Abstract We present Sketch2Growth , an interactive system for generating a family of 3D branching shapes and animating their growth from a single user‐drawn style sketch. The sketch, seen as a 2D idealized representation of the desired shape, is converted into a concise format called the Seed . This representation consists of a Directed Acyclic Graph (DAG) that encodes the recursive branching topology, and a series of Gaussian Mixture Models (GMMs) that capture statistical visual‐style at branching points and in terms of branch curvature. During synthesis, the Seed serves as support for a lightweight, stochastic generator: Branches are recursively expanded by unfolding the DAG and sampling from learned distributions, which maintains the expected correlation between parent and child branches. Further growth and variability of the generated shapes are achieved through similarity‐based looping in the Seed structure (re‐starting from a similar node), stochastic mutations and clamping. Only requiring a light learning process through the GMMs, Sketch2Growth ensures perceptual similarity between the generated shapes and the input style sketch, as validated through a user study. In addition, interactive handles enable users to fake the effect of external forces during growth. As results show, our system allows to easily model and animate the continuous growth of complex branching shapes inspired by trees, plants, corals or anatomical structures.

Bud outgrowth is a major component of plant architectural plasticity and is influenced by light conditions. While the inhibitory effect of low light intensity on branching is well documented, the underlying regulators remain debated and, especially, the role of sugar availability has never been thoroughly evaluated. Here, we combined experiments with a computational approach quantifying carbon source-sink balance in single-axis rose plants to investigate how continuous and transient light limitation regulate bud outgrowth. Continuous low light reduced photosynthesis, leading to decreased sugar availability and inhibited bud outgrowth. In contrast, a transient period of low light followed by high light unexpectedly stimulated bud outgrowth, shortened the delay between outgrowth of successive buds, and produced an over-branched phenotype. This response resulted from a non-reversible reduction in the growth of apical organs appearing under low light, which lowered carbon demand and caused sugar over-accumulation after the return to high light. Manipulating carbon supply and demand through leaf masking, photosynthetic inhibition, and targeted sucrose feeding supported a causal contribution of sugar availability in these contrasting responses. Beyond these findings, key requirements for models simulating branching plasticity were identified and this work provides a basis for predicting branching responses under fluctuating and complex light environments.

Abstract Branching forms are ubiquitous in nature and have evolved repeatedly across scales and species. An important goal of developmental biology remains to identify similarities and differences in the regulatory mechanisms underlying branching. Here, we investigate the branching filaments that form upon spore germination in mosses, using Physcomitrium patens as a model species. To identify the macroscopic rules governing filament patterning, we developed a pipeline to acquire high-resolution 3D images of whole sporelings, reconstruct filament architecture at single-cell resolution, and formalize cell organization using mathematical tree representations. Our quantitative analysis reveals that branch patterning in moss filaments can be captured by a simple probabilistic model in which subapical cells have a near-equal probability of producing – or not producing – a side-branch between successive apical cell divisions. This framework provides a quantitative basis for comparing the developmental rules driving branching morphogenesis within and beyond the plant kingdom.

Plants frequently contain numerous organs, organized in 3D branching systems defining the plant's architecture. Reconstructing the architecture of plants from unstructured observations is challenging because of self-occlusion and spatial proximity between organs, which are often thin structures. To achieve the challenging task, we propose an approach that allows to infer a parameterized representation of the plant's architecture from a given 3D scan of a plant. In addition to the plant's branching structure, this representation contains parametric information for each plant organ, and can therefore be used directly in a variety of tasks. In this data-driven approach, we train a recursive neural network with virtual plants generated using a procedural model. After training, the network allows to infer a parametric tree-like representation based on an input 3D point cloud. Our method is applicable to any plant that can be represented as binary axial tree. We quantitatively evaluate our approach on Chenopodium Album plants on reconstruction, segmentation and skeletonization, which are important problems in plant phenotyping. In addition to carrying out several tasks at once, our method achieves results on-par with strong baselines for each task. We apply our method, trained exclusively on synthetic data, to 3D scans and show that it generalizes well.

