Auditing Haldane Consistency in Reversible Enzyme Kinetics: A Curated Two-Sided Backbone and a Labeled Fold-Error Benchmark

Technical Summary: Auditing Haldane Consistency in Reversible Enzyme Kinetics

Problem Statement
Reversible enzyme kinetic constants provide a quantitative intersection between enzyme kinetics and biochemical thermodynamics. For a reversible reaction in a closed system, microscopic reversibility dictates that the kinetic constants must satisfy the Haldane relation, which fixes the apparent equilibrium constant ($K'_{eq,kin}$) implied by the rate law. However, public databases (e.g., SABIO-RK, BRENDA, TECRDB) often contain heterogeneous data where kinetic parameters and thermodynamic equilibrium constants are reported for different enzyme forms, organisms, or conditions, or where forward and reverse kinetics are not reported in the same study. Consequently, there is a lack of curated, single-study "two-sided" records that allow for an independent audit of whether reported kinetic constants are thermodynamically consistent with measured or estimated equilibrium constants. Furthermore, there is no standardized, calibrated, direction-symmetric reporting scale for quantifying the discrepancy between these two independent windows onto the same reaction.

Methodology
The authors propose a workflow centered on a specific scoring metric and a rigorous curation protocol:

1. The Haldane-Consistency Score ($C_{Haldane}$):
   The paper utilizes the reciprocal cost function defined by Washburn and Zlatanović [1] as a direction-symmetric audit scale:
   $$C_{Haldane} = J(x) = \frac{1}{2}(x + x^{-1}) - 1 = \cosh(\ln x) - 1$$
   where $x = K'_{eq,kin} / K'_{eq,thermo}$.

   • Properties: The score is zero at perfect agreement ($x=1$), symmetric for reciprocal errors ($J(x) = J(1/x)$), and encodes the free-energy discrepancy ($\Delta\Delta G$) in $RT$ units. It is strictly increasing in $|\ln x|$, preserving the ordering of records by fold-error magnitude.
   • Axiomatic Basis: The functional form is uniquely characterized by five axioms (reciprocal symmetry, normalization, a specific composition law, continuity, and unit calibration), distinguishing it from simpler metrics like squared log-error.
   • Reporting Bands: Continuous scores are mapped to descriptive fold-discrepancy bands (e.g., "within twofold" corresponds to $C_{Haldane} \le 0.25$).

2. Curation Protocol:
   A reproducible workflow was established to harvest "two-sided" records:

   • Inclusion Criteria: Records must be reversible uni-uni (or pseudo-uni-uni with fixed cosubstrates) or follow a specific multi-substrate Haldane relation. Crucially, forward and reverse kinetic constants must come from the same enzyme form, organism, and assay context within a single study, paired with a condition-matched thermodynamic comparator (from TECRDB or eQuilibrator).
   • Exclusion Criteria: Thermodynamically constrained global fits are excluded because they satisfy the Haldane relation by construction and offer no independent audit. Records with missing reverse constants, ambiguous identities, or mismatched conditions are discarded.
   • Mechanism Handling: For bi-bi reactions (ordered, rapid-equilibrium random, ping-pong), mechanism-specific Haldane relations are applied to compute $K'_{eq,kin}$, while the scoring metric remains unchanged.

3. Benchmarking Strategy:
   Since real curated records lack ground-truth error labels, a semi-synthetic labeled benchmark was constructed. Starting from 29 "consistent" seed records (within-twofold), the authors injected six distinct error modes (direction swaps, unit transcription errors, proton mismatches, condition mismatches, isoform mismatches, and mechanism/formula mismatches) to generate 1,885 labeled "inconsistent" cases. This allows for the estimation of operating characteristics (e.g., AUC) conditional on the injected error taxonomy.

Key Contributions

• A Calibrated Reporting Scale: The paper establishes $C_{Haldane}$ not as a new ranking criterion, but as a calibrated, direction-symmetric scale that reports fold-error and free-energy discrepancy simultaneously, with a closed-form inverse.
• A Curated Two-Sided Backbone: The authors assembled a dataset of 21 audited single-study records (16 uni-uni, 5 bi-bi) meeting strict inclusion criteria. This is the first public corpus curated to this specific standard.
• A Semi-Synthetic Benchmark: A labeled dataset quantifying the detectability of specific error modes using the proposed score, providing the first operating characteristics for Haldane consistency auditing.
• Reproducibility: All data, code, protocols, and benchmark generators are archived with SHA-256 checksums to ensure exact reproduction of results.

Results

• Demonstration Set: Applied to a small set of manually curated records (including phosphoglucose isomerase, triosephosphate isomerase, and fumarase), the score successfully identified buffer-dependent discrepancies in fumarase (non-phosphate buffers showed 3.8–4.5-fold errors) and confirmed reciprocity in racemase controls.
• The Backbone (21 Records):
  • Distribution: 18 of the 21 records fell within the "within-twofold" band ($C_{Haldane} \le 0.25$).
  • Flagged Records: Three records were flagged as inconsistencies (exceeding the twofold boundary): Pea phosphoriboisomerase (3.09-fold), Clostridium scindens D-psicose 3-epimerase (2.88-fold), and Sinorhizobium D-tagatose 3-epimerase (3.96-fold).
  • Independence: Eight records were "genuinely independent" (kinetics fit without thermodynamic priors vs. separate equilibrium measurements); all eight were within the twofold band.
  • Chemical Scope: The backbone is heavily concentrated in carbohydrate isomerases and epimerases; thus, the results are described as "within-family observations" for this specific chemistry.
• Benchmark Performance:
  • As a continuous classifier, the score achieved an Area Under the ROC Curve (AUC) of 0.784 (95% bootstrap CI: 0.725–0.838).
  • Detectability: Gross errors (unit transcription, proton mismatches) were detected with 100% sensitivity. The least detectable modes were direction swaps (when $K'_{eq} \approx 1$) and condition mismatches (wrong temperature), with sensitivities of 0.76 and 0.11 respectively at the twofold cut.
  • The AUC is invariant under monotone rescaling of $|\ln x|$, confirming it measures the discriminability of fold-error information rather than a unique advantage of the specific $C_{Haldane}$ functional form.

Significance and Claims
The paper modestly claims to provide a reproducible workflow and a descriptive empirical reference distribution for a specific, curated sample of enzyme kinetics.

• Not a General Population Estimate: The authors explicitly state that the results (18/21 within-twofold) cannot yet be extrapolated to all reversible enzyme kinetics, as the sample is chemically narrow (carbohydrate isomerases/epimerases) and lacks ground-truth error labels for the real records.
• Feasibility Demonstration: The study serves as a feasibility demonstration that a two-sided audit is possible and that a significant majority of well-characterized records in this specific class are thermodynamically consistent.
• Curation Targets: The three flagged records are presented not as validated errors, but as "curation targets" requiring further scrutiny, as the discrepancy could arise from unmodeled secondary features (e.g., product inhibition) or data assembly issues.
• Operational Utility: The primary contribution is the establishment of a standardized, calibrated scale ($C_{Haldane}$) and a protocol that can be applied to future data harvests without methodological changes, facilitating the growth of a larger, more diverse audit corpus.