Effective Persistence Length Research Articles

Geometry of Braided DNA Dictates Supercoiling Partition.

During DNA replication, the replisome must rotate relative to the DNA substrate, generating supercoiling that must be partitioned in front of or behind the replisome. Supercoiling partitioned behind the replisome may intertwine (or braid) daughter DNA molecules and restrict chromosome segregation. Supercoiling partitioning and torsional resistance at the replisome should depend on the geometry of the two daughter DNA molecules, determined by their end separations. However, experimental investigation of DNA braiding under well-defined DNA geometry has proven challenging. Here, we present methods to engineer braiding substrates of defined geometry, from minimal to significant end separations. We then directly measured the torque required to braid these substrates using an angular optical trap (AOT) and found that the torque required to initiate the braiding during the first 0.5 turn critically depends on the end separation. Once braiding started, we found that the subsequent effective twist persistence length of DNA braiding is about 20-30 nm, insensitive to the end separations. Our work highlights the crucial role of braiding geometry in dictating supercoiling partitioning and torque build-up during replication. It suggests that dynamic modulation of end separation on the daughter DNA molecules could serve as a mechanism to regulate replication progression in vivo .

Unravelling the Biophysical Properties of Chromatin Proteins and DNA Using Acoustic Force Spectroscopy.

Acoustic force spectroscopy (AFS) is a single-molecule micromanipulation technique that uses sound waves to exert force on surface-tethered DNA molecules in a microfluidic chamber. As large numbers of individual protein-DNA complexes are tracked in parallel, AFS provides insight into the individual properties of such complexes as well as their population averages. In this chapter, we describe in detail how to perform AFS experiments specifically on bare DNA, protein-DNA complexes, and how to extract their (effective) persistence length and contour length from force-extension relations.

Deformability and collision-induced reorientation enhance cell topotaxis in dense microenvironments

In vivo, cells navigate through complex environments filled with obstacles such as other cells and the extracellular matrix. Recently, the term “topotaxis” has been introduced for navigation along topographic cues such as obstacle density gradients. Experimental and mathematical efforts have analyzed topotaxis of single cells in pillared grids with pillar density gradients. A previous model based on active Brownian particles (ABPs) has shown that ABPs perform topotaxis, i.e., drift toward lower pillar densities, due to decreased effective persistence lengths at high pillar densities. The ABP model predicted topotactic drifts of up to 1% of the instantaneous speed, whereas drifts of up to 5% have been observed experimentally. We hypothesized that the discrepancy between the ABP and the experimental observations could be in 1) cell deformability and 2) more complex cell-pillar interactions. Here, we introduce a more detailed model of topotaxis based on the cellular Potts model (CPM). To model persistent cells we use the Act model, which mimics actin-polymerization-driven motility, and a hybrid CPM-ABP model. Model parameters were fitted to simulate the experimentally found motion of Dictyostelium discoideum on a flat surface. For starved D. discoideum, the topotactic drifts predicted by both CPM variants are closer to the experimental results than the previous ABP model due to a larger decrease in persistence length. Furthermore, the Act model outperformed the hybrid model in terms of topotactic efficiency, as it shows a larger reduction in effective persistence time in dense pillar grids. Also pillar adhesion can slow down cells and decrease topotaxis. For slow and less-persistent vegetative D. discoideum cells, both CPMs predicted a similar small topotactic drift. We conclude that deformable cell volume results in higher topotactic drift compared with ABPs, and that feedback of cell-pillar collisions on cell persistence increases drift only in highly persistent cells.

A deep learning framework for quantitative analysis of actin microridges

Microridges are evolutionarily conserved actin-rich protrusions present on the apical surface of squamous epithelial cells. In zebrafish epidermal cells, microridges form self-evolving patterns due to the underlying actomyosin network dynamics. However, their morphological and dynamic characteristics have remained poorly understood owing to a lack of computational methods. We achieved ~95% pixel-level accuracy with a deep learning microridge segmentation strategy enabling quantitative insights into their bio-physical-mechanical characteristics. From the segmented images, we estimated an effective microridge persistence length of ~6.1 μm. We discovered the presence of mechanical fluctuations and found relatively greater stresses stored within patterns of yolk than flank, indicating distinct regulation of their actomyosin networks. Furthermore, spontaneous formations and positional fluctuations of actin clusters within microridges were associated with pattern rearrangements over short length/time-scales. Our framework allows large-scale spatiotemporal analysis of microridges during epithelial development and probing of their responses to chemical and genetic perturbations to unravel the underlying patterning mechanisms.

Path integral description of semiflexible active Brownian polymers.

