Quantifying Side Effects in Multistate Discrete Networks

Developing efficient computational methods to change the state of a cell from an undesirable condition, e.g. diseased, into a desirable, e.g. healthy, condition is an important goal of systems biology. The identification of potential interventions can be achieved through mathematical modeling of the state of a cell by finding appropriate input manipulations in the model that represent external interventions. This paper focuses on quantifying the unwanted or unplanned changes that come along with the application of an intervention to produce a desired effect, which we define as the \emph{side effects} 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 control targets can be represented by a set of nodes and edges that can be manipulated to produce a desired effect on the system. This paper presents practical tools along with applications for the analysis and control of multistate networks. The first result is a polynomial normal form representation for discrete functions that provides a partition of the inputs of the function into canalizing and non-canalizing variables and, within the canalizing ones, we categorize the input variables into layers of canalization. The second theoretical result is 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 stratification of the inputs of the target function where the number of changed transitions depends on the layer of canalization that includes the input to be deleted. Applications from using these formulas to estimate the number of changes in the state space and comparisons with the actual number of changes are also presented.