Abstracts
Dieter Bothe
Continuum thermodynamics of multicomponent fluid mixtures and implications
for modeling electromigration of ionic species (slides)
Shuhao Cao
Adaptive Finite Element Methods for Convection-Reaction Equation
The convection-reaction equation comes from the linearization of the hyperbolic
conservation law that models the transport phenomena. In previous works,
methods like streamline diffusion finite element method (SDFEM) have been
invented to battle the instability coming from the convection term. In this
talk, we opt for a stabilized Discontinuous Galerkin finite element method
(DGFEM) to discretize the convection-reaction equation, and discuss the
difficulty lying in the a posteriori error estimation for this discretization,
especially the trade between the construction of a suitable interpolation
and posing an extra saturation assumption. Several a posteriori error estimators
are constructed, and some numerical examples are presented.
Bob Eisenberg
Ions in Solutions and Channels: the plasma of life
All of biology occurs in ionic solutions that are plasmas in both the physical
and biological meanings of the word. The composition of these ionic mixtures
has profound effects on almost all biological functions, whether on the
length scale of organs like the heart or brain, of the length scale of proteins,
like enzymes and ion channels.
Ion channels are proteins with a hole down their middle that conduct ions
(spherical charges like Na+, K+, Ca2+, and Cl- with diameter ~ 0.2 nm) through
a narrow tunnel of fixed charge ('doping') with diameter ~ 0.6 nm. Ionic
channels control the movement of electric charge and current across biological
membranes and so play a role in biology as significant as the role of transistors
in computers: almost every process in biology is controlled by channels,
one way or the other.
Ionic channels are manipulated with the powerful techniques of molecular
biology in hundreds of laboratories. Atoms (and thus charges) can be substituted
a few at a time and the location of every atom can be determined in favorable
cases. Ionic channels are one of the few living systems of great importance
whose natural biological function can be well described by a tractable set
of equations.
Ions can be studied as complex fluids in the tradition of physical science
although classical treatments as simple fluids have proven inadequate and
must be abandoned in my view. Ion channels can be studied by Poisson-Drift
diffusion equations familiar in plasma and semiconductor physics - called
Poisson Nernst Planck or PNP in biology. They form an adequate model of
current voltage relations in many types of channels under many conditions
if extended to include correlations, and can even describe 'chemical' phenomena
like selectivity with some success.
My collaborators and I have shown how the relevant equations can be derived
(almost) from stochastic differential equations, and how they can be solved
in inverse, variational, and direct problems using models that describe
a wide range of biological situations with only a handful of parameters
that do not change even when concentrations change by a factor of 107. Variational
methods hold particular promise as a way to solve problems outstanding for
more than a century because they describe interactions of 'everything with
everything' else that characterize ions crowded into channels.
An opportunity exists to apply the well established methods of computational
physics to a central problem of computational biology. The plasmas of biology
can be analyzed like the plasmas of physics.
Joe Jerome
Classical Transport Models Beyond PNP: Results and Questions (slides)
We survey and discuss classical models for dilute ions. In particular, we
discuss the Rubinstein model for one-fluid transport, and the Blotekjaer/Baccarani/Wordeman
version of the hydrodynamic model. We also review a gating model, as well
as compatibility aspects of energy transport models. Many of these results
are well-known, but our intent is to coordinate them to give a picture of
classical electrodiffusion, prior to the introduction of some of the topical
models currently under study. Finally, we identify an analytical property
of the `crowded ion' model, which makes this model extremely challenging.
Chiun-Chang Lee
Asymptotic behavior for boundary layers of the charge conserving Poisson-Boltzmann
equation
The charge conserving Poisson-Boltzmann (CCPB) equation is an electrostatic
model that describes electrostatic interactions between molecules in ionic
solutions (electrolytes) and has many applications in electrolyte solutions.
One of the important phenomena is the electrical double layer (EDL) that
appears near the charged surface of electrolyte solutions. To describe the
behavior of the EDL, we study the asymptotic behavior for the boundary layer
of the CCPB equation. First, we shall introduce the CCPB equation without
finite size effects and related boundary condition. Based on these understandings,
the asymptotic behavior for boundary layers of this model will be introduced.
