Motility and swimming: universal description and generic trajectories.

Autonomous locomotion is a ubiquitous phenomenon in biology and in physics of active systems at microscopic scale. This includes prokaryotic, eukaryotic cells (crawling and swimming) and artificial swimmers. An outstanding feature is the ability of these entities to follow complex trajectories, ranging from straight, curved (circular, helical...), to random-like ones. The non-straight nature of these trajectories is often explained as a consequence of the asymmetry of the particle or the medium in which it moves, or due to the presence of bounding walls, etc... Here, we show that for a particle driven by a concentration field of an active species, straight, circular and helical trajectories emerge naturally in the absence of asymmetry of the particle or that of suspending medium. Our proof is based on general considerations, without referring to an explicit form of a model. We show that these three trajectories correspond to self-congruent solutions. Self-congruency means that the states of the system at different moments of time can be made identical by an appropriate combination of rotation and translation of the coordinate space. We show that these solutions are exhibited by spherically symmetric particles as a result of a series of pitchfork bifurcations, leading to spontaneous symmetry breaking in the concentration field driving the particle motility. Self-congruent dynamics in one and two dimensions are analyzed as well. Finally, we present a simple explicit nonlinear exactly solvable model of fully isotropic phoretic particle that shows the transitions from a non-motile state to straight motion to circular motion to helical motion as a series of spontaneous symmetry-breaking bifurcations. Whether a system exhibits or not a given trajectory only depends on the numerical values of parameters entering the model, while asymmetry of swimmer shape, or anisotropy of the suspending medium, or influence of bounding walls are not necessary.

Dynamics and rheology of vesicles under confined Poiseuille flow.

The rheological behavior and dynamics of a vesicle suspension, serving as a simplified model for red blood cells, are explored within a Poiseuille flow under the Stokes limit. Investigating vesicle response has led to the identification of novel solutions that complement previously documented forms like the parachute and slipper shapes. This study has brought to light the existence of alternative configurations, including a fully off-centered form and a multilobe structure. The study unveils the presence of two distinct branches associated with the slipper shape. One branch arises as a consequence of a supercritical bifurcation from the symmetric parachute shape, while the other emerges from a saddle-node bifurcation. Notably, the findings are represented through diagrams that display data collapsing harmoniously based on a combination of independent dimensionless parameters. Delving into the rheological implications, a remarkable observation emerges: the normalized viscosity (i.e. similar to intrinsic viscosity) exhibits a non-monotonic trend as a function of vesicle concentration. Initially, the normalized viscosity diminishes as the concentration increases, followed by a subsequent rise at higher concentrations. Noteworthy is the presence of a minimum value in the normalized viscosity at lower concentrations, aligning well with the concentrations observed in microcirculation scenarios. The intricate behavior of the normalized viscosity can be attributed to a delicate spatial arrangement within the suspension. Importantly, this trend echoes the observations made in a linear shear flow scenario, thereby underscoring the universality of the rheological behavior for confined suspensions.

Flow driven vesicle unbinding under mechanosensitive adhesion.

Ligand receptor based adhesion is the primary mode of interaction of cellular blood constituents with the endothelium. These adhered entities also experience shear flow imposed by the blood which may lead to their detachment due to the viscous lift forces. Here, we have studied the role of the ligand-receptor bond kinetics in the detachment of an adhered vesicle (a simplified cell model) under shear flow. Using boundary integral formulation we performed numerical simulation of a two dimensional vesicle under shear flow for different values of applied shear rates and time scale of bond kinetics. We observe that the vesicle demonstrates three steady state configurations - adhered, pinned and detached for fast enough ligand-receptor kinetics (akin to Lennard-Jones adhesion). However, for slow bond kinetics the pinned state is not observed. We present scaling laws for the critical shear rates corresponding to the transitions among these three states. These results can help with identifying the processes of cell adhesion/detachment in the blood stream, prevalent features during the immune response and cancer metastasis.

Amoeboid Swimming Is Propelled by Molecular Paddling in Lymphocytes.

