Wei

  2. Department of Mathematics, Colorado State University, Fort Collins, Colorado, 80523, USA

    Yongcheng Zhou

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    Jianzhong Wu

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Wei, GW., Zhou, Y. (2017). Variational Methods for Biomolecular Modeling. In: Wu, J. (eds) Variational Methods in Molecular Modeling. Molecular Modeling and Simulation. Springer, Singapore. https://doi.org/10.1007/978-981-10-2502-0_7

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