Reaction-diffusion pattern in shoot apical meristem of plants

An updated model of shoot apical meristem regulation by ERECTA family and CLAVATA3 signaling pathways in Arabidopsis

The shoot apical meristem (SAM) gives rise to the aboveground organs of plants. The size of the SAM is relatively constant due to the balance between stem cell replenishment and cell recruitment into new organs. In angiosperms, the transcription factor WUSCHEL (WUS) promotes stem cell proliferation in the central zone of the SAM. WUS forms a negative feedback loop with a signaling pathway activated by CLAVATA3 (CLV3). In the periphery of the SAM, the ERECTA family receptors (ERfs) constrain WUS and CLV3 expression. Here, we show that four ligands of ERfs redundantly inhibit the expression of these two genes. Transcriptome analysis confirmed that WUS and CLV3 are the main targets of ERf signaling and uncovered new ones. Analysis of promoter reporters indicated that the WUS expression domain mostly overlaps with the CLV3 domain and does not shift along the apical-basal axis in clv3 mutants. Our three-dimensional mathematical model captured gene expression distributions at the single-cell level under various perturbed conditions. Based on our findings, CLV3 regulates cellular levels of WUS mostly through autocrine signaling, and ERfs regulate the spatial expression of WUS, preventing its encroachment into the peripheral zone.

The angiogenic growth of cities

Describing the space-time evolution of urban population is a fundamental challenge in the science of cities, yet a complete theoretical treatment of the underlying dynamics is still missing. Here, we first reconstruct the evolution of London (UK) over 180 years and show that urban growth consists of an initial phase of diffusion-limited growth, followed by the development of the railway transport network and a consequential shift from central to suburban living. Such dynamics-which are analogous to angiogenesis in biological systems-can be described by a minimalist reaction-diffusion model coupled with economic constraints and an adaptive transport network. We then test the generality of our approach by reproducing the evolution of Sydney, Australia, from 1851 to 2011. We show that the rail system coevolves with urban population, displaying hierarchical characteristics that remain constant over time unless large-scale interventions are put in place to alter the modes of transport. These results demonstrate that transport schemes are first-order controls of long-term urbanization patterns and efforts aimed at creating more sustainable and healthier cities require careful consideration of population-transport feedbacks.

Developmental timing in plants

Plants exhibit reproducible timing of developmental events at multiple scales, from switches in cell identity to maturation of the whole plant. Control of developmental timing likely evolved for similar reasons that humans invented clocks: to coordinate events. However, whereas clocks are designed to run independently of conditions, plant developmental timing is strongly dependent on growth and environment. Using simplified models to convey key concepts, we review how growth-dependent and inherent timing mechanisms interact with the environment to control cyclical and progressive developmental transitions in plants.

Size regulation of the lateral organ initiation zone and its role in determining cotyledon number in conifers

Introduction

Unlike monocots and dicots, many conifers, particularly Pinaceae, form three or more cotyledons. These are arranged in a whorl, or ring, at a particular distance from the embryo tip, with cotyledons evenly spaced within the ring. The number of cotyledons, n_c, varies substantially within species, both in clonal cultures and in seed embryos. n_c variability reflects embryo size variability, with larger diameter embryos having higher n_c. Correcting for growth during embryo development, we extract values for the whorl radius at each n_c. This radius, corresponding to the spatial pattern of cotyledon differentiation factors, varies over three-fold for the naturally observed range of n_c. The current work focuses on factors in the patterning mechanism that could produce such a broad variability in whorl radius. Molecularly, work in Arabidopsis has shown that the initiation zone for leaf primordia occurs at a minimum between inhibitor zones of HD-ZIP III at the shoot apical meristem (SAM) tip and KANADI (KAN) encircling this farther from the tip. PIN1-auxin dynamics within this uninhibited ring form auxin maxima, specifying primordia initiation sites. A similar mechanism is indicated in conifer embryos by effects on cotyledon formation with overexpression of HD-ZIP III inhibitors and by interference with PIN1-auxin patterning.

Methods

We develop a mathematical model for HD-ZIP III/KAN spatial localization and use this to characterize the molecular regulation that could generate (a) the three-fold whorl radius variation (and associated n_c variability) observed in conifer cotyledon development, and (b) the HD-ZIP III and KAN shifts induced experimentally in conifer embryos and in Arabidopsis.

Results

This quantitative framework indicates the sensitivity of mechanism components for positioning lateral organs closer to or farther from the tip. Positional shifting is most readily driven by changes to the extent of upstream (meristematic) patterning and changes in HD-ZIP III/KAN mutual inhibition, and less efficiently driven by changes in upstream dosage or the activation of HD-ZIP III. Sharper expression boundaries can also be more resistant to shifting than shallower expression boundaries.

Discussion

The strong variability seen in conifer n_c (commonly from 2 to 10) may reflect a freer variation in regulatory interactions, whereas monocot (n_c = 1) and dicot (n_c = 2) development may require tighter control of such variation. These results provide direction for future quantitative experiments on the positional control of lateral organ initiation, and consequently on plant phyllotaxy and architecture.

