Controlling Confined Collective Organization with Taxis.

Biased locomotion is a common feature of microorganisms, but little is known about its impact on self-organization. Inspired by recent experiments showing a transition to large-scale flows, we study theoretically the dynamics of magnetotactic bacteria confined to a drop. We reveal two symmetry-breaking mechanisms (one local chiral and one global achiral) leading to self-organization into global vortices and a net torque exerted on the drop. The collective behavior is ultimately controlled by the swimmers' microscopic chirality and, strikingly, the system can exhibit oscillations and memorylike features.

Opto-fluidically multiplexed assembly and micro-robotics.

Techniques for high-definition micromanipulations, such as optical tweezers, hold substantial interest across a wide range of disciplines. However, their applicability remains constrained by material properties and laser exposure. And while microfluidic manipulations have been suggested as an alternative, their inherent capabilities are limited and further hindered by practical challenges of implementation and control. Here we show that the iterative application of laser-induced, localized flow fields can be used for the relative positioning of multiple micro-particles, irrespectively of their material properties. Compared to the standing theoretical proposal, our method keeps particles mobile, and we show that their precision manipulation is non-linearly accelerated via the multiplexing of temperature stimuli below the heat diffusion limit. The resulting flow fields are topologically rich and mathematically predictable. They represent unprecedented microfluidic control capabilities that are illustrated by the actuation of humanoid micro-robots with up to 30 degrees of freedom, whose motions are sufficiently well-defined to reliably communicate personal characteristics such as gender, happiness and nervousness. Our results constitute high-definition micro-fluidic manipulations with transformative potential for assembly, micro-manufacturing, the life sciences, robotics and opto-hydraulically actuated micro-factories.

Cortex-driven cytoplasmic flows in elongated cells: fluid mechanics and application to nuclear transport in Drosophila embryos.

The Drosophila melanogaster embryo, an elongated multi-nucleated cell, is a classical model system for eukaryotic development and morphogenesis. Recent work has shown that bulk cytoplasmic flows, driven by cortical contractions along the walls of the embryo, enable the uniform spreading of nuclei along the anterior-posterior axis necessary for proper embryonic development. Here, we propose two mathematical models to characterize cytoplasmic flows driven by tangential cortical contractions in elongated cells. Assuming Newtonian fluid flow at low Reynolds number in a spheroidal cell, we first compute the flow field exactly, thereby bypassing the need for numerical computations. We then apply our results to recent experiments on nuclear transport in cell cycles 4-6 of Drosophila embryo development. By fitting the cortical contractions in our model to measurements, we reveal that experimental cortical flows enable near-optimal axial spreading of nuclei. A second mathematical approach, applicable to general elongated cell geometries, exploits a long-wavelength approximation to produce an even simpler solution, with errors below [Formula: see text] compared with the full model. An application of this long-wavelength result to transport leads to fully analytical solutions for the nuclear concentration that capture the essential physics of the system, including optimal axial spreading of nuclei.

Multiflagellarity leads to the size-independent swimming speed of peritrichous bacteria.

To swim through a viscous fluid, a flagellated bacterium must overcome the fluid drag on its body by rotating a flagellum or a bundle of multiple flagella. Because the drag increases with the size of bacteria, it is expected theoretically that the swimming speed of a bacterium inversely correlates with its body length. Nevertheless, despite extensive research, the fundamental size-speed relation of flagellated bacteria remains unclear with different experiments reporting conflicting results. Here, by critically reviewing the existing evidence and synergizing our own experiments of large sample sizes, hydrodynamic modeling, and simulations, we demonstrate that the average swimming speed of Escherichia coli, a premier model of peritrichous bacteria, is independent of their body length. Our quantitative analysis shows that such a counterintuitive relation is the consequence of the collective flagellar dynamics dictated by the linear correlation between the body length and the number of flagella of bacteria. Notably, our study reveals how bacteria utilize the increasing number of flagella to regulate the flagellar motor load. The collective load sharing among multiple flagella results in a lower load on each flagellar motor and therefore faster flagellar rotation, which compensates for the higher fluid drag on the longer bodies of bacteria. Without this balancing mechanism, the swimming speed of monotrichous bacteria generically decreases with increasing body length, a feature limiting the size variation of the bacteria. Altogether, our study resolves a long-standing controversy over the size-speed relation of flagellated bacteria and provides insights into the functional benefit of multiflagellarity in bacteria.

