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The role of geographical spreaders in infectious pattern formation and front propagation speeds
The pattern formation and spatial spread of infectious populations are investigated using a kernel-based Susceptible–Infectious–Recovered (SIR) model applicable across a wide range of basic reproduction numbers R O . The goal is to examine the role of geographical spreaders on transient spatial pattern formation of infectious populations and the associated maximum invasive front speeds c max . In the simulations conducted here, geographical spreaders are defined as a portion of the infected population Φ experiencing high mobility between identical communities. The spatial organization of the infected population and c max are determined when the infections are randomly initiated in space within multiple communities. For small but finite , scaling analysis and numerical simulations in 1-dimension suggest that when the spreading kernel is Gaussian-shaped, where is the inverse of the infectious duration. This finding for agrees with a diffusion-based representation of mobility in 1-D. Numerical simulations in 2-D across wide-ranging suggest that , the variance of the spatial kernel describing mobility of long-distance geographical spreaders across communities, determines the spatial organization of infections across communities. When (long-distance mobility, where is the minimum spatial extent defining adjacent communities), the infectious population will experience a transient but spatially coherent pattern with a wavelength that can be derived from the spreading kernel properties. Moreover, the 2-D simulations for the bounded kernel suggest that attainment of is also dictated by but the magnitude is not sensitive to unlike diffusion-based models.
Equilibrium-gated pattern formation: How molecular dissociation thermodynamics drive emergent behavior in dissipative polymeric systems
Emergent patterns in biological systems arise through dissipative processes that balance reaction and transport phenomena, producing highly functional properties from self-regulating mechanisms. Synthetic fabrication, by contrast, often relies on user-controlled, multistep methods that lack the self-organizing capabilities of natural systems. Inspired by nature, we sought chemical systems that integrate strongly coupled reaction and transport phenomena, identifying frontal ring-opening metathesis polymerization (FROMP) as a method capable of creating diverse forms and functions through reactive processing. By employing discrete molecular initiators, FROMP allows precise control of key reaction steps—inhibition, initiation, and propagation. Using an integrated computational and experimental framework, we uncover how near-equilibrium inhibition dynamics, coupled with far-from-equilibrium reaction kinetics, drive pattern formation in frontally polymerized synthetic materials. We propose the concept of equilibrium-gated pattern formation, demonstrating how initiator chemistry can be tuned to achieve programmable macroscale properties. Our study reveals a surprising insight: Emergent behavior in FROMP systems arises from the inhibition-dominated regime of resin composition, expanding prior observations that such behavior is confined to a narrow compositional space near the boundary between front quenching and uniform front propagation. We identify a broader compositional window, far from the quenching regime, where emergent behavior reliably manifests. This expanded design space significantly enhances the operational flexibility of reactive systems and their capacity for self-organization. Furthermore, these insights provide a roadmap for designing bioinspired materials with self-organizing capabilities, unlocking possibilities in synthetic manufacturing.
3D pattern formation of a protein–membrane suspension
Many essential cellular processes, including cell division and the establishment of cell polarity during embryogenesis, are regulated by pattern-forming proteins. These proteins often need to bind to a substrate, such as the cell membrane, onto which they interact and form two-dimensional (2D) patterns. It is unclear how the membrane’s continuity and dimensionality impact pattern formation. Here, we address this gap using the MinDE system, a prototypical example of pattern-forming membrane proteins. We show that when the lipid substrate is fragmented into submicrometer-sized diffusive liposomes, adenosine triphosphate-driven protein–protein interactions generate three-dimensional (3D) spatially extended patterns, despite the complete loss of membrane continuity. Remarkably, these 3D patterns emerge at scales four orders of magnitude larger than the individual liposomes. By systematically varying protein concentration, liposome size, and density, we observed and characterized a variety of 3D dynamical patterns not seen on continuous 2D membranes, including traveling waves, dynamical spirals, and a coexistence phase. Simulations and linear stability analysis of a coarse-grained model revealed that the physical properties of the dispersed membrane effectively rescale both the protein–membrane binding rates and diffusion, two key parameters governing pattern formation and wavelength selection. These findings highlight the robustness of Min’s pattern-forming ability, suggesting that protein–membrane suspensions could serve as an adaptable template for studying out-of-equilibrium self-organization in 3D, beyond in vivo contexts.
