## Reaction rates for mesoscopic reaction-diffusion kinetics

## Abstract

The mesoscopic reaction-diffusion master equation (RDME) is a popular modeling framework frequently applied to stochastic reaction-diffusion kinetics in systems biology. The RDME is derived from assumptions about the underlying physical properties of the system, and it may produce unphysical results for models where those assumptions fail. In that case, other more comprehensive models are better suited, such as hard-sphere Brownian dynamics (BD). Although the RDME is a model in its own right, and not inferred from any specific microscale model, it proves useful to attempt to approximate a microscale model by a specific choice of mesoscopic reaction rates. In this paper we derive mesoscopic scale-dependent reaction rates by matching certain statistics of the RDME solution to statistics of the solution of a widely used microscopic BD model: the Smoluchowski model with a Robin boundary condition at the reaction radius of two molecules. We also establish fundamental limits on the range of mesh resolutions for which this approach yields accurate results and show both theoretically and in numerical examples that as we approach the lower fundamental limit, the mesoscopic dynamics approach the microscopic dynamics. Finally, we show that for mesh sizes below the fundamental lower limit, results are less accurate. Thus,more »

- Authors:

- Univ. of California, Santa Barbara, CA (United States)
- Uppsala Univ., Uppsala (Sweden)

- Publication Date:

- Research Org.:
- Univ. of California, Santa Barbara, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1343614

- Alternate Identifier(s):
- OSTI ID: 1180098

- Grant/Contract Number:
- SC0008975

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

- Additional Journal Information:
- Journal Volume: 91; Journal Issue: 2; Journal ID: ISSN 1539-3755

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; 97 MATHEMATICS AND COMPUTING

### Citation Formats

```
Hellander, Stefan, Hellander, Andreas, and Petzold, Linda. Reaction rates for mesoscopic reaction-diffusion kinetics. United States: N. p., 2015.
Web. doi:10.1103/PhysRevE.91.023312.
```

```
Hellander, Stefan, Hellander, Andreas, & Petzold, Linda. Reaction rates for mesoscopic reaction-diffusion kinetics. United States. doi:10.1103/PhysRevE.91.023312.
```

```
Hellander, Stefan, Hellander, Andreas, and Petzold, Linda. Mon .
"Reaction rates for mesoscopic reaction-diffusion kinetics". United States. doi:10.1103/PhysRevE.91.023312. https://www.osti.gov/servlets/purl/1343614.
```

```
@article{osti_1343614,
```

title = {Reaction rates for mesoscopic reaction-diffusion kinetics},

author = {Hellander, Stefan and Hellander, Andreas and Petzold, Linda},

abstractNote = {The mesoscopic reaction-diffusion master equation (RDME) is a popular modeling framework frequently applied to stochastic reaction-diffusion kinetics in systems biology. The RDME is derived from assumptions about the underlying physical properties of the system, and it may produce unphysical results for models where those assumptions fail. In that case, other more comprehensive models are better suited, such as hard-sphere Brownian dynamics (BD). Although the RDME is a model in its own right, and not inferred from any specific microscale model, it proves useful to attempt to approximate a microscale model by a specific choice of mesoscopic reaction rates. In this paper we derive mesoscopic scale-dependent reaction rates by matching certain statistics of the RDME solution to statistics of the solution of a widely used microscopic BD model: the Smoluchowski model with a Robin boundary condition at the reaction radius of two molecules. We also establish fundamental limits on the range of mesh resolutions for which this approach yields accurate results and show both theoretically and in numerical examples that as we approach the lower fundamental limit, the mesoscopic dynamics approach the microscopic dynamics. Finally, we show that for mesh sizes below the fundamental lower limit, results are less accurate. Thus, the lower limit determines the mesh size for which we obtain the most accurate results.},

doi = {10.1103/PhysRevE.91.023312},

journal = {Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics},

number = 2,

volume = 91,

place = {United States},

year = {2015},

month = {2}

}

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