Cooperativity

When a substrate binds to one enzymatic subunit, the rest of the subunits are stimulated and become active. Ligands can either have positive cooperativity, negative cooperativity, or non-cooperativity. An example of positive cooperativity is the binding of oxygen to hemoglobin. One oxygen molecule can bind to the ferrous iron of a heme molecule in each of the four chains of a hemoglobin molecule. Deoxy-hemoglobin has a relatively low affinity for oxygen, but when one molecule binds to a single heme, the oxygen affinity increases, allowing the second molecule to bind more easily, and the third and fourth even more easily. The oxygen affinity of 3-oxy-hemoglobin is ~300 times greater than that of deoxy-hemoglobin. This behavior leads the affinity curve of hemoglobin to be sigmoidal, rather than hyperbolic as with the monomeric myoglobin. By the same process, the ability for hemoglobin to lose oxygen increases as fewer oxygen molecules are bound. ==Subunit cooperativity==

Cooperativity is not only a phenomenon of ligand binding, but also applies anytime energetic interactions make it easier or more difficult for something to happen involving multiple units as opposed to with single units. (That is, easier or more difficult compared with what is expected when only accounting for the addition of multiple units). For example, unwinding of DNA involves cooperativity: Portions of DNA must unwind in order for DNA to carry out replication, transcription and recombination. Positive cooperativity among adjacent DNA nucleotides makes it easier to unwind a whole group of adjacent nucleotides than it is to unwind the same number of nucleotides spread out along the DNA chain.The cooperative unit size is the number of adjacent bases that tend to unwind as a single unit due to the effects of positive cooperativity. This phenomenon applies to other types of chain molecules as well, such as the folding and unfolding of proteins and in the "melting" of phospholipid chains that make up the membranes of cells. Subunit cooperativity is measured on the relative scale known as Hill's Constant. ==Hill equation==

A simple and widely used model for molecular interactions is the Hill equation, which provides a way to quantify cooperative binding by describing the fraction of saturated ligand binding sites as a function of the ligand concentration. ==Hill coefficient==

The Hill coefficient is a measure of ultrasensitivity (i.e. how steep is the response curve). From an operational point of view the Hill coefficient can be estimated as: : n_{H} = \frac{ \log(81)}\ce{\log(EC90/EC10)} . where EC90 and EC10 are the input values needed to produce the 10% and 90% of the maximal response, respectively. ==Response coefficient==

Global sensitivity measures such as the Hill coefficient do not characterise the local behaviours of the s-shaped curves. Instead, these features are well captured by the response coefficient measure defined as: : R(x) = \frac{x}{y}\frac{dy}{dx} In systems biology, such responses are referred to as elasticities. ==Link between Hill coefficient and response coefficient==

Altszyler et al. (2017) have shown that these ultrasensitivity measures can be linked by the following equation: : n_H = 2 \frac{\int^\ce{\log(EC90)}_\ce{\log(EC10)} R_f(I) d(\log I)}\ce{ \log(EC90)-\log(EC10)} = 2 \langle R_f \rangle_\ce{EC10,EC90} where \langle X \rangle_{a,b} denoted the mean value of the variable x over the range [a,b]. ==Ultrasensitivity in function composition==

Consider two coupled ultrasensitive modules, disregarding effects of sequestration of molecular components between layers. In this case, the expression for the system's dose-response curve, , results from the mathematical composition of the functions, f_i, which describe the input/output relationship of isolated modules i=1,2: : F(I_1)=f_2\big(f_1(I_1)\big) Brown et al. (1997) have shown that the local ultrasensitivity of the different layers combines multiplicatively: : R(x)=R_{2}(f_{1}(x) ) . R_{1}(x) . In connection with this result, Ferrell et al. (1997) showed, for Hill-type modules, that the overall cascade global ultrasensitivity had to be less than or equal to the product of the global ultrasensitivity estimations of each cascade's layer, (i.e. the ultrasensitivity of the combination of layers is higher than the product of individual ultrasensitivities), but in many cases the ultimate origin of supramultiplicativity remained elusive. Altszyler et al. (2017) framework naturally suggested a general scenario where supramultiplicative behavior could take place. This could occur when, for a given module, the corresponding Hill's input working range was located in an input region with local ultrasensitivities higher than the global ultrasensitivity of the respective dose-response curve. == References ==