Geometric mapping from rectilinear material orthotropy to isotropy: Insights into plates and shells

Orthotropic shell structures are ubiquitous in biology and engineering, from bacterial cell walls to reinforced domes. We present a rescaling transformation that maps an orthotropic shallow shell to an isotropic one with a different local geometry. The mapping is applicable to any shell section for which the material orthotropy directions match the principal curvature directions, assuming the commonly used Huber form for the orthotropic shear modulus. Using the rescaling transformation, we derive exact expressions for the buckling pressure as well as the linear indentation response of orthotropic cylinders and general ellipsoids of revolution, which we verify against numerical simulations. Our analysis disentangles the separate contributions of geometric and material anisotropy to shell rigidity. In particular, we identify the geometric mean of orthotropic elastic constants as the key quantifier of material stiffness, playing a role akin to the Gaussian curvature which captures the geometric stiffness contribution. Besides providing insights into the mechanical response of orthotropic shells, our work rigorously establishes the validity of isotropic approximations to orthotropic shells and also identifies situations in which these approximations might fail.

The influence of explicit local dynamics on range expansions driven by long-range dispersal

Range expansions are common in natural populations. They can take such forms as an invasive species spreading into a new habitat or a virus spreading from host to host during a pandemic. When the expanding species is capable of dispersing offspring over long distances, population growth is driven by rare but consequential long-range dispersal events that seed satellite colonies far from the densely occupied core of the population. These satellites accelerate growth by accessing unoccupied territory, and also act as reservoirs for maintaining neutral genetic variation present in the originating population, which would ordinarily be lost to drift. Prior theoretical studies of dispersal-driven expansions have shown that the sequential establishment of satellites causes initial genetic diversity to be either lost or maintained to a level determined by the breadth of the distribution of dispersal distances. If the tail of the distribution falls off faster than a critical threshold, diversity is steadily eroded over time; by contrast, broader distributions with a slower falloff allow some initial diversity to be maintained for arbitrarily long times. However, these studies used lattice-based models and assumed an instantaneous saturation of the local carrying capacity after the arrival of a founder. Real-world populations expand in continuous space with complex local dynamics, which potentially allow multiple pioneers to arrive and establish within the same local region. Here, we evaluate the impact of local dynamics on the population growth and the evolution of neutral diversity using a computational model of range expansions with long-range dispersal in continuous space, with explicit local dynamics that can be controlled by altering the mix of local and long-range dispersal events. We found that many qualitative features of population growth and neutral genetic diversity observed in lattice-based models are preserved under more complex local dynamics, but quantitative aspects such as the rate of population growth, the level of maintained diversity, and the rate of decay of diversity all depend strongly on the local dynamics. Besides identifying situations in which modeling the explicit local population dynamics becomes necessary to understand the population structure of jump-driven range expansions, our results show that local dynamics affects different features of the population in distinct ways, and can be more or less consequential depending on the degree and form of long-range dispersal as well as the scale at which the population structure is measured.

Delocalization of interacting directed polymers on a periodic substrate: Localization length and critical exponents from non-Hermitian spectra

We study a classical model of thermally fluctuating polymers confined to two dimensions, experiencing a grooved periodic potential, and subject to pulling forces both along and transverse to the grooves. The equilibrium polymer conformations are described by a mapping to a quantum system with a non-Hermitian Hamiltonian and with fermionic statistics generated by noncrossing interactions among polymers. Using molecular dynamics simulations and analytical calculations, we identify a localized and a delocalized phase of the polymer conformations, separated by a delocalization transition which corresponds (in the quantum description) to the breakdown of a band insulator when driven by an imaginary vector potential. We calculate the average tilt of the many-body system, at arbitrary shear values and filling density of polymer chains, in terms of the complex-valued non-Hermitian band structure. We find the critical shear value, the localization length, and the critical exponent by which the shear modulus diverges in terms of the branch points (exceptional points) in the band structure at which the bandgap closes. We also investigate the combined effects of non-Hermitian delocalization and localization due to both periodicity and disorder, uncovering preliminary evidence that while disorder favors localization at high values, it encourages delocalization at lower values.

