Multicellular growth involves both cell autonomous and non-cell autonomous forces

Similarly to isolated cells, cells in plant tissues are growing under the action of forces that stretch their wall and make it yield. However, for single isolated cells, the only significant forces able to trigger growth are the ones generated by its turgid cytoplasm pushing on the cell wall (Fig. 2A). In a multicellular context, the situation is somehow complicated by the fact that cells are rigidly connected to each other. The deformation of one cell generates physical constraints on its neighbors and vice versa. It has been recognized by Sachs as early as in 1882 (cited by Kutschera [4]) that epidermis cells in plant tissues are experiencing external forces due to the inner turgid cell layers pushing outwards against the surface cells, and inducing tension stresses in the epidermis (Fig. 2B). In a plant tissue, the mechanical stresses undergone by one cell wall are thus not only due to their own turgid cytoplasm, that we call cell autonomous stresses, but also comprise the physical constraints imposed by the neighboring cells, called the non-cell autonomous stresses.

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Figure 2. Origin of forces driving growth in a multicellular tissue.

(A) In the single-cell case, the mechanical (elastic) stresses (, dark blue double arrows) undergone by the cell wall are due to the inner pressure (, light blue single arrows) of the cell. The mechanical equilibrium within this wall is regulated by the cell itself. (B) In a tissular case, (here a shoot apical meristem), mechanical stresses within the outer cell walls of the L1 layer (light red cells), can be modulated by remote cells (here in light green). In this case the stem (light blue cells) plays the role of a base on which the inner cells rely in order to push the L1 layer upward. (C) Three main modalities of growth can be considered in a multicellular context (details on the stresses equilibrium within the outer cell wall are represented in the zooming views). From an initial state (C1) of the growing tissue three scenarios are considered: (C2) & (C3) present cell-autonomous ways where growth of a given cell is triggered by an increase of its inner pressure or a modulation of its wall mechanical properties respectively. (C4) represents a non-cell-autonomous case in which growth of the studied cell is initiated by physical alteration of its neighbors. (C5) All three modifications result in the local outgrowth of the considered region.

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Genes can regulate growth by controlling processes that modulate these stresses. Cell autonomous stresses can be modulated directly by the cell itself. This can be done in two ways: either by increasing the cell's own turgor pressure or by modulating the mechanical properties of its wall (e.g. by changing its elasticity or its yielding threshold). In contrast, non-cell autonomous stresses reflect distant mechanical interaction between cells where genes are being expressed and the ones where the consequences of this expression are observed.

Let us illustrate these different possibilities in the context of a primordium outgrowth on a growing meristematic dome, see Fig. 2C. As discussed above, any specific cell of the epidermis is suject to three types of stresses: i) the stress induced by the cell's inner turgor pressure on the wall ii) the stress resulting from the action of the rest of the tissue on the cell wall and iii) the stress due to the cell wall elastic deformation in reaction to the other stresses. In growing tissue, mechanical equilibrium leads to a balance between these three stress components (Fig. 2C1):(2)

In a cell-autonomous perspective, regulation of the inner pressure of the cell would lead to a change in (Fig. 2.C2) and modifying the mechanical properties of the wall would directly impact the elastic term (Fig. 2.C3). Likewise, modulating (e.g. by changing the turgor pressure of some neighboring cells) would correspond to a non-cell autonomous regulation of growth, (Fig. 2.4). Interestingly, new experimental evidence of the possibility of such a non-cell autonomous regulation has recently been reported by Peaucelle et al. [20].

The situation is complex as, if one of these stress components is affected by some regulation mechanism, all the other stresses will in turn be affected. Moreover, in real plants several of these regulations may be triggered at the same time, leading to even more complex interactions between regulation and growth. Therefore, to understand the growth of multicellular tissues, one needs to model both cell autonomous and non-cell autonomous growth. How can this be achieved?

