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.

7. Perfect Adaptation by Model Reduction¶

What this covers¶

Lectures 7–8 built the E. coli chemotaxis network out of its parts — the Tar receptor complex, its phosphorylation and methylation states, and the pathway down to the flagellar motor — and then ran into a puzzle: the resulting model does not reproduce perfect adaptation, the experimental fact that after a step change in attractant concentration the internal signal eventually returns to exactly its pre-stimulus value, whatever the size of the step. This chapter follows the sequence of four reductions used to find out why, each one adding a single assumption backed — or explicitly not backed — by an experiment, until what is left over adapts perfectly by construction rather than by luck. It closes with the lecture's pivot to the next topic, biological oscillators, via a refresher on fixed points and linear stability built around the autocatalysis example from Problem Set 1. It assumes the biological cast introduced in L7 (Tar, CheA/CheW, CheB, CheR, CheY, CheZ) and the combinatorial receptor-state model built in L8.

The question inherited from L7–L8¶

L7 asked what each molecule in the pathway actually does. L8 assembled a model of all the reactions available to the receptor — every combination of ligand-bound or not, phosphorylated or not, and methylated to varying degrees — and found that it did not reproduce the observed perfect adaptation. The project spanning L8 and this lecture is to strip that full model down to its essential reactions, one assumption at a time, and see where the discrepancy actually comes from. The lecture is explicit that the assumptions along the way are "experimentally justified (or sometimes not)" — worth keeping in mind, because not every step in the reduction rests on the same strength of evidence, and the chapter flags which is which as it goes.

The lecture re-shows the basic behavioural picture as a reminder of what perfect adaptation is a property of: without an attractant, runs are interrupted by tumbles at random; with an attractant gradient, the cell compares the concentration now to a few seconds ago (temporal sensing) and suppresses tumbling while things are improving. A second recurring figure (Spiro, Parkinson & Othmer 1997, fig. 2) attaches timescales to the underlying chemistry: ligand binding and unbinding is fast, phosphorylation ($Y\rightleftharpoons Y_p$ for CheY, $B \rightleftharpoons B_p$ for CheB) is intermediate, and methylation is slow. That separation of timescales is what licenses folding ligand binding into an effective, $L$-dependent rate constant everywhere below: by the time phosphorylation or methylation matters, ligand binding has already equilibrated. The key player throughout, per Spiro et al.'s model, is the Tar–CheA–CheW receptor–kinase complex.

A two-state model, and the safe-zone problem¶

The starting point for this lecture — already the first of the four reductions — keeps only the two highest methylation levels of the receptor, written $2$ and $3$, each of which can be phosphorylated or not. Two processes act on a receptor: phosphorylation and dephosphorylation ($k_{\text{eff}1}(L)$ and $k_{\text{eff}2}(L)$ going up, a constant phosphotransfer rate $k_{\text{pt}}$ coming down), and methylation and demethylation ($k_{\text{eff}4}(L)$ toward higher methylation, $k_{\text{eff}3}(L)$ back). Every rate that carries an $(L)$ is an effective rate: because ligand binding has already equilibrated, $k_{\text{eff}}(L)$ is really a bare rate averaged over how often the receptor happens to be occupied at concentration $L$. Only $k_{\text{pt}}$, which by assumption does not care about occupancy, stays a bare constant.

At whatever steady state the (comparatively fast) methylation reactions settle into, define the fraction of receptors sitting at the higher methylation level:

\[\alpha \equiv \frac{[3]}{[2]+[3]}\]

The net phosphorylation rate seen by the rest of the pathway is the population average of the two methylation states' own rates:

\[k_{\text{phos}} = (1-\alpha)\, k_{\text{eff}1}(L) + \alpha\, k_{\text{eff}2}(L)\]

Perfect adaptation means this net rate returns to its pre-stimulus value once the system has re-settled at the new $L$ — that is, $k_{\text{phos}}$, evaluated with whatever $\alpha(L)$ the new ligand concentration produces, is flat as a function of $L$. That is a question about the shapes of $k_{\text{eff}1}(L)$ and $k_{\text{eff}2}(L)$, and it is not automatic.

Each curve is one methylation state's ligand-dependent phosphorylation rate. Where the "safe zone" exists, the population trades off between the two states as $L$ changes and the weighted average $k_{\text{phos}}$ stays flat — but this depends on the specific shapes of the two curves. For another choice of the same two curves there is no such band anywhere, and the model never adapts perfectly at all. Redrawn schematically from the lecture's plot.

