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# Statistische Physik weicher Materie und biologischer Systeme

Die Arbeitsgruppe bearbeitet ein breites Spektrum von Projekten aus den Bereichen der statistischen Physik weicher Materie und biologischer Systeme und der Optik. Dabei kommen eine Vielzahl von Methoden aus der statistischen Physik und Kontinuumstheorien zum Einsatz.

**Weiche Materie** umfasst eine Vielzahl von Materialien wie Flüssigkristalle, Polymere, kolloidale Dispersionen und Membranen. Sie befindet sich also im Schnittpunkt von Grundlagenforschung, die sich mit der Beschreibung dieser komplexen Syteme mit Hilfe vielfältiger Konzepte befasst, und ihrer Anwendungen in der Technologie und im alltäglichen Leben.

Viele Bestandteile **biologischer Systeme** werden der weichen Materie zugeordnet, die daher mit ihrer Methodenvielfalt eine zentrale Rolle in der Biologischen Physik spielt.

**Optik**, eine Schlüsseltechnologie unseres Jahrhunderts, stellt wesentliche Methoden zur Erforschung der komplexen Strukturen und dynamischen Vorgänge weicher Materie zur Verfügung.

# Aktuelles

- Brennpunkt-Artikel im Physik Journal:
**Die Körperform macht's**

Holger Stark

Physik Journal - Prof. Stark bietet folgende
**Prüfungstermine**an:

Die Studierenden melden sich bitte bei Frau Klemz an (i.d.R. Mo-Fr 10-16h, Raum EW 746). - Seminar: Statistische Physik weicher Materie und biologischer Systeme

# Aktuelle Forschung

**Stability of liquid films covered by a carpet of self-propelled surfactant particles**

**by Andrey Pototsky, Uwe Thiele and Holger Stark**

We consider a carpet of self-propelled particles at the liquid-gas interface of a liquid film on a solid substrate. The particles exert an excess pressure on the interface and also move along the interface while the swimming direction changes due to rotational diffusion. We study the intricate influence of these self-propelled insoluble surfactants on the stability of the film surface and show that depending on the strength of in-surface rotational diffusion and the absolute value of the in-surface swimming velocity, several characteristic instability modes can occur. In particular, rotational diffusion can either stabilize the film or induce instabilities of different character.

Phys. Rev. E. **90**, 030401(R) (2014)

**Dynamic Clustering and Chemotactic Collapse of Self-Phoretic Active Particles**

**by Oliver Pohl and Holger Stark**

Recent experiments with self-phoretic particles at low concentrations show a pronounced dynamic clustering [I. Theurkauff et al.,Phys. Rev. Lett. 108, 268303 (2012)]. We model this situation by taking into account the translational and rotational diffusiophoretic motion, which the active particles perform in their self-generated chemical field. Our Brownian dynamics simulations show pronounced dynamic clustering only when these two phoretic contributions give rise to competing attractive and repulsive interactions, respectively. We identify two dynamic clustering states and characterize them by power-law-exponential distributions. In case of mere attraction a chemotactic collapse occurs directly from the gaslike into the collapsed state, which we also predict by mapping our Langevin dynamics on the Keller-Segel model for bacterial chemotaxis.

Phys. Rev. Lett. **112**, 238303 (2014)

**Self-induced polar order of active Brownian particles in a harmonic trap**

**by Marc Hennes, Katrin Wolff, and Holger Stark**

Hydrodynamically interacting active particles in an external harmonic potential form a self-assembled fluid pump at large enough Péclet numbers. Here, we give a quantitative criterion for the formation of the pump and show that particle orientations align in the self-induced flow field in surprising analogy to ferromagnetic order where the active Péclet number plays the role of inverse temperature. The particle orientations follow a Boltzmann distribution Φ(**p**) ∼ exp(Ap_z) where the ordering mean field A scales with the active Péclet number and polar order parameter. The mean flow field in which the particles’ swimming directions align corresponds to a regularized Stokeslet with strength proportional to swimming speed. Analytic mean-field results are compared with results from Brownian dynamics simulations with hydrodynamic interactions included and are found to capture the self-induced alignment very well.

Phys. Rev. Lett. **112**, 238104 (2014)

**Feedback control of inertial microfluidics using axial control forces**

**by Christopher Prohm and Holger Stark**

Inertial microfluidics is a promising tool for many lab-on-a-chip applications. Particles in channel flows with Reynolds numbers above one undergo cross-streamline migration to a discrete set of equilibrium positions in square and rectangular channel cross sections. This effect has been used extensively for particle sorting and the analysis of particle properties. Using the lattice Boltzmann method, we determined the equilibrium positions in square and rectangular cross sections and classify their types of stability for different Reynolds numbers, particle sizes, and channel aspect ratios. Our findings thereby help to design microfluidic Received 3rd February 2014, channels for particle sorting. Furthermore, we demonstrated how an axial control force, which slows down the particles and shifts the stable equilibrium position towards the channel center. Ultimately, the particles then stay on the centerline for forces exceeding the threshold value. This effect is sensitive to the particle size and channel Reynolds number and therefore suggests an efficient method for particle separation. In combination with a hysteretic feedback scheme, we can even increase the particle throughput.

