Self-Organization in Pattern Formation

Global modules robustly emerge from local interactions and smooth gradients

Modular structure and function are ubiquitous in biology, from the organization of animal brains and bodies to the scale of ecosystems. However, the mechanisms of modularity emergence from non-modular precursors remain unclear. Here we introduce the principle of peak selection, a process by which purely local interactions and smooth gradients can drive the self-organization of discrete global modules. The process combines strengths of the positional and Turing pattern-formation mechanisms into a model for morphogenesis. Applied to the grid-cell system of the brain, peak selection results in the self-organization of functionally distinct modules with discretely spaced spatial periods. Applied to ecological systems, it results in discrete multispecies niches and synchronous spawning across geographically distributed coral colonies. The process exhibits self-scaling with system size and 'topological robustness'1, which renders module emergence and module properties insensitive to most parameters. Peak selection ameliorates the fine-tuning requirement for continuous attractor dynamics in single grid-cell modules and it makes a detail-independent prediction that grid module period ratios should approximate adjacent integer ratios, providing a highly accurate match to the available data. Predictions for grid cells at the transcriptional, connectomic and physiological levels promise to elucidate the interplay of molecules, connectivity and function emergence in brains.

Membrane nanodomains to shape plant cellular functions and signaling

Plasma membrane (PM) nanodomains have emerged as pivotal elements in the regulation of plant cellular functions and signal transduction. These nanoscale membrane regions, enriched in specific lipids and proteins, behave as regulatory/signaling hubs spatially and temporally coordinating critical cellular functions. In this review, we first examine the mechanisms underlying the formation and maintenance of PM nanodomains in plant cells, highlighting the roles of PM lipid composition, protein oligomerization and interactions with cytoskeletal and cell wall components. Then, we discuss how nanodomains act as organizing centers by mediating protein-protein interactions that orchestrate essential processes such as symbiosis, defense against pathogens, ion transport or hormonal and reactive oxygen species (ROS) signaling. Finally, we introduce the concept of nanoenvironments, where localized physicochemical variations are generated in the very close proximity of PM nanodomains, in response to stimuli. After decoding by a dedicated machinery likely localized in the vicinity of nanodomains, this enrichment of secondary messengers, such as ROS or Ca2+, would allow specific downstream cellular responses. This review provides insights into the dynamic nature of nanodomains and proposes future research to better understand their contribution to the intricate signaling networks that govern plant development and stress responses.

Self-organization from organs to embryoids by activin in early amphibian development

Embryonic development is a complex self-organizing process orchestrated by a series of regulatory events at the molecular and cellular levels, resulting in the formation of a fully functional organism. This review focuses on activin protein as a mesoderm-inducing factor and the self-organizing properties it confers. Activin has been detected in both unfertilized eggs and embryos, suggesting its involvement in early developmental processes. To explore its effects, animal cap cells-pluripotent cells from the animal pole of amphibian blastula-stage embryos-were treated with varying concentrations of activin. The results showed that activin induced mesodermal tissues, including blood, muscle, and notochord, in a dose-dependent manner. Co-treatment with activin and retinoic acid further promoted the development of kidney and pancreatic tissues, while activin alone stimulated the formation of beating cardiac tissue. In subsequent experiments, high concentrations of activin conferred an organizer-like activity on animal cap cells. The pretreatment duration affected outcomes: longer exposure induced anterior structures, such as eyes, while shorter exposure resulted in posterior structures, like tails. These findings reflect moderate self-assembly, where cells become increasingly organized. In another experiment, activin was used to create an artificial gradient. Explants cultured on this gradient developed into embryoids with well-defined anteroposterior, dorsoventral, and left-right axes, exemplifying higher-order self-organization. These results demonstrate that controlled activin gradients can drive the formation of nearly complete tadpole-like larvae, effectively recapitulating the processes of early embryogenesis. This system offers valuable insights into the mechanisms underlying axis formation and organogenesis, providing a promising platform for future research in developmental biology.

The geometric basis of epithelial convergent extension

Shape changes of epithelia during animal development, such as convergent extension, are achieved through the concerted mechanical activity of individual cells. While much is known about the corresponding large-scale tissue flow and its genetic drivers, fundamental questions regarding local control of contractile activity on the cellular scale and its embryo-scale coordination remain open. To address these questions, we develop a quantitative, model-based analysis framework to relate cell geometry to local tension in recently obtained time-lapse imaging data of gastrulating Drosophila embryos. This analysis systematically decomposes cell shape changes and T1 rearrangements into internally driven, active, and externally driven, passive, contributions. Our analysis provides evidence that germ band extension is driven by active T1 processes that self-organize through positive feedback acting on tensions. More generally, our findings suggest that epithelial convergent extension results from the controlled transformation of internal force balance geometry which combines the effects of bottom-up local self-organization with the top-down, embryo-scale regulation by gene expression.

Homeotic transformation of tail to limbs: A novel morphogenesis in the framework of self-organization and reprogramming of cell fate during appendage regeneration

Homeotic transformation of tail to hindlimbs in anuran tadpoles is a manifestation of the reprogramming of positional information in the event of tail regeneration. Such discovery of homeosis is of particular interest considering its occurrence in a vertebrate under the influence of a morphogen which represents a self-organizing system in the context of developmental and regenerative studies. This article reviews homeotic transformation of tail to hindlimbs including pelvic girdles induced by retinoic acid (RA) /vitamin A palmitate during tail regeneration under the scope of self-organization and the role of blastema as an organizer. Next, we present a timeline of various findings in this context.

Symmetry breaking and fate divergence during lateral inhibition in Drosophila

Lateral inhibition mediates alternative cell fate decision and produces regular cell fate patterns with fate symmetry breaking (SB) relying on the amplification of small stochastic differences in Notch activity via an intercellular negative-feedback loop. Here, we used quantitative live imaging of endogenous Scute (Sc), a proneural factor, and of a Notch activity reporter to study the emergence of sensory organ precursor cells in the pupal abdomen of Drosophila. SB was observed at low Sc levels and was not preceded by a phase of intermediate Sc expression and Notch activity. Thus, mutual inhibition may only be transient in this context. In support of the intercellular feedback loop model, cell-to-cell variations in Sc levels promoted fate divergence. The size of the apical area of competing cells did not detectably bias this fate choice. Surprisingly, cells that were in direct contact at the time of SB could adopt the sensory organ precursor cell fate, albeit at low frequency (10%). These lateral inhibition defects were corrected by cellular rearrangements, not cell fate change, highlighting the role of cell-cell intercalation in pattern refinement.

