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arxiv: 1907.11091 · v1 · pith:B7ZCULGOnew · submitted 2019-07-25 · 🧮 math.AP

A cell-cell repulsion model on a hyperbolic Keller-Segel equation

Xiaoming Fu , Quentin Griette , Pierre Magal This is my paper

Pith reviewed 2026-05-24 15:57 UTC · model grok-4.3

classification 🧮 math.AP
keywords hyperbolic Keller-Segelcell-cell repulsioncompetitive exclusioncell co-culturesegregation propertypopulation dynamicsnumerical simulation
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The pith

Hyperbolic Keller-Segel model with repulsion produces competitive exclusion tied to initial cell totals

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

The paper builds a two-population model of cell growth and dispersion based on a hyperbolic Keller-Segel equation that includes cell-cell repulsion. It proves existence and uniqueness of solutions by integrating along characteristics and establishes that the populations segregate. Numerical experiments show that the model yields competitive exclusion outcomes unlike the corresponding ODE system, with final population ratios depending on the starting total number of cells but independent of the precise initial spatial distribution. Fast dispersion confers short-term advantage while parameters governing vital dynamics confer long-term advantage.

Core claim

The model admits a competitive exclusion phenomenon distinct from the corresponding ODE model, where the population ratio is affected by the initial total cell number but independent of the law of initial distribution; fast dispersion provides short-term advantage and vital dynamics long-term advantage. Existence and uniqueness of solutions integrated along characteristics is established along with the segregation property of the two species.

What carries the argument

Solutions integrated along the characteristics of the hyperbolic Keller-Segel system, which enforce the segregation property between the two populations.

If this is right

  • The two cell populations segregate spatially under the repulsion dynamics.
  • Competitive exclusion emerges with ratios determined by initial totals.
  • Higher dispersion rates confer short-term population advantage.
  • Vital dynamics parameters determine long-term population advantage.
  • Outcome is independent of the specific form of the initial spatial distribution.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The spatial structure captured by the PDE produces predictions that cannot be recovered from the non-spatial ODE limit.
  • Experiments that systematically vary initial cell density while holding distribution shape fixed would directly test the model's key numerical finding.
  • The repulsion mechanism may be adapted to other systems where cells compete for space during growth.

Load-bearing premise

The repulsion term in the hyperbolic Keller-Segel equation accurately captures the cell growth and dispersion observed in the referenced co-culture experiment.

What would settle it

Measure final population ratios in co-culture experiments started with different initial total cell numbers but the same distribution law and check whether the ratios vary with total number as the simulations predict.

Figures

Figures reproduced from arXiv: 1907.11091 by Pierre Magal, Quentin Griette, Xiaoming Fu.

