Differentiation into well-defined patterns and tissue growth are recognized as key processes in organismal development. However, it is unclear whether patterns are passively, homogeneously dilated by growth or whether they remodel during tissue expansion. Leaf vascular networks are well-fitted to investigate this issue, since leaves are approximately two-dimensional and grow manyfold in size. Here we study experimentally and computationally how vein patterns affect growth. We first model the growing vasculature as a network of viscoelastic rods and consider its response to external mechanical stress. We use the so-called texture tensor to quantify the local network geometry and reveal that growth is heterogeneous, resembling non-affine deformations in composite materials. We then apply mechanical forces to growing leaves after veins have differentiated, which respond by anisotropic growth and reorientation of the network in the direction of external stress. External mechanical stress appears to make growth more homogeneous, in contrast with the model with viscoelastic rods. However, we reconcile the model with experimental data by incorporating randomness in rod thickness and a threshold in the rod growth law, making the rods viscoelastoplastic. Altogether, we show that the higher stiffness of veins leads to their reorientation along external forces, along with a reduction in growth heterogeneity. This process may lead to the reinforcement of leaves against mechanical stress. More generally, our work contributes to a framework whereby growth and patterns are coordinated through the differences in mechanical properties between cell types.
Organismal development relies on both the progressive differentiation of cells according to specific spatial patterns and the growth of tissues and organs towards their target shapes. On the one hand, numerous studies have addressed differentiation mechanisms, leading to a framework where differentiation patterns depend on the establishment of biochemical gradients, see e. g. [1]. On the other hand, it has been shown that simple growth rules can lead to complex morphologies, such as for tumors [2] or ruffled leaves [3–5]. However, the coordination between patterning and growth has received much less attention [6–8]. Are patterns passively stretched by growth like drawings on an inflated rubber balloon, or do patterns remodel during tissue growth? This question is central to the present study. As growth entails dynamic changes in the structural elements that define shape, such as the cytoskeleton or the extra-cellular matrix, it is essential to address the physical properties of these elements and how these properties are controlled at the cellular level [9–17]. In this framework, cell mechanics would provide a direct link between biochemical activity and growth. Accordingly, the question above can be reformulated as follows. Do the patterns of cell differentiation correspond to patterns of changes in mechanical properties? If so, do changes in mechanical properties predict the geometry of the patterns when the organ reaches its target shape? In addition, what would be the functional role of such changes in the geometry of patterns? Here we use a combination of experiments and mechanical modeling of growth to address these questions within the context of leaf vasculature. The leaves of dicotylodonous flowering plants and their vasculature provide a fitting context for the study of patterns on growing tissues. Leaves grow manyfold from a sub-millimetric size to several centimeters [18,19]. They are amenable to genetic [20] or physical manipulation; finally, they can be analyzed quantitatively, being almost two dimensional [21,22]. Vasculature in dicotyledons is an elaborate reticulated network with striking geometrical and statistical properties, as revealed by advanced mathematical quantification [23–29]. Throughout the leaf’s growth, the network multiplies its size by orders of magnitude while maintaining its crucial structural and functional properties [30,31]: due to their rigidity [32], veins are the main carriers of mechanical loads in the mature leaf. On the other hand, veins are responsible for the transport of nutrients and water. With this respect, the leaf’s ability to withstand damage of one vein is often ensured by redundancy: the network is reticulated (featuring loops), allowing for alternative routes. Consequently, the venation network, through its topology and geometry, is thought to optimize both its mechanical [33] and transport properties [34]. Finally, vasculature and leaf development appear to be tightly coupled [35,36]. In many species, the differentiation of ground cells into provascular cells is completed when the leaf is millimetric in size [30]. This process of differentiation is dependent on a biochemical field: the distribution of the phytohormone auxin. The canalization model [37] suggests that the salient features of venation networks are due to instabilities of this field—an initially homogeneous concentration field evolves into a hierarchical network of localized concentrated flow of transported auxin, which eventually becomes the vein system. Canalization has received genetic and molecular support [38–40], while numerical simulations showed that the model accounts for many features of vasculature [41–43]. However, additional hypotheses on transport or on auxin production are needed to account for loops [44,45]. An alternative model [46] proposed that the mechanical stress field regulates differentiation into provascular