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  • tio which measures the relative

    2019-09-24

    tio χ , which measures the relative importance of LY 379268 and D1
    attraction.
    β
    β
    β
    system has a homogeneous (in space) solution, given by
    M
    εr
    t
    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
    ε ε
    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).
    t
    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 .
    n
    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.