Biomathematics
(Biomedical Mathematics, Mathematical Biology)
Department of Mathematics
Florida State University
|
|
|
|
|
|
|
|
Contents
Advisement and Supervisory Committees
Overview
of Biomathematics at Florida State University
Research in biomathematics at Florida State University began more than two decades ago with the pioneering work of DeWitt Sumners that uses knot theory to understand knotted DNA. Later, in the 1990s, Mike Mesterton-Gibbons began working in biomathematics, applying game theory to biological conflicts, and has now moved on to study behavior and group structure in complex social networks. In the mid 1990s Jack Quine, who like Sumners is a pure mathematician by training, began to apply geometric and algebraic methods to the study of the molecular structure of membrane-spanning proteins. Quine and Sumners teamed up in 2000 to start the Biomathematics graduate program at FSU. This was followed a year later by an undergraduate Biomathematics program. Although Sumners has since retired (but still pursues a very active research career), the program has grown since 2000. In addition to Quine and Mesterton-Gibbons, other faculty in the group include Richard Bertram (mathematical neuroscience and physiology), Nick Cogan (biofilms and biofluids), Monica Hurdal (brain imaging and mapping), and Washington Mio (brain mapping and computational anatomy). Research performed by the biomath faculty is funded by the National Institutes of Health and the National Science Foundation.
The Biomathematics graduate program at FSU is designed for curricular flexibility and interdisciplinary training. All students take courses in statistics and biology, as well as either pure or applied mathematics. The focus of the coursework is determined by the student’s background and interest, and the topic of their dissertation research. Many of our students also interact with faculty in other departments or at other universities. For example, some students have worked with researchers at the National High Magnetic Field Laboratory on research on the atomic structure of proteins, some have worked on brain mapping with researchers of the Laboratory of Neuro Imaging at UCLA, and some have performed biological experiments in labs at FSU and elsewhere. Students participate in one or more seminar each semester. These include the Biomathematics Seminar in which talks are given by graduate students, biomathematics faculty, and faculty from other departments with potential biomath applications. There are also specialized seminars run by biomath faculty members. For example, Hurdal runs a seminar on brain mapping and Bertram runs a journal club that focuses primarily on recent research articles in mathematical neuroscience and mathematical physiology.
Return to Contents
Our Students
There are currently 22 Biomathematics graduate students from countries around the world, such as Lebanon, Ethiopia, China, Canada, Mexico, Korea, and Colombia. There are also biomathematics postdoctoral fellows, working either in the department or in faculty research projects. Current student research projects include spatial pattern formation of cortical folds in the brain, a model of eradication of invasive species through the addition of sex-reversed fish to shift the sex ratio of the population over time, mapping brain atrophy in Alzheimer’s disease, shape models for the study of the phenotype of fruit flies, analysis of brain anatomy with methods of spectral geometry, coalition and alliance formation among primates, analysis of bursting electrical oscillations in neurons and pituitary cells, the use of Graphics Processing Units (GPUs) to perform neural network simulations, and models for the production of bird song. Current and prior students have been authors on publications in Journal of Theoretical Biology, Bulletin of Mathematical Biology, the American Journal of Physiology, Journal of Computational Neuroscience, Biophysical Journal, Journal of Magnetic Resonance, Science STKE, Proceedings of the International Conference on Medical Image Computing and Computer Assisted Intervention, and Proceedings of the International Symposium on Biomedical Imaging. Upon graduation, our Biomathematics students have taken postdoctoral positions at the Mathematical Biosciences Institute, New York University, LSU Health Sciences Center, the University of Texas at Austin, McGill University, and Georgia Tech.
Our students have been very successful in finding faculty positions and post-docs. Some of our former student
Here are some recent publications including our students. Publications with students.
The Biomathematics Group at FSU, faculty and students
Return to Contents
FSU faculty members from five departments are involved in this
effort. PhD's
are directed by one or more of the Mathematics faculty, often in
conjunction
with faculty from the other departments.
