Lecture notes. Written from the material of MIT 8.591J 2004, licensed CC BY-NC-SA 4.0. These are notes, not a transcript: the material has been reorganised and rewritten. This adaptation carries the same licence, and the original is linked above.

6. Excitation and Adaptation in Chemotaxis¶

What this covers¶

This chapter follows the lecture that introduces the Spiro–Parkinson–Othmer (1997) model of excitation and adaptation in E. coli chemotaxis. The receptor complex at the heart of the model can be modified in three independent ways at once — ligand binding, phosphorylation, methylation — so the "obvious" chemotaxis diagram is really a lattice of dozens of distinct chemical species. The chapter asks two questions the lecture answers in turn: what assumptions cut that lattice down to something tractable, and how does the resulting model account for the experimental fact that a cell's tumbling frequency snaps down when attractant is added and then, over about a minute, climbs back to baseline even though the attractant never leaves? It assumes the two-component chemotaxis pathway (Tar, CheA, CheW, CheB, CheR, CheY, CheZ) from the preceding lecture.

The receptor complex and its three switches¶

The key player is the Tar–CheA–CheW complex: the aspartate receptor Tar, permanently bound to the histidine kinase CheA and its adaptor CheW. At any moment, three things can vary independently for one copy of this complex:

ligand binding — aspartate bound (\(LT\)) or not (\(T\));
phosphorylation — CheA has autophosphorylated, handing a phosphate onto the complex (\(T_\text{p}\)), or not;
methylation — how many of the receptor's methylation sites carry a methyl group.

Downstream, whichever phosphorylated species it is, the complex passes its phosphate on the same way: to CheY, giving \(\text{CheY}_\text{p}\), which drives the flagellar motor toward clockwise rotation (tumbling), and to CheB, giving \(\text{CheB}_\text{p}\); CheZ then strips the phosphate back off \(\text{CheY}_\text{p}\). Three independent binary/graded modifications on the receptor itself, feeding a common phosphorylation network downstream, is exactly the structure drawn in Figure 2 of Spiro et al.:

The receptor complex's state space: three highest methylation levels (2, 3, 4) times bound/unbound times phosphorylated/not. Every phosphorylated state (front layer) feeds the same phosphotransfer step to CheY and CheB, not drawn here.

Eight assumptions that make the lattice tractable¶

Left unconstrained, three independent modifications compound into a large reaction network. The lecture's model is built on eight simplifications, each removing one source of bookkeeping:

Tar is the only receptor type, and CheW and CheA are always bound to Tar — so a complex's whole identity is its \((T/LT,\ \text{methylation level},\ \text{p or not})\) triple; there is no separate free-CheA or free-CheW population to track.
Methylation happens in a specific order — the receptor's methylation sites fill and empty one at a time along a single sequence, rather than in any of the many possible orders.
Only the three highest methylation states matter — labelled 2, 3 and 4 in the diagram above, rather than the full range the receptor could in principle occupy.
Only \(\text{CheB}_\text{p}\) demethylates — demethylation is tied to CheB's own phosphorylation state, so the methylation dynamics inherit whatever happens to \(\text{CheB}_\text{p}\).
Phosphorylation of CheA does not affect ligand (un)binding — the ligand-binding equilibrium is the same whether or not the complex is phosphorylated.
CheR binding to the complex does not affect ligand (un)binding or phosphorylation of CheA — the methylation machinery does not itself perturb the other two axes.
CheZ is not regulated — \(\text{CheY}_\text{p}\) dephosphorylation runs at a fixed rate, not modulated by the signal.
Phosphotransfer from the complex to CheY or CheB does not depend on occupancy or methylation state — whichever phosphorylated species handed off the phosphate, it does so at the same rate.

Assumptions 5, 6 and 8 in particular decouple the three axes from each other wherever they can, which is exactly what turns a network on three interacting modifications into the separable lattice drawn above: a grid of methylation and ligand-binding states, one copy phosphorylated and one not, connected by the same phosphotransfer step regardless of which grid point it started from.

Two asymmetries hidden in the kinetics¶

Decoupling the axes does not make the complex's biochemistry symmetric along each of them. The lecture singles out three rate asymmetries — read directly off Figure 2 — that are exactly what drives the model's dynamics:

Ligand-bound states generally have lower autophosphorylation rates than unbound states at the same methylation level: binding aspartate makes the kinase worse at its job.
CheR methylates ligand-bound states more rapidly than unbound ones.
Higher methylation states autophosphorylate more easily than lower ones, at the same ligand-binding status: methylation makes the kinase better at its job, in the opposite direction to binding.

