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arxiv: 2507.05960 · v2 · submitted 2025-07-08 · 🧬 q-bio.QM · q-bio.CB· q-bio.PE

Mono- and Polyauxic Growth Kinetics: A Semi-Mechanistic Framework for Complex Biological Dynamics

Gustavo Mockaitis This is my paper

Pith reviewed 2026-05-19 06:10 UTC · model grok-4.3

classification 🧬 q-bio.QM q-bio.CBq-bio.PE
keywords microbial growth kineticspolyauxic growthsemi-mechanistic modelinganaerobic digestionBoltzmann equationGompertz equationkinetic modelingparameter estimation
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The pith

A semi-mechanistic framework reformulates Boltzmann and Gompertz equations to model mono- and polyauxic microbial growth with explicit biological parameters.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a unified approach to kinetic modeling of microbial growth that handles both single-phase and complex multi-phase patterns common in mixed-culture systems. It modifies the Boltzmann and Gompertz equations so the maximum specific reaction rate and lag phase duration become direct, interpretable values rather than hidden constants. Polyauxic growth appears as a sum of sigmoidal phases kept consistent by rules that enforce time order, biological sense, and the ability to estimate parameters uniquely from data. The work supplies a full fitting procedure that combines global search, local refinement, robust loss handling for outliers, and information criteria to pick the right number of phases without excess complexity. Evaluation on anaerobic digestion experiments shows that single-phase models can hide metabolic transitions that matter for process behavior.

Core claim

The paper establishes that microbial growth kinetics, including polyauxic patterns, can be described by reformulating the Boltzmann and Gompertz models into semi-mechanistic versions that directly specify the maximum specific reaction rate and lag phase duration, with polyauxic growth expressed as a weighted sum of these sigmoidal functions under constraints that guarantee parameter identifiability, temporal consistency, and biological plausibility, supported by a two-stage optimization and model selection procedure.

What carries the argument

The weighted sum of constrained sigmoidal phases derived from reformulated Boltzmann and Gompertz equations, which enforces identifiability and plausibility while representing multi-phase growth.

Load-bearing premise

The constraints placed on the sum of sigmoidal phases will keep the resulting parameters both identifiable from data and consistent with real biological behavior in complex co-digestion systems.

What would settle it

Fitting the model to experimental anaerobic digestion data where the best-fit parameters show large uncertainty intervals or the fitted phases violate required time ordering would show that the framework fails to deliver reliable descriptions.

read the original abstract

Kinetic modeling of microbial growth is essential for the design, optimization, and scale-up of industrial bioprocesses. Classical empirical models often lack biologically interpretable parameters or fail to capture complex multiphasic (polyauxic) behaviors, while fully mechanistic models are impractical for systems involving complex substrates and mixed cultures. This study proposes a unified mathematical framework that reformulates the canonical Boltzmann and Gompertz equations into semi-mechanistic forms, explicitly defining the maximum specific reaction rate and lag phase duration. Polyauxic growth is represented as a weighted sum of sigmoidal phases, subject to stringent constraints that ensure parameter identifiability, temporal consistency, and biological plausibility. The methodology integrates a workflow to address nonlinear regression in high-dimensional parameter spaces. A two-stage optimization strategy using Differential Evolution for global search followed by L-BFGS-B for local refinement avoid bias and heuristic parameter initialization. A Charbonnier loss function and the Robust Regression and Outlier Removal procedure are employed to identify and mitigate experimental outliers. Model parsimony is enforced using Akaike (AIC, AICc) and Bayesian (BIC) information criteria to select the optimal number of growth phases and avoid overparameterization. The framework was evaluated using experimental anaerobic digestion datasets, demonstrating that conventional single-phase models can obscure relevant metabolic transitions in co-digestion systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a unified semi-mechanistic framework that reformulates the Boltzmann and Gompertz equations to explicitly define maximum specific reaction rate and lag phase duration. Polyauxic growth is represented as a weighted sum of these sigmoidal phases subject to constraints intended to guarantee identifiability, temporal consistency, and biological plausibility. The workflow uses a two-stage optimizer (Differential Evolution followed by L-BFGS-B), Charbonnier loss with robust outlier removal, and AIC/AICc/BIC selection to determine the number of phases. The approach is evaluated on anaerobic digestion datasets to show that single-phase models can miss relevant metabolic transitions in co-digestion systems.

Significance. If the constraints prove sufficient to yield unique, biologically meaningful parameters on noisy experimental data, the framework would supply a useful middle ground between purely empirical sigmoidal fits and fully mechanistic models that are often intractable for mixed-culture, complex-substrate systems. Explicit per-phase rate and lag parameters could improve interpretability in bioprocess design and scale-up.