Computational framework to model local deformations in tissues

Both cell and non-cell autonomous growth rely on turgor-generated forces that are directly translated into mechanical stresses within the cell walls. Therefore considering a mechanical stress-based growth mechanism ensures that both types of growth are taken into consideration. Depending on their mechanical properties, the cell wall deform in response to the direction and intensity of these stresses. We assume that each small wall region of each cell in the tissue, at any time , has a rest shape, i.e. the shape that the region would have if isolated from the rest of the tissue. Under the effect of the tissue stresses due to turgor pressure and connection to other cells, each small region is elastically deformed with respect to its rest shape. For the sake of simplicity, we assume that the thickness of the wall is kept at a constant value during growth (we do not model the details of the wall remodeling process itself). We also assume that the thickness of the walls has no major mechanical effect at this scale of analysis and therefore can be integrated in the wall's in-plane properties. Then, if the region is chosen sufficiently small, the wall deformation can be assimilated to an affine transformation and represented by a matrix called the deformation gradient (Fig. 3A-B, see Model section for mathematical details). From , it is easy to compute the region strain :(3)

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Figure 3. Formalization of plastic growth of a small region of wall.

A tissue region is in general observed as a deformed object in a real tissue (A) due to local stresses internal to the tissue (light blue arrows). Taken outside its tissue context, without any stress on its borders, the region has a rest shape (B). Note that this rest shape is not actually observed. The transformation matrix to pass from the rest shape to the observed deformed shape is denoted . Due to changes in stress distribution in time, at a subsequent date the stress configuration acting on the region changes (dark blue arrows) and induces a new deformation of the region (C). If the intensity of the elastic deformation between the former rest shape (B) and the new deformed object (C) is above a certain threshold, then plastic growth is triggered: the rest shape is remodeled by the cell by adding material to the wall (D) which reduces the elastic strain. This change is made according to a constitutive rule that describes the material plasticity (see Model section below). As a result, the transformation from the old rest state (B) to the new deformed state has been decomposed as a product of a reversible term and an irreversible term representing growth.

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where denotes the transpose matrix of and the identity matrix. The fact that this strain is the elastic response of the material to tissue stresses is described by a constitutive law of the wall material. In the simplest case, it is a linear relation between elastic strain and stress corresponding to a generalized Hooke's law in 3 dimensions:(4)

where is a matrix representing the stress on the small region and is an order-4 tensor expressing the local elastic properties of the material. In particular, the anisotropy of the material if any is encoded in coefficients.

The forces that act on a region may vary throughout time, notably through either direct or indirect genetic regulation. Regions may be subject to new stress distributions (Fig. 3C), inducing new strains. In the spirit of Lockhart equation for cell volumes (Eq. 1, we assume that if the strains get above some threshold, the walls start to yield and the cell to remodel them. As a consequence, the reference state of the region is modified irreversibly (Fig. 3D). A number of studies have proposed to model this process by using a multiplicative decomposition of the overall deformation [22] and then [23], [24] (Fig. 3B-C-D),(5)

where denotes the irreversible modification of the rest shape of the region, called the growth tensor, and is by a purely elastic deformation corresponding to the reversible part of the process.

The equation of growth

To describe growth we need a constitutive law that relates the rate of change of the matrix , called growth rate tensor, to physical processes. We assume this law to be strain-driven: above a certain deformation threshold, the rest configuration of a region changes at a speed proportional to the strain of the region. In terms of tensors, the simplest form of such a law can be expressed as (see Model section for details):(6)

where the left-hand term defines the relative rate of variation of the reference state, is a constant characterizing the rate at which walls yield (extensibility) and the components of the term correspond to non null tensor components in the directions where the elastic strain is above the threshold values encoded in . Replacing the strain components in Eq.6 by their stress counterpart using Hooke's law, leads us to the following generalized 3D Lockhart-like form:(7)

This equation shows how growth is related to key mechanical variables: the extensibility controlling the rate of growth, the elastic properties of the material , the turgor pressure that appears through the stress it induces in the tissue and , a plastic stress yielding threshold corresponding to the strain yielding threshold . Note that the mechanical properties of the material are taken into consideration through a rigidity tensor which allows the plastic deformation evolution () to be non-collinear to the plastic stress ().

