Excitable Dynamics and Yap-Dependent Mechanical Cues Drive the Segmentation Clock.

The periodic segmentation of the vertebrate body axis into somites, and later vertebrae, relies on a genetic oscillator (the segmentation clock) driving the rhythmic activity of signaling pathways in the presomitic mesoderm (PSM). To understand whether oscillations are an intrinsic property of individual cells or represent a population-level phenomenon, we established culture conditions for stable oscillations at the cellular level. This system was used to demonstrate that oscillations are a collective property of PSM cells that can be actively triggered in vitro by a dynamical quorum sensing signal involving Yap and Notch signaling. Manipulation of Yap-dependent mechanical cues is sufficient to predictably switch isolated PSM cells from a quiescent to an oscillatory state in vitro, a behavior reminiscent of excitability in other systems. Together, our work argues that the segmentation clock behaves as an excitable system, introducing a broader paradigm to study such dynamics in vertebrate morphogenesis.

Rectified random cell motility as a mechanism for embryo elongation.

The body of vertebrate embryos forms by posterior elongation from a terminal growth zone called the tail bud. The tail bud is a source of highly motile cells that eventually constitute the presomitic mesoderm (PSM), a tissue that plays an important role in elongation movements. PSM cells establish an anterior-posterior cell motility gradient that parallels a gradient associated with the degradation of a specific cellular signal (FGF) known to be implicated in cell motility. Here, we combine the electroporation of fluorescent reporters in the PSM with time-lapse imaging in the chicken embryo to quantify cell diffusive movements along the motility gradient. We show that a simple microscopic model for random cell motility induced by FGF activity along with geometric confinement leads to rectified tissue elongation consistent with our observations. A continuum analog of the microscopic model leads to a macroscopic mechano-chemical model for tissue extension that couples FGF activity-induced cell motility and tissue rheology, and is consistent with the experimentally observed speed and extent of elongation. Together, our experimental observations and theoretical models explain how the continuous addition of cells at the tail bud combined with lateral confinement can be converted into oriented movement and drive body elongation.

Cooperative effects driving the multi-periodic dynamics of cyclically sheared amorphous solids.

When subject to cyclic forcing, amorphous solids can reach periodic, repetitive states, where the system behaves plastically, but the particles return to their initial positions after one or more forcing cycles, where the latter response is called multi-periodic. It is known that plasticity in amorphous materials is mediated by local rearrangements called "soft spots" or "shear transformation zones." Experiments and simulations indicate that soft spots can be modeled as hysteretic two-state entities interacting via quadrupolar displacement fields generated when they switch states and that these interactions can give rise to multi-periodic behavior. However, how interactions facilitate multi-periodicity is unknown. Here, we show, using a model of random interacting two-state systems and molecular dynamics simulations, that multi-periodicity arises from oscillations in the magnitudes of the switching field of soft spots, which cause soft spots to be active during some forcing cycles and idle during others. We demonstrate that these oscillations result from cooperative effects facilitated by the frustrated interactions between the soft spots. The presence of such mechanisms has implications for manipulating memory in frustrated hysteretic systems.

Networks and Hierarchies: How Amorphous Materials Learn to Remember.

We consider the slow and athermal deformations of amorphous solids and show how the ensuing sequence of discrete plastic rearrangements can be mapped onto a directed network. The network topology reveals a set of highly connected regions joined by occasional one-way transitions. The highly connected regions include hierarchically organized hysteresis cycles and subcycles. At small to moderate strains this organization leads to near-perfect return point memory. The transitions in the network can be traced back to localized particle rearrangements (soft spots) that interact via Eshelby-type deformation fields. By linking topology to dynamics, the network representations provide new insight into the mechanisms that lead to reversible and irreversible behavior in amorphous solids.

Spontaneous exfoliation of a drying gel.

Wet starch cracks when it dries inhomogeneously, while hot glass cracks when it cools non-uniformly. In both cases, differential shrinkage induced by drying/cooling from the surface causes superficial cracks to grow perpendicular to the surface in different patterns. In contrast with these observations of bulk cracking in brittle materials, when a soft and homogeneously swollen polymer gel dries, differential strains lead to the peeling of a thin layer that spontaneously tears away from the bulk. Continued drying leads to the process repeating itself, forming a peeled-layered structure. The emergent thickness of the exfoliated layer is a function of both the geometry of the original gel and the physical parameters associated with the drying rate and external temperature. We characterize the experimental conditions under which layer peeling can arise, and use simulations to corroborate these observations. Finally, a minimal theory explains the scaling of the peel thickness, consistent with our experiments.

A mechano-sensing mechanism for waving in plant roots.

