Mechanisms of Gene Regulation: Boolean Network Models of the Lactose Operon in Escherichia coli - Mathematical Concepts and Methods in Modern Biology - page 28

L l .
Exercise 1.19. Translate the remaining eight functions inEqs. ( 1.8 ) into polynomials
and simplify as much as possible to obtain the polynomial form of the Boolean model
defined by the set of Eqs. ( 1.8 ). Write down the whole model (it has nine variables
and two parameters!) and save it. You will need this polynomial form of the model
again in Example 1.10 and in Exercise 1.20 below.

Thus, in polynomial form, f A l = ALL l + AL l + LL l + AL + A + L +

Once the system of Boolean Eqs. ( 1.10 ) is translated into a system of polynomial
equations, finding the fixed points becomes a problem of solving a polynomial system
of equations. This can be done by employing a technique from computational algebra
involving Groebner bases. With this technique, it is generally possible to rewrite a
system of polynomial equations in a much simpler form while preserving the set
of solutions. Under some additional conditions, the reduced form of the system of
equations is guaranteed to have a form that makes finding the solutions possible
by back-substitution. Readers familiar with the method of Gaussian elimination for
systems of linear equations will notice that finding a Groebner basis that diagonalizes
the system of equations can be viewed as a generalization of the Gaussian elimination
method for polynomial functions.
For readers with appropriate mathematical background in abstract algebra, includ-
ing the basic theory of rings, fields, ideals, and varieties, an outline of the theory of
Groebner bases, as it relates to solving systems of polynomial equations, is presented
in the online Appendix 1. For what follows, however, we do not assume familiarity
with this theory. Instead, we illustrate how knowing the diagonalized forms of the
polynomial systems (obtained by determining the Groebner basis with the use of
computational software) can help to solve the system of polynomial equations and
determine the fixed points.
Various computer systems includingMacaulay 2,MAGMA, CoCoA, SINGULAR,
and others, have the capability of computing Groebner bases. For our needs, using
the web-based SAGE interface for Macaulay 2 for determining the Groebner basis
for a diagonalized reduction is easy and convenient ( http://www.sagemath.org/ ) . We
suggest that you use the online option first. Select “Try SAGE Online” and register for
a SAGE notebook account. We consider a few examples below. The first two illustrate
how the methods can be used to solve systems of polynomial equations over the real
numbers, a problem that the reader is likely to be more familiar with. We then use the
method to determine the fixed points of the Boolean model of the lac operon defined
by Eqs. ( 1.8 ).
Example 1.8. Consider the system of polynomial equations below where x , y, and
z are real numbers. We want to find the solutions of the system of equations.

x 2 y 2 z 2 + + = 1

x 2 z 2 + = y

x − z = 0 .