Nonreciprocal interactions give rise to fast cilium synchronization in finite systems.

Motile cilia beat in an asymmetric fashion in order to propel the surrounding fluid. When many cilia are located on a surface, their beating can synchronize such that their phases form metachronal waves. Here, we computationally study a model where each cilium is represented as a spherical particle, moving along a tilted trajectory with a position-dependent active driving force and a position-dependent internal drag coefficient. The model thus takes into account all the essential broken symmetries of the ciliary beat. We show that taking into account the near-field hydrodynamic interactions, the effective coupling between cilia even over an entire beating cycle can become nonreciprocal: The phase of a cilium is more strongly affected by an adjacent cilium on one side than by a cilium at the same distance in the opposite direction. As a result, synchronization starts from a seed at the edge of a group of cilia and propagates rapidly across the system, leading to a synchronization time that scales proportionally to the linear dimension of the system. We show that a ciliary carpet is characterized by three different velocities: the velocity of fluid transport, the phase velocity of metachronal waves, and the group velocity of order propagation. Unlike in systems with reciprocal coupling, boundary effects are not detrimental for synchronization, but rather enable the formation of the initial seed.

Theoretical efficiency limits and speed-efficiency trade-off in myosin motors.

Muscle myosin is a non-processive molecular motor that generates mechanical work when cooperating in large ensembles. During its cyle, each individual motor keeps attaching and detaching from the actin filament. The random nature of attachment and detachment inevitably leads to losses and imposes theoretical limits on the energetic efficiency. Here, we numerically determine the theoretical efficiency limit of a classical myosin model with a given number of mechano-chemical states. All parameters that are not bounded by physical limits (like rate limiting steps) are determined by numerical efficiency optimization. We show that the efficiency is limited by the number of states, the stiffness and the rate-limiting kinetic steps. There is a trade-off between speed and efficiency. Slow motors are optimal when most of the available free energy is allocated to the working stroke and the stiffness of their elastic element is high. Fast motors, on the other hand, work better with a lower and asymmetric stiffness and allocate a larger fraction of free energy to the release of ADP. Overall, many features found in myosins coincide with the findings from the model optimization: there are at least 3 bound states, the largest part of the working stroke takes place during the first transition, the ADP affinity is adapted differently in slow and fast myosins and there is an asymmetry in elastic elements.

Spontaneous Chiral Symmetry Breaking and Lane Formation in Ferromagnetic Ferrofluids.

Ferromagnetic ferrofluids are synthetic materials consisting of magnetic nanoplatelets dispersed in an isotropic fluid. Their main characteristics are the formation of stable magnetic domains and the presence of macroscopic magnetization even in the absence of a magnetic field. Here, the authors report on the experimental observation of spontaneous stripe formation in a ferromagnetic ferrofluid in the presence of an oscillating external magnetic field. The striped structure is identified as elongated magnetic domains, which exhibit reorientation upon reversal of the magnetic field. The stripes are oriented perpendicular to the magnetic field and are separated by alternating flow lanes. The velocity profile is measured using a space-time correlation technique that follows the motion of the thermally excited fluctuations in the sample. The highest velocities are found in the depleted regions between individual domains and reach values up to several µm s-1 . The fluid in adjacent lanes moves in the opposite directions despite the applied magnetic field being uniform. The formation of bidirectional flow lanes can be explained by alternating rotation of magnetic nanoparticles in neighboring stripes, which indicates spontaneous breaking of the chiral symmetry in the sample.

Lorentz Reciprocal Theorem in Fluids with Odd Viscosity.

The Lorentz reciprocal theorem-that is used to study various transport phenomena in hydrodynamics-is violated in chiral active fluids that feature odd viscosity with broken time-reversal and parity symmetries. Here, we show that the theorem can be generalized to fluids with odd viscosity by choosing an auxiliary problem with the opposite sign of the odd viscosity. We demonstrate the application of the theorem to two categories of microswimmers. Swimmers with prescribed surface velocity are not affected by odd viscosity, while those with prescribed active forces are. In particular, a torque dipole can lead to directed motion.

A Synthetic Minimal Beating Axoneme.

Cilia and flagella are beating rod-like organelles that enable the directional movement of microorganisms in fluids and fluid transport along the surface of biological organisms or inside organs. The molecular motor axonemal dynein drives their beating by interacting with microtubules. Constructing synthetic beating systems with axonemal dynein capable of mimicking ciliary beating still represents a major challenge. Here, the bottom-up engineering of a sustained beating synthoneme consisting of a pair of microtubules connected by a series of periodic arrays of approximately eight axonemal dyneins is reported. A model leads to the understanding of the motion through the cooperative, cyclic association-dissociation of the molecular motor from the microtubules. The synthoneme represents a bottom-up self-organized bio-molecular machine at the nanoscale with cilia-like properties.

