Reconstruction of the three-dimensional beat pattern underlying swimming behaviors of sperm.

 The eukaryotic flagellum propels sperm cells and simultaneously detects physical and chemical cues that modulate the waveform of the flagellar beat. Most previous studies have characterized the flagellar beat and swimming trajectories in two space dimensions (2D) at a water/glass interface. Here, using refined holographic imaging methods, we report high-quality recordings of three-dimensional (3D) flagellar bending waves. As predicted by theory, we observed that an asymmetric and planar flagellar beat results in a circular swimming path, whereas a symmetric and non-planar flagellar beat results in a twisted-ribbon swimming path. During swimming in 3D, human sperm flagella exhibit torsion waves characterized by maxima at the low curvature regions of the flagellar wave. We suggest that these torsion waves are common in nature and that they are an intrinsic property of beating axonemes. We discuss how 3D beat patterns result in twisted-ribbon swimming paths. This study provides new insight into the axoneme dynamics, the 3D flagellar beat, and the resulting swimming behavior.

The steering gaits of sperm.

Sperm are highly specialized cells, which have been subject to substantial evolutionary pressure. Whereas some sperm features are highly conserved, others have undergone major modifications. Some of these variations are driven by adaptation to mating behaviours or fitness at the organismic level. Others represent alternative solutions to the same task. Sperm must find the egg for fertilization. During this task, sperm rely on long slender appendages termed flagella that serve as sensory antennas, propellers and steering rudders. The beat of the flagellum is periodic. The resulting travelling wave generates the necessary thrust for propulsion in the fluid. Recent studies reveal that, for steering, different species rely on different fundamental features of the beat wave. Here, we discuss some examples of unity and diversity across sperm from different species with a particular emphasis on the steering mechanisms. This article is part of the Theo Murphy meeting issue 'Unity and diversity of cilia in locomotion and transport'.

Nano- and microparticles at fluid and biological interfaces.

Systems with interfaces are abundant in both technological applications and biology. While a fluid interface separates two fluids, membranes separate the inside of vesicles from the outside, the interior of biological cells from the environment, and compartmentalize cells into organelles. The physical properties of interfaces are characterized by interface tension, those of membranes are characterized by bending and stretching elasticity. Amphiphilic molecules like surfactants that are added to a system with two immiscible fluids decrease the interface tension and induce a bending rigidity. Lipid bilayer membranes of vesicles can be stretched or compressed by osmotic pressure; in biological cells, also the presence of a cytoskeleton can induce membrane tension. If the thickness of the interface or the membrane is small compared with its lateral extension, both can be described using two-dimensional mathematical surfaces embedded in three-dimensional space. We review recent work on the interaction of particles with interfaces and membranes. This can be micrometer-sized particles at interfaces that stabilise emulsions or form colloidosomes, as well as typically nanometer-sized particles at membranes, such as viruses, parasites, and engineered drug delivery systems. In both cases, we first discuss the interaction of single particles with interfaces and membranes, e.g. particles in external fields, non-spherical particles, and particles at curved interfaces, followed by interface-mediated interaction between two particles, many-particle interactions, interface and membrane curvature-induced phenomena, and applications.

Physics of microswimmers--single particle motion and collective behavior: a review.

Locomotion and transport of microorganisms in fluids is an essential aspect of life. Search for food, orientation toward light, spreading of off-spring, and the formation of colonies are only possible due to locomotion. Swimming at the microscale occurs at low Reynolds numbers, where fluid friction and viscosity dominates over inertia. Here, evolution achieved propulsion mechanisms, which overcome and even exploit drag. Prominent propulsion mechanisms are rotating helical flagella, exploited by many bacteria, and snake-like or whip-like motion of eukaryotic flagella, utilized by sperm and algae. For artificial microswimmers, alternative concepts to convert chemical energy or heat into directed motion can be employed, which are potentially more efficient. The dynamics of microswimmers comprises many facets, which are all required to achieve locomotion. In this article, we review the physics of locomotion of biological and synthetic microswimmers, and the collective behavior of their assemblies. Starting from individual microswimmers, we describe the various propulsion mechanism of biological and synthetic systems and address the hydrodynamic aspects of swimming. This comprises synchronization and the concerted beating of flagella and cilia. In addition, the swimming behavior next to surfaces is examined. Finally, collective and cooperate phenomena of various types of isotropic and anisotropic swimmers with and without hydrodynamic interactions are discussed.

