Self-organized Pattern Formation Research Articles

Mesoscale heterogeneity is a critical determinant for spiral pattern formation in developing social amoeba

Heterogeneity is a critical determinant for multicellular pattern formation. Although the importance of microscale and macroscale heterogeneity at the single-cell and whole-system levels, respectively, has been well accepted, the presence and functions of mesoscale heterogeneity, such as cell clusters with distinct properties, have been poorly recognized. We investigated the biological importance of mesoscale heterogeneity in signal-relaying abilities (excitability) in the self-organization of spiral waves of intercellular communications by studying the self-organized pattern formation in a population of Dictyostelium discoideum cells, a classical signal-relaying system model. By utilizing pulse-count analysis to evaluate cellular excitability, we successfully visualized the development of mesoscale heterogeneity in excitability, whose spatial scale was comparably large to that of the traveling waves of intercellular communication. Together with perturbation experiments, our detailed analysis of the structural change in mesoscale heterogeneity and associated wave dynamics demonstrated the functional importance of mesoscale heterogeneity in generating the spiral wave pattern, whose experimental observations were first realized. We propose that mesoscale heterogeneity, in addition to microscale and macroscale heterogeneities, is a critical determinant of diverse multicellular pattern formations.

Self-organized pattern formation in the developing mouse neural tube by a temporal relay of BMP signaling.

Developing tissues interpret dynamic changes in morphogen activity to generate cell type diversity. To quantitatively study bone morphogenetic protein (BMP) signaling dynamics in the mouse neural tube, we developed an embryonic stem cell differentiation system tailored for growing tissues. Differentiating cells form striking self-organized patterns of dorsal neural tube cell types driven by sequential phases of BMP signaling that are observed both invitro and invivo. Data-driven biophysical modeling showed that these dynamics result from coupling fast negative feedback with slow positive regulation of signaling by the specification of an endogenous BMP source. Thus, in contrast to relays that propagate morphogen signaling in space, we identify a BMP signaling relay that operates in time. This mechanism allows for a rapid initial concentration-sensitive response that is robustly terminated, thereby regulating balanced sequential cell type generation. Our study provides an experimental and theoretical framework to understand how signaling dynamics are exploited in developing tissues.

Widespread biochemical reaction networks enable Turing patterns without imposed feedback

Understanding self-organized pattern formation is fundamental to biology. In 1952, Alan Turing proposed a pattern-enabling mechanism in reaction-diffusion systems containing chemical species later conceptualized as activators and inhibitors that are involved in feedback loops. However, identifying pattern-enabling regulatory systems with the concept of feedback loops has been a long-standing challenge. To date, very few pattern-enabling circuits have been discovered experimentally. This is in stark contrast to ubiquitous periodic patterns and symmetry in biology. In this work, we systematically study Turing patterns in 23 elementary biochemical networks without assigning any activator or inhibitor. These mass action models describe post-synthesis interactions applicable to most proteins and RNAs in multicellular organisms. Strikingly, we find ten simple reaction networks capable of generating Turing patterns. While these network models are consistent with Turing’s theory mathematically, there is no apparent connection between them and commonly used activator-feedback intuition. Instead, we identify a unifying network motif that enables Turing patterns via regulated degradation pathways with flexible diffusion rate constants of individual molecules. Our work reveals widespread biochemical systems for pattern formation, and it provides an alternative approach to tackle the challenge of identifying pattern-enabling biological systems.

Molybdenum assisted self-organized pattern formation by low energy ion beam sputtering

Molybdenum assisted self-organized pattern formation by low energy ion beam sputtering

Self-enhanced mobility enables vortex pattern formation in living matter.

Ranging from subcellular organelle biogenesis to embryo development, the formation of self-organized structures is a hallmark of living systems. Whereas the emergence of ordered spatial patterns in biology is often driven by intricate chemical signalling that coordinates cellular behaviour and differentiation1-4, purely physical interactions can drive the formation of regular biological patterns such as crystalline vortex arrays in suspensions of spermatozoa5 and bacteria6. Here we discovered a new route to self-organized pattern formation driven by physical interactions, which creates large-scale regular spatial structures with multiscale ordering. Specifically we found that dense bacterial living matter spontaneously developed a lattice of mesoscale, fast-spinning vortices; these vortices each consisted of around 104-105 motile bacterial cells and were arranged in space at greater than centimetre scale and with apparent hexagonal order, whereas individual cells in the vortices moved in coordinated directions with strong polar and vortical order. Single-cell tracking and numerical simulations suggest that the phenomenon is enabled by self-enhanced mobility in the system-that is, the speed of individual cells increasing with cell-generated collective stresses at a given cell density. Stress-induced mobility enhancement and fluidization is prevalent in dense living matter at various scales of length7-9. Our findings demonstrate that self-enhanced mobility offers a simple physical mechanism for pattern formation in living systems and, more generally, in other active matter systems10 near the boundary of fluid- and solid-like behaviours11-17.