These results may provide a viewpoint to see the influence of ionic valences
and concentrations on the boundary layers. On the other hand, using similar
argument, we shall study a modified Poisson-Boltzmann equation with finite
size effects (PB_ns equation). When the small dielectric constant is regarded
as a parameter tending to zero, we obtain two approximation models of the
PB_ns equation. One model is the conventional PB equation, the other model
is a modified PB equation introduced by Borukhov, Andelman, and Orland in
1997. The asymptotic behaviors of three PB type models will be compared.
This is a joint work with YunKyong Hyon, Tai-Chia Lin, and Chun Liu.
Xiaofan Li
A Conservative Scheme for Poisson-Nernst-Planck Equations
A macroscopic model to describe the dynamics of ion transport in ion channels
is the Poisson-Nernst-Planck (PNP) equations. In this talk, we will present
a finite-difference method for solving PNP equations, which is second-order
accurate in both space and time. We use the physical parameters specifically
suited toward the modeling of ion channels. We introduce a simple iterative
scheme to solve the system of nonlinear equations resulting from discretizing
the equations implicitly in time, which converges in a few iterations. We
place emphasis on ensuring numerical methods to have the same physical properties
that the PNP equations themselves also possess, namely conservation of total
ions and correct rates of energy dissipation. Further, we illustrate that,
using realistic values of the physical parameters, the conservation property
is critical in obtaining correct numerical solutions over long time scales.
Jie Liang
Predicting three-dimensional structures, topology, and stabilities
of of bacterial outer-membrane porins and eukaryotic mitochondrial membrane
proteins
Beta-barrel membrane proteins are found in the outer membrane of
gram-negative bacteria, mitochondria, and chloroplasts. They are the basis
of an important class of ion-channels, and are involved in pore formation,
membrane anchoring, and enzyme activity. However, they are sparsely represented
in the protein structure databank.
We have developed a computational method for predicting structures of the
transmembrane (TM) domains of beta-barrel membrane proteins. Based on physical
principles, our method can predict structures of the TM domain of beta-barrel
membrane proteins of novel topology, including those from eukaryotic mitochondria.
Our method is based on a model of physical interactions, a discrete conformational
state space, an empirical potential Bacterial Outer-Membrane and Eukaryotic
Mitochondria function, as well as a model to account for interstrand loop
entropy. We are able to construct three-dimensional atomic structure of
the TM domains from sequences for a set of 23 nonhomologous proteins (resolution
<3.0 A).
In addition, stability determinants and protein-protein interaction sites
can also be predicted. Such predictions on eukaryotic mitochondria outer
membrane protein Tom40 and VDAC are confirmed by independent mutagenesis
and chemical cross-linking studies. These results suggest that our model
captures key components of the organization principles of beta-barrel membrane
protein assembly. The depth dependent transfer free energy of amino acids
allows further insight into the topology and folding of bacterial porins.
Finally, we show how computational prediction can lead to successful engineering
of altered protein-protein interactions and olgomerization state in the
outer membrane protein F (OmpF). Through site-directed mutagenesis based
on computational design, we succeeded in engineering OmpF mutants with dimeric
and monomeric oligomerization states instead of a trimeric state. Moreover,
our results suggest that oligomer dissociation can be separated from the
process of protein unfolding, and the oligomerization proceeds through a
series of interactions involving two distinct regions of the extensive PPI
interface.
Tai-Chia Lin
Stability of PNP type systems for ion transport
To describe ion transport through biological channels, we derive new
PNP (Poisson-Nernst-Planck) type systems and develop mathematical theorems
for these systems. Symmetry and non-symmetry breaking conditions being represented
by their coupling coefficients may affect the stability of these systems.
In this lecture, I will introduce results for the stability of steric PNP
systems and standard PNP systems with boundary layer solutions. Our results
indicate that new PNP type systems may become a useful model to study ion
transport through biological channels.
Weishi Liu
Geometric singular perturbations of Poisson-Nernst-Planck systems and
applications to ion channel problems (slides)
In this talk, we will report our work on Poisson-Nernst-Planck (PNP) type
systems, a class of primitive continuum models for electrodiffusion, mainly
in the content of ionic flow through membrane channels. An important modeling
feature of the PNP type systems studied is the inclusion of hard-sphere
potentials that account for ion size effect. We will focus on hard-sphere
potentials that are ion specific. This complication is critical since ions
with the same charge but different sizes could have significantly different
roles in many important biological functions of living organisms. We will
present an analytical framework that relies on a combination of a powerful
general theory of geometric singular perturbations and of specific structures
of PNP type systems. Beyond existence and uniqueness problems, we are interested
in obtaining concrete characteristics of solutions that have direct implications
to ionic flow properties. A particular attention is paid on effects of the
ion sizes and permanent charges to electrodiffusion and ion channel functions.
