Lecture notes. Written from the material of MIT 8.591J 2004, licensed CC BY-NC-SA 4.0. These are notes, not a transcript: the material has been reorganised and rewritten. This adaptation carries the same licence, and the original is linked above.
2. Cooperativity and the Lambda Switch¶
What this covers¶
Lecture 2 built a general toolkit for equilibrium binding — the Adair equation, detailed balance, and a vocabulary for cooperativity (\(\beta\), the Hill number). This lecture opens by finishing that toolkit and then spends it immediately on a real switch: the decision, inside a newly infected bacterium, between the phage lysing its host and lying dormant as a lysogen. The question the second half answers is deliberately the one on the lecturer's own slide — how do you build a mathematical model that captures the essence of the switch? — and the binding machinery from the first half is exactly what that model is made of. The chapter assumes the Michaelis–Menten scheme and the idea of a quasi-steady state are already familiar; both are recapped only as far as needed to reuse them.
Michaelis–Menten kinetics: the three regimes¶
The enzyme scheme is the usual reversible binding step followed by irreversible product release,
with mass-action kinetics for each species and enzyme conserved, \(E_o = [E] + [ES]\):
started from \([S] = S_o\), \([E]=E_o\), \([ES]=[P]=0\). Plotted against time, the concentrations pass through three regimes: a brief transient in which \([ES]\) builds up from zero; a quasi-steady state in which \([ES]\) has equilibrated on the timescale of \([S]\)'s slower decay, so \(d[ES]/dt \approx 0\) while \([S]\) is still changing; and, once enough substrate is consumed, substrate depletion, where \([ES]\) and \(v\) fall away as \([S]\to 0\).
The quasi-steady-state assumption is what makes the scheme tractable: setting \(d[ES]/dt = 0\) in the equation above gives \(k_1 E_o[S] = (k_1[S] + k_{-1} + k_2)[ES]\), so
and the initial velocity \(v_0 = k_2[ES]\) becomes the familiar
This is a good approximation only when \(S_o \gg E_o\): the quasi-steady state has to be reached while almost none of the substrate has yet been consumed, so that \([S]\) at the start of that regime is still essentially \(S_o\). If enzyme and substrate are comparable in concentration, the transient and the steady state overlap and the formula is no longer trustworthy.
Equilibrium binding: the Adair equation and detailed balance¶
The second topic is a ligand \(S\) binding a protein with \(n\) sites, one at a time,
where \(P_j\) is the species with \(j\) ligands bound. The macroscopic association constant \(K_j = [P_j]/([P_{j-1}][S])\) describes the step from \(j-1\) to \(j\) ligands bound, without saying anything about which of the \(n\) sites is occupied. The average number of ligands bound per protein, \(r\), is the ligand-weighted average over all \(n+1\) species, which gives the Adair equation:
Why \(K_j\) can be read off directly from rate constants — \(K_j = k_{+j}/k_{-j}\) — is a detailed balance argument, not an assumption. Write the chain of binding steps
and set \(d[P_o]/dt = 0\) at equilibrium: \(-k_{+1}[P_o][S] + k_{-1}[P_1] = 0\), so \(K_1 = k_{+1}/k_{-1} = [P_1]/([P_o][S])\) immediately. Substituting that into \(d[P_1]/dt = 0\) cancels the two terms already balanced by the first equation, leaving \(-k_{+2}[P_1][S] + k_{-2}[P_2] = 0\) on its own — the second step balances independently of the first. The pattern repeats down the chain: each step is separately at equilibrium, and each \(K_j\) is exactly the ratio of its own forward and backward rate constants, with no cross terms from neighbouring steps.
Three toy models for two-site binding¶
Three small cases show what the Adair equation does with a specific mechanism plugged in, and they are the ones reused later on the phage operator.
I. Identical, independent sites. Two identical sites, each with intrinsic constant \(K = k_+/k_-\). Statistics alone changes the macroscopic constants: there are two ways to bind the first ligand but only one way to lose it from a singly-bound protein, so \(K_1 = 2K\); conversely there is only one empty site left for the second ligand but two ways to vacate a doubly-bound protein, so \(K_2 = K/2\). The Adair equation collapses to
which is exactly the hyperbolic isotherm of a single site, scaled by two — independence is what lets it factor cleanly.
