Lecture 10: core · exposition · lineage · research essay

Perturbations distinguish mechanisms of adaptation

A specified output load separates candidate adaptive mechanisms. A measurement census and matched frequency perturbations can distinguish candidates that share an adaptive steady output. The inference depends on the observable, disturbance channel and operating range.

CCBS 2026 · Research essay for Lecture 10 · 13 October 2026

The same return to baseline can arise from different mechanisms. An incoherent feedforward loop can cancel the input at steady state. An integral controller can also return to baseline by accumulating the output error. A persistent input step establishes adaptation in both cases. To distinguish these mechanisms, perturb a process for which their predictions differ.

A first-order load on the output separates the two models compared here. The load changes the output equation of both models. In the incoherent feedforward loop it also changes the error-integrating identity, so the adapted output becomes input dependent. In the integral controller it leaves the controller identity intact, so a feasible, attracting rest retains the same reference value. The calculation is developed in §5.

Candidates that pass the steady test can still predict different measurements and dynamics. Section 5 computes a free-output versus total-output distinction caused by binding. Section 10 holds an antithetic controller's equilibrium and integral gain fixed, then compares actuation choices through small frequency perturbations. These tests distinguish specified candidate predictions. They do not uniquely identify a pathway.

A robustness claim must specify what is allowed to change. Alongside the observable and the disturbance class, specify the changes of the system under which the claimed property persists. We call these changes the robustness class. Preserving input invariance and preserving the reference value are different requirements. Section 1 defines both before comparing the literature.

The claim has a stated reach. For the candidate models specified below, displayed algebra establishes the predictions and numerical calculations check their implementation. Beyond those models it holds only where a theorem's hypotheses hold: Francis and Wonham's for linear regulators,⁷ and three later theorems for reaction networks, each under its own. It does not prove that every adaptive network has a physical feedback edge, nor that its memory integrates the error with a constant gain. A chosen change discriminates specified candidates. It does not by itself locate a molecule or recover a network.

1Specify the observable, disturbances and allowed changes

A claim that an output adapts robustly is complete only when it names the output, the inputs it adapts to, and the changes the adaptation survives. Published claims name different ones.

Adaptation is read from a step experiment, and the experiment already fixes two of the three named things. Hold an input at one level until the system rests, step it to another level, and hold it there. The output responds and then returns to its old value while the input stays at its new one. The quantity that is said to return is the claim's observable. The set of inputs, or more generally of constant disturbances, it returns from is the claim's disturbance class. A claim of perfect adaptation says that the steady output is the same for every disturbance in that class.

The observable is a choice, and one network can give three different answers. The Lecture 10 core's two-cycle feedback circuit shows it. An input I modifies a protein A to its active form $A^*$, which modifies a protein B to $B^*$. The active $B^*$ returns $A^*$ to A, and a fixed enzyme E returns $B^*$ to B. Every enzyme-substrate complex is kept: $C_3$ is $A^*$ bound to its substrate B, $C_2$ is $A^*$ bound to $B^*$, the complex that carries most of the memory, and $C_4$ is $B^*$ bound to E. The modified-B total changes at the exact rate $k_{B1}C_3 - k_{B2}C_4$, where $k_{B1}$ and $k_{B2}$ are the catalytic constants of B's two steps (core §7, exact total balances). In the joint regime $C_3 \approx q_{A^*}$ and $C_4 \approx q_E$, with the modified-A total $q_{A^*} = A^* + C_2 + C_3$ and the total $q_E$ of E, that rate becomes $k_{B1}(q_{A^*} - \sigma)$ with $\sigma = (k_{B2}/k_{B1})\,q_E$, so the modified-B total integrates the error of the modified-A total against the set point $\sigma$ (core §7).

Jeynes-Smith and Araujo reached the same distinction in their own search. In their version of the design, free $A^*$ "never tracks the estimated setpoint", while the pool $A^*+C_3$ does, within a tolerance and over a small range of inputs (p. 5).⁸ A claim that "the output adapts" must therefore say which of these quantities it means.

The third named thing is the class of changes the return survives, and it is the one that published claims vary most. A change here is a change of the system itself, made while the inputs keep stepping: a rate constant, an enzyme level, an added sink. Call the class of changes that a claim says its return survives the claim's robustness class. Table 1 reads ten published claims through the three names. The robustness classes range from a finite set of protein titrations to every parameter value of a fixed network.

Scope of the theorem rows, each stated in full where the essay uses it. Yi et al.'s argument is linear, about one equilibrium (Section 4). Araujo and Liotta's topological conditions are necessary, not sufficient across all parameter values (Section 9). Briat et al.'s conclusion is for the stochastic mean, needs ergodicity and bounded moments, and covers the annihilation controller, not reversible binding (Section 10). Gupta and Khammash's deterministic characterization assumes full row rank, an attracting componentwise-positive rest and locally rich perturbations, and proves no stability (Section 5).

"Robust" has two meanings in this literature, and they must be kept apart. In the first, adaptation persists after the change while the set point may move. Araujo and Liotta let the set point be a "'constant' that depends only on network parameters" (SI p. 4).⁴ Khammash grants an incoherent feedforward loop robust perfect adaptation because its adaptation "is robust to changes in the network parameters, even though changes in such parameters affect the steady state value of the output" (p. 515).⁹ In the second, the set point itself holds. Francis and Wonham ask that regulation be "preserved when certain system parameters are perturbed" (p. 457),⁷ and Gupta and Khammash ask that "the output set point ... remain invariant to the largest possible set of network disturbances" (p. 2).¹⁰

Two conventions for the numbers follow from the definition. Every $\Delta$ and $\varepsilon$ in Part III is a ratio of positive outputs: a concentration, an activity or a set point of 1. An output whose set point is zero, such as a signed deviation, takes differences instead. The demo of Section 2 reports the deviation $x^* = X^* - 1$, which equals $\Delta$ there because the old set point is 1.

The thesis concerns tolerance in this strict sense. Under the first meaning of "robust", designs with no memory of the error survive changes downstream of the input. The next section shows a subtractive compensator that stays input invariant under any load, and Section 5 shows an incoherent feedforward loop that stays input invariant when its output's removal constant changes. In both, only the set point moves. A claim that the return survives a change therefore says, in this essay, that the output returns to the same set point after the change, at every input of its class.

Two kinds of evidence carry the tolerance claims that follow. A universal statement, true for every input or every value of a constant, rests on displayed algebra: each design's steady output written as a function of its input and constants. The finite numerical checks of Part III, at one input pair or on a grid, illustrate that algebra and test the code that draws the figures. They prove nothing on their own.

Name the observable, the disturbance class and the robustness class of any adaptation claim, and decide whether its "robust" means that adaptation persists or that the set point holds. For a proposed change, compute the set-point shift $\Delta$ and the input dependence $\varepsilon(u)$.

2Compare input cancellation with error correction

One plant, three designs. What each design's memory reads, the input or the output's error, decides which changes it can answer.