Arabidopsis roots grown on inclined agar surfaces exhibit unusual sinusoidal patterns known as root-waving. The origin of these patterns has been ascribed to both genetic and environmental factors. Here we propose a mechano-sensing model for root-waving, based on a combination of friction induced by gravitropism, the elasticity of the root and the anchoring of the root to the agar by thin hairs, and demonstrate its relevance to previously obtained experimental results. We further test the applicability of this model by performing experiments in which we measure the effect of gradually changing the inclination angles of the agar surfaces on the wavelength and other properties of the growing roots. We find that the observed dynamics is different than the dynamics reported in previous works, but that it can still be explained using the same mechano-sensing considerations. This is supported by the fact that a scaling relation derived from the model describes the observed dependence of the wavelength on the tilt angle for a large range of angles. We also compare the prevalence of waving in different plant species and show that it depends on root thickness as predicted by the model. The results indicate that waving can be explained using mechanics and gravitropism alone and that mechanics may play a greater role in root growth and form than was previously considered.

Yielding in an amorphous solid subject to constant stress at finite temperatures.

Understanding the nature of the yield transition is a long-standing problem in the physics of amorphous solids. Here we use molecular dynamics simulations to study the response of amorphous solids to constant stresses at finite temperatures. We compare amorphous solids that are prepared using fast and slow quenches and show that for thermal systems, the steady-state velocity exhibits a continuous transition from very slow creep to a finite strain rate as a function of the stress. This behavior is observed for both well-annealed and poorly annealed systems. However, the transient dynamics is different in the latter and involves overcoming an energy barrier. Due to the different simulation protocol, the strain rate as a function of stress and temperature follows a scaling relation that is different from the ones that are shown for systems where the strain is controlled. Collapsing the data using this scaling relation allows us to calculate critical exponents for the dynamics close to yield, including an exponent for thermal rounding. We also demonstrate that strain slips due to avalanche events above yield follow standard scaling relations and we extract critical exponents that are comparable to the ones obtained in previous studies that performed simulations of both molecular dynamics and elastoplastic models using strain-rate control.

Topology of the energy landscape of sheared amorphous solids and the irreversibility transition.

Recent experiments and simulations of amorphous solids plastically deformed by an oscillatory drive have found a surprising behavior-for small strain amplitudes the dynamics can be reversible, which is contrary to the usual notion of plasticity as an irreversible form of deformation. This reversibility allows the system to reach limit cycles in which plastic events repeat indefinitely under the oscillatory drive. It was also found that reaching reversible limit cycles can take a large number of driving cycles and it was surmised that the plastic events encountered during the transient period are not encountered again and are thus irreversible. Using a graph representation of the stable configurations of the system and the plastic events connecting them, we show that the notion of reversibility in these systems is more subtle. We find that reversible plastic events are abundant and that a large portion of the plastic events encountered during the transient period are actually reversible in the sense that they can be part of a reversible deformation path. More specifically, we observe that the transition graph can be decomposed into clusters of configurations that are connected by reversible transitions. These clusters are the strongly connected components of the transition graph and their sizes turn out to be power-law distributed. The largest of these are grouped in regions of reversibility, which in turn are confined by regions of irreversibility whose number proliferates at larger strains. Our results provide an explanation for the irreversibility transition-the divergence of the transient period at a critical forcing amplitude. The long transients result from transition between clusters of reversibility in a search for a cluster large enough to contain a limit cycle of a specific amplitude. For large enough amplitudes, the search time becomes very large, since the sizes of the limit cycles become incompatible with the sizes of the regions of reversibility.

Annealing and rejuvenation in a two-dimensional model amorphous solid under oscillatory shear.

We study the annealing and rejuvenation behavior of a two-dimensional amorphous solid model under oscillatory shear. We show that, depending on the cooling protocol used to create the initial configuration, the mean potential energy can either decrease or increase under subyield oscillatory shear. For post-yield oscillatory shear, the mean potential energy increases and is independent on the initial conditions. We explain this behavior by modeling the dynamics using a simple model of forced dynamics on a random energy landscape and show that the model reproduces the qualitative behavior of the mean potential energy and mean-square displacement observed in the particle based simulations. This suggests that some important aspects of the dynamics of amorphous solids can be understood by studying the properties of random energy landscapes and without explicitly taking into account the complex real-space interactions which are involved in plastic deformation.

Forced deterministic dynamics on a random energy landscape: Implications for the physics of amorphous solids.

The dynamics of supercooled liquids and plastically deformed amorphous solids is known to be dominated by the structure of their rough energy landscapes. Recent experiments and simulations on amorphous solids subjected to oscillatory shear at athermal conditions have shown that for small strain amplitudes these systems reach limit cycles of different periodicities after a transient. However, for larger strain amplitudes the transients become longer and for strain amplitudes exceeding a critical value the system reaches a diffusive steady state. This behavior cannot be explained using the current mean-field models of amorphous plasticity. Here we show that this phenomenology can be described and explained using a simple model of forced dynamics on a multidimensional random energy landscape. In this model, the existence of limit cycles can be ascribed to confinement of the dynamics to a small part of the energy landscape which leads to self-intersection of state-space trajectories and the transition to the diffusive regime for larger forcing amplitudes occurs when the forcing overcomes this confinement.

Quantitative theory of a time-correlation function in a one-component glass-forming liquid with anisotropic potential.