Unravelling diverse spatiotemporal orders in chlorine dioxide-iodine-malonic acid reaction-diffusion system through circularly polarized electric field and photo-illumination.

Designing and predicting self-organized pattern formation in out-of-equilibrium chemical and biochemical reactions holds fundamental significance. External perturbations like light and electric fields exert a crucial influence on reaction-diffusion systems involving ionic species. While the separate impacts of light and electric fields have been extensively studied, comprehending their combined effects on spatiotemporal dynamics is paramount for designing versatile spatial orders. Here, we theoretically investigate the spatiotemporal dynamics of chlorine dioxide-iodine-malonic acid reaction-diffusion system under photo-illumination and circularly polarized electric field (CPEF). By applying CPEF at varying intensities and frequencies, we observe the predominant emergence of oscillating hexagonal spot-like patterns from homogeneous stable steady states. Furthermore, our study unveils a spectrum of intriguing spatiotemporal instabilities, encompassing stripe-like patterns, oscillating dumbbell-shaped patterns, spot-like instabilities with square-based symmetry, and irregular chaotic patterns. However, when we introduce periodic photo-illumination to the hexagonal spot-like instabilities induced by CPEF in homogeneous steady states, we observe periodic size fluctuations. Additionally, the stripe-like instabilities undergo alternating transitions between hexagonal spots and stripes. Notably, within the Turing region, the interplay between these two external influences leads to the emergence of distinct superlattice patterns characterized by hexagonal-and square-based symmetry. These patterns include parallel lines of spots, target-like formations, black-eye patterns, and other captivating structures. Remarkably, the simple perturbation of the system through the application of these two external fields offers a versatile tool for generating a wide range of pattern-forming instabilities, thereby opening up exciting possibilities for future experimental validation.

Spatially modulated control of pattern formation in a general nonlocal nonlinear system

Spontaneous symmetry breaking and pattern formation are notable and significant natural phenomena that occur in nonlinear systems. Further advancing such research to Bose–Einstein condensates (BECs) is of much interest for both fundamental physics and practical applications. We investigate the formation and selection of diverse self-organized ground-state patterns, as well as the control of their structural phase transition in a general nonlocal nonlinear system described by nonlocal Gross–Pitaevskii equation (GPE). We show a homogeneous matter wave can undergo roton instability (RI) and their control is addressed by spatially weak modulated nonlinearity, which may result in the formation of various ordered patterns (including square, stripe, annular, and different hexagonal structures, etc.) with and without the trapping potential. In particular, our findings reveal that when a spatially weak modulated nonlinearity is introduced, a hexagonal ground-state pattern (which is the only type of structure observed without such modulation) can transform into several distinct pattern types. In contrast to previous research, the patterns identified in our study can be easily controlled by modifying the nonlinear parameters. Our work opens a pathway for versatile manipulation of self-organized patterns as well as the associated structural phase transitions.

Self-organized submicron structures in photoresist films by UV-laser irradiation at water-confined conditions

A new kind of self-organized pattern formation process has been found during laser irradiation of polymer films in water confinement just below the laser ablation threshold, resulting in a randomly oriented pattern with a period of about 475 nm. The morphology, orientation, period, and amplitude of these patterns are inconsistent with both laser-induced periodic surface structures that typically consist of linear grooves with periods smaller the laser wavelength and wrinkling patterns that feature a much larger period and appear at layered systems. Excimer laser (λ = 248 nm, tp = 25 ns) exposure of 650 nm thick photoresist films on silicon wafers cause the growth of irregular submicron patterns. The pattern morphology that is examined with imaging techniques is correlated to processing parameters. The amplitude of these laser-induced self-organized (LISE) submicron structures are strengthened with pulse number and laser fluence. The experimental results are discussed together with simulations of laser heating the photoresist film in water confinement. The proposed pattern formation mechanism of such laser-induced self-organized submicron structures at temporal excitation of a confined polymer surface comprises the formation of an oriented roughness based on LIPSS that are developed to wrinkled structures due to the transient formation of a soft subsurface area that provides conditions for wrinkling of the water cooled, stiff polymer surface by laser-induced stress fields. Size, amplitude, and morphology of the LISE pattern provide good properties for applications in such fields as wetting, friction, optics, and bioactivity.