Lab Chip **14** , 2115 (2014)

**Controlling inertial focussing using rotational motion**

**by Christopher Prohm, Nikolas Z****ö****ller, and Holger Stark**

In inertial microfluidics lift forces cause a particle to migrate across streamlines to specific positions in the cross section of a microchannel. We control the rotational motion of a particle and demonstrate that this allows to manipulate the lift-force profile and thereby the particle’s equilibrium positions. We perform two-dimensional simulation studies using the method of multi-particle collision dynamics. Particles with unconstrained rotational motion occupy stable equilibrium positions in both halfs of the channel while the center is unstable. When an external torque is applied to the particle, two equilibrium positions annihilate by passing a saddle-node bifurcation and only one stable fixpoint remains so that all particles move to one side of the channel. In contrast, non-rotating particles accumulate in the center and are pushed into one half of the channel when the angular velocity is fixed to a non-zero value.

Eur. Phys. J. E **37,** 36 (2014)

**Hydrodynamics Determines Collective Motion and Phase Behavior of Active Colloids in Quasi-Two-Dimensional Confinement**

**by ****Andreas Zöttl and Holger Stark**We study the collective motion of confined spherical microswimmers such as active colloids which we model by so-called squirmers. To simulate hydrodynamic flow fields including thermal noise, we use the method of multiparticle collision dynamics. We demonstrate that hydrodynamic near fields acting between squirmers as well as between squirmers and bounding walls crucially determine their collective motion. In particular, with increasing density we observe a clear phase separation into a gaslike and cluster phase for neutral squirmers whereas strong pushers and pullers more gradually approach the hexagonal cluster state.

Phys. Rev. Lett.

**112,**118101 (2014)

highlighted as Editors' suggestion and by pro-physik

**How the Motility Pattern of Bacteria Affects Their ****Dispersal and Chemotaxis**

**Johannes Taktikos****, ****Holger Stark and ****Vasily Zaburdaev**

Most bacteria at certain stages of their life cycle are able to move actively; they can swim in a liquid or crawl on various surfaces. A typical path of the moving cell often resembles the trajectory of a random walk. However, bacteria are capable of modifying their apparently random motion in response to changing environmental conditions. As a result, bacteria can migrate towards the source of nutrients or away from harmful chemicals. Surprisingly, many bacterial species that were studied have several distinct motility patterns, which can be theoretically modeled by a unifying random walk approach. We use this approach to quantify the process of cell dispersal in a homogeneous environment and show how the bacterial drift velocity towards the source of attracting chemicals is affected by the motility pattern of the bacteria. Our results open up the possibility of accessing additional information about the intrinsic response of the cells using macroscopic observations of bacteria moving in inhomogeneous environments.

PLOS ONE **8**, e81936 (2013)

**Optimal control of particle separation in inertial microfluidics**

**Christopher Prohm, Fredi ****Tröltzsch**** and Holger Stark**

Recently, inertial mircofluidics has emerged as a promising tool to manipulate complex liquids with possible biomedical applications, for example, to particle separation. Indeed, in experiments different particle types were separated based on their sizes (A.J. Mach, D. Di Carlo, Biotechnol. Bioeng. 107, 302 (2010)). In this article we use a theoretical study to demonstrate how concepts from optimal control theory help to design optimized profiles of control forces that allow to steer particles to almost any position at the outlet of a microfluidic channel. We also show that one specific control force profile is sufficient to guide two types of particles to different locations at the channel outlet, where they can be separated from each other. The particles just differ by their size which determines the strength of the inertial lift forces they experience. Our approach greatly enhances the efficiency of particle separation in the inertial regime.

Eur. Phys. J. E **36,** 118 (2013)

**A Bacterial Swimmer with Two Alternating Speeds of Propagation**

**by ****Matthias Theves, Johannes Taktikos, Vasily Zaburdaev, Holger Stark and Carsten Beta**

We recorded large data sets of swimming trajectories of the soil bacterium Pseudomonas putida. Like other prokaryotic swimmers, P. putida exhibits a motion pattern dominated by persistent runs that are interrupted by turning events. An in-depth analysis of their swimming trajectories revealed that the majority of the turning events is characterized by an angle of a_{1} = 180 (reversals). To a lesser extent, turning angles of a_{2} = 0 are also found. Remarkably, we observed that, upon a reversal, the swimming speed changes by a factor of two on average—a prominent feature of the motion pattern that, to our knowledge, has not been reported before. A theoretical model, based on the experimental values for the average run time and the rotational diffusion, recovers the mean-square displacement of P. putida if the two distinct swimming speeds are taken into account. Compared to a swimmer that moves with a constant intermediate speed, the mean-square displacement is strongly enhanced. We furthermore observed a negative dip in the directional autocorrelation at intermediate times, a feature that is only recovered in an extended model, where the nonexponential shape of the run-time distribution is taken into account.

Biophys. J. **105**, 1915-1924 (2013)

**Nonlinear dynamics of spherical particles in Poiseuille flow under ****creeping-flow condition**

**by ****Sebastian Reddig and Holger Stark**

We study the nonlinear dynamics of spherical colloids under the influence of a pressure driven flow at vanishing Reynolds number. The colloids are confined between two parallel planar walls with a distance comparable to the particle diameter and they interact hydrodynamically via the solvent. We show that the bounded Poiseuille flow gives rise to new classes of trajectories resulting in crossstreamline migration. Two particles moving on these new trajectories exhibit either bound or unbound states. In the first case they oscillate on closed trajectories in the center-of-mass frame. In the second case, they exhibit cross-swapping trajectories in addition to swapping trajectories which were already observed in unbounded or bounded linear shear flow. The different classes of trajectories occur depending on the initial positions of the two particles and their size. We present state diagrams in the lateral positions, where we categorize the trajectories and color code the oscillation frequencies of the bound states. Finally we discuss how the results on the two-particle system help to understand the stability of particle trains composed of several particles.

