  Developing efficient computational methods to assess the impact of external
interventions on the dynamics of a network model is an important problem in
systems biology. This paper focuses on quantifying the global changes that
result from the application of an intervention to produce a desired effect,
which we define as the total effect of the intervention. The type of
mathematical models that we will consider are discrete dynamical systems which
include the widely used Boolean networks and their generalizations. The
potential interventions can be represented by a set of nodes and edges that can
be manipulated to produce a desired effect on the system. We use a class of
regulatory rules called nested canalizing functions that frequently appear in
published models and were inspired by the concept of canalization in
evolutionary biology. In this paper, we provide a polynomial normal form based
on the canalizing properties of regulatory functions. Using this polynomial
normal form, we give a set of formulas for counting the maximum number of
transitions that will change in the state space upon an edge deletion in the
wiring diagram. These formulas rely on the canalizing structure of the target
function since the number of changed transitions depends on the canalizing
layer that includes the input to be deleted. We also present computations on
random networks to compare the exact number of changes with the upper bounds
provided by our formulas. Finally, we provide statistics on the sharpness of
these upper bounds in random networks.
