Mechanism Of Pattern Formation Research Articles

Inferring traits of hyperuniformity from local structures via persistent homology.

Hyperuniformity refers to the suppression of density fluctuations at large scales. Typical for ordered systems, this property also emerges in several disordered physical and biological systems, where it is particularly relevant to understand mechanisms of pattern formation and to exploit peculiar attributes, e.g., interaction with light and transport phenomena. While hyperuniformity is a global property, ideally defined for infinitely extended systems, several disordered correlated systems have finite size. It has been shown in [Phys. Rev. Research 6, 023107 (2024)] that global hyperuniform characteristics systematically correlate with distributions of topological properties representative of local arrangements. In this work, building on this information, we explore and assess the inverse relationship between hyperuniformity and local structures in point patterns as described by persistent homology. Standard machine learning algorithms trained on persistence diagrams are shown to detect hyperuniformity of periodic point patterns with high accuracy. Therefore, we demonstrate that the information on patterns' local structures allows for characterizing whether finite size arrangements are analogous to those realized in hyperuniform patterns. Then, addressing more quantitative aspects, we show that parameters defining hyperuniformity globally can be reconstructed by comparing persistence diagrams of targeted patterns with reference ones. We also explore the generation of patterns entailing given topological properties. The results of this study pave the way for advanced analysis of hyperuniform patterns including local information, and introduce basic concepts for their inverse design.

How the tulip breaking virus creates striped tulips

The beauty of tulips has enchanted mankind for centuries. The striped variety has attracted particular attention for its intricate and unpredictable patterns. A good understanding of the mechanism driving the striped pattern formation of broken tulips has been missing since the 17th century. It has been known since 1928 that these patterned tulips suffer from a viral infection by the tulip breaking virus. Here, we present a mathematical model to understand how a virus infection of the petals can lead to stripes, thereby providing a possible explanation of a 350 year-old mystery. The model, which describes the viral inhibition of pigment expression (anthocyanins) and their interaction with viral reproduction, incorporates a pattern formation mechanism identified as an activator-substrate mechanism, similar to the well-known Turing instability, working together with Wolpert’s positional information mechanism. The model is solved on a growing tulip petal-shaped domain, whereby we introduce a new method to describe the tulip petal growth explicitly. This work shows how a viral infection that inhibits pigment production can lead to beautiful tulip patterns.

Abundant new interaction solutions and nonlinear dynamics for the (3+1)-dimensional Hirota–Satsuma–Ito-like equation

Abstract In this article, the (3+1)-dimensional Hirota–Satsuma–Ito-like equation is investigated by the modified direct method, from which some interaction solutions among lump, stripe solitons, and Jacobi elliptic function wave solutions are obtained, which are crucial in understanding complex behaviors in nonlinear systems where multiple wave types coexist and interact. The corresponding evolution and dynamics for the interaction solutions under different parameters are discussed. Such interactions are key to modeling realistic systems in which multiple phenomena coexist, such as fluid mechanics, plasma physics, and optical systems, where waves can exchange energy and form stable or unstable patterns. These results reported in this article can reveal the theoretical mechanisms of stability, energy transfer, and pattern formation in nonlinear media and may raise the possibility of related experiments and potential applications in nonlinear science fields, such as oceanography, nonlinear optics, and so on.

Pattern Formation Mechanisms of Spatiotemporally Discrete Activator–Inhibitor Model with Self- and Cross-Diffusions

In this paper, we aim to solve the issue of pattern formation mechanisms in a spatiotemporally discrete activator–inhibitor model that incorporates self- and cross-diffusions. We seek to identify the conditions that lead to the emergence of complex patterns and to elucidate the principles governing the dynamic behaviors that result in these patterns. We first construct a corresponding coupled map lattice (CML) model based on the continuous activator–inhibitor reaction–diffusion system. In the reaction stage, we examine the existence, uniqueness, and stability of the homogeneous stationary state and specify the parametric conditions for realizing these properties. Furthermore, by applying the center manifold theorem, we perform a flip bifurcation analysis and confirm that the model is capable of undergoing flip bifurcation. In the diffusion stage, we focus on the analysis of Turing bifurcation and determine the parameter conditions for the emergence of Turing instability. Through numerical simulations, we validate and explain the results of our theoretical analysis. Our study reveals various Turing instability mechanisms by coupling Turing and flip bifurcations, which include pure-self-diffusion-Turing instability, pure-cross-diffusion-Turing instability, flip-self-diffusion-Turing instability, flip-cross-diffusion-Turing instability, and chaos-self-diffusion-Turing instability mechanisms. Under different mechanisms, we illustrate the corresponding Turing patterns and discover a rich variety of pattern types such as labyrinthine, mosaic, alternating mosaic, colorful mottled grid patterns with winding and twisted bands, and patterns with dense patches and twisted bands nested together. Our research provides a theoretical framework and numerical support for understanding the complex dynamical behaviors and pattern formations in activator–inhibitor models with self- and cross-diffusions.

