Quasi-Steady-State Approximations Derived from the Stochastic Model of Enzyme Kinetics

Abstract

In this paper we derive several quasi steady-state approximations (QSSAs) to the stochastic reaction network describing the Michaelis-Menten enzyme kinetics. We show how the different assumptions about chemical species abundance and reaction rates lead to the standard QSSA (sQSSA), the total QSSA (tQSSA), and the reverse QSSA (rQSSA) approximations. These three QSSAs have been widely studied in the literature in deterministic ordinary differential equation (ODE) settings and several sets of conditions for their validity have been proposed. By using multiscaling techniques introduced in [1, 2] we show that these conditions for deterministic QSSAs largely agree with the ones for QSSAs in the large volume limits of the underlying stochastic enzyme kinetic network.In chemistry and biology, we often come across chemical reaction networks where one or more of the species exhibit a different intrinsic time scale and tend to reach an equilibrium state quicker than others. Quasi steady state approximation (QSSA) is a commonly used tool to simplify the description of the dynamics of such systems. In particular, QSSA has been widely applied to the important class of reaction networks known as the Michaelis-Menten models of enzyme kinetics [3,4,5].Traditionally the enzyme kinetics has been studied using systems of ordinary differential equations (ODEs). The ODE approach allows one to analyze various aspects of the enzyme dynamics such as asymptotic stability. However, it ignores the fluctuations of the enzyme reaction network due to intrinsic noise and instead focuses on the averaged dynamics. If accounting for this intrinsic noise is required, the use of an alternative stochastic reaction network approach may be more appropriate, especially when some of the species have low copy numbers or when one is interested in predicting the molecular fluctuations of the system. It is well-known that such molecular fluctuations in the species with small numbers, and stochasticity in general, can lead to interesting dynamics. For instance, in a recent paper [6], Perez et al. gave an account of how intrinsic noise controls and alters the dynamics, and steady state of morphogen-controlled bistable genetic switches. Stochastic models have been strongly advocated by many in recent literature [7,8,9,10,11,12]. In this paper, we consider such stochastic models in the context of QSSA and the Michaelis-Menten enzyme kinetics and relate them to the deterministic ones that are well-known from the chemical physics literature.The QSSAs are very useful from a practical perspective. They not only reduce the model complexity, but also allow us to better relate it to experimental measurements by averaging out the unobservable or difficult-to-measure species. A substantial body of work has been published to justify such QSSA reductions in deterministic models, typically by means of perturbation theory [13,14,15,16,17,18]. In contrast to this approach, we derive here the QSSA reductions using stochastic multiscaling techniques [1,2]. Although our approac...

Total Enzyme-substrate Complex Which Includes Product-destined Enzyme-substrate Complex

“…The issue of substrate mass balance or conservation abounds in literature [12][13][14][15]24]. There was a need to introduce the concept of total substrate concentration.…”

“…This seems to precipitate researches into various concepts with which to validate kinetic parameters: In this regard a lot of research activities are devoted to the determination of kinetic parameters. The validity of such kinetic parameters are assessed on the basis of various quasi-steady-state assumptions [8][9][10][11][12] necessitating in some cases the concept of postcatalytic action total substrate concentration (POSTSC) [13][14][15]. This is in contrast to precatalytic action total substrate concentration (PRESC), at time t = 0.…”

Multiscale Stochastic Reaction–Diffusion Algorithms Combining Markov Chain Models with Stochastic Partial Differential Equations

“…2017; Kang et al. 2019). The idea of model reduction can also be used to develop computational methods which efficiently estimate quantities of interest from stochastic simulations (Cao et al.…”

Mathematical modeling for enzyme inhibitors with slow and fast subsystems

“…where x 2 R m and p ij ¼ b ij À a ij : There are some recently published works that give an important step forwards to understand the model reduction techniques and their applications in biochemical reaction networks and system biology. For example, some methods of model reduction for large-scale biological systems are given in the following references (Bartocci & Li o, 2016;Eilertsen & Schnell, 2020;Moayyedi, 2019;Kang, KhudaBukhsh, Koeppl, & Rempała, 2019;Kapteijn, Gascon, & Nijhuis, 2018;Khazaaleh, 2018;Shin & Nguyen, 2017;L opez Zazueta, Bernard, & Gouz e, 2019;Rahmanzadeh, Asadi, & Atashafrooz, 2020;Snowden, van der Graaf, & Tindall, 2017.…”