Mathematical Modeling

Quantifying strain characteristics, fermentation, and efficacy

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1. D‑Alanine Auxotrophy Validation

Goal: Validate dal knockout and D‑alanine dependence using growth kinetics and dose–response.

📈 Gompertz Growth Kinetics

Wild‑type (WT): μ_max = 0.1958 h⁻¹, X_m = 1.541 OD₆₀₀.
Δdal without D‑Ala: μ_max = 0.0016 h⁻¹ (0.8% of WT), X_m = 0.0785 OD₆₀₀ (5.1% of WT).
Δdal + D‑Ala: μ_max = 0.1908 h⁻¹ (97.4% of WT), X_m = 1.4666 OD₆₀₀ (95.2% of WT).

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📉 Four‑Parameter Logistic Dose–Response

🧪 Biomass: R² = 0.9991, EC₅₀ = 0.190 mM, EC₉₀ = 0.665 mM.

⚡ Growth rate: R² = 0.9989, EC₅₀ = 0.113 mM, EC₉₀ = 0.259 mM.

→ Complete restoration by D‑alanine; EC₉₀ values quantify nutritional requirement.

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2. Sporulation & Butyrate Fermentation Kinetics

⏳ Sporulation – Gompertz Model

Maximum sporulation rate K = 99.24%
Maximum formation rate r = 3.408 %/h
Lag phase λ = 2.11 h→ Adapt quickly, perform rapidly
R² = 0.9978 → excellent fit.

🧫 Butyrate Fermentation – Modified Monod

Model includes μ_max, K_s, product inhibition (K_ip), substrate inhibition (K_is), growth/non‑growth associated coefficients (α, β).
Butyrate production is growth‑associated with weak non‑growth component.
Final butyrate titer matches experimental data.

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This figure shows the dynamic coupling of cell growth, substrate consumption, and product synthesis during butyric acid fermentation. Rapid substrate consumption and product synthesis occur in the logarithmic phase, while both growth and synthesis are inhibited in the late stage by substrate depletion and product accumulation.

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3. Intestinal Barrier Function In Vitro Validation Model

🧬 HDAC8 Inhibition – Non‑competitive Model

10% engineered supernatant shows strong non‑competitive inhibition.
Inhibition constant K_i = 3.30% (high potency).
Average inhibition = 55.68%.
Slc26a3, ZO‑1, Occludin up‑regulated ≥2‑fold.

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4. Pediatric AAD Mouse Model (In Vivo)

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Diarrhea Index

Decreased by 40–60% (meets design goal).

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Tight Junction Proteins

ZO‑1, Occludin ↑ ≥2‑fold (IF/WB).

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Histopathology

Reduced mucosal damage, restored colon architecture.

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Model Summary

The models form a complete chain from strain characterization → sporulation → fermentation → in vitro mechanism → in vivo efficacy, strongly supporting the engineering design and validation of CradleShield.