SHORT COURSES ON THE MATHEMATICS
OF BIOLOGICAL COMPLEXITY
University of Tennessee,
Supported by National Institutes
of Health Award GM59924-01
For related NIH programs see NIGMS Complex
2000 Short Courses
Course 1: June 11-14, 2000
Introduction to the Mathematics of Biological Complexity
Course 2: October 1-4, 2000
Complexity in Evolutionary Biology: Genetic Algorithms,
Cellular Automata and Adaptive Landscapes
Course 3: December 17-20, 2000
Nonlinear Time Series Analysis in Biology
A series of three Short Courses will be held at the
University of Tennessee oriented towards biologically-trained individuals.
The objective is to provide a rapid introduction to the mathematical and
computational topics appropriate for understanding current research in biological
complexity. These Courses will each last four days, starting on a Sunday
morning and continuing through mid-day Wednesday. The Courses will be led
by a group of distinguished faculty with particular expertise in biological
modeling. Each Course will include computer-based workshops in addition to
formal lectures and discussion sessions. Each Course is organized separately,
though some individuals will benefit from attending two or more of them, depending
upon their research interests.
should complete the application form and email or FAX it to the
Director of the Courses as per instructions. Applicants will be notified
of their acceptance by email, and given details regarding travel and lodging
arrangements. Applicants for Course 1 will start to be accepted immediately
with a final deadline of April 30 for this Course - individuals are urged
not to delay in applying. Applicants for Courses 2 and 3 will start to be
accepted 3 months before the start of the Course, with a final application
deadline 2 months before the start of each of these Courses.
Organizer: Dr. Louis J. Gross
- Professor of Ecology and Evolutionary Biology
- Director, The Institute for Environmental Modeling
- University of Tennessee
- Dr. Stephen P. Ellner
- Professor of Statistics and Biomathematics
- North Carolina State University
- Dr. Denise Kirschner
- Assistant Professor of Microbiology and
- University of Michigan Medical
- Course Organizer: Dr. Sergey Gavrilets
- Associate Professor of Ecology and Evolutionary Biology
- University of Tennessee
- Dr. Lee Altenberg
- Department of Information and Computer
- University of Hawaii at Manoa
- Dr. Janko Gravner
- Associate Professor of Mathematics
- University of California, Davis
- Course Organizer: Dr. Daniel Kaplan
- Assistant Professor of Mathematics and Computer Science
- Macalester College
- Dr. Leon Glass
- Centre for Nonlinear Dynamics in Medicine
and Physiology, Department of Physiology
- McGill University
- Max Planck Institute
for the Physics of Complex Systems
Course 1: Introduction to
the Mathematics of Biological Complexity
This Course will provide an overview
of mathematical and computational approaches useful in analyzing complex
biological systems including: continuous dynamical systems, discrete dynamical
systems, matrix approaches including structured population models and Markov
chains, and stochastic process models. This is designed for biologists who
require a rapid, broad overview of modern quantitative techniques that appear
again and again in many biological contexts. The focus will be on modeling
methods, conceptual foundations and biological applications rather than detailed
computations. The concepts will be motivated by numerous biological examples,
chosen from physiology, genetics and infectious diseases. An objective is
to ensure that participants for whom the first short course is appropriate
would have been exposed to the basic mathematical background materials required
for either of the second or the third Courses. Another objective is to enable
attendees to read with comprehension the modeling literature in their own
fields of biological interest.
Topic coverage will include the
below, but will emphasize topics of particular interest to attendees. Accepted
attendees will be polled to ensure that topics of particular interest to
many attendees will be covered.
An overview of calculus with focus on differential
equations in preparation for the section on dynamical systems and diffusion.
An overview of linear algebra in preparation for
the section on Markov Chains and the section on dynamical systems.
An overview of basic probability in preparation
for the section on stochastic models.
How to model it - examples of basic discrete and
continuous models in application to chemostats, population genetics, and
physiology. Application of non-dimensionalization methods to reduce the number
of parameters in a model.
Introduction to dynamical systems - Discrete and
continuous compartment models, phase-plane analysis.
Transport and diffusion - partial differential
equations, the conservation equation, steady-states and traveling waves.
Stochastic models - birth and death processes,
branching processes, Markov Chains. Applications to genetics and evolutionary
Lecture notes and other materials
from Course 1 are at Course 1 Home Page
Course 2: Complexity in Evolutionary Biology:
Genetic Algorithms, Cellular Automata and Adaptive Landscapes
Complex structures, phenotypes and behaviors are
common in many evolutionary processes. A theory of these phenomena might
have important implications for understanding the processes of extinction
and diversification in biological systems. The course will consist of three
parts introducing three different approaches to the studying of complex systems
in biology: Genetic Algorithms, Cellular Automata and Adaptive Landscapes.
Topic coverage will include the below, but will emphasize topics of particular
interest to attendees. Accepted attendees will be polled to ensure that topics
of particular interest to many attendees will be covered.
