The structure of infinitesimal homeostasis in input–output networks

Homeostasis refers to a phenomenon whereby the output x o of a system is approximately constant on variation of an input I . Homeostasis occurs frequently in biochemical networks and in other networks of interacting elements where mathematical models are based on differential equations associated to...

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Published inJournal of mathematical biology Vol. 82; no. 7; p. 62
Main Authors Wang, Yangyang, Huang, Zhengyuan, Antoneli, Fernando, Golubitsky, Martin
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2021
Springer Nature B.V
Subjects
Algorithms
Applications of Mathematics
Combinatorial analysis
Couplings
Differential equations
Feedback
Feedback loops
Graph theory
Homeostasis
Mathematical analysis
Mathematical and Computational Biology
Mathematical models
Mathematics
Mathematics and Statistics
Matrix methods
Matrix theory
Negative feedback
Networks
Polynomials
Biochemical networks
Coupled systems
92C42
34C99
94C15
Homeostasis
Input–output networks
Combinatorial matrix theory
92C40
Perfect adaptation
Online AccessGet full text

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Abstract Homeostasis refers to a phenomenon whereby the output x o of a system is approximately constant on variation of an input I . Homeostasis occurs frequently in biochemical networks and in other networks of interacting elements where mathematical models are based on differential equations associated to the network. These networks can be abstracted as digraphs G with a distinguished input node ι , a different distinguished output node o , and a number of regulatory nodes ρ 1 , … , ρ n . In these models the input–output map x o ( I ) is defined by a stable equilibrium X 0 at I 0 . Stability implies that there is a stable equilibrium X ( I ) for each I near I 0 and infinitesimal homeostasis occurs at I 0 when ( d x o / d I ) ( I 0 ) = 0 . We show that there is an ( n + 1 ) × ( n + 1 ) homeostasis matrix H ( I ) for which d x o / d I = 0 if and only if det ( H ) = 0 . We note that the entries in H are linearized couplings and det ( H ) is a homogeneous polynomial of degree n + 1 in these entries. We use combinatorial matrix theory to factor the polynomial det ( H ) and thereby determine a menu of different types of possible homeostasis associated with each digraph G . Specifically, we prove that each factor corresponds to a subnetwork of G . The factors divide into two combinatorially defined classes: structural and appendage . Structural factors correspond to feedforward motifs and appendage factors correspond to feedback motifs. Finally, we discover an algorithm for determining the homeostasis subnetwork motif corresponding to each factor of det ( H ) without performing numerical simulations on model equations. The algorithm allows us to classify low degree factors of det ( H ) . There are two types of degree 1 homeostasis (negative feedback loops and kinetic or Haldane motifs) and there are two types of degree 2 homeostasis (feedforward loops and a degree two appendage motif).
AbstractList Abstract Homeostasis refers to a phenomenon whereby the output $$x_o$$ x o of a system is approximately constant on variation of an input $${{\mathcal {I}}}$$ I . Homeostasis occurs frequently in biochemical networks and in other networks of interacting elements where mathematical models are based on differential equations associated to the network. These networks can be abstracted as digraphs $${{\mathcal {G}}}$$ G with a distinguished input node $$\iota $$ ι , a different distinguished output node o , and a number of regulatory nodes $$\rho _1,\ldots ,\rho _n$$ ρ 1 , … , ρ n . In these models the input–output map $$x_o({{\mathcal {I}}})$$ x o ( I ) is defined by a stable equilibrium $$X_0$$ X 0 at $${{\mathcal {I}}}_0$$ I 0 . Stability implies that there is a stable equilibrium $$X({{\mathcal {I}}})$$ X ( I ) for each $${{\mathcal {I}}}$$ I near $${{\mathcal {I}}}_0$$ I 0 and infinitesimal homeostasis occurs at $${{\mathcal {I}}}_0$$ I 0 when $$(dx_o/d{{\mathcal {I}}})({{\mathcal {I}}}_0) = 0$$ ( d x o / d I ) ( I 0 ) = 0 . We show that there is an $$(n+1)\times (n+1)$$ ( n + 1 ) × ( n + 1 ) homeostasis matrix $$H({{\mathcal {I}}})$$ H ( I ) for which $$dx_o/d{{\mathcal {I}}}= 0$$ d x o / d I = 0 if and only if $$\det (H) = 0$$ det ( H ) = 0 . We note that the entries in H are linearized couplings and $$\det (H)$$ det ( H ) is a homogeneous polynomial of degree $$n+1$$ n + 1 in these entries. We use combinatorial matrix theory to factor the polynomial $$\det (H)$$ det ( H ) and thereby determine a menu of different types of possible homeostasis associated with each digraph $${{\mathcal {G}}}$$ G . Specifically, we prove that each factor corresponds to a subnetwork of $${{\mathcal {G}}}$$ G . The factors divide into two combinatorially defined classes: structural and appendage . Structural factors correspond to feedforward motifs and appendage factors correspond to feedback motifs. Finally, we discover an algorithm for determining the homeostasis subnetwork motif corresponding to each factor of $$\det (H)$$ det ( H ) without performing numerical simulations on model equations. The algorithm allows us to classify low degree factors of $$\det (H)$$ det ( H ) . There are two types of degree 1 homeostasis (negative feedback loops and kinetic or Haldane motifs) and there are two types of degree 2 homeostasis (feedforward loops and a degree two appendage motif).
