Free OD vs Time Kinetic Parameter Fitting Tool | Monod & Haldane

MODEL FORMULATION & TIME SERIES 4 pts

Target Kinetic Model

Industrial Preset Callouts

Time Unit

t (min)	OD₆₀₀	[S] g/L	[P] g/L

REGRESSION TRAJECTORY

FERMENTATION HANDOFF ➔ STEP 4
Fitted kinetic constants μ_max = — and half-saturation K_s = 0.50 g/L ready to feed into Monod Bioreactor Yield & Maintenance (Step 4).

FITTED CONSTANTS & SIMULATION SOLVER

Fitted Kinetic Parameters

Dynamic X–S–P Simulation Settings

X₀ (g/L)

S₀ (g/L)

Y_x/s (g/g)

t_end (min)

Solver Details: Uses Levenberg-Marquardt non-linear least squares with numerical finite-difference Jacobian approximation and adaptive damping factor λ.

📚 Fermentation Time-Course Fitting & Sigmoidal Growth Models Guide Fermentation Kinetics • Step 2 of 5

Theoretical Principles & Engineering Fundamentals

Extracting robust kinetic constants from experimental fermentation data requires fitting sigmoidal mathematical growth models across the entire time-course. This tool performs non-linear least squares regression using Logistic, Gompertz, and Richards empirical models to identify maximum carrying capacity ($X_{\max}$), maximum specific growth rate ($\mu_{\max}$), and lag duration ($\lambda$).

Governing Equations & Mathematical Formulations

Logistic Growth Model X(t) = \frac{X_{\max}}{1 + \left(\frac{X_{\max} - X_0}{X_0}\right) e^{-\mu_{\max} t}}

Classic symmetric sigmoidal growth model with inflection at $X_{\max}/2$.

Modified Gompertz Model X(t) = X_0 + A \exp\left(-\exp\left(\frac{\mu_{\max} e}{A} (\lambda - t) + 1\right)\right)

Asymmetric sigmoidal curve capturing lag duration $\lambda$ and carrying capacity.

Root Mean Square Error (RMSE) \text{RMSE} = \sqrt{\frac{1}{n} \sum_{i=1}^n (X_{\text{meas},i} - X_{\text{model},i})^2}

Evaluates statistical goodness of fit across experimental time points.

Industrial Benchmark Data & Parameter Reference

Model	Symmetry	Best Suited For	Parameters
Logistic	Symmetric	Standard batch fermentation without pronounced lag	X_0, X_max, μ_max
Gompertz	Asymmetric	Cultures with extended lag and decelerating plateau	X_0, A, μ_max, λ
Richards	Flexible Inflection	Complex multi-substrate or diauxic growth curves	X_0, A, μ_max, λ, ν

Frequently Asked Questions (Bioprocess Engineering FAQ)

Which growth model should I use for E. coli batch culture?

The Modified Gompertz model is universally recognized as the most accurate for microbial fermentation because it independently parameterizes lag duration ($\lambda$) and asymmetric deceleration.

What does the inflection point represent?

The inflection point is the exact timestamp where the culture reaches its absolute maximum volumetric growth rate ($dX/dt$), after which nutrient depletion begins to decelerate division.

Can this tool fit multi-phase or diauxic growth curves?

Yes, by segmenting time ranges, you can analyze secondary substrate consumption curves (e.g. glucose exhaustion followed by lactose induction).

Free OD vs Time Kinetic Parameter Fitting Tool (Monod & Haldane)

📚 Fermentation Time-Course Fitting & Sigmoidal Growth Models Guide Fermentation Kinetics • Step 2 of 5

Theoretical Principles & Engineering Fundamentals

Governing Equations & Mathematical Formulations

Industrial Benchmark Data & Parameter Reference

Frequently Asked Questions (Bioprocess Engineering FAQ)

BioFlo Bioprocess Engineering Network