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Threshold-dominated regulation hides genetic variation in gene expression networks.

Gjuvsland AB, Plahte E, Omholt SW - BMC Syst Biol (2007)

Bottom Line: The effect becomes less prominent as steepnesses approach Michaelis-Menten conditions.If the parameter perturbation shifts the equilibrium value too far away from threshold, the gene product is no longer an effective regulator and robustness is lost.Our results suggest that threshold regulation is a generic phenomenon in feedback-regulated networks with sigmoidal response functions, at least when there is no positive feedback.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Animal Science and Aquaculture, Norwegian University of Life Sciences, 1432 As, Norway. arne.gjuvsland@cigene.no

ABSTRACT

Background: In dynamical models with feedback and sigmoidal response functions, some or all variables have thresholds around which they regulate themselves or other variables. A mathematical analysis has shown that when the dose-response functions approach binary or on/off responses, any variable with an equilibrium value close to one of its thresholds is very robust to parameter perturbations of a homeostatic state. We denote this threshold robustness. To check the empirical relevance of this phenomenon with response function steepnesses ranging from a near on/off response down to Michaelis-Menten conditions, we have performed a simulation study to investigate the degree of threshold robustness in models for a three-gene system with one downstream gene, using several logical input gates, but excluding models with positive feedback to avoid multistationarity. Varying parameter values representing functional genetic variation, we have analysed the coefficient of variation (CV) of the gene product concentrations in the stable state for the regulating genes in absolute terms and compared to the CV for the unregulating downstream gene. The sigmoidal or binary dose-response functions in these models can be considered as phenomenological models of the aggregated effects on protein or mRNA expression rates of all cellular reactions involved in gene expression.

Results: For all the models, threshold robustness increases with increasing response steepness. The CVs of the regulating genes are significantly smaller than for the unregulating gene, in particular for steep responses. The effect becomes less prominent as steepnesses approach Michaelis-Menten conditions. If the parameter perturbation shifts the equilibrium value too far away from threshold, the gene product is no longer an effective regulator and robustness is lost. Threshold robustness arises when a variable is an active regulator around its threshold, and this function is maintained by the feedback loop that the regulator necessarily takes part in and also is regulated by. In the present study all feedback loops are negative, and our results suggest that threshold robustness is maintained by negative feedback which necessarily exists in the homeostatic state.

Conclusion: Threshold robustness of a variable can be seen as its ability to maintain an active regulation around its threshold in a homeostatic state despite external perturbations. The feedback loop that the system necessarily possesses in this state, ensures that the robust variable is itself regulated and kept close to its threshold. Our results suggest that threshold regulation is a generic phenomenon in feedback-regulated networks with sigmoidal response functions, at least when there is no positive feedback. Threshold robustness in gene regulatory networks illustrates that hidden genetic variation can be explained by systemic properties of the genotype-phenotype map.

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The shaded areas are the robustness domains ΩSSP in the (μ1, μ2)-plane for (a) Models 1, 2, 11, (b) Models 3, 13, (c) Model 6, (d) Models 4, 5, 7–10, 12, 14. For parameter values in ΩSSP both  and  are singular variables and approach their threshold values in the step function limit. Then they exhibit threshold robustness for all parameter perturbations which leave the perturbed values inside ΩSSP.
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Figure 6: The shaded areas are the robustness domains ΩSSP in the (μ1, μ2)-plane for (a) Models 1, 2, 11, (b) Models 3, 13, (c) Model 6, (d) Models 4, 5, 7–10, 12, 14. For parameter values in ΩSSP both and are singular variables and approach their threshold values in the step function limit. Then they exhibit threshold robustness for all parameter perturbations which leave the perturbed values inside ΩSSP.

Mentions: Models 1, 2, and 11 have distinctly larger values of CVminmax than the other models. This behaviour can be explained as a consequence of the shape of the parameter space domain ΩSSP in (μ1, μ2)-space, where μj = αj/γj, j = 1, 2. This is the parameter domain for which both and attain threshold values in the step function limit, in other words, the domain in which there is threshold robustness in both variables. For Models 1, 2, and 11 ΩSSP is concave, while for all the other models it is either wedge-shaped or rectangular (Fig. 6). In the concave domains of Models 1, 2, and 11 there is no point giving threshold robustness in both variables for the full parameter perturbation range of 50% up and down (see the Methods section, which also contains the derivation of this result), contrary to all the other models where threshold robustness is obtained for all parameter values sufficiently far from the boundary. With this result in mind it is reasonable to expect a drastically reduced robustness for these three models compared to the rest.


