Abstract
Many chemical reactions in biological cells occur at very low concentrations of constituent molecules. Thus, transcriptional gene-regulation is often controlled by poorly expressed transcription-factors, such as E.coli lac repressor with few tens of copies. Here we study the effects of inherent concentration fluctuations of substrate-molecules on the seminal Michaelis-Menten scheme of biochemical reactions. We present a universal correction to the Michaelis-Menten equation for the reaction-rates. The relevance and validity of this correction for enzymatic reactions and intracellular gene-regulation is demonstrated. Our analytical theory and simulation results confirm that the proposed variance-corrected Michaelis-Menten equation predicts the rate of reactions with remarkable accuracy even in the presence of large non-equilibrium concentration fluctuations. The major advantage of our approach is that it involves only the mean and variance of the substrate-molecule concentration. Our theory is therefore accessible to experiments and not specific to the exact source of the concentration fluctuations.
Highlights
The basic question of enzymology concerns the rate of a reaction, in which a substrate-molecule S first forms a complex SE with an enzyme, and upon catalysis turns into a product P
The rate of a reaction involving these molecules as a substrate follows the Michaelis-Menten equation (MME) (2) with the constant concentration ρ replaced by the mean 〈 ρ(t)〉 of a stochastic, time dependent concentration ρ(t)
We demonstrated that stochastic concentration fluctuations can lead to a significant correction to the Michaelis-Menten equation of reaction kinetics
Summary
The basic question of enzymology concerns the rate of a reaction, in which a substrate-molecule S first forms a complex SE with an enzyme, and upon catalysis turns into a product P. KM = (k−1 + k2)/k1 is the Michaelis-Menten constant, and the maximal rate is vmax = k2 × [E], where [E] is the enzyme concentration. The step in which the SE complex is turned into a product, is in general reversible This can lead to ‘blocking’ of the reaction pathway by www.nature.com/scientificreports/. The second major assumption in the derivation of the MME is that of a quasi-steady state, which says that the concentration of the complex SE in the reaction scheme (1) does not change considerably on the time scale of the product formation[6]. The validity of the quasi-steady state approximation has been discussed, for instance, by Rao and Arkin[7]
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