Plant cells control their volume by regulating the osmotic potential of their cytoplasm and vacuole. Water is attracted into the cell as the result of a cascade of solute exchanges between the cell subcompartments and the cell surroundings, which are governed by chemical, electrostatic and mechanical forces. Due to this multi-physics aspect and to couplings between volume changes and chemical effects, modeling these exchanges remains a challenge that has only been partially addressed. As interest for multi-compartment models grows in the plant cell community, this challenge calls for new modeling strategies. In this paper, we introduce an energy-based approach to couple chemical, electrical and mechanical processes taking place between several subcompartments of a plant cell. The contributions of all physical effects are gathered in an energy function, which allows us to derive the equations satisfied by each variable in a systematic way. We illustrate the properties of this modular, unified approach on the modeling of ion and water transport in a guard cell during stoma opening. We represent the stoma opening process as a quasi-static evolution driven by hydrogen pumps in the plasma and vacuolar membranes, and we show that the new formalism explains why the system varies in a particular direction in response to perturbations. Additional numerical simulations allow us to investigate the role of each hydrogen pump in this process. Altogether, we show that this energy-based approach highlights a hierarchy between the forces involved in the system, and to dissect the role of each physical effect in the complex behavior of the system.

In Arabidopsis thaliana, successful fertilisation relies on the precise guidance of the pollen tube as it navigates through the female tissues to deliver sperm cells to ovules. While prior research has focused on pistil signals directing pollen tubes towards the ovules, the pollen tube growth within the stigmatic epidermis has received limited attention. Our recent work comparing wild-type pollen tube paths on wild-type and katanin1-5 stigmatic cells, revealed a tight connection between pollen tube directionality and mechanical properties of the invaded stigmatic cell. Given that most mechanical properties of the stigmatic tissue are experimentally challenging to access, we used mathematical modelling to investigate the mechanisms underlying early pollen tube guidance through the papilla cell wall. We found that in ktn1-5, the wild-type pollen tube navigates freely across the curved papilla surface, following curves close to geodesics, whereas the wild-type papilla imposes directional guidance. The order of magnitude analysis of the mechanical forces required for pollen tubes to progress at the papilla surface indicates that both the elongated geometry of the papilla and the difference in rigidity of its cell wall layers combine to efficiently orient the pollen tube towards the papilla base.

Uncovering the mechanisms by which developmental patterns consistently arise within species is a major goal of biology. In plants, branching patterns are key determinants of morphological diversity. Similar lateral branching modes convergently evolved in the leafy shoot of vascular plants and bryophytes, driving their independent architectural diversification. While auxin-dependent branch inhibition is shared between both lineages, long-range polar auxin transport plays a central role in branching control in vascular plants, but in bryophytes like the moss Physcomitrium patens, auxin might diffuse through plasmodesmata to regulate branch distribution. However, whether symplasmic auxin diffusion is a biophysically realistic mechanism sufficient to explain observed branch distribution patterns has yet to be assessed. Here, we address this fundamental problem by developing a physics-based, three-dimensional computational model of symplasmic auxin diffusion in the moss shoot, integrating molecular, cell, and tissue scales. Each step of model design was guided by geometry measurements and biological experiments. Our integrative approach demonstrates that branching control based solely on symplasmic diffusion for intercellular auxin movement can account for the observed branching patterns at the whole-shoot level. It also provides mechanistic interpretations of the changes in branch distribution caused by genetic perturbations affecting callose-dependent symplasmic permeability, as well as the unexpected increase in branch spacing robustness during shoot development. Altogether, our findings reveal that branching patterns arising from an auxin diffusion-based regulatory mechanism exhibit a specific developmental signature, not reported in vascular plants, but well exemplified in the moss Physcomitrium.

