Semiflexible Filaments Research Articles

Nonlinear elasticity of semiflexible filament networks.

We develop a continuum theory for equilibrium elasticity of a network of crosslinked semiflexible filaments, spanning the full range between flexible entropy-driven chains to stiff athermal rods. We choose the 3-chain constitutive model of network elasticity over several plausible candidates, and derive analytical expressions for the elastic energy at arbitrary strain, with the corresponding stress-strain relationship. The theory fits well to a wide range of experimental data on simple shear in different filament networks, quantitatively matching the differential shear modulus variation with stress, with only two adjustable parameters (which represent the filament stiffness and the pre-tension in the network, respectively). The general theory accurately describes the crossover between the positive and negative Poynting effect (normal stress on imposed shear) on increasing the stiffness of filaments forming the network. We discuss the network stability (the point of marginal rigidity) and the phenomenon of tensegrity, showing that filament pre-tension on crosslinking into the network determines the magnitude of linear modulus G0.

Trapping, entrainment and synchronization of semiflexible polymers in narrow, asymmetric confinements.

The physical properties of polymeric actin facilitate many mechanical processes within the cell, including cellular deformation and locomotion, whereby the polymers can be confined to a range of different geometries. As actin polymers often form entangled solutions in the cell, we have investigated the effect of confinement on the evolution of entangled semiflexible polymer solutions. Using a microfluidic platform, we examined the physical dynamics of actin polymers confined within narrow (2-4 μm) rectangular channels. Focusing on the entanglement process of two actin polymers, we found that their prolonged entrainment leads to synchronized horizontal undulations and decreased translational diffusion. In the absence of cross-linking molecules or proteins, the long-range entrainment interactions are predominantly controlled by the geometric boundaries. We directly measure the deflection length Λ for an individual polymer, either solitarily confined within a channel or confined in the presence of a second filament, enabling the determination of the change in free energy associated with polymer entanglement. Our results indicate that geometrical confinement can serve as a solitary variable influencing the physical dynamics of entangled semiflexible polymers.

Buckling transition of a semiflexible filament in extensional flow.

An analytical expression for the fluctuation-rounded stretch-coil transition of semiflexible polymers in extensional flows is derived. The competition between elasticity and tension is known to cause a buckling instability in filaments placed near hyperbolic stagnation points and the effect of thermal fluctuations on this transition has yet to receive full quantitative treatment. Motivated by the findings of recent experiments as well as our simulations, we solve for the amplitude of the first buckled mode near the onset of the instability. This reveals a stochastic supercritical bifurcation, which is in excellent agreement with full numerical simulations.

A semi-flexible model prediction for the polymerization force exerted by a living F-actin filament on a fixed wall.

We consider a single living semi-flexible filament with persistence length ℓp in chemical equilibrium with a solution of free monomers at fixed monomer chemical potential μ1 and fixed temperature T. While one end of the filament is chemically active with single monomer (de)polymerization steps, the other end is grafted normally to a rigid wall to mimic a rigid network from which the filament under consideration emerges. A second rigid wall, parallel to the grafting wall, is fixed at distance L < < ℓp from the filament seed. In supercritical conditions where monomer density ρ1 is higher than the critical density ρ1c, the filament tends to polymerize and impinges onto the second surface which, in suitable conditions (non-escaping filament regime), stops the filament growth. We first establish the grand-potential Ω(μ1, T, L) of this system treated as an ideal reactive mixture, and derive some general properties, in particular the filament size distribution and the force exerted by the living filament on the obstacle wall. We apply this formalism to the semi-flexible, living, discrete Wormlike chain model with step size d and persistence length ℓp, hitting a hard wall. Explicit properties require the computation of the mean force f̄i(L) exerted by the wall at L and associated potential f̄i(L)=-dWi(L)/dL on a filament of fixed size i. By original Monte-Carlo calculations for few filament lengths in a wide range of compression, we justify the use of the weak bending universal expressions of Gholami et al. [Phys. Rev. E 74, 041803 (2006)] over the whole non-escaping filament regime. For a filament of size i with contour length Lc = (i - 1) d, this universal form is rapidly growing from zero (non-compression state) to the buckling value fb(Lc,ℓp)=π(2)kBTℓp4Lc (2) over a compression range much narrower than the size d of a monomer. Employing this universal form for living filaments, we find that the average force exerted by a living filament on a wall at distance L is in practice L independent and very close to the value of the stalling force Fs (H)=(kBT/d)ln(ρˆ1) predicted by Hill, this expression being strictly valid in the rigid filament limit. The average filament force results from the product of the cumulative size fraction x=x(L,ℓp,ρˆ1), where the filament is in contact with the wall, times the buckling force on a filament of size Lc ≈ L, namely, Fs (H)=xfb(L;ℓp). The observed L independence of Fs (H) implies that x ∝ L(-2) for given (ℓp,ρˆ1) and x∝lnρˆ1 for given (ℓp, L). At fixed (L,ρˆ1), one also has x∝ℓp (-1) which indicates that the rigid filament limit ℓp → ∞ is a singular limit in which an infinite force has zero weight. Finally, we derive the physically relevant threshold for filament escaping in the case of actin filaments.

