#### Prof. Dr. Jan Kierfeld

Soft Condensed Matter Theory

TU Dortmund University

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**Chemomechanical Simulation of Microtubule Dynamics With Explicit Lateral Bond Dynamics**

Schmidt, M. and Kierfeld, J.*Frontiers in Physics*9 (2021)We introduce and parameterize a chemomechanical model of microtubule dynamics on the dimer level, which is based on the allosteric tubulin model and includes attachment, detachment and hydrolysis of tubulin dimers as well as stretching of lateral bonds, bending at longitudinal junctions, and the possibility of lateral bond rupture and formation. The model is computationally efficient such that we reach sufficiently long simulation times to observe repeated catastrophe and rescue events at realistic tubulin concentrations and hydrolysis rates, which allows us to deduce catastrophe and rescue rates. The chemomechanical model also allows us to gain insight into microscopic features of the GTP-tubulin cap structure and microscopic structural features triggering microtubule catastrophes and rescues. Dilution simulations show qualitative agreement with experiments. We also explore the consequences of a possible feedback of mechanical forces onto the hydrolysis process and the GTP-tubulin cap structure. © Copyright © 2021 Schmidt and Kierfeld.view abstract 10.3389/fphy.2021.673875 **Event-Chain Monte-Carlo Simulations of Dense Soft Matter Systems**

Kampmann, T.A. and Müller, D. and Weise, L.P. and Vorsmann, C.F. and Kierfeld, J.*Frontiers in Physics*9 (2021)We discuss the rejection-free event-chain Monte-Carlo algorithm and several applications to dense soft matter systems. Event-chain Monte-Carlo is an alternative to standard local Markov-chain Monte-Carlo schemes, which are based on detailed balance, for example the well-known Metropolis-Hastings algorithm. Event-chain Monte-Carlo is a Markov chain Monte-Carlo scheme that uses so-called lifting moves to achieve global balance without rejections (maximal global balance). It has been originally developed for hard sphere systems but is applicable to many soft matter systems and particularly suited for dense soft matter systems with hard core interactions, where it gives significant performance gains compared to a local Monte-Carlo simulation. The algorithm can be generalized to deal with soft interactions and with three-particle interactions, as they naturally arise, for example, in bead-spring models of polymers with bending rigidity. We present results for polymer melts, where the event-chain algorithm can be used for an efficient initialization. We then move on to large systems of semiflexible polymers that form bundles by attractive interactions and can serve as model systems for actin filaments in the cytoskeleton. The event chain algorithm shows that these systems form networks of bundles which coarsen similar to a foam. Finally, we present results on liquid crystal systems, where the event-chain algorithm can equilibrate large systems containing additional colloidal disks very efficiently, which reveals the parallel chaining of disks. © Copyright © 2021 Kampmann, Müller, Weise, Vorsmann and Kierfeld.view abstract 10.3389/fphy.2021.635886 **From diffusive mass transfer in Stokes flow to low Reynolds number Marangoni boats**

Ender, H. and Kierfeld, J.*European Physical Journal E*44 (2021)We present a theory for the self-propulsion of symmetric, half-spherical Marangoni boats (soap or camphor boats) at low Reynolds numbers. Propulsion is generated by release (diffusive emission or dissolution) of water-soluble surfactant molecules, which modulate the air–water interfacial tension. Propulsion either requires asymmetric release or spontaneous symmetry breaking by coupling to advection for a perfectly symmetrical swimmer. We study the diffusion–advection problem for a sphere in Stokes flow analytically and numerically both for constant concentration and constant flux boundary conditions. We derive novel results for concentration profiles under constant flux boundary conditions and for the Nusselt number (the dimensionless ratio of total emitted flux and diffusive flux). Based on these results, we analyze the Marangoni boat for small Marangoni propulsion (low Peclet number) and show that two swimming regimes exist, a diffusive regime at low velocities and an advection-dominated regime at high swimmer velocities. We describe both the limit of large Marangoni propulsion (high Peclet number) and the effects from evaporation by approximative analytical theories. The swimming velocity is determined by force balance, and we obtain a general expression for the Marangoni forces, which comprises both direct Marangoni forces from the surface tension gradient along the air–water–swimmer contact line and Marangoni flow forces. We unravel whether the Marangoni flow contribution is exerting a forward or backward force during propulsion. Our main result is the relation between Peclet number and swimming velocity. Spontaneous symmetry breaking and, thus, swimming occur for a perfectly symmetrical swimmer above a critical Peclet number, which becomes small for large system sizes. We find a supercritical swimming bifurcation for a symmetric swimmer and an avoided bifurcation in the presence of an asymmetry. © 2021, The Author(s).view abstract 10.1140/epje/s10189-021-00034-9 **Pressure-deformation relations of elasto-capillary drops (droploons) on capillaries**

Ginot, G. and Kratz, F.S. and Walzel, F. and Farago, J. and Kierfeld, J. and Höhler, R. and Drenckhan, W.*Soft Matter*17 (2021)An increasing number of multi-phase systems exploit complex interfaces in which capillary stresses are coupled with solid-like elastic stresses. Despite growing efforts, simple and reliable experimental characterisation of these interfaces remains a challenge, especially of their dilational properties. Pendant drop techniques are convenient, but suffer from complex shape changes and associated fitting procedures with multiple parameters. Here we show that simple analytical relationships can be derived to describe reliably the pressure-deformation relations of nearly spherical elasto-capillary droplets (“droploons”) attached to a capillary. We consider a model interface in which stresses arising from a constant interfacial tension are superimposed with mechanical extra-stresses arising from the deformation of a solid-like, incompressible interfacial layer of finite thickness described by a neo-Hookean material law. We compare some standard models of liquid-like (Gibbs) and solid-like (Hookean and neo-Hookean elasticity) elastic interfaces which may be used to describe the pressure-deformation relations when the presence of the capillary can be considered negligible. Combining Surface Evolver simulations and direct numerical integration of the drop shape equations, we analyse in depth the influence of the anisotropic deformation imposed by the capillary on the pressure-deformation relation and show that in many experimentally relevant circumstances either the analytical relations of the perfect sphere may be used or a slightly modified relation which takes into account the geometrical change imposed by the capillary. Using the analogy with the stress concentration around a rigid inclusion in an elastic membrane, we provide simple non-dimensional criteria to predict under which conditions the simple analytical expressions can be used to fit pressure-deformation relations to analyse the elastic properties of the interfacesvia“Capillary Pressure Elastometry”. We show that these criteria depend essentially on the drop geometry and deformation, but not on the interfacial elasticity. Moreover, this benchmark case shows for the first time that Surface Evolver is a reliable tool for predictive simulations of elastocapillary interfaces. This opens doors to the treatment of more complex geometries/conditions, where theory is not available for comparison. Our Surface Evolver code is available for download in the ESI. © The Royal Society of Chemistry 2021.view abstract 10.1039/d1sm01109j **Surfactant-loaded capsules as Marangoni microswimmers at the air–water interface: Symmetry breaking and spontaneous propulsion by surfactant diffusion and advection**

Ender, H. and Froin, A.-K. and Rehage, H. and Kierfeld, J.*European Physical Journal E*44 (2021)Abstract: We present a realization of a fast interfacial Marangoni microswimmer by a half-spherical alginate capsule at the air–water interface, which diffusively releases water-soluble spreading molecules (weak surfactants such as polyethylene glycol (PEG)), which act as “fuel” by modulating the air–water interfacial tension. For a number of different fuels, we can observe symmetry breaking and spontaneous propulsion although the alginate particle and emission are isotropic. The propulsion mechanism is similar to soap or camphor boats, which are, however, typically asymmetric in shape or emission to select a swimming direction. We develop a theory of Marangoni boat propulsion starting from low Reynolds numbers by analyzing the coupled problems of surfactant diffusion and advection and fluid flow, which includes surfactant-induced fluid Marangoni flow, and surfactant adsorption at the air–water interface; we also include a possible evaporation of surfactant. The swimming velocity is determined by the balance of drag and Marangoni forces. We show that spontaneous symmetry breaking resulting in propulsion is possible above a critical dimensionless surfactant emission rate (Peclet number). We derive the relation between Peclet number and swimming speed and generalize to higher Reynolds numbers utilizing the concept of the Nusselt number. The theory explains the observed swimming speeds for PEG–alginate capsules, and we unravel the differences to other Marangoni boat systems based on camphor, which are mainly caused by surfactant evaporation from the liquid–air interface. The capsule Marangoni microswimmers also exhibit surfactant-mediated repulsive interactions with walls, which can be qualitatively explained by surfactant accumulation at the wall. Graphic Abstract: [Figure not available: see fulltext.]. © 2021, The Author(s).view abstract 10.1140/epje/s10189-021-00035-8 **Bistability and oscillations in cooperative microtubule and kinetochore dynamics in the mitotic spindle**

