Journal of Mathematical Biology

, Volume 61, Issue 2, pp 231–251

Gene expression dynamics in randomly varying environments

Open Access

DOI: 10.1007/s00285-009-0298-z

Cite this article as:
Smiley, M.W. & Proulx, S.R. J. Math. Biol. (2010) 61: 231. doi:10.1007/s00285-009-0298-z


A simple model of gene regulation in response to stochastically changing environmental conditions is developed and analyzed. The model consists of a differential equation driven by a continuous time 2-state Markov process. The density function of the resulting process converges to a beta distribution. We show that the moments converge to their stationary values exponentially in time. Simulations of a two-stage process where protein production depends on mRNA concentrations are also presented demonstrating that protein concentration tracks the environment whenever the rate of protein turnover is larger than the rate of environmental change. Single-celled organisms are therefore expected to have relatively high mRNA and protein turnover rates for genes that respond to environmental fluctuations.


Gene expression Environmental stochasticity Stochastic process Stationary distribution 

Mathematics Subject Classification (2000)

92C37 92C42 60G10 
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Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Department of MathematicsIowa State UniversityAmesUSA
  2. 2.Ecology Evolution and Marine BiologyUniversity of California, Santa BarbaraSanta BarbaraUSA

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