Elastic properties of magnetosome chains

Magnetotactic bacteria swim and orient in the direction of a magnetic field thanks to the magnetosome chain, a cellular ‘compass needle’ that consists of a string of vesicle-enclosed magnetic nanoparticles aligned on a cytoskeletal filament. Here we investigate the mechanical properties of such a chain, in particular the bending stiffness. We determine the contribution of magnetic interactions to the bending stiffness and the persistence length of the chain. This contribution is comparable to, but typically smaller than the contribution of the semiflexible filament. For a chain of magnetic nanoparticles without a semiflexible filament, the linear configuration is typically metastable and the lowest energy structures are closed chains (flux closure rings) without a net magnetic moment that are thus not functional as a cellular compass. Our calculations show that the presence of the cytoskeletal filament stabilizes the chain against ring closure, either thermodynamically or kinetically, depending on the stiffness of the filament, confirming that such stabilization is one of the roles of this structure in these bacterial cells.

Fragmentation and depolymerization of non-covalently bonded filaments.

Protein molecules often self-assemble by means of non-covalent physical bonds to form extended filaments, such as amyloids, F-actin, intermediate filaments, and many others. The kinetics of filament growth is limited by the disassembly rate, at which inter-protein bonds break due to the thermal motion. Existing models often assume that the thermal dissociation of subunits occurs uniformly along the filament, or even preferentially in the middle, while the well-known propensity of F-actin to depolymerize from one end is mediated by biochemical factors. Here, we show for a very general (and generic) model, using Brownian dynamics simulations and theory, that the breakup location along the filament is strongly controlled by the asymmetry of the binding force about the minimum, as well as by the bending stiffness of the filament. We provide the basic connection between the features of the interaction potential between subunits and the breakup topology. With central-force (that is, fully flexible) bonds, the breakup rate is always maximum in the middle of the chain, whereas for semiflexible or stiff filaments this rate is either a minimum in the middle or flat. The emerging framework provides a unifying understanding of biopolymer fragmentation and depolymerization and recovers earlier results in its different limits.

FiberApp: An Open-Source Software for Tracking and Analyzing Polymers, Filaments, Biomacromolecules, and Fibrous Objects

Biological semiflexible polymers and filaments such as collagen, fibronectin, actin, microtubules, coiled-coil proteins, DNA, siRNA, amyloid fibrils, etc., are ubiquitous in nature. In biology, these systems have a direct relation to critical processes ranging from the movement of actin or assembly of viruses at cellular interfaces to the growth of amyloid plaques in neurodegenerative diseases. In technology and applied sciences, synthetic macromolecules or fibrous objects such as carbon nanotubes are involved in countless applications. Accessing their intrinsic properties at the single molecule level, such as their molecular conformations or intrinsic stiffness, is central to the understanding of these systems, their properties, and the design of related applications. In this Perspective we introduce FiberApp—a new tracking and analysis software based on a cascade of algorithms describing structural and topological features of objects characterized by a very high length-to-width aspect ratio, generally d...

Role of High-Order Hydrodynamic Interaction of Semiflexible Filament Dynamics

Biopolymers like collagen, actin, microtubules, and aggrecan play a structural role in cells and tissues. They are considered semiflexible polymers because their bending stiffness is on the order of, and resists, the Brownian forces that randomize their conformation. The study of the collective dynamics of semiflexible assemblies has come to prominence because it underlies the physics of force-transmission and mechanotransduction in cells and tissues. We had previously proposed modeling a semiflexible filament as a string of beams that bend continuously under Brownian forces (Chandran et al, 2009). This idealization not only captures the high-order nonlinear bending of the filament, but it does so at reduced computational cost compared to current string-of-beads idealizations. We had also proposed solving the relative solvent velocity along the filament as an implicit variable; which is equivalent to including several orders of hydrodynamic interaction and solvent-back reflection in the polymer dynamics (Chandran et al, 2010). In this presentation we compare the predictions of the string-of-beams model with implicit hydrodynamics against that of string-of-beads approaches for new insight on semiflexible polymer dynamics that is produced by the higher-order bending and interaction terms.