When a persistent step moves the vector field off the old rest and the output still returns, the input is held in some other state. The Lecture 10 core gives the bookkeeping for models that settle with the output read from their state: the old rest is no longer a rest, the output is the same at the new one, so some other state differs (core A8). Call that state the memory. At rest it holds the action that cancels the input, in the controller's own units, not the input itself (core A9). A matched subtractive compensator is the exception: it cancels the step inside the same rate equation, so the vector field never moves and no other state has to change. What decides which changes a design survives is what its memory, or its compensation, reads.

Use the continuous-time plant of the core's A9 to give a load and a removal rate physical meaning. This is a continuous-time comparison, separate from the sampled plant in the core's opening experiment. Measure the output $X$ in units of its set point. Let the plant make $X$ at a basal rate $\gamma_0 = 1$, add an input $d$ to that production, and remove $X$ at rate $\gamma$ plus an optional first-order load $\ell$. A controller adds its action $u$:

With $\gamma = \gamma_0$, no load and no control, the plant rests at $X = 1$. The deviation $x = X - 1$ is the signed output deviation of that continuous-time plant. With $u = -k_I z$, equation (2) gives its integral loop (core A9, E1).

There are three designs, and each has a closed-form rest. Proportional feedback acts on the present error, $u = -k_P(X - 1)$. A subtractive compensator measures the input and subtracts it, $u = -c\,d$ with $c = 1$, matched to the plant. Integral feedback lets a memory $z$ change at the rate of the error, $\dot z = X - 1$, and acts through it, $u = -k_I z$. Setting the rates to zero gives, for every input $d \ge 0$ and every removal $\gamma + \ell > 0$,

The proportional loop holds the input as an error. Its rest rises with $d$ at slope $1/(\gamma + \ell + k_P)$, which is positive at every finite gain. It has no memory, so the output itself carries the input.

The compensator reads the input and stores nothing, so it answers only the change it measures. Its rest $1/(\gamma+\ell)$ contains no $d$: it is input invariant for every input and every plant, because it subtracts exactly the input it reads. But the rest equals the set point only while $\gamma + \ell = 1$. A doubled removal rate or a load $\ell = 1$ halves the set point, at every input. The compensator is the clearest case of the first meaning of "robust": adaptation persists and the set point moves.

The integral memory reads the error, so its rest is fixed by its own rate equation. The equation $\dot z = X - 1$ contains no plant constant, so any rest has $X = 1$, whatever $d$, $\gamma$ or $\ell$. The memory holds the action the plant needs, $u^* = -k_I z^* = \gamma + \ell - 1 - d$, and both are signed. The action is negative exactly when $1 + d > \gamma + \ell$, as after an input step on the unloaded plant. It is positive when the change removes more than the input adds, as under a load at $d = 0$, where $u^* = +1$ at $\gamma = \ell = 1$. The memory $z^*$ always has the opposite sign, and at the demo's starting settings $u^* = +0.5$ and $z^* = -0.25$. Whether the loop reaches its rest is a separate test. The characteristic polynomial is $s^2 + (\gamma+\ell)\,s + k_I$, so the rest attracts when $\gamma + \ell > 0$.

The boundary $\gamma + \ell = 0$ separates existence from attraction. With nothing removing $X$, the integral design still has its equilibrium, $X = 1$ and $z = (1 + d)/k_I$, but its eigenvalues are $\pm i\sqrt{k_I}$, so a trajectory circles the equilibrium for ever instead of reaching it. The compensator has no equilibrium at all: its constant addition $\gamma_0 = 1$ remains and nothing removes $X$. The demo below reports both facts from the equations, not from the trace.

From a design's equations, say what its correction reads, write its memory's identity, and predict which of an input step, a changed removal rate and a load it tolerates. Report a rest's existence and its attraction as separate facts.

3Robust regulation requires an internal model under stated hypotheses

Francis and Wonham's internal model principle requires an error-driven model of the disturbance only because it asks regulation to survive a stated class of plant changes. A smaller class asks for less.

The principle answers the question of necessity for linear regulators, under four hypotheses. Francis and Wonham pose the classical regulator problem "in the context of linear, time-invariant, finite-dimensional, multivariable systems": a plant, a separate compensator, and disturbance and reference signals produced by a known generator (p. 457).⁷ The compensator processes plant measurements, the reference, "and possibly the feedforward disturbance signal" (p. 457). It must keep the closed loop stable and drive the regulated error to zero, and a synthesis is called structurally stable when both properties "are preserved when certain system parameters are perturbed" (p. 457). In classroom form the hypotheses are a linear plant, a known class of signals, a stable loop, and regulation that survives small changes of the plant.

Under those hypotheses the controller must feed back the regulated variable and carry a model of the signals in that feedback path. Their Theorem 2 reads: "The synthesis is structurally stable at $A_3$ only if the compensator incorporates an internal model of $A_2$", where $A_2$ generates the exogenous signals and $A_3$ holds the plant parameters allowed to vary (p. 460). For steps the generator is $\dot w = 0$ and the model is an integrator (p. 461). The principle as they state it: "A regulator synthesis is structurally stable only if the controller utilizes feedback of the regulated variable, and incorporates in the feedback path a suitably reduplicated model of the dynamic structure of the exogenous signals" (p. 460).

The robustness class is part of the theorem, and the authors say that shrinking it shrinks the requirement. Directly after Theorem 2 they write: "if some of the elements of $A_3$ are not subject to variation then it may be that a full internal model is not required: a smaller class of parameter variations in general requires less reduplication in the internal model" (p. 460). This is the thesis of this essay in the theorem's own setting. The class of changes the regulation must survive decides how much internal model is needed and that it sits in the error's feedback path.

Model 1 shows the two ends of that statement. The compensator of Section 2 reads the disturbance itself, the feedforward signal the theorem allows, and with a fixed plant it needs no internal model: its rest is input invariant at every $d$. It fails as soon as the plant may change, because its rest $1/(\gamma+\ell)$ moves with the removal. The integral design has the error-driven integrator that the theorem demands for steps, and its rest $X^* = 1$ survives every change of $\gamma$ and $\ell$ that keeps the loop stable. Francis and Wonham also note what a damaged internal model costs. A slightly perturbed integrator leaves "a small steady state offset error", and "the accuracy of regulation depends on the fidelity of the internal model" (p. 461). That is the leak that growth dilution adds to a chemical memory in Section 6.

The hypotheses limit where the theorem applies. It is linear, its perturbations are small, the regulated variable must be readable from the measurements, and a technical condition restricts what the compensator learns from the disturbance directly (p. 460). The extension to nonlinear plants is local, with constant signals (p. 462). A chemical network that adapts may or may not meet these hypotheses, and each case must be checked before the theorem is invoked.

State the robustness class a regulator claim assumes, and the internal model the linear theorem then demands: for constant disturbances and small changes of the plant, an integrator of the regulated error in the feedback path. Check the theorem's hypotheses before applying it to a network.

4An internal model can be a combination of states

A theorem without a robustness hypothesis still finds an internal model in any adapting system that meets its conditions. It does not say whether that model is a separate controller, and the incoherent feedforward loop shows why it cannot.

