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A framework for modelling gene regulation which accommodates non-equilibrium mechanisms.

Ahsendorf T, Wong F, Eils R, Gunawardena J - BMC Biol. (2014)

Bottom Line: At equilibrium, microstate probabilities do not depend on how a microstate is reached but, away from equilibrium, each path to a microstate can contribute to its steady-state probability.Systems that are far from equilibrium thereby become dependent on history and the resulting complexity is a fundamental challenge.As epigenomic data become increasingly available, we anticipate that gene function will come to be represented by graphs, as gene structure has been represented by sequences, and that the methods introduced here will provide a broader foundation for understanding how genes work.

View Article: PubMed Central - PubMed

Affiliation: DKFZ, Heidelberg, D-69120, Germany. tobias.ahsendorf@googlemail.com.

ABSTRACT

Background: Gene regulation has, for the most part, been quantitatively analysed by assuming that regulatory mechanisms operate at thermodynamic equilibrium. This formalism was originally developed to analyse the binding and unbinding of transcription factors from naked DNA in eubacteria. Although widely used, it has made it difficult to understand the role of energy-dissipating, epigenetic mechanisms, such as DNA methylation, nucleosome remodelling and post-translational modification of histones and co-regulators, which act together with transcription factors to regulate gene expression in eukaryotes.

Results: Here, we introduce a graph-based framework that can accommodate non-equilibrium mechanisms. A gene-regulatory system is described as a graph, which specifies the DNA microstates (vertices), the transitions between microstates (edges) and the transition rates (edge labels). The graph yields a stochastic master equation for how microstate probabilities change over time. We show that this framework has broad scope by providing new insights into three very different ad hoc models, of steroid-hormone responsive genes, of inherently bounded chromatin domains and of the yeast PHO5 gene. We find, moreover, surprising complexity in the regulation of PHO5, which has not yet been experimentally explored, and we show that this complexity is an inherent feature of being away from equilibrium. At equilibrium, microstate probabilities do not depend on how a microstate is reached but, away from equilibrium, each path to a microstate can contribute to its steady-state probability. Systems that are far from equilibrium thereby become dependent on history and the resulting complexity is a fundamental challenge. To begin addressing this, we introduce a graph-based concept of independence, which can be applied to sub-systems that are far from equilibrium, and prove that history-dependent complexity can be circumvented when sub-systems operate independently.

Conclusions: As epigenomic data become increasingly available, we anticipate that gene function will come to be represented by graphs, as gene structure has been represented by sequences, and that the methods introduced here will provide a broader foundation for understanding how genes work.

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Calculating microstate probabilities at steady state.(A) On the left, a labelled, directed graph G; on the right, the linear differential equation obtained by taking each edge to be a chemical reaction under mass-action kinetics with the edge label as the rate constant. The resulting matrix is the Laplacian matrix, , of G. (B) Illustration of Equation 7. On the left, a strongly connected graph; on the right, the spanning trees of the graph, each rooted at the circled vertex. Because the graph is strongly connected, each vertex has at least one spanning tree rooted there. The basis vector  is calculated from the spanning trees using Equation 7. Probabilities of microstates are then given by normalising the entries of ρG, as in Equation 4. (C) On the left, the non-strongly connected graph in (A) is shown along with its three strongly connected components (SCCs) demarcated by the dotted lines. The two terminal SCCs are marked with an asterisk and denoted T1 and T2. Each terminal SCC gives rise to a basis vector in  using Equation 7, as in (B), and then forming a normalised vector, as shown by following the curved arrows. Note that vertices that are not in a terminal SCC (i.e., vertices 1, 2 and 3) have zero entries in each basis vector. Any steady state, x∗, can be expressed as a linear combination of these basis vectors, as in Equation 9 SCC, strongly connected component.
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Fig4: Calculating microstate probabilities at steady state.(A) On the left, a labelled, directed graph G; on the right, the linear differential equation obtained by taking each edge to be a chemical reaction under mass-action kinetics with the edge label as the rate constant. The resulting matrix is the Laplacian matrix, , of G. (B) Illustration of Equation 7. On the left, a strongly connected graph; on the right, the spanning trees of the graph, each rooted at the circled vertex. Because the graph is strongly connected, each vertex has at least one spanning tree rooted there. The basis vector is calculated from the spanning trees using Equation 7. Probabilities of microstates are then given by normalising the entries of ρG, as in Equation 4. (C) On the left, the non-strongly connected graph in (A) is shown along with its three strongly connected components (SCCs) demarcated by the dotted lines. The two terminal SCCs are marked with an asterisk and denoted T1 and T2. Each terminal SCC gives rise to a basis vector in using Equation 7, as in (B), and then forming a normalised vector, as shown by following the curved arrows. Note that vertices that are not in a terminal SCC (i.e., vertices 1, 2 and 3) have zero entries in each basis vector. Any steady state, x∗, can be expressed as a linear combination of these basis vectors, as in Equation 9 SCC, strongly connected component.

