# Accurate chemical master equation solution using multi-finite buffers

## Abstract

Here, the discrete chemical master equation (dCME) provides a fundamental framework for studying stochasticity in mesoscopic networks. Because of the multiscale nature of many networks where reaction rates have a large disparity, directly solving dCMEs is intractable due to the exploding size of the state space. It is important to truncate the state space effectively with quantified errors, so accurate solutions can be computed. It is also important to know if all major probabilistic peaks have been computed. Here we introduce the accurate CME (ACME) algorithm for obtaining direct solutions to dCMEs. With multifinite buffers for reducing the state space by $O(n!)$, exact steady-state and time-evolving network probability landscapes can be computed. We further describe a theoretical framework of aggregating microstates into a smaller number of macrostates by decomposing a network into independent aggregated birth and death processes and give an a priori method for rapidly determining steady-state truncation errors. The maximal sizes of the finite buffers for a given error tolerance can also be precomputed without costly trial solutions of dCMEs. We show exactly computed probability landscapes of three multiscale networks, namely, a 6-node toggle switch, 11-node phage-lambda epigenetic circuit, and 16-node MAPK cascade network, the latter two withmore »

- Authors:

- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Univ. of Illinois, Chicago, IL (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE Laboratory Directed Research and Development (LDRD) Program

- OSTI Identifier:
- 1358164

- Report Number(s):
- LA-UR-16-27504

Journal ID: ISSN 1540-3459

- Grant/Contract Number:
- AC52-06NA25396

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Multiscale Modeling & Simulation

- Additional Journal Information:
- Journal Volume: 14; Journal Issue: 2; Journal ID: ISSN 1540-3459

- Publisher:
- SIAM

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY; Computer Science; Mathematics; chemical master equation; stochastic biological networks; state space truncation; steady state probability landscape; time-evolving probability landscapes; first passage time distribu- tion.

### Citation Formats

```
Cao, Youfang, Terebus, Anna, and Liang, Jie.
```*Accurate chemical master equation solution using multi-finite buffers*. United States: N. p., 2016.
Web. doi:10.1137/15M1034180.

```
Cao, Youfang, Terebus, Anna, & Liang, Jie.
```*Accurate chemical master equation solution using multi-finite buffers*. United States. doi:10.1137/15M1034180.

```
Cao, Youfang, Terebus, Anna, and Liang, Jie. Wed .
"Accurate chemical master equation solution using multi-finite buffers". United States.
doi:10.1137/15M1034180. https://www.osti.gov/servlets/purl/1358164.
```

```
@article{osti_1358164,
```

title = {Accurate chemical master equation solution using multi-finite buffers},

author = {Cao, Youfang and Terebus, Anna and Liang, Jie},

abstractNote = {Here, the discrete chemical master equation (dCME) provides a fundamental framework for studying stochasticity in mesoscopic networks. Because of the multiscale nature of many networks where reaction rates have a large disparity, directly solving dCMEs is intractable due to the exploding size of the state space. It is important to truncate the state space effectively with quantified errors, so accurate solutions can be computed. It is also important to know if all major probabilistic peaks have been computed. Here we introduce the accurate CME (ACME) algorithm for obtaining direct solutions to dCMEs. With multifinite buffers for reducing the state space by $O(n!)$, exact steady-state and time-evolving network probability landscapes can be computed. We further describe a theoretical framework of aggregating microstates into a smaller number of macrostates by decomposing a network into independent aggregated birth and death processes and give an a priori method for rapidly determining steady-state truncation errors. The maximal sizes of the finite buffers for a given error tolerance can also be precomputed without costly trial solutions of dCMEs. We show exactly computed probability landscapes of three multiscale networks, namely, a 6-node toggle switch, 11-node phage-lambda epigenetic circuit, and 16-node MAPK cascade network, the latter two with no known solutions. We also show how probabilities of rare events can be computed from first-passage times, another class of unsolved problems challenging for simulation-based techniques due to large separations in time scales. Overall, the ACME method enables accurate and efficient solutions of the dCME for a large class of networks.},

doi = {10.1137/15M1034180},

journal = {Multiscale Modeling & Simulation},

number = 2,

volume = 14,

place = {United States},

year = {Wed Jun 29 00:00:00 EDT 2016},

month = {Wed Jun 29 00:00:00 EDT 2016}

}

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