Reduced surface accumulation of swimming bacteria in viscoelastic polymer fluids.

Surface-associated bacterial communities flourish in nature and in the body of animal hosts with abundant macromolecular polymers. It is unclear how the endowed viscoelasticity of polymeric fluids influences bacterial motile behavior in such environments. Here, we combined experiment and theory to study near-surface swimming of flagellated bacteria in viscoelastic polymer fluids. In contrast to the swimming behavior in Newtonian fluids, we discovered that cells swim in less curved trajectories and display reduced near-surface accumulation. Using a theoretical analysis of the non-Newtonian hydrodynamic forces, we demonstrated the existence of a generic lift force acting on a rotating filament near a rigid surface, which arises from the elastic tension generated along curved flow streamlines. This viscoelastic lift force weakens the hydrodynamic interaction between flagellated swimmers and solid surfaces and contributes to a decrease in surface accumulation. Our findings reveal previously unrecognized facets of bacterial transport and surface exploration in polymer-rich environments that are pertinent to diverse microbial processes and may inform the design of artificial microswimmers capable of navigating through complex geometries.

Microswimmers in vortices: dynamics and trapping.

Biological and artificial microswimmers often self-propel in external flows of vortical nature; relevant examples include algae in small-scale ocean eddies, spermatozoa in uterine peristaltic flows and bacteria in microfluidic devices. A recent experiment has shown that swimming bacteria in model vortices are expelled from the vortex all the way to a well-defined depletion zone (A. Sokolov and I. S. Aranson, Rapid expulsion of microswimmers by a vortical flow. Nat. Commun., 2016, 7, 11114). In this paper, we propose a theoretical model to investigate the dynamics of elongated microswimmers in elementary vortices, namely active particles in two- and three-dimensional rotlets. A deterministic model first reveals the existence of bounded orbits near the centre of the vortex and unbounded orbits elsewhere. We further discover a conserved quantity of motion that allows us to map the phase space according to the type of the orbit (bounded vs unbounded). We next introduce translational and rotational noise into the system. Using a Fokker-Planck formalism, we quantify the quality of trapping near the centre of the vortex by examining the probability of escape and the mean time of escape from the region of deterministically bounded orbits. We finally show how to use these findings to formulate a prediction for the radius of the depletion zone, which compares favourably with the experiments (A. Sokolov and I. S. Aranson, Rapid expulsion of microswimmers by a vortical flow. Nat. Commun., 2016, 7, 11114).

Elastohydrodynamic Synchronization of Rotating Bacterial Flagella.

To rotate continuously without jamming, the flagellar filaments of bacteria need to be locked in phase. While several models have been proposed for eukaryotic flagella, the synchronization of bacterial flagella is less well understood. Starting from a reduced model of flexible and hydrodynamically coupled bacterial flagella, we rigorously coarse grain the equations of motion using the method of multiple scales, and hence show that bacterial flagella generically synchronize to zero phase difference via an elastohydrodynamic mechanism. Remarkably, the far-field rate of synchronization is maximized at an intermediate value of elastic compliance, with surprising implications for bacteria.

Cilia metasurfaces for electronically programmable microfluidic manipulation.

Cilial pumping is a powerful strategy used by biological organisms to control and manipulate fluids at the microscale. However, despite numerous recent advances in optically, magnetically and electrically driven actuation, development of an engineered cilial platform with the potential for applications has remained difficult to realize1-6. Here we report on active metasurfaces of electronically actuated artificial cilia that can create arbitrary flow patterns in liquids near a surface. We first create voltage-actuated cilia that generate non-reciprocal motions to drive surface flows at tens of microns per second at actuation voltages of 1 volt. We then show that a cilia unit cell can locally create a range of elemental flow geometries. By combining these unit cells, we create an active cilia metasurface that can generate and switch between any desired surface flow pattern. Finally, we integrate the cilia with a light-powered complementary metal-oxide-semiconductor (CMOS) clock circuit to demonstrate wireless operation. As a proof of concept, we use this circuit to output voltage pulses with various phase delays to demonstrate improved pumping efficiency using metachronal waves. These powerful results, demonstrated experimentally and confirmed using theoretical computations, illustrate a pathway towards fine-scale microfluidic manipulation, with applications from microfluidic pumping to microrobotic locomotion.

Order and information in the patterns of spinning magnetic micro-disks at the air-water interface.

