Vanderbilt Mathematics
Analysis & Biomathematics Seminar
2003--2004

Wednesday, April 28th, 2004.
Marina Sapir, University of Nebraska Medical Center.
An "Intuitive" Approach for the Classification Problems in Omics Data.
Abstract: Complex problems of data mining in proteomics and genomics invite complex, computationally extensive solutions. However, making the procedure of feature selection and classification transparent can help the extraction of a medical knowledge from data. Human friendly, interpretable decision rule can provide insights for new medical discoveries, and it is more useful for the researchers and medical practitioners. We proposed an integrated "intuitive" approach, which includes transparent algorithms of features selection and classification. The methods designed to model medical thinking. The approach was tested on cancer-related gene array and protein array data. We were able to find features with strong classification ability and to build simple decision rules, having predictive accuracy comparable with the best results, achieved for the data.

Wednesday, April 14th, 2004.
Maria Kiskowski, of the Department of Mathematics, University of Notre Dame.
Cell-Based Model for Chondrogenic Patterning in the Chick Limb Bud.
Abstract: This talk describes a discrete, lattice-based model for behavior of limb bud precartilage mesenchymal cells undergoing chondrogenic pattern formation in micromass cell culture. In our "agent-oriented" model, cells are represented by points on a lattice and are assigned simple rules motivated by experimental findings. The rules include random cell motion, production and lateral deposition of a substratum adhesion molecule (SAM), production and release of a diffusible growth factor ("activator") that stimulates production of the SAM, and another diffusible factor ("inhibitor") that suppresses the activity of the activator. Parameters are identified for which the system exhibits nodular patterns that resemble those of leg cell cultures, including number, distribution and spacing of cell condensations. This reference system was then studied experimentally, including subjecting it to cell dilution, transient exposure to exogenous activator, suppression of inhibitor, and constitutive activation of SAM production. There was good correspondence between in silico and in vitro experimental results.

Wednesday, April 7th, 2004.
Antonio Fasano, of the Dipartimento di Matematica, Università degli Studi di Firenze.
Mass Transfer in Nonisothermal Saturated Solutions and Deposition Phenomena.

Monday, April 5th, 2004, 4:10p.m., 134 Featheringill Hall.
James P. Keener, of the Department of Bioengineering, University of Utah.
Mechanism for the Onset of Fibrillation Following a Heart Attack
Abstract: Each year in the United States, approximately a quater of a million people die as the result of a heart attack before reaching a hospital. In most of these cases, a coronary occlusion led to the sudden onset of fibrillation, a condition, which if not arrested, is fatal.

In this talk, I will use models, mathematical analysis and numerical simulations to describe a possible mechanism for the onset of fibrillation following a coronary occlusion. The mechanism proposed here is substantially different than previously proposed mechanisms for the initiation of reentrant activity (spiral waves, etc.), as it takes into account some of the dynamic processes that are unique to heart attacks.

Wednesday, March 31st, 2004, 3:00p.m., 208 Light Hall.
Peter Detwiler, of the Department of Physiology and Biophysics, University of Washington.
Optical Studies of Calcium Signals in Retinal Photoreceptors and Amacrine Cell Dendrites

Monday, March 29th, 2004, 4:10p.m., SC 1206.
Hans Engler, of the Department of Mathematics, Georgetown University.
Self-similar Asymptotics for Partial Integrodifferential Equations
Abstract: Consider a partial integrodifferential equation of the typical form
u_t-A(0)Δu = A'∗Δu
in Rⁿ, where A(t) is a scalar integral kernel with A(0) ≥ 0. The question will be discussed whether solutions are asymptotic to a limiting profile ψ, in the sense that k(t)u(m(t)x,t) →ψ(x) as t→∞ for suitable scale factors k(t) and m(t). In the talk, I shall identify all possible limits and associated rates k(t) and m(t) and show that these limits actually occur for suitable kernels. A naturally occurring requirement is that the kernel A is regularly varying in the sense of Karamata.

Friday, March 26th, 2004, 3:10p.m., SC 1431.
Stephen H. Davis, of the Department of Engineering Sciences and Applied Mathematics, Northwestern University.
Evolution of Quantum Dots.

Thursday, March 25th, 2004, 3:10p.m., 898B Ingram Cancer Center.
Avner Friedman, of the Mathematical Biosciences Institute, Ohio State University.
Mathematical Models of Cancer.
Abstract: This talk shall describe several models of cancer growth and, in particular, discuss ongoing work on tumor invasion, metastasis, angiogenesis and lymphangiogenesis.

