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Models of biochemical networks are frequently high-dimensional and complex.Reduction methods that preserve important dynamical properties are thereforeessential in their study. Interactions between the nodes in such networks arefrequently modeled using a Hill function, $x^n/(J^n+x^n)$. Reduced ODEs andBoolean networks have been studied extensively when the exponent $n$ is large.However, the case of small constant $J$ appears in practice, but is not wellunderstood. In this paper we provide a mathematical analysis of this limit, andshow that a reduction to a set of piecewise linear ODEs and Boolean networkscan be mathematically justified. The piecewise linear systems have closed formsolutions that closely track those of the fully nonlinear model. On the otherhand, the simpler, Boolean network can be used to study the qualitativebehavior of the original system. We justify the reduction using geometricsingular perturbation theory and compact convergence, and illustrate theresults in networks modeling a genetic switch and a genetic oscillator.

Piecewise linear and Boolean models of chemical reaction networks