Computational Neurobiology Group, EMBL-EBI, Wellcome-Trust Genome Campus, Hinxton, CB10 1SD, UK

Abstract

Background

Modellers using the MWC allosteric framework have often found it difficult to validate their models. Indeed many experiments are not conducted with the notion of alternative conformations in mind and therefore do not (or cannot) measure relevant microscopic constant and parameters. Instead, experimentalists widely use the Adair-Klotz approach in order to describe their experimental data.

Results

We propose a way of computing apparent Adair-Klotz constants from microscopic association constants and allosteric parameters of a generalised concerted model with two different states (

Conclusion

The validation of computational models requires methods to relate model parameters to experimentally observable quantities. We provide such a method for comparing generalised MWC allosteric models to experimentally determined Adair-Klotz constants.

Background

Quantitative descriptions of biological processes are one of the main activities in Life Science research, whether in physiology, biochemistry or molecular and cellular biology. They offer a way of characterising biological systems, measuring subtle effects of perturbations, discriminating between alternative hypotheses, making and testing predictions, and following changes over time. There can be many different ways to describe the same biological process. Phenomenological descriptions provide a way of relating input and outcome of a given process, without requiring a detailed knowledge about the nature of the process or possible intermediate steps. Since they provide a direct link between input and output, they can be easily applied to experimental results. On the other hand, Systems Biology favours more mechanistic representations, that aim at exploring how exactly behaviours of systems emerge from intrinsic properties and interactions of elements at a lower level. Using the former descriptions to build and validate the latter representations may prove a challenge in some cases.

Several types of descriptions may co-exist for a given biological problem. One of these problems is the binding of ligand to a protein with several binding sites, and the apparent cooperativity observed in this context, for which various frameworks have been developed throughout the XX^{th }century

Drawing on observations of oxygen binding to hemoglobin, Hill

where _{H }the "Hill coefficient", intended to be a measure of cooperativity.

Adair

Where _{i }the ^{th }apparent association constant

The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions _{0}]/[_{0}]. If ^{T}/^{R }(note that MWC equations are most often expressed with dissociation constants. However, we will use association constant throughout this paper for the sake of consistency with Hill and Adair-Klotz schemes). If

where [^{R},

Results

Generalisation of the MWC model

The MWC model can be adapted to describe a protein (whether oligomeric or monomeric) with several ligand binding sites possessing different affinities. In that case, microscopic association constants are termed _{i }for the ^{th }binding site.

In this case, the fractional occupancy is described as follows:

where 1 ≤

If not all binding sites are different, but _{i }binding sites have the same affinity

where 1 ≤ _{i }denotes the number of binding sites with affnity _{i}_{i }= _{i }and [

Similarly, it is possible to develop generalisations of the equation for fractional conformational change (

When all _{i }are equal, this corresponds to the original MWC equation

Again, when binding sites are pooled into groups of _{i }binding sites that have the same affnity _{i }_{i }=

In order to compare the numerical outcomes of their models with experimental results, modellers using either the original or the generalised MWC framework need a way of converting microscopic MWC constants into observed Adair-Klotz constants. Here, we derive equations that can be used to compute Adair-Klotz constants and apply them to two special cases of the generalised MWC model presented here.

Obtaining Adair-Klotz constants from microscopic association constants for a protein with four non-equivalent binding sites

Consider a protein _{1 }is defined as follows:

where [_{0}] denotes the concentration of unbound protein, [_{1}] the concentration of protein with exactly one ligand molecule bound and [_{i}] the concentration of protein in the _{i}]), we can re-write the above expression to

Since we treat the four binding sites as non-equivalent, we have to discriminate between them. The first ligand molecule bound to the protein in the _{A }denotes the concentration of protein in the

The balance between unbound protein in the _{1 }to the microscopic association constants (

Substituting for [_{0}] and simplifying, we obtain

In a similar manner we can consider the second association constant, _{2}

Again, distinguishing between the

This reduces to:

We can apply the same reasoning to the third ligand binding event:

which eventually gives:

And, similarly for _{4}:

Note that in the case of four identical binding sites,

Obtaining the ^{th }Adair-Klotz constants from microscopic association constants for a protein with

In general, for a protein with ^{th }binding event by computing the ratio between the concentrations of end products and initial reactants. The equation for the ^{th }apparent association constant thus reads as follows:

As above, both [_{i-1}] and [_{i}] are sums of protein populations in two different states and with ligand molecules bound to combinations of different binding sites. We can again distinguish between

If we now assume that the _{i }is a collection of protein molecules in the

Expressing every _{j1j2...ji }in terms of [_{0}], [_{i }in the following way:

Introducing the following abbreviations

we can obtain the expression for

Now, again, we can use the relationship [_{0}] = _{0}] and eliminate [^{i }and [_{0}] and obtain:

with

If the binding sites can be classed into _{1 }binding sites with affinity _{2 }binding sites with affinity

In the next section, we will consider two proteins with four binding sites each, which constitute extreme cases: In the case of calmodulin, all binding sites are different, so the protein can be seen as having four sub-groups of binding sites containing one binding site each (_{1 }= _{2 }= _{3 }= _{4 }= 1). In the case of hemoglobin, all binding sites are equivalent, so there is only one sub-group of binding sites containing four elements.

