Scaling morphogen gradients during tissue growth by a cell division rule

Morphogens in the evolution of size, shape and patterning

Much of the striking diversity of life on Earth has arisen from variations in the way that the same molecules and networks operate during development to shape and pattern tissues and organs into different morphologies. However, we still understand very little about the potential for diversification exhibited by different, highly conserved mechanisms during evolution, or, conversely, the constraints that they place on evolution. With the aim of steering the field in new directions, we focus on morphogen-mediated patterning and growth as a case study to demonstrate how conserved developmental mechanisms can adapt during evolution to drive morphological diversification and optimise functionality, and to illustrate how evolution algorithms and computational tools can be used alongside experiments to provide insights into how these conserved mechanisms can evolve. We first introduce key conserved properties of morphogen-driven patterning mechanisms, before summarising comparative studies that exemplify how changes in the spatiotemporal expression and signalling levels of morphogens impact the diversification of organ size, shape and patterning in nature. Finally, we detail how theoretical frameworks can be used in conjunction with experiments to probe the role of morphogen-driven patterning mechanisms in evolution. We conclude that morphogen-mediated patterning is an excellent model system and offers a generally applicable framework to investigate the evolution of developmental mechanisms.

Bone morphogenetic protein signaling: the pathway and its regulation

In the mid-1960s, bone morphogenetic proteins (BMPs) were first identified in the extracts of bone to have the remarkable ability to induce heterotopic bone. When the Drosophila gene decapentaplegic (dpp) was first identified to share sequence similarity with mammalian BMP2/BMP4 in the late-1980s, it became clear that secreted BMP ligands can mediate processes other than bone formation. Following this discovery, collaborative efforts between Drosophila geneticists and mammalian biochemists made use of the strengths of their respective model systems to identify BMP signaling components and delineate the pathway. The ability to conduct genetic modifier screens in Drosophila with relative ease was critical in identifying the intracellular signal transducers for BMP signaling and the related transforming growth factor-beta/activin signaling pathway. Such screens also revealed a host of genes that encode other core signaling components and regulators of the pathway. In this review, we provide a historical account of this exciting time of gene discovery and discuss how the field has advanced over the past 30 years. We have learned that while the core BMP pathway is quite simple, composed of 3 components (ligand, receptor, and signal transducer), behind the versatility of this pathway lies multiple layers of regulation that ensures precise tissue-specific signaling output. We provide a sampling of these discoveries and highlight many questions that remain to be answered to fully understand the complexity of BMP signaling.

Maintenance of appropriate size scaling of the C. elegans pharynx by YAP-1

Even slight imbalance between the growth rate of different organs can accumulate to a large deviation from their appropriate size during development. Here, we use live imaging of the pharynx of C. elegans to ask if and how organ size scaling nevertheless remains uniform among individuals. Growth trajectories of hundreds of individuals reveal that pharynxes grow by a near constant volume per larval stage that is independent of their initial size, such that undersized pharynxes catch-up in size during development. Tissue-specific depletion of RAGA-1, an activator of mTOR and growth, shows that maintaining correct pharynx-to-body size proportions involves a bi-directional coupling between pharynx size and body growth. In simulations, this coupling cannot be explained by limitation of food uptake alone, and genetic experiments reveal an involvement of the mechanotransducing transcriptional co-regulator yap-1. Our data suggests that mechanotransduction coordinates pharynx growth with other tissues, ensuring body plan uniformity among individuals.

Gradients of glucose metabolism regulate morphogen signalling required for specifying tonotopic organisation in the chicken cochlea

In vertebrates with elongated auditory organs, mechanosensory hair cells (HCs) are organised such that complex sounds are broken down into their component frequencies along a proximal-to-distal long (tonotopic) axis. Acquisition of unique morphologies at the appropriate position along the chick cochlea, the basilar papilla, requires that nascent HCs determine their tonotopic positions during development. The complex signalling within the auditory organ between a developing HC and its local niche along the cochlea is poorly understood. Using a combination of live imaging and NAD(P)H fluorescence lifetime imaging microscopy, we reveal that there is a gradient in the cellular balance between glycolysis and the pentose phosphate pathway in developing HCs along the tonotopic axis. Perturbing this balance by inhibiting different branches of cytosolic glucose catabolism disrupts developmental morphogen signalling and abolishes the normal tonotopic gradient in HC morphology. These findings highlight a causal link between graded morphogen signalling and metabolic reprogramming in specifying the tonotopic identity of developing HCs.

Tissue interplay during morphogenesis

The process by which biological systems such as cells, tissues and organisms acquire shape has been named as morphogenesis and it is central to a plethora of biological contexts including embryo development, wound healing, or even cancer. Morphogenesis relies in both self-organising properties of the system and in environmental inputs (biochemical and biophysical). The classical view of morphogenesis is based on the study of external biochemical molecules, such as morphogens. However, recent studies are establishing that the mechanical environment is also used by cells to communicate within tissues, suggesting that this mechanical crosstalk is essential to synchronise morphogenetic transitions and self-organisation. In this article we discuss how tissue interaction drive robust morphogenesis, starting from a classical biochemical view, to finalise with more recent advances on how the biophysical properties of a tissue feedback with their surroundings to allow form acquisition. We also comment on how in silico models aid to integrate and predict changes in cell and tissue behaviour. Finally, considering recent advances from the developmental biomechanics field showing that mechanical inputs work as cues that promote morphogenesis, we invite to revisit the concept of morphogen.