Engineering Anisotropy into Organized Nanoscale Matter

Programming the organization of discrete building blocks into periodic and quasi-periodic arrays is challenging. Methods for organizing materials are particularly important at the nanoscale, where the time required for organization processes is practically manageable in experiments, and the resulting structures are of interest for applications spanning catalysis, optics, and plasmonics. While the assembly of isotropic nanoscale objects has been extensively studied and described by empirical design rules, recent synthetic advances have allowed anisotropy to be programmed into macroscopic assemblies made from nanoscale building blocks, opening new opportunities to engineer periodic materials and even quasicrystals with unnatural properties. In this review, we define guidelines for leveraging anisotropy of individual building blocks to direct the organization of nanoscale matter. First, the nature and spatial distribution of local interactions are considered and three design rules that guide particle organization are derived. Subsequently, recent examples from the literature are examined in the context of these design rules. Within the discussion of each rule, we delineate the examples according to the dimensionality (0D-3D) of the building blocks. Finally, we use geometric considerations to propose a general inverse design-based construction strategy that will enable the engineering of colloidal crystals with unprecedented structural control.

The development of hair follicles and nail

The hair follicle and nail unit develop and regenerate through epithelial-mesenchymal interactions. Here, we review some of the key signals and molecular interactions that regulate mammalian hair follicle and nail formation during embryonic development and how these interactions are reutilized to promote their regeneration during adult homeostasis and in response to skin wounding. Finally, we highlight the role of some of these signals in mediating human hair follicle and nail conditions.

The inherent fragility of collective proliferative control

Tissues achieve and maintain their sizes through active feedback, whereby cells collectively regulate proliferation and differentiation so as to facilitate homeostasis and the ability to respond to disturbances. One of the best understood feedback mechanisms-renewal control-achieves remarkable feats of robustness in determining and maintaining desired sizes. Yet in a variety of biologically relevant situations, we show that stochastic effects should cause rare but catastrophic failures of renewal control. We define the circumstances under which this occurs and raise the possibility such events account for important non-genetic steps in the development of cancer. We further suggest that the spontaneous stochastic reversal of these events could explain cases of cancer normalization or dormancy following treatment. Indeed, we show that the kinetics of post-treatment recurrence for many cancers are often better fit by a model of stochastic re-emergence due to loss of collective proliferative control, than by deterministic models of cancer relapse.

Mechanical state transitions in the regulation of tissue form and function

From embryonic development, postnatal growth and adult homeostasis to reparative and disease states, cells and tissues undergo constant changes in genome activity, cell fate, proliferation, movement, metabolism and growth. Importantly, these biological state transitions are coupled to changes in the mechanical and material properties of cells and tissues, termed mechanical state transitions. These mechanical states share features with physical states of matter, liquids and solids. Tissues can switch between mechanical states by changing behavioural dynamics or connectivity between cells. Conversely, these changes in tissue mechanical properties are known to control cell and tissue function, most importantly the ability of cells to move or tissues to deform. Thus, tissue mechanical state transitions are implicated in transmitting information across biological length and time scales, especially during processes of early development, wound healing and diseases such as cancer. This Review will focus on the biological basis of tissue-scale mechanical state transitions, how they emerge from molecular and cellular interactions, and their roles in organismal development, homeostasis, regeneration and disease.

Correspondence between multiple signaling and developmental cellular patterns: a computational perspective

The spatial arrangement of variant phenotypes during stem cell division plays a crucial role in the self-organization of cell tissues. The patterns observed in these cellular assemblies, where multiple phenotypes vie for space and resources, are largely influenced by a mixture of different diffusible chemical signals. This complex process is carried out within a chronological framework of interplaying intracellular and intercellular events. This includes receiving external stimulants, whether secreted by other individuals or provided by the environment, interpreting these environmental signals, and incorporating the information to designate cell fate. Here, given two distinct signaling patterns generated by Turing systems, we investigated the spatial distribution of differentiating cells that use these signals as external cues for modifying the production rates. By proposing a computational map, we show that there is a correspondence between the multiple signaling and developmental cellular patterns. In other words, the model provides an appropriate prediction for the final structure of the differentiated cells in a multi-signal, multi-cell environment. Conversely, when a final snapshot of cellular patterns is given, our algorithm can partially identify the signaling patterns that influenced the formation of the cellular structure, provided that the governing dynamic of the signaling patterns is already known.

Mechanical characterization of regenerating Hydra tissue spheres

Hydra vulgaris, long known for its remarkable regenerative capabilities, is also a long-standing source of inspiration for models of spontaneous patterning. Recently it became clear that early patterning during Hydra regeneration is an integrated mechanochemical process whereby morphogen dynamics is influenced by tissue mechanics. One roadblock to understanding Hydra self-organization is our lack of knowledge about the mechanical properties of these organisms. In this study, we combined microfluidic developments to perform parallelized microaspiration rheological experiments and numerical simulations to characterize these mechanical properties. We found three different behaviors depending on the applied stresses: an elastic response, a viscoelastic response, and tissue rupture. Using models of deformable shells, we quantify their Young's modulus, shear viscosity, and the critical stresses required to switch between behaviors. Based on these experimental results, we propose a description of the tissue mechanics during normal regeneration. Our results provide a first step toward the development of original mechanochemical models of patterning grounded in quantitative experimental data.

