Semiflexible Filaments Research Articles

Symmetry and stability of orientationally ordered collective motions of self-propelled, semiflexible filaments

Ordered, collective motions commonly arise spontaneously in systems of many interacting, active units, ranging from cellular tissues and bacterial colonies to self-propelled colloids and animal flocks. Active phases are especially rich when the active units are sufficiently anisotropic to produce liquid crystalline order and thus active nematic phenomena, with important biophysical examples provided by cytoskeletal filaments including microtubules and actin. Gliding assay experiments have provided a test bed to study the collective motions of these cytoskeletal filaments and unlocked diverse collective active phases, including states with long-range orientational order. However, it is not well understood how such long-range order emerges from the interplay of passive and active aligning mechanisms. We use Brownian dynamics simulations to study the collective motions of semiflexible filaments that self-propel in quasi-two-dimensions, in order to gain insights into the aligning mechanisms at work in these gliding assay systems. We find that, without aligning torques in the microscopic model, long-range orientational order can only be achieved when the filaments are able to overlap. The symmetry (nematic or polar) of the long-range order that first emerges is shown to depend on the energy cost of filament overlap and on filament flexibility. However, our model also predicts that a long-range-ordered active nematic state is merely transient, whereas long-range polar order is the only active dynamical steady state in systems with finite filament rigidity. Published by the American Physical Society 2024

Density and inertia effects on two-dimensional active semiflexible filament suspensions.

We examine the influence of density on the transition between chain and spiral structures in planar assemblies of active semiflexible filaments, utilizing detailed numerical simulations. We focus on how increased density, and higher Péclet numbers, affect the activity-induced transition spiral state in a semiflexible, self-avoiding active chain. Our findings show that increasing the density causes the spiral state to break up, reverting to a motile chain-like shape. This results in a density-dependent reentrant phase transition from spirals back to open chains. We attribute this phenomenon to an inertial effect observed at the single polymer level, where increased persistence length due to inertia has been shown in recent three-dimensional studies to cause polymers to open up. Our two-dimensional simulations further reveal that a reduction in the damping coefficient leads to partial unwinding of the spirals, forming longer arms. In suspension, interactions among these extended arms can trigger a complete unwinding of the spirals, driven by the combined effects of density and inertia.

Inference of the force pattern acting on a semiflexible filament from shape analysis.

The study of the active forces acting on semiflexible filaments networks such as the cytoskeleton requires noninvasive tools able to explore the deformation of single filaments in their natural environment. We propose here a practical method based on the solution of the hydrodynamic beam equationin the presence of transverse forces. We found that the derivative of the local curvature presents discontinuities that match the location of the applied forces, in contrast to the smooth curvature function obtained for the case of compressing longitudinal forces. These patterns can be easily appreciated in a kymograph of the curvature, which also reflects the temporal behavior of the forces. We assessed the method performance with numerical simulations describing the deformation of single microtubules provoked by the action of intracellular active forces.

Transverse Fluctuations Control the Assembly of Semiflexible Filaments

Transverse Fluctuations Control the Assembly of Semiflexible Filaments

Computational modeling of coupled interactions of fluid membranes with embedded filaments

In this work, we present a computational formulation based on continuum mechanics to study the interaction of fluid membranes embedded with semiflexible filaments. This is motivated by systems in membrane biology, such as cytoskeletal networks and protein filaments aiding the cell fission process. We model the membrane as a fluid shell via the Helfrich–Canham energy and the filament as a one-dimensional Cosserat continuum. We assume that the filament is tethered to the surface of the membrane while being allowed to freely float on the surface. The fluidity of the membrane and the novel filament-membrane coupling gives rise to unique computational challenges, which are addressed in this work. Finally, we present validation results and apply the formulation to several problems inspired by cellular biology.