Put together: binding aspartate suppresses autophosphorylation, but it also accelerates methylation, and methylation itself restores autophosphorylation. That closed loop — one modification undermining kinase activity, the other, slower one restoring it — is the whole mechanism of adaptation.

A step in aspartate, on three timescales¶

The lecture walks through what happens to the population of complexes after a step increase in aspartate concentration, and the three timescales involved are well separated:

\(\sim 1\ \text{ms}\): ligand binding is fast, so the ligand-bound fraction \([LT]\) jumps up almost immediately.
\(\sim 5\ \text{s}\): because ligand-bound complexes autophosphorylate poorly, the total number of phosphorylated complexes now falls gradually — \(\text{CheA}_\text{p}\) drops, and with it \(\text{CheB}_\text{p}\). Low \(\text{CheA}_\text{p}\) means low \(\text{CheY}_\text{p}\): tumbling is suppressed. This is excitation.
\(\sim 50\ \text{s}\): with \(\text{CheB}_\text{p}\) now low, demethylation is effectively switched off (assumption 4: only \(\text{CheB}_\text{p}\) demethylates), while ligand-bound complexes are being methylated faster than usual. Methylation climbs, slowly and unopposed. \(\text{CheA}_\text{p}\) and \(\text{CheY}_\text{p}\) stay low for now — tumbling is still suppressed.
Longer still: because higher methylation states autophosphorylate more easily, the now more-heavily-methylated population slowly regains its ability to autophosphorylate even with the ligand still bound. \(\text{CheA}_\text{p}\) and \(\text{CheY}_\text{p}\) climb back to their original levels, and tumbling returns to its pre-stimulus frequency. This is adaptation.

Fast excitation, slow adaptation: kinase activity (solid) falls within seconds of the attractant step and climbs back to baseline over tens of seconds, driven by the slower rise in methylation (dashed) underneath it.

The whole point of the mechanism is that the two responses are on different clocks. The fast phosphorylation response is what lets the cell react to a change at all; the slow methylation response is what lets the same cell stop reacting once it has adjusted, so that a persistent stimulus does not leave it tumbling — or not tumbling — forever. That return to baseline in the presence of a still-bound ligand is what "adaptation" means here.

Where the lecture leaves it¶

The slide carrying this model is titled "fine-tuned model for perfect adaptation" — a name that concedes the model reaches perfect adaptation only for particular parameter choices, not as a structural guarantee. The lecture closes by pointing at Barkai and Leibler's 1997 "Robustness in simple biochemical networks" as the contrasting case: a model in which adaptation does not need to be tuned. The figures making that comparison were not available in this material (see Sources); working out exactly what "fine-tuned" is buying, and whether it can be removed, is where the following lecture picks up.

Sources¶

Slide deck: 08-notes.md (MIT OCW 8.591J, Fall 2004, lecture 8), reconstructed by a model from a PDF with no text layer — treat the prose as paraphrase and every equation as unverified. Covers: the Tar–CheA–CheW complex and its title slide; the eight numbered modelling assumptions; the autophosphorylation/methylation rate asymmetries; the timed walkthrough of the response to a step in aspartate (\(\sim 1\ \text{ms}\), \(\sim 5\ \text{s}\), \(\sim 50\ \text{s}\)); the closing reference to Barkai and Leibler (1997).
Figures extracted from the same source PDF: page 2 (the MCP/CheA/CheB/CheY/CheZ reaction scheme), page 4 (the state lattice reproduced above as an SVG), and page 11 (Figure 4 of Spiro et al., simulated \(\text{CheA}_\text{p}\)/\(\text{CheB}_\text{p}\)/methylation and tumbling-probability timecourses, redrawn schematically above rather than reproduced).
No transcript, written notes, or exercise sheet was supplied for this lecture.
Named but not contained: the primary source, Spiro, P. A., Parkinson, J. S., and Othmer, H. G., "A model of excitation and adaptation in bacterial chemotaxis," PNAS 94, no. 14 (1997): 7263–8, cited throughout via its figures rather than reproduced in full; and Barkai, N. and Leibler, S., "Robustness in simple biochemical networks," Nature 387, no. 6636 (1997): 913–7, whose Figures 2 and 3 were referred to on the closing slides but removed from the converted material for copyright reasons.

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