Editing Cooperative binding (section)

== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[Christian Bohr]] studied [[hemoglobin]] binding to [[oxygen]] under different conditions.<ref name=Bohr1904a>{{cite journal | vauthors = Bohr C | date = 1904 | url = https://www.biodiversitylibrary.org/item/50284#page/760/mode/1up | title = Die Sauerstoffaufnahme des genuinen Blutfarbstoffes und des aus dem Blute dargestellten Hämoglobins | language = de | journal = Zentralblatt Physiol. | volume = 23 | pages = 688–690 }}</ref><ref name=Bohr1904b>{{cite journal | vauthors = Bohr C, Hasselbalch K, Krogh A | author-link2 = Karl Albert Hasselbalch | author-link3 = August Krogh | year = 1904 | title = Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt | journal = Skandinavisches Archiv für Physiologie | volume = 16 | issue = 2| pages = 402–412 | doi = 10.1111/j.1748-1716.1904.tb01382.x | doi-access = free }}</ref> When plotting hemoglobin saturation with oxygen as a function of the [[partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904b/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[Bohr effect]].

[[File:Bohr effect.png|thumb|right|Original figure from [[Christian Bohr]], showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]

A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:

:<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>

If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.

The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[Allosteric regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[Allosteric regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)

Throughout the 20th century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context.<ref name=Wyman1990>{{cite book | vauthors = Wyman J, Gill SJ | year = 1990 | title = Binding and linkage. Functional chemistry of biological molecules | publisher = University Science Books | location = Mill Valley }}</ref>

=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[A V Hill|A.V. Hill]].<ref name=Hill1910>{{cite journal | vauthors = Hill AV | date = 1910 | title = The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves | journal = J Physiol | volume = 40 | pages = iv–vii }}</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[Hill equation (biochemistry)|named after him]]: [[File:Hill Plot.png|thumb|right|Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]

:<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} =  \frac{[X]^n}{K^* + [X]^n} = \frac{[X]^n}{K_d^n + [X]^n}
</math>

where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation), <math>K^*</math> is an empirical dissociation constant, and <math>K_d</math> a microscopic dissociation constant (used in modern forms of the equation, and equivalent to an <math>\mathrm{EC}_{50}</math>). If  <math>n<1</math>, the system exhibits negative cooperativity, whereas cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:

:<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - n\cdot{}\log K_d
</math>

The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>n\cdot\log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.

=== The Adair equation ===
[[Gilbert Smithson Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that binding affinity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>{{cite journal | vauthors = Adair GS | year = 1925 | title = 'The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin | journal = J Biol Chem | volume = 63 | issue = 2 | pages = 529–545 | doi = 10.1016/S0021-9258(18)85018-9 | doi-access = free }}</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:

:<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}[X]^2+3K_{III}[X]^3+4K_{IV}[X]^4}{1+K_I[X]+K_{II}[X]^2+K_{III}[X]^3+K_{IV}[X]^4}
</math>

Or, for any protein with ''n'' ligand binding sites:

:<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>

where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
By combining the Adair treatment with the Hill plot, one arrives at the modern experimental definition of cooperativity (Hill, 1985, Abeliovich, 2005). The resultant Hill coefficient, or more correctly the slope of the Hill plot as calculated from the Adair Equation, can be shown to be the ratio between the variance of the binding number to the variance of the binding number in an equivalent system of non-interacting binding sites.<ref name=Abeliovich2005>{{cite journal | vauthors = Abeliovich H | title = An empirical extremum principle for the hill coefficient in ligand-protein interactions showing negative cooperativity | journal = Biophysical Journal | volume = 89 | issue = 1 | pages = 76–9 | date = July 2005 | pmid = 15834004 | pmc = 1366580 | doi = 10.1529/biophysj.105.060194 | bibcode = 2005BpJ....89...76A }}</ref> Thus, the Hill coefficient defines cooperativity as a statistical dependence of one binding site on the state of other site(s).

=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite journal | vauthors = Klotz IM | title = The application of the law of mass action to binding by proteins; interactions with calcium | journal = Archives of Biochemistry | volume = 9 | pages = 109–17 | date = January 1946 | pmid = 21009581 }}</ref><ref name=Klotz2004>{{cite journal | vauthors = Klotz IM | title = Ligand-receptor complexes: origin and development of the concept | journal = The Journal of Biological Chemistry | volume = 279 | issue = 1 | pages = 1–12 | date = January 2004 | pmid = 14604979 | doi = 10.1074/jbc.X300006200 | doi-access = free }}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:

:<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>

It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}K</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.

The Klotz equation (which is sometimes also called the Adair-Klotz equation) is still often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.<ref name=Dagher2011>{{cite journal | vauthors = Dagher R, Peng S, Gioria S, Fève M, Zeniou M, Zimmermann M, Pigault C, Haiech J, Kilhoffer MC | title = A general strategy to characterize calmodulin-calcium complexes involved in CaM-target recognition: DAPK and EGFR calmodulin binding domains interact with different calmodulin-calcium complexes | journal = Biochimica et Biophysica Acta (BBA) - Molecular Cell Research | volume = 1813 | issue = 5 | pages = 1059–67 | date = May 2011 | pmid = 21115073 | doi = 10.1016/j.bbamcr.2010.11.004 | doi-access = free }}</ref>

=== Pauling equation ===
By the middle of the 20th century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[Linus pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite journal | vauthors = Pauling L | title = The Oxygen Equilibrium of Hemoglobin and Its Structural Interpretation | journal = Proceedings of the National Academy of Sciences of the United States of America | volume = 21 | issue = 4 | pages = 186–91 | date = April 1935 | pmid = 16587956 | pmc = 1076562 | doi = 10.1073/pnas.21.4.186 | bibcode = 1935PNAS...21..186P | doi-access = free }}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localization of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:

:<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>

===The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[Daniel E. Koshland, Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite journal | vauthors = Koshland DE, Némethy G, Filmer D | title = Comparison of experimental binding data and theoretical models in proteins containing subunits | journal = Biochemistry | volume = 5 | issue = 1 | pages = 365–85 | date = January 1966 | pmid = 5938952 | doi = 10.1021/bi00865a047 }}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite journal | vauthors = Koshland DE | title = Application of a Theory of Enzyme Specificity to Protein Synthesis | journal = Proceedings of the National Academy of Sciences of the United States of America | volume = 44 | issue = 2 | pages = 98–104 | date = February 1958 | pmid = 16590179 | pmc = 335371 | doi = 10.1073/pnas.44.2.98 | bibcode = 1958PNAS...44...98K | doi-access = free }}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:

:<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>

Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).

=== 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>