**General Introduction**

One of the major challenges for biology in the 21st century is an understanding
of the organizational principles of biological systems. Mathematics, statistics,
computer science, and engineering will play key roles in meeting this
challenge. Recent years have seen tremendous progress in our understanding
of genetic information in organisms, and its fundamental role in cell
metabolism. Mathematical and statistical tools have been instrumental
in this progress. Beyond an understanding of the processes at these scales,
integrating them with the organism and ecosystem scales presents many
mathematical challenges. Thus, biology will be one of the major drivers
of the development of new mathematics.

The aim of this Short Course is to present the state of the art and the
mathematical challenges at these different scales, from encoded genomic
information all the way to the ecosystem level. Particular emphasis will
be placed on "nontraditional" mathematical approaches. Two panel discussions
will explore the role mathematics and mathematicians can play in life
sciences and biomedical research.

**Phylogenetics**

*Elizabeth S. Allman, University of Maine*

Understanding evolutionary relationships between species is a fundamental
issue in biology. For many years, this was accomplished through painstaking
comparisons of morphological or other features, perhaps with reference
to the fossil record. As biological sequence data has become readily available
in recent years, however, this new data source has allowed increasingly
mathematical approaches to inferring *phylogenetic trees*.

This talk will begin with a survey of the many ideas that have been used
to construct phylogenetic trees from sequence data. Approaches range from
the primarily combinatorial, to probabilistic model-based methods appropriate
for developing statistical viewpoints. Strengths and weaknesses of the
various mathematical approaches currently employed, as well as points
where further development is needed, will be discussed.

The final part of the talk will discuss a thread of research in which
algebraic methods have been adopted to understand some of the probabilistic
models used in phylogenetics. Recent progress on understanding the set
of possible probability distributions arising from a model as an algebraic
variety has helped provide new theoretical results, and may point toward
improved approaches to phylogenetic inference.

**References**

[1] E. Allman and J. Rhodes, Phylogenetic ideals and varieties
for the general Markov model, preprint, http://arXiv:math. AG/0410604.

[2] J. Felsenstein, *Inferring Phylogenies*, Sinauer
Associates, Sunderland, MA, 2004.

[3] O. Gascuel, ed., *Mathematics of Evolution and Phylogeny*,
Oxford University Press, 2005.

[4] L. Pachter and B. Sturmfels, eds., *Algebraic Statistics
for Computational Biology*, Cambridge University Press, 2005, to appear.

[5] C. Semple and M. Steel, *Phylogenetics*, vol. 24
of *Oxford Lecture Series in Mathematics and Its Applications*, Oxford
University Press, 2003.

**Optimal Control of Population and Disease Models**

*Suzanne Lenhart, University of Tennessee and Oak Ridge National Laboratory*

Optimal control theory has been used successfully to make management
or strategy decisions involving biological or medical models. The desired
goal of the control actions depends on the particular scenario. Making
reasonable choices requires understanding and quantifying the trade-offs
between competing goals and features, including their future relevant
consequences.

This talk gives an introduction to optimal control theory for ordinary
differential equations. The applications emphasize biological models,
especially immunology disease models. The background of an iterative numerical
method used to solve the optimality systems of these problems will be
presented.

**References**

[1] S. P. Sethi and G. L. Thompson, *Optimal Control Theory:
Applications to Management Science and Economics*, Kluwer, 2nd Edition,
2000.

[2] D. E. Kirschner, S. Lenhart, and S. Serbin, Optimal control
of the chemotherapy of HIV, *J. Math Biology* **35 **(1997), 7796.

[3] http://www.math.utk.edu/~lenhart/smb2003.v2.html
(webpage for NIH sponsored short course on this topic).

**Interaction-Based Computing Approach to Modeling and Simulations of
Large Biological and Socio-Technical Systems**

*Madhav Marathe, Virginia Bioinformatics Institute and Department of
Computer Science, Virginia Polytechnic Institute and State University*

The lecture will describe an interaction-based modeling and simulation
approach for understanding large biological, information, social, and
technological (BIST) systems.

Such systems consist of large numbers of interacting components that
together produce a global system with properties that are the result of
interactions among the representations of the local system elements. Examples
of biological systems that are suitable for representation and analysis
using the above paradigm include: bio-signaling systems, ecologies, gene
regulatory networks, the public health system, and contagious disease
economics.

