Nonlinear dynamics of stiff polymers

Charged elastic rings: deformation and dynamics

We report the counter-intuitive instability of charged elastic rings, and the persistence of sinusoidal deformations in the lowest-energy configurations by the combination of high-precision numerical simulations and analytical perturbation calculation. We also study the dynamical evolution of the charged ring under random disturbance, and reveal the modulation of the dominant frequencies by the electrostatic force. The purely mechanical analysis of the classical ring system presented in this work yields insights into the subtlety of long-range forces in the organization and dynamics of matter.

Optimal gaits for drag-dominated swimmers with passive elastic joints

In this paper we identify optimal swimming strategies for drag-dominated swimmers with a passive elastic joint. We use resistive force theory to obtain the dynamics of the system. We then use frequency-domain analysis to relate the motion of the passive joint to the motion of the actuated joint. We couple this analysis with elements of the geometric framework introduced in our previous work aimed at identifying useful gaits for systems in drag-dominated environments to identify speed-maximizing and efficiency-maximizing gaits for drag-dominated swimmers with a passive elastic joint.

Propulsion of an elastic filament in a shear-thinning fluid

Some micro-organisms and artificial micro-swimmers propel at low Reynolds numbers (Re) via the interaction of their flexible appendages with the surrounding fluid. While their locomotion has been extensively studied with a Newtonian fluid assumption, in realistic biological environments these micro-swimmers invariably encounter rheologically complex fluids. In particular, many biological fluids such as blood and different types of mucus have shear-thinning viscosities. The influence of this ubiquitous non-Newtonian rheology on the performance of flexible swimmers remains largely unknown. Here, we present a first study to examine how shear-thinning rheology alters the fluid-structure interaction and hence the propulsion performance of elastic swimmers at low Re. Via a simple elastic swimmer actuated magnetically, we demonstrate that shear-thinning rheology can either enhance or hinder elastohydrodynamic propulsion, depending on the intricate interplay between elastic and viscous forces as well as the magnetic actuation. We also use a reduced-order model to elucidate the mechanisms underlying the enhanced and hindered propulsion observed in different physical regimes. These results and improved understanding could guide the design of flexible micro-swimmers in non-Newtonian fluids.

Swirling Instability of the Microtubule Cytoskeleton

In the cellular phenomena of cytoplasmic streaming, molecular motors carrying cargo along a network of microtubules entrain the surrounding fluid. The piconewton forces produced by individual motors are sufficient to deform long microtubules, as are the collective fluid flows generated by many moving motors. Studies of streaming during oocyte development in the fruit fly Drosophila melanogaster have shown a transition from a spatially disordered cytoskeleton, supporting flows with only short-ranged correlations, to an ordered state with a cell-spanning vortical flow. To test the hypothesis that this transition is driven by fluid-structure interactions, we study a discrete-filament model and a coarse-grained continuum theory for motors moving on a deformable cytoskeleton, both of which are shown to exhibit a swirling instability to spontaneous large-scale rotational motion, as observed.

Flow-induced buckling dynamics of sperm flagella

The swimming sperm of many external fertilizing marine organisms face complex fluid flows during their search for egg cells. Aided by chemotaxis, relatively weak flows are known to enhance sperm-egg fertilization rates through hydrodynamic guidance. However, strong flows have the potential to mechanically inhibit flagellar motility through elastohydrodynamic interactions-a phenomenon that remains poorly understood. Here we explore the effects of flow on the buckling dynamics of sperm flagella in an extensional flow through detailed numerical simulations, which are informed by microfluidic experiments and high-speed imaging. Compressional fluid forces lead to rich buckling dynamics of the sperm flagellum beyond a critical dimensionless sperm number, Sp, which represents the ratio of viscous force to elastic force. For nonmotile sperm, the maximum buckling curvature and the number of buckling locations, or buckling mode, increase with increasing sperm number. In contrast, motile sperm exhibit a local flagellar curvature due to the propagation of bending waves along the flagellum. In compressional flow, this preexisting curvature acts as a precursor for buckling, which enhances local curvature without creating new buckling modes and leads to asymmetric beating. However, in extensional flow, flagellar beating remains symmetric with a smaller head yawing amplitude due to tensile forces. The flagellar beating frequency also influences the maximum curvature of motile sperm by facilitating sperm reorientation relative to the compressional axis of the flow near stagnation points. These combined simulations and experiments directly illustrate the microscopic elastohydrodynamic mechanisms responsible for inhibiting flagellar motility in flow and have possible implications for our understanding of external fertilization in dynamic marine systems.

