302 B. Cummins et al.
Switching Systems of Regulatory Networks.A regulatory network involv-
ingNgenes is often modeled by a system of ordinary differential equations
x ̇i=−γixi+fi(x),i=1,...,N, (1)wherexidenotes the concentration level of protein associated with genei, where
each geneiis associated to a node in the regulatory network. The nonlinearity
fiis meant to capturehow production of one gene is regulated by other genes,
butin practiceit is impossible to derivefifrom first principles. Biologists use
a phenomenological choice offi; the default is usually to expressfiin terms of
Hill functions. We build upon ideas of Glass and Kaufmann [ 18 , 19 ] and consider
a particularly simple form of ( 1 ) called aswitching system;
x ̇i=−γixi+Λi(x),i=1,...,N, (2)whereΛiis defined as sums and products of piecewise constant functions of the
variablesxj.
The parameters of the switching systems are directly relatable to the para-
meters of Hill function based models and include decay ratesγi, and for each
regulating edgej→iin the regulatory network, there are three parameters: a
threshold valueθi,jofxj, at which the piecewise constant functionΛichanges
values, andli,j<ui,j, the two values ofΛiin a neighborhood ofθi,j. We note
that becauseΛiis not continuous, classical solutions to ( 2 ) are not guaranteed
to exist. This does not hinder our ability to use switching systems to combinato-
rialize the dynamics of a network. Furthermore, in our perspective the switching
system, rather than being a model on its own right, is only a computational
model to understand the dynamics of the unknown biologically relevant model
that has the form of ( 1 ) where the nonlinearities are sufficiently smooth to guar-
antee existence and uniqueness of solutions.
A general mathematically precise exposition of how switching systems natu-
rally admit discretization of the phase space, dynamics, and the parameter space
is nontrivial and has been developed in [ 20 ]. In this contribution, our emphasis
is on description of DSGRN as a computational tool and we only briefly describe
the underlying theory.
Combinatorial Dynamics. The threshold parameters θ, which denote the
locations of the abrupt changes inΛidefine a decomposition of phase space
into domains, and in each domain system ( 2 ) is readily solvable. Furthermore,
by representing domains as nodes of a State Transition Graph (STG) we can
unambiguously assign edges between nodes that represent the directions of all
solutions between the domains. To be more formal we letX denote the set of
nodes of STG and to emphasize that we are interested in dynamics we represent
the directed graph as a multivalued combinatorial mapF:X⇒Xwhere a node
ξ′∈F(ξ) if and only if there is an edgeξ→ξ′in the directed graph.
Observe thatFdepends on the choice of parameter values. A key insight
of [ 20 ] is thatFis locally constant, that the boundaries of regions whereFis