# An Overview of Network-Based and -Free Approaches for Stochastic Simulation of Biochemical Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Network-Based Approach

**Initialize**: Set the time $t=0$ and set up the initial state vector, propensities, and random number generators.**Execute**: Using a suitable sampling procedure, generate random numbers and, on the basis of these, determine the next reaction to occur and the time interval.**Update**: Update the molecule count, and if needed, recalculate the propensities. Output the system state.**Iterate**: If simulation end time is not reached, go to step 2.

## 3. Network-Free Approach

## 4. Benchmarking Stochastic Simulation

#### 4.1. Simulators

#### 4.2. Models

## 5. Results

#### 5.1. Increasing Numbers of Particles

#### 5.2. Dependency on the Simulation End Time

## 6. Discussion

## 7. Methods

#### 7.1. Model Construction

#### 7.2. Simulations

#### 7.3. Analysis

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Supplementary Figures

#### Appendix A.1. Multi-State Model

**Figure A1.**Multi-state model and its simulation. (

**Left**) Schematic illustration of multi-state model, in which species R binds with L to form an R.L composite. The R.L composite, at a different rate, unbinds to result in R and L. An additional species A can bind with R of the binding model and has a phosphorylation site. Figure generated using RuleBender. (

**Right**) An example time trace of the observables of this model.

#### Appendix A.2. Multi-Site Model

**Figure A2.**Multi-site model and its simulation. (

**Left**) Schematic illustration of multi-site model. In the multi-state model, species A has two additional phosphorylation sites, where both R and L can bind. Figure generated using RuleBender. (

**Right**) An example time trace of the observables of this model.

#### Appendix A.3. EGFR Signaling Model

**Figure A3.**Epidermal growth factor receptor (EGFR) signaling model and its simulation. (

**Left**) Schematic illustration of the EGFR signaling model. Figure generated using RuleBender. (

**Right**) An example time trace of the observables of this model.

#### Appendix A.4. BCR Signaling Model

**Figure A4.**B-cell receptor (BCR) signaling model and its simulation. (

**Left**) Schematic illustration of the BCR signaling model. Figure generated using RuleBender. (

**Right**) An example time trace of the observables of this model.

#### Appendix A.5. FcϵRI Signaling Model

**Figure A5.**FcϵRI signaling model and its simulation. (

**Left**) Schematic illustration of the FcϵRI signaling model. Figure generated using RuleBender. (

**Right**) An example time trace of the observables of this model.

#### Appendix A.6. Fastest Simulators under the Tested Scenarios

**Figure A6.**For each of the tested conditions of all five models, the simulator that took the least amount of time is shown for both (

**Left**) different molecule numbers and (

**Right**) different simulation end times.

#### Appendix A.7. Performance Differences between Random Number Generators

**Figure A7.**Effects due to selection of different random number generators. (

**Left**) Comparison of pseudo-random number generation times for the C runtime rand() function (solid line) and the Mersenne Twister (dashed line). (

**Right**) Difference in performance of BioNetGen using the C runtime rand() function (solid line) and a modified version using the Mersenne Twister (dashed line).

## Appendix B. Supplementary Table

Model | Test Scenario | Number of Molecules | Simulation End Time (s) |
---|---|---|---|

Multi-state | Different molecule numbers | R = 500 to 25,000 | 100 |

L = 100 to 10,000 | |||

A = 500 to 25,000 | |||

Different simulation end times | R = 5000, L = 1000, A = 5000 | 1 to 10,000 | |

Multi-site | Different molecule numbers | R = 500 to 25,000 | 100 |

L = 100 to 10,000 | |||

A = 500 to 25,000 | |||

Different simulation end times | R = 5000, L = 1000, A = 5000 | 1 to 10,000 | |

EGFR | Different molecule numbers | ${egf}_{tot}$ = 1.2 $\times {10}^{4}$ to 6.0 $\times {10}^{6}$ | 100 |

$egf{r}_{tot}$ = 1800 to 9.0 $\times {10}^{5}$ | |||

$Grb{2}_{tot}$ = 1000 to 5.0 $\times {10}^{5}$ | |||

$Sh{c}_{tot}$ = 2700 to 1.35 $\times {10}^{6}$ | |||

$So{s}_{tot}$ = 130 to 6.5 $\times {10}^{4}$ | |||

$Grb2\_So{s}_{tot}$ = 490 to 2.45 $\times {10}^{4}$ | |||

Different simulation end times | $eg{f}_{tot}$= 1.2 $\times {10}^{5}$ | 1 to 1000 | |

