Semiflexible Rod Research Articles

A Theoretical Description of DNA Plectonemes under Tension

Twisting a DNA molecule held under constant tension is accompanied by a transition from a linear to a plectonemic DNA configuration, in which part of the applied twist is absorbed in a superhelical structure. This is seen as a linear shortening of the DNA length with added turns after the transition. So far no theoretical description exists, which consistently describes the slope of the supercoiling curves as well as the torque in the plectonemic regime and its dependency on the applied force and the monovalent ion concentration in solution. Here, we present a simple model, in which the DNA is treated as a semiflexible rod. The energy of the plectonemic structure is calculated considering DNA bending, applied tension and electrostatic repulsion between the DNA strands but excluding fluctuations. We compare the predictions of our simple static theory with experimental supercoiling data, recorded with magnetic tweezers. We obtain an excellent agreement for the supercoiling slopes and the torque as function of force and monovalent ion concentration only if a reduced DNA charge is taken into account. We verify our theory using Monte-Carlo simulations, in which the same energetic terms are used. Surprisingly, the simple static model describes experimental data much better than more sophisticated models considering fluctuations, which considerably overestimate the torque of the plectonemic phase.

Entropic boundary effects on the elasticity of short DNA molecules

We have measured the entropic elasticity of double-stranded-DNA molecules ranging from 247 to 1298 bp in length using axial force-clamp optical tweezers. We show that entropic end effects and excluded-volume forces from surface attachments become significant for such short molecules. The effective persistence length of the shortest molecules decreases by a factor of 2 compared to the established value for long molecules, and excluded-volume forces extend the molecules to about one third of their nominal contour length. We interpret these results in the framework of an inextensible semiflexible rod model.

Macromolecular conformation of chitosan in dilute solution: A new global hydrodynamic approach

Chitosans of different molar masses were prepared by storing freshly prepared samples for up to 6 months at either 4, 25 or 40 °C. The weight-average molar masses, M w and intrinsic viscosities, [ η] were then measured using size exclusion chromatography coupled to multi-angle laser light scattering (SEC-MALLS) and a “rolling ball” viscometer, respectively. The solution conformation of chitosan was then estimated from: (a) the Mark–Houwink–Kuhn–Sakurada (MHKS) power law relationship [ η] = kM w a and (b) the persistence length, L p calculated from a new approach based on equivalent radii [Ortega, A., & Garcia de la Torre, J. (2007). Equivalent radii and ratios of radii from solution properties as indicators of macromolecular conformation, shape, and flexibility. Biomacromolecules, 8, 2464–2475]. Both the MHKS power law exponent ( a = 0.95 ± 0.01) and the persistence length ( L p = 16 ± 2 nm) are consistent with a semi-flexible rod type (or stiff coil) conformation for all 33 chitosans studied. A semi-flexible rod conformation was further supported by the Wales–van Holde ratio, the translational frictional ratio and sedimentation conformation zoning.

On the Deflection and Persistence Lengths of Mesogenic Worm-Like Rods

Persistence length is a key quantity characterizing the conformation of chain-like particles. While it is clear to interpret qualitatively as the distance over which the direction of the chain persists, its quantitative definition is not unique. Very frequently, approximate formulae are used. However, as pointed out in a recent work [P. Cifra, Polymer 45, 5995 (2004)], their predictions are generally not correct when compared to what may be considered the exact estimate, and this especially happens for sufficiently stiff chains. The observed sensitivity of these formulae to the degree of internal flexibility of the chain has stimulated the present work, where, by employing Monte Carlo and Molecular Dynamics computer simulations, the outcomes of exact and approximate formulae are systematically studied for systems of short, semi-flexible rods which form liquid–crystalline phases. While the approximate expressions give consistent results for the isolated chain, their predictions differ in bulk ordered phases. Moreover, compared to the exact expression, the approximate formulae perform overall quite bad. Thus, if the usage of the approximate formulae is questionable in general, it is more so in the particular case of liquid–crystal phases. For mesogenic worm-like rods, another characteristic length was put forward, the deflection length. It was phenomenologically defined as the distance along the chain over which the chain behaves as isolated. It is illustrated how it can be readily evaluated from the exact definition of the persistence length.