The Shintani-Tanaka model is a glass-forming system whose constituents interact via an anisotropic potential depending on the angle of a unit vector carried by each particle. The decay of time-correlation functions of the unit vectors exhibits the characteristics of generic relaxation functions during glass transitions. In particular it exhibits a stretched exponential form, with the stretching index beta depending strongly on the temperature. We construct a quantitative theory of this correlation function by analyzing all the physical processes that contribute to it, separating a rotational from a translational decay channel. These channels exhibit different relaxation times, each with its own temperature dependence. Interestingly, the separate decay function of each of these processes is a temperature-independent function, and is shown to scale (exhibit data collapse) at different temperatures. Taken together with temperature-dependent weights determined a priori by statistical mechanics this allows one to generate the observed correlation function in quantitative agreement with simulations at different temperatures. This underlines the danger of concluding anything about glassy relaxation functions without detailed physical scrutiny.

Critical diffusivity in the reversibility-irreversibility transition of amorphous solids under oscillatory shear.

Recently it was shown that under oscillatory shear at zero temperature an amorphous solid transitions from asymptotically periodic to asymptotically diffusive steady-state at a critical maximal strain amplitude. Current understanding of the physics behind this transition is lacking. Here we show, using computer simulations, evidence that the diffusivity of the vector of coordinates of the particles comprising an amorphous solid, when subject to oscillatory shear, undergoes a second order phase transition at the reversibility-irreversibility transition point. We explain how such a transition is consistent with dissipative forced dynamics on a complex energy landscape, such as is known to exist in amorphous solids. We demonstrate that as the forcing increases, more and more state-space volume becomes accessible to the system, making it less probable for the state-space trajectory of the system to self-intersect and form a limit-cycle, which explains the slowing-down observed at the transition.

Elasticity with arbitrarily shaped inhomogeneity.

A classical problem in elasticity theory involves an inhomogeneity embedded in a material of given stress and shear moduli. The inhomogeneity is a region of arbitrary shape whose stress and shear moduli differ from those of the surrounding medium. In this paper we present a semianalytic method for finding the stress tensor for an infinite plate with such an inhomogeneity. The solution involves two conformal maps, one from the inside and the second from the outside of the unit circle to the inside, and respectively outside, of the inhomogeneity. The method provides a solution by matching the conformal maps on the boundary between the inhomogeneity and the surrounding material. This matching converges well only for relatively mild distortions of the unit circle due to reasons which will be discussed in the article. We provide a comparison of the present result to known previous results.

Aging and relaxation in glass-forming systems.

We propose that there exists a generic class of glass-forming systems that have competing states (of crystalline order or not) which are locally close in energy to the ground state (which is typically unique). Upon cooling, such systems exhibit patches (or clusters) of these competing states which become locally stable in the sense of having a relatively high local shear modulus. It is in between these clusters where aging, relaxation, and plasticity under strain can take place. We demonstrate explicitly that relaxation events that lead to aging occur where the local shear modulus is low (even negative) and result in an increase in the size of local patches of relative order. We examine the aging events closely from two points of view. On the one hand we show that they are very localized in real space, taking place outside the patches of relative order, and from the other point of view we show that they represent transitions from one local minimum in the potential surface to another. This picture offers a direct relation between structure and dynamics, ascribing the slowing down in glass-forming systems to the reduction in relative volume of the amorphous material which is liquidlike. While we agree with the well-known Adam-Gibbs proposition that the slowing down is due to an entropic squeeze (a dramatic decrease in the number of available configurations), we do not agree with the Adam-Gibbs (or the Volger-Fulcher) formulas that predict an infinite relaxation time at a finite temperature. Rather, we propose that generically there should be no singular crisis at any finite temperature: the relaxation time and the associated correlation length (average cluster size) increase at most superexponentially when the temperature is lowered.

Reversibility and criticality in amorphous solids.

The physical processes governing the onset of yield, where a material changes its shape permanently under external deformation, are not yet understood for amorphous solids that are intrinsically disordered. Here, using molecular dynamics simulations and mean-field theory, we show that at a critical strain amplitude the sizes of clusters of atoms undergoing cooperative rearrangements of displacements (avalanches) diverges. We compare this non-equilibrium critical behaviour to the prevailing concept of a 'front depinning' transition that has been used to describe steady-state avalanche behaviour in different materials. We explain why a depinning-like process can result in a transition from periodic to chaotic behaviour and why chaotic motion is not possible in pinned systems. These findings suggest that, at least for highly jammed amorphous systems, the irreversibility transition may be a side effect of depinning that occurs in systems where the disorder is not quenched.

Onset of irreversibility and chaos in amorphous solids under periodic shear.

A fundamental problem in the physics of amorphous materials is understanding the transition from reversible to irreversible plastic behavior and its connection to yield. Currently, continuum material modeling relies on phenomenological yield thresholds, however in many cases the transition from elastic to plastic behavior is gradual, which makes it difficult to identify an exact yield criterion. Here we show that under periodic shear, amorphous solids undergo a transition from repetitive, predictable behavior to chaotic, irregular behavior as a function of the strain amplitude. In both the periodic and chaotic regimes, localized particle rearrangements are observed. We associate the point of transition from repetitive to chaotic behavior with the yield strain and suggest that at least for oscillatory shear, yield in amorphous solids is a result of a "transition to chaos."