Sontag's theorem reaches an internal model from adaptation itself, with no plant-controller split and no robustness requirement. He notes that the Francis and Wonham theory "applies to systems Σ which are already partitioned into a 'plant' plus a 'controller'" (p. 120).¹¹ Some extra condition is still needed, because a system that "simply outputs y = 0 for every possible input signal u does not contain any subsystem generating the signals u" (p. 120). His condition is signal detection: "the output must reflect sudden changes in the input" (p. 120), stated precisely as a uniform relative degree. Under it, with bounded trajectories and a class of signals from a Poisson-stable generator, the system "contains an output-driven internal model" of the signals (Theorem 1, p. 124). The theorem also assumes that certain vector fields constructed from the system are complete and commute with each other (Assumptions 2 and 3, p. 123), which supply global coordinates. Sontag notes that these hold automatically for linear systems and are "quite strong" for nonlinear ones (Remark 3.2, p. 125).

His own chemotaxis example finds the integrator as a combination of states. In a receptor model with unbound and bound receptors $x_1$ and $x_2$, the combination $B = x_1 + x_2$ obeys $\dot B = y$, which he calls "an integrator (internal model of a system which produces constant inputs) driven by the output y" (p. 124). No molecule is that integrator. It is found by a change of coordinates.

The linear necessity argument of Yi and colleagues also produces a coordinate, not a controller. They show that "robust asymptotic tracking, tracking that holds for parameter variations as well as input variations, holds if and only if integral control is a structural property of the system" (p. 4653).¹² The proof linearizes, $\dot x = Ax + bu$ and $y = cx + du$, and uses the zero steady gain $d - cA^{-1}b = 0$ to find a row vector $k$ with $k[A\ b] = [c\ d]$, so that $z = kx$ obeys $\dot z = y$ (p. 4653). Any linear system whose steady gain vanishes has such a coordinate, including the linearization of an incoherent feedforward loop.

The incoherent feedforward loop of the core has exactly such a coordinate. In the loop $\dot x_1 = k_1 w - \delta x_1 x_2$, $\dot x_2 = k_2 w - \gamma x_2$, the combination $z = x_1 - (k_1/k_2)\,x_2$ obeys

an integral of the output's error with a positive gain that depends on the state (core §10). Xiao and Doyle give the same construction for the sniffer, "z acts as a constrained integral variable" (p. 4).¹³ The coordinate fits Sontag's output-driven form, although his Assumptions 2 and 3 have not been checked for this loop (core A1). The identity (4) holds directly, without the theorem. Araujo and Liotta likewise call the feedforward balancing mechanism "a form of integral control that is distinct from the integral feedback control" of their opposer modules (p. 5).⁴

The identity is weighted in laboratory time. Set $\tau(t)=\int_0^t\delta x_2(s)\,ds$. Wherever the degrader is positive, equation (4) becomes $dz/d\tau=x_1^*-x_1$. The change of clock exposes the integral structure. It does not add a molecular feedback edge or establish every hypothesis of an internal-model theorem.

The chemical construction determines the domain of an integral identity. The core's three constructions make this explicit. Saturated removal of a memory, opposed by output-dependent production, gives an approximately additive memory inside a finite concentration regime. A multiplicative memory $\dot Z=\kappa Z(X-X_0)$ gives the exact logarithmic identity $d\ln(Z/Z_{\mathrm{ref}})/dt=\kappa(X-X_0)$ only for $Z>0$, with a positive reference concentration $Z_{\mathrm{ref}}$. The boundary $Z=0$ is absorbing and has no logarithmic coordinate. An antithetic pair instead carries a signed difference of two positive pools, equation (7). These constructions can agree at an adaptive rest while differing in range, boundary behavior and response to dilution.

So "the memory integrates the output's error" does not by itself separate feedback from feedforward. Both designs integrate an error somewhere. What separates them is which changes the integral relation survives. Xiao and Doyle state the criterion: a system has perfect adaptation "for all uncertain dynamics that keep the system stable and the integral variable z unchanged" (p. 3), and the property is "naturally robust to uncertainties that preserve stability of the system and structure of the integral variable" (p. 8). The robustness class therefore determines which of these candidate identities survives the specified changes. Part III computes that for the core's designs.

Tell what an internal-model theorem locates from what it only asserts exists. For an adaptive model, find the coordinate whose rate is a function of the output's error, write its identity, and ask which changes leave that identity intact.

5A first-order load separates the two adaptive mechanisms

The incoherent feedforward loop's coordinate's identity contains the output's own removal law. A first-order load on the output adds a term to it and breaks it, while the sequestering pair's identity contains no plant term and survives.

A change of the output's removal constant keeps the incoherent feedforward loop's identity's form. In equation (4) the rate constant $\delta$ of the output's removal by the degrader appears twice, in the gain and in $x_1^* = k_1\gamma/(k_2\delta)$. Doubling $\delta$ changes both, and the identity still says that $z$ rests only where $x_1 = x_1^*$. So the loop stays input invariant for every input $w > 0$ while its set point halves. In the language of Section 1, the change leaves $\varepsilon(u) = 0$ and makes $\Delta = -50\%$.

A first-order load adds a term that the identity cannot absorb. Let the output also be removed at the rate $\ell x_1$, independent of the input. Then $\dot x_1 = k_1 w - \delta x_1 x_2 - \ell x_1$, and the same coordinate obeys

A rest of $z$ no longer forces $x_1 = x_1^*$, the unloaded set point of equation (4). The loaded rest, written $\bar x_1$ to keep it apart from $x_1^*$, is

so the output depends on the input at every $w > 0$ and every $\ell > 0$. At the core's constants, a load $\ell = 1$ takes the output from 1 to 0.5, and a tenfold input step then takes it to 0.9091, a shift of +81.8% (core A16).

The load class must be named, because other loads behave differently. A removal of the output with the rate law $\ell' x_1 x_2$, proportional to the degrader, is also a load. It belongs to another class: it acts as a change of $\delta$ to $\delta + \ell'$, so the loop stays input invariant while its set point moves. The load class this essay tests is the first-order, input-independent one, the rate law $\ell$ times the output.

The sequestering pair's identity contains no rate law of the plant, so every rest of the pair sits at $X = \mu/\theta$, whatever acts on the output. The pair's controller species are made at the rates $\mu$ and $\theta X$, bind each other reversibly, and the complex is removed. With the totals $q_{Z_1} = Z_1 + C_{Z_1Z_2}$ and $q_{Z_2} = Z_2 + C_{Z_1Z_2}$,

because binding, unbinding and removal of the complex change both totals equally (core §8, exact difference identity). The plant's own reactions, its actuation, its removal and any load, never touch the controller totals. A load changes only where the actuator must rest. With the plant $\dot X = kZ_1 + d - (\gamma + \ell)X$, the actuator rests at $Z_1^* = \big((\gamma+\ell)\mu/\theta - d\big)/k$, which must be positive for the rest to exist. For this one-stage plant the rest is locally stable for every positive constant (core A14). Model 1's integral memory has the same structure, since $\dot z = X - 1$ contains no plant constant (Section 2).