Dynamic magnetic properties of magnetosomes

Magnetosomes are found in magnetotactic bacteria, and consist of linear chains of tens of single-domain magnetic nanoparticles of (magnetite) or (greigite), enclosed in a lipid bilayer membrane. The particles are typically in the size range 30–, and are practically monodisperse. Harnessing the power of biomineralisation could lead to efficient strategies for synthesising semiflexible dipolar filaments, and the development of optimum materials for applications, including in biomedicine. Brownian dynamics simulations of noninteracting magnetosomes, containing ferromagnetic nanoparticles, have been used to determine static properties, and the dynamical response to a weak AC magnetic field. Results are presented for the radius of gyration , the static magnetic susceptibility , the dynamic magnetic susceptibility , and the effective Brownian rotation time , all as functions of N. The results are compared to theoretical predictions of the flexibility, susceptibility, and rotational dynamics of such magnetic filaments.

Ring-linear mixtures of semiflexible rubber bands

During the synthesis of ring polymers, it is quite common to obtain mixtures containing both ring and linear molecules. Recent studies, including experiments and simulations, have shown that even small quantities of linear molecules can significantly influence the shape of ring polymers, ultimately altering their rheological response. To further explore this phenomenon, we investigated blends of semiflexible linear and ring filaments by using disordered assemblies of open and closed rubber bands. We employed x-ray tomography to analyze the structure of these mixtures, focusing on how the length and composition of linear bands influenced the overall mixture. In contrast to the behavior observed in fully-flexible polymers, our findings revealed that increasing the concentration of linear bands could actually decrease the average size of rings within a semiflexible ring-linear mixture. This outcome is attributable to a reduction in inter-ring threading, which naturally occurs as the proportion of rings diminishes. To validate our findings, we conducted molecular dynamics simulations on semiflexible ring-linear polymer mixtures in bulk. These simulations confirmed that our results stem from the semiflexibility of the bands or chains rather than confinement or athermal conditions.

Nonequilibrium diffusion of active particles bound to a semiflexible polymer network: Simulations and fractional Langevin equation.

In a viscoelastic environment, the diffusion of a particle becomes non-Markovian due to the memory effect. An open question concerns quantitatively explaining how self-propulsion particles with directional memory diffuse in such a medium. Based on simulations and analytic theory, we address this issue with active viscoelastic systems where an active particle is connected with multiple semiflexible filaments. Our Langevin dynamics simulations show that the active cross-linker displays superdiffusive and subdiffusive athermal motion with a time-dependent anomalous exponent α. In such viscoelastic feedback, the active particle always exhibits superdiffusion with α = 3/2 at times shorter than the self-propulsion time (τA). At times greater than τA, the subdiffusive motion emerges with α bounded between 1/2 and 3/4. Remarkably, active subdiffusion is reinforced as the active propulsion (Pe) is more vigorous. In the high Pe limit, athermal fluctuation in the stiff filament eventually leads to α = 1/2, which can be misinterpreted with the thermal Rouse motion in a flexible chain. We demonstrate that the motion of active particles cross-linking a network of semiflexible filaments can be governed by a fractional Langevin equation combined with fractional Gaussian noise and an Ornstein-Uhlenbeck noise. We analytically derive the velocity autocorrelation function and mean-squared displacement of the model, explaining their scaling relations as well as the prefactors. We find that there exist the threshold Pe (Pe∗) and crossover times (τ∗ and τ†) above which active viscoelastic dynamics emerge on timescales of τ∗≲ t ≲ τ†. Our study may provide theoretical insight into various nonequilibrium active dynamics in intracellular viscoelastic environments.

Inferring scale-dependent non-equilibrium activity using carbon nanotubes.