Allosteric model of calmodulin

To illustrate the practical relevance of these conversion equations we applied them to a previously proposed MWC model of calmodulin ^{- }^{6 }M, ^{- }^{8 }M, ^{- }^{5 }M, and ^{- }^{8}M. According to this model, ^{7 }M

When the fractional occupancy of calmodulin is plotted against initial free calcium concentration, simulation outcomes seem to agree quite well with experimental observations

To do this, we inserted the parameters of the MWC model into equations 8 to 11 to obtain Adair-Klotz constants. These can be compared to Adair-Klotz constants previously obtained in experimental studies

Apparent Adair-Klotz constants for the calmodulin model

this paper

reported range

_{1}

5.1860 × 10^{5}

1.16 × 10^{5 }^{6 }

_{2}

5.1601 × 10^{5}

1.4 × 10^{5 }^{5 }

_{3}

1.3377 × 10^{5}

2.86 × 10^{4 }^{6 }

_{4}

3.8784 × 10^{4}

1.7 × 10^{3 }^{5 }

Apparent Adair-Klotz constants (in M) for the calmodulin model as computed with our method, and comparison to several experimental reports

Figure

Comparison of the calmodulin model with experimental data

**Comparison of the calmodulin model with experimental data**. Red curve shows the Adair-Klotz equation using the Adair-Klotz constants obtained from the MWC model of calmodulin. Symbols are used to represent data points from various experimental measurements of calmodulin binding to calcium: Circles for Porumb

Allosteric model of Hemoglobin

In a similar manner, the case in which all binding sites are equivalent

Yonetani _{R}, _{T}, and _{1 }to _{4}. Table

Comparison of MWC and Adair-Klotz constants for hemoglobin

this paper

Yonetani

7.68 × 10^{-3}

7.20 × 10^{-3}

0.96 × 10^{-2}

1.05 × 10^{-2}

1.52 × 10^{-2}

1.15 × 10^{-2}

2.32 × 10^{-2}

2.33 × 10^{-2}

Experimental and theoretical determination of Adair-Klotz constants (in torr^{-1}) from MWC constants at pH 7.0. K_{R }= 3.0 × 10^{-2}torr^{-1}, K_{T }= 7.0 × 10^{-3 }torr^{-1}, and _{1 }to _{4 }using the equations presented in _{1 }to _{4 }obtained by Yonetani _{1 }in _{1 }in _{2 }in _{2 }in _{3 }in _{3 }in _{4 }in _{4 }in

Discussion and conclusion

The generalised MWC model proposed here opens up new ways of applying the allosteric framework: Not only to multimers consisting of identical subunits with one ligand binding site on each, but also to proteins with several binding sites of different affinities for the same ligand, be it multimers with more than one binding site on each subunit or monomeric proteins containing several binding sites. This framework has been used for an allosteric model of calmodulin

Other generalisations of the MWC framework have been presented in the past. Mello and Tu

where [_{1}] represents the first ligand, for which _{1 }binding sites exist, and [_{2}] the second ligand, for which there are _{2 }binding sites. For a heterogeneous complex with

The case in which binding sites for a given ligand can be grouped into sets of same affinity is straight-forward, as is the computation of fractional occupancy,

Najdi

In biology, the same question can be tackled at different levels and with different approaches, often based on different underlying theoretical framework. These approaches, however, need to be comparable to allow for cross-validation and for the assembly of different types of data into a comprehensive understanding of a given process. For instance, computational modellers need a way of comparing their models with experimental results to assess the validity of their models. In particular, mechanistic models need to be comparable to data or to the phenomenological models describing them. We offer a way of relating intrinsic association constants in allosteric models to Adair-Klotz constants and thus to bridge the gap between generalised allosteric models and experimental observations.

Apart from enabling modellers to validate their models – as shown here in the two example cases – these conversion equations could also help in model construction by providing ways to constrain parameter space and facilitate the estimation of allosteric parameters, which is very useful in cases where there is little or no additional experimental evidence that could help with their derivation.

Abbreviations

MWC: Monod-Wyman-Changeux; R: relaxed; T: tense.

Authors' contributions

MIS designed the generalised MWC framework and wrote the conversion equations with the help of SJE. All authors contributed to the manuscript. All authors read and approved the final manuscript.