Complex pattern formation by a self-destabilization of established patterns: chemotactic orientation and phyllotaxis as examples

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.

An integral insight into pollen wall development: involvement of physical processes in exine ontogeny in Calycanthus floridus L., with an experimental approach

We aimed to understand the underlying mechanisms of development in the sporopollenin-containing part of the pollen wall, the exine, one of the most complex cell walls in plants. Our hypothesis is that distinct physical processes, phase separation and micellar self-assembly, underpinexine development by taking the molecular building blocks, determined and synthesised by the genome, through several phase transitions. To test this hypothesis, we traced each stage of microspore development in Calycanthus floridus with transmission electron microscopy and then generated in vitro experimental simulations corresponding to every developmental stage. The sequence of structures observed within the periplasmic space around developing microspores starts with spherical units, which are rearranged into columns to then form rod-like units (the young columellae) and, finally, white line centred endexine lamellae. Phase separation precedes each developmental stage. The set of experimental simulations, obtained as self-assembled micellar mesophases formed at the interface between lipid and water compartments, was the same: spherical micelles; columns of spherical micelles; cylindrical micelles; and laminate micelles, separated by gaps, resembling white-lined lamellae. Thus, patterns simulating structures observed at the main stages of exine development in C. floridus were obtained from in vitro experiments, and hence purely physicochemical processes can construct exine-like patterns. This highlights the important part played by physical processes that are not under direct genomic control and share influence on the emerging ultrastructure with the genome during exine development. These findings suggest that a new approach to ontogenetic studies, including a consideration of physical factors, is required for a better understanding of developmental processes.

Formation of aperture sites on the pollen surface as a model for development of distinct cellular domains

Pollen grains are covered by the complex extracellular structure, called exine, which in most species is deposited on the pollen surface non-uniformly. Certain surface areas receive fewer exine deposits and develop into regions whose structure and morphology differ significantly from the rest of pollen wall. These regions are known as pollen apertures. Across species, pollen apertures can vary in their numbers, positions, and morphology, generating highly diverse patterns. The process of aperture formation involves establishment of cell polarity, formation of distinct plasma membrane domains, and deposition of extracellular materials at precise positions. Thus, pollen apertures present an excellent model for studying the development of cellular domains and formation of patterns at the single-cell level. Until very recently, the molecular mechanisms underlying the specification and formation of aperture sites were completely unknown. Here, we review recent advances in understanding of the molecular processes involved in pollen aperture formation, focusing on the molecular players identified through genetic approaches in the model plant Arabidopsis. We discuss a potential working model that describes the process of aperture formation, including specification of domains, creation of their defining features, and protection of these regions from exine deposition.

Changes in morphogen kinetics and pollen grain size are potential mechanisms of aberrant pollen aperture patterning in previously observed and novel mutants of Arabidopsis thaliana

Pollen provides an excellent system to study pattern formation at the single-cell level. Pollen surface is covered by the pollen wall exine, whose deposition is excluded from certain surface areas, the apertures, which vary between the species in their numbers, positions, and morphology. What determines aperture patterns is not understood. Arabidopsis thaliana normally develops three apertures, equally spaced along the pollen equator. However, Arabidopsis mutants whose pollen has higher ploidy and larger volume develop four or more apertures. To explore possible mechanisms responsible for aperture patterning, we developed a mathematical model based on the Gierer-Meinhardt system of equations. This model was able to recapitulate aperture patterns observed in the wild-type and higher-ploidy pollen. We then used this model to further explore geometric and kinetic factors that may influence aperture patterns and found that pollen size, as well as certain kinetic parameters, like diffusion and decay of morphogens, could play a role in formation of aperture patterns. In conjunction with mathematical modeling, we also performed a forward genetic screen in Arabidopsis and discovered two mutants with aperture patterns that had not been previously observed in this species but were predicted by our model. The macaron mutant develops a single ring-like aperture, matching the unusual ring-like pattern produced by the model. The doughnut mutant forms two pore-like apertures at the poles of the pollen grain. Further tests on these novel mutants, motivated by the modeling results, suggested the existence of an area of inhibition around apertures that prevents formation of additional apertures in their vicinity. This work demonstrates the ability of the theoretical model to help focus experimental efforts and to provide fundamental insights into an important biological process.

