Structural conditions on complex networks for the Michaelis–Menten input–output response

Abstract

The Michaelis-Menten (MM) fundamental formula describes how the rate of enzyme catalysis depends on substrate concentration. The familiar hyperbolic relationship was derived by timescale separation for a network of three reactions. The same formula has subsequently been found to describe steady-state input-output responses in many biological contexts, including single-molecule enzyme kinetics, gene regulation, transcription, translation, and force generation. Previous attempts to explain its ubiquity have been limited to networks with regular structure or simplifying parametric assumptions. Here, we exploit the graph-based linear framework for timescale separation to derive general structural conditions under which the MM formula arises. The conditions require a partition of the graph into two parts, akin to a "coarse graining" into the original MM graph, and constraints on where and how the input variable occurs. Other features of the graph, including the numerical values of parameters, can remain arbitrary, thereby explaining the formula's ubiquity. For systems at thermodynamic equilibrium, we derive a necessary and sufficient condition. For systems away from thermodynamic equilibrium, especially those with irreversible reactions, distinct structural conditions arise and a general characterization remains open. Nevertheless, our results accommodate, in much greater generality, all examples known to us in the literature.

Emergent properties of organic matter decomposition by soil enzymes

“…Our study might be generalizable beyond soil systems. Applications of the Michaelis-Menten equation are ubiquitous [e.g., see Table 1 in Wong et al (2018)], and scaling up the associated processes is a common concern in heterogeneous systems. For example, microbe-driven element cycling in marine systems faces a similar problem of heterogeneity (e.g., Follows et al 2007;Ward et al 2014;Moradi et al 2018), and we speculate that an application of the ECA equation (not necessarily the exact same parameter meaning) could help address this challenge.…”

Bayesian inference and comparison of stochastic transcription elongation models

“…The pause-free mean velocities measured during transcription elongation follow Michaelis-Menten kinetics even though the reaction cycle is more complicated than that of a simple enzyme [61]. As such, the inability to resolve the timescale of the substrate binding step is unsurprising [62][63][64].…”

A Cofactor‐Substrate‐Based Supramolecular Fluorescent Probe for the Ultrafast Detection of Nitroreductase under Hypoxic Conditions

“…It is well‐known that the reduction of nitro group by NADH and NTR is a typically double‐substrate enzymatic reaction . While, the new reporter, Zn‐ MPB⊃L‐NO 2 , responded to the NTR concentration mostly unaffected by the NADH supply, and signal communication occurred in a pseudo‐intramolecular manner, simplifying the double‐substrate mechanism into a single‐substrate free collision mechanism. In the presence of various concentrations of L‐NO 2 or Zn‐ MPB⊃L‐NO 2 with a specific content of NTR (Figures S31 and S32), the kinetic parameters calculated by the Michaelis–Menten equation exhibited V max Zn‐ MPB⊃L‐NO2 / V max L‐NO2 and k cat Zn‐ MPB⊃L‐NO2 / k cat L‐NO2 values of about 100.…”