The structure generalizes, under stated hypotheses, to a criterion on the stoichiometry. Gupta and Khammash ask for the largest robustness class in mass-action networks: a set point encoded by two reactions, with every other reaction allowed to change (pp. 3–4).¹⁰ Assuming full row rank of the stoichiometric matrix $\Gamma$, species by reaction (written $S$ in the paper), an attracting positive rest and locally rich perturbations, they show that such maximal robust perfect adaptation requires a combination of species $L = \xi^{\mathsf T} x$, with a coefficient vector $\xi$ (their $q$) satisfying $\xi^{\mathsf T} \Gamma = (\kappa, -1, 0, \dots, 0)$ (p. 4). Its rate involves only the two set-point reactions, every other rate law cancels, and when the two share a monomial factor $m_1(x)$ the identity reads $\dot L = \theta_2\, m_1(x)\,\big[\kappa\theta_1/\theta_2 - x_1^{\,r}\big]$ with $r > 0$: a weighted, transformed error, not the unweighted one. Their sniffer, an incoherent feedforward loop, adapts but is not maximally robust, since its set point $\theta_1\theta_4/(\theta_2\theta_3)$ depends on four constants (p. 2). The theorem proves no stability, and a network with conservation laws, such as the core's two-cycle feedback circuit, does not meet the full-rank hypothesis as written.

Write a memory's identity and read off which changes it survives: those that leave every rate law in it untouched, given a feasible, stable rest. Name the load class before testing it: a first-order load on the output breaks the incoherent feedforward loop's identity, and a load proportional to the degrader only moves the set point.

A readout can change the verdict without changing the mechanism

The load test compares specified rate laws and specified observables. The resolved feedforward mechanism binds free output $X_1$ to free degrader $X_2$ in a complex $C_{12}$. Catalysis removes the output and releases the degrader. Fast binding and mainly free pools supply the bilinear removal law used above. Outside that regime, the first question is whether the measured quantity still adapts.

The exact resting balances give different answers for free output and total output. Let $q_1=X_1+C_{12}$ and $q_2=X_2+C_{12}$. The input produces the two pools at rates $k_1w$ and $k_2w$. The degrader is removed at rate $\gamma$ in both free and bound forms. Removal of a bound degrader releases intact output. With association, dissociation and catalytic constants $k_{\mathrm{on}}$, $k_{\mathrm{off}}$ and $k_{\mathrm{cat}}$, define

At any finite positive rest with $w>0$ and $0<r<1$, the exact total balances and complex balance imply

Free output is input invariant, while total output grows with the input. A sensor that counts only free output and a fluorescence assay that counts both forms would therefore give different verdicts. Neither verdict requires a changed network. The extra $\gamma$ in $K_M$ is essential because removal of a bound degrader competes with dissociation and catalysis. This is an exact steady-state calculation for the declared mechanism. It does not prove nonlinear convergence. The core's finite-binding calculation and the exposition give the mechanism and kinetic-regime audit.

Measure the census before attributing a changed readout to failed adaptation. The two-cycle circuit of Section 1 requires the same discipline: its modified-A total includes both $C_2$ and $C_3$. Chemotaxis couples receptor methylation to CheA-dependent phosphorylation of CheB, as Section 8 explains.¹² That biological pointer does not identify the cell with the abstract two-cycle circuit or establish its joint saturation regime. A proposed perturbation test needs the actual reporter census and a candidate calculation for the actual operating conditions.

6Compare the same perturbations across candidate mechanisms

Applied to six designs from the Lecture 10 core, five changes sort them as Section 5 predicts. Only the designs whose memory identity contains no rate law of the plant tolerate a plant change or a first-order load.

The test applies one input step and five changes to six designs, each at the core's illustrative constants. The designs are Model 1's proportional loop, compensator and integral loop (input $d$ from 0 to 0.5), the chemotaxis teaching model (attractant $L$ from 0 to $5K_I$), the sequestering pair (an input $d$ from 0 to 0.5 added to the production of $X$, an extension made here) and the incoherent feedforward loop ($w$ from 1 to 10). The changes are the input step alone, a constant that sets the reference, a plant constant doubled, a first-order load $\ell = 1$, and growth dilution of every species at $\lambda = 0.1$. The steady states follow from the stated model equations and were checked by integration from the old rest. The chemical models have positive species at the tested rests. The linear controllers instead use signed coordinates. The tested rests are stable. Figure 2 shows the result as the steady output across each input step, before and after each change. Three cells are not defined: the compensator has no constant that sets a reference, and the chemotaxis teaching model has no removal of its output to load or dilute.

A plant change and a first-order load separate the designs whose memory identity contains no plant rate law from the rest. The integral loop and the sequestering pair tolerate both changes, by equations (3) and (7). The chemotaxis teaching model tolerates the plant change because its rest is the activity $a^* = K_B V_R/(V_B - V_R)$ (core A12, E3), which contains no constant of the receptor. Any change of the receptor's offset, its sensitivity to methylation or its attractant affinities leaves $a^*$ where it was, provided the methylation that achieves it stays between zero and the number of sites. The compensator and the incoherent feedforward loop stay input invariant under the plant change, while their set points halve. Under the load the compensator's set point halves, and the incoherent feedforward loop becomes input dependent, by equation (6). The proportional loop is input dependent under every change, by equation (3).

A change of a constant that sets the reference moves every design's set point the same way, so it cannot tell the designs apart. Halving CheR's rate moves the chemotaxis rest from 1/3 to 1/9. Doubling $\mu$ doubles the pair's set point $\mu/\theta$, doubling the reference of Model 1's integral doubles its rest, and doubling the degrader's production $k_2$ halves the feedforward set point. Each design stays input invariant. A titration of a constant that enters the set point therefore produces the same verdict, input invariant with a moved set point, in feedback and feedforward designs alike. Constants that do not enter the set point are tolerated by the feedback designs: doubling the pair's complex removal $\delta_C$, or the integral gain $k_I$, leaves both rests exactly in place.

In these models, which dilute every species at one rate and compensate for none, dilution adds a leak to each memory identity, so none of the candidates tolerates it exactly. Growth dilutes the pair's free controllers, and equation (7) gains the term $-\lambda(Z_1 - Z_2)$ (core A14). At $\lambda = 0.1$ the pair's rest falls 9.8% below $\mu/\theta$ and rises by 5.1% over the input step. A leaking integral memory in Model 1 gives a shift of −0.47% and an input dependence of +2.4%, and the diluted incoherent feedforward loop rises 9.8% over its tenfold step. In these uncompensated models, adaptation under growth is near-perfect at best, and how near depends on how fast the controller turns over against dilution (core A14). The matrix does not exclude other mechanisms that compensate for growth.

The time courses show the same sort, and they show where each memory holds the load. In Figure 3 the integral loop's memory moves to $z^* = -0.5$ under the load, so that its action $u^* = -k_I z^* = +1$ makes up the extra removal, and the pair's actuator doubles. The incoherent feedforward loop's coordinate also comes to rest, but at a point where the output is off its set point, because its rate equation now contains the load. Along the loaded run the pair's identity (7) holds to $4\times10^{-15}$, and the incoherent feedforward loop's identity (4) holds exactly until the load and then misses by the load term, $-71.9$ by $t = 110$.

