Onset of nonlinearity in a stochastic model for auto-chemotactic advancing epithelia

Abstract

We investigate the role of auto-chemotaxis in the growth and motility of an epithelium advancing on a solid substrate. In this process, cells create their own chemoattractant allowing communications among neighbours, thus leading to a signaling pathway. As known, chemotaxis provokes the onset of cellular density gradients and spatial inhomogeneities mostly at the front, a phenomenon able to predict some features revealed in in vitro experiments. A continuous model is proposed where the coupling between the cellular proliferation, the friction on the substrate and chemotaxis is investigated. According to our results, the friction and proliferation stabilize the front whereas auto-chemotaxis is a factor of destabilization. This antagonist role induces a fingering pattern with a selected wavenumber k0. However, in the planar front case, the translational invariance of the experimental set-up gives also a mode at k = 0 and the coupling between these two modes in the nonlinear regime is responsible for the onset of a Hopf-bifurcation. The time-dependent oscillations of patterns observed experimentally can be predicted simply in this continuous non-linear approach. Finally the effects of noise are also investigated below the instability threshold.

Step, dip, and bell-shape traveling waves in a (2 + 1)-chemotaxis model with traction and long-range diffusion

“…It takes into consideration simple and nonlinear cross diffusions, in addition to incorporating two active transport mechanisms, namely chemotaxis and traction. Though traction has been the subject of intensive experimental studies [5,15,16,25], its contribution has not yet been mathematically formulated in chemotactic systems, to the best of our knowledge. The model Eqs.…”

“…We propose Eqs. (3) as a viable generalized chemotaxis model that incorporates several aspects of bacterial collective motion not discussed in previous models [3,15,[28][29][30][31][33][34][35][36][37][38][39][40][41][42].…”

“…Depending on the strength of the bias, cells rearrange themselves and exhibit a different moving pattern. Independently, a theoretical approach to the question was addressed to demonstrate that auto-chemotaxis is a factor of instability [15,16], while friction and proliferation in the system have stabilizing effects. Such an antagonism is at the onset of pattern selection with a well-defined wave number [15].…”

“…Besides the importance of chemotaxis in the cellular realm where it plays a major role in fertilization [2][3][4], intercellular communication [5,6], wound repair [2,3], and pattern formation [3,7], it has also been reported in animal and insect ecology [4], large scale collective behavior [1,[8][9][10][11][12][13][14] and synchronization processes [5,8]. Though the literature is well furnished when it comes to chemotaxis, the various issues associated with a combination of chemical and mechanical constraints cells might feature in their natural habitats has just recently started receiving attention [10,[12][13][14][15][16][17][18][19][20][21][22]. It is known that the latter interactions may lead to interesting holistic dynamical behaviors that to the best of our knowledge remain to be fully understood.…”

“…Independently, a theoretical approach to the question was addressed to demonstrate that auto-chemotaxis is a factor of instability [15,16], while friction and proliferation in the system have stabilizing effects. Such an antagonism is at the onset of pattern selection with a well-defined wave number [15]. On the other side, the fact that cells can become attractive by producing their own chemoattractant implies that in a given population, certain cells are more chemotactic than others, therefore may exert distant actions like traction forces.…”