Correction for 'Droplet breakup against an isolated obstacle' by David J. Meer et al., Soft Matter, 2026, 22, 2809-2822, https://doi.org/10.1039/d5sm01266j.

Multiscale periodic metamaterials have been designed for numerous applications, such as impact absorption, acoustic cloaking, photonic band gaps, and mechanical logic gates. This prior work has focused on optimizing mesoscale structure for desired bulk isotropic properties. In contrast, we seek to develop materials with highly anisotropic elastic properties. To quantify elastic anisotropy, we introduce two rotationally invariant, normalized quantities that characterize the anisotropic response to shear and compression, respectively, A_G and A_C. We find that typical crystalline solids possess average elastic anisotropy A_G ≈ 0.15 and A_C ≈ 0.09. Compared to atomic crystals, jammed granular materials can attain elastic anisotropies that are several orders of magnitude larger. Since grain rearrangements reduce anisotropy in granular materials, to preserve strong elastic anisotropy, we design tessellated granular materials that consist of multiple connected grain-filled voxels, which limit rearrangements and enable highly anisotropic elastic properties. Bulk granular packings with N grains prepared at pressure p have maximal anisotropy for pN^2∼1 and become isotropic in the large-pN^2 limit. We show that homogeneously tessellated granular systems can inherit the elastic response of the constituent voxel configurations with elastic anisotropy up to 100 times that of crystalline compounds over a range of pN^2. We show further methods to tune the elastic anisotropy of tessellations by designing heterogeneously patterned voxel configurations and tessellations that allow large boundary deformations.

Accurate prediction of the liquidus temperature (T-L) of alloys remains a challenge despite numerous theoretical models. Here, we explore and analyze the degree to which machine learning, ML, strategies can be used to predict T-L. We use established literature data on liquidus temperatures of 85,523 binary alloys to train ML models using various feature vectors to represent the alloys. While our results are comparable to previous studies, the persistent similar to 8% error underscores the limitations of current ML models for practical usage. The suboptimal accuracy leads us to question how well-defined the problem is and to what degree fundamental limitations prevent us from attaining more accurate predictions. We identify two major challenges in predicting the liquidus temperature of binary alloys through supervised ML algorithms. One challenge is representing the relevant characteristics of an alloy that determines liquidus temperature through appropriate features. The other fundamental challenge is the discreteness of atomic properties. The difference between two elements and thereby alloy systems is significant, which makes it difficult to learn from one alloy system to predict properties of another. We argue that these problems can be reduced to some extent, however these challenges are common in complex materials science problems and constitute a fundamental challenge in applying supervised ML strategies in this context.

Protein structure is controlled by a high-dimensional energy landscape, which is a function of all of the atomic coordinates of the protein. Can this landscape be accurately described by a low-dimensional representation? We find that residue core identity, a binary N -dimensional encoding indicating whether each of the N amino acids in a protein is buried in the core or not, can predict the protein’s backbone conformation more efficiently than all other representations that we tested. Core identity is 4 times more efficient than previous estimates of the bits per residue needed to encode a protein’s native fold, 2 times more efficient than the C α contact map, and 1.5 times more efficient than the machine-learned embeddings from FoldSeek’s 3Di. Even when the folded structure is unavailable, predicting each residue’s burial from sequence yields a more accurate estimate of fold quality than predicting pairwise contacts from the same sequence information. Thus, this work emphasizes that the problem of determining a protein’s native fold can be re-framed as predicting each residue’s core identity.

Bedload transport occurs when fluid shear stress becomes strong enough to entrain sediment. Recent studies show that the threshold for entrainment varies with the history of applied shear, with beds strengthening under sustained unidirectional flows and weakening when the flow direction is reversed. While stochastic bedload transport models incorporate variability in estimating the onset of grain motion, they typically do not consider time dependence. Here, we investigate how directional stress history affects the ensemble statistics of bedload motion using a rotating sediment bed in a laboratory flume. We subject a uniform sand bed to an initial subcritical conditioning flow, then rotate the bed to angular offsets of 0º, 45º, 90º, 135º, or 180º; before applying erosive flows at Shields stresses ranging from 0.78 to 1.53 times the nominal critical value. Using high-resolution imaging, we identify individual grain trajectories and extract statistics associated with entrainment probability, grain activity, velocity, flight distance, and travel time. Bedload motion statistics vary systematically with prior flow direction: unidirectional conditioning (0º) reduces grain activity while producing slower, more abrupt grain flights, whereas reversed conditioning (135º–180º) results in the opposite outcome. We find that bulk bed statistics—grain activity and entrainment probability—retain signatures of directional stress history across all Shields stresses tested, while trackwise statistics—flight distance and travel time—converge toward unconditioned values above the critical shear. These results indicate that stress history primarily modulates the entrainment probability rather than altering the kinematics of grains undergoing sustained transport, with implications for stochastic bedload models.

