  Boolean network models of molecular regulatory networks have been used
successfully in computational systems biology. The Boolean functions that
appear in published models tend to have special properties, in particular the
property of being nested canalizing, a concept inspired by the concept of
canalization in evolutionary biology. It has been shown that networks comprised
of nested canalizing functions have dynamic properties that make them suitable
for modeling molecular regulatory networks, namely a small number of (large)
attractors, as well as relatively short limit cycles.
  This paper contains a detailed analysis of this class of functions, based on
a novel normal form as polynomial functions over the Boolean field. The concept
of layer is introduced that stratifies variables into different classes
depending on their level of dominance. Using this layer concept a closed form
formula is derived for the number of nested canalizing functions with a given
number of variables. Additional metrics considered include Hamming weight, the
activity number of any variable, and the average sensitivity of the function.
It is also shown that the average sensitivity of any nested canalizing function
is between 0 and 2. This provides a rationale for why nested canalizing
functions are stable, since a random Boolean function in n variables has
average sensitivity n/2. The paper also contains experimental evidence that the
layer number is an important factor in network stability.
