MBES — Michaelis-Menten sensor with first-order drift

Limit of detection, linear range, t₉₀ response time and weekly drift for a whole-cell or enzyme sensor.

Each relation is transcribed from the implementation. Where the code cites no source, the source line says so.

1. $$i = \frac{i_{max}\, S^{h}}{K_{app}^{h} + S^{h}}, \qquad K_{app} = K_M\left(1 + \frac{I}{K_i}\right)$$

   Source: Unsourced engineering default in mbes.ts (2026-10 citation audit).

   The per-analyte constants are unsourced; the table they were attributed to does not exist.

   Symbol	Meaning	Unit
   $i$	steady-state signal current density	A/m²
   $i_{max}$	saturating signal (BOD 0.8)	A/m²
   $S$	analyte concentration (default 50)	mg/L
   $h$	Hill coefficient (BOD 1.2)	1
   $K_M$	half-saturation constant (BOD 100)	mg/L
   $I, K_i$	interferent concentration and its inhibition constant	mg/L

2. $$\mathrm{LOD} = K_{app}\left(\frac{3\sigma_{blank}}{i_{max} - 3\sigma_{blank}}\right)^{1/h}$$

   Source: The 3σ-of-blank detection rule, inverted through the Hill curve; constants unsourced (mbes.ts).

   Symbol	Meaning	Unit
   $\mathrm{LOD}$	concentration whose signal equals three blank standard deviations	mg/L
   $\sigma_{blank}$	blank-current noise (BOD 8×10⁻⁴)	A/m²

   Worked example

   These inputs, passed to the predictor, return the output below. A test re-runs the example on every change, so this number is the one the code produces.

   input	value	unit
   temperatureC	25	°C
   ph	7	pH
   electrodeSpacingCm	2	cm
   hrtHours	24	h
   codMgL	1,000	mg/L

   Output · mbes.limitOfDetectionMgPerL = 0.792 mg/L, within ±0.1 %

   By hand · Default analyte BOD, no interferent: K_app = 100 mg/L, h = 1.2, i_max = 0.8 A/m², 3σ = 0.0024 A/m². LOD = 100 × (0.0024 / 0.7976)^(1/1.2) = 100 × 0.003 009^0.8333 = 0.7919 mg/L.

   Reproduce · Pinned by libs/shared/electrochemistry/src/predictors/worked-examples.test.ts. Run it from the repository root:

   pnpm exec jest -c libs/shared/electrochemistry/jest.config.js worked-examples.test.ts -t "mbes.limitOfDetectionMgPerL"

3. $$S \in \left[\mathrm{LOD},\ c\,K_{app}\right], \qquad c = 3\ (h = 1),\ 1.5\ (h > 1)$$

   Source: Unsourced engineering default in mbes.ts (2026-10 citation audit).

   A rule of thumb, not a fitted linearity criterion.

   Symbol	Meaning	Unit
   $c$	upper-end multiple of K_app	1

4. $$t_{90} = 2.303\,\frac{L^{2}}{D_{eff}}$$

   Source: Diffusion time scale; mbes.ts cites no source for it.

   Symbol	Meaning	Unit
   $t_{90}$	time to 90 % of the steady-state signal	s
   $L$	biofilm thickness (default 0.05 mm)	m
   $D_{eff}$	effective analyte diffusivity in the biofilm (BOD 5×10⁻¹⁰)	m²/s
   $2.303$	−ln(0.1) = 2.302 6, truncated to four figures in the code	1

5. $$S(t) = S_0\, e^{-k_{drift}\, t}$$

   Source: Unsourced engineering default in mbes.ts (2026-10 citation audit).

   Symbol	Meaning	Unit
   $k_{drift}$	first-order fade rate (BOD 0.10)	week⁻¹
   $t$	sensor age	week

• Anode current from the shared Butler-Volmer biofilm kinetics in predict.ts (Geobacter on acetate), with the same Arrhenius, substrate, pH and ohmic corrections as the MFC predictor.
• Linear range from LOD to ≈ 3 · K_M (1.5 · K_M for a cooperative Hill response).
• Effective analyte diffusivity through the biofilm D_eff ≈ 10⁻⁹–10⁻¹⁰ m²/s.
• Drift follows first-order decay of the biofilm / enzyme signal.

• Interferents raise apparent K_M (poorer LOD, narrower range).

• Sensor recalibration / two-point standardisation.
• Diffusional broadening in flow injection — steady state assumed.
• Temperature / pH compensation — fixed K_M / V_max for the stated analyte.

• Modin & Wilén 2012, Water Res 46:6113
• Karube et al. 1977, Biotechnol Bioeng 19:1535

libs/shared/electrochemistry/src/predictors/mbes.ts

heuristic_fixed_fraction — A fixed ±25 % placeholder band. It is NOT a calibrated interval; when served through /api/ml/predict it is replaced by the benchmark band below, or withheld. How served intervals are built

• Benchmark out-of-fold residual quantiles: bes-benchmark-v1 · gkf5 · generated 2026-09-09
• Split-conformal calibration: split-conformal · generated 2026-05-21
• Hierarchical-priors v2 trust snapshot: populated · generated 2026-05-16

Each row is a stratum with n ≥ 30 residuals at nominal level 90%. Coverage is the cross-validated share of held-out points inside the band.

No benchmark cell reaches the minimum n for these systems — intervals are withheld.

The state the lab badge shows for each served parameter and system, from the v2 priors fit.

No v2 prior row for these systems.

pnpm verify:science runs these rules over the committed priors and validation artifacts in pre-commit. Known violations are baselined per rule and slug; a new one blocks the commit. Baseline from hierarchical-priors-v2.json@2026-10-26T00:10:47.721880+00:00.