# Reaction rates for a generalized reaction-diffusion master equation

## Abstract

It has been established that there is an inherent limit to the accuracy of the reaction-diffusion master equation. Specifically, there exists a fundamental lower bound on the mesh size, below which the accuracy deteriorates as the mesh is refined further. In this paper we extend the standard reaction-diffusion master equation to allow molecules occupying neighboring voxels to react, in contrast to the traditional approach in which molecules react only when occupying the same voxel. We derive reaction rates, in two dimensions as well as three dimensions, to obtain an optimal match to the more fine-grained Smoluchowski model, and show in two numerical examples that the extended algorithm is accurate for a wide range of mesh sizes, allowing us to simulate systems that are intractable with the standard reaction-diffusion master equation. In addition, we show that for mesh sizes above the fundamental lower limit of the standard algorithm, the generalized algorithm reduces to the standard algorithm. We derive a lower limit for the generalized algorithm which, in both two dimensions and three dimensions, is on the order of the reaction radius of a reacting pair of molecules.

- Authors:

- Univ. of California, Santa Barbara, CA (United States). Dept. of Computer Science

- Publication Date:

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

- Sponsoring Org.:
- USDOE; National Science Foundation (NSF); National Institutes of Health (NIH); US Army Research Office (ARO)

- OSTI Identifier:
- 1343615

- Alternate Identifier(s):
- OSTI ID: 1235628

- Grant/Contract Number:
- SC0008975; DMS-1001012; R01-GM113241-01; W911NF-09-D-0001; R01-EB014877-01

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physical Review E

- Additional Journal Information:
- Journal Volume: 93; Journal Issue: 1; Journal ID: ISSN 2470-0045

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

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

### Citation Formats

```
Hellander, Stefan, and Petzold, Linda.
```*Reaction rates for a generalized reaction-diffusion master equation*. United States: N. p., 2016.
Web. doi:10.1103/PhysRevE.93.013307.

```
Hellander, Stefan, & Petzold, Linda.
```*Reaction rates for a generalized reaction-diffusion master equation*. United States. doi:10.1103/PhysRevE.93.013307.

```
Hellander, Stefan, and Petzold, Linda. Tue .
"Reaction rates for a generalized reaction-diffusion master equation". United States. doi:10.1103/PhysRevE.93.013307. https://www.osti.gov/servlets/purl/1343615.
```

```
@article{osti_1343615,
```

title = {Reaction rates for a generalized reaction-diffusion master equation},

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

abstractNote = {It has been established that there is an inherent limit to the accuracy of the reaction-diffusion master equation. Specifically, there exists a fundamental lower bound on the mesh size, below which the accuracy deteriorates as the mesh is refined further. In this paper we extend the standard reaction-diffusion master equation to allow molecules occupying neighboring voxels to react, in contrast to the traditional approach in which molecules react only when occupying the same voxel. We derive reaction rates, in two dimensions as well as three dimensions, to obtain an optimal match to the more fine-grained Smoluchowski model, and show in two numerical examples that the extended algorithm is accurate for a wide range of mesh sizes, allowing us to simulate systems that are intractable with the standard reaction-diffusion master equation. In addition, we show that for mesh sizes above the fundamental lower limit of the standard algorithm, the generalized algorithm reduces to the standard algorithm. We derive a lower limit for the generalized algorithm which, in both two dimensions and three dimensions, is on the order of the reaction radius of a reacting pair of molecules.},

doi = {10.1103/PhysRevE.93.013307},

journal = {Physical Review E},

issn = {2470-0045},

number = 1,

volume = 93,

place = {United States},

year = {2016},

month = {1}

}

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