Wormlike Chain Research Articles

POTATO: Automated pipeline for batch analysis of optical tweezers data

Optical tweezers are a single-molecule technique that allows probing of intra- and intermolecular interactions that govern complex biological processes involving molecular motors, protein-nucleic acid interactions, and protein/RNA folding. Recent developments in instrumentation eased and accelerated optical tweezers data acquisition, but analysis of the data remains challenging. Here, to enable high-throughput data analysis, we developed an automated python-based analysis pipeline called POTATO (practical optical tweezers analysis tool). POTATO automatically processes the high-frequency raw data generated by force-ramp experiments and identifies (un)folding events using predefined parameters. After segmentation of the force-distance trajectories at the identified (un)folding events, sections of the curve can be fitted independently to a worm-like chain and freely jointed chain models, and the work applied on the molecule can be calculated by numerical integration. Furthermore, the tool allows plotting of constant force data and fitting of the Gaussian distance distribution over time. All these features are wrapped in a user-friendly graphical interface, which allows researchers without programming knowledge to perform sophisticated data analysis.

Hydrodynamic and Polyelectrolyte Properties of Actin Filaments: Theory and Experiments.

Actin filament’s polyelectrolyte and hydrodynamic properties, their interactions with the biological environment, and external force fields play an essential role in their biological activities in eukaryotic cellular processes. In this article, we introduce a unique approach that combines dynamics and electrophoresis light-scattering experiments, an extended semiflexible worm-like chain model, and an asymmetric polymer length distribution theory to characterize the polyelectrolyte and hydrodynamic properties of actin filaments in aqueous electrolyte solutions. A fitting approach was used to optimize the theories and filament models for hydrodynamic conditions. We used the same sample and experimental conditions and considered several g-actin and polymerization buffers to elucidate the impact of their chemical composition, reducing agents, pH values, and ionic strengths on the filament translational diffusion coefficient, electrophoretic mobility, structure factor, asymmetric length distribution, effective filament diameter, electric charge, zeta potential, and semiflexibility. Compared to those values obtained from molecular structure models, our results revealed a lower value of the effective G-actin charge and a more significant value of the effective filament diameter due to the formation of the double layer of the electrolyte surrounding the filaments. Contrary to the data usually reported from electron micrographs, the lower values of our results for the persistence length and average contour filament length agree with the significant difference in the association rates at the filament ends that shift to sub-micro lengths, which is the maximum of the length distribution.

Mechanical properties of nucleic acids and the non-local twistable wormlike chain model.

Mechanical properties of nucleic acids play an important role in many biological processes that often involve physical deformations of these molecules. At sufficiently long length scales (say, above ∼20-30 base pairs), the mechanics of DNA and RNA double helices is described by a homogeneous Twistable Wormlike Chain (TWLC), a semiflexible polymer model characterized by twist and bending stiffnesses. At shorter scales, this model breaks down for two reasons: the elastic properties become sequence-dependent and the mechanical deformations at distal sites get coupled. We discuss in this paper the origin of the latter effect using the framework of a non-local Twistable Wormlike Chain (nlTWLC). We show, by comparing all-atom simulations data for DNA and RNA double helices, that the non-local couplings are of very similar nature in these two molecules: couplings between distal sites are strong for tilt and twist degrees of freedom and weak for roll. We introduce and analyze a simple double-stranded polymer model that clarifies the origin of this universal distal couplings behavior. In this model, referred to as the ladder model, a nlTWLC description emerges from the coarsening of local (atomic) degrees of freedom into angular variables that describe the twist and bending of the molecule. Different from its local counterpart, the nlTWLC is characterized by a length-scale-dependent elasticity. Our analysis predicts that nucleic acids are mechanically softer at the scale of a few base pairs and are asymptotically stiffer at longer length scales, a behavior that matches experimental data.

Self-propulsion with speed and orientation fluctuation: Exact computation of moments and dynamical bistabilities in displacement.