cells, motivated by the resemblance between the vascular network seen in leaves and the network of cracks in drying mud, which is known to be created by instabilities of the stress field. Numerical simulations of this model [47,48] reproduced many features of the network geometry. However the stress field model of differentiation has not received mechanistic support so far. Here, we do not investigate the process of differentiation of veins, but rather how the vascular network reaches its final geometry. Indeed, after vein formation has ceased the leaf may continue to grow in area by an order of magnitude. Plant growth is driven by the osmotically generated turgor pressure and restrained by cell walls (the extracellular matrix); therefore, mechanical stress can accumulate: for instance, slits made in stems tend to open, indicating that the epidermis is in tension. In leaves, since veins are stiffer than their surrounding environment [32], the vascular network is expected to carry most of the accumulated stress, which might lead to geometrical deformations of the network. This led to the ‘force model’ describing the final geometry of junctions in vasculature [23]: each vein pulls with a force that is proportional to its diameter, and the requirement of local equilibrium at vein junctions leads to a statistical correlation between veins’ diameter and the angles between veins; this correlation was found to hold in the leaves of many cotyledons [23]. More recently, a cell-based mechanical model was developed to describe the time-evolution of the vascular network [49]. The tissue was modeled as a network of viscoelastic cell walls, and vein cells were distinguished from ground cells by their higher rigidity. This yielded realistic venation patterns and reproduced the experimental findings of [23], in line with the force model. However, these studies remain correlative and do not prove that mechanical forces shape the vascular network. Here, we probe the force model by perturbing mechanically a growing leaf and making predictions about the effect of such a perturbation on the vascular network. We use the texture tensor [50] to quantify this effect; we simulate networks on a tissue that grows anisotropically and predict how leaf vasculature is affected by stretching; we apply external forces to growing leaves after veins have differentiated, and compare observations with predictions.
We consider situations in which a leaf grows anisotropically as the result of the application of external forces. More generally, we are interested in the evolution of patterns, here vascular networks, on a growing tissue (Fig 1A): how does the pattern change with growth? Is it merely stretched passively or does its geometry change in a more complex manner? This question is reminiscent of the nature of deformations in elastic solids; in homogeneous solids, elastic deformations are affine, i. e. the local strain is the same as the large-scale strain, whereas in heterogeneous solids, elastic deformations are non-affine, i. e. the local strain differs from large-scale strain [51]. Among biological materials, non-affinity was observed for collagen fibers [52]. Our question therefore amounts to whether growth (an irreversible deformation) is affine or not. Fig 1B shows a portion of a leaf that was subject to external mechanical stress during two weeks of growth. It is clear qualitatively that the network in the region of the leaf that was grown under tension looks stretched while on the other side it is unaffected. Yet one needs a mathematical method to quantify the strength and orientation of this deformation. Deformation is a tensor, meaning that at each point of the leaf, deformation can occur in many directions: imagine that we draw small circles on the leaf. After growth, each circle will become an ellipse. In order to fully characterize growth, one needs to quantify the orientations and areas of the ellipses, as well as their anisotropies (i. e. how elongated the ellipse is). Since we are interested in the quantification of the geometrical properties of a network, we use the texture tensor. The time-derivative of this tensor was proposed as an equivalent of the elastic strain tensor for the quantification of local deformations in materials with a cellular-like structure [50] and has been used to analyse epithelial morphogenesis [53]. It measures the local geometry of a network, and its time-evolution is a measure of the network’s deformation. Using the texture tensor enables capturing both the averaged, continuum-like deformation as well as the local, discrete deformation of the network’s elements. We give here a qualitative description of the tensor’s definition and properties (see Materials and Methods for details). The texture tensor, which we denote by M, is defined for materials that have a network-like structure, and therefore has a natural application in our case. A network (a graph, in mathematical language) is composed of nodes and links that connect between them. In this paper we define the graph by using the areoles (areas surrounded by veins) as the nodes, and we define two areoles as linked (neighbors) if they share a common vein on their boundary. The local texture tensor is defined from the vectors linking the center of an areole to the centers of its neighboring areoles, as sketched geometrically in Fig 1C (see Eq 1 in Methods for the exact definition). Thus, the texture tensor contains information not only about the geometry of a single areole, but also about the local topology. In order to obtain