Dynamics of a polymer chain confined in a membrane.

We present a Brownian dynamics theory with full hydrodynamics (Stokesian dynamics) for a Gaussian polymer chain embedded in a liquid membrane which is surrounded by bulk solvent and walls. The mobility tensors are derived in Fourier space for the two geometries, namely, a free membrane embedded in a bulk fluid, and a membrane sandwiched by the two walls. Within the preaveraging approximation, a new expression for the diffusion coefficient of the polymer is obtained for the free-membrane geometry. We also carry out a Rouse normal mode analysis to obtain the relaxation time and the dynamical structure factor. For large polymer size, both quantities show Zimm-like behavior in the free-membrane case, whereas they are Rouse-like for the sandwiched membrane geometry. We use the scaling argument to discuss the effect of excluded-volume interactions on the polymer relaxation time.

Near-surface structure of a bicontinuous microemulsion with a transition region.

The lamellar ordering of bicontinuous microemulsions adjacent to a planar hydrophilic wall is investigated experimentally by grazing-incidence small-angle neutron scattering and theoretically by computer simulations. It is shown that precise depth information in neutron scattering can be obtained by tuning the scattering length density of the overall microemulsion. Neutron reflectometry completes the characterization. The nucleation of a lamellar phase at the wall is observed, and a perforated lamellar transition region is identified at the lamellar-microemulsion interface. The thickness of the lamellar region is about 400 Å, which corresponds to two bilayers.

Attractive colloidal rods in shear flow.

The effect of shear flow on the isotropic-nematic phase transition of attractive colloidal rods is investigated by a combination of simulations and experiments. The isotropic phase aligns with the flow, while the nematic phase undergoes a collective rotational motion which frustrates the merging of the coexisting regions. The location of binodals, spinodals, and the tumbling-to-aligning transition line in the shear-rate versus concentration plane are investigated. The phase diagrams in the shear-concentration plane for the various strengths of attractions can be mapped onto a master curve by appropriate scaling.

Two-dimensional fluctuating vesicles in linear shear flow.

The stochastic motion of a two-dimensional vesicle in linear shear flow is studied at finite temperature. In the limit of small deformations from a circle, Langevin-type equations of motion are derived, which are highly nonlinear due to the constraint of constant perimeter length. These equations are solved in the low-temperature limit and using a mean-field approach, in which the length constraint is satisfied only on average. The constraint imposes non-trivial correlations between the lowest deformation modes at low temperature. We also simulate a vesicle in a hydrodynamic solvent by using the multi-particle collision dynamics technique, both in the quasi-circular regime and for larger deformations, and compare the stationary deformation correlation functions and the time autocorrelation functions with theoretical predictions. Good agreement between theory and simulations is obtained.

Hydrodynamic screening of star polymers in shear flow.

The mutual effects of the conformations of a star polymer in simple shear flow and the deformation of the solvent flow field are investigated by a hybrid mesoscale simulation technique. We characterize the flow field near the star polymer as a function of its functionality (arm number) f . A strong screening of the imposed flow is found inside the star polymer, which increases with increasing f . To elucidate the importance of hydrodynamic screening, we compare results for hydrodynamic and random solvents. The dependence of the polymer orientation angle on the Weissenberg number shows a power law behavior with super-universal exponent --independent of hydrodynamic and excluded-volume interactions. In contrast, the polymer rotation frequency changes qualitatively when hydrodynamic interactions are switched on.

Star polymers in shear flow.

Linear and star polymers in solution are studied in the presence of shear flow. The solvent is described by a particle-based mesoscopic simulation technique, which accounts for hydrodynamic interactions. The scaling properties of the average gyration tensor, the orientation angle, and the rotation frequency are investigated for various arm lengths and arm numbers. With increasing functionality f, star polymers exhibit a crossover in their flow properties from those of linear polymers to a novel behavior, which resembles the tank-treading motion of elastic capsules.

Forced crumpling of self-avoiding elastic sheets.

