Amoeboid motion in confined geometry Hao Wu, M. Thiébaud, W.-F. Hu, A. Farutin, S. Rafaï, M.-C. Lai, P. Peyla, C. Misbah To cite this version: Hao Wu, M. Thiébaud, W.-F. Hu, A. Farutin, S. Rafaï, et al.. Amoeboid motion in confined geometry. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, American Physical Society, 2015, 92 (5), 10.1103/PhysRevE.92.050701. hal-01230092 HAL Id: hal-01230092 https://hal.archives-ouvertes.fr/hal-01230092 Submitted on 17 Nov 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Amoeboid motion in confined geometry H. Wu,1,2, ∗ M. Thi´ebaud,1,2, ∗ W.-F. Hu,3 A. Farutin,1, 2 S. Rafa¨ı,1,2, † M.-C. Lai,3 P. Peyla,1, 2 and C. Misbah1, 2 1Univ. Grenoble Alpes, LIPHY, F-38000 Grenoble, France 2CNRS, LIPHY, F-38000 Grenoble, France 3Department of Applied Mathematics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan 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.e. either pusher or puller) can be modified by confinement thus suggesting that this is not an intrinsic property of the swimmer. This swimming nature transition stems from intricate internal degrees of freedom of membrane deformation. (ii) The swimming speed might increase with increasing confinement before decreasing again for stronger confinements. (iii) A straight amoeoboid swimmer’s trajectory in the channel can become unstable, and ample lateral excursions of the swimmer prevail. This happens for both pusher and puller-type swimmers. For weak confinement, these excursions are symmetric, while they become asymmetric at stronger confinement, whereby the swimmer is located closer to one of the two walls. In this study, we combine numerical and theoretical analyses. PACS numbers: 47.63.Gd, 47.15.G-,47.63.mf Some unicellular micro-organisms move on solid sur- active research [18–27]. It has been calculated long faces or swim in liquids by deforming their body instead time ago by Katz [28] and more recently pointed out of using flagella or cilia, this is known as amoeboid mo- [21, 23, 25, 27, 29] that swimmers can take advantage of tion. Algae like Eutreptiella Gymnastica [1], amoeba such walls to increase their motility. Understanding the be- as dictyostelium discoideum [2, 3], but also leucocytes havior of microswimmers in confinement can also pave [2, 3] and even cancer cells [4] use this specific way of the way to novel applications in microfluidic devices locomotion. This is a complex movement that recently where properly shaped microstructures can interfere with incited several theoretical studies [5–12] since it is inti- swimming bacteria and guide, concentrate, and arrange mately linked to cell migration involved in several dis- populations of cells [30]. Living microswimmers show a eases. Some experimental results indicate that adhesion large variety of swimming strategies [29] so do theoretical to a solid substratum is not a prerequisite for cells like models aiming at describing their dynamics in confine- amoeba [2] to produce an amoeboid movement during ment. cell migration and suggest that crawling close to a sur- Felderhof [18] has shown that the speed of Taylor-like face and swimming are similar processes. Recently, it was swimmer increases with confinement. Zhu et al. [21] used shown that integrin (a protein involved in adhesion pro- the squirmer model to show that (when only tangential cess) should no longer be viewed as force transducers dur- surface motion are included) the velocity decreases with ing locomotion but as switchable immobilizing anchors confinement and that a pusher crashes into the wall, a that slow down cells in the blood stream before transmi- puller settles in a straight trajectory, and a neutral swim- gration. Indeed, leukocytes migrate by swimming in the mer navigates. When including normal deformation they absence of specific adhesive interactions with the extra- found an increase of velocity with confinement. Liu et cellular environment [13]. al. [25] analyzed a helical flagellum in tube and found When moving, all micro-organisms are sensitive to that except for a small range of tube radii, the swim- their environments. Most microswimmers can follow gra- ming speed, when helix rotation rate is fixed, increases dients of chemicals (chemotaxis), some micro-algae can monotonically as the confinement becomes tighter. Ace- moglu et al. [24] adopted a similar model but, besides move toward light sources (phototaxis) [14] or orient decrease themselves in the gravity