Semiflexible Filaments Research Articles

Reconstituting and Characterizing Actin-Microtubule Composites with Tunable Motor-Driven Dynamics and Mechanics.

The composite cytoskeleton, comprising interacting networks of semiflexible actin filaments and rigid microtubules, restructures and generates forces using motor proteins such as myosin II and kinesin to drive key processes such as migration, cytokinesis, adhesion, and mechanosensing. While actin-microtubule interactions are key to the cytoskeleton's versatility and adaptability, an understanding of their interplay with myosin and kinesin activity is still nascent. This work describes how to engineer tunable three-dimensional composite networks of co-entangled actin filaments and microtubules that undergo active restructuring and ballistic motion, driven by myosin II and kinesin motors, and are tuned by the relative concentrations of actin, microtubules, motor proteins, and passive crosslinkers. Protocols for fluorescence labeling of the microtubules and actin filaments to most effectively visualize composite restructuring and motion using multi-spectral confocal imaging are also detailed. Finally, the results of data analysis methods that can be used to quantitatively characterize non-equilibrium structure, dynamics, and mechanics are presented. Recreating and investigating this tunable biomimetic platform provides valuable insight into how coupled motor activity, composite mechanics, and filament dynamics can lead to myriad cellular processes from mitosis to polarization to mechano-sensation.

General solutions of linear poro-viscoelastic materials in spherical coordinates

The cell cytoskeleton is a dynamic assembly of semi-flexible filaments and motor proteins. The cytoskeleton mechanics is a determining factor in many cellular processes, including cell division, cell motility and migration, mechanotransduction and intracellular transport. Mechanical properties of the cell, which are determined partly by its cytoskeleton, are also used as biomarkers for disease diagnosis and cell sorting. Experimental studies suggest that in whole cell scale, the cell cytoskeleton and its permeating cytosol may be modelled as a two-phase poro-viscoelastic (PVE) material composed of a viscoelastic (VE) network permeated by a viscous cytosol. We present the first general solution to this two-phase system in spherical coordinates, where we assume that both the fluid and network phases are in their linear response regime. Specifically, we use generalized linear incompressible and compressible VE constitutive equations to describe the stress in the fluid and network phases, respectively. We assume a constant permeability that couples the fluid and network displacements. We use these general solutions to study the motion of a rigid sphere moving under a constant force inside a two-phase system, composed of a linear elastic network and a Newtonian fluid. It is shown that the network compressibility introduces a slow relaxation of the sphere and non-monotonic network displacements with time along the direction of the applied force. Our results can be applied to particle-tracking microrheology to differentiate between PVE and VE materials, and to measure the fluid permeability as well as VE properties of the fluid and the network phases.

On Stretching, Bending, Shearing, and Twisting of Actin Filaments I: Variational Models.

Mechanochemical simulations of actomyosin networks are traditionally based on one-dimensional models of actin filaments having zero width. Here, and in the follow up paper (arXiv, DOI 10.48550/arXiv.2203.01284), approaches are presented for more efficient modeling that incorporates stretching, shearing, and twisting of actin filaments. Our modeling of a semiflexible filament with a small but finite width is based on the Cosserat theory of elastic rods, which allows for six degrees of freedom at every point on the filament's backbone. In the variational models presented in this paper, a small and discrete set of parameters is used to describe a smooth filament shape having all degrees of freedom allowed in the Cosserat theory. Two main approaches are introduced: one where polynomial spline functions describe the filament's configuration, and one in which geodesic curves in the space of the configurational degrees of freedom are used. We find that in the latter representation the strain energy function can be calculated without resorting to a small-angle expansion, so it can describe arbitrarily large filament deformations without systematic error. These approaches are validated by a dynamical model of a Cosserat filament, which can be further extended by using multiresolution methods to allow more detailed monomer-based resolution in certain parts of the actin filament, as introduced in the follow up paper. The presented framework is illustrated by showing how torsional compliance in a finite-width filament can induce broken chiral symmetry in the structure of a cross-linked bundle.

