Collective durotaxis of cohesive cell clusters on a stiffness gradient.

Many types of motile cells perform durotaxis, namely directed migration following gradients of substrate stiffness. Recent experiments have revealed that cell monolayers can migrate toward stiffer regions even when individual cells do not-a phenomenon known as collective durotaxis. Here, we address the spontaneous motion of finite cohesive cell monolayers on a stiffness gradient. We theoretically analyze a continuum active polar fluid model that has been tested in recent wetting assays of epithelial tissues and includes two types of active forces (cell-substrate traction and cell-cell contractility). The competition between the two active forces determines whether a cell monolayer spreads or contracts. Here, we show that this model generically predicts collective durotaxis, and that it features a variety of dynamical regimes as a result of the interplay between the spreading state and the global propagation, including sequential contraction and spreading of the monolayer as it moves toward higher stiffness. We solve the model exactly in some relevant cases, which provides both physical insights into the mechanisms of tissue durotaxis and spreading as well as a variety of predictions that could guide the design of future experiments.

Motility and morphodynamics of confined cells.

We introduce a minimal hydrodynamic model of polarization, migration, and deformation of a biological cell confined between two parallel surfaces. In our model, the cell is driven out of equilibrium by an active cytsokeleton force that acts on the membrane. The cell cytoplasm, described as a viscous droplet in the Darcy flow regime, contains a diffusive solute that actively transduces the applied cytoskeleton force. While fairly simple and analytically tractable, this quasi-two-dimensional model predicts a range of compelling dynamic behaviours. A linear stability analysis of the system reveals that solute activity first destabilizes a global polarization-translation mode, prompting cell motility through spontaneous symmetry breaking. At higher activity, the system crosses a series of Hopf bifurcations leading to coupled oscillations of droplet shape and solute concentration profiles. At the nonlinear level, we find traveling-wave solutions associated with unique polarized shapes that resemble experimental observations. Altogether, this model offers an analytical paradigm of active deformable systems in which viscous hydrodynamics are coupled to diffusive force transducers.

Unraveling the hidden complexity of quasideterministic ratchets: Random walks, graphs, and circle maps.

Brownian ratchets are shown to feature a nontrivial vanishing-noise limit where the dynamics is reduced to a stochastic alternation between two deterministic circle maps (quasideterministic ratchets). Motivated by cooperative dynamics of molecular motors, here we solve exactly the problem of two interacting quasideterministic ratchets. We show that the dynamics can be described as a random walk on a graph that is specific to each set of parameters. We compute point by point the exact velocity-force V(f) function as a summation over all paths in the specific graph for each f, revealing a complex structure that features self-similarity and nontrivial continuity properties. From a general perspective, we unveil that the alternation of two simple piecewise linear circle maps unfolds a very rich variety of dynamical complexity, in particular the phenomenon of piecewise chaos, where chaos emerges from the combination of nonchaotic maps. We show convergence of the finite-noise case to our exact solution.

Active wetting of epithelial tissues.

Development, regeneration and cancer involve drastic transitions in tissue morphology. In analogy with the behavior of inert fluids, some of these transitions have been interpreted as wetting transitions. The validity and scope of this analogy are unclear, however, because the active cellular forces that drive tissue wetting have been neither measured nor theoretically accounted for. Here we show that the transition between two-dimensional epithelial monolayers and three-dimensional spheroidal aggregates can be understood as an active wetting transition whose physics differs fundamentally from that of passive wetting phenomena. By combining an active polar fluid model with measurements of physical forces as a function of tissue size, contractility, cell-cell and cell-substrate adhesion, and substrate stiffness, we show that the wetting transition results from the competition between traction forces and contractile intercellular stresses. This competition defines a new intrinsic lengthscale that gives rise to a critical size for the wetting transition in tissues, a striking feature that has no counterpart in classical wetting. Finally, we show that active shape fluctuations are dynamically amplified during tissue dewetting. Overall, we conclude that tissue spreading constitutes a prominent example of active wetting - a novel physical scenario that may explain morphological transitions during tissue morphogenesis and tumor progression.

Author Correction: Formation of helical membrane tubes around microtubules by single-headed kinesin KIF1A.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

A Brownian Ratchet Model Explains the Biased Sidestepping of Single-Headed Kinesin-3 KIF1A.

