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A linear framework for time-scale separation in nonlinear biochemical systems.

Gunawardena J - PLoS ONE (2012)

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: Department of Systems Biology, Harvard Medical School, Boston, Massachusetts, United States of America. jeremy@hms.harvard.edu

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.

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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.
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pone-0036321-g001: 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.

Mentions: 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.


A linear framework for time-scale separation in nonlinear biochemical systems.

Gunawardena J - PLoS ONE (2012)

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.
© Copyright Policy
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC3351455&req=5

pone-0036321-g001: 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.
Mentions: 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.

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: Department of Systems Biology, Harvard Medical School, Boston, Massachusetts, United States of America. jeremy@hms.harvard.edu

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.

Show MeSH
Related in: MedlinePlus