Thin elastic sheets are important materials across length scales ranging from mesoscopic (polymerized membranes, clay platelets, virus capsids) to macroscopic (paper, metal foils). The crumpling of such sheets by external forces is characterized by the formation of a complex pattern of folds. We have investigated the role of self-avoidance, the fact that the sheets cannot self-intersect, for the crumpling process by large-scale computer simulations. At moderate compression, the force-compression relations of crumpled sheets for both self-avoiding and phantom sheets are found to obey universal power-law behaviours. However, self-avoiding sheets are much stiffer than phantom sheets and, for a given compression, develop many more folds. Moreover, self-avoidance is relevant already at very small volume fractions. The fold-length distribution for crumpled sheets is determined, and is found to be well-described by a log-normal distribution. The stiffening owing to self-avoidance is reflected in the changing nature of the sheet-to-sheet contacts from line-like to two-dimensionally extended with increasing compression.

Dynamics of polymers in a particle-based mesoscopic solvent.

We study the dynamics of flexible polymer chains in solution by combining multiparticle-collision dynamics (MPCD), a mesoscale simulation method, and molecular-dynamics simulations. Polymers with and without excluded-volume interactions are considered. With an appropriate choice of the collision time step for the MPCD solvent, hydrodynamic interactions build up properly. For the center-of-mass diffusion coefficient, scaling with respect to polymer length is found to hold already for rather short chains. The center-of-mass velocity autocorrelation function displays a long-time tail which decays algebraically as (Dt)(-3/2) as a function of time t, where D is the diffusion coefficient. The analysis of the intramolecular dynamics in terms of Rouse modes yields excellent agreement between simulation data and results of the Zimm model for the mode-number dependence of the mode-amplitude correlation functions.

Dynamic regimes of fluids simulated by multiparticle-collision dynamics.

We investigate the hydrodynamic properties of a fluid simulated with a mesoscopic solvent model. Two distinct regimes are identified, the "particle regime" in which the dynamics is gaslike and the "collective regime" where the dynamics is fluidlike. This behavior can be characterized by the Schmidt number, which measures the ratio between viscous and diffusive transport. Analytical expressions for the tracer diffusion coefficient, which have been derived on the basis of a molecular-chaos assumption, are found to describe the simulation data very well in the particle regime, but important deviations are found in the collective regime. These deviations are due to hydrodynamic correlations. The model is then extended in order to investigate self-diffusion in colloidal dispersions. We study first the transport properties of heavy pointlike particles in the mesoscopic solvent, as a function of their mass and number density. Second, we introduce excluded-volume interactions among the colloidal particles and determine the dependence of the diffusion coefficient on the colloidal volume fraction for different solvent mean-free paths. In the collective regime, the results are found to be in good agreement with previous theoretical predictions based on Stokes hydrodynamics and the Smoluchowski equation.

Helical polymer in cylindrical confining geometries.

Using an algorithm for simulating equilibrium configurations, we study a fluctuating helical polymer either (i) contained in a cylindrical pore or (ii) wound around a cylindrical rod. We work in the regime where both the contour length and the persistence length of the helical polymer are much larger than the diameter of the cylinder. In case (i) we calculate the free energy of confinement and interpret it in terms of a wormlike chain in a pore with an effective diameter that depends on the parameters of the helix. In case (ii) we consider the possibility that one end of the helical polymer escapes from the rod and wanders away. The average numbers of turns at which the helix escapes or intersects the rod are measured in the simulations, as a function of the pitch p(0) . The behavior for large and small p(0) is explained with simple scaling arguments.

Attraction between DNA molecules mediated by multivalent ions.

The effective force between two parallel DNA molecules is calculated as a function of their mutual separation for different valencies of counterion and salt ions and different salt concentrations. Computer simulations of the primitive model are used and the shape of the DNA molecules is accurately modeled using different geometrical shapes. We find that multivalent ions induce a significant attraction between the DNA molecules whose strength can be tuned by the averaged valency of the ions. The physical origin of the attraction is traced back either to electrostatics or to entropic contributions. For multivalent counterions and monovalent salt ions, we find a salt-enhanced repulsion effect: the force is first attractive but gets repulsive with increasing salt concentration. Furthermore, we show that the multivalent-ion-induced attraction does not necessarily correlate with DNA overcharging.

Adsorption of monovalent and multivalent cations and anions on DNA molecules.