field (gravitaxis) [15], some the flagellum, they included a head and found a et al. other bacteria move along adhesion gradients (hapto- of velocity with confinement. Bilbao [22] treated nu- merically a model inspired by nematode locomotion and taxis) [16, 17], etc. Spatial confinement is another ma- jor environmental constraint which strongly influences found that it navigates more efficiently and moves faster et al. the motion of micro-organisms. As a matter of fact, due to walls. Ledesma [23] reported on a dipolar swimmer in a rigid or elastic tube and found a speed in the low-Reynolds number world, amoeboid motion generally occurs close to surfaces, in small capillaries enhancement due to walls. or in extracellular matrices of biological tissues. Micro- Here, we investigate, by means of numerical and an- organisms swim through permeable boundaries, cell walls alytical modeling, the effect of confinement on the be- or micro-vasculature. Therefore, the effect of walls on havior of an amoeboid swimmer, that is a deformable motile micro-organisms has been a topic of increasingly object subjected to active forces along its inextensible 2 respect to a circular shape (Γ = 0 corresponds to a cir- cle, whereas large Γ signifies a very deflated, and thus amply deformable, swimmer). The strength of confine- ment is defined as C = 2R0/W , with W the channel width. Bounding walls are parallel to x-direction and y denotes the orthogonal one. A set of active forces is distributed on the membrane that reacts with tension forces to preserve the local ar- FIG. 1: (Color online) Snapshots of an Axially Moving Swim- clength. The total force density is given by mer over time (W = 6R0). The dashed profiles show a com- T plete period s of deformation and then a few shapes are ∂ζ represented over a time of the order of 75Ts. Γ=0.085. F = F n − ζcn + t, (1) a ∂s where Fan is the active force to be specified below (which we take to point along the normal n for sim- membrane. Our model swimmer is found to reveal in- plicity), ζ is a Lagrange multiplier that enforces local teresting new features when confined between two walls. membrane incompressibility, c is the curvature, t is the (i) We find that straight trajectories might be unstable unit tangent vector and s is the arclength. We im- independently of the nature of the swimmer (pusher or pose zero total force and torque. In its full general- puller). (ii) For weak confinement, the swimming speed ity the active force can be decomposed into Fourier se- can either increase or decrease depending on the con- k=kmax ikα ries Fa(α, t) = Pk=−kmax Fk(t)e with α = 2πs/L0. finement strength. For strongly confined regimes, the We first consider the case kmax = 3, so that we are velocity decreases in all cases recalling previous results left with two complex amplitudes F2 and F3. Other on different models. (iii) The confined environnement is configurations of the forces have been explored as well shown to induce a transition from one to another type (see below). We consider cyclic strokes represented by of swimmer (i.e. puller or pusher). These behaviors are F2 = F−2 = −A cos(ωt) and F3 = F−3 = A sin(ωt), unique to amoeboid swimming (AS) and point to a non- where A is force amplitude. trivial dynamics owing to the internal degrees of freedom The Stokes equations with boundary conditions (force that evolve in response to various constraints. balance condition, continuity of the fluid velocity and The model Amoeboid swimming is modeled here by membrane incompressibility) are solved using either the taking a 1D closed and inextensible membrane, which boundary integral method (BIM) [31] or the immersed encloses a 2D liquid of certain viscosity η and is sus- boundary method (IBM) [32]. pended in another fluid taken to be of the same viscos- Besides Γ and C, there is an additional dimensionless ity, for simplicity. The extra computational complexity number S = A/(ωη), which is the ratio between the time of dealing with confined geometry restricts our study to scale associated with swimming strokes (Ts =2π/ω) and 2D which draws already rich behaviors. The effective the time scale of fluid flow due to active force (Tc = radius of the swimmer is R0 = pA0/π where A0 is the η/A). Here we take S = 10.0 (shape has enough time to enclosed area. The swimmer has an excess normalized respond to active forces) and explore the effects of Γ and perimeter Γ = L0/(2πR0) − 1 (L0 is the perimeter) with C.