Modelling pattern formation through differential repulsion

Julien Barré; Pierre Degond; Diane Peurichard; Ewelina Zatorska

doi:10.3934/nhm.2020021

Article Contents

2020, Volume 15, Issue 3: 307-352. Doi: 10.3934/nhm.2020021

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Modelling pattern formation through differential repulsion

1.
Institut Denis Poisson, Université d'Orléans, CNRS, Université de Tours, B.P. 6759, 45067 Orléans cedex 2, France, Institut Universitaire de France, Paris, France
2.
Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
3.
INRIA Team Mamba, INRIA Paris, 2 rue Simone Iff, CS 42112, 75589 Paris, France, Sorbonne Université, UMR 7598 LJLL, BC187, 4, Place de Jussieu, F-75252 Paris Cedex 5, France
4.
Department of Mathematics, University College London, Gower Street London WC1E 6BT, UK, United Kingdom

Received: June 2019

Revised: November 2019

Early access: September 2020

Published: September 2020

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Abstract

Motivated by experiments on cell segregation, we present a two-species model of interacting particles, aiming at a quantitative description of this phenomenon. Under precise scaling hypothesis, we derive from the microscopic model a macroscopic one and we analyze it. In particular, we determine the range of parameters for which segregation is expected. We compare our analytical results and numerical simulations of the macroscopic model to direct simulations of the particles, and comment on possible links with experiments.

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Mathematics Subject Classification: Primary: 35Q92, 35B36, 35B32; Secondary: 92C17.

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Figure 1. Scheme of the predictions of the linear stability analysis by acting on the interspecies repulsion force

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Figure 2. Form of the potential $ \tilde{\Phi}^{ST}(x) $ for $ R = 2 $ and $ \frac{\nu^{ST}_c}{\nu^{ST}_d} \frac{\kappa^{ST}}{2} = 1 $. The potential is repulsive on its support

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Figure 3. Values of $ \lambda_1 $ (blue curve), $ \lambda_2 $ (orange curve) and their mean (yellow dotted line) near $ z = 0 $ (z = 0.1), plotted as functions of the scaling parameter $ s $ for $ R = 1 $, $ D^A = D^B = 1, c'^{AA} = c'^{AB} = c'^{BA} = 1, c'^{BB} = 10 $

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Figure 4. (I): Values of $ \lambda_1(z) $ as functions of $ z $ for $ R = 1 $, $ D^A = D^B = 1, \, c^{AA} = c^{AB} = c^{BA} = 1, \, c^{BB} = 10 $ and for different values of $ s $ in the instability regime: $ s = 30 $ (blue curve), $ s = 50 $ (orange curve), $ s = 70 $ (yellow curve), and $ s = 90 $ (red curve). (II): same plots for $ \lambda_2(z) $. (III) Plot of $ z^* $ defined in (36) as a function of parameter $ s $

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Figure 5. Microscopic simulations for Cases 1-4 for parameters described in Table 1. $ A $-cells are represented as red disks, $ B $-cells as green disks. For each subsection, the left figure is obtained starting from a homogeneous distribution of particles, the right one from a segregated initial distribution ($ B $-cells on the half-left of the domain, $ A $-cells on the right)

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Figure 6. Macroscopic simulations for Cases 1-4 for parameters described in table 1 for the final time of simulations equal to $ T = 8000 $

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Figure 7. (I) Simulations of the microscopic and macroscopic models for $ \kappa^{AA} = 4, \kappa^{BB} = \tilde{\kappa}^{AB} = \tilde{\kappa}^{BA} = 1 $ for which $ s^* \approx 2.1 $. Simulations of the microscopic model are performed with $ N_A = N_B = 500 $ (IA) and $ N_A = N_B = 2000 $ (IB) particles. Simulations of the macroscopic model correspond to (IC). We consider 7 values of the interspecies repulsion intensity, from left to right: for $ 2.05 = s<s^* $, for $ s^*<s = \{2.15, 2.2, 2.5, 4, 6, 10 \} $. Type B cells are represented in green, type A cells in red. (II) In (IIA-C), we show the values of the quantifiers at time equilibrium as functions of $ s $. Fig. (IA) shows the mean elongation of the green clusters, (IIB) shows the number of green clusters and (IIC) shows the overlapping amount $ Q $ described by (73). Black curves are obtained with the microscopic model for $ N_A = N_B = 500 $ (corresponding to Figures (IA)), yellow curves are for $ N_A = N_B = 2000 $ and correspond to Figures (IB) and red curves are obtained with the macroscopic model (Figures (IC)). The two bottom figures correspond to a zoom of the corresponding curves close to the transition region for $ s $

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Figure 8. Quantifiers computed on the simulation images as functions of the logarithm of the simulation time for $ \kappa^{AA} = 4, \kappa^{BB} = \tilde{\kappa}^{AB} = \tilde{\kappa}^{BA} = 1 $ for the microscopic model with $ N_A = N_B = 500 $ (green curves), $ N_A = N_B = 2000 $ (blue curves), $ N_A = N_B = 4000 $ (yellow curves) and for the macro model (red curve). (I) Green cluster elongation, (II) Number of cell clusters and (III) Overlapping amount $ Q $. On Figure (I), we superimpose linear fits (dotted lines) for short times and large times, showing the two timescales (two slopes) of the micro model compared to the unique timescale of the macro dynamics (single slope)

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Table 1. Model parameters for the inter- and intra- species forces. The value of parameter $ s^* $ has been computed numerically from the formula (43)

value of $ s $		comment
Case I: $ \kappa^{AA}=\kappa^{BB}= {2} $, $ \tilde{\kappa}^{AB} = \tilde{\kappa}^{BA}= {2} $.
IA	0.5	Stable regime ($ s< s^* \approx 1.01 $)
IB	4	Unstable regime ($ s>s^* \approx 1.01 $)
Case II: $ \kappa^{AA}=\kappa^{BB}= 2 $, $ 1=\tilde{\kappa}^{AB}< \tilde{\kappa}^{BA}= 2 $.
IIA	0.5	Stable regime ($ s< s^* \approx 1.43 $)
IIB	4	Unstable regime ($ s>s^* \approx 1.43 $)
Case III: $ 2=\kappa^{AA}>\kappa^{BB}= 1 $, $ \tilde{\kappa}^{AB} = \tilde{\kappa}^{BA}=2 $.
IIIA	0.5	Stable regime ($ s< s^* \approx 0.72 $)
IIIB	4	Unstable regime ($ s>s^* \approx 0.72 $)
Case IV: $ 2=\kappa^{AA}>\kappa^{BB}= 1 $, $ 1=\tilde{\kappa}^{AB}< \tilde{\kappa}^{BA}= 2 $.
IVA	0.5	Stable regime ($ s< s^* \approx 1.02 $)
IVB	4	Unstable regime ($ s>s^* \approx 1.02 $)

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Table 2. Volume fraction of the green family computed on the simulations of FigS. 5-6 at equilibrium for the microscopic model (left column) and for the macroscopic model (right column)

Case	VF green cells (microscopic model)		VF green cells (macroscopic model)
	Homogeneous IC	Front-like IC	Homogeneous IC	Front-like IC
(IB)	$ 48.2 \% $	$ 50.0 \% $	$ 49.7\% $	$ 50.0\% $
(IIB)	$ 40.2 \% $	$ 42.3 \% $	$ 38.5\% $	$ 42.0\% $
(IIIB)	$ 40.6 \% $	$ 42.4 \% $	$ 44.0\% $	$ 46.0\% $
(IVB)	$ 34.8 \% $	$ 35.0 \% $	$ 35.9\% $	$ 38.0\% $

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