The lecture's own instruction for the working case is to "fine-tune... $k_{\text{eff}1}$ and $k_{\text{eff}2}$ so that $\alpha$ falls in [the] safe zone" — which is exactly the fragility the rest of the reduction sequence is trying to engineer away. A model that adapts only because its rate constants happen to sit in the right range is not a satisfying account of a phenomenon the experiments show is robust.

Second reduction: CheB only demethylates active receptors¶

Additional assumption: CheB removes methyl groups only from phosphorylated receptors, never from unphosphorylated ones. This is the weakest-supported step in the sequence — the lecture says plainly that it is "not possible to directly measure" whether CheB really is restricted to active receptors. The indirect evidence offered instead is that the rate of methylation drops immediately after ligand is added, which is what would happen if the demethylating enzyme's substrate is the population that ligand binding is suppressing. Structurally, the effect is to delete the demethylation arrow from the unphosphorylated chain: unphosphorylated receptors can now only move up in methylation level, never down.

Third reduction: methylation at saturation¶

Additional assumption: $[\text{CheR}] \ll [\text{receptors}]$, so the methylating enzyme CheR is working at saturation and its rate no longer depends on receptor concentration at all — call this constant influx $r_{\text{in}}$. Unlike the last step, this one has a direct check: the Michaelis constant for CheR binding its receptor is much smaller than the receptor concentration, so essentially all of CheR is receptor-bound and $R_{\text{tot}} \sim R_{\text{bnd}}$. The lecture marks this "ok". Structurally, the two separately-tracked methylation-level-2 states disappear from the model, replaced by a constant flux $r_{\text{in}}$ feeding directly into methylation level 3.

Fourth reduction: an exactly adapting module¶

Additional assumption: the demethylation rate itself does not depend on ligand occupancy or on methylation level — $k_{\text{eff}4}$ becomes a genuine constant instead of a function of $L$. This is also checked directly: demethylation kinetics turn out to be "almost independent of level of methylation and ligand binding," and the lecture again marks it "ok". What survives all four steps is a minimal two-state module.

The module left after all four reductions. $C^*$ is the phosphorylated, high-methylation receptor that feeds phosphate onward toward CheY; $C$ is its unphosphorylated counterpart. Both receive the same constant input $r_{\text{in}}$; only $C^*$ leaks out, at the constant rate $k_{\text{eff}4}$. The ligand-gated step $k_{\text{eff}2}(L)$ only moves receptor between $C$ and $C^*$ — it never appears in their combined steady-state balance.

Writing $C^*$ for the phosphorylated, high-methylation receptor and $C$ for its unphosphorylated counterpart, the module's dynamics are linear:

\[\dot C^* = -(k_{\text{pt}}+k_{\text{eff}4})\,C^* + k_{\text{eff}2}(L)\,C + r_{\text{in}}$$ $$\dot C = k_{\text{pt}}\,C^* - k_{\text{eff}2}(L)\,C + r_{\text{in}}\]

Add the two equations. The $k_{\text{eff}2}(L)$ terms cancel exactly, and so do the $k_{\text{pt}}$ terms, leaving

\[\dot C^* + \dot C = -k_{\text{eff}4}\,C^* + 2r_{\text{in}}\]

At steady state this gives

\[[3_{\text{p}}] = C^* = \frac{2 r_{\text{in}}}{k_{\text{eff}4}}\]

— a result with no $L$ in it anywhere. Because $r_{\text{in}}$ was made $L$-independent by the third reduction and $k_{\text{eff}4}$ was made $L$-independent by the fourth, the steady-state level of the active receptor cannot depend on $L$, whatever the ligand-gated activation rate $k_{\text{eff}2}(L)$ is doing on its own. This is perfect adaptation for any value of $L$, not merely inside a fine-tuned safe zone: the fragility from the first reduction is gone because the two rates that could have carried $L$-dependence into the steady state have each, on separate experimental grounds, been argued to carry none.

From switches to oscillators¶

Having closed the chemotaxis question, the lecture turns to what comes next: biological oscillators, motivated by the cell-division cycle and circadian rhythms. The question posed is deliberately informal — what do you need to build an oscillator, and how is that different from the systems already met, switches and the chemotactic network just finished? The answer flagged is not a result but a technique: a graphical way of representing differential equations, to be developed in the next lecture under the name of nullclines. What follows here is a refresher on the pieces that technique will need.

