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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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Experimental data and calculated gene-regulation functions ofPHO5 variants. Each panel corresponds to one of the six variants, as labelled in the top left with high affinity (H, blue), low affinity (L, magenta) or absent (X), using the microstate schematic from Figure 7B. Each panel shows the smoothed and normalised experimental data for that variant scaled to its maximum expression level (blue points) and plotted as normalised CFP for output against normalised YFP for input, overlaid with calculated gene-regulation functions for that variant (red and black curves), plotted as probability of transcription against normalised YFP, which is assumed to be proportional to Pho4 concentration. The red curves show individual fits to each variant, while the black curves show a collective fit to all variants simultaneously. Further details are provided in the text and the Methods. H, high affinity; L, low affinity; X, absent.
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Fig8: Experimental data and calculated gene-regulation functions ofPHO5 variants. Each panel corresponds to one of the six variants, as labelled in the top left with high affinity (H, blue), low affinity (L, magenta) or absent (X), using the microstate schematic from Figure 7B. Each panel shows the smoothed and normalised experimental data for that variant scaled to its maximum expression level (blue points) and plotted as normalised CFP for output against normalised YFP for input, overlaid with calculated gene-regulation functions for that variant (red and black curves), plotted as probability of transcription against normalised YFP, which is assumed to be proportional to Pho4 concentration. The red curves show individual fits to each variant, while the black curves show a collective fit to all variants simultaneously. Further details are provided in the text and the Methods. H, high affinity; L, low affinity; X, absent.

Mentions: Kim and O’Shea estimated the threshold and the maximum expression level of each variant by fitting their experimental data to a Hill function, whose Hill coefficient was found to be nearly 2 for all variants. They then fitted the estimated threshold and maximum values to the calculated gene-regulation function for each variant and found good agreement ([52], Figure 5). We were curious as to how well the gene-regulation function itself would fit the data. This is a more challenging question because the data are noisy and the gene-regulation function is very complicated (see below). To address this, we first smoothed the data. We then used numerical optimisation to find excellent quantitative fits to each variant individually (Figure 8, red curves) but could only undertake a manual fit to all variants collectively, which yielded the parameter values in Equation 16 (Methods). The collective fit was considerably poorer (Figure 8, black curves). While this broadly confirms Kim and O’Shea’s more coarse-grained analysis, it also suggests that the individual variants may exhibit more nuanced behaviours, which are better described by distinct parameter values.Figure 8


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

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

Experimental data and calculated gene-regulation functions ofPHO5 variants. Each panel corresponds to one of the six variants, as labelled in the top left with high affinity (H, blue), low affinity (L, magenta) or absent (X), using the microstate schematic from Figure 7B. Each panel shows the smoothed and normalised experimental data for that variant scaled to its maximum expression level (blue points) and plotted as normalised CFP for output against normalised YFP for input, overlaid with calculated gene-regulation functions for that variant (red and black curves), plotted as probability of transcription against normalised YFP, which is assumed to be proportional to Pho4 concentration. The red curves show individual fits to each variant, while the black curves show a collective fit to all variants simultaneously. Further details are provided in the text and the Methods. H, high affinity; L, low affinity; X, absent.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Fig8: Experimental data and calculated gene-regulation functions ofPHO5 variants. Each panel corresponds to one of the six variants, as labelled in the top left with high affinity (H, blue), low affinity (L, magenta) or absent (X), using the microstate schematic from Figure 7B. Each panel shows the smoothed and normalised experimental data for that variant scaled to its maximum expression level (blue points) and plotted as normalised CFP for output against normalised YFP for input, overlaid with calculated gene-regulation functions for that variant (red and black curves), plotted as probability of transcription against normalised YFP, which is assumed to be proportional to Pho4 concentration. The red curves show individual fits to each variant, while the black curves show a collective fit to all variants simultaneously. Further details are provided in the text and the Methods. H, high affinity; L, low affinity; X, absent.
Mentions: Kim and O’Shea estimated the threshold and the maximum expression level of each variant by fitting their experimental data to a Hill function, whose Hill coefficient was found to be nearly 2 for all variants. They then fitted the estimated threshold and maximum values to the calculated gene-regulation function for each variant and found good agreement ([52], Figure 5). We were curious as to how well the gene-regulation function itself would fit the data. This is a more challenging question because the data are noisy and the gene-regulation function is very complicated (see below). To address this, we first smoothed the data. We then used numerical optimisation to find excellent quantitative fits to each variant individually (Figure 8, red curves) but could only undertake a manual fit to all variants collectively, which yielded the parameter values in Equation 16 (Methods). The collective fit was considerably poorer (Figure 8, black curves). While this broadly confirms Kim and O’Shea’s more coarse-grained analysis, it also suggests that the individual variants may exhibit more nuanced behaviours, which are better described by distinct parameter values.Figure 8

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
Related in: MedlinePlus