Wei-Yuan Kong1, Antonio Mosciatti Jofre1, Manon Quiros2,Marie-B´eatrice Bogeat-Triboulot3, Evelyne Kolb2, and Etienne Couturier11- Laboratoire Mati`ere et Systemes Complexes, Universit´e Paris Diderot CNRS UMR 7057, 10 Rue Alice Domont et Leonie Ducquet, 75205 Paris Cedex 13, France 2- PMMH, CNRS, ESPCI Paris, Universit´e PSL, Sorbonne Universit´e, Universit´e de Paris, F-75005, Paris, France 3- Universit´e de Lorraine, AgroParisTech, INRAE, UMR Silva, 54000 Nancy, FranceAbstractTurgor is the driving force of plant growth, making possible for roots to overcome soil resistance or for stems to counteract gravity. Maintaining a constant growth rate while avoiding the cell content dilution, which would progressively stop the inward water flux, imposes the production or import of osmolytes in proportion to the increase of volume. We coin this phenomenon stationary osmoregulation. The article explores the quantitative consequences of this hypothesis on the interaction of a cylindrical cell growing axially against an obstacle. An instantaneous axial compression of a pressurized cylindrical cell generates a force and a pressure jump which both decrease toward a lower value once water has flowed out of the cell to reach the water potential equilibrium. In a first part, the article derives analytical formula for these force and over-pressure both before and after relaxation. In a second part, we describe how the coupling of the Lockhart’s growth law with the stationary osmoregulation hypothesis predicts a transient slowdownin growth due to contact before a re-acceleration in growth. We finally compare these predictions with the output of an elastic growth model which ignores the osmotic origin of growth: models only match in theearly phase of contact for high stiffness obstacle.
4 Comparison with an elastic growth model The growing cell is now modeled as a growing spring in series with one spring for the obstacle. This morphoelastic model is similar to a model developed for the growth of bacteria in a gel ; we present it here for comparison with the physiological model developed above. The analytical solution of the compression experiment provides the cell spring stiffness. Growth is phenomenologically introduced by increasing the rest cell length (coined target length Ltar). The target length increase rate and the negative force retro-acting on growth can be incorporated in the model thanks to two phenomenological constants calibrated with the analytical solution. Elastic model totally ignores water fluxes; it is thus not easy to incorporate the stationary osmoregulation hypothesis. The elastic growth model (See Annex 6.7) provides an expression for the temporal evolution of the non-dimensionalized observed length, Lobs after the contact: Lobs(t) =
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The solution saturates for an Lobs = Pc Y k . (22) The elastic growth model predicts a saturation of the observed (turgid) length which corresponds to a transient behavior of the physiological growth model more pronounced at high obstacle stiffness. In general both the relaxation rate (i = k/(1 + k) + Y) and the length increment differ with the elastic growth model outputs: L = k + k Pc (1 2) (1 + k) Pc + k Pc Y k + (1 + k) Y . As Pc 1, and Y 1, i equals el and L equals Lobs at the first order of approximation supposing k kcell; both models coincide in the early phase of the interaction ( a few minutes for a 1 cm long Chara corralina internodal cell for an obstacle stiffness superior to 3800 N m1). The morphoelastic model predict the same asymptotic force independently of the obstacle stiffness (Figure 3 (c)). 5 Conclusion
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An apical compression is the simplest way to probe the mechanics of a cylindrical cell. The article derives the forces (before and after relaxation) exerted by the sensor on the compressed cell and it shows that the forces are independent on the pressure at the first order. The initial pressure, equilibrated by the tension in the cell wall, does not contribute to the forces. The forces only depend on the surface modulus Eh multiplied by the radius and the force drop observed during the relaxation is Poisson ratio dependent; the force drop is null when the cell wall is an incompressible material, that is for a Poisson ratio of 0.5, and is maximal for a zero Poisson ratio. A compression also induces a turgor pressure jump which drops strongly during relaxation due to water outflow: before the relaxation, the pressure jump is proportional to the surface modulus divided by the radius while after the relaxation, the pressure jump is proportional to the initial pressure. According to the
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As in the case of isotropic poroelastic gels, the ratio between the force before and the force after relaxation only depends on the Poisson ratio. However the ratio value for cylindrical cells (cid:0) 5 4 (cid:1) /(1 2) is different from that for gels 2(1 ) ; interestingly in both cases the ratio equals one for an incompressible material. As a perspective, the model should be refined to describe the anisotropic properties of plant cell wall which could induce a stronger drop of the force or of the pressure during the relaxation.
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