Program directors: Richard Bertram and Jack Quine
Mathematics faculty:
Richard Bertram (mathematical physiology, protein structure determination)
Nick Cogan (Fluid dynamics, biofilms)
Monica Hurdal (human brain mapping)
Mike Mesterton-Gibbons (Models of animal behavior and social structure)
Washington Mio (pattern analysis, computer vision, biomedical applications)
Jack Quine (protein structure from solid-state NMR data)
De Witt Sumners (Professor Emeritus) (DNA topology, human brain mapping)
Return to Contents
Degree Options
in Biomathematics
Master of Science. This is a two-year
program with 36 semester hours of courses and
seminars. Students develop skills in a number of areas for working on
applications of mathematics to basic research in biology and medicine
and
biotechnology.
Ph.D. Students
do research work in a variety of fields represented by the biomathematics faculty. Ph.
D. students should complete
all
requirements for a Master's degree then pass the preliminary
examinations. The preliminary examination for
Biomathematics consists of
written
examinations on fours semesters of coursework, and a candidacy
examination. Exemptions from the
written examinations can be made on the basis of grades. See
Guidelines
for
Admission to PhD Candidacy.
Return to Contents
The Department
of
Mathematics requirements for exam scores, recommendations and
statements
are necessary for admission. The typical first semester courses in the
program
require knowledge of undergraduate mathematics including at least
multivariate
calculus, ordinary differential equations and linear algebra. A basic
knowledge
of statistics, computer programming, genetics is helpful.
Students intending to get the PhD degree should have taken more
advanced
courses in mathematics, such as advanced calculus (or real analysis),
complex
variables, abstract algebra, or topology.
Biology and programming prerequisites
for the program.
It is
helpful, but not necessary, to have some background and
coursework in Biology. Also some
programming experience is desirable.
Undergraduate courses can be taken to refresh skills in these
subjects,
as described below.
During their course of study, students may take for S/U credit an
undergraduate genetics course, PCB 3063, and read up on basic concepts
in
genetics and molecular biology. The genetics course is given in the
summer B
term (first half of the summer) and in the Fall semester.
Also students may take for S/U credit an undergraduate course in C++
programming. These
courses are available in the
Summer and Fall.
Do not register for undergraduate courses directly, but see the
Academic
Support Coordinator, currently Esther Diaguila. The registration for
C++ is for
one hour credit. These refresher courses do not count for the
requirements of the degree.
Return to Contents
Financial aid. Most
graduate
students in mathematics have support from teaching assistantships. An
early
application is critical for the best chance of financial aid. Also, the
orientation program for new TAs is offered in Summer C-term, and those
awarded
teaching assistantships may be paid a small stipend beginning at that
time.
Advisement and Supervisory Committees.
Students have a faculty advisor to recommend and approve coursework.
For PhD
students, a Supervisory Committee, which determines the program, is
appointed
consisting of at least three faculty members, with at least one from
the Department
of Mathematics and at least one from another participating department.
Substitutions for courses for which the student has prior credit must
be
approved by the advisor.
Return to Contents
Students are expected to have the mathematics prerequisites for all mathematics courses. For Introduction to Mathematical Biophysics and for Computational Biology student should know basic calculus through differential equations, linear algebra, and some computer programming language. More advanced undergraduate courses are prerequisite for advanced mathematics course sequences such as Complex Variables, Topology, Real Analysis and Groups Rings and Vector Spaces.
Core courses taken by all
students:
- MAP 5485 Introduction to Mathematical Biophysics (Biomathematics I) (Fall)
- MAP 5486 Computational Biology (Biomathematics II) (Spring)
- MAP 6437 Biomedical Mathematics Projects Course (Spring) (student projects)
- MAT 6939 Advanced Seminar in
Biomathematics (1 hour credit each semester)
Typical choices are listed below. Other choices can be approved by the director of the program.
