Statistical Physics of Soft Matter and Biological Systems

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We work on a broad spectrum of projects from statistical physics of soft condensed matter and biological systems and from optics using a variety of methods from statistical physics and continuum theories.

Soft matter stands for a broad range of materials such as liquid crystals, polymers, colloidal dispersions and membranes. On the one hand, soft matter is in the center of basic research that describes these complex systems based on many different concepts. On the other hand, soft matter is applied in our everyday life and in technology.

Many biological systems consist of soft materials. Therefore, soft matter and its different methods play an important role in biophysics.

Optics, a key technology of our century, is crucial for exploring the complex structures and dynamical processes in soft matter.

News

• Opening for a POSTDOCTORAL FELLOW:
  "Bacterial locomotion by rotating flagella"
  
  
  
• NECD12 - International Conference of GRK 1558
  NonEquilibrium Collective Dynamics: Bridging the Gap between Hard and Soft Materials
  October 1-4, 2012, Potsdam, Germany
  
  
  
• Seminar: Statistical Physics of Soft Matter and Biological Systems (German and English)
  
  
  
• VW Status Symposium "New trends in simulating biological systems and soft matter"
  
  
  
• Issue of European Physical Journal E dedicated to Georg Maret on the occasion of his 60th birthday
  Cover of the february issue 2009 [pdf] (PDF, 1,5 MB)
  
  
• Überblicks-Artikel im Physik Journal
  Immer in Bewegung bleiben: Die sonderbare Welt der kleinen Reynolds-Zahlen
  Holger Stark
  Physik Journal 6 (11), 31 (2007)
  
  
  

Current research

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)