F-actin bending facilitates net actomyosin contraction By inhibiting expansion with plus-end-located myosin motors

Contraction of actomyosin networks underpins important cellular processes including motility and division. The mechanical origin of actomyosin contraction is not fully-understood. We investigate whether contraction arises on the scale of individual filaments, without needing to invoke network-scale interactions. We derive discrete force-balance and continuum partial differential equations for two symmetric, semi-flexible actin filaments with an attached myosin motor. Assuming the system exists within a homogeneous background material, our method enables computation of the stress tensor, providing a measure of contractility. After deriving the model, we use a combination of asymptotic analysis and numerical solutions to show how F-actin bending facilitates contraction on the scale of two filaments. Rigid filaments exhibit polarity-reversal symmetry as the motor travels from the minus to plus-ends, such that contractile and expansive components cancel. Filament bending induces a geometric asymmetry that brings the filaments closer to parallel as a myosin motor approaches their plus-ends, decreasing the effective spring force opposing motor motion. The reduced spring force enables the motor to move faster close to filament plus-ends, which reduces expansive stress and gives rise to net contraction. Bending-induced geometric asymmetry provides both new understanding of actomyosin contraction mechanics, and a hypothesis that can be tested in experiments.

Ambient Fluid Rheology Modulates Oscillatory Instabilities in Filament-Motor Systems

Semi-flexible filaments interacting with molecular motors and immersed in rheologically complex and viscoelastic media constitute a common motif in biology. Synthetic mimics of filament-motor systems also feature active or field-activated filaments. A feature common to these active assemblies is the spontaneous emergence of stable oscillations as a collective dynamic response. In nature, the frequency of these emergent oscillations is seen to depend strongly on the viscoelastic characteristics of the ambient medium. Motivated by these observations, we study the instabilities and dynamics of a minimal filament-motor system immersed in model viscoelastic fluids. Using a combination of linear stability analysis and full non-linear numerical solutions, we identify steady states, test the linear stability of these states, derive analytical stability boundaries, and investigate emergent oscillatory solutions. We show that the interplay between motor activity, filament and motor elasticity, and fluid viscoelasticity allows for stable oscillations or limit cycles to bifurcate from steady states. When the ambient fluid is Newtonian, frequencies are controlled by motor kinetics at low viscosities, but decay monotonically with viscosity at high viscosities. In viscoelastic fluids that have the same viscosity as the Newtonian fluid, but additionally allow for elastic energy storage, emergent limit cycles are associated with higher frequencies. The increase in frequency depends on the competition between fluid relaxation time-scales and time-scales associated with motor binding and unbinding. Our results suggest that both the stability and oscillatory properties of active systems may be controlled by tailoring the rheological properties and relaxation times of ambient fluidic environments.

Physics-Based Computational Approaches to Compute the Viscoelasticity of Semiflexible Filamentous Biomaterials

This mini-review highlights recent advances on computational approaches that have been used in the characterisation of the viscoelastic response of semiflexible filamentous biomaterials. Special attention is given to the multiscale and coarse-grained approaches that might be used to model the mechanical properties of systems which involve biopolymer assemblies, for instance, actin, collagen, vimentin, microtubules, DNA, viruses, silk, amyloid fibrils, and other protein-based filaments. Besides the basic features of the most commonly used models for semiflexible filaments, I present a brief overview of the numerical approaches that can be used to extract the viscoelasticity of dilute and concentrated solutions, as well as systems with cross-linked networks. Selected examples of simulations that attempt to retrieve the complex shear moduli at experimentally relevant time and length scales, i.e., including not only the fully formed filaments and networks but also their self-assembly kinetics, are also considered.

Theory of Weakly Polydisperse Cytoskeleton Filaments.