The application of the Shannon entropy to study the relationship between information and structures has yielded insights into molecular and material systems. However, the difficulty in directly observing and manipulating atoms and molecules hampers the ability of these systems to serve as model systems for further exploring the links between information and structures. Here, we use, as a model experimental system, hundreds of spinning magnetic micro-disks self-organizing at the air-water interface to generate various spatiotemporal patterns with varying degrees of order. Using the neighbor distance as the information-bearing variable, we demonstrate the links among information, structure, and interactions. We establish a direct link between information and structure without using explicit knowledge of interactions. Last, we show that the Shannon entropy by neighbor distances is a powerful observable in characterizing structural changes. Our findings are relevant for analyzing natural self-organizing systems and for designing collective robots.

Stabilizing viscous extensional flows using reinforcement learning.

The four-roll mill, wherein four identical cylinders undergo rotation of identical magnitude but alternate signs, was originally proposed by G. I. Taylor to create local extensional flows and study their ability to deform small liquid drops. Since an extensional flow has an unstable eigendirection, a drop located at the flow stagnation point will have a tendency to escape. This unstable dynamics can, however, be stabilized using, e.g., a modulation of the rotation rates of the cylinders. Here we use reinforcement learning, a branch of machine learning devoted to the optimal selection of actions based on cumulative rewards, in order to devise a stabilization algorithm for the four-roll mill flow. The flow is modelled as the linear superposition of four two-dimensional rotlets and the drop is treated as a rigid spherical particle smaller than all other length scales in the problem. Unlike previous attempts to devise control, we take a probabilistic approach whereby speed adjustments are drawn from a probability density function whose shape is improved over time via a form of gradient ascent know as actor-critic method. With enough training, our algorithm is able to precisely control the drop and keep it close to the stagnation point for as long as needed. We explore the impact of the physical and learning parameters on the effectiveness of the control and demonstrate the robustness of the algorithm against thermal noise. We finally show that reinforcement learning can provide a control algorithm effective for all initial positions and that can be adapted to limit the magnitude of the flow extension near the position of the drop.

Fluid Mechanics of Mosaic Ciliated Tissues.

In tissues as diverse as amphibian skin and the human airway, the cilia that propel fluid are grouped in sparsely distributed multiciliated cells (MCCs). We investigate fluid transport in this "mosaic" architecture, with emphasis on the trade-offs that may have been responsible for its evolutionary selection. Live imaging of MCCs in embryos of the frog Xenopus laevis shows that cilia bundles behave as active vortices that produce a flow field accurately represented by a local force applied to the fluid. A coarse-grained model that self-consistently couples bundles to the ambient flow reveals that hydrodynamic interactions between MCCs limit their rate of work so that they best shear the tissue at a finite but low area coverage, a result that mirrors findings for other sparse distributions such as cell receptors and leaf stomata.

A mechanism for sarcomere breathing: volume change and advective flow within the myofilament lattice.

During muscle contraction, myosin motors anchored to thick filaments bind to and slide actin thin filaments. These motors rely on energy derived from ATP, supplied, in part, by diffusion from the sarcoplasm to the interior of the lattice of actin and myosin filaments. The radial spacing of filaments in this lattice may change or remain constant during contraction. If the lattice is isovolumetric, it must expand when the muscle shortens. If, however, the spacing is constant or has a different pattern of axial and radial motion, then the lattice changes volume during contraction, driving fluid motion and assisting in the transport of molecules between the contractile lattice and the surrounding intracellular space. We first create an advective-diffusive-reaction flow model and show that the flow into and out of the sarcomere lattice would be significant in the absence of lattice expansion. Advective transport coupled to diffusion has the potential to substantially enhance metabolite exchange within the crowded sarcomere. Using time-resolved x-ray diffraction of contracting muscle, we next show that the contractile lattice is neither isovolumetric nor constant in spacing. Instead, lattice spacing is time varying, depends on activation, and can manifest as an effective time-varying Poisson ratio. The resulting fluid flow in the sarcomere lattice of synchronous insect flight muscles is even greater than expected for constant lattice spacing conditions. Lattice spacing depends on a variety of factors that produce radial force, including cross-bridges, titin-like molecules, and other structural proteins. Volume change and advective transport varies with the phase of muscle stimulation during periodic contraction but remains significant at all conditions. Although varying in magnitude, advective transport will occur in all cases in which the sarcomere is not isovolumetric. Akin to "breathing," advective-diffusive transport in sarcomeres is sufficient to promote metabolite exchange and may play a role in the regulation of contraction itself.