The cilium as a force sensor-myth versus reality.

Cells need to sense their mechanical environment during the growth of developing tissues and maintenance of adult tissues. The concept of force-sensing mechanisms that act through cell-cell and cell-matrix adhesions is now well established and accepted. Additionally, it is widely believed that force sensing can be mediated through cilia. Yet, this hypothesis is still debated. By using primary cilia sensing as a paradigm, we describe the physical requirements for cilium-mediated mechanical sensing and discuss the different hypotheses of how this could work. We review the different mechanosensitive channels within the cilium, their potential mode of action and their biological implications. In addition, we describe the biological contexts in which cilia are acting - in particular, the left-right organizer - and discuss the challenges to discriminate between cilium-mediated chemosensitivity and mechanosensitivity. Throughout, we provide perspectives on how quantitative analysis and physics-based arguments might help to better understand the biological mechanisms by which cells use cilia to probe their mechanical environment.

Minimum Dissipation Theorem for Microswimmers.

We derive a theorem for the lower bound on the energy dissipation rate by a rigid surface-driven active microswimmer of arbitrary shape in a fluid at a low Reynolds number. We show that, for any swimmer, the minimum dissipation at a given velocity can be expressed in terms of the resistance tensors of two passive bodies of the same shape with a no-slip and perfect-slip boundary. To achieve the absolute minimum dissipation, the optimal swimmer needs a surface velocity profile that corresponds to the flow around the perfect-slip body, and a propulsive force density that corresponds to the no-slip body. Using this theorem, we propose an alternative definition of the energetic efficiency of microswimmers that, unlike the commonly used Lighthill efficiency, can never exceed unity. We validate the theory by calculating the efficiency limits of spheroidal swimmers.

Tuning the Properties of Active Microtubule Networks by Depletion Forces.

Suspensions of microtubules and nonadsorbing particles form thick and long bundles due to depletion forces. Such interactions act at the nanometer scale and define the structural and dynamical properties of the resulting networks. In this study, we analyze the depletion forces exerted by two types of nonadsorbing particles, namely, the polymer, poly(ethylene glycol) (PEG), and the block copolymer, Pluronic. We characterize their effects both in passive and active networks by adding motor proteins to the suspensions. By exploiting its bundling effect via entropic forces, we observed that PEG generates a network with thick structures showing a nematic order and larger mesh size. On the other hand, Pluronic builds up a much denser gel-like network without a recognizable mesh structure. This difference is also reflected in the network activity. PEG networks show moderate contraction in lateral directions while Pluronic networks exhibit faster and isotropic contraction. Interestingly, by mixing the two nonadsorbing polymers in different ratios, we observed that the system showed a behavior that exhibited properties of both agents, leading to a robust and fast responsive structure compared to the single-depletant networks. In conclusion, we show how passive osmotic compression modifies the distribution of biopolymers. Its combination with active motors results in a new active material with potential for nanotechnological applications.

Wrinkling Instability in 3D Active Nematics.

In nature, interactions between biopolymers and motor proteins give rise to biologically essential emergent behaviors. Besides cytoskeleton mechanics, active nematics arise from such interactions. Here we present a study on 3D active nematics made of microtubules, kinesin motors, and depleting agent. It shows a rich behavior evolving from a nematically ordered space-filling distribution of microtubule bundles toward a flattened and contracted 2D ribbon that undergoes a wrinkling instability and subsequently transitions into a 3D active turbulent state. The wrinkle wavelength is independent of the ATP concentration and our theoretical model describes its relation with the appearance time. We compare the experimental results with a numerical simulation that confirms the key role of kinesin motors in cross-linking and sliding the microtubules. Our results on the active contraction of the network and the independence of wrinkle wavelength on ATP concentration are important steps forward for the understanding of these 3D systems.

Flagella-like Beating of a Single Microtubule.

Kinesin motors can induce a buckling instability in a microtubule with a fixed minus end. Here we show that by modifying the surface with a protein-repellent functionalization and using clusters of kinesin motors, the microtubule can exhibit persistent oscillatory motion resembling the beating of sperm flagella. The observed period is of the order of 1 min. From the experimental images we theoretically determine a distribution of motor forces that explains the observed shapes using a maximum likelihood approach. A good agreement is achieved with a small number of motor clusters acting simultaneously on a microtubule. The tangential forces exerted by a cluster are mostly in the range 0-8 pN toward the microtubule minus end, indicating the action of 1 or 2 kinesin motors. The lateral forces are distributed symmetrically and mainly below 10 pN, while the lateral velocity has a strong peak around zero. Unlike well-known models for flapping filaments, kinesins are found to have a strong "pinning" effect on the beating filaments. Our results suggest new strategies to utilize molecular motors in dynamic roles that depend sensitively on the stress built-up in the system.

