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

Ibrahim Cheddadi, Michel Génard, Nadia Bertin, Christophe Godin

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Ibrahim Cheddadi, Michel Génard, Nadia Bertin, Christophe Godin. Coupling water fluxes with cell wall mechanics in a multicellular model of plant development. PLoS Computational Biology, 2019, 15 (6), pp.e1007121. 10.1371/journal.pcbi.1007121 . hal-02196768v2

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RESEARCH ARTICLE

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

Ibrahim Cheddadi¹,²*, Michel Génard³, Nadia Bertin³, Christophe Godin²,⁴*

1 Univ. Grenoble Alpes, CNRS, Grenoble INP, TIMC-IMAG, 38000 Grenoble, France, 2 Virtual Plants, INRIA, Montpellier, France, 3 UR 1115 PSH, INRA, F-84914 Avignon Cedex 9, France, 4 Laboratoire Reproduction et Développement des Plantes, Univ Lyon, ENS de Lyon, UCB Lyon 1, CNRS, INRA, INRIA, F-69342, Lyon, France

* ibrahim.cheddadi@univ-grenoble-alpes.fr (IC); christophe.godin@inria.fr (CG)

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## 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.

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OPEN ACCESS

**Citation:** Cheddadi I, Génard M, Bertin N, Godin C (2019) Coupling water fluxes with cell wall mechanics in a multicellular model of plant development. PLoS Comput Biol 15(6): e1007121. https://doi.org/10.1371/journal.pcbi.1007121

**Editor:** Lionel X. Dupuy, The James Hutton Institute, UNITED KINGDOM

**Received:** March 14, 2019

**Accepted:** May 20, 2019

**Published:** June 20, 2019

**Copyright:** © 2019 Cheddadi et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Data Availability Statement:** All relevant data are within the manuscript and its Supporting Information files.

**Funding:** IC received funding from the Agropolis foundation in Montpellier, France. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

**Competing interests:** The authors have declared that no competing interests exist.

## Author summary

Plant cells are surrounded by a rigid wall that prevents cell displacements and rearrangements as in animal tissues. Therefore, plant morphogenesis relies only on cell divisions, shape changes, and local modulation of growth rate. It has long been recognized that cell growth relies on the competition between osmosis that tends to attract water into the cells and wall mechanics that resists to it, but this interplay has never been fully explored in a multicellular model. The goal of this work is to analyze the theoretical consequences of this coupling. We show that the emergent behavior is rich and complex: among other findings, pressure and growth rate heterogeneities are predicted without any ad-hoc assumption; furthermore the model can display a new type of lateral inhibition based on fluxes that could complement and strengthen the efficiency of already known mechanisms such as cell wall loosening.
## Introduction

Plants grow throughout their lifetime at the level of small regions containing undifferentiated cells, the meristems, located at the extremities of their axes. Growth is powered by osmosis that tends to attract water inside the cells. The corresponding increase in volume leads to simultaneous tension in the walls and hydrostatic pressure (so-called turgor pressure) in the cells. Continuous growth occurs thanks to the yielding of the walls to these stretching forces [1-3].

This interplay between growth, water fluxes, wall stress and turgor was first modelled by Lockhart in 1965 [4], in the context of a single elongating cell. Recent models focused on how genes regulate growth at more integrated levels [5-9]. To accompany genetic, molecular, and biophysical analyses of growing tissues, various extensions of Lockhart's model to multicellular tissues have been developed. The resulting models are intrinsically complex as they represent collections from tens to thousands of cells in 2- or 3-dimensions interacting with each other. To cut down the complexity, several approaches abstract organ multicellular structures as polygonal networks of 1D visco-elastic springs either in 2D [7, 10-12] or in 3D [6, 13] submitted to a steady turgor pressure. Other approaches try to represent more realistically the structure of the plant walls by 2D deformable wall elements able to respond locally to turgor pressure by anisotropic growth [8, 14, 15].

Most of these approaches consider turgor as a constant driving force for growth, explicitely or implicitly assuming that fluxes occur much faster than wall synthesis. Cells then regulate the tissue deformations by locally modulating the material structure of their walls (stiffness and anisotropy) [6, 16-20]. However, the situation in real plants is more complex: turgor heterogeneity has been observed at cellular level [21, 22], which challenges the assumption of very fast fluxes. As a matter of fact, the relative importance of fluxes or wall mechanics as limiting factors to growth has fuelled a long standing debate [3, 23] and is still an open question. Moreover, from a physical point of view, pressure is a dynamic quantity that permanently adjusts to both mechanical and hydraulic constraints, which implies that a consistent representation of turgor requires to model both wall mechanics and hydraulic fluxes.

The aim of this article is to explore the potential effect of coupling mechanical and hydraulic processes on the properties of the “living material” that corresponds to multicellular populations of plant cells. To this end, we build a model that describes in a simple manner wall mechanics and cell structure, but do not compromise on the inherent complexity of considering a collection of deformable object hydraulically and mechanically connected.