Schwietert, F. and Kierfeld, J.*New Journal of Physics*22 (2020)In the mitotic spindle microtubules attach to kinetochores via catch bonds during metaphase, and microtubule depolymerization forces give rise to stochastic chromosome oscillations. We investigate the cooperative stochastic microtubule dynamics in spindle models consisting of ensembles of parallel microtubules, which attach to a kinetochore via elastic linkers. We include the dynamic instability of microtubules and forces on microtubules and kinetochores from elastic linkers. A one-sided model, where an external force acts on the kinetochore is solved analytically employing a mean-field approach based on Fokker-Planck equations. The solution establishes a bistable force-velocity relation of the microtubule ensemble in agreement with stochastic simulations. We derive constraints on linker stiffness and microtubule number for bistability. The bistable force-velocity relation of the one-sided spindle model gives rise to oscillations in the two-sided model, which can explain stochastic chromosome oscillations in metaphase (directional instability). We derive constraints on linker stiffness and microtubule number for metaphase chromosome oscillations. Including poleward microtubule flux into the model we can provide an explanation for the experimentally observed suppression of chromosome oscillations in cells with high poleward flux velocities. Chromosome oscillations persist in the presence of polar ejection forces, however, with a reduced amplitude and a phase shift between sister kinetochores. Moreover, polar ejection forces are necessary to align the chromosomes at the spindle equator and stabilize an alternating oscillation pattern of the two kinetochores. Finally, we modify the model such that microtubules can only exert tensile forces on the kinetochore resulting in a tug-of-war between the two microtubule ensembles. Then, induced microtubule catastrophes after reaching the kinetochore are necessary to stimulate oscillations. The model can reproduce experimental results for kinetochore oscillations in PtK1 cells quantitatively. © 2020 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft.view abstract 10.1088/1367-2630/ab7ede **Chaining of hard disks in nematic needles: particle-based simulation of colloidal interactions in liquid crystals**

Müller, D. and Kampmann, T.A. and Kierfeld, J.*Scientific Reports*10 (2020)Colloidal particles suspended in liquid crystals can exhibit various effective anisotropic interactions that can be tuned and utilized in self-assembly processes. We simulate a two-dimensional system of hard disks suspended in a solution of dense hard needles as a model system for colloids suspended in a nematic lyotropic liquid crystal. The novel event-chain Monte Carlo technique enables us to directly measure colloidal interactions in a microscopic simulation with explicit liquid crystal particles in the dense nematic phase. We find a directional short-range attraction for disks along the director, which triggers chaining parallel to the director and seemingly contradicts the standard liquid crystal field theory result of a quadrupolar attraction with a preferred 45 ∘ angle. Our results can be explained by a short-range density-dependent depletion interaction, which has been neglected so far. Directionality and strength of the depletion interaction are caused by the weak planar anchoring of hard rods. The depletion attraction robustly dominates over the quadrupolar elastic attraction if disks come close. Self-assembly of many disks proceeds via intermediate chaining, which demonstrates that in lyotropic liquid crystal colloids depletion interactions play an important role in structure formation processes. © 2020, The Author(s).view abstract 10.1038/s41598-020-69544-4 **Material properties of interfacial films of mucoid and nonmucoid Pseudomonas aeruginosa isolates**

Balmuri, S.R. and Waters, N.G. and Hegemann, J. and Kierfeld, J. and Niepa, T.H.R.*Acta Biomaterialia*118 (2020)Chronic lung infection with bacterial biofilms is a leading cause of death in cystic fibrosis (CF) patients. Pseudomonas aeruginosa, one of the many species colonizing the lung airways, can undergo pathoadaptation, leading to a mucoid phenotype with interesting material properties. We hypothesize that the surface properties and extracellular materials of mucoid P. aeruginosa cells greatly influence the mechanical behavior of their films at fluid interfaces. In this study, we investigate the interfacial properties of films formed by nonmucoid (PANT) and mucoid (PASL) strains of P. aeruginosa isolated from CF patients. We use pendant drop elastometry to analyze the interfacial response of the films formed by PANT and PASL at the hexadecane-water interface. The dynamic rheological analyses of the films highlight the distinctive signature of the mucoid strains at fluid interfaces. The mucoid films exhibit greater relaxation following a compressive strain than a tensile one, while a full hysteresis response is achieved by the nonmucoid films; this indicates that the material properties of the PANT films are conserved under both compression and tension. The wrinkling and shape analyses of the interfacial bacterial films elucidate that the mucoid strain exhibits remarkable viscoelastic properties, enabling the remodeling of the living films and dissipation of the compressive stress. The comparative analysis of the material properties of mucoid and nonmucoid P. aeruginosa cells indicates that mucoid switch can play an important role in protecting the bacteria from interfacial stresses. Further characterization of interfacial bacterial films will provide new insights into the development of methods for controlling interfacial films of bacteria. © 2020view abstract 10.1016/j.actbio.2020.10.010 **Pendant drop tensiometry: A machine learning approach**

Kratz, F.S. and Kierfeld, J.*Journal of Chemical Physics*153 (2020)Modern pendant drop tensiometry relies on the numerical solution of the Young-Laplace equation and allows us to determine the surface tension from a single picture of a pendant drop with high precision. Most of these techniques solve the Young-Laplace equation many times over to find the material parameters that provide a fit to a supplied image of a real droplet. Here, we introduce a machine learning approach to solve this problem in a computationally more efficient way. We train a deep neural network to determine the surface tension of a given droplet shape using a large training set of numerically generated droplet shapes. We show that the deep learning approach is superior to the current state of the art shape fitting approach in speed and precision, in particular if shapes in the training set reflect the sensitivity of the droplet shape with respect to surface tension. In order to derive such an optimized training set, we clarify the role of the Worthington number as a quality indicator in conventional shape fitting and in the machine learning approach. Our approach demonstrates the capabilities of deep neural networks in the material parameter determination from rheological deformation experiments, in general. © 2020 Author(s).view abstract 10.1063/5.0018814 **Snapping elastic disks as microswimmers: Swimming at low Reynolds numbers by shape hysteresis**

Wischnewski, C. and Kierfeld, J.*Soft Matter*16 (2020)We illustrate a concept for shape-changing microswimmers, which exploits the hysteresis of a shape transition of an elastic object, by an elastic disk undergoing cyclic localized swelling. Driving the control parameter of a hysteretic shape transition in a completely time-reversible manner gives rise to a non-time-reversible shape sequence and a net swimming motion if the elastic object is immersed into a viscous fluid. We prove this concept with a microswimmer which is a flat circular elastic disk that undergoes a transition into a dome-like shape by localized swelling of an inner disk. The control parameter of this shape transition is a scalar swelling factor of the disk material. With a fixed outer frame with an additional attractive interaction in the central region, the shape transition between flat and dome-like shape becomes hysteretic and resembles a hysteretic opening and closing of a scallop. Employing Stokesian dynamics simulations of a discretized version of the disk we show that the swimmer is effectively moving into the direction of the opening of the dome in a viscous fluid if the swelling parameter is changed in a time-reversible manner. The swimming mechanism can be qualitatively reproduced by a simple 9-bead model. This journal is © The Royal Society of Chemistry.view abstract 10.1039/d0sm00741b **Profilin and formin constitute a pacemaker system for robust actin filament growth**

