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Robust simplifications of multiscale biochemical networks.

Radulescu O, Gorban AN, Zinovyev A, Lilienbaum A - BMC Syst Biol (2008)

Bottom Line: In these situations reduction to a common level of complexity is needed.We propose a systematic treatment of model reduction of multiscale biochemical networks.For such linear models, we propose a reduction algorithm which is based on a generalized theory of the limiting step that we have developed in 1.

View Article: PubMed Central - HTML - PubMed

Affiliation: IRMAR (CNRS UMR 6025), Université de Rennes 1, Rennes, France. ovidiu.radulescu@univ-rennes1.fr

ABSTRACT

Background: Cellular processes such as metabolism, decision making in development and differentiation, signalling, etc., can be modeled as large networks of biochemical reactions. In order to understand the functioning of these systems, there is a strong need for general model reduction techniques allowing to simplify models without loosing their main properties. In systems biology we also need to compare models or to couple them as parts of larger models. In these situations reduction to a common level of complexity is needed.

Results: We propose a systematic treatment of model reduction of multiscale biochemical networks. First, we consider linear kinetic models, which appear as "pseudo-monomolecular" subsystems of multiscale nonlinear reaction networks. For such linear models, we propose a reduction algorithm which is based on a generalized theory of the limiting step that we have developed in 1. Second, for non-linear systems we develop an algorithm based on dominant solutions of quasi-stationarity equations. For oscillating systems, quasi-stationarity and averaging are combined to eliminate time scales much faster and much slower than the period of the oscillations. In all cases, we obtain robust simplifications and also identify the critical parameters of the model. The methods are demonstrated for simple examples and for a more complex model of NF-kappaB pathway.

Conclusion: Our approach allows critical parameter identification and produces hierarchies of models. Hierarchical modeling is important in "middle-out" approaches when there is need to zoom in and out several levels of complexity. Critical parameter identification is an important issue in systems biology with potential applications to biological control and therapeutics. Our approach also deals naturally with the presence of multiple time scales, which is a general property of systems biology models.

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Model comparison a) Trajectories of various species for the model M (39, 65, 90); quasi-stationary species have concentrations in the lower cluster. b) Production rates of mRNAIκB for two models having the same reactions and species, differing only by one kinetic law. c) Trajectories (signal applied at t = 20). Notice the different behavior of IkBa@csl in ℳ(14, 25, 28).
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Figure 7: Model comparison a) Trajectories of various species for the model M (39, 65, 90); quasi-stationary species have concentrations in the lower cluster. b) Production rates of mRNAIκB for two models having the same reactions and species, differing only by one kinetic law. c) Trajectories (signal applied at t = 20). Notice the different behavior of IkBa@csl in ℳ(14, 25, 28).

Mentions: The intermediate species can be divided into four functional modules: production of mRNAp50, production of mRNAp65, production of mRNAIκB, and min funnel production of the complex p50:p65@csl, see Fig 6. We found three categories of intermediates. There are 10 quasi-stationary species, 3 non-oscillating species and 7 buffered species (species in large excess whose concentrations are practically constant). The elimination of these is entirely justified and has no consequence on the oscillations. There are 5 non-quasistationary, oscillating species. Among these, 4 are low concentration species, representing the states of two promoters (Prop105:RNAP, PropIkBa:RNAP) free and singly occupied by transcription factors FTAx, FTAz, respectively. However, we can safely eliminate them because transcription initiation starts dominantly when both p50:p65@csl and FTAx (or FTAz) are on the promoter, therefore the non-quasistationary promoter states are not important. The last non-quasistationary, oscillating species is p50 who binds to p65 (another slow, but non-oscillating species) to produce p50:p65@csl via the min funnel. Concentrations of all quasi-stationary intermediates are small (see Fig. 7a)), (< 10-4 μM corresponding to less than 30 molecules per cell). The reduction that we propose is fully justified for a deterministic model, but one may ask if deterministic differential equations apply in this case. We have shown elsewhere [33] that deterministic approximation can be applied in two different situations. The first, well known situation is when the numbers of molecules are large; the law of large numbers applies. The second, less known situation, is when some species are in small numbers, but when the reactions involving these species are frequent. An example is the quick binding-unbinding of a transcription factor on a promoter site. In this case, we can consider that various states of the promoter are at stochastic equilibrium (meaning they have reached a time invariant probability distribution). Under some conditions (the intermediate reactions should be pseudo-monomolecular), stochastic averaging [69] of the remaining equations (describing the promoter activity) with respect to the invariant distribution is equivalent to applying quasi-stationarity to the fast concentrations in the deterministic approach.


