TY - JOUR
T1 - Global dynamics for switching systems and their extensions by linear differential equations
AU - Huttinga, Zane
AU - Cummins, Bree
AU - Gedeon, Tomáš
AU - Mischaikow, Konstantin
N1 - Funding Information:
T.G. was partially supportedby National Science Foundation (NSF) grants DMS-1226213 DMS-1361240 , DARPA D12AP200025 and NIH R01 grant 1R01AG040020-01 . B.C. was supported by DARPA D12AP200025 and Z.H. was partially supported by the NIGMS Award P20GM103474 .
Funding Information:
Research reported in this publication was supported by the National Institute of General Medical Sciences of the National Institutes of Health under Award Number P20GM103474 . The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
Publisher Copyright:
© 2017 Elsevier B.V.
PY - 2018/3/15
Y1 - 2018/3/15
N2 - Switching systems use piecewise constant nonlinearities to model gene regulatory networks. This choice provides advantages in the analysis of behavior and allows the global description of dynamics in terms of Morse graphs associated to nodes of a parameter graph. The parameter graph captures spatial characteristics of a decomposition of parameter space into domains with identical Morse graphs. However, there are many cellular processes that do not exhibit threshold-like behavior and thus are not well described by a switching system. We consider a class of extensions of switching systems formed by a mixture of switching interactions and chains of variables governed by linear differential equations. We show that the parameter graphs associated to the switching system and any of its extensions are identical. For each parameter graph node, there is an order-preserving map from the Morse graph of the switching system to the Morse graph of any of its extensions. We provide counterexamples that show why possible stronger relationships between the Morse graphs are not valid.
AB - Switching systems use piecewise constant nonlinearities to model gene regulatory networks. This choice provides advantages in the analysis of behavior and allows the global description of dynamics in terms of Morse graphs associated to nodes of a parameter graph. The parameter graph captures spatial characteristics of a decomposition of parameter space into domains with identical Morse graphs. However, there are many cellular processes that do not exhibit threshold-like behavior and thus are not well described by a switching system. We consider a class of extensions of switching systems formed by a mixture of switching interactions and chains of variables governed by linear differential equations. We show that the parameter graphs associated to the switching system and any of its extensions are identical. For each parameter graph node, there is an order-preserving map from the Morse graph of the switching system to the Morse graph of any of its extensions. We provide counterexamples that show why possible stronger relationships between the Morse graphs are not valid.
KW - Gene regulation
KW - Morse graphs
KW - Switching systems
KW - Transcription/translation model
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U2 - 10.1016/j.physd.2017.11.003
DO - 10.1016/j.physd.2017.11.003
M3 - Article
AN - SCOPUS:85036647065
SN - 0167-2789
VL - 367
SP - 19
EP - 37
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
ER -