Pattern Formation In Systems Research Articles

Induction of patterns through crowding in a cross-diffusion model

In this paper we focus on pattern formation in systems of interacting populations. We show that if one considers these populations to be “crowded” in a way that is defined below, then cross-diffusion terms appear naturally. Moreover, we show that these additional cross-diffusion terms can generate stable spatial patterns that are not manifest in the corresponding standard “dilute” formulation. This result demonstrates the need for care when choosing standard Fickian diffusion as the default in applications to population dynamics.

Gradient elasticity in Swift–Hohenberg and phase-field crystal models

The Swift–Hohenberg (SH) and phase-field crystal (PFC) models are minimal yet powerful approaches for studying phenomena such as pattern formation, collective order, and defects via smooth order parameters. They are based on a free-energy functional that inherently includes elasticity effects. This study addresses how gradient elasticity (GE), a theory that accounts for elasticity effects at microscopic scales by introducing additional characteristic lengths, is incorporated into SH and PFC models. After presenting the fundamentals of these theories and models, we first calculate the characteristic lengths for various lattice symmetries in an approximated setting. We then discuss numerical simulations of stress fields at dislocations and comparisons with analytic solutions within first and second strain-gradient elasticity. Effective GE characteristic lengths for the elastic fields induced by dislocations are found to depend on the free-energy parameters in the same manner as the phase correlation length, thus unveiling how they change with the quenching depth. The findings presented in this study enable a thorough discussion and analysis of small-scale elasticity effects in pattern formation and crystalline systems using SH and PFC models and, importantly, complete the elasticity analysis therein. Additionally, we provide a microscopic foundation for GE in the context of order-disorder phase transitions.

Lane formation in an active particle model with chirality for pedestrian traffic

We analyze the pattern formation in systems of active particles with chiral forces in the context of pedestrian dynamics. To describe the interparticle interactions, we use the standard social force model and supplement it with a new type of force that reflects chirality. We perform numerical simulations of two pedestrian flows moving in opposite directions along a corridor. We observe two dynamic phase transitions that occur for varying number densities of particles and strengths of the chirality force: one from disordered motion to multi-lane motion and another from multi-lane to two-lane motion. We develop a qualitative theory that describes the demarcation lines for these phase transitions in the phase diagram chirality-density. The results of our analysis agree fairly well with the simulation data. A comparison with previously reported experimental data has been provided. Our findings may find applications in urban and transportation-planning problems.

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.

Stability of Pattern Formation in Systems with Dynamic Source Regions.

We explain the principles of gene expression pattern stabilization in systems of interacting, diffusible morphogens, with dynamically established source regions. Using a reaction-diffusion model with a step-function production term, we identify the phase transition between low-precision indeterminate patterning and the phase in which a traveling, well-defined contact zone between two domains is formed. Our model analytically explains single- and two-gene domain dynamics and provides pattern stability conditions for all possible two-gene regulatory network motifs.

Gierer-Meinhardt nonlinear system for pattern formation: An analytical and computational approach

Gierer-Meinhardt nonlinear system for pattern formation: An analytical and computational approach

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.

Turing patterns in systems with high-order interactions

Turing theory of pattern formation is among the most popular theoretical means to account for the variety of spatio-temporal structures observed in Nature and, for this reason, finds applications in many different fields. While Turing patterns have been thoroughly investigated on continuous support and on networks, only a few attempts have been made towards their characterization in systems with higher-order interactions. In this paper, we propose a way to include group interactions in reaction–diffusion systems, and we study their effects on the formation of Turing patterns. To achieve this goal, we rewrite the problem originally studied by Turing in a general form that accounts for a microscopic description of interactions of any order in the form of a hypergraph, and we prove that the interplay between the different orders of interaction may either enhance or repress the emergence of Turing patterns. Our results shed light on the mechanisms of pattern-formation in systems with many-body interactions and pave the way for further extensions of Turing original framework.

Distributed information encoding and decoding using self-organized spatial patterns

SummaryDynamical systems often generate distinct outputs according to different initial conditions, and one can infer the corresponding input configuration given an output. This property captures the essence of information encoding and decoding. Here, we demonstrate the use of self-organized patterns that generate high-dimensional outputs, combined with machine learning, to achieve distributed information encoding and decoding. Our approach exploits a critical property of many natural pattern-formation systems: in repeated realizations, each initial configuration generates similar but not identical output patterns due to randomness in the patterning process. However, for sufficiently small randomness, different groups of patterns that arise from different initial configurations can be distinguished from one another. Modulating the pattern-generation and machine learning model training can tune the tradeoff between encoding capacity and security. We further show that this strategy is scalable by implementing the encoding and decoding of all characters of the standard English keyboard.