Branching structure of genealogies in spatially growing populations and its implications for population genetics inference

Spatial models where growth is limited to the population edge have been instrumental to understanding the population dynamics and the clone size distribution in growing cellular populations, such as microbial colonies and avascular tumours. A complete characterization of the coalescence process generated by spatial growth is still lacking, limiting our ability to apply classic population genetics inference to spatially growing populations. Here, we start filling this gap by investigating the statistical properties of the cell lineages generated by the two dimensional Eden model, leveraging their physical analogy with directed polymers. Our analysis provides quantitative estimates for population measurements that can easily be assessed via sequencing, such as the average number of segregating sites and the clone size distribution of a subsample of the population. Our results not only reveal remarkable features of the genealogies generated during growth, but also highlight new properties that can be misinterpreted as signs of selection if non-spatial models are inappropriately applied.

Indentation responses of pressurized ellipsoidal and cylindrical elastic shells: Insights from shallow-shell theory

Pressurized elastic shells are ubiquitous in nature and technology, from the outer walls of yeast and bacterial cells to artificial pressure vessels. Indentation measurements simultaneously probe the internal pressure and elastic properties of thin shells and serve as a useful tool for strength testing and for inferring internal biological functions of living cells. We study the effects of geometry and pressure-induced stress on the indentation stiffness of ellipsoidal and cylindrical elastic shells using shallow-shell theory. We show that the linear indentation response reduces to a single integral with two dimensionless parameters that encode the asphericity and internal pressure. This integral can be numerically evaluated in all regimes and is used to generate compact analytical expressions for the indentation stiffness in various regimes of technological and biological importance. Our results provide theoretical support for previous scaling and numerical results describing the stiffness of ellipsoids, reveal a new pressure scale that dictates the large-pressure response, and give new insights to the linear indentation response of pressurized cylinders.

Topological Protection Can Arise from Thermal Fluctuations and Interactions

Topological quantum and classical materials can exhibit robust properties that are protected against disorder, for example, for noninteracting particles and linear waves. Here, we demonstrate how to construct topologically protected states that arise from the combination of strong interactions and thermal fluctuations inherent to soft materials or miniaturized mechanical structures. Specifically, we consider fluctuating lines under tension (e.g., polymer or vortex lines), subject to a class of spatially modulated substrate potentials. At equilibrium, the lines acquire a collective tilt proportional to an integer topological invariant called the Chern number. This quantized tilt is robust against substrate disorder, as verified by classical Langevin dynamics simulations. This robustness arises because excitations in this system of thermally fluctuating lines are gapped by virtue of interline interactions. We establish the topological underpinning of this pattern via a mapping that we develop between the interacting-lines system and a hitherto unexplored generalization of Thouless pumping to imaginary time. Our work points to a new class of classical topological phenomena in which the topological signature manifests itself in a structural property observed at finite temperature rather than a transport measurement.

Spatial soft sweeps: Patterns of adaptation in populations with long-range dispersal

Adaptation in extended populations often occurs through multiple independent mutations responding in parallel to a common selection pressure. As the mutations spread concurrently through the population, they leave behind characteristic patterns of polymorphism near selected loci—so-called soft sweeps—which remain visible after adaptation is complete. These patterns are well-understood in two limits of the spreading dynamics of beneficial mutations: the panmictic case with complete absence of spatial structure, and spreading via short-ranged or diffusive dispersal events, which tessellates space into distinct compact regions each descended from a unique mutation. However, spreading behaviour in most natural populations is not exclusively panmictic or diffusive, but incorporates both short-range and long-range dispersal events. Here, we characterize the spatial patterns of soft sweeps driven by dispersal events whose jump distances are broadly distributed, using lattice-based simulations and scaling arguments. We find that mutant clones adopt a distinctive structure consisting of compact cores surrounded by fragmented “haloes” which mingle with haloes from other clones. As long-range dispersal becomes more prominent, the progression from diffusive to panmictic behaviour is marked by two transitions separating regimes with differing relative sizes of halo to core. We analyze the implications of the core-halo structure for the statistics of soft sweep detection in small genomic samples from the population, and find opposing effects of long-range dispersal on the expected diversity in global samples compared to local samples from geographic subregions of the range. We also discuss consequences of the standing genetic variation induced by the soft sweep on future adaptation and mixing.

