Hydrodynamics of chiral squirmers.

Many microorganisms take a chiral path while swimming in an ambient fluid. In this paper we study the combined behavior of two chiral swimmers using the well-known squirmer model taking into account chiral asymmetries. In contrast to the simple squirmer model, which has an axisymmetric distribution of slip velocity, the chiral squirmer has additional asymmetries in the surface slip, which contribute to both translations and rotations of the motion. As a result, swimming trajectories can become helical and chiral asymmetries arise in the flow patterns. We study the swimming trajectories of a pair of chiral squirmers that interact hydrodynamically. This interaction can lead to attraction and repulsion, and in some cases even to bounded states where the swimmers continue to periodically orbit around a common average trajectory. Such bound states are a signature of the chiral nature of the swimmers. Our study could be relevant to the collective movements of ciliated microorganisms.

Phase separation provides a mechanism to reduce noise in cells.

Expression of proteins inside cells is noisy, causing variability in protein concentration among identical cells. A central problem in cellular control is how cells cope with this inherent noise. Compartmentalization of proteins through phase separation has been suggested as a potential mechanism to reduce noise, but systematic studies to support this idea have been missing. In this study, we used a physical model that links noise in protein concentration to theory of phase separation to show that liquid droplets can effectively reduce noise. We provide experimental support for noise reduction by phase separation using engineered proteins that form liquid-like compartments in mammalian cells. Thus, phase separation can play an important role in biological signal processing and control.

Motor regulation results in distal forces that bend partially disintegrated Chlamydomonas axonemes into circular arcs.

The bending of cilia and flagella is driven by forces generated by dynein motor proteins. These forces slide adjacent microtubule doublets within the axoneme, the motile cytoskeletal structure. To create regular, oscillatory beating patterns, the activities of the axonemal dyneins must be coordinated both spatially and temporally. It is thought that coordination is mediated by stresses or strains, which build up within the moving axoneme, and somehow regulate dynein activity. During experimentation with axonemes subjected to mild proteolysis, we observed pairs of doublets associating with each other and forming bends with almost constant curvature. By modeling the statics of a pair of filaments, we show that the activity of the motors concentrates at the distal tips of the doublets. Furthermore, we show that this distribution of motor activity accords with models in which curvature, or curvature-induced normal forces, regulates the activity of the motors. These observations, together with our theoretical analysis, provide evidence that dynein activity can be regulated by curvature or normal forces, which may, therefore, play a role in coordinating the beating of cilia and flagella.

Active chiral processes in thin films.

We develop a generic description of thin active films that captures key features of flow and rotation patterns emerging from the activity of chiral motors which introduce torque dipoles. We highlight the role of the spin rotation field and show that fluid flows can occur in two ways: by coupling of the spin rotation rate to the velocity field via a surface or by spatial gradients of the spin rotation rate. We discuss our results in the context of patches of bacteria on solid surfaces and groups of rotating cilia. Our theory could apply to active chiral processes in the cell cytoskeleton and in epithelia.

Active chiral fluids.

Active processes in biological systems often exhibit chiral asymmetries. Examples are the chirality of cytoskeletal filaments which interact with motor proteins, the chirality of the beat of cilia and flagella as well as the helical trajectories of many biological microswimmers. Here, we derive constitutive material equations for active fluids which account for the effects of active chiral processes. We identify active contributions to the antisymmetric part of the stress as well as active angular momentum fluxes. We discuss four types of elementary chiral motors and their effects on a surrounding fluid. We show that large-scale chiral flows can result from the collective behavior of such motors even in cases where isolated motors do not create a hydrodynamic far field.

Tissue dynamics with permeation.

Animal tissues are complex assemblies of cells, extracellular matrix (ECM), and permeating interstitial fluid. Whereas key aspects of the multicellular dynamics can be captured by a one-component continuum description, cell division and apoptosis imply material turnover between different components that can lead to additional mechanical conditions on the tissue dynamics. We extend our previous description of tissues in order to account for a cell/ECM phase and the permeating interstitial fluid independently. In line with our earlier work, we consider the cell/ECM phase to behave as an elastic solid in the absence of cell division and apoptosis. In addition, we consider the interstitial fluid as ideal on the relevant length scales, i.e., we ignore viscous stresses in the interstitial fluid. Friction between the fluid and the cell/ECM phase leads to a Darcy-like relation for the interstitial fluid velocity and introduces a new characteristic length scale. We discuss the dynamics of a tissue confined in a chamber with a permeable piston close to the homeostatic state where cell division and apoptosis balance, and we calculate the rescaled effective diffusion coefficient for cells. For different mass densities of the cell/ECM component and the interstitial fluid, a treadmilling steady state due to gravitational forces can be found.

A mean-field approach to elastically coupled hair bundles.

We study the dynamics of oscillatory hair bundles which are coupled elastically in their deflection variable and are subject to noise. We present a stochastic description capturing the dynamics of the hair bundles' mean field. In particular, the presented derivation elucidates the origin of the previously described noise reduction by coupling. By comparison of simulations of the approximate dynamics and the full system, we verify our results. Furthermore, we demonstrate that the specific type of coupling considered implies coupling-induced changes in the dynamics beyond mere noise reduction.