Robust simplifications of multiscale biochemical networks.

Radulescu O, Gorban AN, Zinovyev A, Lilienbaum A - BMC Syst Biol (2008)

Model comparison a) Trajectories of various species for the model M (39, 65, 90); quasi-stationary species have concentrations in the lower cluster. b) Production rates of mRNAIκB for two models having the same reactions and species, differing only by one kinetic law. c) Trajectories (signal applied at t = 20). Notice the different behavior of IkBa@csl in ℳ(14, 25, 28).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC2654786&req=5

Figure 7: Model comparison a) Trajectories of various species for the model M (39, 65, 90); quasi-stationary species have concentrations in the lower cluster. b) Production rates of mRNAIκB for two models having the same reactions and species, differing only by one kinetic law. c) Trajectories (signal applied at t = 20). Notice the different behavior of IkBa@csl in ℳ(14, 25, 28).
Mentions: The intermediate species can be divided into four functional modules: production of mRNAp50, production of mRNAp65, production of mRNAIκB, and min funnel production of the complex p50:p65@csl, see Fig 6. We found three categories of intermediates. There are 10 quasi-stationary species, 3 non-oscillating species and 7 buffered species (species in large excess whose concentrations are practically constant). The elimination of these is entirely justified and has no consequence on the oscillations. There are 5 non-quasistationary, oscillating species. Among these, 4 are low concentration species, representing the states of two promoters (Prop105:RNAP, PropIkBa:RNAP) free and singly occupied by transcription factors FTAx, FTAz, respectively. However, we can safely eliminate them because transcription initiation starts dominantly when both p50:p65@csl and FTAx (or FTAz) are on the promoter, therefore the non-quasistationary promoter states are not important. The last non-quasistationary, oscillating species is p50 who binds to p65 (another slow, but non-oscillating species) to produce p50:p65@csl via the min funnel. Concentrations of all quasi-stationary intermediates are small (see Fig. 7a)), (< 10-4 μM corresponding to less than 30 molecules per cell). The reduction that we propose is fully justified for a deterministic model, but one may ask if deterministic differential equations apply in this case. We have shown elsewhere [33] that deterministic approximation can be applied in two different situations. The first, well known situation is when the numbers of molecules are large; the law of large numbers applies. The second, less known situation, is when some species are in small numbers, but when the reactions involving these species are frequent. An example is the quick binding-unbinding of a transcription factor on a promoter site. In this case, we can consider that various states of the promoter are at stochastic equilibrium (meaning they have reached a time invariant probability distribution). Under some conditions (the intermediate reactions should be pseudo-monomolecular), stochastic averaging [69] of the remaining equations (describing the promoter activity) with respect to the invariant distribution is equivalent to applying quasi-stationarity to the fast concentrations in the deterministic approach.

Bottom Line: In these situations reduction to a common level of complexity is needed.We propose a systematic treatment of model reduction of multiscale biochemical networks.For such linear models, we propose a reduction algorithm which is based on a generalized theory of the limiting step that we have developed in 1.

View Article: PubMed Central - HTML - PubMed

Affiliation: IRMAR (CNRS UMR 6025), Université de Rennes 1, Rennes, France. ovidiu.radulescu@univ-rennes1.fr

ABSTRACT

Background: Cellular processes such as metabolism, decision making in development and differentiation, signalling, etc., can be modeled as large networks of biochemical reactions. In order to understand the functioning of these systems, there is a strong need for general model reduction techniques allowing to simplify models without loosing their main properties. In systems biology we also need to compare models or to couple them as parts of larger models. In these situations reduction to a common level of complexity is needed.

Results: We propose a systematic treatment of model reduction of multiscale biochemical networks. First, we consider linear kinetic models, which appear as "pseudo-monomolecular" subsystems of multiscale nonlinear reaction networks. For such linear models, we propose a reduction algorithm which is based on a generalized theory of the limiting step that we have developed in 1. Second, for non-linear systems we develop an algorithm based on dominant solutions of quasi-stationarity equations. For oscillating systems, quasi-stationarity and averaging are combined to eliminate time scales much faster and much slower than the period of the oscillations. In all cases, we obtain robust simplifications and also identify the critical parameters of the model. The methods are demonstrated for simple examples and for a more complex model of NF-kappaB pathway.

Conclusion: Our approach allows critical parameter identification and produces hierarchies of models. Hierarchical modeling is important in "middle-out" approaches when there is need to zoom in and out several levels of complexity. Critical parameter identification is an important issue in systems biology with potential applications to biological control and therapeutics. Our approach also deals naturally with the presence of multiple time scales, which is a general property of systems biology models.

Show MeSH