Pollen is renowned for its ability to form beautiful and complex patterns on its surface. One of the most prominent patterns on the pollen surface is formed by apertures, the regions that lack deposition of the pollen wall exine and develop at precise locations which often vary between the species. How aperture patterns are created is an intriguing and poorly understood question. We developed a mathematical model that aims to explore the mechanisms responsible for the aperture patterning in the pollen of the model plant Arabidopsis. Our model showed that size of the pollen grain could be solely responsible for the increase in aperture number observed in the pollen of some Arabidopsis mutants. Additionally, kinetic parameters, such as diffusion and decay of aperture factors, could also influence aperture number. We coupled our mathematical modeling with a forward genetic screen of a mutagenized population of Arabidopsis. This screen discovered novel mutants with aperture patterns that had been predicted by our mathematical model. Further experiments on these mutants provided additional support to the modeling predictions. These results demonstrate that mathematical modeling could be a powerful tool for understanding the mechanisms responsible for patterning of pollen grains.

Mutually inhibitory Ras-PI(3,4)P2 feedback loops mediate cell migration

Signal transduction and cytoskeleton networks in a wide variety of cells display excitability, but the mechanisms are poorly understood. Here, we show that during random migration and in response to chemoattractants, cells maintain complementary spatial and temporal distributions of Ras activity and phosphatidylinositol (3,4)-bisphosphate [PI(3,4)P₂]. In addition, depletion of PI(3,4)P₂ by disruption of the 5-phosphatase, Dd5P4, or by recruitment of 4-phosphatase INPP4B to the plasma membrane, leads to elevated Ras activity, cell spreading, and altered migratory behavior. Furthermore, RasGAP2 and RapGAP3 bind to PI(3,4)P₂, and the phenotypes of cells lacking these genes mimic those with low PI(3,4)P₂ levels, providing a molecular mechanism. These findings suggest that Ras activity drives PI(3,4)P₂ down, causing the PI(3,4)P₂-binding GAPs to dissociate from the membrane, further activating Ras, completing a positive-feedback loop essential for excitability. Consistently, a computational model incorporating such a feedback loop in an excitable network model accurately simulates the dynamic distributions of active Ras and PI(3,4)P₂ as well as cell migratory behavior. The mutually inhibitory Ras-PI(3,4)P₂ mechanisms we uncovered here provide a framework for Ras regulation that may play a key role in many physiological processes.

A stochastic multicellular model identifies biological watermarks from disorders in self-organized patterns of phyllotaxis

Exploration of developmental mechanisms classically relies on analysis of pattern regularities. Whether disorders induced by biological noise may carry information on building principles of developmental systems is an important debated question. Here, we addressed theoretically this question using phyllotaxis, the geometric arrangement of plant aerial organs, as a model system. Phyllotaxis arises from reiterative organogenesis driven by lateral inhibitions at the shoot apex. Motivated by recurrent observations of disorders in phyllotaxis patterns, we revisited in depth the classical deterministic view of phyllotaxis. We developed a stochastic model of primordia initiation at the shoot apex, integrating locality and stochasticity in the patterning system. This stochastic model recapitulates phyllotactic patterns, both regular and irregular, and makes quantitative predictions on the nature of disorders arising from noise. We further show that disorders in phyllotaxis instruct us on the parameters governing phyllotaxis dynamics, thus that disorders can reveal biological watermarks of developmental systems.

DOI:http://dx.doi.org/10.7554/eLife.14093.001

Plants grow throughout their lifetime, forming new flowers and leaves at the tips of their stems through a patterning process called phyllotaxis, which occurs in spirals for a vast number of plant species. The classical view suggests that the positioning of each new leaf or flower bud at the tip of a growing stem is based on a small set of principles. This includes the idea that buds produce inhibitory signals that prevent other buds from forming too close to each other. When computational models of phyllotaxis follow these ‘deterministic’ principles, they are able to recreate the spiral pattern the buds form on a growing stem.

