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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Format: | Technical Report |
Language: | en_US |
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2018
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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. |
first_indexed | 2024-09-23T12:28:50Z |
format | Technical Report |
id | mit-1721.1/118380 |
institution | Massachusetts Institute of Technology |
language | en_US |
last_indexed | 2024-09-23T12:28:50Z |
publishDate | 2018 |
record_format | dspace |
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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