Flagellar ultrastructure suppresses buckling instabilities and enables mammalian sperm navigation in high-viscosity media

Eukaryotic flagellar swimming is driven by a slender motile unit, the axoneme, which possesses an internal structure that is essentially conserved in a tremendous diversity of sperm. Mammalian sperm, however, which are internal fertilizers, also exhibit distinctive accessory structures that further dress the axoneme and alter its mechanical response. This raises the following two fundamental questions. What is the functional significance of these structures? How do they affect the flagellar waveform and ultimately cell swimming? Hence we build on previous work to develop a mathematical mechanical model of a virtual human sperm to examine the impact of mammalian sperm accessory structures on flagellar dynamics and motility. Our findings demonstrate that the accessory structures reinforce the flagellum, preventing waveform compression and symmetry-breaking buckling instabilities when the viscosity of the surrounding medium is increased. This is in agreement with previous observations of internal and external fertilizers, such as human and sea urchin spermatozoa. In turn, possession of accessory structures entails that the progressive motion during a flagellar beat cycle can be enhanced as viscosity is increased within physiological bounds. Hence the flagella of internal fertilizers, complete with accessory structures, are predicted to be advantageous in viscous physiological media compared with watery media for the fundamental role of delivering a genetic payload to the egg.

The asymptotic coarse-graining formulation of slender-rods, bio-filaments and flagella

The inertialess fluid-structure interactions of active and passive inextensible filaments and slender-rods are ubiquitous in nature, from the dynamics of semi-flexible polymers and cytoskeletal filaments to cellular mechanics and flagella. The coupling between the geometry of deformation and the physical interaction governing the dynamics of bio-filaments is complex. Governing equations negotiate elastohydrodynamical interactions with non-holonomic constraints arising from the filament inextensibility. Such elastohydrodynamic systems are structurally convoluted, prone to numerical errors, thus requiring penalization methods and high-order spatio-temporal propagators. The asymptotic coarse-graining formulation presented here exploits the momentum balance in the asymptotic limit of small rod-like elements which are integrated semi-analytically. This greatly simplifies the elastohydrodynamic interactions and overcomes previous numerical instability. The resulting matricial system is straightforward and intuitive to implement, and allows for a fast and efficient computation, more than a hundred times faster than previous schemes. Only basic knowledge of systems of linear equations is required, and implementation achieved with any solver of choice. Generalizations for complex interaction of multiple rods, Brownian polymer dynamics, active filaments and non-local hydrodynamics are also straightforward. We demonstrate these in four examples commonly found in biological systems, including the dynamics of filaments and flagella. Three of these systems are novel in the literature. We additionally provide a Matlab code that can be used as a basis for further generalizations.

Effect of external shear flow on sperm motility

The trajectory of sperm in the presence of background flow is of utmost importance for the success of fertilization, as sperm encounter background flow of different magnitude and direction on their way to the egg. Here, we have studied the effect of an unbounded simple shear flow as well as a Poiseuille flow on the sperm trajectory. In the presence of a simple shear flow, the sperm moves on an elliptical trajectory in the reference frame advecting with the local background flow. The length of the major-axis of this elliptical trajectory decreases with the shear rate. The flexibility of the flagellum and consequently the length of the major axis of the elliptical trajectories increases with the sperm number. The sperm number is a dimensionless number representing the ratio of viscous force to elastic force. The sperm moves downstream or upstream depending on the strength of background Poiseuille flow. In contrast to the simple shear flow, the sperm also moves toward the centerline in a Poiseuille flow. Far away from the centerline, the cross-stream migration velocity of the sperm increases as the transverse distance of the sperm from the centerline decreases. Close to the centerline, on the other hand, the cross-stream migration velocity decreases as the sperm further approaches the center. The cross-stream migration velocity of the sperm also increases with the sperm number.

Nonlinear multimode buckling dynamics examined with semiflexible paramagnetic filaments

We present the contractile buckling dynamics of superparamagnetic filaments using experimental, theoretical, and simulation approaches. Under the influence of an orthogonal magnetic field, flexible magnetic filaments exhibit higher-order buckling dynamics that can be identified as occurring in three stages: initiation, development, and decay. Unlike initiation and decay stages where the balance between magnetic interactions and elastic forces is dominant, in the development stage, the influence of hydrodynamic drag results in transient buckling dynamics that is nonlinear along the filament contour. The inhomogeneous temporal evolution of the buckling wavelength is analyzed and the contractions under various conditions are compared.

Spontaneous oscillations of elastic filaments induced by molecular motors

It is known from the wave-like motion of microtubules in motility assays that the piconewton forces that motors produce can be sufficient to bend the filaments. In cellular phenomena such as cytosplasmic streaming, molecular motors translocate along cytoskeletal filaments, carrying cargo which entrains fluid. When large numbers of such forced filaments interact through the surrounding fluid, as in particular stages of oocyte development in Drosophila melanogaster, complex dynamics are observed, but the detailed mechanics underlying them has remained unclear. Motivated by these observations, we study here perhaps the simplest model for these phenomena: an elastic filament, pinned at one end, acted on by a molecular motor treated as a point force. Because the force acts tangential to the filament, no matter what its shape, this 'follower-force' problem is intrinsically non-variational, and thereby differs fundamentally from Euler buckling, where the force has a fixed direction, and which, in the low-Reynolds-number regime, ultimately leads to a stationary, energy-minimizing shape. Through a combination of linear stability theory, analytical study of a solvable simplified 'two-link' model and numerical studies of the full elastohydrodynamic equations of motion, we elucidate the Hopf bifurcation that occurs with increasing forcing of a filament, leading to flapping motion analogous to the high-Reynolds-number oscillations of a garden hose with a free end.