Semiflexible polymers comprised of active Brownian particles (ABPOs) exhibit intriguing activity-driven conformational and dynamical features. Analytically, the generic properties of ABPOs can be obtained in a mean-field description applying the Gaussian semiflexible polymer model. In this article, we derive a path integral representation of the stationary-state distribution function of such ABPOs, based on the stationary-state distribution function of the normal mode amplitudes following from the Langevin equation of motion. The path integral includes characteristic semiflexible polymer contributions from entropy and bending energy, with activity dependent coefficients, and, in addition, activity-induced torsional and higher order correlations along the polymer contour. Focusing on a semiflexible polymer approximation, we determine various properties such as the tangent-vector correlation function, effective persistence length, and the mean-square end-to-end distance. The latter reflects the characteristic features of ABPOs, and good quantitative agreement is obtained with the full solution for larger activities, specifically for flexible polymers. Moreover, the approximation indicates the relevance of torsional and higher order contour correlations for the ABPO conformations. In general, the ABPO path integral illustrates how colored noise (active fluctuations) affects semiflexible polymer conformations in comparison to white noise thermal fluctuations.

Pressure and diffusion of active matter with inertia.

It has been discovered that active matter generates novel physical quantities such as the swim pressure. This quantity arises from the exchange of extra momentum between active particles and the boundaries of the system. Given its origin, this quantity can exist at different scales; hence microorganisms and larger organisms like fish or birds generate their own swim pressure. For larger organisms or for high swimming speeds, inertia cannot necessarily be neglected; hence in this paper, we start by calculating analytically the effect of finite translational and rotational particles' inertia on the diffusion of a system of noninteracting spherical active Brownian particles. From this analysis, an enhanced diffusion coefficient due to rotational inertia is obtained, and an alternative effective persistence length and an alternative reorientation time, both sensitive to rotational inertia, are also identified. Afterwards, and to see the implications of finite inertia on bulk properties, the pressure of this system is elucidated by calculating its respective swim and Reynolds pressures. It is found that their sum becomes asymptotically sensitive to the square root of its rotational inertia. To validate our analytical results, Langevin dynamics simulations are also performed showing an excellent agreement between our theoretical predictions and the numerical results.

Measuring the Conformation and Persistence Length of Single-Stranded DNA Using a DNA Origami Structure.

Measuring the mechanical properties of single-stranded DNA (ssDNA) is a challenge that has been addressed by different methods lately. The short persistence length, fragile structure, and the appearance of stem loops complicate the measurement, and this leads to a large variability in the measured values. Here we describe an innovative method based on DNA origami for studying the biophysical properties of ssDNA. By synthesizing a DNA origami structure that consists of two rigid rods with an ssDNA segment between them, we developed a method to characterize the effective persistence length of a random-sequence ssDNA while allowing the formation of stem loops. We imaged the structure with an atomic force microscope (AFM); the rigid rods provide a means for the exact identification of the ssDNA ends. This leads to an accurate determination of the end-to-end distance of each ssDNA segment, and by fitting the measured distribution to the ideal chain polymer model we measured an effective persistence length of 1.98 ± 0.72 nm. This method enables one to measure short or long strands of ssDNA, and it can cope with the formation of stem loops that are often formed along ssDNA. We envision that this method can be used for measuring stem loops for determining the effect of repetitive nucleotide sequences and environmental conditions on the mechanical properties of ssDNA and the effect of interacting proteins with ssDNA. We further noted that the method can be extended to nanoprobes for measuring the interactions of specific DNA sequences, because the DNA origami rods (or similar structures) can hold multiple fluorescent probes that can be easily detected.

Form factor for distorted semi-flexible polymer chains.

The statistical presence of kinks which form defects in semi-flexible polymer chains leads to a polydispersity in the effective persistence length. The form factor of a distorted semi-flexible polymer results as an average over this persistence polydispersity. It turns out that the scattering behavior of an ensemble of distorted semi-flexible polymer chains is quite well approximated by a form factor of an undistorted chain with a Rg-equivalent persistence length. An apparent length polydispersity is observed for short distorted chains. The Rg-equivalent persistence length is significantly smaller than the innate persistence length multiplied by the fraction of regular monomers. The results are compared to related work on DNA from the literature.

Unraveling the Biophysical Properties of Chromatin Proteins and DNA Using Acoustic Force Spectroscopy.

Acoustic Force Spectroscopy (AFS) is a single-molecule micromanipulation technique that uses sound waves to exert force on surface-tethered DNA molecules in a microfluidic chamber. As large numbers of individual protein-DNA complexes are tracked in parallel, AFS provides insight into the individual properties of such complexes as well as their population averages. In this chapter, we describe in detail how to perform AFS experiments specifically on bare DNA, protein-DNA complexes, and how to extract their (effective) persistence length and contour length from force-extension relations.

