Fig 1.
Overview of software packages for stochastic modeling and their capabilities.
Fig 2.
(a) CERENA can be used to study (multi-compartment) chemical reaction networks. (b) The reaction network can be defined in MATLAB, or alternatively, imported from SBML. (c) The system of equations for different modeling approaches implemented in CERENA is generated, and optionally stored as MATLAB functions for numerical simulation using MATLAB ODE solvers. Furthermore, the representation of the system can be exported to the estimation toolbox Data2Dynamics. (d) The symbolic representation of the system of equations together with the initial conditions is stored in a MATLAB script. (e) Based on the symbolic representation, 1st and 2nd order sensitivity equations are derived. MEX-files, which use CVODES and IDAS packages of SUNDIALS for the numerical simulation of the models, are compiled. (f) The generated MEX-files are used for numerical simulation, and can be integrated with other software for parameter estimation. (g) Various aspects of the simulation results can be visualized using CERENA.
Fig 3.
Simulation of the three-stage model of gene expression.
(a) Schematic of the three-stage model of gene expression. (b) Mean (left) and variance (right) of the number of protein molecules obtained using different orders of SSE, MM and MCM. (c) Marginal probabilities of promotor states (left), the mean of protein molecule numbers conditioned on the promotor state (middle), and the variance of protein molecule numbers conditioned on the promotor state (right) predicted by MCM of order 3. (b,c) FSP results serve as the reference solution. Low dispersion closure was used for MM and MCM. MM2, MM3, MCM2 and MCM3 denote the second- and third-order MM and the second- and third-order MCM.
Fig 4.
Approximation error of MM and MCM of various orders with various moment closures for the three-stage model of gene expression.
Relative errors of mean and variance of the protein concentration at the steady state are depicted for different truncation orders and moment closures. The truncation order m means that moments up to order m are simulated. For moment orders and closures for which the numerical simulation could not be completed, i.e. derivative matching, no approximation error is reported.
Fig 5.
Complexity of different descriptions of the three-stage model of gene expression.
Number of state-variables (top) and computation time (bottom). Runtimes are shown for the numerical simulation using CVODES/IDAS wrappers implemented in CERENA and MATLAB solver ode15s, as well as for StochDynTools. The computation times were calculated by averaging over at least 10 simulations. For MM and MCM low dispersion closure was used.
Fig 6.
Simulation results for the JAK-STAT signaling pathway.
(a) Schematic of the simplified JAK-STAT signaling pathway. The intermediate states npSTAT1 to npSTAT5 are used to model the delayed export of STAT from the nucleus. (b) The mean (left) and variance (right) of dimerized phosphorylated STAT concentration, obtained using several methods. SSA simulation results serve as the reference solution.
Fig 7.
Computation time for different sensitivity analysis methods.
The objective function gradient for MM2 simulation is evaluated for an increasing number of parameters. The computation times of finite differences, forward sensitivity analysis, and adjoint sensitivity analysis are shown.