In living systems, irreversible, yet stochastic, molecular interactions form multiscale structures (such as cytoskeletal networks), which mediate processes (such as cytokinesis and cellular motility) in a close relationship between the structure and function. However, owing to a lack of methods to quantify non-equilibrium activity, their dynamics remain poorly characterized. Here, by measuring the time-reversal asymmetry encoded in the conformational dynamics of filamentous single-walled carbon nanotubes embedded in the actomyosin network of Xenopus egg extract, we characterize the multiscale dynamics of non-equilibrium activity encoded in bending-mode amplitudes. Our method is sensitive to distinct perturbations to the actomyosin network and the concentration ratio of adenosine triphosphate to adenosine diphosphate. Thus, our method can dissect the functional coupling of microscopic dynamics to the emergence of larger scale non-equilibrium activity. We relate the spatiotemporal scales of non-equilibrium activity to the key physical parameters of a semiflexible filament embedded in a non-equilibrium viscoelastic environment. Our analysis provides a general tool to characterize steady-state non-equilibrium activity in high-dimensional spaces.

Bending fluctuations in semiflexible, inextensible, slender filaments in Stokes flow: Toward a spectral discretization.

Semiflexible slender filaments are ubiquitous in nature and cell biology, including in the cytoskeleton, where reorganization of actin filaments allows the cell to move and divide. Most methods for simulating semiflexible inextensible fibers/polymers are based on discrete (bead-link or blob-link) models, which become prohibitively expensive in the slender limit when hydrodynamics is accounted for. In this paper, we develop a novel coarse-grained approach for simulating fluctuating slender filaments with hydrodynamic interactions. Our approach is tailored to relatively stiff fibers whose persistence length is comparable to or larger than their length and is based on three major contributions. First, we discretize the filament centerline using a coarse non-uniform Chebyshev grid, on which we formulate a discrete constrained Gibbs-Boltzmann (GB) equilibrium distribution and overdamped Langevin equation for the evolution of unit-length tangent vectors. Second, we define the hydrodynamic mobility at each point on the filament as an integral of the Rotne-Prager-Yamakawa kernel along the centerline and apply a spectrally accurate "slender-body" quadrature to accurately resolve the hydrodynamics. Third, we propose a novel midpoint temporal integrator, which can correctly capture the Ito drift terms that arise in the overdamped Langevin equation. For two separate examples, we verify that the equilibrium distribution for the Chebyshev grid is a good approximation of the blob-link one and that our temporal integrator for overdamped Langevin dynamics samples the equilibrium GB distribution for sufficiently small time step sizes. We also study the dynamics of relaxation of an initially straight filament and find that as few as 12 Chebyshev nodes provide a good approximation to the dynamics while allowing a time step size two orders of magnitude larger than a resolved blob-link simulation. We conclude by applying our approach to a suspension of cross-linked semiflexible fibers (neglecting hydrodynamic interactions between fibers), where we study how semiflexible fluctuations affect bundling dynamics. We find that semiflexible filaments bundle faster than rigid filaments even when the persistence length is large, but show that semiflexible bending fluctuations only further accelerate agglomeration when the persistence length and fiber length are of the same order.

Effects of direction reversals on patterns of active filaments

Active matter systems provide fascinating examples of pattern formation and collective motility without counterpart in equilibrium systems. Here, we employ Brownian dynamics simulations to study the collective motion and self-organization in systems of self-propelled semiflexible filaments, inspired by the gliding motility of filamentous Cyanobacteria. Specifically, we investigate the influence of stochastic direction reversals on the patterns. We explore pattern formation and dynamics by modulating three relevant physical parameters, the bending stiffness, the activity, and the reversal rate. In the absence of reversals, our results show rich dynamical behavior including spiral formation and collective motion of aligned clusters of various sizes, depending on the bending stiffness and self-propulsion force. The presence of reversals diminishes spiral formation and reduces the sizes of clusters or suppresses clustering entirely. This homogenizing effect of direction reversals can be understood as reversals providing an additional mechanism to either unwind spirals or to resolve clusters.

Tuning the depolymerization and restructuring of actin networks via thymosin and cofilin.