Features of genomic organization in a nucleotide-resolution molecular model of the Escherichia coli chromosome.

We describe structural models of the Escherichia coli chromosome in which the positions of all 4.6 million nucleotides of each DNA strand are resolved. Models consistent with two basic chromosomal orientations, differing in their positioning of the origin of replication, have been constructed. In both types of model, the chromosome is partitioned into plectoneme-abundant and plectoneme-free regions, with plectoneme lengths and branching patterns matching experimental distributions, and with spatial distributions of highly-transcribed chromosomal regions matching recent experimental measurements of the distribution of RNA polymerases. Physical analysis of the models indicates that the effective persistence length of the DNA and relative contributions of twist and writhe to the chromosome's negative supercoiling are in good correspondence with experimental estimates. The models exhibit characteristics similar to those of ‘fractal globules,’ and even the most genomically-distant parts of the chromosome can be physically connected, through paths combining linear diffusion and inter-segmental transfer, by an average of only ∼10 000 bp. Finally, macrodomain structures and the spatial distributions of co-expressed genes are analyzed: the latter are shown to depend strongly on the overall orientation of the chromosome. We anticipate that the models will prove useful in exploring other static and dynamic features of the bacterial chromosome.

Cyclization of short DNA fragments

From the per unit length free energy for DNA under tension, we have calculated an effective contour length dependent persistence length for short DNA. This effective persistence length results from the enhanced fluctuations in short DNA. It decreases for shorter DNA, making shorter DNA more flexible. The results of the J-factor calculated using this effective persistence length are in good agreement with experimental data.

Phosphorylation Increases Persistence Length and End-to-End Distance of a Segment of Tau Protein

Intrinsically disordered regions of proteins, which lack unique tertiary structure under physiological conditions, are enriched in phosphorylation sites and in significant local bias toward the polyproline II conformation. The overrepresented coincidence of this posttranslational regulatory signal and local conformational bias within unstructured regions raises a question: can phosphorylation serve to manipulate the conformational preferences of a disordered protein? In this study, we use time-resolved fluorescence resonance energy transfer and a, to our knowledge, novel data analysis method to directly measure the end-to-end distance distribution of a phosphorylatable peptide derived from the human microtubule associated protein tau. Our results show that phosphorylation at threonine or serine extends the end-to-end distance and increases the effective persistence length of the tested model peptides. Unexpectedly, the extension is independent of salt concentration, suggestive of a nonelectrostatic origin. The phosphorylation extension and stiffening effect provides a peptide-scale physical interpretation for the posttranslational regulation of the highly abundant protein-protein interactions found in disordered proteins, as well as a potential insight into the regulatory mechanism of the tau protein’s microtubule binding activity.

Liquid crystals of self-assembled DNA bottlebrushes.

Early theories for bottlebrush polymers have suggested that the so-called main-chain stiffening effect caused by the presence of a dense corona of side chains along a central main chain should lead to an increased ratio of effective persistence length (lp,eff) over the effective thickness (Deff) and, hence, ultimately to lyotropic liquid crystalline behavior. More recent theories and simulations suggest that lp,eff ∼ Deff, such that no liquid crystalline behavior is induced by bottlebrushes. In this paper we investigate experimentally how lyotropic liquid crystalline behavior of a semiflexible polymer is affected by a dense coating of side chains. We use semiflexible DNA as the main chain. A genetically engineered diblock protein polymer C4K12 is used to physically adsorb long side chains on the DNA. The C4K12 protein polymer consists of a positively charged binding block (12 lysines, K12) and a hydrophilic random coil block of 400 amino acids (C4). From light scattering we find that, at low ionic strength (10 mM Tris-HCl), the thickness of the self-assembled DNA bottlebrushes is on the order of 30 nm and the effective grafting density is 1 side chain per 2.7 nm of DNA main chain. We find these self-assembled DNA bottlebrushes form birefringent lyotropic liquid crystalline phases at DNA concentrations as low as 8 mg/mL, roughly 1 order of magnitude lower than for bare DNA. Using small-angle X-ray scattering (SAXS) we show that, at DNA concentrations of 12 mg/mL, there is a transition to a hexagonal phase. We also show that, while the effective persistence length increases due to the bottlebrush coating, the effective thickness of the bottlebrush increases even more, such that in our case the bottlebrush coating reduces the effective aspect ratio of the DNA. This is in agreement with theoretical estimates that show that, in most cases of practical interest, a bottlebrush coating will lead to a decrease of the effective aspect ratio, whereas, only for bottlebrushes with extremely long side chains at very high grafting densities, a bottlebrush coating may be expected to lead to an increase of the effective aspect ratio.