When a species is spread out over a large geographic range, different regions may adapt to the same selection pressure by acquiring distinct beneficial mutations. The resulting pattern of genetic variation in the population is called a soft sweep. Dispersal strongly influences soft sweep patterns, as it determines how a mutation that arose in one region might spread to others. Although most plant and animal populations experience some amount of dispersal over very long distances, the impact of such long-range dispersal events on soft sweep patterns remains poorly understood. We use computer simulations and mathematical analysis to study patterns of genetic variation in a model of soft sweeps including long-range dispersal. We show that long-range dispersal leaves distinct signatures in the genetic makeup of the population, which can be detected in genetic samples from individuals across the range. Our results are important for correctly interpreting patterns of genetic diversity in populations that have undergone recent adaptation.

Sonic Landau Levels and Synthetic Gauge Fields in Mechanical Metamaterials

Mechanical strain can lead to a synthetic gauge field that controls the dynamics of electrons in graphene sheets as well as light in photonic crystals. Here, we show how to engineer an analogous synthetic gauge field for lattice vibrations. Our approach relies on one of two strategies: shearing a honeycomb lattice of masses and springs or patterning its local material stiffness. As a result, vibrational spectra with discrete Landau levels are generated. Upon tuning the strength of the gauge field, we can control the density of states and transverse spatial confinement of sound in the metamaterial. We also show how this gauge field can be used to design waveguides in which sound propagates with robustness against disorder as a consequence of the change in topological polarization that occurs along a domain wall. By introducing dissipation, we can selectively enhance the domain-wall-bound topological sound mode, a feature that may potentially be exploited for the design of sound amplification by stimulated emission of radiation (SASER, the mechanical analogs of lasers).

Mechanical strain sensing implicated in cell shape recovery in Escherichia coli

The shapes of most bacteria are imparted by the structures of their peptidoglycan cell walls, which are determined by many dynamic processes that can be described on various length-scales ranging from short-range glycan insertions to cellular-scale elasticity.1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 Understanding the mechanisms that maintain stable, rod-like morphologies in certain bacteria has proved to be challenging due to an incomplete understanding of the feedback between growth and the elastic and geometric properties of the cell wall.3, 4, 12, 13, 14 Here we probe the effects of mechanical strain on cell shape by modeling the mechanical strains caused by bending and differential growth of the cell wall. We show that the spatial coupling of growth to regions of high mechanical strain can explain the plastic response of cells to bending4 and quantitatively predict the rate at which bent cells straighten. By growing filamentous E. coli cells in donut-shaped microchambers, we find that the cells recovered their straight, native rod-shaped morphologies when released from captivity at a rate consistent with the theoretical prediction. We then measure the localization of MreB, an actin homolog crucial to cell wall synthesis, inside confinement and during the straightening process and find that it cannot explain the plastic response to bending or the observed straightening rate. Our results implicate mechanical strain-sensing, implemented by components of the elongasome yet to be fully characterized, as an important component of robust shape regulation in E. coli.