BCH 5205 Structure and Function of Enzymes (Fall)
PCB 5525 Molecular Biology (Fall)
BSC 5936 Membrane Biophysics (Spring)
BCH 5887 Macromolecular X-ray Crystallography
BCH 5887 Biomolecular NMR Spectroscopy
MAD 5932 Methods in Interdisciplinary Applications (Spring)
STA 5236 Distribution Theory (Fall)
STA 5327 Statistical Inference (Spring)
STA 5807 Topics in Stochastic Processes (Summer)
Mathematics courses, additional courses from the following, all together to total 36 hours of listed courses (not including seminar) of which at least 5 courses are in the Department of Mathematics. (Students for the PhD degree should begin taking one of the two semester sequences indicated below. These are the basis for written preliminary examinations, of which the student will take two.)
MTG 5326, 5327 Topology I, II (Fall, Spring)
MAS 5307, 5308 Groups Rings Vector Spaces I, II (Fall, Spring)
MAA 5406, 5407 Complex Variables I, II (Fall, Spring)
MAA 5616, 5617 Measure and Integration I, II (Fall, Spring)
MAD 5403, 5404 Foundations of Computational Mathematics I, II (Fall, Spring)
MAP 5345, 5346 Elementary Partial Differential Equations I, II (Fall, Spring)
MAD
5738, 5739 Numerical
Solution of Partial Differential Equations I, II (Fall, Spring)
MAP 5165 Methods in Applied Mathematics I (Fall)
MAD 5305 Graph Theory
MAA 6416 Topics in Stochastic Calculus (odd-year Springs)
other approved graduate mathematics course
Return to Contents
Course descriptions and prerequisites
Introduction to Mathematical Biophysics
(Biomathematics I),
MAP
5485 (Fall, see Fall
2005 course online)
Most students will take this course in their first semester.
The goal of the course is to introduce students from a variety of
disciplines to some of the many uses of mathematics in modern molecular
biology
and to the use of symbolic and numerical packages for doing the
computations.
Mathematical tools in Biophysics: symbolic and numerical packages for
matrix
computations, rotation matrices, Euclidean motions, lattices,
continuous and
discrete curves in space, torsion angles, gram and distance matrices,
graphs,
trees and strings. Applications such as: protein secondary structure,
structure
determination by crystallography and NMR, writhing twisting and
knotting of
DNA, sequence alignment, HP model of protein folding.
Prerequisites: Calculus,
linear
algebra.
Return to Contents
Computational Biology (Biomathematics II) MAP 5486 (Spring)
Several applications of mathematics to biology will be discussed. Computational methods will be used, in
conjunction with qualitative tools from dynamical systems theory to
analyze the
models. Topics include the construction
and analysis of neuron models, intracellular calcium dynamics, minimal
models
of excitable systems, fast and slow time scales, models of circadian
gene
dynamics, stability properties of delay differential equations, models
of the
cell cycle, stochastic models of ion channel activity, and stochastic
resonance.
Prerequisites: MAP 5485
Introduction to Mathematical Biophysics, MAT 5932,
Methods of Applied Math I, or equivalent knowledge of ODEs and
dynamical
systems. Knowledge of a
computer programming language is expected.
Return to Contents
Elementary
Partial Differential Equations I, II MAP 5345, 5346 (Fall, Spring)
MAP 5345.
Elementary Partial
Differential Equations I (3). Prerequisites: MAC 2313; MAP 2302 or
3305.
Separation of variables; Fourier series; Sturm-Liouville problems;
multidimensional
initial boundary value problems; nonhomogeneous problems; Bessel
functions and
Legendre polynomials.
MAP 5346. Elementary Partial Differential Equations II (3). Prerequisite: MAP 4341 or 5345. Solution of first order quasi-linear partial differential equations; classification and reduction to normal form of linear second order equations; Greens function; infinite domain problems; the wave equation; radiation condition; spherical harmonics.
Foundations
of Computational Mathematics I, II
MAD 5304, 5304 (Fall, Spring)
MAD 5403.
Foundations of
Computational Mathematics I. Analysis and implementation of numerical
algorithms. Matrix analysis, conditioning, errors, direct and iterative
solution of linear systems, rootfinding, systems of nonlinear
equations,
numerical optimization.