Wednesday, March 24th, 2004.
Srinivas Ravi V. Iyengar, of the Department of Pharmacology and Biological Chemistry, Mount Sinai School of Medicine.
Computational Analysis of Cellular Signaling Networks.

Wednesday, March 17th, 2004.
Stephan Luckhaus, of the Fakultät für Mathematik und Informatik, Universität Leipzig.
A Lattice Model for Competition between Malignant and Normal Cells Focusing on Lateral Contact Inhibition.

Wednesday, March 10th, 2004, 3:10p.m., SC 1206.
Dieter Bothe, of the Department of Chemical Engineering, University of Paderborn.
Two-Phase Flows with Mass Transfer Across Deforming Phase Boundaries.
Abstract: Multiphase flows provide the basis for many chemical processes of industrial importance. An important prototype of such a process is the reactive mass transfer of, say, oxygen from rising gas bubbles to the ambient liquid. In this case, transport processes occur on different length scales ranging from macro scale (length of rise paths) via meso scale (bubble diameter) down to micro scale (thickness of concentration layers).

Even well established industrial methods need to be further optimized due to increasing economical as well as ecological constraints. Besides experimental investigations, the necessary intensification requires numerical simulations based on mathematical modeling. Here, we use the continuum mechanical balances of mass, momentum and species mass with main emphasis on the deformable phase boundary. This yields free boundary problems for the Navier-Stokes equations and convection-diffusion equations for the involved chemical components, complemented by interfacial jump conditions. In an abstract formulation this leads to quasi-linear evolution systems in Banach spaces, tractable by means of maximal regularity theory.

Any numerical treatment requires an adequate discrete description of the dynamical interface. One approach is given by so-called volume tracking methods that employ conservation of phase-specific mass. The main problem any simulation technique has to cope with, is the appearance of a full hierarchy of length and time scales. Therefore, efficient numerical computations call for parallel computing techniques and adaptive grids.

This talk aims to show the full range from mathematical modeling via analysis to numerical simulation of two-phase flows.

Wednesday, March 3rd, 2004.
Gianghui Wang, of the Chinese Academy of Sciences.
Dual Mixed Finite Element Method for Contact Problem in Elasticity.
Abstract: Based on a mixed variational formulation, a new dual mixed variational formulation was presented for contact problem in elasticity. According to symmetrical character of stress tensor, finite element subspaces were reasonably chosen. The existence and uniqueness of the finite element solution of this new variational formulation were proved, and the relative error bound was derived.

Monday, March 1st, 2004, 4:10p.m., SC 1308.
Christoph Walker, of the University of Zürich.
Continuous Coalescence and Breakage Processes.
Abstract: We consider a new model for the time evolution of a system of particles, which can merge to form larger particles or split into smaller ones. The basic feature is that a maximal particle size is taken into account. This requires a reformulation of coalescence of large particles. We study the well-posedness of the resulting system of unaccountably many differential equations in the non-diffusive as well as in the diffusive case. In the former case, we also investigate the long time behavior.

