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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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Strongly connected graphs and components. Outlines of hypothetical graphs are shown, omitting some vertices and edges and all labels. (A) A strongly connected graph in which any pair of vertices can be joined, both ways, by a path of contiguous edges in the same direction (central motif). (B) A graph that is not strongly connected can always be decomposed into maximal strongly connected sub-graphs, called strongly connected components (SCCs). The graph shown here has four SCCs demarcated by the dotted lines. In the macroscopic interpretation of one-dimensional chemistry, matter can only flow in one direction between SCCs, so that it eventually accumulates only on the terminal SCCs (marked with an asterisk). In the microscopic interpretation, microstates that are not in a terminal SCC have zero steady-state probability.
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Fig2: Strongly connected graphs and components. Outlines of hypothetical graphs are shown, omitting some vertices and edges and all labels. (A) A strongly connected graph in which any pair of vertices can be joined, both ways, by a path of contiguous edges in the same direction (central motif). (B) A graph that is not strongly connected can always be decomposed into maximal strongly connected sub-graphs, called strongly connected components (SCCs). The graph shown here has four SCCs demarcated by the dotted lines. In the macroscopic interpretation of one-dimensional chemistry, matter can only flow in one direction between SCCs, so that it eventually accumulates only on the terminal SCCs (marked with an asterisk). In the microscopic interpretation, microstates that are not in a terminal SCC have zero steady-state probability.

Mentions: It is easy to imagine that, no matter what initial concentrations of matter are distributed over the vertices, the one-dimensional chemistry will eventually reach a steady state, in which production and consumption of each species are in balance and the concentration of each species is unchanging. Such a steady state occurs no matter what the structure of the graph. In a general graph, the steady state can depend on the initial concentrations that were chosen at time 0, so that there is a memory of these initial conditions (see the section ‘Formation of an inherently bounded chromatin domain’). However, if the graph is strongly connected, such memory is lost and the steady state becomes independent of the initial conditions and depends only on the structure of the graph. A strongly connected graph is one in which any pair of vertices are connected, both ways, by a path of consecutive edges that all point in the same direction (Figure 2A). In effect, any two vertices can communicate with each other in both directions. Strong connectivity depends only on the edges and not on the labels.Figure 2


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

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

Strongly connected graphs and components. Outlines of hypothetical graphs are shown, omitting some vertices and edges and all labels. (A) A strongly connected graph in which any pair of vertices can be joined, both ways, by a path of contiguous edges in the same direction (central motif). (B) A graph that is not strongly connected can always be decomposed into maximal strongly connected sub-graphs, called strongly connected components (SCCs). The graph shown here has four SCCs demarcated by the dotted lines. In the macroscopic interpretation of one-dimensional chemistry, matter can only flow in one direction between SCCs, so that it eventually accumulates only on the terminal SCCs (marked with an asterisk). In the microscopic interpretation, microstates that are not in a terminal SCC have zero steady-state probability.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Fig2: Strongly connected graphs and components. Outlines of hypothetical graphs are shown, omitting some vertices and edges and all labels. (A) A strongly connected graph in which any pair of vertices can be joined, both ways, by a path of contiguous edges in the same direction (central motif). (B) A graph that is not strongly connected can always be decomposed into maximal strongly connected sub-graphs, called strongly connected components (SCCs). The graph shown here has four SCCs demarcated by the dotted lines. In the macroscopic interpretation of one-dimensional chemistry, matter can only flow in one direction between SCCs, so that it eventually accumulates only on the terminal SCCs (marked with an asterisk). In the microscopic interpretation, microstates that are not in a terminal SCC have zero steady-state probability.
Mentions: It is easy to imagine that, no matter what initial concentrations of matter are distributed over the vertices, the one-dimensional chemistry will eventually reach a steady state, in which production and consumption of each species are in balance and the concentration of each species is unchanging. Such a steady state occurs no matter what the structure of the graph. In a general graph, the steady state can depend on the initial concentrations that were chosen at time 0, so that there is a memory of these initial conditions (see the section ‘Formation of an inherently bounded chromatin domain’). However, if the graph is strongly connected, such memory is lost and the steady state becomes independent of the initial conditions and depends only on the structure of the graph. A strongly connected graph is one in which any pair of vertices are connected, both ways, by a path of consecutive edges that all point in the same direction (Figure 2A). In effect, any two vertices can communicate with each other in both directions. Strong connectivity depends only on the edges and not on the labels.Figure 2

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