Minimal Design of the Elephant Trunk as an Active Filament.

One of the key problems in active materials is the control of shape through actuation. A fascinating example of such control is the elephant trunk, a long, muscular, and extremely dexterous organ with multiple vital functions. The elephant trunk is an object of fascination for biologists, physicists, and children alike. Its versatility relies on the intricate interplay of multiple unique physical mechanisms and biological design principles. Here, we explore these principles using the theory of active filaments and build, theoretically, computationally, and experimentally, a minimal model that explains and accomplishes some of the spectacular features of the elephant trunk.

A multiscale computational framework for the development of spines in molluscan shells.

From mathematical models of growth to computer simulations of pigmentation, the study of shell formation has given rise to an abundant number of models, working at various scales. Yet, attempts to combine those models have remained sparse, due to the challenge of combining categorically different approaches. In this paper, we propose a framework to streamline the process of combining the molecular and tissue scales of shell formation. We choose these levels as a proxy to link the genotype level, which is better described by molecular models, and the phenotype level, which is better described by tissue-level mechanics. We also show how to connect observations on shell populations to the approach, resulting in collections of molecular parameters that may be associated with different populations of real shell specimens. The approach is as follows: we use a Quality-Diversity algorithm, a type of black-box optimization algorithm, to explore the range of concentration profiles emerging as solutions of a molecular model, and that define growth patterns for the mechanical model. At the same time, the mechanical model is simulated over a wide range of growth patterns, resulting in a variety of spine shapes. While time-consuming, these steps only need to be performed once and then function as look-up tables. Actual pictures of shell spines can then be matched against the list of existing spine shapes, yielding a potential growth pattern which, in turn, gives us matching molecular parameters. The framework is modular, such that models can be easily swapped without changing the overall working of the method. As a demonstration of the approach, we solve specific molecular and mechanical models, adapted from available theoretical studies on molluscan shells, and apply the multiscale framework to evaluate the characteristics of spines from three distinct populations of Turbo sazae.

An automated method for tendon image segmentation on ultrasound using grey-level co-occurrence matrix features and hidden Gaussian Markov random fields.

BACKGROUND: Despite knowledge of qualitative changes that occur on ultrasound in tendinopathy, there is currently no objective and reliable means to quantify the severity or prognosis of tendinopathy on ultrasound. OBJECTIVE: The primary objective of this study is to produce a quantitative and automated means of inferring potential structural changes in tendinopathy by developing and implementing an algorithm which performs a texture based segmentation of tendon ultrasound (US) images. METHOD: A model-based segmentation approach is used which combines Gaussian mixture models, Markov random field theory and grey-level co-occurrence (GLCM) features. The algorithm is trained and tested on 49 longitudinal B-mode ultrasound images of the Achilles tendons which are labelled as tendinopathic (24) or healthy (25). Hyperparameters are tuned, using a training set of 25 images, to optimise a decision tree based classification of the images from texture class proportions. We segment and classify the remaining test images using the decision tree. RESULTS: Our approach successfully detects a difference in the texture profiles of tendinopathic and healthy tendons, with 22/24 of the test images accurately classified based on a simple texture proportion cut-off threshold. Results for the tendinopathic images are also collated to gain insight into the topology of structural changes that occur with tendinopathy. It is evident that distinct textures, which are predominantly present in tendinopathic tendons, appear most commonly near the transverse boundary of the tendon, though there was a large variability among diseased tendons. CONCLUSION: The GLCM based segmentation of tendons under ultrasound resulted in distinct segmentations between healthy and tendinopathic tendons and provides a potential tool to objectively quantify damage in tendinopathy.

Mechanics reveals the role of peristome geometry in prey capture in carnivorous pitcher plants (Nepenthes).

Carnivorous pitcher plants (Nepenthes) are a striking example of a natural pitfall trap. The trap's slippery rim, or peristome, plays a critical role in insect capture via an aquaplaning mechanism that is well documented. While the peristome has received significant research attention, the conspicuous variation in peristome geometry across the genus remains unexplored. We examined the mechanics of prey capture using Nepenthes pitcher plants with divergent peristome geometries. Inspired by living material, we developed a mathematical model that links the peristomes' three-dimensional geometries to the physics of prey capture under the laws of Newtonian mechanics. Linking form and function enables us to test hypotheses related to the function of features such as shape and ornamentation, orientation in a gravitational field, and the presence of "teeth," while analysis of the energetic costs and gains of a given geometry provides a means of inferring potential evolutionary pathways. In a separate modeling approach, we show how prey size may correlate with peristome dimensions for optimal capture. Our modeling framework provides a physical platform to understand how divergence in peristome morphology may have evolved in the genus Nepenthes in response to shifts in prey diversity, availability, and size.

