How five verdicts found their regimes

How the field came to model the sign of regulation, ultrasensitivity, multistability, adaptation and oscillation with effective rate laws, where binding and catalysis re-entered, and why each published verdict keeps its place.

CCBS · Lecture 9 · lineage · 7 October 2026

The core of Lecture 9 builds five behaviors from binding and catalysis: the sign of regulation, ultrasensitivity, multistability, adaptation and oscillation. This page tells how the field came to model those behaviors with effective rate laws, what each step of that history asked and measured, and where binding and catalysis re-entered the description. The sequence is conceptual, and each source carries its own date. It does not claim one uninterrupted line of influence.

The historical comparisons live here in full. Each published verdict is read for the question it asked, what its authors built or measured, the model they expected to hold, and the regime in which the verdict holds. The exposition derives the course's own results that this page quotes.

Four kinds of evidence are kept apart throughout: a measured curve, a reconstruction fitted to data, a model prediction at a source's own parameters, and a present-day calculation made for this course.

1Five verdicts, each holding under the conditions it assumed

Each of the five behaviors acquired a verdict from an effective rate law. An effective rate law gives the rate of a process without resolving all of its intermediate steps. It may depend on several concentrations. The simplest depend on one regulator concentration, such as a Michaelis-Menten or Hill law standing in for the binding layer that produces it. A verdict here means a published conclusion about what a mechanism can or cannot do. Activator and repressor were treated as equivalent designs. A sharp switch in a covalent-modification cycle was traced to saturated enzymes. A toggle switch was said to need cooperative repression, and so was a ring oscillator. Negative feedback was credited with robustness, and integral feedback with perfect adaptation. The incoherent feedforward loop was written as an exact cancellation of its input.

Every one of these verdicts followed correctly from its assumptions. Each assumed something about the binding layer that the effective rate law does not show: a ligand in excess of its receptor, a promoter in trace against its repressor, a bound repressor spared from removal, an enzyme saturated by its substrate. The history below is about the assumptions, and about what happened when an experiment or an analysis left them.

A verdict can hold in three different ways. A theorem about a reduced model holds exactly under its hypotheses. A rate law reached as a limit of a binding mechanism holds approximately, and better the deeper the system sits in that limit. A measurement supports a mechanism over the range it tested.

Four questions recover the assumptions from any source. Which molecular state carries the activity? Which amount does the experiment or the cell control, a free concentration or a total? Which states does removal act on, free species only or every state alike? Lecture 8 calls the answer to this third question the removal variant. How tight is binding against the totals? The answers are the parameter conditions under which a result holds. Lecture 8's binding-catalysis model makes all four explicit: binding at equilibrium allocates each total among free species and complexes, and catalysis and removal change the totals. The binding part of the conditions is a regime in Lecture 8's sense, a choice of which species dominates each conserved total.

The table pairs each verdict with the condition it assumed and with the work in which binding and catalysis re-entered.

Read this way, the verdicts and the re-entries in the table do not contradict each other. Each holds in the regime that its assumptions select. Part I follows the verdicts, Part II the re-entries, and Part III shows how one modeling framework, the binding-catalysis model class, places them all.

Before weighing a published verdict about one of the five behaviors, write down the four conditions it assumed: the active state, the controlled amount, the removal variant and the binding strength against the totals.

2A saturation curve and a fitted exponent became the rate laws of regulation

The Michaelis-Menten hyperbola entered as a statement about excess. Michaelis and Menten's 1913 invertase paper computed the enzyme-substrate complex from mass action with a dissociation constant. They then replaced free sucrose by total sucrose, because only a vanishingly small fraction of it is bound by enzyme.¹ The hyperbola is the limit approached as the substrate total comes to exceed the enzyme total, and it is accurate while binding depletes a negligible share of the substrate.

Hill's sigmoid entered as a fitting device. In 1910 A. V. Hill fitted the oxygen dissociation curves of haemoglobin in several salt solutions with $y=100\,Kx^n/(1+Kx^n)$, in his letters. The fitted exponents ranged from about 1.4 to 3.2. Hill wrote that his object was to see whether an equation of this type could satisfy all the observations, rather than to give $n$ and $K$ a direct physical meaning.² Two constants then summarized a whole curve.

The two forms became the production laws of gene regulation. A repressed gene is written as $\beta/(1+(u/K)^n)$, with maximal rate $\beta$, regulator concentration $u$, half-maximal input $K$ and Hill coefficient $n$. An activated gene is written as its complement, and both inherit the two histories. The input $u$ is a free concentration, which equals the regulator's total only when the regulator far exceeds its binding sites. The exponent is a fitted steepness. Weiss's 1997 review made the point about the exponent plain: for a receptor with several sites, the Hill coefficient estimates the number of sites only under marked positive cooperativity, and it is better read as an interaction coefficient.³

Both inheritances point back to the binding layer. A free input equal to the total input describes a regime, the one in which the regulator's free species dominates its own total. An exponent above 1 needs a mechanism, which the law does not name. The rest of this page follows each behavior to the point where the excess failed or the unnamed mechanism mattered.

Read a Michaelis-Menten or Hill law with the excess condition that makes its input a free concentration, and read a fitted Hill coefficient as a summary of steepness rather than a count of binding sites.

3Activator and repressor were compared as equivalent designs

Savageau asked in 1977 why some genes are switched on by an activator and others by removing a repressor. Both designs turn a gene on when a signal appears. He asked whether the difference is a historical accident between functionally equivalent solutions to one regulatory problem, or a property that selection maintains.⁴

His method was a controlled comparison carried out in mathematics. Comparing the lactose and maltose operons directly confounds the mode of control with everything else that differs between them. Savageau instead compared two model systems identical in every respect except the type of control mechanism, a comparison hard to arrange in experiments but open to mathematical analysis.⁴

The answer was the demand rule. Repressor control correlates with low demand for expression of the regulated genes, and activator control with high demand. The proposed mechanism is mutational. The same kinds of regulatory mutation tend to make a repressor-controlled system constitutive and an activator-controlled system super-repressed. Selection keeps the mode that suits the demand, and the other mode is lost through genetic drift. He supported the rule with a survey of inducible catabolic pathways, repressible biosynthetic pathways, inducible drug resistance and prophage induction.⁴

The comparison took the two designs to compute the same response. Matched on everything but the mode of control, they differed only in how they fail under mutation and in what selection favours. The lecturer's 2026 draft names this as the premise shared by the analyses that followed: the behavioral capacity of the two architectures was taken to be the same.⁵ The premise holds in the limit where the signal is in excess of the regulator and the regulator in excess of the gene, which is the input of the effective rate law of §2.

