With the advent of high-resolution methods of single-molecule spectroscopy, it is now possible to directly observe and manipulate the behavior of individual enzymes in the course of a chemical reaction. Chemistry at the single-molecule level is, however, inherently stochastic and, at times, extremely unintuitive. In this paper, we explain why, and under what circumstances, an increase in the rate at which an enzyme unproductively departs from a bound substrate will—unexpectedly—lead to an acceleration in the rate of product formation. The far-reaching implications of this effect are discussed. Keywords: single enzyme, enzyme kinetics, renewal theory Abstract The Michaelis—Menten equation provides a hundred-year-old prediction by which any increase in the rate of substrate unbinding will decrease the rate of enzymatic turnover.

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Thus, transcriptional gene-regulation is often controlled by poorly expressed transcription-factors, such as E. 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. 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 reaction, commonly written as was first described and analysed by Henri 1 , 2. The derivation of the MME relies on a series of assumptions. First, the step in which the SE complex is turned into a product, is in general reversible. However, if the products are immediately removed from the system by some other reaction, or their concentration is otherwise small, the backward reaction can be neglected. 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. Third, and perhaps most fundamentally, the whole derivation of the MME is based on deterministic ordinary differential equations, in which the total substrate and enzyme concentrations enter as parameters.

In practice, this means that the solution of constituent molecules must be well-mixed by fast diffusion to avoid local concentration differences. Moreover, there must be large numbers of constituent particles in a large reaction volume in order to be able to define concentrations and to neglect the fluctuations due to the inherently stochastic nature of the reaction and substrate import or synthesis.

The breakdown of the MME due to the finiteness of the reaction volume were predicted and discussed by Grima 8. Theoretical results for rates of single-enzyme reactions were reviewed in ref. In this article, we show that concentration fluctuations due to stochastic substrate production and degradation can drastically change the reaction-rate from the prediction of Eq.

Their formula for the reaction-rate is remarkable because it is a very simple function of the mean compartmental substrate concentration alone, even if the process effects a non-zero variance.

This is probably due to the Poissonian nature of the input process in their model. Effects of substrate concentration fluctuations on turnover times of single-enzyme molecules were discussed, for instance, in refs 16 , However, no explicit, universal correction to the Michaelis-Menten formula due to substrate fluctuations appears to exist in the literature.

In the next section, we derive mathematically exact upper and lower bounds for the reaction-rates of enzymatic reactions under such fluctuations and propose a first order correction to the Michaelis-Menten equation.

As we will demonstrate, this variance correction works exceptionally well. Results Variance-corrected Michaelis-Menten equation It is a quite common feature that the concentrations of molecules that are genetically expressed change in time in an abrupt manner: The concentration can be almost constant for a long period of time, although the cell growth causes some gradual change.

This behavior contrasts the slow concentration drifts and only small fluctuations around the mean when there is a large number of molecules present in a reaction volume such as a biological the cell. More precisely, out of an experiment with a large sample of cells, the number of reactions that have occurred up to time t for each cell and the rate computed from that information exhibits large cell-to-cell variations.

The inequality is clearly the sharpest that can be expressed as a function of the mean and variance of the substrate concentration alone. Any further refinement requires the knowledge of higher moments of the concentration distribution. As the MME 2 sets the optimal upper bound, given deterministic dynamics, we have thereby accurately bounded the effects of stochastic substrate fluctuations on the rate of the Michaelis-Menten reaction scheme 1 from both above and below.

The bounding of the Michaelis-Menten reaction-rate is our first important result. The natural candidate for a refined equation for the reaction-rate is the one given by a second order Taylor approximation for small fluctuations around the mean.

We call this the variance-corrected Michaelis-Menten equation VCMME ; It has the desired properties of reducing to the MME in the deterministic limit and falling between the optimal variance bounds for all values of the mean concentration, its variance and the constant KM. Figure 1A shows that the variance-corrected approximation is able to predict the rate of a reaction catalysed by a single enzyme in a cellular compartment with bursty input of substrate-molecules: The only difference to the usual reaction scheme 1 in this example is that the substrate-molecules enter the reaction volume in batches.

With growing batch size, the concentration fluctuations increase and the contribution of the variance correction in Eq.

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## Ever-fluctuating single enzyme molecules: Michaelis-Menten equation revisited

Several hundred single GUS molecules were separated in large arrays of 62 ultrasmall reaction chambers etched into the surface of a fused silica slide to observe their individual substrate turnover rates in parallel by fluorescence microscopy. Individual GUS molecules feature long-lived but divergent activity states, and their mean activity is consistent with classic Michaelis—Menten kinetics. The large number of single molecule substrate turnover rates is representative of the activity distribution within an entire enzyme population. Partially evolved GUS displays a much broader activity distribution among individual enzyme molecules than wild-type GUS. The broader activity distribution indicates a functional division of work between individual molecules in a population of partially evolved enzymes that—as so-called generalists—are characterized by their promiscuous activity with many different substrates. Introduction Jump To Analyzing the catalytic mechanisms of enzymes and their evolution is crucial to understanding the biochemical reactions of life. Exactly years ago, the landmark work of Michaelis and Menten provided a conceptual framework for analyzing enzyme kinetics in bulk solution.

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Thus, transcriptional gene-regulation is often controlled by poorly expressed transcription-factors, such as E. 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.

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## Role of substrate unbinding in Michaelis–Menten enzymatic reactions

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