Schedule for: 16w5118 - Ion Transport: Electrodiffusion, Electrohydrodynamics and Homogenization

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Life is different because it is inherited. All life comes from a blueprint (genes) that can only make proteins. Proteins are studied by more than one hundred thousand scientists and physicians every day because they are so important in health and disease. The function of proteins is on the macroscopic scale, but atomic details control that function, as is shown in a multitude of experiments. The structure of proteins is so important that governments spend billions studying them. Structures are known in exquisite detail determined by crystallographic measurement of more than 105 different proteins. But the forces that govern the movement and function of proteins are not visible in the structure. Mathematics is needed to compute both function and forces so comparison with experiment can be made. Experiments report numbers, typically sets of numbers in the form of graphs. Verbal models, however beautifully written in the biological tradition, do not provide numerical outputs, and so it is difficult to tell which verbal model better fits data. The mathematics of molecular biology must be multiscale because atomic details control macroscopic function. The device approach of the engineering and English physiological tradition provides the dimensional reduction needed to solve the multiscale problem. Mathematical analysis of hundreds of experiments (reported in some fifty papers) has been successful in showing how some properties of an important class of proteins—ion channels— work. Ion channels are natural nanovalves as important to animals as Field Effect Transistors (FETs) are to computers. I will present the Fermi Poisson approach started by Jinn Liang Liu. The Fermi distribution is used to describe the saturation of space produced by crowded spherical ions. The Poisson equation (and continuity of current) is used to describe long range electrodynamics. Short range correlations are approximated by the Santangelo equation. A fully consistent mathematical description reproduces macroscopic properties of bulk solutions of sodium and calcium chloride solutions. It also describes several different channels (with quite different atomic detailed structures) quite well in a wide range of conditions using a handful of parameters never changed. It is not clear why the model works as well it does, nor is it clear how well the model will work on other channels, transporters or proteins.

We have developed a continuum-molecular theory --- Poisson-Nernst-Planck-Fermi theory --- in the last three years for simulating ionic flows in biological ion channels under physiological or experimental conditions by treating ions and water of any diameter as hard spheres with interstitial voids and polarization of water. The theory can compute electric and steric potentials from all atoms in a protein and all ions and water molecules in channel pore while keeping electrolyte solutions in the extra- and intracellular baths as a continuum dielectric medium. The PNPF model has been verified with the experimental data of gramicidin A, sodium/calcium exchanger, and transient receptor potential channels all in real structures from Protein Data Bank. It was also verified with the experimental data of L-type calcium channels and single ion activities in bulk solutions and with Monte Carlo results of electric double layer.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

In this talk we discuss ion exchange through porous membrane as a model for hemodialysis. A simplified one-dimensional problem has been derived based on matched asymptotic analysis and we present results to show the effect of the electric fields and membrane porosity on the solute transport between blood and dialysate regions. The resulted equations appear to be of mixed types and fully-coupled. Numerical simulations are carried out to probe the nonlinear structure of the partial differential equations (PDE) system and insight is gained on the control of solute distribution and transport across the membrane via a number of system parameters.

The Poisson-Nernst-Planck (PNP) theory is one of the most widely used analytical methods to describe electrokinetic phenomena for electrolytes. The model, however, considers isolated charges and thus is valid only for dilute ion concentrations. The key importance of concentrated electrolytes in application has led to the development of a large family of generalized PNP models. In particular, the Bikerman model that takes into account the finite size of the ions has been one of the most commonly used extension to PNP. In this talk, we derive a thermodynamically consistent mean-field model for concentrated solutions. Our model recovers the Bikerman term, but shows that it is inconsistent in the sense that additional terms of equal magnitude should be taken into account. Furthermore, our study shows that the Bikerman approach inherently fails to describe finite-size effects at the highly concentrated regime, and presents a supplementary approach in this regime. The result is a modelling framework that is valid over the whole range of concentrations - from dilute electrolyte solutions to highly concentrated solution, such as ionic liquids. Importantly, the new model predicts distinct transport properties which are not governed by Einstein-Stokes relations, but are rather effected by inter-diffusion and even the emergence of nano-structure. This is a joint work with Doron Elad and Arik Yochelis.

