Limits...
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.

Show MeSH

Related in: MedlinePlus

Similarity measures (weakly cooperative Adair-Klotz model, h = 2). Time dependence of the similarity measures for h = 2 case: . This and all other figures in the manuscript were generated with P0 = 2 and L0 = 5. In this figure weakly cooperative Adair-Klotz model has been considered with αi = βi = 1s-1 for i = 1, ..., h. The parameters for the Hill model were optimized so that δ(t) is largest possible (β/α = 0.5 and α = 0.5s-1). The time t is expressed in units of s. The full line is for Δ = δ, while the dashed and the dotted lines are for  and  respectively.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC3104946&req=5

Figure 1: Similarity measures (weakly cooperative Adair-Klotz model, h = 2). Time dependence of the similarity measures for h = 2 case: . This and all other figures in the manuscript were generated with P0 = 2 and L0 = 5. In this figure weakly cooperative Adair-Klotz model has been considered with αi = βi = 1s-1 for i = 1, ..., h. The parameters for the Hill model were optimized so that δ(t) is largest possible (β/α = 0.5 and α = 0.5s-1). The time t is expressed in units of s. The full line is for Δ = δ, while the dashed and the dotted lines are for and respectively.

Mentions: The three similarity measures have been computed numerically by solving the master equations for the models. Figure 1 shows how the similarity measures depend on time in the situation where it is expected that Hill's model cannot approximate the dynamics of Adair-Klotz's model, i.e. when all reaction rates are equal and Adair-Klotz's reaction system cannot be described as cooperative. The similarity is perfect at t = 0 by construction, since in principle both systems are prepared in identical states. The similarity starts decreasing since the intermediate states become populated. This can be seen from the fact that the dashed line goes up, starting from zero. Please note that after some time the intermediate states become de-populated since the dashed line goes down after the initial peak around t ≈ 0.25. The choice of reaction rates for the Adair-Klotz model clearly makes the intermediate states long lived. In such a case it is not possible to find the parameters α and β such that the fundamental (main) similarity measure is large.


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

Konkoli Z - Theor Biol Med Model (2011)

Similarity measures (weakly cooperative Adair-Klotz model, h = 2). Time dependence of the similarity measures for h = 2 case: . This and all other figures in the manuscript were generated with P0 = 2 and L0 = 5. In this figure weakly cooperative Adair-Klotz model has been considered with αi = βi = 1s-1 for i = 1, ..., h. The parameters for the Hill model were optimized so that δ(t) is largest possible (β/α = 0.5 and α = 0.5s-1). The time t is expressed in units of s. The full line is for Δ = δ, while the dashed and the dotted lines are for  and  respectively.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 1: Similarity measures (weakly cooperative Adair-Klotz model, h = 2). Time dependence of the similarity measures for h = 2 case: . This and all other figures in the manuscript were generated with P0 = 2 and L0 = 5. In this figure weakly cooperative Adair-Klotz model has been considered with αi = βi = 1s-1 for i = 1, ..., h. The parameters for the Hill model were optimized so that δ(t) is largest possible (β/α = 0.5 and α = 0.5s-1). The time t is expressed in units of s. The full line is for Δ = δ, while the dashed and the dotted lines are for and respectively.
Mentions: The three similarity measures have been computed numerically by solving the master equations for the models. Figure 1 shows how the similarity measures depend on time in the situation where it is expected that Hill's model cannot approximate the dynamics of Adair-Klotz's model, i.e. when all reaction rates are equal and Adair-Klotz's reaction system cannot be described as cooperative. The similarity is perfect at t = 0 by construction, since in principle both systems are prepared in identical states. The similarity starts decreasing since the intermediate states become populated. This can be seen from the fact that the dashed line goes up, starting from zero. Please note that after some time the intermediate states become de-populated since the dashed line goes down after the initial peak around t ≈ 0.25. The choice of reaction rates for the Adair-Klotz model clearly makes the intermediate states long lived. In such a case it is not possible to find the parameters α and β such that the fundamental (main) similarity measure is large.

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
Related in: MedlinePlus