Nucleation of stem cell domains in a bistable activator-inhibitor model of the shoot apical meristem

Shoot apical meristems (SAMs) give rise to all above-ground tissues of a plant. Expansion of meristematic tissue is derived from the growth and division of stem cells that reside in a central zone of the SAM. This reservoir of stem cells is maintained by expression of a transcription factor WUSCHEL that is responsible for the development of stem cells in the central zone. WUSCHEL expression is self-activating and downregulated by a signaling pathway initiated by CLAVATA proteins, which are upregulated by WUSCHEL. This classic activator-inhibitor network can generate localized patterns of WUSCHEL activity by a Turing instability provided certain constraints on reaction rates and diffusion constants of WUSCHEL and CLAVATA are satisfied, and most existing mathematical models of nucleation and confinement of stem cells in the SAM rely on Turing's mechanism. However, Turing patterns have certain properties that are inconsistent with observed patterns of stem cell differentiation in the SAM. As an alternative mechanism, we propose a model for stem cell confinement based on a bistable-switch in WUSCHEL-CLAVATA interactions. We study the bistable-switch mechanism for pattern formation in a spatially continuous domain and in a discrete cellularized tissue in the presence of a non-uniform field of a rapidly diffusing hormone. By comparing domain formation by Turing and bistable-switch mechanisms in these contexts, we show that bistable switching provides a superior account of nucleation and confinement of the stem cell domain under reasonable assumptions on reaction rates and diffusion constants.

Secondary cell wall patterning-connecting the dots, pits and helices

All plant cells are encased in primary cell walls that determine plant morphology, but also protect the cells against the environment. Certain cells also produce a secondary wall that supports mechanically demanding processes, such as maintaining plant body stature and water transport inside plants. Both these walls are primarily composed of polysaccharides that are arranged in certain patterns to support cell functions. A key requisite for patterned cell walls is the arrangement of cortical microtubules that may direct the delivery of wall polymers and/or cell wall producing enzymes to certain plasma membrane locations. Microtubules also steer the synthesis of cellulose-the load-bearing structure in cell walls-at the plasma membrane. The organization and behaviour of the microtubule array are thus of fundamental importance to cell wall patterns. These aspects are controlled by the coordinated effort of small GTPases that probably coordinate a Turing's reaction-diffusion mechanism to drive microtubule patterns. Here, we give an overview on how wall patterns form in the water-transporting xylem vessels of plants. We discuss systems that have been used to dissect mechanisms that underpin the xylem wall patterns, emphasizing the VND6 and VND7 inducible systems, and outline challenges that lay ahead in this field.

Phyllotaxis: from classical knowledge to molecular genetics

Plant organs are repetitively generated at the shoot apical meristem (SAM) in recognizable patterns. This phenomenon, known as phyllotaxis, has long fascinated scientists from different disciplines. While we have an enriched body of knowledge on phyllotactic patterns, parameters, and transitions, only in the past 20 years, however, have we started to identify genes and elucidate genetic pathways that involved in phyllotaxis. In this review, I first summarize the classical knowledge of phyllotaxis from a morphological perspective. I then discuss recent advances in the regulation of phyllotaxis, from a molecular genetics perspective. I show that the morphological beauty of phyllotaxis we appreciate is the manifestation of many regulators, in addition to the critical role of auxin as a patterning signal, exerting their respective effects in a coordinated fashion either directly or indirectly in the SAM.

The human EDAR 370V/A polymorphism affects tooth root morphology potentially through the modification of a reaction-diffusion system

Morphological variations in human teeth have long been recognized and, in particular, the spatial and temporal distribution of two patterns of dental features in Asia, i.e., Sinodonty and Sundadonty, have contributed to our understanding of the human migration history. However, the molecular mechanisms underlying such dental variations have not yet been completely elucidated. Recent studies have clarified that a nonsynonymous variant in the ectodysplasin A receptor gene (EDAR 370V/A; rs3827760) contributes to crown traits related to Sinodonty. In this study, we examined the association between the EDAR polymorphism and tooth root traits by using computed tomography images and identified that the effects of the EDAR variant on the number and shape of roots differed depending on the tooth type. In addition, to better understand tooth root morphogenesis, a computational analysis for patterns of tooth roots was performed, assuming a reaction-diffusion system. The computational study suggested that the complicated effects of the EDAR polymorphism could be explained when it is considered that EDAR modifies the syntheses of multiple related molecules working in the reaction-diffusion dynamics. In this study, we shed light on the molecular mechanisms of tooth root morphogenesis, which are less understood in comparison to those of tooth crown morphogenesis.

Geographic tongue as a reaction-diffusion system

Geographic tongue or benign migratory glossitis is a condition of an unknown cause characterized by chronic lesions that slowly migrate across the surface of the tongue. The condition's characteristic wavefronts suggest that it can be modeled as a reaction-diffusion system. Here, we present a model for geographic tongue pattern evolution using reaction-diffusion equations applied to portions of spheroids and paraboloids that approximate a tongue shape. We demonstrate that the observed patterns of geographic tongue lesions can be explained by propagating reaction-diffusion waves on these variably curved surfaces.