Self-assembled micro-patterns in uphill-diffusion solution system.

In this work we present self-organized regular patterns in a solution system through uphill-diffusion. Micrometer thick organic semiconductor solution is sandwiched between a substrate and cover-plate. Self-assembled regular patterns can be observed on the substrate after solvent evaporation. Different micro-patterns and pattern defects were displayed and analyzed. Mechanisms of defect formation, mode selection process during patten generation, and pattern sedimentation onto substrate from solution were proposed. Organic thin film transistors were fabricated with the assembled line patterns which demonstrate a promising way to produce patterned micro/nano materials.

Controlled Protein-Membrane Interactions Modulate Self-Organization of Min Protein Patterns.

Self-organizing protein patterns are crucial for living systems, governing important cellular processes such as polarization and division. While the field of protein self-organization has reached a point where basic pattern-forming mechanisms can be reconstituted in vitro using purified proteins, understanding how cells can dynamically switch and modulate these patterns, especially when transiently needed, remains an interesting frontier. Here, we demonstrate the efficient regulation of self-organizing protein patterns through the modulation of simple biophysical membrane parameters. Our investigation focuses on the impact of membrane affinity changes on Min protein patterns at lipid membranes composed of Escherichia coli lipids or minimal lipid compositions, and we present three major results. First, we observed the emergence of a diverse array of pattern phenotypes, ranging from waves over flower-shaped patterns to snowflake-like structures. Second, we demonstrated the dependency of these patterns on the density of protein-membrane linkers. Finally, we demonstrate thatthe shape of snowflake-like patterns isfine-tuned by membrane charge. Our results demonstrate the significant influence of membrane linkage as a straightforward biophysical parameter governing protein pattern formation. Our research points towards a simple yet intriguing mechanism by which cells can adeptly tune and switch protein patterns on the mesoscale.

Fluctuation-driven morphological patterning: An unconventional approach to morphogenesis

Recent experimental investigations into regeneration revealed a remarkable phenomenon: the morphological transformation of a tissue fragment from the incipient spherical configuration to a tubelike structure—the hallmark of a mature —has the dynamical characteristics of a first-order phase transition, with calcium field fluctuations within the tissue playing an essential role. This morphological transition was shown to be generated by activation over a barrier within an effective morphological potential that underlies morphogenesis. Inspired by this intriguing insight, we propose an unconventional mechanism where stochastic fluctuations drive the emergence of morphological patterns. Thus, the inherent fluctuations determine the nature of the dynamics and are not incidental noise in the background of the otherwise deterministic dynamics. Instead, they play an important role as a driving force that defines the attributes of the pattern formation dynamics and the nature of the transition itself. Here, we present a simple model that captures the essence of this mechanism for morphological pattern formation. Specifically, we consider a one-dimensional tissue arranged as a closed contour embedded in a two-dimensional space, where the local curvature of the contour is coupled to a non-negative scalar field. An effective temperature parameter regulates the strength of the fluctuations in the system. The tissue exhibits fluctuations near a circular shape at sufficiently low coupling strengths, but as the coupling strength exceeds some critical value, the circular state becomes unstable. The nature of the transition to the new state, namely whether it is a first-order-like or second-order-like transition, depends on the effective temperature and the effective cutoff on the wavelength of the spatial variations in the system. The former sets the level of noise, while the latter determines the height of the potential barrier between the initial circular state and the final deformed state of the tissue. It is also found that entropic barriers separate various metastable states of the system. Published by the American Physical Society 2024

Spatial patterns of the Brusselator model with asymmetric Lévy diffusion

Abstract The formation of spatial patterns is an important issue in reaction–diffusion systems. Previous studies have mainly focused on the spatial patterns in reaction–diffusion models equipped with symmetric diffusion (such as normal or fractional Laplace diffusion), namely, assuming that spatial environments of the systems are homogeneous. However, the complexity and heterogeneity of spatial environments of biochemical reactions in vivo can lead to asymmetric diffusion of reactants. Naturally, there arises an open question of how the asymmetric diffusion affects dynamical behaviors of biochemical reaction systems. To answer this, we build a general asymmetric Lévy diffusion model based on the theory of a continuous time random walk. In addition, we investigate the two-species Brusselator model with asymmetric Lévy diffusion, and obtain a general condition for the formation of Turing and wave patterns. More interestingly, we find that even though the Brusselator model with symmetric diffusion cannot produce steady spatial patterns for some parameters, the asymmetry of Lévy diffusion for this model can produce wave patterns. This is different from the previous result that wave instability requires at least a three-species model. In addition, the asymmetry of Lévy diffusion can significantly affect the amplitude and frequency of the spatial patterns. Our results enrich our knowledge of the mechanisms of pattern formation.

