Table Of Contenthttp://dx.doi.org/10.1090/psapm/064
AMS SHORT COURSE LECTURE NOTES
Introductory Survey Lectures
published as a subseries of
Proceedings of Symposia in Applied Mathematics
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Proceedings of Symposia in
APPLIED MATHEMATICS
Volume 64
Modeling and
Simulation of
Biological Networks
American Mathematical Society
Short Course
January 10-11, 2006
San Antonio, Texas
Reinhard C Laubenbacher
Editor
tfSEMAT/
American Mathematical Society
!$ Providence, Rhode Island
Editorial Board
Mary Pugh Lenya Ryzhik Eitan Tadmor (Chair)
LECTURE NOTES PREPARED FOR THE
AMERICAN MATHEMATICAL SOCIETY SHORT COURSE
MODELING AND SIMULATION OF BIOLOGICAL NETWORKS
HELD IN SAN ANTONIO, TEXAS
JANUARY 10-11, 2006
The AMS Short Course Series is sponsored by the Society's Program Committee for
National Meetings. The series is under the direction of the Short Course Subcommittee
of the Program Committee for National Meetings.
2000 Mathematics Subject Classification. Primary 92B05.
Library of Congress Cataloging-in-Publication Data
American Mathematical Society. Short Course, Modeling and Simulation of Biological Networks
(2006 : San Antonio, Tex.)
Modeling and simulation of biological networks / Reinhard C. Laubenbacher, editor.
p. cm.—(Proceedings of symposia in applied mathematics ; v. 64)
Includes bibliographical references and index.
ISBN 978-0-8218-3964-5 (alk. paper)
1. Biology—Mathematical models—Congresses. 2. Computational biology—Congresses.
I. Laubenbacher, Reinhard. II. Title.
QH323.5 .A45 2006
570.1'5118—dc22 2007060770
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10 9 8 7 6 5 4 3 2 1 12 11 10 09 08 07
Contents
Introduction to Modeling and Simulation of Biological Networks vii
An introduction to reconstructing ancestral genomes
LIOR PACHTER 1
Phylogenetics
ELIZABETH S. ALLMAN AND JOHN A. RHODES 21
Polynomial dynamical systems in systems biology
BRANDILYN STIGLER 53
An introduction to optimal control applied to immunology problems
SUZANNE LENHART AND JOHN T. WORKMAN 85
Modeling and simulation of large biological, information and socio-technical
systems: An interaction-based approach
CHRISTOPHER L. BARRETT, KEITH BISSET, STEPHEN EUBANK,
MADHAV V. MARATHE, V. S. ANIL KUMAR, AND HENNING S.
MORTVEIT 101
Index 149
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Introduction to
Modeling and Simulation of Biological Networks
"All processes in organisms, from the interaction of molecules to the complex
functions of the brain and other whole organs, strictly obey these physical laws.
Where organisms differ from inanimate matter is in the organization of their systems
and especially in the possession of coded information [M]."
It is the task of computational biology to help elucidate those differences. This
process has barely begun, and many researchers are testing computational tools that
have been used successfully in other fields for their efficacy in helping to understand
many biological systems. Mathematical and statistical network modeling is an im
portant step toward uncovering the organizational principles and dynamic behavior
of biological networks. Undoubtedly, new mathematical tools will be needed, how
ever, to meet this challenge. The workhorse of this effort at present comprises
the standard tools from applied mathematics, which have proven to be successful
for many problems. But new areas of mathematics not traditionally considered
applicable are contributing powerful tools.
The advent of "digital biology" has provided a rich application area for dis
crete mathematics. One type of problem faced by life scientists is computational:
organize data into models that are explanatory and predictive. But another type
of problem is conceptual. An important problem to make biological phenomena
treatable with quantitative methods is the need for a language to express concepts
such as "self-organization" or "robustness" of biological systems. There is no telling
what mathematical specialty has the right tools for this task. Both the National
Science Foundation and the National Institutes of Health are investing heavily in
fostering the synthesis between biology and mathematics. Progress can be made
only through a close collaboration between life scientists and quantitative scien
tists, in particular mathematicians. Altogether, the mathematical sciences face an
exciting and stimulating challenge. In [C] the author argues that the relationship
between mathematics and biology in the twenty-first century might rival that of
mathematics and physics in the twentieth. In [S] we can find examples of new
theorems inspired by biological problems.
The AMS Short Course Modeling and Simulation of Biological Networks at the
2006 Joint Annual Meetings, on which this volume is based, was intended to intro
duce this topic to a broad mathematical audience. The aim of the course and of
this volume is to explain some of the biology and the computational and mathe
matical challenges we are facing. The different chapters provide examples of how
these challenges are met, with particular emphasis on nontraditional mathemati
cal approaches. The volume features a broad spectrum of networks across scales,
ranging from biochemical networks within a single cell to epidemiological networks
vii
Vlll INTRODUCTION
encompassing whole cities. Also, the volume is broad in the range of mathematical
tools used in solving problems involving these networks.