Quantitative analysis of surface wave patterns of Min proteins

The Min protein system is arguably the best-studied model system for biological pattern formation. It exhibits pole-to-pole oscillations in E. coli bacteria as well as a variety of surface wave patterns in in vitro reconstitutions. Such Min surface wave patterns pose particular challenges to quantification as they are typically only semi-periodic and non-stationary. Here, we present a methodology for quantitatively analysing such Min patterns, aiming for reproducibility, user-independence, and easy usage. After introducing pattern-feature definitions and image-processing concepts, we present an analysis pipeline where we use autocorrelation analysis to extract global parameters such as the average spatial wavelength and oscillation period. Subsequently, we describe a method that uses flow-field analysis to extract local properties such as the wave propagation velocity. We provide descriptions on how to practically implement these quantification tools and provide Python code that can directly be used to perform analysis of Min patterns.

Dynamics and mechanism of a deswelling transition of the sponge phase in a bilayer membrane system

In surfactant solutions, membranes are self-organized and divide water into two domains, an ``inside'' and an ``outside.'' When the membrane phase transforms into a dilute sponge (${L}_{1}$) phase, the ${L}_{1}$ phase is also partitioned into inside and outside domains. Such systems are critical not only for material exchange mechanisms in biology but also for the formation of boundaries between topologically distinct regions. Nevertheless, the mechanism and associated behavior of the system at the transition remain unclear. Here, we use both experimental observations and numerical simulations to investigate the phase transition dynamics. It is found that domains of inside and outside are formed which appear connected to each other and that they nucleate as droplets despite the ${L}_{1}$ phase being the majority phase. Although such behavior is significantly different from what is seen in ordinary phase separation, we successfully reproduced all typical behavior seen in the experiments with numerical simulations. For the formation of the connected domains, it is found that the correlation length associated with the concentration needs to be larger than that of the inside/outside patterns. This offers a means to control pattern formation in systems with a parity mechanism, such as chirality.

Dissecting cell fate dynamics in pediatric glioblastoma through the lens of complex systems and cellular cybernetics.

Cancers are complex dynamic ecosystems. Reductionist approaches to science are inadequate in characterizing their self-organized patterns and collective emergent behaviors. Since current approaches to single-cell analysis in cancer systems rely primarily on single time-point multiomics, many of the temporal features and causal adaptive behaviors in cancer dynamics are vastly ignored. As such, tools and concepts from the interdisciplinary paradigm of complex systems theory are introduced herein to decode the cellular cybernetics of cancer differentiation dynamics and behavioral patterns. An intuition for the attractors and complex networks underlying cancer processes such as cell fate decision-making, multiscale pattern formation systems, and epigenetic state-transitions is developed. The applications of complex systems physics in paving targeted therapies and causal pattern discovery in precision oncology are discussed. Pediatric high-grade gliomas are discussed as a model-system to demonstrate that cancers are complex adaptive systems, in which the emergence and selection of heterogeneous cellular states and phenotypic plasticity are driven by complex multiscale network dynamics. In specific, pediatric glioblastoma (GBM) is used as a proof-of-concept model to illustrate the applications of the complex systems framework in understanding GBM cell fate decisions and decoding their adaptive cellular dynamics. The scope of these tools in forecasting cancer cell fate dynamics in the emerging field of computational oncology and patient-centered systems medicine is highlighted.

