Authors: Alan VelizCuba Ajit Kumar Krešimir Josić
Publish Date: 2014/11/21
Volume: 76, Issue: 12, Pages: 2945-2984
Abstract
Models of biochemical networks are frequently complex and highdimensional Reduction methods that preserve important dynamical properties are therefore essential for their study Interactions in biochemical networks are frequently modeled using Hill functions xn/Jn+xn Reduced ODEs and Boolean approximations of such model networks have been studied extensively when the exponent n is large However while the case of small constant J appears in practice it is not well understood We provide a mathematical analysis of this limit and show that a reduction to a set of piecewise linear ODEs and Boolean networks can be mathematically justified The piecewise linear systems have closedform solutions that closely track those of the fully nonlinear model The simpler Boolean network can be used to study the qualitative behavior of the original system We justify the reduction using geometric singular perturbation theory and compact convergence and illustrate the results in network models of a toggle switch and an oscillatorHere we present a heuristic justification of the use of Eq 1 The ideas follow those presented in Goldbeter and Koshland 1981 Goldbeter 1991 Novak et al 1998 Novak et al 2001 Tyson et al 2003 Aguda 2006 As mentioned in the Introduction this is only heuristic in generalPlots of righthand side of Eq 22 for three different values of J as functions of x Other parameters A = 1 I = 5 This figure suggests that differential equations of the form Eq 22 can be approximated by linear ODEs in the interior of the domain
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