Cytoskeleton filaments have the extraordinary ability to change conformations dynamically in response to alterations of the number density of actins/tubulin, the number density and type of binding agents, and the electrolyte concentration. This property is crucial for eukaryotic cells to achieve specific biological functions in different cellular compartments. Conventional approaches to biopolymers’ solution break down for cytoskeleton filaments because they entail several approximations to treat their polyelectrolyte and mechanical properties. In this article, we introduce a novel density functional theory for polydisperse, semiflexible cytoskeleton filaments. The approach accounts for the equilibrium polymerization kinetics, length and orientation filament distributions, as well as the electrostatic interaction between filaments and the electrolyte. This is essential for cytoskeleton polymerization in different cell compartments generating filaments of different lengths, sometimes long enough to become semiflexible. We characterized the thermodynamics properties of actin filaments in electrolyte aqueous solutions. We calculated the free energy, pressure, chemical potential, and second virial coefficient for each filament conformation. We also calculated the phase diagram of actin filaments’ solution and compared with the corresponding results in in vitro experiments.

Thermal fluctuation spectrum of flexoelectric viscoelastic semiflexible filaments and polymers: A line liquid crystal model

AbstractIn this paper, we generalize the internal viscosity model developed by Professor Williams to semiflexible polymers, biofilaments, and worm‐like micelles, where molecular dissipation is generated by bending. Current models for viscoelasticity with internal viscosity in semiflexible polymers and filaments are based on generalizations of the worm‐like model, but they neglect potential electromechanical couplings such as flexoelectricity. In this paper, inspired by the early work of Professor Williams, we develop a model for worm‐like viscoelastic flexoelectric filaments based on the “line liquid crystal model”. The electroelastic free energy and entropy production are formulated and used to derive the shape equation for these filaments undergoing thermal fluctuations. The resulting time relaxation spectrum is a useful tool to characterize experimentally viscoelastic material parameters. We show that flexoelectricity or polarization‐induced bending softens the filaments. The predicted time relaxation spectrum shows that at longer wavelength modes, the filament behaves like a rigid rod in a viscous solvent, but at shorter wavelengths, it reaches a plateau defined by the bending time scale. The key effect of flexoelectricity is to shift the entire spectrum to higher values, slowing down the response. The model itself is validated using the worm‐like chain and the viscoelasticity of liquid crystals, and the predictions are shown to be in qualitative consistency with the data. Since filament flexoelectricity is associated with 1D sensor‐actuator functionalities, the presented model has many potential novel applications in reduced geometries.

Extreme mechanics of colloidal polymers under compression: Buckling, creep, and break-up

Self-assembling patchy colloidal particles form a promising platform to create designer soft materials. To dress such systems with mechanical functionality, one can take inspiration from biological structures such as the cell's cytoskeleton, which consists of semiflexible filaments, whose mechanical behavior give the cell its unique mechanical properties. Here we present mechanical experiments on analogs of biological fibers, semiflexible ``colloidal polymers'' made from bonded patchy colloidal particles. We use optical tweezers to probe their extreme mechanics under increasingly high compressions and we reveal a rich nonlinear mechanical response involving buckling, viscoelastic creep, and ultimately break-up. We characterize and model this response using elastic and viscoelastic models involving Euler buckling and stress relaxation. This allows us to identify the critical Euler buckling force, and relate the critical bending at break-up to the finite patch size of the colloids. These results demonstrate the crucial role of the patch-patch interactions in the mechanics of self-assembled colloidal materials, and they provide mechanical relationships that are essential to design functional colloidal architectures inspired by nature.

Active dynamics and tunable mechanics of actin-microtubule composites

The composite cytoskeleton, comprising interacting networks of semiflexible actin filaments and rigid microtubules, restructures and generates forces using motor proteins such as myosin II to drive key processes including migration, cytokinesis, adhesion, and mechanosensing. While actin-microtubule interactions are key to the cytoskeleton's versatility and adaptability, an understanding of their interplay with actomyosin activity is still nascent. I will discuss our recent studies investigating the dynamics, structure and microrheological properties of in vitro actin-microtubule composites driven by myosin II minifilaments.