We consider the influence of active speed fluctuations on the dynamics of a d-dimensional active Brownian particle performing a persistent stochastic motion. The speed fluctuation brings about a dynamical anisotropy even in the absence of shape anisotropy. We use the Laplace transform of the Fokker-Planck equationto obtain exact expressions for time-dependent dynamical moments. Our results agree with direct numerical simulations and show several dynamical crossovers determined by the activity, persistence, and speed fluctuation. The dynamical anisotropy leads to a subdiffusive scaling in the parallel component of displacement fluctuation at intermediate times. The kurtosis remains positive at short times determined by the speed fluctuation, crossing over to a negative minimum at intermediate times governed by the persistence before vanishing asymptotically. The probability distribution of particle displacement obtained from numerical simulations in two dimensions shows two crossovers between compact and extended trajectories via two bimodal distributions at intervening times. While the speed fluctuation dominates the first crossover, the second crossover is controlled by persistence like in the wormlike chain model of semiflexible polymers.

A fit-less approach to the elasticity of the handles in optical tweezers experiments.

The elastic properties of the double-stranded DNA handles used in optical tweezers experiments on biomolecules are customarily modeled by an extensible worm-like chain model. Fitting such a model to experimental data, however, is no trivial task, as the function depends on four parameters in a highly non-linear fashion. We hereby propose a method to bypass the fitting procedure and obtain an empirical force vs. extension curve that accurately reproduces the elasticity of the handles.

Entropic surface segregation from athermal polymer blends: Polymer flexibility vs bulkiness.

We examine athermal binary blends composed of conformationally asymmetric polymers of equal molecular volume next to a surface of width ξ. The self-consistent field theory (SCFT) of Gaussian chains predicts that the more compact polymer with the shorter average end-to-end length, R0, is entropically favored at the surface. Here, we extend the SCFT to worm-like chains with small persistence lengths, ℓp, relative to their contour lengths, ℓc, for which R0≈2ℓpℓc. In the limit of ℓp ≪ ξ, we recover the Gaussian-chain prediction where the segregation depends only on the product ℓpℓc, but for realistic polymer/air surfaces with ξ ∼ ℓp, the segregation depends separately on the two quantities. Although the surface continues to favor flexible polymers with smaller ℓp and bulky polymers with shorter ℓc, the effect of bulkiness is more pronounced. This imbalance can, under specific conditions, lead to anomalous surface segregation of the more extended polymer. For this to happen, the polymer must be bulkier and stiffer, with a stiffness that is sufficient to produce a larger R0 yet not so rigid as to reverse the surface affinity that favors bulky polymers.

Nonlinear mechanics of human mitotic chromosomes

In preparation for mitotic cell division, the nuclear DNA of human cells is compacted into individualized, X-shaped chromosomes1. This metamorphosis is driven mainly by the combined action of condensins and topoisomerase IIα (TOP2A)2,3, and has been observed using microscopy for over a century. Nevertheless, very little is known about the structural organization of a mitotic chromosome. Here we introduce a workflow to interrogate the organization of human chromosomes based on optical trapping and manipulation. This allows high-resolution force measurements and fluorescence visualization of native metaphase chromosomes to be conducted under tightly controlled experimental conditions. We have used this method to extensively characterize chromosome mechanics and structure. Notably, we find that under increasing mechanical load, chromosomes exhibit nonlinear stiffening behaviour, distinct from that predicted by classical polymer models4. To explain this anomalous stiffening, we introduce a hierarchical worm-like chain model that describes the chromosome as a heterogeneous assembly of nonlinear worm-like chains. Moreover, through inducible degradation of TOP2A5 specifically in mitosis, we provide evidence that TOP2A has a role in the preservation of chromosome compaction. The methods described here open the door to a wide array of investigations into the structure and dynamics of both normal and disease-associated chromosomes.

The Regulatory Roles of Intrinsically Disordered Linker in VRN1-DNA Phase Separation.

