Non-reciprocity induces resonances in a two-field Cahn-Hilliard model.

We consider a non-reciprocally coupled two-field Cahn-Hilliard system that has been shown to allow for oscillatory behaviour and suppression of coarsening. After introducing the model, we first review the linear stability of steady uniform states and show that all instability thresholds are identical to the ones for a corresponding two-species reaction-diffusion system. Next, we consider a specific interaction of linear modes-a 'Hopf-Turing' resonance-and derive the corresponding amplitude equations using a weakly nonlinear approach. We discuss the weakly nonlinear results and finally compare them with fully nonlinear simulations for a specific conserved amended FitzHugh-Nagumo system. We conclude with a discussion of the limitations of the employed weakly nonlinear approach. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

Shape-morphing architectures actuated by Janus fibers.

We describe a combined experimental and theoretical investigation of shape-morphing structures assembled by actuating composite (Janus) fibers, taking into account multiple relevant factors affecting shape transformations, such as strain rate, composition, and geometry of the structures. Starting with simple bending experiments, we demonstrate the ways to attain multiple out-of-plane shapes of closed rings and square frames. Through combining theory and simulation, we examine how the mechanical properties of Janus fibers affect shape transitions. This allows us to control shape changes and to attain target 3D shapes by precise tuning of the material properties and geometry of the fibers. Our results open new perspectives of design of advanced mechanical metamaterials capable to create elaborate structures through sophisticated actuation modes.

Biomechanical modeling of invasive breast carcinoma under a dynamic change in cell phenotype: collective migration of large groups of cells.

According to recent studies, cancer is an evolving complex ecosystem. It means that tumor cells are well differentiated and involved in heterotypic interactions with their microenvironment competing for available resources to proliferate and survive. In this paper, we propose a chemo-mechanical model for the growth of specific subtypes of an invasive breast carcinoma. The model suggests that a carcinoma is a heterogeneous entity comprising cells of different phenotypes, which perform different functions in a tumor. Every cell is represented by an elastic polygon changing its form and size under pressure from the tissue. The mechanical model is based on the elastic potential energy of the tissue including the effects of contractile forces within the cell perimeter and the elastic resistance to stretching or compressing the cell with respect to the reference area. A tissue can evolve via mechanisms of cell division and intercalation. The phenotype of each cell is determined by its environment and can dynamically change via an epithelial-mesenchymal transition and vice versa. The phenotype defines the cell adhesion to the adjacent tissue and the ability to divide. In this part, we focus on the forms of collective migration of large groups of cells. Numerical simulations show the different architectural subtypes of invasive carcinoma. For each communication, we examine the dynamics of the cell population and evaluate the complexity of the pattern in terms of the synergistic paradigm. The patterns are compared with the morphological structures previously identified in clinical studies.

A theoretical model of collective cell polarization and alignment.

Collective cell polarization and alignment play important roles in tissue morphogenesis, wound healing and cancer metastasis. How cells sense the direction and position in these processes, however, has not been fully understood. Here we construct a theoretical model based on describing cell layer as a nemato-elastic medium, by which the cell polarization, cell alignment and cell active contraction are explicitly expressed as functions of components of the nematic order parameter. To determine the order parameter we derive two sets of governing equations, one for the force equilibrium of the system, and the other for the minimization of the system's free energy including the energy of cell polarization and alignment. By solving these coupled governing equations, we can predict the effects of substrate stiffness, geometries of cell layers, external forces and myosin activity on the direction- and position-dependent cell aspect ratio and cell orientation. Moreover, the axisymmetric problem with cells on a ring-like pattern is solved analytically, and the analytical solution for cell aspect ratio are governed by parameter groups which include the stiffness of the cell and the substrate, the strength of myosin activity and the external forces. Our predictions of the cell aspect ratio and orientation are generally comparable to experimental observations. These results show that the pattern of cell polarization is determined by the anisotropic degree of active contractile stress, and suggest a stress-driven polarization mechanism that enables cells to sense their spatial positions to develop direction- and position-dependent behavior. This, in turn, sheds light on the ways to control pattern formation in tissue engineering for potential biomedical applications.

Viscous dissipation and dynamics of defects in an active nematic interface⋆.