A mathematical model for understanding synergistic regulations and paradoxical feedbacks in the shoot apical meristem

The shoot apical meristem (SAM) is the primary stem cell niche in plant shoots. Stem cells in the SAM are controlled by an intricate regulatory network, including negative feedback between WUSCHEL (WUS) and CLAVATA3 (CLV3). Recently, we identified a group of signals, Epidermal Patterning Factor-Like (EPFL) proteins, that are produced at the peripheral region and are important for SAM homeostasis. Here, we present a mathematical model for the SAM regulatory network. The model revealed that the SAM uses EPFL and signals such as HAIRY MERISTEM from the middle in a synergistic manner to constrain both WUS and CLV3. We found that interconnected negative and positive feedbacks between WUS and CLV3 ensure stable WUS expression in the SAM when facing perturbations, and the positive feedback loop also maintains distinct cell populations containing WUS on and CLV3 on cells in the apical-basal direction. Furthermore, systematic perturbations of the parameters revealed a tradeoff between optimizations of multiple patterning features. Our results provide a holistic view of the regulation of SAM patterning in multiple dimensions. They give insights into how Arabidopsis integrates signals from lateral and apical-basal axes to control the SAM patterning, and they shed light into design principles that may be widely useful for understanding regulatory networks of stem cell niche.

Mathematical modeling of plant cell fate transitions controlled by hormonal signals

Coordination of fate transition and cell division is crucial to maintain the plant architecture and to achieve efficient production of plant organs. In this paper, we analysed the stem cell dynamics at the shoot apical meristem (SAM) that is one of the plant stem cells locations. We designed a mathematical model to elucidate the impact of hormonal signaling on the fate transition rates between different zones corresponding to slowly dividing stem cells and fast dividing transit amplifying cells. The model is based on a simplified two-dimensional disc geometry of the SAM and accounts for a continuous displacement towards the periphery of cells produced in the central zone. Coupling growth and hormonal signaling results in a nonlinear system of reaction-diffusion equations on a growing domain with the growth rate depending on the model components. The model is tested by simulating perturbations in the level of key transcription factors that maintain SAM homeostasis. The model provides new insights on how the transcription factor HECATE is integrated in the regulatory network that governs stem cell differentiation.

Role for the shoot apical meristem in the specification of juvenile leaf identity in Arabidopsis

The extent to which the shoot apical meristem (SAM) controls developmental decisions, rather than interpreting them, is a longstanding issue in plant development. Previous work suggests that vegetative phase change is regulated by signals intrinsic and extrinsic to the SAM, but the relative importance of these signals for this process is unknown. We investigated this question by examining the effect of meristem-deficient mutations on vegetative phase change and on the expression of key regulators of this process, miR156 and its targets, SPL transcription factors. We found that the precocious phenotypes of meristem-deficient mutants are a consequence of reduced miR156 accumulation. Tissue-specific manipulation of miR156 levels revealed that the SAM functions as an essential pool of miR156 early in shoot development, but that its effect on leaf identity declines with age. We also found that SPL genes control meristem size by repressing WUSCHEL expression via a novel genetic pathway.

Theoretical models for branch formation in plants

Various branch architectures are observed in living organisms including plants. Branch formation has traditionally been an area of interest in the field of developmental biology, and theoretical approaches are now commonly used to understand the complex mechanisms involved. In this review article, we provide an overview of theoretical approaches including mathematical models and computer simulations for studying plant branch formation. These approaches cover a wide range of topics. In particular, we focus on the importance of positional information in branch formation, which has been especially revealed by theoretical research in plants including computations of developmental processes.

A Rho-based reaction-diffusion system governs cell wall patterning in metaxylem vessels

Rho GTPases play crucial roles in cell polarity and pattern formation. ROPs, Rho of plant GTPases, are widely involved in cell wall patterning in plants, yet the molecular mechanism underlying their action remains unknown. Arabidopsis ROP11 is locally activated to form plasma membrane domains, which direct formation of cell wall pits in metaxylem vessel cells through interaction with cortical microtubules. Here, we show that the pattern formation of cell wall pits is governed by ROP activation via a reaction-diffusion mechanism. Genetic analysis and reconstructive assays revealed that ROPGEF4/7 and ROPGAP3/4, which encode ROP activators and inactivators, respectively, regulated the formation of ROP-activated domains; these in turn determined the pattern of cell wall pits. Mathematical modelling showed that ROP-activation cycle generated ROP domains by reaction-diffusion mechanism. The model predicted that a positive feedback and slow diffusion of ROP11-ROPGEF4 complex were required to generate ROP-activated domains. ROPGEF4 formed a dimer that interacted with activated ROP11 in vivo, which could provide positive feedback for ROP activation. ROPGEF4 was highly stable on the plasma membrane and inhibited ROP11 diffusion. Our study indicated that ROP-based reaction-diffusion system self-organizes ROP-activated domains, thereby determines the pit pattern of metaxylem vessels.

Topological graph description of multicellular dynamics based on vertex model

Vertex models are generally powerful tools for exploring biological insights into multicellular dynamics. In these models, a multicellular structure is represented by a network, which is dynamically rearranged using topological operations. Remarkably, the topological dynamics of the network are important in guaranteeing the results from the models and their biological implications. However, it remains unclear whether the entire pattern of multicellular topological dynamics can be accurately expressed by a set of operators in the models. Surprisingly, vertex models have been empirically used for several decades without any mathematical verification. In this study, we propose a rigorous two-/three-dimensional (2D/3D) vertex model to describe multicellular topological dynamics. To do this, we classify several types of vertex models from a graph-theoretic perspective. Based on the classification, mathematical analyses reveal several conditions that enable us to apply the operators accurately without topological errors. Under these conditions, the operators can completely express the entire pattern of multicellular topological dynamics. From these results, we newly propose rigorous 2D/3D vertex models that can be applied to general multicellular dynamics, and we clarify several points to verify the results obtained from previous models.