Funk, J. and Merino, F. and Venkova, L. and Heydenreich, L. and Kierfeld, J. and Vargas, P. and Raunser, S. and Piel, M. and Bieling, P.*eLife*8 (2019)The actin cytoskeleton drives many essential biological processes, from cell morphogenesis to motility. Assembly of functional actin networks requires control over the speed at which actin filaments grow. How this can be achieved at the high and variable levels of soluble actin subunits found in cells is unclear. Here we reconstitute assembly of mammalian, non-muscle actin filaments from physiological concentrations of profilin-actin. We discover that under these conditions, filament growth is limited by profilin dissociating from the filament end and the speed of elongation becomes insensitive to the concentration of soluble subunits. Profilin release can be directly promoted by formin actin polymerases even at saturating profilin-actin concentrations. We demonstrate that mammalian cells indeed operate at the limit to actin filament growth imposed by profilin and formins. Our results reveal how synergy between profilin and formins generates robust filament growth rates that are resilient to changes in the soluble subunit concentration. © 2019, eLife Sciences Publications Ltd. All rights reserved.view abstract 10.7554/eLife.50963 **Shallow shell theory of the buckling energy barrier: From the Pogorelov state to softening and imperfection sensitivity close to the buckling pressure**

Baumgarten, L. and Kierfeld, J.*Physical Review E*99 (2019)We study the axisymmetric response of a complete spherical shell under homogeneous compressive pressure p to an additional point force. For a pressure p below the classical critical buckling pressure pc, indentation by a point force does not lead to spontaneous buckling but an energy barrier has to be overcome. The states at the maximum of the energy barrier represent a subcritical branch of unstable stationary points, which are the transition states to a snap-through buckled state. Starting from nonlinear shallow shell theory, we obtain a closed analytical expression for the energy barrier height, which facilitates its effective numerical evaluation as a function of pressure by continuation techniques. We find a clear crossover between two regimes: For p/pc?1 the postbuckling barrier state is a mirror-inverted Pogorelov dimple, and for (1-p/pc)?1 the barrier state is a shallow dimple with indentations smaller than shell thickness and exhibits extended oscillations, which are well described by linear response. We find systematic expansions of the nonlinear shallow shell equations about the Pogorelov mirror-inverted dimple for p/pc?1 and the linear response state for (1-p/pc)?1, which enable us to derive asymptotic analytical results for the energy barrier landscape in both regimes. Upon approaching the buckling bifurcation at pc from below, we find a softening of an ideal spherical shell. The stiffness for the linear response to point forces vanishes ∞(1-p/pc)1/2; the buckling energy barrier vanishes ∞(1-p/pc)3/2; and the shell indentation in the barrier state vanishes ∞(1-p/pc)1/2. This makes shells sensitive to imperfections which can strongly reduce pc in an avoided buckling bifurcation. We find the same softening scaling in the vicinity of the reduced critical buckling pressure also in the presence of imperfections. We can also show that the effect of axisymmetric imperfections on the buckling instability is identical to the effect of a point force that is preindenting the shell. In the Pogorelov limit, the energy barrier maximum diverges ∞(p/pc)-3 and the corresponding indentation diverges ∞(p/pc)-2. Numerical prefactors for proportionalities both in the softening and the Pogorelov regime are calculated analytically. This also enables us to obtain results for the critical unbuckling pressure and the Maxwell pressure. © 2019 American Physical Society.view abstract 10.1103/PhysRevE.99.022803 **Buckling of thermally fluctuating spherical shells: Parameter renormalization and thermally activated barrier crossing**

Baumgarten, L. and Kierfeld, J.*Physical Review E*97 (2018)We study the influence of thermal fluctuations on the buckling behavior of thin elastic capsules with spherical rest shape. Above a critical uniform pressure, an elastic capsule becomes mechanically unstable and spontaneously buckles into a shape with an axisymmetric dimple. Thermal fluctuations affect the buckling instability by two mechanisms. On the one hand, thermal fluctuations can renormalize the capsule's elastic properties and its pressure because of anharmonic couplings between normal displacement modes of different wavelengths. This effectively lowers its critical buckling pressure [Košmrlj and Nelson, Phys. Rev. X 7, 011002 (2017)2160-330810.1103/PhysRevX.7.011002]. On the other hand, buckled shapes are energetically favorable already at pressures below the classical buckling pressure. At these pressures, however, buckling requires to overcome an energy barrier, which only vanishes at the critical buckling pressure. In the presence of thermal fluctuations, the capsule can spontaneously overcome an energy barrier of the order of the thermal energy by thermal activation already at pressures below the critical buckling pressure. We revisit parameter renormalization by thermal fluctuations and formulate a buckling criterion based on scale-dependent renormalized parameters to obtain a temperature-dependent critical buckling pressure. Then we quantify the pressure-dependent energy barrier for buckling below the critical buckling pressure using numerical energy minimization and analytical arguments. This allows us to obtain the temperature-dependent critical pressure for buckling by thermal activation over this energy barrier. Remarkably, both parameter renormalization and thermal activation lead to the same parameter dependence of the critical buckling pressure on temperature, capsule radius and thickness, and Young's modulus. Finally, we study the combined effect of parameter renormalization and thermal activation by using renormalized parameters for the energy barrier in thermal activation to obtain our final result for the temperature-dependent critical pressure, which is significantly below the results if only parameter renormalization or only thermal activation is considered. © 2018 American Physical Society.view abstract 10.1103/PhysRevE.97.052801 **Elastic capsules at liquid-liquid interfaces**

Hegemann, J. and Boltz, H.-H. and Kierfeld, J.*Soft Matter*14 (2018)We investigate the deformation of elastic microcapsules adsorbed at liquid-liquid interfaces. An initially spherical elastic capsule at a liquid-liquid interface undergoes circumferential stretching due to the liquid-liquid surface tension and becomes lens- or discus-shaped, depending on its bending rigidity. The resulting elastic capsule deformation is qualitatively similar, but distinct from the deformation of a liquid droplet into a liquid lens at a liquid-liquid interface. We discuss the deformed shapes of droplets and capsules adsorbed at liquid-liquid interfaces for a whole range of different surface elasticities: from droplets (only surface tension) deforming into liquid lenses, droplets with a Hookean membrane (finite stretching modulus, zero bending modulus) deforming into elastic lenses, to microcapsules (finite stretching and bending modulus) deforming into rounded elastic lenses. We calculate capsule shapes at liquid-liquid interfaces numerically using shape equations from nonlinear elastic shell theory. Finally, we present theoretical results for the contact angle (or the capsule height) and the maximal capsule curvature at the three phase contact line. These results can be used to infer information about the elastic moduli from optical measurements. During capsule deformation into a lens-like shape, surface energy of the liquid-liquid interface is converted into elastic energy of the capsule shell giving rise to an overall adsorption energy gain by deformation. Soft hollow capsules exhibit a pronounced increase of the adsorption energy as compared to filled soft particles and, thus, are attractive candidates as foam and emulsion stabilizers. © 2018 The Royal Society of Chemistry.view abstract 10.1039/c8sm00316e **Pendant capsule elastometry**