Information content and optimization of self-organized developmental systems

A key feature of many developmental systems is their ability to self-organize spatial patterns of functionally distinct cell fates. To ensure proper biological function, such patterns must be established reproducibly, by controlling and even harnessing intrinsic and extrinsic fluctuations. While the relevant molecular processes are increasingly well understood, we lack a principled framework to quantify the performance of such stochastic self-organizing systems. To that end, we introduce an information-theoretic measure for self-organized fate specification during embryonic development. We show that the proposed measure assesses the total information content of fate patterns and decomposes it into interpretable contributions corresponding to the positional and correlational information. By optimizing the proposed measure, our framework provides a normative theory for developmental circuits, which we demonstrate on lateral inhibition, cell type proportioning, and reaction-diffusion models of self-organization. This paves a way toward a classification of developmental systems based on a common information-theoretic language, thereby organizing the zoo of implicated chemical and mechanical signaling processes.

Collective Cell Radial Ordered Migration in Spatial Confinement

Collective cells, a typical active matter system, exhibit complex coordinated behaviors fundamental for various developmental and physiological processes. The present work discovers a collective radial ordered migration behavior of NIH3T3 fibroblasts that depends on persistent top-down regulation with 2D spatial confinement. Remarkably, individual cells move in a weak-oriented, diffusive-like rather than strong-oriented ballistic manner. Despite this, the collective movement is spatiotemporal heterogeneous and radial ordering at supracellular scale, manifesting as a radial ordered wavefront originated from the boundary and propagated toward the center of pattern. Combining bottom-up cell-to-extracellular matrix (ECM) interaction strategy, numerical simulations based on a developed mechanical model well reproduce and explain above observations. The model further predicts the independence of geometric features on this ordering behavior, which is validated by experiments. These results together indicate such radial ordered collective migration is ascribed to the couple of top-down regulation with spatial restriction and bottom-up cellular endogenous nature.

Creative processes during vertebrate organ morphogenesis: Biophysical self-organization at the supracellular scale

Here, we review recent developments in the literature that provide insight into self-organization at supracellular scales in vertebrate organ morphogenesis. We briefly present a historical and conceptual analysis of the term "self-organization." Based on this analysis, we suggest that self-organizing processes, at their root, possess a form of causal relationship, reciprocal causality, that is markedly distinct from linear causal chains. We survey the extent to which reciprocal causality can be used to interpret or clarify supracellular studies in development and disease. Finally, we explore how reciprocal causality can exist across length-scales, identifying situations where multiple scales require simultaneous analysis.

The Gene: An appraisal

The gene can be described as the foundational concept of modern biology. As such, it has spilled over into daily discourse, yet it is acknowledged among biologists to be ill-defined. Here, following a short history of the gene, I analyse critically its role in inheritance, evolution, development, and morphogenesis. Wilhelm Johannsen's genotype-conception, formulated in 1910, has been adopted as the foundation stone of genetics, giving the gene a higher degree of prominence than is justified by the evidence. An analysis of the results of the Long-Term Evolution Experiment (LTEE) with E. coli bacteria, grown over 60,000 generations, does not support spontaneous gene mutation as the source of variance for natural selection. From this it follows that the gene is not Mendel's unit of inheritance: that must be Johannsen's transmission-conception at the gamete phenotype level, a form of inheritance that Johannsen did not consider. Alternatively, I contend that biology viewed on the bases of thermodynamics, complex system dynamics, and self-organisation, provides a new framework for the foundations of biology. In this framework, the gene plays a passive role as a vital information store: it is the phenotype that plays the active role in inheritance, evolution, development, and morphogenesis.

PatU3 plays a central role in coordinating cell division and differentiation in pattern formation of filamentous cyanobacterium Nostoc sp. PCC 7120

Spatial periodic signal for cell differentiation in some multicellular organisms is generated according to Turing's principle for pattern formation. How a dividing cell responds to the signal of differentiation is addressed with the filamentous cyanobacterium Nostoc sp. PCC 7120, which forms the patterned distribution of heterocysts. We show that differentiation of a dividing cell was delayed until its division was completed and only one daughter cell became heterocyst. A mutant of patU3, which encodes an inhibitor of heterocyst formation, showed no such delay and formed heterocyst pairs from the daughter cells of cell division or dumbbell-shaped heterocysts from the cells undergoing cytokinesis. The patA mutant, which forms heterocysts only at the filament ends, restored intercalary heterocysts by a single nucleotide mutation of patU3, and double mutants of patU3/patA and patU3/hetF had the phenotypes of the patU3 mutant. We provide evidence that HetF, which can degrade PatU3, is recruited to cell divisome through its C-terminal domain. A HetF mutant with its N-terminal peptidase domain but lacking the C-terminal domain could not prevent the formation of heterocyst pairs, suggesting that the divisome recruitment of HetF is needed to sequester HetF for the delay of differentiation in dividing cells. Our study demonstrates that PatU3 plays a key role in cell-division coupled control of differentiation.

Novel Aspects in Pattern Formation Arise from Coupling Turing Reaction-Diffusion and Chemotaxis

Recent experimental studies on primary hair follicle formation and feather bud morphogenesis indicate a coupling between Turing-type diffusion driven instability and chemotactic patterning. Inspired by these findings we develop and analyse a mathematical model that couples chemotaxis to a reaction-diffusion system exhibiting diffusion-driven (Turing) instability. While both systems, reaction-diffusion systems and chemotaxis, can independently generate spatial patterns, we were interested in how the coupling impacts the stability of the system, parameter region for patterning, pattern geometry, as well as the dynamics of pattern formation. We conduct a classical linear stability analysis for different model structures, and confirm our results by numerical analysis of the system. Our results show that the coupling generally increases the robustness of the patterning process by enlarging the pattern region in the parameter space. Concerning time scale and pattern regularity, we find that an increase in the chemosensitivity can speed up the patterning process for parameters inside and outside of the Turing space, but generally reduces spatial regularity of the pattern. Interestingly, our analysis indicates that pattern formation can also occur when neither the Turing nor the chemotaxis system can independently generate pattern. On the other hand, for some parameter settings, the coupling of the two processes can extinguish the pattern formation, rather than reinforce it. These theoretical findings can be used to corroborate the biological findings on morphogenesis and guide future experimental studies. From a mathematical point of view, this work sheds a light on coupling classical pattern formation systems from the parameter space perspective.