The interaction-based computer simulations are based on a mathematical
and computational theory called Sequential Dynamical Systems (SDS). SDS
theory forms the formal basis for describing complex simulations, by composing
simpler ones. SDS is a new class of discrete, finite dynamical systems.
SDS-based "formal simulations" potentially provide a rigorous, useful
new setting for a theory of interaction-based computation. The setting
is natural for comprehension of distributed systems characterized by interdependent,
but separately functioning sub-parts.

I will focus on the mathematical and computational aspects of SDS. Applicability
of these concepts will be described in the context of large-scale biological
and epidemiological simulations being developed by our group at the Virginia
Bioinformatics Institute.

**Modeling and Simulation of Biochemical Networks**

*Pedro Mendes, Virginia Bioinformatics Institute*

Cellular behavior and function passes through an intricate network of
enzyme-catalyzed reactions. These networks and the phenomena that depend
on them can be modeled through several mathematical frameworks. The best-established
one uses the machinery of calculus, especially differential equations.
A review of this framework for modeling biochemical networks will be made,
starting with its origins in physical chemistry and enzymology. Particular
emphasis will be placed on the biochemical concepts and their mathematical
counterparts. Illustrations of the concepts discussed will be made using
biochemical modeling software.

**Reconstructing Ancestral Genomes**

*Lior Pachter, Department of Mathematics, University of California,
Berkeley*

Recent advances in high-throughput genomics technologies have resulted
in the sequencing of large numbers of (near) complete genomes. These genome
sequences are being mined for important functional elements, such as genes,
however and they are also being compared and contrasted in order to identify
sequences that have been conserved over time. Such sequences frequently
point to biologically important elements. In cases where DNA sequences
from different organisms can be determined to have originated from a common
ancestor, it is natural to try to infer the ancestral sequences. The reconstruction
of the ancestral genome can lead to insights about genome evolution, and
the origin and diversity of functional elements. There are a number of
interesting mathematical questions associated with reconstructing ancestral
genomes. What are the appropriate statistical models for evolution that
allow for making inferences about ancestral sequences? How many present
day genomes are necessary to reconstruct an ancient genome? How can insertions
and deletions be accounted for? We will review progress on these questions,
and highlight interesting mathematical results and problems.

**References**

[1] M. Blanchette, E. D. Green, W. Miller, and D. Haussler,
Reconstructing large regions of an ancestral mammalian genome in silico,
*Genome Research*, **14**:24122423, 2004.

[2] L. Pachter and B. Sturmfels, *Algebraic Statistics for
Computational Biology*, Cambridge University Press, 2005.

[3] ------ , The mathematics of phylogenomics, http://arxiv.
org/abs/math.ST/0409132, 2005.

[4] L. Pachter and S. Snir, Phylogenetic profiling of insertions
and deletions in vertebrate genomes, 2005.

**A Computational Algebra Approach to Systems Biology**

*Brandilyn Stigler, Mathematical Biosciences Institute*

**Synopsis:** One goal of systems biology is to predict and modify
the behavior of biological networks by accurately monitoring and modeling
their responses to certain types of perturbations. The construction of
mathematical models based on observation of these responses, referred
to as reverse engineering, is an important step in elucidating the structure
and dynamics of such networks. Continuous models, described by systems
of differential equations, are an example of a framework that has been
used to reverse engineer biochemical networks. Of increasing interest
is the use of discrete models, given by systems of polynomials defined
over finite fields, which may provide a qualitative description of the
network.

In this talk I will provide an overview of existing reverse-engineering
methods, as well as discuss some modeling issues that arise from biological
systems. A discrete modeling approach, rooted in computational algebra,
to reverse engineer networks from experimental time series data will also
be introduced. The discrete method uses algorithmic tools, including Gröbner-basis
techniques, to build the set of all polynomial models that fit time series
data and to select minimal models from this set. The effectiveness of
the algorithm will be demonstrated on simulated networks. As it is important
to identify the type of data set that are best suited to build accurate
models, properties of data that make them suitable for the algebraic method
will also be discussed.

**Reading List**

[1] P. D'haeseleer, S. Liang, and R. Somogyi, Genetic network
inference: From co-expression clustering to reverse engineering, *Bioinformatics*
**16** (2000) 707726.

[2] H. de Jong, Modeling and simulation of genetic regulatory
systems: A literature review, *J. Computational Biology* **9**
(2002) 67103.

[3] H. Kitano, Systems biology: A brief overview, *Science*
**295** (2002), 16621664.

[4] R. Laubenbacher and B. Stigler, A computational algebra
approach to the reverse engineering of gene regulatory networks, *J.
Theoretical Biology* **229** (2004) 523537.

[5] R. May, Uses and abuses of mathematics in biology, *Science
***303** (2004), 79079