A linear framework for time-scale separation in nonlinear biochemical systems.
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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.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.
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Affiliation: Department of Systems Biology, Harvard Medical School, Boston, Massachusetts, United States of America. jeremy@hms.harvard.edu
ABSTRACT
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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. Related in: MedlinePlus |
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Mentions: As discussed above, different mod-forms may elicit different downstream responses. The expression in (12) can be thought of as the steady-state probability of finding the substrate in mod-form Si. If Si elicits a quantitative level of effect given by , then the overall response of the PTM substrate can be estimated as an average over this probability distribution,(13)It is assumed here, as part of the time-scale separation depicted in Figure 1, that the downstream response is operating sufficiently slowly to average over the different mod-forms. This is similar to the calculation of the rate of gene expression as a function of transcription factor concentrations (Figure 3A). In the case of PTM, formulas like (13) are functions of free enzyme concentrations, which are set by the enzyme mechanisms and cannot be readily estimated or approximated. However, since enzymes are assumed to be neither synthesised nor degraded (see below for how this restriction can be addressed), there is a conservation law for each enzyme. These may be explicitly written down in terms of the symbolic rate constants,(14)to give p nonlinear algebraic equations for the p unknown free-enzyme concentrations. The essential nonlinearity in the biochemistry makes its appearance in these equations. For instance, for given total amounts of enzymes and substrate, there may be multiple solutions to (14), giving rise to multistability, [46], [57]. Equations (12) and (14) together provide a complete mathematical description of the substrate’s behaviour at steady state. |
View Article: PubMed Central - PubMed
Affiliation: Department of Systems Biology, Harvard Medical School, Boston, Massachusetts, United States of America. jeremy@hms.harvard.edu