In a conformational nonequilibrium steady state (cNESS), enzyme turnover is modulated by the underlying conformational dynamics. Based on a discrete kinetic network model, we use the integrated probability flux balance method to derive the cNESS turnover rate for a conformation-modulated enzymatic reaction. The traditional Michaelis-Menten (MM) rate equation is extended to a generalized form, which includes non-MM corrections induced by conformational population currents within combined cyclic kinetic loops. When conformational detailed balance is satisfied, the turnover rate reduces to the MM functional form, explaining its validity for many enzymatic systems. For the first time, a one-to-one correspondence is established between non-MM terms and combined cyclic loops with unbalanced conformational currents. Cooperativity resulting from nonequilibrium conformational dynamics has been observed in enzymatic reactions, and we provide a novel, rigorous means of predicting and characterizing such behavior. Our generalized MM equation affords a systematic approach for exploring cNESS enzyme kinetics.
Enzyme kinetics is very often characterised by the irreversible Michaelis-Menten (MM) equation. However, in open chemical reaction networks such as metabolic pathways, this approach can lead to significant kinetic and thermodynamic inconsistencies. Based on recent developments in nonequilibrium chemical thermodynamics, we present a validity criterion solely expressed in terms of the equilibrium constant of the enzyme-catalysed reaction. When satisfied, it guarantees the ability of the irreversible MM equation to generate kinetic and thermodynamic data that are quantitatively reliable for reaso…
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-regul…
We present a model of an ATP-fueled molecular machine which push a polymer through a pore channel. The machine acts between two levels (working-waiting), and the working one remains active for a fixed time giving a constant force. The machine activation rate can be put in relationship with the available ATP concentration in the solution, which gives the necessary energy supply. The translocation time shows a monotonic behavior as a function of the activation frequency and the velocity follows a Michaelis-Menten law that arises naturally in this description. The estimation of the stall force of…
Recent fluorescence spectroscopy measurements of the turnover time distribution of single-enzyme turnover kinetics of $β$-galactosidase provide evidence of Michaelis-Menten kinetics at low substrate concentration. However, at high substrate concentrations, the dimensionless variance of the turnover time distribution shows systematic deviations from the Michaelis-Menten prediction. This difference is attributed to conformational fluctuations in both the enzyme and the enzyme-substrate complex and to the possibility of both parallel and off-pathway kinetics. Here, we use the chemical master equa…
The sequence of amino acid monomers in the primary structure of a protein is decided by the corresponding sequence of codons (triplets of nucleic acid monomers) on the template messenger RNA (mRNA). The polymerization of a protein, by incorporation of the successive amino acid monomers, is carried out by a molecular machine called ribosome. We develop a stochastic kinetic model that captures the possibilities of mis-reading of mRNA codon and prior mis-charging of a tRNA. By a combination of analytical and numerical methods we obtain the distribution of the times taken for incorporation of the …
Cytoskeletal motor proteins move on filamentous tracks by converting input chemical energy that they derive by catalyzing the hydrolysis of ATP. The ATPase site is the analog of an engine and hydrolysis of ATP is the analog of burning of chemical fuel. Moreover, the functional role of a segment of the motor is analogous to that of the transmission system of an automobile that consists of shaft, gear, clutch, etc. The operation of the engine is intrinsically "noisy" and the motor faces a molecular "hailstorm" in the aqueous medium. In this commemorative article, we celebrate the centenary of Mi…
In this paper we study noise-induced bistability in a specific circuit with many biological implications, namely a single-step enzymatic cycle described by Michaelis Menten equations with quasi-steady state assumption. We study the system both with a Master Equation formalism, and with the Fokker-Planck continuous approximation, characterizing the conditions in which the continuous approach is a good approximation of the exact discrete model. An analysis of the stationary distribution in both cases shows that bimodality can not occur in such a system. We discuss which additional requirements c…
We study the effect of restart, and retry, on the mean completion time of a generic process. The need to do so arises in various branches of the sciences and we show that it can naturally be addressed by taking advantage of the classical reaction scheme of Michaelis and Menten. Stopping a process in its midst, only to start it all over again, may prolong, leave unchanged, or even shorten the time taken for its completion. Here we are interested in the optimal restart problem, i.e., in finding a restart rate which brings the mean completion time of a process to a minimum. We derive the governin…