Actin is a key component of the cytoskeleton responsible for vital functions ranging from motility to structural support of cells. This multifunctionality is due, in large part, to the dynamic polymerization of actin monomers into semi-flexible filaments, the depolymerization of filaments into monomers, and the changing lengths of filaments. Polymerization drives cell migration, while depolymerization and turnover mediate cell growth and restructuring. Thymosin-β4 and cofilin are two key actin-binding proteins that promote depolymerization by sequestering actin monomers or severing filaments, respectively. Here, we incorporate thymosin and cofilin into entangled actin networks and monitor the subsequent in situ depolymerization and restructuring via confocal microscopy. We use Fourier image analysis techniques to characterize the time-varying dynamics and restructuring of the networks and map their dependence on the concentrations of actin, cofilin, and thymosin. Our results not only shed light on cellular processes that rely on actin polymerization and depolymerization but also may inspire the engineering of non-equilibrium materials that leverage depolymerization as a route to dynamically transition from gel-like to fluid-like states.

Whey protein isolate nanofibrils formed with phosphoric acid: Formation, structural characteristics, and emulsion stability

In this study, phosphoric acid was used to replace hydrochloric acid which is widely reported in the production of whey protein isolate fibrils (WPF). The pH, temperature, and protein concentration for the phosphoric acid-induced fibrillation were optimized (4% w/v, pH 2.5, 90 °C). Under this condition, the mean particle size and molecular weight decreased significantly while the ζ-potential increased slightly, it suggested that the fibrillation was accompanied by hydrolysis. WPF was already formed via intermolecular interactions at 6 h, which exhibited a semiflexible and unbranched filament of about 10 nm in diameter. A few single-filament flexible fibrils (about 5 nm in diameter) formed at 24 h. Thioflavin T analysis supported by Intrinsic fluorescence showed that the use of phosphoric acid accelerated the fibrillation. The ANS fluorescence and Second-derivative UV–vis spectra indicated the increase in surface hydrophobicity, peaking after 12 h of heating. Compared with the hydrochloric acid induced, WPF induced by phosphoric acid exhibited a shorter lag phase and maintained a more stable dispersion state with a nanoscale particle size. The great interfacial properties and proper aspect ratio (length/diameter) provided the contribution that WPF effectively absorbed at the oil-water interface. Moreover, the emulsion stability of the WPF improved with increasing salt concentration. The phosphoric acid-induced WPF had the best emulsion stability at pH 5.5 which was further enhanced by proper heating. • The whey protein isolate fibrils (WPF) were formed via phosphoric acid. • The optimal preparation of WPF was at pH 2.5, 4%, 90 °C. • The surface hydrophobicity of WPF-12 (WPF formed at 12 h) reached the highest. • The WPF-12 exhibited excellent emulsion stability at pH 5.5.

Kinesin-driven active cytoskeletal composite restructuring.

The cytoskeleton is an intricate dynamic network that enables vital processes in the cell such as mitosis, movement, and growth. It is composed of protein filaments including semi-flexible actin filaments and rigid microtubules, as well as motor proteins, such as kinesin. The versatile restructurability of the cytoskeleton makes it highly intriguing for designing materials that combine tunability, flexibility, and resilience. Applications can span from soft tissue repair to biodegradable plastics and rubbers. Here, we create actin-microtubule composites, driven by kinesin motors, and adapted by different types of cross-linkers. We observe a range of restructuring of the composites, from formation of homogeneous clusters to heterogeneous threadlike networks. We quantify dynamics of our active composites by using particle image velocimetry (PIV) on time series videos of fluorescently labeled actin and microtubules, and by performing single particle tracking (SPT) on fluorescently labeled NeutrAvidin. Additionally, by imaging spherical hydrogels in the networks, then measuring the deformation of the spheres, we are able to measure forces generated by actin-microtubule composites being reconfigured by kinesin motors.

The Arp2/3 complex contributes to the regeneration of branched actin filament networks.