II. Non-identical, independent sites. Two distinct sites with their own constants \(K\) and \(K^*\), still not interacting, simply add:
III. Identical, interacting sites. Now the two sites are chemically identical but the occupancy of one changes the affinity of the other: \(K_1 = 2K\) as before, but the second step carries a different intrinsic constant \(K^* = k_+^*/k_-^*\), so \(K_2 = K^*/2\). The Adair equation becomes
Reading cooperativity off the isotherm¶
Case III is the one that matters, because it is the minimal model with cooperativity in it. Writing \(r\) as a fractional saturation \(Y = r/2\) (out of a maximum of two sites), substituting \(x = K[S]\) and \(\beta = K^*/K\), and using \(KK^*[S]^2 = (K[S])(K^*[S]) = \beta x^2\) collapses the case-III expression to
\(\beta\) is the whole story of cooperativity here: \(\beta > 1\) means the second ligand binds more readily once the first is bound (positive cooperativity), \(\beta < 1\) means it binds less readily (negative cooperativity), and \(\beta = 1\) recovers the independent case. The curvature of \(Y(x)\) tracks this: differentiating twice shows the sign of \(Y''\) at \(x=0\) is the sign of \(\beta - 2\). For \(\beta \le 2\) the curve is concave for every \(x \ge 0\) — a plain saturating hyperbola. For \(\beta > 2\), \(Y\) is convex near \(x=0\) and concave for large \(x\), which forces an inflection point in between: that inflection is what makes the curve sigmoidal, and \(\beta = 2\) is exactly the threshold at which it first appears. The slide's plot of the effective Hill number \(n_H(x)\) for \(\beta = 1, 2, 10, 100\) traces the same fact from a different angle: at \(\beta=1\), \(n_H\equiv 1\) everywhere (no cooperativity to detect); as \(\beta\) grows, \(n_H\) rises above 1 near the midpoint of the transition and the isotherm sharpens, approaching — without reaching, for a real, finite dimer — the ideal step-function response of a perfectly cooperative site.
From binding polynomials to a genetic switch¶
That machinery is the whole content of Lecture 2. This lecture puts it to work on a specific molecular decision: bacteriophage \(\lambda\), a virus with a \(48{,}512\)-base-pair genome, injects its DNA into E. coli and then has to choose between two fates. Immediately after injection, phage genes are transcribed and translated using the host's own machinery; which set of phage proteins gets expressed decides the outcome — lysis (the phage replicates and kills the host cell) or lysogeny (the phage genome is retained, quiescent, inside the host chromosome). A lysogen is immune to a second infection: repressor dimers already present bind the incoming phage's DNA and shut its genes off, and it is a high, steady concentration of repressor that keeps the cell in the lysogenic state. (Source: Ptashne, A Genetic Switch: Phage Lambda, 3rd ed., Cold Spring Harbor Laboratory Press, 2004 — the images the lecture showed here are not reproduced in the converted slides.)
The operator OR: three sites, three control logics¶
The decision is written into a short stretch of DNA, the right operator \(O_R\), which contains three repressor-binding sites — \(O_{R}3\), \(O_{R}2\), \(O_{R}1\) — sandwiched between two promoters that run in opposite directions: \(P_{RM}\) (leftward, transcribing the repressor gene \(cI\)) and \(P_R\) (rightward, transcribing \(cro\)). The key physical fact is that only one RNA polymerase fits over this region at a time — repressor and polymerase compete for overlapping or adjacent space, so a repressor dimer parked on one site can block, or in one case help, polymerase at a promoter several base pairs away.
With a single repressor dimer bound, the slides give three cases, one per site:
- Dimer at \(O_{R}2\): negative control of \(P_R\) (blocks RNAp from transcribing \(cro\)) and positive control of \(P_{RM}\) (a dimer here actually helps RNAp bind and transcribe \(cI\)).
- Dimer at \(O_{R}1\): negative control of \(P_R\), and — on its own — a weak negative effect on \(P_{RM}\) too, since \(O_{R}1\) is too far from \(P_{RM}\) for a lone dimer there to help it directly.
- Dimer at \(O_{R}3\): negative control of \(P_{RM}\) (blocks the repressor's own gene) and positive control of \(P_R\) (a dimer here actually allows RNAp to transcribe \(cro\)).