properties averaged at the scale of a few areoles, we also define the averaged texture tensor from a spatial smoothing (with a constant Gaussian weight) of the local tensor. Since the texture tensor is a symmetric 2nd order tensor, it describes an ellipse, which is a measure of the local shape of the network. The area of the ellipse (the determinant of texture tensor det (M) ) quantifies the size of areoles, while anisotropy corresponds to the ratio of the greater axis to smaller axis of the ellipse (ratio of eigenvalues of M); note that in this definition anisotropy is always larger than unity. We now turn to testing the model in experiments. To do so, we chose to work with leaves in which the vein network has already formed, so as to avoid a direct coupling with differentiation mechanisms. We sought a species such that (i) a large number of veins would improve the statistics and (ii) veins are apparent on photographs to allow for a non-perturbative time-lapse analysis of the geometry of the network. It turned out that bay laurel (Laurus nobilis) was appropriate as seen in Fig 1A. Each leaf was loaded by a U-shaped spring, glued to two points on its edge, typically 3mm apart (Fig 1). The loaded leaf was allowed to grow for 15 days, during which it multiplied its area by about one order of magnitude. The vascular system of the entire leaf was repeatedly photographed. The images were processed and the geometry and topology of the vascular network were extracted. The unstretched half of the leaf was considered as a control. The robust qualitative results were observed in a dozen bay leaves. The detailed mathematical analysis was preformed on three bay leaves. Qualitatively similar results were obtained with tobacco leaves (Nicotiana benthamiana, S2 Fig). We sought to reconcile simulations with experimental data. The broad distribution of non-affinity (q) with no stretching (Fig 6C) suggests that the venation network is affected by noise. Indeed, the ‘force model’ was observed to hold only approximately and vein thickness is broadly distributed [23]. We therefore modified the initial state of the rod network by adding noise in rod thickness; each value of thickness was multiplied by a random number uniformly distributed between 1 − r and 1 + r. We started the simulations from this state and observed non-affine growth. The distribution of q is shown in Fig 7A for a noise of r = 40% (ratio of standard deviation of thickness to its average), a value that was chosen to match the observed distribution on the unstretched side of the leaf (Fig 6C). Consequently, a frozen noise in vein thickness is sufficient to retrieve observations of non-affinity in unstretched leaves. Finally, non-affinity decreases with vein thickness (S3 Fig) meaning that areoles surrounded by thick veins tend to grow less than their neighborhood in the presence of noise. However, when we added external stress to the simulation, we found again that the distribution of q was broadened (Fig 7A), in contrast with the experimental trend. Thus, we hypothesized that the growth equation, according to which vein elongation rate is proportional to vein tension, was not sufficient to model the system. The narrowing of non-affinity distribution in stretching experiments suggests that high tensions have relatively more effects on vein growth. Accordingly, elongation rate should be a concave function of vein tension. We then recalled that the commonly accepted plant growth law, the Lockhart equation [10,16,56], is nonlinear and concave: elongation occurs only above a threshold stress and is then an affine function of stress. We incorporated this into our model and added a threshold to the growth equation (Eq 3 in Methods) in the form ν h i l i 0 d l i 0 d t = max 0, T i - h i η where ν is the effective viscosity, l i 0 is the rest-length of the i-th rod, Ti is the tension in the i-th rod, and η is the threshold stress for elongation. We first chose the value of η equal to η0 = 6 that corresponds to the average vein stress at the first step of simulations. We repeated the stretching simulations with this new growth law and we found that the non-affinity (q) distribution narrowed under unidirectional external stress (Fig 7B), as in experiments. Moreover the distributions of q for σx = 0 and σx = 2Ptur are quite similar to experimental distributions. In addition, we found that other growth laws (quadratic, with a maximum, with a saturation) yield a broadening of the distribution of q when tensile external stress is applied (S4 Fig). Finally we investigated the robustness of the model by studying the sensitivity of this behavior to the value of the stress threshold. The control parameter was the normalized stress threshold η⋆ = η/η0, η⋆ = 1 corresponding to our first successful trial. We did not consider values of η⋆ < 0. 1 as the model converges to the initial model with no threshold, as well as η⋆ > 1. 3 as the tissue stopped growing because the tension in all rods remains below the threshold. With no external stress, increasing the threshold broadens the non-affinity distribution, as shown in Fig 7C. With high external stress (σ = 2Ptur), the distribution of q is insensitive to α as the behavior of the system is then dominated by external stress. Importantly, the distribution of q is broader with no stress, as in experiments, when η⋆ > 0. 7. We therefore conclude that our model reproduces experimental observations when including noise and a Lockhart-like growth law as long as the mean vein tension is not much higher than the threshold tension.
