University of Tennessee, Knoxville

Supported by National Institutes of Health Award GM59924-01

For related NIH programs see NIGMS Complex Systems Initiatives

 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.



The Faculty

The Courses

Further Information

Application Form


Interested individuals are invited to apply for admission to these Courses. Attendance will be by invitation only and will be limited to forty participants for each of the Courses. Preference will be given to individuals without extensive prior mathematical training, particularly for the first of the Courses.

These Courses are supported by a grant from the National Institutes of Health, which will cover most local expenses, including hotel, meals and the registration fee, for participants from colleges and non-profit organizations. Additionally, travel grants are available for those who cannot obtain travel funds from other sources. Preference for these grants will be given to graduate students and those who can obtain some support from their home institution, but grants of full travel support are also available. A special airfare discount is available through Delta airlines, including a discount off the supersaver fares. All those requesting travel grants will be expected to make travel arrangements at the lowest possible fare through our local travel agent. Generally, individuals from private, for-profit institutions will be expected to cover their own expenses. Discounted lodging rates are available at the hotel for these Courses (Holiday Inn, World's Fair Park) which is adjacent to the University of Tennessee Conference Center where all Courses session will be held.

Applicants 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.


The registration fee for each Course is $300, which includes all Course materials as well as all meals during the Course (breakfast, lunch and dinner for Sunday through Tuesday and breakfast and lunch on Wednesday). There is an additional non-refundable deposit of $50 for all participants required within two weeks of acceptance. The negotiated lodging rate is $69 per day plus tax for a single room. Shuttles will be provided for transportation between the airport and the hotel.

The Faculty:

Course 1:

Course Organizer: Dr. Louis J. Gross
Professor of Ecology and Evolutionary Biology and Mathematics
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 Immunology
University of Michigan Medical School

Course 2:

Course Organizer: Dr. Sergey Gavrilets
Associate Professor of Ecology and Evolutionary Biology and Mathematics
University of Tennessee
Dr. Lee Altenberg
Department of Information and Computer Sciences
University of Hawaii at Manoa
Dr. Janko Gravner
Associate Professor of Mathematics
University of California, Davis

Course 3:

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
Dr. Thomas Schreiber
Max Planck Institute for the Physics of Complex Systems

The Courses:

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 theory.

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.

An Overview of Nonlinear Dynamics: Dynamical systems. Maps and flows. The Poincare surface of section. Equilibria, limit cycles, tori, and strange attractors. Routes to chaos.
Embedding 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.
Self-similarity and Dimension: Attractor geometry and the spectrum of fractal dimensions. Correlation dimension. Advanced algorithms. How long is long enough? Traps and pitfalls.
Instability 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.
Statistical Methods: Testing for nonlinearity in time series. The method of surrogate data.
Filtering, Forecasting, and Controlling Chaos: Noise reduction in dynamical systems. Nonlinear prediction of chaotic time series. Stabilization of unstable orbits.

A tentative schedule of Course 3 is here.

Introductory material and other items related to Course 3 are at Course 3 Home Page

Further Information:

Any questions about the Courses should be directed to

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Last Modified: September 12, 2000