Cytokinin promotes growth cessation in the Arabidopsis root

The Arabidopsis root offers good opportunities to investigate how regulated cellular growth shapes different tissues and organs, a key question in developmental biology. Along the root's longitudinal axis, cells sequentially occupy different developmental states. Proliferative meristematic cells give rise to differentiating cells, which rapidly elongate in the elongation zone, then mature and stop growing in the differentiation zone. The phytohormone cytokinin contributes to this zonation by positioning the boundary between the meristem and the elongation zone, called the transition zone. However, the cellular growth profile underlying root zonation is not well understood, and the cellular mechanisms that mediate growth cessation remain unclear. By using time-lapse imaging, genetics, and computational analysis, we analyze the effect of cytokinin on root zonation and cellular growth. We found that cytokinin promotes growth cessation in the distal (shootward) elongation zone in conjunction with accelerating the transition from elongation to differentiation. We estimated cell-wall stiffness by using osmotic treatment experiments and found that cytokinin-mediated growth cessation is associated with cell-wall stiffening and requires the action of an auxin influx carrier, AUX1. Our measurement of growth and cell-wall mechanical properties at a cellular resolution reveal mechanisms via which cytokinin influences cell behavior to shape tissue patterns.

Lévy Walk Dynamics in an External Constant Force Field in Non-Static Media

Based on the recognition of the huge change of the transport properties for diffusion particles in non-static media, we consider a Lévy walk model subjected to an external constant force in non-static media. Since the physical and comoving coordinates of non-static media are related by scale factor, we equivalently transfer the process from physical coordinate into comoving coordinate and derive the master equation governing the probability density function of the position of the particles in comoving coordinate. Utilizing the Hermite orthogonal polynomial expansions, some statistical properties are obtained, including the asymptotic behaviors of the first two moments in both coordinates and kurtosis. For some representative types of non-static media and Lévy walks, the striking and interesting phenomena originating from the interplay between non-static media, external force, and intrinsic stochastic motion are observed. The stationary distribution are also analyzed for some cases through numerical simulations.

New styrylquinoxaline: synthesis, structural, biological evaluation, ADMET prediction and molecular docking investigations

The organic compound (E)-3-(4-methylstyryl)quinoxalin-2(1H)-one (SQO) with molecular formula C₁₇H₁₄N₂O was synthesized and analyzed using single crystal X-ray diffraction, ¹H, ¹³C NMR and FTIR spectroscopic techniques. The geometric parameters of the molecule was optimized by density-functional theory (DFT) choosing B3LYP with 6-31++G(d,p) basis set. For compatibility, the theoretical structure and experimental structure were overlapped with each other. Frontier molecular orbitals of the title compound were made, and energy gap between HOMO and LUMO was calculated. Molecular electrostatic potential map was generated finding electrophilic and nucleophilic attack centers using DFT method. Hirshfeld surface analysis (HSA) confirms active regions at the circumference of N1 atoms and O1 atoms that form intermolecular N1-H1···O1 hydrogen bond. The acute oral toxicity study was carried out according to OECD guideline, which approve that the compound SQO was non-toxic. In addition, this quinoxaline derivative was evaluated for its in vitro antidiabetic activity against α-glucosidase and α-amylase enzymes and for antioxidant activity by utilizing several tests as 1,1-diphenyl-2-picryl hydrazyl, (2,2'-azino-bis(3-ethyl benzthiazoline-6-sulfonicacid), reducing power test (FRAP) and hydrogen peroxide activity H₂O₂. The molecular docking studies were performed to investigate the antidiabetic activity of SQO and compared with the experimental results. SQO is a potent antidiabetic from both the experimental and molecular docking results. Finally, the physicochemical, pharmacokinetic and toxicological properties of SQO have been evaluated by using in silico absorption, distribution, metabolism, excretion and toxicity analysis prediction.

Morphogen gradient scaling by recycling of intracellular Dpp

Morphogen gradients are fundamental to establish morphological patterns in developing tissues¹. During development, gradients scale to remain proportional to the size of growing organs^2,3. Scaling is a universal gear adjusting patterns to size in living organisms^3-8, yet its mechanisms remain unclear. Here, focusing on the Dpp gradient in the Drosophila wing disc, we unravel a cell biological basis behind scaling. From small to large discs, scaling of the Dpp gradient is achieved by increasing the contribution of the internalized Dpp molecules to Dpp transport: to expand the gradient, endocytosed molecules are re-exocytosed to spread extracellularly. To regulate the contribution of endocytosed Dpp to the spreading extracellular pool during tissue growth, it is the Dpp binding rates that are progressively modulated by the extracellular factor Pentagone, driving scaling. Thus, for some morphogens, evolution may act on endocytic trafficking to regulate the range of the gradient and its scaling, which could allow adaptation of shape and pattern to different sizes of organs in different species.