$egf{r}_{tot}$ = 1.8 $\times {10}^{5}$ | |||

$Grb{2}_{tot}$ = 1.0 $\times {10}^{5}$ | |||

$Sh{c}_{tot}$ = 2.7 $\times {10}^{5}$ | |||

$So{s}_{tot}$ = 1.3 $\times {10}^{4}$ | |||

$Grb2\_So{s}_{tot}$ = 4.9 $\times {10}^{4}$ | |||

BCR | Different molecule numbers | $p1$ = 3000 to 7.5 $\times {10}^{5}$ | 100 |

Different simulation end times | $p1$ = 30,000 | 1 to 1000 | |

FcϵRI | Different molecule numbers | $Li{g}_{tot}$ = 6000 to 600,000 | 100 |

$Re{c}_{tot}$ = 400 to 40,000 | |||

$Ly{n}_{tot}$ = 30 to 3000 | |||

$Sy{k}_{tot}$ = 400 to 40,000 | |||

Different simulation end times | $Li{g}_{tot}$ = 60,000 | 1 to 1000 | |

$Re{c}_{tot}$ = 4000 | |||

$Ly{n}_{tot}$ = 300 | |||

$Sy{k}_{tot}$ = 4000 |

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**Figure 1.**Execution times of the simulators for different number of molecules in the tested models, namely, (

**A**) multi-state model, (

**B**) multi-site model, (

**C**) epidermal growth factor receptor (EGFR) signaling model, (

**D**) B-cell receptor (BCR) signaling model, and (

**E**) The high-affinity human IgE receptor (FcϵRI) signaling model. In all the models for this test condition, the simulation end time was set to 100 s.

**Figure 2.**Execution times of the simulators for different simulation end times in the tested models, namely, (

**A**) multi-state model, (

**B**) multi-site model, (

**C**) EGFR signaling model, (

**D**) BCR signaling model, and (

**E**) FcϵRI signaling model. In all the models for this test condition, the initial number of particles was fixed (see Appendix B, Table A1).

**Table 1.**Simulators used in this study. Stochastic simulation algorithm (SSA) used in each of the simulators is listed along with the language they are implemented with.

Approach | Simulator | SSA Method | Language | Version | Reference |
---|---|---|---|---|---|

Network-based | BioNetGen | SDM ^{*} | Perl and C++ | 2.3.1 | [17] |

COPASI_D | DM ^{**} | C++ | 4.21 (Build 166) | [6] | |

COPASI_GB | NRM ^{***} | C++ | 4.21 (Build 166) | [6] | |

Dizzy | DM | Java | 1.11.4 | [9] | |

Gillespie2 | DM | C | Rev: 56 | [10] | |

pSSAlib_SPDM | SPDM ^{#} | C++ | 2.0.0 | [13] | |

pSSAlib_SSACR | CR ^{##} | C++ | 2.0.0 | [13] | |

RoadRunner | DM | C | 1.4.24 | [12] | |

SGNS2 | NRM | C++ | 2.1.170 | [11] | |

StochKit2 | CR | C++ | 2.0.13 | [33] | |

StochPy | DM | Python | 2.3 | [8] | |

Network-free | DYNSTOC | — | C | 1.2.0 | [25] |

KaSim | — | OCaml | 3.5 | [22] | |

NFsim | — | C++ | 1.11 | [19] | |

RuleMonkey | — | C | 2.0.25 | [39] |

^{*}Sorting direct method ;

^{**}Direct method ;

^{***}Next reaction method;

^{#}Sorting partial propensity direct method;

^{##}Composition rejection method.

**Table 2.**Models used in this study. The network derivation time with BioNetGen is also shown for each of the models.

Model | No. of Species | No. of Rules | No. of Reactions | Derivation Time (s) |
---|---|---|---|---|

Multi-state [17,25] | 6 | 4 | 8 | 0.0 |

Multi-site [39] | 66 | 12 | 288 | 0.3 |

EGFR ^{*} signaling [47] | 356 | 23 | 3749 | 11.6 |

BCR ^{**} signaling [48] | 1122 | 72 | 24,388 | 33.17 |

FcϵRI ^{***} signaling ($\gamma $) [49] | 3744 | 24 | 58,276 | 163.8 |

^{*}Epidermal growth factor receptor ;

^{**}B-cell receptor ;

^{***}The high-affinity human IgE receptor.

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Gupta, A.; Mendes, P.
An Overview of Network-Based and -Free Approaches for Stochastic Simulation of Biochemical Systems. *Computation* **2018**, *6*, 9.
https://doi.org/10.3390/computation6010009

**AMA Style**

Gupta A, Mendes P.
An Overview of Network-Based and -Free Approaches for Stochastic Simulation of Biochemical Systems. *Computation*. 2018; 6(1):9.
https://doi.org/10.3390/computation6010009

**Chicago/Turabian Style**

Gupta, Abhishekh, and Pedro Mendes.
2018. "An Overview of Network-Based and -Free Approaches for Stochastic Simulation of Biochemical Systems" *Computation* 6, no. 1: 9.
https://doi.org/10.3390/computation6010009