Cooperative binding makes the two genes take turns¶
The three-toy-model section is not incidental background: repressor's binding across \(O_{R}1\), \(O_{R}2\), \(O_{R}3\) is exactly case III, generalized to three sites. The intrinsic constants for repressor decrease roughly tenfold from site to site, \(K_{O_R1} \sim 10\,K_{O_R2} \sim 10\,K_{O_R3}\) — \(O_R1\) is bound first on affinity alone — but the cooperative constant for the second step is far larger than the intrinsic one, \(K^*_{O_R2} \gg K_{O_R2}\): once a dimer sits at \(O_R1\), a second dimer at the adjacent \(O_R2\) is strongly favoured by direct protein–protein contact between the two dimers. At the low repressor concentrations that hold in a stable lysogen, \(O_R1\) and \(O_R2\) are effectively filled together — turning \(P_R\) off and \(P_{RM}\) on — while \(O_R3\) stays empty. Only once repressor accumulates further does it also occupy \(O_R3\), which shuts down \(P_{RM}\): the repressor gene turns off its own promoter, capping how much repressor gets made. That is the negative feedback the slides point to for keeping \([\text{repressor}]\) at a steady level in the lysogenic state, and it is disturbed when the switch is thrown by UV damage.
Cro, the protein whose accumulation drives the cell toward lysis, binds the same three sites non-cooperatively and with the opposite preference: \(K_{O_R3} \sim 10\,K_{O_R2} \sim 10\,K_{O_R1}\), so Cro fills \(O_R3\) first (case II's independent-sites picture, not case III) — shutting off \(P_{RM}\) and repressor synthesis directly, without needing the cooperative assist that repressor relies on to reach \(O_R2\).
Why cooperativity: a sharp, all-or-nothing decision¶
The slides compare percent repression against repressor concentration for a single, isolated operator versus the full \(\lambda\) \(O_R\) system, and separately compare an isotherm with Hill number \(n_H=1\) against one with \(n_H=3\): the cooperative system switches over a much narrower range of repressor concentration than a single non-cooperative site would. The lecture's own gloss on why is that the switch stacks several layers of cooperativity on top of one another — repressor first has to dimerize (itself a cooperative step, two monomers to one active binding unit), and the dimers then bind \(O_R1\)–\(O_R2\) cooperatively as above. Each layer sharpens the response further, which is what turns a graded, analog binding curve into something closer to a genuine switch: a cell is pushed toward one committed fate rather than settling on some intermediate mixture of both.
Where the model goes next¶
The lecture closes by posing, without yet answering, the question that motivates building an actual dynamical model of this switch: how do you turn this binding picture into a model that predicts which fate a given cell takes? It points to the mechanism of that model rather than building it here, citing Arkin, Ross, and McAdams, "Stochastic kinetic analysis of developmental pathway bifurcation in phage \(\lambda\)-infected Escherichia coli cells," Genetics 149(4) (1998): 1633–48 — a full stochastic treatment of exactly this decision, left as a pointer for where the course goes next rather than material covered in this lecture.
Sources¶
- Slides:
01-review-lecture-2.md(Michaelis–Menten kinetics; Adair equation; detailed balance; the three two-site binding models; cooperativity and the Hill number; introduction to phage biology and the lysis–lysogeny decision) and02-the-lysis-lysogeny-decision-is-a-genetic-switch.md(the \(O_R\) operator, the three single-dimer cases, repressor and Cro binding cooperativity, and the closing pointer to a mathematical model), both from 8.591J Systems Biology (MIT OCW, Fall 2004), lecture 3 notes. - No transcript, written notes, or exercise set was supplied for this lecture; the exposition above is built from the slides alone. Both slide files are a model's reconstruction of a PDF with no extractable text layer, and each carries its own fidelity note flagging every displayed equation as unverified against the original — treat the equations here as a faithful-effort transcription rather than a checked source.
- Named but not contained in the slides: Ptashne, A Genetic Switch: Phage Lambda, 3rd ed. (Cold Spring Harbor Laboratory Press, 2004), the source of the phage and operator diagrams the images were removed from; and Arkin, Ross & McAdams, Genetics 149(4) (1998): 1633–48, cited as where a full model of the switch is built.
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