An important goal of computational studies of protein-protein interfaces (PPIs) is to predict the binding site between two monomers that form a heterodimer. The simplest version of this problem is to rigidly redock the bound forms of the monomers, which involves generating computational models of the heterodimer and then scoring them to determine the most nativelike models. PPI scoring functions have been assessed previously using rank- and classification-based metrics; however, these methods are sensitive to the number and quality of models in the scoring function training set. We assess the accuracy of seven physical, statistical, or deep-learning-based PPI scoring functions by comparing their scores of computational models of PPIs to a measure of structural similarity to the x-ray crystal structure (i.e., the DockQ score) for a nonredundant set of heterodimers from the Protein Data Bank. For each heterodimer, we generate redocked models uniformly sampled over DockQ and calculate the Spearman correlation between the PPI scores and DockQ. For some targets, the scores and DockQ are highly correlated; however, for many targets, there are weak correlations. Several physical features explain the difference between difficult- and easy-to-score targets. Strong correlations exist between the score and DockQ for targets with highly intertwined monomers and many interface contacts. We also develop a new score based on only two physical features that matches the performance of current PPI scoring functions. In addition, we address the more general problem of flexible-body docking by generating and docking intermediate monomer conformations between their bound and unbound forms. We score the docked models and find that the Spearman correlations between the PPI scores and DockQ decrease strongly as the monomers are deformed from their bound conformations. These results emphasize that PPI docking predictions can be improved by focusing on correlations between the PPI score and DockQ and incorporating more discriminating physical features into PPI scoring functions.

In recent decades, there has been increasing effort toward modeling bedload transport using statistical ensembles for grain velocities, flight distance, and travel times. The forms of these probability distributions are integral to estimating sediment flux, with implications for predicting sediment transport in both natural and engineered systems. In this study, we analyze a high-resolution dataset of grain motions collected from laboratory flume experiments to extract these distributions. We recorded videos of a sand bed exposed to increasing fluid shear ranging from below the nominal threshold stress for grain motion to well above (τ * /τ *cr = 0.78−1.53). We extracted trajectories of a population of sand grains and used a hidden Markov model to predict the states of motion and rest from observed grain velocities. We characterize probability distributions for the activity, velocity, acceleration, flight distance, and travel times of moving grains and largely find agreement with previous studies. Streamwise velocities are well fit by an exponential distribution, accelerations display a Laplace distribution, flight travel times also show an exponential distribution, while flight distances are described by a Weibull distribution. We note, however, that these distributions described our data more poorly for bedload motion at and below the expected threshold of motion, especially for flight travel times, suggesting that bedload dynamics differ in this most intermittent regime. These experimental results provide a reference dataset for validating numerical simulations or analytical solutions for stochastic sediment transport models, especially for flow conditions at and below the nominal critical Shields stress.

We describe combined experiments and simulations of single droplet breakup during flow-driven interactions with a circular obstacle in a quasi-two-dimensional microfluidic chamber. Due to a lack of in-plane confinement, the droplets can also slip past the obstacle without breaking. Droplets are more likely to break when they have a higher flow velocity, larger size (relative to the obstacle radius R), smaller surface tension, and for head-on collisions with the obstacle. We also observe that droplet-obstacle collisions are more likely to result in breakup when the height of the sample chamber is increased. We define a nondimensional breakup number Bk ∼ Ca that accounts for changes in the likelihood of droplet break up with variations in these parameters, where Ca is the Capillary number. As Bk increases, we find in both experiments and discrete element method (DEM) simulations of the deformable particle model that the behavior changes from droplets never breaking (Bk ≪ 1) to always breaking for Bk ≫ 1, with a rapid change in the probability of droplet breakup near Bk = 1. We also find that Bk ∼ S4/3, where S characterizes the symmetry of the collision, which implies that the minimum symmetry required for breakup is controlled by a characteristic distance h ∼ R.

