Analysis of binding and catalysis via reaction-order geometry
Reaction order is the structure of how binding regulates catalysis. Reaction-order polyhedra turn a binding-catalysis network into a geometric object you can read without simulating it.
Lecture 8 argued that a cell is binding regulating catalysis. This lecture is the analysis that argument was for. The reaction order of a catalytic step with respect to a species is the log-log slope of its rate, and because the binding layer is at equilibrium those slopes take a small number of values. The set of achievable orders is a polyhedron, its vertices are the archetypal behaviors, and a trajectory of the network is a path across it. That is a holistic description: the whole behavior at once, rather than one simulation at a time.
This lecture is new in 2026 or substantially re-scoped, so parts of it are not yet carried over from the 2025 notes. The teaching team should treat this list as the commissioning brief, and the lecturer as the backlog.
- Define reaction order carefully as a log-derivative, and show it is the natural coordinate for a binding equilibrium.
- Build the polyhedron for a small network, name its vertices, and interpret each as a regime.
- Behavior as a trajectory across regimes; what changes at a facet.
- Homotopy continuation for binding-catalysis steady states, with runnable code.
- Provenance: this is the lecturer's own research programme (reaction-order geometry; the essay 'The Order of a Reaction'), not 2025 course material.
Reaction-order polyhedra
The log-log slope as structure, the polyhedron of achievable orders, and the three archetypes.
1Reaction-order polyhedra
The framework asks what a reaction network can be, in the space of stoichiometric behavior — reaction orders form a polyhedron, and behavior is a point in it.
The ROP framework (reaction-order polyhedra) is the holistic-analysis counterpart to the per-circuit perspective. Instead of simulating one instance, it characterizes the set of behaviors a binding reaction can exhibit: the exponents in a monomial description span a polyhedron in reaction-order space, and a given binding-catalysis network lives on a trajectory through regimes of that space. The three archetypal behaviors of a binding reaction (saturation, bottleneck, ultrasensitivity — the week-3 results) reappear as distinguished regions of the polyhedron, and analysis becomes an exercise in locating the regime rather than integrating the ODE.
2Three archetypes, regimes, homotopy
Computation of binding–catalysis networks via homotopy continuation: follow the solution manifold as parameters change, and the regimes — and the transitions between them — become visible.
The computational arm: polynomial systems of binding–catalysis networks are solved by homotopy continuation — track the solutions of the steady-state equations as parameters move, so the regime structure and bifurcation points are computed rather than guessed. This is the frontier of the course, and it connects to the essays' Structure is Sparsity question: what is actually obtainable about such a network from data, and where the line between structure and parameters really falls.3 For the student blocks: treat this as the "beyond the core" block — the core is the three archetypes and the idea of regime-location; the honest caveat is that the framework is recent and the lecturer's own, so every claim in this block should be checked against the framework's papers before it is taught.
Pick one: (a) the counting argument applied to a system you know (how many gates would it need as NAND logic — and how does the cell do it instead?); (b) derive (2) yourself for a three-gene example and compute its decision boundary; (c) find a multistable fate-decision circuit and explain which "curvature fix" it uses; (d) read the ROP material and state one thing you found compelling and one you doubt. One argument, one figure, one source.
†References & note sources
Where this page's claims and numbers live.
- F. Xiao, CCBS 2025 Lecture 8 (computation biomachines half: logic gates, GRN-encoded LTU, Turing universality, counting critique, binding-based computation). PDF This lecture traces here.
- F. Xiao, CCBS 2025 Lecture 10 — Computation (bio)machine (the NAND reactions, the ~10³-enzyme counting table, LTU derivation and titration GRN, binding vs catalysis counts, toggle/multistability/SANI/tristability, Multi Fate example). PDF This lecture traces here.
- F. Xiao, Structure is Sparsity and the ROP framework — the essays; the reaction-order-polyhedra repository.A supplement; not in the 2025 handwritten notes.
- Zhu, R., et al. (Elowitz lab), Science 2022 — "Multi Fate": combinatorial encoding and multistability in cell fate.An example for this lecture.
- J. Collins et al., Nature 2000 — the toggle switch. The canonical bistable circuit.