Inhibition of Wnt/β‐catenin Signaling Pathway on Melanogenesis‐ Insights from Zebrafish (Danio rerio)

IntroductionVertebrate color patterns are composed of specialized pigment‐producing cells, the chromatophores. Melanin is the main pigment type that colors the skin and skin appendages in mammals. Melanin is produced by specialized cells called melanocytes. Melanocytes have embryonic neural crest cells origin and their development and functions are ubiquitous across vertebrates, residing in dermal and epidermal skin layers. Studies of melanogenesis have provided experimentalists with a valuable system for understanding pattern formation in the developing embryo. In addition, studying these mechanisms may have important consequences for the clinical sciences. Alcohol is a widely abused teratogenic substance. Fetal alcohol spectrum disorder (FASD) is described as birth defects associated with prenatal alcohol exposure (PAE). PAE is likely one of the leading causes of birth defects but pathophysiology of FASD associated body pigmentation disorders are needed to be investigated further. Zebrafish (Danio rerio) (ZF) is identified as a good model to study FASD and melanocyte biology. Wnt signaling pathway is a conserved signal transduction pathway which is important in embryonic development and it regulates many aspects of melanogenesis. The current study was designed to investigate the effect of Alcohol‐Wnt signaling pathway activity on ZF melanogenesis.Materials and MethodsWild‐type (WT) ZF embryos were treated with Wnt pathway inhibitor (LiCl) and combined chemical treatment (LiCl and 1% alcohol) at 10 hours post fertilization (hpf). The differential effects of above substances on early development, differentiation, patterning of melanocytes were investigated using cell biology and imaging techniques.ResultsMelanocytes of LiCl exposed samples showed defects in migration, arrangement and survival. Ectopic melanocytes and developmental aberrations of eye pigmentation were noticed in both chemical treatments. Significant variation of the melanocyte cell size, shape and body pigmentation patterning was observed among the LiCl and combined treated samples compared to the control.ConclusionEmbryonic LiCl and alcohol exposures affect the development and patterning of the melanocytes in ZF. The molecular mechanisms underlying the above phenotypes will be investigated further to understand the FASD associated pigmentation defects.

Competitive Hybridization of a Microarray Identifies CMKLR1 as an Up-Regulated Gene in Human Bone Marrow-Derived Mesenchymal Stem Cells Compared to Human Embryonic Fibroblasts.

Mesenchymal stem cells (MSCs) have been widely applied to the regeneration of damaged tissue and the modulation of immune response. The purity of MSC preparation and the delivery of MSCs to a target region are critical factors for success in therapeutic application. In order to define the molecular identity of an MSC, the gene expression pattern of a human bone marrow-derived mesenchymal stem cell (hBMSC) was compared with that of a human embryonic fibroblast (hEF) by competitive hybridization of a microarray. A total of 270 and 173 genes were two-fold up- and down-regulated with FDR < 0.05 in the hBMSC compared to the hEF, respectively. The overexpressed genes in the hBMSC over the hEF, including transcription factors, were enriched for biological processes such as axial pattern formation, face morphogenesis and skeletal system development, which could be expected from the differentiation potential of MSCs. CD70 and CD339 were identified as additional CD markers that were up-regulated in the hBMSC over the hEF. The differential expression of CD70 and CD339 might be exploited to distinguish hEF and hBMSC. CMKLR1, a chemokine receptor, was up-regulated in the hBMSC compared to the hEF. RARRES2, a CMKLR1 ligand, stimulated specific migration of the hBMSC, but not of the hEF. RARRES2 manifested as ~two-fold less effective than SDF-1α in the directional migration of the hBMSC. The expression of CMKLR1 was decreased upon the osteoblastic differentiation of the hBMSC. However, the RARRES2-loaded 10% HA-silk scaffold did not recruit endogenous cells to the scaffold in vivo. The RARRES2–CMKLR1 axis could be employed in recruiting systemically delivered or endogenous MSCs to a specific target lesion.

Smart City's Security Model for Management of Image Data on Cloud

In modern world of advancement, cloud technology is taking its grip. COVID-19 also enhance the use of the cloud services. The cloud concept comes with the new advancement of technology, but with that data breaches and data hacks are also increasing. In the cloud-based environment, we suggested the model of security enhancement which, make use pf role-based security model where the files are shared according to the role and the access of such file is controlled using TPA validation and apart for that the concept of authentication of user is also suggested using the Pattern Lock based graphical system for pattern formation. The pattern obtained are validated using the various of online password or pattern strength validation tools, results which are obtained proves that pattern obtained through the proposed concept performs better against the attack.

