I will give an overview of my work in mathematical cell biology.  First I will discuss topics related to polarity, specifically in the context of cell movement.  This and numerous other cell functions require identification of a “front” and “back” (e.g. polarity).   In some cases this can form spontaneously and in others sufficiently large stimuli are required.  I will discuss a mechanistic theory for how cells might transition between these behaviors by modulating their sensitivity to external stimuli.  In order to address this and analyze the systems being presented, I will describe a new non-linear bifurcation technique, the Local Perturbation Analysis, for analyzing complex, spatial biochemical networks.  This methodology fills a void between simple (but limited) stability techniques and more thorough (but in many cases impractical) non-linear PDE analysis techniques.  Additionally, I will discuss work related to early development of the mammalian embryo.  A vital first step in this process is the formation of an early placenta prior to implantation.  I will discuss a multi-scale stochastic model of this spatial patterning event and show that genetic expression noise is both necessary and sufficient for this event to occur robustly. 

Driving nanomagnets by spin-polarized currents offers exciting prospects in magnetoelectronics, but the response of the magnets to such currents remains poorly understood. For a single domain ferromagnet, I will show that an averaged equation describing the diffusion of energy on a graph captures the low-damping dynamics of these systems. Specifically, I obtain analytical expressions for the critical spin-polarized currents needed to induce stable precessional states and magnetization switching in the zero temperature system as well as for the mean times of thermally assisted magnetization reversals in the finite temperature system, giving explicit expressions for the effective energy barriers conjectured to exist. I will then outline the problem of extending the analysis to spatially non-unifrom magnets, modeled by an infinite dimensional Hamiltonian system.

I will discuss the singularities of the zero-locus of a polynomial equation. A particular focus will be payed to comparing different singularities. I will discuss two different approaches to this question: analytic (characteristic zero) and algebraic (positive characteristic).

The theory of minimal surfaces arose historically from work of J. L. Lagrange and physical observations of J. Plateau almost 200 years ago. Rigorous mathematical theory was developed in the 20th century. In more recent times the theory has found important applications to diverse areas of geometry and relativity. In this talk, which is aimed at a general mathematical audience, we will introduce the subject and describe a few recent applications of the theory.

Traveling waves in actin have recently been reported in many cell types. Fish keratocyte cells, which usually exhibit rapid and steady motility, exhibit traveling waves of protrusion when plated on highly adhesive surfaces. We hypothesize that waving arises from a competition between actin polymerization and mature adhesions for VASP, a protein that associates with growing actin barbed ends. We developed a mathematical model of actin protrusion coupled with membrane tension, adhesions and VASP. The model is formulated as a system of partial differential equations with a nonlocal integral term and noise. Simulations of this model lead to a number of predictions, for example, that overexpression of VASP prevents waving, but depletion of VASP does not increase the fraction of cells that wave. The model also predicts that VASP exhibits a traveling wave whose peak is out of phase with the F-actin wave. Further experiments confirmed these predictions and provided quantitative data to estimate the model parameters. We thus conclude that the waves are the result of competition between actin and adhesions for VASP, rather than a regulatory biochemical oscillator or mechanical tag-of-war. We hypothesize that this waving behavior contributes to adaptation of cell motility mechanisms in perturbed environments.