Abstract
The ultrasensitivity of a dose response function can be quantifiably defined using the generalized Hill coefficient of the function. We examined an upper bound for the Hill coefficient of the composition of two functions, namely the product of their individual Hill coefficients. We proved that this upper bound holds for compositions of Hill functions, and that there are instances of counterexamples that exist for more general sigmoidal functions. Additionally, we tested computationally other types of sigmoidal functions, such as the logistic and inverse trigonometric functions, and we provided computational evidence that in these cases the inequality also holds. We show that in large generality there is a limit to how ultrasensitive the composition of two functions can be, which has applications to understanding signaling cascades in biochemical reactions.
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Acknowledgements
The authors would like to thank Alvaro Fletcher for his contributions to this manuscript, including initial discussions and ideas for what became the current work. We would also like to thank Reagan Hsu, Anh Du Tran, and the anonymous reviewers of this manuscript. This research was partially supported by NSF-Simons Center grant DMS1763272 and Simons Foundation grant 594598.
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Pajoh-Casco, M., Vinujudson, A. & Enciso, G. Bounds on the Ultrasensitivity of Biochemical Reaction Cascades. Bull Math Biol 86, 59 (2024). https://doi.org/10.1007/s11538-024-01287-z
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DOI: https://doi.org/10.1007/s11538-024-01287-z