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A Linear Framework for Time-Scale Separation in Nonlinear Biochemical Systems

A Linear Framework for Time-Scale Separation in Nonlinear Biochemical Systems

  • Jeremy Gunawardena
PLOS
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Corrections

3 Dec 2013: Gunawardena J (2013) Correction: A Linear Framework for Time-Scale Separation in Nonlinear Biochemical Systems. PLOS ONE 8(12): 10.1371/annotation/6830fba7-6e52-48f9-9f55-75ee37c75b5a. https://doi.org/10.1371/annotation/6830fba7-6e52-48f9-9f55-75ee37c75b5a View correction

24 Jun 2013: Gunawardena J (2013) Correction: A Linear Framework for Time-Scale Separation in Nonlinear Biochemical Systems. PLOS ONE 8(6): 10.1371/annotation/fa4c5f9f-4071-4b32-864f-b82c2e4e973b. https://doi.org/10.1371/annotation/fa4c5f9f-4071-4b32-864f-b82c2e4e973b View correction

Abstract

Cellular physiology is implemented by formidably complex biochemical systems with highly nonlinear dynamics, presenting a challenge for both experiment and theory. Time-scale separation has been one of the few theoretical methods for distilling general principles from such complexity. It has provided essential insights in areas such as enzyme kinetics, allosteric enzymes, G-protein coupled receptors, ion channels, gene regulation and post-translational modification. In each case, internal molecular complexity has been eliminated, leading to rational algebraic expressions among the remaining components. This has yielded familiar formulas such as those of Michaelis-Menten in enzyme kinetics, Monod-Wyman-Changeux in allostery and Ackers-Johnson-Shea in gene regulation. Here we show that these calculations are all instances of a single graph-theoretic framework. Despite the biochemical nonlinearity to which it is applied, this framework is entirely linear, yet requires no approximation. We show that elimination of internal complexity is feasible when the relevant graph is strongly connected. The framework provides a new methodology with the potential to subdue combinatorial explosion at the molecular level.

Introduction

The overwhelming molecular complexity of biological systems presents a formidable scientific challenge. The mere number of protein-coding genes barely captures this complexity, [1]. Transcription factor binding to DNA to regulate gene expression and protein post-translational modification, to mention just two well-studied mechanisms, enable combinatorial construction of vast numbers of molecular states, [2]. How such complexity evolves and how it gives rise to robust cellular physiology are among the central questions in biology.

One of the few conceptual methods for rising above this complexity, and thereby distilling general principles, has been time-scale separation (Figure 1). A system of interest (dashed box) is identified, which, for a particular behaviour being studied, is assumed to contain all the components relevant to that behaviour. A sub-system (box) within the larger system is taken to be operating sufficiently fast that it may be assumed to have reached a steady state or, as a special case of that, a state of thermodynamic equilibrium. The larger system and its environment adjust on slower time-scales to the steady-state of the sub-system. The components within the sub-system may be viewed as “fast variables”, while those additional components within the larger system are “slow variables”. Those components in the environment that might be influenced by the overall system are taken to be operating on the slowest time scale. Such assumptions often enable the internal states of the sub-system to be eliminated, thereby simplifying the description of the larger system’s behaviour.

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Figure 1. Schematic illustration of time-scale separation.

A system is shown within the dashed box, which is assumed to contain all the components relevant to a given behaviour, so that it is partially uncoupled from its environment: it influences its environment (arrows leading outwards) but is not in turn influenced by the environment. Within the system is a smaller sub-system (box) which may be fully coupled to the larger system (bi-directional arrows). The components in the sub-system (blue dots) are taken to be operating sufficiently fast that they may be assumed to have reached a steady state, or a state of thermodynamic equilibrium, to which the remaining components in the larger system (green dots), and those in the environment that are influenced by the system (magenta dots), adjust on slower time scales. The Michaelis-Menten formula in (2) is derived from a time-scale separation of this kind.

https://doi.org/10.1371/journal.pone.0036321.g001

Time-scale separation was first introduced at the molecular level in the famous work of Michaelis and Menten on enzyme kinetics, [3], [4]. They considered the following biochemical reaction scheme, in which an enzyme, E, reversibly binds to a substrate, S, to form an intermediate enzyme-substrate complex, ES, which then irreversibly breaks up to form the product of the reaction, P, and release the enzyme:(1)A time-scale separation was assumed in which the free enzyme, E, and the enzyme-substrate complex, ES, were regarded as fast variables, while S and P were regarded as slow variables. (As a matter of historical accuracy, Michaelis and Menten made a simpler rapid equilibrium assumption. The so-called “quasi steady-state” assumption used here, and now universally employed, was first introduced by Briggs and Haldane, [5].) A simple algebraic calculation leads to the Michaelis-Menten rate formula(2)in which the aggregated parameters and are determined by the underlying rate constants for the reactions in (1) and the total amount of enzyme that is present; see equation (9) below. Here, [X] denotes the concentration of the chemical species X.

This example has two characteristic features. First, the algebra has eliminated the fast variables, E and ES, leaving a formula that involves only the slow variable, S, on the right hand side. In this case, P does not appear on the right hand side because of the irreversibility of (1). Second, the expression on the right hand side is rational in the concentrations of the slow variables: it is a ratio of two polynomials in [S]. In this simple case, both numerator and denominator are first-order (linear) in [S].

Time-scale separations have been used in many distinct areas of biology, including enzyme kinetics, allosteric enzymes, G-protein coupled receptors, ligand-gated ion channels, gene regulation in both prokaryotes and eukaryotes and protein post-translational modification (see below). The characteristic features noted above, of elimination leading to a rational expression, are shared by all of them. Among these rational expressions are the familiar formulas of Michaelis-Menten in enzyme kinetics, Monod-Wyman-Changeux and Koshland-Némethy-Filmer in allostery and Shea-Ackers-Johnson in prokaryotic gene regulation. However, the corresponding analyses have been largely independent and ad hoc. In this paper we introduce a graph-theoretic framework that underlies all of these analyses. Despite the biochemical nonlinearity of the systems to which it is applied, the framework is entirely linear, yet it is not an approximation. Moreover, it is symbolic in the values of all rate constants, so that it may be used without