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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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Ligand binding.A. Gene regulation, with two transcription factors (TFs), L1 and L2, binding to a promoter. A labelled, directed graph can be constructed as described in the text, with the microstates being denoted here by the bitstrings . In this example there are no changes of conformational state in the scaffold, as would be the case if there was DNA looping or displacement of nucleosomes. The rate of mRNA transcription by RNA polymerase (RNAP) is assumed to depend on the pattern of TF binding as shown and the overall rate is calculated as an average over the probabilities of finding the promoter in each of the patterns. Assuming the system is at an equilibrium, x, these probabilities are given by the ratios, , which can be calculated using the second elimination formula in (4). B. An allosteric homodimeric protein is shown in two conformational states, relaxed (R) and tense (T). The labelled, directed graph has both conformational state changes in the scaffold as well as ligand binding and unbinding. The microstates are denoted here . Labels have been omitted for clarity. For allosteric enzymes, the overall rate of product formation is usually assumed to be proportional to the fraction of sites that are bound by ligand (fractional saturation), as shown. This can be calculated as described in Supporting Information S1.
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pone-0036321-g003: Ligand binding.A. Gene regulation, with two transcription factors (TFs), L1 and L2, binding to a promoter. A labelled, directed graph can be constructed as described in the text, with the microstates being denoted here by the bitstrings . In this example there are no changes of conformational state in the scaffold, as would be the case if there was DNA looping or displacement of nucleosomes. The rate of mRNA transcription by RNA polymerase (RNAP) is assumed to depend on the pattern of TF binding as shown and the overall rate is calculated as an average over the probabilities of finding the promoter in each of the patterns. Assuming the system is at an equilibrium, x, these probabilities are given by the ratios, , which can be calculated using the second elimination formula in (4). B. An allosteric homodimeric protein is shown in two conformational states, relaxed (R) and tense (T). The labelled, directed graph has both conformational state changes in the scaffold as well as ligand binding and unbinding. The microstates are denoted here . Labels have been omitted for clarity. For allosteric enzymes, the overall rate of product formation is usually assumed to be proportional to the fraction of sites that are bound by ligand (fractional saturation), as shown. This can be calculated as described in Supporting Information S1.

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.


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

Gunawardena J - PLoS ONE (2012)

Ligand binding.A. Gene regulation, with two transcription factors (TFs), L1 and L2, binding to a promoter. A labelled, directed graph can be constructed as described in the text, with the microstates being denoted here by the bitstrings . In this example there are no changes of conformational state in the scaffold, as would be the case if there was DNA looping or displacement of nucleosomes. The rate of mRNA transcription by RNA polymerase (RNAP) is assumed to depend on the pattern of TF binding as shown and the overall rate is calculated as an average over the probabilities of finding the promoter in each of the patterns. Assuming the system is at an equilibrium, x, these probabilities are given by the ratios, , which can be calculated using the second elimination formula in (4). B. An allosteric homodimeric protein is shown in two conformational states, relaxed (R) and tense (T). The labelled, directed graph has both conformational state changes in the scaffold as well as ligand binding and unbinding. The microstates are denoted here . Labels have been omitted for clarity. For allosteric enzymes, the overall rate of product formation is usually assumed to be proportional to the fraction of sites that are bound by ligand (fractional saturation), as shown. This can be calculated as described in Supporting Information S1.
© Copyright Policy
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC3351455&req=5

pone-0036321-g003: Ligand binding.A. Gene regulation, with two transcription factors (TFs), L1 and L2, binding to a promoter. A labelled, directed graph can be constructed as described in the text, with the microstates being denoted here by the bitstrings . In this example there are no changes of conformational state in the scaffold, as would be the case if there was DNA looping or displacement of nucleosomes. The rate of mRNA transcription by RNA polymerase (RNAP) is assumed to depend on the pattern of TF binding as shown and the overall rate is calculated as an average over the probabilities of finding the promoter in each of the patterns. Assuming the system is at an equilibrium, x, these probabilities are given by the ratios, , which can be calculated using the second elimination formula in (4). B. An allosteric homodimeric protein is shown in two conformational states, relaxed (R) and tense (T). The labelled, directed graph has both conformational state changes in the scaffold as well as ligand binding and unbinding. The microstates are denoted here . Labels have been omitted for clarity. For allosteric enzymes, the overall rate of product formation is usually assumed to be proportional to the fraction of sites that are bound by ligand (fractional saturation), as shown. This can be calculated as described in Supporting Information S1.
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.

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