Broken detailed balance and non-equilibrium dynamics in living systems: a review

Living systems operate far from thermodynamic equilibrium. Enzymatic activity can induce broken detailed balance at the molecular scale. This molecular scale breaking of detailed balance is crucial to achieve biological functions such as high-fidelity transcription and translation, sensing, adaptation, biochemical patterning, and force generation. While biological systems such as motor enzymes violate detailed balance at the molecular scale, it remains unclear how non-equilibrium dynamics manifests at the mesoscale in systems that are driven through the collective activity of many motors. Indeed, in several cellular systems the presence of non-equilibrium dynamics is not always evident at large scales. For example, in the cytoskeleton or in chromosomes one can observe stationary stochastic processes that appear at first glance thermally driven. This raises the question how non-equilibrium fluctuations can be discerned from thermal noise. We discuss approaches that have recently been developed to address this question, including methods based on measuring the extent to which the system violates the fluctuation-dissipation theorem. We also review applications of this approach to reconstituted cytoskeletal networks, the cytoplasm of living cells, and cell membranes. Furthermore, we discuss a more recent approach to detect actively driven dynamics, which is based on inferring broken detailed balance. This constitutes a non-invasive method that uses time-lapse microscopy data, and can be applied to a broad range of systems in cells and tissue. We discuss the ideas underlying this method and its application to several examples including flagella, primary cilia, and cytoskeletal networks. Finally, we briefly discuss recent developments in stochastic thermodynamics and non-equilibrium statistical mechanics, which offer new perspectives to understand the physics of living systems.

The counterbend dynamics of cross-linked filament bundles and flagella

Cross-linked filament bundles, such as in cilia and flagella, are ubiquitous in biology. They are considered in textbooks as simple filaments with larger stiffness. Recent observations of flagellar counterbend, however, show that induction of curvature in one section of a passive flagellum instigates a compensatory counter-curvature elsewhere, exposing the intricate role of the diminutive cross-linking proteins at large scales. We show that this effect, a material property of the cross-linking mechanics, modifies the bundle dynamics and induces a bimodal L² - L³ length-dependent material response that departs from the Euler-Bernoulli theory. Hence, the use of simpler theories to analyse experiments can result in paradoxical interpretations. Remarkably, the counterbend dynamics instigates counter-waves in opposition to driven oscillations in distant parts of the bundle, with potential impact on the regulation of flagellar bending waves. These results have a range of physical and biological applications, including the empirical disentanglement of material quantities via counterbend dynamics.

Maximizing propulsive thrust of a driven filament at low Reynolds number via variable flexibility

At low Reynolds numbers the locomotive capability of a body can be dramatically hindered by the absence of inertia. In this work, we show how propulsive performance in this regime can be significantly enhanced by employing spatially varying flexibility. As a prototypical example, we consider the propulsive thrust generated by a filament periodically driven at one end. The rigid case leads to zero propulsion, as so constrained by Purcell's scallop theorem, while for uniform filaments there exists a bending stiffness maximizing the propulsive force at a given frequency; here we demonstrate explicitly how considerable further improvement can be achieved by simply varying the stiffness along the filament. The optimal flexibility distribution is strongly configuration-dependent: while increasing the flexibility towards the tail-end enhances the propulsion of a clamped filament, for a hinged filament decreasing the flexibility towards the tail-end is instead favorable. The results reveal new design principles for maximizing propulsion at low Reynolds numbers, potentially useful for developing synthetic micro-swimmers requiring large propulsive force for various biomedical applications.

Stenosis triggers spread of helical Pseudomonas biofilms in cylindrical flow systems

Biofilms are multicellular bacterial structures that adhere to surfaces and often endow the bacterial population with tolerance to antibiotics and other environmental insults. Biofilms frequently colonize the tubing of medical devices through mechanisms that are poorly understood. Here we studied the helicoidal spread of Pseudomonas putida biofilms through cylindrical conduits of varied diameters in slow laminar flow regimes. Numerical simulations of such flows reveal vortical motion at stenoses and junctions, which enhances bacterial adhesion and fosters formation of filamentous structures. Formation of long, downstream-flowing bacterial threads that stem from narrowings and connections was detected experimentally, as predicted by our model. Accumulation of bacterial biomass makes the resulting filaments undergo a helical instability. These incipient helices then coarsened until constrained by the tubing walls, and spread along the whole tube length without obstructing the flow. A three-dimensional discrete filament model supports this coarsening mechanism and yields simulations of helix dynamics in accordance with our experimental observations. These findings describe an unanticipated mechanism for bacterial spreading in tubing networks which might be involved in some hospital-acquired infections and bacterial contamination of catheters.

