The Number of Equilibrium Points of Perturbed Nonlinear Positive Dynamical Systems (Extended Version)

The number of equilibrium points of a dynamical system dictates important qualitative properties such as the ability of the system to store different memory states, and may be significantly affected by state-dependent perturbations. In this paper, we develop a methodology based on tools from degree...

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Main Authors: McBride, Cameron, Del Vecchio, Domitilla
Format: Technical Report
Language:en_US
Published: 2018
Subjects:
Online Access:http://hdl.handle.net/1721.1/118380
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author McBride, Cameron
Del Vecchio, Domitilla
author_facet McBride, Cameron
Del Vecchio, Domitilla
author_sort McBride, Cameron
collection MIT
description The number of equilibrium points of a dynamical system dictates important qualitative properties such as the ability of the system to store different memory states, and may be significantly affected by state-dependent perturbations. In this paper, we develop a methodology based on tools from degree theory to determine whether the number of equilibrium points in a positive dynamical system changes due to structured state-dependent perturbations. Positive dynamical systems are particularly well suited to describe biological systems where the states are always positive. We prove two main theorems that utilize the determinant of the system's Jacobian to find algebraic conditions on the parameters determining whether the number of equilibrium points is guaranteed either to change or to remain the same when a nominal system is compared to its perturbed counterpart. We demonstrate the application of the theoretical results to genetic circuits where state-dependent perturbations arise due to fluctuations in cellular resources. These fluctuations constitute a major problem for predicting the behavior of genetic circuits. Our results allow us to determine whether such fluctuations change the genetic circuit's intended number of steady states.
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spelling mit-1721.1/1183802019-08-05T03:06:06Z The Number of Equilibrium Points of Perturbed Nonlinear Positive Dynamical Systems (Extended Version) McBride, Cameron Del Vecchio, Domitilla genetic circuits homotopy methods degree theory nonlinear dynamical systems multistability systems biology The number of equilibrium points of a dynamical system dictates important qualitative properties such as the ability of the system to store different memory states, and may be significantly affected by state-dependent perturbations. In this paper, we develop a methodology based on tools from degree theory to determine whether the number of equilibrium points in a positive dynamical system changes due to structured state-dependent perturbations. Positive dynamical systems are particularly well suited to describe biological systems where the states are always positive. We prove two main theorems that utilize the determinant of the system's Jacobian to find algebraic conditions on the parameters determining whether the number of equilibrium points is guaranteed either to change or to remain the same when a nominal system is compared to its perturbed counterpart. We demonstrate the application of the theoretical results to genetic circuits where state-dependent perturbations arise due to fluctuations in cellular resources. These fluctuations constitute a major problem for predicting the behavior of genetic circuits. Our results allow us to determine whether such fluctuations change the genetic circuit's intended number of steady states. This work was supported by National Science Foundation under award no. 1521925 2018-10-05T20:39:28Z 2018-10-05T20:39:28Z 2019-07-26 Technical Report http://hdl.handle.net/1721.1/118380 en_US Attribution-NonCommercial-NoDerivs 3.0 United States http://creativecommons.org/licenses/by-nc-nd/3.0/us/ application/pdf
spellingShingle genetic circuits
homotopy methods
degree theory
nonlinear dynamical systems
multistability
systems biology
McBride, Cameron
Del Vecchio, Domitilla
The Number of Equilibrium Points of Perturbed Nonlinear Positive Dynamical Systems (Extended Version)
title The Number of Equilibrium Points of Perturbed Nonlinear Positive Dynamical Systems (Extended Version)
title_full The Number of Equilibrium Points of Perturbed Nonlinear Positive Dynamical Systems (Extended Version)
title_fullStr The Number of Equilibrium Points of Perturbed Nonlinear Positive Dynamical Systems (Extended Version)
title_full_unstemmed The Number of Equilibrium Points of Perturbed Nonlinear Positive Dynamical Systems (Extended Version)
title_short The Number of Equilibrium Points of Perturbed Nonlinear Positive Dynamical Systems (Extended Version)
title_sort number of equilibrium points of perturbed nonlinear positive dynamical systems extended version
topic genetic circuits
homotopy methods
degree theory
nonlinear dynamical systems
multistability
systems biology
url http://hdl.handle.net/1721.1/118380
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