Characterization of single semiflexible filaments under geometric constraints

Probing physical properties of single amyloid fibrils using nanofluidic channels

Amyloid fibril formation is central to the pathology of many diseases, including neurodegenerative disorders such as Alzheimer's and Parkinson's disease. Amyloid fibrils can also have functional and scaffolding roles, for example in bacterial biofilms, and have also been exploited as useful biomaterials. Despite being linear protein homopolymers, amyloid fibrils can exhibit significant structural and morphological polymorphism, making it relevant to study them on the single fibril level. We here introduce the concept of nanofluidic channel analysis to the study of single, fluorescently-labeled amyloid fibrils in solution, monitoring the extension and emission intensity of individual fibrils confined in nanochannels with a depth of 300 nm and a width that gradually increases from 300 to 3000 nm. The change in fibril extension with channel width permitted accurate determination of the persistence length of individual fibrils using Odijk's theory for strongly confined polymers. The technique was applied to amyloid fibrils prepared from the Alzheimer's related peptide amyloid-β(1-42) and the Parkinson's related protein α-synuclein, obtaining mean persistence lengths of 5.9 ± 4.5 μm and 3.0 ± 1.6 μm, respectively. The broad distributions of fibril persistence lengths indicate that amyloid fibril polymorphism can manifest in their physical properties. Interestingly, the α-synuclein fibrils had lower persistence lengths than the amyloid-β(1-42) fibrils, despite being thicker. Furthermore, there was no obvious within-sample correlation between the fluorescence emission intensity per unit length of the labelled fibrils and their persistence lengths, suggesting that stiffness may not be proportional to thickness. We foresee that the nanofluidics methodology established here will be a useful tool to study amyloid fibrils on the single fibril level to gain information on heterogeneity in their physical properties and interactions.

Conformation of Flexible and Semiflexible Chains Confined in Nanoposts Array of Various Geometries

The conformation and distribution of a flexible and semiflexible chain confined in an array of nanoposts arranged in parallel way in a square-lattice projection of their cross-section was investigated using coarse-grained molecular dynamics simulations. The geometry of the nanopost array was varied at the constant post diameter dp and the ensuing modifications of the chain conformation were compared with the structural behavior of the chain in the series of nanopost arrays with the constant post separation Sp as well as with the constant distance between two adjacent post walls (passage width) wp. The free energy arguments based on an approximation of the array of nanopost to a composite of quasi-channels of diameter dc and quasi-slits of height wp provide semiqualitative explanations for the observed structural behavior of both chains. At constant post separation and passage width, the occupation number displays a monotonic decrease with the increasing geometry ratio dc/wp or volume fraction of posts, while a maximum is observed at constant post diameter. The latter finding is attributed to a relaxed conformation of the chains at small dc/wp ratio, which results from a combination of wide interstitial volumes and wide passage apertures. This maximum is approximately positioned at the same dc/wp value for both flexible and semiflexible chains. The chain expansion from a single interstitial volume into more interstitial volumes also starts at the same value of dc/wp ratio for both chains. The dependence of the axial chain extension on the dc/wp ratio turns out to be controlled by the diameter of the interstitial space and by the number of monomers in the individual interstitial volumes. If these two factors act in the same way on the axial extension of chain fragments in interstitial volumes the monotonic increase of the axial chain extension with the dc/wp in the nanopost arrays is observed. At constant wp, however, these two factors act in opposite way and the axial chain extension plotted against the dc/wp ratio exhibits a maximum. In the case of constant post diameter, the characteristic hump in the single chain structure factor whose position correlates with the post separation is found only in the structure factor of the flexible chain confined in the nanopost array of certain value of Sp. The structure factor of the flexible chain contains more information on the monomer organization and mutual correlations than the structure factor of the semiflexible chain. The stiffer chain confined in the nanopost array is composed of low number of statistical segments important for the presence of respective hierarchical regimes in the structure factor.

Stochastic curvature of enclosed semiflexible polymers

The conformational states of a semiflexible polymer enclosed in a compact domain of typical size a are studied as stochastic realizations of paths defined by the Frenet equations under the assumption that stochastic "curvature" satisfies a white noise fluctuation theorem. This approach allows us to derive the Hermans-Ullman equation, where we exploit a multipolar decomposition that allows us to show that the positional probability density function is well described by a telegrapher's equation whenever 2a/ℓ_{p}>1, where ℓ_{p} is the persistence length. We also develop a Monte Carlo algorithm for use in computer simulations in order to study the conformational states in a compact domain. In addition, the case of a semiflexible polymer enclosed in a square domain of side a is presented as an explicit example of the formulated theory and algorithm. In this case, we show the existence of a polymer shape transition similar to the one found by Spakowitz and Wang [Phys. Rev. Lett. 91, 166102 (2003)PRLTAO0031-900710.1103/PhysRevLett.91.166102] where in this case the critical persistence length is ℓ_{p}^{*}≃a/8 such that the mean-square end-to-end distance exhibits an oscillating behavior for values ℓ_{p}>ℓ_{p}^{*}, whereas for ℓ_{p}<ℓ_{p}^{*} it behaves monotonically increasing.