The driving force for active physical and biological systems is determined by both the underlying landscape and the non-equilibrium curl flux. While landscape can be quantified in the experiments by the histograms of the collecting trajectories of the observables, the experimental flux quantification is still challenging. In this work, we studied the single molecule enzyme dynamics and observed the deviation in kinetics from the conventional Michaelis-Menton reaction rate. We identified and quantified the non-equilibrium flux as the origin of such non-Michaelis-Menton enzyme rate behavior. Thi…
In this study, through phenomenological comparison of the velocity-force data of processive motor proteins, including conventional kinesin, cytoplasmic dynein and myosin V, we found that, the ratio between motor velocities of two different ATP concentrations is almost invariant for any substall, superstall or negative external loads. Therefore, the velocity of motor can be well approximated by a Michaelis-Menten like formula $V=\atp k(F)L/(\atp +K_M)$, with $L$ the step size, and $k(F)$ the external load $F$ dependent rate of one mechanochemical cycle of motor motion in saturated ATP solution.…
Recent fluorescence spectroscopy measurements of single-enzyme kinetics have shown that enzymatic turnovers form a renewal stochastic process in which the inverse of the mean waiting time between turnovers follows the Michaelis-Menten equation. Under typical physiological conditions, however, tens to thousands of enzymes react in catalyzing thousands to millions of substrates. We study enzyme kinetics at these physiologically relevant conditions through a master equation including stochasticity and molecular discreteness. From the exact solution of the master equation we find that the waiting …
Dynamic cooperativity in monomeric enzymes is characterized in terms of a non-Michaelis-Menten kinetic behaviour. The latter is believed to be associated with mechanisms that include multiple reaction pathways due to enzymatic conformational fluctuations. Recent advances in single-molecule fluorescence spectroscopy have provided new fundamental insights on the possible mechanisms underlying reactions catalyzed by fluctuating enzymes. Here, we present a bottom-up approach to understand enzyme turnover kinetics at physiologically relevant mesoscopic concentrations informed by mechanisms extracte…
It is well known in enzyme kinetics that the Michaelis-Menten (MM) equation is applicable only to enzymes in the steady state. We show that the result obtained in the previous work [Phys. Rev. Lett. 107, 218301 (2011)] is inconsistent with the MM equation, not because the authors considered the enzyme system at mesoscopic concentrations but because they considered the enzyme system in the non-stationary state. The substrate concentration dependence of the mean turnover time is, in fact, consistent with the MM equation in the steady state, regardless of the number of enzymes in the system.
The celebrated Michaelis-Menten (MM) expression provides a fundamental relation between the rate of enzyme catalysis and substrate concentration. The validity of this classical expression is, however, restricted to macroscopic amounts of enzymes and substrates and, thus, to processes with negligible fluctuations. Recent experiments have measured fluctuations in the catalytic rate to reveal that the MM equation, though valid for bulk amounts, is not obeyed at the molecular level. In this mini-review, we show how new statistical measures of fluctuations in the catalytic rate identify a regime in…
The hyperbolic dependence of catalytic rate on substrate concentration is a classical result in enzyme kinetics, quantified by the celebrated Michaelis-Menten equation. The ubiquity of this relation in diverse chemical and biological contexts has recently been rationalized by a graph-theoretic analysis of deterministic reaction networks. Experiments, however, have revealed that "molecular noise" - intrinsic stochasticity at the molecular scale - leads to significant deviations from classical results and to unexpected effects like "molecular memory", i.e., the breakdown of statistical independe…
A common kinetic framework for studies of whole-cell catalysis is vital for understanding and optimizing bioflow reactors. In this work, we demonstrate the applicability of a flow-adapted version of Michaelis-Menten kinetics to a catalytic bacterial biofilm. A three-electrode microfluidic electrochemical flow cell measured increased turnover rates by as much as 50% from a Geobacter sulfurreducens biofilm as flow rate was varied. Based on parameters from the applied kinetic framework, flow-induced increases to turnover rate, catalytic efficiency and device reaction capacity could be linked to a…
Proteolysis of the multimeric blood coagulation protein von Willebrand Factor (VWF) by ADAMTS13 is crucial for prevention of microvascular thrombosis. ADAMTS13 cleaves VWF within the mechanosensitive A2 domain, which is believed to open under shear flow. Here, we combine Fluorescence Correlation Spectroscopy (FCS) and a microfluidic shear cell to monitor real-time kinetics of full-length VWF proteolysis as a function of shear stress. For comparison, we also measure the Michaelis-Menten kinetics of ADAMTS13 cleavage of wild-type VWF in the absence of shear but partially denaturing conditions. U…