Hegemann, J. and Knoche, S. and Egger, S. and Kott, M. and Demand, S. and Unverfehrt, A. and Rehage, H. and Kierfeld, J.*Journal of Colloid and Interface Science*513 (2018)We provide a C/C++ software for the shape analysis of deflated elastic capsules in a pendant capsule geometry, which is based on an elastic description of the capsule material as a quasi two-dimensional elastic membrane using shell theory. Pendant capsule elastometry provides a new in situ and non-contact method for interfacial rheology of elastic capsules that goes beyond determination of the Gibbs- or dilational modulus from area-dependent measurements of the surface tension using pendant drop tensiometry, which can only give a rough estimate of the elastic capsule properties as they are based on a purely liquid interface model. Given an elastic model of the capsule membrane, pendant capsule elastometry determines optimal elastic moduli by fitting numerically generated axisymmetric shapes optimally to an experimental image. For each digitized image of a deflated capsule elastic moduli can be determined, if another image of its undeformed reference shape is provided. Within this paper, we focus on nonlinear Hookean elasticity because of its low computational cost and its wide applicability, but also discuss and implement alternative constitutive laws. For Hookean elasticity, Young's surface modulus (or, alternatively, area compression modulus) and Poisson's ratio are determined; for Mooney-Rivlin elasticity, the Rivlin modulus and a dimensionless shape parameter are determined; for neo-Hookean elasticity, only the Rivlin modulus is determined, using a fixed dimensionless shape parameter. Comparing results for different models we find that nonlinear Hookean elasticity is adequate for most capsules. If series of images are available, these moduli can be evaluated as a function of the capsule volume to analyze hysteresis or aging effects depending on the deformation history or to detect viscoelastic effects for different volume change rates. An additional wrinkling wavelength measurement allows the user to determine the bending modulus, from which the layer thickness can be derived. We verify the method by analyzing several materials, compare the results to available rheological measurements, and review several applications. We make the software available under the GPL license at github.com/jhegemann/opencapsule. © 2017view abstract 10.1016/j.jcis.2017.11.048 **Spheroidal and conical shapes of ferrofluid-filled capsules in magnetic fields**

Wischnewski, C. and Kierfeld, J.*Physical Review Fluids*3 (2018)We investigate the deformation of soft spherical elastic capsules filled with a ferrofluid in external uniform magnetic fields at fixed volume by a combination of numerical and analytical approaches. We develop a numerical iterative solution strategy based on nonlinear elastic shape equations to calculate the stretched capsule shape numerically and a coupled finite element and boundary element method to solve the corresponding magnetostatic problem and employ analytical linear response theory, approximative energy minimization, and slender-body theory. The observed deformation behavior is qualitatively similar to the deformation of ferrofluid droplets in uniform magnetic fields. Homogeneous magnetic fields elongate the capsule and a discontinuous shape transition from a spheroidal shape to a conical shape takes place at a critical field strength. We investigate how capsule elasticity modifies this hysteretic shape transition. We show that conical capsule shapes are possible but involve diverging stretch factors at the tips, which gives rise to rupture for real capsule materials. In a slender-body approximation we find that the critical susceptibility above which conical shapes occur for ferrofluid capsules is the same as for droplets. At small fields capsules remain spheroidal and we characterize the deformation of spheroidal capsules both analytically and numerically. Finally, we determine whether wrinkling of a spheroidal capsule occurs during elongation in a magnetic field and how it modifies the stretching behavior. We find the nontrivial dependence between the extent of the wrinkled region and capsule elongation. Our results can be helpful in quantitatively determining capsule or ferrofluid material properties from magnetic deformation experiments. All results also apply to elastic capsules filled with a dielectric liquid in an external uniform electric field. © 2018 American Physical Society.view abstract 10.1103/PhysRevFluids.3.043603 **Strong Deformation of Ferrofluid-Filled Elastic Alginate Capsules in Inhomogenous Magnetic Fields**

Wischnewski, C. and Zwar, E. and Rehage, H. and Kierfeld, J.*Langmuir*34 (2018)We present a new system based on alginate gels for the encapsulation of a ferrofluid drop, which allows us to create millimeter-sized elastic capsules that are highly deformable by inhomogeneous magnetic fields. We use a combination of experimental and theoretical work in order to characterize and quantify the deformation behavior of these ferrofluid-filled capsules. We introduce a novel method for the direct encapsulation of unpolar liquids by sodium alginate. By adding 1-hexanol to the unpolar liquid, we can dissolve sufficient amounts of CaCl2 in the resulting mixture for ionotropic gelation of sodium alginate. The addition of polar alcohol molecules allows us to encapsulate a ferrofluid as a single phase rather than an emulsion without impairing ferrofluid stability. This encapsulation method increases the amount of encapsulated magnetic nanoparticles resulting in high deformations of approximately 30% (in height-to-width ratio) in inhomogeneous magnetic field with magnetic field variations of 50 mT over the size of the capsule. This offers possible applications of capsules as actuators, switches, or valves in confined spaces like microfluidic devices. We determine both elastic moduli of the capsule shell, Young's modulus and Poisson's ratio, by employing two independent mechanical methods, spinning capsule measurements and capsule compression between parallel plates. We then show that the observed magnetic deformation can be fully understood from magnetic forces exerted by the ferrofluid on the capsule shell if the magnetic field distribution and magnetization properties of the ferrofluid are known. We perform a detailed analysis of the magnetic deformation by employing a theoretical model based on nonlinear elasticity theory. Using an iterative solution scheme that couples a finite element/boundary element method for the magnetic field calculation to the solution of the elastic shape equations, we achieve quantitative agreement between theory and experiment for deformed capsule shapes using the Young modulus from mechanical characterization and the surface Poisson ratio as a fit parameter. This detailed analysis confirms the results from mechanical characterization that the surface Poisson ratio of the alginate shell is close to unity, that is, deformations of the alginate shell are almost area conserving. © 2018 American Chemical Society.view abstract 10.1021/acs.langmuir.8b02357 **Adsorption of finite semiflexible polymers and their loop and tail distributions**

Kampmann, T.A. and Kierfeld, J.*Journal of Chemical Physics*147 (2017)We discuss the adsorption of semiflexible polymers to a planar attractive wall and focus on the questions of the adsorption threshold for polymers of finite length and their loop and tail distributions using both Monte Carlo simulations and analytical arguments. For the adsorption threshold, we find three regimes: (i) a flexible or Gaussian regime if the persistence length is smaller than the adsorption potential range, (ii) a semiflexible regime if the persistence length is larger than the potential range, and (iii) for finite polymers, a novel crossover to a rigid rod regime if the deflection length exceeds the contour length. In the flexible and semiflexible regimes, finite size corrections arise because the correlation length exceeds the contour length. In the rigid rod regime, however, it is essential how the global orientational or translational degrees of freedom are restricted by grafting or confinement. We discuss finite size corrections for polymers grafted to the adsorbing surface and for polymers confined by a second (parallel) hard wall. Based on these results, we obtain a method to analyze adsorption data for finite semiflexible polymers such as filamentous actin. For the loop and tail distributions, we find power laws with an exponential decay on length scales exceeding the correlation length. We derive and confirm the loop and tail power law exponents for flexible and semiflexible polymers. This allows us to explain that, close to the transition, semiflexible polymers have significantly smaller loops and both flexible and semiflexible polymers desorb by expanding their tail length. The tail distribution allows us to extract the free energy per length of adsorption for actin filaments from experimental data [D. Welch et al., Soft Matter 11, 7507 (2015)]. © 2017 Author(s).view abstract 10.1063/1.4990418 **Event-chain Monte Carlo algorithms for three- and many-particle interactions**

Harland, J. and Michel, M. and Kampmann, T.A. and Kierfeld, J.*EPL*117 (2017)We generalize the rejection-free event-chain Monte Carlo algorithm from many-particle systems with pairwise interactions to systems with arbitrary three- or many-particle interactions. We introduce generalized lifting probabilities between particles and obtain a general set of equations for lifting probabilities, the solution of which guarantees maximal global balance. We validate the resulting three-particle event-chain Monte Carlo algorithms on three different systems by comparison with conventional local Monte Carlo simulations: i) a test system of three particles with a three-particle interaction that depends on the enclosed triangle area; ii) a hard-needle system in two dimensions, where needle interactions constitute three-particle interactions of the needle end points; iii) a semiflexible polymer chain with a bending energy, which constitutes a three-particle interaction of neighboring chain beads. The examples demonstrate that the generalization to many-particle interactions broadens the applicability of event-chain algorithms considerably. Copyright © 2017 EPLA.view abstract 10.1209/0295-5075/117/30001 **Microcapsule Buckling Triggered by Compression-Induced Interfacial Phase Change**

