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Question Number 207150 by Wuji last updated on 07/May/24

Expression of protein is controlled by an external S. the protein also controls its own exprexxion by a negative feedback. The fol<lwing system of ODEs represents the dynamics of the system, with m and p representing mRNA and protein respectively, prove that for any value of S≥0, the steady state of the system is spiral sink (dm/dt)=(S/(1+S))+(1/(1+P))−m (dp/dt)=m−p

$$\mathrm{Expression}\:\mathrm{of}\:\mathrm{protein}\:\mathrm{is}\:\mathrm{controlled} \\ $$$$\mathrm{by}\:\mathrm{an}\:\mathrm{external}\:\mathrm{S}.\:\mathrm{the}\:\mathrm{protein}\:\mathrm{also}\:\mathrm{controls} \\ $$$$\mathrm{its}\:\mathrm{own}\:\mathrm{exprexxion}\:\mathrm{by}\:\mathrm{a}\:\mathrm{negative}\:\mathrm{feedback}. \\ $$$$\mathrm{The}\:\mathrm{fol}<\mathrm{lwing}\:\mathrm{system}\:\mathrm{of}\:\mathrm{ODEs}\:\mathrm{represents} \\ $$$$\mathrm{the}\:\mathrm{dynamics}\:\mathrm{of}\:\mathrm{the}\:\mathrm{system},\:\mathrm{with} \\ $$$$\mathrm{m}\:\mathrm{and}\:\mathrm{p}\:\mathrm{representing}\:\mathrm{mRNA}\:\mathrm{and} \\ $$$$\mathrm{protein}\:\mathrm{respectively},\:\mathrm{prove}\:\mathrm{that}\:\mathrm{for} \\ $$$$\mathrm{any}\:\mathrm{value}\:\mathrm{of}\:\mathrm{S}\geqslant\mathrm{0},\:\mathrm{the}\:\mathrm{steady}\:\mathrm{state}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{system}\:\mathrm{is}\:\mathrm{spiral}\:\mathrm{sink} \\ $$$$\frac{\mathrm{dm}}{\mathrm{dt}}=\frac{\mathrm{S}}{\mathrm{1}+\mathrm{S}}+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{P}}−\mathrm{m}\:\:\frac{\mathrm{dp}}{\mathrm{dt}}=\mathrm{m}−\mathrm{p} \\ $$

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