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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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Graph structures satisfying detailed balance. Labels have been omitted for clarity. (A) A sequence of reversible edges, as considered by Ong et al. [46]. (B) A tree of reversible edges. A tree is characterised by having no cycle of reversible edges and is an example of a general graph structure that always satisfies detailed balance, irrespective of the kinds of edges in the graph and the labels on these edges (Methods).
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Fig5: Graph structures satisfying detailed balance. Labels have been omitted for clarity. (A) A sequence of reversible edges, as considered by Ong et al. [46]. (B) A tree of reversible edges. A tree is characterised by having no cycle of reversible edges and is an example of a general graph structure that always satisfies detailed balance, irrespective of the kinds of edges in the graph and the labels on these edges (Methods).

Mentions: Thermodynamic equilibrium requires detailed balance but a graph can satisfy detailed balance without being at equilibrium. For instance, certain graph structures in which each edge is reversible, such as a sequence structure (Figure 5A) or, more generally, a tree structure (Figure 5B), always satisfy detailed balance (Methods). In such a graph the edges may involve dissipative mechanisms. However, although an edge is accompanied by a reverse edge , these edges may not arise from an underlying biochemical reaction operating reversibly but from two separate dissipative reactions, such as phosphorylation and dephosphorylation, each acting irreversibly. The ratio a/b would no longer have a thermodynamic interpretation in terms of a free energy difference, as in Equation 6.Figure 5


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

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

Graph structures satisfying detailed balance. Labels have been omitted for clarity. (A) A sequence of reversible edges, as considered by Ong et al. [46]. (B) A tree of reversible edges. A tree is characterised by having no cycle of reversible edges and is an example of a general graph structure that always satisfies detailed balance, irrespective of the kinds of edges in the graph and the labels on these edges (Methods).
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Fig5: Graph structures satisfying detailed balance. Labels have been omitted for clarity. (A) A sequence of reversible edges, as considered by Ong et al. [46]. (B) A tree of reversible edges. A tree is characterised by having no cycle of reversible edges and is an example of a general graph structure that always satisfies detailed balance, irrespective of the kinds of edges in the graph and the labels on these edges (Methods).
Mentions: Thermodynamic equilibrium requires detailed balance but a graph can satisfy detailed balance without being at equilibrium. For instance, certain graph structures in which each edge is reversible, such as a sequence structure (Figure 5A) or, more generally, a tree structure (Figure 5B), always satisfy detailed balance (Methods). In such a graph the edges may involve dissipative mechanisms. However, although an edge is accompanied by a reverse edge , these edges may not arise from an underlying biochemical reaction operating reversibly but from two separate dissipative reactions, such as phosphorylation and dephosphorylation, each acting irreversibly. The ratio a/b would no longer have a thermodynamic interpretation in terms of a free energy difference, as in Equation 6.Figure 5

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