Mechanics of transient semi-flexible networks: Soft-elasticity, stress relaxation and remodeling

Networks of semi-flexible (or athermal) filaments cross-linked by flexible chains are found in a variety of biopolymers such as soft connective tissues, the cell’s cytoskeleton or the wall of plant cells. They can also be synthetized in the lab to create liquid crystal elastomers-like gels as well as tissue mimetics. While the elasticity of these networks has been explored, the visco-elastic response that originate from the existence of reversible and dynamic cross-links is still poorly understood. We here develop a model for these networks by taking a multiscale, statistical mechanics approach where the network is decomposed into its most basic building blocks: elastic rods (to describe semi-flexible filaments) and the flexible chains used to cross-link them. The topology of this assembly is represented by a hairy rod model for which we express the non-affine kinematics, and evolution equations for both cross-linkers and rods conformation. The mechanical response of this hairy rod is then expressed by an elastic potential that is built as a function of the basic elasticity of its components. The resulting model is able to capture salient features of the mechanics of such networks, including nonlinear elasticity (and in particular a liquid crystal-like soft-elastic response), creep and stress relaxation, as well as rate- and history-dependent network remodeling. The theory can thus be potentially used to better understand the rich response of these complex, yet ubiquitous networks and guide their development in the laboratory.

Single-walled carbon nanotube reptation dynamics in submicron sized pores from randomly packed mono-sized colloids

Studying the Brownian motion of fibers and semi-flexible filaments in porous media is the key to understanding the transport and mechanical properties in a variety of systems. The motion of semi-flexible filaments in gel-like porous media including polymer networks and cell cytoskeleton has been studied theoretically and experimentally, whereas the motion of these materials in packed-colloid porous media, advanced foams, and rock-like systems has not been thoroughly studied. Here we use video microscopy to directly visualize the reptation and transport of intrinsically fluorescent, semiflexible, semiconducting single-walled carbon nanotubes (SWCNTs) in the sub-micron pores of packed colloids as fixed obstacles of packed-colloid porous media. By visualizing the filament motion and Brownian diffusion at different locations in the pore structures, we study how the properties of the environment, like the pore shape and pore structure of the porous media, affect SWCNT mobility. These results show that the porous media structure controls SWCNT reorientation during Brownian diffusion. In packed-colloid pores, SWCNTs diffuse along straight pores and bend across pores; conversely, in gel pores, SWCNTs consistently diffuse into curved pores, displaying a faster parallel motion. In both gel and packed-colloid porous media, SWCNT finite stiffness enhances SWCNT rotational diffusion and prevents jamming, allowing for inter-pore diffusion.

Obstacle-induced giant jammed aggregation of active semiflexible filaments.

Filaments driven by bound motor proteins and chains of self-propelled colloidal particles are a typical example of active polymers (APs). Due to deformability, APs exhibit very rich dynamic behaviors and collective assembling structures. Here, we are concerned with a basic question: how APs behave near a single obstacle? We find that, in the presence of a big single obstacle, the assembly of APs becomes a two-state system, i.e. APs either gather nearly completely together into a giant jammed aggregate (GJA) on the surface of the obstacle or distribute freely in space. No partial aggregation is observed. Such a complete aggregation/collection is unexpected since it happens on a smooth convex surface instead of, e.g., a concave wedge. We find that the formation of a GJA experiences a process of nucleation and the curves of the transition between the GJA and the non-aggregate state form hysteresis-like loops. Statistical analysis of massive data on the growing time, chirality and angular velocity of both the GJAs and the corresponding nuclei shows the strong random nature of the phenomenon. Our results provide new insights into the behavior of APs in contact with porous media and also a reference for the design and application of polymeric active materials.

Rigidity Dictates Spontaneous Helix Formation of Thermoresponsive Colloidal Chains in Poor Solvent.