Conformations, hydrodynamic interactions, and instabilities of sedimenting semiflexible filaments.

The conformations and dynamics of semiflexible filaments subject to a homogeneous external (gravitational) field, e.g., in a centrifuge, are studied numerically and analytically. The competition between hydrodynamic drag and bending elasticity generates new shapes and dynamical features. We show that the shape of a semiflexible filament undergoes instabilities as the external field increases. We identify two transitions that correspond to the excitation of higher bending modes. In particular, for strong fields the filament stabilizes in a non-planar shape, resulting in a sideways drift or in helical trajectories. For two interacting filaments, we find the same transitions, with the important consequence that the new non-planar shapes have an effective hydrodynamic repulsion, in contrast to the planar shapes which attract themselves even when their osculating planes are rotated with respect to each other. For the case of planar filaments, we show analytically and numerically that the relative velocity is not necessarily due to a different drag of the individual filaments, but to the hydrodynamic interactions induced by their shape asymmetry.

Dynamic Actomyosin Network Morphology in 3D Model of Cytokinetic Ring Assembly

At the onset of cytokinesis in fission yeast (a model organism for studying mitosis), the locations of actin filament nucleation by formins shift from the cell tips to the cell middle. These actin filaments are captured by myosin motors bound to medial cortical nodes and bundled by crosslinking proteins, in order to form a ring. The myosin motors exert forces on the actin filaments, resulting in net pulling of nodes and filaments towards the cell equator, while cross-linking interactions help align actin filaments into a single bundle. We used these mechanisms in a 3D computational model of ring assembly, with semiflexible actin filaments growing from formins at the cell tips and at the medial cell cortex, capturing and pulling of filaments by nodes, and cross-linking among filaments through attractive interactions. The model was used to predict profiles of actin density at the cortex, morphologies of condensing node-filament networks, including the transition from actin cables to ring in different conditions of myosin activity and strength of crosslinking. Results show that cross-linking interactions can lead to confinement of actin filaments that grow from medial locations of the cell cortex. We show that the ring formation region in parameter space lies close to regions leading to clumps, meshworks or double rings, and stars/cables. Since boundaries between regions are not sharp, transient structures that resemble clumps, stars and meshworks can appear in the process of ring assembly. The presence of transient nonmedial cables linking the medial region to the cell tips can further modify these boundaries. These results are consistent with prior experiments with mutations in actin filament turnover regulators, myosin motor activity and changes in the concentration of cross linkers that alter the morphology of the condensing network.

Bond breaking dynamics in semiflexible networks under load.

We examine the bond-breaking dynamics of transiently cross-linked semiflexible networks using a single filament model in which that filament is peeled from an array of cross-linkers. We examine the effect of quenched disorder in the placement of the linkers along the filament and the effect of stochastic bond-breaking (assuming Bell model unbinding kinetics) on the dynamics of filament cross-linker dissociation and the statistics of ripping events. We find that bond forces decay exponentially away from the point of loading and that bond breaking proceeds sequentially down the linker array from the point of loading in a series of stochastic ripping events. We compare these theoretical predictions to the observed trajectories of large beads in a cross-linked microtubule network and identify the observed jumps of the bead with the linker rupture events predicted by the single filament model.

The role of filament length, finite-extensibility and motor force dispersity in stress relaxation and buckling mechanisms in non-sarcomeric active gels.

After relaxing some assumptions we apply a single-chain mean-field mathematical model recently introduced [RSC Adv. (2014)] to describe the role of molecular motors in the mechanical properties of active gels. The model allows physics that are not available in models postulated on coarser levels of description. Moreover it proposes a level of description that allows the prediction of observables at time scales too difficult to achieve in multi-chain simulations for realistic filament lengths and densities. We model the semiflexible filaments that compose the active gel as bead-spring chains; molecular motors are accounted for by using a mean-field approach, in which filaments undergo transitions of one motor attachment state depending on the state of the probe filament. The level of description includes the end-to-end distance and attachment state of the filaments, and the motor-generated forces, as stochastic state variables which evolve according to a proposed differential Chapman-Kolmogorov equation. The motor-generated forces are drawn from a stationary distribution of motor stall forces. We consider bead-spring chains with multiple beads, explore the effect of finite-extensibility of the strands and incorporate into the model motor force distributions that have been measured experimentally. The model can no longer be solved analytically but is amenable to numerical simulation. This version of the model allows a more quantitative description of buckling dynamics [Lenz et. al. PRL, 2012, 108, 238107] and the dynamic modulus of active gels. The effect of finite extensibility of the filament strands on the dynamic modulus was also found to be in agreement with the microrheology experiments of Mizuno et. al., [Science, 2007, 315, 370-373].

