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Systems BiologyApplied and Environmental Microbiology
Peer-reviewed journal article

Monod model is insufficient to explain biomass growth in nitrogen-limited yeast fermentation

David HenriquesiD, Eva Balsa‐CantoiD

Abstract

The yeast Saccharomyces cerevisiae is an essential microorganism in food biotechnology; particularly, in wine and beer making. During wine fermentation, yeasts transform sugars present in the grape juice into ethanol and carbon dioxide. The process occurs in batch conditions and is, for the most part, an anaerobic process. Previous studies linked limited-nitrogen conditions with problematic fermentations, with negative consequences for the performance of the process and the quality of the final product. It is, therefore, of the highest interest to anticipate such problems through mathematical models. Here we propose a model to explain fermentations under nitrogen-limited anaerobic conditions. We separated the biomass formation into two phases: growth and carbohydrate accumulation. Growth was modelled using the well-known Monod equation while carbohydrate accumulation was modelled by an empirical function, analogous to a proportional controller activated by the limitation of available nitrogen. We also proposed to formulate the fermentation rate as a function of the total protein content when relevant data are available. The final model was used to successfully explain experiments taken from the literature, performed under normal and nitrogen-limited conditions. Our results revealed that Monod model is insufficient to explain biomass formation kinetics in nitrogen-limited fermentations of S. cerevisiae. The goodness-of-fit of the herewith proposed model is superior to that of previously published models, offering the means to predict, and thus control fermentations. ImportanceProblematic fermentations still occur in the winemaking industrial practise. Problems include sluggish rates of fermentation, which have been linked to insufficient levels of assimilable nitrogen. Data and relevant models can help anticipate poor fermentation performance. In this work, we proposed a model to predict biomass growth and fermentation rate under nitrogen-limited conditions and tested its performance with previously published experimental data. Our results show that the well-known Monod equation does not suffice to explain biomass formation.

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Claude Opus 5.5 · xhigh effort · 8 Nov 2026 · from the full text of version 1 (10 Jun 2021) · CC BY 4.0

Adding carbohydrate accumulation to Monod growth and tying fermentation rate to protein lets a yeast kinetic model capture post-nitrogen-depletion biomass gains that Monod alone misses.

Stuck and sluggish wine fermentations are tied to insufficient assimilable nitrogen, yet Monod and logistic growth laws impose a biomass ceiling and miss the biomass rise after ammonium depletion, while transporter-based models need many fitted parameters. This work supplies a compact kinetic model separating protein and carbohydrate biomass that remains identifiable from routine fermentation measurements.

Key results

  1. ShownA two-term growth law (Monod growth on nitrogen plus carbohydrate accumulation after depletion) reproduces the secondary biomass rise after nitrogen exhaustion. Pure Monod (MM) and the earlier active-biomass model (MC) cannot reproduce it.

    ExpS biomass normalised RMSE (NRMSE): MP 1.91% vs MM 18.60% and MC 18.56% (approximately 9.7 times lower); MP biomass R^2 over 0.97 in all cases vs around 0.57 for MM and MC on ExpS; MM and MC converge to a constant biomass value. · 2.4 Biomass formation; Figure legends

  2. ShownAcross both calibration data sets, the protein-linked model MP gives the lowest overall error and information criterion, and its extra biomass-composition parameters do not appear to cause overfitting.

    All models fit within 9% NRMSE, but MP NRMSE was 3.9 and 1.8 times lower than MM and MC; corrected Akaike (AICc) differences vs MM and MC were < −100 (computed on shared variables only), beyond the Δ > 10 no-support threshold; MX ranked next closest to MP. · 2.3 Calibration of candidate models; 4.5 Model selection; Tables

  3. ShownScaling hexose uptake with protein content rather than total biomass improves fits to sugar consumption and ethanol production, especially when initial nitrogen is low.

    MP outperformed MX (same parameter count) on all measured extracellular variables. All models reached R^2 > 0.8 for glucose, ethanol and glycerol, but in low-nitrogen ExpV, MX, MC and MM predicted faster glucose uptake and ethanol output than observed. Only MP exceeded R^2 0.9 for every state and condition. · 2.5 Fermentation rate; Figure legends

  4. ShownMP is structurally identifiable without any protein measurements, provided biomass, sugar, nitrogen and ethanol are tracked over time.

