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

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

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

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

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

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

Uptake and secretion fluxes across EFMs.

<p>(a) Oxygen uptake (scaled by glucose uptake). Flux values are shown by colors in the rate/yield spectrum (same points as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006010#pcbi.1006010.g002" target="_blank">Fig 2b</a>). The EFMs with the highest growth rates consume intermediate levels of oxygen. The other diagrams show <b>(b)</b> acetate secretion, <b>(c)</b> lactate secretion and <b>(d)</b> succinate secretion, each scaled by glucose uptake. Acetate secretion and <i>O</i>₂ uptake versus biomass yield are shown in Figure 9 in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006010#pcbi.1006010.s001" target="_blank">S1 Text</a>.</p

Metabolic enzyme cost explains variable trade-offs between microbial growth rate and yield

<div><p>Microbes may maximize the number of daughter cells per time or per amount of nutrients consumed. These two strategies correspond, respectively, to the use of enzyme-efficient or substrate-efficient metabolic pathways. In reality, fast growth is often associated with wasteful, yield-inefficient metabolism, and a general thermodynamic trade-off between growth rate and biomass yield has been proposed to explain this. We studied growth rate/yield trade-offs by using a novel modeling framework, Enzyme-Flux Cost Minimization (EFCM) and by assuming that the growth rate depends directly on the enzyme investment per rate of biomass production. In a comprehensive mathematical model of core metabolism in <i>E. coli</i>, we screened all elementary flux modes leading to cell synthesis, characterized them by the growth rates and yields they provide, and studied the shape of the resulting rate/yield Pareto front. By varying the model parameters, we found that the rate/yield trade-off is not universal, but depends on metabolic kinetics and environmental conditions. A prominent trade-off emerges under oxygen-limited growth, where yield-inefficient pathways support a 2-to-3 times higher growth rate than yield-efficient pathways. EFCM can be widely used to predict optimal metabolic states and growth rates under varying nutrient levels, perturbations of enzyme parameters, and single or multiple gene knockouts.</p></div