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Safe uses of Hill's model: an exact comparison with the Adair-Klotz model.

Konkoli Z - Theor Biol Med Model (2011)

Bottom Line: There are very few studies investigating the situations in which the model can be safely used.A strongly cooperative Adair-Klotz model can be replaced by a suitable Hill model in such a way that any property computed from the two models, even the one describing stochastic features, is approximately the same.The quantitative analysis showed that boundaries of the regions in the parameter space where the models behave in the same way exhibit a rather rich structure.

View Article: PubMed Central - HTML - PubMed

Affiliation: Chalmers University of Technology, Department of Microtechnology and Nanoscience, Bionano Systems Laboratory, Sweden. zorank@chalmers.se

ABSTRACT

Background: The Hill function and the related Hill model are used frequently to study processes in the living cell. There are very few studies investigating the situations in which the model can be safely used. For example, it has been shown, at the mean field level, that the dose response curve obtained from a Hill model agrees well with the dose response curves obtained from a more complicated Adair-Klotz model, provided that the parameters of the Adair-Klotz model describe strongly cooperative binding. However, it has not been established whether such findings can be extended to other properties and non-mean field (stochastic) versions of the same, or other, models.

Results: In this work a rather generic quantitative framework for approaching such a problem is suggested. The main idea is to focus on comparing the particle number distribution functions for Hill's and Adair-Klotz's models instead of investigating a particular property (e.g. the dose response curve). The approach is valid for any model that can be mathematically related to the Hill model. The Adair-Klotz model is used to illustrate the technique. One main and two auxiliary similarity measures were introduced to compare the distributions in a quantitative way. Both time dependent and the equilibrium properties of the similarity measures were studied.

Conclusions: A strongly cooperative Adair-Klotz model can be replaced by a suitable Hill model in such a way that any property computed from the two models, even the one describing stochastic features, is approximately the same. The quantitative analysis showed that boundaries of the regions in the parameter space where the models behave in the same way exhibit a rather rich structure.

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Validity region of a K1 ≫ K2 ≫ K3 ≫ K4 parameterization. The plots depicts the boundary of the δmax(K1, K2, K3, K4) > 0.9 region in (K1, ξ) plane with the parameterization K2 = K1/ξ, K3 = K1/ξ2, and K4 = K1/ξ3. (K0 has been optimized as in the previous figures.)
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Figure 6: Validity region of a K1 ≫ K2 ≫ K3 ≫ K4 parameterization. The plots depicts the boundary of the δmax(K1, K2, K3, K4) > 0.9 region in (K1, ξ) plane with the parameterization K2 = K1/ξ, K3 = K1/ξ2, and K4 = K1/ξ3. (K0 has been optimized as in the previous figures.)

Mentions: Figure 6 is a contour plot that depicts how δmax depends on K1 and ξ for h = 4. The figure shows that many parameter choices that are chemically interesting do lead to a high value of the fundamental similarity measure (the grey region in the plot). Since there is no upper limit for ξ, for any value of K1, it is possible to choose ξ so that the reaction is chemically operational: for large ξ the product becomes very small. However, there is rather large region close to the origin (the white region in the plot) where the Hill model is not a good replacement for the Adair-Klotz model. The minimal value of ξ that guarantees a good match needs to be adjusted depending on a value of K1. Interestingly, for K1 ≳ 65 any value of ξ will lead to large δmax. Unfortunately, it was not possible to generate similar figures for h ≥ 5 owing to the limitations of the computer hardware.


Safe uses of Hill's model: an exact comparison with the Adair-Klotz model.

Konkoli Z - Theor Biol Med Model (2011)

Validity region of a K1 ≫ K2 ≫ K3 ≫ K4 parameterization. The plots depicts the boundary of the δmax(K1, K2, K3, K4) > 0.9 region in (K1, ξ) plane with the parameterization K2 = K1/ξ, K3 = K1/ξ2, and K4 = K1/ξ3. (K0 has been optimized as in the previous figures.)
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3104946&req=5

Figure 6: Validity region of a K1 ≫ K2 ≫ K3 ≫ K4 parameterization. The plots depicts the boundary of the δmax(K1, K2, K3, K4) > 0.9 region in (K1, ξ) plane with the parameterization K2 = K1/ξ, K3 = K1/ξ2, and K4 = K1/ξ3. (K0 has been optimized as in the previous figures.)
Mentions: Figure 6 is a contour plot that depicts how δmax depends on K1 and ξ for h = 4. The figure shows that many parameter choices that are chemically interesting do lead to a high value of the fundamental similarity measure (the grey region in the plot). Since there is no upper limit for ξ, for any value of K1, it is possible to choose ξ so that the reaction is chemically operational: for large ξ the product becomes very small. However, there is rather large region close to the origin (the white region in the plot) where the Hill model is not a good replacement for the Adair-Klotz model. The minimal value of ξ that guarantees a good match needs to be adjusted depending on a value of K1. Interestingly, for K1 ≳ 65 any value of ξ will lead to large δmax. Unfortunately, it was not possible to generate similar figures for h ≥ 5 owing to the limitations of the computer hardware.

Bottom Line: There are very few studies investigating the situations in which the model can be safely used.A strongly cooperative Adair-Klotz model can be replaced by a suitable Hill model in such a way that any property computed from the two models, even the one describing stochastic features, is approximately the same.The quantitative analysis showed that boundaries of the regions in the parameter space where the models behave in the same way exhibit a rather rich structure.

View Article: PubMed Central - HTML - PubMed

Affiliation: Chalmers University of Technology, Department of Microtechnology and Nanoscience, Bionano Systems Laboratory, Sweden. zorank@chalmers.se

ABSTRACT

Background: The Hill function and the related Hill model are used frequently to study processes in the living cell. There are very few studies investigating the situations in which the model can be safely used. For example, it has been shown, at the mean field level, that the dose response curve obtained from a Hill model agrees well with the dose response curves obtained from a more complicated Adair-Klotz model, provided that the parameters of the Adair-Klotz model describe strongly cooperative binding. However, it has not been established whether such findings can be extended to other properties and non-mean field (stochastic) versions of the same, or other, models.

Results: In this work a rather generic quantitative framework for approaching such a problem is suggested. The main idea is to focus on comparing the particle number distribution functions for Hill's and Adair-Klotz's models instead of investigating a particular property (e.g. the dose response curve). The approach is valid for any model that can be mathematically related to the Hill model. The Adair-Klotz model is used to illustrate the technique. One main and two auxiliary similarity measures were introduced to compare the distributions in a quantitative way. Both time dependent and the equilibrium properties of the similarity measures were studied.

Conclusions: A strongly cooperative Adair-Klotz model can be replaced by a suitable Hill model in such a way that any property computed from the two models, even the one describing stochastic features, is approximately the same. The quantitative analysis showed that boundaries of the regions in the parameter space where the models behave in the same way exhibit a rather rich structure.

Show MeSH