Hydrodynamics and direction change of tumbling bacteria.

The bacterium Escherichia coli (E. coli) swims in viscous fluids by rotating several helical flagellar filaments, which are gathered in a bundle behind the cell during 'runs' wherein the cell moves steadily forward. In between runs, the cell undergoes quick 'tumble' events, during which at least one flagellum reverses its rotation direction and separates from the bundle, resulting in erratic motion in place and a random reorientation of the cell. Alternating between runs and tumbles allows cells to sample space by stochastically changing their propulsion direction after each tumble. The change of direction during a tumble is not uniformly distributed but is skewed towards smaller angles with an average of about 62°-68°, as first measured by Berg and Brown (1972). Here we develop a theoretical approach to model the angular distribution of swimming E. coli cells during tumbles. We first use past experimental imaging results to construct a kinematic description of the dynamics of the flagellar filaments during a tumble. We then employ low-Reynolds number hydrodynamics to compute the consequences of the kinematic model on the force and torque balance of the cell and to deduce the overall change in orientation. The results of our model are in good agreement with experimental observations. We find that the main change of direction occurs during the 'bundling' part of the process wherein, at the end of a tumble, the dispersed flagellar filaments are brought back together in the helical bundle, which we confirm using a simplified forced-sphere model.

The bank of swimming organisms at the micron scale (BOSO-Micro).

Unicellular microscopic organisms living in aqueous environments outnumber all other creatures on Earth. A large proportion of them are able to self-propel in fluids with a vast diversity of swimming gaits and motility patterns. In this paper we present a biophysical survey of the available experimental data produced to date on the characteristics of motile behaviour in unicellular microswimmers. We assemble from the available literature empirical data on the motility of four broad categories of organisms: bacteria (and archaea), flagellated eukaryotes, spermatozoa and ciliates. Whenever possible, we gather the following biological, morphological, kinematic and dynamical parameters: species, geometry and size of the organisms, swimming speeds, actuation frequencies, actuation amplitudes, number of flagella and properties of the surrounding fluid. We then organise the data using the established fluid mechanics principles for propulsion at low Reynolds number. Specifically, we use theoretical biophysical models for the locomotion of cells within the same taxonomic groups of organisms as a means of rationalising the raw material we have assembled, while demonstrating the variability for organisms of different species within the same group. The material gathered in our work is an attempt to summarise the available experimental data in the field, providing a convenient and practical reference point for future studies.

Energetics of synchronization for model flagella and cilia.

Synchronization is often observed in the swimming of flagellated cells, either for multiple appendages on the same organism or between the flagella of nearby cells. Beating cilia are also seen to synchronize their dynamics. In 1951, Taylor showed that the observed in-phase beating of the flagella of coswimming spermatozoa was consistent with minimization of the energy dissipated in the surrounding fluid. Here we revisit Taylor's hypothesis for three models of flagella and cilia: (1) Taylor's waving sheets with both longitudinal and transverse modes, as relevant for flexible flagella, (2) spheres orbiting above a no-slip surface to model interacting flexible cilia, and (3) whirling rods above a no-slip surface to address the interaction of nodal cilia. By calculating the flow fields explicitly, we show that the rate of working of the model flagella or cilia is minimized in our three models for (1) a phase difference depending on the separation of the sheets and precise waving kinematics, (2) in-phase or opposite-phase motion depending on the relative position and orientation of the spheres, and (3) in-phase whirling of the rods. These results will be useful in future models probing the dynamics of synchronization in these setups.

Fluid flow in the sarcomere.

A highly organized and densely packed lattice of molecular machinery within the sarcomeres of muscle cells powers contraction. Although many of the proteins that drive contraction have been studied extensively, the mechanical impact of fluid shearing within the lattice of molecular machinery has received minimal attention. It was recently proposed that fluid flow augments substrate transport in the sarcomere, however, this analysis used analytical models of fluid flow in the molecular machinery that could not capture its full complexity. By building a finite element model of the sarcomere, we estimate the explicit flow field, and contrast it with analytical models. Our results demonstrate that viscous drag forces on sliding filaments are surprisingly small in contrast to the forces generated by single myosin molecular motors. This model also indicates that the energetic cost of fluid flow through viscous shearing with lattice proteins is likely minimal. The model also highlights a steep velocity gradient between sliding filaments and demonstrates that the maximal radial fluid velocity occurs near the tips of the filaments. To our knowledge, this is the first computational analysis of fluid flow within the highly structured sarcomere.