Homeostasis refers to a phenomenon whereby the output xo of a system is approximately constant on variation of an input I. Homeostasis occurs frequently in biochemical networks and in other networks of interacting elements where mathematical models are based on differential equations associated to the network. These networks can be abstracted as digraphs G with a distinguished input node ι, a different distinguished output node o, and a number of regulatory nodes ρ1,…,ρn. In these models the input–output map xo(I) is defined by a stable equilibrium X0 at I0. Stability implies that there is a stable equilibrium X(I) for each I near I0 and infinitesimal homeostasis occurs at I0 when (dxo/dI)(I0)=0. We show that there is an (n+1)×(n+1)homeostasis matrixH(I) for which dxo/dI=0 if and only if det(H)=0. We note that the entries in H are linearized couplings and det(H) is a homogeneous polynomial of degree n+1 in these entries. We use combinatorial matrix theory to factor the polynomial det(H) and thereby determine a menu of different types of possible homeostasis associated with each digraph G. Specifically, we prove that each factor corresponds to a subnetwork of G. The factors divide into two combinatorially defined classes: structural and appendage. Structural factors correspond to feedforward motifs and appendage factors correspond to feedback motifs. Finally, we discover an algorithm for determining the homeostasis subnetwork motif corresponding to each factor of det(H) without performing numerical simulations on model equations. The algorithm allows us to classify low degree factors of det(H). There are two types of degree 1 homeostasis (negative feedback loops and kinetic or Haldane motifs) and there are two types of degree 2 homeostasis (feedforward loops and a degree two appendage motif).
Homeostasis refers to a phenomenon whereby the output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_o$$\end{document} x o of a system is approximately constant on variation of an input \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {I}}}$$\end{document} I . Homeostasis occurs frequently in biochemical networks and in other networks of interacting elements where mathematical models are based on differential equations associated to the network. These networks can be abstracted as digraphs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}$$\end{document} G with a distinguished input node \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\iota $$\end{document} ι , a different distinguished output node o , and a number of regulatory nodes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _1,\ldots ,\rho _n$$\end{document} ρ 1 , … , ρ n . In these models the input–output map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_o({{\mathcal {I}}})$$\end{document} x o ( I ) is defined by a stable equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_0$$\end{document} X 0 at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {I}}}_0$$\end{document} I 0 . Stability implies that there is a stable equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X({{\mathcal {I}}})$$\end{document} X ( I ) for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {I}}}$$\end{document} I near \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {I}}}_0$$\end{document} I 0 and infinitesimal homeostasis occurs at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {I}}}_0$$\end{document} I 0 when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(dx_o/d{{\mathcal {I}}})({{\mathcal {I}}}_0) = 0$$\end{document} ( d x o / d I ) ( I 0 ) = 0 . We show that there is an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n+1)\times (n+1)$$\end{document} ( n + 1 ) × ( n + 1 ) homeostasis matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H({{\mathcal {I}}})$$\end{document} H ( I ) for which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$dx_o/d{{\mathcal {I}}}= 0$$\end{document} d x o / d I = 0 if and only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\det (H) = 0$$\end{document} det ( H ) = 0 . We note that the entries in H are linearized couplings and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\det (H)$$\end{document} det ( H ) is a homogeneous polynomial of degree \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n+1$$\end{document} n + 1 in these entries. We use combinatorial matrix theory to factor the polynomial \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\det (H)$$\end{document} det ( H ) and thereby determine a menu of different types of possible homeostasis associated with each digraph \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}$$\end{document} G . Specifically, we prove that each factor corresponds to a subnetwork of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}$$\end{document} G . The factors divide into two combinatorially defined classes: structural and appendage . Structural factors correspond to feedforward motifs and appendage factors correspond to feedback motifs. Finally, we discover an algorithm for determining the homeostasis subnetwork motif corresponding to each factor of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\det (H)$$\end{document} det ( H ) without performing numerical simulations on model equations. The algorithm allows us to classify low degree factors of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\det (H)$$\end{document} det ( H ) . There are two types of degree 1 homeostasis (negative feedback loops and kinetic or Haldane motifs) and there are two types of degree 2 homeostasis (feedforward loops and a degree two appendage motif).
