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arxiv: 2604.19852 · v1 · submitted 2026-04-21 · 🧬 q-bio.CB

Multi-stage volume exclusion models for cell proliferation

John Carlo Dimaculangan , Cameron A. Smith , Christian A. Yates This is my paper

Pith reviewed 2026-05-10 01:04 UTC · model grok-4.3

classification 🧬 q-bio.CB
keywords cell proliferationvolume exclusionagent-based modelmulti-stage cell cyclemean-field approximationpartial differential equationcell invasiontravelling waves
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The pith

Multi-stage cell cycle models with myopic proliferation and volume exclusion can be approximated by mean-field PDEs for cell growth and invasion.

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

The paper develops stochastic on-lattice agent-based models incorporating volume exclusion, random movement, multi-stage cell cycle representations for realistic division time distributions, and a new myopic proliferation rule where cells sense nearby space before dividing. Corresponding continuum PDEs are derived under the mean-field approximation for each model variant. Simulations explore how these proliferation rules affect population dynamics in growth-to-confluence assays with uniform initial conditions and in travelling wave invasion scenarios. The work compares PDE solutions to averages over many agent-based model realizations to assess the approximation quality. Sympathetic readers care because this links detailed stochastic cell behaviors to simpler continuum descriptions useful for predicting tissue growth in biology.

Core claim

We develop stochastic, on-lattice agent-based models which incorporate volume exclusion, random movement, and multi-stage representations of the cell cycle. The multi-stage framework enables a more realistic representation of true cell cycle time distributions. We also introduce a novel form of myopic behaviour, in which cells sense their local environment when attempting to proliferate. For each ABM, we derive a corresponding continuum partial differential equation description under the mean-field approximation. Using numerical simulations, we investigate how different proliferation mechanisms influence population-level dynamics in both the discrete and continuum models in growth-to-conflue

What carries the argument

The multi-stage cell cycle framework in volume-excluding on-lattice agent-based models with myopic proliferation sensing.

Load-bearing premise

The mean-field approximation holds so that correlations between nearby cells do not significantly affect the average population behavior.

What would settle it

A large mismatch between the solutions of the derived PDEs and the averaged outcomes from repeated simulations of the corresponding agent-based models in either confluence growth or invasion wave metrics.

Figures

Figures reproduced from arXiv: 2604.19852 by Cameron A. Smith, Christian A. Yates, John Carlo Dimaculangan.

Figure 1
Figure 1. Figure 1: FIG. 1: Schematic of how a cell chooses a neighbouring site during proliferation and [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Single realisations of various ABMs simulated over [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: (a) Density evolutions over [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Example showing how a myopic cell chooses a neighbouring site during a [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Single realisations of various ABMs including myopic behaviour simulated over [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Average density evolutions over [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Comparing the numerical solutions of the PDEs ( [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Plotting the PCFs [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Example occupancy matrices, [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Comparing the solutions to PDE ( [PITH_FULL_IMAGE:figures/full_fig_p030_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: As in Figure [PITH_FULL_IMAGE:figures/full_fig_p031_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: We simulate the Myopic Remain ABM with [PITH_FULL_IMAGE:figures/full_fig_p032_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: ABM simulations of travelling wavefronts on a [PITH_FULL_IMAGE:figures/full_fig_p037_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Wave-speeds for the ABM simulations in Figure [PITH_FULL_IMAGE:figures/full_fig_p038_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: Comparing the wave-fronts at time [PITH_FULL_IMAGE:figures/full_fig_p039_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: Comparing ABM simulations to PDE solutions over [PITH_FULL_IMAGE:figures/full_fig_p040_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: PDE solutions at [PITH_FULL_IMAGE:figures/full_fig_p041_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: Comparing ABM and PDE wave-speeds. We calculated the PDE wave [PITH_FULL_IMAGE:figures/full_fig_p042_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19: Plotting the PCFs [PITH_FULL_IMAGE:figures/full_fig_p050_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20: Plotting average PCFs for Myopic models, as in Figure [PITH_FULL_IMAGE:figures/full_fig_p051_20.png] view at source ↗
read the original abstract

Cell proliferation and cell movement are fundamentally stochastic processes which lead to variability in the growth and spatial structure of cell populations in many biological settings, such as cell invasion, wound healing, and tumour growth. We develop stochastic, on-lattice agent-based models (ABMs) which incorporate volume exclusion, random movement, and multi-stage representations of the cell cycle. The multi-stage framework enables a more realistic representation of true cell cycle time distributions. We also introduce a novel form of myopic behaviour, in which cells sense their local environment when attempting to proliferate. For each ABM, we derive a corresponding continuum partial differential equation (PDE) description under the mean-field approximation. Using numerical simulations, we investigate how different proliferation mechanisms influence population-level dynamics in both the discrete and continuum models. In particular, we consider biologically relevant contexts of growth-to-confluence assays (using uniform initial conditions) and travelling wave behaviour associated with cell invasion. We examine how the PDE solutions compare with the behaviour of the corresponding ABMs averaged over many realisations.

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 / 2 minor

Summary. The manuscript develops stochastic on-lattice agent-based models (ABMs) for cell proliferation incorporating volume exclusion, random movement, multi-stage cell-cycle representations (to produce non-exponential division-time distributions), and a novel myopic proliferation rule in which cells sense local occupancy before attempting division. For each ABM variant the authors derive a corresponding continuum PDE under the standard mean-field closure. Numerical simulations then compare ensemble-averaged ABM trajectories against PDE solutions for two biologically relevant settings: growth-to-confluence under uniform initial conditions and travelling-wave invasion.