Mentions: Suppose we are given a graph G, as in Figure 4A, with vertices indexed 1,…,n. We typically use the index 1 for the reference microstate with no TFs bound and choose the order of the other microstates arbitrarily. The notation signifies the edge with label a from source vertex i to target vertex j. A dynamics can be imposed on G in two equivalent ways. In the macroscopic interpretation, the vertices are chemical species and the edges are chemical reactions, which convert source species to target species. The edge labels are rate constants for the corresponding reactions, assuming mass-action kinetics. Since each reaction is uni-molecular, with only one substrate and one product, this one-dimensional chemistry yields a linear dynamics (Figure 4A), (1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \frac{d}{dt}x(t) = \mathcal{L}(G)\cdot x(t), $$ \end{document}ddtx(t)=ℒ(G)·x(t),Figure 4


A framework for modelling gene regulation which accommodates non-equilibrium mechanisms.

Ahsendorf T, Wong F, Eils R, Gunawardena J - BMC Biol. (2014)

Calculating microstate probabilities at steady state.(A) On the left, a labelled, directed graph G; on the right, the linear differential equation obtained by taking each edge to be a chemical reaction under mass-action kinetics with the edge label as the rate constant. The resulting matrix is the Laplacian matrix, , of G. (B) Illustration of Equation 7. On the left, a strongly connected graph; on the right, the spanning trees of the graph, each rooted at the circled vertex. Because the graph is strongly connected, each vertex has at least one spanning tree rooted there. The basis vector  is calculated from the spanning trees using Equation 7. Probabilities of microstates are then given by normalising the entries of ρG, as in Equation 4. (C) On the left, the non-strongly connected graph in (A) is shown along with its three strongly connected components (SCCs) demarcated by the dotted lines. The two terminal SCCs are marked with an asterisk and denoted T1 and T2. Each terminal SCC gives rise to a basis vector in  using Equation 7, as in (B), and then forming a normalised vector, as shown by following the curved arrows. Note that vertices that are not in a terminal SCC (i.e., vertices 1, 2 and 3) have zero entries in each basis vector. Any steady state, x∗, can be expressed as a linear combination of these basis vectors, as in Equation 9 SCC, strongly connected component.
© Copyright Policy - open-access
Related In: Results  -  Collection