Monday, March 1st, 2004, 4:00p.m., 208 Light Hall.
Arnold Mandell, of the Cielo Institute, Emory University.
A New Rational Design Technique for Peptide Allosteric Modulators of D2 Dopamine and M1 Muscarinic Receptors.
Abstract: In the absence of x-ray or nmr-derived knowledge of receptor structure or a pharmacophore analogy, today's dominant peptide drug design methods seeks to speed up Darwinian time using high throughput screening of random peptide libraries. Our more deterministic, information-based approach to receptor-targeted peptide design exploits one-dimensional signal processing approaches to patterns in the quantitative physical property-transformed primary sequences involved in polymeric polypeptide interactions. The force/properties of potential physical relevance include van der Waals, charge interactions, hydrogen bonding, and hydrophobic and related properties, such as accessible surface area, partial specific volume and lipophoic contact surface. Hydrophobicity, hydrophobic free energy in kcal/mol, is >2 orders of magnitude stonger than van der Waals forces, extending > 20 nm in aqueous solution. Relations between protein x-ray secondary structure and sequential patterns of amino acid side chain hydrophobic variation (as rotations, "wheels," or Fourier wavelengths ("modes")) are well established. Generally: antiparallel b = 2.0 aa, b-strand, = 2.2 aa, 310 helix = 3.0 aa, a-helix = 3.6 aa, p-helix = 4.4 aa. Extramembraneous receptor loops as "disordered" sequences contain a multiplicity of potential modes (Dyson). Hydrophobic mode-matched aggregation between b-strands has been called "hydrophobic zippers" (Dill) and between matching helical patterns is referred to as "hydrophobic knots" (Lumry). A multiplicity of conformational matches have been found between receptor antibodies and their extracellular domains (Beattie). Computationally, hydrophobic wavelengths available for mode matched peptide binding are masked by the sequential patterns of hydrophobic transmembrane segments in G-protein coupled receptors. Decomposing the lagged autocorrelation/autocovariance matrix representation of receptor hydrophobic sequences (exploiting the Khinchine relation between Fourier representations and autocorrelation functions) allowed for the computational removal of the transmembrane pattern. The residual hydrophobic eigenvectors served as templates for human, long form, D2-dopamine and M1 receptor-targeted peptide designs via amino acid pool weighted, random hydrophobic category amino acid assignment. Wavelet transformations were used to confirm the dominant wavelengths and locate them in the receptor sequence, mostly in the sequence locations of the extracellular loops. D2-targeted peptides were designed, synthesized and tested in receptor stably transfected CHO and LtK cells. D2 peptides were assessed using extracellular acidification rates, EAR, and inhibition of forskolin-stimulated cAMP responses. These peptides evidenced indirect agonist and/or positive allosteric effects with an 83% hit rate. In rats, bilateral nucleus accumbens administration of D2 peptides significantly augmented and/or potentiated amphetamine-induced increases in exploratory behavior and disruption of prepulse inhibition, PPI. Eigenvector-generated peptides in D-amino acid, mode preserving, inverse sequence (retro-inverso) form, were behaviorally active following parenteral administration. Peptide design targeting the M1 muscarinic receptor and evaluated by EAR demonstrated similar results with mode matches in the sequence regions of the extracellular loops and had a = 50% hit rate. The Results of crossover EAR studies of M1 targeted peptides on M2 transfected LtK cells, D2 targeted peptides on M1 transfected LtK cells and M1 peptides on D2 transfected LtK cells were consistented with receptor subtype-targeted peptide specificity. Generalizing these techniques to globular protein targets, these algorithms generated "miniantibodies" with = 35% positive ELISA test-indication of b-galactosidase binding and mode matched with known bGAL antibody accessible binding sites.

Thursday, February 26th, 2004, 2:30p.m., SC 1432.
Henghui Zou, of the University of Alabama.
A Cauchy-Liouville Theorem for Quasilinear Elliptic Equations.
Abstract: Let n ≥2 be an integer and m>1 be a positive number. Consider the non-homogeneous degenerate elliptic equation Δ_mu+f(u) = 0, u ≥ 0, x∈Ω, where Ω is a domain (connected open set) in Rⁿ, Δ_mu= div(|∇u|^m-2∇u) is the well-known m-Laplace operator and f(u) is a non-negative function in C([0,∞))∩ C¹((0,∞)).

For the above equation, we shall prove Liouville type (non-existence) results on unbounded domains and universal a priori estimates on arbitrary domains. A canonical prototype model is Δ_mu+u^p-1=0, u ≥ 0, p>1, x∈Ω.

Wednesday, February 25th, 2004.
Minchul Kang, of the School of Mathematics, University of Minnesota.
The Variety of Cytosolic Calcium Responses and Possible Roles of PLC and PKC.
Abstract: Many cells can show various and complex [Ca²⁺]_i responses to many extra-cellular stimuli by mobilizing the stored calcium in endoplasmic reticulum (ER). Albeit these diversities, temporal [Ca²⁺]_i responses fall into some typical categories: Puff, transient calcium spike, baseline spikes, and sinusoidal oscillation, and possibly more. It has been generally accepted that iterative inhibition on both ligand receptors and Phospholipase C (PLC) by protein kinase C (PKC) are responsible for sinusoidal calcium oscillation, but the separate role of each inhibition is still unknown. On the other hand, PKC independent CICR (calcium induce calcium release) mechanism of ER is believed to be responsible for baseline spikes although some evidences showed that baseline spikes could be observed under high PKC activity or under low ligand concentration. A model of glutamatergic system that includes PKC feedback kinetics and CICR mechanism was derived to validate two distinct mechanisms of [Ca²⁺]_i oscillation. Also, to investigate underlying mechanisms of the variety of calcium responses in cytosol, simpler model was derived by using scaling and reduction technique. Using this model, it was demonstrated that baseline spikes can be generated under high protein kinase C activity or under low ligand concentration switching from calcium controlling CICR mechanism to repetitive PKC inhibition is responsible for transition between baseline spiking and sinusoidal oscillation. Moreover, it was shown that three different modes of intracellular calcium transient and accordingly, five modes of cytosolic calcium responses (puff, two transient modes and two oscillatory modes) could be identified in the two parameter space of PKC and PLC activities.