Efficient data-driven regression for reduced-order modeling of spatial pattern formation
We present an efficient data-driven regression approach for constructing reduced-order models (ROMs) of reaction-diffusion systems exhibiting pattern formation. The ROMs are learned non-intrusively from available training data of physically accurate numerical simulations. The method can be applied to general nonlinear systems through the use of polynomial model form, while not requiring knowledge of the underlying physical model, governing equations, or numerical solvers. The process of learning ROMs is posed as a low-cost least-squares problem in a reduced-order subspace identified via Proper Orthogonal Decomposition (POD). Numerical experiments on classical pattern-forming systems–including the Schnakenberg and Mimura–Tsujikawa models–demonstrate that higher-order surrogate models significantly improve prediction accuracy while maintaining low computational cost. The proposed method provides a flexible, non-intrusive model reduction framework, well suited for the analysis of complex spatio-temporal pattern formation phenomena.
Transient pattern formation in an active matter contact poisoning model
Abstract One of the most notable features in repulsive particle based active matter systems is motility-induced-phase separation (MIPS) where a dense, often crystalline phase and low density fluid coexist. Most active matter studies involve time-dependent activity; however, there are many active systems where individual particles transition from living or moving to dead or nonmotile due to lack of fuel, infection, or poisoning. Here we consider an active matter particle system at densities where MIPS does not occur. When we add a small number of infected particles that can poison other particles, rendering them nonmotile, we find a rich variety of time dependent pattern formation, including MIPS, a wetting phase, and a fragmented state formed when mobile particles plow through a nonmotile packing. We map the patterns as a function of time scaled by epidemic duration, and show that the pattern formation is robust for a wide range of poisoning rates and activity levels. We also show that pattern formation does not occur in a random death model, but requires the promotion of nucleation by contact poisoning. Our results should be relevant to biological and active matter systems where there is some form of poisoning, death, or transition to nonmotility.
Phase Pattern Formation in Grain Boundaries
Here, in this Letter, we derive conditions that predict the existence of two-phase periodic-pattern grain boundary structures that are stable against coarsening. While previous research has established that elastic effects can lead to phase pattern formation on crystal surfaces, the possibility of stable grain boundary structures composed of alternating grain boundary phases has not been previously analyzed. Our theory identifies the specific combination of grain boundary and materials properties that enable the emergence of patterned grain boundary states and shows that the dislocation content of grain boundary phase junctions, absent in surface phenomena, weakens the stability of the patterned structures. The predictions of the theory are tested using a model copper grain boundary that exhibits multiple phases and two-phase pattern formation. We discuss how, similarly to surfaces, elastic effects associated with grain boundary phase junctions have profound implications for how grain boundary phases transform.
Electrostatically driven pattern formation in mixed charged–neutral multicomponent elastic membranes
Multicomponent crystalline and amorphous elastic shells exhibit heterogeneous surface patterns that provide distinctive functionalities in cellular environments. Such patterning typically arises from the competition between short-range attractive and long-range repulsive interactions in membranes. Here, we demonstrate that the intrinsic competition between electrostatic repulsion and elastic deformation is sufficient to drive spontaneous surface patterning in elastic shells, requiring no additional attractive interactions. Using numerical simulations, we demonstrate pattern formation in mechanically homogeneous membranes with heterogeneous surface charge composition across different topologies, including spheres, discs, and flat periodic membranes. We also examine patterns in crystalline and amorphous shells of coassembled charged and neutral components with different bending rigidities. At low charge fraction, discrete charged surface domains form. At intermediate charge fraction, the competition between electrostatics and elasticity leads to elongated domains (rods) of the charged component, which results in lamellar patterns at nearly equal fraction of the charged and neutral components. At high charge fraction, nanodomains of the neutral component form. Amorphous shells exhibit similar progressions but with disordered structures rather than ordered lamellar patterns. These pattern morphologies are observed in both the closed shells and flat membranes. As salt concentration increases, all patterns coarsen due to the screening of electrostatic interactions.
Subsurface Cooling and Sea Surface Temperature Pattern Formation Over the Equatorial Pacific
Abstract The equatorial cold tongue region has not warmed up in response to historical radiative forcing in the real world, contrary to the strong warming often simulated by climate models. Here we demonstrate that climate models fail to represent one or both of the key processes driving observed sea surface temperature (SST) pattern formation: a realistic surface wind stress pattern shaping subsurface cooling through wind‐driven circulation changes, and effective connectivity between subsurface and surface temperatures via upwelling and mixing. Consequently, none of the models approximate the observed lack of cold tongue SST warming and strengthening of zonal SST gradient across the equatorial Pacific. Furthermore, those that come closest achieve this due to interhemispheric warming differences rather than equatorial dynamics as observed. Addressing different origins of subsurface cooling in observations and simulations, and how they connect to SST, will lead to improved understanding of tropical Pacific SST changes to date and how they will evolve in the future.