The Unreasonable Effectiveness of Reaction Diffusion in Vertebrate Skin Color Patterning

In 1952, Alan Turing published the reaction-diffusion (RD) mathematical framework, laying the foundations of morphogenesis as a self-organized process emerging from physicochemical first principles. Regrettably, this approach has been widely doubted in the field of developmental biology. First, we summarize Turing's line of thoughts to alleviate the misconception that RD is an artificial mathematical construct. Second, we discuss why phenomenological RD models are particularly effective for understanding skin color patterning at the meso/macroscopic scales, without the need to parameterize the profusion of variables at lower scales. More specifically, we discuss how RD models (a) recapitulate the diversity of actual skin patterns, (b) capture the underlying dynamics of cellular interactions, (c) interact with tissue size and shape, (d) can lead to ordered sequential patterning, (e) generate cellular automaton dynamics in lizards and snakes, (f) predict actual patterns beyond their statistical features, and (g) are robust to model variations. Third, we discuss the utility of linear stability analysis and perform numerical simulations to demonstrate how deterministic RD emerges from the underlying chaotic microscopic agents.

Effects of square spatial periodic forcing on oscillatory hexagon patterns in coupled reaction-diffusion systems

One of the central issues in pattern formation is understanding the response of pattern-forming systems to an external stimulus. While significant progress has been made in systems with only one instability, much less is known about the response of complex patterns arising from the interaction of two or more instabilities. In this paper, we consider the effects of square spatial periodic forcing on oscillatory hexagon patterns in a two-layer coupled reaction diffusion system which undergoes both Turing and Hopf instabilities. Two different types of additive forcings, namely direct and indirect forcing, have been applied. It is shown that the coupled system exhibits different responses towards the spatial forcing under different forcing types. In the indirect case, the oscillatory hexagon pattern transitions into other oscillatory Turing patterns or resonant Turing patterns, depending on the forcing wavenumber and strength. In the direct forcing case, only non-resonant Turing patterns can be obtained. Our results may provide new insight into the modification and control of spatio-temporal patterns in multilayered systems, especially in biological and ecological systems.

Design principles of Cdr2 node patterns in fission yeast cells

Pattern-forming networks have diverse roles in cell biology. Rod-shaped fission yeast cells use pattern formation to control the localization of mitotic signaling proteins and the cytokinetic ring. During interphase, the kinase Cdr2 forms membrane-bound multiprotein complexes termed nodes, which are positioned in the cell middle due in part to the node inhibitor Pom1 enriched at cell tips. Node positioning is important for timely cell cycle progression and positioning of the cytokinetic ring. Here, we combined experimental and modeling approaches to investigate pattern formation by the Pom1-Cdr2 system. We found that Cdr2 nodes accumulate near the nucleus, and Cdr2 undergoes nucleocytoplasmic shuttling when cortical anchoring is reduced. We generated particle-based simulations based on tip inhibition, nuclear positioning, and cortical anchoring. We tested model predictions by investigating Pom1-Cdr2 localization patterns after perturbing each positioning mechanism, including in both anucleate and multinucleated cells. Experiments show that tip inhibition and cortical anchoring alone are sufficient for the assembly and positioning of nodes in the absence of the nucleus, but that the nucleus and Pom1 facilitate the formation of unexpected node patterns in multinucleated cells. These findings have implications for spatial control of cytokinesis by nodes and for spatial patterning in other biological systems.

Membrane nanodomains: Dynamic nanobuilding blocks of polarized cell growth

Cell polarity is intimately linked to numerous biological processes, such as oriented plant cell division, particular asymmetric division, cell differentiation, cell and tissue morphogenesis, and transport of hormones and nutrients. Cell polarity is typically initiated by a polarizing cue that regulates the spatiotemporal dynamic of polarity molecules, leading to the establishment and maintenance of polar domains at the plasma membrane. Despite considerable progress in identifying key polarity regulators in plants, the molecular and cellular mechanisms underlying cell polarity formation have yet to be fully elucidated. Recent work suggests a critical role for membrane protein/lipid nanodomains in polarized morphogenesis in plants. One outstanding question is how the spatiotemporal dynamics of signaling nanodomains are controlled to achieve robust cell polarization. In this review, we first summarize the current state of knowledge on potential regulatory mechanisms of nanodomain dynamics, with a special focus on Rho-like GTPases from plants. We then discuss the pavement cell system as an example of how cells may integrate multiple signals and nanodomain-involved feedback mechanisms to achieve robust polarity. A mechanistic understanding of nanodomains' roles in plant cell polarity is still in the early stages and will remain an exciting area for future investigations.

Design principles of Cdr2 node patterns in fission yeast cells

Pattern forming networks have diverse roles in cell biology. Rod-shaped fission yeast cells use pattern formation to control the localization of mitotic signaling proteins and the cytokinetic ring. During interphase, the kinase Cdr2 forms membrane-bound multiprotein complexes termed nodes, which are positioned in the cell middle due in part to the node inhibitor Pom1 enriched at cell tips. Node positioning is important for timely cell cycle progression and positioning of the cytokinetic ring. Here, we combined experimental and modeling approaches to investigate pattern formation by the Pom1-Cdr2 system. We found that Cdr2 nodes accumulate near the nucleus, and Cdr2 undergoes nucleocytoplasmic shuttling when cortical anchoring is reduced. We generated particle-based simulations based on tip inhibition, nuclear positioning, and cortical anchoring. We tested model predictions by investigating Pom1-Cdr2 localization patterns after perturbing each positioning mechanism, including in both anucleate and multinucleated cells. Experiments show that tip inhibition and cortical anchoring alone are sufficient for the assembly and positioning of nodes in the absence of the nucleus, but that the nucleus and Pom1 facilitate the formation of unexpected node patterns in multinucleated cells. These findings have implications for spatial control of cytokinesis by nodes and for spatial patterning in other biological systems.

