Lecture notes. Written from the material of MIT 8.591J 2014, 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.

24. Negative Autoregulation and the Repressilator¶

What this covers¶

Why does wiring up a gene that represses itself not automatically give you a biological clock, and what does it actually take? This chapter works through the negative-autoregulation models used to answer that question, the stability argument (Poincaré–Bendixson, linearization, eigenvalues) that tells you when a set of ODEs can and cannot oscillate, and the Elowitz–Leibler repressilator as the worked case where the argument was used to design a real circuit. It assumes you have already seen linear stability analysis for a system of two linear ODEs (trace/determinant, eigenvalues) and know what a Hill-function repression term looks like.

Why build an oscillator at all¶

Oscillators matter, first, because they are the basis for time-keeping: a pendulum clock is just an oscillation that lets a winding mechanism advance a little on every swing, and every modern clock, however high its frequency, is still built on some oscillatory dynamic underneath. Oscillators are also just interesting dynamically in their own right.

Biology already has an oscillator built from a gene network: the circadian clock, which keeps track of the day/night cycle and is entrained by it. It is useful for an organism to know roughly where in the day it is, independent of instantaneous light levels — a cloud crossing the sun is not a reliable signal that night has fallen, and an organism that shut down on every cloud would be at a real disadvantage. Circadian oscillators are not the subject of this lecture, but it is worth knowing that in some systems the oscillation survives in vitro, with the gene-expression machinery removed entirely — just the protein components cycling in a test tube. That was a genuinely surprising result when it was first published.

A single auto-repressing gene cannot oscillate¶

Start with the simplest possible negative feedback: one gene whose protein represses its own expression. A verbal argument for oscillation is easy to construct: start with a lot of protein, so expression is strongly repressed and the concentration falls; once it has fallen enough, repression relaxes, expression resumes, and the concentration climbs back to where it started — and now you're back at the beginning, so the cycle should repeat forever, with no decay in amplitude.

That argument should not be trusted. Just because you can tell a story in which something happens does not mean a particular equation reproduces it — the value of writing an actual equation is that it forces every assumption into the open, and then you can ask, of that specific equation, whether it oscillates or not. The simplest non-dimensionalized model of a negative auto-regulatory loop is

\[\dot p = \frac{\alpha}{1+p^n} - p ,\]

where $\alpha$ bundles together the expression strength, the protein's lifetime, and the repressor's binding affinity, and $n$ is the cooperativity (Hill coefficient) of the repression.

Without doing any analysis, this equation is already ruled out. For a fixed value of $p$, this equation gives exactly one value of $\dot p$ — it is single-valued. But any oscillation has to pass back and forth through the same concentration, with the derivative sometimes rising and sometimes falling at that same point; that is a multi-valued relationship between $p$ and $\dot p$, which a first-order autonomous ODE in one variable simply cannot produce. This is special to differential equations with a single variable: a discrete-time difference equation can oscillate, and a second-order equation — position and velocity, as in a mass on a spring — can too, because it has two interacting dynamical variables. One variable, one derivative, no oscillation, regardless of what function sits on the right-hand side.

Adding the mRNA: a two-variable model¶

The one-variable argument does not survive once you make the model even slightly more realistic. A gene is first transcribed into mRNA, and the mRNA is then translated into protein, so write $m$ for mRNA concentration and $p$ for protein concentration:

\[\dot m = \frac{\alpha}{1+p^n} - m , \qquad \dot p = \beta(m-p).\]

The protein represses transcription of its own mRNA; the mRNA decays; the mRNA's translation makes protein; the protein also decays, at a rate scaled by $\beta$. Two non-dimensionalization choices are buried in this equation and are worth making explicit:

The unit of time is the mRNA lifetime — that's why there is no coefficient in front of the $-m$ term. $\beta$ is then the ratio of the mRNA lifetime to the protein lifetime, introduced precisely because the two lifetimes are usually different. (The Elowitz–Leibler paper itself states this ratio backwards, as protein lifetime over mRNA lifetime — worth checking against the algebra rather than the caption if you go back to it.) Since proteins are typically far more stable than mRNA, $\beta$ is typically much less than 1.
The unit of protein concentration is $K$, the dissociation constant — the protein concentration that gives half-maximal repression, so $p=1$ means half repression by definition. The unit of mRNA concentration is more subtle: it is fixed by the requirement that the $p-m$ term above carry no extra coefficient, and that requirement turns out to depend on three things — the translation efficiency (how much protein one mRNA molecule produces per unit time), $\beta$, and $K$ itself. This is a good example of why a clean non-dimensional equation is not the same as an equation whose terms you can interpret at a glance: mathematically simple, biologically you have to be careful about what "$m=1$" is actually saying.

