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.

Heterogeneous ATP patterns in microvascular networks.

ATP is not only an energy carrier but also serves as an important signalling molecule in many physiological processes. Abnormal ATP level in blood vessel is known to be related to several pathologies, such as inflammation, hypoxia and atherosclerosis. Using advanced numerical methods, we analysed ATP released by red blood cells (RBCs) and its degradation by endothelial cells (ECs) in a cat mesentery-inspired vascular network, accounting for RBC mutual interaction and interactions with vascular walls. Our analysis revealed a heterogeneous ATP distribution in the network, with higher concentrations in the cell-free layer, concentration peaks around bifurcations and heterogeneity among vessels of the same level. These patterns arise from the spatio-temporal organization of RBCs induced by the network geometry. It is further shown that an alteration of hematocrit and flow strength significantly affects ATP level as well as heterogeneity in the network. These findings constitute a first building block to elucidate the intricate nature of ATP patterns in vascular networks and the far reaching consequences for other biochemical signalling, such as calcium, by ECs.

Anomalous Diffusion of Deformable Particles in a Honeycomb Network.

Transport of deformable particles in a honeycomb network is studied numerically. It is shown that the particle deformability has a strong impact on their distribution in the network. For sufficiently soft particles, we observe a short memory behavior from one bifurcation to the next, and the overall behavior consists in a random partition of particles, exhibiting a diffusionlike transport. On the contrary, stiff enough particles undergo a biased distribution whereby they follow a deterministic partition at bifurcations, due to long memory. This leads to a lateral ballistic drift in the network at small concentration and anomalous superdiffusion at larger concentration, even though the network is ordered. A further increase of concentration enhances particle-particle interactions which shorten the memory effect, turning the particle anomalous diffusion into a classical diffusion. We expect the drifting and diffusive regime transition to be generic for deformable particles.

Mathematical modeling of intracellular calcium in presence of receptor: a homeostatic model for endothelial cell.

Calcium is a ubiquitous molecule and second messenger that regulates many cellular functions ranging from exocytosis to cell proliferation at different time scales. In the vasculature, a constant adenosine triphosphate (ATP) concentration is maintained because of ATP released by red blood cells (RBCs). These ATP molecules continuously react with purinergic receptors on the surface of endothelial cells (ECs). Consequently, a cascade of chemical reactions are triggered that result in a transient cytoplasmic calcium (Ca[Formula: see text]), followed by return to its basal concentration. The mathematical models proposed in the literature are able to reproduce the transient peak. However, the trailing concentration is always higher than the basal cytoplasmic Ca[Formula: see text] concentrations, and the Ca[Formula: see text] concentration in endoplasmic reticulum (ER) remains lower than its initial concentration. This means that the intracellular homeostasis is not recovered. We propose, herein, a minimal model of calcium kinetics. We find that the desensitization of EC surface receptors due to phosphorylation and recycling plays a vital role in maintaining calcium homeostasis in the presence of a constant stimulus (ATP). The model is able to capture several experimental observations such as refilling of Ca[Formula: see text] in the ER, variation of cytoplasmic Ca[Formula: see text] transient peak in ECs, the resting cytoplasmic Ca[Formula: see text] concentration, the effect of removing ATP from the plasma on Ca[Formula: see text] homeostasis, and the saturation of cytoplasmic Ca[Formula: see text] transient peak with increase in ATP concentration. Direct confrontation with several experimental results is conducted. This work paves the way for systematic studies on coupling between blood flow and chemical signaling, and should contribute to a better understanding of the relation between (patho)physiological conditions and Ca[Formula: see text] kinetics.

Human endothelial cells in high glucose: New clues from culture in 3D microfluidic chips.

Several studies have demonstrated the role of high glucose in promoting endothelial dysfunction utilizing traditional two-dimensional (2D) culture systems, which, however, do not replicate the complex organization of the endothelium within a vessel constantly exposed to flow. Here we describe the response to high glucose of micro- and macro-vascular human endothelial cells (EC) cultured in biomimetic microchannels fabricated through soft lithography and perfused to generate shear stress. In 3D macrovascular EC exposed to a shear stress of 0.4 Pa respond to high glucose with cytoskeletal remodeling and alterations in cell shape. Under the same experimental conditions, these effects are more pronounced in microvascular cells that show massive cytoskeletal disassembly and apoptosis after culture in high glucose. However, when exposed to a shear stress of 4 Pa, which is physiological in the microvasculature, human dermal microvascular endothelial cells (HDMEC) show alterations of the cytoskeleton but no apoptosis. This result emphasizes the sensitivity of HDMEC to different regimens of flow. No significant variations in the thickness of glycocalyx were detected in both human endothelial cells from the umbilical vein and HDMEC exposed to high glucose in 3D, whereas clear differences emerge between cells cultured in static 2D versus microfluidic channels. We conclude that culture in microfluidic microchannels unveils unique insights into endothelial dysfunction by high glucose.