In real plants, however, the spiral pattern is not always perfect. The observed disturbances in the pattern are believed to reflect the presence of random fluctuations – regarded as noise – in phyllotaxis. Here, using numerical simulations, Refahi et al. noticed that the patterns of inhibitory signals in a shoot tip pre-determine the locations of several competing sites where buds could form in a robust manner. However, random fluctuations in the way cells perceive these inhibitory signals could greatly disturb the timing of organ formation and affect phyllotaxis patterns.

Building on this, Refahi et al. created a new computational model of bud patterning that takes into account some randomness in how cells perceive the inhibitory signals released by existing buds. The model can accurately recreate the classical spiral patterns of buds and also produces occasional disrupted patterns that are similar to those seen in real plants. Unexpectedly, Refahi et al. show that these ‘errors’ reveal key information about how the signals that control phyllotaxis might work.

These findings open up new avenues of research into the role of noise in phyllotaxis. The model can be used to predict how altering the activities of genes or varying plant growth conditions might disturb this patterning process. Furthermore, the work highlights how the structure of disturbances in a biological system can shed new light on how the system works.

DOI:http://dx.doi.org/10.7554/eLife.14093.002

On the evolution of morphogenetic models: mechano-chemical interactions and an integrated view of cell differentiation, growth, pattern formation and morphogenesis

In the 1950s, embryology was conceptualized as four relatively independent problems: cell differentiation, growth, pattern formation and morphogenesis. The mechanisms underlying the first three traditionally have been viewed as being chemical in nature, whereas those underlying morphogenesis have usually been discussed in terms of mechanics. Often, morphogenesis and its mechanical processes have been regarded as subordinate to chemical ones. However, a growing body of evidence indicates that the biomechanics of cells and tissues affect in striking ways those phenomena often thought of as mainly under the control of cell-cell signalling. This accumulation of data has led to a revival of the mechano-transduction concept in particular, and of complexity in general, causing us now to consider whether we should retain the traditional conceptualization of development. The researchers' semantic preferences for the terms 'patterning', 'pattern formation' or 'morphogenesis' can be used to describe three main 'schools of thought' which emerged in the late 1970s. In the 'molecular school', the term patterning is deeply tied to the positional information concept. In the 'chemical school', the term 'pattern formation' regularly implies reaction-diffusion models. In the 'mechanical school', the term 'morphogenesis' is more frequently used in relation to mechanical instabilities. Major differences among these three schools pertain to the concept of self-organization, and models can be classified as morphostatic or morphodynamic. Various examples illustrate the distorted picture that arises from the distinction among differentiation, growth, pattern formation and morphogenesis, based on the idea that the underlying mechanisms are respectively chemical or mechanical. Emerging quantitative approaches integrate the concepts and methods of complex sciences and emphasize the interplay between hierarchical levels of organization via mechano-chemical interactions. They draw upon recent improvements in mathematical and numerical morphogenetic models and upon considerable progress in collecting new quantitative data. This review highlights a variety of such models, which exhibit important advances, such as hybrid, stochastic and multiscale simulations.

PAR-3 oligomerization may provide an actin-independent mechanism to maintain distinct par protein domains in the early Caenorhabditis elegans embryo

Par proteins establish discrete intracellular spatial domains to polarize many different cell types. In the single-cell embryo of the nematode worm Caenorhabditis elegans, the segregation of Par proteins is crucial for proper division and cell fate specification. Actomyosin-based cortical flows drive the initial formation of anterior and posterior Par domains, but cortical actin is not required for the maintenance of these domains. Here we develop a model of interactions between the Par proteins that includes both mutual inhibition and PAR-3 oligomerization. We show that this model gives rise to a bistable switch mechanism, allowing the Par proteins to occupy distinct anterior and posterior domains seen in the early C. elegans embryo, independent of dynamics or asymmetries in the actin cortex. The model predicts a sharp loss of cortical Par protein asymmetries during gradual depletion of the Par protein PAR-6, and we confirm this prediction experimentally. Together, these results suggest both mutual inhibition and PAR-3 oligomerization are sufficient to maintain distinct Par protein domains in the early C. elegans embryo.