Fixed points. The origin is not a fixed point: with no protein present there is no repression, so mRNA is still being made ($\alpha/(1+0)=\alpha\neq 0$). Setting both derivatives to zero gives $m_{\mathrm{eq}}=p_{\mathrm{eq}}$, and — writing $p_0$ for this common value, to avoid presupposing whether it is a stable equilibrium or just a fixed point —

\[p_0 = \frac{\alpha}{1+p_0^{\,n}} .\]

For example, with $\alpha=10$ and $n=2$, $p_0=2$ solves this exactly: $10/(1+2^2)=10/5=2$.

When can a two-variable system oscillate? Poincaré–Bendixson¶

Whether this system oscillates is no longer decidable "for free" the way the one-variable case was — you actually have to compute something. But there is still a clean criterion available in two dimensions, the Poincaré–Bendixson criterion. Suppose you can draw a bounded region in the $(m,p)$ plane that every trajectory enters and never leaves — here that's true, because trajectories can't cross the axes (some mRNA generates protein, some protein generates mRNA, so neither species alone can stay pinned at the boundary) and can't run off to infinity (once both concentrations are large enough, degradation dominates and pulls them back in). Given such a trapping region containing a single interior fixed point, the question "does the system oscillate?" collapses to the question "is that fixed point stable?" A stable fixed point means every trajectory spirals in and settles there — no sustained oscillation. An unstable fixed point pushes trajectories away from it, while the outer boundary of the trapping region pushes trajectories in from outside; because two-dimensional trajectories can never cross each other (crossing would mean two different values of $\dot m,\dot p$ at the same point — the same single-valuedness argument as before, just in two dimensions), the flow has nowhere to go except to settle onto a closed loop sandwiched between the two — a limit cycle.

Trajectories cannot cross in two dimensions. An unstable fixed point pushes the flow outward (spiral) while the trapping boundary of a Poincaré–Bendixson region pushes it inward; the only place left for the flow to go is a closed orbit between the two — the limit cycle.

It matters that this is a two-dimensional argument. The no-crossing property of trajectories holds in any dimension, but in two dimensions it is an extremely strong constraint — a trajectory that can't cross itself or any other trajectory has very little room to move. In three or more dimensions there is a whole extra direction for trajectories to use to avoid each other, which is essentially why chaotic behaviour in differential equations needs at least three dimensions.

Linearizing around the fixed point¶

To find the stability of $p_0$, linearize. Write $\tilde m = m-m_0$, $\tilde p = p-p_0$ for small deviations from the fixed point (with $m_0=p_0$), and take the Jacobian of $(\dot m,\dot p) = (f(m,p),g(m,p))$ at that point:

\[A=\begin{pmatrix}\partial f/\partial m & \partial f/\partial p\\ \partial g/\partial m & \partial g/\partial p\end{pmatrix}_{(m_0,p_0)} = \begin{pmatrix}-1 & x\\ \beta & -\beta\end{pmatrix},\qquad x = \left.\frac{\partial}{\partial p}\!\left(\frac{\alpha}{1+p^n}\right)\right|_{p_0} = -\,\frac{n\alpha\,p_0^{\,n-1}}{(1+p_0^{\,n})^2} .\]

Note $x<0$ — repression is, well, repressive.