Extensive embryonic patterning without cellular differentiation primes the plant epidermis for efficient post-embryonic stomatal activities

Plant leaves feature epidermal stomata that are organized in stereotyped patterns. How does the pattern originate? We provide transcriptomic, imaging, and genetic evidence that Arabidopsis embryos engage known stomatal fate and patterning factors to create regularly spaced stomatal precursor cells. Analysis of embryos from 36 plant species indicates that this trait is widespread among angiosperms. Embryonic stomatal patterning in Arabidopsis is established in three stages: first, broad SPEECHLESS (SPCH) expression; second, coalescence of SPCH and its targets into discrete domains; and third, one round of asymmetric division to create stomatal precursors. Lineage progression is then halted until after germination. We show that the embryonic stomatal pattern enables fast stomatal differentiation and photosynthetic activity upon germination, but it also guides the formation of additional stomata as the leaf expands. In addition, key stomatal regulators are prevented from driving the fate transitions they can induce after germination, identifying stage-specific layers of regulation that control lineage progression during embryogenesis.

Directing Min protein patterns with advective bulk flow

The Min proteins constitute the best-studied model system for pattern formation in cell biology. We theoretically predict and experimentally show that the propagation direction of in vitro Min protein patterns can be controlled by a hydrodynamic flow of the bulk solution. We find downstream propagation of Min wave patterns for low MinE:MinD concentration ratios, upstream propagation for large ratios, but multistability of both propagation directions in between. Whereas downstream propagation can be described by a minimal model that disregards MinE conformational switching, upstream propagation can be reproduced by a reduced switch model, where increased MinD bulk concentrations on the upstream side promote protein attachment. Our study demonstrates that a differential flow, where bulk flow advects protein concentrations in the bulk, but not on the surface, can control surface-pattern propagation. This suggests that flow can be used to probe molecular features and to constrain mathematical models for pattern-forming systems.

Jag1-Notch cis-interaction determines cell fate segregation in pancreatic development

The Notch ligands Jag1 and Dll1 guide differentiation of multipotent pancreatic progenitor cells (MPCs) into unipotent pro-acinar cells (PACs) and bipotent duct/endocrine progenitors (BPs). Ligand-mediated trans-activation of Notch receptors induces oscillating expression of the transcription factor Hes1, while ligand-receptor cis-interaction indirectly represses Hes1 activation. Despite Dll1 and Jag1 both displaying cis- and trans-interactions, the two mutants have different phenotypes for reasons not fully understood. Here, we present a mathematical model that recapitulates the spatiotemporal differentiation of MPCs into PACs and BPs. The model correctly captures cell fate changes in Notch pathway knockout mice and small molecule inhibitor studies, and a requirement for oscillatory Hes1 expression to maintain the multipotent state. Crucially, the model entails cell-autonomous attenuation of Notch signaling by Jag1-mediated cis-inhibition in MPC differentiation. The model sheds light on the underlying mechanisms, suggesting that cis-interaction is crucial for exiting the multipotent state, while trans-interaction is required for adopting the bipotent fate.

Modulation of self-organizing circuits at deforming membranes by intracellular and extracellular factors

Mechanical forces exerted to the plasma membrane induce cell shape changes. These transient shape changes trigger, among others, enrichment of curvature-sensitive molecules at deforming membrane sites. Strikingly, some curvature-sensing molecules not only detect membrane deformation but can also alter the amplitude of forces that caused to shape changes in the first place. This dual ability of sensing and inducing membrane deformation leads to the formation of curvature-dependent self-organizing signaling circuits. How these cell-autonomous circuits are affected by auxiliary parameters from inside and outside of the cell has remained largely elusive. Here, we explore how such factors modulate self-organization at the micro-scale and its emerging properties at the macroscale.

Resolving trabecular metaphyseal bone profiles downstream of the growth plate adds value to bone histomorphometry in mouse models

INTRODUCTION

Histomorphometry of rodent metaphyseal trabecular bone, by histology or microCT, is generally restricted to the mature secondary spongiosa, excluding the primary spongiosa nearest the growth plate by imposing an 'offset'. This analyses the bulk static properties of a defined segment of secondary spongiosa, usually regardless of proximity to the growth plate. Here we assess the value of trabecular morphometry that is spatially resolved according to the distance 'downstream' of-and thus time since formation at-the growth plate. Pursuant to this, we also investigate the validity of including mixed primary-secondary spongiosal trabecular bone, extending the analysed volume 'upstream' by reducing the offset. Both the addition of spatiotemporal resolution and the extension of the analysed volume have potential to enhance the sensitivity of detection of trabecular changes and to resolve changes occurring at different times and locations.

METHOD

Two experimental mouse studies of trabecular bone are used as examples of different factors influencing metaphyseal trabecular bone: (1) ovariectomy (OVX) and pharmacological prevention of osteopenia and (2) limb disuse induced by sciatic neurectomy (SN). In a third study into offset rescaling, we also examine the relationship between age, tibia length, and primary spongiosal thickness.

RESULTS

Bone changes induced by either OVX or SN that were early or weak and marginal were more pronounced in the mixed primary-secondary upstream spongiosal region than in the downstream secondary spongiosa. A spatially resolved evaluation of the entire trabecular region found that significant differences between experimental and control bones remained undiminished either right up to or to within 100 μm from the growth plate. Intriguingly, our data revealed a remarkably linear downstream profile for fractal dimension in trabecular bone, arguing for an underlying homogeneity of the (re)modelling process throughout the entire metaphysis and against strict anatomical categorization into primary and secondary spongiosal regions. Finally, we find that a correlation between tibia length and primary spongiosal depth is well conserved except in very early and late life.

CONCLUSIONS

These data indicate that the spatially resolved analysis of metaphyseal trabecular bone at different distances from the growth plate and/or times since formation adds a valuable dimension to histomorphometric analysis. They also question any rationale for rejecting primary spongiosal bone, in principle, from metaphyseal trabecular morphometry.

Periodic spatial patterning with a single morphogen

During multicellular development, periodic spatial patterning systems generate repetitive structures, such as digits, vertebrae, and teeth. Turing patterning provides a foundational paradigm for understanding such systems. The simplest Turing systems are believed to require at least two morphogens to generate periodic patterns. Here, using mathematical modeling, we show that a simpler circuit, including only a single diffusible morphogen, is sufficient to generate long-range, spatially periodic patterns that propagate outward from transient initiating perturbations and remain stable after the perturbation is removed. Furthermore, an additional bistable intracellular feedback or operation on a growing cell lattice can make patterning robust to noise. Together, these results show that a single morphogen can be sufficient for robust spatial pattern formation and should provide a foundation for engineering pattern formation in the emerging field of synthetic developmental biology.