Cytokinin stabilizes WUSCHEL by acting on the protein domains required for nuclear enrichment and transcription

Concentration-dependent transcriptional regulation and the spatial regulation of transcription factor levels are poorly studied in plant development. WUSCHEL, a stem cell-promoting homeodomain transcription factor, accumulates at a higher level in the rib meristem than in the overlying central zone, which harbors stem cells in the shoot apical meristems of Arabidopsis thaliana. The differential accumulation of WUSCHEL in adjacent cells is critical for the spatial regulation and levels of CLAVATA3, a negative regulator of WUSCHEL transcription. Earlier studies have revealed that DNA-dependent dimerization, subcellular partitioning and protein destabilization control WUSCHEL protein levels and spatial accumulation. Moreover, the destabilization of WUSCHEL may also depend on the protein concentration. However, the roles of extrinsic spatial cues in maintaining differential accumulation of WUS are not understood. Through transient manipulation of hormone levels, hormone response patterns and analysis of the receptor mutants, we show that cytokinin signaling in the rib meristem acts through the transcriptional regulatory domains, the acidic domain and the WUSCHEL-box, to stabilize the WUS protein. Furthermore, we show that the same WUSCHEL-box functions as a degron sequence in cytokinin deficient regions in the central zone, leading to the destabilization of WUSCHEL. The coupled functions of the WUSCHEL-box in nuclear retention as described earlier, together with cytokinin sensing, reinforce higher nuclear accumulation of WUSCHEL in the rib meristem. In contrast a sub-threshold level may expose the WUSCHEL-box to destabilizing signals in the central zone. Thus, the cytokinin signaling acts as an asymmetric spatial cue in stabilizing the WUSCHEL protein to lead to its differential accumulation in neighboring cells, which is critical for concentration-dependent spatial regulation of CLAVATA3 transcription and meristem maintenance. Furthermore, our work shows that cytokinin response is regulated independently of the WUSCHEL function which may provide robustness to the regulation of WUSCHEL concentration.

Stem cell regulation is critical for the development of all organisms, and plants have particularly unique stem cell populations that are maintained throughout their lifespan at the tips of both the shoots and roots. Proper spatial and temporal regulation of gene expression by mobile proteins is essential for maintaining these stem cell populations. Here we show that in the shoot, the mobile stem cell promoting factor WUSCHEL is stabilized at the protein level by the plant hormone cytokinin. This stabilization occurs in a tightly restricted spatial context, and movement of WUSCHEL outside of this region results in WUSCHEL instability that leads to its degradation. The specific regions on the WUSCHEL protein that respond to the cytokinin signaling are the same regions that are essential for both proper WUSCHEL localization in the nucleus and regulation of its target genes. This spatially specific response to cytokinin results in differential accumulation of WUSCHEL in space, and reveals an intrinsic link between protein stability and the regulation of target genes to maintain a stable population of stem cells.

Two-stage patterning dynamics in conifer cotyledon whorl morphogenesis

Background and Aims

Conifer embryos, unlike those of monocots or dicots, have variable numbers of cotyledons, even within the same species. Cotyledons form in a single whorl on a dome-shaped embryo. The closely spaced cotyledons are not found outside this ring, indicating a radial control on where they can form. Polar transport of the hormone auxin affects outgrowth of distinct cotyledons, but not the radial aspect of the whorl or the within-whorl spacing between cotyledons. A quantitative model of plant growth regulator patterning is needed to understand the dynamics of this complex morphogenetic process.

Methods

A two-stage reaction-diffusion model is developed for the spatial patterning of growth regulators on the embryo surface, with a radial pattern (P1) constraining the shorter-wavelength cotyledon pattern (P2) to a whorl. These patterns drive three-dimensional (3-D) morphogenesis by catalysing local surface growth.

Key Results

Growth driven by P2 generates single whorls across the experimentally observed range of two to 11 cotyledons, as well as the circularly symmetric response to auxin transport interference. These computations are the first corroboration of earlier theoretical proposals for hierarchical control of whorl formation. The model generates the linear relationship between cotyledon number and embryo diameter observed experimentally. This accounts for normal integer cotyledon number selection, as well as the less common cotyledon fusings and splittings observed experimentally. Flattening of the embryo during development may affect the upward outgrowth angle of the cotyledons.

Conclusions

Cotyledon morphogenesis is more complex geometrically in conifers than in angiosperms, involving 2-D patterning which deforms a surface in three dimensions. This work develops a quantitative framework for understanding the growth and patterning dynamics involved in conifer cotyledon development, and applies more generally to the morphogenesis of whorls with many primordia.

Combining Turing and 3D vertex models reproduces autonomous multicellular morphogenesis with undulation, tubulation, and branching

This study demonstrates computational simulations of multicellular deformation coupled with chemical patterning in the three-dimensional (3D) space. To address these aspects, we proposes a novel mathematical model, where a reaction–diffusion system is discretely expressed at a single cell level and combined with a 3D vertex model. To investigate complex phenomena emerging from the coupling of patterning and deformation, as an example, we employed an activator–inhibitor system and converted the activator concentration of individual cells into their growth rate. Despite the simplicity of the model, by growing a monolayer cell vesicle, the coupling system provided rich morphological dynamics such as undulation, tubulation, and branching. Interestingly, the morphological variety depends on the difference in time scales between patterning and deformation, and can be partially understood by the intrinsic hysteresis in the activator-inhibitor system with domain growth. Importantly, the model can be applied to 3D multicellular dynamics that couple the reaction–diffusion patterning with various cell behaviors, such as deformation, rearrangement, division, apoptosis, differentiation, and proliferation. Thus, the results demonstrate the significant advantage of the proposed model as well as the biophysical importance of exploring spatiotemporal dynamics of the coupling phenomena of patterning and deformation in 3D space.