Lecture 9 inherits the method and then moves one total. Its comparison of two models matches them at a reference condition before changing anything, which is Savageau's controlled comparison. It then moves a total that the cell controls. §8 shows what that move revealed.

Compare two designs by matching them on everything except the difference in question, and then ask which amount the match held fixed.

4A sharp switch was traced to two saturated enzymes

Goldbeter and Koshland asked in 1981 how a pathway can switch on and off with a small change in its effector. Cooperative interactions, and an effector acting at more than one step of a pathway, were known answers. Their paper examined covalent modification instead: one converter enzyme modifies a protein, a second reverses the modification, and the protein's total is conserved.⁶

Their model kept saturation and set depletion aside. Earlier analyses of such cycles had assumed first-order kinetics and negligible enzyme-substrate complexes. Goldbeter and Koshland let the converter enzymes saturate while keeping them scarce against the substrate. When both enzymes run near saturation, a small change in the ratio of their activities flips the modified fraction. They named this zero-order ultrasensitivity.⁶

The result was a theoretical regime, stated with its conditions. The paper is a calculation, and it reports no measured switch. It notes that enzyme-to-substrate ratios of 0.1 or less change the sensitivity little at its parameters. It states the condition as one or more converter enzymes operating outside the first-order region.⁶ A steep response then had two candidate sources, cooperative binding and catalytic saturation, and both were summarized by a Hill coefficient.

Lecture 9's analysis sharpens the condition into a regime with two parts. One saturated enzyme caps the Hill coefficient at 2, and an abundant phosphatase caps it again by holding the modified substrate. An unbounded switch needs both enzymes saturated and both scarce, and it switches where their capacities match. These are model calculations made for the course. Each part is a dominance statement about the substrate and enzyme totals.

Treat zero-order ultrasensitivity as one regime of a modification cycle, with both enzymes saturated and both scarce, rather than as the only route to a switch.

5The first synthetic circuits were designed with Hill laws

In 2000 two groups asked whether ordinary regulatory parts could be wired into a memory and into a clock. Known switches and oscillators, such as the phage λ switch and circadian clocks, used specialized components. Gardner, Cantor and Collins, and Elowitz and Leibler, each built a circuit in E. coli from well-characterized repressors.^{7, 8}

The toggle switch was measured to hold two states. Gardner and colleagues joined two repressible promoters in mutual inhibition. Five of the six variants they constructed were bistable. Cells flipped by a pulse of inducer or heat stayed in the new state, and one variant held both states for 22 hours.⁷

The toggle's design model wrote each repression as a Hill law. In their Box 1 the synthesis rates are lumped parameters, and two exponents, which they call $\beta$ and $\gamma$, describe cooperativity from repressor multimerization and multiple operator sites. The nullclines cross three times because of their sigmoidal shape, which arises for $\beta,\gamma>1$. To obtain bistability, they concluded, at least one repressor must act with cooperativity greater than one, and reducing the cooperativity shrinks the bistable region.⁷ Inside the model the statement is exact.

The repressilator was measured to oscillate with a period of about 150 minutes. Elowitz and Leibler joined LacI, TetR and λ cI in a ring with a fluorescent readout. Single cells oscillated noisily, with periods around 150 minutes, about three times the cell-division time.⁸

The repressilator's model was meant to find a class of behavior, not to describe the cell. Their Box 1 says so directly: too little was known to describe the system precisely, so the model aimed to identify possible classes of dynamic behaviour and the parameters to adjust. It used Hill coefficient 2. Oscillations were favoured by strong promoters, tight repression, cooperative repression and comparable protein and mRNA lifetimes. The paper's word is favoured, not required. Its stochastic version obtained cooperativity by giving each promoter two operator sites.⁸

The design model and the measurement agree in scale. At the Box 1 parameters, with Hill coefficient 2 and an mRNA stage, the model's steady state is unstable and its predicted period is about 125 minutes, a present-day calculation for this course. The measured period was about 150 minutes.⁸ Cooperative repression with the promoter in trace is the regime in which this design model holds.

Separate what a synthetic-circuit paper built and measured from the design model it used, and read the model's conditions as guidance for that circuit's parts.

6Analysis made cooperativity a theorem about Hill laws and protected repressors

Four analyses made the requirement exact, each for its own class of models. They span thirty years. Take them in order.

Tyson and Othmer bounded biochemical feedback loops in 1978. For a straight-through loop of $n$ steps, sustained oscillation needs a degree of cooperativity above $\sec^n(\pi/n)$, which is 8 for three steps. They added that time delays, coupled loops or spatial localization could allow oscillation at smaller cooperativity or shorter loops.⁹

Cherry and Adler made the toggle condition exact in 2000. They defined the largest log-slope of a repression function, which they identified with an effective power function or reaction order. For two mutually repressing genes, a product of the two largest log-slopes above 1 suffices for multiple equilibria at some removal rates, and a product of at least 1 is necessary. Michaelis-Menten repression has largest log-slope 1, so it cannot make a switch.¹⁰ For Hill laws the product is $\beta\gamma$, so the Gardner toggle can hold two states only when $\beta\gamma>1$.

Müller and colleagues closed the ring with explicit promoter binding in 2006. Their generalized repressilator kept gene-repressor complexes and degraded only mRNA and free protein, so bound repressor is spared. With three genes and no leaky transcription, the central equilibrium is stable whenever the Hill coefficient is below 4/3.¹¹ Promoter binding enters their stability condition only through a factor that rescales the ratio of degradation rates.

