Reply to the 'Comment on "Dynamics and rheology of vesicles under confined Poiseuille flow"' by G. Coupier and T. Podgorski, , 2024, , DOI: 10.1039/D3SM01679J.

In this answer, we provide our arguments in support of the possibility to observe the single file-organization of red blood cells in microvessels and the resulting unexpectedly weak increase of blood viscosity with increasing hematocrit, the physiological relevance of which was questioned in the comment. The key element is that the equivalent diameter in 3D for the maximal hematocrit corresponding to a single file of red blood cells is about 10 µm and not 20 µm, as in 2D. In addition, the viscosity contrast (ratio between the cell internal and external viscosities) value must be chosen in our 2D simulation in a such a way that the effective viscosity (a linear combination of the internal, external and membrane viscosities) be close to that of a real RBC.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Shear thinning and shear thickening of a confined suspension of vesicles.

Widely regarded as an interesting model system for studying flow properties of blood, vesicles are closed membranes of phospholipids that mimic the cytoplasmic membranes of red blood cells. In this study we analyze the rheology of a suspension of vesicles in a confined geometry: the suspension, bound by two planar rigid walls on each side, is subject to a shear flow. Flow properties are then analyzed as a function of shear rate γ[over ̇], the concentration of the suspension ϕ, and the viscosity contrast λ=η_{in}/η_{out}, where η_{in} and η_{out} are the fluid viscosities of the inner and outer fluids, respectively.

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.

Effect of Gaussian curvature modulus on the shape of deformed hollow spherical objects.

A popular description of soft membranes uses the surface curvature energy introduced by Helfrich, which includes a spontaneous curvature parameter. In this paper we show how the Helfrich formula can also be of interest for a wider class of spherical elastic surfaces, namely with shear elasticity, and likely to model other deformable hollow objects. The key point is that when a stress-free state with spherical symmetry exists before subsequent deformation, its straightforwardly determined curvature ("geometrical spontaneous curvature") differs most of the time from the Helfrich spontaneous curvature parameter that should be considered in order to have the model being correctly used.

Amoeboid motion in confined geometry.

Many eukaryotic cells undergo frequent shape changes (described as amoeboid motion) that enable them to move forward. We investigate the effect of confinement on a minimal model of amoeboid swimmer. A complex picture emerges: (i) The swimmer's nature (i.

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.

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.

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.

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.

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.

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).

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.

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.