J. Chem. Phys. 138, 234902 (2013)

**Sedimentation and polar order of active bottom-heavy particles **

by Katrin Wolff, Aljoscha M. Hahn, and Holger Stark

by Katrin Wolff, Aljoscha M. Hahn, and Holger Stark

Self-propelled particles in an external gravitational field have been shown to display both an increased sedimentation length and polar order even without particle interactions. Here, we investigate self-propelled particles which additionally are bottom-heavy, that is they feel a torque aligning them to swim against the gravitational field. For bottom-heavy particles the gravitational field has the two opposite effects of i) sedimentation and ii) upward alignment of the particles’ swimming direction. We perform a multipole expansion of the one-particle distribution of non-interacting particles with respect to orientation and derive expressions for sedimentation length and mean particle orientation which we check against Brownian Dynamics simulations. For large strength of gravity or small particle speeds and aligning torque, we observe sedimentation with increased sedimentation length compared with passive colloids but also active colloids without bottom-heaviness. Increasing, for example, swimming speed the sedimentation profile is inverted and the particles swim towards the top wall of the enclosing box. We find maximal orientational order at intermediate swimming speeds for both cases of particles with bottom-heaviness and those without. Ordering unsurprisingly is increased for the bottom-heavy particles, but this difference disappears at higher levels of activity and for very high activities ordering goes to zero in both cases.

Eur. Phys. J. E **36,** 43 (2013)

**Rectification of self-propelled particles by symmetric barriers**

**by** **Andrey Pototsky, Aljoscha M. Hahn and Holger ****Stark**

The motion of self-propelled particles can be rectified by asymmetric or ratchetlike periodic patterns in space. Here we show that a nonzero average drift can already be induced in a periodic potential with symmetric barriers when the self-propulsion velocity is also symmetric and periodically modulated but phase-shifted against the potential. In the adiabatic limit of slow rotational diffusion we determine the mean drift analytically and discuss the influence of temperature. In the presence of asymmetric barriers, modulating the self-propulsion can largely enhance the mean drift or even reverse it.

Phys. Rev. E **87**, 042124 (2013)

Rotation-Induced Polymorphic Transitions in Bacterial Flagella

**by ****Reinhard Vogel and Holger ****Stark**

Bacteria propel themselves with the help of rotating helical flagella. They change their swimming direction during tumbling events in order to increase, for example, their supply of nutrients (chemotaxis). During tumbling a bacterial flagellum assumes different polymorphic states. Based on a continuum model for the motor-flagellum system, we demonstrate that a changing motor torque can initiate these polymorphic transformations. In particular, we investigate the run-and-stop tumble strategy of Rhodobacter sphaeroides which uses a coiled-to-normal transition in its single flagellum. We also show that torque reversal in single-flagellated Escherichia coli generates a normal-to-curly I transition as observed for tumbling E. coli that swim with a bundle of several flagella.

Phys. Rev. Lett. **110**, 158104 (2013)

**Periodic and quasiperiodic motion of an elongated microswimmer ****in Poiseuille flow**

**by ****Andreas Zöttl and Holger ****Stark**

We study the dynamics of a prolate spheroidal microswimmer in Poiseuille flow for different flow geometries. When moving between two parallel plates or in a cylindrical microchannel, the swimmer performs either periodic swinging or periodic tumbling motion. Although the trajectories of spherical and elongated swimmers are qualitatively similar, the swinging and tumbling frequency strongly depends on the aspect ratio of the swimmer. In channels with reduced symmetry the swimmers perform quasiperiodic motion which we demonstrate explicitly for swimming in a channel with elliptical cross-section.

Eur. Phys. J. E** 36**, 4 (2013)

**Phason-induced dynamics of colloidal particles ****on quasicrystalline substrates**

**by ****Justus A. Kromer, Michael Schmiedeberg, Johannes Roth, and Holger ****Stark**

Phasons are special hydrodynamic modes that occur in quasicrystals. The trajectories of particles due to a phasonic drift were recently studied by Kromer et al. (Phys. Rev. Lett. 108, 218301 (2012)) for the case where the particles stay in the minima of a quasicrystalline potential. Here, we study the mean motion of colloidal particles in quasicrystalline laser fields when a phasonic drift or displacement is applied and also consider the cases where the colloids cannot follow the potential minima. While the mean square displacement is similar to the one of particles in a random potential with randomly changing potential wells, there also is a net drift of the colloids that reverses its direction when the phasonic drift velocity is increased. Furthermore, we explore the dynamics of the structural changes in a laser-induced quasicrystal during the rearrangement process that is caused by a steady phasonic drift or an instantaneous phasonic displacement.

Eur. Phys. J. E **36**, 25 (2013)

**Clustering and mobility of hard rods in a quasicrystalline substrate potential**

**by Philipp Kählitz, Martin Schoen, and ****Holger Stark**

Recently, we have studied the self-assembly of hard needles in a quasicrystalline substrate potential with decagonal symmetry [P. Kählitz and H. Stark, J. Chem. Phys. 136, 174705 (2012)10.1063/1.4711086]. We have identified new structure formation using Monte Carlo simulations. However, hard needles have a zero width. To investigate how the excluded volume of rod-shaped particles influences their phase ordering, we extend here our studies to spherocylinders. We determine phase diagrams and plot them in the relevant variables, strength of substrate potential versus area fraction. At increasing area fraction η short rods form clusters that ultimately destroy directional ordering along the decagonal symmetry directions while surface-induced positional order exists for all η. In contrast, long rods show directional order in the whole density range. However, at high area fractions they assemble into compact clusters which destroy positional ordering. Finally, we also study the rod mobility using the kinetic Monte Carlo method and discuss an unexpected mobility enhancement with increasing density. All these features crucially depend on the non-zero excluded volume of the spherocylinders.