Salmon, A.R. and Parker, R.M. and Groombridge, A.S. and Maestro, A. and Coulston, R.J. and Hegemann, J. and Kierfeld, J. and Scherman, O.A. and Abell, C.*Langmuir*32 (2016)There is an emerging trend toward the fabrication of microcapsules at liquid interfaces. In order to control the parameters of such capsules, the interfacial processes governing their formation must be understood. Here, poly(vinyl alcohol) films are assembled at the interface of water-in-oil microfluidic droplets. The polymer is cross-linked using cucurbit[8]uril ternary supramolecular complexes. It is shown that compression-induced phase change causes the onset of buckling in the interfacial film. On evaporative compression, the interfacial film both increases in density and thickens, until it reaches a critical density and a phase change occurs. We show that this increase in density can be simply related to the film Poisson ratio and area compression. This description captures fundamentals of many compressive interfacial phase changes and can also explain the observation of a fixed thickness-to-radius ratio at buckling, (TR)buck. © 2016 American Chemical Society.view abstract 10.1021/acs.langmuir.6b03011 **Stationary shapes of deformable particles moving at low Reynolds numbers**

Boltz, H. H. and Kierfeld, J.*European Physical Journal-special Topics*225 (2016)We introduce an iterative solution scheme in order to calculate stationary shapes of deformable elastic capsules which are steadily moving through a viscous fluid at low Reynolds numbers. The iterative solution scheme couples hydrodynamic boundary integral methods and elastic shape equations to find the stationary axisymmetric shape and the velocity of an elastic capsule moving in a viscous fluid governed by the Stokes equation. We use this approach to systematically study dynamical shape transitions of capsules with Hookean stretching and bending energies and spherical resting shape sedimenting under the influence of gravity or centrifugal forces. We find three types of possible axisymmetric stationary shapes for sedimenting capsules with fixed volume: a pseudospherical state, a pear-shaped state, and buckled shapes. Capsule shapes are controlled by two dimensionless parameters, the Foppl-von-Karman number characterizing the elastic properties and a Bond number characterizing the driving force. For increasing gravitational force the spherical shape transforms into a pear shape. For very large bending rigidity (very small Foppl-von-Karman number) this transition is discontinuous with shape hysteresis. The corresponding transition line terminates, however, in a critical point, such that the discontinuous transition is not present at typical Foppl-von-Karman numbers of synthetic capsules. In an additional bifurcation, buckled shapes occur upon increasing the gravitational force.view abstract 10.1140/epjst/e2016-60059-7 **Elasticity of interfacial rafts of hard particles with soft shells**

Knoche, S. and Kierfeld, J.*Langmuir*31 (2015)We study an elasticity model for compressed protein monolayers or particle rafts at a liquid interface. Based on the microscopic view of hard-core particles with soft shells, a bead-spring model is formulated and analyzed in terms of continuum elasticity theory. The theory can be applied, for example, to hydrophobin-coated air-water interfaces or, more generally, to liquid interfaces coated with an adsorbed monolayer of interacting hard-core particles. We derive constitutive relations for such particle rafts and describe the buckling of compressed planar liquid interfaces as well as their apparent Poisson ratio. We also use the constitutive relations to obtain shape equations for pendant or buoyant capsules attached to a capillary, and to compute deflated shapes of such capsules. A comparison with capsules obeying the usual Hookean elasticity (without hard cores) reveals that the hard cores trigger capsule wrinkling. Furthermore, it is shown that a shape analysis of deflated capsules with hard-core/soft-shell elasticity gives apparent elastic moduli which can be much higher than the original values if Hookean elasticity is assumed. © 2015 American Chemical Society.view abstract 10.1021/acs.langmuir.5b00083 **Monte Carlo simulation of dense polymer melts using event chain algorithms**

Kampmann, T.A. and Boltz, H.-H. and Kierfeld, J.*Journal of Chemical Physics*143 (2015)We propose an efficient Monte Carlo algorithm for the off-lattice simulation of dense hard sphere polymer melts using cluster moves, called event chains, which allow for a rejection-free treatment of the excluded volume. Event chains also allow for an efficient preparation of initial configurations in polymer melts. We parallelize the event chain Monte Carlo algorithm to further increase simulation speeds and suggest additional local topology-changing moves ("swap" moves) to accelerate equilibration. By comparison with other Monte Carlo and molecular dynamics simulations, we verify that the event chain algorithm reproduces the correct equilibrium behavior of polymer chains in the melt. By comparing intrapolymer diffusion time scales, we show that event chain Monte Carlo algorithms can achieve simulation speeds comparable to optimized molecular dynamics simulations. The event chain Monte Carlo algorithm exhibits Rouse dynamics on short time scales. In the absence of swap moves, we find reptation dynamics on intermediate time scales for long chains. © 2015 AIP Publishing LLC.view abstract 10.1063/1.4927084 **Parallelized event chain algorithm for dense hard sphere and polymer systems**

Kampmann, T.A. and Boltz, H.-H. and Kierfeld, J.*Journal of Computational Physics*281 (2015)We combine parallelization and cluster Monte Carlo for hard sphere systems and present a parallelized event chain algorithm for the hard disk system in two dimensions. For parallelization we use a spatial partitioning approach into simulation cells. We find that it is crucial for correctness to ensure detailed balance on the level of Monte Carlo sweeps by drawing the starting sphere of event chains within each simulation cell with replacement. We analyze the performance gains for the parallelized event chain and find a criterion for an optimal degree of parallelization. Because of the cluster nature of event chain moves massive parallelization will not be optimal. Finally, we discuss first applications of the event chain algorithm to dense polymer systems, i.e., bundle-forming solutions of attractive semiflexible polymers. © 2014 Elsevier Inc.view abstract 10.1016/j.jcp.2014.10.059 **Shapes of sedimenting soft elastic capsules in a viscous fluid**

Boltz, H.-H. and Kierfeld, J.*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*92 (2015)Soft elastic capsules which are driven through a viscous fluid undergo shape deformation coupled to their motion. We introduce an iterative solution scheme which couples hydrodynamic boundary integral methods and elastic shape equations to find the stationary axisymmetric shape and the velocity of an elastic capsule moving in a viscous fluid at low Reynolds numbers. We use this approach to systematically study dynamical shape transitions of capsules with Hookean stretching and bending energies and spherical rest shape sedimenting under the influence of gravity or centrifugal forces. We find three types of possible axisymmetric stationary shapes for sedimenting capsules with fixed volume: a pseudospherical state, a pear-shaped state, and buckled shapes. Capsule shapes are controlled by two dimensionless parameters, the Föppl-von-Kármán number characterizing the elastic properties and a Bond number characterizing the driving force. For increasing gravitational force the spherical shape transforms into a pear shape. For very large bending rigidity (very small Föppl-von-Kármán number) this transition is discontinuous with shape hysteresis. The corresponding transition line terminates, however, in a critical point, such that the discontinuous transition is not present at typical Föppl-von-Kármán numbers of synthetic capsules. In an additional bifurcation, buckled shapes occur upon increasing the gravitational force. This type of instability should be observable for generic synthetic capsules. All shape bifurcations can be resolved in the force-velocity relation of sedimenting capsules, where up to three capsule shapes with different velocities can occur for the same driving force. All three types of possible axisymmetric stationary shapes are stable with respect to rotation during sedimentation. Additionally, we study capsules pushed or pulled by a point force, where we always find capsule shapes to transform smoothly without bifurcations. © 2015 American Physical Society.view abstract 10.1103/PhysRevE.92.033003 **Depinning of stiff directed lines in random media**

Boltz, H.-H. and Kierfeld, J.*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*90 (2014)Driven elastic manifolds in random media exhibit a depinning transition to a state with nonvanishing velocity at a critical driving force. We study the depinning of stiff directed lines, which are governed by a bending rigidity rather than line tension. Their equation of motion is the (quenched) Herring-Mullins equation, which also describes surface growth governed by surface diffusion. Stiff directed lines are particularly interesting as there is a localization transition in the static problem at a finite temperature and the commonly exploited time ordering of states by means of Middleton's theorems [Phys. Rev. Lett. 68, 670 (1992)PRLTAO0031-900710.1103/PhysRevLett.68.670] is not applicable. We employ analytical arguments and numerical simulations to determine the critical exponents and compare our findings with previous works and functional renormalization group results, which we extend to the different line elasticity. We see evidence for two distinct correlation length exponents. © 2014 American Physical Society.view abstract 10.1103/PhysRevE.90.012101 **Effects of microtubule mechanics on hydrolysis and catastrophes**