Choose the change that tells two candidate designs apart. A plant change or a first-order load on the output separates a memory identity that contains the output's removal law from one that does not, and the load also separates the incoherent feedforward loop, which becomes input dependent, from the compensator, which stays input invariant with a moved set point. Recognize the changes that cannot: an input step, and a titration of a constant that sets the reference. In the stated models, read uniform dilution as a leak that none of the identities survives.

7Separate model discrimination from a necessity theorem

The conclusion holds at three strengths: established among the candidate designs, a theorem beyond them only under named hypotheses, and never a location of a molecule or a network.

Among the core's designs the conclusion is established by algebra and checked by computation. The designs that tolerate a plant change or a first-order load are exactly those whose memory identity contains no rate law of the plant: equations (3) and (7), and the chemotaxis rest $a^*$. The feedforward designs carry input invariance through a coordinate or a measurement, and lose the set point under the same changes. This is an inference among specified candidate mechanisms under declared perturbations.

Beyond the candidates, four theorems extend it, each only under its own hypotheses. Francis and Wonham's principle covers linear regulators that see the regulated error, with small changes of the plant (Section 3).⁷ Araujo and Liotta show that a network whose stimulus enters at the output node itself "cannot invoke the balancing mechanism to achieve RPA, and must instead rely on opposition" (p. 4, and their Corollary 2, SI p. 13).⁴ A first-order load is such a stimulus, so in their topological setting only the error-integrating opposer module can tolerate it. Gupta and Khammash's criterion covers the largest robustness class in mass-action networks, under the hypotheses stated in Section 5.¹⁰ Hirono, Gupta and Khammash show that integral feedback "is necessary (and sufficient) for kinetics-independent RPA in deterministic chemical reaction systems" (p. 013017-25), where kinetics-independent means that adaptation survives any change of the rate laws that keeps the system stable (p. 013017-7).¹⁴ Hirono and colleagues also say that the internal model principle is a family of results, not one: "There can be several instantiations of the IMP depending on the type of disturbances allowed and the nature of the robustness desired" (p. 013017-12).

An error integral may be weighted or approximate, and the claim allows both. Yi and colleagues' chemotaxis coordinate is a weighted approximation to total methylation, valid under four assumptions (pp. 4650–4651).¹² The incoherent feedforward loop's coordinate's gain $\delta x_2$ and Gupta and Khammash's factor $m_1(x)$ depend on the state. The core's zero-order integrators hold only inside a regime of saturated enzymes (core §4 and §7). None of the conclusions here needs a constant gain or a single molecule that integrates.

A chosen change discriminates candidates. It does not locate a molecule or a network. Model 1's integral memory, the chemotaxis methylation and the sequestering pair behave alike in every column of Figure 2 where each is defined. They tolerate the input step, the plant change and, where it is defined, the load. A reference change moves the set point of each, and dilution, where it is defined, breaks each. The robustness class cannot tell them apart. That takes experiments of another kind: removing the memory, as a cheR deletion does, or pushing the input past the memory's range (core A17).

A failed recovery is evidence against a feedback memory only when the mechanism's own conditions held. A feedback design can fail to return to its set point after a load for three reasons that leave its identity exact, and a fourth that changes it. The new rest may need a negative actuator, so that a controller pool grows without bound (core A14, windup). The rest may be unstable: with an mRNA stage the core's annihilation controller, at annihilation constant $\eta = 100$, is unstable for actuation gains from 2.0038 to 49.904 (core A14). The actuator may saturate. These three leave the identity exact. Leakage is different: dilution adds a term to the identity itself, as in Section 6. The refutation criteria below are stated with those conditions in place.

State the necessity of a memory with its strength: established among named candidates, a theorem under named hypotheses, or neither. Before reading a failed recovery as evidence against feedback, check reachability, stability, saturation and leakage.

8Chemotaxis: random twofold variation and chosen protein titrations kept the precision and moved the level

Barkai and Leibler varied a model's parameters, and Alon and colleagues titrated chosen proteins. Both found the precision of adaptation robust and the steady level and adaptation time not. That pattern does not by itself place the memory. The models derive an error-integrating memory, and one mutant and one deletion support parts of that mechanism.

Four objects carry the name "chemotaxis" and stay apart here. The pathway is the biology: attractant binds the receptor, the receptor-coupled kinase CheA phosphorylates the response regulator CheY that makes the motor tumble, CheR and CheB add and remove methyl groups, and CheA also phosphorylates CheB. Barkai and Leibler's reduced module is their eq. (1), $dE_m/dt = V_{\max}^{R} - V_{\max}^{B}\,A/(K_b + A)$, written for the modified-enzyme concentration $E_m$ and the activity concentration $A$ (p. 916).¹ Yi and colleagues analyze a weighted coordinate approximating total methylation under four assumptions (pp. 4650–4651).¹² The course's teaching model is a nonlinear model of the mean methylation per receptor that adopts the activity-only rate form (core A12, E2). Model 2 in Section 6 is that teaching model.

Barkai and Leibler's robustness class was a random sample of every biochemical parameter, and it found the precision robust and the adaptation time not. Their altered model systems were "obtained by randomly increasing or decreasing by a factor of two all biochemical parameters of the reference system", 6,157 of them, and most single rate constants "may be changed by several orders of magnitude without inducing a significant deviation from perfect adaptation" (p. 915). The adaptation time varied substantially, and they concluded that "Robustness is thus a characteristic of specific network properties and not of the network as a whole" (p. 915). The steady activity is not robust either: it "depends on the enzyme concentrations" (p. 916). The sample is finite and twofold. It is not every parameter value.

The mechanism they proposed is an error-reading memory, and they state its robustness class themselves. Robust adaptation follows "when the rates of the modification and the reverse-modification reactions depend solely on the system activity, A, and not explicitly on the concentrations Em and E", and they read this as "a feed-back system, in which the output A determines the rates of modification reactions" (pp. 915–916). From eq. (1) the activity rests at $A_{\mathrm{st}} = K_b V_{\max}^{R}/(V_{\max}^{B} - V_{\max}^{R})$. Its consequence for the class: "the system will respond and adapt to any environmental change that affects this activity" (p. 916). That is the property Section 5 derives for an identity with no plant term, stated for chemotaxis in 1997.

Alon and colleagues tested a finite set of protein changes against one stimulus, and their observable was a ratio within each strain. They "systematically varied the concentrations of the chemotaxis-network proteins" (p. 169) and stimulated cells with 1 mM L-aspartate.² Precision is "the ratio between the steady-state tumbling frequency of unstimulated and stimulated cells" (p. 169), measured within one strain. Over CheR between about 0.4 and 6 times wild type, the adaptation time, a half-recovery time, "varied more than 20-fold, from 23 ± 2 min to ~1 min", and the steady tumbling frequency changed about threefold, while precision held (p. 169). Table 1 adds CheB at 0.4 and 12 times wild type, a truncated CheB, CheY at 0.2 times, a deletion of CheZ and a fivefold rise of the whole operon (p. 171).

