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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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Formation of an inherently bounded chromatin domain [47,48].(A) An array of nucleosomes is shown, with nucleation taking place at the right-hand end. White nucleosomes are unmarked, black nucleosomes are marked and grey nucleosomes are either marked or unmarked. Nucleation, at rate k+, is confined to the nucleation site; propagation, also at rate k+, allows a marked nucleosome to propagate the mark to one of its two immediate (unmarked) neighbours; turnover, at rate k_, allows any marked nucleosome, including the nucleation site, to become unmarked. (B) Directed graph for the model with three nucleosomes. Each microstate shows its marking pattern as a bit string with 0 denoting unmarked and 1 denoting marked. The microstates are enumerated by considering the bit string as a number in base 2 notation and adding 1. The edges correspond to nucleation, propagation and turnover, as above. Labels have been omitted for clarity but an edge that increases, respectively decreases, the number of bits has label k+, respectively k_. (C) On the left, an extension of the model to include mark stabilisation, with a stably marked nucleosome shown in magenta. A stabilised mark is no longer subject to turnover. This leads to the non-strongly connected graph shown on the right for an array of two nucleosomes, in which the digit 2 in the microstate description signifies a stabilised mark. Edges that change digit 1 to digit 2 have label k∗, while the other edges are labelled as in (B). The strongly connected components (SCCs) are indicated by dotted outlines, with the two terminal SCCs identified by an asterisk.
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Fig6: Formation of an inherently bounded chromatin domain [47,48].(A) An array of nucleosomes is shown, with nucleation taking place at the right-hand end. White nucleosomes are unmarked, black nucleosomes are marked and grey nucleosomes are either marked or unmarked. Nucleation, at rate k+, is confined to the nucleation site; propagation, also at rate k+, allows a marked nucleosome to propagate the mark to one of its two immediate (unmarked) neighbours; turnover, at rate k_, allows any marked nucleosome, including the nucleation site, to become unmarked. (B) Directed graph for the model with three nucleosomes. Each microstate shows its marking pattern as a bit string with 0 denoting unmarked and 1 denoting marked. The microstates are enumerated by considering the bit string as a number in base 2 notation and adding 1. The edges correspond to nucleation, propagation and turnover, as above. Labels have been omitted for clarity but an edge that increases, respectively decreases, the number of bits has label k+, respectively k_. (C) On the left, an extension of the model to include mark stabilisation, with a stably marked nucleosome shown in magenta. A stabilised mark is no longer subject to turnover. This leads to the non-strongly connected graph shown on the right for an array of two nucleosomes, in which the digit 2 in the microstate description signifies a stabilised mark. Edges that change digit 1 to digit 2 have label k∗, while the other edges are labelled as in (B). The strongly connected components (SCCs) are indicated by dotted outlines, with the two terminal SCCs identified by an asterisk.

Mentions: Graphs arising in gene regulation may not always be strongly connected (see the section ‘Formation of an inherently bounded chromatin domain’ and Figure 6C). Steady-state probabilities for non-strongly connected graphs can be calculated by considering the SCCs of G (Figures 2B and 4C). The SCCs inherit connections from the underlying graph but these connections can never form a cycle, for otherwise the SCCs would collapse into each other. It is therefore possible to identify terminal SCCs, from which there are no outgoing connections. The terminal SCCs yield steady states in the following way.Figure 6


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

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

Formation of an inherently bounded chromatin domain [47,48].(A) An array of nucleosomes is shown, with nucleation taking place at the right-hand end. White nucleosomes are unmarked, black nucleosomes are marked and grey nucleosomes are either marked or unmarked. Nucleation, at rate k+, is confined to the nucleation site; propagation, also at rate k+, allows a marked nucleosome to propagate the mark to one of its two immediate (unmarked) neighbours; turnover, at rate k_, allows any marked nucleosome, including the nucleation site, to become unmarked. (B) Directed graph for the model with three nucleosomes. Each microstate shows its marking pattern as a bit string with 0 denoting unmarked and 1 denoting marked. The microstates are enumerated by considering the bit string as a number in base 2 notation and adding 1. The edges correspond to nucleation, propagation and turnover, as above. Labels have been omitted for clarity but an edge that increases, respectively decreases, the number of bits has label k+, respectively k_. (C) On the left, an extension of the model to include mark stabilisation, with a stably marked nucleosome shown in magenta. A stabilised mark is no longer subject to turnover. This leads to the non-strongly connected graph shown on the right for an array of two nucleosomes, in which the digit 2 in the microstate description signifies a stabilised mark. Edges that change digit 1 to digit 2 have label k∗, while the other edges are labelled as in (B). The strongly connected components (SCCs) are indicated by dotted outlines, with the two terminal SCCs identified by an asterisk.
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Related In: Results  -  Collection

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Fig6: Formation of an inherently bounded chromatin domain [47,48].(A) An array of nucleosomes is shown, with nucleation taking place at the right-hand end. White nucleosomes are unmarked, black nucleosomes are marked and grey nucleosomes are either marked or unmarked. Nucleation, at rate k+, is confined to the nucleation site; propagation, also at rate k+, allows a marked nucleosome to propagate the mark to one of its two immediate (unmarked) neighbours; turnover, at rate k_, allows any marked nucleosome, including the nucleation site, to become unmarked. (B) Directed graph for the model with three nucleosomes. Each microstate shows its marking pattern as a bit string with 0 denoting unmarked and 1 denoting marked. The microstates are enumerated by considering the bit string as a number in base 2 notation and adding 1. The edges correspond to nucleation, propagation and turnover, as above. Labels have been omitted for clarity but an edge that increases, respectively decreases, the number of bits has label k+, respectively k_. (C) On the left, an extension of the model to include mark stabilisation, with a stably marked nucleosome shown in magenta. A stabilised mark is no longer subject to turnover. This leads to the non-strongly connected graph shown on the right for an array of two nucleosomes, in which the digit 2 in the microstate description signifies a stabilised mark. Edges that change digit 1 to digit 2 have label k∗, while the other edges are labelled as in (B). The strongly connected components (SCCs) are indicated by dotted outlines, with the two terminal SCCs identified by an asterisk.
Mentions: Graphs arising in gene regulation may not always be strongly connected (see the section ‘Formation of an inherently bounded chromatin domain’ and Figure 6C). Steady-state probabilities for non-strongly connected graphs can be calculated by considering the SCCs of G (Figures 2B and 4C). The SCCs inherit connections from the underlying graph but these connections can never form a cycle, for otherwise the SCCs would collapse into each other. It is therefore possible to identify terminal SCCs, from which there are no outgoing connections. The terminal SCCs yield steady states in the following way.Figure 6

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