Diterpenes/Diterpenoids and Their Derivatives as Potential Bioactive Leads against Dengue Virus: A Computational and Network Pharmacology Study

Dengue fever is a dangerous infectious endemic disease that affects over 100 nations worldwide, from Africa to the Western Pacific, and is caused by the dengue virus, which is transmitted to humans by an insect bite of Aedes aegypti. Millions of citizens have died as a result of dengue fever and dengue hemorrhagic fever across the globe. Envelope (E), serine protease (NS3), RNA-directed RNA polymerase (NS5), and non-structural protein 1 (NS1) are mostly required for cell proliferation and survival. Some of the diterpenoids and their derivatives produced by nature possess anti-dengue viral properties. The goal of the computational study was to scrutinize the effectiveness of diterpenoids and their derivatives against dengue viral proteins through in silico study. Methods: molecular docking was performed to analyze the binding affinity of compounds against four viral proteins: the envelope (E) protein, the NS1 protein, the NS3 protein, and the NS5 protein. Results: among the selected drug candidates, triptolide, stevioside, alepterolic acid, sphaeropsidin A, methyl dodovisate A, andrographolide, caesalacetal, and pyrimethamine have demonstrated moderate to good binding affinities (-8.0 to -9.4 kcal/mol) toward the selected proteins: E protein, NS3, NS5, and NS1 whereas pyrimethamine exerts -7.5, -6.3, -7.8, and -6.6 kcal/mol with viral proteins, respectively. Interestingly, the binding affinities of these lead compounds were better than those of an FDA-approved anti-viral medication (pyrimethamine), which is underused in dengue fever. Conclusion: we can conclude that diterpenoids can be considered as a possible anti-dengue medication option. However, in vivo investigation is recommended to back up the conclusions of this study.

Ediacara growing pains: Modular addition and development in Dickinsonia costata

Constraining patterns of growth using directly observable and quantifiable characteristics can reveal a wealth of information regarding the biology of the Ediacara Biota - the oldest macroscopic, complex community forming organisms in the fossil record. However, these rely on individuals captured at an instant in time at various growth stages, and so different interpretations can be derived from the same material. Here we leverage newly discovered and well-preserved Dickinsonia costata Sprigg 1947 from South Australia, combined with hundreds of previously described specimens, to test competing hypotheses for the location of module addition. We find considerable variation in the relationship between the total number of modules and body size that cannot be explained solely by expansion and contraction of individuals. Patterns derived assuming new modules differentiated at the anterior result in numerous examples where the oldest module(s) must decrease in size with overall growth, potentially falsifying this hypothesis. Observed polarity as well as the consistent posterior location of defects and indentations support module formation at this end in D. costata. Regardless, changes in repeated units with growth share similarities with those regulated by morphogen gradients in metazoans today, suggesting that these genetic pathways were operating in Ediacaran animals.

Scalable control of developmental timetables by epigenetic switching networks

During development, progenitor cells follow timetables for differentiation that span many cell generations. These developmental timetables are robustly encoded by the embryo, yet scalably adjustable by evolution, facilitating variation in organism size and form. Epigenetic switches, involving rate-limiting activation steps at regulatory gene loci, control gene activation timing in diverse contexts, and could profoundly impact the dynamics of gene regulatory networks controlling developmental lineage specification. Here, we develop a mathematical framework to model regulatory networks with genes controlled by epigenetic switches. Using this framework, we show that such epigenetic switching networks uphold developmental timetables that robustly span many cell generations, and enable the generation of differentiated cells in precisely defined numbers and fractions. Changes to epigenetic switching networks can readily alter the timing of developmental events within a timetable, or alter the overall speed at which timetables unfold, enabling scalable control over differentiated population sizes. With their robust, yet flexibly adjustable nature, epigenetic switching networks could represent central targets on which evolution acts to manufacture diversity in organism size and form.

Reconstitution of Morphogen Signaling Gradients in Cultured Cells

Development of multicellular organisms depends on the proper establishment of signaling information in space and time. Secreted molecules called morphogens form concentration gradients in space and provide positional information to differentiating cells within the organism. Although the key molecular components of morphogen pathways have been identified, how the architectures and key parameters of morphogen pathways control the properties of signaling gradients, such as their size, speed, and robustness to perturbations, remains challenging to study in developing embryos. Reconstituting morphogen gradients in cell culture provides an alternative approach to address this question. Here we describe the methodology for reconstituting Sonic Hedgehog (SHH) signaling gradients in mouse fibroblast cells. The protocol includes the design of morphogen sending and receiving cell lines, the setup of radial and linear gradients, the quantitative time-lapse imaging, and the data analysis. Similar approaches could potentially be applied to other cell-cell communication pathways.

The development of body and organ shape

Background

Organisms show an incredibly diverse array of body and organ shapes that are both unique to their taxon and important for adapting to their environment. Achieving these specific shapes involves coordinating the many processes that transform single cells into complex organs, and regulating their growth so that they can function within a fully-formed body.

Main text

Conceptually, body and organ shape can be separated in two categories, although in practice these categories need not be mutually exclusive. Body shape results from the extent to which organs, or parts of organs, grow relative to each other. The patterns of relative organ size are characterized using allometry. Organ shape, on the other hand, is defined as the geometric features of an organ’s component parts excluding its size. Characterization of organ shape is frequently described by the relative position of homologous features, known as landmarks, distributed throughout the organ. These descriptions fall into the domain of geometric morphometrics.