For a $2\times2$ system, the fixed point is stable exactly when the trace of $A$ is negative and the determinant is positive (this trace/determinant shortcut is special to two dimensions; the general rule, needed below, is that every eigenvalue must have negative real part). Here:

\[\operatorname{tr}A = -1-\beta < 0 \quad\text{always, since } \beta>0,$$ $$\det A = \beta - x\beta = \beta(1-x) > 0 \quad\text{always, since } x<0 \text{ makes } 1-x>1.\]

Both conditions hold for every choice of $\alpha$, $n$, $\beta>0$ — there is no parameter regime in which this fixed point is unstable. So the two-variable (mRNA + protein) negative-autoregulation model can never sustain a limit-cycle oscillation, for any parameters. That's a stronger statement than the one-variable case, and unlike the one-variable case it required an actual calculation to see — it isn't visible "for free." (Stability doesn't mean the approach to the fixed point is monotonic: the eigenvalues could still be complex, giving a decaying, ringing approach. What's ruled out is a sustained oscillation.)

Limit cycles versus neutrally stable orbits¶

A limit cycle has a characteristic amplitude and period that don't depend on where you start — in that sense it behaves like a stable fixed point, just for an orbit instead of a point, and this is what makes limit-cycle oscillations the mathematically robust, "nice" kind. Contrast this with a neutrally stable orbit, which shows up when the linearization gives purely imaginary eigenvalues: the orbits then form closed loops around the fixed point whose size depends on the initial condition — there's a whole family of them, not one preferred amplitude. These are considered less interesting biologically because they are fragile: a small change to the model's parameters typically turns them into decaying spirals, growing spirals, or genuine limit cycles. This is exactly the character of the oscillations in the Lotka–Volterra predator–prey model, revisited later in the course — not limit cycles, but neutrally stable orbits.

What negative autoregulation needs in order to oscillate¶

None of this means negative autoregulation can never produce oscillations, experimentally or computationally — only that the bare two-variable model above cannot. What's missing is delay. Two ways to add it:

An explicit time lag — make repression depend on the mRNA or protein level from some time in the past rather than the instantaneous level, reflecting the real time it takes to transcribe, translate, and fold a protein.
More intermediate steps — mRNA is made, then a peptide chain, then the chain folds, then the folded proteins have to multimerize before they can repress. Writing all of these steps down explicitly raises the dimension of the system, and for reasonable parameters that alone can produce oscillations from nothing but negative autoregulation.

Jeff Hasty's group (UC San Diego) has spent roughly the last decade building exactly this kind of delay-based single-gene oscillator in E. coli, moving back and forth between models and cells. One thing modeling gets right immediately: leakage — residual expression even while the gene is supposedly repressed — actively works against oscillation, so a good design uses especially tight promoters with low leaky expression.

The repressilator: Elowitz and Leibler's design¶

Elowitz and Leibler's question was whether you could actually build an oscillator by wiring together components that had no history of working together, rather than repurposing something like the circadian clock. Their design was a ring of three mutual repressors — call them $x$, $y$, $z$ — with $x$ repressing $y$, $y$ repressing $z$, and $z$ repressing $x$. Before building it, they built a model to guide the design: a week of thinking is cheaper than a year of experimental biology.

Two lessons the modeling gave them, both consistent with the discussion above:

Match the lifetimes of mRNA and protein. This plays the same role as the delay elements discussed above — if one species decays much faster than the other, the faster process effectively drops out of the dynamics. It's hard to lengthen an mRNA's lifetime much in bacteria, so instead they shortened the protein lifetime, by tagging the $x,y,z$ transcription factors for faster degradation.
Minimize leakage. They used synthetic promoters engineered for a large ratio of "on" expression to "off" (repressed) expression, since leakage inhibits oscillation.

Bulk experiment. They synchronized a population of cells with a pulse of IPTG and tracked fluorescence from one of the three proteins. The population signal showed a single damped-looking cycle rather than sustained oscillation. The explanation is desynchronization: individual cells' oscillators start roughly in phase after the pulse, but random noise causes their phases to drift apart over time, so an average over many out-of-phase single-cell oscillators looks like a damped signal even if every individual cell keeps oscillating.