Phosphorylation Modulates Conformational Bias of a Disordered Peptide

Disordered proteins, proteins lacking a stable three-dimensional structure, are often enriched in a broadly conserved sequence composition and patterning that imbues them with residual conformational bias toward the polyproline-II (PII) conformation. Since post-translational modification often effects biological function, we ask how phosphorylation impacts PII bias and overall geometry of a short, phosphorylatable peptide. We developed a method to use time-correlated single-photon counting data to measure Forster Resonance Energy Transfer (FRET), then subsequently calculate polymer properties of the peptide. We explore the impact of phosphorylation on effective persistence length and end-to-end distance, and compare the results of our experiments to behaviors modeled by a hard-sphere collision simulation.

Single Molecule Studies of RPA's Sequential Binding to ssDNA Reveals a Highly Stiff and Stable State Induced by the Binding of Zinc

Replication protein A (RPA) is a prototypical modular eukaryotic single-stranded DNA (ssDNA) binding protein that has essential biological function in virtually all DNA transactions in cell. In this work we used magnetic tweezers to study the sequential engagement of RPA to single large ssDNA molecule in real time. Using mechanical manipulation of single ssDNA tethers, we found that binding of RPA causes ssDNA to rapidly extend and elongate reaching a steady state within several minutes. Analysis of RPA's effect on the force-extension curve of ssDNA reveals a increased effective persistence length at RPA binding, which indicates a moderate stiffening effect of RPA on ssDNA backbone. Further analysis of RPA's occupation fraction on ssDNA at varying RPA concentration shows that RPA's ssDNA binding is largely non-cooperative with an interesting non-monotonic salt concentration dependent binding affinity. Upon examination of the effect of biometals, we discovered a unique role of zinc on RPA's ssDNA binding: zinc in micromolar concentrations drives formation of a significantly stiffer and more compact RPA coated ssDNA state. These results together provide new mechanochemical insights to the influences and the mechanisms of action of RPA on large single ssDNA substrate.

Resolving Internal Motional Correlations to Complete the Conformational Entropy Meter.

Conformational entropy (SΩ) has long been used to theoretically characterize the dynamics of proteins, DNA, and other polymers. Though recent advances enabled its calculation also from simulations and nuclear magnetic resonance (NMR) relaxation experiments, correlated molecular motion has hitherto greatly hindered both numerical and experimental determination, requiring demanding empirical and computational calibrations. Herein, we show that these motional correlations can be estimated directly from the temperature-dependent SΩ series that reveal effective persistence lengths of the polymers, which we demonstrate by measuring SΩ of amphiphilic molecules in model lipid systems by spin-labeling electron paramagnetic resonance (EPR) spectroscopy. We validate our correlation-corrected SΩ meter against the basic biophysical interactions underlying biomembrane formation and stability, against the changes in enthalpy and diffusion coefficients upon phase transitions, and against the energetics of fatty acid dissociation. As the method can be directly applied to conformational analysis of proteins and other polymers, as well as adapted to NMR or polarized fluorescence techniques, we believe that the approach can greatly enrich the scope of experimentally available statistical thermodynamics, offering new physical insights into the behavior of biomolecules.

Lateral Motion and Bending of Microtubules Studied with a New Single-Filament Tracking Routine in Living Cells

The cytoskeleton is involved in numerous cellular processes such as migration, division, and contraction and provides the tracks for transport driven by molecular motors. Therefore, it is very important to quantify the mechanical behavior of the cytoskeletal filaments to get a better insight into cell mechanics and organization. It has been demonstrated that relevant mechanical properties of microtubules can be extracted from the analysis of their motion and shape fluctuations. However, tracking individual filaments in living cells is extremely complex due, for example, to the high and heterogeneous background. We introduce a believed new tracking algorithm that allows recovering the coordinates of fluorescent microtubules with ∼9 nm precision in in vitro conditions. To illustrate potential applications of this algorithm, we studied the curvature distributions of fluorescent microtubules in living cells. By performing a Fourier analysis of the microtubule shapes, we found that the curvatures followed a thermal-like distribution as previously reported with an effective persistence length of ∼20 μm, a value significantly smaller than that measured in vitro. We also verified that the microtubule-associated protein XTP or the depolymerization of the actin network do not affect this value; however, the disruption of intermediate filaments decreased the persistence length. Also, we recovered trajectories of microtubule segments in actin or intermediate filament-depleted cells, and observed a significant increase of their motion with respect to untreated cells showing that these filaments contribute to the overall organization of the microtubule network. Moreover, the analysis of trajectories of microtubule segments in untreated cells showed that these filaments presented a slower but more directional motion in the cortex with respect to the perinuclear region, and suggests that the tracking routine would allow mapping the microtubule dynamical organization in cells.

