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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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The linear framework.A. A labelled, directed graph, G, gives rise to a system of linear differential equations by treating each edge as a first-order chemical reaction under mass-action kinetics, with the label as rate constant. The corresponding matrix is the Laplacian of G. B. In a strongly connected graph (note the difference to the one in A), there are spanning trees rooted at each vertex, the roots being circled. The MTT gives an element of  according to the formula in the box, as explained in the text. C. In a general directed graph, G, two distinct vertices are in the same strongly connected component (SCC) if each can be reached from the other by a path of directed edges. The SCCs form a directed graph, , in which two SCCs are linked by a directed edge if some vertex of the first SCC has an edge to some vertex of the second SCC.  has no directed cycles, allowing initial and terminal SCCs to be identified.
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pone-0036321-g002: The linear framework.A. A labelled, directed graph, G, gives rise to a system of linear differential equations by treating each edge as a first-order chemical reaction under mass-action kinetics, with the label as rate constant. The corresponding matrix is the Laplacian of G. B. In a strongly connected graph (note the difference to the one in A), there are spanning trees rooted at each vertex, the roots being circled. The MTT gives an element of according to the formula in the box, as explained in the text. C. In a general directed graph, G, two distinct vertices are in the same strongly connected component (SCC) if each can be reached from the other by a path of directed edges. The SCCs form a directed graph, , in which two SCCs are linked by a directed edge if some vertex of the first SCC has an edge to some vertex of the second SCC. has no directed cycles, allowing initial and terminal SCCs to be identified.

Mentions: We start from a graph, G, consisting of vertices, , with labelled, directed edges, , and no self loops, (Figure 2A). The vertices represent components of a system, on which a dynamics is defined by treating each edge as if it were a chemical reaction under mass-action kinetics, with the label as rate constant. We can imagine that an amount (or, equivalently, a concentration) of each component is placed on the corresponding vertex and that these amounts are transported across the edges in the direction of the arrows at rates that are proportional to the amounts on the source vertices. The constant of proportionality is the label on the edge. For instance, if the amounts of the components are denoted , then the edge in Figure 2A contributes to the amount of component 1 at a rate , and so on. Since each edge has only one source vertex, the reactions are all first-order and the dynamics are therefore linear.


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

Gunawardena J - PLoS ONE (2012)

The linear framework.A. A labelled, directed graph, G, gives rise to a system of linear differential equations by treating each edge as a first-order chemical reaction under mass-action kinetics, with the label as rate constant. The corresponding matrix is the Laplacian of G. B. In a strongly connected graph (note the difference to the one in A), there are spanning trees rooted at each vertex, the roots being circled. The MTT gives an element of  according to the formula in the box, as explained in the text. C. In a general directed graph, G, two distinct vertices are in the same strongly connected component (SCC) if each can be reached from the other by a path of directed edges. The SCCs form a directed graph, , in which two SCCs are linked by a directed edge if some vertex of the first SCC has an edge to some vertex of the second SCC.  has no directed cycles, allowing initial and terminal SCCs to be identified.
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC3351455&req=5

pone-0036321-g002: The linear framework.A. A labelled, directed graph, G, gives rise to a system of linear differential equations by treating each edge as a first-order chemical reaction under mass-action kinetics, with the label as rate constant. The corresponding matrix is the Laplacian of G. B. In a strongly connected graph (note the difference to the one in A), there are spanning trees rooted at each vertex, the roots being circled. The MTT gives an element of according to the formula in the box, as explained in the text. C. In a general directed graph, G, two distinct vertices are in the same strongly connected component (SCC) if each can be reached from the other by a path of directed edges. The SCCs form a directed graph, , in which two SCCs are linked by a directed edge if some vertex of the first SCC has an edge to some vertex of the second SCC. has no directed cycles, allowing initial and terminal SCCs to be identified.
Mentions: We start from a graph, G, consisting of vertices, , with labelled, directed edges, , and no self loops, (Figure 2A). The vertices represent components of a system, on which a dynamics is defined by treating each edge as if it were a chemical reaction under mass-action kinetics, with the label as rate constant. We can imagine that an amount (or, equivalently, a concentration) of each component is placed on the corresponding vertex and that these amounts are transported across the edges in the direction of the arrows at rates that are proportional to the amounts on the source vertices. The constant of proportionality is the label on the edge. For instance, if the amounts of the components are denoted , then the edge in Figure 2A contributes to the amount of component 1 at a rate , and so on. Since each edge has only one source vertex, the reactions are all first-order and the dynamics are therefore linear.

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