Blossey, Cardelli and Phillips stated the requirement in general form in 2008. Their stochastic gate model oscillated without cooperativity. They wrote that cooperativity is required to bring about oscillations in a deterministic gate dynamics, and they applied Müller's bound to show that their deterministic ring with Hill exponent 1 does not oscillate.¹² In their gate equations a transcription factor is not consumed when it binds a gate, so its free amount equals its total.

By 2006 the requirement read as a consensus. Lipshtat and colleagues opened a paper on the toggle by noting that numerous studies had concluded that cooperative binding is a necessary condition for its two stable states.¹³

Each theorem holds exactly for its class of models. One class is repression in a free input, which is the promoter in trace. The other is explicit binding with bound repressor spared from removal. Within either class no later work contradicts these theorems. §10 and §12 follow what happened outside them.

Attach its model class to every necessity claim: "cooperativity is required" is a theorem about repression in a free input, or about rings that spare bound repressor, and it holds exactly there.

7Feedforward adaptation was written as an exact cancellation

Adaptation asks a circuit to return to its old output after a sustained change of input. Feedback is one answer. The incoherent feedforward loop is another: the input drives the output and also a controller that opposes it, so that a ratio cancels the input.

Tyson, Chen and Novák wrote the canonical form in 2003. In their sniffer a signal makes both the response and a controller, and the controller removes the response. In their letters $\dot R=k_1S-k_2XR$ and $\dot X=k_3S-k_4X$, so the steady response $k_1k_4/(k_2k_3)$ does not depend on the signal.¹⁴ The removal term is bilinear, a mass-action law between controller and response. In binding-catalysis terms it holds while both stay mostly free.

The motif census of 2002 put the incoherent loop on the map. Milo and colleagues defined network motifs as patterns of interconnection that occur far more often than in randomized networks, and the feedforward loop was among them.¹⁵ In E. coli, about 85% of feedforward loops are coherent.¹⁶ Mangan and Alon then showed that incoherent loops act as sign-sensitive accelerators and can generate pulses, which is a transient, not yet perfect adaptation.¹⁷

Ma and colleagues located the condition inside the rate laws in 2009. Searching three-node enzyme networks for adaptation, they found that the working incoherent loops share a proportioner node whose steady level tracks the input. In their Michaelis-Menten equations this needs the proportioner's production saturated and its removal linear.¹⁸ The cancellation is exact inside those regimes of the effective rate laws.

The cancellation does not say which molecules implement it. A controller can degrade the output, or it can repress the output's gene. Both compute the same ratio, and their effective rate laws make them equivalent. §14 follows the designs that tested each implementation.

Read a perfect-adaptation claim as an exact cancellation inside one rate law, and ask which binding regime makes that rate law hold for the molecules that implement it.

8A total titrates the repressor's free state but not the activator's complex

Two lines of work reopened the comparison of §3. A measurement showed a repressor titrated by the copies of its own target, and the lecturer's theory showed why only the repressor can be titrated into a threshold. Take the measurement first.

Brewster and colleagues asked in 2014 what happens when one repressor is shared among many copies of its gene. Most models of transcription picture a single promoter copy. Real genes sit on multicopy plasmids or in several chromosomal copies, and a transcription factor binds many sites.¹⁹

They measured that expression escapes repression once gene copies outnumber repressors. The Lac repressor's target sat on a multicopy plasmid, or was integrated at five chromosomal sites, which should give about ten gene copies by the end of the cell cycle. The fold-change in expression stayed near 1 while repressor copies were fewer than gene copies. As repressor copies passed the gene copies it dropped sharply to ordinary repression. A thermodynamic model predicted these curves without free parameters.¹⁹ A Hill law in the repressor concentration has no promoter total in it, so it cannot represent this measurement.

The lecturer's 2022 thesis located the difference between the two signs in the active state. Take one binding pair with free species E and S and complex $C_{ES}$. If S activates, the active form is the complex, and its reaction order in the total $q_S$ stays between 0 and 1. If S represses, the active form is the free species E, whose order in $q_S$ can become hypersensitive and tend to $-\infty$ under tight binding when the two totals are close.²⁰

The 2026 manuscripts carried the contrast to induced regulators. The draft shows that a repressor inhibited by its signal can reach an arbitrarily large reaction order in the signal total, while the activator architectures stay at most linear. Under the Michaelis-Menten assumptions, signal in excess and regulator in excess of the gene, every behavioral distinction among them disappears.⁵ The supplement adds that both designs are routinely summarized by one increasing Hill curve once basal level, range, midpoint and steepness are matched, and that one fitted induction curve does not identify a mechanism.²¹

Savageau's premise and the new contrast hold in different regimes. With the signal in excess, activator and repressor compute the same function, and the comparison of §3 holds. With the signal's total under the cell's control and binding tight, the repressor switches where the signal total passes the repressor total, while the activator stays graded. Lecture 9's activator-repressor example realizes both regimes at matched parameters, as a model calculation. Matched in free inducer, the two circuits coincide. In the inducer total, the repressor's threshold follows its own total while the activator's half-rise barely moves.

Ask which state carries the activity before reading a sign of regulation: a total can titrate a free repressor into a threshold, while an activator's complex rises at most in proportion to its input total.

9Titration and negative cooperativity put the threshold at a total

Two lines of work moved the threshold of an ultrasensitive response from a fitted constant to a total. Take titration first, then negative cooperativity.

Molecular titration made equivalence the threshold

Buchler and Louis asked in 2008 how sequestration into an inactive complex shapes a response. Homodimer cooperativity gives a Hill coefficient of at most 2, which they called mild. A titrant that binds an active protein into an inactive complex acts instead as a sink, and the protein stays inactive until its total passes the titrant's. In the course's notation, for an active protein A and titrant B, the threshold sits near $q_A\approx q_B$, and the ratio $q_B/K_d$ sets its sharpness.²²

They showed that degradation changes the switch inside a cell. When monomers and complex are degraded at one rate and the complex dissociates faster than it is degraded, the in vivo sharpness equals the in vitro one. Fast degradation of the complex weakens the switch, and passive dilution of every state strengthens it.²² The removal variant was already part of the mechanism.

