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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 independence between turnover events. Here, we show, through a new method of analysis, that memory and non-hyperbolicity have a common source in an initial, and observably long, transient peculiar to stochastic reaction networks of multiple enzymes. Networks of single enzymes do not admit such transients. The transient yields, asymptotically, to a steady-state in which memory vanishes and hyperbolicity is recovered. We propose new statistical measures, defined in terms of turnover times, to distinguish between the transient and steady-states and apply these to experimental data from a landmark experiment that first observed molecular memory in a single enzyme with multiple binding sites. Our study shows that catalysis at the molecular level with more than one enzyme always contains a non-classical regime and provides insight on how the classical limit is attained. [ABSTRACT FROM AUTHOR]

We study the temporal fluctuations in catalytic rates for single enzyme reactions undergoing slow transitions between two active states. We use a first passage time distribution formalism to obtain the closed-form analytical expressions of the mean reaction time and the randomness parameter for reaction schemes where conformational fluctuations are present between two free enzyme conformers. Our studies confirm that the sole presence of free enzyme fluctuations yields a non Michaelis-Menten equation and can lead to dynamic cooperativity. The randomness parameter, which is a measure of the dynamic disorder in the system, converges to unity at a high substrate concentration. If slow fluctuations are present between the enzyme-substrate conformers (off-pathway mechanism), dynamic disorder is present at a high substrate concentration. Our results confirm that the dynamic disorder at a high substrate concentration is determined only by the slowfluctuations between the enzyme-substrate conformers and the randomness parameter is greater than unity. Slow conformational fluctuations between free enzymes are responsible for the emergence of dynamic cooperativity in single enzymes. Our theoretical findings are well supported by comparison with experimental data on the single enzyme beta-galactosidase. [ABSTRACT FROM AUTHOR]

To understand the behaviour of complex systems, it is often necessary to use models that describe the dynamics of subnetworks. It has previously been established using projection methods that such subnetwork dynamics generically involves memory of the past and that the memory functions can be calculated explicitly for biochemical reaction networks made up of unary and binary reactions. However, many established network models involve also Michaelis-Menten kinetics, to describe, e.g., enzymatic reactions. We show that the projection approach to subnetwork dynamics can be extended to such networks, thus significantly broadening its range of applicability. To derive the extension, we construct a larger network that represents enzymes and enzyme complexes explicitly, obtain the projected equations, and finally take the limit of fast enzyme reactions that gives back Michaelis-Menten kinetics. The crucial point is that this limit can be taken in closed form. The outcome is a simple procedure that allows one to obtain a description of subnetwork dynamics, including memory functions, starting directly from any given network of unary, binary, and Michaelis-Menten reactions. Numerical tests show that this closed form enzyme elimination gives a much more accurate description of the subnetwork dynamics than the simpler method that represents enzymes explicitly and is also more efficient computationally. [ABSTRACT FROM AUTHOR]

This work aims at the development of a mathematical and computational approach that enables quantification of the inherent sources of stochasticity and of the corresponding sensitivities in stochastic simulations of chemical reaction networks. The approach is based on reformulating the system dynamics as being generated by independent standardized Poisson processes. This reformulation affords a straightforward identification of individual realizations for the stochastic dynamics of each reaction channel, and consequently a quantitative characterization of the inherent sources of stochasticity in the system. By relying on the Sobol-Hoeffding decomposition, the reformulation enables us to perform an orthogonal decomposition of the solution variance. Thus, by judiciously exploiting the inherent stochasticity of the system, one is able to quantify the variance-based sensitivities associated with individual reaction channels, as well as the importance of channel interactions. Implementation of the algorithms is illustrated in light of simulations of simplified systems, including the birth-death, Schlögl, and Michaelis-Menten models. [ABSTRACT FROM AUTHOR]

Using the (slow-scale) linear noise approximation, we give parameter-independent bounds to the substrate and product intrinsic noise variance for the stochastic Michaelis-Menten approximation at steady state. [ABSTRACT FROM AUTHOR]

Recent studies in single-molecule enzyme kinetics reveal that the turnover statistics of a single enzyme is governed by the waiting time distribution that decays as mono-exponential at low substrate concentration and multi-exponential at high substrate concentration. The multi-exponentiality arises due to protein conformational fluctuations, which act on the time scale longer than or comparable to the catalytic reaction step, thereby inducing temporal fluctuations in the catalytic rate resulting in dynamic disorder. In this work, we study the turnover statistics of a single enzyme in the presence of inhibitors to show that the multi-exponentiality in the waiting time distribution can arise even when protein conformational fluctuations do not influence the catalytic rate. From the Michaelis-Menten mechanism of inhibited enzymes, we derive exact expressions for the waiting time distribution for competitive, uncompetitive, and mixed inhibitions to quantitatively show that the presence of inhibitors can induce dynamic disorder in all three modes of inhibitions resulting in temporal fluctuations in the reaction rate. In the presence of inhibitors, dynamic disorder arises due to transitions between active and inhibited states of enzymes, which occur on time scale longer than or comparable to the catalytic step. In this limit, the randomness parameter (dimensionless variance) is greater than unity indicating the presence of dynamic disorder in all three modes of inhibitions. In the opposite limit, when the time scale of the catalytic step is longer than the time scale of transitions between active and inhibited enzymatic states, the randomness parameter is unity, implying no dynamic disorder in the reaction pathway. [ABSTRACT FROM AUTHOR]