br Fig Alternative model Evolution of the
Fig. 7. Alternative model. Evolution of the density of cells from the initial state n0 = 0.1 in the case where ϕ(n) = ne−n . All parameters have been chosen as in Fig. 4.
fect and its modulation at high cell densities. We have analyzed theoretically the conditions which lead to instability and pattern formation. The numerical solutions confirm the theoretical analy-sis and qualitatively reproduce the observed patterns, in spite of its relatively low parametrization. Contrarily to other modeling ap-proaches, the salient feature is not cell-scaffold adhesion, which we reduce to a constant diffusion term, but instead chemotaxis. We thus hypothesize that it is a key phenomenon responsible for these aggregates.
The numerical simulations of the model do not only show qual-itative accordance with the experimental results. Indeed, the sys-tem (1) has spatially inhomogeneous solutions with spheroidal patterns, and it also describes how different kinds of patterns can arise: few, elongated structures for a small diffusion value or nu-merous, mainly round, small Stearamide for a stronger chemotactic sensitivity.
The simplicity of the model induces two main limitations:
• it does not seem to offer a satisfying flexibility for the size of patterns, essentially fixed by the geometry of .
• it is not suitable to explain the post-aggregation phase, and in particular the increase in the number of spheres as observed experimentally 10 days after initial pattern formation.
In fact, the distributions (in size) of spheroids for the biological images and the numerical simulations do not match well: while the standard deviation is of the order of the average size for ex-periments as evidenced by Fig. 2 in the Introduction, we find that standard deviation is about one third of the mean size in simula-tions.
In the Keller–Segel model, the variability is essentially captured only by the Laplacian eigenfunctions which themselves are com-pletely characterized by the domain geometry. A natural direction of research for a better matching is to model cell-scaffold adhesion more finely than with a diffusion term, incorporating anisotropies (such as in Painter, 2009), or even randomness, in the extracellular matrix density.
As for the second point, statistical estimates obtained from im-ages taken later during the experiment (not shown here) indeed evidence a growth in both the size and number of spheroids. This is not reproduced by numerical simulations. In fact, the pat-terns that formed then typically continue to merge, probably until the cells are all packed in very few aggregates. This phenomenon for this type of model is explained in detail in (Potapov and Hillen, 2005).
We insist that a minimally-parametrized model such as ours is more amenable to mathematical analysis and also paves the way for works aiming at a more quantitative prediction of the typ-ical size and number of spheroids. To go further in this direc-tion, one should look for the actual modes along which instabil-ities will be observed in a time-dependent setting, in the spirit of Madzvamuse et al. (2010).
The authors acknowledge partial funding from the ANR blanche project Kibord ANR-13-BS01-0004 funded by the French Ministry of Research. B.P. has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 740623). F.B. has re-ceived funding for international mobility from Universit Franco-italienne.
Appendix A. Explicit computation of the modes
Since has a particular shape, eigenvalues and eigenfunctions can actually be explicitly computed. We first consider the case of the 2D simulations, namely when is a disk of radius a. It is then standard that all eigenfunctions can be obtained after separation of variables in polar coordinates ψ (x, y) = f (r)g(θ ), the equation − ψ = λψ with Neumann boundary conditions is equivalent to