Mammalian cells developed two main migration modes. The slow mesenchymatous mode, like crawling of fibroblasts, relies on maturation of adhesion complexes and actin fiber traction, whereas the fast amoeboid mode, observed exclusively for leukocytes and cancer cells, is characterized by weak adhesion, highly dynamic cell shapes, and ubiquitous motility on two-dimensional and in three-dimensional solid matrix. In both cases, interactions with the substrate by adhesion or friction are widely accepted as a prerequisite for mammalian cell motility, which precludes swimming. We show here experimental and computational evidence that leukocytes do swim, and that efficient propulsion is not fueled by waves of cell deformation but by a rearward and inhomogeneous treadmilling of the cell external membrane. Our model consists of a molecular paddling by transmembrane proteins linked to and advected by the actin cortex, whereas freely diffusing transmembrane proteins hinder swimming. Furthermore, continuous paddling is enabled by a combination of external treadmilling and selective recycling by internal vesicular transport of cortex-bound transmembrane proteins. This mechanism explains observations that swimming is five times slower than the retrograde flow of cortex and also that lymphocytes are motile in nonadherent confined environments. Resultantly, the ubiquitous ability of mammalian amoeboid cells to migrate in two dimensions or three dimensions and with or without adhesion can be explained for lymphocytes by a single machinery of heterogeneous membrane treadmilling.

Amoeboid swimming in a compliant channel.

Several prokaryotes and eukaryotic cells swim in the presence of deformable and rigid surfaces that form confinement. The most commonly observed examples from biological systems are motility of leukocytes and pathogens present within the blood suspension through a microvascular network, and locomotion of eukaryotic cells such as immune system cells and cancerous cells through interstices between soft interstitial cells and the extracellular matrix within the interstitial tissue. This motivated us to investigate numerically the flow dynamics of amoeboid swimming in a flexible channel. The effects of wall stiffness and channel confinement on the flow dynamics and swimmer motion are studied. The swimmer motion through the flexible channel is substantially decelerated compared to the rigid channel. The strong confinement in the amply flexible channel imprisons the swimmer by severely restricting its forward motion. The swimmer velocity in a stiff channel displays nonmonotonic variation with the confinement while it shows monotonic reduction in a highly flexible channel. The physical rationale behind such distinct velocity behaviour in flexible and rigid channels is illustrated using an instantaneous flow field and flow history displayed by the swimmer. This behavior follows from a subtle interplay between the shape changes exhibited by the swimmer and the wall compliance. This study may aid in understanding the influence of elasticity of the surrounding environment on cell motility in immunological surveillance and invasiveness of cancer cells.

Crawling in a Fluid.

There is increasing evidence that mammalian cells not only crawl on substrates but can also swim in fluids. To elucidate the mechanisms of the onset of motility of cells in suspension, a model which couples actin and myosin kinetics to fluid flow is proposed and solved for a spherical shape. The swimming speed is extracted in terms of key parameters. We analytically find super- and subcritical bifurcations from a nonmotile to a motile state and also spontaneous polarity oscillations that arise from a Hopf bifurcation. Relaxing the spherical assumption, the obtained shapes show appealing trends.

Emerging Attractor in Wavy Poiseuille Flows Triggers Sorting of Biological Cells.

Microflows constitute an important instrument to control particle dynamics. A prominent example is the sorting of biological cells, which relies on the ability of deformable cells to move transversely to flow lines. A classic result is that soft microparticles migrate in flows through straight microchannels to an attractor at their center. Here, we show that flows through wavy channels fundamentally change the overall picture. They lead to the emergence of a second, coexisting attractor for soft particles. Its emergence and off-center location depends on the boundary modulation and the particle properties. The related cross-stream migration of soft particles is explained by analytical considerations, Stokesian dynamics simulations in unbounded flows, and Lattice-Boltzmann simulations in bounded flows. The novel off-center attractor can be used, for instance, in diagnostics, for separating cells of different size and elasticity, which is often an indicator of their health status.

Effect of Cytoskeleton Elasticity on Amoeboid Swimming.