For nearly 450 million years, mycorrhizal fungi have constructed networks to collect and trade nutrient resources with plant roots1,2. Owing to their dependence on host-derived carbon, these fungi face conflicting trade-offs in building networks that balance construction costs against geographical coverage and long-distance resource transport to and from roots3. How they navigate these design challenges is unclear4. Here, to monitor the construction of living trade networks, we built a custom-designed robot for high-throughput time-lapse imaging that could track over 500,000 fungal nodes simultaneously. We then measured around 100,000 cytoplasmic flow trajectories inside the networks. We found that mycorrhizal fungi build networks as self-regulating travelling waves-pulses of growing tips pull an expanding wave of nutrient-absorbing mycelium, the density of which is self-regulated by fusion. This design offers a solution to conflicting trade demands because relatively small carbon investments fuel fungal range expansions beyond nutrient-depletion zones, fostering exploration for plant partners and nutrients. Over time, networks maintained highly constant transport efficiencies back to roots, while simultaneously adding loops that shorten paths to potential new trade partners. Fungi further enhance transport flux by both widening hyphal tubes and driving faster flows along 'trunk routes' of the network5. Our findings provide evidence that symbiotic fungi control network-level structure and flows to meet trade demands, and illuminate the design principles of a symbiotic supply-chain network shaped by millions of years of natural selection.

Plant morphogenesis relies on dynamic growth deformations at the cell and tissue scales driven by osmotic fluxes. A mechanistic understanding of this phenomenon demands a physical framework that integrates cell imbibition, tissue mechanics, and water fluxes, as well as their biophysical and molecular regulations, within a theory of plant active matter capturing the open-system and out-of-equilibrium properties of tissues. Building on historical insights into growth geometry, physics, and mechanics, combined with recent experimental results, we outline the key challenges in modelling plant growth and propose steps towards a unified physical theory of plant morphogenesis, in which biological regulation, mechanical forces, and water fluxes interact to shape biological form through the fundamental principles of living matter.

In the past 50 years, the formalism of L-systems has been successfully used and developed to model the growth of filamentous and branching biological forms. These simulations take place in classical 2-D or 3-D Euclidean spaces. However, various biological forms actually grow in curved, non-Euclidean, spaces. This is, for example, the case of vein networks growing within curved leaf blades, of unicellular filaments, such as pollen tubes, growing on curved surfaces to fertilise distant ovules, of teeth patterns growing on folded epithelia of animals, of diffusion of chemical or mechanical signals at the surface of plant or animal tissues, etc. To model these forms growing in curved spaces, we thus extended the formalism of L-systems to non-Euclidean spaces. In a first step, we show that this extension can be carried out by integrating concepts of differential geometry in the notion of turtle geometry. We then illustrate how this extension can be applied to model and program the development of both mathematical and biological forms on curved surfaces embedded in our Euclidean space. We provide various examples applied to plant development. We finally show that this approach can be extended to more abstract spaces, called abstract Riemannian spaces, that are not embedded into any higher-dimensional space, while being intrinsically curved. We suggest that this abstract extension can be used to provide a new approach for effective modelling of growth of branching systems within non-uniform substrates and illustrate this idea on a few conceptual examples.

Neuronal stem cells generate a limited and consistent number of neuronal progenies, each possessing distinct morphologies and functions, which are crucial for optimal brain function. Our study focused on a neuroblast (NB) lineage in Drosophila known as Lin A/15, which generates motoneurons (MNs) and glia. Intriguingly, Lin A/15 NB dedicates 40% of its time to producing immature MNs (iMNs) that are subsequently eliminated through apoptosis. Two RNA-binding proteins, Imp and Syp, play crucial roles in this process. Imp+ MNs survive, while Imp−, Syp+ MNs undergo apoptosis. Genetic experiments show that Imp promotes survival, whereas Syp promotes cell death in iMNs. Late-born MNs, which fail to express a functional code of transcription factors (mTFs) that control their morphological fate, are subject to elimination. Manipulating the expression of Imp and Syp in Lin A/15 NB and progeny leads to a shift of TF code in late-born MNs toward that of early-born MNs, and their survival. Additionally, introducing the TF code of early-born MNs into late-born MNs also promoted their survival. These findings demonstrate that the differential expression of Imp and Syp in iMNs links precise neuronal generation and distinct identities through the regulation of mTFs. Both Imp and Syp are conserved in vertebrates, suggesting that they play a fundamental role in precise neurogenesis across species.