The above equation describes the plastic deformation evolution of a small region of the tissue, typically a part of a wall, during a small amount of time. To compute the deformation of the whole tissue during development, we need to integrate these local deformations over the whole tissue and throughout time, so that all these elements are assembled in a symplastic manner in the deformed tissue. This computation is made by minimizing the global mechanical energy in the tissue (see Model section). The strain and stress configuration of each region will thus be chosen so that the mechanical energy is minimal among all possible combinations of local elastic deformations that preserve the integrity of the tissue and the adjacency of cell walls. By contrast, the rest configuration of each individual small region is not necessarily compatible with that of other regions [25], i.e. there may not be physical continuity between rest configurations. The integration is carried out using a finite element method - FEM - (see Model section below).

A comparative analysis of the putative mechanisms behind organogenesis at the shoot apical meristem

We next used our modeling framework to analyze organogenesis at the shoot apical meristem (SAM). The SAM is a population of stem cells that continuously initiates new stem tissues and lateral organs, thus generating all the aerial parts of the plant. We first constructed a model of the SAM as a dome made up of polyhedra representing the 3-D cells and rigidly connected to each other (Fig. 4A). The faces of these polyhedra represent cell walls and are composed of 2-D elastic triangular elements whose mechanical properties are represented by tensors . The stiffness of these elastic triangles is set higher in the epidermis walls than in the inner walls and may be either isotropic or anisotropic for epidermis triangles. We assume that cells are inflated with a uniform turgor pressure , and that triangle mechanical properties are all initially isotropic and uniform.

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Figure 4. Growth regulation mechanisms and their impact on shape development.

(A) Face, top and inside view of an artificial dome made of cells with mechanical properties. The transversal cut shows the inner cells. The basal faces of cells shown in blue here are constrained to keep in a horizontal plane. (B-E) Growth of a multi-cellular dome. In all the simulations, the gray scale code on the initial dome represents regions with different rigidities. A different color code is then used on the other steps to figure mechanical stress intensity, c.f. color scale on the top right corner. (B) Homogeneous dome: all cells are isotropic with identical elasticity, plasticity threshold and growth speed. (C) Mechanical anisotropy is imposed on the lower half of epidermis to model the effect of microtubules circumferential orientation. Axial growth emerges. (D) Analysis of the extent of the anisotropic zone on growth. From left to right: Initial state of the simulation with circumferential anisotropy imposed up to 80% of the dome height: The resulting growth is axial. Initial state with a dome anisotropy limited to 40% of the dome height: The corresponding growth is globular. (E) Growth with a gradient of circumferential anisotropy from the bottom to the top of the dome: The resulting growth is inbetween purely axial and isotropic. (F-J) Creation of a lateral dome. (F) The rigidity of the cells in a small region at the flank of the meristem is decreased (cell autonomous regulation). During growth a lateral bump starts to form. The simulated dome is shown at two time points (middle and right). (G) Transversal cuts of a dome showing tentative generations of a bump with non-cell autonomous stresses: (G-1) Decreasing wall rigidity (10-fold) in a group of inner cells (blue cells with i.e. low mechanical stress): No visible bump emerges; (G-3) Increasing the turgor pressure (3-fold) in the same group of cells (red cells i.e. high mechanical stress): A shallow bump emerges and inner tissues are compressed inside. Compare with the reference situation (G-2) corresponding to a transversal cut of F middle. (H) Similar to F, but cells surrounding the primordium region are made stiffer. A well marked dome appears (middle and right). (I) Similar to F, but cells surrounding the primordium region are made stiffer in the bump ortho-radial direction only (anisotropy in boundary region). (J) Simulation similar to H, combining a smaller decrease of rigidity with an increase of the walls synthesis rate (namely extensibility) in the primordium. Movies corresponding to each simulation are available as Supporting Information.