Minimal Morphoelastic Models of Solid Tumour Spheroids: A Tutorial.

Tumour spheroids have been the focus of a variety of mathematical models, ranging from Greenspan's classical study of the 1970 s through to contemporary agent-based models. Of the many factors that regulate spheroid growth, mechanical effects are perhaps some of the least studied, both theoretically and experimentally, though experimental enquiry has established their significance to tumour growth dynamics. In this tutorial, we formulate a hierarchy of mathematical models of increasing complexity to explore the role of mechanics in spheroid growth, all the while seeking to retain desirable simplicity and analytical tractability. Beginning with the theory of morphoelasticity, which combines solid mechanics and growth, we successively refine our assumptions to develop a somewhat minimal model of mechanically regulated spheroid growth that is free from many unphysical and undesirable behaviours. In doing so, we will see how iterating upon simple models can provide rigorous guarantees of emergent behaviour, which are often precluded by existing, more complex modelling approaches. Perhaps surprisingly, we also demonstrate that the final model considered in this tutorial agrees favourably with classical experimental results, highlighting the potential for simple models to provide mechanistic insight whilst also serving as mathematical examples.

The physical basis of mollusk shell chiral coiling.

Snails are model organisms for studying the genetic, molecular, and developmental bases of left-right asymmetry in Bilateria. However, the development of their typical helicospiral shell, present for the last 540 million years in environments as different as the abyss or our gardens, remains poorly understood. Conversely, ammonites typically have a bilaterally symmetric, planispiraly coiled shell, with only 1% of 3,000 genera displaying either a helicospiral or a meandering asymmetric shell. A comparative analysis suggests that the development of chiral shells in these mollusks is different and that, unlike snails, ammonites with asymmetric shells probably had a bilaterally symmetric body diagnostic of cephalopods. We propose a mathematical model for the growth of shells, taking into account the physical interaction during development between the soft mollusk body and its hard shell. Our model shows that a growth mismatch between the secreted shell tube and a bilaterally symmetric body in ammonites can generate mechanical forces that are balanced by a twist of the body, breaking shell symmetry. In gastropods, where a twist is intrinsic to the body, the same model predicts that helicospiral shells are the most likely shell forms. Our model explains a large diversity of forms and shows that, although molluscan shells are incrementally secreted at their opening, the path followed by the shell edge and the resulting form are partly governed by the mechanics of the body inside the shell, a perspective that explains many aspects of their development and evolution.

Multiscale integration of environmental stimuli in plant tropism produces complex behaviors.

Plant tropism refers to the directed movement of an organ or organism in response to external stimuli. Typically, these stimuli induce hormone transport that triggers cell growth or deformation. In turn, these local cellular changes create mechanical forces on the plant tissue that are balanced by an overall deformation of the organ, hence changing its orientation with respect to the stimuli. This complex feedback mechanism takes place in a three-dimensional growing plant with varying stimuli depending on the environment. We model this multiscale process in filamentary organs for an arbitrary stimulus by explicitly linking hormone transport to local tissue deformation leading to the generation of mechanical forces and the deformation of the organ in three dimensions. We show, as examples, that the gravitropic, phototropic, nutational, and thigmotropic dynamic responses can be easily captured by this framework. Further, the integration of evolving stimuli and/or multiple contradictory stimuli can lead to complex behavior such as sun following, canopy escape, and plant twining.

The role of topology and mechanics in uniaxially growing cell networks.

In biological systems, the growth of cells, tissues and organs is influenced by mechanical cues. Locally, cell growth leads to a mechanically heterogeneous environment as cells pull and push their neighbours in a cell network. Despite this local heterogeneity, at the tissue level, the cell network is remarkably robust, as it is not easily perturbed by changes in the mechanical environment or the network connectivity. Through a network model, we relate global tissue structure (i.e. the cell network topology) and local growth mechanisms (growth laws) to the overall tissue response. Within this framework, we investigate the two main mechanical growth laws that have been proposed: stress-driven or strain-driven growth. We show that in order to create a robust and stable tissue environment, networks with predominantly series connections are naturally driven by stress-driven growth, whereas networks with predominantly parallel connections are associated with strain-driven growth.

Mechanics unlocks the morphogenetic puzzle of interlocking bivalved shells.