Threshold-dominated regulation hides genetic variation in gene expression networks.

Gjuvsland AB, Plahte E, Omholt SW - BMC Syst Biol (2007)

The shaded areas are the robustness domains ΩSSP in the (μ1, μ2)-plane for (a) Models 1, 2, 11, (b) Models 3, 13, (c) Model 6, (d) Models 4, 5, 7–10, 12, 14. For parameter values in ΩSSP both  and  are singular variables and approach their threshold values in the step function limit. Then they exhibit threshold robustness for all parameter perturbations which leave the perturbed values inside ΩSSP.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC2238762&req=5

Figure 6: The shaded areas are the robustness domains ΩSSP in the (μ1, μ2)-plane for (a) Models 1, 2, 11, (b) Models 3, 13, (c) Model 6, (d) Models 4, 5, 7–10, 12, 14. For parameter values in ΩSSP both and are singular variables and approach their threshold values in the step function limit. Then they exhibit threshold robustness for all parameter perturbations which leave the perturbed values inside ΩSSP.
Mentions: Models 1, 2, and 11 have distinctly larger values of CVminmax than the other models. This behaviour can be explained as a consequence of the shape of the parameter space domain ΩSSP in (μ1, μ2)-space, where μj = αj/γj, j = 1, 2. This is the parameter domain for which both and attain threshold values in the step function limit, in other words, the domain in which there is threshold robustness in both variables. For Models 1, 2, and 11 ΩSSP is concave, while for all the other models it is either wedge-shaped or rectangular (Fig. 6). In the concave domains of Models 1, 2, and 11 there is no point giving threshold robustness in both variables for the full parameter perturbation range of 50% up and down (see the Methods section, which also contains the derivation of this result), contrary to all the other models where threshold robustness is obtained for all parameter values sufficiently far from the boundary. With this result in mind it is reasonable to expect a drastically reduced robustness for these three models compared to the rest.

Bottom Line: The effect becomes less prominent as steepnesses approach Michaelis-Menten conditions.If the parameter perturbation shifts the equilibrium value too far away from threshold, the gene product is no longer an effective regulator and robustness is lost.Our results suggest that threshold regulation is a generic phenomenon in feedback-regulated networks with sigmoidal response functions, at least when there is no positive feedback.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Animal Science and Aquaculture, Norwegian University of Life Sciences, 1432 As, Norway. arne.gjuvsland@cigene.no

ABSTRACT

Background: In dynamical models with feedback and sigmoidal response functions, some or all variables have thresholds around which they regulate themselves or other variables. A mathematical analysis has shown that when the dose-response functions approach binary or on/off responses, any variable with an equilibrium value close to one of its thresholds is very robust to parameter perturbations of a homeostatic state. We denote this threshold robustness. To check the empirical relevance of this phenomenon with response function steepnesses ranging from a near on/off response down to Michaelis-Menten conditions, we have performed a simulation study to investigate the degree of threshold robustness in models for a three-gene system with one downstream gene, using several logical input gates, but excluding models with positive feedback to avoid multistationarity. Varying parameter values representing functional genetic variation, we have analysed the coefficient of variation (CV) of the gene product concentrations in the stable state for the regulating genes in absolute terms and compared to the CV for the unregulating downstream gene. The sigmoidal or binary dose-response functions in these models can be considered as phenomenological models of the aggregated effects on protein or mRNA expression rates of all cellular reactions involved in gene expression.

Results: For all the models, threshold robustness increases with increasing response steepness. The CVs of the regulating genes are significantly smaller than for the unregulating gene, in particular for steep responses. The effect becomes less prominent as steepnesses approach Michaelis-Menten conditions. If the parameter perturbation shifts the equilibrium value too far away from threshold, the gene product is no longer an effective regulator and robustness is lost. Threshold robustness arises when a variable is an active regulator around its threshold, and this function is maintained by the feedback loop that the regulator necessarily takes part in and also is regulated by. In the present study all feedback loops are negative, and our results suggest that threshold robustness is maintained by negative feedback which necessarily exists in the homeostatic state.

Conclusion: Threshold robustness of a variable can be seen as its ability to maintain an active regulation around its threshold in a homeostatic state despite external perturbations. The feedback loop that the system necessarily possesses in this state, ensures that the robust variable is itself regulated and kept close to its threshold. Our results suggest that threshold regulation is a generic phenomenon in feedback-regulated networks with sigmoidal response functions, at least when there is no positive feedback. Threshold robustness in gene regulatory networks illustrates that hidden genetic variation can be explained by systemic properties of the genotype-phenotype map.

Show MeSH