With a small parameter $\varepsilon$, Poisson-Nernst-Planck (PNP) systems over two-dimensional (2D) annular domains have steady state boundary layer solutions with radial symmetry, which profiles form boundary layers near boundary curves and become flat in the interior domain as $\varepsilon$ approaches zero. For the stability of 2D boundary layer solutions to (time-dependent) PNP systems, we estimate the solution of the linearized problem of the PNP system (with respect to the boundary layer solution) with global electroneutrality. We prove that the $H_{x}^{-1}$ norm of the solution of the linearized problem decays exponentially (in time) with exponent independent of $\varepsilon$ if the coefficient of the Robin boundary condition of electrostatic potential has a suitable positive lower bound. The main difficulty is that the gradients of 2D boundary layer solutions on boundary curves may blow up as $\varepsilon$ tends to zero. The main idea of our argument is to transform the perturbed problem into another parabolic system with a new and useful energy law for the proof of the exponential decay estimate.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Solvation is an elementary process in nature and its understanding is a prerequisite for the study of more complex processes, such as ion channel transport, protein specification, protein-drug binding, and signal transduction. Although there has been much advance in solvation analysis in the past decade, protein-ligand binding prediction, which is at the heart of drug design, remains a grand challenge. We discuss a number of mathematical techniques for automatic and blind prediction of molecular solvation and protein-ligand binding free energies. Our approaches include multiscale modeling, variation PDE, differential geometry, graph Laplacian, uncertainty quantification, and machine learning. Extensive comparison with experimental data confirms the superiority of mathematical methodologies.

We review recent advances in particle simulations, continuum theory and multiscale modeling for equilibrium and transport properties of Coulomb many-body systems in soft matter and electrochemical energy devices. The properties of dielectric and correlation effects near material interfaces are discussed under image-charge based Monte Carlo simulations. These properties are also modeled by modified Poisson-Nernst-Planck/Poisson-Boltzmann equations incorporated with a Green's function governed by a generalized Debye-Hückel equation. Numerical methods for both particle simulations and continuum equations are present with a comparison to show attractive performance of the new model and numerical algorithms.

Gating currents of the voltage sensor involve back-and forth movements of positively charged arginines through the hydrophobic plug of the gating pore. Transient movements of the permanent charge of the arginines induce structural changes and polarization charge nearby. The moving permanent charge induces current flow everywhere. Everything interacts with everything else in this structural model so everything must interact with everything else in the mathematics, as everything does in the structure. Energy variational methods EnVarA are used to compute gating currents in which all movements of charge and mass satisfy conservation laws of current and mass. Conservation laws are partial differential equations in space and time. Ordinary differential equations cannot capture such interactions with one set of parameters. Indeed, they may inadvertently violate conservation of current. Conservation of current is particularly important since small violations (<0.01%) quickly (microseconds) produce forces that destroy molecules. Our model reproduces signature properties of gating current: (1) equality of ON and OFF charge (2) saturating voltage dependence and (3) many (but not all) details of the shape of charge movement as a function of voltage, time, and solution composition. The model computes gating current flowing in the baths produced by arginines moving in the voltage sensor. The movement of arginines induces current flow everywhere producing ‘capacitive’ pile ups at the ends of the channel. Such pile-ups at charged interfaces are well studied in measurements and theories of physical chemistry but they are not typically included in models of gating current or ion channels. The pile-ups of charge change local electric fields, and they store charge in series with the charge storage of the arginines of the voltage sensor. Implications are being investigated. Joint with Robert S. Eisenberg, Chun Liu, Francisco Bezanilla

We discuss recent progress in developing computationally derived transfer free energy scales for beta barrel outer membrane proteins and show how they can be used to detect functional residues, to identify structural deformation, and to understand asymmetric insertion and folding mechanism of beta barrel membrane protein. We further show how artificial outer membrane protein can be designed based on motif analysis and stability calculation using transfer free energies, along with experimental validations using CD, fluorescence, dye leakage, and single chanel measurements. We then discuss progress in developing a mechanistic model of pH-dependent gating of outer membrane protein G (OmpG). Through analysis of 100,000 independently genereated samples of 3D conformations of multiplly interacting loops of OmpG, we uncover characteristic loop configuration for pH dependent conducting states and discuss implications for pH-induced functional switch of OmpG. More information can be found at: www.uic.edu/~jliang (Joint work with Meishan Lin, Wei Tian, Alan Perez-Rathke, Linda Kenney, Ge Zhang, Tim Keiderling, Christina Chisholm, and Min Chen)

Many PDE-based models of collective cell behaviour implicitly assume that the population of cells is ‘well mixed’. This is called a spatial mean-field assumption. In reality, populations often have a more complex spatial structure, such as clusters and/or spatial segregation of cells. This spatial structure is both a cause and an effect of interactions among neighbouring cells and can make a significant difference to model predictions about, for example, cell densities and invasion speeds. I will describe an individual-based model of collective cell behaviour that is based on interactions between pairs of cells. I will show how a neighbour-dependent directional bias can be included in the model and how spatial moment dynamics can be used to give a continuum-level description of the population that retains spatial structure beyond the mean-field.