Selective 3-dimensional patterning during phase separation of a continuously laminated layer

Control over the physical properties of materials is ubiquitously required in many fields. One means by which this can be achieved is controlling the internal structure of multi-component materials with an eye to enhancing mechanical properties. Here, we focus on self-organized pattern formation in phase separating materials, where microscopic patterns with a smooth, continuous connection may be realized. We propose a feasible method to control pattern formation using phase separation combined with continuous ``lamination'' of material, when material is continuously and homogeneously layered on top of a base. We find that a random droplet pattern, a lamellar pattern, and a cylindrical pattern are formed depending on the lamination rate $V$. We clarify the dynamics of pattern formation, focusing on the mechanism. This study may lead to the creation of new functional materials through artificial pattern control.

Effect of surface energy anisotropy on hole growth in a single-crystalline thin film: A phase-field study

Prediction of the temporal evolution of a pre-existing hole on a solid-state thin film has important implications for self-organized nanoscale pattern formation. In this work, we employ a three-dimensional phase-field model to elucidate the mechanism of corner instability during capillary-driven hole growth in a single-crystalline free-standing thin film. We perform a systematic study to demonstrate the effect of crystalline anisotropy in surface energy, surface diffusivity, and film thickness on hole growth. Our simulations reveal that the instability during hole growth arises due to saturation of the rim height at the corner of the hole that is directly related to the arc length of the corner, specified by the anisotropy in surface energy. Our investigation further reveals the presence of a time-invariant shape of the corner with the onset of corner instability.

Self-Wrinkling in Polyacrylamide Hydrogel Bilayers.

Swelling of a gel film attached to a soft substrate can induce surface instability, which results in the formation of highly ordered patterns such as wrinkles and folds. This phenomenon has been exploited to fabricate functional devices and rationalize morphogenesis. However, obtaining centimeter-scale patterns without immersing the film in a solvent remains challenging. Here, we show that wrinkles with wavelengths of up to a few centimeters can be spontaneously created during the open-air fabrication of film-substrate bilayers of polyacrylamide (PAAm) hydrogels. When the film of an aqueous pregel solution of acrylamide prepared on the PAAm hydrogel substrate undergoes open-air gelation, hexagonally packed dimples initially emerge on the surface, which later evolve into randomly oriented wrinkles. The formation of such self-organized patterns can be attributed to the surface instability resulting from autonomous water transport in the bilayer system during open-air fabrication. The temporal evolution of the patterns can be ascribed to an increase in overstress in the hydrogel film due to continued water uptake. The wrinkle wavelength can be controlled in the centimeter-scale range by adjusting the film thickness of the aqueous pregel solution. Our self-wrinkling method provides a simple mechanism for the generation of swelling-induced centimeter-scale wrinkles without requiring an external solvent, which is unachievable with conventional approaches.

Cellular reaction gene regulation network for swarm robots with pattern formation maneuvering control.

Self-organized pattern formation enables swarm robots to interact with local environments to self-organize into intricate structures generated by gene regulatory network (GRN) control methods without global knowledge. Previous studies have reported that it is challenging to maintain pattern formation stability during maneuvering in the environment due to local morphogenetic reaction rules. Motivated by the mechanism of the GRN in multi-cellular organisms, we propose a novel cellular reaction gene regulatory network (CR-GRN) for pattern formation maneuvering control. In CR-GRN, a cellular reaction network is creatively proposed to depict the robots, environment, virtual target pattern, and their interaction to generate emergent swarm behavior in multi-robot systems. A novel diffusion equation is proposed to simulate the process of morphogen diffusion among cells to ensure stable adaptive pattern generation. In addition, genes, proteins, and morphogens are used to define the internal and external states of cells and form a feedback regulation network. Simulation experiments are conducted to validate the proposed method. The results show that the CR-GRN can satisfy the requirements of turning curvature and maintain the robot's uniformity based on the proposed algorithm. This proves that robots using the CR-GRN can cooperate more effectively to cope in a complicated environment, and maintain a stable formation during maneuvering.

