D. Brian Walton

Research Associate
VIGRE postdoctoral fellow

Address:	University of Washington Department of Applied Mathematics Box 352420 Seattle WA 98195-2420
Office:	Guggenheim 408C
Phone:	(206) 685-9298
Fax:	(206) 685-1440
E-Mail:	walton@amath.washington.edu

Quick Links

Current Class: AMath 383 (Fall 2003) An Introduction to Continuous Modeling

Research Interests:

Recent rapid advances in quantitative biology lead to the need for mathematically founded methods in analyzing these data and for models that predict behaviors. I am particularly interested in developing probabilistic models for experimental systems typically classified as biophysics, and more particularly, experiments relating to motor proteins. Such systems are fundamentally noisy because of thermal fluctuations, so that stochastic processes arise naturally in studying their behaviors.

One such motor protein, kinesin, is responsible for transporting essential cellular products that are stored in membrane-bound vesicles to distant regions of a cell. To do this, kinesin attaches to a microtubule and essentially walks along its lattice. Recent experiments allow biophysicists to track the progress of individual kinesin proteins as they walk along a microtubule by observing an attached microscopic sphere. I have developed a hidden Markov model filter for analyzing data coming from such experiments, working with biophysicist Koen Visscher to understand the biology and physics of the experiment properly.

Dissertation: "Analysis of Single-Molecule Kinesin Assay Data by Hidden Markov Model Filtering," The University of Arizona, 2002. (PDF version)

Recently, I have begun considering another protein related to kinesin which is called mitotic centromere-associated kinesin (MCAK). Instead of transporting cargo along a microtubule, MCAK finds the ends of the microtubule and then proceeds to disassemble the microtubule. I am particularly interested in modeling how this protein reaches the microtubule ends and then how it contributes to the depolymerization of the microtubule.

Of course, other topics catch my attention. Working with Koen Visscher, I explored a topic referred to as "Noise Suppression by Noise" (Vilar and Rubi, Phys. Rev. Lett. 86, 950 (2001)). This has been typically regarded as an unintuitive phenomenon. Essentially, imagine an input/output device where the output is a function of the input but with additional noise, and where the size of the noise depends on the particular value of the input. Noise suppression says that the size of noise at the output can be reduced by increasing the fluctuations at the input. We demonstrated that noise suppression simply corresponds to spending a sufficiently large fraction of the time in low-noise input states that the average size of noise is reduced. More precisely, we explicitly compute the power spectrum of the output signal and provide exact conditions for a decrease in the spectrum.

D. Brian Walton and Koen Visscher, "Noise Suppression and Spectral Decomposition for State-Dependent Noise in the Presence of a Stationary Fluctuating Input", submitted 2003 (PDF preprint)

Teaching Duties:

At the University of Washington, I have taught a course titled "An Introduction to Continuous Modeling" in the Department of Applied Mathematics three times (Fall 2002, Spring 2003, and Fall 2003). This class uses case studies in modeling with differential equations to teach junior-level mathematical sciences students techniques in developing and analyzing differential equations for various systems. When I have taught this course, I have focused on population models, competition models, predator-prey models, Lanchester combat models, physical models with a potential energy, and traffic flow. This is a fun course to teach, and students are introduced to a variety of tools developed for dynamical systems such as bifurcations, phase-plane analysis, linear stability of equilibria, and dimensional analysis. This course also involves students preparing and writing a term project applying differential equations to a model system of their choice.

Prior to coming to the UW, I taught three courses at the University of Arizona in the Math department. I taught the series for College Algebra (Math 116 and 117 at the time) as well as the introductory calculus course Elements of Calculus (Math 113). I also was an advanced TA for the graduate course in probability (Math 565), helping other graduate students in problem sessions.

This web-page was written by D. Brian Walton.