Conclusion

In this review, we discuss the methods of characterizing body and organ shape, the developmental programs thought to underlie each, highlight when and how the mechanisms regulating body and organ shape might overlap, and provide our perspective on future avenues of research.

Reaction-diffusion and reaction-subdiffusion equations on arbitrarily evolving domains

Reaction-diffusion equations are widely used as the governing evolution equations for modeling many physical, chemical, and biological processes. Here we derive reaction-diffusion equations to model transport with reactions on a one-dimensional domain that is evolving. The model equations, which have been derived from generalized continuous time random walks, can incorporate complexities such as subdiffusive transport and inhomogeneous domain stretching and shrinking. Inhomogeneously growing domains are frequently encountered in biological phenomena involving stochastic transport, such as tumor growth and morphogen gradient formation. A method for constructing analytic expressions for short-time moments of the position of the particles is developed and moments calculated from this approach are shown to compare favorably with results from random walk simulations and numerical integration of the reaction transport equation. The results show the important role played by the initial condition. In particular, it strongly affects the time dependence of the moments in the short-time regime by introducing additional drift and diffusion terms. We also discuss how our reaction transport equation could be applied to study the spreading of a population on an evolving interface. From a more general perspective, our findings help to mitigate the scarcity of analytic results for reaction-diffusion problems in geometries displaying nonuniform growth. They are also expected to pave the way for further results, including the treatment of first-passage problems associated with encounter-controlled reactions in such domains.

A dynamic cell recruitment process drives growth of the Drosophila wing by overscaling the vestigial expression pattern

Organs mainly attain their size by cell growth and proliferation, but sometimes also grow through recruitment of undifferentiated cells. Here we investigate the participation of cell recruitment in establishing the pattern of Vestigial (Vg), the product of the wing selector gene in Drosophila. We find that the Vg pattern overscales along the dorsal-ventral (DV) axis of the wing imaginal disc, i.e., it expands faster than the DV length of the pouch. The overscaling of the Vg pattern cannot be explained by differential proliferation, apoptosis, or oriented-cell divisions, but can be recapitulated by a mathematical model that explicitly considers cell recruitment. When impairing cell recruitment genetically, we find that the Vg pattern almost perfectly scales and adult wings are approximately 20% smaller. Conversely, impairing cell proliferation results in very small wings, suggesting that cell recruitment and cell proliferation additively contribute to organ growth in this system. Furthermore, using fluorescent reporter tools, we provide direct evidence that cell recruitment is initiated between early and mid third-instar larval development. Altogether, our work quantitatively shows when, how, and by how much cell recruitment shapes the Vg pattern and drives growth of the Drosophila wing.

Regulation of size and scale in vertebrate spinal cord development

All vertebrates have a spinal cord with dimensions and shape specific to their species. Yet how species‐specific organ size and shape are achieved is a fundamental unresolved question in biology. The formation and sculpting of organs begins during embryonic development. As it develops, the spinal cord extends in anterior–posterior direction in synchrony with the overall growth of the body. The dorsoventral (DV) and apicobasal lengths of the spinal cord neuroepithelium also change, while at the same time a characteristic pattern of neural progenitor subtypes along the DV axis is established and elaborated. At the basis of these changes in tissue size and shape are biophysical determinants, such as the change in cell number, cell size and shape, and anisotropic tissue growth. These processes are controlled by global tissue‐scale regulators, such as morphogen signaling gradients as well as mechanical forces. Current challenges in the field are to uncover how these tissue‐scale regulatory mechanisms are translated to the cellular and molecular level, and how regulation of distinct cellular processes gives rise to an overall defined size. Addressing these questions will help not only to achieve a better understanding of how size is controlled, but also of how tissue size is coordinated with the specification of pattern.

This article is categorized under:

Establishment of Spatial and Temporal Patterns > Regulation of Size, Proportion, and Timing

Signaling Pathways > Global Signaling Mechanisms

Nervous System Development > Vertebrates: General Principles

BMP Signaling Gradient Scaling in the Zebrafish Pectoral Fin

Secreted growth factors can act as morphogens that form spatial concentration gradients in developing organs, thereby controlling growth and patterning. For some morphogens, adaptation of the gradients to tissue size allows morphological patterns to remain proportioned as the organs grow. In the zebrafish pectoral fin, we found that BMP signaling forms a two-dimensional gradient. The length of the gradient scales with tissue length and its amplitude increases with fin size according to a power-law. Gradient scaling and amplitude power-laws are signatures of growth control by time derivatives of morphogenetic signaling: cell division correlates with the fold change over time of the cellular signaling levels. We show that Smoc1 regulates BMP gradient scaling and growth in the fin. Smoc1 scales the gradient by means of a feedback loop: Smoc1 is a BMP agonist and BMP signaling represses Smoc1 expression. Our work uncovers a layer of morphogen regulation during vertebrate appendage development.

