Vol. 10, No. 5, 2017

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ISSN: 1944-4184 (e-only)
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Relations between the conditions of admitting cycles in Boolean and ODE network systems

Yunjiao Wang, Bamidele Omidiran, Franklin Kigwe and Kiran Chilakamarri

Vol. 10 (2017), No. 5, 813–831
Abstract

Boolean (BL) systems and coupled ordinary differential equations (ODEs) are popular models for studying biological networks. BL systems can be set up without detailed reaction mechanisms and rate constants and provide qualitatively useful information, but they cannot capture the continuous dynamics of biological systems. On the other hand, ODEs are able to capture the continuous dynamic features of biological networks and provide more information on how the activities of components depend on other components and parameter values. However, a useful coupled ODE model requires details about interactions and parameter values. The introduction of the relationships between the two types of models will enable us to leverage their advantages and better understand the target network systems. In this paper, we investigate the relations between the conditions of the existence of limit cycles in ODE networks and their homologous discrete systems. We prove that for a single feedback loop, as long as the corresponding governing functions of the homologous continuous and discrete systems have the same upper and lower asymptotes, the limit cycle borne via Hopf bifurcation corresponds to the cycle of the discrete system. However, for some coupled feedback loops, besides having the same upper and lower asymptotes, parameters such as the decay rates also play crucial roles.

This paper is dedicated to our dear friend Professor Kiran Chilakamarri who passed away due to a sudden illness in 2015.

Keywords
feedback loops, limit cycles, Boolean networks, coupled differential equations
Mathematical Subject Classification 2010
Primary: 37G99
Milestones
Received: 18 March 2016
Revised: 11 August 2016
Accepted: 17 August 2016
Published: 14 May 2017

Communicated by Richard Rebarber
Authors
Yunjiao Wang
Department of Mathematics
Texas Southern University
3100 Cleburne Street
Houston, TX 77004
United States
Bamidele Omidiran
Department of Economics
University of Pennsylvania
Philadelphia, PA 19104
United States
Franklin Kigwe
Department of Engineering
Texas Southern University
Houston, TX 77004
United States
Kiran Chilakamarri
Department of Mathematics
Texas Southern University
3100 Cleburne Street
Houston, TX 77004
United States