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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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The product graph construction. The corresponding basis vector in the respective Laplacian kernel is shown below each graph. For legibility, the vertices of the product graph are denoted i,j, rather than (i,j). All three graphs are strongly connected. The basis vector for the Laplacian kernel of graph G was calculated in Figure 4B, while that for graph H follows directly from Equation 7. The basis vector for the Laplacian kernel of G×H is given by the Kronecker product formula in Equation 14, as described in the text.
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Fig9: The product graph construction. The corresponding basis vector in the respective Laplacian kernel is shown below each graph. For legibility, the vertices of the product graph are denoted i,j, rather than (i,j). All three graphs are strongly connected. The basis vector for the Laplacian kernel of graph G was calculated in Figure 4B, while that for graph H follows directly from Equation 7. The basis vector for the Laplacian kernel of G×H is given by the Kronecker product formula in Equation 14, as described in the text.

Mentions: We used the product graph construction to capture the concept of independence. Let G and H be any two graphs which represent two modules within a gene regulation system. We make no assumptions about the graphs, which do not have to be at equilibrium and do not have to be strongly connected. The product graph G×H is constructed as follows (Figure 9). It has vertices (i,j), where i is a vertex in G and j is a vertex in H. The vertices are enumerated lexicographically, so that (i,j)<(i′,j′) if either i<i′ or i=i′ and j<j′. For each labelled edge in G and for every vertex j in H, the labelled edge is created in G×H. The retention of the same label a on these edges ensures that the transition from (i1,j) to (i2,j) occurs independently of j and always at the same rate, which captures the independence assumption. Similarly, for each labelled edge in H and for every vertex i in G, the labelled edge is created in G×H. These are the only edges in G×H.Figure 9


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

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

The product graph construction. The corresponding basis vector in the respective Laplacian kernel is shown below each graph. For legibility, the vertices of the product graph are denoted i,j, rather than (i,j). All three graphs are strongly connected. The basis vector for the Laplacian kernel of graph G was calculated in Figure 4B, while that for graph H follows directly from Equation 7. The basis vector for the Laplacian kernel of G×H is given by the Kronecker product formula in Equation 14, as described in the text.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Fig9: The product graph construction. The corresponding basis vector in the respective Laplacian kernel is shown below each graph. For legibility, the vertices of the product graph are denoted i,j, rather than (i,j). All three graphs are strongly connected. The basis vector for the Laplacian kernel of graph G was calculated in Figure 4B, while that for graph H follows directly from Equation 7. The basis vector for the Laplacian kernel of G×H is given by the Kronecker product formula in Equation 14, as described in the text.
Mentions: We used the product graph construction to capture the concept of independence. Let G and H be any two graphs which represent two modules within a gene regulation system. We make no assumptions about the graphs, which do not have to be at equilibrium and do not have to be strongly connected. The product graph G×H is constructed as follows (Figure 9). It has vertices (i,j), where i is a vertex in G and j is a vertex in H. The vertices are enumerated lexicographically, so that (i,j)<(i′,j′) if either i<i′ or i=i′ and j<j′. For each labelled edge in G and for every vertex j in H, the labelled edge is created in G×H. The retention of the same label a on these edges ensures that the transition from (i1,j) to (i2,j) occurs independently of j and always at the same rate, which captures the independence assumption. Similarly, for each labelled edge in H and for every vertex i in G, the labelled edge is created in G×H. These are the only edges in G×H.Figure 9

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