A conceptual molecular network for chemotactic behaviors characterized by feedback of molecules cycling between the membrane and the cytosol

Cell chemotaxis has been characterized as the formation of a front-back axis that is triggered by a gradient of chemoattractant; however, chemotaxis is accompanied by more complicated behaviors. These include migration in a straight line with a stable axis [the stable single-axis (SSA) pattern] and repeated splitting of the leading edge of the cell into two regions, followed by the "choice" of one of these as the new leading edge [the split and choice (S&C) pattern]. Indeed, transition between these two behaviors can be observed in individual cells. However, the conceptual framework of the network of signaling molecules that generates these patterns remains to be clarified. We confirmed theoretically that a system that has positive and negative feedback loops involving the reciprocal cycling between the membrane and the cytosol of molecules that promote membrane protrusion or retraction generates SSA and S&C patterns of migratory behavior under similar conditions. We also predicted properties of the instabilities of such a system, which are essential for the generation of these behaviors, and we verified their existence in chemotaxing cells. Our research provides a simple model of network structure for chemotactic behaviors, including cell polarization.

Back to the future: evolution of computational models in plant morphogenesis

There has been a recent surge of studies in plant biology that combine experimental data with computational modeling. Here, we categorize a diversity of theoretical models and emphasize the need to tailor modeling approaches to the questions at hand. Models can start from biophysical or purely heuristic basic principles, and can focus at several levels of biological organization. Recent examples illustrate that this entire spectrum can be useful to understand plant development, and point to a future direction where more approaches are combined in fruitful ways--either by proving the same result with different basic principles or by exploring interactions across levels, in the so-called multilevel models.

Feedback inhibition of Jak/STAT signaling by apontic is required to limit an invasive cell population

In both normal development and in a variety of pathological conditions, epithelial cells can acquire migratory and invasive properties. Border cells in the Drosophila ovary provide a genetically tractable model for elucidating the mechanisms controlling such behaviors. Here we report the identification of a mutant, apontic (apt), in which the migratory population expanded and separation from the epithelium was impeded. This phenotype resembled gain-of-function of JAK/STAT activity. Gain-of-function of APT also mimicked loss of function of STAT and its key downstream target, SLBO. APT expression was induced by STAT, which bound directly to sites in the apt gene. The data suggest that a regulatory circuit between STAT, APT, and SLBO functions to convert an initially graded signal into an all-or-nothing activation of JAK/STAT and thus to proper cell specification and migration. These findings are supported by a mathematical model, which accurately simulates wild-type and mutant phenotypes.

Modeling homeoprotein intercellular transfer unveils a parsimonious mechanism for gradient and boundary formation in early brain development

Morphogens are molecules inducing morphogenetic responses from cells and cell ensembles. The concept of morphogen is related to that of positional value, as the generation of morphological and physiological characteristics is function of position. Based on the observation that homeoproteins, a category of transcription factors with morphogenetic functions, traffic between abutting cells and, very often, regulate their own expression, we develop here a biophysical model of homeoprotein propagation and study the associated mathematical equations. This mode of cell signaling can generate domains of homeoprotein expression. We study both the transient and steady-state regimes and, in this latter regime, we obtain various morphogenetic gradients, depending on the value of some parameters, such as morphogen synthesis, degradation rates and efficiency of intercellular passage. The same equations, applied to pairs of homeoproteins with auto-activation and reciprocal inhibition properties, account for border formation. They also allow us to compute how specific perturbations can either be buffered or lead to modifications in the position of borders between adjacent areas. The model developed here, based on experimental data, and avoids theoretical obstacles associated with pluricellularity. It extends the idea that Bicoid homeoprotein is a morphogen in the fly embryo syncitium to most homeoproteins and to pluricellular systems. Because the position of borders between brain areas is of primary physiological importance, our model might lead to original views regarding epigenetic inter-individual variations and the origin of neurological and psychiatric diseases. In addition, it provides new hypotheses regarding the molecular basis of brain evolution.