Helical Superstructure of Intermediate Filaments

Intermediate filaments are the least explored among the large cytoskeletal elements. We show here that they display conformational anomalies in narrow microfluidic channels. Their unusual behavior can be understood as the consequence of a previously undetected, large-scale helically curved superstructure. Confinement in a channel orders the otherwise soft, strongly fluctuating helical filaments and enhances their structural correlations, giving rise to experimentally detectable, strongly oscillating tangent correlation functions. We propose an explanation for the detected intrinsic curving phenomenon-an elastic shape instability that we call autocoiling. The mechanism involves self-induced filament buckling via a surface stress located at the outside of the cross section. The results agree with ultrastructural findings and rationalize for the commonly observed looped intermediate filament shapes. Beyond curvature, explaining the molecular origin of the detected helical torsion remains an interesting challenge.

Density fields for branching, stiff networks in rigid confining regions

We develop a formalism to describe the equilibrium distributions for segments of confined branched networks consisting of stiff filaments. This is applicable to certain situations of cytoskeleton in cells, such as for example actin filaments with branching due to the Arp2/3 complex. We develop a grand ensemble formalism that enables the computation of segment density and polarisation profiles within the confines of the cell. This is expressed in terms of the solution to nonlinear integral equations for auxiliary functions. We find three specific classes of behaviour depending on filament length, degree of branching and the ratio of persistence length to the dimensions of the geometry. Our method allows a numerical approach for semi-flexible filaments that are networked.

Structural Behavior of a Semiflexible Polymer Chain in an Array of Nanoposts

The structural properties of a flexible and semiflexible circular chain confined in an array of parallel nanoposts with a square lattice cross-sectional projection were studied using coarse-grained molecular dynamics simulations. To address the effect of the circular topology, a comparison with linear analogs was also carried out. In the interpretation of the chain structural properties, the geometry of the post array is considered as a combination of a channel approximating the interstitial volume with the diameter dc and a slit approximating the passage aperture with the width wp. The number of interstitial volumes occupied by a chain monotonically increases with the decreasing ratio dc/wp regardless of the way the geometry of the post array is varied. However, depending on how the array geometry is modified, the chain span along the posts displays a monotonic (constant post separation) or a non-monotonic behavior (constant passage width) when plotted as a function of the post diameter. In the case of monotonic trend, the width of interstitial spaces increases with the increasing chain occupation number, while, in the case of non-monotonic trend, the width of interstitial spaces decreases with the increasing chain occupation number. In comparison with linear topology, for circular topology, the stiffness affects more significantly the relative chain extension along the posts and less significantly the occupation number. The geometrical parameters of the post arrays are stored in the single-chain structure factors. The characteristic humps are recognized in the structure factor which ensue from the local increase in the density of segments in the circular chains presented in an interstitial volume or from the correlation of parallel chain fragments separated by a row of posts. Although the orientation correlations provide qualitative information about the chain topology and the character of confinement within a single interstitial volume, information about the array periodicity is missing.

Comparison of a stripe and slab confinement for ring and linear macromolecules in nanochannel

The combined effects of the channel asymmetry and the closed chain topology on the chain extension, structure factor, and the orientation correlations were studied using coarse-grained molecular dynamics simulations for moderate chain lengths. These effects are related to applications in linearization experiments with a DNA molecule in nanofluidic devices. According to the aspect ratio, the channels are classified as a stripe or slabs. The chain segments do not have any freedom to move in the direction of the narrowest stripe size, being approximately the same size as the segment size. The chains of both ring and linear topologies are extended more in a stripe than in a slab; this effect is strengthened for a ring. For a ring in a stripe, the extension-confinement strength dependence leads to effective Flory exponents even larger than 3/4, which is characteristic for a self-avoiding two-dimensional chain. While the chain extension-confinement strength dependence for both topologies conforms to the de Gennes regime in a stripe, a linear chain undergoes gradual transition to the pseudoideal regime as the slab height increases in the slab-like confinement. For a confined circle, the onset of the pseudoideal regime is shifted to larger slab heights. The structure factor confirms the absence of the pseudoideal and extended de Gennes regime in a stripe and the transition from the extended to the pseudoideal regime of a circular and linear chain upon increasing the slab heights.