Active Curved Polymers Form Vortex Patterns on Membranes

Recent in vitro experiments with FtsZ polymers show self-organization into different dynamic patterns, including structures reminiscent of the bacterial Z ring. We model FtsZ polymers as active particles moving along chiral, circular paths by Brownian dynamics simulations and a Boltzmann approach. Our two conceptually different methods point to a generic phase behavior. At intermediate particle densities, we find self-organization into vortex structures including closed rings. Moreover, we show that the dynamics at the onset of pattern formation is described by a generalized complex Ginzburg-Landau equation.

Efficient simulation of semiflexible polymers

Using a recently developed bead-spring model for semiflexible polymers that takes into account their natural extensibility, we report an efficient algorithm to simulate the dynamics for polymers like double-stranded DNA (dsDNA) in the absence of hydrodynamic interactions. The dsDNA is modeled with one bead-spring element per base pair, and the polymer dynamics is described by the Langevin equation. The key to efficiency is that we describe the equations of motion for the polymer in terms of the amplitudes of the polymer's fluctuation modes, as opposed to the use of the physical positions of the beads. We show that, within an accuracy tolerance level of 5% of several key observables, the model allows for single Langevin time steps of ≈1.6, 8, 16, and 16 ps for a dsDNA model chain consisting of 64, 128, 256, and 512 base pairs (i.e., chains of 0.55, 1.11, 2.24, and 4.48 persistence lengths), respectively. Correspondingly, in 1 h, a standard desktop computer can simulate 0.23, 0.56, 0.56, and 0.26 ms of these dsDNA chains, respectively. We compare our results to those obtained from other methods, in particular, the (inextensible discretized) wormlike chain (WLC) model. Importantly, we demonstrate that at the same level of discretization, i.e., when each discretization element is one base pair long, our algorithm gains about five to six orders of magnitude in the size of time steps over the inextensible WLC model. Further, we show that our model can be mapped one on one to a discretized version of the extensible WLC model, implying that the speed-up we achieve in our model must hold equally well for the latter. We also demonstrate the use of the method by simulating efficiently the tumbling behavior of a dsDNA segment in a shear flow.

Effects of thermal noise on the transitional dynamics of an inextensible elastic filament in stagnation flow

We investigate the dynamics of a single inextensible elastic filament subject to anisotropic friction in a viscous stagnation-point flow, by employing both a continuum model represented by Langevin type stochastic partial differential equations (SPDEs) and a dissipative particle dynamics (DPD) method. Unlike previous works, the filament is free to rotate and the tension along the filament is determined by the local inextensible constraint. The kinematics of the filament is recorded and studied with normal modes analysis. The results show that the filament displays an instability induced by negative tension, which is analogous to Euler buckling of a beam. Symmetry breaking of normal modes dynamics and stretch-coil transitions are observed above the threshold of the buckling instability point. Furthermore, both temporal and spatial noise are amplified resulting from the interaction of thermal fluctuations and nonlinear filament dynamics. Specifically, the spatial noise is amplified with even normal modes being excited due to symmetry breaking, while the temporal noise is amplified with increasing time correlation length and variance.

Slack dynamics on an unfurling string

An arch will grow on a rapidly deployed thin string in contact with a rigid plane. We present a qualitative model for the growing structure involving the amplification, rectification, and advection of slack in the presence of a steady stress field, validate our assumptions with numerical experiments, and pose new questions about the spatially developing motions of thin objects.

Surface traction and the dynamics of elastic rods at low Reynolds number

Molecular and cell biological processes often use proteins and structures that are significantly longer in one dimension than they are in the other two, for example, DNA, actin, and bacterial flagella. The dynamics of these structures are the consequence of the balance between the elastic forces from the structure itself and viscous forces from the surrounding fluid. Typically, the motion of these filamentary objects is described using variations of the Kirchhoff rod equations with resistive forces from the fluid treated as body forces acting on the centerline. In reality, though, these forces are applied to the surface of the filament; however, the standard derivation of the Kirchhoff equations ignores surface traction stresses. Here, we rederive the Kirchhoff rod equations in the presence of resistive traction stresses and determine the conditions under which treating the drag forces as body forces is reasonable. We show that in most biologically relevant cases the standard implementation of resistive forces into the Kirchhoff rod equations is applicable; however, we note one particular biological system where the Kirchhoff rod formalism may not apply.

Experiments and theory of undulatory locomotion in a simple structured medium

Undulatory locomotion of micro-organisms through geometrically complex, fluidic environments is ubiquitous in nature and requires the organism to negotiate both hydrodynamic effects and geometrical constraints. To understand locomotion through such media, we experimentally investigate swimming of the nematode Caenorhabditis elegans through fluid-filled arrays of micro-pillars and conduct numerical simulations based on a mechanical model of the worm that incorporates hydrodynamic and contact interactions with the lattice. We show that the nematode's path, speed and gait are significantly altered by the presence of the obstacles and depend strongly on lattice spacing. These changes and their dependence on lattice spacing are captured, both qualitatively and quantitatively, by our purely mechanical model. Using the model, we demonstrate that purely mechanical interactions between the swimmer and obstacles can produce complex trajectories, gait changes and velocity fluctuations, yielding some of the life-like dynamics exhibited by the real nematode. Our results show that mechanics, rather than biological sensing and behaviour, can explain some of the observed changes in the worm's locomotory dynamics.

