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We modeled cell division in both an explicit and implicit manner see Information S1 Section 5. Modeling cell division explicitly can be seen as a complicated time event where timing and assignments depend on distinct distributions.

## Applied Stochastic Models in Business and Industry

We consider gamma distributed event waiting times for cell division. At a cell division event the molecular content of the mother cell is binomially partitioned over its two daughter cells. This is in contrast with earlier work from Kierzek et al. Commonly used model definition formats e. In such a scenario, each simulation starts with the results of the previous simulation.

For these reasons, modeling cell division explicitly is inconvenient and non-trivial, especially for graphical user interface GUI based simulators. In the implicit cell division model growth rate is incorporated as a first-order rate constant that continuously dilutes cellular components into new cells.

The differences between modeling cell division explicitly and implicitly are illustrated in Figure 7. Without incorporating cell division explicitly the dynamics of transcription factors, mRNA, and proteins are different Figures 7A—F. Most apparent are the differences for the protein copy numbers, which quickly reached a steady state Figure 7C. In contrast, explicit modeling of cell division caused large variations in the protein copy numbers Figure 7F.

As a result, the distribution of mean protein copy numbers was significantly different if cell division was explicitly incorporated Figure 7G. These differences are also observable for transcription factors active, inactive and mRNA not shown.

Modeling details of cell division periods: Gamma-distributed with scale parameter is Distributions of protein copy numbers for modeling cell division explicitly and implicitly G. The model is further described in Information S1 Section 5. The StochPy software features and speed performance were benchmarked against widely used, existing stochastic software to make a fair and broad comparison of available tools for stochastic simulations. Here, we will discuss and compare these tools against StochPy through a feature comparison.

A comprehensive feature comparison is provided in Table 1. Explicit stochastic simulation output. An important distinction between StochPy and most other stochastic simulators is in their output: Explicit rather than fixed-interval output is used and has several implications. Firstly, fixed-interval output does not allow the calculation of event waiting times, a unique feature provided by StochPy.

Secondly, handling of fixed-interval output requires expert knowledge of the modeler how many fixed intervals are sufficient to do a certain type of analysis properly. Thirdly, determining the minimal number of fixed intervals necessary is time-consuming. In the StochPy software one can exploit the convergence of higher moments to determine the minimal number of time steps necessary to accurately determine certain model properties.

However, the benefit of using fixed-interval output is that less data can be stored which gives this approach a significant speed advantage. This can be useful if, for instance, only moments and time series are of interest and not probability distributions. SBML support and simulating diverse stochastic models. As a consequence, many models e.

Analysis of stochastic data. Besides various unique numerical analysis techniques StochPy also provides preprogrammed plotting functions. Calculation of event waiting times, propensity probability distributions, co- variances, and auto-correlations for one or more generated trajectories are currently unique numerical analysis techniques of StochPy.

In addition, analysis of time series of species and propensities, probability distributions of species amounts, and moments can be done within StochPy. While not shown in the case studies, StochPy can calculate and visualize the average of multiple time trajectories for time series, auto-correlations, and distributions.

Flexible environment for interactive modeling. By integrating its functionality with industrial strength Python scientific libraries e. Matplotlib [18] , NumPy [26] and SciPy [27] sophisticated user-defined analysis methods can be seamlessly applied to the output of StochPy simulations. The direct solver of StochPy was benchmarked against the direct solvers of two widely used and high-performance stochastic tools i. Note that determining the minimal number of fixed-intervals necessary to perform a particular analysis requires doing multiple simulations as shown in Figure 2 , which makes this fixed-interval approach less efficient and more time-consuming than Table 2 reports.

The first conclusion from this benchmark is that no single solver was the fastest in any case. Secondly, StochPy is the only stochastic simulator that was able to correctly simulate all stochastic models tested in this benchmark. Thirdly, for different numbers of simulation time steps, significant differences in simulation time were found for different solvers. This effect is negligible for relatively long simulations. For instance, CAIN needed about 4 minutes to parse the largest model tested parsing time was omitted from the benchmark , while both StochKit2 and StochPy were able to parse this model within seconds.

### Smith, J. MacGregor

While exploiting these solvers in StochPy appears slower than the native application, this time only includes parsing of the simulation output for post-simulation analysis. This can take a significant amount of time for large data sets. Here, decisions are made depending on the simulation time and the output of the solver. Insights into time series or moments can be easily obtained with solvers that provide fixed-interval output, whereas solvers that provide explicit output are, in principle, better suited for determining probability distributions of molecule copy numbers and event waiting times.

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. Stochastic modeling in systems biology demands a certain level of flexibility in simulation, management of stochastic models and the handling of simulation data.

Depending on the size of the system of interest and its degrees of time-scale separation, the different SSAs each have their particular dis- advantages. The differences in simulation time between stochastic simulation packages are often due to the fixed-interval reporting of simulation data versus the use of explicit output. To achieve the accuracy of explicit solvers the differences in simulation time greatly reduce, and ultimately boil down to, differences in the programming languages.

In systems biology applications, often the pure simulation data rather than the fixed-interval simulation data is of interest. The pure simulation data allows for the accurate determination of various time and copy number associated probability measures. We presented StochPy as a versatile modeling package for stochastic simulation of molecular control networks inside living cells that provides solvers which return explicit stochastic simulation output.

## Stochastic Models in Preventive Maintenance Policies

We highlighted this by implementing both the solvers of CAIN and StochKit2 that return only fixed-interval output, which can be useful for obtaining insight into time series and moments. The high-level statistical and plotting functions of StochPy allow for quick and interactive model interrogation at the command-line. Annotated scripts and input files used to generate modeling results. Stochastic models and a script to simulate the models used for benchmarking StochPy. Additional information about the different models that were used as examples to demonstrate the capabilities of StochPy.

The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. National Center for Biotechnology Information , U. PLoS One. Published online Nov Timo R. Maarleveld , 1 , 2 , 3 Brett G. Olivier , 1 and Frank J.

Brett G.