Biomacromolecules often form condensates to function in cells. VRN1 is a transcriptional repressor that plays a key role in plant vernalization. Containing two DNA-binding domains connected by an intrinsically disordered linker (IDL), VRN1 was shown to undergo liquid-like phase separation with DNA, and the length and charge pattern of IDL play major regulatory roles. However, the underlying mechanism remains elusive. Using a polymer chain model and lattice-based Monte-Carlo simulations, we comprehensively investigated how the IDL regulates VRN1 and DNA phase separation. Using a worm-like chain model, we showed that the IDL controls the binding affinity of VRN1 to DNA, by modulating the effective local concentration of the VRN1 DNA-binding domains. The predicted binding affinities, under different IDL lengths, were in good agreement with previously reported experimental results. Our simulation of the phase diagrams of the VRN1 variants with neutral IDLs and DNA revealed that the ability of phase separation first increased and then decreased, along with the increase in the linker length. The strongest phase separation ability was achieved when the linker length was between 40 and 80 residues long. Adding charged patches to the IDL resulted in robust phase separation that changed little with IDL length variations. Our study provides mechanism insights on how IDL regulates VRN1 and DNA phase separation, and why naturally occurring VRN1-like proteins evolve to contain the charge segregated IDL sequences, which may also shed light on the molecular mechanisms of other IDL-regulated phase separation processes in living cells.

A Review of Polymer Physics and Potential Applications to Radiation Biophysics and Neurodegenerative Disorders

Polymers are an essential aspect of molecular biology and biochemistry. The most significant of macromolecules involved in biological processes and phenomena are polymers. In this article, we provide a comprehensive summary of the proposed physical models for the configuration of polymers. Such physical models include the freely-jointed chain, freely-rotating chain, worm-like chain, and the Gaussian chain model. We then connect these models to the existing models regarding the radiation biophysics of DNA damage, as well as to the damage of RNA molecules, and provide an insight into future areas of research in the subject areas. The conclusion is that polymer physics and the Linear-Quadratic model can be used for future biophysical research in cancer and neurological disorders. Through such connections, we hope to provide a potential future insight with regards to biophysical research in cancer and neurodegenerative disorders.

Fiber Bending Flexibility Evaluation by Worm-like Chain Model

Fiber Bending Flexibility Evaluation by Worm-like Chain Model

Self-consistent field theory of chiral nematic worm-like chains.

Many macromolecules of biological and technological interest are both chiral and semi-flexible. DNA and collagen are good examples. Such molecules often form chiral nematic (or cholesteric) phases, as is well-documented in collagen and chitin. This work presents a method for studying cholesteric phases in the highly successful self-consistent field theory of worm-like chains, offering a new way of studying many biologically relevant molecules. The method involves an effective Hamiltonian with a chiral term inspired by the Oseen-Frank (OF) model of liquid crystals. This method is then used to examine the formation of cholesteric phases in chiral-nematic worm-like chains as a function of polymer flexibility, as well as the optimal cholesteric pitch and distribution of polymer segment orientations. Our approach not only allows for the determination of the isotropic-cholesteric transition and segment distributions, beyond what the OF model promises, but also explicitly incorporates polymer flexibility into the study of the cholesteric phase, offering a more complete understanding of the behavior of semiflexible chiral-nematic polymers.

Concentration-Driven Self-Assembly of PS-b-PLA Bottlebrush Diblock Copolymers in Solution.