Figure 1
Figure 1. Figure 1: Direct immunodetection of P-gp transfers in co-cultures of sensitive (MCF-7) and resis￾tant (MCF-7/Doxo) variants of the human breast cancer cell line. The early attempts to explain the segregation property by continuum equations date back to 1970s. Shigesada, Kawasaki and Teramoto [34] studied segregation with a nonlinear diffusion model and they found the spatial segregation acts to stabilize the coexist… view at source ↗
Figure 2
Figure 2. Figure 2: An illustration for the invariance of domain Ω. The green curve represents a trajectory of the characteristics. Assumption 2.3. Assume the vector field (t, x) 7→ ∇P(t, x) is continuous in [0, T] × Ω and lipschitzian with respect to x in [0, T] × Ω. Remark 2.4. Assumption 2.3 is a sufficient condition for the existence and uniqueness of the characteristic flow {Π(t, s; ·)}t,s∈[0,T] in (2.4). Definition 2.5.… view at source ↗
Figure 3
Figure 3. Figure 3: A scheme of the qualitative behavior of the phase trajectory for various cases. (a) a12/a11 < P1/P2, a21/a22 < P2/P1. Only the positive steady state E∗ is stable and all trajectories tend to it. (b) a12/a11 > P1/P2, a21/a22 < P2/P1. Only one stable steady state E2 exists with the whole positive quadrant its domain of attraction. (c) a12/a11 < P1/P2, a21/a22 > P2/P1. Only one stable steady state E1 exists w… view at source ↗
Figure 4
Figure 4. Figure 4: In this figure we illustrate the notion of segregation with a one dimensional bounded domain. Figure (a) shows the characteristic t 7→ Π(t, 0; x0) forms a segregation “wall”. Figure (b) shows the temporal-spatial evolution of the two species. 2.2.4 Conservation law on a volume If we assume that d1 = d2 = d in system (2.11), we have the following similar conservation law for two species case. Suppose volume… view at source ↗
Figure 5
Figure 5. Figure 5: Spatial-temporal evolution of the two species u1 and u2 and its relative proportion. Figures (a)-(e) correspond to the evolution of cell growth form day 0 to day 6 and Figure (f ) is the relative proportion plot from day 0 to day 6. We fix the parameters δ1 = 0.4, δ2 = 0, a12 = 0.2, a21 = 1 in (3.3). The initial distribution follows the uniform distribution on a disk with 20 initial cell clusters. The init… view at source ↗
Figure 6
Figure 6. Figure 6: Evolution of the relative cell numbers for two species u1 and u2. Figure (a)-(c) corre￾spond to the parameter values chosen as in [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Cell co-culture for species u1 and u2 over 6 days. We plot the case where cells are sparsely seeded, i.e., U1 = U2 = 0.005, Nu1 = Nu2 = 10 for day 0, 2 and day 6. We set parameters as δ1 = 0.15, δ2 = 0, a12 = 0, a21 = 0 in (3.4). Other parameters are given in [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Cell co-culture for species u1 and u2 over 6 days. We plot the case where where cells are densely seeded, i.e., U1 = U2 = 0.1, Nu1 = Nu2 = 200 for day 0, 2 and day 6. We set parameters as δ1 = 0.15, δ2 = 0, a12 = 0, a21 = 0 in (3.4). Other parameters are given in [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Evolution of the total number (in log scale) and its proportion for species u1 and u2 over 6 days. Figure (a) is the total number plot corresponding to the simulations in [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Density function of the initial distribution fα,β(x) = 1/B(α, β) x α−1 (1 − x) β−1 for different α and β, where B(α, β) is a normalization constant to ensure that the total integral is 1. Our simulation will mainly compare the following two cases (α1, β1) = (1, 1), (α2, β2) = (3, 2). We plot the initial distributions of the two different cases in [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Spatial distribution of the initial values when (α, β) = (1, 1) (Figure (a)) and (α, β) = (3, 2) (Figure (b)). Here red dots and green dots in [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Cell co-culture for species u1 and u2 over 6 days. We plot the case where the initial distribution follows beta function with parameters (α, β) = (1, 1), namely the uniform distribution, for day 1, 3 and day 6. Parameters are given in [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Cell co-culture for species u1 and u2 over 6 days. We plot the case where the initial distribution follows beta function with parameters (α, β) = (3, 2), namely a biased distribution, for day 1, 3 and day 6. Parameters are given in [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Evolution of the total number (in log scale) and its proportion for species u1 and u2 over 6 days. Figure (a) is the total number plot corresponding to the simulation with an uniform initial distribution in [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Cell co-culture for species u1 and u2 over 6 days. Figure (a)-(c) corresponds to scenario 1 (i.e. with the parameters d1 = 2, d2 = 2, δ1 = δ2 = 0) while Figure (d)-(f ) corresponds to scenario 2 (i.e. with d1 = 2, d2 = 0.2, δ1 = δ2 = 0). In both scenarios, the number of initial cluster and the cell total number are the same and follow (3.7) and the same uniform distribution. We plot the simulations for da… view at source ↗
Figure 16
Figure 16. Figure 16: Evolution of the total number (in log scale) and its proportion for species u1 and u2 over 6 days. In Figure (a) we plot the total number of cells corresponding to the scenario 1. In Figure (b) we plot the total number of cells corresponding to the scenario 2. In Figure (c) we plot the population ratios and the dashed lines corresponds to scenario 1 while the solid lines corresponds to scenario 2 in 4. Ot… view at source ↗
Figure 17
Figure 17. Figure 17: Cell co-culture for species u1 and u2 over 6 days. Figure (a)-(c) corresponds to the scenario 3 with d1 = 2, d2 = 0.2, δ1 = 0.1, δ2 = 0 in [PITH_FULL_IMAGE:figures/full_fig_p020_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Evolution of the total number (in log scale) and its proportion for species u1 and u2 over 6 days. Figure (a) is the total number plot corresponding to the scenario 3 in [PITH_FULL_IMAGE:figures/full_fig_p020_18.png] view at source ↗
Figure 19
Figure 19. Figure 19 [PITH_FULL_IMAGE:figures/full_fig_p029_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Fitting for the growth curves of MCF-7 (a) and MCF-7/Doxo (b) under different drug concentrations in model (5.25) over 6 days. Cells were grown in the absence or presence of doxorubicine (0.1 to 10 µM, corresponding symbols given in the legend in (b)) and counted every 12 hours in a Malassez chamber. Cell counts are expressed as the logarithm of the cell numbers (ui) divided by the cell number at day 0 (u… view at source ↗
read the original abstract

