# Poster Session and Reception

Tuesday, July 21, 2015 - 4:05pm - 6:00pm

Lind 400

**Time-Domain Matched Interface and Boundary Methods for Transverse Electric Modes with Complex Dispersive Iinterfaces**

Duc Nguyen (University of Alabama)

The material is dispersive when its permittivity or permeability are functions of frequency. Therefore, the dispersive material is often used to simulate the electromagnetic waves movements in the complex environment such as in soils, rock, ice, snow, and biological tissue. As a result, it plays an important role in numerous electromagnetic applications. For instance, the ground penetrating radar (GPR) and microwave imaging for early detection of breast cancer are involved in dealing with dispersive soil and dispersive tissue respectively. It is known that the transverse electric (TE) Maxwell’s equations with the presence of the dispersive media produce non-smooth and discontinuous solutions. We formulate the interface auxiliary differential equations (IADEs) to acquire evanescent changes of the field regularities along the interface. A novel matched interface boundary time-domain (MIBTD) based on the leapfrog scheme is proposed to rigorously implement the time-dependent jump conditions. Numerical tests indicate the second order of accuracy is achieved in both $L_\infty$ and $L_2$.**A New Finite Element and Finite Difference Hybrid Method for Computing Electrostatics of Ionic Solvated Biomolecule**

Dexuan Xie (University of Wisconsin)

The Poisson-Boltzmann equation (PBE) is one widely-used implicit solvent continuum model for calculating electrostatics of ionic solvated biomolecule. In this poster, a new finite element and finite difference hybrid method is presented to solve PBE efficiently based on a special seven-overlapped box partition with one central box containing the solute region and surrounded by six neighboring boxes. In particular, an efficient finite element solver is applied to the central box while a fast preconditioned conjugate gradient method using a multigrid V-cycle preconditioning is constructed for solving a system of finite difference equations defined on a uniform mesh of each neighboring box. Moreover, the PBE domain, the box partition, and an interface fitted tetrahedral mesh of the central box can be generated adaptively for a given PQR file of a biomolecule. This new hybrid PBE solver is programmed in C, Fortran, and Python as a software tool for predicting electrostatics of a biomolecule in a symmetric 1:1 ionic solvent. Numerical results on two test models with analytical solutions and 12 proteins validate this new software tool, and demonstrate its high performance in terms of CPU time and memory usage.

The related paper has been published on Journal of Computational Physics, Vol. 298, pages 636 - 651, 2015. It can be downloaded from the website http://authors.elsevier.com/a/1RKI7508HRsXx

This project is a joined work with my student, Jinyong Ying, under the support by NSF award DMS-1226259.**Variational Multiscale Modeling of Biomolecular Complexes**

Kelin Xia (Michigan State University)**Parameter Optimization in Differential Geometry-based Solvation Models**

Bao Wang (Michigan State University)

Differential geometry based solvation model is a new class of implicit solvent model, which construct the solute solvent separation based on the energy minimization principle, the model is formulated as the coupled generalized Poisson Boltzmann and generalized Laplace Beltrami equations. The model automatically circumvents lots of unphysical surface definition, meanwhile the numerical method for solving the system is easier than the classical Poisson Boltzmann model. However, the parameter optimization is an issue we encounter in solving the model. We present a convex optimization based parameter optimization framework in the model, which guarantees the stability of the numerical solution to the coupled system, also the best solvation free energy prediction is provided.**Molecular Based Mathematical Biology**

Shan Zhao (University of Alabama)

Molecular Based Mathematical Biology is a unique Open Access electronic journal that publishes all mathematical and quantitative results concerning the molecular-level biological sciences. One of major features of biological sciences in the 21st century will be their transition from phenomenological and descriptive sciences to quantitative and predictive ones. Mathematics will be a major driven force for the transition. MBMB is devoted to the mathematical modeling, computation, and analysis of molecular-level biological sciences.**Differential Geometry Based Ion Transport Models**

Guowei Wei (Michigan State University)**An Online Server for Electrostatic Analysis**

Zhixiong Zhao (Michigan State University)**Two Faces of Ephaptic Coupling in Cardiac Arrhythmias**

Ning Wei (University of Minnesota, Twin Cities)

Introduction: Introduction: Decreased expressions of connexion 43 (Cx43) and disrupted heterogeneity are the common features in animal heart failure models. Ephaptic coupling, which relies on the presence of intercalated disc (or cleft) between adjacent cells, was found to substantially increase conduction velocity in cardiac tissue with sufficiently low Cx43 expressions. However, interaction between heterogeneous Cx43 expressions and ephaptic coupling on ventricular arrhythmias is still unclear.

Methods: Methods: We develop a 2D bidomain model while incorporating ephaptic coupling. Cleft is modelled as a narrow compartment with resistive

connections to extracellular space. Adjacent cells are ephaptically coupled

through a shared cleft connecting active membranes at the ends of cells.

Genotypes of cells are allocated stochastically according to binomial distribution defined by the Cx43KO content. Simulations are performed 100 times for every Cx43KO content and ephaptic coupling strength in Matlab. Membrane currents are simulated using Luo-Rudy dynamic guinea pig ven-

tricular model 2007, with localized sodium channels distribution.

Results: Results: When Cx43KO content is 100%, we observe a smaller amplitude travelling action potential (SAP) as as result of ephaptic coupling, with wave front peaked at -40 mv. We further identify several modes of cardiac propagation in which ephaptic and gap junctions-mediated mechanisms alternate. In the co-culturing 2D narrow strands, we discover that ephaptic coupling can lower the possibilities of block when Cx43KO con-

tent is greater than 60%. However, in 2D wide strands, the possibilities of

block are increased with the ephaptic mechanism when Cx43KO content is between 40% and 60%.