Pattern formation in odd viscoelastic fluids
Nonreciprocal interactions fueled by local energy consumption can be found in biological and synthetic active matter at scales where viscoelastic forces are important. Such systems can be described by “odd” viscoelasticity, which assumes fewer material symmetries than traditional theories. Here we study odd viscoelasticity analytically and using lattice Boltzmann simulations. We identify a pattern-forming instability which produces an oscillating array of fluid vortices, and we elucidate which features govern the growth rate, wavelength, and saturation of the vortices. Our observation of pattern formation through odd mechanical response can inform models of biological patterning and guide engineering of odd dynamics in soft active matter systems. Published by the American Physical Society 2024
Microstructural pattern formation during liquid metal dealloying: Phase-field simulations and theoretical analyses
In recent years, liquid metal dealloying has emerged as a promising material processing method to generate micro- and nanoscale bicontinuous or porous structures. Most previous studies focused on the experimental characterization of the dealloying process and on the properties of the dealloyed materials, leaving the theoretical study incomplete to fully understand the fundamental mechanisms of the liquid metal dealloying process. In this paper, we use theoretical models and phase-field simulations to clarify the kinetics and pattern formation during liquid metal dealloying. Our investigation starts from a theoretical analysis of the 1D dissolution of a binary precursor alloy, which reveals that the 1D dissolution process involves two regimes. In the first regime, due to the low solubility of one of the elements in the melt, it accumulates at the solid-liquid interface, which reduces the dissolution kinetics. In the second regime, the interface kinetics reaches a stationary regime where both elements of the precursor alloy dissolve into the melt. Previous works revealed that in the early dealloying stage, the dealloying front is destabilized by an interfacial spinodal decomposition, which triggers the formation of interconnected ligaments. Here we extend this line of work by proposing a linear stability analysis able to predict the initial length scale of the ligaments formed in the initial stage of the dealloying. Combining this analysis with the 1D dissolution model proposed here enables us to better understand the initial conditions (composition of the precursor alloy and the melt) leading to a planar dissolution without interface destabilization. Finally, we report a strong influence of solid-state diffusion on dealloying. Although the solid-state diffusivity is four to five orders of magnitude smaller than in the liquid phase, it is found to affect both dissolution kinetics and ligament morphologies.
Ratchet effects in cyclic pattern formation systems with competing interactions
Ratchet effects can appear for particles interacting with an asymmetric potential under ac driving or for a thermal system in which a substrate is periodically flashed. Here, we show that a different type of collective ratchet effect can arise for a pattern-forming system coupled to an asymmetric substrate when the interaction potential between the particles is periodically oscillated in order to cycle the system through different patterns. We consider particles with competing short-range attraction and long-range repulsion subjected to time-dependent oscillations of the ratio between the attractive and repulsive interaction terms, which causes the system to cycle periodically between crystal and bubble states. In the presence of the substrate, this system exhibits both a positive and a reversed ratchet effect, and we show that there is a maximum in the ratchet efficiency as a function of interaction strength, ac drive frequency, and particle density. In conclusion, our results could be realized for a variety of pattern-forming systems on asymmetric substrates where the pattern type or particle interactions can be oscillated.
A multi-phase-field model of topological pattern formation during electrochemical dealloying of binary alloys
Porous metal structures produced by electrochemical dealloying have been extensively studied for use in various applications that require high surface areas. A new comprehensive multi-phase-field (MPF) model is proposed to study topological porous patterns formed by spontaneously etching a bulk binary alloy that involves electrochemical reactions, bulk and surface diffusion, ion transport, applied electrode potential, and charge conservation. The governing equations for the alloy-porous cluster-electrolyte system account for a generalized Butler-Volmer electrochemical reaction and are in accordance with the classical nucleation theory. Based on a quantitative examination of the effects of electrode potential and precursor composition, the simulation results reproduce typical phenomena including passive surface dealloying, active porosity evolution, critical potential, and characteristic length scale in two-dimension (2-D) and three-dimension (3-D). To consider more complicated dealloying systems, the evolutions of a bimodal porous metal structure, nanocomposite, and nested porous network with a structured hierarchy are investigated. Finally, the proposed model can be a useful tool for understanding and predicting the morphology evolution of diverse porous structures during electrochemical dealloying.