Longitudinal response of confined semiflexible polymers

The longitudinal response of single semiflexible polymers to sudden changes in externally applied forces is known to be controlled by the propagation and relaxation of backbone tension. Under many experimental circumstances, realized, for example, in nanofluidic devices or in polymeric networks or solutions, these polymers are effectively confined in a channel- or tubelike geometry. By means of heuristic scaling laws and rigorous analytical theory, we analyze the tension dynamics of confined semiflexible polymers for various generic experimental setups. It turns out that in contrast to the well-known linear response, the influence of confinement on the nonlinear dynamics can largely be described as that of an effective prestress. We also study the free relaxation of an initially confined chain, finding a surprising superlinear ~t(9/8) growth law for the change in end-to-end distance at short times.

Propulsion by passive filaments and active flagella near boundaries

Confinement and wall effects are known to affect the kinematics and propulsive characteristics of swimming microorganisms. When a solid body is dragged through a viscous fluid at constant velocity, the presence of a wall increases fluid drag, and thus the net force required to maintain speed has to increase. In contrast, recent optical trapping experiments have revealed that the propulsive force generated by human spermatozoa is decreased by the presence of boundaries. Here, we use a series of simple models to analytically elucidate the propulsive effects of a solid boundary on passively actuated filaments and model flagella. For passive flexible filaments actuated periodically at one end, the presence of the wall is shown to increase the propulsive forces generated by the filaments in the case of displacement-driven actuation, while it decreases the force in the case of force-driven actuation. In the case of active filaments as models for eukaryotic flagella, we demonstrate that the manner in which a solid wall affects propulsion cannot be known a priori, but is instead a nontrivial function of the flagellum frequency, wavelength, its material characteristics, the manner in which the molecular motors self-organize to produce oscillations (prescribed activity model or self-organized axonemal beating model), and the boundary conditions applied experimentally to the tethered flagellum. In particular, we show that in some cases, the increase in fluid friction induced by the wall can lead to a change in the waveform expressed by the flagella, which results in a decrease in their propulsive force.

Linear viscoelasticity of a single semiflexible polymer with internal friction

The linear viscoelastic behaviors of single semiflexible chains with internal friction are studied based on the wormlike-chain model. It is shown that the frequency dependence of the complex compliance in the high frequency limit is the same as that of the Voigt model. This asymptotic behavior appears also for the Rouse model with internal friction. We derive the characteristic times for both the high frequency limit and the low frequency limit and compare the results with those obtained by Khatri et al.

Dynamics of a semiflexible polar filament in Stokes flow

In this work, the dynamics and transport of a polarly driven filament is examined using a continuum slender-body model. Immersed in a viscous fluid, the filament gains polar propulsion from the motor proteins (anchored on the motility assay) while experiencing a viscous drag from the bottom wall. Results from the linear analysis on a straight polar filament illustrate the necessity of spatial inhomogeneity in the polar forcing for the buckling instability. The ensuing buckling leads to filament deformation, undulation, and change of its direction of motion in the numerical simulations. Repeated filament buckling in two types of motor protein concentration landscape results in diffusive transport of a polar filament on scales much larger than the mean-free path and the average duration between filament buckling events.

Nonlinear instability in flagellar dynamics: a novel modulation mechanism in sperm migration?

Throughout biology, cells and organisms use flagella and cilia to propel fluid and achieve motility. The beating of these organelles, and the corresponding ability to sense, respond to and modulate this beat is central to many processes in health and disease. While the mechanics of flagellum-fluid interaction has been the subject of extensive mathematical studies, these models have been restricted to being geometrically linear or weakly nonlinear, despite the high curvatures observed physiologically. We study the effect of geometrical nonlinearity, focusing on the spermatozoon flagellum. For a wide range of physiologically relevant parameters, the nonlinear model predicts that flagellar compression by the internal forces initiates an effective buckling behaviour, leading to a symmetry-breaking bifurcation that causes profound and complicated changes in the waveform and swimming trajectory, as well as the breakdown of the linear theory. The emergent waveform also induces curved swimming in an otherwise symmetric system, with the swimming trajectory being sensitive to head shape-no signalling or asymmetric forces are required. We conclude that nonlinear models are essential in understanding the flagellar waveform in migratory human sperm; these models will also be invaluable in understanding motile flagella and cilia in other systems.

