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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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Labelled, directed edges for graphs.(A, B) Binding transitions. (C–J) Non-binding transitions. Each example shows a source (left) and a target (right) microstate connected by a labelled edge (curved, barbed arrow). Grey ovals signify background components that make up the microstate. A nominal transcription start site is shown. The magenta shape in (C), (D), (G), (H) and (I) depicts a component of the source microstate that is specifically involved in the reaction represented by the edge. A small dashed arrow signifies an enzymatic action by a component in the source microstate (magenta shape), which remains bound after catalysis. The yellow disc depicts RNA polymerase with a nascent mRNA molecule in the elongating state. The edge-label formula in (B) comes from the rapid equilibrium assumption discussed in the text and is derived in the Methods. 3D, three dimensional; TF, transcription factor; Me, methylation; P, phosphorylation; Ub, ubiquitination.
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Fig3: Labelled, directed edges for graphs.(A, B) Binding transitions. (C–J) Non-binding transitions. Each example shows a source (left) and a target (right) microstate connected by a labelled edge (curved, barbed arrow). Grey ovals signify background components that make up the microstate. A nominal transcription start site is shown. The magenta shape in (C), (D), (G), (H) and (I) depicts a component of the source microstate that is specifically involved in the reaction represented by the edge. A small dashed arrow signifies an enzymatic action by a component in the source microstate (magenta shape), which remains bound after catalysis. The yellow disc depicts RNA polymerase with a nascent mRNA molecule in the elongating state. The edge-label formula in (B) comes from the rapid equilibrium assumption discussed in the text and is derived in the Methods. 3D, three dimensional; TF, transcription factor; Me, methylation; P, phosphorylation; Ub, ubiquitination.

Mentions: These represent the binding of a component L to a microstate (Figure 3A). The label is a=k[ L], where k is an on-rate and [ L] is the free concentration of L. We follow the thermodynamic formalism and assume, first, that components are neither synthesised nor degraded over the timescale of interest so that their total amounts are conserved quantities and, second, that the depletion of L can be ignored, so that the binding of a single molecule of L does not appreciably change its free concentration, [ L]. In other words, [ L]≈Ltot. Non-specific binding to DNA can significantly reduce the free concentration and if this is thought to jeopardise the no-depletion assumption, a more elaborate analysis is needed [36,44].Figure 3


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

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

Labelled, directed edges for graphs.(A, B) Binding transitions. (C–J) Non-binding transitions. Each example shows a source (left) and a target (right) microstate connected by a labelled edge (curved, barbed arrow). Grey ovals signify background components that make up the microstate. A nominal transcription start site is shown. The magenta shape in (C), (D), (G), (H) and (I) depicts a component of the source microstate that is specifically involved in the reaction represented by the edge. A small dashed arrow signifies an enzymatic action by a component in the source microstate (magenta shape), which remains bound after catalysis. The yellow disc depicts RNA polymerase with a nascent mRNA molecule in the elongating state. The edge-label formula in (B) comes from the rapid equilibrium assumption discussed in the text and is derived in the Methods. 3D, three dimensional; TF, transcription factor; Me, methylation; P, phosphorylation; Ub, ubiquitination.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Fig3: Labelled, directed edges for graphs.(A, B) Binding transitions. (C–J) Non-binding transitions. Each example shows a source (left) and a target (right) microstate connected by a labelled edge (curved, barbed arrow). Grey ovals signify background components that make up the microstate. A nominal transcription start site is shown. The magenta shape in (C), (D), (G), (H) and (I) depicts a component of the source microstate that is specifically involved in the reaction represented by the edge. A small dashed arrow signifies an enzymatic action by a component in the source microstate (magenta shape), which remains bound after catalysis. The yellow disc depicts RNA polymerase with a nascent mRNA molecule in the elongating state. The edge-label formula in (B) comes from the rapid equilibrium assumption discussed in the text and is derived in the Methods. 3D, three dimensional; TF, transcription factor; Me, methylation; P, phosphorylation; Ub, ubiquitination.
Mentions: These represent the binding of a component L to a microstate (Figure 3A). The label is a=k[ L], where k is an on-rate and [ L] is the free concentration of L. We follow the thermodynamic formalism and assume, first, that components are neither synthesised nor degraded over the timescale of interest so that their total amounts are conserved quantities and, second, that the depletion of L can be ignored, so that the binding of a single molecule of L does not appreciably change its free concentration, [ L]. In other words, [ L]≈Ltot. Non-specific binding to DNA can significantly reduce the free concentration and if this is thought to jeopardise the no-depletion assumption, a more elaborate analysis is needed [36,44].Figure 3

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