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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): Domains of spatial wavelengths q = (k − k0)/k0 in the nonlinear regime defined by μ = (Λ − Λ0)/Λ0. In grey, unstable modes, in green stable modes. The left-hand frontier between grey and green domains corresponds to the Hopf bifurcation. Right panel (B): Schematic representation of the interface profile ζ and the average front velocity, for μ = 0.1. The interface oscillates in time between the blue and red curves, the curve in purple indicates the mean velocity.
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f6: Left panel (A): Domains of spatial wavelengths q = (k − k0)/k0 in the nonlinear regime defined by μ = (Λ − Λ0)/Λ0. In grey, unstable modes, in green stable modes. The left-hand frontier between grey and green domains corresponds to the Hopf bifurcation. Right panel (B): Schematic representation of the interface profile ζ and the average front velocity, for μ = 0.1. The interface oscillates in time between the blue and red curves, the curve in purple indicates the mean velocity.

Mentions: The last terms of the first equation (15) represent the advection of the pattern, , by the change in the front velocity B. We also notice that B has only a nonzero value for inhomogeneous values of the amplitudes A and B (see Section Hopf-Bifurcation). Equations (15) are similar to those established previously4344 and such coupling has been called Hopf-Turing bifurcation42. Exact solutions for A and B in equation (15) can be easily found: A = A0eiqX with amplitude: and B = 0 (see the domain in green in Fig. 6A). A stability analysis of these solutions is necessary knowing that their domain of existence is constrained by the inequality q2 < 1/4. Once perturbed according to A = (A0 + r)eiqX+ϕ and B =ψ, where r, ϕ and ψ are small real functions of wavenumber p and growth rate η, an algebraic equation is established (see Section Hopf-Bifurcation) for the growth rate η giving modes growing exponentially in time for q > 0, a range of stable modes for negative q values, bounded by a Hopf bifurcation for q = qh (see the schematic representation of the analysis in Fig. 6A where unstable modes are in the grey domain, stable modes are in the green domain and the Hopf-Turing modes are at the frontier on left). For , waves appear and superpose to the static spatial mode qh affecting simultaneously the spatially periodic oscillation A and the average velocity B of the front. The interface oscillates between the blue and red curves of Fig. 6B, but also the average velocity of the front (represented in purple). For a number p of the wave, the frequency for the time-dependence is equal to ωhp with .


Onset of nonlinearity in a stochastic model for auto-chemotactic advancing epithelia
Left panel (A): Domains of spatial wavelengths q = (k − k0)/k0 in the nonlinear regime defined by μ = (Λ − Λ0)/Λ0. In grey, unstable modes, in green stable modes. The left-hand frontier between grey and green domains corresponds to the Hopf bifurcation. Right panel (B): Schematic representation of the interface profile ζ and the average front velocity, for μ = 0.1. The interface oscillates in time between the blue and red curves, the curve in purple indicates the mean velocity.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f6: Left panel (A): Domains of spatial wavelengths q = (k − k0)/k0 in the nonlinear regime defined by μ = (Λ − Λ0)/Λ0. In grey, unstable modes, in green stable modes. The left-hand frontier between grey and green domains corresponds to the Hopf bifurcation. Right panel (B): Schematic representation of the interface profile ζ and the average front velocity, for μ = 0.1. The interface oscillates in time between the blue and red curves, the curve in purple indicates the mean velocity.
Mentions: The last terms of the first equation (15) represent the advection of the pattern, , by the change in the front velocity B. We also notice that B has only a nonzero value for inhomogeneous values of the amplitudes A and B (see Section Hopf-Bifurcation). Equations (15) are similar to those established previously4344 and such coupling has been called Hopf-Turing bifurcation42. Exact solutions for A and B in equation (15) can be easily found: A = A0eiqX with amplitude: and B = 0 (see the domain in green in Fig. 6A). A stability analysis of these solutions is necessary knowing that their domain of existence is constrained by the inequality q2 < 1/4. Once perturbed according to A = (A0 + r)eiqX+ϕ and B =ψ, where r, ϕ and ψ are small real functions of wavenumber p and growth rate η, an algebraic equation is established (see Section Hopf-Bifurcation) for the growth rate η giving modes growing exponentially in time for q > 0, a range of stable modes for negative q values, bounded by a Hopf bifurcation for q = qh (see the schematic representation of the analysis in Fig. 6A where unstable modes are in the grey domain, stable modes are in the green domain and the Hopf-Turing modes are at the frontier on left). For , waves appear and superpose to the static spatial mode qh affecting simultaneously the spatially periodic oscillation A and the average velocity B of the front. The interface oscillates between the blue and red curves of Fig. 6B, but also the average velocity of the front (represented in purple). For a number p of the wave, the frequency for the time-dependence is equal to ωhp with .

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.