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In this initial configuration, the turgor pressure induces a stress that puts all the cell walls under tension. If the plastic growth threshold is reached, the dome grows isotropically in all the directions (Fig. 4B). This plastic deformation is accompanied by a decrease of inner pressure in cells as their volume increases due to growth. In vivo, this would lead to a difference in water potential between the cells within the dome and surrounding tissues, drawing new water from below the dome inside the meristematic cells. It is generally assumed that water uptake is combined with osmolyte regulation to maintain turgor relatively constant in the cell during growth [4]. As these processes are considered fast compared with growth, we simply assume in our model that, otherwise stated, the turgor pressure is continuously kept constant at .

A key feature of meristematic activity is the generation of cylindrical stems and roots. In principle there are several ways to generate axial structures, but there is overwhelming evidence that this is due to a switch from isotropic to anisotropic growth over several cell files. Indeed, when cells leave the meristematic dome, they start to generate cellulose microfibrils in highly ordered arrays, oriented along the circumference of the meristem [3], [26]. This microfibril organization creates a high circumferential rigidity that favors growth in the perpendicular axial direction. This mechanical anisotropy can be reproduced in the 3D model by imposing higher rigidity in circumferential direction using anisotropic tensors (Fig. 4C). Interestingly, the ability of the microfibrils to create well formed axes depends on the amount of microtubule anisotropy in the dome (Fig. 4D dark gray). If too small, a bulge forms at the tip (Fig. 4D). Intermediate shapes can be obtained by using a less sharp transition between isotropic and anisotropic regions, using e.g. a gradient of anisotropy between the bottom and the top of the dome (Fig. 4E).

Next, using this mechanical model of the SAM, we tested different scenarios for organ emergence. As discussed above, several processes can in principle account for this phenomenon, in a cell- or non-cell-autonomous way, including changes in turgor pressure, modifications in wall stiffness and modifications in the rate of wall synthesis.

Starting from the growing dome of Fig. 4E, we first tested the possibility to grow an organ on the dome assuming cell autonomous regulation. For this, we lowered the outer periclinal wall rigidity in a small region close to the tip of the main dome (Fig. 4F). This created a bulging zone in the corresponding region of the tissue. We then tested the alternative possibility to create a primordium in a non-cell autonomous manner, by relaxing the wall rigidity up to 10-fold in a group of cells immediately below the previous surface region. No visible bulges could be obtained in this way (Fig. 4G–1; compare with Fig. 4G–2 corresponding to a transversal cut obtained from the simulation of Fig. 4F middle that shows a much larger bump at the same stage of development). To further explore the ability of non-cell autonomous stresses to trigger bump outgrowth locally, we also tested the possibility to obtain a bulge by increasing (by up to a 3-fold factor) the turgor pressure in the same group of inner cells. Here again, no clear bulge could be obtained (Fig. 4G–3). In this latter case, we could observe in the transversal cut that the tissues were actually compressed internally. We concluded from this series of simulations that bumps can more easily be generated in the context of cell autonomous regulation.

In this context however, we could observe that the growing bumps were not clearly separated from the main dome (e.g. Fig. 4F). We therefore constrained a ring of cells at the surface, around the bump location, to be very rigid. This resulted in a well formed bump growing on the top of the initial dome (Fig. 4H). A similar separation between the two organs can be obtained by stiffening the cells of the ring only in the circumferential direction. Interestingly, the cells of the ring are also supporting high stress but are left free to grow in the axial direction of the new bump (compare right images in H and I). A well marked bump can also be created by a less important decrease of the primordium zone rigidity compensated by a local increase of the cell growth rate in this region (Fig. 4J).

The simulations showed how simple scenarios can in principle explain important morphogenetic processes during plant development, including the formation of cylindrical stems and roots and the outgrowth of lateral organs at the SAM. Importantly, although these simulations can lead to comparable shapes, they make different predictions regarding the mechanical properties and resulting growth patterns.