Prerequisites: Linear
algebra, competence in a programming language suitable for numeric
computation.
MAD 5404
Foundations of
Computational Mathematics. Interpolation, quadrature, approximation
theory,
numerical methods for ordinary differential equations and partial
differential
equations.
Prerequisite:
MAD 5403.
Numerical
Solution of Partial Differential Equations I, II. MAD 5738, 5739
(Fall, Spring)
Prerequisites:
MAD 5708; MAP
4342 or 5346. Finite difference methods for parabolic, elliptic, and
hyperbolic
problems; consistency, convergence, stability.
Methods
of Applied Mathematics I
MAT
5932 (Fall)
Linear systems
of ODE, phase plane, limit cycles,
bifurcations.
Biomedical
Mathematics Projects Course MAP 6437
(Spring)
The goal of the course give students an opportunity to apply and
supplement
knowledge gained from coursework to real problems in biology or
medicine. Students will give two class presentations concerning
their
research and will present a written report at the end of the
semester.
Prerequisites: This is the projects
course for the Master's degree.
Students should have three semesters of coursework in Biomathematics.
Return to Contents
Distribution Theory STA 5326 (Fall)
Axioms and basic properties of probability, Combinatorial
probability,
Conditional probability and independence, Applications of the Law of
Total
Probability and Bayes Theorem, Random variables, Cumulative
distribution, density,
and mass functions, Distributions of functions of a random variable,
Expected
values, Computations using indicator random variables, Moments and
moment
generating functions, Common families of distributions, Location and
scale
families. Exponential families, Joint and conditional distributions,
Bivariate
transformations, Covariance and correlation, Hierarchical Models,
Variance and
Conditional variance. Introduction to Brownian Motion
Prerequisites.
Three semesters of calculus and an undergraduate course in probability
(or some
exposure to probability plus a sufficiently strong math background).
Return to Contents
Statistical Inference STA 5327 (Spring)
Statistical inference viewed at a measure-theoretic level.
Prerequisites.
STA 5326, Distribution Theory
Return to Contents
Molecular Biology, BCH 5425 (Spring)
Course discusses gene organization and replication; control of gene
expression
in transcription and translation; application of recombinant DNA
techniques. Prerequisites:
Introductory biochemistry or consent of instructor.
Return to Contents
Structure and Function of Enzymes, BCH 5505 (Fall)
Course addresses elements of protein structure and structural motifs, structure determination methods; protein folding and stability; enzyme kinetics and mechanisms; structure-function relationships.
Prerequisites: Pre- or co-requisite: BCH 4053 General Biochemistry I or equivalent.
Return to Contents
Bioinformatics, BSC
5936 (Spring)
Sequences alignment and analysis, phylogenetics, evolutionary trees.
Prerequisite: A course in genetics
Molecular Biology, PCB 5525 (Fall)
Introduction to molecular biology and molecular genetics. The
emphasis will
be on the activities of DNA, RNA, regulation of gene expression, gene
cloning,
bioinformatics, and biotechnology.
Prerequisites: PCB
3063, or the equivalent, or permission of the instructor.
Return to Contents
Membrane Biophysics, BSC 5936
(Spring)
The primary objective of this course is to train the graduate
student with
the necessary mathematical, physiological, and molecular background
that he or
she will need to be able to design competitive research in the field of
membrane biophysics. This course is an integrated approach to
modern biophysics
with an emphasis on neural applications. Modern biophysics
requires a
strong working knowledge of physical laws, molecular approaches,
physiological
responses, structural proteins, and the mechanics of the equipment used
to
measure the physical properties of biological membranes. It is a
tandem
objective of this course that the student will be able to apply this
working
knowledge to a deep comprehension of the primary literature.
Towards this
end, the class will collectively build a literature resource that can
be drawn
upon for a firm foundation for comprehensive research directives in two
fields
1) Ion Channels, and 2) Biophysical Methods.
Methods
in Interdisciplinary Applications (new Spring 2005),
MAP 5932 (Spring)
Regression, time series.
Software: R (free software)
Return to Contents