**Trypanosome Motion Represents an Adaptation to theCrowded Environment of the Vertebrate Bloodstream**

**by Niko Heddergott, Timothy Krüger, Sujin B. Babu, Ai Wei, Erik Stellermanns, Sravanti Uppaluri, Thomas Pfohl,**

**Holger Stark,**

**Markus Engstler**

Blood is a remarkable habitat: it is highly viscous, contains a dense packaging of cells and perpetually flows at velocities varying over three orders of magnitude. Only few pathogens endure the harsh physical conditions within the vertebrate bloodstream and prosper despite being constantly attacked by host antibodies. African trypanosomes are strictly extracellular blood parasites, which evade the immune response through a system of antigenic variation and incessant motility. How the flagellates actually swim in blood remains to be elucidated. Here, we show that the mode and dynamics of trypanosome locomotion are a trait of life within a crowded environment. Using high-speed fluorescence microscopy and ordered micro-pillar arrays we show that the parasites mode of motility is adapted to the density of cells in blood. Trypanosomes are pulled forward by the planar beat of the single flagellum. Hydrodynamic flow across the asymmetrically shaped cell body translates into its rotational movement. Importantly, the presence of particles with the shape, size and spacing of blood cells is required and sufficient for trypanosomes to reach maximum forward velocity. If the density of obstacles, however, is further increased to resemble collagen networks or tissue spaces, the parasites reverse their flagellar beat and consequently swim backwards, in this way avoiding getting trapped. In the absence of obstacles, this flagellar beat reversal occurs randomly resulting in irregular waveforms and apparent cell tumbling. Thus, the swimming behavior of trypanosomes is a surprising example of micro-adaptation to life at low Reynolds numbers. For a precise physical interpretation, we compare our high-resolution microscopic data to results from a simulation technique that combines the method of multi-particle collision dynamics with a triangulated surface model. The simulation produces a rotating cell body and a helical swimming path, providing a functioning simulation method for a microorganism with a complex swimming

strategy.

PLoS Pathogen **8**, e1003023 (2012)

**Inertial microfluidics with multi-particle collision dynamics****by Christopher Prohm, Michael Gierlak,**** and Holger Stark**

Using the method of multi-particle collision dynamics (MPCD), we investigate inertial focussing in microfluidic channels that gives rise to the Segré-Silberberg effect. At intermediate Reynolds numbers, we model the motion of a spherical colloid in a circular microchannel under pressure-driven flow. We determine the radial distribution function and show how its width and the location of its maximum are strongly influenced by the colloid size and the Reynolds number of the Poiseuille flow. We demonstrate that MPCD is well suited for calculating mean values for the lift force acting on the colloid in the cross-sectional plane and for its mean axial velocity. We introduce a Langevin equation for the cross-sectional motion whose steady state is the Boltzmann distribution that contains the integrated lift force as potential energy. It perfectly coincides with the simulated radial distribution function.

Eur. Phys. J. E **35**, 80 (2012)

**Active Brownian particles in two-dimensional traps****by Andrey Pototsky and Holger Stark**

We consider a population of self-propelled Brownian particles in 2D traps. For noninteracting particles the stationary distribution for position and orientation is found analytically for small and large rotational diffusivities. These results are used to map the system of interacting active particles onto a system of passive particles in a modified trapping potential which we then formulate as a dynamic density functional theory. Our approach is supported by Brownian dynamics simulations of the original and the effective model.

Europhys. Lett. **98**, 50004 (2012)

**What phasons look like: Particle trajectories in a quasicrystalline potential****by Justus A. Kromer, Michael Schmiedeberg, Johannes Roth, and Holger Stark**

Among the distinctive features of quasicrystals—structures with long-range order but without periodicity—are phasons. Phasons are hydrodynamic modes that, like phonons, do not cost free energy in the long-wavelength limit. For light-induced colloidal quasicrystals, we analyze the collective rearrangements of the colloids that occur when the phasonic displacement of the light field is changed. The colloidal model system is employed to study the link between the continuous description of phasonic modes in quasicrystals and collective phasonic flips of atoms. We introduce characteristic areas of reduced phononic and phasonic displacements and use them to predict individual colloidal trajectories. In principle, our method can be employed with all quasicrystalline systems in order to derive collective rearrangements of particles from the continuous description of phasons.

Phys. Rev. Lett. **108**, 218301 (2012)

Highlighted article and selected as an Editors' Suggestion

See also Synopsis in Physics: Phasons passing by

**Nonlinear dynamics of a microswimmer in Poiseuille flow****by Andreas Zöttl and Holger Stark**

We study the three-dimensional dynamics of a spherical microswimmer in cylindrical Poiseuille flow which can be mapped onto a Hamiltonian system. Swinging and tumbling trajectories are identified. In 2D they are equivalent to oscillating and circling solutions of a mathematical pendulum. Hydrodynamic interactions between the swimmer and confining channel walls lead to dissipative dynamics and result in stable trajectories, different for pullers and pushers. We demonstrate this behavior in the dipole approximation of the swimmer and with simulations using the method of multiparticle collision dynamics.