Buchler and Cross measured the switch in yeast in 2009. A basic leucine zipper activator sequestered by a dominant-negative inhibitor gave apparent Hill coefficients up to 12. The threshold and the steepness were set by the inhibitor's abundance, as their model predicted.²³

Negative cooperativity made the ligand total the input

Ha and Ferrell asked in 2016 what happens when a ligand cannot be assumed in excess. The classical analysis of a two-site receptor assumes many more ligand molecules than receptors, which suits oxygen and haemoglobin or a drug in the circulation. In that regime negative cooperativity makes a response more graded. In intracellular signaling, they noted, an upstream regulator may be as scarce as its target.²⁴

Their theory found the opposite effect in the ligand total. Suppose only the doubly bound receptor is active and binding is tight. Negative cooperativity then makes the first sites a stoichiometric buffer that soaks up the first ligand without a response. The response turns on sharply once the buffer is full, and its effective Hill exponent approaches about 8.04 as the cooperativity $c=K_1/K_2$ tends to zero. That number is a model limit.²⁴

Their experiments measured the threshold with DNA strands. A receptor strand held at 1000 nM bound two ligand strands, whose overlap set the cooperativity. Pairs designed for negative cooperativity gave increasingly marked thresholds, with $c$ fitted as the only adjustable parameter, for example $c=0.022\pm0.014$ for one pair. Varying the receptor total gave the predicted rise and fall of the complete complex, in good agreement with theory.²⁴

Both lines place the threshold at a total. Titration puts it where the input total passes the titrant total. Negative cooperativity puts it where the ligand total fills the receptor's first sites. A Hill function fitted at one partner total stores the threshold as a constant, so it misses the switch when that total moves. In Lecture 9's receptor example, a Hill function fitted at one receptor total misses the half-rise about threefold at a third of that total or three times it, a model calculation.

Locate a threshold at a total match, expect it to move when the partner's total moves, and check which states are removed before trusting its sharpness in a growing cell.

10Depletion of free repressor made toggles without cooperative binding

The paper that made the toggle condition exact also named a way around it. Cherry and Adler noted in 2000 that the Michaelis-Menten form of repression rests on ignoring the depletion of free repressor by binding. Keeping the depletion, with free and bound repressor decaying at one rate, gives a repression function whose largest log-slope exceeds 1 exactly when the operator total exceeds the dissociation constant.¹⁰

They stated the conditions and two cautions. Simple mass action with one repressor site per gene copy can make a switch if depletion of free repressor is significant. That needs more than one operator copy per cell and a small dissociation constant.¹⁰ With a single operator site per cell the Michaelis-Menten form is exact, so the route needs several sites. And the small dissociation constant it needs might make the switch fragile to stochastic fluctuations.

Lipshtat and colleagues found the removal variant decisive in 2006. Rate equations for a toggle with explicit binding and no cooperativity had one positive steady state when bound repressor was never degraded. With degradation of bound repressor, bistability appeared already in the rate equations.¹³

Chen and Arkin built such a switch in 2012. They noted that synthetic bistable circuits had so far relied on cooperativity, with few possible exceptions. A sigma factor and its anti-sigma factor, combined with positive feedback, gave a bistable switch in E. coli, and the sigma factor showed no intrinsic cooperativity.²⁵

Santos-Moreno and colleagues explained a CRISPRi toggle by the same route in 2020. The lack of cooperativity in CRISPR interference had been flagged as an obstacle to multistable circuits. Their CRISPRi toggle was bistable. They accounted for it by depletion of free repressor, here through nonspecific binding of dCas9 to the many PAM sites of the genome.²⁶

One mechanism holds both verdicts. A monomeric repressor on a promoter pool reduces to Hill repression with coefficient 1 when the pool is in trace, and that toggle has one stable state. Once the pool exceeds the dissociation constant and bound repressor is removed with free repressor, the same toggle can hold two.

Lecture 9's analysis locates the boundary for the reduced toggle. For the symmetric toggle with binding at equilibrium and no leak, two stable states need a promoter total above the dissociation constant together with a minimum production strength. On every point of a 36-point parameter grid, a toggle whose bound repressor is spared, or whose promoter is in trace, held a single stable state. These are model calculations, and they sharpen Cherry and Adler's 2000 route rather than replace it.

For a two-gene switch, look for a partner pool larger than its dissociation constant whose bound state is removed with the free state: that pool can supply the steepness that a Hill coefficient would.

11One shared dimerization pool held many stable states

Zhu and colleagues asked in 2022 for multistability that can be expanded. Multicellular organisms hold thousands of molecularly distinct, heritable cell states, which they attribute to multistable regulatory circuits. Synthetic multistable circuits in mammalian cells had been limited to two states, or had used architectures that could not easily grow.²⁷

MultiFate was built and measured to hold seven states. Its transcription factors share one dimerization domain. Homodimers activate their own genes and heterodimers are inactive, so each factor inhibits the others by taking their partners. MultiFate-3 held seven distinct states, each stable for at least 18 days.²⁷

Its model kept the binding pool and an effective law for transcription. Box 1 writes the dynamics in total concentrations, with every species removed at one rate regardless of its dimerization state. Dimerization sits at equilibrium with one dissociation constant, counted with the paper's statistical factor of two for a heterodimer, and transcription is a Hill function of the homodimer.²⁷ The supplement reports multistability even with transcriptional Hill coefficient 1. It traces the extra steepness to homodimerization when dimerization is weak, and to titration by heterodimers when dimerization is strong.²⁷

The course's reanalysis splits the reported count at larger N into stable states and saddles. At the parameters used for MultiFate-2 and MultiFate-3, the paper's model reached a maximum of 256 attractors at nine factors, most of them reported robust to noise.²⁷ A numerical enumeration of the reduced equations, made for this course and checked against direct integration and Jacobian eigenvalues, finds 130 stable states there and 126 symmetric four-ON fixed points that are saddles. Together they match the reported 256. The authors' counting code was not inspected, so this split is the course's reading of the number, not an account of how it was produced.