The kinesin-3 motor KIF1A is involved in long-ranged axonal transport in neurons. To ensure vesicular delivery, motors need to navigate the microtubule lattice and overcome possible roadblocks along the way. The single-headed form of KIF1A is a highly diffusive motor that has been shown to be a prototype of a Brownian motor by virtue of a weakly bound diffusive state to the microtubule. Recently, groups of single-headed KIF1A motors were found to be able to sidestep along the microtubule lattice, creating left-handed helical membrane tubes when pulling on giant unilamellar vesicles in vitro. A possible hypothesis is that the diffusive state enables the motor to explore the microtubule lattice and switch protofilaments, leading to a left-handed helical motion. Here, we study the longitudinal rotation of microtubules driven by single-headed KIF1A motors using fluorescence-interference contrast microscopy. We find an average rotational pitch of ≃1.5µm, which is remarkably robust to changes in the gliding velocity, ATP concentration, microtubule length, and motor density. Our experimental results are compared to stochastic simulations of Brownian motors moving on a two-dimensional continuum ratchet potential, which quantitatively agree with the fluorescence-interference contrast experiments. We find that single-headed KIF1A sidestepping can be explained as a consequence of the intrinsic handedness and polarity of the microtubule lattice in combination with the diffusive mechanochemical cycle of the motor.

Active Fingering Instability in Tissue Spreading.

During the spreading of epithelial tissues, the advancing tissue front often develops fingerlike protrusions. Their resemblance to traditional viscous fingering patterns in driven fluids suggests that epithelial fingers could arise from an interfacial instability. However, the existence and physical mechanism of such a putative instability remain unclear. Here, based on an active polar fluid model for epithelial spreading, we analytically predict a generic instability of the tissue front. On the one hand, active cellular traction forces impose a velocity gradient that leads to an accelerated front, which is, thus, unstable to long-wavelength perturbations. On the other hand, contractile intercellular stresses typically dominate over surface tension in stabilizing short-wavelength perturbations. Finally, the finite range of hydrodynamic interactions in the tissue selects a wavelength for the fingering pattern, which is, thus, given by the smallest between the tissue size and the hydrodynamic screening length. Overall, we show that spreading epithelia experience an active fingering instability based on a simple kinematic mechanism. Moreover, our results underscore the crucial role of long-range hydrodynamic interactions in the dynamics of tissue morphology.

Role of Substrate Stiffness in Tissue Spreading: Wetting Transition and Tissue Durotaxis.

Living tissues undergo wetting transitions: On a surface, they can either form a dropletlike cell aggregate or spread as a monolayer of migrating cells. Tissue wetting depends not only on the chemical but also on the mechanical properties of the substrate. Here, we study the role of substrate stiffness in tissue spreading, which we describe by means of an active polar fluid model. Taking into account that cells exert larger active traction forces on stiffer substrates, we predict a tissue wetting transition at a critical substrate stiffness that decreases with tissue size. On substrates with a stiffness gradient, we find that the tissue spreads faster on the stiffer side. Furthermore, we show that the tissue can wet the substrate on the stiffer side while dewetting from the softer side. We also show that, by means of viscous forces transmitted across the tissue, the stiffer-side interface can transiently drag the softer-side interface toward increasing stiffness, against its spreading tendency. These two effects result in directed tissue migration up the stiffness gradient. This phenomenon-tissue durotaxis-can thus emerge both from dewetting on the soft side and from hydrodynamic interactions between the tissue interfaces. Overall, our work unveils mechanisms whereby substrate stiffness impacts the collective migration and the active wetting properties of living tissues, which are relevant in development, regeneration, and cancer.

Mixed-order phase transition in a colloidal crystal.

Mixed-order phase transitions display a discontinuity in the order parameter like first-order transitions yet feature critical behavior like second-order transitions. Such transitions have been predicted for a broad range of equilibrium and nonequilibrium systems, but their experimental observation has remained elusive. Here, we analytically predict and experimentally realize a mixed-order equilibrium phase transition. Specifically, a discontinuous solid-solid transition in a 2D crystal of paramagnetic colloidal particles is induced by a magnetic field [Formula: see text] At the transition field [Formula: see text], the energy landscape of the system becomes completely flat, which causes diverging fluctuations and correlation length [Formula: see text] Mean-field critical exponents are predicted, since the upper critical dimension of the transition is [Formula: see text] Our colloidal system provides an experimental test bed to probe the unconventional properties of mixed-order phase transitions.

Noise focusing in neuronal tissues: Symmetry breaking and localization in excitable networks with quenched disorder.

We introduce a coarse-grained stochastic model for the spontaneous activity of neuronal cultures to explain the phenomenon of noise focusing, which entails localization of the noise activity in excitable networks with metric correlations. The system is modeled as a continuum excitable medium with a state-dependent spatial coupling that accounts for the dynamics of synaptic connections. The most salient feature is the emergence at the mesoscale of a vector field V(r), which acts as an advective carrier of the noise. This entails an explicit symmetry breaking of isotropy and homogeneity that stems from the amplification of the quenched fluctuations of the network by the activity avalanches, concomitant with the excitable dynamics. We discuss the microscopic interpretation of V(r) and propose an explicit construction of it. The coarse-grained model shows excellent agreement with simulations at the network level. The generic nature of the observed phenomena is discussed.

