Variational Methods for Biomolecular Modeling

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 coarsegraining or multiscale molecular simulations are outlined.

Water models for biomolecular simulations

“…An early example of this type of approach are ‘Langevin dipole’ family of models based on point dipoles . In this very broad class of ‘all at once’ are also models based directly on the use of the fundamental variational principles, including classical density functional approaches, integral equation formalism and (3D‐)Reference Interaction Site Model (RISM), where the coupling between the polar and nonpolar components of the solvation energy is automatically accounted for. Another broad class of models that aim to strike a good balance between accuracy and speed are hybrid explicit/implicit models where some parts of the solvent are treated explicitly, for example ions .…”

Modeling the Effects of Calcium Overload on Mitochondrial Ultrastructural Remodeling

“…The spontaneous pattern formation and evolution of a two-component system can be described by a number of models, including the Canham–Helfrich curvature functional model, the Ginzburg–Landau theory, and the Cahn–Hilliard (CH) free energy [ 55 - 58 ]. Among them, the Canham–Helfrich model is typically used to describe cell membrane pattern formation in a water-based environment, driven by the curvature energy per unit area of the closed lipid bilayer, osmotic pressure, and surface tension [ 59 ].…”

Inclusion of Water Multipoles into the Implicit Solvation Framework Leads to Accuracy Gains

“…Models aiming to go beyond the popular PE for two dielectric media (solute cavity/solvent) consider various forms of position-dependent dielectric function, − but still neglect important corrections from the multipole moments of water molecules beyond the dipole; these effects are also missing from several more sophisticated “beyond PE” solvent models based on point dipoles. − Examples of other “beyond PE” models include RISM (3D-RISM), − integral equation formalism, and explicit/implicit hybrid solvent models that consider the nearest to solute layers of solvent at the atomic level, ,,,− including semiexplicit assembly methods. , These models, useful in their respective domains, account for many of the explicit water effects “all at once”. Approaches based directly on the fundamental variational principles ,,,,− are arguably among the most conceptually advanced, physics-based implicit solvent models. Recently, approaches based on deep neural networks (DNNs) began to show promise in improving the accuracy of description of complex solvation effects, − including a strategy in which the initial prediction by a physics-based implicit solvent model is further refined by a DNN correction …”

“… 136 , 137 These models, useful in their respective domains, account for many of the explicit water effects “all at once”. Approaches based directly on the fundamental variational principles 34 , 36 , 107 , 115 , 138 − 141 are arguably among the most conceptually advanced, physics-based implicit solvent models. Recently, approaches based on deep neural networks (DNNs) began to show promise in improving the accuracy of description of complex solvation effects, 142 − 149 including a strategy in which the initial prediction by a physics-based implicit solvent model is further refined by a DNN correction.…”