Cosserat curve model for superelasticity of helices

Superelasticity behavior of helices has been the focus of recent research in micro-/nano-engineering, while the traditional Kirchhoff rod model restricts itself in the bending and torsion conditions. With the aid of the concept of a Cosserat curve, a novel theoretical basis has been established for statics and dynamics of helices with essential extension and shear, which is able to quantitatively analyze the superelastic mechanical properties. Except for a good agreement with the experimental observation, numerical solutions have shown that we cannot only predict two important properties of the superelasticity characteristics: the breaking force and the stretch of the coil wire under the axial loading, but also precisely describe and explain the Hooke's constant and torque in the entire stretching and breaking processes. The present work has provided useful information for the future experimental investigation on superelasticity as well as its application in meta-/quantum devices.

Effect of intrinsic curvature on semiflexible polymers

Recently many important biopolymers have been found to possess intrinsic curvature. Tubulin protofilaments in animal cells, FtsZ filaments in bacteria and double stranded DNA are examples. We examine how intrinsic curvature influences the conformational statistics of such polymers. We give exact results for the tangent-tangent spatial correlation function C(r)=<t(s).t(s+r)>, both in two and three dimensions. Contrary to expectation, C(r) does not show any oscillatory behavior, rather decays exponentially and the effective persistence length has strong length dependence for short polymers. We also compute the distribution function P(R) of the end to end distance R and show how curved chains can be distinguished from wormlike chains using loop formation probability.

Dynamic labyrinthine pattern in an active liquid film

We report the generation of a dynamic labyrinthine pattern in an active alcohol film. A dynamic labyrinthine pattern is formed along the contact line of air/pentanol/aqueous three phases. The contact line shows a clear time-dependent change with regard to both perimeter and area of a domain. An autocorrelation analysis of time development of the dynamics of the perimeter and area revealed a strong geometric correlation between neighboring patterns. The pattern showed autoregressive behavior. The behavior of the dynamic pattern is strikingly different from those of stationary labyrinthine patterns. The essential aspects of the observed dynamic pattern are reproduced by a diffusion-controlled geometric model.

Tension dynamics and viscoelasticity of extensible wormlike chains

The dynamic response of prestressed semiflexible biopolymers is characterized by the propagation and relaxation of tension, which arises due to the near inextensibility of a stiff backbone. It is coupled to the dynamics of contour length stored in thermal undulations but also to the local relaxation of elongational strain. We present a systematic theory of tension dynamics for stiff yet extensible wormlike chains. Our results show that even moderate prestress gives rise to distinct Rouse-like extensibility signatures in the high-frequency viscoelastic response.

Hydrodynamic interactions between two semiflexible inextensible filaments in Stokes flow

Hydrodynamic interactions between two semiflexible inextensible filaments are shown to have a significant impact on filament buckling and their subsequent motion in Stokesian fluids. In linear shear flow, hydrodynamic interactions lead to filament shear dispersion that depends on the filament aspect ratio and the initial filament separation. In linear extensional flow, hydrodynamic interactions lead to complex filament dynamics around the stagnation point. These results suggest that hydrodynamic interactions need to be taken into account to determine the self-diffusion of non-Brownian semiflexible filaments in a cellular flow [Y.-N. Young and M. J. Shelley, Phys. Rev. Lett. 99, 058303 (2007)].

Freely relaxing polymers remember how they were straightened

The relaxation of initially straight semiflexible polymers has been discussed mainly with respect to the longest relaxation time. The biologically relevant nonequilibrium dynamics on shorter times is comparatively poorly understood, partly because "initially straight" can be realized in manifold ways. Combining Brownian dynamics simulations and systematic theory, we demonstrate how different experimental preparations give rise to specific short-time and universal long-time dynamics. We also discuss boundary effects and the onset of the stretch-coil transition.

Models of protein linear molecular motors for dynamic nanodevices

Protein molecular motors are natural nano-machines that convert the chemical energy from the hydrolysis of adenosine triphosphate into mechanical work. These efficient machines are central to many biological processes, including cellular motion, muscle contraction and cell division. The remarkable energetic efficiency of the protein molecular motors coupled with their nano-scale has prompted an increasing number of studies focusing on their integration in hybrid micro- and nanodevices, in particular using linear molecular motors. The translation of these tentative devices into technologically and economically feasible ones requires an engineering, design-orientated approach based on a structured formalism, preferably mathematical. This contribution reviews the present state of the art in the modelling of protein linear molecular motors, as relevant to the future design-orientated development of hybrid dynamic nanodevices.

Origin of contractile force during cell division of bacteria

When a bacterium divides, its cell wall at the division site grows radially inward like the shutter of a camera and guillotines the cell into two halves. The wall is pulled upon from inside by a polymeric ring, which itself shrinks in radius. The ring is made of an intracellular protein FtsZ (filamenting temperature sensitive Z) and thus is called the Z ring. It is not understood how the Z ring generates the required contractile force. We propose a theoretical model and simulate it to show how the natural curvature of the FtsZ filaments and lateral attraction among them may facilitate force generation.