Nonlinear amplitude dynamics in flagellar beating.

The physical basis of flagellar and ciliary beating is a major problem in biology which is still far from completely understood. The fundamental cytoskeleton structure of cilia and flagella is the axoneme, a cylindrical array of microtubule doublets connected by passive cross-linkers and dynein motor proteins. The complex interplay of these elements leads to the generation of self-organized bending waves. Although many mathematical models have been proposed to understand this process, few attempts have been made to assess the role of dyneins on the nonlinear nature of the axoneme. Here, we investigate the nonlinear dynamics of flagella by considering an axonemal sliding control mechanism for dynein activity. This approach unveils the nonlinear selection of the oscillation amplitudes, which are typically either missed or prescribed in mathematical models. The explicit set of nonlinear equations are derived and solved numerically. Our analysis reveals the spatio-temporal dynamics of dynein populations and flagellum shape for different regimes of motor activity, medium viscosity and flagellum elasticity. Unstable modes saturate via the coupling of dynein kinetics and flagellum shape without the need of invoking a nonlinear axonemal response. Hence, our work reveals a novel mechanism for the saturation of unstable modes in axonemal beating.

Fluidization and Active Thinning by Molecular Kinetics in Active Gels.

We derive the constitutive equations of an active polar gel from a model for the dynamics of elastic molecules that link polar elements. Molecular binding kinetics induces the fluidization of the material, giving rise to Maxwell viscoelasticity and, provided that detailed balance is broken, to the generation of active stresses. We give explicit expressions for the transport coefficients of active gels in terms of molecular properties, including nonlinear contributions on the departure from equilibrium. In particular, when activity favors linker unbinding, we predict a decrease of viscosity with activity-active thinning-of kinetic origin, which could explain some experimental results on the cell cortex. By bridging the molecular and hydrodynamic scales, our results could help understand the interplay between molecular perturbations and the mechanics of cells and tissues.

Formation of metastable phases by spinodal decomposition.

Metastable phases may be spontaneously formed from other metastable phases through nucleation. Here we demonstrate the spontaneous formation of a metastable phase from an unstable equilibrium by spinodal decomposition, which leads to a transient coexistence of stable and metastable phases. This phenomenon is generic within the recently introduced scenario of the landscape-inversion phase transitions, which we experimentally realize as a structural transition in a colloidal crystal. This transition exhibits a rich repertoire of new phase-ordering phenomena, including the coexistence of two equilibrium phases connected by two physically different interfaces. In addition, this scenario enables the control of sizes and lifetimes of metastable domains. Our findings open a new setting that broadens the fundamental understanding of phase-ordering kinetics, and yield new prospects of applications in materials science.

Bleb Nucleation through Membrane Peeling.

We study the nucleation of blebs, i.e., protrusions arising from a local detachment of the membrane from the cortex of a cell. Based on a simple model of elastic linkers with force-dependent kinetics, we show that bleb nucleation is governed by membrane peeling. By this mechanism, the growth or shrinkage of a detached membrane patch is completely determined by the linker kinetics, regardless of the energetic cost of the detachment. We predict the critical nucleation radius for membrane peeling and the corresponding effective energy barrier. These may be typically smaller than those predicted by classical nucleation theory, implying a much faster nucleation. We also perform simulations of a continuum stochastic model of membrane-cortex adhesion to obtain the statistics of bleb nucleation times as a function of the stress on the membrane. The determinant role of membrane peeling changes our understanding of bleb nucleation and opens new directions in the study of blebs.

Formation of helical membrane tubes around microtubules by single-headed kinesin KIF1A.

The kinesin-3 motor KIF1A is in charge of vesicular transport in neuronal axons. Its single-headed form is known to be very inefficient due to the presence of a diffusive state in the mechanochemical cycle. However, recent theoretical studies have suggested that these motors could largely enhance force generation by working in teams. Here we test this prediction by challenging single-headed KIF1A to extract membrane tubes from giant vesicles along microtubule filaments in a minimal in vitro system. Remarkably, not only KIF1A motors are able to extract tubes but they feature a novel phenomenon: tubes are wound around microtubules forming tubular helices. This finding reveals an unforeseen combination of cooperative force generation and self-organized manoeuvreing capability, suggesting that the diffusive state may be a key ingredient for collective motor performance under demanding traffic conditions. Hence, we conclude that KIF1A is a genuinely cooperative motor, possibly explaining its specificity to axonal trafficking.