Modeling convergent scale-by-scale skin color patterning in multiple species of lizards

Skin color patterning in vertebrates emerges at the macroscale from microscopic cell-cell interactions among chromatophores. Taking advantage of the convergent scale-by-scale skin color patterning dynamics in five divergent species of lizards, we quantify the respective efficiencies of stochastic (Lenz-Ising and cellular automata, sCA) and deterministic reaction-diffusion (RD) models to predict individual patterns and their statistical attributes. First, we show that all models capture the underlying microscopic system well enough to predict, with similar efficiencies, neighborhood statistics of adult patterns. Second, we show that RD robustly generates, in all species, a substantial gain in scale-by-scale predictability of individual adult patterns without the need to parametrize the system down to its many cellular and molecular variables. Third, using 3D numerical simulations and Lyapunov spectrum analyses, we quantitatively demonstrate that, given the non-linearity of the dynamical system, uncertainties in color measurements at the juvenile stage and in skin geometry variation explain most, if not all, of the residual unpredictability of adult individual scale-by-scale patterns. We suggest that the efficiency of RD is due to its intrinsic ability to exploit mesoscopic information such as continuous scale colors and the relations among growth, scales geometries, and the pattern length scale. Our results indicate that convergent evolution of CA patterning dynamics, leading to dissimilar macroscopic patterns in different species, is facilitated by their spontaneous emergence under a large range of RD parameters, as long as a Turing instability occurs in a skin domain with periodic thickness. VIDEO ABSTRACT.

Cell shape anisotropy contributes to self-organized feather pattern fidelity in birds

Developing tissues can self-organize into a variety of patterned structures through the stabilization of stochastic fluctuations in their molecular and cellular properties. While molecular factors and cell dynamics contributing to self-organization have been identified in vivo, events channeling self-organized systems such that they achieve stable pattern outcomes remain unknown. Here, we described natural variation in the fidelity of self-organized arrays formed by feather follicle precursors in bird embryos. By surveying skin cells prior to and during tissue self-organization and performing species-specific ex vivo drug treatments and mechanical stress tests, we demonstrated that pattern fidelity depends on the initial amplitude of cell anisotropy in regions of the developing dermis competent to produce a pattern. Using live imaging, we showed that cell shape anisotropy is associated with a limited increase in cell motility for sharp and precisely located primordia formation, and thus, proper pattern geometry. These results evidence a mechanism through which initial tissue properties ensure stability in self-organization and thus, reproducible pattern production.

A study of natural variation in feather pattern geometry and a combination of pharmacological and mechanical manipulations ex vivo provides evidence for a mechanism by which initial tissue properties ensure stability in self-organization and species-specific pattern production.

Thermodynamic principles for system biology and the patterns of flower pigmentation

The thermodynamic principles for system biology are reviewed and formulated, and then basic patterns of flower pigmentation are interpreted. Main thoughts: (1) any biological trait (color or function of a cell) is logically related to a thermodynamic system (or physiological system, signaling network of the cell), (2) the striped, speckled and circle are three basic patterns of flower pigmentation, the development of flowers is an irreversible process, (3) the patterns of flower pigmentation are formed in flower development, (4) the flower cells can change its color in a period of development and this process is controlled thermodynamically, (5) there is giant space of physiology within an organism and within its numerous thermal states can appear under different conditions. In this theory, the dominant inheritance means that a gene contributes great to the thermodynamic stability of a trait related system; different genes can be interacted or integrated thermodynamically according to their contribution to the stability of its related system. By combination of Turing theory and our views, complex patterns of pigmentation could be explained theoretically.

Color Variability Constrains Detection of Geometrically Perfect Mirror Symmetry

Symmetry in nature is a result of biological self-organization, driven by evolutionary processes. Detected by the visual systems of various species, from invertebrates to primates, symmetry determines survival relevant choice behaviors and supports adaptive function by reducing stimulus uncertainty. Symmetry also provides a major structural key to bio-inspired artificial vision and shape or movement simulations. In this psychophysical study, local variations in color covering the whole spectrum of visible wavelengths are compared to local variations in luminance contrast across an axis of geometrically perfect vertical mirror symmetry. The chromatic variations are found to delay response time to shape symmetry to a significantly larger extent than achromatic variations. This effect depends on the degree of variability, i.e., stimulus complexity. In both cases, we observe linear increase in response time as a function of local color variations across the vertical axis of symmetry. These results are directly explained by the difference in computational complexity between the two major (magnocellular vs. parvocellular) visual pathways involved in filtering the contrast (luminance vs. luminance and color) of the shapes. It is concluded that color variability across an axis of symmetry proves detrimental to the rapid detection of symmetry, and, presumably, other structural shape regularities. The results have implications for vision-inspired artificial intelligence and robotics exploiting functional principles of human vision for gesture and movement detection, or geometric shape simulation for recognition systems, where symmetry is often a critical property.

Strength of interactions in the Notch gene regulatory network determines patterning and fate in the notochord

Development of multicellular organisms requires the generation of gene expression patterns that determines cell fate and organ shape. Groups of genetic interactions known as Gene Regulatory Networks (GRNs) play a key role in the generation of such patterns. However, how the topology and parameters of GRNs determine patterning in vivo remains unclear due to the complexity of most experimental systems. To address this, we use the zebrafish notochord, an organ where coin-shaped precursor cells are initially arranged in a simple unidimensional geometry. These cells then differentiate into vacuolated and sheath cells. Using newly developed transgenic tools together with in vivo imaging, we identify jag1a and her6/her9 as the main components of a Notch GRN that generates a lateral inhibition pattern and determines cell fate. Making use of this experimental system and mathematical modeling we show that lateral inhibition patterning is promoted when ligand-receptor interactions are stronger within the same cell than in neighboring cells. Altogether, we establish the zebrafish notochord as an experimental system to study pattern generation, and identify and characterize how the properties of GRNs determine self-organization of gene patterning and cell fate.

