### Abstract

Microscopic randomness and the small volumes of living cells combine to generate random fluctuations in molecule concentrations called “noise”. Here I investigate the effect of noise on biochemical reactions obeying Michaelis–Menten kinetics, concluding that substrate noise causes these reactions to slow. I derive a general expression for the time evolution of the joint probability density of chemical species in arbitrarily connected networks of non-linear chemical reactions in small volumes. This equation is a generalization of the chemical master equation (CME), a common tool for investigating stochastic chemical kinetics, extended to reaction networks occurring in small volumes, such as living cells. I apply this equation to a generalized Michaelis-Menten reaction in an open system, deriving the following general result: 〈p〉≤p¯ and 〈s〉≥s¯, where s¯ and p¯ denote the deterministic steady-state concentration of reactant and product species, respectively, and 〈s〉 and 〈p〉 denote the steady-state ensemble average over independent realizations of a stochastic reaction. Under biologically realistic conditions, namely when substrate is degraded or diluted by cell division, 〈p〉≤p¯. Consequently, noise slows the rate of in vivo Michaelis–Menten reactions. These predictions are validated by extensive stochastic simulations using Gillespie's exact stochastic simulation algorithm. I specify the conditions under which these effects occur and when they vanish, therefore reconciling discrepancies among previous theoretical investigations of stochastic biochemical reactions. Stochastic slowdown of reaction flux caused by molecular noise in living cells may have functional consequences, which the present theory may be used to quantify.

Original language | English (US) |
---|---|

Pages (from-to) | 21-31 |

Number of pages | 11 |

Journal | Journal of Theoretical Biology |

Volume | 430 |

DOIs | |

State | Published - Oct 7 2017 |

### Fingerprint

### Keywords

- Diffusion equation
- Gene expression
- Kinetics
- Non-linear
- Reaction networks
- Stochastic process

### ASJC Scopus subject areas

- Statistics and Probability
- Medicine(all)
- Modeling and Simulation
- Immunology and Microbiology(all)
- Biochemistry, Genetics and Molecular Biology(all)
- Agricultural and Biological Sciences(all)
- Applied Mathematics

### Cite this

**Noise slows the rate of Michaelis–Menten reactions.** / Van Dyken, James.

Research output: Contribution to journal › Article

*Journal of Theoretical Biology*, vol. 430, pp. 21-31. https://doi.org/10.1016/j.jtbi.2017.06.039

}

TY - JOUR

T1 - Noise slows the rate of Michaelis–Menten reactions

AU - Van Dyken, James

PY - 2017/10/7

Y1 - 2017/10/7

N2 - Microscopic randomness and the small volumes of living cells combine to generate random fluctuations in molecule concentrations called “noise”. Here I investigate the effect of noise on biochemical reactions obeying Michaelis–Menten kinetics, concluding that substrate noise causes these reactions to slow. I derive a general expression for the time evolution of the joint probability density of chemical species in arbitrarily connected networks of non-linear chemical reactions in small volumes. This equation is a generalization of the chemical master equation (CME), a common tool for investigating stochastic chemical kinetics, extended to reaction networks occurring in small volumes, such as living cells. I apply this equation to a generalized Michaelis-Menten reaction in an open system, deriving the following general result: 〈p〉≤p¯ and 〈s〉≥s¯, where s¯ and p¯ denote the deterministic steady-state concentration of reactant and product species, respectively, and 〈s〉 and 〈p〉 denote the steady-state ensemble average over independent realizations of a stochastic reaction. Under biologically realistic conditions, namely when substrate is degraded or diluted by cell division, 〈p〉≤p¯. Consequently, noise slows the rate of in vivo Michaelis–Menten reactions. These predictions are validated by extensive stochastic simulations using Gillespie's exact stochastic simulation algorithm. I specify the conditions under which these effects occur and when they vanish, therefore reconciling discrepancies among previous theoretical investigations of stochastic biochemical reactions. Stochastic slowdown of reaction flux caused by molecular noise in living cells may have functional consequences, which the present theory may be used to quantify.

AB - Microscopic randomness and the small volumes of living cells combine to generate random fluctuations in molecule concentrations called “noise”. Here I investigate the effect of noise on biochemical reactions obeying Michaelis–Menten kinetics, concluding that substrate noise causes these reactions to slow. I derive a general expression for the time evolution of the joint probability density of chemical species in arbitrarily connected networks of non-linear chemical reactions in small volumes. This equation is a generalization of the chemical master equation (CME), a common tool for investigating stochastic chemical kinetics, extended to reaction networks occurring in small volumes, such as living cells. I apply this equation to a generalized Michaelis-Menten reaction in an open system, deriving the following general result: 〈p〉≤p¯ and 〈s〉≥s¯, where s¯ and p¯ denote the deterministic steady-state concentration of reactant and product species, respectively, and 〈s〉 and 〈p〉 denote the steady-state ensemble average over independent realizations of a stochastic reaction. Under biologically realistic conditions, namely when substrate is degraded or diluted by cell division, 〈p〉≤p¯. Consequently, noise slows the rate of in vivo Michaelis–Menten reactions. These predictions are validated by extensive stochastic simulations using Gillespie's exact stochastic simulation algorithm. I specify the conditions under which these effects occur and when they vanish, therefore reconciling discrepancies among previous theoretical investigations of stochastic biochemical reactions. Stochastic slowdown of reaction flux caused by molecular noise in living cells may have functional consequences, which the present theory may be used to quantify.

KW - Diffusion equation

KW - Gene expression

KW - Kinetics

KW - Non-linear

KW - Reaction networks

KW - Stochastic process

UR - http://www.scopus.com/inward/record.url?scp=85022206160&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85022206160&partnerID=8YFLogxK

U2 - 10.1016/j.jtbi.2017.06.039

DO - 10.1016/j.jtbi.2017.06.039

M3 - Article

C2 - 28676416

AN - SCOPUS:85022206160

VL - 430

SP - 21

EP - 31

JO - Journal of Theoretical Biology

JF - Journal of Theoretical Biology

SN - 0022-5193

ER -