Recently, it has been reported that the cells of the immune system, as well as Dictyostelium amoebae, can swim in a bulk fluid by changing their shape repeatedly. We refer to this motion as amoeboid swimming. Here, we explore how the propulsion and the deformation of the cell emerge as an interplay between the active forces that the cell employs to activate the shape changes and the passive, viscoelastic response of the cell membrane, the cytoskeleton, and the surrounding environment. We introduce a model in which the cell is represented by an elastic capsule enclosing a viscous liquid. The motion of the cell is activated by time-dependent forces distributed along its surface. The model is solved numerically using the boundary integral formulation. The cell can swim in a fluid medium using cyclic deformations or strokes. We measure the swimming velocity of the cell as a function of the force amplitude, the stroke frequency, and the viscoelastic properties of the cell and the medium. We show that an increase in the shear modulus leads both to a regular slowdown of the swimming, which is more pronounced for more deflated swimmers, and to a tendency toward cell buckling. For a given stroke frequency, the swimming velocity shows a quadratic dependence on force amplitude for small forces, as expected, but saturates for large forces. We propose a scaling relationship for the dependence of swimming velocity on the relevant parameters that qualitatively reproduces the numerical results and allows us to define regimes in which the cell motility is dominated by elastic response or by the effective cortex viscosity. This leads to an estimate of the effective cortex viscosity of 103 Pa ⋅ s for which the two effects are comparable, which is close to that provided by several experiments.

Blood Crystal: Emergent Order of Red Blood Cells Under Wall-Confined Shear Flow.

Driven or active suspensions can display fascinating collective behavior, where coherent motions or structures arise on a scale much larger than that of the constituent particles. Here, we report numerical simulations and an analytical model revealing that deformable particles and, in particular, red blood cells (RBCs) assemble into regular patterns in a confined shear flow. The pattern wavelength concurs well with our experimental observations. The order is of a pure hydrodynamic and inertialess origin, and it emerges from a subtle interplay between (i) hydrodynamic repulsion by the bounding walls that drives deformable cells towards the channel midplane and (ii) intercellular hydrodynamic interactions that can be attractive or repulsive depending on cell-cell separation. Various crystal-like structures arise depending on the RBC concentration and confinement. Hardened RBCs in experiments and rigid particles in simulations remain disordered under the same conditions where deformable RBCs form regular patterns, highlighting the intimate link between particle deformability and the emergence of order.

Three-bead steering microswimmers.

The self-propelled microswimmers have recently attracted considerable attention as model systems for biological cell migration as well as artificial micromachines. A simple and well-studied microswimmer model consists of three identical spherical beads joined by two springs in a linear fashion with active oscillatory forces being applied on the beads to generate self-propulsion. We have extended this linear microswimmer configuration to a triangular geometry where the three beads are connected by three identical springs in an equilateral triangular manner. The active forces acting on each spring can lead to autonomous steering motion; i.e., allowing the swimmer to move along arbitrary paths. We explore the microswimmer dynamics analytically and pinpoint its rich character depending on the nature of the active forces. The microswimmers can translate along a straight trajectory, rotate at a fixed location, as well as perform a simultaneous translation and rotation resulting in complex curved trajectories. The sinusoidal active forces on the three springs of the microswimmer contain naturally four operating parameters which are more than required for the steering motion. We identify the minimal operating parameters which are essential for the motion of the microswimmer along any given arbitrary trajectory. Therefore, along with providing insights into the mechanics of the complex motion of the natural and artificial microswimmers, the triangular three-bead microswimmer can be utilized as a model for targeted drug delivery systems and autonomous underwater vehicles where intricate trajectories are involved.

Dynamics of flexible fibers and vesicles in Poiseuille flow at low Reynolds number.

The dynamics of flexible fibers and vesicles in unbounded planar Poiseuille flow at low Reynolds number is shown to exhibit similar basic features, when their equilibrium (moderate) aspect ratio is the same and vesicle viscosity contrast is relatively high. Tumbling, lateral migration, accumulation and shape evolution of these two types of flexible objects are analyzed numerically. The linear dependence of the accumulation position on relative bending rigidity, and other universal scalings are derived from the local shear flow approximation.