Modeling of granulocyte cytoskeletal responses following fMLP challenging

Formyl peptides released from Gram-negative bacteria ligate a group of specific mammalian receptors, expressed mainly on granulocytes, monocytes, and macrophages. Receptor ligation activates different transduction cascades, eventually leading to the release of reactive oxygen species and other bactericidal chemical species, and the activation of the actin cytoskeleton with extension of lamellipodia and migration toward the sites of maximal formyl peptide concentration. In vitro, under conditions of nongradient formyl peptide concentrations, lamellipodia form all around the cell contour (chemokinesis). In granulocytes challenged under these conditions with N-formyl-methionyl-leucyl-phenylalanine, (i) the power spectrum of the contour of activated cells shows a peak at a specific periodicity, indicating that the lamellipodial extension is not completely random but stochastically conforms to a deterministic scheme, and (ii) the morphological response (percent of cells exhibiting chemokinesis) tends to reach a maximum at certain drug concentrations, then declining at higher concentrations. Accordingly, the logarithm of the drug concentration-polarizing effect curve is bell-shaped. Herein we illustrate theoretical models for the simulation of these two components of the chemokinetic responses. We show that the main traits of the general morphology and arrangement of lamellipodia may be simulated by an algorithm that starting from a situation of random distribution of active receptors on the cell membrane, encompasses in the successive calculation cycles both a local autocatalytic enhancement of the actin polymerization and a relative inhibition of the actin polymerization at some distance from the more active polymerization foci. In addition, a drug log concentration-polarizing effect bell-shaped curve may be simulated by assuming that the N-formyl-methionyl-leucyl-phenylalanine, while binding with high affinity to the specific receptor, is also able to bind to another lower affinity receptor that may effect depolarizing actions or, more generally, metabolic blocking effects. Under these conditions, at low drug concentrations the polarizing effect brought about by the ligation of the specific receptor is largely predominant. However, as the drug concentration increases and the specific receptors approach saturation, the inhibitory effects become more and more powerful and the net polarizing effect is reduced.

Phosphoinositides and Rho proteins spatially regulate actin polymerization to initiate and maintain directed movement in a one-dimensional model of a motile cell

Gradient sensing, polarization, and chemotaxis of motile cells involve the actin cytoskeleton, and regulatory modules, including the phosphoinositides (PIs), their kinases/phosphatases, and small GTPases (Rho proteins). Here we model their individual components (PIP1, PIP2, PIP3; PTEN, PI3K, PI5K; Cdc42, Rac, Rho; Arp2/3, and actin), their interconversions, interactions, and modular functions in the context of a one-dimensional dynamic model for protrusive cell motility, with parameter values derived from in vitro and in vivo studies. In response to a spatially graded stimulus, the model produces stable amplified internal profiles of regulatory components, and initiates persistent motility (consistent with experimental observations). By connecting the modules, we find that Rho GTPases work as a spatial switch, and that the PIs filter noise, and define the front versus back. Relatively fast PI diffusion also leads to selection of a unique pattern of Rho distribution from a collection of possible patterns. We use the model to explore the importance of specific hypothesized interactions, to explore mutant phenotypes, and to study the role of actin polymerization in the maintenance of the PI asymmetry. We also suggest a mechanism to explain the spatial exclusion of Cdc42 and PTEN and the inability of cells lacking active Cdc42 to properly detect chemoattractant gradients.

Stochastic approaches in phyllotaxis

Theoretical models of phyllotaxis are based on geometric regularities appearing at the level of the shoot apical meristem (SAM). However, one cannot forget the presence of perturbed patterns in many plants. Disorganized patterns found in mutants of Arabidopsis and Antirrhinum bring new theoretical problems that cannot be solved by using models developed to analyse regular phyllotactic patterns. One way to take into account the perturbed patterns is to use a probabilistic approach to phyllotaxis. This review will focus mainly on recent probabilistic approaches that can be used to analyse perturbed patterns found in the plant kingdom in general and in phyllotactic mutants in particular. More precisely, it will be shown how probabilistic approaches can be used to determine the degree of order of phyllotactic patterns. By using particular tests, it is possible to statistically differentiate between whorled and distichous patterns (aggregated dispersion), spiral patterns (uniform dispersion), and random patterns (random dispersion). The elaboration of a general probabilistic model of phyllotaxis represents a new challenge for both theoretical and experimental research.