Self-organization in spatial ecology

Biologists have long known that populations of organisms - microbes, plants, animals - can self-organize into emergent patterns. Yet, the fact that such patterns can arise with remarkable symmetry at the scale of entire ecosystems remains astonishing, even as aerial imagery has documented their existence across all continents. As the enormous scale of landscape patterns makes them experimentally intractable, ecologists have relied on theoretical modelling - typically rooted in physics - to investigate the underlying pattern-forming mechanisms. Such models have succeeded in generating mechanistic hypotheses and indicate that self-organized spatial patterns can reflect the health of an ecosystem. However, most of these hypotheses remain untested. This essay reflects on our current understanding of the causes and consequences of ecosystem-scale pattern formation.

The effect of diffusion on multistability and stochastic sensitivity in spatially extended systems

A spatially-extended reaction-diffusion model is considered. The work is focused on self-organization mechanisms of diffusion instability and Turing pattern formation. Pattern generation and multistability of the system is demonstrated for varying diffusion intensity. In the stochastic variant of the model, the sensitivity of coexisting patterns to noise is studied. The stochastic sensitivity function technique is used for the analysis of noise-induced transitions between patterns. Application of stochastic sensitivity functions is discussed on examples.

Geometrical incompatibility regulated pattern selection and morphological evolution in growing spherical soft tissues

Surface morphological patterns are widely observed in natural systems, which are closely correlated to vital biological functions and inspire surface morphology designs in soft matter systems. Geometrical incompatibility widely exists in biological tissues across different length scales and plays an important role in growth-induced pattern selection and morphological evolution of soft tissues. However, the underlying physical mechanism of growth-induced pattern formation and post-buckling evolution in geometrically incompatible spherical soft tissues remain elusive. Here, the effect of geometrical incompatibility on the growth-induced pattern selection and post-buckling evolution are investigated through swelling experiment, theoretical analysis and numerical simulation. The results show that not only the instability pattern but also the instability threshold can be regulated by manipulating geometric incompatibility. Notably, when the geometrical incompatibility parameter exceeds a critical value, spontaneous instability is observed before growth. With continuous growth, the core–shell soft sphere buckles into a periodic buckyball pattern and evolves toward a bean-shaped pattern, and then undergoes a wrinkle-to-fold transition into a labyrinth topography. Our results demonstrate, both experimentally and theoretically, that geometrical incompatibility can guide the growth-induced pattern formation and morphological evolution effectively. This study not only enhances our understanding of the growth-induced pattern selection and morphological evolution in spherical soft tissues, but also provides an inspiring insight for the fabrication of morphological patterns on curved surfaces.

Formation mechanism of radial and circular cracks promoted by delamination in drying silica colloidal deposits.

Cracks with radial and circular patterns are appealing in nature and industry. Although morphologies and propagation conditions of cracks are extensively studied, the formation mechanism of crack pattern by the interaction of channel fracture and interfacial delamination remains elusive. Here, we present the transition of radial to coexisting radial and circular crack patterns when the thickness of colloidal deposits on both hard and soft substrates exceeds a critical value, through the colloidal volume fraction dependence. In addition, a thickness-dependent phase diagram from radial crack to coexistence of radial and circular cracks was constructed with respect to the radius and the volume fractions of silica colloidal deposits. A phase-field fracture model is developed to elucidate how the formation of radial cracks is facilitated by simultaneous delamination. The warping-induced radial tensile stress at the bottom surface of the striped deposit is proportional to the thickness. It leads to subsequent nucleation and growth of circular cracks in thick deposits. This work provides insight into the formation mechanism of complex crack patterns in drying colloidal deposits and revolutionizes the design space of crack-based micro-nano structures.

Acoustofluidic Diversity Achieved by Multiple Modes of Acoustic Waves Generated on Piezoelectric-Film-Coated Aluminum Sheets.