Part 1. Evolutionary Computation (Lee Altenberg)
Evolutionary Computation (EC) is a computer-based
method of problem solving which imitates basic processes of biological evolution
(recombination, mutation, selection, etc) to find good solutions within
large spaces of candidate solutions. This is done by the creation within
a computer of a population of "individuals" (trial solutions) represented
by "chromosomes" or other data structures (such as trees in Genetic Programming).
The data structures may encode the values for the different parameters
being optimized, the components of an algorithm being optimized, or elements
of the space being searched. An initial population is sampled from the
search space, typically with no prior information. From there on, the genetic
operators are used to generate new sample points from the previous ones,
and a fitness measure is used to select the previous samples to use as
"parents" for the new samples. Iterated application of genetic operators
and selection improve the population through a process of simulated evolution.
EC is used to heuristically find solutions where analytical and other Monte-Carlo
search methods fail. The list of successful EC applications includes biocomputing/bioinformatics
(protein and RNA folding, sequence alignments), cellular programming, scheduling
and timetabling problems (e.g. classes and exams), mechanical engineering,
and non-linear filtering. This section of the course will be at a level
sufficient for participants to leave with an understanding of the material
in Holland (1992), Mitchell (1996), and Toquenaga and Wade (1996).
Part 2. Cellular Automata (Janko Gravner)
Cellular automata (CA) form a general class of
models of dynamical systems which are appealingly simple and yet capture
a rich variety of behavior. These properties have made them a favorite tool
for studying and modeling the generic behavior of complex dynamical systems.
In constructing a CA one assumes that the system under consideration is
distributed in space, and that nearby regions ("cells") have more to do
with each other than with regions far apart. The time dependence of cell
variables is given by an iterative rule that specifies what is happening
at a given instance of time in a particular cell as a function of the state
of this cell and other "neighboring" cells. Perhaps the most famous example
of CA is Conway's Game of Life, designed to capture the basic phenomenology
in reproduction and death of biological organisms. CA have been used for
simulating polymer folding, pattern formation, ecological interactions,
excitable media, crystal and tumor growth, and spread of epidemics. This
course will be oriented towards examples which will illustrate modeling,
experimental and mathematical aspects of CA research, and will be at a level
sufficient for participants to leave with an understanding of the material
in Wolfram (1984), Ermentrout and Edelstein-Keshet (1993), Duchting (1990),
Bar-Yam (1997), and Gaylord and Wellin (1994).
Part 3. Adaptive Landscapes (Sergey Gavrilets)
The metaphor of adaptive landscapes (AL) was introduced
by S. Wright (1932) to visualize the complex relationships between genotype
and fitness. AL can be thought of as a surface in a multidimensional space,
with an individual being a point of the surface, and a population being a
cloud of points. Different factors operating in natural populations (selection,
mutation, recombination etc) change both the location and structure of this
cloud. Several specific forms of AL have been considered: rugged (e.g. Wright
1932; Kauffman 1993), flat (e.g. Kimura 1983; Derrida and Peliti 1991), single-peak
(e.g. Fisher 1930), holey (e.g. Dobzhansky 1937; Gavrilets 1997). In evolutionary
biology AL have been extensively used for modeling adaptation, molecular
evolution and speciation. Random walks on AL have been a general metaphor
for thinking about optimization in general (Kauffman 1993). Different applications
of the metaphor of AL will be illustrated.
Lecture notes and other materials from Course
2 are at Course
2 Home Page
Course 3: Nonlinear Time Series Analysis in Biology.
This Course will emphasize approaches to model and analyze irregular
and chaotic time series from medicine, epidemiology, and physiology. Most
complicated systems in biology are nonlinear. As a result, data from complicated
biological systems are increasingly being analyzed using tools and concepts
that were developed (1) within the framework of nonlinear dynamics, and (2)
with an eye towards understanding deterministic chaos.
A brief outline of the Course follows, but will emphasize topics
of particular interest to attendees. Accepted attendees will be polled to
ensure that topics of particular interest to many attendees will be covered.
of Nonlinear Dynamics: Dynamical systems. Maps and flows. The Poincare
surface of section. Equilibria, limit cycles, tori, and strange attractors.
Routes to chaos.
Methods: Phase-space reconstruction. Embedding theorems. Delay coordinates.
The choice of an appropriate time lag. Derivative coordinates. Determining
the embedding dimension for phase-space reconstruction.
and Dimension: Attractor geometry and the spectrum of fractal dimensions.
Correlation dimension. Advanced algorithms. How long is long enough? Traps
and Lyapunov Exponents: Sensitive dependence on initial conditions. Exponential
divergence. Numerical calculation of Lyapunov exponents and the Lyapunov
spectrum. Detecting and rejecting spurious Lyapunov exponents.
Methods: Testing for nonlinearity in time series. The method of surrogate
Forecasting, and Controlling Chaos: Noise reduction in dynamical systems.
Nonlinear prediction of chaotic time series. Stabilization of unstable
A tentative schedule of Course 3 is here.
Introductory material and other items related to Course 3 are
at Course 3 Home
Any questions about the Courses should be
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Last Modified: September 12, 2000