Bottlebrush polymers are a class of semiflexible, hierarchical macromolecules with unique potential for shape-, architecture-, and composition-based structure–property design. It is now well-established that in dilute to semidilute solution, bottlebrush homopolymers adopt a wormlike conformation, which decreases in extension (persistence length) as the concentration and molecular overlap increase. By comparison, the solution phase self-assembly of bottlebrush diblock copolymers (BBCP) in a good solvent remains poorly understood, despite critical relevance for solution processing of ordered phases and photonic crystals. In this work, we combine small-angle X-ray scattering, coarse-grained simulation, and polymer synthesis to map the equilibrium phase behavior and conformation of a set of large, nearly symmetric PS-b-PLA bottlebrush diblock copolymers in toluene. Three BBCP are synthesized, with side chains of number-averaged molecular weights of 4500 (PS) and 4200 g/mol (PLA) and total backbone degrees of polymerization of 100, 255, and 400 repeat units. The grafting density is one side chain per backbone repeat unit. With increasing concentration in solution, all three polymers progress through a similar structural transition: from dispersed, wormlike chains with concentration-dependent (decreasing) extension, through the onset of disordered PS/PLA compositional fluctuations, to the formation of a long-range ordered lamellar phase. With increasing concentration in the microphase-separated regimes, the domain spacing increases as individual chains partially re-extend due to block immiscibility. Increases in the backbone degree of polymerization lead to changes in the scattering profiles which are consistent with the increased segregation strength. Coarse-grained simulations using an implicit side-chain model are performed, and concentration-dependent self-assembly behavior is qualitatively matched to experiments. Finally, using the polymer with the largest backbone length, we demonstrate that lamellar phases develop a well-defined photonic band gap in solution, which can be tuned across the visible spectrum by varying polymer concentration.

Energy/entropy partition of force at DNA stretching.

We compute by molecular simulation the energy/entropic partition of the force in a stretched double-stranded (ds)DNA molecule that is not yet available from the single-molecule measurements. Simulation using the coarse-grained wormlike chain (WLC) model predicts a gradual decrease in the internal (bending) energy of DNA at stretching. The ensuing negative energy contribution to force fU is outweighed by the positive entropy contribution fS . The ratio fU /f, used to assess the polymer elasticity, is about -1 at the moderate extension of DNA. At the high extension, the extra energy expenses due to the contour length elongation make the ratio fU /f less negative. The simulation findings of the hybrid energy/entropy nature of DNA elasticity at weak and moderate forces are supported by computations using the thermoelastic method mimicking the polymer experiments in bulk. It is contended that the observation of the negative energy elasticity in DNA can be generalized to other semiflexible polymers described by the WLC model.

Quantitative Elasticity of Flexible Polymer Chains Using Interferometer-Based AFM.

We estimate the elasticity of single polymer chains using atomic force microscope (AFM)-based oscillatory experiments. An accurate estimate of elasticity using AFM is limited by assumptions in describing the dynamics of an oscillating cantilever. Here, we use a home-built fiber-interferometry-based detection system that allows a simple and universal point-mass description of cantilever oscillations. By oscillating the cantilever base and detecting changes in cantilever oscillations with an interferometer, we extracted stiffness versus extension profiles for polymers. For polyethylene glycol (PEG) in a good solvent, stiffness–extension data showed significant deviation from conventional force–extension curves (FECs) measured in constant velocity pulling experiments. Furthermore, modeling stiffness data with an entropic worm-like chain (WLC) model yielded a persistence length of (0.5 ± 0.2 nm) compared to anomaly low value (0.12 nm ± 0.01) in conventional pulling experiments. This value also matched well with equilibrium measurements performed using magnetic tweezers. In contrast, polystyrene (PS) in a poor solvent, like water, showed no deviation between the two experiments. However, the stiffness profile for PS in good solvent (8M Urea) showed significant deviation from conventional force–extension curves. We obtained a persistence length of (0.8 ± 0.2 nm) compared to (0.22 nm ± 0.01) in pulling experiments. Our unambiguous measurements using interferometer yield physically acceptable values of persistence length. It validates the WLC model in good solvents but suggests caution for its use in poor solvents.