Brachiopods and mollusks are 2 shell-bearing phyla that diverged from a common shell-less ancestor more than 540 million years ago. Brachiopods and bivalve mollusks have also convergently evolved a bivalved shell that displays an apparently mundane, yet striking feature from a developmental point of view: When the shell is closed, the 2 valve edges meet each other in a commissure that forms a continuum with no gaps or overlaps despite the fact that each valve, secreted by 2 mantle lobes, may present antisymmetric ornamental patterns of varying regularity and size. Interlocking is maintained throughout the entirety of development, even when the shell edge exhibits significant irregularity due to injury or other environmental influences, which suggests a dynamic physical process of pattern formation that cannot be genetically specified. Here, we derive a mathematical framework, based on the physics of shell growth, to explain how this interlocking pattern is created and regulated by mechanical instabilities. By close consideration of the geometry and mechanics of 2 lobes of the mantle, constrained both by the rigid shell that they secrete and by each other, we uncover the mechanistic basis for the interlocking pattern. Our modeling framework recovers and explains a large diversity of shell forms and highlights how parametric variations in the growth process result in morphological variation. Beyond the basic interlocking mechanism, we also consider the intricate and striking multiscale-patterned edge in certain brachiopods. We show that this pattern can be explained as a secondary instability that matches morphological trends and data.

A computational framework for the morpho-elastic development of molluskan shells by surface and volume growth.

Mollusk shells are an ideal model system for understanding the morpho-elastic basis of morphological evolution of invertebrates' exoskeletons. During the formation of the shell, the mantle tissue secretes proteins and minerals that calcify to form a new incremental layer of the exoskeleton. Most of the existing literature on the morphology of mollusks is descriptive. The mathematical understanding of the underlying coupling between pre-existing shell morphology, de novo surface deposition and morpho-elastic volume growth is at a nascent stage, primarily limited to reduced geometric representations. Here, we propose a general, three-dimensional computational framework coupling pre-existing morphology, incremental surface growth by accretion, and morpho-elastic volume growth. We exercise this framework by applying it to explain the stepwise morphogenesis of seashells during growth: new material surfaces are laid down by accretive growth on the mantle whose form is determined by its morpho-elastic growth. Calcification of the newest surfaces extends the shell as well as creates a new scaffold that constrains the next growth step. We study the effects of surface and volumetric growth rates, and of previously deposited shell geometries on the resulting modes of mantle deformation, and therefore of the developing shell's morphology. Connections are made to a range of complex shells ornamentations.

Are Homeostatic States Stable? Dynamical Stability in Morphoelasticity.

Biological growth is often driven by mechanical cues, such as changes in external pressure or tensile loading. Moreover, it is well known that many living tissues actively maintain a preferred level of mechanical internal stress, called the mechanical homeostasis. The tissue-level feedback mechanism by which changes in the local mechanical stresses affect growth is called a growth law within the theory of morphoelasticity, a theory for understanding the coupling between mechanics and geometry in growing and evolving biological materials. This coupling between growth and mechanics occurs naturally in macroscopic tubular structures, which are common in biology (e.g., arteries, plant stems, airways). We study a continuous tubular system with spatially heterogeneous residual stress via a novel discretization approach which allows us to obtain precise results about the stability of equilibrium states of the homeostasis-driven growing dynamical system. This method allows us to show explicitly that the stability of the homeostatic state depends nontrivially on the anisotropy of the growth response. The key role of anisotropy may provide a foundation for experimental testing of homeostasis-driven growth laws.

Post-buckling behaviour of a growing elastic rod.

We consider mechanically-induced pattern formation within the framework of a growing, planar, elastic rod attached to an elastic foundation. Through a combination of weakly nonlinear analysis and numerical methods, we identify how the shape and type of buckling (super- or subcritical) depend on material parameters, and a complex phase-space of transition from super- to subcritical is uncovered. We then examine the effect of heterogeneity on buckling and post-buckling behaviour, in the context of a heterogeneous substrate adhesion, elastic stiffness, or growth. We show how the same functional form of heterogeneity in different properties is manifest in a vastly differing post-buckled shape. Finally, a fourth form of heterogeneity, an imperfect foundation, is incorporated and shown to have a more dramatic impact on the buckling instability, a difference that can be qualitatively understood via the weakly nonlinear analysis.

The Fluid Mechanics of Ureteroscope Irrigation.

PURPOSE: To develop a physical understanding of ureterorenoscopy irrigation, we derive mathematical models from basic physical principles and compare these predictions with the results of benchtop experiments. Mathematical modeling can be used to understand the role of inlet pressure, tip deflection, the presence of working tools, geometric properties of the instruments used, and material properties of the irrigation fluid on resulting flow rate. MATERIALS AND METHODS: We develop theoretical models to describe irrigation flow in an idealized setup and compare with benchtop experiments for flow through a straight scope, a scope with a deflected tip, and a scope with a working tool inserted. The benchtop experiments were performed using Boston Scientific LithoVue ureteroscope and a variety of Boston Scientific working tools. Standard ureteroscope working channels have circular cross sections, but using theoretical models we investigate whether modifications to the cross-sectional geometry can enhance flow rates. RESULTS: The theoretical flow predictions are confirmed by experimental results. Tip deflection is shown to have a negligible effect on flow rate, but the presence of working tools decreases flow significantly (for a fixed driving pressure). Flow rate is predicted to improve when tools are placed at the edge of the channel, rather than the center, and modifying the cross-sectional shape from a circle to an ellipse can further increase flow rate. CONCLUSIONS: A mathematical framework is formulated and shown to accurately predict the properties of ureteroscope irrigation flow. The theoretical approach has significant potential in quantifying irrigation flow and improving ureteroscope design.

