Multistability and oscillations

A plethora of examples, all read as binding-catalysis networks: toggle switches, MultiFate, the repressilator, relaxation oscillators, circadian clocks, the cell cycle, replication initiation.

CCBS · EST 5053 · Fall 2026 · lecture 9 · core page · 2026-10-08 (Thu) 10:40–12:15 · students present, lecturer-backed

With the framework in place, this lecture is a gallery. Each example is a circuit biology actually uses, and the exercise each time is the same: write it as binding and catalysis, simulate it, and read off the property that matters. For switches that is how many stable states there are and how deep the basins run. For oscillators it is whether the oscillation exists at all, what sets the period, and how robust that period is. The biological motivation is cell fate control and development on the switch side, and timekeeping on the oscillator side.

1The LEGO of bioregulation

Beyond the three archetypes: Hill functions, competitive binding, and the design question — what can a cell build from a constrained set of blocks?

The homework of the 2025 lecture sets the LEGO question explicitly: the building blocks are the elementary reaction + its regimes; the bricks are Hill-type regulation, competitive binding, and the titration sponge; and the exercise is to say what a given combination does.¹ The design discipline is stronger than it sounds: any circuit the students propose in later weeks can be checked — is the production term S-shaped enough, are the parameters in a dominance regime, is there a hidden parameter coincidence?

2Multimodality, extinction, ergodicity

Three consequences of discreteness: multi-stability becomes multimodality, zero is an absorbing risk, and time averages betray single cells.

• Multimodality. A bistable deterministic system has two attractors; a stochastic one has two peaks in its stationary distribution — and the rate of switching between them is finite. The toggle switch from lecture 4, in single cells, samples both states (the fate decisions below exploit exactly this).
• Extinction & the absorbing zero. When n hits 0 in the master equation, the production term cannot bring it back — extinction is a real thermodynamic risk in small volumes, and the quasi-steady "leak" terms that ODE models carry are precisely the ones that hide it.
• Ergodicity — time averages vs population averages. A single cell that switches between states over time averages differently from a population snapshot; ergodicity is the assumption that the two agree, and when a system is slowly switching they don't. The design consequence: cross-sectional (single-cell fixed) data and longitudinal (single-cell tracked) data are different instruments, and measures of "noise" depend on which one you used.

3Cell fate: 2ᴺ stable states

A cell doesn't deduce; it transits. N genes with multistability give O(2^N) cell states — combinatorial encoding, not logic.

The lecture's answer to "what do cells actually compute": not logical deduction, not arbitrary maps, but responses to environments = distinct gene-expression states — cell-state transitions. The math of that claim is multistability:

• Toggle switch (Collins 2000): mutual repression, monotone nullclines → one intersection; "needs some more curvature." The failure mode is instructive: a flat monotone picture cannot hold a bistable design.
• The curvature fix: Hill-2 (dimer) repression G_A ~ G_tot/(1 + (B_tot/K_B)²) → Jacobian [[−1,−2],[−2,−1]] style, strong mutual repression destabilizes the middle point; weak/competitive binding (the SANI motif — self-activation + mutual inhibition) restores it.
• Tristability: homodimer activation + heterodimer inhibition (A+A⇌A₂, B+B⇌B₂, A+B⇌C_AB inactive, A₂ activating, B₂ activating) gives a quadratic nullcline β(A_tot−B_tot)²/(K_AK_GA) = δA_tot → more intersections → three stable states. Two genes → three fates.
• The scaling: ~2^N; 10 genes → 100–1000 states. The Elowitz-lab Multi Fate 2022 example (Zhu et al. Science): cell differentiation as computation — ~10²–10³ cell types encoded by ~10 genes via combinatorial encoding of high/low combinations.⁸

4What each motif buys

A one-definition tour of the gallery: toggle, oscillator, positive/negative feedback, feedforward, pulse-front — and the cost of each.

• Toggle switch (mutual repression): two stable states, memory. Needs enough cooperativity (Hill ≥ 2) or it collapses, by the nullcline argument of this lecture.
• Negative feedback: reduces sensitivity, speeds relaxation, can oscillate with delay (repressilator-style; the linear test says stable, with lecture 5's stochastic counterpoint).
• Positive feedback: amplification and decision-making; costs bistability (and noise-latching, lecture 5).
• Feedforward: the tool of lecture 10: incoherent feedforward gives perfect adaptation; coherent feedforward gives pulse-detection and filtering.
• Pulse/ultrasensitive front: time-scale separation as the miniaturization of a differentiation step.

†References & note sources

Where this page's claims and numbers live.