Variational Methods for Biomolecular Modeling

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.Comment: 36 page

An alternative view on the Bateman-Luke variational principle

A new derivation of the Bernoulli equation for water waves in three-dimensional rotating and translating coordinate systems is given. An alternative view on the Bateman-Luke variational principle is presented. The variational principle recovers the boundary value problem governing the motion of potential water waves in a container undergoing prescribed rigid-body motion in three dimensions. A mathematical theory is presented for the problem of three-dimensional interactions between potential surface waves and a floating structure with interior potential fluid sloshing. The complete set of equations of motion for the exterior gravity-driven water waves, and the exact nonlinear hydrodynamic equations of motion for the linear momentum and angular momentum of the floating structure containing fluid, are derived from a second variational principle

Multi-function based modeling of 3D heterogeneous wound scaffolds for improved wound healing

This paper presents a new multi-function based modeling of 3D heterogeneous porous wound scaffolds to improve wound healing process for complex deep acute or chronic wounds. An imaging-based approach is developed to extract 3D wound geometry and recognize wound features. Linear healing fashion of the wound margin towards the wound center is mimicked. Blending process is thus applied to the extracted geometry to partition the scaffold into a number of uniformly gradient healing regions. Computer models of 3D engineered porous wound scaffolds are then developed for solid freeform modeling and fabrication. Spatial variation over biomaterial and loaded bio-molecule concentration is developed based on wound healing requirements. Release of bio-molecules over the uniform healing regions is controlled by varying their amount and entrapping biomaterial concentration. Thus, localized controlled release is developed to improve wound healing. A prototype multi-syringe single nozzle deposition system is used to fabricate a sample scaffold. Proposed methodology is implemented and illustrative examples are presented in this paper

Analytical Solutions to General Anti-Plane Shear Problems In Finite Elasticity

This paper presents a pure complementary energy variational method for solving anti-plane shear problem in finite elasticity. Based on the canonical duality-triality theory developed by the author, the nonlinear/nonconex partial differential equation for the large deformation problem is converted into an algebraic equation in dual space, which can, in principle, be solved to obtain a complete set of stress solutions. Therefore, a general analytical solution form of the deformation is obtained subjected to a compatibility condition. Applications are illustrated by examples with both convex and nonconvex stored strain energies governed by quadratic-exponential and power-law material models, respectively. Results show that the nonconvex variational problem could have multiple solutions at each material point, the complementary gap function and the triality theory can be used to identify both global and local extremal solutions, while the popular (poly-, quasi-, and rank-one) convexities provide only local minimal criteria, the Legendre-Hadamard condition does not guarantee uniqueness of solutions. This paper demonstrates again that the pure complementary energy principle and the triality theory play important roles in finite deformation theory and nonconvex analysis.Comment: 23 pages, 4 figures. Mathematics and Mechanics of Solids, 201

A variational principle for fluid sloshing with vorticity, dynamically coupled to vessel motion

A variational principle is derived for two-dimensional incompressible rotational fluid flow with a free surface in a moving vessel when both the vessel and fluid motion are to be determined. The fluid is represented by a stream function and the vessel motion is represented by a path in the planar Euclidean group. Novelties in the formulation include how the pressure boundary condition is treated, the introduction of a stream function into the Euler-Poincar\'e variations, the derivation of free surface variations, and how the equations for the vessel path in the Euclidean group, coupled to the fluid motion, are generated automatically.Comment: 19 pages, 3 figure

Spinor representation of surfaces and complex stresses on membranes and interfaces

Variational principles are developed within the framework of a spinor representation of the surface geometry to examine the equilibrium properties of a membrane or interface. This is a far-reaching generalization of the Weierstrass-Enneper representation for minimal surfaces, introduced by mathematicians in the nineties, permitting the relaxation of the vanishing mean curvature constraint. In this representation the surface geometry is described by a spinor field, satisfying a two-dimensional Dirac equation, coupled through a potential associated with the mean curvature. As an application, the mesoscopic model for a fluid membrane as a surface described by the Canham-Helfrich energy quadratic in the mean curvature is examined. An explicit construction is provided of the conserved complex-valued stress tensor characterizing this surface.Comment: 17 page