tio which measures the relative
tio χ , which measures the relative importance of LY 379268 and D1
system has a homogeneous (in space) solution, given by
the (linear) stability of which we now investigate in detail. We now assume that the perturbation is initiated at time t0 > 0 and we focus on an interval of the form (t0, t0 + t ) whose size t is of the order of the kinetics time-scale ε. As such, it is small when compared to the growth time-scale r1 . Note that in the origi-
nal time variable, this amounts to considering a time interval of or-der β1 (small when compared to 1r ). In particular, neglecting terms of order O((r0ε)), we can approximate all functions of time by their value at t0.
with μ of real part (μ) > 0, which imposes that μ is an eigen-value of Ak(t). Let us denote μ+k (t ) to be either the largest real eigenvalue of Ak(t) or the real part of its complex conjugate eigen-values.
We now look for su cient and necessary conditions ensuring that μ+k (t ) > 0. For a given time t > 0, we compute
It is thus easy to check that Ak(t) has an eigenvalue with positive real part if and only if det(Ak (t )) < 0, which is equivalent to
Since we are interested in perturbations other than those in the direction of the first homogeneous eigenfunction ψ 1 (and since the above polynomial is positive at 0), a necessary and su cient con-dition to have μ+k (t ) > 0 is for λk > 0 to satisfy ¯ λk < λ(t )
In other words, looking for perturbations in a direction other than that of the eigenfunction ψ 1 which is homogeneous, a per-turbation at time t0 will yield Turing instability if and only if
neglecting small terms in O((r0ε)). As in the Introduction, we call this condition IC(t0).
The perturbations happen along the modes ψ k which for simple geometries as in our case can be explicitly computed, see Appendix A.
3.1. On the condition for Turing instability In condition (6), the right-hand side ¯ (t ) has the same mono- λ tonicity as ϕ as a function of n¯(t0 ) = Mer0t0 . Recall that the function ϕ increases and then decreases in both cases ϕ(n) =
n nmax .
Assume that M is fixed such that initially, ϕ(n¯(t )) increases with time (i.e., ϕ (M) > 0). Consequently, the condition IC(t) in (6) might not initially be satisfied but it is more likely to be as time increases. The typical dynamics expected from this condi-tion is thus an increase of n(t, · ), c(t, · ) very close to n¯(t ), c¯(t ), up until the Turing instability condition becomes satisfied. Patterns then form very quickly (on the time scale of ε).
3.1.2. Dependence on the initial mass
For a small initial mass M, the right-hand side in condition IC(t) in (6) is small, meaning that Turing instabilities are expected only
at a large time t. This provides an explanation for the fact that pat-terns are not initially observed. In the limit when M is very small, one should wait for a very long time before IC(t) starts being sat-isfied.
As M increases, Turing instability is more likely to occur before D4, until the cell density reaches the region in which ϕ decreases. When M is too large, IC(t) is not satisfied initially and will never be later on: no patterns should be obtained. This in accordance with experimental observations, and explains why we choose a nonlin-ear sensitivity function instead of the more classical linear one. A linear choice would indeed predict Turing instabilities for high val-ues of M.
We note that the dependence of IC(t) in (6) as a function of A is such that if IC(t) is satisfied at some time t for a given A, then it should be for a larger A as well. In other words, if MCF7 cells have created patterns at day D4, then so should have the MCF7-sh-wisp2 (all other parameters being equal). This is not in agree-ment with the experimental findings with M = 10 000, value for which MCF7 cells have created spheroids, but not MCF7-sh-wisp2. We however stress that predictions of our continuum PDE model should be taken with care when dealing with the lowest possible densities and first days of experiment.