Müller, N. and Kierfeld, J.*Physical Biology*11 (2014)We introduce a model for microtubule (MT) mechanics containing lateral bonds between dimmers in neighboring protofilaments, bending rigidity of dimers, and repulsive interactions between protofilaments modeling steric constraints to investigate the influence of mechanical forces on hydrolysis and catastrophes. We use the allosteric dimer model, where tubulin dimers are characterized by an equilibrium bending angle, which changes from 0°to 22°by hydrolysis of a dimer. This also affects the lateral interaction and bending energies and, thus, the mechanical equilibrium state of the MT. As hydrolysis gives rise to conformational changes in dimers, mechanical forces also influence the hydrolysis rates by mechanical energy changes modulating the hydrolysis rate. The interaction via the MT mechanics then gives rise to correlation effects in the hydrolysis dynamics, which have not been taken into account before. Assuming a dominant influence of mechanical energies on hydrolysis rates, we investigate the most probable hydrolysis pathways both for vectorial and random hydrolysis. Investigating the stability with respect to lateral bond rupture, we identify initiation configurations for catastrophes along the hydrolysis pathways and values for a lateral bond rupture force. If we allow for rupturing of lateral bonds between dimers in neighboring protofilaments above this threshold force, our model exhibits avalanche-like catastrophe events. © 2014 IOP Publishing Ltd Printed in the UK.view abstract 10.1088/1478-3975/11/4/046001 **Feedback mechanism for microtubule length regulation by stathmin gradients**

Zeitz, M. and Kierfeld, J.*Biophysical Journal*107 (2014)We formulate and analyze a theoretical model for the regulation of microtubule (MT) polymerization dynamics by the signaling proteins Rac1 and stathmin. In cells, the MT growth rate is inhibited by cytosolic stathmin, which, in turn, is inactivated by Rac1. Growing MTs activate Rac1 at the cell edge, which closes a positive feedback loop. We investigate both tubulin sequestering and catastrophe promotion as mechanisms for MT growth inhibition by stathmin. For a homogeneous stathmin concentration in the absence of Rac1, we find a switchlike regulation of the MT mean length by stathmin. For constitutively active Rac1 at the cell edge, stathmin is deactivated locally, which establishes a spatial gradient of active stathmin. In this gradient, we find a stationary bimodal MT-length distribution for both mechanisms of MT growth inhibition by stathmin. One subpopulation of the bimodal length distribution can be identified with fast-growing and long pioneering MTs in the region near the cell edge, which have been observed experimentally. The feedback loop is closed through Rac1 activation by MTs. For tubulin sequestering by stathmin, this establishes a bistable switch with two stable states: one stable state corresponds to upregulated MT mean length and bimodal MT length distributions, i.e., pioneering MTs; the other stable state corresponds to an interrupted feedback with short MTs. Stochastic effects as well as external perturbations can trigger switching events. For catastrophe-promoting stathmin, we do not find bistability. © 2014 Biophysical Society.view abstract 10.1016/j.bpj.2014.10.056 **Osmotic buckling of spherical capsules**

Knoche, S. and Kierfeld, J.*Soft Matter*10 (2014)We study the buckling of elastic spherical shells under osmotic pressure with the osmolyte concentration of the exterior solution as a control parameter. We compare our results for the bifurcation behavior with results for buckling under mechanical pressure control, that is, with an empty capsule interior. We find striking differences for the buckling states between osmotic and mechanical buckling. Mechanical pressure control always leads to fully collapsed states with opposite sides in contact, whereas uncollapsed states with a single finite dimple are generic for osmotic pressure control. For sufficiently large interior osmolyte concentrations, osmotic pressure control is qualitatively similar to buckling under volume control with the volume prescribed by the osmolyte concentrations inside and outside the shell. We present a quantitative theory which also captures the influence of shell elasticity on the relationship between osmotic pressure and volume. These findings are relevant for the control of buckled shapes in applications. We show how the osmolyte concentration can be used to control the volume of buckled shells. An accurate analytical formula is derived for the relationship between the osmotic pressure, the elastic moduli and the volume of buckled capsules. This also allows use of elastic capsules as osmotic pressure sensors or deduction of elastic properties and the internal osmolyte concentration from shape changes in response to osmotic pressure changes. We apply our findings to published experimental data on polyelectrolyte capsules. This journal is © the Partner Organisations 2014.view abstract 10.1039/c4sm01205d **Secondary polygonal instability of buckled spherical shells**

Knoche, S. and Kierfeld, J.*EPL*106 (2014)When a spherical elastic capsule is deflated, it first buckles axisymmetrically and subsequently loses its axisymmetry in a secondary instability, where the dimple acquires a polygonal shape. We explain this secondary polygonal buckling in terms of wrinkles developing at the inner side of the dimple edge in response to compressive hoop stress. Analyzing the axisymmetric buckled shape, we find a compressive hoop stress with parabolic stress profile at the dimple edge. We further show that there exists a critical value for this hoop stress, where it becomes favorable for the membrane to buckle out of its axisymmetric shape, thus releasing the compression. The instability mechanism is analogous to the formation of wrinkles under compressive stress. A simplified stability analysis allows us to quantify the critical stress for secondary buckling. Applying this secondary buckling criterion to the axisymmetric shapes, we can determine the critical volume for secondary buckling. Our analytical result is in close agreement with existing numerical data. © 2014 EPLA.view abstract 10.1209/0295-5075/106/24004 **The secondary buckling transition: Wrinkling of buckled spherical shells**

Knoche, S. and Kierfeld, J.*European Physical Journal E*37 (2014)Abstract: We theoretically explain the complete sequence of shapes of deflated spherical shells. Decreasing the volume, the shell remains spherical initially, then undergoes the classical buckling instability, where an axisymmetric dimple appears, and, finally, loses its axisymmetry by wrinkles developing in the vicinity of the dimple edge in a secondary buckling transition. We describe the first axisymmetric buckling transition by numerical integration of the complete set of shape equations and an approximate analytic model due to Pogorelov. In the buckled shape, both approaches exhibit a locally compressive hoop stress in a region where experiments and simulations show the development of polygonal wrinkles, along the dimple edge. In a simplified model based on the stability equations of shallow shells, a critical value for the compressive hoop stress is derived, for which the compressed circumferential fibres will buckle out of their circular shape in order to release the compression. By applying this wrinkling criterion to the solutions of the axisymmetric models, we can calculate the critical volume for the secondary buckling transition. Using the Pogorelov approach, we also obtain an analytical expression for the critical volume at the secondary buckling transition: The critical volume difference scales linearly with the bending stiffness, whereas the critical volume reduction at the classical axisymmetric buckling transition scales with the square root of the bending stiffness. These results are confirmed by another stability analysis in the framework of Donnel, Mushtari and Vlasov (DMV) shell theory, and by numerical simulations available in the literature. Graphical abstract: [Figure not available: see fulltext.] © 2014, EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg.view abstract 10.1140/epje/i2014-14062-9 **Wrinkling of random and regular semiflexible polymer networks**

Müller, P. and Kierfeld, J.*Physical Review Letters*112 (2014)We investigate wrinkling of two-dimensional random and triangular semiflexible polymer networks under shear. Both types of semiflexible networks exhibit wrinkling above a small critical shear angle, which scales with an exponent of the bending modulus between 1.9 and 2.0. Random networks exhibit hysteresis at the wrinkling threshold. Wrinkling lowers the total elastic energy by up to 20% and strongly affects the elastic properties of all semiflexible networks such as the crossover between bending and stretching dominated behavior. In random networks, we also find evidence for metastable wrinkled configurations. While the disordered microstructure of random networks affects the scaling behavior of wrinkle amplitudes, it has little effect on wrinkle wavelength. Therefore, wrinkles represent a robust, microstructure-independent assay of shear strain or elastic properties. © 2014 American Physical Society.view abstract 10.1103/PhysRevLett.112.094303 **Bifurcation of velocity distributions in cooperative transport of filaments by fast and slow motors**