Adsorption of monovalent and multivalent cations and anions on a deoxyribose nucleic acid (DNA) molecule from a salt solution is investigated by computer simulation. The ions are modeled as charged hard spheres, the DNA molecule as a point charge pattern following the double-helical phosphate strands. The geometrical shape of the DNA molecules is modeled on different levels ranging from a simple cylindrical shape to structured models which include the major and minor grooves between the phosphate strands. The densities of the ions adsorbed on the phosphate strands in the major and in the minor grooves are calculated. First, we find that the adsorption pattern on the DNA surface depends strongly on its geometrical shape: counterions adsorb preferentially along the phosphate strands for a cylindrical model shape, but in the minor groove for a geometrically structured model. Second, we find that an addition of monovalent salt ions results in an increase of the charge density in the minor groove while the total charge density of ions adsorbed in the major groove stays unchanged. The adsorbed ion densities are highly structured along the minor groove while they are almost smeared along the major groove. Furthermore, for a fixed amount of added salt, the major-groove cationic charge is independent of the counterion valency. For increasing salt concentration the major groove is neutralized while the total charge adsorbed in the minor groove is constant. DNA overcharging is detected for multivalent salts. Simulations for larger ion radii, which mimic the effect of ion hydration, indicate an increased adsorbtion of cations in the major groove.

Budding of crystalline domains in fluid membranes.

Crystalline domains embedded in fluid membrane vesicles are studied by Monte Carlo simulations of dynamically triangulated surfaces and by scaling arguments. A budding transition from a caplike state to a budded shape is observed for increasing spontaneous curvature C0 of the crystalline domain as well as increasing line tension lambda. The location of the budding transition is determined as a function of C0, lambda, and the radius R(A) of the crystalline domain. In contrast to previous theoretical predictions, it is found that budding occurs at a value of the spontaneous curvature C0, that is always a decreasing function of the domain size R(A). Several characteristic scaling regimes are predicted. The distribution of five- and sevenfold disclinations as the budding transition is approached is determined, and the dynamics of the generation of defects is studied.

Elastic properties of polymer interfaces: aggregation of pure diblock, mixed diblock, and triblock copolymers.

Block copolymers adsorbing to an interface between two immiscible homopolymers modify the elastic constants of this interface. Within self-consistent field calculations for Gaussian chains, we determine how the bending constants vary in dependence on the block copolymer concentration and architecture. Four phenomena are discussed. (i) When a tricritical or isotropic Lifshitz critical point is approached in a ternary mixture by varying the concentration of diblock copolymers or changing temperature, the elastic constants vanish. We determine the corresponding power laws, and show that the de Gennes-Taupin criterium for the stability of lamellar phases against undulations and the Ginzburg-Landau criterium for bulk fluctuations yield identical predictions for the validity of the mean-field approximation. (ii) Addition of a small amount of diblock copolymers modifies the bending rigidity. If the diblock copolymers are comparable in length to the homopolymers, adsorption of the diblocks reduces the bending rigidity. If the diblocks are much longer, they increase the bending rigidity. Only for an extreme ratio of chain lengths (>100), the predictions for polymers tethered to an infinitely thin, impenetrable sheet become accurate. (iii) Mixtures of short and long symmetric diblock copolymers are studied, as well as mixtures of two asymmetric diblocks, which are obtained by exchanging the long and short ends. The saddle-splay modulus is found to be the same in both mixtures, while the bending rigidity is significantly smaller in the latter case. (iv) The role of the block copolymer architecture is studied by comparing the effect of triblock copolymers with the effect of diblocks with the same overall length and composition. We propose that triblock copolymers are a very efficient way to control the spontaneous curvature of an interface.

Mesoscopic solvent simulations: multiparticle-collision dynamics of three-dimensional flows.

A recently developed mesoscopic solvent model with multiparticle-collision dynamics is applied to three-dimensional solvent flows in a channel with and without a spherical obstacle. The advantage of a gravitationally driven flow of the solvent over the flow induced by a pressure gradient in the calculation of the solvent viscosity is demonstrated. Three different algorithms for stochastic collision steps are investigated and compared. In particular, we have examined an alternative algorithm with relative velocities drawn from a Maxwell-Boltzmann distribution at each collision step. This algorithm increases the numerical efficiency of the mesoscopic model for solvent flows with low and intermediate Reynolds numbers. Our simulation results for the recirculation length of stationary vortices behind a spherical obstacle are in good agreement with the previous experimental measurements.