We consider active flow and dynamics of topological defects in an active nematic interfacial layer confined between immissible viscous fluid layers. The velocity of defects is determined by asymptotic matching of solutions in the defect core and the far field. Self-propulsion of positive defects along the direction of their "comet tails" is identified as the principal deterministic component of defect dynamics, while topological and hydrodynamic interactions among mobile defects is responsible for quasi-random jitter.

Discrete model of periodic pattern formation through a combined autocrine-juxtacrine cell signaling.

We model the formation of periodic patterns of gene expression in epithelial cell sheets driven by autocrine signaling coupled to juxtacrine lateral inhibition. The mathematical model is based on a continuous description of the extracellular matrix and a discrete cell-level description of the layer of cells, coupling the dynamics of diffusible ligands to the threshold-controlled cell-autonomous regulation with randomly fluctuating production rates. The results of numerical simulations indicate that propagating signaling waves emerge in a certain parametric domain, leading to the formation of a variety of either periodic or irregular patterns. For some selections of parameters, a propagating stripe of uniform expression leaves in its wake stationary periodic arrays. Coupling of autocrine and juxtacrine cell communication is essential for the pattern regularity and for the selection of expression patterns. Moreover, weak but non-vanishing noise levels are essential for the formation of regular patterns. Additional autocrine and cell-autonomous regulatory interactions can be introduced to increase the spacing of a periodic pattern.

EGFR-dependent network interactions that pattern Drosophila eggshell appendages.

Similar to other organisms, Drosophila uses its Epidermal Growth Factor Receptor (EGFR) multiple times throughout development. One crucial EGFR-dependent event is patterning of the follicular epithelium during oogenesis. In addition to providing inductive cues necessary for body axes specification, patterning of the follicle cells initiates the formation of two respiratory eggshell appendages. Each appendage is derived from a primordium comprising a patch of cells expressing broad (br) and an adjacent stripe of cells expressing rhomboid (rho). Several mechanisms of eggshell patterning have been proposed in the past, but none of them can explain the highly coordinated expression of br and rho. To address some of the outstanding issues in this system, we synthesized the existing information into a revised mathematical model of follicle cell patterning. Based on the computational model analysis, we propose that dorsal appendage primordia are established by sequential action of feed-forward loops and juxtacrine signals activated by the gradient of EGFR signaling. The model describes pattern formation in a large number of mutants and points to several unanswered questions related to the dynamic interaction of the EGFR and Notch pathways.

A cell-level biomechanical model of Drosophila dorsal closure.

We report a model describing the various stages of dorsal closure of Drosophila. Inspired by experimental observations, we represent the amnioserosa by 81 hexagonal cells that are coupled mechanically through the position of the nodes and the elastic forces on the edges. In addition, each cell has radial spokes representing actin filaments on which myosin motors can attach and exert contractile forces on the nodes, the attachment being controlled by a signaling molecule. Thus, the model couples dissipative cell and tissue motion with kinetic equations describing the myosin and signal dynamics. In the early phase, amnioserosa cells oscillate as a result of coupling among the chemical signaling, myosin attachment/detachment, and mechanical deformation of neighboring cells. In the slow phase, we test two ratcheting mechanisms suggested by experiments: an internal ratchet by the apical and junctional myosin condensates, and an external one by the supracellular actin cables encircling the amnioserosa. Within the range of parameters tested, the model predictions suggest the former as the main contributor to cell and tissue area reduction in this stage. In the fast phase of dorsal closure, cell pulsation is arrested, and the cell and tissue areas contract consistently. This is realized in the model by gradually shrinking the resting length of the spokes. Overall, the model captures the key features of dorsal closure through the three distinct phases, and its predictions are in good agreement with observations.

Cytoskeleton fluidization versus resolidification: prestress effect.

The differential elastic modulus of an active actomyosin network is computed as a function of applied stress, taking into account both thermal and motor contributions to filament compliance in the low-frequency domain. It is shown that, due to a dual nature of motor activity, increasing motor concentration may either stiffen the network due to stronger prestress or soften it due to motor agitation, in accordance with experimental data. Prestress anisotropy, which may be induced by redistribution of motors triggered by external force, causes anisotropy of the elastic moduli. This helps to explain the contradictory phenomena of cell fluidization and resolidification in response to transient stretch observed in recent experiments.

