Abstract

Biochemical Systems Theory (BST) represents nonlinear biochemical rate laws by local power-law approximations in logarithmic concentration coordinates. First-order coefficients are elasticities, whereas higher-order derivatives describe local log-synergism and its variation. Ordinary higher derivatives, however, are not tensorial under nonlinear reparameterizations and can mix biochemical response structure with coordinate artifacts. We formulate a covariant hierarchy, cBST1--cBST3, on a positive operating-point space equipped with a declared reference connection. cBST1 recovers classical elasticities in a flat logarithmic chart, cBST2 is the covariant Hessian of the log-response, and cBST3 is the symmetrized covariant derivative of cBST2. The framework is quantitatively evaluated using three representative rate laws from the curated yeast glycolysis model BIOMD0000000064: glucose transport, glucose phosphorylation, and phosphofructokinase. For 10,000 finite log-concentration perturbations at each of five radii, cBST2 reduced the cBST1 log-rate root-mean-square error by 96.6--98.6% at the largest tested radius, and cBST3 provided a further 68.4--98.1% reduction. The contracted cBST2 and cBST3 terms strongly predicted the corresponding lower-order truncation errors. Under the nonlinear transformation q_i=sinh(u_i), covariant contractions agreed across coordinates to within a 95th-percentile relative error of 1.2x10^-13, whereas ordinary higher derivatives showed order-unity coordinate mismatches. Supplementary analytic tests recovered the expected second-, third-, and fourth-order truncation-error scaling. These results show that cBST1--cBST3 are not only coordinate-consistent descriptors but also practical diagnostics of where local power-law approximations require higher-order correction.