The role of elastic stresses on leaf venation morphogenesis

We explore the possible role of elastic mismatch between epidermis and mesophyll as a driving force for the development of leaf venation. The current prevalent 'canalization' hypothesis for the formation of veins claims that the transport of the hormone auxin out of the leaves triggers cell differentiation to form veins. Although there is evidence that auxin plays a fundamental role in vein formation, the simple canalization mechanism may not be enough to explain some features observed in the vascular system of leaves, in particular, the abundance of vein loops. We present a model based on the existence of mechanical instabilities that leads very naturally to hierarchical patterns with a large number of closed loops. When applied to the structure of high order veins, the numerical results show the same qualitative features as actual venation patterns and, furthermore, have the same statistical properties. We argue that the agreement between actual and simulated patterns provides strong evidence for the role of mechanical effects on venation development.Comment: 10 figures, published in PLoS Computational Biolog

A model for hierarchical patterns under mechanical stresses

We present a model for mechanically-induced pattern formation in growing biological tissues and discuss its application to the development of leaf venation networks. Drawing an analogy with phase transitions in solids, we use a phase field method to describe the transition between two states of the tissue, e.g. the differentiation of leaf veins, and consider a layered system where mechanical stresses are generated by differential growth. We present analytical and numerical results for one-dimensional systems, showing that a combination of growth and irreversibility gives rise to hierarchical patterns. Two-dimensional simulations suggest that such a mechanism could account for the hierarchical, reticulate structure of leaf venation networks, yet point to the need for a more detailed treatment of the coupling between growth and mechanical stresses.Comment: To appear in Philosophical Magazine. 18 pages, 8 figure

Transient morphing and optimal shape design of synthetic and natural active structures

Living organisms often display shape morphing capabilities allowing them to efficiently perform tasks that are fundamental for survival. Understanding the way biological activity is exploited to perform shape changes has a deep impact both on natural sciences and technology, often through a process of reverse engineering. In this thesis, we examine four instances of shape morphing both in synthetic and natural, active structures. In the first Chapter, we analyze the transient shaping of a linear poroelastic plate and investigate how mechanical parameters, strains, and stresses influence the swelling dynamics. We obtain an approximate analytical solution for the case of stress-free evolutions and investigate the effect of stresses in the case of an axisymmetric plate. We show that compressive stresses promote faster swelling with respect to the stress-free case, and vice-versa. In the the second Chapter, we address the question of devising efficient morphing strategies for the attainment of specific shape changes in active structures. We set up an optimal control problem which selects, among the activation patterns producing a prescribed shape change, the one minimizing an objective functional, designed to quantify the complexity of the activation. We provide analytical insights for the case of affine shape changes and, with the aid of numerics, we explore the outcome of different objective functionals in a more general context. Chapter 3 is devoted to the study of active reconfigurations in axons, slender cylindrical structures of neurons, which are responsible for the transmission of electro-chemical signals. Axons are able to actively regulate their thickness trough a contractile coating, named cortex, surrounding the cytoplasm (axoplasm). Here, we develop a continuum model describing the interplay between the cortex contractility and the axoplasm elastic response inherited by a network of microtubules. The validity of our modelling assumptions are supported by an excellent match between numerical simulations and experiments. Finally, in the last Chapter, we develop a teleological model to interpret leaves morphogenesis by accounting for the simultaneous growth of both the venation pattern and the blade. Inspired by previous works in the relevant literature, we develop a continuum model by which leaves growth is driven by a gradient flow maximizing the net power absorbed by the leaf. The numerical solution of the ensuing equations provides preliminary results showing some qualitative agreement with features of existing leaves

Mechanical Stress Induces Remodeling of Vascular Networks in Growing Leaves

International audienceDifferentiation 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 vascu-lar 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

Computational Morphodynamics: A Modeling Framework to Understand Plant Growth

Computational morphodynamics utilizes computer modeling to understand the development of living organisms over space and time. Results from biological experiments are used to construct accurate and predictive models of growth. These models are then used to make novel predictions that provide further insight into the processes involved, which can be tested experimentally to either confirm or rule out the validity of the computational models. This review highlights two fundamental challenges: (a) to understand the feedback between mechanics of growth and chemical or molecular signaling, and (b) to design models that span and integrate single cell behavior with tissue development. We review different approaches to model plant growth and discuss a variety of model types that can be implemented to demonstrate how the interplay between computational modeling and experimentation can be used to explore the morphodynamics of plant development

Shaping leaf vein pattern by auxin and mechanical feedback

This article comments on: Kneuper I, Teale W, Dawson JE, Tsugeki R, Katifori E, Palme K, Ditengou FA. 2021. Auxin biosynthesis and cellular efflux act together to regulate leaf vein patterning. Journal of Experimental Botany 72, 1151–1165. Auxin is an essential factor for the specification of veins in plant organs. The distribution of auxin in tissues depends on several physiological processes including auxin biosynthesis and transport. By using empirical data and theoretical analysis, Kneuper et al. (2021) explored a role for these processes in the establishment of leaf vasculature in Arabidopsis. They propose that the formation of early vein patterns may essentially be described in terms of auxin biosynthesis, its transport, and growth-dependent mechanical feedback from surrounding tissues

Numerical approach to centrality of optimal transportation networks

We study hierarchical properties of optimal transportation networks with biological background. The networks are obtained as minimizers of an energy functional which involves a metabolic cost term of a power-law form with exponent $\gamma>0$. In the range $\gamma\in (0,1)$, most relevant for biological applications, the functional is non-convex and its local minima correspond to loop-free graphs (trees). We propose a numerical scheme that performs energy descent by searching the discrete set of local minimizers, combined with a Monte-Carlo approach. We verify the performance of the scheme in the borderline case $\gamma=1$, where the functional is convex. For~a~particular example of a leaf-shaped planar graph, we evaluate the global reaching centrality (GRC) of the (local) minimizers in dependence on the value of $\gamma\in (0,1]$. We observe that the GRC, which can be understood as a measure of hierarchical organization of the graph, monotonically increases with increasing $\gamma$. To our best knowledge, this is the first quantification of the influence of the value of the metabolic exponent on the hierarchical organization of the (almost) optimal transportation network.Comment: 7 pages, 4 figures; submitted to the Proceedings of the ENUMATH 2023 conferenc

Branching Boogaloo: Botanical Adventures in Multi-Mediated Morphologies

FormaLeaf is a software interface for exploring leaf morphology using parallel string rewriting grammars called L-systems. Scanned images of dicotyledonous angiosperm leaves removed from plants around Bard’s campus are displayed on the left and analyzed using the computer vision library OpenCV. Morphometrical information and terminological labels are reported in a side-panel. “Slider mode” allows the user to control the structural template and growth parameters of the generated L-system leaf displayed on the right. “Vision mode” shows the input and generated leaves as the computer ‘sees’ them. “Search mode” attempts to automatically produce a formally defined graphical representation of the input by evaluating the visual similarity of a generated pool of candidate leaves. The system seeks to derive a possible internal structural configuration for venation based purely off a visual analysis of external shape. The iterations of the generated L-system leaves when viewed in succession appear as a hypothetical development sequence. FormaLeaf was written in Processing