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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 = 2. The contour plot that depicts how long time limit of δ∞ = limt→∞ δ(t) depends on the dissociation constants K1 = β1/α1 and K2 = β2/α2; δ∞ = f(K0, K1, K2). For a fixed pair (K1, K2) the Hill model dissociation constant K0 = β/α is optimized to make δ∞ as large as possible, making the Hill's model dissociation constant dependent on Adair-Klotz's model dissociation constants in a well defined way; K0 = g(K1, K2) leading to the function δ∞ = f(g(K1, K2), K1, K2) = δmax(K1, K2) that is depicted in the plot.
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Figure 4: Equilibrium state similarity measure for h = 2. The contour plot that depicts how long time limit of δ∞ = limt→∞ δ(t) depends on the dissociation constants K1 = β1/α1 and K2 = β2/α2; δ∞ = f(K0, K1, K2). For a fixed pair (K1, K2) the Hill model dissociation constant K0 = β/α is optimized to make δ∞ as large as possible, making the Hill's model dissociation constant dependent on Adair-Klotz's model dissociation constants in a well defined way; K0 = g(K1, K2) leading to the function δ∞ = f(g(K1, K2), K1, K2) = δmax(K1, K2) that is depicted in the plot.

Mentions: Figure 4 shows how δmax depends on the values of the Adair-Klotz model reaction rates for the case h = 2. The figure depicts contours where δmax = const in the (K1, K2) plane. The first interesting region is in the range 0 ≤ K1 ≲ 45 and below the full curve. In this range (the grey region below the full curve) K1 ≫ K2 guarantees high similarity measure values. This analysis confirms the previous mean field study [5] where it was shown that choosing K1 ≫ K2 leads to similar dose response curves. In the present article it has been shown that the results holds for any observable (average numbers, variances, etc). The second interesting region is for K1 ≳ 45. In that region the fundamental similarity measure is large for any K2. Cases with relatively large values of K2 are not interesting chemically, since such reactions would be chemically non-functional: K1K2 ≫ 1 would lead to the situation where the fraction of final products (complexes) in the system would be vanishingly small. However, a reaction with K1 ≳ 45 and K2 ≪ 1 could be functional provided K1K2 ~1.


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 = 2. The contour plot that depicts how long time limit of δ∞ = limt→∞ δ(t) depends on the dissociation constants K1 = β1/α1 and K2 = β2/α2; δ∞ = f(K0, K1, K2). For a fixed pair (K1, K2) the Hill model dissociation constant K0 = β/α is optimized to make δ∞ as large as possible, making the Hill's model dissociation constant dependent on Adair-Klotz's model dissociation constants in a well defined way; K0 = g(K1, K2) leading to the function δ∞ = f(g(K1, K2), K1, K2) = δmax(K1, K2) that is depicted in the plot.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 4: Equilibrium state similarity measure for h = 2. The contour plot that depicts how long time limit of δ∞ = limt→∞ δ(t) depends on the dissociation constants K1 = β1/α1 and K2 = β2/α2; δ∞ = f(K0, K1, K2). For a fixed pair (K1, K2) the Hill model dissociation constant K0 = β/α is optimized to make δ∞ as large as possible, making the Hill's model dissociation constant dependent on Adair-Klotz's model dissociation constants in a well defined way; K0 = g(K1, K2) leading to the function δ∞ = f(g(K1, K2), K1, K2) = δmax(K1, K2) that is depicted in the plot.
Mentions: Figure 4 shows how δmax depends on the values of the Adair-Klotz model reaction rates for the case h = 2. The figure depicts contours where δmax = const in the (K1, K2) plane. The first interesting region is in the range 0 ≤ K1 ≲ 45 and below the full curve. In this range (the grey region below the full curve) K1 ≫ K2 guarantees high similarity measure values. This analysis confirms the previous mean field study [5] where it was shown that choosing K1 ≫ K2 leads to similar dose response curves. In the present article it has been shown that the results holds for any observable (average numbers, variances, etc). The second interesting region is for K1 ≳ 45. In that region the fundamental similarity measure is large for any K2. Cases with relatively large values of K2 are not interesting chemically, since such reactions would be chemically non-functional: K1K2 ≫ 1 would lead to the situation where the fraction of final products (complexes) in the system would be vanishingly small. However, a reaction with K1 ≳ 45 and K2 ≪ 1 could be functional provided K1K2 ~1.

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