Rossi Francesco, Duteil Nastassia Pouradier, Yakoby Nir, Piccoli Benedetto
Aix Marseille Université, CNRS, ENSAM, Université de Toulon, LSIS UMR 7296, 13397, Marseille, France.
Department of Mathematical Sciences and CCIB, Rutgers University - Camden, Camden, NJ.
Proc IEEE Conf Decis Control. 2016 Dec;2016:1614-1619. doi: 10.1109/CDC.2016.7798496. Epub 2016 Dec 29.
Among the main actors of organism development there are morphogens, which are signaling molecules diffusing in the developing organism and acting on cells to produce local responses. Growth is thus determined by the distribution of such signal. Meanwhile, the diffusion of the signal is itself affected by the changes in shape and size of the organism. In other words, there is a complete coupling between the diffusion of the signal and the change of the shapes. In this paper, we introduce a mathematical model to investigate such coupling. The shape is given by a manifold, that varies in time as the result of a deformation given by a transport equation. The signal is represented by a density, diffusing on the manifold via a diffusion equation. We show the non-commutativity of the transport and diffusion evolution by introducing a new concept of Lie bracket between the diffusion and the transport operator. We also provide numerical simulations showing this phenomenon.
在生物体发育的主要因素中，存在形态发生素，它们是在发育中的生物体中扩散并作用于细胞以产生局部反应的信号分子。因此，生长由这种信号的分布决定。同时，信号的扩散本身又受到生物体形状和大小变化的影响。换句话说，信号扩散与形状变化之间存在完全的耦合。在本文中，我们引入一个数学模型来研究这种耦合。形状由一个流形给出，该流形由于由一个输运方程给出的变形而随时间变化。信号由一个密度表示，通过一个扩散方程在流形上扩散。我们通过引入扩散算子和输运算子之间李括号的新概念，展示了输运和扩散演化的不可交换性。我们还提供了显示这种现象的数值模拟。