Stretch-coil transition and transport of fibers in cellular flows

It is shown that a slender elastic fiber moving in a Stokesian fluid can be susceptible to a buckling instability--termed the "stretch-coil" instability--when moving in the neighborhood of a hyperbolic stagnation point of the flow. When the stagnation point is part of an extended cellular flow, it is found that immersed fibers can move as random walkers across time-independent closed-streamline flow. It is also found that the flow is segregated into transport regions around hyperbolic stagnation points and their manifolds, and closed entrapment regions around elliptic points.

Stretching dynamics of semiflexible polymers

We analyze the nonequilibrium dynamics of single inextensible semiflexible biopolymers as stretching forces are applied at the ends. Based on different (contradicting) heuristic arguments, various scaling laws have been proposed for the propagation speed of the backbone tension which is induced in response to stretching. Here, we employ a newly developed unified theory to systematically substantiate, restrict, and extend these approaches. Introducing the practically relevant scenario of a chain equilibrated under some prestretching force f (pre) that is suddenly exposed to a different external force f (ext) at the ends, we give a concise physical explanation of the underlying relaxation processes by means of an intuitive blob picture. We discuss the corresponding intermediate asymptotics, derive results for experimentally relevant observables, and support our conclusions by numerical solutions of the coarse-grained equations of motion for the tension.

Tension dynamics in semiflexible polymers. I. Coarse-grained equations of motion

Based on the wormlike chain model, a coarse-grained description of the nonlinear dynamics of a weakly bending semiflexible polymer is developed. By means of a multiple-scale perturbation analysis, a length-scale separation inherent to the weakly bending limit is exploited to reveal the deterministic nature of the spatio temporal relaxation of the backbone tension and to deduce the corresponding coarse-grained equation of motion. From this partial integro-differential equation, some detailed analytical predictions for the nonlinear response of a weakly bending polymer are derived in an accompanying paper [O. Hallatschek, following paper, Phys. Rev. E 75, 031906 (2007)].

Dynamics of semiflexible polymers in a flow field

We present a method to investigate the dynamics of a single semiflexible polymer, subject to anisotropic friction in a viscous fluid. In contrast to previous approaches, we do not rely on a discrete bead-rod model, but introduce a suitable normal mode decomposition of a continuous space curve. By means of a perturbation expansion for stiff filaments, we derive a closed set of coupled Langevin equations in mode space for the nonlinear dynamics in two dimensions, taking into account exactly the local constraint of inextensibility. The stochastic differential equations obtained this way are solved numerically, with parameters adjusted to describe the motion of actin filaments. As an example, we show results for the tumbling motion in shear flow.

Dynamics of inextensible semiflexible filaments

We address the problem of the fluctuating dynamics of an isolated inextensible and semiflexible filament of length L . By focusing explicitly on two key properties of the filaments--their connectivity and inextensibility--we obtain insights into their dynamics. We calculate the dynamics, and we calculate the correlation functions for chains of finite length taking into account the boundary conditions at the chain ends. We obtain closed-form expressions that separate the internal bending modes from the global rotation modes.

Force and flexibility of flailing myxobacteria

Myxococcus xanthus is a common Gram-negative bacterium that moves by a process called gliding motility. In myxobacteria, two distinct mechanisms for gliding have been discovered. S-type motility requires the extension, attachment, and retraction of type IV pili. The other mechanism, designated as A-type motility, may be driven by the secretion and swelling of slime; however, experiments to confirm or refute this model are still lacking and the force exerted by this mechanism has not been measured. A previously published experiment found that when an M. xanthus cell became stuck at one end, the cell underwent flailing motions. Based on this experiment, I propose an elastic model that can estimate the force produced by the A-motility engine and the bending modulus of a single myxobacterial cell. The model estimates a bending modulus of 3 x 10(-14) erg cm and a force between 50-150 pN. This force is comparable to that predicted by slime extrusion, and the bending modulus is 30-fold smaller than that measured in Bacillus subtilis. This model suggests experiments that can further quantify this process.

Propagation and relaxation of tension in stiff polymers

We present a unified theory for the longitudinal dynamic response of a stiff polymer in solution to various external perturbations (mechanical excitations, hydrodynamic flows, electrical fields, temperature quenches, etc.) that can be represented as sudden changes of ambient/boundary conditions. The theory relies on a comprehensive analysis of the nonequilibrium propagation and relaxation of backbone stresses in a wormlike chain. We recover and substantially extend previous results based on heuristic arguments. New experimental implications are pointed out.

General equilibrium shape equations of polymer chains

The general equilibrium shape equations of polymer chains are analytically derived in this paper. This provides a unified description for many models, such as the well-known wormlike chain (WLC) model, the wormlike rod chain (WLRC) model, carbon nanotubes, and so on. Using the WLC model, we find that the pitch-to-radius ratio of coils, 4.443, agrees with Z-DNA, and the pitch-to-radius ratio from WLRC agrees with the data of B-DNA qualitatively. Using the general shape equations, we discuss a chiral model in which the solutions of straight, helical, and circular biopolymers are given, respectively. We also find that the model suggested by Helfrich [Langmuir 7, 567 (1991)] is very appropriate to describe B-DNA (or other biopolymers) if we choose the four phenomenological parameters as A=50 nm , C=60 nm(2) , alpha=40 nm(3) , and beta=50 nm(2) .