Read through the three names, the observed combination of preserved precision and a shifted baseline is shared by the candidate mechanisms compared here. That combination alone does not identify the memory. Precision compares a strain with itself, so it measures the input dependence $\varepsilon$ for the one tested step, and never the set-point shift $\Delta$ between strains. CheR and CheB set the reference, as $A_{\mathrm{st}}$ shows, and Figure 2's reference column leaves the error integrators and the incoherent feedforward loop alike input invariant with a moved set point. CheY and CheZ act downstream of the activity that methylation reads, so they change the map from activity to tumbling, and a design that keeps its precision under them is input invariant with a moved set point, whatever its memory. The paper asked a different question, "To differentiate between the fine-tuned and robust scenarios" (p. 169), and answered it.

Three other pieces of evidence support an error-integrating memory in chemotaxis: the models derive it, and two experiments support parts of the mechanism. The first is the rate law: in the models, demethylation that depends on the activity alone gives the identity of eq. (1), whose rest contains no receptor constant, and Yi and colleagues found the model's adaptation equally robust to receptor level, rate constants and activation probabilities (pp. 4649–4650). The second is a mutant. A truncated CheB that CheA cannot phosphorylate still gave exact adaptation, which "seems to exclude CheB phosphorylation as the mechanism of exact adaptation" (p. 170). It separates that phosphorylation feedback from demethylation that depends on the active conformation, and is consistent with the latter. The third is the deletion: a strain without CheR "did not tumble and could not adapt" (p. 169). That shows the memory's writer is needed. Each experiment is a specified change of the system and supports part of the mechanism, but neither uniquely identifies the error-integrating coordinate or the network's topology.

Sort a paper's perturbations by what they can discriminate. A change of the plant upstream of the integrated variable can separate a memory identity free of plant rate laws from one that contains the output's removal law. A titration of a reference constant, or a change downstream of the integrated variable, cannot.

9Searches over network designs fixed the input class, so both routes qualified

The searches that sorted adaptive networks into feedback and feedforward classes defined robustness against one input at one node. Under that class both routes adapt, exactly as the thesis predicts.

Ma and colleagues defined robustness as a count of sampled parameter sets that adapt to one input step. They stepped the input at node A from 0.5 to 0.6, asked for a sensitivity above 1 and a precision above 10, and judged "a circuit ... to be more robust if a larger number of parameter sets yield the behavior defined above" (p. 762), from 10,000 parameter sets sampled log-uniformly per network (p. 772).³ Of 16,038 three-node networks, 395 adapted robustly, and all contain a negative feedback loop with a buffering node or an incoherent feedforward loop with a proportioner node. In the feedforward class, node B "monitors the input and exerts an opposing force on node C" (p. 766). The robustness class was sampled parameters under an input step, and under it both memories qualify.

Araujo and Liotta asked for adaptation without fine-tuned parameters in a network of fixed topology, still for a stimulus at one input node. Their output node must return "regardless of the magnitude of the stimulus delivered to the input node, with no requirement for special (fine-tuned) parameter choices" (p. 11), while the set point is allowed to depend on the parameters (SI p. 4).⁴ Every such network decomposes into opposer modules, which compute "an integral of a tracking error", and balancer modules, of which Ma and colleagues' proportioners are special cases (pp. 4, 6, 10). These topological conditions are necessary, and the authors note that they "are not sufficient by themselves to guarantee the implementation of RPA across all possible parameter regimes" (p. 12). Their framework already contains the load test. When the stimulus enters at the output node itself, balancing is impossible and only opposition remains (Corollary 2, SI p. 13).

Later work named the input-node assumption as a choice. Gupta and Khammash note that earlier quantitative studies let disturbances enter "via a prespecified input node", and that biological disturbances "do not have a single well-defined entry point" (p. 1).¹⁰ Xiao and Doyle draw the design consequence. A controller built from species outside an unknown plant "always gives rise to a feedback system", while an incoherent feedforward loop "can only be obtained by a design problem where we are given how the disturbance enters the target variable" (p. 7).¹³ Jeynes-Smith and Araujo, searching the two-node reduction of Ma and colleagues' feedback design with every complex, add the first named thing: the quantity that adapts is the pool $A^*+C_3$, within ±1% and over a limited input range, not the free output (pp. 4–5).⁸

Read a design search's definition of robustness before reading its classes: say where its disturbance enters, which changes it samples, and whether its set point may move. A search whose disturbance enters only at an input node cannot separate the feedforward class from the feedback class.

10Steady adaptation leaves actuation dynamics and noise undecided

A downstream perturbation tests the antithetic identity. Once several implementations pass that test, a matched frequency perturbation can separate their dynamics. Intrinsic molecular noise requires a further stochastic test.

Briat, Gupta and Khammash claim tracking against every network parameter except the two that set the reference. In their stochastic setting, if the closed loop is ergodic, the controller imposes tracking "regardless of the initial conditions or model parameters (other than μ/θ). A sudden change in these parameters will result in the same steady-state value after a transient" (p. 20).¹⁵ Their worked example claims adaptation "with respect to any change in the parameters of the gene expression network" (p. 22), and their theorem needs an open loop that is already stable, "because our controller can only act positively on the open-loop network" (p. 21). The claim is about the mean of the regulated species. Their Theorem 1 assumes a closed loop that "is ergodic and has uniformly bounded first- and second-order moments" (p. 20), and their controller is the annihilation reaction, in which one molecule of each controller species removes the other. A pair that binds reversibly, as in the core's Model 4, keeps the same difference identity but is not covered by the theorem as stated (core A4).

Aoki and colleagues tested a changed removal rate of the regulated protein, the downstream class. In their circuit the sigma factor SigW drives both AraC, which closes the loop by regulating the anti-sigma factor RsiW, and a coexpressed reporter, sfGFP. An induced protease, MfLon, "recognizes cognate Pdt degradation tags appended to the C termini of AraC and sfGFP and increases protein degradation" (p. 536).⁵ The perturbation therefore loads the sensed output, not only the reporter.

The measured class agrees with the inferred memory, approximately and within the tested conditions. The memory is the difference of the two controller species, the identity (7), which contains no rate law of the plant and so predicts that an added protease is rejected. The authors name what keeps the agreement approximate. Their theory assumed unsaturated production and "negligible degradation of controller proteins Z1 and Z2. In living cells, these assumptions are not satisfied" (p. 535). Their claim that every such controller must contain the antithetic motif concerns a stochastic controller class with a split between controller and plant (Box 1), not every deterministic adaptive network.

Dilution is the change that the antithetic memory does not tolerate exactly. Qian and Del Vecchio show that dilution makes an integral memory leak, so that "the adaptation error can be arbitrarily large" (p. 4), and prove that the error becomes negligible "if the reactions implementing the integral controller are all much faster than dilution" (p. 1).¹⁶ Olsman, Xiao and Doyle find the trade from the other side: controller degradation "decreases the fragility of the system at the cost of introducing steady-state error" (p. 283), and that error is parameter dependent enough that it "may undermine the whole purpose of the circuit" (p. 284).¹⁷ Section 6 gave the core's number: a 9.8% shortfall at $\lambda = 0.1$.

Hold the steady prediction fixed, then perturb at different frequencies

Actuation choices can share the same adaptive steady output and differ in dynamics. Reference-only actuation makes the output's production increase with $Z_1$. Sensor-only actuation makes that production decrease with $Z_2$. Sensor-and-reference actuation uses both. All three retain the difference identity (7). Filo and colleagues show how sensor actuation introduces a proportional contribution alongside integral action, which changes the dynamic response.²¹ This motivates a second discriminating experiment after a constant perturbation has left the implementations indistinguishable.