Unified framework for localized patterns in reaction–diffusion systems; the Gray–Scott and Gierer–Meinhardt cases

A recent study of canonical activator-inhibitor Schnakenberg-like models posed on an infinite line is extended to include models, such as Gray-Scott, with bistability of homogeneous equilibria. A homotopy is studied that takes a Schnakenberg-like glycolysis model to the Gray-Scott model. Numerical continuation is used to understand the complete sequence of transitions to two-parameter bifurcation diagrams within the localized pattern parameter regime as the homotopy parameter varies. Several distinct codimension-two bifurcations are discovered including cusp and quadruple zero points for homogeneous steady states, a degenerate heteroclinic connection and a change in connectedness of the homoclinic snaking structure. The analysis is repeated for the Gierer-Meinhardt system, which lies outside the canonical framework. Similar transitions are found under homotopy between bifurcation diagrams for the case where there is a constant feed in the active field, to it being in the inactive field. Wider implications of the results are discussed for other pattern-formation systems arising as models of natural phenomena. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Snowdrift game induces pattern formation in systems of self-propelled particles.

Evolutionary games between species are known to lead to intriguing spatiotemporal patterns in systems of diffusing agents. However, the role of interspecies interactions is hardly studied when agents are (self-)propelled, as is the case in many biological systems. Here, we combine aspects from active matter and evolutionary game theory and study a system of two species whose individuals are (self-)propelled and interact through a snowdrift game. We derive hydrodynamic equations for the density and velocity fields of both species from which we identify parameter regimes in which one or both species form macroscopic orientational order as well as regimes of propagating wave patterns. Interestingly, we find simultaneous wave patterns in both species that result from the interplay between alignment and snowdrift interactions-a feedback mechanism that we call game-induced pattern formation. We test these results in agent-based simulations and confirm the different regimes of order and spatiotemporal patterns as well as game-induced pattern formation.

Spatiotemporal instabilities and pattern formation in systems of diffusively coupled Izhikevich neurons

Neurons are often connected, spatially and temporally, in phenomenal ways that promote wave propagation. Therefore, it is essential to analyze the emergent spatiotemporal patterns to understand the working mechanism of brain activity, especially in cortical areas. Here, we present an explicit mathematical analysis, corroborated by numerical results, to identify and investigate the spatiotemporal, non-uniform, patterns that emerge due to instability in an extended homogeneous 2D spatial domain, using the excitable Izhikevich neuron model. We examine diffusive instability and perform bifurcation and fixed-point analyses to characterize the patterns and their stability. Then, we derive analytically the amplitude equations that establish the activities of reaction-diffusion structures. We report on the emergence of diverse spatial structures including hexagonal and mixed-type patterns by providing a systematic mathematical approach, including variations in correlated oscillations, pattern variations and amplitude fluctuations. Our work shows that the emergence of spatiotemporal behavior, commonly found in excitable systems, has the potential to contribute significantly to the study of diffusively-coupled biophysical systems at large.

Quantification of Myxococcus xanthus Aggregation and Rippling Behaviors: Deep-Learning Transformation of Phase-Contrast into Fluorescence Microscopy Images.

Myxococcus xanthus bacteria are a model system for understanding pattern formation and collective cell behaviors. When starving, cells aggregate into fruiting bodies to form metabolically inert spores. During predation, cells self-organize into traveling cell-density waves termed ripples. Both phase-contrast and fluorescence microscopy are used to observe these patterns but each has its limitations. Phase-contrast images have higher contrast, but the resulting image intensities lose their correlation with cell density. The intensities of fluorescence microscopy images, on the other hand, are well-correlated with cell density, enabling better segmentation of aggregates and better visualization of streaming patterns in between aggregates; however, fluorescence microscopy requires the engineering of cells to express fluorescent proteins and can be phototoxic to cells. To combine the advantages of both imaging methodologies, we develop a generative adversarial network that converts phase-contrast into synthesized fluorescent images. By including an additional histogram-equalized output to the state-of-the-art pix2pixHD algorithm, our model generates accurate images of aggregates and streams, enabling the estimation of aggregate positions and sizes, but with small shifts of their boundaries. Further training on ripple patterns enables accurate estimation of the rippling wavelength. Our methods are thus applicable for many other phenotypic behaviors and pattern formation studies.

Global existence for vector valued fractional reaction-diffusion equations

In this paper we study the initial value problem for infinite dimensional fractional non-autonomous reaction-diffusion equations. Applying general time-splitting methods, we prove the existence of solutions globally defined in time using convex sets as invariant regions. We expose examples where biological and pattern formation systems, under suitable assumptions, achieve global existence. We also analyze the asymptotic behavior of solutions.