Positional information and tissue scaling during development and regeneration

In order to contribute to the appropriate tissues during development, cells need to know their position within the embryo. This positional information is conveyed by gradients of signaling molecules, termed morphogens, that are produced in specific regions of the embryo and induce concentration-dependent responses in target tissues. Positional information is remarkably robust, and embryos often develop with the correct proportions even if large parts of the embryo are removed. In this Review, we discuss classical embryological experiments and modern quantitative analyses that have led to mechanistic insights into how morphogen gradients adapt, scale and properly pattern differently sized domains. We analyze these experimental findings in the context of mathematical models and synthesize general principles that apply to multiple systems across species and developmental stages.

Standard and fractional Ornstein-Uhlenbeck process on a growing domain

We study normal diffusive and subdiffusive processes in a harmonic potential (Ornstein-Uhlenbeck process) on a uniformly growing or contracting domain. Our starting point is a recently derived fractional Fokker-Planck equation, which covers both the case of Brownian diffusion and the case of a subdiffusive continuous-time random walk (CTRW). We find a high sensitivity of the random walk properties to the details of the domain growth rate, which gives rise to a variety of regimes with extremely different behaviors. At the origin of this rich phenomenology is the fact that the walkers still move while they wait to jump, since they are dragged by the deterministic drift arising from the domain growth. Thus, the increasingly long waiting times associated with the aging of the subdiffusive CTRW imply that, in the time interval between two consecutive jumps, the walkers might travel over much longer distances than in the normal diffusive case. This gives rise to seemingly counterintuitive effects. For example, on a static domain, both Brownian diffusion and subdiffusive CTRWs yield a stationary particle distribution with finite width when a harmonic potential is at play, thus indicating a confinement of the diffusing particle. However, for a sufficiently fast growing or contracting domain, this qualitative behavior breaks down, and differences between the Brownian case and the subdiffusive case are found. In the case of Brownian particles, a sufficiently fast exponential domain growth is needed to break the confinement induced by the harmonic force; in contrast, for subdiffusive particles such a breakdown may already take place for a sufficiently fast power-law domain growth. Our analytic and numerical results for both types of diffusion are fully confirmed by random walk simulations.

Critical Point in Self-Organized Tissue Growth

We present a theory of pattern formation in growing domains inspired by biological examples of tissue development. Gradients of signaling molecules regulate growth, while growth changes these graded chemical patterns by dilution and advection. We identify a critical point of this feedback dynamics, which is characterized by spatially homogeneous growth and proportional scaling of patterns with tissue length. We apply this theory to the biological model system of the developing wing of the fruit fly Drosophila melanogaster and quantitatively identify signatures of the critical point.

The topological requirements for robust perfect adaptation in networks of any size

Robustness, and the ability to function and thrive amid changing and unfavorable environments, is a fundamental requirement for living systems. Until now it has been an open question how large and complex biological networks can exhibit robust behaviors, such as perfect adaptation to a variable stimulus, since complexity is generally associated with fragility. Here we report that all networks that exhibit robust perfect adaptation (RPA) to a persistent change in stimulus are decomposable into well-defined modules, of which there exist two distinct classes. These two modular classes represent a topological basis for all RPA-capable networks, and generate the full set of topological realizations of the internal model principle for RPA in complex, self-organizing, evolvable bionetworks. This unexpected result supports the notion that evolutionary processes are empowered by simple and scalable modular design principles that promote robust performance no matter how large or complex the underlying networks become.

Robust perfect adaptation (RPA), the ability of a system to return to its pre-stimulus state in the presence of a new signal, enables organisms to respond to further changes in stimuli. Here, the authors identify the modular structure of the full set of network topologies that can confer RPA on complex networks.

Scaling of Morphogenetic Patterns

Mathematical studies of morphogenetic pattern formation are commonly performed by using reaction-diffusion equations that describe the dynamics of morphogen concentration. Various features of the modeled patterns, including their ability to scale, are analyzed to justify constructed models and to understand the processes responsible for these features in nature. In this chapter, we introduce a method for evaluation of scaling for patterns arising in mathematical models and demonstrate its use by applying it to a set of different models. We introduce a quantity representing the sensitivity of a pattern to changes in the size of the domain, where it forms, and we show how to use it to perform a formal analysis of scaling for chemical patterns forming in continuous systems.

Spatially Different Tissue-Scale Diffusivity Shapes ANGUSTIFOLIA3 Gradient in Growing Leaves

The spatial gradient of signaling molecules is pivotal for establishing developmental patterns of multicellular organisms. It has long been proposed that these gradients could arise from the pure diffusion process of signaling molecules between cells, but whether this simplest mechanism establishes the formation of the tissue-scale gradient remains unclear. Plasmodesmata are unique channel structures in plants that connect neighboring cells for molecular transport. In this study, we measured cellular- and tissue-scale kinetics of molecular transport through plasmodesmata in Arabidopsis thaliana developing leaf primordia by fluorescence recovery assays. These trans-scale measurements revealed biophysical properties of diffusive molecular transport through plasmodesmata and revealed that the tissue-scale diffusivity, but not the cellular-scale diffusivity, is spatially different along the leaf proximal-to-distal axis. We found that the gradient in cell size along the developmental axis underlies this spatially different tissue-scale diffusivity. We then asked how this diffusion-based framework functions in establishing a signaling gradient of endogenous molecules. ANGUSTIFOLIA3 (AN3) is a transcriptional co-activator, and as we have shown here, it forms a long-range signaling gradient along the leaf proximal-to-distal axis to determine a cell-proliferation domain. By genetically engineering AN3 mobility, we assessed each contribution of cell-to-cell movement and tissue growth to the distribution of the AN3 gradient. We constructed a diffusion-based theoretical model using these quantitative data to analyze the AN3 gradient formation and demonstrated that it could be achieved solely by the diffusive molecular transport in a growing tissue. Our results indicate that the spatially different tissue-scale diffusivity is a core mechanism for AN3 gradient formation. This provides evidence that the pure diffusion process establishes the formation of the long-range signaling gradient in leaf development.