The article is organized as follows (see Fig 1): we first recall the Lockhart-Ortega model and its main properties. Then we explore two simple extensions of this model: first we relax the constraint of uniaxial growth in the case of a single polygonal cell; then we study how two cells hydraulically connected interact with each other. Finally we describe our multicellular and multidimensional model and numerically explore its properties.

A table of notations is provided in Supplementary Information (S1 Table).

## Models and results

### The Lockhart model

In 1965, Lockhart [4] derived the elongation of a plant cell by coupling osmosis-based fluxes and visco-plastic wall mechanics. Ortega [24] extended this seminal model to include the elastic properties of the cell walls. We recall here the main properties of this model, see Fig 1a for the geometrical configuration.

**Cell wall elongation.** It is expressed as a rheological law [4, 24]: let $l$ be the length of the cell, the total strain rate of the walls $\dot{\epsilon} = \frac{1}{l} \frac{dl}{dt}$ is decomposed into the sum of a plastic and an
<figure><img src="image_4.png" /><figcaption>**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 proportional to the elastic deformation *ε*<sup>e</sup>; *E* is the elastic modulus.</figcaption></figure>

https://doi.org/10.1371/journal.pcbi.1007121.g001

elastic strain rate:

$$ \dot{\epsilon} = \phi^w (P - P^Y)_+ + \frac{1}{\bar{E}} \frac{dP}{dt}, \qquad (1) $$

where *P* is the hydrostatic pressure, the extensibility *φ*<sup>w</sup> (inverse of a viscosity) describes the ability of the cell to synthesize wall material, and *E* is an effective elastic modulus. Here, *φ*<sup>w</sup> and *E* both depend on cell wall thickness. The notation (*x*)<sub>+</sub> denotes *x* if *x* > 0 and 0 otherwise for any real number *x*.

**Water uptake.** Lockhart described water uptake by the cell as a flux through a semi-permeable membrane characterized by its surface *A* and its permeability *L*<sup>a</sup>. Assuming the membrane is perfectly impermeable to solutes, the rate of volume change is the result of a difference between the water potential *Ψ* of the cell and *Ψ*<sub>ext</sub> of its exterior [25]:

$$ \frac{dV}{dt} = AL^a(\Psi_{ext} - \Psi), \qquad (2) $$

The cell water potential *Ψ* = *P* − *π* results from the antagonistic effect of the cell hydrostatic pressure *P* that tends to expel water from the cell and its osmotic pressure *π* that tends to attract water inside the cell. In the case of a single solute of concentration *c*, we have *π* = *RTc* where *R* is the ideal gas constant and *T* the temperature. Let us denote *φ*<sup>a</sup> = *Al*<sup>a</sup>*V* which has the
same dimension as $\phi^w$. Assuming that the fluxes occur mostly on the lateral surface, the ratio $A/V$ is constant in the configuration of an elongating cell (see S1 Text). After dividing (2) by $V$, we get the following expression for the relative growth rate $\dot{\gamma} = \frac{1}{V} \frac{dV}{dt}$:

$$ \dot{\gamma} = \phi^a(P^M - P). \qquad (3) $$

where $P^M = \Psi_{ext} + \pi$ quantifies the power of the osmotic pump: it is positive if $\pi$ is high enough to overcome the possibly negative water potential $\Psi_{ext}$ of the exterior of the cell. Growth ($\dot{\gamma} > 0$) implies $P < P^M$ and hence $P^M$ is an upper bound for turgor, above which the cell would lose water to the exterior. The additional condition for growth $P > P^Y$ (see above) requires $P^M > P^Y$: growth is possible only when the osmotic pump is able to overcome the mechanical resistance of the walls.

In order to keep the analysis as simple as possible, we take here and in the remaining of the article $P^M$ constant with time and homogeneous among the cells, which corresponds for instance to constant $\pi$ and $\Psi_{ext}$. This choice will be commented in the discussion section.

**Coupling hydraulics and mechanics for a single cell.** Equating the expressions of strain rate $\dot{\epsilon}$ from (1) and relative growth rate $\dot{\gamma}$ from (3) ensures that the requirements for water uptake and yield of the cell wall are simultaneously satisfied. This means that turgor $P$, that is present in both equations, has to be adjusted to satisfy both hydraulic and mechanical constraints. The resolution of the model is detailed in SI (see S1 Text). The time dependent solutions can be analytically determined and we find that $P$ and $\dot{\gamma}$ converge towards a stationary solution ($P^*, \dot{\gamma}^*$): first, $P^*$ writes

$$ P^* = \alpha^a P^M + (1 - \alpha^a) P^Y, \qquad (4) $$

where

$$ \alpha^a = \frac{\phi^a}{\phi^a + \phi^w} \in [0, 1] \qquad (5) $$

measures the relative importance of $\phi^a$ compared to $\phi^w$. In the limit $\phi^a \ll \phi^w$ ($\alpha^a = 0$), any excess of turgor above the threshold is relaxed by cell wall synthesis and turgor is minimal at $P = P^Y$. Conversely, in the limit $\phi^w \ll \phi^a$ ($\alpha^a = 1$), the wall synthesis is not able to relax turgor, which reaches then its maximal value $P = P^M$. Second, the expression of the relative growth rate is:

$$ \dot{\gamma}^* = \frac{\phi^a \phi^w}{\phi^a + \phi^w} (P^M - P^Y), \qquad (6) $$

or equivalently: $P^M - P^Y = (\frac{1}{\phi^a} + \frac{1}{\phi^w}) \dot{\gamma}^*$ This equation is the analog of Ohm's law $\Delta U = (R_1 + R_2)I$ with two resistors $R_1 = 1/\phi^a$ and $R_2 = 1/\phi^w$ in series: growth can be limited by either hydraulic conductivity or wall synthesis.

**Link with wall rheology.** Wall expansion law (1) can be equivalently described as a function of wall stress $\sigma$ rather than cell turgor $P$: in the geometry of the Lockhart-Ortega model, we find (see S1 Text for the calculations) $P = 2 \frac{w}{h} \sigma$, where $w$ is the width of the walls and $h$ their height. Thanks to this relation, (1) translates into $\dot{\epsilon} = \frac{1}{E} \frac{d\sigma}{dt} + \Phi^w(\sigma - \sigma^Y)_+$, where $E = \frac{h}{2w} \bar{E}$ (resp. $\Phi^w = \frac{2w}{h} \phi^w$) is the intrinsic elastic modulus (resp. extensibility) of the walls. Let $\epsilon^e = \sigma/E$ be the so-called elastic deformation of the walls. It is dimensionless and can be measured from the image analysis of experiments, without the knowledge of the elastic modulus.
The wall rheology is then described as follows:

$$ \dot{\epsilon} = \frac{d\epsilon^e}{dt} + \Phi^w E(\epsilon^e - \epsilon^Y)_+, \qquad (7) $$

where $\epsilon^Y = \sigma^Y/E$ is the threshold elastic deformation. Note that $\frac{1}{\Phi^w E}$ can be interpreted as the characteristic time of wall synthesis.

## Multidimensional and multicellular models

A multicellular extension of the Lockhart-Ortega model adapted to the study of morphogenesis requires first to relax the constraint of uniaxial growth and allow multidimensional geometries, and second is complexified by the possibility of fluxes between cells. We study separately the effect of each of these extensions before presenting the complete model.

**First extension: Multidimensional growth.** In order to keep the analysis as simple as possible, we study here the expansion of a single 2D cell whose shape is a regular polygon with $n$ edges (see Fig 1c). This model allows to evaluate the effect of a varying surface/volume ratio compared to the Lockhart-Ortega model where this ratio is constant. The fluxes are described in the same way as for Lockhart's model (2) but wall synthesis is described with (7), as a function of elastic deformation instead of turgor. We find (see S2 Text for detailed calculations) that the relation between cell turgor and wall stress becomes $P = \frac{w}{R \cos(\pi/n)} \sigma$ where $R$ is the cell radius. In contrast with the Lockhart-Ortega model, the ratio $P/\sigma$ is no more constant as the cell grows, and the turgor vanishes at long times if the stress remains in the order of magnitude of the threshold. Note also that for a given stress the turgor decreases with the number of edges $n$. Therefore, the yield turgor $P^Y$ depends both on $n$ and $R$ and is not a well defined parameter. It suggests also that cells with less neighbors should have a higher turgor, as experimentally observed in [21, 22].

The prediction of growth rate requires a numerical resolution of the model (see S2 Text). The parameters are chosen to ensure a turgor of the order of 0.5 MPa and a relative growth rate of the order of 2% per hour, using the predictions (4) and (6). First let's examine the case of a cell of initial radius $R = 10\mu m$ for which wall synthesis is the limiting factor to growth (case $\alpha^a = 0.9$ in S2 Text). We find that it results initially in an accelerating growth (the bigger the cell, the faster the growth), much faster than predicted by the Lockhart model, during which the elastic deformation of the walls can reach values up to 20%. The ratio area/surface = $1/R$ decreases with growth and there is less and less water available compared to the volume; as a consequence, the relative growth rate vanishes at long times after this initial accelerating phase.

In the case where the fluxes are already limiting in the initial state (case $\alpha^a = 0.1$ in S2 Text), the initial behavior is closer to the predictions of the Lockhart model but the relative growth rate still vanishes at long times.

Altogether, these results show that a non constant surface/volume ratio deeply modifies the behavior of the model compared to the Lockhart model. In particular, flux and wall synthesis as limiting factors for growth are no more equivalent.

**Second extension: Multicellular growth.** Then, we study a simple multicellular extension of the Lockhart-Ortega model where two elongating cells $i = 0, 1$ are in contact through one of their walls (see Fig 1b). The cells can absorb water from their lateral surface and in the meantime exchange water with each other through their common wall. We look for stationary solutions: $\frac{dP_i}{dt} = 0$ and $\frac{1}{V_i} \frac{dV_i}{dt} = Cst$.