Actin filaments are one of the indispensable components of almost all cell types. These semi-flexible filaments constitute an important part of the cytoskeleton and give the cell its mechanical properties. Among various structures that actin filaments form, branched actin network plays a crucial role in many cellular processes such as cell motility and endocytosis. Branched actin filaments are formed when an activated Arp2/3 complex binds to the side of an existing filament (mother) and generates a new filament branching off the mother. These filaments are subjected to force when they are pushing against the membrane or as a result of myosin motors pulling on them. Furthermore, biochemical conditions, namely actin-binding proteins, can regulate the dynamics of branched actin networks, notably the stability of branch junctions. Here we use a microfluidics approach to reconstitute individual branched actin filaments from purified proteins in-vitro. Thanks to microfluidics, we can grow filaments branching off from the side of the mother filaments and pull on them with different orientations, varying continuously between 0 and 180 degrees. Such an approach allows us to characterize the effect of force orientation, as well as force magnitude, on debranching in the presence of different regulatory proteins. Furthermore, we are able to follow the fate of Arp2/3 after debranching. Contrary to previous observations, we find specific physiological conditions in which Arp2/3 can remain attached to the mother filament after branch departure. Unexpectedly, in this type of debranching, the remaining Arp2/3 complex is able to nucleate a new branch and contribute to the turnover of the network.

Coiling of semiflexible paramagnetic colloidal chains.

Semiflexible filaments deform into a variety of configurations that dictate different phenomena manifesting at low Reynolds number. Harnessing the elasticity of these filaments to perform transport-related processes at the microfluidic scale requires structures that can be directly manipulated to attain controllable geometric features during their deformation. The configuration of semiflexible chains assembled from paramagnetic colloids can be readily controlled upon the application of external time-varying magnetic fields. In circularly rotating magnetic fields, these chains undergo coiling dynamics in which their ends close into loops that wrap inward, analogous to the curling of long nylon filaments under shear. The coiling is promising for the precise loading and targeted transport of small materials, however effective implementation requires an understanding of the role that field parameters and chain properties play on the coiling features. Here, we investigate the formation of coils in semiflexible paramagnetic chains using numerical simulations. We demonstrate that the size and shape of the initial coils are governed by the Mason and elastoviscous numbers, related to the field parameters and the chain bending stiffness. The size of the initial coil follows a nonmonotonic behavior with Mason number from which two regions are identified: (1) an elasticity-dependent nonlinear regime in which the coil size decreases with increasing field strength and for which loop shape tends to be circular, and (2) an elasticity-independent linear regime where the size increases with field strength and the shape become more elliptical. From the time scales associated to these regimes, we identify distinct coiling mechanisms for each case that relate the coiling dynamics to two other configurational dynamics of paramagnetic chains: wagging and folding behaviors.

Structural Features and Nonlinear Rheology of Self-Assembled Networks of Cross-Linked Semiflexible Polymers.

Disordered networks of semiflexible filaments are common support structures in biology. Familiar examples include fibrous matrices in blood clots, bacterial biofilms, and essential components of cells and tissues of plants, animals, and fungi. Despite the ubiquity of these networks in biomaterials, we have only a limited understanding of the relationship between their structural features and their highly strain-sensitive mechanical properties. In this work, we perform simulations of three-dimensional networks produced by the irreversible formation of cross-links between linker-decorated semiflexible filaments. We characterize the structure of networks formed by a simple diffusion-dependent assembly process and measure their associated steady-state rheological features at finite temperature over a range of applied prestrains that encompass the strain-stiffening transition. We quantify the dependence of network connectivity on cross-linker availability and detail the associated connectivity dependence of both linear elasticity and nonlinear strain-stiffening behavior, drawing comparisons with prior experimental measurements of the cross-linker concentration-dependent elasticity of actin gels.