MOTIVATION:Advances in high-throughput chromatin conformation capture have provided insight into the three-dimensional structure and organization of chromatin. While bulk Hi-C experiments capture spatio-temporally averaged chromatin interactions across millions of cells, single-cell Hi-C experiments report on the chromatin interactions of individual cells. Supervised and unsupervised algorithms have been developed to embed single-cell Hi-C maps and identify different cell types. However, single-cell Hi-C maps are often difficult to cluster due to their high sparsity, with state-of-the-art algorithms achieving a maximum Adjusted Rand Index (ARI) of only ≲0.4 on several datasets. RESULTS:We introduce a novel unsupervised algorithm, Single-cell Clustering Using Diagonal Diffusion Operators (SCUDDO), to embed and cluster single-cell Hi-C maps. We evaluate SCUDDO on four previously difficult-to-cluster single-cell Hi-C datasets, and show that it can outperform other current algorithms in ARI by ≳0.2. Further, SCUDDO outperforms all other tested algorithms even when we restrict the number of intrachromosomal maps for each cell type and when we use only a small fraction of contacts in each Hi-C map. Thus, SCUDDO can capture the underlying latent features of single-cell Hi-C maps and provide accurate labelling of cell types even when cell types are not known a priori. AVAILABILITY AND IMPLEMENTATION:SCUDDO is freely available at https://www.github.com/lmaisuradze/scuddo as well as https://doi.org/10.6084/m9.figshare.31759915. The tested datasets are publicly available and can be downloaded from the Gene Expression Omnibus.

Bedload transport occurs when the shear stress, or non-dimensional Shields stress, imparted by a fluid onto a sediment bed exceeds a critical value for sediment entrainment. The history of fluid stress imparted onto a sediment bed influences this critical Shields stress, with bed strengthening occurring under unidirectional flows and bed weakening occurring when the flow direction is reversed. In this study, we examine directional strengthening and weakening in a sediment bed for multiple fluid stress orientations using a rotating bed of sand in a laboratory flume. This sediment bed is exposed to an initial subcritical conditioning flow followed by a subsequent erosive flow at an offset angle of , , , , or . We identify the particle trajectories of a population of sediment grains to measure their velocity, activity, and associated bulk statistics. We confirm bed strengthening (i.e., lower grain velocity and activity) in the unidirectional case, especially for flows at or below the nominal critical Shields stress. As the angular offset increases between the conditioning and erosive flows, both grain velocity and activity increase, with the greatest bed weakening at offsets of and . Our results confirm that stress history is stored anisotropically in the sediment bed, supporting mechanisms such as shear jamming where an anisotropic granular fabric develops in response to shear. These results inform our understanding of how subcritical and critical fluid-imposed stresses can modify the grain contact and force networks in geophysical contexts.

Interactions between fluids and granular materials are prevalent on the Earth's surface. In the case of fluid flow over a sediment bed, the fluid imparts a shear stress to the granular materials. When the applied shear stress is above a critical value, the grains become entrained in the fluid flow. Prior experimental studies have shown that granular beds subjected to a sub-critical fluid flow can strengthen in the same direction as the sub-critical flow. In contrast, granular beds can become weaker in the direction opposite to the sub-critical fluid flow. To investigate the grain-scale mechanisms that control directional strengthening and weakening, we perform discrete element method (DEM) simulations of granular beds subjected to model fluid flows in two (2D) and three (3D) dimensions with varied inter-particle static friction coefficients and conditioning flow speeds. In these studies, the sub-critical grain motion does not cause significant bed compaction. Instead, we find that the strength of a granular bed in a particular direction is highly correlated with the fraction of surface grains that can be dislodged by a fluid force applied in that direction. Further, the anisotropic bed strength only persists over a finite time scale that is set by the Shields number. We also show that inter-particle static friction is not required for bed strength anisotropy, but varying the friction affects the magnitude of the anisotropy. This research enhances the grain-scale understanding of erosion of granular beds caused by fluid flows and underscores the importance of tracking the history of the fabric of the bed surface since it couples strongly to bed strength.

We describe size-varying cylindrical particles made from silicone elastomers that can serve as building blocks for robotic granular materials. The particle size variation, which is achieved by inflation, gives rise to changes in stiffness under compression. We design and fabricate inflatable particles that can become stiffer or softer during inflation, depending on key parameters of the particle geometry, such as the ratio of the fillet radius to the wall thickness, r/t. We also conduct numerical simulations of the inflatable particles and show that they only soften during inflation when localization of large strains occurs in the regime r/t -> 0. This work introduces novel particle systems with tunable size and stiffness that can be implemented in numerous soft robotic applications.

Proteins fold to a specific functional conformation with a densely packed hydrophobic core that controls their stability. We develop a geometric, yet all-atom model for proteins that explains the universal core packing fraction of ϕ_c=0.55 found in experimental measurements. We show that as the hydrophobic interactions increase relative to the temperature, a novel jamming transition occurs when the core packing fraction exceeds ϕ_c. The model also recapitulates the global structure of proteins since it can accurately refold to native-like structures from partially unfolded states.