The formation of helical motifs typically requires specific directional interactions. Here, we demonstrate that isotropic interparticle attraction can drive self-assembly of colloidal chains into thermo-reversible helices, for chains with a critical level of backbone rigidity. We prepare thermoresponsive colloidal chains by cross-linking PNIPAM microgel-coated polystyrene colloids ("monomers"), aligned in an AC electric field. We control the chain rigidity by varying cross-linking time. Above the LCST of PNIPAM, there is an effective attraction between monomers so that the colloidal chains are in a bad solvent. On heating, the chains decrease in size. For the most rigid chains, the decrease is modest and is not accompanied by a change in shape. Much less rigid chains form relatively compact structures, resulting in a large increase in the local monomer density. Unusually, chains with intermediate rigidity spontaneously assemble into helical structures. The chain helicity increases with temperature and plateaus above the collapse transition temperature of the microgel particles. We simulate a minimal model that captures the spontaneous emergence of the helical conformations of the polymeric chain and provides insight into this shape transition. Our work suggests that a purely mechanical instability for semiflexible filaments can drive helix formation, without the need to invoke directional interactions.

Simulations of dynamically cross-linked actin networks: Morphology, rheology, and hydrodynamic interactions.

Cross-linked actin networks are the primary component of the cell cytoskeleton and have been the subject of numerous experimental and modeling studies. While these studies have demonstrated that the networks are viscoelastic materials, evolving from elastic solids on short timescales to viscous fluids on long ones, questions remain about the duration of each asymptotic regime, the role of the surrounding fluid, and the behavior of the networks on intermediate timescales. Here we perform detailed simulations of passively cross-linked non-Brownian actin networks to quantify the principal timescales involved in the elastoviscous behavior, study the role of nonlocal hydrodynamic interactions, and parameterize continuum models from discrete stochastic simulations. To do this, we extend our recent computational framework for semiflexible filament suspensions, which is based on nonlocal slender body theory, to actin networks with dynamic cross linkers and finite filament lifetime. We introduce a model where the cross linkers are elastic springs with sticky ends stochastically binding to and unbinding from the elastic filaments, which randomly turn over at a characteristic rate. We show that, depending on the parameters, the network evolves to a steady state morphology that is either an isotropic actin mesh or a mesh with embedded actin bundles. For different degrees of bundling, we numerically apply small-amplitude oscillatory shear deformation to extract three timescales from networks of hundreds of filaments and cross linkers. We analyze the dependence of these timescales, which range from the order of hundredths of a second to the actin turnover time of several seconds, on the dynamic nature of the links, solvent viscosity, and filament bending stiffness. We show that the network is mostly elastic on the short time scale, with the elasticity coming mainly from the cross links, and viscous on the long time scale, with the effective viscosity originating primarily from stretching and breaking of the cross links. We show that the influence of nonlocal hydrodynamic interactions depends on the network morphology: for homogeneous meshworks, nonlocal hydrodynamics gives only a small correction to the viscous behavior, but for bundled networks it both hinders the formation of bundles and significantly lowers the resistance to shear once bundles are formed. We use our results to construct three-timescale generalized Maxwell models of the networks.

Multiscale Microrheology Using Fluctuating Filaments as Stealth Probes.

The mechanical properties of soft materials can be probed on small length scales by microrheology. A common approach tracks fluctuations of micrometer-sized beads embedded in the medium to be characterized. This approach yields results that depend on probe size when the medium has structure on comparable length scales. Here, we introduce filament-based microrheology using high-aspect-ratio semiflexible filaments as probes. Such quasi-1D probes are much less invasive than beads due to their small cross sections. Moreover, by imaging transverse bending modes, we simultaneously determine the micromechanical response of the medium on multiple length scales corresponding to the mode wavelengths. We use semiflexible single-walled carbon nanotubes as probes that can be accurately and rapidly imaged based on their stable near-IR fluorescence. We find that the viscoelastic properties of sucrose, polyethylene oxide, and hyaluronic acid solutions measured in this way are in good agreement with those measured by conventional micro- and macrorheology.