Passive Control of Viscous Flow via Elastic Snap-Through.

We demonstrate the passive control of viscous flow in a channel by using an elastic arch embedded in the flow. Depending on the fluid flux, the arch may "snap" between two states-constricting and unconstricting-that differ in hydraulic conductivity by up to an order of magnitude. We use a combination of experiments at a macroscopic scale and theory to study the constricting and unconstricting states, and determine the critical flux required to transition between them. We show that such a device may be precisely tuned for use in a range of applications, and, in particular, has potential as a passive microfluidic fuse to prevent excessive fluxes in rigid-walled channels.

Predictability in community dynamics.

The coupling between community composition and climate change spans a gradient from no lags to strong lags. The no-lag hypothesis is the foundation of many ecophysiological models, correlative species distribution modelling and climate reconstruction approaches. Simple lag hypotheses have become prominent in disequilibrium ecology, proposing that communities track climate change following a fixed function or with a time delay. However, more complex dynamics are possible and may lead to memory effects and alternate unstable states. We develop graphical and analytic methods for assessing these scenarios and show that these dynamics can appear in even simple models. The overall implications are that (1) complex community dynamics may be common and (2) detailed knowledge of past climate change and community states will often be necessary yet sometimes insufficient to make predictions of a community's future state.

Morphomechanics and Developmental Constraints in the Evolution of Ammonites Shell Form.

The idea that physical processes involved in biological development underlie morphogenetic rules and channel morphological evolution has been central to the rise of evolutionary developmental biology. Here, we explore this idea in the context of seashell morphogenesis. We show that a morphomechanical model predicts the effects of variations in shell shape on the ornamental pattern in ammonites, a now extinct group of cephalopods with external chambered shell. Our model shows that several seemingly unrelated characteristics of synchronous, ontogenetic, intraspecific, and evolutionary variations in ornamental patterns among various ammonite species may all be understood from the fact that the mechanical forces underlying the oscillatory behavior of the shell secreting system scale with the cross-sectional curvature of the shell aperture. This simple morphogenetic rule, emerging from biophysical interactions during shell formation, introduced a non-random component in the production of phenotypic variation and channeled the morphological evolution of ammonites over millions of years. As such, it provides a paradigm for the concept of "developmental constraints."

Morphomechanical Innovation Drives Explosive Seed Dispersal.

How mechanical and biological processes are coordinated across cells, tissues, and organs to produce complex traits is a key question in biology. Cardamine hirsuta, a relative of Arabidopsis thaliana, uses an explosive mechanism to disperse its seeds. We show that this trait evolved through morphomechanical innovations at different spatial scales. At the organ scale, tension within the fruit wall generates the elastic energy required for explosion. This tension is produced by differential contraction of fruit wall tissues through an active mechanism involving turgor pressure, cell geometry, and wall properties of the epidermis. Explosive release of this tension is controlled at the cellular scale by asymmetric lignin deposition within endocarp b cells-a striking pattern that is strictly associated with explosive pod shatter across the Brassicaceae plant family. By bridging these different scales, we present an integrated mechanism for explosive seed dispersal that links evolutionary novelty with complex trait innovation. VIDEO ABSTRACT.

The shallow shell approach to Pogorelov's problem and the breakdown of 'mirror buckling'.

We present a detailed asymptotic analysis of the point indentation of an unpressurized, spherical elastic shell. Previous analyses of this classic problem have assumed that for sufficiently large indentation depths, such a shell deforms by 'mirror buckling'-a portion of the shell inverts to become a spherical cap with equal but opposite curvature to the undeformed shell. The energy of deformation is then localized in a ridge in which the deformed and undeformed portions of the shell join together, commonly referred to as Pogorelov's ridge. Rather than using an energy formulation, we revisit this problem from the point of view of the shallow shell equations and perform an asymptotic analysis that exploits the largeness of the indentation depth. This reveals first that the stress profile associated with mirror buckling is singular as the indenter is approached. This consequence of point indentation means that mirror buckling must be modified to incorporate the shell's bending stiffness close to the indenter and gives rise to an intricate asymptotic structure with seven different spatial regions. This is in contrast with the three regions (mirror-buckled, ridge and undeformed) that are usually assumed and yields new insight into the large compressive hoop stress that ultimately causes the secondary buckling of the shell.