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.

Hydrodynamic instability and flow reduction in polymer brush coated channels.

A polymer brush is a passive medium. At equilibrium the knowledge of its chemical composition and thickness is enough for a full system characterization. However, when the brush is exposed to fluid flow it reveals a much more intriguing nature, in which filamentous protrusions and the way they interact among themselves and with the surrounding fluid are of outmost importance. Here we investigate such a rich behavior via numerical simulations. We focus on the brush hydrodynamic response at low Reynolds numbers, observing a significant fluid flow reduction inside a polymer-brush coated channel. We find that the reduction of the flow inside the channel is significantly larger than what would happen if the brush effect consisted only in reducing the effective channel width. This amplified reduction is understood as being due to the morphological instability of the brush-liquid interface which is shown to have an elastic origin: the mechanical stress acting on the brush due to the imposed flow is partially released by the interface modulation. In turn, this modulation dissipates more energy than a flat interface in the surrounding fluid, causing a reduction of flow velocity. Our results and interpretations provide an explanation for recent experimental measurements.

Red blood cells under flow show maximal ATP release for specific hematocrit.

ATP release by red blood cells (RBCs) under shear stress (SS) plays a pivotal role in endothelial biochemical signaling cascades. The aim of this study is to investigate through numerical simulation how RBC spatiotemporal organization depends on flow and geometrical conditions to generate ATP patterns. Numerical simulations were conducted in a straight channel by considering both plasma and explicit presence of RBCs, their shape deformation and cell-cell interaction, and ATP release by RBCs. Two ATP release pathways through cell membrane are taken into account: pannexin 1 channel, sensitive to SS, and cystic fibrosis transmembrane conductance regulator, which responds to cell deformation. Several flow and hematocrit conditions are explored. The problem is solved by the lattice Boltzmann method. Application of SS to the RBC suspension triggers a nontrivial spatial RBC organization and ATP patterns. ATP localizes preferentially in the vicinity of the cell-free layer close to channel wall. Conditions for maximal ATP release per cell are identified, which depend on vessel size and hematocrit Ht. Increasing further Ht beyond optimum enhances the total ATP release but should degrade oxygen transport capacity, a compromise between an efficient ATP release and minimal blood dissipation. Moreover, ATP is boosted in capillaries, suggesting a vasomotor activity coordination throughout the resistance network.

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.

The effect of shear stress reduction on endothelial cells: A microfluidic study of the actin cytoskeleton.

Reduced blood flow, as occurring in ischemia or resulting from exposure to microgravity such as encountered in space flights, induces a decrease in the level of shear stress sensed by endothelial cells forming the inner part of blood vessels. In the present study, we use a microvasculature-on-a-chip device in order to investigate in vitro the effect of such a reduction in shear stress on shear-adapted endothelial cells. We find that, within 1 h of exposition to reduced wall shear stress, human umbilical vein endothelial cells undergo reorganization of their actin skeleton with a decrease in the number of stress fibers and actin being recruited into the cells' peripheral band, indicating a fairly fast change in the cells' phenotype due to altered flow.

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.

Chaotic Swimming of Phoretic Particles.

The swimming of a rigid phoretic particle in an isotropic fluid is studied numerically as a function of the dimensionless solute emission rate (or Péclet number Pe). The particle sets into motion at a critical Pe. Whereas the particle trajectory is straight at a small enough Pe, it is found that it loses its stability at a critical Pe in favor of a meandering motion. When Pe is increased further, the particle meanders at a short scale but its trajectory wraps into a circle at a larger scale. Increasing even further, Pe causes the swimmer to escape momentarily the circular trajectory in favor of chaotic motion, which lasts for a certain time, before regaining a circular trajectory, and so on. The chaotic bursts become more and more frequent as Pe increases, until the trajectory becomes fully chaotic, via the intermittency scenario. The statistics of the trajectory is found to be of the run-and-tumble-like nature at a short enough time and of diffusive nature at a long time without any source of noise.

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.

Paroxysmal Permeability Disorders: Development of a Microfluidic Device to Assess Endothelial Barrier Function.