Rebound and scattering of motile Chlamydomonas algae in confined chambers.

Motivated by recent experiments demonstrating that motile algae get trapped in draining foams, we study the trajectories of microorganisms confined in model foam channels (section of a Plateau border). We track single Chlamydomonas reinhardtii cells confined in a thin three-circle microfluidic chamber and show that their spatial distribution exhibits strong corner accumulation. Using empirical scattering laws observed in previous experiments (scattering with a constant scattering angle), we next develop a two-dimension geometrical model and compute the phase space of trapped and periodic trajectories of swimmers inside a three-circles billiard. We find that the majority of cell trajectories end up in a corner, providing a geometrical mechanism for corner accumulation. Incorporating the distribution of scattering angles observed in our experiments and including hydrodynamic interactions between the cells and the surfaces into the geometrical model enables us to reproduce the experimental probability density function of micro-swimmers in microfluidic chambers. Both our experiments and models demonstrate therefore that motility leads generically to trapping in complex geometries.

Hydrodynamic synchronization in strong confinement.

Cellular appendages conferring motility, such as flagella and cilia, are known to synchronise their periodic beats. The origin of synchronization is a combination of long-range hydrodynamic interactions with physical mechanisms allowing the phases of these biological oscillators to evolve. Two of such mechanisms have been identified by previous work, the elastic compliance of the periodic orbit or oscillations driven by phase-dependent biological forcing, both of which can lead generically to stable phase locking. In order to help uncover the physical mechanism for hydrodynamic synchronization most essential overall in biology, we theoretically investigate in this paper the effect of strong confinement on the effectiveness of hydrodynamic synchronization. Following past work, we use minimal models of cilia where appendages are modeled as rigid spheres forced to move along circular trajectories near a rigid surface. Strong confinement is modeled by adding a second nearby surface, parallel to the first one, where the distance between the surfaces is much smaller than the typical distance between the cilia, which results in a qualitative change in the nature of hydrodynamic interactions. We calculate separately the impact of hydrodynamic confinement on the synchronization dynamics of the elastic compliance and the force modulation mechanisms and compare our results to the usual case with a single surface. Applying our results to the biologically relevant situation of nodal cilia, we show that force modulation is a mechanism that leads to phase-locked states under strong confinement that are very similar to those without confinement as a difference with the elastic compliance mechanism. Our results point therefore to the robustness of force modulation for synchronization, an important feature for biological dynamics that therefore suggests it could be the most essential physical mechanism overall in arrays of nodal cilia. We further examine the distinct biologically relevant situation of primary cilia and show in that case that the difference in robustness of the mechanisms is not as pronounced but still favors the force modulation.

Swirling Instability of the Microtubule Cytoskeleton.

In the cellular phenomena of cytoplasmic streaming, molecular motors carrying cargo along a network of microtubules entrain the surrounding fluid. The piconewton forces produced by individual motors are sufficient to deform long microtubules, as are the collective fluid flows generated by many moving motors. Studies of streaming during oocyte development in the fruit fly Drosophila melanogaster have shown a transition from a spatially disordered cytoskeleton, supporting flows with only short-ranged correlations, to an ordered state with a cell-spanning vortical flow. To test the hypothesis that this transition is driven by fluid-structure interactions, we study a discrete-filament model and a coarse-grained continuum theory for motors moving on a deformable cytoskeleton, both of which are shown to exhibit a swirling instability to spontaneous large-scale rotational motion, as observed.

Cilia density and flow velocity affect alignment of motile cilia from brain cells.

In many organs, thousands of microscopic 'motile cilia' beat in a coordinated fashion generating fluid flow. Physiologically, these flows are important in both development and homeostasis of ciliated tissues. Combining experiments and simulations, we studied how cilia from brain tissue align their beating direction. We subjected cilia to a broad range of shear stresses, similar to the fluid flow that cilia themselves generate, in a microfluidic setup. In contrast to previous studies, we found that cilia from mouse ependyma respond and align to these physiological shear stress at all maturation stages. Cilia align more easily earlier in maturation, and we correlated this property with the increase in multiciliated cell density during maturation. Our numerical simulations show that cilia in densely packed clusters are hydrodynamically screened from the external flow, in agreement with our experimental observation. Cilia carpets create a hydrodynamic screening that reduces the susceptibility of individual cilia to external flows.