Li, X. and Lipowsky, R. and Kierfeld, J.*Biophysical Journal*104 (2013)Several intracellular processes are governed by two different species of molecular motors, fast and slow ones, that both move in the same direction along the filaments but with different velocities. The transport of filaments arising from the cooperative action of these motors has been recently studied by three in vitro experiments, in which the filament velocity was measured for varying fraction of the fast motors adsorbed onto substrate surfaces in a gliding assay. As the fast motor fraction was increased, two experiments found a smooth change whereas the third one observed an abrupt increase of the filament velocity. Here, we show that all of these experimental results reflect the competition between fast and slow motors and can be understood in terms of an underlying saddle-node bifurcation. The comparison between theory and experiment leads to predictions for the detachment forces of the two motor species. Our theoretical study shows the existence of three different motility regimes: 1), fast transport with a single velocity; 2), slow transport with a single velocity; and 3), bistable transport, where the filament velocity stochastically switches between fast and slow transport. We determine the parameter regions for these regimes in terms of motility diagrams as a function of the surface fraction of fast motors and microscopic single-motor parameters. An abrupt increase of the filament velocity for an increasing fraction of fast motors is associated with the occurrence of bistable transport. © 2013 Biophysical Society.view abstract 10.1016/j.bpj.2012.11.3834 **Controlling adsorption of semiflexible polymers on planar and curved substrates**

Kampmann, T.A. and Boltz, H.-H. and Kierfeld, J.*Journal of Chemical Physics*139 (2013)We study the adsorption of semiflexible polymers such as polyelectrolytes or DNA on planar and curved substrates, e.g., spheres or washboard substrates via short-range potentials using extensive Monte Carlo simulations, scaling arguments, and analytical transfer matrix techniques. We show that the adsorption threshold of stiff or semiflexible polymers on a planar substrate can be controlled by polymer stiffness: adsorption requires the highest potential strength if the persistence length of the polymer matches the range of the adsorption potential. On curved substrates, i.e., an adsorbing sphere or an adsorbing washboard surface, the adsorption can be additionally controlled by the curvature of the surface structure. The additional bending energy in the adsorbed state leads to an increase of the critical adsorption strength, which depends on the curvature radii of the substrate structure. For an adsorbing sphere, this gives rise to an optimal polymer stiffness for adsorption, i.e., a local minimum in the critical potential strength for adsorption, which can be controlled by curvature. For two- and three-dimensional washboard substrates, we identify the range of persistence lengths and the mechanisms for an effective control of the adsorption threshold by the substrate curvature. © 2013 AIP Publishing LLC.view abstract 10.1063/1.4813021 **Cooperative dynamics of microtubule ensembles: Polymerization forces and rescue-induced oscillations**

Zelinski, B. and Kierfeld, J.*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*87 (2013)We investigate the cooperative dynamics of an ensemble of N microtubules growing against an elastic barrier. Microtubules undergo so-called catastrophes, which are abrupt stochastic transitions from a growing to a shrinking state, and rescues, which are transitions back to the growing state. Microtubules can exert pushing or polymerization forces on an obstacle, such as an elastic barrier, if the growing end is in contact with the obstacle. We use dynamical mean-field theory and stochastic simulations to analyze a model where each microtubule undergoes catastrophes and rescues and where microtubules interact by force sharing. For zero rescue rate, cooperative growth terminates in a collective catastrophe. The maximal polymerization force before catastrophes grows linearly with N for small N or a stiff elastic barrier, in agreement with available experimental results, whereas it crosses over to a logarithmic dependence for larger N or a soft elastic barrier. For a nonzero rescue rate and a soft elastic barrier, the dynamics becomes oscillatory with both collective catastrophe and rescue events, which are part of a robust limit cycle. Both the average and maximal polymerization forces then grow linearly with N, and we investigate their dependence on tubulin on-rates and rescue rates, which can be involved in cellular regulation mechanisms. We further investigate the robustness of the collective catastrophe and rescue oscillations with respect to different catastrophe models. © 2013 American Physical Society.view abstract 10.1103/PhysRevE.87.012703 **Elastometry of deflated capsules: Elastic moduli from shape and wrinkle analysis**

Knoche, S. and Vella, D. and Aumaitre, E. and Degen, P. and Rehage, H. and Cicuta, P. and Kierfeld, J.*Langmuir*29 (2013)Elastic capsules, prepared from droplets or bubbles attached to a capillary (as in a pendant drop tensiometer), can be deflated by suction through the capillary. We study this deflation and show that a combined analysis of the shape and wrinkling characteristics enables us to determine the elastic properties in situ. Shape contours are analyzed and fitted using shape equations derived from nonlinear membrane-shell theory to give the elastic modulus, Poisson ratio and stress distribution of the membrane. We include wrinkles, which generically form upon deflation, within the shape analysis. Measuring the wavelength of wrinkles and using the calculated stress distribution gives the bending stiffness of the membrane. We compare this method with previous approaches using the Laplace-Young equation and illustrate the method on two very different capsule materials: polymerized octadecyltrichlorosilane (OTS) capsules and hydrophobin (HFBII) coated bubbles. Our results are in agreement with the available rheological data. For hydrophobin coated bubbles, the method reveals an interesting nonlinear behavior consistent with the hydrophobin molecules having a rigid core surrounded by a softer shell. © 2013 American Chemical Society.view abstract 10.1021/la402322g **Stiff directed lines in random media**

Boltz, H.-H. and Kierfeld, J.*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*88 (2013)We investigate the localization of stiff directed lines with bending energy by a short-range random potential. We apply perturbative arguments, Flory scaling arguments, a variational replica calculation, and functional renormalization to show that a stiff directed line in 1+d dimensions undergoes a localization transition with increasing disorder for d>2/3. We demonstrate that this transition is accessible by numerical transfer matrix calculations in 1+1 dimensions and analyze the properties of the disorder-dominated phase in detail. On the basis of the two-replica problem, we propose a relation between the localization of stiff directed lines in 1+d dimensions and of directed lines under tension in 1+3d dimensions, which is strongly supported by identical free-energy distributions. This shows that pair interactions in the replicated Hamiltonian determine the nature of directed line localization transitions with consequences for the critical behavior of the Kardar-Parisi-Zhang equation. We support the proposed relation to directed lines via multifractal analysis, revealing an analogous Anderson transition-like scenario and a matching correlation length exponent. Furthermore, we quantify how the persistence length of the stiff directed line is reduced by disorder. © 2013 American Physical Society.view abstract 10.1103/PhysRevE.88.012103 **Critical motor number for fractional steps of cytoskeletal filaments in gliding assays**

Li, X. and Lipowsky, R. and Kierfeld, J.*PLoS ONE*7 (2012)In gliding assays, filaments are pulled by molecular motors that are immobilized on a solid surface. By varying the motor density on the surface, one can control the number N of motors that pull simultaneously on a single filament. Here, such gliding assays are studied theoretically using Brownian (or Langevin) dynamics simulations and taking the local force balance between motors and filaments as well as the force-dependent velocity of the motors into account. We focus on the filament stepping dynamics and investigate how single motor properties such as stalk elasticity and step size determine the presence or absence of fractional steps of the filaments. We show that each gliding assay can be characterized by a critical motor number, Nc. Because of thermal fluctuations, fractional filament steps are only detectable as long as N < Nc. The corresponding fractional filament step size is ℓ/N where ℓ is the step size of a single motor. We first apply our computational approach to microtubules pulled by kinesin-1 motors. For elastic motor stalks that behave as linear springs with a zero rest length, the critical motor number is found to be Nc=4, and the corresponding distributions of the filament step sizes are in good agreement with the available experimental data. In general, the critical motor number Nc depends on the elastic stalk properties and is reduced to Nc=3 for linear springs with a nonzero rest length. Furthermore, Nc is shown to depend quadratically on the motor step size ℓ. Therefore, gliding assays consisting of actin filaments and myosin-V are predicted to exhibit fractional filament steps up to motor number N=31. Finally, we show that fractional filament steps are also detectable for a fixed average motor number 〈N〉 as determined by the surface density (or coverage) of the motors on the substrate surface. © 2012 Li et al.view abstract 10.1371/journal.pone.0043219 **Dynamics and length distribution of microtubules under force and confinement**

