Applied Mathematics Colloquia Series

The Applied Mathematics Colloquium is held on Fridays at 04:00 PM, in Mathematics 501 (refreshments served at 3:30pm in Math 401N).

Fall 2008

September

09/12
Arthur F. Voter
Theoretical Division
Los Alamos National Laboratory

Accelerated Molecular Dynamics Methods

A significant problem in the atomistic simulation of materials, as well as in other areas of chemistry and physics where atomistic simulations are used, is that molecular dynamics simulations are limited to nanoseconds, while important reactions and diffusive events often occur on time scales of microseconds and longer. Although rate constants for these infrequent events can be computed directly using transition state theory (with dynamical corrections, if desired, to give exact rates), this requires first knowing the transition state. Often, however, we cannot even guess what events will occur. For example, in vapor-deposited metallic surface growth, surprisingly complicated exchange events are pervasive. In this talk, I will discuss the accelerated molecular dynamics approach, which we have been developing over the last decade, for treating these complex infrequent-event systems. The idea is to directly accelerate the dynamics to achieve longer times without prior knowledge of the available reaction paths. In some cases, we can achieve time scales with these methods that are many orders of magnitude beyond what is accessible to molecular dynamics. I will give an introduction to the three main methods in this class, hyperdynamics, parallel-replica dynamics, and temperature-accelerated dynamics, and discuss their relative merits. I will present some illustrative and recent applications to materials problems, such as metallic surface growth, grain boundary slip under applied shear, radiation damage annealing in MgO, and tensile tests on metallic nanowires. I will also discuss some of the ongoing challenges in making these methods as powerful and generally useful as possible.

Refreshments served in Math 401N at 3:30
09/26
Jon Pelletier
Department of Geosciences
The University of Arizona

Modeling the Evolution of Large River Systems

Our understanding of the erosion of rivers is improving greatly due to recent advances in the use of numerical modeling and new techniques for quantifying erosion rates isotopically. Isotopic techniques provide hard data on how quickly rivers erode over a given time frame (usually short compared to the time frame of interest) and numerical modeling allows us to extrapolate the isotopic data further back in time using empirical equations that describe the mechanical interactions of water, sediment, and bedrock. In this talk I will describe the methods and equations used in studying the evolution of large river systems, and present numerical models illustrating the evolution of the Sierra Nevada and Grand Canyon over the past few tens of millions of years.

Refreshments served in Math 401N at 3:30

October

10/17
Sharon Crook
Department of Mathematics and Statistics
Arizona State University

Modeling Activity-Dependent Changes in Dendritic Spine Structure

Recent evidence indicates that the morphology and density of dendritic spines are regulated during synaptic plasticity. This activity-dependent structural plasticity exists over a vast range of time scales, from minutes to days or weeks. In this work, performed in collaboration with Steve Baer, we extend previous modeling studies to include calcium-mediated spine restructuring. The models are based on the standard dimensionless cable equation for the changes in membrane potential in a passive dendrite. Additional equations characterize the activity-dependent changes in spine type along the dendrite. We use computational studies to investigate the interactions between the many activity-dependent spines and to reveal the impact of their collective dynamics on the output properties of the dendrite.

Refreshments served in Math 401N at 3:30
10/31
Sergei L. Kosakovsky Pond
School of Medicine
University of California, San Diego

Evolutionary Fingerprinting of Genes

Over time, natural selection molds every gene into a unique mosaic of sites evolving rapidly and resisting change--an 'evolutionary fingerprint' of the gene. We introduce a metric, called the evolutionary selection distance (ESD), to identify similarities and idiosyncrasies in selection pressures inferred from sequence alignments, including a direct comparison of heterologous genes. Using a broad survey of viral genes, we apply a variety of computational techniques and machine learning techniques based on ESD to classify genes by the similarity of their evolutionary fingerprints, identify genes with distinctive evolutionary features, and correlate evolution with phylogenetic, functional, and taxonomic information. We demonstrate that the data follow a pattern of evolutionary continuum rather than a few rigid evolutionary modes; genes within the same functional group tend to exhibit similar evolutionary patterns, both within a single viral genome and between different viruses; similarity in selection pressures mirrors phylogenetic relationships among hepatitis C virus subtypes, but not HIV-1 subtypes; and that evolutionary patterns in the hemagglutinin gene of the Influenza A virus are largely determined by the host with a few notable exceptions. By comparing genes at the level of the evolutionary processes rather than the pattern of sequence variation, we can compare both closely and distantly related genes, potentially revealing the guiding principles underlying the evolution of genetic diversity.