Strongly Bent DNA: Reconciling Theory and Experiment

The majority of double stranded DNA (dsDNA) in living cells is strongly bent. There is little doubt that standard models that assume harmonic bending energy (Hooke's law, worm-like chain) with effective persistence length of around 50 nm accurately describe the weak bending of DNA. However, it is not clear if these commonly used approximations are still accurate for the poorly understood strong bending regime, which is most relevant to biology. The available experimental evidence is controversial, with the most recent cyclization experiments of ∼100 base-pair DNA fragments pointing to strongly bent dsDNA being considerably more flexible than expected. We present a general concept of polymer bending that naturally allows for a transition from the weak, harmonic deformation regime into the “softer” mode at large deformations. In this model, convexity properties of the bending energy potential of individual monomer sub-units of the polymer give rise to the transition between two bending modes. Namely, if the effective monomer deformation energy as a function of bending angle has a non-convex (concave up) region, then the deformation energy of the polymer is a linear function of the bending angle. The dsDNA effective bending potential determined from the angular distribution of base-pairs from PDB structures of DNA-protein complexes is in agreement with our model: it is harmonic in the weak bending regime and non-convex for strong bends. We propose a simple expression that replaces the harmonic approximation in the strong bending regime. The theory shows quantitative agreement with recent cyclization experiments of short dsDNA fragments. Atomistic molecular dynamics simulations point to short-range Lennard-Jones interactions as responsible for the onset of the linear bending regime of dsDNA.

Mesoscale simulation of semiflexible chains. I. Endpoint distribution and chain dynamics

The endpoint distribution and dynamics of semiflexible fibers are studied by numerical simulation. A brief overview is given over the analytical theory of flexible and semiflexible polymers. In particular, a closed expression is given for the relaxation spectrum of wormlike chains, which determines polymer diffusion and rheology. Next a simulation model for wormlike chains with full hydrodynamic interaction is described, and relations for the bending and torsion modulus are given. Two methods are introduced to include torsion stiffness into the model. The model is validated by simulating single chains in a heat bath, and comparing the endpoint distribution of the chains with established Monte Carlo results. It is concluded that torsion stiffness leads to a slightly shorter effective persistence length for a given bending stiffness. To further validate the simulation model, polymer diffusion is studied for fixed persistence length and varying polymer length N. The diffusion constant shows crossover from Rouse (D [proportionality] N(-1)) to reptation behaviour (D [proportionality] N(-2)). The terminal relaxation time obtained from the monomer displacement is consistent with the theory of wormlike chains. The probability for chain crossing has also been studied. This probability is so low that it does not influence the present results.

Semi-flexible polymer chains in quasi-one-dimensional confinement: a Monte Carlo study on the square lattice

Single semi-flexible polymer chains are modeled as self-avoiding walks (SAWs) on the square lattice with every 90° kink requiring an energy eb. While for eb = 0 this is the ordinary SAW, varying the parameter qb = exp(−eb/kBT) allows the variation of the effective persistence length p over about two decades. Using the pruned-enriched Rosenbluth method (PERM), chain lengths up to about N = 105 steps can be studied. In previous work it has already been shown that for contour lengths L = Nb (the bond length b is the lattice spacing) of order p a smooth crossover from rods to two-dimensional self-avoiding walks occurs, with radii R ∝ p1/4L3/4, the Gaussian regime predicted by the Kratky–Porod model for worm-like chains being completely absent. In the present study, confinement of such chains in strips of width D is considered, varying D from 4 to 320 lattice spacings. It is shown that for narrow strips (D < p) the effective persistence length of the chains (in the direction parallel to the confining boundaries) scales like p2/D, and R‖ ∝ L (with a pre-factor of order unity). For very wide strips, D ≫ p, the two-dimensional SAW behavior prevails for chain lengths up to Lcross ∝ p(D/p)4/3, while for L ≫ Lcross the chain is a string of blobs of diameter D, i.e. R‖ ∝ L(p/D)1/3. In the regime D < p, the chain is a sequence of straight sequences with length of the order p2/D parallel to the boundary, separated by sequences with length < D perpendicular to the boundary; thus Odijk's deflection length plays no role for discrete bond angles.