Computing phenomenologic Adair-Klotz constants from microscopic MWC parameters
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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.
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 (R and T), with an arbitrary number of non-equivalent ligand binding sites. We apply this framework to compute Adair-Klotz constants from existing models of calmodulin and hemoglobin, two extreme cases of the general framework.
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.
KeywordsAssociation Constant Ligand Molecule Ligand Binding Site Conversion Equation Apparent Binding Constant
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 XXth century .
where K denotes an apparent association constant, [X] denotes ligand concentration, and n H the "Hill coefficient", intended to be a measure of cooperativity.
Where n denotes the number of binding sites and K i the i th apparent association constant
where [X] denotes ligand concentration, and with K R , L and c as described in the paragraph above. In this paper, we first propose a generalised MWC framework that can be applied to proteins whose ligand binding sites have different affnities. We then develop a set of equations that uses the parameters of such a generalised MWC model to compute apparent association constants according to the Adair-Klotz model. We show how these can be used in order to compare model results with experimental data using two examples which constitute extreme cases of the general framework, calmodulin and hemoglobin.
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 Open image in new window and Open image in new window , and their ratio is denoted by c i for the i th binding site.
where 1 ≤ i, j ≤ n, Open image in new window and L and [X] as described above.
where 1 ≤ i, j ≤ k, m i denotes the number of binding sites with affnity Open image in new window (note that Σ i m i = n), and L, c i and [X] as described above.
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
Note that in the case of four identical binding sites, Open image in new window and Open image in new window , and the above expressions reduce to conversion equations for identical binding sites reported by Edelstein .
Obtaining the i th Adair-Klotz constants from microscopic association constants for a protein with n non-equivalent binding sites
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 (m1 = m2 = m3 = m4 = 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 . According to this model, calmodulin can exist in two different states, R (that corresponds to the open state, stabilised by binding of calcium) and T (that correspond to the closed, often mistakenly called "apo", state). Each of these states can bind four calcium ions. The four different binding sites were designated A, B, C, and D (A and B on the N-terminal domain, C and D on the C-terminal domain, with no sequential order being implied within the domains). Each of the states and each of the reactions was explicitly modelled, with distinct dissociation constants for each of the sites. The dissociation constants for the R state were Open image in new window = 8.32 × 10- 6 M, Open image in new window = 1.66 × 10- 8 M, Open image in new window = 1.74 × 10- 5 M, and Open image in new window = 1.45 × 10- 8M. According to this model, L = 20670, and c = 0.00396 for all four binding sites . The calmodulin concentration used for the model was 2 × 107 M , and simulations were run using COPASI .
When the fractional occupancy of calmodulin is plotted against initial free calcium concentration, simulation outcomes seem to agree quite well with experimental observations , but such a plot does not provide a direct way of quantifying this agreement.
Allosteric model of Hemoglobin
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 , and could be useful in the analysis of a wide range of other proteins.
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, R.
Najdi et al.  have proposed a generalised MWC (GMWC) model for a protein binding to several ligand types and regulated by multiple allosteric activators or inhibitors. This model can be combined with the model presented here by replacing the term that denotes substrate concentration and affinity for each ligand in  by the appropriate sum: in the notation employed by , this would mean replacing Open image in new window by Open image in new window for each ligand. Such a combined model could then cater for proteins that bind to several ligand types (with non-identical binding sites per ligand) and that are regulated by multiple allosteric activators or inhibitors.
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.
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