Amoeboid swimming in a channel.

Several micro-organisms, such as bacteria, algae, or spermatozoa, use flagellar or ciliary activity to swim in a fluid, while many other micro-organisms instead use ample shape deformation, described as amoeboid, to propel themselves either by crawling on a substrate or swimming. Many eukaryotic cells were believed to require an underlying substratum to migrate (crawl) by using membrane deformation (like blebbing or generation of lamellipodia) but there is now increasing evidence that a large variety of cells (including those of the immune system) can migrate without the assistance of focal adhesion, allowing them to swim as efficiently as they can crawl. This paper details the analysis of amoeboid swimming in a confined fluid by modeling the swimmer as an inextensible membrane deploying local active forces (with zero total force and torque). The swimmer displays a rich behavior: it may settle into a straight trajectory in the channel or navigate from one wall to the other depending on its confinement. The nature of the swimmer is also found to be affected by confinement: the swimmer can behave, on average over one swimming cycle, as a pusher at low confinement, and becomes a puller at higher confinement, or vice versa. The swimmer's nature is thus not an intrinsic property. The scaling of the swimmer velocity V with the force amplitude A is analyzed in detail showing that at small enough A, V∼A(2)/η(2) (where η is the viscosity of the ambient fluid), whereas at large enough A, V is independent of the force and is determined solely by the stroke cycle frequency and the swimmer size. This finding starkly contrasts with models where motion is based on ciliary and flagellar activity, where V∼A/η. To conclude, two definitions of efficiency as put forward in the literature are analyzed with distinct outcomes. We find that one type of efficiency has an optimum as a function of confinement while the other does not. Future perspectives are outlined.

Viscoelastic transient of confined red blood cells.

The unique ability of a red blood cell to flow through extremely small microcapillaries depends on the viscoelastic properties of its membrane. Here, we study in vitro the response time upon flow startup exhibited by red blood cells confined into microchannels. We show that the characteristic transient time depends on the imposed flow strength, and that such a dependence gives access to both the effective viscosity and the elastic modulus controlling the temporal response of red cells. A simple theoretical analysis of our experimental data, validated by numerical simulations, further allows us to compute an estimate for the two-dimensional membrane viscosity of red blood cells, η(mem)(2D) ∼ 10(-7) N ⋅ s ⋅ m(-1). By comparing our results with those from previous studies, we discuss and clarify the origin of the discrepancies found in the literature regarding the determination of η(mem)(2D), and reconcile seemingly conflicting conclusions from previous works.

Symmetry breaking and cross-streamline migration of three-dimensional vesicles in an axial Poiseuille flow.

We analyze numerically the problem of spontaneous symmetry breaking and migration of a three-dimensional vesicle [a model for red blood cells (RBCs)] in axisymmetric Poiseuille flow. We explore the three relevant dimensionless parameters: (i) capillary number, Ca, measuring the ratio between the flow strength over the membrane bending mode, (ii) the ratio of viscosities of internal and external liquids, λ, and (iii) the reduced volume, ν=[V/(4/3)π]/(A/4π)3/2 (A and V are the area and volume of the vesicle). The overall picture turns out to be quite complex. We find that the parachute shape undergoes spontaneous symmetry-breaking bifurcations into a croissant shape and then into slipper shape. Regarding migration, we find complex scenarios depending on parameters: The vesicles either migrate towards the center, or migrate indefinitely away from it, or stop at some intermediate position. We also find coexisting solutions, in which the migration is inwards or outwards depending on the initial position. The revealed complexity can be exploited in the problem of cell sorting out and can help understanding the evolution of RBCs' in vivo circulation.

Amoeboid swimming: a generic self-propulsion of cells in fluids by means of membrane deformations.