Formation of self-organized patterns on aluminum substrates by atmospheric pressure plasma jet for surface engineering applications

Aluminum (Al-2024) substrates were exposed to an atmospheric pressure plasma jet (APPJ) produced by a rotating arc device in air. For short nozzle-to-substrate distances, transient streamers were transferred to alter large-scale metallic surfaces. Through judicious control of the operating conditions, in particular the plasma absorbed power, air flux, distance from the jet head, scanning speed, and number of passes, self-organized patterns appeared on the aluminum substrates. Confocal surface topography, scanning electron microscopy, x-ray diffraction, and energy-dispersive x-ray spectroscopy revealed bump-like, heavily-oxidized aluminum microstructures grown on top of a layered metallic surface. Depending on the operating conditions, the bumps are either isolated or linked as serpentines. The plasma-treated surfaces with self-organized, multilayered and multiscale features identified in this work highlights a promising structure for many surface engineering applications.

Bridging scales in a multiscale pattern-forming system

Self-organized pattern formation is vital for many biological processes. Reaction-diffusion models have advanced our understanding of how biological systems develop spatial structures, starting from homogeneity. However, biological processes inherently involve multiple spatial and temporal scales and transition from one pattern to another over time, rather than progressing from homogeneity to a pattern. To deal with such multiscale systems, coarse-graining methods are needed that allow the dynamics to be reduced to the relevant degrees of freedom at large scales, but without losing information about the patterns at small scales. Here, we present a semiphenomenological approach which exploits mass conservation in pattern formation, and enables reconstruction of information about patterns from the large-scale dynamics. The basic idea is to partition the domain into distinct regions (coarse grain) and determine instantaneous dispersion relations in each region, which ultimately inform about local pattern-forming instabilities. We illustrate our approach by studying the Min system, a paradigmatic model for protein pattern formation. By performing simulations, we first show that the Min system produces multiscale patterns in a spatially heterogeneous geometry. This prediction is confirmed experimentally by in vitro reconstitution of the Min system. Using a recently developed theoretical framework for mass-conserving reaction-diffusion systems, we show that the spatiotemporal evolution of the total protein densities on large scales reliably predicts the pattern-forming dynamics. Our approach provides an alternative and versatile theoretical framework for complex systems where analytical coarse-graining methods are not applicable, and can, in principle, be applied to a wide range of systems with an underlying conservation law.

Analysis of emergent patterns in crossing flows of pedestrians reveals an invariant of 'stripe' formation in human data.

When two streams of pedestrians cross at an angle, striped patterns spontaneously emerge as a result of local pedestrian interactions. This clear case of self-organized pattern formation remains to be elucidated. In counterflows, with a crossing angle of 180°, alternating lanes of traffic are commonly observed moving in opposite directions, whereas in crossing flows at an angle of 90°, diagonal stripes have been reported. Naka (1977) hypothesized that stripe orientation is perpendicular to the bisector of the crossing angle. However, studies of crossing flows at acute and obtuse angles remain underdeveloped. We tested the bisector hypothesis in experiments on small groups (18-19 participants each) crossing at seven angles (30° intervals), and analyzed the geometric properties of stripes. We present two novel computational methods for analyzing striped patterns in pedestrian data: (i) an edge-cutting algorithm, which detects the dynamic formation of stripes and allows us to measure local properties of individual stripes; and (ii) a pattern-matching technique, based on the Gabor function, which allows us to estimate global properties (orientation and wavelength) of the striped pattern at a time T. We find an invariant property: stripes in the two groups are parallel and perpendicular to the bisector at all crossing angles. In contrast, other properties depend on the crossing angle: stripe spacing (wavelength), stripe size (number of pedestrians per stripe), and crossing time all decrease as the crossing angle increases from 30° to 180°, whereas the number of stripes increases with crossing angle. We also observe that the width of individual stripes is dynamically squeezed as the two groups cross each other. The findings thus support the bisector hypothesis at a wide range of crossing angles, although the theoretical reasons for this invariant remain unclear. The present results provide empirical constraints on theoretical studies and computational models of crossing flows.