Background: Paroxysmal Permeability Disorders (PPDs) are pathological conditions caused by periodic short lasting increase of endothelial permeability, in the absence of inflammatory, degenerative, ischemic vascular injury. PPDs include primary angioedema, idiopathic systemic capillary leak syndrome and some rare forms of localized retroperitoneal-mediastinal edema. Aim: to validate a microfluidic device to study endothelial permeability in flow conditions. Materials and Methods: we designed a microchannel network (the smallest channel is 30µm square section). Human Umbilical Vein Endothelial Cells (HUVECs) were cultured under constant shear stress in the networks. Endothelial permeability assessment was based on interaction of biotinylated fibronectin used as a matrix for HUVECs and FITC-conjugated avidin. The increase in endothelial permeability was identified as changes in fluorescence intensity detected by confocal fluorescent microscopy. Results: The microchannels were constantly perfused with a steady flow of culture medium, ensuring a physiologically relevant level of shear stress at the wall of ~0.2 Pa. Our preliminary results demonstrated that circulation of culture medium or plasma from healthy volunteers was associated with low fluorescence of fibronectin matrix. When bradykinin diluted in culture medium was perfused, an increase in average fluorescence was detected. Conclusion: Our microvasculature model is suitable to study endothelial functions in physiological flow conditions and in the presence of factors like bradykinin known as mediator of several PPDs. Therefore, it can be a promising tool to better understand the mechanisms underlying disorders of endothelial permeability.

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.

Shear stress in the microvasculature: influence of red blood cell morphology and endothelial wall undulation.

The effect of red blood cells and the undulation of the endothelium on the shear stress in the microvasculature is studied numerically using the lattice Boltzmann-immersed boundary method. The results demonstrate a significant effect of both the undulation of the endothelium and red blood cells on wall shear stress. Our results also reveal that morphological alterations of red blood cells, as occur in certain pathologies, can significantly affect the values of wall shear stress. The resulting fluctuations in wall shear stress greatly exceed the nominal values, emphasizing the importance of the particulate nature of blood as well as a more realistic description of vessel wall geometry in the study of hemodynamic forces. We find that within the channel widths investigated, which correspond to those found in the microvasculature, the inverse minimum distance normalized to the channel width between the red blood cell and the wall is predictive of the maximum wall shear stress observed in straight channels with a flowing red blood cell. We find that the maximum wall shear stress varies several factors more over a range of capillary numbers (dimensionless number relating strength of flow to membrane elasticity) and reduced areas (measure of deflation of the red blood cell) than the minimum wall shear stress. We see that waviness reduces variation in minimum and maximum shear stresses among different capillary and reduced areas.

Multi-wavelength photo-acoustic microscopy in the frequency domain for simultaneous excitation and detection of dyes.

An optical-resolution photoacoustic microscope with modulated CW laser diodes allowing multi-channel imaging is presented that can be used for both imaging biological tissues and for targeted photo-dynamic therapy (PDT) varying the optical power and exposure time. The effects of this therapy are immediately monitored in order to optimize the time of irradiation. After the description of the experimental setup, in vitro and in vivo applications are presented on a synthetic sample and on the mouse ear using hemoglobin as endogenous and methylene blue as exogenous dye for imaging and PDT, respectively.

ATP Release by Red Blood Cells under Flow: Model and Simulations.

ATP is a major player as a signaling molecule in blood microcirculation. It is released by red blood cells (RBCs) when they are subjected to shear stresses large enough to induce a sufficient shape deformation. This prominent feature of chemical response to shear stress and RBC deformation constitutes an important link between vessel geometry, flow conditions, and the mechanical properties of RBCs, which are all contributing factors affecting the chemical signals in the process of vasomotor modulation of the precapillary vessel networks. Several in vitro experiments have reported on ATP release by RBCs due to mechanical stress. These studies have considered both intact RBCs as well as cells within which suspected pathways of ATP release have been inhibited. This has provided profound insights to help elucidate the basic governing key elements, yet how the ATP release process takes place in the (intermediate) microcirculation zone is not well understood. We propose here an analytical model of ATP release. The ATP concentration is coupled in a consistent way to RBC dynamics. The release of ATP, or the lack thereof, is assumed to depend on both the local shear stress and the shape change of the membrane. The full chemo-mechanical coupling problem is written in a lattice-Boltzmann formulation and solved numerically in different geometries (straight channels and bifurcations mimicking vessel networks) and under two kinds of imposed flows (shear and Poiseuille flows). Our model remarkably reproduces existing experimental results. It also pinpoints the major contribution of ATP release when cells traverse network bifurcations. This study may aid in further identifying the interplay between mechanical properties and chemical signaling processes involved in blood microcirculation.

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.