Dynamics of Dpp signaling and proliferation control.

Morphogens, such as Decapentaplegic (Dpp) in the fly imaginal discs, form graded concentration profiles that control patterning and growth of developing organs. In the imaginal discs, proliferative growth is homogeneous in space, posing the conundrum of how morphogen concentration gradients could control position-independent growth. To understand the mechanism of proliferation control by the Dpp gradient, we quantified Dpp concentration and signaling levels during wing disc growth. Both Dpp concentration and signaling gradients scale with tissue size during development. On average, cells divide when Dpp signaling levels have increased by 50%. Our observations are consistent with a growth control mechanism based on temporal changes of cellular morphogen signaling levels. For a scaling gradient, this mechanism generates position-independent growth rates.

Mechanics and remodelling of cell packings in epithelia.

Epithelia are sheets of cells that are dynamically remodelled by cell division and cell death during development. Here we describe the cell shapes and packings as networks of polygons: stable and stationary network configurations obey force balance and are represented as local minima of a potential function. We characterize the physical properties of this vertex model, including the set of ground states, and the energetics of topological rearrangements. We furthermore discuss a quasistatic description of cell division that allows us to study the mechanics and dynamics of tissue remodelling during growth. The biophysics of cells and their rearrangements can account for the morphology of cell packings observed in experiments.

The remarkable cochlear amplifier.

This composite article is intended to give the experts in the field of cochlear mechanics an opportunity to voice their personal opinion on the one mechanism they believe dominates cochlear amplification in mammals. A collection of these ideas are presented here for the auditory community and others interested in the cochlear amplifier. Each expert has given their own personal view on the topic and at the end of their commentary they have suggested several experiments that would be required for the decisive mechanism underlying the cochlear amplifier. These experiments are presently lacking but if successfully performed would have an enormous impact on our understanding of the cochlear amplifier.

High-precision tracking of sperm swimming fine structure provides strong test of resistive force theory.

The shape of the flagellar beat determines the path along which a sperm cell swims. If the flagellum bends periodically about a curved mean shape then the sperm will follow a path with non-zero curvature. To test a simple hydrodynamic theory of flagellar propulsion known as resistive force theory, we conducted high-precision measurements of the head and flagellum motions during circular swimming of bull spermatozoa near a surface. We found that the fine structure of sperm swimming represented by the rapid wiggling of the sperm head around an averaged path is, to high accuracy, accounted for by resistive force theory and results from balancing forces and torques generated by the beating flagellum. We determined the anisotropy ratio between the normal and tangential hydrodynamic friction coefficients of the flagellum to be 1.81+/-0.07 (mean+/-s.d.). On time scales longer than the flagellar beat cycle, sperm cells followed circular paths of non-zero curvature. Our data show that path curvature is approximately equal to twice the average curvature of the flagellum, consistent with quantitative predictions of resistive force theory. Hence, this theory accurately predicts the complex trajectories of sperm cells from the detailed shape of their flagellar beat across different time scales.

Quantification of growth asymmetries in developing epithelia.

Many developmental processes of multicellular organisms involve the patterning and growth of two-dimensional tissues, so called epithelia. We have quantified the growth of the wing imaginal disk, which is the precursor of the adult wing, of the fruit fly Drosophila melanogaster. We find that growth follows a simple rule with exponentially decreasing area growth rate. Anisotropies of growth can be precisely determined by comparing experimental results to a continuum theory. Growth anisotropies are to good approximation constant in space and time. They are weak in wild-type wing disks but threefold increased in GFP-Dpp disks in which the morphogen Dpp is overexpressed. Our findings indicate that morphogens such as Dpp control tissue shape via oriented cell divisions that generate anisotropic growth.

Spontaneous movements and linear response of a noisy oscillator.

A deterministic system that operates in the vicinity of a Hopf bifurcation can be described by a single equation of a complex variable, called the normal form. Proximity to the bifurcation ensures that on the stable side of the bifurcation (i.e. on the side where a stable fixed point exists), the linear-response function of the system is peaked at the frequency that is characteristic of the oscillatory instability. Fluctuations, which are present in many systems, conceal the Hopf bifurcation and lead to noisy oscillations. Spontaneous hair bundle oscillations by sensory hair cells from the vertebrate ear provide an instructive example of such noisy oscillations. By starting from a simplified description of hair bundle motility based on two degrees of freedom, we discuss the interplay of nonlinearity and noise in the supercritical Hopf normal form. Specifically, we show here that the linear-response function obeys the same functional form as for the noiseless system on the stable side of the bifurcation but with effective, renormalized parameters. Moreover, we demonstrate in specific cases how to relate analytically the parameters of the normal form with added noise to effective parameters. The latter parameters can be measured experimentally in the power spectrum of spontaneous activity and linear-response function to external stimuli. In other cases, numerical solutions were used to determine the effects of noise and nonlinearities on these effective parameters. Finally, we relate our results to experimentally observed spontaneous hair bundle oscillations and responses to periodic stimuli.