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.

Shape control of lipid bilayer membranes by confined actin bundles

In living cells, lipid membranes and biopolymers determine each other's conformation in a delicate force balance. Cellular polymers such as actin filaments are strongly confined by the plasma membrane in cell protrusions such as lamellipodia and filopodia. Conversely, protrusion formation is facilitated by actin-driven membrane deformation and these protrusions are maintained by dense actin networks or bundles of actin filaments. Here we investigate the mechanical interplay between actin bundles and lipid bilayer membranes by reconstituting a minimal model system based on cell-sized liposomes with encapsulated actin filaments bundled by fascin. To address the competition between the deformability of the membrane and the enclosed actin bundles, we tune the bundle stiffness (through the fascin-to-actin molar ratio) and the membrane rigidity (through protein decoration). Using confocal microscopy and quantitative image analysis, we show that actin bundles deform the liposomes into a rich set of morphologies. For liposomes having a small membrane bending rigidity, the actin bundles tend to generate finger-like membrane protrusions that resemble cellular filopodia. Stiffer bundles formed at high crosslink density stay straight in the liposome body, whereas softer bundles formed at low crosslink density are bent and kinked. When the membrane has a large bending rigidity, membrane protrusions are suppressed. In this case, membrane enclosure forces the actin bundles to organize into cortical rings, to minimize the energy cost associated with filament bending. Our results highlight the importance of taking into account mechanical interactions between the actin cytoskeleton and the membrane to understand cell shape control.

Force-Extension for DNA in a Nanoslit: Mapping between the 3D and 2D Limits

The force-extension relation for a semiflexible polymer confined in a nanoslit is investigated. Both the effective correlation length and force-extension relation change as the chain goes from 3D (large slit heights) to 2D (tight confinement). At low forces, correlations along the polymer give an effective dimensionality. The strong force limit can be interpolated with the weak force limit for two regimes: when confinement dominates over extensile force and vice versa. These interpolations give good agreement with simulations for all slit heights and forces. We thus generalize the Marko-Siggia force-extension relation for DNA and other semiflexible biopolymers in nanoconfinement.

Stripe to slab confinement for the linearization of macromolecules in nanochannels

We investigated the recently suggested advantageous analysis of chain linearization experiments with macromolecules confined in a stripe-like channel (Huang and Battacharya, EPL, 2014, 106, 18004) using Monte Carlo simulations. The enhanced chain extension in a stripe, which is due to the significant excluded volume interactions between the monomers in two dimensions, weakens considerably on transition to an experimentally feasible slit-like channel. Based on the chain extension-confinement strength dependence and the structure factor behavior for a chain in a stripe, we infer the excluded volume regime (de Gennes regime) typical for two-dimensional systems. On widening of the stripe in a direction perpendicular to the stripe plane, i.e. on the transition to the slab geometry, the advantageous chain extension decreases and a Gaussian regime is observed for not very long semiflexible chains. The evidence for pseudo-ideality in confined chains is based on four indicators: the extension curves, variation of the extension with the persistence length P, estimated limits for the regimes in the investigated systems, and the structure factor behavior. The slab behavior can be observed when the two-dimensional stripe (originally of a one-monomer thickness) reaches a reduced thickness D larger than approximately D/P ≈ 0.2 in the third dimension. This maximum height of a slab at which the advantage of a stripe is retained is very low and has implications for DNA linearization experiments.

Orientational correlations in confined DNA

We study how the orientational correlations of DNA confined to nanochannels depend on the channel diameter D by means of Monte Carlo simulations and a mean-field theory. This theory describes DNA conformations in the experimentally relevant regime where the Flory-de Gennes theory does not apply. We show how local correlations determine the dependence of the end-to-end distance of the DNA molecule upon D. Tapered nanochannels provide the necessary resolution in D to study experimentally how the extension of confined DNA molecules depends upon D. Our experimental and theoretical results are in qualitative agreement.

Mobility Gradient Induces Cross-Streamline Migration of Semiflexible Polymers

Many aspects of modern material science and biology rely on the strategic manipulation and understanding of polymer dynamics in confining micro- and nanoflow. We directly observe and analyze nonequilibrium structural and dynamic properties of individual semiflexible actin filaments in pressure-driven microfluidic channel flow using fluorescence microscopy. Different conformational shapes, such as filaments fluctuating in an elongated manner, parabolically bent, as well as tumbling, are identified. With increasing flow velocity, a strong center-of-mass migration toward the channel walls is observed. This significant migration effect can be explained by a shear rate dependent spatial diffusivity due to a gradient in chain mobility of the semiflexible polymers.