We set for both cells a common value of $P^M$, $L^a$ and $\phi^w$, while the value of the yield turgors $P_i^Y$ can differ; this corresponds for instance to a heterogeneity of wall elastic modulus or yield
deformation. For the sake of convenience, we refer to fluxes between cells as symplasmic fluxes, characterized by a water conductivity $L^s$, and to fluxes from the water source as apoplasmic fluxes, characterized by the water conductivity $L^a$. The biological relevance of this choice is addressed in the discussion section. Assuming that the symplasmic fluxes occur through plasmodesmata that are permeable to both water and solutes, the flux equation writes

$$ \frac{dV_i}{dt} = A_i L_i^a (P^M - P_i) + A_{01} L^s (P_j - P_i), \quad i \in \{0, 1\}, $$

where $j = 1 - i$, and $A_{01}$ is the surface of the common wall of cells 0 and 1. We introduce the number $\phi^s = 2A_{01}L^s/V_i$ which has the same dimension as $\phi^a$ and $\phi^w$. In order to allow an analytical resolution of this set of equations, we assume $\phi^s$ to be constant with time, and consider it in this section as a parameter of the model. Thus, we have

$$ \dot{\gamma}_i = \phi^a (P^M - P_i) + \frac{\phi^s}{2} (P_j - P_i). \quad (8) $$

We introduce the dimensionless number

$$ \alpha^s = \frac{\phi^s}{\phi^s + \phi^a} \in [0, 1] \quad (9) $$

which represents the relative importance of symplastic fluxes with respect to apoplasmic ones. We combine this flux equation with the growth Eq (1) and find analytical solutions for any values of the parameters (see S3 Text). We use here the following set of control parameters:

$$ P^M, P_i^Y, \dot{\gamma}_0^*, \alpha^a, \alpha^s, $$

and fix the value $\dot{\gamma}_0^* = 2\% \cdot h^{-1}$; this way, the parameters space to explore is reduced to ($P^M, P^Y, \alpha^a, \alpha^s$). When $\alpha^s = 0$, the cells are completely isolated one from another and reach turgors $P_i^*$ and growth rates $\dot{\gamma}_i^*$ as predicted by the Lockhart model ((4) and (6)). In particular, the condition $P^M > P_i^Y$ ensures that each cell is growing. When $\alpha^s > 0$, the fluxes between cells modify this behavior. We restrict to the case $P_0^Y < P_1^Y < P^M$, which corresponds to less mechanical constraints on cell 0 than cell 1; therefore we can expect $P_1 > P_0$ and $\dot{\gamma}_1 < \dot{\gamma}_0$. The calculations show a complex non linear behavior that is illustrated in Fig 2, in which the parameters subspace ($\alpha^a, \alpha^s$) is explored for given values of $P_i^Y$ and $P^M$ (detailed calculations are provided in S3 Text). Let $\Delta P^Y = P_1^Y - P_0^Y > 0$ be the difference of the two yield turgors and $\bar{P}^Y = 0.5(P_0^Y + P_1^Y)$ their average; we also introduce the dimensionless number

$$ \rho = \frac{\Delta P^Y}{2(P^M - \bar{P}^Y)}. \quad (10) $$

Note that the hypothesis $P_0^Y < P_1^Y < P^M$ is equivalent to $\rho \in ]0, 1[$.

We find that the subspace ($\alpha^a, \alpha^s$) can be divided in two main regions separated by the curve $\alpha^s = \frac{1-\rho}{1-\alpha^a}$ (see Fig 2a): surprisingly, in the region $\alpha^s > \frac{1-\rho}{1-\alpha^a}$, only cell 0 is growing ($\dot{\gamma}_0^* > 0$, $\dot{\gamma}_1^* = 0$, and equivalently $P_0^* > P_0^Y, P_1^* < P_1^Y$). Hence, the growth of cell 1 is inhibited by fluxes with cell 0. Conversely, in the region $\alpha^s < \frac{1-\rho}{1-\alpha^a}$ both cells are growing ($\dot{\gamma}_i^* > 0$ and equivalently $P_i^* > P_i^Y$). The size of the region $\alpha^s > \frac{1-\rho}{1-\alpha^a}$ increases with $\rho$ and fills the whole square $[0, 1] \times [0, 1]$ when $\rho \to 1$; such values can be reached when $\Delta P^Y$ is large and / or $P^M$ is close to $\bar{P}^Y$.

