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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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Equilibrium state similarity measure for h = 3. The plot depicts equilibrium state similarity measure for h = 3 case. For each triple (K1, K2, K3) an optimal value is found for K0 that maximizes δ∞. In such a way δ∞ = δmax(K1, K2, K3). The lines plotted in both panels denote the δ∞ = 0.9 boundaries. For a given curve, the region with δ∞ > 0.9 is always to the right of the curve. Panel (a): the reaction rates parameter space is projected on to (K1, K2) plane with K3 fixed at the values indicated in the panel. Panel (b): the parameter space is projected on the (K2, K3) plane with several choices for K1 as indicated in the panel.
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Figure 5: Equilibrium state similarity measure for h = 3. The plot depicts equilibrium state similarity measure for h = 3 case. For each triple (K1, K2, K3) an optimal value is found for K0 that maximizes δ∞. In such a way δ∞ = δmax(K1, K2, K3). The lines plotted in both panels denote the δ∞ = 0.9 boundaries. For a given curve, the region with δ∞ > 0.9 is always to the right of the curve. Panel (a): the reaction rates parameter space is projected on to (K1, K2) plane with K3 fixed at the values indicated in the panel. Panel (b): the parameter space is projected on the (K2, K3) plane with several choices for K1 as indicated in the panel.

Mentions: Figure 5 shows similar kind of analysis as done for Figure 4 but for the first higher value of the Hill coefficient, h = 3. Unfortunately, because the structure of the parameter space is more complicated, it is not possible to use a single contour plot. Instead, various hyperplanes in the parameter space are studied. Panel (a) depicts the regions in the (K1, K2) plane where δmax = 0.9 for different choices of K3. The region with δmax > 0.9 is always to the right of each curve. For example, in the grey region in panel (a), for K3 = 1000, it is always true that δmax > 0.9. On the one hand, it can be seen that increase in K3 reduces the area where the fundamental similarity measure is large. On the other hand, for a fixed value of K3, and for a chemically functioning reactions (K1K2 ~1), choosing K1 ≫ K2 makes the fundamental similarity measure large. Likewise, panel (b) indicates that to obtain a large value for the fundamental similarity measure K1 should be as large as possible. For a given value of K1 one should take K2 ≫ K3. In brief, one can say that K1 ≫ K2 ≫ K3 ensures that δmax is large but the plot shows that there are many subtle details associated with such a statement. Again, this confirms the previous finding in [5] that K1 ≫ K2 ≫ K3 results in similar dose response curves for both models, but please note that the statement made in here is much more general.


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

Konkoli Z - Theor Biol Med Model (2011)

Equilibrium state similarity measure for h = 3. The plot depicts equilibrium state similarity measure for h = 3 case. For each triple (K1, K2, K3) an optimal value is found for K0 that maximizes δ∞. In such a way δ∞ = δmax(K1, K2, K3). The lines plotted in both panels denote the δ∞ = 0.9 boundaries. For a given curve, the region with δ∞ > 0.9 is always to the right of the curve. Panel (a): the reaction rates parameter space is projected on to (K1, K2) plane with K3 fixed at the values indicated in the panel. Panel (b): the parameter space is projected on the (K2, K3) plane with several choices for K1 as indicated in the panel.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 5: Equilibrium state similarity measure for h = 3. The plot depicts equilibrium state similarity measure for h = 3 case. For each triple (K1, K2, K3) an optimal value is found for K0 that maximizes δ∞. In such a way δ∞ = δmax(K1, K2, K3). The lines plotted in both panels denote the δ∞ = 0.9 boundaries. For a given curve, the region with δ∞ > 0.9 is always to the right of the curve. Panel (a): the reaction rates parameter space is projected on to (K1, K2) plane with K3 fixed at the values indicated in the panel. Panel (b): the parameter space is projected on the (K2, K3) plane with several choices for K1 as indicated in the panel.
Mentions: Figure 5 shows similar kind of analysis as done for Figure 4 but for the first higher value of the Hill coefficient, h = 3. Unfortunately, because the structure of the parameter space is more complicated, it is not possible to use a single contour plot. Instead, various hyperplanes in the parameter space are studied. Panel (a) depicts the regions in the (K1, K2) plane where δmax = 0.9 for different choices of K3. The region with δmax > 0.9 is always to the right of each curve. For example, in the grey region in panel (a), for K3 = 1000, it is always true that δmax > 0.9. On the one hand, it can be seen that increase in K3 reduces the area where the fundamental similarity measure is large. On the other hand, for a fixed value of K3, and for a chemically functioning reactions (K1K2 ~1), choosing K1 ≫ K2 makes the fundamental similarity measure large. Likewise, panel (b) indicates that to obtain a large value for the fundamental similarity measure K1 should be as large as possible. For a given value of K1 one should take K2 ≫ K3. In brief, one can say that K1 ≫ K2 ≫ K3 ensures that δmax is large but the plot shows that there are many subtle details associated with such a statement. Again, this confirms the previous finding in [5] that K1 ≫ K2 ≫ K3 results in similar dose response curves for both models, but please note that the statement made in here is much more general.

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