Editing Cooperativity (section)

==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, {{mvar|F}}, results from the mathematical composition of the functions, <math>f_i</math>, which describe the input/output relationship of isolated modules <math>i=1,2</math>:

:<math> F(I_1)=f_2\big(f_1(I_1)\big) </math>

Brown et al. (1997)<ref name="brown1997">{{cite journal | vauthors = Brown GC, Hoek JB, Kholodenko BN | title = Why do protein kinase cascades have more than one level? | journal = Trends in Biochemical Sciences | volume = 22 | issue = 8 | pages = 288 | date = August 1997 | pmid = 9270298 | doi = 10.1016/s0968-0004(97)82216-5 }}</ref><ref name="Altszyler_2017">{{cite journal | vauthors = Altszyler E, Ventura AC, Colman-Lerner A, Chernomoretz A | title = Ultrasensitivity in signaling cascades revisited: Linking local and global ultrasensitivity estimations | journal = PLOS ONE | volume = 12 | issue = 6 | pages = e0180083 | date = 29 June 2017 | pmid = 28662096 | pmc = 5491127 | doi = 10.1371/journal.pone.0180083 | arxiv = 1608.08007 | bibcode = 2017PLoSO..1280083A | doi-access = free }} [[File:CC-BY icon.svg|50x50px]] This article contains quotations from this source, which is available under the [[creativecommons:by/4.0/|Creative Commons Attribution 4.0 International (CC BY 4.0)]] license.</ref> have shown that the local ultrasensitivity of the different layers combines multiplicatively:

:<math> R(x)=R_{2}(f_{1}(x) ) .  R_{1}(x) </math>.

In connection with this result, Ferrell et al. (1997)<ref name="ferrell1997">{{cite journal | vauthors = Ferrell JE | title = How responses get more switch-like as you move down a protein kinase cascade | journal = Trends in Biochemical Sciences | volume = 22 | issue = 8 | pages = 288–9 | date = August 1997 | pmid = 9270299 | doi = 10.1016/s0968-0004(97)82217-7 }}</ref> 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,<ref name="Altszyler_2017"/>

:<math> n \leq n_{1} . n_{2} </math>,

where <math>n_1</math> and <math>n_2</math> are the Hill coefficient of modules 1 and 2 respectively.

Altszyler et al. (2017)<ref name="Altszyler_2017"/> have shown that the cascade's global ultrasensitivity can be analytically calculated:

:<math> n  = 2 \overbrace{\langle R_2 \rangle_{X10_2,X90_2}}^{\nu_2} \overbrace{\langle R_1 \rangle_{X10_1,X90_1}}^{\nu_1} = 2 \,\nu_2 \, \nu_1</math>

where <math>X10_i</math> and <math>X90_i</math> delimited the Hill input's working range of the composite system, i.e. the input values for the i-layer so that the last layer (corresponding to <math>i=2</math> in this case) reached the 10% and 90% of it maximal output level. It followed this equation that the system's Hill coefficient {{mvar|n}} could be written as the product of two factors, <math>\nu_1</math>  and <math>\nu_2</math>, which characterized local average sensitivities over the relevant input region for each layer: <math>[X10_i,X90_i ]</math>, with <math>i=1,2</math> in this case.

For the more general case of a cascade of {{mvar|N}} modules, the Hill coefficient can be expressed as:

:<math> n =\nu_{N} \, \nu_{N-1}...\nu_{1}  </math>,

===Supramultiplicativity===
Several authors have reported the existence of supramultiplicative behavior in signaling cascades <ref name="altszylerUltrasens2014">{{cite journal | vauthors = Altszyler E, Ventura A, Colman-Lerner A, Chernomoretz A | title = Impact of upstream and downstream constraints on a signaling module's ultrasensitivity | journal = Physical Biology | volume = 11 | issue = 6 | pages = 066003 | date = October 2014 | pmid = 25313165 | pmc = 4233326 | doi = 10.1088/1478-3975/11/6/066003 | bibcode = 2014PhBio..11f6003A }}</ref><ref name="racz2008">{{cite journal | vauthors = Rácz E, Slepchenko BM | title = On sensitivity amplification in intracellular signaling cascades | journal = Physical Biology | volume = 5 | issue = 3 | pages = 036004 | date = July 2008 | pmid = 18663279 | pmc = 2675913 | doi = 10.1088/1478-3975/5/3/036004 | bibcode = 2008PhBio...5c6004R }}</ref> (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)<ref name="Altszyler_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.