More quantitatively, Fig 2d and 2e) show that $\dot{\gamma}_1^*$ is always below $\dot{\gamma}_1^*$, while $\dot{\gamma}_0^*$ is always above $\dot{\gamma}_0^*$ and can reach up to twice this value. Furthermore, maximal values of $\dot{\gamma}_0^*$ coincide with
<figure><img src="image_5.png" /><figcaption>**Fig 2.** Analytical resolution of the two cells model, properties of the solution in the parameters space $\alpha^a \times \alpha^e$; **a)** delimitation of the two zones $\dot{\gamma}_1 = 0$ and $\dot{\gamma}_1 > 0$: the red thick solid line $\alpha^e(\alpha^a) = \frac{1-p}{1-\alpha^a}$ corresponds to $\rho = 0.75$. The two black thin dashed lines correspond to the values $\rho = 0.5$ and 0.99. **b-c)** Turgors $P_0$ and $P_1$ for $\rho = 0.75$. **d-e)** relative growth rates $\dot{\gamma}_i/\dot{\gamma}_i^*$ for $\rho = 0.75$.</figcaption></figure>

https://doi.org/10.1371/journal.pcbi.1007121.g002
minimal values of $\dot{\gamma}_1$: this confirms quantitatively that the growth of the cell with less favorable mechanical conditions is slowed down if not inhibited by the growth of its neighbor. This shows also that the growth rate heterogeneity is amplified by fluxes.

Turgor heterogeneity is also affected by fluxes (see Fig 2b and 2c): when $\alpha^s$ is close to zero, the cells are hydraulically isolated and their turgors vary with $\alpha^a$ as predicted by Lockhart model (4), this is where the turgor heterogeneity is maximal. Conversely, when $\alpha^s$ is close to 1, there is no hydraulic resistance between the two cells and the two turgors are equal. Between these two limits, $P_0$ is only slightly affected and remains in the $[P_0^Y, P^M]$ interval; conversely, $P_1$ is dramatically affected as it shifts from the interval $[P_1^Y, P^M]$ when $\alpha^s = 0$ to the interval $[P_0^Y, P^M]$ when $\alpha^s = 1$. Therefore, as $P_0^Y < P_1^Y$, there is a region where $P_1 < P_1^Y$ which corresponds to the region $\alpha^s > \frac{1-\rho}{1-\alpha^a}$, where cell 1 is not growing.

Finally, we have seen that intercellular fluxes tend to increase (resp. decrease) growth rate (resp. turgor) heterogeneities; the cell with less mechanical constraints takes control and imposes its turgor, which can lead the other one to stop growing. The growing cell then benefits from the water resources of the other cell and its growth is all the more increased.

**Extension to a chain of cells.** To further explore and quantify the spatial range of this process, we extended our two-cells model (see S4 Text for detailed equations) to a chain of $2N + 1$ cells where the central cell has twice softer walls, and focussed on the propagation of the inhibition of growth on its neighbors. We numerically solved the corresponding system of differential equations; the exploration of the parameters space showed that the number $2N_i$ of inhibited cells scales with $\sqrt{I^s}$. We computed the prefactor $c$ (such that $N_i \approx c\sqrt{I^s}$) for values of $(\alpha^a, P^M) \in [0.05, 0.35] \times [0.51, 0.85]$ (the interval for $P^M$ is in MPa) and plotted its value in the $(\alpha^a, P^M)$ space (See S4 Text). This shows that the propagation of the inhibition is favored by low values of $\alpha^a$ and $P^M - P^Y$.

**Generalization: A multidimensional and multicellular model of growth.** We consider (see Fig 1d) a collection of $N$ cells that form a (non necessarily regular) 2D mesh with a fixed topology (distribution of neighbors) as is the case with plant tissues when no division occurs.

The cell walls rheology is described by the visco-elasto-plastic law (7) of the Ortega model and the fluxes toward a cell $i$ are described as in the simple multicellular model presented above:

$$ \frac{dV_i}{dt} = A_i L_i^a (P^M - P_i) + \sum_{j \in n(i)} A_{ij} L_{ij}^s (P_j - P_i), \quad (11) $$

where $n(i)$ is the set of neighbors of cell $i$, $A_{ij}$ is the area of the common wall with cell $j$, $L_{ij}^s$ its permeability (it is symmetric: $L_{ij}^s = L_{ji}^s$), and $L_i^a$ is the permeability of the lateral walls to the supply of water.

The last missing part to obtain a closed set of equation is the mechanical equilibrium, that allows to link cells turgors, walls tensions, and geometry. Contrary to the cases studied above, no explicit expression of turgors as a function of stresses can be obtained and the equilibrium has to be solved at each time step. Let $P_i$ be the turgor pressure in each cell $i$. The tissue being at every moment in a quasi-static equilibrium, pressure forces on wall edges and elastic forces within walls balance exactly at each vertex $v$:

$$ \frac{1}{2} \sum_{k \in f(v)} \Delta_k P S_k n_k + \sum_{k \in f(v)} E_k \epsilon_k^e s_k e_{k,v} = 0, \quad (12) $$

where $f(v)$ is the set of faces adjacent to junction $v$, $\Delta_k P = P_{k_1} - P_{k_2}$ is the pressure jump across face $k$, with $k_1 < k_2$ being indices of the cells across face $k$, $S_k = hl_k$ is the area of the face $k$ on
which pressure is exerted, $n_k$ is the normal vector to face $k$, oriented from cell $k_1$ to cell $k_2$, and
$s_k = hw$ is the cross-section area of the face, on which the elastic stress is exerted; finally, $e_{k,v}$ is
the unit vector in the direction of face $k$, oriented from junction $v$ to the other end of face $k$.