Additives for vaccine storage to improve thermal stability of adenoviruses from hours to months

Up to 80% of the cost of vaccination programmes is due to the cold chain problem (that is, keeping vaccines cold). Inexpensive, biocompatible additives to slow down the degradation of virus particles would address the problem. Here we propose and characterize additives that, already at very low concentrations, improve the storage time of adenovirus type 5. Anionic gold nanoparticles (10⁻⁸–10⁻⁶ M) or polyethylene glycol (PEG, molecular weight ∼8,000 Da, 10⁻⁷–10⁻⁴ M) increase the half-life of a green fluorescent protein expressing adenovirus from ∼48 h to 21 days at 37 °C (from 7 to >30 days at room temperature). They replicate the known stabilizing effect of sucrose, but at several orders of magnitude lower concentrations. PEG and sucrose maintained immunogenicity in vivo for viruses stored for 10 days at 37 °C. To achieve rational design of viral-vaccine stabilizers, our approach is aided by simplified quantitative models based on a single rate-limiting step.

Keeping viral vaccines cold from the manufacturers to patients is problematic and costly. Here, Krol and others show additives that can significantly improve at very low concentrations the storage of adenovirus type 5 at ambient and elevated temperature.

Topological Mechanics of Origami and Kirigami

Origami and kirigami have emerged as potential tools for the design of mechanical metamaterials whose properties such as curvature, Poisson ratio, and existence of metastable states can be tuned using purely geometric criteria. A major obstacle to exploiting this property is the scarcity of tools to identify and program the flexibility of fold patterns. We exploit a recent connection between spring networks and quantum topological states to design origami with localized folding motions at boundaries and study them both experimentally and theoretically. These folding motions exist due to an underlying topological invariant rather than a local imbalance between constraints and degrees of freedom. We give a simple example of a quasi-1D folding pattern that realizes such topological states. We also demonstrate how to generalize these topological design principles to two dimensions. A striking consequence is that a domain wall between two topologically distinct, mechanically rigid structures is deformable even when constraints locally match the degrees of freedom.

Selective buckling via states of self-stress in topological metamaterials

States of self-stress--tensions and compressions of structural elements that result in zero net forces--play an important role in determining the load-bearing ability of structures ranging from bridges to metamaterials with tunable mechanical properties. We exploit a class of recently introduced states of self-stress analogous to topological quantum states to sculpt localized buckling regions in the interior of periodic cellular metamaterials. Although the topological states of self-stress arise in the linear response of an idealized mechanical frame of harmonic springs connected by freely hinged joints, they leave a distinct signature in the nonlinear buckling behavior of a cellular material built out of elastic beams with rigid joints. The salient feature of these localized buckling regions is that they are indistinguishable from their surroundings as far as material parameters or connectivity of their constituent elements are concerned. Furthermore, they are robust against a wide range of structural perturbations. We demonstrate the effectiveness of this topological design through analytical and numerical calculations as well as buckling experiments performed on two- and three-dimensional metamaterials built out of stacked kagome lattices.

Theory of interacting dislocations on cylinders

We study the mechanics and statistical physics of dislocations interacting on cylinders, motivated by the elongation of rod-shaped bacterial cell walls and cylindrical assemblies of colloidal particles subject to external stresses. The interaction energy and forces between dislocations are solved analytically, and analyzed asymptotically. The results of continuum elastic theory agree well with numerical simulations on finite lattices even for relatively small systems. Isolated dislocations on a cylinder act like grain boundaries. With colloidal crystals in mind, we show that saddle points are created by a Peach-Koehler force on the dislocations in the circumferential direction, causing dislocation pairs to unbind. The thermal nucleation rate of dislocation unbinding is calculated, for an arbitrary mobility tensor and external stress, including the case of a twist-induced Peach-Koehler force along the cylinder axis. Surprisingly rich phenomena arise for dislocations on cylinders, despite their vanishing Gaussian curvature.

Delayed buckling and guided folding of inhomogeneous capsules

Colloidal capsules can sustain an external osmotic pressure; however, for a sufficiently large pressure, they will ultimately buckle. This process can be strongly influenced by structural inhomogeneities in the capsule shells. We explore how the time delay before the onset of buckling decreases as the shells are made more inhomogeneous; this behavior can be quantitatively understood by coupling shell theory with Darcy's law. In addition, we show that the shell inhomogeneity can dramatically change the folding pathway taken by a capsule after it buckles.