Protein friction and filament bending facilitate contraction of disordered actomyosin networks

We use mathematical modeling and computation to investigate how protein friction facilitates contraction of disordered actomyosin networks. We simulate two-dimensional networks using an agent-based model, consisting of a system of force-balance equations for myosin motor proteins and semiflexible actin filaments. A major advantage of our approach is that it enables direct calculation of the network stress tensor, which provides a quantitative measure of contractility. Exploiting this, we use repeated simulations of disordered networks to confirm that both protein friction and actin filament bending are required for contraction. We then use simulations of elementary two-filament systems to show that filament bending flexibility can facilitate contraction on the microscopic scale. Finally, we show that actin filament turnover is necessary to sustain contraction and prevent filament aggregation. Simulations with and without turnover also exhibit contractile pulses. However, these pulses are aperiodic, suggesting that periodic pulsation can only arise because of additional regulatory mechanisms or more complex mechanical behavior.

Braiding Dynamics in Semiflexible Filament Bundles under Oscillatory Forcing.

We examine the nonequilibrium production of topological defects—braids—in semiflexible filament bundles under cycles of compression and tension. During these cycles, the period of compression facilitates the thermally activated pair production of braid/anti-braid pairs, which then may separate when the bundle is under tension. As a result, appropriately tuned alternating periods of compression and extension should lead to the proliferation of braid defects in a bundle so that the linear density of these pairs far exceeds that expected in the thermal equilibrium. Secondly, we examine the slow extension of braided bundles under tension, showing that their end-to-end length creeps nonmonotonically under a fixed force due to braid deformation and the motion of the braid pair along the bundle. We conclude with a few speculations regarding experiments on semiflexible filament bundles and their networks.

Multiscale sterilizable 3D printed auricular templates to guide cartilaginous framework sizing and sculpture during autologous microtia reconstruction

SummaryMicrotia reconstruction using autologous costal cartilage can be one of the most challenging tasks in reconstructive surgery. An intraoperative guide using 2-dimentional drawing of the contralateral ear on an x-ray film remains the current standard of care. In this paper, we present the use of computer-aided design and desktop 3D printing to fabricate low cost, sterilizable auricular carving templates to serve as a peri-operative reference for microtia reconstruction. The design was made as a single component which incorporated the usual anatomic reference points of the ear based on Nagata technique as a Stereo-lithography file format (. STL) for 3D printing. The templates were created in sizes ranging from 55 mm to 70 mm with a 2 mm increment with an average production cost of 0.26 US dollars per material per template and about 4.5 US dollars for the whole set. Individual templates were then 3D-printed using a thermoplastic polyurethane (TPU 95A) semiflexible filament on a desktop fused deposition modeling, Ultimaker 2 + 3D printer. The produced template tolerated the sterilization process with no structural changes as compared to its pre-sterilization condition. In conclusion, we present cost-effective, sterilizable, multiscale auricular templates to guide the pre- and intra-operative carving of the cartilaginous framework during microtia reconstruction with more accuracy in a time efficient manner, thereby overcoming the drawbacks of using the traditional x-ray film. The templates are readily accessible and sharable for free through open-source software and can be directly 3D-printed using an affordable desktop 3D printer.

Hydrodynamic interactions significantly alter the dynamics of actin networks and result in a length scale dependent loss modulus

Actin, the primary component of the cytoskeleton, is the most studied semiflexible filament. Yet, the dynamics of actin filamentous network is still a subject of debate. Here we show that hydrodynamic interactions may significantly alter the time scale of actin network deformation. The alteration may be easily in the range of 2–20 fold depending on the structural conformations and scales of interest. We show that for a single fiber, hydrodynamic interactions between the cytoskeletal mesh-sized segments can change the net force by up to 7 folds. We also demonstrate that cytoskeletal relaxation times are underestimated if hydrodynamic interaction effects are ignored, but bending mode shapes are not appreciably influenced. Ignoring hydrodynamic interactions can result in up to 20-fold overestimation of shear loss modulus in the 2 μm range we investigated. Moreover, in agreement with experimental studies, our models explain a highly length scale dependent loss modulus. Taken together, our data suggest that including hydrodynamic interactions is key to proper modeling and analysis of actin dynamics at any scales and dimensions, and therefore must not be neglected in future models and experimental analyses of cytoskeletal dynamics.