Phys. Rev. Lett. **108**, 218104 (2012)

**Phase ordering of hard needles on a quasicrystalline substrate****by Philipp Kählitz and Holger Stark**

Quasicrystals possess long-range positional and orientational order. However, they cannot be periodic in space due to their non-crystallographic symmetries such as a 10-fold rotational axis. We perform Monte Carlo simulations of two-dimensional hard-needle systems subject to a quasiperiodic substrate potential. We determine phase diagrams as a function of density and potential strength for two needle lengths. With increasing potential strength short needles tend to form isolated clusters that display directional order along the decagonal directions. Long needles create interacting clusters that stabilize the nematic phase. At large potential strengths the clusters position themselves on two interwoven Fibonacci sequences perpendicular to the cluster orientation. Alternatively, one obtains extended domains of needle clusters which are aligned along all decagonal symmetry directions.

J. Chem. Phys. **136**, 174705 (2012)

**Collective dynamics of model microorganisms with chemotactic signaling****by Johannes Taktikos, Vasily Zaburdaev, and Holger Stark**

Various microorganisms use chemotaxis for signaling among individuals — a common strategy for communication that is responsible for the formation of microcolonies. We model the microorganisms as autochemotactic active random walkers and describe them by an appropriate Langevin dynamics. It consists of rotational diffusion of the walker’s velocity direction and a deterministic torque that aligns the velocity direction along the gradient of a self-generated chemical field. To account for finite size, each microorganism is treated as a soft disk. Its velocity is modified when it overlaps with other walkers according to a linear force-velocity relation and a harmonic repulsion force. We analyze two-walker collisions by presenting typical trajectories and by determining a state diagram that distinguishes between free walker, metastable, and bounded cluster states. We mention an analogy to Kramer’s escape problem. Finally, we investigate relevant properties of many-walker systems and describe characteristics of cluster formation in unbounded geometry and in confinement.

Phys. Rev. E **85**, 051901 (2012)

**Motor-driven bacterial flagella and buckling instabilities****by Reinhard Vogel and Holger Stark**

Many types of bacteria swim by rotating a bundle of helical filaments also called flagella. Each filament is driven by a rotary motor and a very flexible hook transmits the motor torque to the filament. We model it by discretizing Kirchhoff’s elastic-rod theory and develop a coarse-grained approach for driving the helical filament by a motor torque. A rotating flagellum generates a thrust force, which pushes the cell body forward and which increases with the motor torque. We fix the rotating flagellum in space and show that it buckles under the thrust force at a critical motor torque. Buckling becomes visible as a supercritical Hopf bifurcation in the thrust force. A second buckling transition occurs at an even higher motor torque. We attach the flagellum to a spherical cell body and also observe the first buckling transition during locomotion. By changing the size of the cell body, we vary the necessary thrust force and thereby obtain a characteristic relation between the critical thrust force and motor torque. We present a elaborate analytical model for the buckling transition based on a helical rod which quantitatively reproduces the critical force-torque relation. Real values for motor torque, cell body size, and the geometry of the helical filament suggest that buckling should occur in single bacterial flagella. We also find that the orientation of pulling flagella along the driving torque is not stable and comment on the biological relevance for marine bacteria.

Eur. Phys. J. E **35**, 15 (2012)

**Modeling a self-propelled autochemotactic walker****by Johannes Taktikos, Vasily Zaburdaev, and Holger Stark**

We develop a minimal model for the stochastic dynamics of microorganisms where individuals communicate via autochemotaxis. This means that microorganisms, such as bacteria, amoebae, or cells, follow the gradient of a chemical that they produce themselves to attract or repel each other. A microorganism is represented as a self-propelled particle or walker with constant speed while its velocity direction diffuses on the unit circle. We study the autochemotactic response of a single self-propelled walker whose dynamics is non-Markovian. We show that its long-time dynamics is always diffusive by deriving analytic expressions for its diffusion coefficient in the weak- and strong-coupling case. We confirm our findings by numerical simulations.

Phys. Rev. E **84**, 041924 (2011)

**Modelling bacterial flagellar growth****by Maximilian Schmitt and Holger Stark**

The growth of bacterial flagellar filaments is a self-assembly process where flagellin molecules are transported through the narrow core of the flagellum and are added at the distal end. To model this situation, we generalize a growth process based on the TASEP model by allowing particles to move both forward and backward on the lattice. The bias in the forward and backward jump rates determines the lattice tip speed, which we analyze and also compare to simulations. For positive bias, the system is in a non-equilibrium steady state and exhibits boundary-induced phase transitions. The tip speed is constant. In the no-bias case we find that the length of the lattice grows as N(t)∝sqrt(t), whereas for negative drift *N*(*t*)∝ln *t*. The latter result agrees with experimental data of bacterial flagellar growth.

Europhys. Lett. **96**, 28001 (2011)

**Active colloidal suspensions exhibit polar order under gravity****by Mihaela Enculescu and Holger Stark**

Recently, the steady sedimentation profile of a dilute suspension of chemically powered colloids under gravitational field was studied experimentally [J. Palacci et al, Phys. Rev. Lett. **105**, 088304 (2010)]. It was found that the sedimentation length increases quadratically with the swimming speed of the active Brownian particles. Here we investigate theoretically the sedimentation of self-propelled particles undergoing translational and rotational diffusion. We find that with increasing sedimentation length the swimming directions of the particles develop polar order against the gravitational field. We suggest realistic parameter values to observe this ordering. Finally, we formulate a dynamic density functional theory for active suspensions under the condition that a non-equilibrium steady state exists.

