Bo Li

This work concerns both the classical and the ionic size-modified Poisson–Boltzmann (PB) models of the continuum electrostatics for an ionic solution. A unified approach is developed to analyze the minimizers of the PB electrostatic free-energy functionals of ionic concentrations and the solutions to the corresponding PB and the generalized PB equations. Key results of the analysis are the uniform positive bounds for the equilibrium concentrations and the uniform bounds for the solutions of the PB equations. Penalized and constraint-free PB energy functionals are constructed that can be used for solving the underlying variational problems and partial differential equations by machine learning with application to complex charged molecular systems. In addition to the existence and uniqueness of minimizers of such new functionals, uniform bounds with respect to the penalization parameters are obtained for such minimizers. The convergence of the penalized models is finally established.

Bacterial colonies growing on solid surfaces can exhibit robust expansion kinetics, with constant radial growth and saturating vertical expansion, suggesting a common developmental program. Here, we study this process for Escherichia coli cells using a combination of modeling and experiments. We show that linear radial colony expansion is set by the verticalization of interior cells due to mechanical constraints rather than radial nutrient gradients as commonly assumed. In contrast, vertical expansion slows down from an initial linear regime even while radial expansion continues linearly. This vertical slowdown is due to limitation of cell growth caused by vertical nutrient gradients, exacerbated by concurrent oxygen depletion. Starvation in the colony interior results in a distinct death zone which sets in as vertical expansion slows down, with the death zone increasing in size along with the expanding colony. Thus, our study reveals complex heterogeneity within simple monoclonal bacterial colonies, especially along the vertical dimension. The intricate dynamics of such emergent behavior can be understood quantitatively from an interplay of mechanical constraints and nutrient gradients arising from obligatory metabolic processes.

We study finite-difference approximations of both Poisson and Poisson-Boltzmann (PB) electrostatic energy functionals for periodic structures constrained by Gauss' law and a class of local algorithms for minimizing the finite-difference discretization of such functionals. The variable of Poisson energy is the vector field of electric displacement and that for the PB energy consists of an electric displacement and ionic concentrations. The displacement is discretized at midpoints of edges of grid boxes while the concentrations are discretize at grid points. The local algorithm is an iteration over all the grid boxes that locally minimizes the energy on each grid box, keeping Gauss' law satisfied. We prove that the energy functionals admit unique minimizers that are solutions to the corresponding Poisson's and charge-conserved PB equation, respectively. Local equilibrium conditions are identified to characterize the finite-difference minimizers of the discretized energy functionals. These conditions are the curl free for the Poisson case and the discrete Boltzmann distributions for the PB case, respectively. Next, we obtain the uniform bound with respect to the grid size h and O(h2)-error estimates in maximum norm for the finite-difference minimizers. The local algorithms are detailed, and a new local algorithm with shift is proposed to treat the general case of a variable coefficient for the Poisson energy. We prove the convergence of all these local algorithms, using the characterization of the finite-difference minimizers. Finally, we present numerical tests to demonstrate the results of our analysis.

The variational implicit-solvent model (VISM) is an efficient approach to biomolecular interactions, where electrostatic interactions are crucial. The total VISM free energy of a dielectric boundary (i.e. solute–solvent interface) consists of the interfacial energy, solute–solvent interaction energy and dielectric electrostatic energy. The last part is the maximum value of the classical and concave Poisson–Boltzmann (PB) energy functional of electrostatic potentials, with the maximizer being the equilibrium electrostatic potential governed by the PB equation. For the consistency of energy minimization and computational stability, here we propose alternatively to minimize the convex Legendre-transformed Poisson–Boltzmann (LTPB) electrostatic energy functional of all dielectric displacements constrained by Gauss’ Law in the solute region. Both integrable and discrete solute charge densities are treated, and the duality of the LTPB and PB functionals is established. A penalty method is designed for the constrained minimization of the LTPB functional. In application to biomolecular interactions, we minimize the total VISM free energy iteratively, while in each step of such iteration, minimize the LTPB energy. Convergence of such a min–min algorithm is shown. Our numerical results on the solvation of a single ion indicate that the LTPB performs better than the PB formulation, providing possibilities for efficient biomolecular simulations.

