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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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Similarity measures (strongly cooperative Adair-Klotz model, h = 2). Generated in the same way as Figure 1, but with different values for the reaction rates. The particular choice of the reaction rates makes the intermediate states weakly populated: α1 = 1s-1, β1 = 10s-1, α2 = 10s-1, and β2 = 1s-1. The parameters for the Hill model were optimized in the same way as for the Figure 1 resulting in α = 0.5s-1 and β = 0.25s-1. δ(t) stays relatively close to one indicating a good match. The dashed curve stays low, which indicates that intermediate states are short lived. The dotted line stays close to one indicating that the distributions have a similar shape.
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Figure 3: Similarity measures (strongly cooperative Adair-Klotz model, h = 2). Generated in the same way as Figure 1, but with different values for the reaction rates. The particular choice of the reaction rates makes the intermediate states weakly populated: α1 = 1s-1, β1 = 10s-1, α2 = 10s-1, and β2 = 1s-1. The parameters for the Hill model were optimized in the same way as for the Figure 1 resulting in α = 0.5s-1 and β = 0.25s-1. δ(t) stays relatively close to one indicating a good match. The dashed curve stays low, which indicates that intermediate states are short lived. The dotted line stays close to one indicating that the distributions have a similar shape.

Mentions: For the case in which intermediate states are short lived, one intuitively expects that Hill's model could be a useful substitute for Adair-Klotz's model. Figure 3 depicts the dependence of the similarity measures on time, for systems that are expected to behave in a similar way. In particular, the reaction rates for the Adair-Klotz model used were chosen in such a way that the intermediate states are short lived. Indeed, the value of stays very close to 0. The shapes similarity measure stays very close to one, finally leading to large values for the fundamental similarity measure δ(t). This is an important finding since it indicates that Hill's model can be used to investigate an arbitrary observable, e.g., not just the average number of free ligands, but also the noise characteristics of that quantity. Naturally, such a claim comes with the implicit constraint that the observable should be interpreted in the context of Hill's model state space. For example, quantities such as the number of free receptor proteins, or the number of fully occupied receptors, fall in this category. However, any quantity that would involve counting the number of intermediates does not.


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

Konkoli Z - Theor Biol Med Model (2011)

Similarity measures (strongly cooperative Adair-Klotz model, h = 2). Generated in the same way as Figure 1, but with different values for the reaction rates. The particular choice of the reaction rates makes the intermediate states weakly populated: α1 = 1s-1, β1 = 10s-1, α2 = 10s-1, and β2 = 1s-1. The parameters for the Hill model were optimized in the same way as for the Figure 1 resulting in α = 0.5s-1 and β = 0.25s-1. δ(t) stays relatively close to one indicating a good match. The dashed curve stays low, which indicates that intermediate states are short lived. The dotted line stays close to one indicating that the distributions have a similar shape.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 3: Similarity measures (strongly cooperative Adair-Klotz model, h = 2). Generated in the same way as Figure 1, but with different values for the reaction rates. The particular choice of the reaction rates makes the intermediate states weakly populated: α1 = 1s-1, β1 = 10s-1, α2 = 10s-1, and β2 = 1s-1. The parameters for the Hill model were optimized in the same way as for the Figure 1 resulting in α = 0.5s-1 and β = 0.25s-1. δ(t) stays relatively close to one indicating a good match. The dashed curve stays low, which indicates that intermediate states are short lived. The dotted line stays close to one indicating that the distributions have a similar shape.
Mentions: For the case in which intermediate states are short lived, one intuitively expects that Hill's model could be a useful substitute for Adair-Klotz's model. Figure 3 depicts the dependence of the similarity measures on time, for systems that are expected to behave in a similar way. In particular, the reaction rates for the Adair-Klotz model used were chosen in such a way that the intermediate states are short lived. Indeed, the value of stays very close to 0. The shapes similarity measure stays very close to one, finally leading to large values for the fundamental similarity measure δ(t). This is an important finding since it indicates that Hill's model can be used to investigate an arbitrary observable, e.g., not just the average number of free ligands, but also the noise characteristics of that quantity. Naturally, such a claim comes with the implicit constraint that the observable should be interpreted in the context of Hill's model state space. For example, quantities such as the number of free receptor proteins, or the number of fully occupied receptors, fall in this category. However, any quantity that would involve counting the number of intermediates does not.

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