Continuous-time random-walk model for anomalous diffusion in expanding media

Expanding media are typical in many different fields, e.g., in biology and cosmology. In general, a medium expansion (contraction) brings about dramatic changes in the behavior of diffusive transport properties such as the set of positional moments and the Green's function. Here, we focus on the characterization of such effects when the diffusion process is described by the continuous-time random-walk (CTRW) model. As is well known, when the medium is static this model yields anomalous diffusion for a proper choice of the probability density function (pdf) for the jump length and the waiting time, but the behavior may change drastically if a medium expansion is superimposed on the intrinsic random motion of the diffusing particle. For the case where the jump length and the waiting time pdfs are long-tailed, we derive a general bifractional diffusion equation which reduces to a normal diffusion equation in the appropriate limit. We then study some particular cases of interest, including Lévy flights and subdiffusive CTRWs. In the former case, we find an analytical exact solution for the Green's function (propagator). When the expansion is sufficiently fast, the contribution of the diffusive transport becomes irrelevant at long times and the propagator tends to a stationary profile in the comoving reference frame. In contrast, for a contracting medium a competition between the spreading effect of diffusion and the concentrating effect of contraction arises. In the specific case of a subdiffusive CTRW in an exponentially contracting medium, the latter effect prevails for sufficiently long times, and all the particles are eventually localized at a single point in physical space. This "big crunch" effect, totally absent in the case of normal diffusion, stems from inefficient particle spreading due to subdiffusion. We also derive a hierarchy of differential equations for the moments of the transport process described by the subdiffusive CTRW model in an expanding medium. From this hierarchy, the full time evolution of the second-order moment is obtained for some specific types of expansion. In the case of an exponential expansion, exact recurrence relations for the Laplace-transformed moments are obtained, whence the long-time behavior of moments of arbitrary order is subsequently inferred. Our analytical and numerical results for both Lévy flights and subdiffusive CTRWs confirm the intuitive expectation that the medium expansion hinders the mixing of diffusive particles occupying separate regions. In the case of Lévy flights, we quantify this effect by means of the so-called "Lévy horizon."

Finite cell-size effects on protein variability in Turing patterned tissues

Herein we present a framework to characterize different sources of protein expression variability in Turing patterned tissues. In this context, we introduce the concept of granular noise to account for the unavoidable fluctuations due to finite cell-size effects and show that the nearest-neighbours autocorrelation function provides the means to measure it. To test our findings, we perform in silico experiments of growing tissues driven by a generic activator-inhibitor dynamics. Our results show that the relative importance of different sources of noise depends on the ratio between the characteristic size of cells and that of the pattern domains and on the ratio between the pattern amplitude and the effective intensity of the biochemical fluctuations. Importantly, our framework provides the tools to measure and distinguish different stochastic contributions during patterning: granularity versus biochemical noise. In addition, our analysis identifies the protein species that buffer the stochasticity the best and, consequently, it can help to determine key instructive signals in systems driven by a Turing instability. Altogether, we expect our study to be relevant in developmental processes leading to the formation of periodic patterns in tissues.

Gene expression boundary scaling and organ size regulation in the Drosophila embryo

How the shape and size of tissues and organs is regulated during development is a major question in developmental biology. Such regulation relies upon both intrinsic cues (such as signaling networks) and extrinsic inputs (such as from neighboring tissues). Here, we focus on pattern formation and organ development during Drosophila embryogenesis. In particular, we outline the importance of both biochemical and mechanical tissue-tissue interactions in size regulation. We describe how the Drosophila embryo can potentially provide novel insights into how shape and size are regulated during development. We focus on gene expression boundary scaling in the early embryo and how size is regulated in three organs (hindgut, trachea, and ventral nerve cord) later in development, with particular focus on the role of tissue-tissue interactions. Overall, we demonstrate that Drosophila embryogenesis provides a suitable model system for studying spatial and temporal scaling and size control in vivo.

Collective Space-Sensing Coordinates Pattern Scaling in Engineered Bacteria

Scale invariance refers to the maintenance of a constant ratio of developing organ size to body size. Although common, its underlying mechanisms remain poorly understood. Here, we examined scaling in engineered Escherichia coli that can form self-organized core-ring patterns in colonies. We found that the ring width exhibits perfect scale invariance to the colony size. Our analysis revealed a collective space-sensing mechanism, which entails sequential actions of an integral feedback loop and an incoherent feedforward loop. The integral feedback is implemented by the accumulation of a diffusive chemical produced by a colony. This accumulation, combined with nutrient consumption, sets the timing for ring initiation. The incoherent feedforward is implemented by the opposing effects of the domain size on the rate and duration of ring maturation. This mechanism emphasizes a role of timing control in achieving robust pattern scaling and provides a new perspective in examining the phenomenon in natural systems.

