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null (Ed.)We introduce a mathematical modeling framework for the conformational dynamics of charged molecules (i.e., solutes) in an aqueous solvent (i.e., water or salted water). The solvent is treated as an incompressible fluid, and its fluctuating motion is described by the Stokes equation with the Landau–Lifschitz stochastic stress. The motion of the solutesolvent interface (i.e., the dielectric boundary) is determined by the fluid velocity together with the balance of the viscous force,hydrostatic pressure, surface tension, solutesolvent van der Waals interaction force, and electrostatic force. The electrostatic interactions are described by the dielectric Poisson–Boltzmann theory.Within such a framework, we derive a generalized Rayleigh–Plesset equation, a nonlinear stochastic ordinary differential equation (SODE), for the radius of a spherical charged molecule, such as anion. The spherical average of the stochastic stress leads to a multiplicative noise. We design and test numerical methods for solving the SODE and use the equation, together with explicit solvent molecular dynamics simulations, to study the effective radius of a single ion. Potentially, our general modeling framework can be used to efficiently determine the solutesolvent interfacial structures and predict the free energies of more complex molecular systems.more » « less

null (Ed.)We develop a hybrid approach that combines the Monte Carlo (MC)method, a variational implicitsolvent model (VISM), and a binary levelset method forthe simulation of biomolecular binding in an aqueous solvent. The solvation free energy for the biomolecular complex is estimated by minimizing the VISM freeenergy functional of all possible solute−solvent interfaces that are used as dielectric boundaries. This functional consists of the solute volumetric, solute−solvent interfacial, solute−solvent van der Waals interaction, and electrostatic free energy. A technique of shifting the dielectric boundary is used to accurately predict the electrostatic part of the solvation free energy.Minimizing such a functional in each MC move is made possible by our new and fast binary levelset method. This method is based on the approximation of surface area by the convolution of an indicator function with a compactly supported kernel and is implemented by simple flips of numerical grid cells locally around the solute−solvent interface. We apply our approach to the p53MDM2 system for which the two molecules are approximated by rigid bodies. Our efficient approach captures some of the poses before the final bound state. All atom molecular dynamics simulations with most of such poses quickly reach the final bound state.Our work is a new step toward realistic simulations of biomolecular interactions. With further improvement of coarse graining and MC sampling, and combined with other models, our hybrid approach can be used to study the freeenergy landscape and kinetic pathways of ligand binding to proteins.more » « less

null (Ed.)We study analytically and numerically the minimizers for the CahnHilliard energy functional with a symmetric quartic doublewell potential and under a strong anchoring condition(i.e., the Dirichlet condition) on the boundary of an underlying bounded domain. We show a bifurcation phenomenon determined by the boundary value and a parameter that describes the thickness of a transition layer separating two phases of an underlying system of binary mixtures. For the case that the boundary value is exactly the average of the two pure phases, if the bifurcation parameter is larger than or equal to a critical value, then the minimizer is unique and is exactly the homogeneous state. Otherwise, there are exactly two symmetric minimizers. The critical bifurcation value is inversely proportional to the first eigenvalue of the negative Laplace operator with the zero Dirichlet boundary condition. For a boundary value that is larger (or smaller) than that of the average of the two pure phases, the symmetry is broken and there is only one minimizer. We also obtain the bounds and morphological properties of the minimizers under additional assumptions on the domain.Our analysis utilizes the notion of the Nehari manifold and connects it to the eigenvalue problem for the negative Laplacian with the homogeneous boundary condition. We numerically minimize the functional E by solving the gradientflow equation of E, i.e., the AllenCahn equation, with the designated boundary conditions, and with random initial values. We present our numerical simulations and discuss them in the context of our analytical results.more » « less