Flags, landscapes and signaling: contact-mediated inter-cellular interactions enable plasticity in fate determination driven by positional information

Multicellular organisms exhibit a high degree of structural organization with specific cell types always occurring in characteristic locations. The conventional framework for describing the emergence of such consistent spatial patterns is provided by Wolpert's "French flag" paradigm. According to this view, intra-cellular genetic regulatory mechanisms use positional information provided by morphogen concentration gradients to differentially express distinct fates, resulting in a characteristic pattern of differentiated cells. However, recent experiments have shown that suppression of inter-cellular interactions can alter these spatial patterns, suggesting that cell fates are not exclusively determined by the regulation of gene expression by local morphogen concentration. Using an explicit model where adjacent cells communicate by Notch signaling, we provide a mechanistic description of how contact-mediated interactions allow information from the cellular environment to be incorporated into cell fate decisions. Viewing cellular differentiation in terms of trajectories along an epigenetic landscape (as first enunciated by Waddington), our results suggest that the contours of the landscape are moulded differently in a cell position-dependent manner, not only by the global signal provided by the morphogen but also by the local environment via cell-cell interactions. We show that our results are robust with respect to different choices of coupling between the inter-cellular signaling apparatus and the intra-cellular gene regulatory dynamics. Indeed, we show that the broad features can be observed even in abstract spin models. Our work reconciles interaction-mediated self-organized pattern formation with boundary-organized mechanisms involving signals that break symmetry.

Explaining multiple patches of aquatic vegetation through linear stability analysis

Aquatic vegetation is ubiquitous in lowland rivers, and it is typically present in the shape of spatial self-organized patches of biomass. In this work, we mathematically define the threshold conditions for the incipient formation of self-organized vegetated patterns in the shape of central or multiple row patches. The analysis is carried out through a linear stability analysis whereby the 2D eco-hydrodynamic model is linearized and the growth rate of small-scale perturbations is evaluated considering a basic state represented by an initially uniformly vegetated and straight channel having a certain aspect ratio and Froude number. Results illustrate that, for given vegetation properties, instability arises when both the Froude number and the aspect ratio are higher than a given threshold; in this case, self-organization occurs, and spatial patterns of patches will develop according to the wavelength associated to the maximum growth rate. Moreover, instability and self-organization take place when the undisturbed vegetation density is lower than upper bound; this suggests that densely vegetated channels, as in the case of rivers populated by invasive species, will not experience the formation of any spatial patterns.

반도체 공정용 RF 용량성 결합 플라즈마 장비에서 발생하는 Striation 현상

This study provides the conditions and characteristics of a radial striation in a capacitively coupled plasmas, which is caused by self-organized pattern formation. With the variation of the gas pressure and the RF applied voltage, we found that the striation occurs in the parameter space near the breakdown boundary of the glow discharge. By observing the time-averaged plasma parameters, it was found that the striation is triggered by the local non-uniformity of electron heating. The radial electron flux distribution shows a strong shear with the evolving striations.

Localized states with nontrivial symmetries: Localized labyrinthine patterns.

The formation of self-organized patterns and localized states are ubiquitous in Nature. Localized states containing trivial symmetries such as stripes, hexagons, or squares have been profusely studied. Disordered patterns with nontrivial symmetries such as labyrinthine patterns are observed in different physical contexts. Here we report stable localized disordered patterns in spatially extended dissipative systems. These two- and three-dimensional localized structures consist of an isolated labyrinth embedded in a homogeneous steady state. Their partial bifurcation diagram allows us to explain this phenomenon as a manifestation of a pinning-depinning transition. We illustrate our findings on the Swift-Hohenberg-type of equationsand other well-established models for plant ecology, nonlinear optics, and reaction-diffusion systems.

Contact-mediated cellular communication supplements positional information to regulate spatial patterning during development.

Development in multicellular organisms is marked by a high degree of spatial organization of the cells attaining distinct fates in the embryo. Recent experiments showing that suppression of intercellular interactions can alter the spatial patterns arising during development suggest that cell fates cannot be determined by the exclusive regulation of differential gene expression by morphogen gradients (the conventional view encapsulated in the French flag model). Using a mathematical model that describes the receptor-ligand interaction between cells in close physical proximity, we show that such intercellular signaling can regulate the process of selective gene expression within each cell, allowing information from the cellular neighborhood to influence the process by which the thresholds of morphogen concentration that dictate cell fates adaptively emerge. This results in local modulations of the positional cues provided by the global field set up by the morphogen, allowing interaction-mediated self-organized pattern formation to complement boundary-organized mechanisms in the context of development.