The growth of a stable stationary structure: coordinating cell behavior and patterning at the shoot apical meristem

Plants are characterized by their ability to produce new organs post-embryonically throughout their entire life cycle. In particular development of all above-ground organs relies almost entirely on the function of the shoot apical meristem (SAM). The SAM performs a dual role by maintaining a pool of undifferentiated cells and simultaneously driving cell differentiation to initiate organogenesis. Both processes require strict coordination between individual cells which leads to formation of reproducible morphological and molecular patterns within SAM. The patterns are formed and maintained in large part due to spatio-temporal variation in signaling of plant hormones auxin and cytokinin resulting in tissue-specific transcriptional regulation. Integration of these mechanisms into computational models further identifies the key regulatory interactions involved in SAM function.

Morphological Plant Modeling: Unleashing Geometric and Topological Potential within the Plant Sciences

The geometries and topologies of leaves, flowers, roots, shoots, and their arrangements have fascinated plant biologists and mathematicians alike. As such, plant morphology is inherently mathematical in that it describes plant form and architecture with geometrical and topological techniques. Gaining an understanding of how to modify plant morphology, through molecular biology and breeding, aided by a mathematical perspective, is critical to improving agriculture, and the monitoring of ecosystems is vital to modeling a future with fewer natural resources. In this white paper, we begin with an overview in quantifying the form of plants and mathematical models of patterning in plants. We then explore the fundamental challenges that remain unanswered concerning plant morphology, from the barriers preventing the prediction of phenotype from genotype to modeling the movement of leaves in air streams. We end with a discussion concerning the education of plant morphology synthesizing biological and mathematical approaches and ways to facilitate research advances through outreach, cross-disciplinary training, and open science. Unleashing the potential of geometric and topological approaches in the plant sciences promises to transform our understanding of both plants and mathematics.

Anisotomous dichotomy results from an unequal bifurcation of the original shoot apical meristem in Diphasiastrum digitatum (Lycopodiaceae)

PREMISE OF THE STUDY

Two types of dichotomy are recognized in Lycopodiaceae: isotomous (equal) and anisotomous (unequal). Anisotomous dichotomy (anisotomy) has been hypothesized to result from unequal growth of an equal bifurcation of the original shoot apical meristem (SAM). Diphasiastrum digitatum (Lycopodiaceae) exhibits anisotomy at various locations. We thus used D. digitatum to test this classic hypothesis about anisotomy.

METHODS

Transverse areas of original and derived SAMs of anisotomy exhibited by the rhizome and the vertical aerial vegetative stem were measured using scanning electron microscopy. The difference between half of the original SAM and one derived SAM in terms of transverse area were compared using paired t-tests.

KEY RESULTS

During the anisotomy exhibited by the rhizome SAM, 77.4% of the transverse area of the original rhizome SAM contributed to the derived rhizome SAM. During the first anisotomy exhibited by the vertical aerial vegetative stem SAM, 66.2% of the transverse area of the original vertical aerial vegetative stem SAM contributed to the derived vertical aerial vegetative stem SAM. During the second anisotomy exhibited by the vertical aerial vegetative stem SAM, 49.4% of the transverse area of the original vertical aerial vegetative stem SAM contributed to the derived vertical aerial vegetative stem SAM. Nonetheless, the shape of the two derived SAMs differed though they did not differ in size.

CONCLUSIONS

In D. digitatum, anisotomy results from an unequal bifurcation of the original SAM. This finding sheds light on plant body architecture evolution as well as plant organ (megaphyllous leaf) evolution.

Morphological Plant Modeling: Unleashing Geometric and Topological Potential within the Plant Sciences

The geometries and topologies of leaves, flowers, roots, shoots, and their arrangements have fascinated plant biologists and mathematicians alike. As such, plant morphology is inherently mathematical in that it describes plant form and architecture with geometrical and topological techniques. Gaining an understanding of how to modify plant morphology, through molecular biology and breeding, aided by a mathematical perspective, is critical to improving agriculture, and the monitoring of ecosystems is vital to modeling a future with fewer natural resources. In this white paper, we begin with an overview in quantifying the form of plants and mathematical models of patterning in plants. We then explore the fundamental challenges that remain unanswered concerning plant morphology, from the barriers preventing the prediction of phenotype from genotype to modeling the movement of leaves in air streams. We end with a discussion concerning the education of plant morphology synthesizing biological and mathematical approaches and ways to facilitate research advances through outreach, cross-disciplinary training, and open science. Unleashing the potential of geometric and topological approaches in the plant sciences promises to transform our understanding of both plants and mathematics.