Excitation of multiple acoustic wave modes on a single chip is beneficial to implement diversified acoustofluidic functions. Conventional acoustic wave devices made of bulk LiNbO3 substrates generally generate few acoustic wave modes once the crystal-cut and electrode pattern are defined, limiting the realization of acoustofluidic diversity. In this paper, we demonstrated diversity of acoustofluidic behaviors using multiple modes of acoustic waves generated on piezoelectric-thin-film-coated aluminum sheets. Multiple acoustic wave modes were excited by varying the ratios between IDT pitch/wavelength and substrate thickness. Through systematic investigation of fluidic actuation behaviors and performances using these acoustic wave modes, we demonstrated fluidic actuation diversities using various acoustic wave modes and showed that the Rayleigh mode, pseudo-Rayleigh mode, and A0 mode of Lamb wave generally have better fluidic actuation performance than those of Sezawa mode and higher-order modes of Lamb wave, providing guidance for high-performance acoustofluidic actuation platform design. Additionally, we demonstrated diversified particle patterning functions, either on two sides of acoustic wave device or on a glass sheet by coupling acoustic waves into the glass using the gel. The pattern formation mechanisms were investigated through finite element simulations of acoustic pressure fields under different experimental configurations.

Spatial patterning and species coexistence: A case study using concentric circular vegetation patches in saline land

Spatial patterns in plant community structures within stressed ecosystems have drawn much attention in the field of ecology. However, the mechanisms underlying spatial formation and its impact on species coexistence and diversity remain controversial. In this study, we investigated concentric circular vegetation patches in coastal saline land, and analysed the spatial patterning of plant communities and associated soil physicochemical properties. Thereafter, we tested how the soil conditioned by plant communities from different locations within the vegetation patches influence the species growth and inter-specific competition. Our results show soil salinity enlarges in a centrifugal manner in horizontal direction in all patches. Soil salinity decreased and species diversity increased along with the increase of patch size. In addition, we found significant shifts in both the composition of plant communities and in soil physicochemical properties from outer to center. The results indicate that the pioneer species Suaeda salsa facilitated the subsequent species. However Suaeda salsa was inhibited and became inferior competitor in the soil conditioned by the subsequent species. We infer that the less-visible spatial patterns of soil physicochemical properties at small scales create ecological niches for specialized species, allowing them to coexist but not mix. We suggest that a trade-off between tolerance to salt stress and competitive ability under ameliorated conditions may underlie mechanisms of pattern formation in small scale. Our findings lend support to the idea that soil stress constraints community assembly and triggers spatial patterns, which, in turn, buffer the stress on plant communities and enhance species diversity.

Spatial distribution of self-seeded air lasers induced by the femtosecond laser filament plasma

The femtosecond laser filament-induced air laser plays a significant role for the remote sensing of air pollutants. The spatial distributions of air laser intensity were investigated experimentally in previous studies. However, the mechanism of the air laser propagation properties inside the filament plasma has not been quite clear yet. Moreover, few studies have been dedicated to the reproduction of the air laser profile from nitrogen molecules propagating in the filament plasma based on the numerical simulation method. In this study, the lasing action of the air laser from the transition of the first negative (0,0) band of nitrogen ions at 391 nm was simulated during the femtosecond laser filamentation. The beam profile of the air laser changes from a Gaussian or super-Gaussian shape to an outer ring structure by increasing the filament length or nitrogen ion density, which is in accord with the previous experimental result. A multiple-diffraction effect has been proposed to clarify the mechanism of the outer rings beam pattern formation, which is induced by the dynamical interaction between the lasing effect and diffraction effect of the air laser propagating inside the filament plasma. In addition, the amplified air laser power as a function of both the filament length and nitrogen ion density was investigated. Our study would pave the way to improve the energy conversion efficiency and directivity of remote air lasers, which would be significant for remote sensing applications.

Formation mechanism of bright and dark concentric-ring pattern in dielectric barrier discharge

In this work, a bright and dark concentric-ring pattern is reported in a dielectric barrier discharge for the first time. The spatiotemporal dynamics of the bright and dark concentric-ring pattern are investigated with an intensified charge-coupled device and photomultiplier tubes. The results indicate that the bright and dark concentric-ring pattern is composed of three concentric-ring sublattices. These are bright concentric-ring structures, dark concentric-ring structures and wider concentric-ring structures, respectively. The bright concentric-ring structures and dark concentric-ring structures are alternately distributed. The bright concentric-ring structures are located at the centre of the wider concentric-ring structures. The wider concentric-ring structures first form from the outer edge and gradually develop to the centre. The essence of all three concentric-ring structures is the individual discharge filaments. The optical emission spectra of different sublattices are acquired and analysed. It is found that the plasma parameters of the three concentric-ring sublattices are different. Finally, the formation mechanism of the bright and dark concentric-ring pattern is discussed.

Nature of instability in flow-driven porous anodic oxide.