Chapters: The first two chapters, one by Elizabeth Allman and John Rhodes,
and the other by Lior Pachter, focus on the "coded information" that Mayr refers
to above. They discuss mathematical tools that help analyze genome-level infor
mation, locate genes in newly sequenced genomes, and organize evolutionary infor
mation. The mathematical areas involved include statistics, discrete mathematics
and algebraic geometry.
The third chapter, by Brandilyn Stigler, discusses the biochemical networks
that translate genome-level information into cellular metabolism, using the exam
ple of gene regulatory networks. Constructing network-level mathematical models
poses unique challenges and constitutes one of the frontiers of research in math
ematical biology. The main mathematical areas discussed in these chapters are
dynamical systems theory and computational algebra.
Cells assemble to form organisms, and organisms assemble to form populations.
The fourth chapter, by Suzanne Lenhart and John Workman, describes a control-
theoretic approach to problems in immunology, such as drug delivery. The final
chapter, by Christopher Barrett, Keith Bisset, Stephen Eubank, Madhav Marathe,
A. Kumar, and H. Mortveit, discusses an interaction-based approach to modeling in
population biology and epidemiology, as well as mathematical problems associated
with this modeling paradigm.
The Short Course lectures were complemented by two panel discussions: 1. The
New Face of Mathematical Biology. Over the last decade mathematical biology has
dramatically changed, in particular with the advent of high-throughput genomics
and the need for mathematical and statistical methods to align and annotate the
large number of complete genomes that are becoming available. This discussion
focused on new areas of research and the central role that mathematics, including
pure mathematics, can play in the life sciences. 2. Opportunities in Mathematical
Biology. This discussion focused on new opportunities for mathematics students
and researchers in this field. If the mathematics community embraces the fact that
the twenty-first century clearly will be the century of biology, then the life sciences
can play a role in twenty-first-century mathematics similar to the role of physics in
the twentieth.
Acknowledgement. The editor thanks the AMS staff involved in the produc
tion of this volume and the organization of the AMS Short Course preceding it for
their extremely helpful and professional services.
References
[C] J. E. Cohen, Mathematics is biology's next microscope, only better; biology is mathematics'
next physics, only better, PLoS Biology 2 (12) 2004.
[M] E. Mayr, Toward a new philosophy of biology, Harvard Univ. Press, Cambridge, MA, 1988.
[S] B. Sturmfels, Can biology lead to new theorems?, Clay Mathematical Institute 2005 Annual
Report.
Reinhard Laubenbacher
VIRGINIA BIOINFORMATICS INSTITUTE AT VIRGINIA TECH
http://dx.doi.org/10.1090/psapm/064/2359647
Proceedings of Symposia in Applied Mathematics
Volume 64, 2007
An Introduction to Reconstructing Ancestral Genomes
Lior Pachter
ABSTRACT. 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. They are also being compared and contrasted in order to identify other
functional sequences, such as those involved in the regulation of genes. 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 an
cestral sequences. The reconstruction of ancestral genomes can lead to insights
about genome evolution, and the origins and diversity of function. There are
a number of interesting foundational questions associated with reconstructing
ancestral genomes: Which statistical models for evolution should be used for
making inferences about ancestral sequences? How should extant genomes be
compared in order to facilitate ancestral reconstruction? Which portions of
ancestral genomes can be reconstructed reliably, and what are the limits of
ancestral reconstruction? We discuss recent progress on some of these ques
tions, offer some of our own opinions, and highlight interesting mathematics,
statistics, and computer science problems.
1. What is comparative genomics?
These notes summarize a lecture at a special session of the American Math
ematical Society on mathematical biology, during which we discussed the central
problem of comparative genomics, namely how to reconstruct the ancestral genomes
that evolved into the present-day extant genomes. This is fundamentally a statis
tics problem, because with a few exceptions, it is not possible to sequence the
genomes of ancestral species, and one can only infer ancestral genomes from the
multitude of genomes that can be sampled at the present time. The problem is a
grand scientific challenge that has only begun to be tackled in recent years, now
that whole genomes are being sequenced for the first time. Our aim is to introduce
the reader to the statistical (and related mathematical) elements of the methods
of comparative genomics, while providing a glimpse of the exciting results that are
emerging from first generation attempts to reconstruct ancestral genomes. Due to
the complex interdisciplinary scope of the subject, we have been forced to omit a lot
of detail and many interesting topics, but we hope that the curious mathematical
reader may find some threads worthy of further exploration.
1991 Mathematics Subject Classification. Primary 92D15, 62P10; Secondary 94C15,68W30.
Key words and phrases. Comparative genomics, statistics, algebra, combinatorics.
The author was supported in part by NSF Grant CCF-0347992.
©2007 American Mathematical Society
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