Information Hiding Based on Statistical Features of Self-Organizing Patterns

A computational technique for the determination of optimal hiding conditions of a digital image in a self-organizing pattern is presented in this paper. Three statistical features of the developing pattern (the Wada index based on the weighted and truncated Shannon entropy, the mean of the brightness of the pattern, and the p-value of the Kolmogorov-Smirnov criterion for the normality testing of the distribution function) are used for that purpose. The transition from the small-scale chaos of the initial conditions to the large-scale chaos of the developed pattern is observed during the evolution of the self-organizing system. Computational experiments are performed with the stripe-type patterns, spot-type patterns, and unstable patterns. It appears that optimal image hiding conditions are secured when the Wada index stabilizes after the initial decline, the mean of the brightness of the pattern remains stable before dropping down significantly below the average, and the p-value indicates that the distribution becomes Gaussian.

Pattern formation features might explain homoplasy: fertile surfaces in higher fungi as an example

Fungi show a high degree of morphological convergence. Regarded for a long time as an obstacle for phylogenetic studies, homoplasy has also been proposed as a source of information about underlying morphogenetic patterning mechanisms. The "local-activation and long-range inhibition principle" (LALIP), underlying the famous reaction-diffusion model proposed by Alan Turing in 1952, appears to be one of the universal phenomena that can explain the ontogenetic origin of seriate patterns in living organisms. Reproductive structures of fungi in the class Agaricomycetes show a highly periodic structure resulting in, for example, poroid, odontoid, lamellate or labyrinthic hymenophores. In this paper, we claim that self-organized patterns might underlie the basic ontogenetic processes of these structures. Simulations based on LALIP-driven models and covering a wide range of parameters show an absolute mutual correspondence with the morphospace explored by extant agaricomycetes. This could not only explain geometric particularities but could also account for the limited possibilities displayed by hymenial configurations, thus making homoplasy a direct consequence of the limited morphospace resulting from the proposed patterning dynamics.

Emergence of a geometric pattern of cell fates from tissue-scale mechanics in the Drosophila eye

Pattern formation of biological structures involves the arrangement of different types of cells in an ordered spatial configuration. In this study, we investigate the mechanism of patterning the Drosophila eye epithelium into a precise triangular grid of photoreceptor clusters called ommatidia. Previous studies had led to a long-standing biochemical model whereby a reaction-diffusion process is templated by recently formed ommatidia to propagate a molecular prepattern across the eye. Here, we find that the templating mechanism is instead, mechanochemical in origin; newly born columns of differentiating ommatidia serve as a template to spatially pattern flows that move epithelial cells into position to form each new column of ommatidia. Cell flow is generated by a source and sink, corresponding to narrow zones of cell dilation and contraction respectively, that straddle the growing wavefront of ommatidia. The newly formed lattice grid of ommatidia cells are immobile, deflecting, and focusing the flow of other cells. Thus, the self-organization of a regular pattern of cell fates in an epithelium is mechanically driven.

Systems for intricate patterning of the vertebrate anatomy

Periodic patterns form intricate arrays in the vertebrate anatomy, notably the hair and feather follicles of the skin, but also internally the villi of the gut and the many branches of the lung, kidney, mammary and salivary glands. These tissues are composite structures, being composed of adjoined epithelium and mesenchyme, and the patterns that arise within them require interaction between these two tissue layers. In embryonic development, cells change both their distribution and state in a periodic manner, defining the size and relative positions of these specialized structures. Their placement is determined by simple spacing mechanisms, with substantial evidence pointing to a variety of local enhancement/lateral inhibition systems underlying the breaking of symmetry. The nature of the cellular processes involved, however, has been less clear. While much attention has focused on intercellular soluble signals, such as protein growth factors, experimental evidence has grown for contributions of cell movement or mechanical forces to symmetry breaking. In the mesenchyme, unlike the epithelium, cells may move freely and can self-organize into aggregates by chemotaxis, or through generation and response to mechanical strain on their surrounding matrix. Different modes of self-organization may coexist, either coordinated into a single system or with hierarchical relationships. Consideration of a broad range of distinct biological processes is required to advance understanding of biological pattern formation. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Mechano-chemical feedback mediated competition for BMP signalling leads to pattern formation

Developmental patterning is thought to be regulated by conserved signalling pathways. Initial patterns are often broad before refining to only those cells that commit to a particular fate. However, the mechanisms by which pattern refinement takes place remain to be addressed. Using the posterior crossvein (PCV) of the Drosophila pupal wing as a model, into which bone morphogenetic protein (BMP) ligand is extracellularly transported to instruct vein patterning, we investigate how pattern refinement is regulated. We found that BMP signalling induces apical enrichment of Myosin II in developing crossvein cells to regulate apical constriction. Live imaging of cellular behaviour indicates that changes in cell shape are dynamic and transient, only being maintained in those cells that retain vein fate competence after refinement. Disrupting cell shape changes throughout the PCV inhibits pattern refinement. In contrast, disrupting cell shape in only a subset of vein cells can result in a loss of BMP signalling. We propose that mechano-chemical feedback leads to competition for the developmental signal which plays a critical role in pattern refinement.

Self-Organization Provides Cell Fate Commitment in MSC Sheet Condensed Areas via ROCK-Dependent Mechanism

Multipotent mesenchymal stem/stromal cells (MSC) are one of the crucial regulators of regeneration and tissue repair and possess an intrinsic program from self-organization mediated by condensation, migration and self-patterning. The ability to self-organize has been successfully exploited in tissue engineering approaches using cell sheets (CS) and their modifications. In this study, we used CS as a model of human MSC spontaneous self-organization to demonstrate its structural, transcriptomic impact and multipotent stromal cell commitment. We used CS formation to visualize MSC self-organization and evaluated the role of the Rho-GTPase pathway in spontaneous condensation, resulting in a significant anisotropy of the cell density within the construct. Differentiation assays were carried out using conventional protocols, and microdissection and RNA-sequencing were applied to establish putative targets behind the observed phenomena. The differentiation of MSC to bone and cartilage, but not to adipocytes in CS, occurred more effectively than in the monolayer. RNA-sequencing indicated transcriptional shifts involving the activation of the Rho-GTPase pathway and repression of SREBP, which was concordant with the lack of adipogenesis in CS. Eventually, we used an inhibitory analysis to validate our findings and suggested a model where the self-organization of MSC defined their commitment and cell fate via ROCK1/2 and SREBP as major effectors under the putative switching control of AMP kinase.

