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

View Article: PubMed Central - PubMed

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.

No MeSH data available.


Left panel (A): Density profile of the morphogens as a function of the distance from the front position y = 0, in the experimental cell. Velocities are increasing from 3 to 7 while setting N0 = 1. Right panel (B): The front velocity as a function of combined chemotactic parameters, when varying the interface pressure and setting α = 1 (see equation (8)).
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f3: Left panel (A): Density profile of the morphogens as a function of the distance from the front position y = 0, in the experimental cell. Velocities are increasing from 3 to 7 while setting N0 = 1. Right panel (B): The front velocity as a function of combined chemotactic parameters, when varying the interface pressure and setting α = 1 (see equation (8)).

Mentions: Figure 3 summarizes the results for a particular choice of the parameters. The density (Fig. 3A) and pressure fields vary exponentially with the distance from the front, only the morphogen concentration is given, having its maximum at the front. Notice that fast fronts maintain the morphogen concentration on large distances. As U decreases, the morphogen concentration is more localized around the leading edge but for higher velocities (so higher cellular proliferation), one may observe a spreading of the morphogen concentration. In the water, the decrease is abrupt, but this event does not affect the cells. Figure 3B shows the front velocity as a function of the combined parameters Λ(1 + N0) and αPi. From equation (8), we recover the two limiting cases: A front driven only by chemotaxis without proliferation, having velocity: ) and a proliferative front with velocity: U = −αPi. Let us consider now the stability of such fronts. The stability analysis of such fronts is performed by introducing a sinusoidal perturbation induced by a wavy frontier of the cellular domain: ζ = Ut + εeikxeΩt (in analogy with the Fig. 2, middle panel) and expanding all fields according to:


Onset of nonlinearity in a stochastic model for auto-chemotactic advancing epithelia
Left panel (A): Density profile of the morphogens as a function of the distance from the front position y = 0, in the experimental cell. Velocities are increasing from 3 to 7 while setting N0 = 1. Right panel (B): The front velocity as a function of combined chemotactic parameters, when varying the interface pressure and setting α = 1 (see equation (8)).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC5037363&req=5

f3: Left panel (A): Density profile of the morphogens as a function of the distance from the front position y = 0, in the experimental cell. Velocities are increasing from 3 to 7 while setting N0 = 1. Right panel (B): The front velocity as a function of combined chemotactic parameters, when varying the interface pressure and setting α = 1 (see equation (8)).
Mentions: Figure 3 summarizes the results for a particular choice of the parameters. The density (Fig. 3A) and pressure fields vary exponentially with the distance from the front, only the morphogen concentration is given, having its maximum at the front. Notice that fast fronts maintain the morphogen concentration on large distances. As U decreases, the morphogen concentration is more localized around the leading edge but for higher velocities (so higher cellular proliferation), one may observe a spreading of the morphogen concentration. In the water, the decrease is abrupt, but this event does not affect the cells. Figure 3B shows the front velocity as a function of the combined parameters Λ(1 + N0) and αPi. From equation (8), we recover the two limiting cases: A front driven only by chemotaxis without proliferation, having velocity: ) and a proliferative front with velocity: U = −αPi. Let us consider now the stability of such fronts. The stability analysis of such fronts is performed by introducing a sinusoidal perturbation induced by a wavy frontier of the cellular domain: ζ = Ut + εeikxeΩt (in analogy with the Fig. 2, middle panel) and expanding all fields according to:

View Article: PubMed Central - PubMed

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.

No MeSH data available.