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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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Regulation of yeastPHO5, adapted from Figures one and four b of [52].(A) Schematic of the experimental set-up. A doxycycline-inducible (Dox), YFP-tagged Pho4, modified to be constitutively active (SA1-4) and constitutively nuclear (PA6), stimulates expression of CFP from a partial PHO5 promoter, with three nucleosomes (-3, -2 and -1) and two Pho4 binding sites, a low-affinity exposed site between nucleosomes -2 and -3 (UASp1) and a high-affinity site occluded by nucleosome -2 (UASp2). The TATA box is occluded by nucleosome -1. (B) The labelled, directed graph of this system, showing the microstates (left) and the labels (bottom), in the notation used by Kim and O’Shea. Label a corresponds to Pho4 binding through a Hill function, which arises through the rapid equilibrium mechanism of Figure 3B. Labels b and c correspond to Pho4 unbinding (Figure 3C) from, respectively, UASp1 and UASp2. Labels d (kremod) and e (kreass) correspond to disassembly and assembly, respectively, of nucleosomes (Figure 3F), which introduce the non-equilibrium and irreversible features of the graph. Nucleosome -3 has been ignored in the graph. For other features, see the cited paper CFP, cyan fluorescent protein; YFP, yellow fluorescent protein.
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Fig7: Regulation of yeastPHO5, adapted from Figures one and four b of [52].(A) Schematic of the experimental set-up. A doxycycline-inducible (Dox), YFP-tagged Pho4, modified to be constitutively active (SA1-4) and constitutively nuclear (PA6), stimulates expression of CFP from a partial PHO5 promoter, with three nucleosomes (-3, -2 and -1) and two Pho4 binding sites, a low-affinity exposed site between nucleosomes -2 and -3 (UASp1) and a high-affinity site occluded by nucleosome -2 (UASp2). The TATA box is occluded by nucleosome -1. (B) The labelled, directed graph of this system, showing the microstates (left) and the labels (bottom), in the notation used by Kim and O’Shea. Label a corresponds to Pho4 binding through a Hill function, which arises through the rapid equilibrium mechanism of Figure 3B. Labels b and c correspond to Pho4 unbinding (Figure 3C) from, respectively, UASp1 and UASp2. Labels d (kremod) and e (kreass) correspond to disassembly and assembly, respectively, of nucleosomes (Figure 3F), which introduce the non-equilibrium and irreversible features of the graph. Nucleosome -3 has been ignored in the graph. For other features, see the cited paper CFP, cyan fluorescent protein; YFP, yellow fluorescent protein.

Mentions: We now turn back to gene regulation and to one of the very few models in which a non-equilibrium mechanism has been rigorously analysed without assuming detailed balance. Pho5 is an acid phosphatase in Saccharomyces cerevisiae that is expressed under phosphate-starvation conditions. Kim and O’Shea undertook a quantitative analysis of PHO5 regulation by the transcription factor Pho4, using a construct detached from the phosphate-response pathway [52] (Figure 7A).Figure 7


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

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

Regulation of yeastPHO5, adapted from Figures one and four b of [52].(A) Schematic of the experimental set-up. A doxycycline-inducible (Dox), YFP-tagged Pho4, modified to be constitutively active (SA1-4) and constitutively nuclear (PA6), stimulates expression of CFP from a partial PHO5 promoter, with three nucleosomes (-3, -2 and -1) and two Pho4 binding sites, a low-affinity exposed site between nucleosomes -2 and -3 (UASp1) and a high-affinity site occluded by nucleosome -2 (UASp2). The TATA box is occluded by nucleosome -1. (B) The labelled, directed graph of this system, showing the microstates (left) and the labels (bottom), in the notation used by Kim and O’Shea. Label a corresponds to Pho4 binding through a Hill function, which arises through the rapid equilibrium mechanism of Figure 3B. Labels b and c correspond to Pho4 unbinding (Figure 3C) from, respectively, UASp1 and UASp2. Labels d (kremod) and e (kreass) correspond to disassembly and assembly, respectively, of nucleosomes (Figure 3F), which introduce the non-equilibrium and irreversible features of the graph. Nucleosome -3 has been ignored in the graph. For other features, see the cited paper CFP, cyan fluorescent protein; YFP, yellow fluorescent protein.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Fig7: Regulation of yeastPHO5, adapted from Figures one and four b of [52].(A) Schematic of the experimental set-up. A doxycycline-inducible (Dox), YFP-tagged Pho4, modified to be constitutively active (SA1-4) and constitutively nuclear (PA6), stimulates expression of CFP from a partial PHO5 promoter, with three nucleosomes (-3, -2 and -1) and two Pho4 binding sites, a low-affinity exposed site between nucleosomes -2 and -3 (UASp1) and a high-affinity site occluded by nucleosome -2 (UASp2). The TATA box is occluded by nucleosome -1. (B) The labelled, directed graph of this system, showing the microstates (left) and the labels (bottom), in the notation used by Kim and O’Shea. Label a corresponds to Pho4 binding through a Hill function, which arises through the rapid equilibrium mechanism of Figure 3B. Labels b and c correspond to Pho4 unbinding (Figure 3C) from, respectively, UASp1 and UASp2. Labels d (kremod) and e (kreass) correspond to disassembly and assembly, respectively, of nucleosomes (Figure 3F), which introduce the non-equilibrium and irreversible features of the graph. Nucleosome -3 has been ignored in the graph. For other features, see the cited paper CFP, cyan fluorescent protein; YFP, yellow fluorescent protein.
Mentions: We now turn back to gene regulation and to one of the very few models in which a non-equilibrium mechanism has been rigorously analysed without assuming detailed balance. Pho5 is an acid phosphatase in Saccharomyces cerevisiae that is expressed under phosphate-starvation conditions. Kim and O’Shea undertook a quantitative analysis of PHO5 regulation by the transcription factor Pho4, using a construct detached from the phosphate-response pathway [52] (Figure 7A).Figure 7

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