Polarization and Movement of Keratocytes: A Multiscale Modelling Approach

Eukariotic cell motility is a complex phenomenon, in which the cytoskeleton and its major constituent, actin, play an essential role. Actin forms polymers of long, stiff filaments that are cross-linked into an anisotropic network inside a thin sheet-like cellular protrusion, the lamellipod. At the leading edge of this structure, polymerization of actin filaments creates the force that pushes out the membrane and leads to translocation of a motile cell. Dynamics of the actin network account for changes in cell shape, crawling motion and turning of the cell in response to external cues. Regulating the dynamics of the cytoskeleton, and playing a central role in signal transduction in the cell, are Cdc42, Rac and Rho (GTPases of the rho family, collectively known as the small G-proteins) and the actin nucleating complex, Arp2/3. In this paper, we use a multiscale modelling approach in a 2D model of a motile cell. We describe the mutual interactions of the small G-proteins, and their effects on capping and side-branching of actin filaments. We incorporate the pushing exerted by oriented actin filament ends on the cell edge, and a Rho-dependent contraction force. Combining these biochemical and mechanical aspects, we investigate the dynamics of a model epidermal fish keratocyte through in silico experiments. Our model gives insight into how, in response to some cue, a cell can polarize, form a leading edge, and move; concomitantly it explains how a keratocyte cell can maintain its shape and polarity, even after removal of the initial stimulus, and how it can change direction quickly in response to changes in its environment. We show that establishment of polarity stems from interactions of Cdc42, Rac and Rho, while maintenance and robustness of polarity is due to the rapid cytosolic diffusion of the inactive (GDI-bound) forms of the small G-proteins. Our model produces a cell shape that closely resembles the keratocytes and correct speeds for biologically reasonable parameter values. Movies of the simulations can be obtained from http://theory.bio.uu.nl/stan/keratocyte.

Contact pressure models for spiral phyllotaxis and their computer simulation

We present a simple, biologically motivated model for the creation of phyllotactical patterns in capitula and their computer simulation. An in-depth investigation of Ridley's contact pressure model is performed and a refinement of contact pressure between primordia on the basis of local centroidal Voronoi relaxation is presented. Using this method in combination with Hofmeisters rule for placing new primordia creates stable patterns with Fibonacci spirals of high degree for a large range of initial conditions.

Patterning of proneuronal and inter-proneuronal domains by hairy- and enhancer of split-related genes in zebrafish neuroectoderm

In teleosts and amphibians, the proneuronal domains, which give rise to primary-motor, primary-inter and Rohon-Beard (RB) neurons, are established at the beginning of neurogenesis as three longitudinal stripes along the anteroposterior axis in the dorsal ectoderm. The proneuronal domains are prefigured by the expression of basic helix-loop-helix (bHLH) proneural genes,and separated by domains (inter-proneuronal domains) that do not express the proneural genes. Little is known about how the formation of these domains is spatially regulated. We have found that the zebrafish hairy- and enhancer of split-related (Her) genes her3 and her9are expressed in the inter-proneuronal domains, and are required for their formation. her3 and her9 expression was not regulated by Notch signaling, but rather controlled by positional cues, in which Bmp signaling is involved. Inhibition of Her3 or Her9 by antisense morpholino oligonucleotides led to ectopic expression of the proneural genes in part of the inter-proneuronal domains. Combined inhibition of Her3 and Her9 induced ubiquitous expression of proneural and neuronal genes in the neural plate, and abolished the formation of the inter-proneuronal domains. Furthermore,inhibition of Her3/Her9 and Notch signaling led to ubiquitous and homogeneous expression of proneural and neuronal genes in the neural plate, revealing that Her3/Her9 and Notch signaling have distinct roles in neurogenesis. These data indicate that her3 and her9 function as prepattern genes that link the positional dorsoventral polarity information in the posterior neuroectoderm to the spatial regulation of neurogenesis.