In order to address the coordination between patterns and growth during the course of organismal development, we studied the response of leaf vasculature to external stress. More specifically, we investigated whether leaf vasculature is merely dilated by growth, like a drawing on a balloon that is inflated, or whether growth is non-affine. We combined the simulations of a two-dimensional mechanical model of vasculature with the experimental manipulation of leaves in which veins have formed. The main assumption of the model was that veins are much stiffer than ground tissues. The application of anisotropic external mechanical stress resulted in elongated areoles in simulations; on average, the long axis of the areole corresponded to the direction of the maximal stress. To quantify this effect, we used the texture tensor, which is a good measure of the local geometry of the network. We found that, overall, the anisotropy of the texture tensor increased with the level of stress. We then used the texture tensor to quantify experiments on leaves. While measuring the geometry of leaves before stress application, we retrieved known features of leaf geometry and growth. On the one hand, areoles are bigger and more anisotropic near the base of the leaf, which might be ascribed to an enhanced growth, reflecting the gradient in maturation along the leaf axis that occurs at the later stages of leaf development [22,57]. On the other hand, the anisotropy of areoles follows the left-right symmetry of the leaf and its local geometry; the major axis of areoles parallels secondary veins and the margin, while on average, it is aligned with the leaf axis. When external mechanical stress is applied to the leaf, areoles become elongated in the direction of the largest stress, as in simulations. Nevertheless, the elongation of areoles might only be a passive consequence of the largest growth in the direction of external force. To test this possibility, we used the texture tensor to quantify non-affinity. We found that, both in simulations and experiments, the local change in texture tensor differed from the average change, demonstrating that growth is heterogeneous and non-affine. However the simulated distributions differed in behavior from experimental distributions. Therefore we modified the model by incorporating noise in thickness and a threshold in the growth law. Both this model and experiments featured heterogeneity in growth, which was reduced upon external stretching. A threshold in the growth law was introduced by Lockhart [56] to describe experimental data showing that a minimum turgor pressure was needed for growth to occur. This model is well-supported in situations with growth along one axis, as in single cells or in cylindrical plant organs [10,16]. Our results further support this model in a two-dimensional setting. While we cannot exclude more complex hypotheses involving biochemical feedbacks, it is more parsimonious to ascribe our observations to the vein mechanics that induce non-affine growth. It is still left to find out whether they can be explained by a simple viscoelastic behavior of the veins as implemented in the model, or whether they also involve a more sophisticated regulation process. In the former case, it suffices that veins have a specific ‘mechanical identity’, being stiffer than ground tissues, as is obvious in mature leaves [32]. If additional regulation existed, it might be manifested, for example, by softening of cell walls in correlation with stress, or by preferential thickening of veins that carry higher loads. However none of the growth laws that we tried yields results that agreed with observations, except the one with a threshold. In this context, one should note that the effect of stress exists only when the leaf is growing: we did not observe any measurable effect when we applied stress to mature leaves that do not grow in area, or to areas that stopped growing within a growing leaf. Pursuing this direction, we wondered whether non-affinity was also applicable to the earlier stages of leaf development. We thus examined leaf primordia in Arabidopsis thaliana. While this species does not fulfill the requirements stated above for an experimental investigation of the effects of external forces, many molecular and genetic resources are available, such as a reporter for early vascular identity (pVH1: : GUS, see Materials and Methods). Using this reporter, we visualized veins in dissected leaf primordia (S5 Fig); the midvein appears to be smooth and almost straight before tertiary veins have formed, while at later stages it features kinks at the junctions with secondary veins (S5 Fig). This observation indicates that the shape of the midvein does not change according to a simple dilation of the leaf but rather that growth is inhomogeneous and influenced by the local geometry of vasculature, consistently with our observations on older leaves. This might seem at odds with the work in [8], showing that growth fields in early leaves can be accounted for by the affine dilation of an initial polarity pattern, but this work considered younger primordia: tertiary veins appear only at the end of the periods monitored there. To conclude, we showed that, in leaves in which the vasculature has formed, veins reorient in the direction of applied external forces, and that the geometry of the midvein suggests that this also applies to leaves in which vasculature is differentiating. It would be interesting to investigate whether this is relevant to vasculature in animals [58,59], to veins in insect wings, or more generally to netted patterns of differentiation in other growing tissues. Our results further support the force model [23,49], according to which most of the mechanical load is carried by the veins (or equivalently, that the veins are stiffer) and that the tension in each vein is proportional to its thickness. Our results may imply that the network changes so as to become reinforced in the direction of the main stress. This reinforcement would be reminiscent of Wolff’s law according to which bone remodels so as to resist changes in mechanical stress, or of the reorientation of cortical microtubules in plant cells according to the direction of highest stress [60–62]. Similarly to these studies, applying external stress helped us identify a response to internal stress, which can result from differential growth. However, we note that the reduction in growth heterogeneity with higher anisotropy of mechanical stress differs from work in the shoot apex showing that the reorientation of cortical microtubules according to external forces induces growth heterogeneity at the cell scale [63]. These mechanisms operating at different scales might reflect a form of homeostasis, in which the tissue becomes anisotropically stiffer so as to resist the effect of external forces, and which would also underlie the coordination between patterning and tissue growth.