In Arabidopsis thaliana , successful fertilization relies on the precise guidance of the pollen tube tip as it navigates through the female pistil tissues to deliver non-motile sperm cells to ovules. While prior studies have unveiled the role of the pistil in directing pollen tubes to ovules, growth guidance mechanisms within the stigmatic epidermis during the initial phase of the pollen tube’s journey remains elusive. A recent analysis comparing wild-type (WT) pollen tube paths in WT and ktn1-5 stigmatic cells revealed a tight connection between directed pollen tube growth and the mechanical properties of the invaded stigmatic cell. Building upon these observations, we constructed here a mathematical model to explore the mechanisms guiding early pollen tube growth through the papilla cell wall (CW). We found that in ktn1-5 , the pollen tube moves freely on the curved papilla surface, following geodesics, while the WT papilla exerts directional guidance on the pollen tube. An order of magnitude analysis of the mechanical forces involved in pollen tube growth in papillae suggests a guidance mechanism, where the elongated papilla geometry and the CW elasticity combine to efficiently direct pollen tube growth towards the papilla base.

In multicellular organisms, tissue outgrowth creates a new water sink, modifying local hydraulic patterns. Although water fluxes are often considered passive by-products of development, their contribution to morphogenesis remains largely unexplored. Here, we mapped cell volumetric growth across the shoot apex in Arabidopsis thaliana. We found that, as organs grow, a subpopulation of cells at the organ-meristem boundary shrinks. Growth simulations using a model that integrates hydraulics and mechanics revealed water fluxes and predicted a water deficit for boundary cells. In planta, a water-soluble dye preferentially allocated to fast-growing tissues and failed to enter the boundary domain. Cell shrinkage next to fast-growing domains was also robust to different growth conditions and different topographies. Finally, a molecular signature of water deficit at the boundary confirmed our conclusion. Taken together, we propose that the differential sink strength of emerging organs prescribes the hydraulic patterns that define boundary domains at the shoot apex. Combining cell volumetric analysis, growth simulation, and in planta water flow tracing, the authors reveal the correlation between water flux distribution and cell growth rate in Arabidopsis shoot apex, suggesting the potential contribution of hydraulic patterns to morphogenesis.

The simple random walk on $\mathbb{Z}^p$ shows two drastically different behaviours depending on the value of $p$: it is recurrent when $p\in\{1,2\}$ while it escapes (with a rate increasing with $p$) as soon as $p\geq3$. This classical example illustrates that the asymptotic properties of a random walk provides some information on the structure of its state space. This paper aims to explore analogous questions on space made up of combinatorial objects with no algebraic structure. We take as a model for this problem the space of unordered unlabeled rooted trees endowed with Zhang edit distance. To this end, it defines the canonical unbiased random walk on the space of trees and provides an efficient algorithm to evaluate its escape rate. Compared to Zhang algorithm, it is incremental and computes the edit distance along the random walk approximately 100 times faster on trees of size $500$ on average. The escape rate of the random walk on trees is precisely estimated using intensive numerical simulations, out of reasonable reach without the incremental algorithm.

Biological organisms have an immense diversity of forms. Some of them exhibit conspicuous and fascinating fractal structures that present self-similar patterns at all scales. How such structures are produced by biological processes is intriguing. In a recent publication, we used a multi-scale modelling approach to understand how gene activity can produce macroscopic cauliflower curds. Our work provides a plausible explanation for the appearance of fractal-like structures in plants, linking gene activity with development.

In multicellular organisms, localized tissue outgrowth creates a new water sink thereby modifying hydraulic patterns at the organ level. These fluxes are often considered passive by-products of development and their patterning and potential contribution to morphogenesis remains largely unexplored. Here, we generated a complete map of cell volumetric growth and deformation across the shoot apex in Arabidopsis thaliana . Within the organ-meristem boundary, we found that a subpopulation of cells next to fast-growing cells experiences volumetric shrinkage. To understand this process, we used a vertex-based model integrating mechanics and hydraulics, informed by the measured growth rates. Organ outgrowth simulations revealed the emerging water fluxes and predicted water deficit with volume loss for a few cells at the boundary. Consistently, in planta, a water-soluble dye is preferentially allocated to fast-growing tissues and fails to enter the boundary domain. Analysis of intact meristems further validated our model by revealing cell shrinkage next to fast-growing cells in different contexts of tissue surface curvature and cell deformation. A molecular signature of water deficit at the boundary further confirmed our conclusion. Taken together, we propose a model where the differential sink strength of emerging organs prescribes the hydraulic patterns that define the boundary domain at the shoot apex.