= Negative feedback
<Negative feedback> closes a causal loop whose net effect opposes an initiating change. In <signal transduction>, activation followed by a downstream inhibition of the earlier activating stage is one example of <negative feedback>. Loop sign alone does not establish <asymptotic stability> or complete <signaling adaptation>; see the worked <signaling feedback> example.
A useful biological example is <chemotaxis feedback regulation>: an <Escherichia coli> <biological cell> adjusts its response to a sustained attractant instead of remaining at its immediate response. Increased <CheA> activity promotes <CheB> <phosphorylation>; <CheB> reverses <receptor methylation>, opposing their activation. After attractant addition, reduced activity instead favors <receptor methylation> by <CheR>, restoring activity. This is a return from <receptor> activity to the machinery regulating that activity. The response need not return exactly at every dose. <Precision and Kinetics of Adaptation in Bacterial Chemotaxis by Yigal Meir and others>{ref}
A motivating measurement is the <chemotaxis FRET assay>. <CheY> and <CheZ> carrying yellow and cyan fluorescent tags were observed in immobilized <Escherichia coli> <biological cells> on a coverslip in a flow chamber. Adding and removing attractant changed the yellow-to-cyan emission <ratio>, reporting the association of phosphorylated <CheY> with <CheZ>. This measures a molecular readout upstream of motor behavior. Comparisons with <biological cells> lacking <CheR>, <CheB>, or both tested the dependence on <receptor> modification; these perturbations also change starting activity and sensitivity, so they are not pure changes of one <feedback gain>. <Receptor sensitivity in bacterial chemotaxis by Victor Sourjik and others>{ref}
The later <chemotaxis feedback regulation> experiment used only the <Tar receptor> and nonmetabolizable methylaspartate, supplied and removed by flow. Long recordings separated slow recovery from incomplete <signaling adaptation>; correction for bleaching addressed drift in the optical readout. Activity recovered incompletely after sufficiently large additions, with detectable imprecision already at $10\,\mathrm{\mu M}$ in that preparation. Restricting the <receptor> type and preventing consumption of the stimulus help interpret the perturbation, but limit transfer to other <receptors> and conditions. The readout infers <CheA> activity through <CheY>-<CheZ> association, rather than directly measuring every molecular step. <Precision and Kinetics of Adaptation in Bacterial Chemotaxis by Yigal Meir and others>{ref}
<Negative feedback> can also overshoot or sustain <oscillations> if its return arrives too late. The worked <signaling feedback> example gives the contrasting instantaneous case; the <delayed negative feedback model> below isolates the effect of timing. Loop sign describes causation, while <asymptotic stability> describes what happens to displacements over time.
\Image[/negative-feedback-delay-modes.png]
{title=<Negative feedback> with an instantaneous or delayed return}
{description=Original synthetic <delay differential equation> modes, not measured <chemotaxis> responses. All variables and time are normalized. The same negative return coefficient is used throughout; only its delay changes. Each curve has its own matching history. A growing <oscillation> signals instability of this linear <mathematical model>, not a bounded biological rhythm.}
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