The measured states are a separate result. MultiFate-3 held seven states in cells. The reduced model also has seven stable states with three factors.

Competition for a shared partner has its own theory. As Cinquin and Page restate it, Cinquin and Demongeot showed in 2005 that switch proteins competing for a common partner can keep $k$ of them on together only if a competition parameter is below $1/k^2$. Cinquin and Page generalized the rule in 2007.²⁸ MultiFate's cap on how many factors stay ON together belongs to this family.

MultiFate and Hill mutual repression are two architectures, not two limits of one mechanism. MultiFate has self-activation and competitive dimerization, and the symmetric Hill model of mutual repression has neither. For product-form, symmetric Hill repression without self-activation, the largest count over the course's tested grid was at most $N+1$, while MultiFate's numerically enumerated count grew as sums of binomial coefficients. The comparison places two architectures inside one framework. The same-mechanism limit for multistability is the promoter-pool toggle of §10.

Count stable states from the closed dynamics of the actual architecture, and keep a model's count of fixed points apart from its count of stable ones.

12A monomer ring oscillated once bound repressor left with free repressor

The ring was revisited five times with binding made explicit, in models, in vitro and in cells. Take the five in order of date.

Loinger and Biham asked in 2007 how the repressilator behaves when its molecules are few. They wrote deterministic rate equations with the same explicit binding of repressors to promoters that their stochastic simulations used.²⁹

They found three deterministic results. The Michaelis-Menten equations with Hill coefficient 1 never oscillate. Rate equations with a separate population of bound repressors, none of it degraded, have a single steady state that is stable for any parameters. With bound repressor degraded, the same ring of monomeric repressors on single promoter sites oscillates for a broad range of parameters, with or without mRNA.²⁹

They traced the oscillation to degradation of bound repressor. Cell division supplies it, since only one of the two daughter DNAs keeps a bound repressor. They read the effect as a nonlinear degradation term, whose rate falls as the repressor rises, and they did not describe it as titration.²⁹ Theirs is an earlier deterministic monomer oscillator, and the course claims no priority for the possibility.

In 2011 a ring whose thresholds were set by titration ran outside cells. Kim and Winfree built in vitro transcriptional switches whose thresholds were set by the totals of DNA activator and inhibitor strands. A ring of three such switches ran four cycles.³⁰

Potvin-Trottier and colleagues rebuilt the repressilator for precision in 2016. They removed degradation tags and reintroduced a sponge of TetR binding sites. The streamlined circuit peaked every 14 generations and kept its phase for hundreds of generations. They read the circuit as a relaxation oscillator, for which cooperative repression matters less than it does for harmonic oscillation.³¹

Their models separate a stochastic verdict from a deterministic one. Box 1 states that single cells can oscillate without the traditional requirements of cooperative repression and feedback delays. Extended Data Fig. 4d shows the case: a minimal model with Hill coefficient 1 oscillates stochastically, while its deterministic equations give damped oscillations. Their supplement adds that equal dilution of repressors and binding sites is key to effective titration, and it attributes the sponge's main effect to a raised repression threshold, doubting that added steepness was what helped.³¹

Clamons and Murray asked whether CRISPR interference, which lacks obvious cooperativity, can build dynamic circuits. Their idealized model assumes dCas in excess, instantaneous binding and irreversible one-to-one binding to DNA, so a guide represses one promoter until the pool is full. The total guide, bound and free, is diluted at one rate. Their analysis takes the central steady state far from saturation, which needs the leak from an occupied promoter to stay below dilution, $\alpha_0<\gamma$. On that branch the central state of a three-guide ring loses stability when $(\alpha-\alpha_0)/2>\gamma$, where, in their letters, $\alpha$ and $\alpha_0$ are the guide production rates of a free and an occupied promoter and $\gamma$ is the dilution rate.³² With a larger leak, their production law puts the central state at full occupancy, where it is stable however large $\alpha$ is.

They concluded that cooperativity is necessary only when genes bind in a Hill-like fashion. Linear binding with sharp saturation, they wrote, is another path to useful instability. Their fuller model predicted a period of about seven hours, set by dilution. That period is a model prediction.³²

A CRISPRi ring was then measured to oscillate. Santos-Moreno and colleagues built the three-node CRISPRlator in E. coli. It oscillated with a period of 10 to 12 hours, and an open-ring control did not. Their models addressed the toggle, not the ring.²⁶

Before accepting or rejecting a monomer oscillator, ask how bound repressor is removed: spared, the rings analyzed here settle, while removed with free repressor at a finite promoter pool they can oscillate.

13Feedback traded gain for robustness, and integral feedback added memory

Feedback was engineered for constancy long before biologists asked how cells adapt. The history has three steps: an amplifier made insensitive to its own parts, feedback recognized in cells, and a memory added to remove the error that feedback leaves. Lecture 10, the adaptation biomachine, develops the mechanisms. This section records who saw what, and the conditions under which each result holds.

Black spent surplus gain to buy constancy

Black's problem was drift in high-gain telephone amplifiers. Long-distance lines chained repeater amplifiers, and small variations in battery potentials and currents added up to serious transmission changes.³³ In his own later account, someone had to readjust the filament current every hour of every day.³⁴

His fix fed the output back to throw gain away. Black built amplifiers with, say, 40 decibels more gain than needed and fed the output back on the input so as to throw away the excess. When the loop transmission is large, the amplification approaches the value set by the feedback path and becomes approximately independent of the amplifier's own gain. Gain variations and internally generated distortion shrink by the same factor as the gain. He measured gain changes below 0.01 decibel for a plate-voltage change from 240 to 260 volts, where a conventional amplifier changed by about 0.7.³³

The ferry story is Black's recollection, fifty years on. In 1977 he dated the idea to 2 August 1927, on the Lackawanna ferry across the Hudson.³⁴ Mindell reads the story as feedback's myth of origin, one that effaces its sources, and notes that the patent filed in 1928 was granted only in 1937.³⁵

Finite gain shrinks a disturbance and leaves a residual. In Black's analysis, noise and distortion generated inside the amplifier are divided by a factor set by the loop transmission, large but finite.³³ The constancy he bought holds in the regime of large loop gain and improves as the gain grows. It is never exact.

