Traditionally astronomy, physics, and
engineering have been the heaviest and almost exclusive users of advanced
mathematical techniques other than statistical methods. In recent
years however, applications to other areas, among them applications
of mathematics to biology, have been mushrooming. The reason for this
timing is simple: problems in the physical sciences often lead to
elegant and parsimonious mathematical models. In contrast, living
creatures tend to be complicated and unpredictable, which makes mathematical
models of them messy and intractable by traditional methods. Fortunately,
with the tremendous power of computers that are sitting on practically
everybody's desktop, these messy models of biological systems can
now be studied numerically, often with enlightening results.
One area that especially interests me
are game-theoretic models of animal interactions. Game theory is a
branch of mathematics that investigates situations of conflict between
two or more players and tries to predict their optimal behavior. In
biological applications, the players are organisms competing for food,
mates, or other resources. Success in a game is usually measured in
the number of offspring a given organism produces. Game-theoretic
models of such situations are developed to explain or predict which
behavioral patterns for conflict resolution will evolve under what
circumstances. The predictions coming out of the mathematical model
can be tested empirically by observing actual animal behavior and/or
by running computer simulations.
Game-theoretic models have been very
successful in explaining why animal contests tend to be settled by
ritualized displays rather than aggressive fights. But if a fight
does occur, then which contestant will more often initiate it, the
likely loser or the likely winner? Together with Molly R. Morris I
am working on the development of game-theoretic models and the design
of experiments that will shed light on this natural but almost completely
unexplored question.
A second force that drives recent developments in biomathematics is
the unprecedented proliferation of biological data, especially genomic
data. For example, the human genome alone consists of approximately
three billion base pairs, which are commonly represented by the letters
A,C,G,T. In order to extract biologically useful information from
these huge data sets, powerful computer algorithms are needed. The
new field of bioinformatics, also known as computational biology or
computational genomics, is devoted to the design, analysis, and fine-tuning
of such algorithms. For more information about important
topics in bioinformatics,
click here.
A particularly powerful technique for
making biological inferences from genomic data are so-called multiple
alignments of corresponding genomic or amino acid sequences from several
different organisms. By comparing amino acids or nucleotides in corresponding
loci, biologists can infer phylogenetic relationships of the organisms
involved or find regions in the proteins that are highly conserved
by evolution and thus apparently crucial for the function of the given
protein. The multiple alignment problem is the problem of finding
the best (with respect to a given scoring scheme) multiple alignment
of a given set of sequences. The problem is nontrivial because evolutionary
changes involve not only replacement of one nucleotide by another,
but also insertions and deletions. In fact, as results by myself and
other authors show, the problem is in general computationally intractible
in the sense that it is not possible to find an algorithm that runs
reasonably fast and always finds the best alignment. These results
show the importance of designing good approximation algorithms that
always find a multiple alignment that is not much worse than the best
one or of heuristic algorithms that in most cases find a reasonably
good multiple alignment. For more information on the multiple
sequence alignment problem and other complexity issues in
bioinformatics,
click
here.