Model for probing membrane-cortex adhesion by micropipette aspiration and fluctuation spectroscopy.

We propose a model for membrane-cortex adhesion that couples membrane deformations, hydrodynamics, and kinetics of membrane-cortex ligands. In its simplest form, the model gives explicit predictions for the critical pressure for membrane detachment and for the value of adhesion energy. We show that these quantities exhibit a significant dependence on the active acto-myosin stresses. The model provides a simple framework to access quantitative information on cortical activity by means of micropipette experiments. We also extend the model to incorporate fluctuations and show that detailed information on the stability of membrane-cortex coupling can be obtained by a combination of micropipette aspiration and fluctuation spectroscopy measurements.

Landscape-inversion phase transition in dipolar colloids: tuning the structure and dynamics of 2D crystals.

We study the 2D crystalline phases of paramagnetic colloidal particles with dipolar interactions and constrained on a periodic substrate. Combining theory, simulation, and experiments, we demonstrate a new scenario of first-order phase transitions that occurs via a complete inversion of the energy landscape, featuring nonconventional properties that allow for (i) tuning of crystal symmetry, (ii) control of dynamical properties of different crystalline orders via tuning of their relative stability with an external magnetic field, (iii) an equivalent but independent control of the same dynamic properties via temporal modulations of that field, and (iv) nonstandard phase-ordering kinetics involving spontaneous formation of transient metastable domains.

Cooperative action of KIF1A Brownian motors with finite dwell time.

We study in detail the cooperative action of small groups of KIF1A motors in its monomeric (single-headed) form within an arrangement relevant to vesicle traffic or membrane tube extraction. It has been recently shown that under these circumstances, the presence of a finite dwell time in the motor cycle contributes to remarkably enhance collective force generation [D. Oriola and J. Casademunt, Phys. Rev. Lett. 111, 048103 (2013)]. We analyze this mechanism in detail by means of a two-state noise-driven ratchet model with hard-core repulsive interactions. We obtain staircase-shaped velocity-force curves and show that motors self-organize in clusters with a nontrivial force distribution that conveys a large part of the load to the central motors. Under heavy loads, large clusters adopt a synchronic mode of totally asymmetric steps. We also find a dramatic increase of the collective efficiency with the number of motors. Finally, we complete the study by addressing different interactions that impose spatial constraints such as rigid coupling and raft-induced confinement. Our results reinforce the hypothesis that the specificity of KIF1A to axonal vesicular transport may be deeply related to its high cooperativity.

Cooperative force generation of KIF1A Brownian motors.

KIF1A is a kinesin motor protein that can work processively in a monomeric (single-headed) form by using a noise-driven ratchet mechanism. Here, we show that the combination of a passive diffusive state and finite-time kinetics of adenosine triphosphate hydrolysis provides a powerful mechanism of cooperative force generation, implying for instance that ∼10 monomeric KIF1As can team up to become ∼100 times stronger than a single one. Consequently, we propose that KIF1A could outperform conventional (double-headed) kinesin collectively and thus explain its specificity in axonal trafficking. We elucidate the cooperativity mechanism with a lattice model that includes multiparticle transitions.

Dynamics and mechanical stability of the developing dorsoventral organizer of the wing imaginal disc.

Shaping the primordia during development relies on forces and mechanisms able to control cell segregation. In the imaginal discs of Drosophila the cellular populations that will give rise to the dorsal and ventral parts on the wing blade are segregated and do not intermingle. A cellular population that becomes specified by the boundary of the dorsal and ventral cellular domains, the so-called organizer, controls this process. In this paper we study the dynamics and stability of the dorsal-ventral organizer of the wing imaginal disc of Drosophila as cell proliferation advances. Our approach is based on a vertex model to perform in silico experiments that are fully dynamical and take into account the available experimental data such as: cell packing properties, orientation of the cellular divisions, response upon membrane ablation, and robustness to mechanical perturbations induced by fast growing clones. Our results shed light on the complex interplay between the cytoskeleton mechanics, the cell cycle, the cell growth, and the cellular interactions in order to shape the dorsal-ventral organizer as a robust source of positional information and a lineage controller. Specifically, we elucidate the necessary and sufficient ingredients that enforce its functionality: distinctive mechanical properties, including increased tension, longer cell cycle duration, and a cleavage criterion that satisfies the Hertwig rule. Our results provide novel insights into the developmental mechanisms that drive the dynamics of the DV organizer and set a definition of the so-called Notch fence model in quantitative terms.