Diffusive ferromagnetic roller gas.

An ensemble of actively rotating ferromagnetic particles is used to realize an active roller gas. Here, we investigate the diffusive properties of such a gas in experiments and simulations. We reveal that ferromagnetic rollers demonstrate a normal (Fickian) diffusion with a characteristic linear growth of the mean-squared displacement, while statistics of displacements stay non-Gaussian. At short times the system has a bimodal distribution of the displacements that transitions with time to a quasi-Gaussian distribution (Gaussian core with overpopulated tails) for a range of studied particle number densities. Inert particles introduced into the active roller gas exhibit similar diffusive behavior. The results provide insights into diffusive properties of active colloidal systems with activity originating from spinning degrees of freedom.

Working stroke of the kinesin-14, ncd, comprises two substeps of different direction.

Single-molecule experiments have been used with great success to explore the mechanochemical cycles of processive motor proteins such as kinesin-1, but it has proven difficult to apply these approaches to nonprocessive motors. Therefore, the mechanochemical cycle of kinesin-14 (ncd) is still under debate. Here, we use the readout from the collective activity of multiple motors to derive information about the mechanochemical cycle of individual ncd motors. In gliding motility assays we performed 3D imaging based on fluorescence interference contrast microscopy combined with nanometer tracking to simultaneously study the translation and rotation of microtubules. Microtubules gliding on ncd-coated surfaces rotated around their longitudinal axes in an [ATP]- and [ADP]-dependent manner. Combined with a simple mechanical model, these observations suggest that the working stroke of ncd consists of an initial small movement of its stalk in a lateral direction when ADP is released and a second, main component of the working stroke, in a longitudinal direction upon ATP binding.

Magnetically driven omnidirectional artificial microswimmers.

We present an experimental realisation of two new artificial microswimmers that swim at low Reynolds number. The swimmers are externally driven with a periodically modulated magnetic field that induces an alternating attractive/repulsive interaction between the swimmer parts. The field sequence also modulates the drag on the swimmer components, making the working cycle non-reciprocal. The resulting net translational displacement leads to velocities of up to 2 micrometers per second. The swimmers can be made omnidirectional, meaning that the same magnetic field sequence can drive swimmers in any direction in the sample plane. Although the direction of their swimming is determined by the momentary orientation of the swimmer, their motion can be guided by solid boundaries. We demonstrate their omnidirectionality by letting them travel through a circular microfluidic channel. We use simple scaling arguments as well as more detailed numerical simulations to explain the measured velocity as a function of the actuation frequency.

Cargo Transport by Two Coupled Myosin Va Motors on Actin Filaments and Bundles.

Myosin Va (myoVa) is a processive, actin-based molecular motor essential for intracellular cargo transport. When a cargo is transported by an ensemble of myoVa motors, each motor faces significant physical barriers and directional challenges created by the complex actin cytoskeleton, a network of actin filaments and actin bundles. The principles that govern the interaction of multiple motors attached to the same cargo are still poorly understood. To understand the mechanical interactions between multiple motors, we developed a simple in vitro model in which two individual myoVa motors labeled with different-colored Qdots are linked via a third Qdot that acts as a cargo. The velocity of this two-motor complex was reduced by 27% as compared to a single motor, whereas run length was increased by only 37%, much less than expected from multimotor transport models. Therefore, at low ATP, which allowed us to identify individual motor steps, we investigated the intermotor dynamics within the two-motor complex. The randomness of stepping leads to a buildup of tension in the linkage between motors-which in turn slows down the leading motor-and increases the frequency of backward steps and the detachment rate. We establish a direct relationship between the velocity reduction and the distribution of intermotor distances. The analysis of run lengths and dwell times for the two-motor complex, which has only one motor engaged with the actin track, reveals that half of the runs are terminated by almost simultaneous detachment of both motors. This finding challenges the assumptions of conventional multimotor models based on consecutive motor detachment. Similar, but even more drastic, results were observed with two-motor complexes on actin bundles, which showed a run length that was even shorter than that of a single motor.

Dynamic Assembly of Magnetic Colloidal Vortices.

Magnetic colloids in external time-dependent fields are subject to complex induced many-body interactions governing their self-assembly into a variety of equilibrium and out-of-equilibrium structures such as chains, networks, suspended membranes, and colloidal foams. Here, we report experiments, simulations, and theory probing the dynamic assembly of superparamagnetic colloids in precessing external magnetic fields. Within a range of field frequencies, we observe dynamic large-scale structures such as ordered phases composed of precessing chains, ribbons, and rotating fluidic vortices. We show that the structure formation is inherently coupled to the buildup of torque, which originates from internal relaxation of induced dipoles and from transient correlations among the particles as a result of short-lived chain formation. We discuss in detail the physical properties of the vortex phase and demonstrate its potential in particle-coating applications.