Self-organized porous anodic oxide films are formed by the electrochemical oxidation of reactive metal aluminum in acidic solutions in which the oxide is soluble. Recently, viscous flow models have shown using linear stability analysis that the instability results from a trade-off between the destabilizing effect of viscous flow of oxide and the stabilizing effect of oxide formation, which provides the wavelength selection mechanism for pattern formation. Anion adsorption on surface growth sites causes nonuniform compressive stress at the oxide-solution interface, which drives the flow. This anodic instability is analogous to the classical Marangoni instability induced by surface tension gradients. In this work, nature of the instability beyond the stability threshold is determined using a weakly nonlinear analysis. For the growth of well-developed pores beyond the threshold, a subcritical nature of the instability is essential. However, our weakly nonlinear analysis shows that the solutions emerging from neutral stability are supercritical in nature at all wavenumbers for the practical range of anodizing control parameters investigated. We also determine the region where the model is Hadamard stable, a necessary condition for well-posedness.

Drivers of β-diversity and its component-dependence along the natural restoration in an extremely heterogeneous forest ecosystem

β-diversity usually refers to species turnover along a gradient or changes in species diversity between communities. It is a key concept in understanding the mechanisms of biodiversity pattern formation. However, the component dependence and the driving factors of β-diversity during natural restoration remain unclear. In this study, a series of forest dynamic plots (FDPs) were established through the natural restoration stages in degraded karst forests, including the shrub-canopy mixed forest stage (SC), the secondary-growth forest stage (SG), and the old-growth forest stage (OG). After comparing species composition during natural restoration and its dependence on species turnover and nestedness, it was found that natural restoration significantly influenced the species composition by altering the species turnover in degraded karst forests rather than the nestedness. The contribution of turnover to β-diversity gradually increased, while the contribution of nestedness decreases gradually. Furthermore, the PER-SIMPER method was used to quantify the first-order assembly process that drives variations in species composition during restoration stages. To identify the factors influencing β-diversity, both variation partitioning analysis and a random forest model were employed. The results suggested that differences in species compositions across restoration stages were mainly driven by stochastic processes, and the deterministic processes were strengthened during the natural restoration. In addition, topographic factors contributed more to β-diversity than soil nutrient factors. This study highlighted changes in component dependence of degraded karst forests during natural restoration and emphasized the importance of a more comprehensive framework in assessing the restoration of degraded forest ecosystems. More attention should be paid to topographic structure in understanding biodiversity patterns, especially in highly heterogeneous forest ecosystems.

Mechanism for axial pattern formation of concentrated suspension in a horizontal rotating cylinder

We have performed numerical simulations to investigate the phenomenon of axial pattern formation exhibited by a non-neutrally buoyant concentrated suspension. Continuum modelling of the concentrated suspension is done using the suspension balance model to identify the underlying mechanism of the phenomenon. We demonstrate that axial concentration variations become amplified to axial bands owing to the influence of the second normal stress difference ( $N_2$ ), and the first normal difference ( $N_1$ ) accentuates the effect of $N_2$ . We demonstrate that the end walls of the rotating cylinder are necessary to prevent the smearing out of axial bands but are not a direct cause of the phenomenon.

Active Density Pattern Formation in Bacterial Binary Mixtures

Microbial communities often exhibit complex spatial structures that are important for their functioning, ecology, and evolution. While the role of biochemical interactions is extensively studied, how physical interactions contribute to structuring typically multispecies and phenotypically diverse bacterial communities is unclear. Here, we examined how motility, a major bacterial trait, affects the organization of model binary mixtures of motile and nonmotile bacteria, revealing a novel type of active self-organization. Motile bacteria induce large-scale, fluctuating density patterns in nonmotile bacteria across a wide range of physiologically relevant cell densities. Combining experiments and quantitative modeling, we demonstrate that this pattern formation solely results from physical interactions and relies on two key ingredients: By swimming in circles at surfaces, the motile bacteria generate recirculating hydrodynamic flows that advect nonmotile cells, and the breaking of vertical symmetry by gravity permits local density accumulation. The density patterns fluctuate as the advection landscape slowly rearrange with motile cells configurations on the surfaces. This new nonequilibrium mechanism for pattern formation in bacteria belongs to a different class compared to previous models for self-organization in self-propelled systems, which rely on localized traffic jamming in two dimensions. It is instead similar to fluctuation-dominated phase ordering in active nematics. The behavior also drives the formation of large scale and highly structured aggregates of nonmotile cells that express adhesins in the mixture, and it showcases how motile species activity can shape the spatial organization of complex microbial communities. Published by the American Physical Society 2024