Trains, tails and loops of partially adsorbed semi-flexible filaments.

Polymer adsorption is a fundamental problem in statistical mechanics that has direct relevance to diverse disciplines ranging from biological lubrication to stability of colloidal suspensions. We combine experiments with computer simulations to investigate depletion induced adsorption of semi-flexible polymers onto a hard-wall. Three dimensional filament configurations of partially adsorbed F-actin polymers are visualized with total internal reflection fluorescence microscopy. This information is used to determine the location of the adsorption/desorption transition and extract the statistics of trains, tails and loops of partially adsorbed filament configurations. In contrast to long flexible filaments which primarily desorb by the formation of loops, the desorption of stiff, finite-sized filaments is largely driven by fluctuating filament tails. Simulations quantitatively reproduce our experimental data and allow us to extract universal laws that explain scaling of the adsorption-desorption transition with relevant microscopic parameters. Our results demonstrate how the adhesion strength, filament stiffness, length, as well as the configurational space accessible to the desorbed filament can be used to design the characteristics of filament adsorption and thus engineer properties of composite biopolymeric materials.

Dynamic Network Morphology and Tension Buildup in a 3D Model of Cytokinetic Ring Assembly

During fission yeast cytokinesis, actin filaments nucleated by cortical formin Cdc12 are captured by myosin motors bound to a band of cortical nodes and bundled by cross-linking proteins. The myosin motors exert forces on the actin filaments, resulting in a net pulling of the nodes into a contractile ring, while cross-linking interactions help align actin filaments and nodes into a single bundle. We used these mechanisms in a three-dimensional computational model of contractile ring assembly, with semiflexible actin filaments growing from formins at cortical nodes, capturing of filaments by neighboring nodes, and cross-linking among filaments through attractive interactions. The model was used to predict profiles of actin filament density at the cell cortex, morphologies of condensing node-filament networks, and regimes of cortical tension by varying the node pulling force and strength of cross-linking among actin filaments. Results show that cross-linking interactions can lead to confinement of actin filaments at the simulated cortical boundary. We show that the ring-formation region in parameter space lies close to regions leading to clumps, meshworks or double rings, and stars/cables. Since boundaries between regions are not sharp, transient structures that resemble clumps, stars, and meshworks can appear in the process of ring assembly. These results are consistent with prior experiments with mutations in actin-filament turnover regulators, myosin motor activity, and changes in the concentration of cross-linkers that alter the morphology of the condensing network. Transient star shapes appear in some simulations, and these morphologies offer an explanation for star structures observed in prior experimental images. Finally, we quantify tension along actin filaments and forces on nodes during ring assembly and show that the mechanisms describing ring assembly can also drive ring constriction once the ring is formed.

INSIGHTS INTO THE MECHANICS OF CYTOKINETIC RING ASSEMBLY USING 3D MODELING.

During fission yeast cytokinesis, actin filaments nucleated by cortical formin Cdc12 are captured by myosin motors bound to a band of cortical nodes. The myosin motors exert forces that pull nodes together into a contractile ring. Cross-linking interactions help align actin filaments and nodes into a single bundle. Mutations in the myosin motor domain and changes in the concentration of cross-linkers alpha-actinin and fimbrin alter the morphology of the condensing network, leading to clumps, rings or extended meshworks. How the contractile tension developing during ring formation depends on the interplay between network morphology, myosin motor activity, cross-linking and actin filament turnover remains to be elucidated. We addressed this question using a 3D computational model in which semiflexible actin filaments (represented as beads connected by springs) grow from formins, can be captured by myosin in neighboring nodes, and get cross-linked with one another through an attractive interaction. We identify regimes of tension generation between connected nodes under a wide set of conditions regarding myosin dynamics and strength of cross-linking between actin filaments. We find conditions that maximize circumferential tension, correlate them with network morphology and propose experiments to test these predictions. This work addresses "Morphogenesis of soft and living matter" using computational modeling to simulate cytokinetic ring assembly from the key molecular mechanisms of viscoelastic cross-linked actin networks that include active molecular motors.

Designing highly tunable semiflexible filament networks.