A case study: Morphogenesis of a flower bud

The simulations presented above were based on abstract versions of real meristems. We therefore next applied our computational framework to perform simulations from realistic templates. Hereby, we used the floral meristem of Arabidopsis thaliana - which has been very well characterized - as a case study. As a reference for model construction, we used a series of confocal stacks of the same young growing floral primordium taken at 24 h intervals from early stage 1 to stage 2 (3 time-points, Fig. 5A-B-C). Using the Mars-Alt pipeline [19], the individual cells were identified and cell lineages were tracked in the thus segmented reconstructions (Fig. 5D-E-F). The confocal images and 4D reconstructions suggest that in vivo, the primordium first grows out from the meristem as a small radial symmetric globular structure in a direction normal to the surface of the meristem. Then the global direction of primordium growth changes progressively and the initial symmetry around the normal to the surface breaks as the the abaxial region expands more rapidly than the adaxial region (Fig. 5G-H-I). The sepals appear with different growth rates as the abaxial and adaxial sepals grow much faster than the two lateral ones. A recent analysis of gene expression in the flower bud showed complex spatio-temporal expression patterns with as many as 16 different domains expressing different combinations of transcription factors [27]. This raises two main questions: (i) how many different gene activities would in theory be required to produce this structure? (ii) are mechanical-related actions of these genes alone sufficient to reflect for the observed shape evolution?

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Figure 5. First stages of development of a flower bud.

Upper part: (A-B-C) Transversal sections in the young outgrowing flower bud at time points separated by 24 h. (D-E-F) Automatic 3D segmentation of the corresponding confocal images using the MARS-ALT pipeline [19]. (G-H-I) The analysis of growth patterns shows that growth at the abaxial side is faster than at the adaxial side, causing the floral meristem to bend towards the SAM. Lower part: Different attempts were made to regulate the mechanical parameters in time so as to reproduce this differential growth behavior. On the left:representation of the zones used in the simulation (CZ  =  Central Zone, Fr  =  Frontier, Pr  =  Primordium, Ad  =  Adaxial zone, Ab  =  Abaxial zone, Pe  =  Periphery). For all the simulations, the rigidity was decreased (light gray) in Pr (relative to CZ and Pe, and in the anisotropic zone Fr, the direction of maximum rigidity was set ortho-radially to Pr. With such an initial configuration, a globular and symmetric dome emerges normal to the surface (J-K). Then by tuning the mechanical properties of the Ad/Ab regions we could obtain different asymmetric developments: increasing the rigidity of Ad cells (medium gray) resulted in a restricted development of the upper part of the primordium (L-M) while, by contrast, an increased rigidity of the Ab cells (medium gray) shifted the primordium development upwards (N-O) as expected. Finally a growing dome with correct development of the Ad/Ab regions could be obtained when the abaxial cells where also imposed a high degree of anisotropy (orientation shown by the thick black bars oriented circumferentially in the Ab, (P-Q)). The table under the snapshots illustrates the relative variations of Elastic modulus used for each case. The x and y coordinates respectively refer to the axial and circumferential directions, as exposed on sub-figures (J) and (K). Numerical values used in the simulations and corresponding movies are available as Supporting Information.

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To address these questions we investigated with our mechanical model how to reproduce, at first qualitatively, the developmental pattern described above. Hereby we distinguished two phases: (a) the initial outgrowth of the bud and (b) the formation of the sepal primordia. For this purpose, we made a number of assumptions based on the literature and on the first set of models described above.

First, we simulated the outgrowth of the flower bud using the following itinerary through successive scenarios:

