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

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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.


Phase-diagram Λ versus the averaged front velocity U for the parameter values corresponding to Fig. 4, on left: α = N0 = 1, σ = 10−3. The fully stable domain is in white (without noise), in green the estimated weakly nonlinear domains with steady advancing periodic patterns (represented below) and Hopf bifurcation, in yellow, the domain of spatio-temporal pattern, not treated here as the fully unstable domain represented in red. In the inset, schematic representation of the steady pattern, the pressure at the interface is fixed to Pi = −6.76, giving U = 7 and k0 = 5.
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f5: Phase-diagram Λ versus the averaged front velocity U for the parameter values corresponding to Fig. 4, on left: α = N0 = 1, σ = 10−3. The fully stable domain is in white (without noise), in green the estimated weakly nonlinear domains with steady advancing periodic patterns (represented below) and Hopf bifurcation, in yellow, the domain of spatio-temporal pattern, not treated here as the fully unstable domain represented in red. In the inset, schematic representation of the steady pattern, the pressure at the interface is fixed to Pi = −6.76, giving U = 7 and k0 = 5.

Mentions: where . How to select the nonlinearity in equation (13)? Above threshold, the first nonlinearity is −β2u2 where β is a numerical value and the negative sign indicates a saturation of the velocity amplitude. There is no symmetry u− > → −u, since the velocity correction u cannot behave symmetrically in the front direction (upstream) and in the opposite direction (downstream). Defining eliminates the unknown constant β from the equation for and we recover equation(13). However a more fancy way consists in applying the Galilean invariance symmetry (1/2∂u2/∂x): x → x − vt and u → u + v. Such consideration allows to avoid the very tedious nonlinear analysis from the free-boundary equations. This quadratic nonlinearity also imposes the scaling of the parameters: Choosing /μ/ of order ε2, from the dispersion relation, we deduce that the lengths scale as 1/ε while, from the right-hand-side of equation (13), the velocity u scales also as 1/ε. The left-hand-side gives us the time-scale as 1/ε2. As shown in Fig. 1, chemotaxis, which is responsible for diffuse fronts, increases stochasticity and the disorder. Restricting to the biochemical noise in our model, stochasticity enters at the level of the boundary conditions: The jump condition for both concentration and gradient of concentration and the chemotactic migration constant Λ. Stochasticity from the pressure and the friction of the cells at the substrate are neglected. We are then face simultaneously to a multiplicative noise η1, indicating a noisy threshold for bifurcation and an additive noise η2 arising from all other sources. Both sources of noise are chosen Gaussian in space, white in time and are not correlated so 〈ηi(x, t)ηj(x′, t′)〉 = Cj(x)δijδ(t − t′)δ(x − x′). Restricting on the vicinity of the interface, close but below the bifurcation threshold μ < 0, (which corresponds to the white zone of parameters in Fig. 5), we transform equation (13) into a stochastic equation adding η1 and η2. It reads:


Onset of nonlinearity in a stochastic model for auto-chemotactic advancing epithelia
Phase-diagram Λ versus the averaged front velocity U for the parameter values corresponding to Fig. 4, on left: α = N0 = 1, σ = 10−3. The fully stable domain is in white (without noise), in green the estimated weakly nonlinear domains with steady advancing periodic patterns (represented below) and Hopf bifurcation, in yellow, the domain of spatio-temporal pattern, not treated here as the fully unstable domain represented in red. In the inset, schematic representation of the steady pattern, the pressure at the interface is fixed to Pi = −6.76, giving U = 7 and k0 = 5.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f5: Phase-diagram Λ versus the averaged front velocity U for the parameter values corresponding to Fig. 4, on left: α = N0 = 1, σ = 10−3. The fully stable domain is in white (without noise), in green the estimated weakly nonlinear domains with steady advancing periodic patterns (represented below) and Hopf bifurcation, in yellow, the domain of spatio-temporal pattern, not treated here as the fully unstable domain represented in red. In the inset, schematic representation of the steady pattern, the pressure at the interface is fixed to Pi = −6.76, giving U = 7 and k0 = 5.
Mentions: where . How to select the nonlinearity in equation (13)? Above threshold, the first nonlinearity is −β2u2 where β is a numerical value and the negative sign indicates a saturation of the velocity amplitude. There is no symmetry u− > → −u, since the velocity correction u cannot behave symmetrically in the front direction (upstream) and in the opposite direction (downstream). Defining eliminates the unknown constant β from the equation for and we recover equation(13). However a more fancy way consists in applying the Galilean invariance symmetry (1/2∂u2/∂x): x → x − vt and u → u + v. Such consideration allows to avoid the very tedious nonlinear analysis from the free-boundary equations. This quadratic nonlinearity also imposes the scaling of the parameters: Choosing /μ/ of order ε2, from the dispersion relation, we deduce that the lengths scale as 1/ε while, from the right-hand-side of equation (13), the velocity u scales also as 1/ε. The left-hand-side gives us the time-scale as 1/ε2. As shown in Fig. 1, chemotaxis, which is responsible for diffuse fronts, increases stochasticity and the disorder. Restricting to the biochemical noise in our model, stochasticity enters at the level of the boundary conditions: The jump condition for both concentration and gradient of concentration and the chemotactic migration constant Λ. Stochasticity from the pressure and the friction of the cells at the substrate are neglected. We are then face simultaneously to a multiplicative noise η1, indicating a noisy threshold for bifurcation and an additive noise η2 arising from all other sources. Both sources of noise are chosen Gaussian in space, white in time and are not correlated so 〈ηi(x, t)ηj(x′, t′)〉 = Cj(x)δijδ(t − t′)δ(x − x′). Restricting on the vicinity of the interface, close but below the bifurcation threshold μ < 0, (which corresponds to the white zone of parameters in Fig. 5), we transform equation (13) into a stochastic equation adding η1 and η2. It reads:

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&thinsp;=&thinsp;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.