## Coupling mechanical and hydraulic models

In the Lockhart-Ortega model, the compatibility between wall enlargement and cell volume
variation is automatically enforced through the geometrical constraint of uni-directional
growth that leads to the identity between the relative growth rate of the cell and the strain rate
of the walls. In contrast, in the multicellular model, this identity is no longer true. One has to
solve the closed set of Eqs (7)–(11) and (12) with respect to the unknowns X, P, and $\varepsilon^e$.

Despite its apparent simplicity, the problem to be solved is not straightforward as water fluxes
induce potentially long range interactions. In this respect, it differs from most vertex-based
models (e.g [11, 26]) where turgor is an input of the model. The numerical resolution required
the development of an original algorithm (see S5 Text) implemented in an in-house code.

## Numerical experiments: Growth of primordia in the shoot apical meristem (SAM)

The properties of this model cannot be as thoroughly studied as those of the simpler models
presented above, first because of the numerical cost of the resolution, but above all because it
allows an infinite variety of geometries and spatial distribution of its parameters. We present
here a numerical experiment that illustrates on the one hand how the properties of the simple
multidimensional and multicellular submodels are combined in the generalized model; in turn
the study of these models helps us to anticipate the properties of the generalized model. And
on the other hand, we show that this model is readily applicable to the study of systems of bio-
logical interest.

Growth heterogeneities can be triggered by the local modulation of the mechanical properties
of the cell walls [27]. In SAMs, new organs are initiated by a local increase in growth rate that
leads to the appearance of small bumps. Measurements show that physico-chemical properties
of walls are modified so that mechanical anisotropy and elastic modulus are decreased. Our 2D
model is used to represent a cross section of a SAM and we explore what effect a local softening
of the walls has on growth rate and turgor heterogeneities; based on our previous analysis of the
model in simple configurations, we expect that the growth heterogeneities will be maximal for
parameters such that the growth is restricted by fluxes rather than wall synthesis ($\alpha^a$ close to 0),
cell-cell conductivity is large ($\alpha^s$ close to 1), and the walls deformations are just above the growth
threshold, which can be enforced by a low value of the osmotic pressure (yet large enough to
ensure growth). The set of parameters (REF) is chosen according to these criteria; then we
explore the effect of a higher $\alpha^a$ ((ALPHA+) set) and lower cell-cell conductivity ((CC-) set)
that should both decrease the growth heterogeneities, and also test the effect of a lower osmotic
pressure ((PM-) set) that should conversely increase the growth heterogeneity. See S6 Text for
detailed explanations on the values of the parameters corresponding to these sets.

We build a mesh made primarily of hexagons (see Fig 3a) and first let it grow with homogeneous parameters until the elastic regime ends and plastic growth occurs. Then we divide by two the elastic modulus of a small group of cells (marked with a white star in Fig 3a) that will be referred to as “bump cells” thereafter. First, Fig 3b shows that the multicellular system grows globally in the same way as the single hexagonal cell studied above; it diverges from the Lockhart predictions because the ratio A/V of the cells is not constant: the (ALPHA+) simulations exhibit a very large initial growth rate that decreases only when the cells are so large that water fluxes become limiting. The (PM-) set leads to a roughly twice lower growth rate
<figure><img src="image_6.png" /><figcaption>**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+).</figcaption></figure>

https://doi.org/10.1371/journal.pcbi.1007121.g003
than (REF). The set (CC-) leads to the same dynamics at the tissue level as (REF), because the total influx of water is not affected by fluxes between cells in this setup.

Then we turn to the observation of heterogeneities: we focus on the differences between the bump region and the rest of the tissue. For all the parameters sets, Fig 3c shows that turgor is in general lower in bump cells, but the gap varies depending on the parameters, as it has been predicted by the study of the two-cells model: compared to (REF), the heterogeneity in turgor is increased by a lower cell-cell conductivity (set CC-), and decreased by a larger value of $\alpha^a$ (set ALPHA+). Decreasing the value of $P^M$ (set PM-) does not alter much the turgor heterogeneity compared to (REF). The maps of turgor (Fig 3e, 3g, 3i and 3k) confirm visually these observations.