Theoretical model for DNA looping by positively charged condensing agents

Many positively charged molecules are known to loop and condense DNA. These positively charged condensing agents bind to the negatively charged DNA and are thought to use electrostatic interactions to bend the DNA. Here, we propose a general theory for this charge-assisted looping process, develop a computational model to support the theory, and test the model against experimental data. To computationally model charge-assisted looping, we first discretize the DNA into a piecewise linear curve. Then, we apply the worm-like chain model to the curve to simulate thermal fluctuations of the DNA.

Unraveling looping efficiency of stochastic Cosserat polymers

Understanding looping probabilities, including the particular case of ring closure or cyclization, of fluctuating polymers (e.g., DNA) is important in many applications in molecular biology and chemistry. In a continuum limit the configuration of a polymer is a curve in the group $\mathit{SE}$(3) of rigid body displacements, whose energy can be modeled via the Cosserat theory of elastic rods. Cosserat rods are a more detailed version of the classic wormlike-chain (WLC) model, which we show to be more appropriate in short-length scale, or stiff, regimes, where the contributions of extension and shear deformations are not negligible and lead to noteworthy high values for the cyclization probabilities (or $J$-factors). We therefore observe that the Cosserat framework is a candidate for gaining a better understanding of the enhanced cyclization of short DNA molecules reported in various experiments, which is not satisfactorily explained by WLC-type models. Characterizing the stochastic fluctuations about minimizers of the energy by means of Laplace expansions in a (real) path integral formulation, we develop efficient analytical approximations for the two cases of full looping, in which both end-to-end relative translation and rotation are prescribed, and of marginal looping probabilities, where only end-to-end translation is prescribed. For isotropic Cosserat rods, certain looping boundary value problems admit nonisolated families of critical points of the energy due to an associated continuous symmetry. For the first time, taking inspiration from (imaginary) path integral techniques, a quantum mechanical probabilistic treatment of Goldstone modes in statistical rod mechanics sheds light on $J$-factor computations for isotropic rods in the semiclassical context. All the results are achieved exploiting appropriate Jacobi fields arising from Gaussian path integrals and show good agreement when compared with intense Monte Carlo simulations for the target examples.

Estimation of Effective Concentrations Enforced by Complex Linker Architectures from Conformational Ensembles.

Proteins and protein assemblies often tether interaction partners to strengthen interactions, to regulate activity through auto-inhibition or -activation, or to boost enzyme catalysis. Tethered reactions are regulated by the architecture of the tether, which defines an effective concentration of the interactor. Effective concentrations can be estimated theoretically for simple linkers via polymer models, but there is currently no general method for estimating effective concentrations for complex linker architectures consisting of both flexible and folded domains. We describe how effective concentrations can be estimated computationally for any protein linker architecture by defining a realistic conformational ensemble. We benchmark against prediction from a worm-like chain and values measured by competition experiments and find minor differences likely due to excluded volume effects. Systematic variation of the properties of flexible and folded segments shows that the effective concentration is mainly determined by the combination of the total length of flexible segments and the distance between the termini of the folded domains. We show that a folded domain in a disordered linker can increase the effective concentration beyond what can be achieved by a fully disordered linker by focusing the end-to-end distance at the appropriate spacing. This suggests that complex linker architecture may have advantages over simple flexible linkers and emphasizes that annotation as a linker should depend on the molecular context.

Conformational characteristics of regioselectively PEG/PS-grafted cellulosic bottlebrushes in solution: cross-sectional structure and main-chain stiffness