Intermediate filaments in small configuration spaces

Intermediate filaments play a key role in cell mechanics. Apart from their great importance from a biomedical point of view, they also act as a very suitable micrometer-sized model system for semiflexible polymers. We perform a statistical analysis of the thermal fluctuations of individual filaments confined in microchannels. The small channel width and the resulting deflections at the walls give rise to a reduction of the configuration space by about 2 orders of magnitude. This circumstance enables us to precisely measure the intrinsic persistence length of vimentin intermediate filaments and to show that they behave as ideal wormlike chains; we observe that small fluctuations in perpendicular planes decouple. Furthermore, the inclusion of results for confined actin filaments demonstrates that the Odijk confinement regime is valid over at least 1 order of magnitude in persistence length.

Semiflexible polymer conformation, distribution and migration in microcapillary flows

The flow behavior of a semiflexible polymer in microchannels is studied using multiparticle collision dynamics, a particle-based hydrodynamic simulation technique. Conformations, distributions, and radial cross-streamline migration are investigated for various bending rigidities, with persistence lengths L(p) in the range 0.5 ≤ L(p)/L(r) ≤ 30. The flow behavior is governed by the competition between a hydrodynamic lift force and steric repulsion from the wall, which lead to migration away from the wall, and a locally varying flow induced orientation, which drives the polymer away from the channel center and towards the wall. The different dependences of these effects on the polymer bending rigidity and the flow velocity results in a complex dynamical behavior. However, a generic effect is the appearance of a maximum in the monomer and the center-of-mass distributions, which occurs at the channel center for small flow velocities, but moves off-center at higher velocities.

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.

Fluctuations of a long, semiflexible polymer in a narrow channel

We consider an inextensible, semiflexible polymer or wormlike chain, with persistence length P and contour length L, fluctuating in a cylindrical channel of diameter D. In the regime D<<P<<L , corresponding to a long, tightly confined polymer, the average length of the channel <R(∥)> occupied by the polymer and the mean-square deviation from the average vary as <R(∥)>=[1-α(∘)(D/P)(2/3)]L and <ΔR(∥)(2)>=β(∘)(D(2)P)L , respectively, where α(∘) and β(∘) are dimensionless amplitudes. In earlier work we determined α(∘) and the analogous amplitude α(square) for a channel with a rectangular cross section from simulations of very long chains. In this paper, we estimate β(∘) and β(square) from the simulations. The estimates are compared with exact analytical results for a semiflexible polymer confined in the transverse direction by a parabolic potential instead of a channel and with a recent experiment. For the parabolic confining potential we also obtain a simple analytic result for the distribution of R(∥) or radial distribution function, which is asymptotically exact for large L and has the skewed shape seen experimentally.

Buckling of stiff polymer rings in weak spherical confinement

Confinement is a versatile and well-established tool to study the properties of polymers either to understand biological processes or to develop new nanobiomaterials. We investigate the conformations of a semiflexible polymer ring in weak spherical confinement imposed by an impenetrable shell. We develop an analytic argument for the dominating polymer trajectory depending on polymer flexibility considering elastic and entropic contributions. Monte Carlo simulations are performed to assess polymer ring conformations in probability densities and by the shape measures asphericity and nature of asphericity. Comparison of the analytic argument with the mean asphericity and the mean nature of asphericity confirm our reasoning to explain polymer ring conformations in the stiff regime, where elastic response prevails.

In situ formation, manipulation, and imaging of droplet-encapsulated fibrin networks

The protein fibrin plays a principal role in blood clotting and forms robust three dimensional networks. Here, microfluidic devices have been tailored to strategically generate and study these bionetworks by confinement in nanoliter volumes. The required protein components are initially encapsulated in separate droplets, which are subsequently merged by electrocoalescence. Next, distinct droplet microenvironments are created as the merged droplets experience one of two conditions: either they traverse a microfluidic pathway continuously, or they "park" to fully evolve an isotropic network before experiencing controlled deformations. High resolution fluorescence microscopy is used to image the fibrin networks in the microchannels. Aggregation (i.e."clotting") is significantly affected by the complicated flow fields in moving droplets. In stopped-flow conditions, an isotropic droplet-spanning network forms after a suitable ripening time. Subsequent network deformation, induced by the geometric structure of the microfluidic channel, is found to be elastic at low rates of deformation. A shape transition is identified for droplets experiencing rates of deformation higher than an identified threshold value. In this condition, significant densification of protein within the droplet due to hydrodynamic forces is observed. These results demonstrate that flow fields considerably affect fibrin in different circumstances exquisitely controlled using microfluidic tools.