Synchrony in reaction–diffusion models of morphogenesis: applications to curvature-dependent proliferation and zero-diffusion front waves

Abstract

The paper presents the classical age-dependent approach of the morphogenesis in the framework of the von Foerster equation, in which we introduce a new constraint and study a new feature: (i) the new constraint concerns cell proliferation along the contour lines of the cell density, depending on the local curvature such as it favours the amplification of the concavities (like in the gastrulation process) and (ii) the new feature consists of considering, on the cell density surface, a remarkable line (the null mean Gaussian curvature line), on which the normal diffusion vanishes, favouring local coexistence of diffusing morphogens, metabolites or cells, and hence the auto-assemblages of these entities. Two applications to biological multi-agents systems are studied, gastrulation and feather morphogenesis.

Demography and Diffusion in Epidemics: Malaria and Black Death Spread

“…For improving this fit, contagion parameters b and f can be chosen depending on space, e.g. maximum in zones, which constitute overlaps between domains where diffusion of infective vectors and hosts (A i and G) is minimum and domains where concentration of susceptible (A s and S) is maximum, ensuring locally a large coexistence time, hence a high contagion rate between these interacting subpopulations (Abbas et al 2009). …”

“…These lines correspond to regions where the mean Gaussian curvatures on surfaces of concentration S and A i , defined respectively by q 2 S/qx 2 q 2 S/qy 2 -(q 2 S/ qxqy) 2 and q 2 A i /qx 2 q 2 A i /qy 2 -(q 2 A i /qxqy) 2 , vanish. Figure 3 shows the possibility of Demography and Diffusion in Epidemics 285 such an intersection on only one tangency point or two intersection points (left) and on whole zero-diffusion curves asymptotically confounded (right) for a convenient value of the ratio between the diffusion coefficients D S /D Ai (Abbas et al 2009). …”

“…the spawning places of Anopheles gambiae-the malaria vector-are located on backwater perimeter, whose length is equal to 0 in absence of rain (stable dry season), tends to infinity when backwater is progressively fulfilled by water (fractal transient phase during the season transition) and diminishes until 2pR, where R is the radius of the final backwater mare (stable rainy season). During the susceptible host and infective Anopheles spread, the maximum of contagion is observed on the common zones of least diffusion of both hosts and vectors, which can be asymptotically confounded: as for the morphogens interaction in morphogenesis, the common zero-diffusion domain allows a maximum of contagious contacts between interacting species (Abbas et al 2009). …”

From biological and clinical experiments to mathematical models

“…Article 9 is 'Synchrony in reaction-diffusion models of morphogenesis: applications to curvature-dependent proliferation and zero-diffusion front waves' by Abbas et al (2009). This article relates to probabilistic dynamics (Kolmogoroff-Sinai entropy), which are used to describe cell dynamics incorporating the fundamental cell cycle.…”

Modelling and Image Processing of Constriction and Proliferation in the Gastrulation Process of Drosophila melanogaster

“…In both these cases, a couple of morphogens acting often simultaneously i n opposit e (e. g. a coupl e of acti vat or and inhibitor like BMP-7 and BMP-2 in feather morphogenesis in the chicken [24][25][26][27] can induce the chemotactic motion of fibroblats or the biosynthesis of the elements constituting the auto-assemblage (like proteins and phospholipids). The fact that for a certain value of their viscosity ratio, the morphogens can coexist for a relatively long time in a precise location can greatly favor the birth of anatomic organ boundaries.…”