License 1 - License 2
Show All Figures
getmorefigures.php?uid=PMC4288563&req=5

Fig4: Calculating microstate probabilities at steady state.(A) On the left, a labelled, directed graph G; on the right, the linear differential equation obtained by taking each edge to be a chemical reaction under mass-action kinetics with the edge label as the rate constant. The resulting matrix is the Laplacian matrix, , of G. (B) Illustration of Equation 7. On the left, a strongly connected graph; on the right, the spanning trees of the graph, each rooted at the circled vertex. Because the graph is strongly connected, each vertex has at least one spanning tree rooted there. The basis vector is calculated from the spanning trees using Equation 7. Probabilities of microstates are then given by normalising the entries of ρG, as in Equation 4. (C) On the left, the non-strongly connected graph in (A) is shown along with its three strongly connected components (SCCs) demarcated by the dotted lines. The two terminal SCCs are marked with an asterisk and denoted T1 and T2. Each terminal SCC gives rise to a basis vector in using Equation 7, as in (B), and then forming a normalised vector, as shown by following the curved arrows. Note that vertices that are not in a terminal SCC (i.e., vertices 1, 2 and 3) have zero entries in each basis vector. Any steady state, x∗, can be expressed as a linear combination of these basis vectors, as in Equation 9 SCC, strongly connected component.
Mentions: Suppose we are given a graph G, as in Figure 4A, with vertices indexed 1,…,n. We typically use the index 1 for the reference microstate with no TFs bound and choose the order of the other microstates arbitrarily. The notation signifies the edge with label a from source vertex i to target vertex j. A dynamics can be imposed on G in two equivalent ways. In the macroscopic interpretation, the vertices are chemical species and the edges are chemical reactions, which convert source species to target species. The edge labels are rate constants for the corresponding reactions, assuming mass-action kinetics. Since each reaction is uni-molecular, with only one substrate and one product, this one-dimensional chemistry yields a linear dynamics (Figure 4A), (1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \frac{d}{dt}x(t) = \mathcal{L}(G)\cdot x(t), $$ \end{document}ddtx(t)=ℒ(G)·x(t),Figure 4

Bottom Line: At equilibrium, microstate probabilities do not depend on how a microstate is reached but, away from equilibrium, each path to a microstate can contribute to its steady-state probability.Systems that are far from equilibrium thereby become dependent on history and the resulting complexity is a fundamental challenge.As epigenomic data become increasingly available, we anticipate that gene function will come to be represented by graphs, as gene structure has been represented by sequences, and that the methods introduced here will provide a broader foundation for understanding how genes work.

View Article: PubMed Central - PubMed

Affiliation: DKFZ, Heidelberg, D-69120, Germany. tobias.ahsendorf@googlemail.com.

ABSTRACT

Background: Gene regulation has, for the most part, been quantitatively analysed by assuming that regulatory mechanisms operate at thermodynamic equilibrium. This formalism was originally developed to analyse the binding and unbinding of transcription factors from naked DNA in eubacteria. Although widely used, it has made it difficult to understand the role of energy-dissipating, epigenetic mechanisms, such as DNA methylation, nucleosome remodelling and post-translational modification of histones and co-regulators, which act together with transcription factors to regulate gene expression in eukaryotes.

Results: Here, we introduce a graph-based framework that can accommodate non-equilibrium mechanisms. A gene-regulatory system is described as a graph, which specifies the DNA microstates (vertices), the transitions between microstates (edges) and the transition rates (edge labels). The graph yields a stochastic master equation for how microstate probabilities change over time. We show that this framework has broad scope by providing new insights into three very different ad hoc models, of steroid-hormone responsive genes, of inherently bounded chromatin domains and of the yeast PHO5 gene. We find, moreover, surprising complexity in the regulation of PHO5, which has not yet been experimentally explored, and we show that this complexity is an inherent feature of being away from equilibrium. At equilibrium, microstate probabilities do not depend on how a microstate is reached but, away from equilibrium, each path to a microstate can contribute to its steady-state probability. Systems that are far from equilibrium thereby become dependent on history and the resulting complexity is a fundamental challenge. To begin addressing this, we introduce a graph-based concept of independence, which can be applied to sub-systems that are far from equilibrium, and prove that history-dependent complexity can be circumvented when sub-systems operate independently.

Conclusions: As epigenomic data become increasingly available, we anticipate that gene function will come to be represented by graphs, as gene structure has been represented by sequences, and that the methods introduced here will provide a broader foundation for understanding how genes work.

Show MeSH