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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.

3. The Lambda Phage Bistable Switch

What this covers

This chapter follows the second half of Lecture 3 of MIT 8.591J (2004): how the lysis/lysogeny decision made by bacteriophage \(\lambda\) is turned into a testable mass-action model, and how the single equation that model produces can be read for stability directly off a picture, without solving it. It assumes the reader already has mass-action kinetics and the idea of a stable versus unstable fixed point of a first-order ODE; nothing about the phage biology is assumed beyond what is stated here.

From a genetic decision to a reaction scheme

Phage \(\lambda\), on infecting E. coli, either lyses the cell right away or integrates and sits dormant (lysogeny); UV light can later flip a dormant lysogen back into lysis. That is the biological decision the lecture sets out to model, following Hasty et al.

The reaction scheme is six steps, and the slide itself splits them into two groups by timescale:

Fast (binding equilibria):

\[2\text{X} \rightleftharpoons \text{X}_2 \qquad (K_1)\]
\[\text{D} + \text{X}_2 \rightleftharpoons \text{DX}_2 \qquad (K_2)\]
\[\text{D} + \text{X}_2 \rightleftharpoons \text{DX}_2^{*} \qquad (K_3)\]
\[\text{DX}_2 + \text{X}_2 \rightleftharpoons \text{DX}_2\text{DX}_2 \qquad (K_4)\]

Here X is the free repressor monomer, X\(_2\) its dimer, D the operator DNA, DX\(_2\) the dimer bound at one operator site (OR2), DX\(_2^{*}\) the dimer bound at a competing site (OR3), and DX\(_2\)DX\(_2\) the state where a second dimer has bound and looped the two operator sites together.

Slow (synthesis and decay):

\[\text{DX}_2 + \text{P} \xrightarrow{k_t} \text{DX}_2 + \text{P} + n\text{X}\]
\[\text{X} \xrightarrow{k_d} \text{A}\]

Polymerase P transcribes \(n\) new copies of the monomer from the OR2-bound state, and the monomer degrades. The slide calls the fast/slow split itself "the most important step in modeling": because the four binding reactions equilibrate far faster than transcription and degradation, they can be replaced by their mass-action equilibrium relations, collapsing the whole scheme to one slow ODE for the free monomer.

One equation, in dimensionless form

Eliminating the fast variables this way gives a single equation for the monomer level \(x\), already written in dimensionless, relative variables:

\[\frac{dx}{dt} = \frac{\alpha x^2}{1+(1+\sigma_1)x^2+\sigma_2 x^4} - \gamma x + 1\]

where

\[\sigma_1 = \frac{K_3}{K_2}, \qquad \sigma_2 = \frac{K_4}{K_2}\]

are relative binding constants (OR3 relative to OR2, and the looped complex relative to OR2), and

\[\alpha = \frac{n k_t p_0 d_T}{r} \sim \text{synthesis rate}/\text{basal rate}, \qquad \gamma = \frac{k_d}{r\sqrt{K_1 K_2}} \sim \text{degradation rate}/\text{basal rate}\]

for some basal rate \(r\). The lecture's point is the move as much as the result: choosing elegant, dimensionless, relative variables is what turns six reactions and six rate constants into one equation with two shape parameters (\(\sigma_1,\sigma_2\)) and two rate parameters (\(\alpha,\gamma\)).

The \(x^4\) in the denominator is not decorative: it means the synthesis term eventually loses to the denominator as \(x\) grows, so it rises and then falls rather than simply saturating — the signature of the looped, doubly-bound state DX\(_2\)DX\(_2\) shutting transcription back down once \(X_2\) becomes abundant.

Reading stability off a picture

Rather than solving the ODE, plot its two terms separately against \(x\): "creation" (\(\alpha x^2/(1+(1+\sigma_1)x^2+\sigma_2x^4) + 1\), the leaky synthesis term) and "destruction" (\(\gamma x\), a straight line through the origin). Fixed points are the crossings, and their stability follows from the direction of crossing alone: where creation falls from above the destruction line to below it as \(x\) increases, a perturbation is pulled back — stable. Where the crossing runs the other way, a perturbation is pushed further away — unstable.

x rate destruction creation low x high x
Medium degradation rate: creation and destruction cross three times. The two filled points are stable; the open point between them is unstable, and the flow runs away from it toward whichever stable state is nearer. Steepening the destruction line (raising the degradation rate) leaves only the low crossing; flattening it leaves only the high crossing.

Because \(\gamma\) sets the slope of the destruction line, sweeping it sweeps the number of crossings:

  • High \(\gamma\) (steep line): one crossing, at low \(x\) — one stable low state.
  • Medium \(\gamma\): three crossings — two stable states separated by one unstable one. This is the bistable switch.
  • Low \(\gamma\) (shallow line): one crossing, at high \(x\) — one stable high state.

This is lysis versus lysogeny restated as a dynamical fact: bistability of \(x\) (repressor level) is the mathematical form of "the same phage genome can commit to two different fates," and \(\gamma\) — the degradation rate relative to synthesis — is the single parameter that moves the system between the three regimes.

Testing the model against something built, not found

The slide's own next question is how to check this against biology rather than just the algebra. Its answer is synthetic biology: build the designed circuit from scratch in a cell, rather than working only with the naturally occurring switch, and see whether it behaves as the model predicts. It cites two examples of this done for related genetic switches:

  • Isaacs et al., "Prediction and measurement of an autoregulatory genetic module," PNAS 100, 7714 (2003).
  • Gardner et al., "Construction of a genetic toggle switch in Escherichia coli," Nature 403, 399 (2000).

From there the slide turns to what building such a circuit requires, listing the toolbox of the genetic engineer — restriction enzymes, plasmids, PCR, and fluorescent proteins — as the opening of the next topic, without saying more about any of them.

Sources

  • Slides only: MIT 8.591J Systems Biology (Fall 2004), Lecture 3 summary, computational-biology/mit-ocw/8591j-2004/lectures/04-notes.md. No transcript, written notes, or exercises were supplied for this lecture.
  • That file is itself a model-reconstructed conversion of a PDF with no text layer (fidelity: reconstructed); its own header warns that the prose is a paraphrase and every equation is unverified against the original slide. This chapter reproduces the reaction scheme, the governing ODE, and the graphical-stability discussion exactly as given there, and the diagram above redraws the stability-analysis figure on that same slide (three panels: high, medium, and low degradation rate) as a single medium-rate case, in the notation the slide uses (\(\alpha\), \(\gamma\), \(\sigma_1\), \(\sigma_2\)).
  • The two citations at the end (Isaacs et al. 2003; Gardner et al. 2000) are given on the slide itself and are not expanded on beyond the citation.
  • The slide's closing list — restriction enzymes, plasmids, PCR, fluorescent proteins — is the start of the next lecture's material and is not developed further in this source.

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