i In the first scenario, we tested the possibility to initiate organ outgrowth by a combination of local wall softening in the incipient primordium and anisotropic wall stiffening in the organ boundaries. A number of experimental evidence supports such a scenario. The young primordium is characterized by relatively high concentrations of the plant hormone auxin [28], [29]. There is evidence, that auxin influences cell wall loosening by the activation of specific enzymes. For instance, in the hypocotyl ABP1 (Auxin Binding Protein 1) controls the level of expression of glycosyl hydrolases known to modify hemicellulose within the cell wall, [30]. At the apex, atomic force microscopy measurements suggested that auxin accumulation leads to cell wall softening via a PMEI3-dependent pathway [31]. Finally, transcription factors such as AINTEGUMENTA and MONOPTEROS which are homogeneously expressed in the young outgrowing flower bud are thought to regulate the expression of expansins and xyloglucan modifying enzymes [9], [10]. Organ boundaries are also characterized by specific gene expression patterns. In particular the CUP-SHAPED COTYLEDON transcription factors are strongly expressed between the meristem and the primordium and genetic studies show that they repress growth in this region (e.g. [32], [33]). In addition, Hamant et al. [3] found that in the same region, cells are likely to have highly anisotropic wall structure. Based on these observations and on preliminary simulations shown on Fig. 4I, we assumed the existence of a band of anisotropic cells around the primordium's upper half. As a result, the model produced a bulge normal to the surface with a quasi symmetric shape (Fig. 5J-K).

ii The next morphologically significant step in organogenesis happens when differential growth behavior between the adaxial and the abaxial regions emerges. At stage 2 (or P2) cells of the abaxial side of the primordium start to expand faster in the meridional direction than the ones from the adaxial region, (Fig. 5G-I, see also [34]). To account for that differential growth in our simulations, we assumed that mechanical properties are differentially regulated between the abaxial/adaxial regions. Although there is no strict evidence for differences in mechanical properties, both regions are characterized by specific gene expression patterns early on. For instance, transcription factors such as FIL, KANADI, YABBI or ARF3 and ARF4 are part of interlocking pathways necessary for organ polarity and proper development [35], [36], for reviews see also e.g. [27] and references therein. To account for the faster growth rate in the abaxial region, we first assumed that the abaxially expressed genes caused a lower rigidity in that region (Fig. 5L-M). This indeed resulted in a developmental abaxial-adaxial asymmetry. However, the organ tended to develop downwards, opposite to what is observed in vivo.

iii To correct this behavior, we set rigidities in the primordium zone in the opposite way, i.e. with higher rigidity now in the abaxial region. The summit/adaxial region was now developing faster toward the upper direction but, contrary to observations in actual development (Fig. 5G-I), with little shape asymmetry (Fig. 5N-O). Indeed, in vivo observations carried out by e.g. Long et al. [34] and Fernandez et al. [19] were reporting anisotropic growth in the abaxial side of the floral primordium. The discrepancies in simulations of Fig. 5N-O could finally be corrected by setting a high rigidity anisotropy in the abaxial cells. (Fig. 5P) The dome asymmetry was then correctly oriented upward (Fig. 5Q), with fast development of the abaxial region upwards as expected.

Following on the simulated flower bud growth, we next simulated the outgrowth of sepal primordia (Fig. 6A-B). To initialize the FEM simulations we constructed a triangle-mesh of the floral meristem at stage 2 based on the segmented image. The vertices of triangles coincide with the cell vertices (Fig. 6C). We then reused the mechanical setting of this model to grow the four sepal primordia on the outgrowing dome obtained in phase (a). In vivo, this stage is characterized by auxin accumulation at the four sites of sepal initiation [37], followed by the activation of genes that promote organ outgrowth - in particular AINTEGUMENTA - first in the abaxial and adaxial sepals, then in the lateral ones (Fig. 6B). Simultaneously, organ boundary genes are activated at the flower meristem boundary and in between the primordia. As for the flower bud simulations presented above, we translated these gene expression domains into zones with different mechanical properties, i.e. wall loosening in the primordia and a anisotropic stiffening in the boundaries as described above. Again the rigidity in the abaxial regions was set higher than in the adaxial ones. We also assumed that the initiation of lateral sepals was slightly delayed with respect to the adaxial/abaxial pair. Based on initial geometric structures obtained from confocal microscopy, the resulting simulation was able to reproduce the developmental dynamics of a flower bud and its first lateral organs based on a spatio-temporal synchronization of the changes in mechanical properties of regions (Fig. 6D).

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Figure 6. (A) Transverse sections of confocal images showing floral bud development between stage 1 and early stage 3.