Refresher: fixed points and linear stability¶

The running example, revisited from Problem Set 1, is autocatalysis:

\[A + X \underset{k_{-1}}{\overset{k_{+1}}{\rightleftharpoons}} 2X\]

With $x = [X]$ and $a = [A]$ held constant (an enormous surplus of $A$), the rate equation is a single first-order ODE of the form $\dot x = f(x)$:

\[\dot x = k_1 a x - k_{-1}x^2\]

The lecture's recipe for any such equation is two steps: find the fixed points, then check their stability.

Fixed points. Setting $f(x^*)=0$ and factoring, $f(x) = x(k_1 a - k_{-1}x)$, gives two roots:

\[x_1^* = 0 \qquad\qquad x_2^* = \frac{k_1 a}{k_{-1}}\]

Stability. Perturb slightly away from a fixed point, $\eta(t) = x(t) - x^*$, and Taylor-expand:

\[f(x^*+\eta) = f(x^*) + \eta f'(x^*) + O(\eta^2) \approx \eta f'(x^*)\]

Since $f(x^*)=0$, this leaves $\dot\eta \approx \eta f'(x^*)$, a linear equation with exponential solution $\eta(t) \sim \exp(f'(x^*)\,t)$. So the sign of $f'(x^*)$ decides everything: the perturbation grows if $f'(x^*)>0$ (unstable fixed point) and decays if $f'(x^*)<0$ (stable fixed point). For $f(x) = k_1ax - k_{-1}x^2$, $f'(x) = k_1a - 2k_{-1}x$, so $f'(x_1^*) = k_1a > 0$ (unstable) and $f'(x_2^*) = k_1a - 2k_1a = -k_1a < 0$ (stable): one unstable fixed point at the origin, one stable fixed point at $x_2^*$.

Flow along the phase line: any perturbation away from $x^*_1=0$ grows, and every trajectory on either side of $x_2^*$ is carried back toward it — matching the lecture's "guestimate of dynamics" sketch of $[X]$ against time for different starting conditions.

Setting up a two-variable system¶

The lecture's last move is to point out that the fourth-reduction module is itself already an example of the next technique, just written in biological variables. Relabelling $x \equiv C^*$, $y\equiv C$, the two equations above become a generic linear system in two variables:

\[\dot x = ax + by + r_{\text{in}} \qquad\qquad \dot y = cx + dy + r_{\text{in}}\]

with $a \equiv -(k_{\text{pt}}+k_{\text{eff}4})$, $b\equiv k_{\text{eff}2}$, $c \equiv k_{\text{pt}}$, and $d \equiv -b$. Nothing new is being modelled here — it is the same chemotaxis module reused as the worked example for the graphical, two-variable machinery (nullclines) that the next lecture builds.

Sources¶

Slides: Wrapping up E. coli Chemotaxis (L7 & L8), lectures/09-notes/01-wrapping-up-e-coli-chemotaxis-l7-l8.md — the L7/L8 recap, the timescale figure (Spiro, Parkinson & Othmer 1997, fig. 2), the four-state box of the first reduction and its safe-zone plots, and the second and third reductions.
Slides: Fourth reduction, lectures/09-notes/02-fourth-reduction.md — the fourth reduction and its exact steady state, the pivot to oscillators, the Problem Set 1 autocatalysis refresher, and the two-variable rewrite of the reduced module.
No transcript, written notes, or exercise sheet was supplied for this lecture, so nothing from a spoken account or a problem set appears here beyond what the slides themselves state. "Problem Set 1" is named on the slide as the source of the autocatalysis example, but its actual questions were not supplied and are not reproduced; the next lecture's nullcline technique, referenced at the end, is likewise named but not covered here.
Both source files are machine reconstructions of a PDF deck with no text layer (their own front matter reads "fidelity: reconstructed... every equation is unverified"); the reaction diagrams in particular have been read here for their qualitative structure — which reactions are present or absent at each step — rather than trusted digit-for-digit.
Figures cited on the slides but not reproduced in the source markdown: Spiro, Parkinson & Othmer, "A model of excitation and adaptation in bacterial chemotaxis," PNAS 94 (1997): 7263–8; Mittal, Budrene, Brenner & Van Oudenaarden, "Motility of Escherichia coli cells in clusters formed by chemotactic aggregation," PNAS 100 (2003): 13259–63.

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