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.

18. Genetic Switch in Phage Lambda

What this covers

This chapter sets up a single case study: how the decision between lysis and lysogeny in bacteriophage \(\lambda\) is modelled, in Hasty, Pradines, Dolnik and Collins, Noise-based switches and amplifiers for gene expression, PNAS 97, 2075–2080 (2000), as a bistable switch built from one gene regulating itself. It assumes the equilibrium-binding and mass-action machinery from earlier in the course — dissociation constants, cooperative binding of a dimer to DNA. The surviving material covers the biological setup and the reaction scheme; it stops before the rate equation the lecture set out to derive (see Sources).

The autoregulatory loop at \(P_{RM}\)

The decision between the two phage lifestyles is controlled by the promoter \(P_{RM}\), which drives a single gene, cI. The cI gene product is a repressor protein that dimerizes and then binds back to the operator DNA in front of its own gene, so the protein controls the transcription of its own message — the loop that makes a switch possible at all.

The wild-type \(\lambda\) operator has three binding sites (OR1, OR2, OR3); the model considered here keeps only two of them. Hasty et al. assume that:

• occupying OR2 turns transcription on,
• occupying OR3 turns transcription off.

The binding and production scheme

Writing \(X\) for the repressor monomer, \(D\) for the operator DNA, and \(P\) for RNA polymerase, the lecture's reaction scheme (its equation [III.1]) is

\[2X \rightleftharpoons_{K_1} X_2$$ $$D + X_2 \rightleftharpoons_{K_2} DX_2$$ $$D + X_2 \rightleftharpoons_{K_3} DX_2^{*}$$ $$DX_2 + X_2 \rightleftharpoons_{K_4} DX_2X_2$$ $$DX_2 + P \xrightarrow{k_t} DX_2 + P + nX\]

Reading it as a set of promoter states: the repressor first dimerizes (\(K_1\)), and the dimer \(X_2\) then competes for the two sites — binding OR2 gives \(DX_2\) (the transcriptionally active state), binding OR3 gives the distinct complex \(DX_2^{*}\) (silent), and a second dimer can occupy the remaining site on top of \(DX_2\) to give \(DX_2X_2\) (both sites occupied, also silent). Only the \(DX_2\) state produces protein: the last line says that, whenever the promoter is in that state, RNA polymerase turns out \(n\) new copies of \(X\) per transcription event, without \(DX_2\) or \(P\) being consumed. Because a dimer, not a monomer, has to bind before transcription switches on, the production rate is a nonlinear (cooperative) function of the repressor concentration rather than a linear one — this is the ingredient the rest of the argument turns on.

Reading a switch off a creation/destruction diagram

Creation rate (sigmoidal, from cooperative dimer binding) against destruction rate (a straight line) as functions of repressor concentration $x$. Where the curves cross, production balances loss: a fixed point. The two filled points are stable — the arrows show nearby concentrations moving toward them — and the open point between them is unstable, so it is the threshold that decides which stable state the system settles into.

The same picture, drawn for different destruction rates, gives the three regimes shown in the lecture: with a high destruction rate the curves cross only once, at a low value, and the system has a single low steady state; with a low destruction rate they cross once at a high value, giving a single high steady state; in between, for a medium destruction rate, they cross three times, giving two stable states separated by the unstable one drawn above. That middle regime is the switch: which of the two stable states the system ends up in depends on which side of the unstable point it starts on, not on any change in the reaction rates themselves.

Sources

• Transcript: recordings/l3-syllabus-transcript.md (MIT OCW 8.591J, 2004), section III, "A Genetic Switch in Lambda Phage" — the autoregulatory loop, the OR2/OR3 roles, and reaction scheme [III.1] all come from this fragment. The transcript is a model's reconstruction of a PDF with no text layer (flagged fidelity: reconstructed in the file), so the equations are unverified against the original, and the extracted text stops mid-equation right after scheme [III.1] — the lecture's stated goal, deriving Hasty et al.'s kinetic equation [7], is not present in what survived.
• Figure: recordings/l3-syllabus-transcript/figures/p006-1.jpeg, extracted from page 6 of the same source PDF (labels and captions read directly off the image). The lecture's own Figure 5 — a schematic of the promoter's bound/unbound states — is referenced in the text but its image link in the source is a placeholder, not the actual figure, so it is not reproduced here; the state labelling in "The binding and production scheme" above is read off the reaction scheme and the OR2/OR3 description instead.
• No slides, written notes or exercises were supplied for this chapter.

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