Computational Modeling of Auxin: A Foundation for Plant Engineering

Since the development of agriculture, humans have relied on the cultivation of plants to satisfy our increasing demand for food, natural products, and other raw materials. As we understand more about plant development, we can better manipulate plants to fulfill our particular needs. Auxins are a class of simple metabolites that coordinate many developmental activities like growth and the appearance of functional structures in plants. Computational modeling of auxin has proven to be an excellent tool in elucidating many mechanisms that underlie these developmental events. Due to the complexity of these mechanisms, current modeling efforts are concerned only with single phenomena focused on narrow spatial and developmental contexts; but a general model of plant development could be assembled by integrating the insights from all of them. In this perspective, we summarize the current collection of auxin-driven computational models, focusing on how they could come together into a single model for plant development. A model of this nature would allow researchers to test hypotheses in silico and yield accurate predictions about the behavior of a plant under a given set of physical and biochemical constraints. It would also provide a solid foundation toward the establishment of plant engineering, a proposed discipline intended to enable the design and production of plants that exhibit an arbitrarily defined set of features.

Centering the Organizing Center in the Arabidopsis thaliana Shoot Apical Meristem by a Combination of Cytokinin Signaling and Self-Organization

Plants have the ability to continously generate new organs by maintaining populations of stem cells throught their lives. The shoot apical meristem (SAM) provides a stable environment for the maintenance of stem cells. All cells inside the SAM divide, yet boundaries and patterns are maintained. Experimental evidence indicates that patterning is independent of cell lineage, thus a dynamic self-regulatory mechanism is required. A pivotal role in the organization of the SAM is played by the WUSCHEL gene (WUS). An important question in this regard is that how WUS expression is positioned in the SAM via a cell-lineage independent signaling mechanism. In this study we demonstrate via mathematical modeling that a combination of an inhibitor of the Cytokinin (CK) receptor, Arabidopsis histidine kinase 4 (AHK4) and two morphogens originating from the top cell layer, can plausibly account for the cell lineage-independent centering of WUS expression within SAM. Furthermore, our laser ablation and microsurgical experiments support the hypothesis that patterning in SAM occurs at the level of CK reception and signaling. The model suggests that the interplay between CK signaling, WUS/CLV feedback loop and boundary signals can account for positioning of the WUS expression, and provides directions for further experimental investigation.

Coupling intercellular molecular signalling with multicellular deformation for simulating three-dimensional tissue morphogenesis

During morphogenesis, three-dimensional (3D) multicellular structures emerge from biochemical and mechanical interplays among cells. In particular, by organizing their gradient within tissues, the diffusible signalling molecules play an essential role in producing the spatio-temporal patterns of cell status such as the differentiation states. Notably, this biochemical patterning can be dynamically coupled with multicellular deformations by signal-dependent cell activities such as contraction, adhesion, migration, proliferation and apoptosis. However, the mechanism by which these cellular activities mediate the interactions between multicellular deformations and patterning is still unknown. Herein, we propose a novel framework of a 3D vertex model to express molecular signalling among the mechanically deforming cells. By specifying a density of signalling molecules for each cell, we express their transport between neighbouring cells. By simulating signal-dependent epithelial growth, we found various types of tissue morphogenesis such as arrest, expansion, invagination and evagination. In the expansion phase, growth molecules were widely diffused with increasing tissue volume, which diluted the growth molecules in order to support the autonomous suppression of tissue growth. These results indicate that the proposed model successfully expresses 3D multicellular deformations dynamically coupled with biochemical patterning. We expect our proposed model to be a useful tool for predicting new phenomena emerging from mechanochemical coupling in multicellular morphogenesis.

Domain nucleation and confinement in agent-controlled bistable systems

We report a mechanism of pattern formation in growing bistable systems coupled indirectly. A modified Fujita et al. model is studied as an example of a reaction-diffusion system of nondiffusive activator and inhibitor molecules immersed in the medium of a fast diffusive agent. Here we show that, as the system grows, a new domain nucleates spontaneously in the area where the local level of the agent becomes critical. Newly nucleated domains are stable and the pattern formation is different from Turing's mechanism in monostable systems. Domains are spatially confined by the agent even if the activator and inhibitor molecules diffuse. With the spatial extension of the system, a larger domain may undergo a wave number instability, and the concentrations of active molecules within the neighboring elements of a domain can become sharply different. The mechanism reported in this work could be generic for pattern formation systems involving multistability, growth, and indirect coupling.

Modelling the development and arrangement of the primary vascular structure in plants

BACKGROUND AND AIMS

The process of vascular development in plants results in the formation of a specific array of bundles that run throughout the plant in a characteristic spatial arrangement. Although much is known about the genes involved in the specification of procambium, phloem and xylem, the dynamic processes and interactions that define the development of the radial arrangement of such tissues remain elusive.

METHODS

This study presents a spatially explicit reaction-diffusion model defining a set of logical and functional rules to simulate the differentiation of procambium, phloem and xylem and their spatial patterns, starting from a homogeneous group of undifferentiated cells.

KEY RESULTS

Simulation results showed that the model is capable of reproducing most vascular patterns observed in plants, from primitive and simple structures made up of a single strand of vascular bundles (protostele), to more complex and evolved structures, with separated vascular bundles arranged in an ordered pattern within the plant section (e.g. eustele).

CONCLUSIONS

The results presented demonstrate, as a proof of concept, that a common genetic-molecular machinery can be the basis of different spatial patterns of plant vascular development. Moreover, the model has the potential to become a useful tool to test different hypotheses of genetic and molecular interactions involved in the specification of vascular tissues.

