# On a theory of stability for nonlinear stochastic chemical reaction networks

## Abstract

We present elements of a stability theory for small, stochastic, nonlinear chemical reaction networks. Steady state probability distributions are computed with zero-information (ZI) closure, a closure algorithm that solves chemical master equations of small arbitrary nonlinear reactions. Stochastic models can be linearized around the steady state with ZI-closure, and the eigenvalues of the Jacobian matrix can be readily computed. Eigenvalues govern the relaxation of fluctuation autocorrelation functions at steady state. Autocorrelation functions reveal the time scales of phenomena underlying the dynamics of nonlinear reaction networks. In accord with the fluctuation-dissipation theorem, these functions are found to be congruent to response functions to small perturbations. Significant differences are observed in the stability of nonlinear reacting systems between deterministic and stochastic modeling formalisms.

- Authors:

- Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave. SE, Minneapolis, Minnesota 55455 (United States)

- Publication Date:

- OSTI Identifier:
- 22415757

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Chemical Physics

- Additional Journal Information:
- Journal Volume: 142; Journal Issue: 18; Other Information: (c) 2015 Author(s); Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9606

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CHEMICAL REACTIONS; COMPUTERIZED SIMULATION; CORRELATION FUNCTIONS; DISTURBANCES; EIGENVALUES; EQUATIONS; FLUCTUATIONS; MATRICES; NONLINEAR PROBLEMS; PROBABILITY; RELAXATION; RESPONSE FUNCTIONS; STEADY-STATE CONDITIONS; STOCHASTIC PROCESSES

### Citation Formats

```
Smadbeck, Patrick, and Kaznessis, Yiannis N.
```*On a theory of stability for nonlinear stochastic chemical reaction networks*. United States: N. p., 2015.
Web. doi:10.1063/1.4919834.

```
Smadbeck, Patrick, & Kaznessis, Yiannis N.
```*On a theory of stability for nonlinear stochastic chemical reaction networks*. United States. doi:10.1063/1.4919834.

```
Smadbeck, Patrick, and Kaznessis, Yiannis N. Thu .
"On a theory of stability for nonlinear stochastic chemical reaction networks". United States. doi:10.1063/1.4919834.
```

```
@article{osti_22415757,
```

title = {On a theory of stability for nonlinear stochastic chemical reaction networks},

author = {Smadbeck, Patrick and Kaznessis, Yiannis N.},

abstractNote = {We present elements of a stability theory for small, stochastic, nonlinear chemical reaction networks. Steady state probability distributions are computed with zero-information (ZI) closure, a closure algorithm that solves chemical master equations of small arbitrary nonlinear reactions. Stochastic models can be linearized around the steady state with ZI-closure, and the eigenvalues of the Jacobian matrix can be readily computed. Eigenvalues govern the relaxation of fluctuation autocorrelation functions at steady state. Autocorrelation functions reveal the time scales of phenomena underlying the dynamics of nonlinear reaction networks. In accord with the fluctuation-dissipation theorem, these functions are found to be congruent to response functions to small perturbations. Significant differences are observed in the stability of nonlinear reacting systems between deterministic and stochastic modeling formalisms.},

doi = {10.1063/1.4919834},

journal = {Journal of Chemical Physics},

issn = {0021-9606},

number = 18,

volume = 142,

place = {United States},

year = {2015},

month = {5}

}