Phys. Rev. Lett. **107**, 058301 (2011)

**Modeling the bacterial flagellum by an elastic network of rigid bodies****by Christoph Speier, Reinhard Vogel, and Holger Stark**

Bacteria such as *Escherichia coli* propel themselves by rotating a bundle of helical filaments, each driven by a rotary motor embedded in the cell membrane. Each filament is an assembly of thousands of copies of the protein flagellin which assumes two different states. We model the filament by an elastic network of rigid bodies that form bonds with one another according to a scheme suggested by Namba and Vondervistz (1997 *Q. Rev. Biophys.* **30** 1–65) and add additional binding sites at the inner part of the rigid body. Our model reproduces the helical parameters of the 12 possible polymorphic configurations very well. We demonstrate that its energetical ground state corresponds to the normal helical form, usually observed in nature, only when inner and outer binding sites of the rigid body have a large axial displacement. This finding correlates directly to the elongated shape of the flagellin molecule. An Ising Hamiltonian in our model directly addresses the two states of the flagellin protein. It contains an external field that represents external parameters which allow us to alter the ground state of the filament.

Phys. Biol. **8** 046009 (2011)

**Langevin dynamics deciphers the motility pattern of swimming parasites****by Vasily Zaburdaev, Sravanti Uppaluri, Thomas Pfohl, Markus Engstler, Rudolf Friedrich, and Holger Stark**

The parasite African trypanosome swims in the bloodstream of mammals and causes the highly dangerous human sleeping sickness. Cell motility is essential for the parasite's survival within the mammalian host. We present an analysis of the random-walk pattern of a swimming trypanosome in a well-controlled environment. From experimental time-autocorrelation functions for the direction of motion we identify two relaxation times that dier by an order of magnitude. They originate from the rapid deformations of the cell body and a slower rotational diusion of the average swimming direction. Velocity uctuations are athermal and increase for faster trypanosomes whose trajectories are also straighter. We demonstrate that such a complex behavior is fully captured by two decoupled Langevin equations that decipher the complex trajectory pattern by referring it to the microscopic details of cell behavior. Moreover, the model provides a prediction for the shorter relaxation time beyond experimental resolution.

Phys. Rev. Lett. **106**, 208103 (2011)

**Metachronal waves in a chain of rowers with hydrodynamic interactions****by Christopher Wollin and Holger Stark**

Filaments on the surface of a microorganism such as Paramecium or Ophalina beat highly synchronized and form so-called metachronal waves that travel along the surfaces. In order to study under what principal conditions these waves form, we introduce a chain of beads, called rowers, each periodically driven by an external force on a straight line segment. To implement hydrodynamic interactions between the beads, they are considered point-like. Two beads synchronize in antiphase or in phase depending on the positive or negative curvature of their driving-force potential. Concentrating on in-phase synchronizing rowers, we find that they display only transient synchronization in a bulk fluid. On the other hand, metachronal waves with wavelengths of 7-10 rower distances emerge, when we restrict the range of hydrodynamic interactions either artificially to nearest neighbors or by the presence of a bounding surface as in any relevant biological system.

Eur. Phys. J. E **34**, 42--1-10 (2011)

**Force-extension curves of bacterial flagella****by Reinhard Vogel and Holger Stark**

Bacterial flagella assume different helical shapes during the tumbling phase of a bacterium but also in response to varying environmental conditions. Force-extension measurements by Darnton and Berg explicitly demonstrate a transformation from the coiled to the normal helical state (N.C. Darnton, H.C. Berg, Biophys. J. 92, 2230 (2007)). We here develop an elastic model for the flagellum based on Kirchhoff's theory of an elastic rod that describes such a polymorphic transformation and use resistive force theory to couple the flagellum to the aqueous environment. We present Brownian-dynamics simulations that quantitatively reproduce the force-extension curves and study how the ratio $\Gamma$ of torsional to bending rigidity and the extensional rate influence the response of the flagellum. An upper bound for $\Gamma$ is given. Using clamped flagella, we show in an adiabatic approximation that the mean extension, where a local coiled-to-normal transition occurs first, depends on the logarithm of the extensional rate.

Eur. Phys. J. E **33**, 259-271 (2010)

**Archimedean-like colloidal tilings on substrates with decagonal and tetradecagonal symmetry****by Michael Schmiedeberg, Jules Mikhael, Sebastian Rausch, Johannes Roth, Laurent Helden, Clemens Bechinger, and Holger Stark**

Two-dimensional colloidal suspensions subjected to laser interference patterns with decagonal symmetry can form an Archimedean-like tiling phase where rows of squares and triangles order aperiodically along one direction (J. Mikhael *et al.*, Nature **454**, 501 (2008)). In experiments as well as in Monte Carlo and Brownian dynamics simulations, we identify a similar phase when the laser field possesses tetradecagonal symmetry. We characterize the structure of both Archimedean-like tilings in detail and point out how the tilings differ from each other. Furthermore, we also estimate specific particle densities where the Archimedean-like tiling phases occur. Finally, using Brownian dynamics simulations we demonstrate how phasonic distortions of the decagonal laser field influence the Archimedean-like tiling. In particular, the domain size of the tiling can be enlarged by phasonic drifts and constant gradients in the phasonic displacement. We demonstrate that the latter occurs when the interfering laser beams are not ideally adjusted.