Microorganisms, such as bacteria, algae, or spermatozoa, are able to propel themselves forward thanks to flagella or cilia activity. By contrast, other organisms employ pronounced changes of the membrane shape to achieve propulsion, a prototypical example being the Eutreptiella gymnastica. Cells of the immune system as well as dictyostelium amoebas, traditionally believed to crawl on a substratum, can also swim in a similar way. We develop a model for these organisms: the swimmer is mimicked by a closed incompressible membrane with force density distribution (with zero total force and torque). It is shown that fast propulsion can be achieved with adequate shape adaptations. This swimming is found to consist of an entangled pusher-puller state. The autopropulsion distance over one cycle is a universal linear function of a simple geometrical dimensionless quantity A/V(2/3) (V and A are the cell volume and its membrane area). This study captures the peculiar motion of Eutreptiella gymnastica with simple force distribution.

Analytical and numerical study of three main migration laws for vesicles under flow.

Blood flow shows nontrivial spatiotemporal organization of the suspended entities under the action of a complex cross-streamline migration, that renders understanding of blood circulation and blood processing in lab-on-chip technologies a challenging issue. Cross-streamline migration has three main sources: (i) hydrodynamic lift force due to walls, (ii) gradients of the shear rate (as in Poiseuille flow), and (iii) hydrodynamic interactions among cells. We derive analytically these three laws of migration for a vesicle (a model for an erythrocyte) showing good agreement with numerical simulations and experiments. In an unbounded Poiseuille flow, the situation turns out to be quite complex. We predict that a vesicle may migrate either towards the center or away from it, or even show both behaviors for the same parameters, depending on initial position. This finding can both help understanding healthy and pathological erythrocyte behavior in blood circulation and be exploited in biotechnologies for cell sorting out.

Vesicle dynamics under weak flows: application to large excess area.

Dynamics of a vesicle under simple shear flow is studied in the limit of small capillary number. A perturbative approach is used to derive the equation of vesicle dynamics. The expansions are shown to converge for significantly deflated vesicles (with excess area from the sphere as high as 2). In particular, we provide an explicit analytical expression for the tank-treading to tumbling bifurcation point. This expression is valid for excess areas up to 2.5. The results are compared with full 3D numerical simulations. The proposed method can be used for analytical or numerical solution of vesicle dynamics under weak flow of general form.

Shape diagram of vesicles in Poiseuille flow.

Soft bodies flowing in a channel often exhibit parachutelike shapes usually attributed to an increase of hydrodynamic constraint (viscous stress and/or confinement). We show that the presence of a fluid membrane leads to the reverse phenomenon and build a phase diagram of shapes-which are classified as bullet, croissant, and parachute-in channels of varying aspect ratio. Unexpectedly, shapes are relatively wider in the narrowest direction of the channel. We highlight the role of flow patterns on the membrane in this response to the asymmetry of stress distribution.

Squaring, parity breaking, and S tumbling of vesicles under shear flow.

The numerical study of 3D vesicles with a reduced volume equal to that of human red blood cells leads to the discovery of three types of dynamics: (i) squaring motion, in which the angle between the direction of the longest distance and the flow velocity undergoes discontinuous jumps over time, (ii) spontaneous parity breaking of the shape leading to cross-streamline migration, and (iii) S tumbling where the vesicle tumbles, exhibiting a pronounced S-like shape with a waisted morphology in the center. We report on the phase diagram within a wide range of relevant parameters. Our estimates reveal that healthy and pathological red blood cells are also prone to these types of motion, which may affect blood microcirculation and impact oxygen transport.

Symmetry breaking of vesicle shapes in Poiseuille flow.

Vesicle behavior under unbounded axial Poiseuille flow is studied analytically. Our study reveals subtle features of the dynamics. It is established that there exists a stable off-centerline steady-state solution for low enough flow strength. This solution appears as a symmetry-breaking bifurcation upon lowering the flow strength and includes slipper shapes, which are characteristic of red blood cells in the microvasculature. A stable axisymmetric solution exists for any flow strength provided the excess area is small enough. It is shown that the mechanism of the symmetry breaking depends on the geometry of the flow: The bifurcation is subcritical in axial Poiseuille flow and supercritical in planar flow.