Calculations for the streaming flow around deformable bodies are of interest in a number of areas including physiology and chemical engineering. Electroporation is an experimental technique in which the application of a large electric pulse results in the dielectric breakdown of a cell membrane; the formation of a pore allows the introduction of once-impermeant molecules into the cell. Illumination of liposomes is another technique where excitation of the fluorescently labeled lipids to leads to the formation of pores that widen in order to relieve mechanical tension. In order to make progress in modeling these phenomena, and eventually use the techniques for medical applications, it is useful to know the details of the flow patterns and quantify the rate at which electric or mechanical energy is dissipated by viscous losses. A fairly complete analytical description is possible in this situation because even though the initial deformations can be large, pore widening and closure occur over a long time scale while the cell is nearly spherical.

We focus on numerical methods for the diffusion equation for the random genetic drift that occurs at a single unlinked locus with two or more alleles. It is governed by a degenerated convection-dominated parabolic equation. Due to the degeneration and convection, Dirac singularities will always be developed at boundary points as time e- volves, which is just the so-called fixation phenomenon. In order to find a complete solution which should keep the conservation of total probability and expectation, three different schemes based on finite volume methods are used to solve the equation numerically: one is a upwind scheme, the other two are different central schemes. We observed that all the methods are stable and can keep the total prob- ability, but have totally different long-time behaviors concerning with the conservation of expectation. We prove that any extra infinitesi- mal diffusion leads to a same artificial steady state. So upwind scheme does not work due to its intrinsic numerical viscosity. We find one of the central schemes introduces a numerical viscosity term too, which is beyond the common understanding in the convection-diffusion com- munity. Careful analysis is presented to prove that the other cen- tral scheme does work. Our study shows that the numerical methods should be carefully chosen and any method with intrinsic numerical viscosity must be avoided. This is a joint work with Chun Liu, David Waxman and Shixin Xu.

Pore network models are widely used in materials science to derive macroscopic properties of a material, based on physical phenomena at the pore level. One open question is whether they are also applicable to polymer electrolyte membranes (PEM), where water and protons flow through nanoscopic pores. Can the properties of a soft porous medium be reproduced in which pores change their shape dynamically with different operating conditions? This contribution is the first attempt at applying advanced PEM pore network models to study the relation between electro-osmotic coefficients at the pore level and electro-osmotic drag coefficients at the macroscopic level, as measured in experiments. Strengths and shortcomings of this approach are described, along with future goals. These include simulations of dynamic water uptake, transport across the PEM surface, and scaling laws for conductivity, permeability and electro-osmotic drag. References [1] P. Berg, M. Stornes, Towards a consistent interpretation of electro-osmotic drag in polymer electrolyte membranes, Fuel Cells DOI: 10.1002/fuce.201500210, 1-10 (2016) [2] P. Berg, S.-J. Kimmerle, A. Novruzi, Modeling, shape analysis and computation of the equilibrium pore shape near a PEM-PEM intersection, J. Math. Anal. Appl. 410, 241-256 (2014) [3] M. Eikerling, P. Berg, Poroelectroelastic theory of water sorption and swelling in polymer electrolyte membranes, Soft Matter 7, 5976-5990 (2011) [4] P. Berg, J. Findlay, Analytical solution of PNP-Stokes equations in a cylindrical channel, Proc. Roy. Soc. A 467, 3157-3169 (2011)

This talk concerns new continuum phenomenological model for epitaxial thin film growth with three different forms of the Ehrlich-Schwoebel current. Two of these forms were first proposed by Politi and Villain and then studied by Evans, Thiel and Bartelt. The other one is completely new. Following the techniques used in Li and Liu, we present rigorous analysis of the well-posedness, regularity and time stability for the new model. We also studied both the global and the local behavior of the surface roughness in the growth process. The new model differs from other known models in that it features a linear convex part and a nonlinear concave part, and thus by using a convex-concave time splitting scheme, one can naturally build unconditionally stable semi-implicit numerical discretizations with linear implicit parts, which is much easier to implement than conventional models requiring nonlinear implicit parts. Despite this fundamental difference in the model, numerical experiments show that the nonlinear morphological instability of the new model agrees well with results of other models, which indicates that the new model correctly captures the essential morphological states in the thin film growth process.

We want to look at the macro end of neurovascular coupling (or other coupled cell model) where PNP is approximated. How do we make the jump from Debye length to cell length to artery length? What is the best way of developing a truly multi-scale model?

Additional talks, group discussion and/or hiking

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.