Eur. Phys. J. E **32**, 25-34 (2010)

**Proliferation of anomalous symmetries in colloidal monolayers subjected to quasiperiodic light fields****by Jules Mikhael, Michael Schmiedeberg, Sebastian Rausch, Johannes Roth, Holger Stark, and Clemens Bechinger**

Quasicrystals provide a fascinating class of materials with intriguing properties. Despite a strong potential for numerous technical applications, the conditions under which quasicrystals form are still poorly understood. Currently, it is not clear why most quasicrystals hold 5- or 10-fold symmetry but no single example with 7- or 9-fold symmetry has ever been observed. Here we report on geometrical constraints which impede the formation of quasicrystals with certain symmetries in a colloidal model system. Experimentally, colloidal quasicrystals are created by subjecting micron-sized particles to two-dimensional quasiperiodic potential landscapes created by

*n*= 5 or seven laser beams. Our results clearly demonstrate that quasicrystalline order is much easier established for

*n*= 5 compared to

*n*= 7. With increasing laser intensity we observe that the colloids first adopt quasiperiodic order at local areas which then laterally grow until an extended quasicrystalline layer forms. As nucleation sites where quasiperiodicity originates, we identify highly symmetric motifs in the laser pattern. We find that their density strongly varies with

*n*and surprisingly is smallest exactly for those quasicrystalline symmetries which have never been observed in atomic systems. Since such high-symmetry motifs also exist in atomic quasicrystals where they act as preferential adsorption sites, this suggests that it is indeed the deficiency of such motifs which accounts for the absence of materials with e.g., 7-fold symmetry.

PNAS

**107**, 7214-7218 (2010)

**Simulation of a Model Microswimmer****by Matthew T. Downton and Holger Stark**

We discuss the modelling of a microswimmer that operates in a 'squirmer' mode, by means of stochastic rotation dynamics. The squirmer that we model can easily be tuned between a 'pusher' and a 'puller'. We examine the flows produced by the squirmer and find that there is good agreement between both the predicted and simulated velocities of locomotion and the resulting flow field.

J. Phys.: Cond. Matter **21**, 204101 (2009)

**Fluid transport at low Reynolds numbers with magnetically actuated artificial cilia****by Erik M. Gauger, Matthew T. Downton, and Holger Stark**

By numerical modeling we investigate fluid transport in low-Reynolds-number flow achieved with a special elastic filament or artifical cilium attached to a planar surface. The filament is made of superparamagnetic particles linked together by DNA double strands. An external magnetic field induces dipolar interactions between the beads of the filament which provides a convenient way of actuating the cilium in a well-controlled manner. The filament has recently been used to successfully construct the first artificial micro-swimmer (R. Dreyfus *et al.*, Nature **437**, 862 (2005)). In our numerical study we introduce a measure, which we call pumping performance, to quantify the fluid transport induced by the magnetically actuated cilium and identify an optimum stroke pattern of the filament. It consists of a slow transport stroke and a fast recovery stroke. Our detailed parameter study also reveals that for sufficiently large magnetic fields the artificial cilium is mainly governed by the Mason number that compares frictional to magnetic forces. Initial studies on multi-cilia systems show that the pumping performance is very sensitive to the imposed phase lag between neighboring cilia, *i.e.*, to the details of the initiated metachronal wave.

Eur. Phys. J. E **28**, 231-242 (2009)

**Dry and wet interfaces: Influence of solvent particles on molecular recognition****by Johannes Taktikos and Hans Behringer**

We present a coarse-grained lattice model to study the influence of water on the recognition process of two rigid proteins. The basic model is formulated in terms of the hydrophobic effect. We then investigate several modifications of our basic model showing that the selectivity of the recognition process can be enhanced by considering the explicit influence of single solvent particles. When the number of cavities at the interface of a protein-protein complex is fixed as an intrinsic geometric constraint, there typically exists a characteristic fraction that should be filled with water molecules such that the selectivity exhibits a maximum. In addition the optimum fraction depends on the hydrophobicity of the interface so that one

has to distinguish between dry and wet interfaces.

Phys. Rev. E **79**, 041908 (2009)

**Beating kinematics of magnetically actuated cilia**

by Matthew T. Downton and Holger Stark

We study the beating kinematics and pumping performance of a magnetically actuated artificial cilium attached to a surface using a bead spring model. Several different beating patterns for the external field are considered along with the possiblity of defects in the filament at isolated points. Hydrodynamic interactions between the beads are included by a modified Rotne-Prage tensor such that the no-slip boundary condition at the surface is satisfied. We find that the correct positioning of defects along the filament length can lead to significant increases in the pumping performance of a planar beating pattern. Even more efficient for pumping fluid are three-dimensional beating strokes which bring the filament close to the surface during the return part of the stroke.

Europhys. Lett. **85**, 44002 (2009)

**Random Walks with Random Velocities**

**by Vasily Zaburdaev, Michael Schmiedeberg, and Holger Stark**

We consider a random walk model that takes into account the velocity distribution of random walkers. Random motion with alternating velocities is inherent to various physical and biological systems. Moreover, the velocity distribution is often the first characteristic that is experimentally accessible. Here, we derive transport equations describing the dispersal process in the model and solve them analytically. The asymptotic properties of solutions are presented in the form of a phase diagram that shows all possible scaling regimes,including superdiffusive, ballistic, and superballistic motion. The theoretical results of this work are in excellent agreement with accompanying numerical simulations.

