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Math 319: Applied Probability and Stochastic Processes for Biology

# Applied Probability and Stochastic Processes for Biology

## MATH 319 / BIOL 319-419 / PHOL 419 / EBME 419 / EECS 319

### Catalog Description

Applications of probability and stochastic processes to biological
systems. Mathematical topics will include: introduction to discrete and
continuous probability spaces (including numerical generation of pseudo
random samples from specified probability distributions), Markov
processes in discrete and continuous time with discrete and continuous
sample spaces, point processes including homogeneous and inhomogeneous
Poisson processes and Markov chains on graphs, and diffusion processes
including Brownian motion and the Ornstein-Uhlenbeck process.
Biological topics will be determined by the interests of the students
and the instructor. Likely topics include: stochastic ion channels,
molecular motors and stochastic ratchets, actin and tubulin
polymerization, random walk models for neural spike trains, bacterial
chemotaxis, signaling and genetic regulatory networks, and stochastic
predator-prey dynamics. The emphasis will be on practical simulation
and analysis of stochastic phenomena in biological systems. Numerical
methods will be developed using both MATLAB and the R statistical
package. Student projects will comprise a major part of the course.
Offered as BIOL 319/419, MATH 319, EECS 319, EBME 419, PHOL 419.
Students enrolled for graduate credit will have additional expectations
related to the course projects.
### New for Spring 2011

Thanks to a generous award from UCITE's Glennan Fellows Program, the
course will be significantly revised to emphasize the use of MCell.
MCell is a specialized numerical platform designed for state of the art
**M**onte Carlo simulations of **Cell**ular microphysiology,
developed jointly by the Pittsburgh Supercomputer Center
(http://www.mcell.psc.edu/) and the Computational Neurobiology
Laboratory at the Salk Institute for Biological Studies
(http://www.mcell.cnl.salk.edu/). The course syllabus has been totally
redesigned so that students will learn the practical details of
simulating and visualizing stochastic microscale biological systems at
the same time as they learn the mathematical framework for stochastic
modeling. MCell will play a major role in the course along with some
use of Matlab or similar platforms for data visualization and analysis.
Class will meet two days a week in a lecture hall and one day a week in
the Department of Mathematics' computer laboratory. Students will be
expected to complete a course project using MCell, and will have access
to the CWRU High Performance Computing Cluster as part of the course.
Also, there is a class trip to visit the MCell developers at the
Pittsburgh Supercomputer Center in the works.
Course description, how the course is being updated, rules and regulations etc.
An attractive flyer.
Under construction. Check back here for reading assignments etc.
Check back to the Google Docs page for classwork assignments and other course information.
These tutorials were adapted for the CWRU course from materials provided by the developers of MCell. For the tutorials provided by the developers
click here.
### Background & Topics

Mathematical models of biological systems frequently involve systems of
ordinary or partial differential equations. While these deterministic
models can give important insights into biological behavior, they fail
to include the effects of chance fluctuations on biological dynamics.
This course will explore applications of probability theory and
stochastic processes in biological systems. It is a natural extension of
the biological dynamics courses (BIOL 300 or BIOL 306) or a first course
in differential equations (MATH 224 or 228) and any of these can serve
as a prerequisite. Students should be comfortable with multivariable
calculus (MATH 223 or 227) and linear algebra (MATH 201 or MATH 307).
While the mathematical content will be appropriate for a 300 level
undergraduate course, the emphasis will be on applications and practical
matters such as numerically simulating biological stochastic phenomena
using numerical platforms such as the R statistical package, MATLAB,
scientific/numerical python, and MCell.
Mathematical topics to be covered include applications of:
- Discrete and continuous probability spaces. Expectation, independence, conditional probability. Bayes' theorem.
- Numerical techniques for generating samples from different probability distributions.
- Random walks: Markov processes in discrete time with discrete and continuous space variables.
- Diffusion processes: Markov processes in continuous time and space obtained as the limit of a random walk; Wiener and Ornstein-Uhlenbeck processes.
- Point processes: Poisson, inhomogeneous Poisson, Markov chains on graphs.
- Numerical methods for generating each type of process, include Gillespie's algorithm for exact stochastic simulation of coupled chemical reactions.
- Statistical analysis of time series; Power spectra of random processes.
- Kurtz's theorem: a continuous time Markov model for a set of chemical reactions converges to the right ODE in the limit of a large, well-mixed volume.
- ``Keizer's paradox'' and the correspondence of master equation
and mass action chemical kinetics models.

Biological applications will be determined by the mutual interests of the students and the instructor. Suggestions are welcomed. A list of tentative topics includes:
- Stochastic membrane ion channel kinetics.
- Simulation of biochemical and genetic regulatory networks.
- Stochastic predatory-prey models.
- Molecular motors and stochastic ratchets.
- Dynamics of actin and tubulin polymerization, actin treadmilling, cell motility, dynamic instability and tubulin ``catastrophes".
- Stochastic treatment of Michaelis Menten kinetics.
- Random walk models for neural spike trains.
- Limits to visual perception at low light levels.
- Bacterial random walks and chemotaxis.

### Matlab References

For more information, please contact Dr. Thomas.
Updated: February 6, 2011