Fig 3d shows the time evolution of $\dot{\gamma}/\dot{\gamma}^*$ where $\dot{\gamma}^*$ is the relative growth rate predicted by the Lockhart model (see (6)); its value is 2% $h^{-1}$ for (REF), (CC-) and (ALPHA+), and 0.5% $h^{-1}$ for (PM-). In the considered time frame, the relative growth rate of bump cells is always higher except for (ALPHA+): after an initial fast increase where bump cells grow faster, the tendency is inversed at $t \approx 20h$ because the bump cells have grown so much that fluxes become limiting. In the (REF) simulation, while the growth rate of non bump cells is almost constant and close to $\dot{\gamma}^*$, the growth rate of the bump cells is up to 6 times $\dot{\gamma}^*$ at the beginning of the simulation and progressively decreases toward $\dot{\gamma}^*$. As a result of this large discrepancy, the bump region can be clearly distinguished from the rest of the tissue (Fig 3e and 3f). In (CC-), the growth rate of the non bump cells is close to that of (REF), but the growth rate of the bump cells is much lower (Fig 3d). As a result, the global shape remains convex and the bump is not clearly detached from the rest of the tissue (Fig 3i and 3j). Note that (CC-) corresponds to a lower value of $\alpha^s$ compared to (REF), which corresponded to a lower growth heterogeneity with the two-cells model studied above; this is also confirmed by the lower cell-cell fluxes towards the bump cells for (CC-), see the arrows in Fig 3e and 3i. The (ALPHA+) simulation exhibits also a convex shape (Fig 3k and 3l); it corresponds to a larger value of $\alpha^a$ than (REF), and similarly to the two-cells model studied above, the growth rate heterogeneity is lower than (REF). Finally, the set (PM-) corresponds to an increase of the dimensionless parameter $\rho$ (see (10)), and accordingly to an increase in growth rate heterogeneity as can be seen with Fig 3d. Consequently, the bump region can be clearly distinguished from the rest of the tissue, even better than (REF) (Fig 3g and 3h); moreover, the growth of the cells close to the bump seems to be inhibited by fluxes as explained in the two-cells model described above and further explored below.

**Flux-based lateral inhibition predicted by the model.** As we saw, cells that benefit from better mechanical conditions for growth (in the present case a lower elastic modulus) have a lower turgor than the other cells, and therefore attract water from them. Not only does it amplify their growth but it also inhibits the growth of their neighbors. Such a lateral inhibition mechanism is important for morphogenesis, as it allows very large growth rate heterogeneities and the appearance of well differentiated shapes (in the present case the appearance of a bump on the surface of the meristem). The efficiency of this mechanism varies depending on the position in the parameters space, as predicted by the study of the simplified models (see Fig 2): for instance it is increased if the cell-cell conductivity $L^s$ (or equivalently $\alpha^s$) is increased (see Fig 4a-4d); even the whole tissue can be inhibited. Inhibited cells can also relax the tension of their walls and decrease their volume (see Fig 4a).

# Discussion

## A minimal model with a complex and rich behavior

The model proposed in this article is a minimal multicellular and multidimensional extension of the Lockhart 1-D single cell model; it can be regarded as a conceptual tool to study the
<figure><img src="image_7.png" /><figcaption>**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 $\alpha^s = 0.95$, (PM-) with $\alpha^s = 0.99$. $V_0$ 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 $\alpha^s = 0.95$, $t = 10h$ for (PM-) and $\alpha^s = 0.99$.</figcaption></figure>

https://doi.org/10.1371/journal.pcbi.1007121.g004

interplay between fluxes and wall mechanics in a multicellular tissue. Wall expansion is mod-
eled with a visco-elasto-plastic rheological law, while fluxes derive from water potential gradi-
ents. These two contributions are integrated into the mechanical equilibrium and interact
through the pressure term. Contrary to most previous approaches, turgor is not an input of the
model but a variable that adjusts simultaneously to mechanical, hydraulic, and geometrical
constraints. First of all, this leads to a physically consistent representation of turgor: for
instance, the model predicts that cells with softer walls have a lower turgor. Moreover, this has
deep implications at tissue level: in the previous example, lower turgor is associated with a
faster growth which can be itself amplified by fluxes that follow decreasing pressure gradients.

Thanks to the simplicity of the model, the predicted behavior can be analyzed and inter-
preted with two submodels built from the Lockhart model: first, in a 1-D system, cells are
only elongating and their surface-to-volume ratio is constant. We thus extended the Lockhart
model in two dimensions, where cells have more degree of freedom to change their shape. In
particular their allometric surface-to-volume ratio may then vary. This new possibility induces
additional complexity in the tissue development as the rate of growth of cell surfaces may
become a limiting factor for growing cells. Second, a 1-D multicellular submodel was build
with two or more side-by-side cells; it was used to study the growth of competing cells with
heterogeneous properties. Key ingredients here are the wall synthesis threshold, the fact that fluxes and growth can relax turgor, and cell to cell fluxes that allow long range interactions.