Organization and dynamics of cross-linked actin filaments in confined environments

The organization of the actin cytoskeleton is impacted by the interplay between physical confinement, features of cross-linking proteins, and deformations of semiflexible actin filaments. Some cross-linking proteins preferentially bind filaments in parallel, although others bind more indiscriminately. However, a quantitative understanding of how the mode of binding influences the assembly of actin networks in confined environments is lacking. Here we employ coarse-grained computer simulations to study the dynamics and organization of semiflexible actin filaments in confined regions upon the addition of cross-linkers. We characterize how the emergent behavior is influenced by the system shape, the number and type of cross-linking proteins, and the length of filaments. Structures include isolated clusters of filaments, highly connected filament bundles, and networks of interconnected bundles and loops. Elongation of one dimension of the system promotes the formation of long bundles that align with the elongated axis. Dynamics are governed by rapid cross-linking into aggregates, followed by a slower change in their shape and connectivity. Cross-linking decreases the average bending energy of short or sparsely connected filaments by suppressing shape fluctuations. However, it increases the average bending energy in highly connected networks because filament bundles become deformed, and small numbers of filaments exhibit long-lived, highly unfavorable configurations. Indiscriminate cross-linking promotes the formation of high-energy configurations due to the increased likelihood of unfavorable, difficult-to-relax configurations at early times. Taken together, this work demonstrates physical mechanisms by which cross-linker binding and physical confinement impact the emergent behavior of actin networks, which is relevant both in cells and in synthetic environments.

Distinct impacts of polar and nematic self-propulsion on active unjamming

Though jamming transitions are long studied in condensed matter physics and granular systems, much less is known about active jamming (or unjamming), which commonly takes place in living materials. In this paper, we explore, by molecular dynamic simulations, the jamming-unjamming transition in a dense system of active semiflexible filaments. In particular, we characterize the distinct impact of polar vs nematic driving for different filament rigidities and at varying densities. Our results show that high densities of dynamic active filaments can be achieved by only changing the nature of the active force, nematic or polar. Interestingly, while polar driving is more effective at unjamming the system at high densities below confluency, we find that at even higher densities, nematic driving enhances unjamming compared to its polar counterpart. The effect of varying the rigidity of filaments is also significantly different in the two cases: While for nematic driving, lowering the bending rigidity unjams the system, we find an intriguing reentrant jamming-unjamming-jamming transition for polar driving as the filament rigidity is lowered. While the first transition (unjamming) is driven by softening due to reduced rigidity, the second transition (jamming) is a cooperative effect of ordering and coincides with the emergence of nematic order in the system. Together, through a generic model of self-propelled flexible filaments, our results demonstrate how tuning the nature of self-propulsion and flexibility can be employed by active materials to achieve high densities without getting jammed.

Compression-induced buckling of a semiflexible filament in two and three dimensions.

The ability of biomolecules to exert forces on their surroundings or resist compression from the environment is essential in a variety of biologically relevant contexts. For filaments in the low-temperature limit and under a constant compressive force, Euler buckling theory predicts a sudden transition from a compressed state to a bent state in these slender rods. In this paper, we use a mean-field theory to show that if a semiflexible chain is compressed at a finite temperature with a fixed end-to-end distance (permitting fluctuations in the compressive forces), it exhibits a continuous phase transition to a buckled state at a critical level of compression. We determine a quantitatively accurate prediction of the transverse position distribution function of the midpoint of the chain that indicates this transition. We find that the mean compressive forces are non-monotonic as the extension of the filament varies, consistent with the observation that strongly buckled filaments are less able to bear an external load. We also find that for the fixed extension (isometric) ensemble, the buckling transition does not coincide with the local minimum of the mean force (in contrast to Euler buckling). We also show that the theory is highly sensitive to fluctuations in length in two dimensions and the buckling transition can still be accurately recovered by accounting for those fluctuations. These predictions may be useful in understanding the behavior of filamentous biomolecules compressed by fluctuating forces, relevant in a variety of biological contexts.