## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 19 (2014), paper no. 59, 53 pp.

### Sensitivity analysis for stochastic chemical reaction networks with multiple time-scales

Ankit Gupta and Mustafa Khammash

#### Abstract

Stochastic models for chemical reaction networks have become very popular in recent years. For such models, the estimation of parameter sensitivities is an important and challenging problem. Sensitivity values help in analyzing the network, understanding its robustness properties and also in identifying the key reactions for a given outcome. Most of the methods that exist in the literature for the estimation of parameter sensitivities, rely on Monte Carlo simulations using Gillespie's stochastic simulation algorithm or its variants. It is well-known that such simulation methods can be prohibitively expensive when the network contains reactions firing at different time-scales, which is a feature of many important biochemical networks. For such networks, it is often possible to exploit the time-scale separation and approximately capture the original dynamics by simulating a "reduced" model, which is obtained by eliminating the fast reactions in a certain way. The aim of this paper is to tie these model reduction techniques with sensitivity analysis. We prove that under some conditions, the sensitivity values for the reduced model can be used to approximately recover the sensitivity values for the original model. Through an example we illustrate how our result can help in sharply reducing the computational costs for the estimation of parameter sensitivities for reaction networks with multiple time-scales. To prove our result, we use coupling arguments based on the random time change representation of Kurtz. We also exploit certain connections between the distributions of the occupation times of Markov chains and multi-dimensional wave equations.

#### Article information

**Source**

Electron. J. Probab., Volume 19 (2014), paper no. 59, 53 pp.

**Dates**

Accepted: 5 July 2014

First available in Project Euclid: 4 June 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1465065701

**Digital Object Identifier**

doi:10.1214/EJP.v19-3246

**Mathematical Reviews number (MathSciNet)**

MR3238779

**Zentralblatt MATH identifier**

1327.60137

**Subjects**

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Secondary: 60J22: Computational methods in Markov chains [See also 65C40] 60J27: Continuous-time Markov processes on discrete state spaces 60H35: Computational methods for stochastic equations [See also 65C30] 65C05: Monte Carlo methods

**Keywords**

parameter sensitivity chemical reaction network time-scale separation multiscale network reduced models random time change coupling

**Rights**

This work is licensed under a Creative Commons Attribution 3.0 License.

#### Citation

Gupta, Ankit; Khammash, Mustafa. Sensitivity analysis for stochastic chemical reaction networks with multiple time-scales. Electron. J. Probab. 19 (2014), paper no. 59, 53 pp. doi:10.1214/EJP.v19-3246. https://projecteuclid.org/euclid.ejp/1465065701