Modelling planar polarity of epithelia: the role of signal relay in collective cell polarization.

Collective cell polarization is an important characteristic of tissues. Epithelia commonly display cellular structures that are polarized within the plane of the tissue. Establishment of this planar cell polarity requires mechanisms that locally align polarized structures between neighbouring cells, as well as cues that provide global information about alignment relative to an axis of a tissue. In the Drosophila ovary, the cadherin Fat2 is required to orient actin filaments located at the basal side of follicle cells perpendicular to the long axis of the egg chamber. The mechanisms directing this orientation of actin filaments, however, remain unknown. Here we show, using genetic mosaic analysis, that fat2 is not essential for the local alignment of actin filaments between neighbouring cells. Moreover, we provide evidence that Fat2 is involved in the propagation of a cue specifying the orientation of actin filaments relative to the tissue axis. Monte Carlo simulations of actin filament orientation resemble the results of the genetic mosaic analysis, if it is assumed that a polarity signal can propagate from a signal source only through a connected chain of wild-type cells. Our results suggest that Fat2 is required for propagating global polarity information within the follicle epithelium through direct cell-cell contact. Our computational model might be more generally applicable to study collective cell polarization in tissues.

Breathing instability versus drift instability in a two-component reaction-diffusion system.

We investigate the stability of an excited domain in a two-component reaction-diffusion system in two dimensions and correct the previous results obtained by one of the authors [L. M. Pismen, Patterns and Interfaces in Dissipative Dynamics (Springer, Berlin, 2006); L. M. Pismen, Phys. Rev. Lett. 86, 548 (2001)].

Motor-driven effective temperature and viscoelastic response of active matter.

We consider dynamic response of a cytoskeletal network to both thermal and motor-induced fluctuations. The latter are viewed in two independent ways, as either additive or multiplicative colored noise. Due to a natural upper frequency limit of the motor agitation, the response of a living cell is similar to that of an equilibrium system in the high-frequency domain. At lower frequencies, the role of motor agitation manifests itself in intensified network fluctuations, which is equivalent to effective growth of the environment temperature. The effective temperature becomes frequency dependent, which signifies violation of the conventional fluctuation-dissipation theorem. The motor action affects the dynamic shear modulus in two opposite ways: by stiffening the network through filament prestress and softening it through increased agitation. The latter tendency is isolated when only single-headed motors are present. The theory is in good agreement with experimental measurements of the amplitude of the shear modulus under these conditions.

Capillarity driven instability of a soft solid.

We report the observation of a Plateau instability in a thin filament of solid gel with a very small elastic modulus. A longitudinal undulation of the surface of the cylinder reduces its area thereby triggering capillary instability, but is counterbalanced by elastic forces following the deformation. This competition leads to a nontrivial instability threshold for a solid cylinder. The ratio of surface tension to elastic modulus defines a characteristic length scale. The onset of linear instability is when the radius of the cylinder is one-sixth of this length scale, in agreement with theory presented here.

Droplet motion driven by surface freezing or melting: a mesoscopic hydrodynamic approach.

A fluid droplet may exhibit self-propelled motion by modifying the wetting properties of the substrate. We propose a model for droplet propagation upon a terraced landscape of ordered layers formed as a result of surface freezing driven by the contact angle dependence on the terrace thickness. Simultaneous melting or freezing of the terrace edge results in a joint droplet-terrace motion. The model is tested numerically and compared to experimental observations on long-chain alkane systems in the vicinity of the surface melting point.

Spinodal dewetting in a volatile liquid film.

The coexistence of film domains of different thickness in an evaporating film of a polar liquid on a solid substrate is studied using the multiscale expansion technique. The propagation speed of a straight-line front is computed both in the quasistationary approximation and in the comoving frame. The limit of long-scale zigzag instability is computed. The instability is observed during evaporation only and exists in a range of propagation velocities bounded both from below and from above; during condensation, the propagating front is always stable. Computations for a circular front yield a critical nucleation radius for a thin film and confirm the anomalous dependence of the evaporation rate on the droplet radius observed in recent experiments.