## A potentially new type of lateral inhibition mechanism

Depending on mechanical and hydraulic parameters of tissue regions, the model exhibits different growth regimes corresponding to either uniform or differential growth. One unexpected consequence of such an hydraulic-mechanical coupling at the tissue level is the observation that in certain regions of the parameters space where cell-to-cell hydraulic exchanges are non-limiting, growing tissue may exert an inhibiting influence on the growth of neighboring regions. This may be interpreted as a lateral inhibition mechanism. It has for long been recognized that lateral inhibitory mechanisms play a key role in setting some morphogenetic patterns in procaryotes (e.g. [28]), animals (e.g. [29, 30]) or plants (e.g. [31, 32]). Lateral inhibition operates in these systems via chemical signals, such as delta-notch in animals or auxin in plants. Our model predicts the existence of a novel type of lateral inhibition mechanism based on the coupling between mechanics and water fluxes. Previous observations of tissue growth suggest that such a phenomenon may occur in real tissues. In the shoot apical meristem for instance, detailed quantification of growth with cellular resolution indicates that the region surrounding primordia growth may have a negative growth rate ([33], Figs 2G and 3K). According to our model, this decrease of volume in boundary regions might be due to the primordium growth attracting locally most of the water supply and depriving lateral regions from water, and thus conforts the hypothesis of a new hydraulic-mechanical component of primordium lateral inhibition, beyond already identified auxin and cytokinin signals [34].

## Model simplifications and further potential extensions

Throughout the development of the model, we made several key choices concerning the abstraction of a multicellular plant tissue. First, our model was developed in 2-D for reasons of computational efficiency. In principle, it can be extended in 3-D, though at the expense of more complex formalism and implementation. Second, the current model considers that water transport is performed in the plant tissue through two conceptually different pathways [1]. Water can first move within the apoplasmic compartment between the cells and finally enter a cell. Water can also move locally from cell to cell. This movement includes itself conceptually both symplasmic movements (water circulates between cells through plasmodesmata without crossing membranes) and movements from cell to cell with intermediate steps in the wall (water is for example exported locally out of the cell by water transporters like aquaporins into the wall and immediately re-imported by water transporters into neighboring cells). For the sake of simplicity in this first analysis, we represented the apoplasm as a single abstract compartment able to exchange water with every cell. To analyze precisely the effect of water transporters and their genetic regulation or to assess the impact of wall resistance to water movement in the processes, explicit spatial representation of the apoplasm, of plasmodesmata and of membrane water transporters could be integrated into the model in the future.

Finally, we considered a simplified situation here by imposing constant cell osmolarity. Allowing osmolarity variations (for instance higher values in faster growing regions) may impact turgor distribution (e.g [35]). However, this should not affect the ability of the system to build up growth heterogeneities. Similarly, we further simplified our model by keeping constant the apoplasmic water potential. Relaxing this hypothesis would increase cell-cell water fluxes (via the apoplasm) and could also shift the model in the direction of the flux-limiting regime. This would therefore favor regimes where growth heterogeneities are amplified by fluxes.
**Biological relevance of the model**

This model may impact our understanding of various biological questions at the interface between mechanics and hydraulics in plants thanks to its emergent properties that are far more complex and rich than the Lockhart model it is based on. We showed here the impact of fluxes on turgor and growth rate heterogeneities at tissue level and how they can impact morphogenesis. In a recent study [22], we focused on heterogeneities at cell level and compared the model to experimental measurements; in particular, we correctly predicted that the number of cell neighbors is negatively correlated with cell turgor. Finally, based on its ability to provide quantitative insights in growing multicellular systems, this model could contribute to revisit the long-standing debate initiated by Boyer and Cosgrove regarding the relative importance of fluxes and wall softening in the limitation of growth in plants.

**Supporting information**

**S1 Table. List of main notations.**

(PDF)

**S1 Text. Calculations for the Lockhart / Ortega model.** The relation between turgor *P* and stress σ is derived from the mechanical equilibrium and the time-dependent solution is explicitly computed.

(PDF)

**S2 Text. Calculations for a single polygonal cell.** The relation between turgor *P* and stress σ is derived from the mechanical equilibrium as a function of cell size *R* and number of edges *n*. The time-dependent solution is numerically computed.

(PDF)

**S3 Text. Calculations for the two-cells model.** Detailed calculations of the explicit solution of the steady-state equations.

(PDF)

**S4 Text. Calculations for a chain of connected cells.** Detailed corresponding equations and numerical computation of the time-dependent solution.

(PDF)

**S5 Text. Numerical resolution of the 2D multicellular model.** Detailed presentation of the numerical algorithm.

(PDF)

**S6 Text. Parameters used for the bump simulations.** Detailed explanations for the choice and definitions of the parameters sets used for the bump simulations.

(PDF)

**Author Contributions**

**Conceptualization:** Ibrahim Cheddadi, Michel Génard, Christophe Godin.

**Formal analysis:** Ibrahim Cheddadi, Christophe Godin.

**Funding acquisition:** Nadia Bertin.

**Investigation:** Ibrahim Cheddadi, Christophe Godin.

**Methodology:** Ibrahim Cheddadi.
**Project administration:** Christophe Godin.

**Software:** Ibrahim Cheddadi.

**Supervision:** Christophe Godin.

**Validation:** Ibrahim Cheddadi.

**Visualization:** Ibrahim Cheddadi.

**Writing – original draft:** Ibrahim Cheddadi, Christophe Godin.

**Writing – review & editing:** Ibrahim Cheddadi, Michel Génard, Nadia Bertin, Christophe Godin.

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