Deformations and growth are associated with the mathematical concept of a second rank tensorial field. The texture tensor was proposed in [50] for quantifying local geometry in materials with cellular-like geometry; the time-derivative of the texture tensor allows the quantification of geometry. In this paper we define a graph by taking the areoles to be the graph’s nodes, and defining two areoles as connected if they share a common vein. One can also define the dual graph, whose nodes are the vein junctions, linked by veins. This method gives similar, yet more noisy, results. For each node i, located at r → i, the local texture tensor is defined as M i = 1 N ∑ k (r → k - r → i) ⊗ (r → k - r → i) (1) where the summation runs over all the neighbors if the site i, N is the number of r → i’s neighbors, and ⊗ denotes the standard 2D tensor product, defined by (u → ⊗ v →) α β = u α v β where α, β are Cartesian coordinate indices. In experiments, the texture tensor is undefined for areoles that are on the boundary of the leaf. The process is sketched geometrically in Fig 1C. This gives the local texture tensor, which is defined only on the graph’s nodes r → i. To get the averaged texture tensor, M (r →), which is a continuous field defined everywhere on the leaf, the local tensor is averaged over the whole leaf with a Gaussian weight centered at r →. The width of the Gaussian, σ, is chosen so that the area πσ2 is 30 times the mean area of an areole. This value of 30 was determined to reveal general trends, but the results were insensitive to the width of the Gaussian, in a range around this value. When the averaged tensor is used for an areole, we take its value M (r → i) at the areole center r → i. At each point the texture tensor field describes an ellipse, which is a measure of the local shape of the network. The determinant measures the area of the ellipse and the directions of the tensor’s eigenvectors indicate the ellipse’s orientation. The eigenvalues are the lengths of the ellipse’s axes, and we define the anisotropy of the tensor to be the ratio of the larger to the smaller axis. Note that by definition the anisotropy is always larger than 1. At each time step, we compare the local texture tensor Mi (t) of each areole i to the averaged texture tensor M (r → i, t), using the ratio of their determinants. If dilation were locally homogeneous, or equivalently if growth were affine, this ratio would be independent of time, because the geometry of the network would be the same up to a magnification factor. Therefore we define the non-affinity index of areole i between time t1 and t2 as q i = det M i (t 2) det M (r → i, t 2) / det M i (t 1) det M (r → i, t 1). (2) If the network was affinely dilated, then q = 1 in all areoles because the ratio of detMi (t) to det M (r → i, t) would be time-independent. The deviation of q from unity quantifies the differences between the local and averaged behavior of the areole. An equivalent index can be defined using areal growth [64], see S6 Fig, that is related to the coefficient of variability of growth introduced in [63]. As described earlier, the simulations were built upon the work of F. Corson et al [49,54]. We give here a brief description of the model, and refer the reader to [49] for details. Corson’s model consists of an array of interconnected viscoelastic rods, modeling the cell walls, in a two-dimensional periodic boundary condition space. The difference between cell walls of the background tissue and cell walls of the vascular tissue is manifested in their elastic properties—vein cell walls are stiffer when oriented with the direction of the vein. The simulations were divided to two stages: In the creation stage, a ‘reference’ network was created using Corson’s model, which yields networks statistically similar to real venation networks [49]. In the reorganization stage, creation of new veins was arrested, and the network was transformed into an ‘effective’ network, where each vein was replaced by a viscoelastic rod, with the same thickness hi and rest length l i 0, given by those of the vein that it represents. The background tissue was erased. The process is shown in S7 Fig. In order to have an ideal initial configuration, we further optimized vein thickness so that the tension carried by each vein is proportional to its thickness. The linear viscoelastic behavior of the rods is manifested in the change of the rods’ rest length, given by T i = μ h i l i l i 0 - 1 = ν h i l i 0 d l i 0 d t (3) where Ti is the tension in the i-th rod, μ is the vein’s Young modulus, ν is its viscosity, and l i, l i 0 are its length