Dichotomous branching: the plant form and integrity upon the apical meristem bifurcation

The division of the apical meristem into two independently functioning axes is defined as dichotomous branching. This type of branching typically occurs in non-vascular and non-seed vascular plants, whereas in seed plants it presents a primary growth form only in several taxa. Dichotomy is a complex process, which requires a re-organization of the meristem structure and causes changes in the apex geometry and activity. However, the mechanisms governing the repetitive apex divisions are hardly known. Here, an overview of dichotomous branching is presented, occurring in structurally different apices of phylogenetically distant plants, and in various organs (e.g., shoots, roots, rhizophores). Additionally, morphogenetic effects of dichotomy are reviewed, including its impact on organogenesis and mechanical constraints. At the end, the hormonal and genetic regulation of the dichotomous branching is discussed.

Dynamics of cell-fate determination and patterning in the vascular bundles of Arabidopsis thaliana

Plant vascular meristems are sets of pluripotent cells that enable radial growth by giving rise to vascular tissues and are therefore crucial to plant development. However, the overall dynamics of cellular determination and patterning in and around vascular meristems is still unexplored. We study this process in the shoot vascular tissue of Arabidopsis thaliana, which is organized in vascular bundles that contain three basic cell types (procambium, xylem and phloem). A set of molecules involved in this process has now been identified and partially characterized, but it is not yet clear how the regulatory interactions among them, in conjunction with cellular communication processes, give rise to the steady patterns that accompany cell-fate determination and arrangement within vascular bundles. We put forward a dynamic model factoring in the interactions between molecules (genes, peptides, mRNA and hormones) that have been reported to be central in this process, as well as the relevant communication mechanisms. When a few proposed interactions (unverified, but based on related data) are postulated, the model reproduces the hormonal and molecular patterns expected for the three regions within vascular bundles. In order to test the model, we simulated mutant and hormone-depleted systems and compared the results with experimentally reported phenotypes. The proposed model provides a formal framework integrating a set of growing experimental data and renders a dynamic account of how the collective action of hormones, genes, and other molecules may result in the specification of the three main cell types within shoot vascular bundles. It also offers a tool to test the necessity and sufficiency of particular interactions and conditions for vascular patterning and yields novel predictions that may be experimentally tested. Finally, this model provides a reference for further studies comparing the overall dynamics of tissue organization and formation by meristems in other plant organs and species.

Tissue organization of fasciated lateral roots of Arabidopsis mutants suggestive of the robust nature of outer layer patterning

In three temperature-sensitive mutants of Arabidopsis, root redifferentiation 1 (rrd1), rrd2, and root initiation defective 4 (rid4), formation of fasciated lateral roots was previously observed under high temperature conditions of 28°C. When lateral roots were induced from explants of very young seedlings of these mutants by culture with exogenously supplied auxin at 28°C, expansion of lateral root primordia leading to lateral root fasciation occurred reproducibly and semi-synchronously with a high frequency. This experimental system allowed us to examine how radial organization of root tissues is altered in association with expansion of primordia. Analysis with various tissue-specific reporter genes indicated that in the fasciated lateral roots, cell files of the stele are increased markedly while the numbers of cortical and epidermal cell layers are not changed. This suggests that radial organization during root primordium development involves a mechanism that makes outer layer patterning more robust than inner layer patterning against unusual enlargement of the morphogenetic field.

Dynamic regulation of growing domains for elongating and branching morphogenesis in plants

With their continuous growth, understanding how plant shapes form is fundamentally linked to understanding how growth rates are controlled across different regions of the plant. Much of a plant's architecture is generated in shoots and roots, where fast growth in tips contrasts with slow growth in supporting stalks. Shapes can be determined by where the boundaries between fast- and slow-growing regions are positioned, determining whether tips elongate, branch, or cease to grow. Across plants, there is a diversity in the cell wall chemistry through which growth operates. However, prototypical morphologies, such as tip growth and branching, suggest there are common dynamic constraints in localizing chemical growth catalysts. We have used Turing-type reaction-diffusion mechanisms to model this spatial localization and the resulting growth trajectories, characterizing the chemistry-growth feedback necessary for maintaining tip growth and for inducing branching. The mechanism defining the boundaries between fast- and slow-growing regions not only affects tip shape, it must be able to form new boundaries when the pattern-forming dynamics break symmetry, for instance in the branching of a tip. In previous work, we used an arbitrary concentration threshold to switch between two dynamic regimes of the growth catalyst in order to define growth boundaries. Here, we present a chemical dynamic basis for this threshold, in which feedback between two pattern-forming mechanisms controls the extent of the regions in which fast growth occurs. This provides a general self-contained mechanism for growth control in plant morphogenesis (not relying on external cues) which can account for both simple tip extension and symmetry-breaking branching phenomena.

