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

Let chemoattractant reduce the activity per methylated receptor by a factor $f$, with $0<f\le 1$, taking the initial factor as one. The active output is $a=fm$. If $V_B$ changes to $V_B^{\prime }$ but receptor demethylation still acts on total methylated receptor, the new output isDemanding $a_{\ast }^{\prime }=a_{\ast }=K_{B}r/(V_B[B]-r)$ givesReduced receptor demethylation lets methylated receptors accumulate to $m_{\ast }^{\prime }=m_{\ast }/f$, compensating exactly for their reduced activity. During the transition the activity first falls, then rises as receptor methylation accumulates. This compensation works only if the required methylated concentration is accessible; in particular $m_{\ast }/f<E_T$ in a finite receptor pool.

The required relation contains both enzyme concentrations. At fixed rates, increasing CheR raises the baseline receptor methylation and tumbling output, while increasing CheB lowers them. After such a change, the previous fine tuning generally no longer gives perfect adaptation. For example the required reverse capacity is $b^{\prime }=fb+(1-f)r$, so changing $r$ without retuning $b^{\prime }$ changes the adapted output ratio. It is the precision of adaptation that is fragile here, in addition to the baseline itself changing. Changing all relevant capacities together in a way preserving the relation is a special exception, not generic robustness.