Conclusions: Conclusions: Our findings reveal an antiarrhythmic role of ephaptic coupling when Cx43KO content is greater than 60%. In particular, as Cx43KO content is increased to 100%, the SAP phenomenon may reveal marked experimental improvements in cardiac signal propagation from myocardial infarct to healthy regions via ephaptic coupling. However, ephaptic mechanism potentiates arrhythmias when Cx43KO content is between 40% and

60%. The detailed effects of ephaptic coupling can only be demonstrated with localized sodium.**Three-dimensional Multi-scale Models of Deformable Platelets and Fibrin Network**

Zhiliang Xu (University of Notre Dame)**Simulating Chemical Diffusion and Osmosis-induced Flow with a Moving Elastic Interface**

Lingxing Yao (Case Western Reserve University)

Based on the model proposed by Yoichiro, Liu, and Eisenberg, here we design a stable and accurate numerical scheme to simulate osmotic flow in living cells. In biological cells, cell membrane is not only permeable to water flow, but also permit ion exchange with the help of ion channels along the membrane. This will lead to osmotic pressure on the two sides of the membrane, which plays a important role in cell migration, as pointed in literature (see Figure 1). In our model system, cell membranes, which are permeable to both water and ionic flows, divide the domain into intracellular and extracellular regions. The cell membrane moves with fluid flow it is embedded in, while its elastic force and osmotic force due to ions will in turn affect fluid properties. The whole system then consists of fluid-structure interactions, coupled with ionic electrodiffusion on a domain with moving interfaces.**A Matched Interface & Boundary Method for Parabolic Equations**

David Bramer (Michigan State University)

A framework for treating parabolic partial differential equations(PDEs) with discontinuous co-effcients and complex interface geometries is developed numerically in three dimensions(3D). By combining the matched interface and boundary method(MIB) and Crank-Nicolson(CN) the authors create a numerical method capable of handling complex interface geometries in 3D which is second-order in both time and space. CN ensures unconditional stability which unlike other numeric parabolic methods has no Courant-Freidrichs-Lewy(CFL) condition. This allows large time steps while maintaining high spatial resolution. To our knowledge this is the first method that combines MIB and CN to solve such problems.**The DC Shift of Cortical Spreading Depression**

Rosemary O'Connell (University of Minnesota, Twin Cities)

Cortical spreading depression (CSD) is a pathological phenomenon which interferes with normal brain activity as neurons depolarize for several minutes. We present a multiphasic continuum model to describe the dynamics of voltages, ion concentrations, and cell volume during CSD. The model incorporates the flow of sodium, potassium, and chloride ions in neurons, glia, and the extracellular space through electrodiffusion and across cell membranes through various channels, pumps, and transporters. Cell volume changes as a result of osmosis. With this model, we explore the effects of several parameters on the propagation speed and extracellular DC shift associated with CSD. Our simulations reveal an intricate connection between the DC shift and glial cells, acting as potassium buffers.**Calculating Energy Barriers for Steps of Fusion**

Rolf Ryham (Fordham University)

We use continuum mechanics to calculate the entire least energy pathway of membrane fusion. The static continuum stalk structure agrees with experimental stalk architecture and the string method is used to determine the dynamics of two bilayers as they pass through intermediate states; from stalk formation, to pore creation, and through fusion pore enlargement. Hemifusion requires a small amount of energy independently of lipid composition, while hemifusion diaphragm expansions is spontaneous for distal monolayers containing at least two lipid components, given sufficiently negative diaphragm spontaneous curvature. Conversely, diaphragms formed from single component distal monolayers do not expand without the continual injection of energy. We identify a diaphragm radius below which central pore expansion is spontaneous, independent of lipid composition.**Free Energy Dissipative Discretizations of the Poisson-Nernst-Planck System**

Michael White (University of Minnesota, Twin Cities)**Coupling a Mechanosensitive Channel with a Vesicle Under Shear Flow**

Yuan-Nan Young (New Jersey Institute of Technology)**Dimension Reduction of Langevin Dynamics**

Lina Ma (The Pennsylvania State University)

Molecular dynamics (MD) is a powerful tool in modeling of biomolecules, and the conventional temporal scale is femto seconds. Due to the multi scale nature of biology, coarse-grained models become necessary. In this work we present dimension reduction of Langevin dynamics, which include an integrator scheme that fits classic results in both high and low friction regime, and a coarse-grained model that significantly improve the numerical stability.**A 3D Implementation of the Electro-Neutral Model**

Adam Stinchcombe (University of Michigan)

Mori and Peskin developed a model of cellular electrical activity that includes the electrodiffusion of ions and the physics of space-charge layers near cellular membranes. We present an implementation of a finite volume cut-cell method in three spatial dimensions. High performance is obtained by parallel computation with NVIDIA CUDA. Being able to consider many different ion species, a distribution of ion channels, spatial and compartment dependent diffusivities, and arbitrary membrane geometry, we are seeking interesting applications.**New Numerical Algorithms for Solving Size Modified Poisson Boltzmann Equation**

Jiao Li (Changsha University of Science and Technology)

The size modified Poisson Boltzmann equation (SMPBE) is one important variant of the classic Poisson Boltzmann equation (PBE) to reflect ionic size effects. We propose two algorithms for solving SMPBE based on our solution decomposition: One is a finite element solver and the other is a hybrid solver. We then program them as a software package for calculating electrostatics of a biomolecule in an ionic solvent based on the state-of-the-art finite element library DOLFIN from the FEniCS project. Numerical results validate the new software and demonstrate its high performance.