Limits...
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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Log-log sensitivity of the damping time and of the period of the oscillations with respect to variations of different parameters of the modelℳ(5, 8, 15). The parameters are multiplied by a scale s ∈ (1/50, 50). The log(timescales) are represented as functions of log(s). Period and damping time are not represented on intervals of parameter values where oscillations are over-damped (the ratio of the damping time to the period is smaller than 1.75). Damping time is infinite and not represented for intervals of parameter values where oscillations are self-sustained. The latter intervals are limited by Hopf bifurcations where the damping time diverges.
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Figure 4: Log-log sensitivity of the damping time and of the period of the oscillations with respect to variations of different parameters of the modelℳ(5, 8, 15). The parameters are multiplied by a scale s ∈ (1/50, 50). The log(timescales) are represented as functions of log(s). Period and damping time are not represented on intervals of parameter values where oscillations are over-damped (the ratio of the damping time to the period is smaller than 1.75). Damping time is infinite and not represented for intervals of parameter values where oscillations are self-sustained. The latter intervals are limited by Hopf bifurcations where the damping time diverges.

Mentions: As an example, we detect critical monomials in the simplest reduced model ℳ(5, 8, 15), first with respect to damping time and then with respect to the period of the oscillations. Deciding rigorously what large sensitivity means is not easy. In [34] we proposed a criterion which applies to properties that are homogeneous of degree ±1 in the kinetic constants, in particular, to characteristic times. Let τ be the studied quantity and k the parameter (monomial). We say that k is critical if , where A > 0 is some fixed constant and k0 some central value of the parameter. The sensitivity study is presented in Fig. 4. The relation between parameters of the initial and the reduced models is represented in Fig. 5. Damping time of the oscillations is most sensitive to parameters k14p1, k18, k20p, k21p1, k22, k26, C0. By changing these parameters, the oscillations can be modified from damped to self-sustained. The above parameters are the critical monomials from which we get the critical parameters (with respect to damping time) of the unreduced model: k23, k18, k16, k20, k17, k3, k9, k4, k22, k26, C0. The degrees of the critical monomials represent logarithmic sensitivities, therefore they provide both sign an strength of the influence of the critical parameters on the studied property. For instance, from k21p1 = k3k9(k4)-1 we can say that damping time can be increased (produce sustained oscillations) by reducing k3, or by reducing k9, or by increasing k4), see also Fig. 3.


Robust simplifications of multiscale biochemical networks.

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

Log-log sensitivity of the damping time and of the period of the oscillations with respect to variations of different parameters of the modelℳ(5, 8, 15). The parameters are multiplied by a scale s ∈ (1/50, 50). The log(timescales) are represented as functions of log(s). Period and damping time are not represented on intervals of parameter values where oscillations are over-damped (the ratio of the damping time to the period is smaller than 1.75). Damping time is infinite and not represented for intervals of parameter values where oscillations are self-sustained. The latter intervals are limited by Hopf bifurcations where the damping time diverges.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 4: Log-log sensitivity of the damping time and of the period of the oscillations with respect to variations of different parameters of the modelℳ(5, 8, 15). The parameters are multiplied by a scale s ∈ (1/50, 50). The log(timescales) are represented as functions of log(s). Period and damping time are not represented on intervals of parameter values where oscillations are over-damped (the ratio of the damping time to the period is smaller than 1.75). Damping time is infinite and not represented for intervals of parameter values where oscillations are self-sustained. The latter intervals are limited by Hopf bifurcations where the damping time diverges.
Mentions: As an example, we detect critical monomials in the simplest reduced model ℳ(5, 8, 15), first with respect to damping time and then with respect to the period of the oscillations. Deciding rigorously what large sensitivity means is not easy. In [34] we proposed a criterion which applies to properties that are homogeneous of degree ±1 in the kinetic constants, in particular, to characteristic times. Let τ be the studied quantity and k the parameter (monomial). We say that k is critical if , where A > 0 is some fixed constant and k0 some central value of the parameter. The sensitivity study is presented in Fig. 4. The relation between parameters of the initial and the reduced models is represented in Fig. 5. Damping time of the oscillations is most sensitive to parameters k14p1, k18, k20p, k21p1, k22, k26, C0. By changing these parameters, the oscillations can be modified from damped to self-sustained. The above parameters are the critical monomials from which we get the critical parameters (with respect to damping time) of the unreduced model: k23, k18, k16, k20, k17, k3, k9, k4, k22, k26, C0. The degrees of the critical monomials represent logarithmic sensitivities, therefore they provide both sign an strength of the influence of the critical parameters on the studied property. For instance, from k21p1 = k3k9(k4)-1 we can say that damping time can be increased (produce sustained oscillations) by reducing k3, or by reducing k9, or by increasing k4), see also Fig. 3.

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