Phys. Rev. E **78**, 011119 (2008)

**Colloidal ordering on a 2D quasicrystalline substrate**

**by Michael Schmiedeberg and Holger Stark**

By using Monte Carlo simulations, we study the complex phase behavior of charged-stabilized colloidal particles in a two-dimensional substrate potential with quasicrystalline decagonal symmetry. In the regime where the strengths of the substrate and colloidal pair potential are comparable, we identify a novel and unexpected quasicrystalline phase with pure 20-fold bond order and a disordered structure without any apparent rotational symmetry. Furthermore, we demonstrate how phasonic displacements in the substrate potential induce phasonic flips in the colloidal monolayer.

Phys. Rev. Lett.** 101**, 218302 (2008)

**Immer in Bewegung bleiben: Die sonderbare Welt der kleinen Reynoldszahlen**

**by Holger Stark**

Die Natur hat ausgeklügelte Mechanismen entwickelt, mit denen sich Mikroorganismen wie Bakterien und Spermien in wässriger Lösung fortbewegen oder mit denen sich auf Mikrometerskala Flüssigkeit transportieren lässt, z. B. beim Abtransport von Schleim in der Lunge. Ein tieferes Verständnis der physikalischen Grundlagen dieser Mechanismen ist nicht nur von physiologischer Bedeutung, sondern hilft auch dabei, Mikrometer große künstliche Schwimmer zu konstruieren oder in der technologisch wichtigen Mikrofluidik winzige Flüssigkeitsmengen zu bewegen und zu mischen.

Physik Journal **6**, 31-37 (2007)

**Non-Central Forces in Crystals of Charged Colloids**

**by D. Reinke, H. Stark, H.-H. von Grünberg, A.B. Schofield, G. Maret, and U. Gasser**

The elastic properties of fcc crystals consisting of charged stabilized colloidal particles are determined from real space imaging experiments using confocal microscopy. The normal modes and the force constants of the crystal are obtained from the fluctuations of the particles around their lattice sites using the equipartition theorem. We show that the Cauchy relation is not fulfilled and that only noncentral many-body forces can account for the elastic properties.

Phys. Rev. Lett. **98**, 038301 (2007)

**Melting of a Colloidal Absorbate on a 1D Quasicrystalline Substrate**

**by M. Schmiedeberg, J. Roth, and H. Stark**

Using Monte-Carlo simulations and an extended Landau-Alexander-MCTague theory, we demonstrate that colloids in a one-dimensional quasicrystalline potential order in triangular and rhombic-alpha crystalline phases.

Increasing the strength of the potential further, a new type of light-induced melting is discovered that has its origin in the non-periodicity of the potential. In contrast to reentrant melting in periodic potentials, the quasicrystalline potenial melts the crystalline phases even when they already exist at zero potential.

Phys. Rev. Lett. **97**, 158304 (2006)

**Direct Observation of Hydrodynamic Rotation-Translation Coupling between two colloidal Spheres**

**by S. Martin, M. Reichert, H. Stark, and T. Gisler**

By combining optical tweezers with polarization microscopy, the hydrodynamic coupling between position and orientation fluctuations in a pair of colloidal spheres has been measured.

Imaging of birefringent particles under crossed polarizers allows for the simultaneous determination of the positions and orientations of both particles. The temporal cross-correlation function between random displacements of one particle and orientation fluctuations of its neighbor allows for the quantification of the hydrodynamic rotation-translation coupling between the spheres. Our results are in good agreement with predictions for the hydrodynamic mobility tensors calculated in the creeping-flow limit of the Navier-Stokes equation.

Phys. Rev. Lett. **97**, 248301 (2006)

**Numerical Study of a Microscopic Artificial Swimmer**

**by E. Gauger and H. Stark**

We present a detailed numerical study of a microscopic artificial swimmer realized recently by Dreyfus (em et al.) in experiments [R. Dreyfus (em et al.), Nature, **437**, 862 (2005)].

It consists of an elastic filament composes of superparamagnetic particles that are linked together by DNA strands. Attached to a load particle, the resulting swimmer is actuated by an oscillating external magnetic field so that it performs a non-reciprocal motion in order to move forward.

We model the superparamagnetic filament by a bead-spring configuration that resists bending like a rigid rod and whose beads experience friction with surrounding fluid and hydrodynamic interactions with each other.

Phys. Rev. E **74**, 021907 (2006)

**Surmounting Barriers: The Benefit of Hydrodynamic Interactions**

**by C. Lutz, M. Reichert, H. Stark, and C. Bechinger**

We experimentally and theoretically investigate the collective behavior of three colloidal particles that are driven by a constant force along a toroidal trap. Due to hydrodynamic interactions, a characteristic limit cycle is observed. When we additionally apply a periodic sawtooth potential, we find a novel caterpillar-like motional sequence that is dominated by hydrodynamic interactions and promotes the surmounting of potential barriers by the particles.

Europhys. Lett. **74**, 719 (2006)

**Superdiffusion in a Honeycomb Billiard**

**by M. Schmiedeberg and H. Stark**

We investigate particle transport in the honeycomb billiard that consists of connected channels placed on the edges of a honeycomb structure. The spreading of particles is superdiffusive due to the existence of ballistic trajectories which we term perfect paths.

Simulations give a time exponent of 1.72 for the mean square displacement and a starlike, i.e., anisotropic particle distribution. We present an analytical treatment based on the formalism of continuous-time random walks and explain both the time exponent and the anisotropic distribution.

Phys. Rev. E **73**, 031113 (2006)