and rest-length, correspondingly. The network was grown in quasi-static conditions, at each time step minimizing the elastic energy of the network, which is given by E e l = ∑ i ∈ veins 1 2 μ h i l i l i 0 - 1 2 - P t u r S - E a n i, (4) where is the turgor pressure, and S is the total area of the network. While Corson’s model was restricted to isotropic stress, we introduced an external stress by an anisotropic term in the energy: E a n i = W H ϵ i j σ i j = H 0 (W - W 0) σ x x + W 0 (H - H 0) σ y y (5) where W, H, W0, H0 are the network’s width and height, reference width and reference hight, respectively. The definition of the reference width and hight is done by calculating the equilibrium configuration of the network without the term Eq (5) in the energy equation. The rod model was implemented in C. The energy is minimized according to the BFGS algorithm using the NLopt library. The system of ordinary differential equations is solved using the GNU Scientific Library. All parameters were set to 1 except for μ = 300. Thus the typical strain was around 0. 02 in the initial conditions. The energy was minimized every Δt = 10−5. The experimental set-up consists of attaching a U-shaped steel wire stretcher to a growing leaf, using epoxy glue. After polymerization, the glue was attached to the leaves’ trichomes. The leaves showed no pathologic behavior in response to the glue as could be checked in leaves where two glue drops were deposited with no spring. The applied stress is of same order of magnitude as the turgor pressure. We present a rough estimation: The order of magnitude of the stress is σ ≈ F/S where F ≈ 1 grams ≈10N and S is the surface of the stretched area, perpendicular to the applied force. We estimate the affected area to be about 1 cm wide. The leaf thickness is of the order of 1mm. Therefore we have σ = F S ≈ 10 N 10 - 3 m 10 - 2 m = 10 6 P a = 10 atm which is of the same order of magnitude as the turgor pressure. In the numerical model, the external stress was in the range 0 < σ/Ptur < 2. In order to quantify the vascular network, the leaf was photographed using a commercial digital camera (Nikon CoolPix 8800VR), with strong back-light. The different optical properties of the vascular network allow it to be easily distinguishable from the rest of the leaf. The vascular network was then extracted from the image either by semi-automated image processing methods (written in Matlab) or manually. During several repetitions we noticed that the effect of external force is much more pronounced when stretching close to the base of the leaf, which might be ascribed to the fact that in later stages of development, growth is concentrated near the base of the leaf [22,57]. We used Arabidopis thaliana Col-0 transgenic plants expressing β-glucuronidase under the control of the promoter of VASCULAR HIGHWAY 1 (pVH1: : GUS), an early vascular marker [65]. Plants were grown in soil in long day (16hrs day/8hrs night) conditions and at 20–22°C and harvested two weeks after sowing. The plants were stained for GUS activity in 10mM sodium phosphate buffer (pH 7), 10mM EDTA, 0. 1% Triton X-100,0. 5g/L X-glucuronic acid, and 10mM ferri- and ferro-cyanide for 24hrs at 37°C and subsequently cleared in 70%-100% ethanol for 2 days. Leaves were dissected and mounted in 70% glycerol and pictured with a Zeiss Axiophoto microscope and Axiovision software.
The development of an organism involves a coordination between the differentiation of cells in well-defined spatial patterns and the growth of tissues towards their target shapes. While extensive research has addressed each of these key processes, their coordination has received less attention. In particular, when a pattern has formed and the tissue continues growing, is the pattern passively dilated like a drawing on an inflated balloon, or does the pattern remodel during tissue expansion? We address this question in the context of leaf vasculature and examine the role of mechanics in leaf growth. We model the growing vascular network and identify quantities that compare network growth to background tissue growth. We apply this quantification to mature leaves that are stretched mechanically; we find that vasculature does not dilate passively and that veins reorient in the direction of external forces. This is reminiscent of the reinforcement of bones or of the cytoskeleton so as to resist to mechanical stress. In a developmental context, this might be an essential process to match patterns and growth.
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Mechanical Stress Induces Remodeling of Vascular Networks in Growing Leaves