Semiflexible polymers can generate a range of filamentous networks significantly different in structure from those seen in conventional polymer solutions. Our coarse-grained simulations with an implicit cross-linker potential show that networks of branching bundles, knotted morphologies, and structural chirality can be generated by a generalized approach independent of specific cross-linkers. Network structure depends primarily on filament flexibility and separation, with significant connectivity increase after percolation. Results should guide the design of engineered semiflexible polymers.

Rheology of semiflexible bundle networks with transient linkers.

We present a theoretical and computational analysis of the rheology of networks made up of bundles of semiflexible filaments bound by transient cross-linkers. Such systems are ubiquitous in the cytoskeleton and can be formed in vitro using filamentous actin and various cross-linkers. We find that their high-frequency rheology is characterized by a scaling behavior that is quite distinct from that of networks of the well-studied single semiflexible filaments. This regime can be understood theoretically in terms of a length-scale-dependent bending modulus for bundles. Next, we observe new dissipative dynamics associated with the shear-induced disruption of the network at intermediate frequencies. Finally, at low frequencies, we encounter a region of non-Newtonian rheology characterized by power-law scaling. This regime is dominated by bundle dissolution and large-scale rearrangements of the network driven by equilibrium thermal fluctuations.

High-frequency affine mechanics and nonaffine relaxation in a model cytoskeleton.

The cytoskeleton is a network of crosslinked, semiflexible filaments, and it has been suggested that it has properties of a glassy state. Here we employ optical-trap-based microrheology to apply forces to a model cytoskeleton and measure the high-bandwidth response at an anterior point. Simulating the highly nonlinear and anisotropic stress-strain propagation assuming affinity, we found that theoretical predictions for the quasistatic response of semiflexible polymers are only realized at high frequencies inaccessible to conventional rheometers. We give a theoretical basis for determining the frequency when both affinity and quasistaticity are valid, and we discuss with experimental evidence that the relaxations at lower frequencies can be characterized by the experimentally obtained nonaffinity parameter.

Scale-Dependent Nonaffine Elasticity of Semiflexible Polymer Networks

The cytoskeleton of eukaryotic cells provides mechanical support and governs intracellular transport. These functions rely on the complex mechanical properties of networks of semiflexible protein filaments. We study the impact of local network deformations on the scale-dependent mobility of probe particles in entangled networks of actin filaments using high-bandwidth microrheology. We find that micron-sized particles in these networks experience two opposing noncontinuum elastic effects: entropic depletion reduces the effective network rigidity, while local nonaffine deformations of the network substantially enhance the rigidity at low frequencies, eventually leading to a size-independent response and strong violation of the generalized Stokes formula. We show that a simple model of lateral bending of filaments embedded in a viscoelastic background leads to an intermediate scaling regime for the apparent elastic modulus ${G}^{\ensuremath{'}}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{9/16}$, closely matching the experiments. These results demonstrate that nonaffine bending deformations can be dominant for the mobility of objects of the size of vesicles and organelles in the cell.

A single-chain model for active gels I: active dumbbell model

We have developed a single-chain theory to describe the dynamics of active gels. Active gels are networks of semiflexible polymer filaments driven by motor proteins that convert chemical energy from the hydrolysis of adenosine triphosphate to mechanical work and motion. In active gels molecular motors create active cross-links between the semiflexible filaments. We model the semiflexible filaments as bead-spring chains; the active interactions between filaments are accounted for using a mean-field approach, in which filaments have prescribed probabilities to undergo a transition from one motor attachment state into the other depending on the state of the probe filament. The level of description of the model includes the change in the end-to-end distance of the filaments, the attachment state of the filaments, and the motor-generated forces, as stochastic state variables which evolve according to a proposed differential Chapman–Kolmogorov equation. The motor-generated forces are drawn from a stationary distribution of motor stall forces that can be measured experimentally. The general formulation of the model allows accounting for physics that is not possible, or not practical, to include in available models that have been postulated on coarser levels of description. However, in this introductory manuscript, we make several assumptions to simplify the mathematics and to obtain analytical results, from which insight into the microscopic mechanisms underlying the dynamics of active gels can be gained. We treat the filaments as one-dimensional dumbbells, approximate the elasticity of the semiflexible filaments with a Hookean spring law, and assume that the transition rates are independent of the tension in the filaments. We show that even in this simplified form, the model can predict the buckling of individual filaments that is thought to be the underlying mechanism in the self-contraction of non-sarcomeric actin–myosin bundles [Lenz et al., Phys. Rev. Lett., 2012, 108, 238107]. The active dumbbell model can also explain the violation of the fluctuation–dissipation theorem observed in microrheology experiments on active gels [Mizuno et al., Science, 2007, 315, 370–373].