A fair local comparison holds the operating point and low-frequency integral gain fixed. Choose a concentration scale $q_{\mathrm{ref}}$ and a removal-rate scale $\gamma_{\mathrm{ref}}$. Let $X^*=Z_1^*=Z_2^*=q_{\mathrm{ref}}$, $\theta=\eta q_{\mathrm{ref}}=\gamma_{\mathrm{ref}}$, and $\mu=q_{\mathrm{ref}}\gamma_{\mathrm{ref}}$. The common production flux is $h^*=q_{\mathrm{ref}}\gamma_{\mathrm{ref}}$. Define the actuation slopes $g_1=\partial h/\partial Z_1$ and $g_2=-\partial h/\partial Z_2$ at that rest. The three slope pairs $(g_1,g_2)/\gamma_{\mathrm{ref}}$ are $(2,0)$, $(0,2)$ and $(1,1)$, respectively. The positive law

realizes all three slope pairs at the declared common rest. The comparison concerns small perturbations of this rest. It does not equate the three laws away from it.

The linearized controller exposes what this experiment measures. Write the production perturbation as $\delta u=-C(s)\delta X$, where $s$ is complex frequency. With $\bar s=s/\gamma_{\mathrm{ref}}$ and $\bar C=C/\gamma_{\mathrm{ref}}$, the three controller transfer functions are

Each has the same low-frequency coefficient $K_I=\lim_{s\to0}sC(s)=\gamma_{\mathrm{ref}}^2$. Here $K_I$ is an integral gain with units of inverse time squared. It is unrelated to the ligand-affinity constant of the chemotaxis model. Let a one-stage plant relax at $\gamma_{\mathrm{ref}}$. For the two-stage plant, an upstream species $M$ drives the output at that same rate, and both species relax at that rate. The upstream species has the matched rest $M^*=q_{\mathrm{ref}}$. Their normalized production-to-output transfer functions are $\bar P=\gamma_{\mathrm{ref}}P=1/(\bar s+1)$ and $\bar P=1/(\bar s+1)^2$. The loop transfer function is $L=PC=\bar P\bar C$. Define the sensitivity function $S=1/(1+L)$.

The same zero sensitivity integral permits different peaks and stability margins. All six full equilibria in Figure 5 are locally stable. The loop transfer functions have no poles in the open right half-plane and satisfy $sL(s)\to0$ at high frequency. The Bode sensitivity integral therefore gives $\int_0^\infty\ln|S(i\omega)|\,d\omega=0$ in every case.²³ This balances attenuation and amplification against linear frequency. The area under a graph with a logarithmic horizontal axis is not that integral. Neither the zero integral nor the adaptive steady output determines the maximum sensitivity $M_S=\sup_{\omega\ge0}|S(i\omega)|$ or the phase margin, the extra phase lag at unit loop gain that reaches the instability boundary.

The injection point determines the measured disturbance response. Figure 5 normalizes output perturbations by $q_{\mathrm{ref}}$ and imposed flux perturbations by $h^*$. Add a small bounded sinusoidal production flux $w(t)$ upstream. The output response is $\delta X=PSw$. Instead perturb the output's removal constant by $\delta\gamma_X(t)$. To first order this adds the output flux $-X^*\delta\gamma_X(t)$ downstream. In the two-stage plant the response is $\delta X=-X^*QS\delta\gamma_X$, with $Q=1/(s+\gamma_{\mathrm{ref}})$. The upstream species does not filter this downstream injection. Both experiments use the same sensitivity function. They use different transfers from disturbance to readout, as Figure 5 shows. A fitted transfer can reject a candidate whose predicted response differs. Other mechanisms may share that transfer.

A small frequency experiment cannot diagnose every finite-amplitude failure. The calculation assumes a feasible stable rest, a small perturbation, and negligible controller dilution. Increase the perturbation until the required action exceeds an actuator's available range, and those assumptions can fail. The integral identity can remain exact while a controller pool grows without bound because the target is unreachable. This is windup. The core's full frequency analysis separates the local comparison from the feasibility and nonlinear calculations. A small-signal resonance and a finite-amplitude loss of range are different observations.

Intrinsic molecular noise is a separate source of evidence. A prescribed noisy production signal uses the transfer $PS$. A prescribed noisy output-removal signal uses $QS$ and its stated flux conversion. Their output variances depend on the imposed spectra as well as those transfers. Intrinsic molecular noise instead originates in every stochastic reaction, including sensing, reference production, annihilation and plant turnover. Its variance requires those reaction channels, their intensities and correlations. A common sensitivity integral cannot replace that calculation. Kell, Ripsman and Hilfinger use a linear noise approximation and stochastic simulations to show that sensor-only actuation can reduce intrinsic noise while retaining adaptation in their specified models.²² Their result also distinguishes the idealized reference-only actuation penalty from controllers with small leakage. It is not a universal theorem that adaptation amplifies noise.

The published experiments and the model comparison answer different questions. Filo and colleagues measured AraC fused to mScarlet-I after an aTc perturbation of their genetic circuit. Their experimental comparison used sensor actuation, filtered proportional control and open-loop circuits. They explicitly did not construct an intein-based controller with reference actuation for a direct experimental comparison (printed p. 10).²¹ The reference-versus-sensor advantage is supported by their theoretical and computational analysis. Figure 5 proposes a matched local-frequency comparison. It does not reproduce an experimental comparison in that paper.

Use successive tests for successive questions. A constant downstream perturbation tests the steady identity. A matched small frequency perturbation tests actuation and plant dynamics through its specified injection channel. A stochastic reaction model tests intrinsic noise. Agreement in one test does not supply the result of the others.

11Input scale and Ras: classes that do not place a memory

Fold-change detection is invariance of a whole response to the scale of the input, a property of the input class. The Ras experiments tested input level only. Neither class separates a memory identity that contains the output's removal law from one that does not.

Fold-change detection asks more than adaptation, and it is a property of the input class. Goentoro and colleagues defined it as a response for which a step from level 1 to 2 "provides exactly the same output" as a step from 2 to 4 (p. 894), and showed that an incoherent feedforward loop has it because "the repressor Y acts as a memory of the activator X" (p. 896).¹⁸ They also note that "Not all circuits with precise adaptation, however, are expected to show fold-change detection" (p. 897). Shoval and colleagues showed that fold-change detection entails exact adaptation, and that a stable, exactly adapting system has it when its equations are unchanged by scaling the input and an internal variable together (pp. 15995, 15997).¹⁹

Both feedforward and feedback memories can carry fold-change detection, but neither the core's incoherent feedforward loop, a sniffer, nor its linear integral loop has it. Shoval and colleagues find that the sniffer, in which the memory speeds the output's removal, "does not show FCD", and that integral feedback "does not provide FCD in its commonly used linear form" (pp. 15997–15998). The feedback design that has it is a nonlinear pair, with memory $x$, output $y$ and input $u$: $\dot x = x\,(y - y_0)$ with $\dot y = u/x - y$ (their eqs. 11 and 12, p. 15998). Its memory equation "multiplies the error $y - y_0$ by $x$" and its output equation "uses a ratio-based controller $u/x$", one for each of their two scaling conditions (p. 15998), so the pair supplies the symmetry, not the multiplicative memory alone. That memory also integrates the error in its logarithm, which is the multiplicative route of the core (core §3b). Input scale is one more class that does not separate the two routes.

