Monomials and Basin Cylinders for Network Dynamics

Abstract

We describe methods to identify cylinder sets inside a basin of attraction for Boolean dynamics of biological networks. Such sets are used for designing regulatory interventions that make the system evolve towards a chosen attractor, for example initiating apoptosis in a cancer cell. We describe two algebraic methods for identifying cylinders inside a basin of attraction, one based on the Groebner fan that finds monomials that define cylinders and the other on primary decomposition. Both methods are applied to current examples of gene networks.

Computational methods for asynchronous basins

“…Ways to find control variables, or nodes for intervention, are developed in [3]. One way is to find the prime decomposition (see [7]) of the ideal I Bex,A which is the algebraic way to find simple cylindrical subsets of the variety B ex,A .…”

“…Then the prime ideal P = 〈 s j 1 − a 1 , …, s j c − a c 〉 in the ring ℂ[ s ] will show up in the prime or minimal decomposition ∩ j P j of the radical ideal $\sqrt{Ī}$. The details are in Theorem 2.2 of [3]. …”