Zelinski, B. and Müller, N. and Kierfeld, J.*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*86 (2012)We investigate the microtubule polymerization dynamics with catastrophe and rescue events for three different confinement scenarios, which mimic typical cellular environments: (i) The microtubule is confined by rigid and fixed walls, (ii) it grows under constant force, and (iii) it grows against an elastic obstacle with a linearly increasing force. We use realistic catastrophe models and analyze the microtubule dynamics, the resulting microtubule length distributions, and force generation by stochastic and mean field calculations; in addition, we perform stochastic simulations. Freely growing microtubules exhibit a phase of bounded growth with finite microtubule length and a phase of unbounded growth. The main results for the three confinement scenarios are as follows: (i) In confinement by fixed rigid walls, we find exponentially decreasing or increasing stationary microtubule length distributions instead of bounded or unbounded phases, respectively. We introduce a realistic model for wall-induced catastrophes and investigate the behavior of the average length as a function of microtubule growth parameters. (ii) Under a constant force, the boundary between bounded and unbounded growth is shifted to higher tubulin concentrations and rescue rates. The critical force f c for the transition from unbounded to bounded growth increases logarithmically with tubulin concentration and the rescue rate, and it is smaller than the stall force. (iii) For microtubule growth against an elastic obstacle, the microtubule length and polymerization force can be regulated by microtubule growth parameters. For zero rescue rate, we find that the average polymerization force depends logarithmically on the tubulin concentration and is always smaller than the stall force in the absence of catastrophes and rescues. For a nonzero rescue rate, we find a sharply peaked steady-state length distribution, which is tightly controlled by microtubule growth parameters. The corresponding average microtubule length self-organizes such that the average polymerization force equals the critical force f c for the transition from unbounded to bounded growth. We also investigate the force dynamics if growth parameters are perturbed in dilution experiments. Finally, we show the robustness of our results against changes of catastrophe models and load distribution factors. © 2012 American Physical Society.view abstract 10.1103/PhysRevE.86.041918 **Localization transition of stiff directed lines in random media**

Boltz, H.-H. and Kierfeld, J.*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*86 (2012)We investigate the localization of stiff directed lines with bending energy by a short-range random potential. Using perturbative arguments, Flory arguments, and a replica calculation, we show that a stiff directed line in 1+d dimensions undergoes a localization transition with increasing disorder for d>2/3. We demonstrate that this transition is accessible by numerical transfer matrix calculations in 1+1 dimensions and analyze the properties of the disorder-dominated phase. On the basis of the two-replica problem, we propose a relation between the localization of stiff directed lines in 1+d dimensions and of directed lines under tension in 1+3d dimensions, which is strongly supported by identical free energy distributions. This shows that pair interactions in the replicated Hamiltonian determine the nature of directed line localization transitions with consequences for the critical behavior of the Kardar-Parisi-Zhang (KPZ) equation. Furthermore, we quantify how the persistence length of the stiff directed line is reduced by disorder. © 2012 American Physical Society.view abstract 10.1103/PhysRevE.86.060102 **Buckling of spherical capsules**

Knoche, S. and Kierfeld, J.*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*84 (2011)We investigate buckling of soft elastic capsules under negative pressure or for reduced capsule volume. Based on nonlinear shell theory and the assumption of a hyperelastic capsule membrane, shape equations for axisymmetric and initially spherical capsules are derived and solved numerically. A rich bifurcation behavior is found, which is presented in terms of bifurcation diagrams. The energetically preferred stable configuration is deduced from a least-energy principle both for prescribed volume and prescribed pressure. We find that buckled shapes are energetically favorable already at smaller negative pressures and larger critical volumes than predicted by the classical buckling instability. By preventing self-intersection for strongly reduced volume, we obtain a complete picture of the buckling process and can follow the shape from the initial undeformed state through the buckling instability into the fully collapsed state. Interestingly, the sequences of bifurcations and stable capsule shapes differ for prescribed volume and prescribed pressure. In the buckled state, we find a relation between curvatures at the indentation rim and the bending modulus, which can be used to determine elastic moduli from experimental shape analysis. © 2011 American Physical Society.view abstract 10.1103/PhysRevE.84.046608 **Stall force of polymerizing microtubules and filament bundles**

Krawczyk, J. and Kierfeld, J.*EPL*93 (2011)We investigate stall force and polymerization kinetics of rigid protofilaments in a microtubule or interacting filaments in bundles under an external load force in the framework of a discrete growth model. We introduce the concecpt of polymerization cycles to describe the stochastic growth kinetics, which allows us to derive an exact expression for the stall force. We find that the stall force is independent of ensemble geometry and load distribution. Furthermore, the stall force is proportional to the number of filaments and increases linearly with the strength of lateral filament interactions. These results are corroborated by simulations, which also show a strong influence of ensemble geometry on growth kinetics below the stall force. Copyright © EPLA, 2011.view abstract 10.1209/0295-5075/93/28006 **Coupling of actin hydrolysis and polymerization: Reduced description with two nucleotide states**

Li, X. and Lipowsky, R. and Kierfeld, J.*EPL*89 (2010)The polymerization of actin filaments is coupled to the hydrolysis of adenosine triphosphate (ATP), which involves both the cleavage of ATP and the release of inorganic phosphate. We describe hydrolysis by a reduced two-state model with a cooperative cleavage mechanism, where the cleavage rate depends on the state of the neighboring actin protomer in a filament. We obtain theoretical predictions of experimentally accessible steady-state quantities such as the size of the ATP-actin cap, the size distribution of ATP-actin islands, and the cleavage flux for cooperative cleavage mechanisms. © 2010 Europhysics Letters Association.view abstract 10.1209/0295-5075/89/38010 **Modelling semiflexible polymers: Shape analysis, buckling instabilities, and force generation**

Kierfeld, J. and Baczynski, K. and Gutjahr, P. and Kühne, T. and Lipowsky, R.*Soft Matter*6 (2010)The behavior of semiflexible polymers and filaments is governed by their bending energy. The corresponding bending rigidity gives rise to material properties that are distinct from those of flexible polymers governed by entropy. In particular, bending rigidity plays an important role for the shapes of these polymers and their ability to withstand and transmit forces. Recent theoretical studies and modelling approaches are briefly reviewed and used for a systematic analysis of shapes of adsorbed semiflexible polymers and buckling instabilities. Semiflexible polymers and filaments exhibit a buckling instability which is modified by thermal fluctuations and provides upper bounds on the generation of polymerization forces. Growing bundles of polymers or filaments can generate force via adhesive interactions. The latter mechanism remains effective even after single filaments have attained a buckled state. © 2010 The Royal Society of Chemistry.view abstract 10.1039/c002035b **Semiflexible polymer rings on topographically and chemically structured surfaces**

Gutjahr, P. and Lipowsky, R. and Kierfeld, J.*Soft Matter*6 (2010)We investigate morphologies of semiflexible polymer rings, such as circular DNA, which are adsorbed onto topographically or chemically structured substrate surfaces. We classify all equilibrium morphologies for two striped surface structures, (i) topographical surface grooves and (ii) chemically structured surface domains. For both types of stripes, we find four equilibrium shapes: a round toroidal and a confined elongated shape as well as two shapes containing bulges. We determine the complete bifurcation diagram of these morphologies as a function of their contour length and the ratio of adhesive strength to bending rigidity. For more complex geometries consisting of several stripes we find a cascade of transitions between elongated shapes. Finally, we compare our findings to ring condensation by attractive interactions. © 2010 The Royal Society of Chemistry.view abstract 10.1039/c0sm00381f

#### buckling

#### modelling and simulation

#### polymers