Cells were read as feedback amplifiers

Negative autoregulation is common in bacteria and changes their dynamics. Rosenfeld, Elowitz and Alon found it in over 40% of known E. coli transcription factors. In a TetR–GFP fusion that represses its own gene, it cut the rise time to about a fifth of a cell cycle.³⁶

Khammash connected Black's amplifier to gene regulation in 2016. Linearized, a gene that represses itself obeys the same equations as the feedback amplifier, and its robustness comes from high gain and feedback, as the amplifier's does. Expression is an amplifier that makes many proteins from one DNA copy, and autoregulation spends some of that gain on robustness to parameter changes. The residual sensitivity of the steady state falls as the repression steepens, but it stays finite.³⁷

A binding threshold can supply that gain. §9 traced how titration makes a response steep without a large Hill coefficient. Inside a negative feedback loop, the same steepness narrows the output's response to a disturbance. This is the route Lecture 9 takes from thresholds to adaptation.

Integral feedback removed the residual with memory

Bacterial chemotaxis returns to its old tumbling rate, robustly and within measurement error. Barkai and Leibler argued in 1997 that the precision of adaptation follows from the network's connectivity rather than from fine-tuned parameters, and their model adapts exactly.³⁸ Alon and colleagues tested this in E. coli in 1999. Wild-type cells returned to their unstimulated tumbling frequency with a measured precision of 0.98 ± 0.05. Adaptation time and steady-state behaviour varied with the levels of the chemotaxis proteins, while the precision did not. The receptors' methylation, added by CheR and removed by CheB, carries the adaptation.³⁹

Yi, Huang, Simon and Doyle named the mechanism integral feedback in 2000. They showed that integral control is structurally inherent in the Barkai-Leibler model, with the methylation state as the integrated variable, and argued that integral control in some form is necessary for robust perfect adaptation.⁴⁰ The integrated variable is a memory: it accumulates the error until the error is gone. A finite-gain loop, however steep, has no such store. In Khammash's control language, for constant disturbances, integral feedback supplies the unbounded zero-frequency gain that finite gain cannot.³⁷

Ma and colleagues found that feedback adaptation needs an integrating node. Their 2009 search of enzyme networks found negative feedback with a buffering node to be one of only two robust adaptive topologies, and noted that it implements integral control. Its integrator relies on zero-order ultrasensitivity, with the node's two enzymes at saturation.¹⁸ The saturation regime of §4 here serves as memory.

Jeynes-Smith and Araujo reduced the loop to two nodes and wrote its complexes out. In their two-node Opposer model, an ultrasensitive switch inside a negative feedback loop becomes a narrow tolerance around the adapted set point. With the enzyme-substrate complexes explicit, the free active output $A^*$ did not track a nonzero set point anywhere in their parameter search. What tracked it was the partial pool $A^*_S=A^*+C_3$, in their letters, where $C_3$ is active A bound to inactive B. It did so over a limited range of inputs and within their numerical tolerance.⁴¹ Their model is a generic enzymatic loop, not a chemotaxis model. Which pool adapts is again a question about binding, as in §14.

Sequestration became the comparator of a designed integrator

Briat, Gupta and Khammash designed an integrator from two molecules in 2016. A reference species is made at a constant rate, a sensor species is made in proportion to the output, and the two annihilate one to one. Each annihilation removes one of each, so their difference is preserved and integrates the error between reference and output. Their guarantee holds under stated conditions on the open-loop network, and the deterministic version of the controller can oscillate for some parameters.⁴²

The comparator can also be written with binding. Reversible sequestration of the two species, with the complex removed, also takes one of each per binding event. The binding terms cancel in their difference, so the memory survives.

Aoki and colleagues built the controller in E. coli in 2019. The sigma factor SigW served as reference and actuator, and its anti-sigma factor RsiW, which sequesters it, served as sensor. After a protease was induced as a disturbance, the closed loop showed virtually no change in fluorescence, while the open loop fell by more than 40% in most conditions. The theory assumed negligible degradation of the two controller proteins, which a growing cell does not satisfy, so the adaptation there is near-perfect rather than exact.⁴³

Moving the actuation lowered noise in their analysis. Kell, Ripsman and Hilfinger studied the idealized controller on a minimal production-removal plant in 2023. The linear noise approximation predicts that reference actuation raises noise above matched open-loop levels, and an exact stochastic search over more than a million parameter sets found no violation for mean copy numbers of at least one. A two-stage gene-expression model gave the same kind of bound. Letting the sensor species repress the output's production instead can suppress noise below open loop, still with perfect adaptation. The existing ergodicity theorem covers only the original design, so the variant's stability rests on simulation.⁴⁴

A hidden proportional term explained a faster response. Filo, Hou and Khammash showed in a 2023 preprint that sensor actuation adds a proportional element to the integrator. With it, the response can be sped up without overshoot or oscillation, which the reference-actuated design cannot match. This is a modeling result.⁴⁵

The sensor-actuated controller was then built and tested on its own. Filo and colleagues built it in E. coli from TetR and a split intein, and it almost completely rejected a disturbance. They compared it with open-loop and filtered-proportional circuits, and they built no reference-actuated intein controller for a direct comparison. They credit the variant's first appearance to Briat and colleagues' supplement.⁴⁶

Xiao and Doyle asked in 2018 how a reaction network can hold an integral of error at all. A concentration cannot go negative, so no single species can carry an unconstrained, signed integral of the error, as a capacitor or a register can. They listed three ways around that constraint. A species can integrate in a constrained form, growing in proportion to itself times the error, which adds a steady state at zero. A combination of species can serve as a transformed coordinate, as in the incoherent feedforward sniffer and in the sequestration controller's difference. And an integrator can appear as an approximation, when saturated enzymes make a rate independent of its substrate.⁴⁷

The table sorts the mechanisms of this section by what stores the error.

