The Cha / King–Altman algorithm

EnzymeRates derives rate equations symbolically with the King–Altman schematic method (King and Altman, 1956), extended by Cha's rapid-equilibrium treatment (Cha, 1968). What makes this practical is Julia's metaprogramming: the whole derivation runs at compile time inside @generated functions. The first time rate_equation is evaluated for a given mechanism, the package performs the full symbolic derivation once and writes the resulting closed-form expression directly into the compiled method. Every later call is just that expression — a plain arithmetic formula with no symbolic machinery left behind, no heap allocations, and a runtime on the order of 100 ns on a modern processor. The fitter can therefore evaluate the rate equation millions of times per cross-validation fold and never pay the symbolic cost again.

Overview of the algorithm

The mechanism graph is partitioned into rapid-equilibrium (RE) segments — sets of enzyme forms linked by RE steps — connected to each other by SS steps.

Within each RE segment, the package picks the form with the fewest bound metabolites as the segment root (the free-enzyme reference form), with ties broken toward no covalent residual and then by name. Every other form in the segment gets an alpha factor — its relative abundance expressed as a ratio to the root. Referencing to the free-enzyme form is what produces the readable 1 + S / K_S_E + P / K_P_E denominator in the final equation.

Across segments, the SS steps form an inter-segment rate matrix. The Cha numerator and denominator come from cofactor determinants of that matrix, with each diagonal entry weighted by the sigma sum of alpha factors for its segment.

Keeping the equation finite at zero concentration

The derived rate equation is division-free in the concentrations: no metabolite concentration ever appears in a denominator, so the equation contains no 1/[S]-style terms anywhere.

This matters for fitting real data. Datasets often include measurements where a substrate or product concentration is exactly zero — a saturation curve that starts at zero substrate, or an initial-rate measurement taken before any product has formed. An equation with a 1/[S] term would diverge at those points and the fit would fail. Because the rate equation has no such terms, it stays finite at every concentration, including zero, and those data points are fit like any other.