    GenSSI2 generating-series analysis of the fourteen-parameter model showed unique parameter estimation is possible from time courses of biomass, glucose uptake, nitrogen uptake and ethanol; protein data are not required to identify MP. · 2.3 Calibration of candidate models; 4.3 Structural identifiability analysis

  5. SuggestedIn MP, which by design attributes biomass gained after nitrogen depletion to carbohydrate and ties protein and mRNA to nitrogen-driven growth, the fits agree with the idea that secondary growth is mostly carbohydrate accumulation.

    MP trajectories show carbohydrates accumulating after YAN depletion at around 20 hours; for ExpS and high-nitrogen ExpV, fits to YAN, dry-weight biomass, protein and mRNA are consistent with carbohydrate-driven secondary growth. · 2.4 Biomass formation; 3 Discussion; Figure legends

Methods, data and reproducibility

Six-ODE kinetic model of S. cerevisiae batch fermentation: growth sums a Monod term on assimilable nitrogen (YAN) and a proportional-controller-like carbohydrate accumulation term; Michaelis-Menten hexose uptake with ethanol inhibition scales with protein. Fit by weighted least squares and compared with Monod-type models by NRMSE and AICc.

  • ExpS: published batch fermentation time course (biomass, YAN, hexoses, ethanol, glycerol), described as nitrogen sufficient and fitted on its own · Saccharomyces cerevisiae · n = one experiment
  • ExpV: published fermentations at high and low nitrogen (50 and 300mg YAN), fitted jointly with shared parameters · Saccharomyces cerevisiae · n = two experiments
  • Exp1 (per Methods): published nitrogen-limited batch fermentation (165 mg NH3, 50g glucose) with biomass, biomass protein, glycerol, ethanol and ammonia · Saccharomyces cerevisiae
  • Exp2 to Exp5 (per Methods): published fermentations at 11°C, 15°C, 30°C and 35°C, combining high or normal sugar with low or normal nitrogen · Saccharomyces cerevisiae · n = four data-sets
Methods & tools
AMIGO2 toolbox (parameter estimation and model selection), GenSSI2 toolbox (structural identifiability analysis), CVODES ODE solver, Enhanced scatter search (eSS) hybrid metaheuristic, Corrected Akaike Information Criterion (AICc), Normalised root mean square error (NRMSE), Monod growth kinetics, Michaelis-Menten hexose transport with non-competitive ethanol inhibition
Code
https://sites.google.com/site/amigo2toolbox/examples
Data & accessions
https://sites.google.com/site/amigo2toolbox/examples
Software & environment
AMIGO2 toolbox with a compiled ODE system solved by CVODES; eSS metaheuristic for estimation; GenSSI2 for identifiability. No software versions or computing platform reported.
Reproducibility gaps
Full equations, parameter bounds and fitted values are in Supplementary Text S1, which was not part of the provided text. Fitted data are reused from earlier publications without accession IDs. No tool versions are given.
Limitations (5)
  • The authors limit applicability to controlled fermentations with a single starter and a well-characterized must: YAN and carbon sources must be measured at the start and tracked, with biomass and target products monitored over time. · stated by the authors
  • The authors state that unique parameter estimation requires biomass, glucose, nitrogen and ethanol to be measured throughout fermentation, and note that Monod parameters fitted without nitrogen and sugar uptake data lose identifiability and predictive power. · stated by the authors
  • Because no experimental error estimates were available, least-squares residuals were normalized by each state's maximum measured value instead of by measurement variance. · stated by the authors
  • The data descriptions conflict. Results use ExpS (one experiment, called nitrogen sufficient) and ExpV (two experiments), while Methods describe five experiments (Exp1 to Exp5) spanning 11 to 35°C. It is unclear which data underlie each fit. · noted in this summary
  • All models were judged only on goodness of fit to the data used to calibrate them. No held-out experiment or cross-condition forecast is reported, so the claimed value for predicting and controlling winemaking fermentations is untested. · noted in this summary

Who should read:Read this if you model wine or other batch yeast fermentations and need a low-parameter growth law that captures biomass gains after nitrogen depletion.

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