The Ras experiments tested the input class, and the model selection rested on the transients. Takeda and colleagues stepped cAMP over concentrations "ranging from 1.0 × 10⁻² to 1.0 × 10³ nM", and the reporter returned to its prestimulus level within 35 s at every concentration.⁶ The reporter is cytosolic RBD-GFP, whose intensity falls when it binds Ras-GTP at the membrane. They fitted a feedforward model and an integral-feedback model to the same data, "21 data points" for "nine free model parameters", and rejected the feedback model because its "adaptation kinetics" depended strongly "on the size of the stimulus step". Latrunculin B removed actin-dependent feedback and left adaptation intact. They also allowed that "adaptation is achieved downstream from the receptors and upstream from RasGEF".

Read through the three names, the inference is a model selection among two candidates, and a downstream change would test it. Steps of input level are the class to which both fitted models adapt exactly, so adaptation itself could not decide, and the decision came from the shape of the transients within the two fitted models. Latrunculin B removes one feedback route, but it neither loads activated Ras nor changes its removal. The thesis proposes the change that would test those two fitted mechanisms: a first-order load on Ras-GTP, removing it independently of the input. In the fitted feedforward model, whose error-integrating coordinate contains activated Ras's own removal through RasGAP, such a load would move the set point. In the fitted integral-feedback model it would not. This is a proposed test of the two fitted mechanisms, not a decisive test of every possible topology. Whether the reporter itself, which binds Ras-GTP, acts as such a load is the question of the Lecture 10 worked extension.

Name the change that would test a published feedforward inference: a first-order load or an input-independent change of the output's removal, applied with the input held at several levels. Do not read input-level invariance, or fold-change detection, as evidence for either route.

12Choose a perturbation with different model predictions

Among candidate mechanisms, the change an experiment applies separates a memory identity that contains the output's removal law from one that does not. The strongest comparison is a specified load with the reference held fixed. Its outcome can reject a candidate model only after checking that the model could still reach its set point.

Each change an experiment can apply decides some questions and not others. Table 3 collects them for the three kinds of design whose predictions differ. The matched compensator stores nothing. The incoherent feedforward loop's error-integrating coordinate contains the output's removal law. The error integrators, the integral loop and the sequestering pair, have identities that contain no plant rate law. A doubled removal constant separates the error integrators from both others, and a first-order load separates all three. The other rows treat the designs alike or decide a different question. Once candidates agree in the steady tests, the frequency test of Section 10 can still distinguish their actuation and plant dynamics.

An experiment must separate measurement choice, local dynamics and loss of operating range. Table 4 gives a protocol for the candidates that remain after the steady test. A binding-aware readout checks what was measured. A matched sinusoidal perturbation checks a transfer through a declared channel. A larger perturbation checks the range in which the local prediction remains applicable. Intrinsic noise then asks a stochastic question that none of these deterministic tests answers by itself.

Among the candidates, and within the cited theorems' settings, nothing rejects a first-order load without an identity free of plant rate laws, so the first criterion is not met. Among the core's designs, the two that tolerate the load, the integral loop and the sequestering pair, both have one, equations (3) and (7). The theorems of Section 7 exclude the alternative in their own settings. Absolute concentration robustness, the other structural robustness known for reaction networks, concerns a different class: one species takes the same value "in every positive steady state the system might admit", whatever the conserved totals (Shinar and Feinberg, p. 1389).²⁰ As Xiao and Doyle note, perfect adaptation in general "could reject disturbances other than the total concentration" (p. 5).¹³ A network outside the candidates and the theorems' settings is not covered by the claim, so it could not refute it either.

No measured response read here contradicts a named fitted model with its conditions checked, so the literature criterion is not met either. In chemotaxis the titrations could not discriminate and agree with the inferred error-reading memory, and the truncated-CheB result agrees with activity-dependent demethylation (Section 8). In the synthetic controller the protease test is the downstream class, and the closed loop rejected it as the antithetic identity predicts, approximately and within a roughly threefold output range (Section 10). The Ras inference has not met the proposed test: its robustness class was the input level, which both fitted models pass (Section 11).

Four questions remain open in this analysis, and each names the change or the hypothesis it needs. The first is the proposed load test of the two fitted Ras mechanisms in Dictyostelium, with the reporter's own binding to activated Ras accounted for. The second is a measurement of the set-point shift $\Delta$, not only of the input dependence $\varepsilon$, under a change of the receptor alone, a plant change upstream of the variable that methylation integrates. Alon and colleagues changed receptors only together with CheR, CheB, CheY and CheZ. The third is how a fitted biological robustness class extends beyond the small local perturbations computed here. Section 10 supplies one controlled frequency comparison. It does not establish nonlinear adaptation to arbitrary time-varying disturbances or prove intrinsic-noise bounds for an unmodeled pathway. The fourth is a criterion for networks with conservation laws: this essay has not applied one to the core's two-cycle feedback circuit, which Gupta and Khammash's full-rank hypothesis, as stated, does not cover.

Reaction orders and perturbation tests answer successive questions. Binding allocations determine which effective laws a mechanism can realize. A perturbation test then asks which of those laws continues to restore the specified output. Lecture 11 develops the first calculation. The comparison here supplies the second.

A discriminating perturbation makes candidate mechanisms predict different observations. A constant first-order load separates the memory identities compared here. A measurement census can distinguish changed binding allocation from changed free output. A matched small frequency perturbation can distinguish actuation and plant dynamics after their steady outputs agree. Each comparison needs its own observable, disturbance channel and operating conditions.

The steady necessity claim keeps its original scope. Among the stated candidates, tolerance of a first-order load or changed output-removal rate requires a memory identity with no plant rate law. Beyond those candidates, necessity follows only where a cited theorem's hypotheses hold. The frequency comparison adds candidate predictions. It does not strengthen that necessity claim or uniquely locate the memory.

Choose a perturbation from Table 3 that splits the steady predictions. Specify the reporter census. If candidates still agree, choose a matched dynamic test from Table 4 and state its injection point. Measure the set-point shift as well as the input dependence. Check reachability, stability, saturation and leakage before interpreting a failed recovery. A test rejects a candidate prediction when its conditions held. It does not identify a unique molecule or network.

References

Every source was read in the full text under the project's archive, at the pages cited. Local links point into that archive, which is not redistributed. The Lecture 10 core is cited by section as "core".

The perturbation matrix and load-test model predictions are computed by analysis/robustness_classes.py and checked by analysis/verify_essay.py beside this file. Figure 5 and Table 2 use the matched local models of analysis/frequency_discrimination.py, checked against the full controller-and-plant Jacobian. These model calculations are not fits to experimental data. Figure 4 instead shows measurements transcribed from Alon and colleagues' Table 1.