Overdamped stress relaxation in buckled rods

We present a comprehensive theoretical analysis of the stress relaxation in a multiply but weakly buckled incompressible rod in a viscous solvent. For the bulk, two interesting parameter regimes of generic self-similar intermediate asymptotics are distinguished, which give rise to approximate and exact power-law solutions, respectively. For the case of open boundary conditions the corresponding nontrivial boundary-layer scenarios are derived by a multiple-scale perturbation ("adiabatic") method. Our results compare well with--and provide the theoretical explanation for--previous results from numerical simulations, and they suggest directions for further fruitful numerical and experimental investigations.

Dynamics of a flexible magnetic chain in a rotating magnetic field

The model of an elastic magnetic rod is applied for a study of a behavior of the flexible magnetic particle chain in a rotating magnetic field. By numerical simulation it is shown that behavior of a flexible magnetic chain is characterized by the existence of a critical frequency beyond which the dynamics of the rod is periodic with subsequent stages of bending and straightening. The value of the critical frequency found is explained by a simple model. Below the critical frequency the chain is bent and rotates synchronously with a field. It is illustrated that in particular cases the considered model reproduces phenomena observed experimentally and numerically for the magnetic particle chains in magnetorheological suspensions. It is emphasized that the present approach gives the general framework for the description of different phenomena in magnetorheological suspensions.

Drag reduction through self-similar bending of a flexible body

The classical theory of high-speed flow predicts that a moving rigid object experiences a drag proportional to the square of its speed. However, this reasoning does not apply if the object in the flow is flexible, because its shape then becomes a function of its speed--for example, the rolling up of broad tree leaves in a stiff wind. The reconfiguration of bodies by fluid forces is common in nature, and can result in a substantial drag reduction that is beneficial for many organisms. Experimental studies of such flow-structure interactions generally lack a theoretical interpretation that unifies the body and flow mechanics. Here we use a flexible fibre immersed in a flowing soap film to measure the drag reduction that arises from bending of the fibre by the flow. Using a model that couples hydrodynamics to bending, we predict a reduced drag growth compared to the classical theory. The fibre undergoes a bending transition, producing shapes that are self-similar; for such configurations, the drag scales with the length of self-similarity, rather than the fibre profile width. These predictions are supported by our experimental data.

Near equilibrium dynamics of nonhomogeneous Kirchhoff filaments in viscous media

We study the near equilibrium dynamics of nonhomogeneous elastic filaments in viscous media using the Kirchhoff model of rods. Viscosity is incorporated in the model as an external force, which we approximate by the resistance felt by an infinite cylinder immersed in a slowly moving fluid. We use the recently developed method of Goriely and Tabor [Phys. Rev. Lett. 77, 3537 (1996); Physica D 105, 20 (1997); 105, 45 (1997)] to study the dynamics in the vicinity of the simplest equilibrium solution for a closed rod with nonhomogeneous distribution of mass, namely, the planar ring configuration. We show that small variations of the mass density along the rod are sufficient to couple the symmetric modes of the homogeneous rod problem, producing asymmetric deformations that modify substantially the dynamical coiling, even at quite low Reynolds number. The higher-density segments of the rod tend to become more rigid and less coiled. We comment on possible applications to DNA.

Buckling instabilities of one-layered growing tissues

Growth and folding in one-layered model tissue sheets are studied in a stochastic, lattice-free single cell model which considers the discrete cellular structure of the tissue, and in a coarse grained analytical approach. The polarity of the one-layered tissue is considered by a bending term. Cell division gives rise to a locally increasing metric. An exponential and a power-law growth regime are identified. In both regimes, folding occurs as soon as the bending contribution becomes too small to compensate the destabilizing effect of the cell proliferation. The potential biological relevance is discussed.

Bistable helices

We extend elasticity theory of filaments to encompass systems, such as bacterial flagella, that display competition between two helical structures of opposite chirality. A general, fully intrinsic formulation of the dynamics of bend and twist degrees of freedom is developed using the natural frame of space curves, spanning from the inviscid limit to the viscously overdamped regime applicable to cellular biology. Aspects of front propagation found in flagella are discussed.

Transport of torsional stress in DNA

It is well known that transcription can induce torsional stress in DNA, affecting the activity of nearby genes or even inducing structural transitions in the DNA duplex. It has long been assumed that the generation of significant torsional stress requires the DNA to be anchored, forming a limited topological domain, because otherwise it would spin almost freely about its axis. Previous estimates of the rotational drag have, however, neglected the role of small natural bends in the helix backbone. We show how these bends can increase the drag several thousandfold relative to prior estimates, allowing significant torsional stress even in linear unanchored DNA. The model helps explain several puzzling experimental results on structural transitions induced by transcription of DNA.