Editing Cooperative binding (section)

===  The MWC model ===
[[File:MWC structure.png|thumb|right|Monod-Wyman-Changeux model reaction scheme of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC energy.png|thumb|right|Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[MWC model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite journal | vauthors = Monod J, Wyman J, Changeux JP | journal = Journal of Molecular Biology | volume = 12 | pages = 88–118 | date = May 1965 | pmid = 14343300 | doi = 10.1016/S0022-2836(65)80285-6 | title = On the nature of allosteric transitions: A plausible model }}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states  - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".

The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. With <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:

:<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n} 
</math>

The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The slope of the Hill plot also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and the y-axis allow to determine the affinities of both states for the ligand. 
 
[[File:Hill Plot MWC model.png|thumb|right|Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well as the microscopic dissociation constants of R and T states.]]

In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant or what is experimentally measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:

:<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>

A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite journal | vauthors = Rubin MM, Changeux JP | title = On the nature of allosteric transitions: implications of non-exclusive ligand binding | journal = Journal of Molecular Biology | volume = 21 | issue = 2 | pages = 265–74 | date = November 1966 | pmid = 5972463 | doi = 10.1016/0022-2836(66)90097-0 }}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreover, the extents of the cooperativity of binding and the cooperativity of activation can be very different: an extreme case is provide by the bacteria flagella motor with a Hill coefficient of 1.7 for the binding and 10.3 for the activation.<ref name=Cluzel2000>{{cite journal | vauthors = Cluzel P, Surette M, Leibler S | title = An ultrasensitive bacterial motor revealed by monitoring signaling proteins in single cells | journal = Science | volume = 287 | issue = 5458 | pages = 1652–5 | date = March 2000 | pmid = 10698740 | doi = 10.1126/science.287.5458.1652 | bibcode = 2000Sci...287.1652C }}</ref><ref name=Sourjick2002>{{cite journal | vauthors = Sourjik V, Berg HC | title = Binding of the Escherichia coli response regulator CheY to its target measured in vivo by fluorescence resonance energy transfer | journal = Proceedings of the National Academy of Sciences of the United States of America | volume = 99 | issue = 20 | pages = 12669–74 | date = October 2002 | pmid = 12232047 | pmc = 130518 | doi = 10.1073/pnas.192463199 | bibcode = 2002PNAS...9912669S | doi-access = free }}</ref> The supra-linearity of the response is sometimes called [[ultrasensitivity]].

If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilizes the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called [[allosteric modulator]]s.

Since its inception, the MWC framework has been extended and generalized. Variations have been proposed, for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite journal | vauthors = Edelstein SJ, Schaad O, Henry E, Bertrand D, Changeux JP | title = A kinetic mechanism for nicotinic acetylcholine receptors based on multiple allosteric transitions | journal = Biological Cybernetics | volume = 75 | issue = 5 | pages = 361–79 | date = November 1996 | pmid = 8983160 | doi = 10.1007/s004220050302 | citeseerx = 10.1.1.17.3066 | s2cid = 6240168 }}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite journal | vauthors = Mello BA, Tu Y | title = An allosteric model for heterogeneous receptor complexes: understanding bacterial chemotaxis responses to multiple stimuli | journal = Proceedings of the National Academy of Sciences of the United States of America | volume = 102 | issue = 48 | pages = 17354–9 | date = November 2005 | pmid = 16293695 | pmc = 1297673 | doi = 10.1073/pnas.0506961102 | bibcode = 2005PNAS..10217354M | doi-access = free }}</ref><ref name=Najdi2006>{{cite journal | vauthors = Najdi TS, Yang CR, Shapiro BE, Hatfield GW, Mjolsness ED | title = Application of a generalized MWC model for the mathematical simulation of metabolic pathways regulated by allosteric enzymes | journal = Journal of Bioinformatics and Computational Biology | volume = 4 | issue = 2 | pages = 335–55 | date = April 2006 | pmid = 16819787 | doi = 10.1142/S0219720006001862 | citeseerx = 10.1.1.121.9382 }}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite journal | vauthors = Stefan MI, Edelstein SJ, Le Novère N | title = Computing phenomenologic Adair-Klotz constants from microscopic MWC parameters | journal = BMC Systems Biology | volume = 3 | pages = 68 | date = July 2009 | pmid = 19602261 | pmc = 2732593 | doi = 10.1186/1752-0509-3-68 | doi-access = free }}</ref>