The metabolic biomachine: flux balance and flux exponent control

The machine's plumbing, written as a linear program: what goes in, what comes out, who pays for it, and how the exponents control the flux.

CCBS · EST 5053 · Fall 2026 · lecture 13 · core page · 2026-10-22 (Thu) 10:40–12:15 · students present, lecturer-backed

Metabolism is the part of the cell that is genuinely a factory, and the factory has a mathematics of its own. Stoichiometry plus a steady-state assumption gives a linear program whose solution is a flux distribution, and that is flux balance analysis, the workhorse of metabolic engineering. The lecture then connects it back to the course's own language: control of flux by reaction orders, which is the metabolic-control-analysis question asked in binding-catalysis coordinates.

Still to be written into this core page

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.

  • A toy FBA worked end to end with code: build S, set bounds, maximize growth, read the flux distribution.
  • Flux exponent control: control coefficients as log-derivatives, the summation theorem, and why this is the same reaction-order geometry as Lecture 11.
  • Genome-scale models: what they get right, and what a purely stoichiometric model can never say.
  • Metabolic engineering as the industry payoff: a designed pathway, its bottleneck, and the fix.
Part 1

Flux balance

Stoichiometry, the steady state, and the linear program.

1Stoichiometry and flux balance

A metabolic network is a stoichiometric matrix; steady state is S·v = 0; feasibility + an objective = flux balance analysis.

Same formalism as lecture 3, different question. Write the metabolic network's stoichiometric matrix S (species × reactions) and impose steady state S·v = 0 with bounds on fluxes; choose an objective (growth rate, yield of a product) — that linear program is flux balance analysis. Its beauty: it needs no rate constants, only stoichiometry (which is why it works at genome scale): the answer is the cell's best feasible flux distribution. The 2025 growth lecture runs the same bookkeeping in physiological form: the supply-demand picture Jsupply vs Jconsumption must balance at a steady state, and the crossing point is the realized flux.2

2Where the energy comes from

ATP is the cell's currency, and its yield is an accounting identity: per glucose, the electron transport chain's proton-motive force converts the lion's share.

The industry school's first law: life works by spending ATP, and ATP is bought with a redox budget. The metabolic accounting: glycolysis produces 2 ATP per glucose; oxidative phosphorylation produces the rest (the textbook ~30–32 figure, best treated as an OoM quantity: tens, not hundreds, per molecule). The bioenergetic landscape is the same OoM reasoning as lectures 1 and 2 — the difference between fermentation and respiration is the position of the redox ladder, worth one estimate with bionumbers′ numbers next to it. For the course, the operative insight: every design in lecture 15 and every regulation in lecture 10 has an energy price, and the price is denominated in ATP and NADH — cell-scale bookkeeping, machine-scale economics.

†References & note sources

Where this page's claims and numbers live.

  1. F. Xiao, CCBS 2025 Lecture 9 — Introduction to Protein Design (slides: Rosetta, de novo binder pipeline, SARS-CoV-2/LCB1, RFdiffusion, ProteinMPNN, AlphaFold2/ColabFold/MMseqs2, hands-on recipes). PDF This lecture traces here, including the two run_inference recipes.
  2. F. Xiao, CCBS 2025 Lecture 11 — Growth machine (growth law, chemostat/Monod, consumer–resource, proteome partition λ = εφ_R, flux balance, upshift/downshift, Cobb–Douglas/Solow economics analogy, V̇ = λV frontier). PDF This lecture traces here.
  3. L. Cao et al., Science 2020 — SARS-CoV-2 miniprotein inhibitors (LCB1). The case study for this lecture.
  4. J. Watson et al., Nature 2023 — RFdiffusion; J. Ingraham et al., Science 2023 — ProteinMPNN; J. Jumper et al., Nature 2021 — AlphaFold2; M. Steinegger & J. Söding, Nat. Biotechnol. 2017 — MMseqs2. Toolkit references (cited on the lecture slides).
  5. M. Scott, C. Gunderson, E. Mateescu, Z. Zhang, T. Hwa, Science 2011 — the proteome-partition growth law.The backbone result for this lecture.
  6. R. Phillips & J. Kondev, Cell biology by the numbers (book.bionumbers.org) — metabolic yields and energy scales.The numbers for this lecture.