Proteins are composed of chains of amino acids that fold into complex three-dimensional structures. Several key features, such as the radius of gyration, fraction of core amino acids f_{core}, packing fraction 〈ϕ〉 of core amino acids, and structure factor S(q) define the structure of folded proteins. It is well-known that folded proteins are compact with a radius of gyration R_{g}(N)∼N^{ν} that obeys power-law scaling with the number of amino acids N and ν∼1/3, f_{core}≈0.09, and 〈ϕ〉≈0.55. We also investigate the internal scaling of the radius of gyration R_{g}(n) versus the chemical separation n between amino acids for subchains of length n and show that it does not obey simple power-law scaling with ν∼1/3. Instead, R_{g}(n)∼n^{ν_{1,2}} with a larger exponent ν_{1}>1/3 for small n and a smaller exponent ν_{2}<1/3 for large n. To develop a minimal model for proteins that recapitulates these defining structural features, we carry out collapse simulations for a series of coarse-grained models with increasing complexity. We show that a model, which coarse-grains amino acids into a single spherical backbone bead and several variable-sized side-chain beads and enforces bend- and dihedral-angle constraints for the backbone, recapitulates R_{g}(n), f_{core}, 〈ϕ〉, and S(q) for more than 2500 x-ray crystal structures of proteins.

Substrates that compute using vibration rather than electricity offer the potential of creating and deploying computers into electronics-denying environments. Another advantage of these materials is that some of them can compute multiple functions in the same place and at the same time, providing computational results at different frequencies. These so-called polycomputational materials may eventually compete with more traditional computers in terms of computational density because there is no currently known upper bound on how many functions can be simultaneously computed by a vibrational substrate. However, three challenges remain for polycomputational materials: how to ensure that the different functions are computed independently; developing evolutionary algorithms that allow for embedding increasingly more functions into these materials in silico; and validating the evolved in silico materials as physical materials. Here we report progress on all three of these issues.

Soft robots can achieve exceptional adaptability through tunable morphological and mechanical properties. Incorporating materials with dynamically adjustable characteristics can enhance this versatility further. Granular meta-materials, consisting of discrete particles with individually variable properties, offer a promising approach to bulk property adaptation by adjusting the properties of constituent particles. This work introduces variable size and variable stiffness (VS2) particles, in which both particle size and stiffness are independently modulated through concentric pneumatic chambers. We characterize the achievable workspace, mapping particle responses to independent chamber inflation. To demonstrate their use in a granular assembly, we arrange an array of VS2 particles in a hexagonal packing and validate that behavior in packed configurations aligns with free-space characterizations. This study establishes a foundation for adaptive granular materials and provides a platform for further computational and experimental exploration of 2D and 3D granular metamaterials with tunable properties.

Unconventional computing may overcome some of the limitations of traditional silicon-based systems using alternative materials and computational mechanisms. However, due to their complex underlying dynamics and high-dimensional parameter space, the design of these materials such that they perform computation is non-intuitive, making AI-driven design attractive. It has been shown that evolutionary algorithms can tune the structural properties of grains within a granular material such that it computes logical functions. In recent years, programmable granular metamaterials have been developed so that multiple physical properties of individual grains can be altered independently. This raises the question of whether allowing evolutionary algorithms to tune more grain features within a granular material frustrates or facilitates its ability to embed computation. In this work, we show that the latter is the case, when grain sizes and stiffnesses are co-evolved to embed Boolean logic gates, compared to evolving just sizes or stiffnesses alone. We report physical verification of evolved designs, taking a further step toward the provision of alternatives to electronic computing.

Under an externally applied load, granular packings form force chains that depend on the contact network and moduli of the grains. In this work, we investigate packings of variable modulus (VM) particles, where we can direct force chains by changing the Young's modulus of individual particles within the packing on demand. Each VM particle is made of a silicone shell that encapsulates a core made of a low-melting-point metallic alloy (Field's metal). By sending an electric current through a co-located copper heater, the Field's metal internal to each particle can be melted via Joule heating, which softens the particle. As the particle cools to room temperature, the alloy solidifies and the particle recovers its original modulus. To optimize the mechanical response of granular packings containing both soft and stiff particles, we employ an evolutionary algorithm coupled with discrete element method simulations to predict the patterns of particle moduli that will yield specific force outputs on the assembly boundaries. The predicted patterns of particle moduli from the simulations were realized in experiments using quasi-2D assemblies of VM particles and the force outputs on the assembly boundaries were measured using photoelastic techniques. These studies represent a step towards making robotic granular metamaterials that can dynamically adapt their mechanical properties in response to different environmental conditions or perform specific tasks on demand.