Mechanistic stochastic model of histone modification pattern formation

BACKGROUND: The activity of a single gene is influenced by the composition of the chromatin in which it is embedded. Nucleosome turnover, conformational dynamics, and covalent histone modifications each induce changes in the structure of chromatin and its affinity for regulatory proteins. The dynamics of histone modifications and the persistence of modification patterns for long periods are still largely unknown. RESULTS: In this study, we present a stochastic mathematical model that describes the molecular mechanisms of histone modification pattern formation along a single gene, with non-phenomenological, physical parameters. We find that diffusion and recruitment properties of histone modifying enzymes together with chromatin connectivity allow for a rich repertoire of stochastic histone modification dynamics and pattern formation. We demonstrate that histone modification patterns at a single gene can be established or removed within a few minutes through diffusion and weak recruitment mechanisms of histone modification spreading. Moreover, we show that strong synergism between diffusion and weak recruitment mechanisms leads to nearly irreversible transitions in histone modification patterns providing stable patterns. In the absence of chromatin connectivity spontaneous and dynamic histone modification boundaries can be formed that are highly unstable, and spontaneous fluctuations cause them to diffuse randomly. Chromatin connectivity destabilizes this synergistic system and introduces bistability, illustrating state switching between opposing modification states of the model gene. The observed bistable long-range and localized pattern formation are critical effectors of gene expression regulation. CONCLUSION: This study illustrates how the cooperative interactions between regulatory proteins and the chromatin state generate complex stochastic dynamics of gene expression regulation

Time series of bursty and non-bursty transcription.

<p>StochPy plots of simulating stochastic gene expression. (A) long lifetimes of both the ON and OFF state. (B) bursty transcription. (C) short lifetimes of both the ON and OFF state. (D) non-bursty transcription.</p

StochPy simulation output.

<p>An example of explicit simulation output of StochPy is shown in a table. It reports the number of molecules of each molecular species and the reaction propensities at each time point when a reaction occurs. The time differences between consecutive rows indicate waiting times between reaction events. In the last column, the waiting times for reaction 4, , are given and they correspond to the time period between consecutive instances of activity of reaction 4.</p

Fixed interval versus explicit simulation output.

<p>Hundred stochastic simulations until t = 60.000 min ( time steps) were done with min⁻¹, min⁻¹, min⁻¹, and min⁻¹. (A) Accuracy of mean and standard deviation estimates as function of the number of fixed intervals. (B) Simulation time with fixed-interval output increases with the number of fixed intervals. Fixed-interval simulations were done with the StochPy interface to StochKit2 and include the time to calculate the associated probability distributions. (C) The stationary mRNA distribution for fixed intervals (red error bars, 1.96 ) vs. explicit output (blue 95% confidence interval). Note that 1.96 corresponds to a 95% confidence interval. (D) The stationary mRNA distribution for fixed intervals (red error bars, 1.96 ) vs. explicit output (blue 95% confidence interval).</p

mRNA copy number and event waiting times distributions.

<p>StochPy plots of simulating stochastic gene expression with StochPy simulations (step, markers, colored) and analytical solutions (solid, black). (A) probability distribution of the mRNA copy numbers. (B) probability distribution of the mRNA synthesis event waiting times.</p

Stochastic modeling Decision Tree.

<p>Both fixed-interval and explicit output have their advantages and disadvantages. The decision whether to use fixed-interval or explicit output depends on the type of analysis.</p

Feature comparison between StochPy and existing (stochastic) software.

<p>Summary of features offered in StochPy and other stochastic modeling software.</p><p>•: Feature is present.</p>○<p>: Feature is partially present or requires additional dependencies.</p><p>Notes: 1. Limited ability to parse kinetic laws: Complicated expressions may not parsed. 2. Not all SBML documents can be converted into the StochKit2 model format. 3. Provided as an add-on functionality of StochKit2, whereas with limited options compared to the default installation of StochKit2. 4. Only if proprietary software (MATLAB) is installed.</p

Speed performance benchmark between StochPy and existing (stochastic) software.

<p>Results of benchmarking the direct method of StochPy. Simulation time was divided by the simulation time of the StochPy solvers: StochPy’s solver was faster if the reported ratio’s are larger than one and vice versa. A “−” indicates that short and long simulations were done to illustrate the potential difference between them. N/A is shown if the simulator was not possible to perform the simulation. For parallel simulations, 100 trajectories were done. In each comparison the number of fixed intervals was equal to the number of time steps in the simulation. Simulations were done on a Intel Core i5-2430M CPU 2.40 GHz×4 64-bit with Ubuntu 12.04 LTS as operating system. Stochastic models and a script to simulate these models within StochPy are available in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0079345#pone.0079345.s003" target="_blank">Scripts S2</a>.</p><p>Notes:</p>1<p>StochPy with interfaces to CAIN and StochKit2. Simulation time includes time to parse results into StochPy.</p>2<p>Cain cannot parse events, so the user most specify them in the GUI.</p>3<p>Optimal theoretical result without including time to merge the output of all sequential simulations.</p

Modeling single-cell transcription and translation with and without cell division.

<p>StochPy plots of simulating stochastic gene expression. Modeling details of cell division periods: Gamma-distributed with scale parameter is 60.0 and shape parameter is 1.0. Implicit and explicit time series of transcription factor copy numbers (A and D), mRNA copy numbers (B and E), and protein copy numbers (C and F). Distributions of protein copy numbers for modeling cell division explicitly and implicitly (G). The model is further described in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0079345#pone.0079345.s004" target="_blank">Information S1</a> Section 5.</p