A Model of the Spatio-temporal Dynamics of Drosophila Eye Disc Development

Patterning and growth are linked during early development and have to be tightly controlled to result in a functional tissue or organ. During the development of the Drosophila eye, this linkage is particularly clear: the growth of the eye primordium mainly results from proliferating cells ahead of the morphogenetic furrow (MF), a moving signaling wave that sweeps across the tissue from the posterior to the anterior side, that induces proliferating cells anterior to it to differentiate and become cell cycle quiescent in its wake. Therefore, final eye disc size depends on the proliferation rate of undifferentiated cells and on the speed with which the MF sweeps across the eye disc. We developed a spatio-temporal model of the growing eye disc based on the regulatory interactions controlled by the signals Decapentaplegic (Dpp), Hedgehog (Hh) and the transcription factor Homothorax (Hth) and explored how the signaling patterns affect the movement of the MF and impact on eye disc growth. We used published and new quantitative data to parameterize the model. In particular, two crucial parameter values, the degradation rate of Hth and the diffusion coefficient of Hh, were measured. The model is able to reproduce the linear movement of the MF and the termination of growth of the primordium. We further show that the model can explain several mutant phenotypes, but fails to reproduce the previously observed scaling of the Dpp gradient in the anterior compartment.

Integrated live imaging and molecular profiling of embryoid bodies reveals a synchronized progression of early differentiation

Embryonic stem cells can spontaneously differentiate into cell types of all germ layers within embryoid bodies (EBs) in a highly variable manner. Whether there exists an intrinsic differentiation program common to all EBs is unknown. Here, we present a novel combination of high-throughput live two-photon imaging and gene expression profiling to study early differentiation dynamics spontaneously occurring within developing EBs. Onset timing of Brachyury-GFP was highly variable across EBs, while the spatial patterns as well as the dynamics of mesendodermal progression following onset were remarkably similar. We therefore defined a 'developmental clock' using the Brachyury-GFP signal onset timing. Mapping snapshot gene expression measurements to this clock revealed their temporal trends, indicating that loss of pluripotency, formation of primitive streak and mesodermal lineage progression are synchronized in EBs. Exogenous activation of Wnt or BMP signaling accelerated the intrinsic clock. CHIR down-regulated Wnt3, allowing insights into dependency mechanisms between canonical Wnt signaling and multiple genes. Our findings reveal a developmental clock characteristic of an early differentiation program common to all EBs, further establishing them as an in vitro developmental model.

Developmental Pattern Formation in Phases

Cells in developing organs undergo a series of changes in their transcriptional state until a complete repertoire of cell types is specified. These changes in cell identity, together with the control of tissue growth, determine the pattern of gene expression in the tissue. Recent studies explore the dynamics of pattern formation during development and provide new insights into the control mechanisms. Changes in morphogen signalling and transcriptional networks control the specification of cell types. This is often followed by a distinct second phase, where pattern is elaborated by tissue growth. Here, we discuss the transitions between distinct phases in pattern formation. We consider the implications of the underlying mechanisms for understanding how reproducible patterns form during development.

New Insights into Mechanisms and Functions of Nuclear Size Regulation

Nuclear size is generally maintained within a defined range in a given cell type. Changes in cell size that occur during cell growth, development, and differentiation are accompanied by dynamic nuclear size adjustments in order to establish appropriate nuclear-to-cytoplasmic volume relationships. It has long been recognized that aberrations in nuclear size are associated with certain disease states, most notably cancer. Nuclear size and morphology must impact nuclear and cellular functions. Understanding these functional implications requires an understanding of the mechanisms that control nuclear size. In this review, we first provide a general overview of the diverse cellular structures and activities that contribute to nuclear size control, including structural components of the nucleus, effects of DNA amount and chromatin compaction, signaling, and transport pathways that impinge on the nucleus, extranuclear structures, and cell cycle state. We then detail some of the key mechanistic findings about nuclear size regulation that have been gleaned from a variety of model organisms. Lastly, we review studies that have implicated nuclear size in the regulation of cell and nuclear function and speculate on the potential functional significance of nuclear size in chromatin organization, gene expression, nuclear mechanics, and disease. With many fundamental cell biological questions remaining to be answered, the field of nuclear size regulation is still wide open.

Read-Out of Dynamic Morphogen Gradients on Growing Domains

Quantitative data from the Drosophila wing imaginal disc reveals that the amplitude of the Decapentaplegic (Dpp) morphogen gradient increases continuously. It is an open question how cells can determine their relative position within a domain based on a continuously increasing gradient. Here we show that pre-steady state diffusion-based dispersal of morphogens results in a zone within the growing domain where the concentration remains constant over the patterning period. The position of the zone that is predicted based on quantitative data for the Dpp morphogen corresponds to where the Dpp-dependent gene expression boundaries of spalt (sal) and daughters against dpp (dad) emerge. The model also suggests that genes that are scaling and are expressed at lateral positions are either under the control of a different read-out mechanism or under the control of a different morphogen. The patterning mechanism explains the extraordinary robustness that is observed for variations in Dpp production, and offers an explanation for the dual role of Dpp in controlling patterning and growth. Pre-steady-state dynamics are pervasive in morphogen-controlled systems, thus making this a probable general mechanism for the scaled read-out of morphogen gradients in growing developmental systems.