Single-cell experiment. Plating cells on an agar pad and imaging them individually as they oscillated and divided settled the question: the cells really do oscillate — the first demonstration that you could assemble unrelated regulatory parts into a working oscillator. But the oscillation was far from clean: only about 40% of the cells showed clear oscillation, and even those had substantial noise in period and amplitude. The paper attributes some of this to the low copy numbers of the relevant genes and proteins, which makes the system intrinsically noisy (a stochastic effect that a deterministic ODE model cannot capture). This observation is plausibly what pushed Elowitz toward a later, more influential paper specifically about noise in gene networks — not required reading for this course, but worth knowing it exists.

The symmetric protein-only repressilator model¶

The full model — three mRNAs and three proteins — is a six-dimensional system, too large to work through by hand in one sitting. Following the intuition-building move Elowitz himself used, simplify to a symmetric, protein-only model: assume the three genes are identical (same $\alpha$, $n$, $K$), even though in the real construct the three synthetic promoters differ.

\[\dot p_1 = \frac{\alpha}{1+p_3^{\,n}} - p_1,\qquad \dot p_2 = \frac{\alpha}{1+p_1^{\,n}} - p_2,\qquad \dot p_3 = \frac{\alpha}{1+p_2^{\,n}} - p_3 .\]

By symmetry the interior fixed point again has $p_1=p_2=p_3=p_0$ with $p_0=\alpha/(1+p_0^{\,n})$, the same equation as before.

One caution before linearizing: with three variables there is no Poincaré–Bendixson theorem to fall back on. The no-crossing argument that pinned two-dimensional flows down so tightly no longer applies once there's a third direction available, so "the interior fixed point is unstable" does not automatically imply "the system has a limit cycle" the way it did in two dimensions — that has to be checked (e.g. by simulation), not assumed. It turns out to be true for this particular model, but that is a fact about this model, not a general theorem.

The Jacobian, thanks to the cyclic symmetry, is the circulant matrix

\[A = \begin{pmatrix}-1 & 0 & x\\ x & -1 & 0\\ 0 & x & -1\end{pmatrix},\qquad x = -\,\frac{n\alpha\,p_0^{\,n-1}}{(1+p_0^{\,n})^2}\]

— the same $x$ as before, now coupling each protein to its repressor around the ring. Stability in three dimensions requires every eigenvalue of $A$ to have negative real part (the trace/determinant shortcut used earlier does not generalize). Expanding $\det(A-\lambda I)=0$ for this matrix, most of the terms vanish and it collapses to the remarkably simple

\[(1+\lambda)^3 = x^3 .\]

Writing $\mu = 1+\lambda$, this equation has exactly three roots, equally spaced by $120°$ around a circle of radius $|x|$ in the complex plane: since $x$ is real and negative, one root is $\mu=x$ itself, sitting on the negative real axis, and the other two form a complex-conjugate pair $120°$ away from it on either side.

The shaded region is where the eigenvalue λ has negative real part. The real root µ=x always lands in it; the complex pair sits at real part |x|/2, which crosses the stability line Re(1+λ)=1 exactly when |x|=2 — the whole stability question for the ring reduces to this one number.

To recover $\lambda$ itself, shift every point one unit to the left: $\lambda=\mu-1$. Stability needs every $\mu$ to have real part less than 1. The real root $\mu=x$ is negative, so it's never a problem. The complex pair sits at real part $|x|\cos 60° = |x|/2$ (the short side of a 30-60-90 triangle whose hypotenuse is $|x|$). So the entire stability question comes down to comparing $|x|/2$ with $1$:

\[\text{stable (no oscillation)} \iff |x|<2, \qquad \text{unstable (oscillation)} \iff |x|>2 .\]

Strong-expression limit ($\alpha\gg1$). The fixed point condition $p_0(1+p_0^{\,n})=\alpha$ gives $p_0^{\,n+1}\approx\alpha$. Substituting,

\[|x| = \frac{n\alpha\,p_0^{\,n-1}}{(1+p_0^{\,n})^2} \approx \frac{n\alpha\,p_0^{\,n-1}}{p_0^{\,2n}} = \frac{n\alpha}{p_0^{\,n+1}} \approx \frac{n\alpha}{\alpha} = n .\]

So $|x|\to n$ as $\alpha\to\infty$, and the stability condition $|x|<2$ collapses to $n<2$: however strong the expression, oscillation still requires cooperativity $n>2$. This is not obviously intuitive — one might expect strong enough expression on its own to be enough, and it isn't; raw expression strength cannot substitute for cooperative repression. (There isn't a fully satisfying intuitive explanation for this beyond it being a statement about the slopes of the repression curves near the fixed point.)

