Biology Reference
In-Depth Information
Booleanmodels ideal for an early (below-calculus level) introduction tomathematical
models, removing the need for calculus or other mathematical prerequisites. For
mathematics students, such models can be introduced in low-level finite mathematics
or discrete mathematics courses and used to provide an early demonstration of the
important link between mathematics and biology.
At the more advanced mathematics level, Boolean models can be generalized to
finite dynamical systems (FDS) and used to provide an introduction to some serious
theoretical mathematical questions or as a path to questions appropriate for student
research projects. In this chapter we examined the questions of determining the fixed
points of FDS. This leads to a question of solving systems of polynomial equations
over a finite field, for which the theory of Groebner bases provides a practical solution.
The algorithm is essentially a generalization of the well-known process of Gaussian
elimination for solving systems of linear equations.
The actual implementation of the method requires the use of specialized software
(as even the verification that a given set of polynomials is a Groebner basis for an
ideal is labor intensive and virtually impossible to do by hand). Although there are
several open-source computational algebra systems that compute Groebner bases
(e.g., Macaulay 2, MAGMA, CoCoA, SINGULAR, and others), most such systems
require download and installation. For the purposes of this chapter the web-based
SAGE interface toMacaulay 2 is appropriate, as it requires only a few straightforward
commands. The students can then focus on the output and its interpretationwith regard
to the question of solving polynomial systems of equations.
For mathematics students, we see the use of the chapter material to be threefold.
On one hand, it introduces them to a newmodeling approach that is currently not taught
in any of the mainstream undergraduate mathematics courses. On the other hand, FDS
models provide links to important mathematical theory and results in abstract alge-
bra and algebraic geometry that can be further pursued in advanced-level courses or
as independent student research projects. Finally, the topic provides evidence for the
important connections between modern biology and modern mathematics, and can be
used to highlight mathematical and systems biology as career paths for mathematics
majors.
For biology students, the chapter can be used as an introduction to mathematical
modeling without calculus prerequisites. Our experience indicates that the
“just-in-time” approach for developing the necessary mathematical concepts as a way
to formalize specific aspects of the biology works well for Boolean models. It allows
students to focus on the logical links that determine the variable interactions instead of
on the detailed kinetics needed for calculus-based models. Concurrent or subsequent
introduction to such models in calculus or differential equations courses will allow
students to reinforce the conceptual framework, further improve their mathematical
sophistication, and solidify the retention of basic ideas.
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