Coupling water fluxes with cell wall mechanics in a multicellular model of plant development

Abstract

The growth of plant organs is a complex process powered by osmosis that attracts water inside the cells; this influx induces simultaneously an elastic extension of the walls and pressure in the cells, called turgor pressure; above a threshold, the walls yield and the cells grow. Based on Lockhart's seminal work, various models of plant morphogenesis have been proposed, either for single cells, or focusing on the wall mechanical properties. However, the synergistic coupling of fluxes and wall mechanics has not yet been fully addressed in a multicellular model. This work lays the foundations of such a model, by simplifying as much as possible each process and putting emphasis on the coupling itself. Its emergent properties are rich and can help to understand plant morphogenesis. In particular, we show that the model can display a new type of lateral inhibitory mechanism that amplifies growth heterogeneities due e.g to cell wall loosening.

Fig 1. Hierarchy of models presented in this article.

The cells are given a height h as illustrated in c). The walls that hold stresses (in green) have a thickness w. a) Lockhart-Ortega model: uniaxial growth in the x direction of a cell of length l. b) two cells extension, both growing along x; the lighter shade of green corresponds to a lower elastic modulus; c) 2D extension of a single cell growth; d) Multicellular, multidimensional model; left: fluxes, right: mechanical equilibrium; the stress σ is proportionnal to the elastic deformation ε^e; E is the elastic modulus.

Fig 2

Analytical resolution of the two cells model, properties of the solution in the parameters space α^a × α^s; a) delimitation of the two zones ${\dot{\gamma }}_1=0$ and ${\dot{\gamma }}_1>0$: the red thick solid line ${\alpha }^s\left({\alpha }^a\right)=\frac{1-\rho }{1-{\alpha }^a}$ corresponds to ρ = 0.75. The two black thin dashed lines correspond to the values ρ = 0.5 and 0.99. b-c) Turgors P₀ and P₁ for ρ = 0.75. d-e) relative growth rates ${\dot{\gamma }}_i/{\dot{\gamma }}_i^{*}$ for ρ = 0.75.

Fig 3. Growth of tissue with heterogeneous mechanical parameters, see S6 Text.

a) (a) Initial state for (REF): walls are under tension because of turgor and have reached their yield deformation. At t = 0, the walls of the cells marked with a white star are softened (the elastic modulus is divided by two). (b) Time evolution of the total volume. The dashtype of the lines distinguishes the parameters sets; the same dashtype convention is used in (c) and (d). (c) Time evolution of turgor pressure of bump cells (red) and other cells (blue). (d) Time evolution of relative growth rate of bump cells (red) and other cells (blue). (e-l) Turgor and relative growth rate maps of parameters sets (REF) ((e-f)), (PM-) ((g-h)), (CC-) ((i-j)), and (ALPHA+) ((k-l)), at the time when the volume of the bump cells has increased by a factor 5: t = 51h for (REF), t = 33h for (PM-), t = 80h for (CC-), t = 14.8h for (ALPHA+). The arrows represent the intensity and direction of cell-cell water fluxes; the scale for arrows is the same for (REF), (PM-) and (CC-) and close to 4 times higher for (ALPHA+).

Fig 4

Evidence of lateral inhibition: left: a) time evolution of the volume of two cells on the boundary of the bump (marked with a green dot on the maps b, c, d) with the sets of parameters (REF), (PM-), (PM-) with α^s = 0.95, (PM-) with α^s = 0.99. V₀ is the volume of the cells at t = 0. b,c,d) maps of relative growth rate at t = 33h for (PM-), t = 20h for (PM-) and α^s = 0.95, t = 10h for (PM-) and α^s = 0.99.