Wednesday, February 18th, 2004, 4:10p.m., SC 1432.
Gustavo Carrero, of the Department of Mathematics, University of Alberta.
A Mathematical Model for the Compartmentalization of Splicing Factors.
Abstract: The dynamic nature of nuclear architecture within eukaryotic cells is an interesting problem in cellular biology. Specifically, the origin, maintenance and disappearance of speckles, which are heterogeneously distributed nuclear compartments enriched with pre-mRNA splicing factors, is unknown. It has been hypothesized that a process of self-aggregation among dephosphorylated splicing factors, modulated by a phosphorylation-dephosphorylation cycle, is responsible for the formation and disappearance of speckles. Also, it is thought that the existence of an underlying nuclear structure plays a major role in the organization of splicing factors. In this talk, it will be explained how these hypotheses and a diffusion-approximation approach allow for the derivation of a fourth order aggregation-diffusion model that describes a possible mechanism underlying the organization of splicing factors in speckles. A linear stability analysis, supplemented by numerical simulations, will show how the self-interaction among dephosphorylated splicing factors can result in spatial patterns that are caused by instabilities about homogeneous steady states.

Wednesday, February 18th, 2004.
Lixin Shen, of the Computational Analysis and Modeling, Louisiana Tech University.
A Three-Level Finite Difference Scheme for Solving a Dual-Phase-Lagging Heat Transport Equation in Spherical Coordinates.
Abstract: Heat transport through thin films or micro-objects is of vital importance in microtechnology applications. For instance, metal thin films are important components of microelectronic devices. The reduction of the device size to microscale has the advantage of enhancing the switching speed of the device. On the other hand, size reduction increases the rate of heat generation, which leads to a high thermal load on the microelectronic devices. Heat transfer at the microscale is also important for the thermal possessing of materials with a pulsed-laser. Examples in metal processing are laser micromachining, laser patterning, laser processing of diamond films from carbon ion implanted copper substrates, and laser surface hardening. In thermal processing of materials, microvoids may be found owing to thermal expansion. When such defects begin in the workpiece, their thermal energy in the neighborhood of the defects may be amplified, resulting in severe material damage and, consequently, total failure of the thermal processing. A detailed understanding of the way in which the local defects dissipate the thermal energy is then necessary not only to avoid the damage but also to improve the efficiency of the thermal processing.

The heat transport equation at the microscale is different from the traditional heat diffusion equation because a second-order derivative of temperature with respect to time and a third-order mixed derivative of temperature with respect to space and time are introduced. In this study, we consider the heat transport equation in three-dimensional spherical coordinates and develop a three-level finite difference scheme for solving the heat transport equation in a microsphere. Stability of the scheme is proved. It is shown that the scheme is unconditionally stable. The scheme is then employed to investigate the temperature rise in a gold sphere subjected to a short-pulse laser. Numerical results are obtained for the cases that the laser irradiation is symmetric on the surface of the sphere, and the laser irradiation is from the top to a portion of the surface of the sphere.

At the end of this talk, Dependability Modeling and Analysis of a High Availability Beowulf Cluster will be introduced briefly, which is my research work in the Computer Science program, Louisiana Tech University.

Wednesday, February 11th, 2004. (3:10 - 4:00 in room SC 1432)
Daphne Manoussaki, of the Applied Mathematics and Computing Laboratory, Technical University of Crete .
A Mathematical Study of Blood Vessel Formation.
Abstract: Blood vessels form as the cells that make them up follow chemical or mechanical cues. In order to study the role of the mechanical and chemical forces during different stages of blood vessel formation, we present a mathematical model consisting of partial differential equations. The model describes the traction forces which the cells exert onto their fibrilar environment and the cellular response to chemical signals.

