This paper concerns the analytic structure of the self-consistent field theory (SCFT) energy functional of multicomponent block copolymer systems which contain more than two chemically distinct blocks. The SCFT has enjoyed considered success and wide usage in investigation of the complex phase behavior of block copolymers. It is well-known that the physical solutions of the SCFT equations are saddle points, however, the analytic structure of the SCFT energy functional has received little attention over the years. A recent work by Fredrickson and collaborators [see the monograph by Fredrickson, The Equilibrium Theory of Inhomogeneous Polymers, (2006), pp. 203–209] has analysed the mathematical structure of the field energy functional for polymeric systems, and clarified the index-1 saddle point nature of the problem caused by the incompressible constraint. In this paper, our goals are to draw further attention to multicomponent block copolymers utilizing the Hubbard-Stratonovich transformation used by Fredrickson and co-workers. We firstly show that the saddle point character of the SCFT energy functional of multicomponent block copolymer systems may be high index, not only produced by the incompressible constraint, but also by the Flory-Huggins interaction parameters. Our analysis will be beneficial to many theoretical studies, such as the nucleation theory of ordered phases, the mesoscopic dynamics. As an application, we utilize the discovery to develop the gradient-based iterative schemes to solve the SCFT equations, and illustrate its performance through several numerical experiments taking ABC star triblock copolymers as an example.

The conventional Poisson-Nernst-Planck equations do not account for the finite size of ions explicitly. This leads to solutions featuring unrealistically high ionic concentrations in the regions subject to external potentials, in particular, near highly charged surfaces. A modified form of the Poisson-Nernst-Planck equations accounts for steric effects and results in solutions with finite ion concentrations. Here, we evaluate numerical methods for solving the modified Poisson-Nernst-Planck equations by modeling electric field-driven transport of ions through a nanopore. We describe a novel, robust finite element solver that combines the applications of the Newton’s method to the nonlinear Galerkin form of the equations, augmented with stabilization terms to appropriately handle the drift-diffusion processes.

To make direct comparison with particle-based simulations possible, our method is specifically designed to produce solutions under periodic boundary conditions and to conserve the number of ions in the solution domain. We test our finite element solver on a set of challenging numerical experiments that include calculations of the ion distribution in a volume confined between two charged plates, calculations of the ionic current though a nanopore subject to an external electric field, and modeling the effect of a DNA molecule on the ion concentration and nanopore current.

Many molecular events are associated with small or large conformational changes occurring in the corresponding proteins. Modeling such changes is a challenge and requires significant amount of computing time. From point of view of electrostatics, these changes can be viewed as a reorganization of local charges and dipoles in response to the changes of the electrostatic field, if the cause is insertion or deletion of a charged amino acid. Here we report a large scale investigation of modeling the changes of the folding energy due to single mutations involving charged group. This allows the changes of the folding energy to be considered mostly electrostatics in origin and to be calculated with DelPhi assigning residue-specific value of the internal dielectric constant of protein. The predicted energy changes are benchmarked against experimentally measured changes of the folding energy on a set of 257 single mutations. The best fit between experimental values and predicted changes is used to find out the effective value of the internal dielectric constant for each type of amino acid. The predicted folding free energy changes with the optimal, amino acid specific, dielectric constants are within RMSD=0.86 kcal/mol from experimentally measured changes.

Here we report a web server, the DelPhi web server, which utilizes DelPhi program to calculate electrostatic energies and the corresponding electrostatic potential and ionic distributions, and dielectric map. The server provides extra services to fix structural defects, as missing atoms in the structural file and allows for generation of missing hydrogen atoms. The hydrogen placement and the corresponding DelPhi calculations can be done with user selected force field parameters being either Charmm22, Amber98 or OPLS. Upon completion of the calculations, the user is given option to download fixed and protonated structural file, together with the parameter and Delphi output files for further analysis. Utilizing Jmol viewer, the user can see the corresponding structural file, to manipulate it and to change the presentation. In addition, if the potential map is requested to be calculated, the potential can be mapped onto the molecule surface. The DelPhi web server is available from http://compbio.clemson.edu/delphi_webserver.

Experimental data shows that the binding of human prolactin (hPRL) to human prolactin receptor (hPRLr-ECD) is strongly pH-dependent, while the binding of the same receptor to human growth hormone (hGH) is pH-independent. Here we carry in silico analysis of the molecular effects causing such a difference and reveal the role of individual amino acids. It is shown that the computational modeling correctly predicts experimentally determined pKa’s of histidine residues in an unbound state in the majority of the cases and the pH-dependence of the binding free energy. Structural analysis carried in conjunction with calculated pH-dependence of the binding revealed that the main reason for pH-dependence of the binding of hPRL-hPRLr-ECD is a number of salt-bridges across the interface of the complex, while no salt-bridges are formed in the hGH-hPRlr-ECD. Specifically, most of the salt-bridges involve histidine residues and this is the reason for the pH-dependence across a physiological range of pH. The analysis not only revealed the molecular mechanism of the pH-dependence of the hPRL-hPRLr-ECD, but also provided critical insight into the underlying physic-chemical mechanism.