Generic theory of colloidal transport.

We discuss the motion of colloidal particles relative to a two-component fluid consisting of solvent and solute. Particle motion can result from i) net body forces on the particle due to external fields such as gravity; ii) slip velocities on the particle surface due to surface dissipative phenomena. The perturbations of the hydrodynamic flow field exhibit characteristic differences in cases i) and ii) which reflect different patterns of momentum flux corresponding to the existence of net forces, force dipoles or force quadrupoles. In the absence of external fields, gradients of concentration or pressure do not generate net forces on a colloidal particle. Such gradients can nevertheless induce relative motion between particle and fluid. We present a generic description of surface dissipative phenomena based on the linear response of surface fluxes driven by conjugate surface forces. In this framework we discuss different transport scenarios including self-propulsion via surface slip that is induced by active processes on the particle surface. We clarify the nature of force balances in such situations.

Thermal and non-thermal fluctuations in active polar gels.

We discuss general features of noise and fluctuations in active polar gels close to and away from equilibrium. We use the single-component hydrodynamic theory of active polar gels built by Kruse and coworkers to describe the cytoskeleton in cells. Close to equilibrium, we calculate the response function of the gel to external fields and introduce Langevin forces in the constitutive equations with correlation functions respecting the fluctuation-dissipation theorem. We then discuss the breakage of the fluctuation-dissipation theorem due to an external field such as the activity of the motors. Active gels away from equilibrium are considered at the scaling level. As an example of application of the theory, we calculate the density correlation function (the dynamic structure factor) of a compressible active polar gel and discuss possible instabilities.

The chirality of ciliary beats.

Many eukaryotic cells possess cilia which are motile, whip-like appendages that can oscillate and thereby induce motion and fluid flows. These organelles contain a highly conserved structure called the axoneme, whose characteristic architecture is based on a cylindrical arrangement of nine doublets of microtubules. Complex bending waves emerge from the interplay of active internal forces generated by dynein motor proteins within the structure. These bending waves are typically chiral and often exhibit a sense of rotation. In order to study how the shape of the beat emerges from the axonemal structure, we present a three-dimensional description of ciliary dynamics based on the self-organization of dynein motors and microtubules. Taking into account both bending and twisting of the cilium, we determine self-organized beating patterns and find that modes with both a clockwise and anticlockwise sense of rotation exist. Because of the axonemal chirality, only one of these modes is selected dynamically for given parameter values and properties of dynein motors. This physical mechanism, which underlies the selection of a beating pattern with specific sense of rotation, triggers the breaking of the left-right symmetry of developing embryos which is induced by asymmetric fluid flows that are generated by rotating cilia.

Morphogen transport in epithelia.

We present a general theoretical framework to discuss mechanisms of morphogen transport and gradient formation in a cell layer. Trafficking events on the cellular scale lead to transport on larger scales. We discuss in particular the case of transcytosis where morphogens undergo repeated rounds of internalization into cells and recycling. Based on a description on the cellular scale, we derive effective nonlinear transport equations in one and two dimensions which are valid on larger scales. We derive analytic expressions for the concentration dependence of the effective diffusion coefficient and the effective degradation rate. We discuss the effects of a directional bias on morphogen transport and those of the coupling of the morphogen and receptor kinetics. Furthermore, we discuss general properties of cellular transport processes such as the robustness of gradients and relate our results to recent experiments on the morphogen Decapentaplegic (Dpp) that acts in the wing disk of the fruit fly Drosophila.

Dynamics and mechanics of motor-filament systems.

Motivated by the cytoskeleton of eukaryotic cells, we develop a general framework for describing the large-scale dynamics of an active filament network. In the cytoskeleton, active cross-links are formed by motor proteins that are able to induce relative motion between filaments. Starting from pair-wise interactions of filaments via such active processes, our framework is based on momentum conservation and an analysis of the momentum flux. This allows us to calculate the stresses in the filament network generated by the action of motor proteins. We derive effective theories for the filament dynamics which can be related to continuum theories of active polar gels. As an example, we discuss the stability of homogenous isotropic filament distributions in two spatial dimensions.

Contractility and retrograde flow in lamellipodium motion.

We present a phenomenological description of cell locomotion on a solid substrate. The material properties of the actin cytoskeleton in the lamellipodium are described by the constitutive equations of a viscous polar gel with intrinsic activity. The polymerization of the gel takes place in a localized region near the leading edge. Using a simple two-dimensional description, we calculate in the steady state the thickness profile of the lamellipodium which at the rear connects to the cell body; we also calculate the flow profiles and the forces exerted on the substrate. The cell velocity is estimated as a function of externally applied forces. Our description is consistent with experimentally observed properties of motile cells such as the existence of a retrograde flow in the lamellipodium and a dipolar force distribution exerted by the cell on the substrate.