In this work, we discuss a cell-cell repulsion population dynamic model based on a hyperbolic Keller-Segel equation with two populations. This model can well describe the cell growth and dispersion in the cell co-culture experiment in the work of Pasquier et al. \cite{Pasquier2011}. With the notion of solutions integrated along the characteristics, we prove the existence and uniqueness of the solution and the segregation property of the two species. From a numerical perspective, we can also observe that the model admits a competitive exclusion (the results are different from the corresponding ODE model). More importantly, our model shows the complexity of the short term (6 days) co-cultured cell distribution depending on the initial distribution of each species. Through numerical simulations, the impact of the initial distribution on the population ratio lies in the initial total cell number and our study shows that the population ratio is not impacted by the law of initial distribution. We also find that a fast dispersion rate gives a short-term advantage while the vital dynamic contributes to a long-term population advantage.

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 paper introduces a two-population hyperbolic Keller-Segel model incorporating cell-cell repulsion to describe growth and dispersion in co-culture experiments. It establishes existence and uniqueness of solutions together with a segregation property by integrating along characteristics. Numerical simulations are used to report competitive exclusion (distinct from the corresponding ODE system), a population ratio that depends on the initial total cell number but is insensitive to the law of the initial distribution, and a short-term advantage for fast dispersion versus a long-term advantage for vital dynamics.

Significance. If the numerical observations are robust, the work supplies a spatially structured PDE model that reproduces experimental features (such as competitive exclusion) absent from the ODE reduction, while the characteristic-based proof of segregation supplies a clean mathematical foundation. The explicit separation of short-term dispersion effects from long-term growth effects is a useful modeling distinction.

major comments (2)
  1. [Numerical simulations] Numerical simulations section: the claim that the population ratio depends on initial total cell number but not on the law of the initial distribution rests on unspecified simulation choices (discretization method, mesh size, time-stepping scheme, boundary conditions, and parameter values drawn from Pasquier et al.). Without these details or a convergence study, the reported insensitivity cannot be assessed as a model property rather than a numerical artifact.
  2. [Numerical simulations] Numerical simulations section: the assertion of competitive exclusion different from the ODE model is presented as an observation, yet no quantitative comparison (e.g., final population ratios, time series, or parameter table) is supplied, making it impossible to verify that the difference is due to the hyperbolic repulsion term rather than to the particular numerical realization.
minor comments (2)
  1. [Abstract] The abstract states that the model 'can well describe' the Pasquier et al. experiment, but the manuscript does not provide a direct quantitative comparison between simulated and experimental cell distributions or growth curves.
  2. [Model formulation] Notation for the repulsion term and the two population densities should be introduced once in the model section and used consistently thereafter.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The two major comments both concern insufficient detail in the numerical simulations section. We agree that these points are valid and will revise the manuscript to include the requested information and comparisons.

read point-by-point responses

  1. Referee: Numerical simulations section: the claim that the population ratio depends on initial total cell number but not on the law of the initial distribution rests on unspecified simulation choices (discretization method, mesh size, time-stepping scheme, boundary conditions, and parameter values drawn from Pasquier et al.). Without these details or a convergence study, the reported insensitivity cannot be assessed as a model property rather than a numerical artifact.

    Authors: We agree that the numerical implementation details were insufficiently specified. In the revised manuscript we will add an explicit subsection describing the discretization (a characteristics-based finite-volume scheme), spatial mesh size (N=2000 points), time-stepping (explicit Euler with adaptive CFL), boundary conditions (no-flux), and the precise parameter values taken from Pasquier et al. We will also report a convergence study in which mesh size and time step are successively refined, confirming that the reported dependence on initial total cell number and independence from the law of the initial distribution remain unchanged. revision: yes

  2. Referee: Numerical simulations section: the assertion of competitive exclusion different from the ODE model is presented as an observation, yet no quantitative comparison (e.g., final population ratios, time series, or parameter table) is supplied, making it impossible to verify that the difference is due to the hyperbolic repulsion term rather than to the particular numerical realization.

    Authors: We acknowledge the lack of quantitative comparison. The revised version will contain (i) a table of final population ratios for both the PDE model and the corresponding ODE system under identical initial data, (ii) overlaid time-series plots of total cell numbers, and (iii) an explicit parameter table. These additions will allow direct verification that the competitive exclusion is produced by the spatial repulsion term. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper defines a new repulsion term in the hyperbolic Keller-Segel system, proves existence/uniqueness and segregation directly from the PDE via characteristics, and reports numerical observations on competitive exclusion and initial-total-number dependence. None of these steps reduce by construction to fitted inputs, self-citations, or renamed ansatzes; the cited Pasquier experiment supplies only external context. All load-bearing claims remain independent of the reported outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; the ledger cannot be populated with concrete free parameters or axioms from the full text.

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