Abaxial sepals start to grow out first (middle and right image). (B) Growth patterns and gene expression profiles. The respective development of the different zone is indicated by small bars at the meristem surface. This growth pattern is accompanied by a change in gene expression patterns. At stage one, the floral bud is characterized by adaxially (light blue) and abaxially (dark blue) expressed genes. Other genes such as LFY and ANT are first expressed throughout the young flower. When the sepals start to grow out abaxial and adaxial domains are again established in these young organs (resp. dark and light pink), characterized by specific expression patterns (e.g. REV or FIL). Other genes, such as ANT or AHP6 will finally remain active throughout the pink zones that will generate the sepals (dark and light pink). Boundary zones, characterized by genes like CUC (red) separate the primordia from the meristem proper, where genes like STM (green) are active. For review of expression patterns see [27]. (C) Creation of a 3D geometric model of a flower bud. From left to right: confocal image; automatic cell segmentation using Mars-Alt pipeline [19]; construction of a mesh based on cell vertices; transverse section of the mesh showing the geometric representation of the inner layers. (D) Mechanical simulation of a flower bud development and its regulation by genes. Progression in the flower bud development is shown at three different stages, from primordia initiation to early stage 3 (see Supporting Movie S5).

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Kinematics variables & relations

The positions of material points of are tracked throughout time. The deformation field relates the coordinates of a material point in the current, “observed” () and reference configurations () respectively noted and : (8)

This field is assumed smooth enough so that its gradient, mapping small variations of length between the material configuration and the current configuration, is defined: (9)

To model growth, the most common approach is to use a multiplicative decomposition of the deformation gradient [22]–[24](10)

where the global deformation gradient is decomposed into a reversible part and and irreversible part . This decomposition leads to an intermediate, “grown”, configuration (), where material coordinates are noted . thus maps growth-related deformations between and and maps purely elastic deformations between and , see Fig. 7. In that perspective, growth is regarded as the time evolution of the irreversible part () of the deformation gradient field. Geometrically, the time evolution of such deformation gradient fields is estimated through their related velocity gradient fields. In the grown configuration , the growth-related part of the velocity gradient field reads as follows:(11)

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Figure 7. Schematic representation of the different configurations at different time and the deformations between them.

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Conservation equations

Constitutive equations

To compute the time evolution of the meristem under pressure, we need to define how the cell walls elastically deform and how they grow.

Growth.

The growth of the wall can be regarded as creep that can be modeled using the Maxwell model of viscoelasticity. In this model, reversible and irreversible phenomena are embodied, respectively, as a purely elastic spring (characterized by an effective spring constant ) and as a purely viscous dashpot (characterized by an effective viscosity ), the two being connected in series. Adding a friction force in parallel to the viscous drag enables us to take into account the threshold phenomenon, see Fig. 8. Under loading forces, the mechanical equilibrium of such a system leads to:(17)

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Figure 8. 1D version of a unit element of the biomechanical model.

a) the system in its resting configuration (). b) the system deformed by a the loading forces, at mechanical equilibrium (). Orange, blue and gray arrows represent respectively the loading (turgor-related) force, the elastic forces and the sum of viscous drag and friction force.

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The first line of Eq.17 refers to the mechanical equilibrium at point on Fig. 8−b and links the elastic stretching of the material to its loading force. The second line refers to the mechanical equilibrium at point and links the creeping rate of the material to the difference between its elastic stretching and the threshold related force. Using the measure of strain in the system exposed at Eq. 17 leads us to the following expression:(18)

where we assumed that the friction force is proportional to the initial rest length of the system () and that the loading force can be expressed as a pressure force with the section on which the pressure is applied. We also introduced the effective Young's modulus of the material in the considered direction and a coefficient characterizing the extensibility of the material in that direction. Extrapolating Eq. 18 in 3D with large deformation leads to the constitutive growth equation:(19)

where is the threshold matrix strain has to overcome in order to induce growth and represents the matrix version of the primitive of the Heaviside function, defined as followed:(20)

Discretization