Refreshments served in Math 401N at 3:30

November

11/07
Dmitry Pelinovsky
Department of Mathematics
McMaster University

Localized Modes in Nonlinear Schrodinger Lattices

I will give a review of recent results on existence and stability of discrete solitons in the framework of the discrete nonlinear Schrodinger equation. In particular, I focus on orbital and asymptotic stability of discrete solitons, periodic oscillations of solitons in the presence of diffraction management, and existence of traveling waves in nonlinear Schrodinger lattices.

Refreshments served in Math 401N at 3:30
11/14
Larry A. Taber
Department of Biomedical Engineering
Washington University, St. Louis

On a Fundamental Principle for Morphomechanics

Embryogenesis involves a carefully coordinated series of morphogenetic events, which are carried out by a relatively limited number of basic cellular processes, including migration, multiplication, and the stretching and folding of epithelia (cell sheets). These events are regulated by a dynamic interaction between genetic and environmental factors (chemical and mechanical), with adjustments being made continually through feedback mechanisms. The nature of this interaction remains a central question of developmental biology. Today, most researchers agree that mechanics plays a significant role in regulating morphogenesis (e.g., through mechanotransduction).

Here, computational models are used to explore the possibility that morphogenesis is regulated, in part, by feedback from mechanical stress. Comparing theoretical and experimental results, we consider the following questions: Is the behavior of developing tissues governed by a fundamental principle for morphogenesis? Can such a principle be expressed mathematically? While it is clear that developing tissues must obey the quantitative laws of physics, it is not clear that they also obey quantitative laws of biology. To date, these issues have been discussed primarily in philosophical terms, with many investigators arguing that mathematical laws in biology simply cannot exist.

The results from this study suggest that cellular responses depend on the rate of loading caused by external or internally generated forces. Morphomechanical laws are proposed for slow, medium, and fast stress rates, and the feasibility of these laws is illustrated by models for stretching of epithelia and axons, invagination of sea urchin embryos, growth of arteries, wound healing, and early heart development. This study represents an initial attempt to formulate a general principle for the mechanics of soft tissue growth and development.

Refreshments served in Math 401N at 3:30
11/21
Maria-Carme T. Calderer
School of Mathematics
University of Minnesota

Elastic and Ferroelectric Properties of Liquid Crystals: Modeling and Analysis

Since the development and commercialization of the first nematic liquid crystal display devices in the middle of the last century, mathematical modeling and analysis of liquid crystals has experienced significant progress. Liquid crystals are phases intermediate between solid and liquid; they occur in synthetic as well as in organic compounds. The Kevlar fiber is an example of a highly employed liquid crystal polymer; many virus and bacteria colonies as well as biological tissues present liquid crystal ordering.

Liquid crystals of small molecular weight consist of rigid, rod-like molecules that tend to follow preferential directions of alignment. Their interaction with electric and magnetic fields is at the core of application to display devices. Recently developed liquid crystals exhibit more complicated molecular shapes able to sustain permanent dipoles that result in ferroelectric coupling with applied electromagnetic fields. The speed of switching of such devices is about 103 to 104 times that of the nematic cell. Equilibrium states of ferroelectric liquid crystals result from minimizing the total energy subject to packing and electrostatic constraints. I will present an application of such a theory to predicting shape of material filaments.

Liquid crystal elastomers are nonlinear elastic solids that may also present liquid crystal phases. One remarkable feature is their capability to undergo unusually large deformations along preferential directions. Upon analyzing mathematical issues of such models, I will address their gel states and discuss the potential matrix role in modeling cell motility in the brain.

Refreshments served in Math 401N at 3:30