Significance. If the mean-field PDEs remain quantitatively faithful, the work supplies a practical route to embed realistic cell-cycle timing and local sensing into continuum models of invasion and wound healing, while retaining the computational advantages of PDEs. The explicit ABM-to-PDE derivation and side-by-side numerical tests constitute a clear strength; the multi-stage formulation directly addresses a known limitation of single-stage exponential waiting times.

major comments (2)
  1. [PDE derivation for myopic proliferation] The mean-field closure used to obtain the PDE proliferation term (standard independence assumption P(occupied_i and occupied_j) = P_i P_j) is applied to the myopic rule, which by construction correlates a cell's division decision with the occupancy of its sensing neighbourhood. Volume exclusion further enforces hard-core correlations. The manuscript does not report any diagnostic (e.g., measured pair-correlation functions from the ABM, comparison of effective proliferation rates, or higher-moment closures) to confirm that the closure remains accurate near fronts or at moderate-to-high densities where these correlations are strongest.
  2. [Numerical results (growth-to-confluence and travelling waves)] In the numerical comparison sections, ABM results are presented as averages over realisations but without error bars, standard deviations, or convergence tests with respect to lattice size, number of realisations, or sensing radius. Consequently it is impossible to judge whether apparent agreement with the PDE is statistically robust or merely qualitative, especially for the travelling-wave speed and the approach to confluence.
minor comments (2)
  1. The abstract and introduction would benefit from a concise statement of the precise sensing radius and number of cell-cycle stages used in the main figures, rather than deferring all parameter values to the methods.
  2. Figure legends should explicitly state the initial density, lattice size, and number of ABM realisations for each panel to allow direct replication of the reported comparisons.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful comments on our manuscript. We address each of the major comments below and outline the revisions we plan to make to strengthen the paper.

read point-by-point responses

  1. Referee: [PDE derivation for myopic proliferation] The mean-field closure used to obtain the PDE proliferation term (standard independence assumption P(occupied_i and occupied_j) = P_i P_j) is applied to the myopic rule, which by construction correlates a cell's division decision with the occupancy of its sensing neighbourhood. Volume exclusion further enforces hard-core correlations. The manuscript does not report any diagnostic (e.g., measured pair-correlation functions from the ABM, comparison of effective proliferation rates, or higher-moment closures) to confirm that the closure remains accurate near fronts or at moderate-to-high densities where these correlations are strongest.

    Authors: We agree that the mean-field approximation may be affected by correlations induced by the myopic proliferation rule and volume exclusion. The manuscript presents the derivation under the standard mean-field closure and shows that the resulting PDEs provide a good approximation to the averaged ABM dynamics in the regimes considered. To directly address this point, we will add diagnostics in the revised manuscript, including computed pair-correlation functions from ABM simulations at various densities and near fronts, to evaluate the validity of the independence assumption. If significant deviations are found, we will discuss their implications for the approximation. revision: yes

  2. Referee: [Numerical results (growth-to-confluence and travelling waves)] In the numerical comparison sections, ABM results are presented as averages over realisations but without error bars, standard deviations, or convergence tests with respect to lattice size, number of realisations, or sensing radius. Consequently it is impossible to judge whether apparent agreement with the PDE is statistically robust or merely qualitative, especially for the travelling-wave speed and the approach to confluence.

    Authors: We acknowledge that the absence of error bars and convergence tests limits the assessment of the numerical agreement. In the revised version, we will include error bars or standard deviations on the ensemble-averaged ABM results. Additionally, we will perform and report convergence tests with respect to the number of realizations, lattice size, and sensing radius to demonstrate the statistical robustness of the comparisons, particularly for travelling wave speeds and confluence dynamics. revision: yes

Circularity Check

0 steps flagged

No circularity: standard mean-field derivation from ABMs with independent numerical validation

full rationale

The paper constructs on-lattice ABMs incorporating volume exclusion, random movement, multi-stage cell cycles, and a novel myopic proliferation rule, then applies the standard mean-field closure to obtain continuum PDEs. These PDEs are not obtained by fitting parameters to the same data being predicted, nor by self-definition, renaming, or load-bearing self-citation chains. The subsequent numerical comparisons between averaged ABM realizations and PDE solutions constitute an external check rather than a tautology. No quoted step reduces the claimed result to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review provides insufficient detail to enumerate specific fitted values or invented entities; the central modelling step relies on the mean-field closure and the assumption that multi-stage timing better matches biological distributions.

axioms (1)
  • domain assumption Mean-field approximation accurately captures the average behaviour of the stochastic ABM
    Invoked to obtain the continuum PDE description from the discrete agent rules

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Reference graph

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    In Figure 13, we can see that the wave-fronts which form in each ABM have similar shapes and are very steep

    Low motility We begin by considering a regime in which the motility rate,rm, is of the same order as the proliferation rate,rp (previously the low motility regime). In Figure 13, we can see that the wave-fronts which form in each ABM have similar shapes and are very steep. This is because cell proliferation is the major factor driving the travelling wave ...

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    High motility We now consider a parameter regime in which the motility rate is high relative to the proliferation rate. In Figure 16, we see that even after300time units, the Exponential and Reset models have not yet reached a steady state wavefront. We also note that the wavefronts across all models under this high motility scheme are shallower than in t...

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