We study our model analytically and numerically. Our results suggest that during the initial stages of the patterning process, cellular traction forces alone can drive the formation of the blood vessel network. Our simulations predict networks that compare well with the patterns observed experimentally. In subsequent stages, blood vessel tree growth is driven by the cellular response to chemical gradients, however, we find that mechanical forces may again be important in the formation of well defined vascular structures.

Wednesday, February 4th, 2004.
Jan Prüss, of the Fachbereich Mathematik und Informatik, Martin-Luther-Universität Halle-Wittenberg and the Department of Mathematics, Vanderbilt University.
The Cahn-Hilliard Equation with Dynamic Boundary Condition.

Wednesday, January 28th, 2004.
Peter Thomas, of the Computational Neurobiology Laboratory, Salk Institute for Biological Studies.
Inside the Mind of the Amoeba: Simulation and Analysis of Biochemical Signal Transduction Channels.
Abstract: In lieu of nervous systems, single-celled organismsrely on complex networks of biocehmical reactions to extract information about their environments and determine responses to chemical messages. A wealth of quantitative biological data from bioinformatics to fluorescence microscopy has created the possibility of building detailed biophysically realistic working models of the information processing occuring inside cells, in analogy to the extensive progress made by modeling realistic neural networks. We are developing conceptual, analytic and numerical tools to shed light on several aspects of this problem.

I. Chemical reactants are localized within subcellular volumes, invalidating "well-mixed" ODE formulations of cellular control networks. In collaboration with cell biologists and theoretical biophysicists at UCSD we have constructed a finite-element model for solving arbitrary boundary-coupled PDEs as a platform for studying spatially heterogeneous signal-transduction networks, and used it to develop a model for the orienting response of a eukaryotic cell during directed cell movement (chemotaxis).

II. Stochastic effects rising from small copy numbers of chemical reactants force us to consider the consequences of fluctuations about the mean behavior described by deterministic ODE or PDE descriptions of chemical networks. For example, the amount of information a cell can extract from a chemical signal is governed by the size of fluctuations in local concentration and in the stochastic binding of signaling molecules to its receptors. In collaboration with signal-processing engineers at UCSD we have constructed a Monte Carlo simulator of molecule-by-molecule signal transduction and have obtained lower bounds on the information capacity of a simplified biochemical signal relay, and investigated how the capacity depends on key parameters such as system geometry and reaction rates.

III. Biochemical reaction networks may be represented as Markov flows on graphs of connected chemical states; these graphs are often too complicated for ready simulation or comparison to experiment. In collaboration with experimental biochemists at Caltech we are currently developing criteria for evaluating optimal sectioning of reaction graphs to provide canonical "coarse-graining" rules for complex reation networks.

Wednesday, January 21st, 2004, 3:10p.m., SC 1117.
Viktoria R. T. Hsu, of the Department of Applied Mathematics, University of Washington.
Electro-Diffusion in Cell Membranes, a Quasi Steady-State Approach.
Abstract: Most mathematical models for signal generation in single neurons, such as the classic Hodgkin-Huxley model, assume the single neuron is bathed in an infinite buffer solution. Thus the composition of the bath never changes. This assumption is appropriate for the comparison of model results to in vitro studies, because in these studies the cell preparation is actually bathed in a relatively fixed environment. In their current state, such models are not able to take into account large changes in the external environment of a cell, as occur when metabolite levels are depleted (ischemia). Ischemia have been linked to ailments like epileptic seizures and heart attacks.

My goal is to improve current neuron models such that changing extracellular conditions can be taken into account in a single-cell micro-environment. In this talk, I lay the foundation for a physically consistent model for signal generation and ion transport based on the quasi steady-state approximation to an electrodiffusion system. In the first part of the presentation, an efficient numerical method for the solution of 1D Poisson-Nernst-Planck (PNP) systems is introduced. In the second part of the talk, this numerical method is applied to solving the consecutive steady-state dynamics of a two-compartment system of ions. The results of this approach are compared to the full PDE in order to demonstrate the sensibility of the steady-state assumption. Finally, the quasi steady-state approach is compared to a Hodgkin-Huxley type model for a cell with intact gated channels (passive transport) but no ion pumps (active transport).

In the near future of this project, I shall incorporate active ion transport, applied currents, and cell volume dynamics. In the long term, I would like to consider tissues (networks of cells), and incorporate connections between ion transport and other signaling mechanisms.

Previous semesters:

• Fall 2003
• Spring 2003
• Fall 2002
• Spring 2002
• Fall 2001