Benzhou Lu
Finite element simulation of ion permeation in 3D ion channel systems
based on their atomic structure
Modeling based on molecular structure can naturally incorporate structural
information and atomic properties, and use the least number of fitting parameters.
However, real 3D ion channel is particularly difficult to simulate due to
the multiscale nature of the transport process, the complex geometry/boundary
of the channel protein system, and the singular charge distribution inside
the channel protein(s). For these reasons, there are so far only a very
few software publicly available in this important area of biology. I'll
talk about a software platform and methods we recently developed for a complete
simulation procedure for ion transport in a channel. The governing model
is focused on the Poisson-Nernst-Planck equations, but a size -modified
PNP and a variable dielectric Poisson-Boltzmann (a special case of PNP)
models as well as some of their effects will also be discussed. A parallel
finite element solver and stable algorithms are developed. Two other useful
programs are for meshing and visualization. Qualified molecular meshing
is essential and was a bottleneck issue for finite/boundary element modelings
of biomolecular systems. We recently developed a robust molecular surface
meshing tool, TMSmesh, which can handle complex and arbitrarily large biomolecular
system. The visualization system, VCMM, is specifically designed to facilitate
researches in molecular continuum modeling community. Applications are demonstrated
in some channel systems for simulating such as current-voltage characteristics
(curves), conductance, and certain size effects to permeation. Some systems
are of challenging sizes for the simulation community. The results are compared
with those obtained with Brownian Dynamics simulations and experiments.
Maximilian Metti
Applications and Discretizations of the PNP Equations (slides)
Many devices involving charged particles or electric current can
be modeled using the PNP equations. We explore some of these devices as
applications of the PNP system to engineering and biological contexts, with
an emphasis on mathematical modeling and device functionality. Further consideration
is given to discrete formulations of the PNP system and a numerical approach
for computing a solution.
Rolf Ryham
Very weak solutions for Poisson-Nernst-Planck system (slides)
We formulate a notion of very weak solution for the Poisson-Nernst-Planck
system. A local monotonicity formula is derived for stationary, very weak
solutions and is used to prove an interior regularity result for a system
with multiple species and variable coefficients. Stationary, very weak solutions
of the Keller-Segel model are also considered and shown to be regular in
two dimensions and counter examples are given in higher dimensions.
Yuan-Nan Young
Modeling the electro-hydrodynamics of a leaky lipid bilayer membrane:
Continuum vs coarse-grained modeling
In this work we first present recent results from studying the electro-hydrodynamics
of a "leaky" lipid bilayer membrane. Within the continuum framework
the stability of a lipid bilayer membrane under an electric field (both
DC and AC) is investigated. The nonlinear dynamics is further investigated
to elucidate the novel electrohydrodynamics of a lipid bilayer membrane.
These results show membrane conductance is essential to both linear instability
and nonlinear dynamics of the membrane. Finally we present a coarse-grained
algorithm that utilizes the fast multipole method (FMM) to consider the
non-local hydrodynamic interactions and hopefully the electrostatic interaction
between transmembrane proteins and the lipid bilayer membrane. From these
results we draw conclusions for future directions to combine the two approaches
into a multi-scale model.
Zhenli Xu
Self-Consistent Continuum Theory and Monte Carlo Simulations for Coulomb
Many-Body Systems in Inhomogeneous Environments
In this talk, I will present recent work on modeling and simulations
of nanoscale electrostatic systems in inhomogeneous dielectric media with
strong many-body correlation effects. We consider Monte Carlo simulations
and continuum models by self-consistent field theory for electrolytes including
dielectric-boundary, ion-correlation, and excluded-volume effects. For particle
simulations, we developed efficient algorithm for treating dielectric interfaces.
For continuum theory, we derived self-energy-modified Poisson-Boltzmann
equations for equilibrium systems and Poisson-Nernst-Planck equations for
charge transport. We studied the asymptotic properties of the models, discussed
efficient algorithms for these PDE models. By both continuum and particle
simulations, we attempt to understand many-body properties of systems with
dielectric interfaces, arising from soft matter and biological applications.