Cell polarity and movement are fundamental to many biological functions. Experimental and theoretical studies have indicated that interactions of certain proteins lead to the cell polarization which plays a key role in controlling the cell movement. We study the cell polarity and movement based on a class of biophysical models that consist of reaction-diffusion equations for different proteins and the dynamics of a moving cell boundary. Such a moving boundary is often simulated by a phase-field model. We first apply the matched asymptotic analysis to give a rigorous derivation of the sharp-interface model of the cell boundary from a phase-field model. We then develop a robust numerical approach that combines the level-set method to track the sharp boundary of a moving cell and accurate discretization techniques for solving the reaction-diffusion equations on the moving cell region. Our extensive numerical simulations predict the cell polarization under various kinds of stimuli and capture both the linear and the circular trajectories of a moving cell for a long period of time. In particular, we have identified some key parameters controlling different cell trajectories that are less accurately predicted by reduced models. Our work has linked different models and also developed tools that can be adapted for the challenging three-dimensional simulations.

Bacterial colony growth on hard agar is commonplace in microbiology; yet, what occurs inside a growing colony is complex even in the simplest cases. Robust colony expansion kinetics featuring a linear radial growth and a saturating vertical growth indicates a common developmental program which is elucidated here for Escherichia coli cells using a combination of modeling and experiments. Radial colony expansion is found to be limited by mechanical factors rather than nutrients as commonly assumed. In contrast, vertical expansion is limited by glucose depletion inside the colony, an effect compounded by reduced growth yield due to anaerobiosis. Carbon starvation in the colony interior results in substantial cell death within 1-2 days, with a distinct death zone that expands with the growing colony. Overall, the development of simple colonies lacking EPS production and differentiation is dictated by an interplay of mechanical constraints and emergent nutrient gradients arising from obligatory metabolic processes.

Water fluctuates in a hydrophobic confinement, forming multiple dry and wet hydration states through evaporation and condensation. Transitions between such states are critical to both thermodynamics and kinetics of solute molecular processes, such as protein folding and protein–ligand binding and unbinding. To efficiently predict such dry–wet transition paths, we develop a hybrid approach that combines a variational implicit solvation model, a generalized string method for minimum free-energy paths, and the level-set numerical implementation. This approach is applied to three molecular systems: two hydrophobic plates, a carbon nanotube, and a synthetic host molecule Cucurbit[7]uril. Without an explicit description of individual water molecules, our mesoscale approach effectively captures multiple dry and wet hydration states, multiple dry–wet transition paths, such as those geometrically symmetric and asymmetric paths, and transition states, providing activation energy barriers between different states. Further analysis shows that energy barriers depend on mesoscopic lengths, such as the separation distance between the two plates and the cross section diameter of the nanotube, and that the electrostatic interactions strongly influence the dry–wet transitions. With the inclusion of solute atomic motion, general collective variables as reaction coordinates, and the finite-temperature string method, together with an improved treatment of continuum electrostatics, our approach can be further developed to sample an ensemble of transition paths, providing more accurate predictions of the transition kinetics.

We develop a hybrid approach that combines the Monte Carlo (MC) method, a variational implicit-solvent model (VISM), and a binary level-set method for the simulation of biomolecular binding in an aqueous solvent. The solvation free energy for the biomolecular complex is estimated by minimizing the VISM free-energy 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 level-set 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 p53-MDM2 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 free-energy landscape and kinetic pathways of ligand binding to proteins.

We analyze the passage from discrete (i.e., point) charges and the corresponding discrete electrostatic energies to a continuum charge density and the corresponding continuum electrostatic energy i...