Each verdict holds in its regime here too. High-gain feedback gives approximate disturbance rejection, with a residual that shrinks as the loop gain grows. Integral feedback gives zero steady error for constant disturbances under four conditions: the loop is stable, the controller operates within a feasible range, the integrating variable is not itself lost to dilution or degradation, and the disturbance leaves the set point alone. In the antithetic controller the set point is the ratio of the reference production rate to the sensing rate. Lower noise and faster responses are separate properties. Lecture 10 asks what memory a loop needs in order to adapt everywhere.

Tell finite-gain feedback, which divides a disturbance's effect by a large but finite factor, from integral feedback, which stores the accumulated error and removes it, and check stability and controller dilution before calling an adaptation perfect.

14Each feedforward implementation adapted inside its own binding regime

The 2025 course posed the question as homework. Two networks share the reaction-order structure of perfect adaptation, one degrading the output and one repressing its gene. The problem asked whether the two implementations behave differently, and whether each fails when the input is too large or too small. The published designs below answer it piece by piece.

Segall-Shapiro, Sontag and Voigt built the repression design in 2018. Theory predicted copy-number independence if the negative regulation is perfectly non-cooperative, so they used TALE repressors, which bind as monomers to a single operator. Their stabilized promoters gave near-identical expression across plasmids from 3 to 100 copies per cell, at a cost in output strength.⁴⁸

Their model needed the repressor in excess. The supplementary model assumes the repressor well above its dissociation constant, and the key design parameter is the repressor level at the lowest copy number.⁴⁸ That is the regime in which Lecture 9's repression design adapts: the repressor above both its dissociation constant and its promoter total.

Bleris and colleagues built transcriptional and miRNA-based loops in mammalian cells in 2011. Every loop they built and measured adapted to the amount of its DNA template, and the miRNA version showed an excellent adaptive response.⁴⁹

Their model predicted where the miRNA loop would fail. At high copy number and low levels of the RNA-induced silencing complex, the model's silencing complex saturates and adaptation is lost.⁴⁹ That is a model prediction, not a measured failure. In Lecture 9's terms it is the degradation design leaving its regime as the degrader saturates.

Du and colleagues stated the degradation window outright. In their miRNA modules, catalytic degradation must exceed a minimum for total mRNA to stay invariant to gene dosage. Their model, written in dimensionless rates measured against mRNA removal, finds total mRNA roughly compensated only over an intermediate window of dosage. The dosage, which sets the regulator level, must be high enough for catalytic degradation to dominate, and low enough that the regulator stays below the model's effective binding scale, a scale that changes when removal of the complex is included. Above that scale total mRNA again follows dosage.⁵⁰

Each effective rate law misses a different part of the story. The Hill repression law already predicts the failure at low input, because a repressor below its dissociation constant represses weakly. What it misses is depletion of free repressor by a finite promoter pool, which moves the window and can make the output over-adapt. The Michaelis-Menten degradation law misses that the complex holds a share of the output, which is why total output fails at high input while free output still returns in Lecture 9's adaptation example.

Place each adaptation design on its binding pair's map: a degrader works while it stays free and below its dissociation constant, a repressor while it exceeds both its dissociation constant and its promoter total, so test both ends of the input range.

15The lecturer's programme made the regime the unit of description

The lecturer's papers from 2020 to 2026 described a mechanism by its regimes. Lecture 8's lineage tells that programme in full. Three of its pieces carry the placement made in the next two sections.

The 2022 thesis bounded what each state of a binding pair can do. As the totals vary, the reaction orders of the free species and of the complex fill polyhedra whose vertices are the dominance regimes. The activator-repressor dichotomy of §8 is one reading of them.²⁰

The 2026 validity work made a law's regime explicit. Liu, Ren and Xiao note that the functions of biocircuits are often valid only in specific parameter regimes, which are rarely made explicit. They derive validity conditions from regimes. For the Michaelis-Menten law the condition is that the free substrate wins its own total.⁵¹

Lecture 8 made the regimes a complete list for a written mechanism. Each conservation sum has a largest term at every parameter point, so enumerating the feasible choices of winners exhausts the dominance possibilities, ties and singular regimes included. That list lets a published verdict be matched to a regime rather than argued against.

Describe a mechanism by its full list of regimes, each with a validity condition, so that a published verdict can be matched to the regime its assumptions select.

16The tight-binding threshold places the monomer repressilator

The published results on monomer rings differ in three conditions. They differ in binding strength against the promoter total, in the removal variant, and in cooperativity. The ring in question is the one of §12 with binding at equilibrium: each repressor binds the promoter sites of the next gene one to one, only free sites make repressor, and free and bound repressor are diluted together.

Lecture 9's analysis gives the boundary in the first condition. In that reduced model, with no leak and uniform removal, some production strength destabilizes the symmetric ring exactly when $q_P/K>7+4\sqrt3\approx 13.9$, where $q_P$ is the promoter total and $K$ the dissociation constant. This page calls it the tight-binding threshold. The exposition derives it from the loop gain per node, the logarithmic slope of one gene's production in its repressor's total at the steady state, which must exceed 2 for a ring of three.

One fast-binding map holds the published rings, once their parameters are projected onto it. Figure 1 draws, against $q_P/K$, the window of production strength $a=k_P/\lambda$ in which the symmetric steady state is unstable. Here $k_P$ is the production rate of one free promoter site and $\lambda$ the dilution rate. The window closes at the tight-binding threshold.