A concrete case, $\alpha=2$. Here $p_0(1+p_0^{\,n})=2$ is solved by $p_0=1$ for every $n$, since $1\cdot(1+1)=2$ regardless of $n$. At $p_0=1$, $x=-n\alpha/(1+1)^2=-n/2$, so $|x|=n/2$, and the stability condition $|x|<2$ becomes $n<4$: with weaker maximal expression, oscillation now requires a more strongly cooperative repressor, $n>4$.

A trap: "any ring of repressors has to oscillate"¶

There's an appealing Boolean argument that seems to settle the question for free. Represent each gene as simply on or off. Start at $(0,1,0)$: $y$ is on, repressing $z$ (already off); once $y$'s target relaxes and $x$ turns on, you get $(1,1,0)$, and now $x$ starts repressing $y$, giving $(1,0,0)$ — and tracking this through, the "on" state visibly marches around the ring forever. It looks like any three mutually repressing genes are structurally guaranteed to oscillate.

The stability analysis above shows this is false. Without enough cooperativity — $n$ below the threshold set by $|x|<2$ — the continuous system just relaxes to the symmetric interior fixed point and stays there; no oscillation at all. Simply wiring up three mutual repressors is not sufficient. You need transcription factors that actually multimerize and repress cooperatively, with a large enough Hill coefficient, to have a real chance of getting the ring to oscillate.

A methodological note: true, but not the mechanism¶

A natural follow-up question: doesn't cooperativity (multimerization) itself introduce a hidden delay, since dimerizing takes some extra time after the monomer proteins are made? That may well be true of real cells — but it is not what produces instability in this model, because the model as written contains no time delay of any kind. Cooperativity here changes only the steepness of the repression curve (how sharply the Hill function turns on), not any time lag. It's worth holding on to the general point: a statement can be true of a real system and still not be the mechanism actually doing the explanatory work in a specific piece of analysis. It is easy — and common — to conflate the two.

Sources¶

MIT 8.591J Systems Biology (Fall 2014), lecture recording transcript computational-biology/mit-ocw/8591j-2014/recordings/recordings/xnnxlsy-f-s.md (CC BY-NC-SA 4.0). Transcript-only source — no slides or board images were supplied, so every equation and figure above is reconstructed from what was said aloud (timestamps below), not copied from a board.
Motivation for oscillators, circadian aside: 00:00–03:32.
One-variable auto-repression model and the single-valuedness argument: 03:32–12:43.
Two-variable (mRNA + protein) model, non-dimensionalization, $\beta$: 12:43–22:31.
Poincaré–Bendixson, direction of rotation, noise-induced-oscillation aside: 22:31–24:56.
Fixed point, linearization, trace/determinant, stability result: 24:56–43:40.
Limit cycles vs. neutrally stable orbits, Lotka–Volterra aside, delay/leakage discussion: 43:40–47:22.
Repressilator design rationale and experiments (bulk and single-cell): 47:22–54:08.
Symmetric protein-only repressilator model, eigenvalues, complex-plane argument: 54:08–1:14:32.
Boolean-logic trap and the cooperativity/delay methodological aside: 1:14:32–1:21:09.
Named but not supplied as source material, so not reproduced here beyond what was said about them: M.B. Elowitz and S. Leibler's repressilator paper (the paper under discussion throughout, including its stated design rationale, its own — reversed — definition of $\beta$, and its remark that "more realistic models may exhibit other complex types of dynamic behavior"); a later Elowitz paper on noise in gene networks (named as optional reading); Jeff Hasty's papers on delay-based single-gene oscillators in E. coli; the circadian-oscillator literature referenced in passing; the Lotka–Volterra predator–prey model (a forward reference to later in the course); the Poincaré–Bendixson theorem itself (used, but not cited to a text).

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