Homeostasis refers to a phenomenon whereby the output x o of a system is approximately constant on variation of an input I . Homeostasis occurs frequently in biochemical networks and in other networks of interacting elements where mathematical models are based on differential equations associated to the network. These networks can be abstracted as digraphs G with a distinguished input node ι , a different distinguished output node o , and a number of regulatory nodes ρ 1 , … , ρ n . In these models the input–output map x o ( I ) is defined by a stable equilibrium X 0 at I 0 . Stability implies that there is a stable equilibrium X ( I ) for each I near I 0 and infinitesimal homeostasis occurs at I 0 when ( d x o / d I ) ( I 0 ) = 0 . We show that there is an ( n + 1 ) × ( n + 1 ) homeostasis matrix H ( I ) for which d x o / d I = 0 if and only if det ( H ) = 0 . We note that the entries in H are linearized couplings and det ( H ) is a homogeneous polynomial of degree n + 1 in these entries. We use combinatorial matrix theory to factor the polynomial det ( H ) and thereby determine a menu of different types of possible homeostasis associated with each digraph G . Specifically, we prove that each factor corresponds to a subnetwork of G . The factors divide into two combinatorially defined classes: structural and appendage . Structural factors correspond to feedforward motifs and appendage factors correspond to feedback motifs. Finally, we discover an algorithm for determining the homeostasis subnetwork motif corresponding to each factor of det ( H ) without performing numerical simulations on model equations. The algorithm allows us to classify low degree factors of det ( H ) . There are two types of degree 1 homeostasis (negative feedback loops and kinetic or Haldane motifs) and there are two types of degree 2 homeostasis (feedforward loops and a degree two appendage motif).
ArticleNumber 62
Author Wang, Yangyang
Golubitsky, Martin
Huang, Zhengyuan
Antoneli, Fernando
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  organization: The Ohio State University
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  surname: Golubitsky
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Issue 7
Keywords Biochemical networks
Coupled systems
92C42
34C99
94C15
Homeostasis
Input–output networks
Combinatorial matrix theory
92C40
Perfect adaptation
Language English
License Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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Snippet Homeostasis refers to a phenomenon whereby the output x o of a system is approximately constant on variation of an input I . Homeostasis occurs frequently in...
Abstract Homeostasis refers to a phenomenon whereby the output $$x_o$$ x o of a system is approximately constant on variation of an input $${{\mathcal {I}}}$$...
Homeostasis refers to a phenomenon whereby the output xo of a system is approximately constant on variation of an input I. Homeostasis occurs frequently in...
Homeostasis refers to a phenomenon whereby the output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}...
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StartPage 62
SubjectTerms Algorithms
Applications of Mathematics
Combinatorial analysis
Couplings
Differential equations
Feedback
Feedback loops
Graph theory
Homeostasis
Mathematical analysis
Mathematical and Computational Biology
Mathematical models
Mathematics
Mathematics and Statistics
Matrix methods
Matrix theory
Negative feedback
Networks
Polynomials
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Title The structure of infinitesimal homeostasis in input–output networks
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