Symmetry and scale orient Min protein patterns in shaped bacterial sculptures

The boundary of a cell defines the shape and scale of its subcellular organization. However, the effects of the cell's spatial boundaries as well as the geometry sensing and scale adaptation of intracellular molecular networks remain largely unexplored. Here, we show that living bacterial cells can be 'sculpted' into defined shapes, such as squares and rectangles, which are used to explore the spatial adaptation of Min proteins that oscillate pole-to-pole in rod-shaped Escherichia coli to assist cell division. In a wide geometric parameter space, ranging from 2 × 1 × 1 to 11 × 6 × 1 μm(3), Min proteins exhibit versatile oscillation patterns, sustaining rotational, longitudinal, diagonal, stripe and even transversal modes. These patterns are found to directly capture the symmetry and scale of the cell boundary, and the Min concentration gradients scale with the cell size within a characteristic length range of 3-6 μm. Numerical simulations reveal that local microscopic Turing kinetics of Min proteins can yield global symmetry selection, gradient scaling and an adaptive range, when and only when facilitated by the three-dimensional confinement of the cell boundary. These findings cannot be explained by previous geometry-sensing models based on the longest distance, membrane area or curvature, and reveal that spatial boundaries can facilitate simple molecular interactions to result in far more versatile functions than previously understood.

The wing and the eye: a parsimonious theory for scaling and growth control?

How a developing organ grows and patterns to its final shape is an important question in developmental biology. Studies of growth and patterning in the Drosophila wing imaginal disc have identified a key player, the morphogen Decapentaplegic (Dpp). These studies provided insights into our understanding of growth control and scaling: expansion of the Dpp gradient correlated with the growth of the tissue. A recent report on growth of a Drosophila organ other than the wing, the eye imaginal disc, prompts a reconsideration of our models of growth control. Despite striking differences between the two, the Dpp gradient scales with the target tissues of both organs and the growth of both the wing and the eye is controlled by Dpp. The goal of this review is to discuss whether a parsimonious model of scaling and growth control can explain the relationship between the Dpp gradient and growth in these two different developmental systems.

Fundamental origins and limits for scaling a maternal morphogen gradient

Tissue expansion and patterning are integral to development, but it is unknown quantitatively how a mother accumulates molecular resources to invest in the future of instructing robust embryonic patterning. Here we develop a model, Tissue Expansion-Modulated Maternal Morphogen Scaling (TEM³S), to study scaled anterior-posterior patterning in Drosophila embryos. Using both ovaries and embryos, we measure a core quantity of the model, the scaling power of the Bicoid (Bcd) morphogen gradient’s amplitude n_A. We also evaluate directly model-derived predictions about Bcd gradient and patterning properties. Our results show that scaling of the Bcd gradient in the embryo originates from, and is constrained fundamentally by, a dynamic relationship between maternal tissue expansion and bcd gene copy number expansion in the ovary. This delicate connection between the two transitioning stages of a life cycle, stemming from a finite value of n_A ~ 3, underscores a key feature of developmental systems depicted by TEM³S.

Mobility of signaling molecules: the key to deciphering plant organogenesis

Signaling molecules move between cells to form a characteristic distribution pattern within a developing organ; thereafter, they spatiotemporally regulate organ development. A key question in this process is how the signaling molecules robustly form the precise distribution on a tissue scale in a reproducible manner. Despite of an increasing number of quantitative studies regarding the mobility of signaling molecules, the detail mechanism of organogenesis via intercellular signaling is still unclear. We here review the potential advantages of plant development to address this question, focusing on the cytoplasmic continuity of plant cells through the plasmodesmata. The plant system would provide a unique opportunity to define the simple transportation mode of diffusion process, and, hence, the mechanism of organogenesis via intercellular signaling. Based on the advances in the understanding of intercellular signaling at the molecular level and in the quantitative imaging techniques, we discuss our current challenges in measuring the mobility of signaling molecules for deciphering plant organogenesis.

Dynamic scaling of morphogen gradients on growing domains

Developmental mechanisms are highly conserved, yet act in embryos of very different sizes. How scaling is achieved has remained elusive. Here we identify a generally applicable mechanism for dynamic scaling on growing domains and show that it quantitatively agrees with data from the Drosophila wing imaginal disc. We show that for the measured parameter ranges, the Dpp gradient does not reach steady state during Drosophila wing development. We further show that both, pre-steady-state dynamics and advection of cell-bound ligand in a growing tissue can, in principle, enable scaling, even for non-uniform tissue growth. For the parameter values that have been established for the Dpp morphogen in the Drosophila wing imaginal disc, we show that scaling is mainly a result of the pre-steady-state dynamics. Pre-steady-state dynamics are pervasive in morphogen-controlled systems, thus making this a probable general mechanism for dynamic scaling of morphogen gradients in growing developmental systems.