Past exam of the biological physics course of the University of Oxford / 2016 / 7 / b / Solution

Write $m=[E_m]$, $r=V_R[R]$ and $b=V_B[B]$. Saturated receptor methylation provides a constant positive flux, so the complete balance isAt a finite steady state, $r(K_B+m_{\ast })=b{m}_{\ast }$ and thereforeThe derivative of the right side is $-b{K}_B/(K_B+m{)}^2<0$, so this fixed point is stable in the scalar model. With total receptor concentration $E_T$, it must additionally lie between zero and $E_T$ and leave sufficient substrate for the assumed saturated receptor methylation. If $r\ge b$, the formal equation has no finite positive fixed point: finite receptor capacity then matters.

For an chemoattractant step, activity and hence tumbling frequency fall rapidly before receptor methylation can change appreciably. Slower chemical modification can then restore receptor signaling and make the tumbling frequency rise toward a new steady value. Perfect adaptation means return of that output to its original baseline despite the persistent chemoattractant step; it does not mean that receptor receptor methylation also returns to its original level.

The balance written here alone does not specify how ligand changes activity or receptor demethylation, and does not yet establish perfect adaptation. If activity is simply multiplied by a ligand-dependent fraction while this $m$ balance remains unchanged, $m_{\ast }$ is unchanged and the activity remains reduced. The following two models add different mechanisms that can compensate for that direct activity reduction.