The gene: An appraisal

The gene can be described as the foundational concept of modern biology. As such, it has spilled over into daily discourse, yet it is acknowledged among biologists to be ill-defined. Here, following a short history of the gene, I analyse critically its role in inheritance, evolution, development, and morphogenesis. Wilhelm Johannsen's genotype-conception, formulated in 1910, has been adopted as the foundation stone of genetics, giving the gene a higher degree of prominence than is justified by the evidence. An analysis of the results of the Long-Term Evolution Experiment (LTEE) with E. coli bacteria, grown over 60,000 generations, does not support spontaneous gene mutation as the source of variance for natural selection. From this it follows that the gene is not Mendel's unit of inheritance: that must be Johannsen's transmission-conception at the gamete phenotype level, a form of inheritance that Johannsen did not consider. Alternatively, I contend that biology viewed on the bases of thermodynamics, complex system dynamics and self-organisation, provides a new framework for the foundations of biology. In this framework, the gene plays a passive role as a vital information store: it is the phenotype that plays the active role in inheritance, evolution, development, and morphogenesis.

Self-organized protein patterns: The MinCDE and ParABS systems

Self-organized protein patterns are of tremendous importance for biological decision-making processes. Protein patterns have been shown to identify the site of future cell division, establish cell polarity, and organize faithful DNA segregation. Intriguingly, several key concepts of pattern formation and regulation apply to a variety of different protein systems. Herein, we explore recent advances in the understanding of two prokaryotic pattern-forming systems: the MinCDE system, positioning the FtsZ ring precisely at the midcell, and the ParABS system, distributing newly synthesized DNA along with the cell. Despite differences in biological functionality, these two systems have remarkably similar molecular components, mechanisms, and strategies to achieve biological robustness.

Towards a physical understanding of developmental patterning

The temporal coordination of events at cellular and tissue scales is essential for the proper development of organisms, and involves cell-intrinsic processes that can be coupled by local cellular signalling and instructed by global signalling, thereby creating spatial patterns of cellular states that change over time. The timing and structure of these patterns determine how an organism develops. Traditional developmental genetic methods have revealed the complex molecular circuits regulating these processes but are limited in their ability to predict and understand the emergent spatio-temporal dynamics. Increasingly, approaches from physics are now being used to help capture the dynamics of the system by providing simplified, generic descriptions. Combined with advances in imaging and computational power, such approaches aim to provide insight into timing and patterning in developing systems.

From embryos to embryoids: How external signals and self-organization drive embryonic development

Embryonic development has been traditionally seen as an inductive process directed by exogenous maternal inputs and extra-embryonic signals. Increasing evidence, however, is showing that, in addition to exogenous signals, the development of the embryo involves endogenous self-organization. Recently, this self-organizing potential has been highlighted by a number of stem cell models known as embryoids that can recapitulate different aspects of embryogenesis in vitro. Here, we review the self-organizing behaviors observed in different embryoid models and seek to reconcile this new evidence with classical knowledge of developmental biology. This analysis leads to reexamine embryonic development as a guided self-organizing process, where patterning and morphogenesis are controlled by a combination of exogenous signals and endogenous self-organization. Finally, we discuss the multidisciplinary approach required to investigate the genetic and cellular basis of self-organization.

Unscrambling exit site patterns on the endoplasmic reticulum as a quenched demixing process

The endoplasmic reticulum (ER) is a vital organelle in mammalian cells with a complex morphology. Consisting of sheet-like cisternae in the cell center, the peripheral ER forms a vast tubular network on which a dispersed pattern of a few hundred specialized domains (ER exit sites (ERESs)) is maintained. Molecular details of cargo sorting and vesicle formation at individual ERESs, fueling the early secretory pathway, have been studied in some detail. The emergence of spatially extended ERES patterns, however, has remained poorly understood. Here, we show that these patterns are determined by the underlying ER morphology, suggesting ERESs to emerge from a demixing process that is quenched by the ER network topology. In particular, we observed fewer but larger ERESs when transforming the ER network to more sheet-like morphologies. In contrast, little to no changes with respect to native ERES patterns were observed when fragmenting the ER, indicating that hampering the diffusion-mediated coarse graining of domains is key for native ERES patterns. Model simulations support the notion of effective diffusion barriers impeding the coarse graining and maturation of ERES patterns. We speculate that tuning a simple demixing mechanism by the ER topology allows for a robust but flexible adaption of ERES patterns, ensuring a properly working early secretory pathway in a variety of conditions.

Dynamic spatiotemporal coordination of neural stem cell fate decisions occurs through local feedback in the adult vertebrate brain

Neural stem cell (NSC) populations persist in the adult vertebrate brain over a lifetime, and their homeostasis is controlled at the population level through unknown mechanisms. Here, we combine dynamic imaging of entire NSC populations in their in vivo niche over several weeks with pharmacological manipulations, mathematical modeling, and spatial statistics and demonstrate that NSCs use spatiotemporally resolved local feedback signals to coordinate their decision to divide in adult zebrafish brains. These involve Notch-mediated short-range inhibition from transient neural progenitors and a dispersion effect from the dividing NSCs themselves exerted with a delay of 9–12 days. Simulations from a stochastic NSC lattice model capturing these interactions demonstrate that these signals are linked by lineage progression and control the spatiotemporal distribution of output neurons. These results highlight how local and temporally delayed interactions occurring between brain germinal cells generate self-propagating dynamics that maintain NSC population homeostasis and coordinate specific spatiotemporal correlations.

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NSC activation events are spatiotemporally coordinated within adult NSC populations

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This involves inhibition by neural progenitors (relying on Notch) and by dividing NSCs

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A dynamic lattice model shows that these interactions are linked by lineage progression

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NSCs dynamics generate an intrinsic niche that maintains the NSC population long-term