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.

Mammalian cells developed two main migration modes. The slow mesenchymatous mode, like crawling of fibroblasts, relies on maturation of adhesion complexes and actin fibre traction, while the fast amoeboid mode, observed exclusively for leukocytes and cancer cells, is characterized by weak adhesion, highly dynamic cell shapes, and ubiquitous motility on 2D and in 3D 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 fuelled 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 non-adherent confined environments. Resultantly, the ubiquitous ability of mammalian amoeboid cells to migrate in 2D or 3D, and with or without adhesion, can be explained for lymphocytes by a single machinery of heterogeneous membrane treadmilling. SIGNIFICANCE STATEMENT Leukocytes have a ubiquitous capacity to migrate on or in solid matrices, and with or without adhesion, which is instrumental to fight infections. The precise mechanisms sustaining migration remain however arguable. It is for instance widely accepted that leukocytes cannot crawl on 2D substrates without adhesion. In contrast, we showed that human lymphocytes swim on nonadherent 2D substrates and in suspension. Furthermore, our experiments and modelling suggest that propulsion rely hardly on cell body deformations and predominantly on molecular paddling by transmembrane proteins protruding outside the cell. For physics, this study reveals a new type of micro-swimmer, and for biology it suggests that leukocytes ubiquitous crawling may have evolved from an early machinery of swimming shared by various eukaryotic cells.

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Planar Cell Polarity (PCP) is an evolutionarily conserved characteristic of animal tissues marked by coordinated polarization of cells or structures in the plane of a tissue. In insect wing epithelium, for instance, PCP is characterized by en masse orientation of hairs orthogonal to its apical-basal axis and pointing along the proximal-distal axis of the organ. Directional cue for PCP has been proposed to be generated by complex sets of interactions amongst three proteins - Fat (Ft), Dachsous (Ds) and Four-jointed (Fj). Ft and Ds are two atypical cadherins, which are phosphorylated by Fj, a Golgi kinase. Ft and Ds from adjacent cells bind heterophilically via their tandem cadherin repeats, and their binding affinities are regulated by Fj. Further, in the wing epithelium, sub-cellular levels of Ft-Ds heterodimers are seen to be elevated at the distal edges of individual cells, prefiguring their PCP. Mechanisms generating this sub-cellular asymmetry of Ft-Ds heterodimer in proximal and distal edges of cells, however, have not been resolved yet. Using a mathematical modeling approach, here we provide a framework for generation of this sub-cellular asymmetry of Ft-Ds heterodimer. First, we explain how the known interactions within Ft-Ds-Fj system translate into sub-cellular asymmetry of Ft-Ds heterodimer. Second, we show that this asymmetric localization of Ft-Ds heterodimer is lost when tissue-level gradient of Fj is flattened, or when phosphorylation of Ft by Fj is abolished, but not when tissue-level gradient of Ds is flattened or when phosphorylation of Ds is abrogated. Finally, we show that distal enrichment of Ds also amplifies Ft-Ds asymmetry. These observations reveal that gradient of Fj expression, phosphorylation of Ft by Fj and sub-cellular distal accumulation of Ds are three critical elements required for generating sub-cellular asymmetry of Ft-Ds heterodimer. Our model integrates the known experimental data and presents testable predictions for future studies.

Epithelial morphogenesis is marked by dynamic changes in cell sizes until the final adult organ is formed. Using a combination of quantitative, mathematical simulation and genetic techniques, this paper reveals a novel mechanism of epithelial cell size regulation based on heterophilic cell–cell adhesion between two atypical cadherins, Ft and Ds.

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Planar Cell Polarity (PCP), characterized by asymmetric localization of proteins at the cell membrane within the epithelial plane, plays essential roles in embryonic development and physiological functions. The significance of PCP can be appreciated by the outcomes of its failure in the form of defects in neural tube formation, tracheal malfunctions, organ shape misregulation, hair follicle misalignment etc. Extensive experimental works on PCP in fruit fly Drosophila melanogaster have classified the proteins involved in PCP into a ‘core’ module, acting locally by inter-cellular protein interactions, and a ‘global’ module, responsible for the alignment of cell polarities with that of the tissue axis. Despite the involvement of different molecular players the asymmetric localization of the proteins of the two modules on cell membrane primarily involves inter-cellular dimer formations. We have developed a continuum model of the localization of PCP proteins on the cell membrane and its regulation via intra- and inter-cellular protein-protein interactions. We have identified the conditions for the asymmetric protein localization, or PCP establishment, for uniform and graded protein expression levels in the tissue. We have found that in the absence of any tissue level expression gradient polarized state of the tissue does not arise. However, in the presence of tissue-level expression gradients of proteins the polarized state remains stable. We have also looked at the influence of the loss of PCP proteins from a select region of the tissue on the polarization of the cells outside of that region. This continuum theory of planar cell polarity can be coupled with active-matter hydrodynamics to study cell flows and their regulation by genetic machinery.