Cellulosic bottlebrushes with polystyrene (PS) and poly(ethylene glycol) (PEG) side chains at the O-6 and O-2,3 positions, respectively (PEG-PS-cellulose), were synthesized and characterized in diluted solution to reveal the second structure of heterografted bottlebrushes. The regioselectivity and degree of substitution were evaluated by 1H- and 13C-nuclear magnetic resonance spectroscopy and size-exclusion chromatography (SEC). The cross-sectional structure of PEG-PS-cellulose was evaluated from the cross-sectional radius of gyration determined by the small angle X-ray scattering technique as a function of the molecular weight of the PS side chain. As a result, PEG-PS-cellulose was found to show a core-shell-corona structure, in which PEG and PS side chains formed a homogeneous shell layer surrounding the cellulosic core and the outer segments of PS chains formed an outer corona layer. The stiffness parameter (λ−1) of the main chain was analyzed by the SEC–multiangle light scattering technique along with the Kratky-Porod wormlike chain model. In comparison with a previously reported cellulosic bottlebrush with a PS side chain at the O-6 position, it is suggested that the observed increase in λ−1 with increasing molecular weight of PS is mainly derived from the interaction among PS side chains located in an outer layer, while the PEG side chains at the O-2,3 position effectively suppressed the internal rotation of the cellulosic main chain. Cellulosic bottlebrush regioselectively possessing poly(ethylene glycol) (PEG) and polystyrene (PS) side chains (PEG-PS-cellulose) was synthesized, and its secondary structure in dilute solution was investigated with SAXS and SEC-MALS. The relationship between the cross-sectional mean squared radius of gyration (〈Sc2〉) and molecular weight of PS chain (MWPS) showed that PEG-PS-cellulose has a core-shell-corona structure in cross section. The dependency of main-chain stiffness (λ−1) on MWPS was discussed on the basis of the interactions of the PS and PEG side chains as well as the restricted rotation of the cellulosic main chain.

Two-phase dynamics of DNA supercoiling based on DNA polymer physics

DNA supercoils are generated in genome regulation processes such as transcription and replication and provide mechanical feedback to such processes. Under tension, a DNA supercoil can present a coexistence state of plectonemic and stretched phases. Experiments have revealed the dynamic behaviors of plectonemes, e.g., diffusion, nucleation, and hopping. To represent these dynamics with conformational changes, we demonstrated first the fast dynamics on the DNA to reach torque equilibrium within the plectonemic and stretched phases, and then identified the two-phase boundaries as collective slow variables to describe the essential dynamics. According to the timescale separation demonstrated here, we developed a two-phase model on the dynamics of DNA supercoiling, which can capture physiologically relevant events across timescales of several orders of magnitudes. In this model, we systematically characterized the slow dynamics between the two phases and compared the numerical results with those from the DNA polymer physics-based worm-like chain model. The supercoiling dynamics, including the nucleation, diffusion, and hopping of plectonemes, have been well represented and reproduced, using the two-phase dynamic model, at trivial computational costs. Our current developments, therefore, can be implemented to explore multiscale physical mechanisms of the DNA supercoiling-dependent physiological processes.

Transverse Magnetic Tweezers Allowing Coincident Epi-Fluorescence Microscopy on Horizontally Extended DNA.

Longitudinal magnetic tweezers (L-MT) have seen wide-scale adoption as the tool of choice for stretching and twisting a single DNA molecule. They are also used to probe topological changes in DNA as a result of protein binding and enzymatic activity. However, in the longitudinal configuration, the DNA molecule is extended perpendicular to the imaging plane. As a result, it is only possible to infer biological activity from the motion of the tethered paramagnetic microsphere. Described here is a "transverse" magnetic tweezers (T-MT) geometry featuring simultaneous control of DNA extension and spatially coincident video-rate epi-fluorescence imaging. Unlike in L-MT, DNA tethers in T-MT are extended parallel to the imaging plane between two micron-sized spheres, and importantly protein targets on the DNA can be localized using fluorescent nanoparticles. The T-MT can manipulate a long DNA construct at molecular extensions approaching the contour length defined by B-DNA helical geometry, and the measured entropic elasticity agrees with the wormlike chain model (force <35 pN). By incorporating a torsionally constrained DNA tether, the T-MT would allow both the relative extension and twist of the tether to be manipulated, while viewing far-red emitting fluorophore-labeled targets. This T-MT design has the potential to enable the study of DNA binding and remodeling processes under conditions of constant force and defined torsional stress.