Cytokinin signaling as a positional cue for patterning the apical-basal axis of the growing Arabidopsis shoot meristem

The transcription factor WUSCHEL (WUS) acts from a well-defined domain within the Arabidopsis thaliana shoot apical meristem (SAM) to maintain a stem cell niche. A negative-feedback loop involving the CLAVATA (CLV) signaling pathway regulates the number of WUS-expressing cells and provides the current paradigm for the homeostatic maintenance of stem cell numbers. Despite the continual turnover of cells in the SAM during development, the WUS domain remains patterned at a fixed distance below the shoot apex. Recent work has uncovered a positive-feedback loop between WUS function and the plant hormone cytokinin. Furthermore, loss of function of the cytokinin biosynthetic gene, LONELY GUY (LOG), results in a wus-like phenotype in rice. Herein, we find the Arabidopsis LOG4 gene is expressed in the SAM epidermis. We use this to develop a computational model representing a growing SAM to suggest the plausibility that apically derived cytokinin and CLV signaling, together, act as positional cues for patterning the WUS domain within the stem cell niche. Furthermore, model simulations backed by experimental data suggest a previously unknown negative feedback between WUS function and cytokinin biosynthesis in the Arabidopsis SAM epidermis. These results suggest a plausible dynamic feedback principle by which the SAM stem cell niche is patterned.

On evaluating models in Computational Morphodynamics

Recent advances in experimental plant biology have led to an increased potential to investigate plant development at a systems level. The emerging research field of Computational Morphodynamics has the aim to lead this development by combining dynamic spatial experimental data with computational models of molecular networks, growth, and mechanics in a multicellular context. The increased number of published models may lead to a diversification of our understanding of the systems, and methods for evaluating, comparing, and sharing models are main challenges for the future. We will discuss this problem using ideas originating from physics and use recent computational models of plant development as examples.

Dynamic network-based epistasis analysis: boolean examples

In this article we focus on how the hierarchical and single-path assumptions of epistasis analysis can bias the inference of gene regulatory networks. Here we emphasize the critical importance of dynamic analyses, and specifically illustrate the use of Boolean network models. Epistasis in a broad sense refers to gene interactions, however, as originally proposed by Bateson, epistasis is defined as the blocking of a particular allelic effect due to the effect of another allele at a different locus (herein, classical epistasis). Classical epistasis analysis has proven powerful and useful, allowing researchers to infer and assign directionality to gene interactions. As larger data sets are becoming available, the analysis of classical epistasis is being complemented with computer science tools and system biology approaches. We show that when the hierarchical and single-path assumptions are not met in classical epistasis analysis, the access to relevant information and the correct inference of gene interaction topologies is hindered, and it becomes necessary to consider the temporal dynamics of gene interactions. The use of dynamical networks can overcome these limitations. We particularly focus on the use of Boolean networks that, like classical epistasis analysis, relies on logical formalisms, and hence can complement classical epistasis analysis and relax its assumptions. We develop a couple of theoretical examples and analyze them from a dynamic Boolean network model perspective. Boolean networks could help to guide additional experiments and discern among alternative regulatory schemes that would be impossible or difficult to infer without the elimination of these assumption from the classical epistasis analysis. We also use examples from the literature to show how a Boolean network-based approach has resolved ambiguities and guided epistasis analysis. Our article complements previous accounts, not only by focusing on the implications of the hierarchical and single-path assumption, but also by demonstrating the importance of considering temporal dynamics, and specifically introducing the usefulness of Boolean network models and also reviewing some key properties of network approaches.

Strategy for shoot meristem proliferation in plants

Shoot apical meristem (SAM) of plants harbors stem cells capable of generating the aerial tissues including reproductive organs. Therefore, it is very important for plants to control SAM proliferation and its density as a survival strategy. The SAM is regulated by the dynamics of a specific gene network, such as the WUS-CLV interaction of A. thaliana. By using a mathematical model, we previously proposed six possible SAM patterns in terms of the manner and frequency of stem cell proliferation. Two of these SAM patterns are predicted to generate either dichotomous or axillary shoot branch. Dichotomous shoot branches caused by this mechanism are characteristic of the earliest vascular plants, such as Cooksonia and Rhynia, but are observed in only a small minority of plant species of the present day. On the other hand, axillary branches are observed in the majority of plant species and are induced by a different dynamics of the feedback regulation between auxin and the asymmetric distribution of PIN auxin efflux carriers. During evolution, some plants may have adopted this auxin-PIN system to more strictly control SAM proliferation.

The Function of the CLE Peptides in Plant Development and Plant-Microbe Interactions

The CLAVATA3 (CLV3)/ENDOSPERM SURROUNDING REGION (ESR) (CLE) peptides consist of 12 or 13 amino acids, including hydroxylated proline residues that may or may not contain sugar modifications, and function in a non-cell-autonomous fashion. The CLE gene was first reported in Zea mays (maize) as an endosperm-specific gene, ESR, in 1997 (Opsahl-Ferstad et al., 1997). CLE genes encode secreted peptides that function in the extracellular space as intercellular signaling molecules and bind to cellular surface receptor-like proteins to transmit a signal. CLE peptides regulate various physiological and developmental processes and its signaling pathway are conserved in diverse land plants. Recent CLE functional studies have pointed to their significance in regulating meristematic activity in plant meristems, through the CLE-receptor kinase-WOX signaling node. CLV3 and CLE40 are responsible for maintenance of shoot apical meristem (SAM) and root apical meristem (RAM) function, regulating homeodomain transcription factors, WUSCHEL (WUS) and WUSCHEL-related homeobox 5 (WOX5), respectively. CLE and WOX form an interconnected and self-correcting feedback loop to provide robustness to stem cell homeostasis. CLE peptides are required for certain plant-microbe interactions, such as those that occur during legume symbiosis and phytopathogenic nematode infection. Understanding the molecular properties of CLE peptides may provide insight into plant cell-cell communication, and therefore also into plant-microbe interactions.