  Identifying features of molecular regulatory networks is an important problem
in systems biology. It has been shown that the combinatorial logic of such
networks can be captured in many cases by special functions called nested
canalyzing in the context of discrete dynamic network models. It was also shown
that the dynamics of networks constructed from such functions has very special
properties that are consistent with what is known about molecular networks, and
that simplify analysis. It is important to know how restrictive this class of
functions is, for instance for the purpose of network reverse-engineering. This
paper contains a formula for the number of such functions and a comparison to
the class of all functions. In particular, it is shown that, as the number of
variables becomes large, the ratio of the number of nested canalyzing functions
to the number of all functions converges to zero. This shows that the class of
nested canalyzing functions is indeed very restrictive, indicating that
molecular networks have very special properties. The principal tool used for
this investigation is a description of these functions as polynomials and a
parameterization of the class of all such polynomials in terms of relations on
their coefficients.
