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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 coefficient of variation as function of the Hill coefficient for the most robust parameter sets for gene 1 and 2 across all 14 models. (a)-(b): Minimum of CV1k and CV2k, respectively, over all 81 parameter sets. (c): CVminmax as function of the Hill coefficient for each of the 14 models. An explanation for the remarkably high values for Models 1, 2, and 11 for high Hill coefficient values is given in the text. (d) The ratio of minimum of CV2k to minimum of CV1k as function of the Hill coefficient for each of the 14 models.
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Figure 5: The coefficient of variation as function of the Hill coefficient for the most robust parameter sets for gene 1 and 2 across all 14 models. (a)-(b): Minimum of CV1k and CV2k, respectively, over all 81 parameter sets. (c): CVminmax as function of the Hill coefficient for each of the 14 models. An explanation for the remarkably high values for Models 1, 2, and 11 for high Hill coefficient values is given in the text. (d) The ratio of minimum of CV2k to minimum of CV1k as function of the Hill coefficient for each of the 14 models.

Mentions: For all 14 models a decrease in p reduces robustness of both singular variables, defined as the minima of CV1k and CV2k over all 81 parameter sets (Fig. 5). But even at p = 1 there is less variation in the equilibrium values than in the perturbed parameter values. The cases of highest robustness for each of the 14 models and all steepness values show large variation among the models (Fig. 5). But for all models, even for the least robust Models 1 and 2, robustness increases with increasing Hill exponent, and is always smaller than CVuni. In Fig. 5c the differences among the models are accentuated for high Hill exponents. For each model the number CVminmax plotted along the vertical axis is computed as follows: for each of the 81 parameter sets first compute = max{CV1k, CV2k}, then find CVminmax = min{}. For each model this procedure selects the parameter set for which both CV1 and CV2 are small, giving CVminmax as a measure of the highest robustness when both genes 1 and 2 are taken into account.


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

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

The coefficient of variation as function of the Hill coefficient for the most robust parameter sets for gene 1 and 2 across all 14 models. (a)-(b): Minimum of CV1k and CV2k, respectively, over all 81 parameter sets. (c): CVminmax as function of the Hill coefficient for each of the 14 models. An explanation for the remarkably high values for Models 1, 2, and 11 for high Hill coefficient values is given in the text. (d) The ratio of minimum of CV2k to minimum of CV1k as function of the Hill coefficient for each of the 14 models.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 5: The coefficient of variation as function of the Hill coefficient for the most robust parameter sets for gene 1 and 2 across all 14 models. (a)-(b): Minimum of CV1k and CV2k, respectively, over all 81 parameter sets. (c): CVminmax as function of the Hill coefficient for each of the 14 models. An explanation for the remarkably high values for Models 1, 2, and 11 for high Hill coefficient values is given in the text. (d) The ratio of minimum of CV2k to minimum of CV1k as function of the Hill coefficient for each of the 14 models.
Mentions: For all 14 models a decrease in p reduces robustness of both singular variables, defined as the minima of CV1k and CV2k over all 81 parameter sets (Fig. 5). But even at p = 1 there is less variation in the equilibrium values than in the perturbed parameter values. The cases of highest robustness for each of the 14 models and all steepness values show large variation among the models (Fig. 5). But for all models, even for the least robust Models 1 and 2, robustness increases with increasing Hill exponent, and is always smaller than CVuni. In Fig. 5c the differences among the models are accentuated for high Hill exponents. For each model the number CVminmax plotted along the vertical axis is computed as follows: for each of the 81 parameter sets first compute = max{CV1k, CV2k}, then find CVminmax = min{}. For each model this procedure selects the parameter set for which both CV1 and CV2 are small, giving CVminmax as a measure of the highest robustness when both genes 1 and 2 are taken into account.

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