major comments (2)
  1. The central claim that the 'stringent constraints' on the weighted sum of sigmoidal phases ensure parameter identifiability, temporal consistency, and biological plausibility is load-bearing, yet the manuscript provides neither the explicit mathematical statements of these constraints (e.g., ordering of lag durations, bounds on phase weights, or non-overlap conditions) nor a demonstration that they remain active and non-degenerate under the Charbonnier loss and the two-stage optimizer when applied to real co-digestion time series.
  2. Evaluation section: the assertion that conventional single-phase models obscure metabolic transitions is not accompanied by quantitative fit statistics (R², RMSE, or residual distributions) or direct comparison of AIC/BIC values between the selected multi-phase model and the single-phase baseline on the same datasets; without these numbers it is impossible to judge whether the additional phases recover distinct metabolic events or merely improve empirical flexibility.
minor comments (1)
  1. The abstract refers to a 'Robust Regression and Outlier Removal procedure' without naming the specific algorithm or outlier threshold; this detail should be added for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed review. The comments identify key areas where additional clarity and quantitative support will strengthen the manuscript. We will revise accordingly to make the constraint formulations explicit and to supply direct model comparison metrics. Point-by-point responses follow.

read point-by-point responses

  1. Referee: The central claim that the 'stringent constraints' on the weighted sum of sigmoidal phases ensure parameter identifiability, temporal consistency, and biological plausibility is load-bearing, yet the manuscript provides neither the explicit mathematical statements of these constraints (e.g., ordering of lag durations, bounds on phase weights, or non-overlap conditions) nor a demonstration that they remain active and non-degenerate under the Charbonnier loss and the two-stage optimizer when applied to real co-digestion time series.

    Authors: We acknowledge that the explicit mathematical statements of the constraints and a direct demonstration of their action were insufficiently detailed. In the revised manuscript we will insert a new subsection in Methods that states the constraints formally: strict lag ordering (λ_i < λ_{i+1}), weight summation (∑ w_i = 1 with w_i > 0), and non-overlap conditions that enforce temporal separation of phases. We will also add a supplementary analysis that applies the two-stage optimizer with and without these constraints to the anaerobic digestion time series, reporting condition numbers of the Hessian and recovered parameter uniqueness to show that the constraints remain active and non-degenerate under the Charbonnier loss. revision: yes

  2. Referee: Evaluation section: the assertion that conventional single-phase models obscure metabolic transitions is not accompanied by quantitative fit statistics (R², RMSE, or residual distributions) or direct comparison of AIC/BIC values between the selected multi-phase model and the single-phase baseline on the same datasets; without these numbers it is impossible to judge whether the additional phases recover distinct metabolic events or merely improve empirical flexibility.

    Authors: We agree that quantitative metrics are required for a rigorous assessment. The revised Evaluation section will contain a table reporting R², RMSE, and summary residual statistics for both the single-phase baseline and the selected multi-phase model on each co-digestion dataset. We will also tabulate AIC, AICc, and BIC values for the single-phase model versus the parsimonious multi-phase model chosen by the information criteria, together with residual-distribution plots that illustrate the reduction in systematic structure once the additional phases are included. revision: yes

Circularity Check

0 steps flagged

No significant circularity; framework is a reparametrization plus standard constrained fitting.

full rationale

The paper reformulates Boltzmann/Gompertz equations with explicit max-rate and lag parameters, then represents polyauxic growth as a constrained weighted sum of sigmoids. These are fitted to experimental data via DE + L-BFGS-B with Charbonnier loss, AIC/BIC selection, and outlier removal. No step claims a first-principles prediction or derivation that reduces to its own fitted inputs by construction; the semi-mechanistic forms are explicit reparametrizations whose parameters are estimated from the target observations in the usual manner for kinetic models. The constraints address identifiability during fitting rather than creating a self-referential loop. The derivation chain is self-contained against external benchmarks and does not rely on load-bearing self-citations or ansatzes smuggled from prior work.

Axiom & Free-Parameter Ledger

4 free parameters · 2 axioms · 0 invented entities

The framework rests on several fitted parameters for rates, lags, and phase weights plus domain assumptions about sigmoidal shapes; no new entities are postulated.

free parameters (4)
  • number of growth phases
    Chosen via AIC/AICc/BIC to enforce parsimony
  • phase weights
    Fitted coefficients in the weighted sum for polyauixic representation
  • maximum specific reaction rate
    Explicitly defined parameter but estimated from data
  • lag phase duration
    Explicitly defined parameter but estimated from data
axioms (2)
  • domain assumption Boltzmann and Gompertz sigmoidal forms can be rewritten to expose biologically interpretable parameters
    Invoked when reformulating the canonical equations into semi-mechanistic versions
  • ad hoc to paper Stringent constraints on the weighted sum guarantee identifiability, temporal consistency, and biological plausibility
    Stated as a requirement for the polyauixic representation

pith-pipeline@v0.9.0 · 5780 in / 1674 out tokens · 67510 ms · 2026-05-19T06:10:28.437297+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

  1. [1]

    Jason, A deterministic model for monophasic growth of batch cultures of bacteria, Antonie Van Leeuwenhoek 49 (1983) 513–536

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    work page doi:10.1007/bf00399845 1983

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    Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing , volume =

    Y. Benjamini, Y. Hochberg, Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing, J R Stat Soc Series B Stat Methodol 57 (1995) 289–300. https://doi.org/10.1111/j.2517-6161.1995.tb02031.x

    work page doi:10.1111/j.2517-6161.1995.tb02031.x 1995