Loinger and Biham's oscillating ring sits inside the window. Their Fig. 2 used one promoter site with unbinding over binding rate $\alpha_1/\alpha_0=0.02$ molecules. Projected onto the fast-binding model, that gives $q_P/K=50$ and production strength $g/d\approx 17$.²⁹ At that tightness the reduced model's window runs from $a\approx 2.4$ to $86$. The map speaks for the reduced model. Their own equations, with finite binding and unbinding rates, were checked separately: integrating their Eq. 5 at these parameters gives a period of 2261 s, against the 2260 s they report, and the course's fast-binding model gives 2339 s. Both are present-day calculations.

Clamons and Murray's condition is the window's edge at irreversible binding. Irreversible binding means $q_P/K\to\infty$, where the lower edge of the window tends to $a=2$. Their condition $(\alpha-\alpha_0)/2>\gamma$, which assumes a central state below saturation and hence $\alpha_0<\gamma$, is that edge, with production strength $(\alpha-\alpha_0)/\gamma$. Their production law rewrites as a background $\alpha_0$ from every promoter plus an increment $\alpha-\alpha_0$ from each free one, which is the course's form after a change of coefficients. With no leak, the course's check finds their ring and the binding-catalysis ring in this limit identical, trajectory for trajectory.

Their linear segment is a steep response on logarithmic axes. Below equivalence their free promoter fraction is $p=1-r$, with $r$ the guide total over the promoter total. Its logarithmic slope, $r/(1-r)$, exceeds 1 past half occupancy and grows without bound near equivalence. Titration is the mechanism, and the steep logarithmic response is its consequence.

The measured CRISPRi ring shares the binding assumptions of that edge. Guide-loaded dCas binds its target one to one and, for practical purposes, does not unbind on its own. Those are the binding assumptions of Clamons and Murray's idealized model, in which dilution removes bound and free guide alike.³² If the cell dilutes bound complexes along with free ones, Santos-Moreno's CRISPRlator sits near the edge at irreversible binding.²⁶ Its parameters were not reported in a form that pins one point on the map.

The Hill-law verdicts sit on the trace-promoter side. With the promoter in trace, repression is a Hill law with coefficient 1, and no production strength destabilizes the ring. Loinger and Biham's Michaelis-Menten equations, Potvin-Trottier's deterministic minimal model and Blossey's gate equations all hold there.

The protected-repressor verdicts sit off the plane. With bound repressor spared from removal, Lecture 9's analysis finds the loop gain per node to be exactly $1-p^*<1$, where $p^*$ is the free fraction of sites at the steady state. No point of the plane is then unstable. Müller's model and Loinger and Biham's rate equations without degradation of bound repressor hold there.

Cooperative rings need a third axis. Cooperative repression, as in Elowitz and Leibler's design model, raises the loop gain per node without any promoter pool, and the plane does not draw it. Potvin-Trottier's streamlined circuit kept the original cooperative repressors and added a TetR sponge, so it belongs on that axis too.

The threshold is what places them, so placement is the addition. An earlier deterministic monomer oscillator exists, and so do the analytic condition at irreversible binding and a measured CRISPRi ring. The tight-binding threshold adds the boundary at finite binding strength, and that boundary puts all of them, and the Hill theorems, on one map.

Place any repressor-ring result by three conditions, binding strength against the promoter total, the removal variant and cooperativity, and predict whether its steady state can lose stability before simulating it.

17Every verdict holds its place, which gives design and a guarantee

The record establishes each verdict together with the conditions under which it holds. The table collects them, behavior by behavior. Each row holds in its regime: exactly for a theorem about a reduced model, approximately for a limit of a binding mechanism, and over the tested range for a measurement. None of them needs overturning.

The placement is the course's addition. The course does not claim the first monomer oscillator or the first sequestration switch. Cherry and Adler named depletion of free repressor as a route to a switch in 2000, Buchler and Louis developed molecular titration in 2008, and Loinger and Biham ran a deterministic monomer ring in 2007. What Lecture 9 adds is that one modeling framework, the binding-catalysis model class, places the published results on all five behaviors, each at the parameter conditions under which it was derived. The five architectures are different networks within that class, not one network at five parameter settings.

The placement inverts into design. A wanted behavior names its regime, and the regime names what to build and what to measure. For a threshold in an induced gene, choose a repressor that binds its inducer much more tightly than its operator, and set its total near the wanted threshold. For adaptation, choose degradation or repression by the end of the input range that must be covered. For a monomer oscillator, seek tight one-to-one repressors, enough promoter sites to pass the tight-binding threshold, and removal that does not spare bound repressor.

The placement is complete for the written mechanism, within a stated limit. If the written binding-catalysis mechanism is the intended one, Lecture 8's enumeration of dominance regimes over the conservation sums lists every regime, so a complete regime analysis misses none. The guarantee covers the regimes, ties and singular regimes included, under Lecture 8's equilibrium assumptions. It does not cover attractors. Which state or cycle the closed loop reaches inside a regime still comes from the closed dynamics. The monomer repressilator shows why: just beyond the upper edge of its window, Lecture 9's analysis finds rest and a large cycle coexisting.

Three readings of the record deserve distrust. The first is a necessity claim stated without its model class. The second is a fitted Hill coefficient read as a count of binding sites. The third is a model prediction read as a measurement, such as a model's count of states read as the number a cell can hold, or a failure seen only in a model.

Three questions stay open. No source used here settles which architecture a natural circuit uses: which induced repressors bind their inducer more tightly than their operator, or which natural feedforward loops degrade their output rather than repress it. Whether DNA-bound repressors are protected from active degradation in cells decides how much of the oscillation window survives a protease. Whether natural titration pools keep time at the scale that the tight-binding threshold allows remains to be tested.

Start from a wanted behavior, find the published verdicts that hold in its regime, and turn the regime's conditions into the experiments, directions and parts the design needs.

References

Every number on this page is either quoted from a source at the page given, or marked as a present-day calculation. The calculations behind Figure 1 and §16, including the integration of Loinger and Biham's Eq. 5, are kept with the Lecture 9 analysis records. Model predictions of the core are marked as such where they are used.