Two classes of ODE models with switch-like behavior

Abstract

In cases where the same real-world system can be modeled both by an ODE system ⅅ and a Boolean system 𝔹, it is of interest to identify conditions under which the two systems will be consistent, that is, will make qualitatively equivalent predictions. In this note we introduce two broad classes of relatively simple models that provide a convenient framework for studying such questions. In contrast to the widely known class of Glass networks, the right-hand sides of our ODEs are Lipschitz-continuous. We prove that if 𝔹 has certain structures, consistency between ⅅ and 𝔹 is implied by sufficient separation of time scales in one class of our models. Namely, if the trajectories of 𝔹 are “one-stepping” then we prove a strong form of consistency and if 𝔹 has a certain monotonicity property then there is a weaker consistency between ⅅ and 𝔹. These results appear to point to more general structure properties that favor consistency between ODE and Boolean models.

Approximating network dynamics: Some open problems

“…All one-stepping Boolean functions are monotonestepping, but not vice versa. For precise details, see Definition 4 of [20]. These results conform to expectations that one might form based on simulation studies for gene regulatory networks reported in [1].…”

“…A class of toy models. In order to gain insight into general mechanisms that favor consistency or strong consistency with Boolean models, two classes D 1 and D 2 of toy ODE models were introduced and studied in [20]. The dynamics of the flows M for these systems are relatively easy to understand, and each class is rich enough so that every Boolean network N with n variables becomes a natural candidate for a Boolean approximation to one of these flows.…”

“…These constructions are partially based on the ones proposed in [32]. In the interest of greater transparency, here we only outline the basic mechanism; for the technical details we refer the reader to [20].…”

“…We can then define a sequence (T t ) t∈N = (T t ( x(0))) t∈N by taking T j as the midpoints of the intervals between two successive switching times if there is no last such time, or in terms of fixed increments after such a last switching time. In [20], M and N were considered (strongly) consistent if the resulting sequence satisfied the requirements of Definition 4.1.…”

“…, f n ) of N is what we called one-stepping, which means that for every s the Boolean vectors s and f ( s) differ in at most one coordinate. Hence the following theorem of [20] is in a sense the strongest possible one that can be obtained about strong consistency:…”