Chemically controlled pattern formation in self-oscillating elastic shells
Significance Mechanochemical processes drive various biological functions including morphogenesis and morphological changes in responsive synthetic hydrogels. By allowing reactive components to function in an elastic closed shell, autonomous chemical reactions generate microcompartments capable of expanding or contracting or induce buckled patterns. This capability can drive specific functions such as enhancing the rate of diffusion of compartmentalized chemicals or releasing cargos at specific rates. Here, we demonstrate the coupling of mechanical response to autonomous chemical reactions in elastic shells which swell and deswell to generate specific dynamic patterns and morphologies. The reverse feedback from the mechanical input triggers the chemical reaction. Our work inspires the design of responsive shells to enhance the functionality of microcompartments and nanoreactors.
Collective effects and pattern formation for directional locking of disks moving through obstacle arrays
We examine directional locking effects in an assembly of disks driven through a square array of obstacles as the angle of drive rotates from 0° to 90°. For increasing disk densities, the system exhibits a series of different dynamic patterns along certain locking directions, including one-dimensional or multiple-row chain phases and density-modulated phases. For nonlocking driving directions, the disks form disordered patterns or clusters. Here, when the obstacles are small or far apart, a large number of locking phases appear; however, as the number of disks increases, the number of possible locking phases drops due to the increasing frequency of collisions between the disks and obstacles. For dense arrays or large obstacles, we find an increased clogging effect in which immobile and moving disks coexist.
Elastic forces drive nonequilibrium pattern formation in a model of nanocrystal ion exchange
The widely used process of nanocrystal ion exchange operates out of thermodynamic equilibrium and can require mixing components of varying sizes. Here we use theory and computer simulation to study a simple model which captures these two basic features of ion exchange reactions. We show that a strong driving force for exchange among different-sized species creates nonequilibrium patterns within model nanocrystals. We further demonstrate that such patterns can be thermodynamically stable within core/shell nanocrystals. These results help us understand the heterostructures formed in ion-exchanged nanocrystals and suggest strategies for leveraging elasticity to design patterns in nanoscale materials. Chemical transformations, such as ion exchange, are commonly employed to modify nanocrystal compositions. Yet the mechanisms of these transformations, which often operate far from equilibrium and entail mixing diverse chemical species, remain poorly understood. Here we explore an idealized model for ion exchange in which a chemical potential drives compositional defects to accumulate at a crystal’s surface. These impurities subsequently diffuse inward. We find that the nature of interactions between sites in a compositionally impure crystal strongly impacts exchange trajectories. In particular, elastic deformations which accompany lattice-mismatched species promote spatially modulated patterns in the composition. These same patterns can be produced at equilibrium in core/shell nanocrystals, whose structure mimics transient motifs observed in nonequilibrium trajectories. Moreover, the core of such nanocrystals undergoes a phase transition—from modulated to unstructured—as the thickness or stiffness of the shell is decreased. Our results help explain the varied patterns observed in heterostructured nanocrystals produced by ion exchange and suggest principles for the rational design of compositionally patterned nanomaterials.
Pattern formation and flocking for particles near the jamming transition on resource gradient substrates
Here, we numerically examine a bidisperse system of active and passive particles coupled to a resource substrate. The active particles deplete the resource at a fixed rate and move toward regions with higher resources, while all of the particles interact sterically with each other. We show that at high densities, this system exhibits a rich variety of pattern-forming phases along with directed motion or flocking as a function of the relative rates of resource absorption and consumption as well as the active to passive particle ratio. These include partial phase separation into rivers of active particles flowing through passive clusters, strongly phase separated states where the active particles induce crystallization of the passive particles, mixed jammed states, and fluctuating mixed fluid phases. For higher resource recovery rates, we demonstrate that the active particles can undergo motility-induced phase separation, while at high densities, there can be a coherent flock containing only active particles or a solid mixture of active and passive particles. The directed flocking motion typically shows a transient in which the flow switches among different directions before settling into one direction, and there is a critical density below which flocking does not occur. We map out the different phases as function of system density, resource absorption and recovery rates, and the ratio of active to passive particles.