Non reciprocal skewed rolling of a colloidal wheel due to induced chirality.

We use a rotating magnetic field to assemble an oblate cluster of paramagnetic colloidal particles. If the field is rotating about a horizontal axis, the cluster acts as a colloidal wheel rolling across the supporting glass surface. The motion is reversible upon switching the direction of rotation. Surprisingly, the reversibility is lost if the axis of field rotation is tilted with respect to the surface. The wheel then rolls in a direction that is not perpendicular to the field rotation axis. We explain the skewed rotation with an interplay between a magnetic driving torque, magnetic anisotropy and an anisotropy in the hydrodynamic mobility tensor in the vicinity of a surface. The opposing forward and backward drive induce opposite chirality in the degrees of freedom of the mechanically achiral colloidal wheel.

The winch model can explain both coordinated and uncoordinated stepping of cytoplasmic dynein.

Cytoplasmic dynein moves processively along microtubules, but the mechanism of how its heads use the energy from ATP hydrolysis, coupled to a linker swing, to achieve directed motion, is still unclear. In this article, we present a theoretical model based on the winch mechanism in which the principal direction of the linker stroke is toward the microtubule-binding domain. When mechanically coupling two identical heads (each with postulated elastic properties and a minimal ATPase cycle), the model reproduces stepping with 8-nm steps (even though the motor itself is much larger), interhead coordination, and processivity, as reported for mammalian dyneins. Furthermore, when we loosen the elastic connection between the heads, the model still shows processive directional stepping, but it becomes uncoordinated and the stepping pattern shows a greater variability, which reproduces the properties of yeast dyneins. Their slower chemical kinetics allows processive motility and a high stall force without the need for coordination.

Finding the ciliary beating pattern with optimal efficiency.

We introduce a measure for energetic efficiency of biological cilia acting individually or collectively and numerically determine the optimal beating patterns according to this criterion. Maximizing the efficiency of a single cilium leads to curly, often symmetric, and somewhat counterintuitive patterns. However, when looking at a densely ciliated surface, the optimal patterns become remarkably similar to what is observed in microorganisms like Paramecium. The optimal beating pattern then consists of a fast effective stroke and a slow sweeping recovery stroke. Metachronal coordination is essential for efficient pumping and the highest efficiency is achieved with antiplectic waves. Efficiency also increases with an increasing density of cilia up to the point where crowding becomes a problem. We finally relate the pumping efficiency of cilia to the swimming efficiency of a spherical microorganism and show that the experimentally estimated efficiency of Paramecium is surprisingly close to the theoretically possible optimum.

Near-field hydrodynamic interactions determine travelling wave directions of collectively beating cilia.

Cilia can beat collectively in the form of a metachronal wave, and we investigate how near-field hydrodynamic interactions between cilia can influence the collective response of the beating cilia. Based on the theoretical framework developed in the work of Meng et al. (Meng et al. 2021 Proc. Natl Acad. Sci. USA 118, e2102828118), we find that the first harmonic mode in the driving force acting on each individual cilium can determine the direction of the metachronal wave after considering the finite size of the beating trajectories, which is confirmed by our agent-based numerical simulations. The stable wave patterns, e.g. the travelling direction, can be controlled by the driving forces acting on the cilia, based on which one can change the flow field generated by the cilia. This work can not only help to understand the role of the hydrodynamic interactions in the collective behaviours of cilia, but can also guide future designs of artificial cilia beating in the desired dynamic mode.

Magnetic field directed assembly of magnetic non-spherical microparticles.

This study reports on the fabrication and assembly of anisotropic microparticles as versatile building blocks for directed magnetic assemblies. Although spherical microparticles have received extensive attention, the assembly of non-spherical magnetic microparticles remains underexplored. Herein, we present a fabrication approach that utilizes photolithography and soft lithography to create prism-shaped magnetic microparticles. In order to investigate their assembly, a switching rotating magnetic field was employed. To support our experimental findings, a numerical model which takes into account the magnetic dipole moments induced by the field of other particles was developed. This model helps in understanding the forces and torques governing particle behavior during assembly. Simulations were conducted using the numerical model to complement our experimental findings. In the two particle experiments, attractive magnetic interactions led to various configurations depending on initial positions. For three particles, a tip-to-tip configuration suggested closed or stable ring-like structures. Our work highlights the feasibility of producing highly responsive, non-spherical magnetic microparticles and their potential for assemblies. The versatile fabrication method, coupled with the added degree of freedom conferred by prismatic shapes, opens promising avenues for applications in biology and material science.