Single-Molecule Observation of Anomalous Electrohydrodynamic Orientation of Microtubules

We use fluorescence microscopy to measure the orientation and shape of microtubules-which serve as a model system for semiflexible rods-that are electrophoretically driven. Surprisingly, a bimodal orientation distribution is observed, with microtubules in either parallel or perpendicular orientations to the electric field. The occupancy of these states varies nonmonotonically with the microtubule length L and the electric field E. We also observe a surprising bending deformation of microtubules. Interestingly, all data collapse onto a universal scaling curve when the average alignment is plotted as a function of B proportional, variantEL3, which reflects the ratio between the driving force and a restoring elastic force. Our results have important implications for the interpretation of electrical birefringence experiments and, more generally, for a better understanding of the electrokinetics of rods.

Conserved Asp-137 Imparts Flexibility to Tropomyosin and Affects Function

Tropomyosin (Tm) is an alpha-helical coiled-coil that controls muscle contraction by sterically regulating the myosin-actin interaction. Tm moves between three states on F-actin as either a uniform or a non-uniform semi-flexible rod. Tm is stabilized by hydrophobic residues in the "a" and "d" positions of the heptad repeat. The highly conserved Asp-137 is unusual in that it introduces a negative charge on each chain in a position typically occupied by hydrophobic residues. The occurrence of two charged residues in the hydrophobic region is expected to destabilize the region and impart flexibility. To determine whether this region is unstable, we have substituted hydrophobic Leu for Asp-137 and studied changes in Tm susceptibility to limited proteolysis by trypsin and changes in regulation. We found that native and Tm controls that contain Asp-137 were readily cleaved at Arg-133 with t 1/2 of 5 min. In contrast, the Leu-137 mutant was not cleaved under the same conditions. Actin stabilized Tm, causing a 10-fold reduction in the rate of cleavage at Arg-133. The actin-myosin subfragment S1 ATPase activity was greater for the Leu mutant compared with controls in the absence of troponin and in the presence of troponin and Ca2+. We conclude that the highly conserved Asp-137 destabilizes the middle of Tm, resulting in a more flexible region that is important for the cooperative activation of the thin filament by myosin. We thus have shown a link between the dynamic properties of Tm and its function.

Buckling of stiff polymers: Influence of thermal fluctuations

The buckling of biopolymers is a frequently studied phenomenon The influence of thermal fluctuations on the buckling transition is, however, often ignored and not completely understood. A quantitative theory of the buckling of a wormlike chain based on a semiclassical approximation of the partition function is presented. The contribution of thermal fluctuations to the force-extension relation that allows one to go beyond the classical Euler buckling is derived in the linear and nonlinear regimes as well. It is shown that the thermal fluctuations in the nonlinear buckling regime increase the end-to-end distance of the semiflexible rod if it is confined to two dimensions as opposed to the three-dimensional case. The transition to a buckled state softens at finite temperature. We derive the scaling behavior of the transition shift with increasing ratio of contour length versus persistence length.

Influence of charge and flexibility on smectic phase formation in filamentous virus suspensions

We present experimental measurements of the cholesteric-smectic phase transition of suspensions of charged semiflexible rods as a function of rod flexibility and surface charge. The rod particles consist of the bacteriophage M13 and closely related mutants, which are structurally identical to M13, but vary either in contour length and therefore ratio of persistence length to contour length, or surface charge. Surface charge is altered in two ways; by changing solution pH and by comparing M13 with fd virus, a virus which differs from M13 only by the substitution of a single charged amino acid for a neutral one per viral coat protein. Phase diagrams are measured as a function of particle length, particle charge, and ionic strength. The experimental results are compared with existing theoretical predictions for the phase behavior of flexible rods and charged rods.

Simulations of dynamics and viscoelasticity in highly entangled solutions of semiflexible rods

Brownian dynamics simulations are used to study highly entangled solutions of semiflexible polymers. Bending fluctuations of semiflexible rods are significantly affected by entanglement only above a concentration c(**), where c(**) approximately 10(3)L(-3) for chains of length L similar to their persistence length. For c>c(**), the tube radius R(e) approaches a dependence R(e) proportional to c(-35) and the linear viscoelastic response develops an elastic plateau that is absent for c<c(**). Experiments on isotropic solutions of F-actin are shown to span concentrations near c(**) for which the predicted asymptotic scaling of the plateau modulus, G proportional to c(75), is not yet valid.

Vorticity banding in rodlike virus suspensions

Vorticity banding under steady shear flow is observed in a suspension of semiflexible colloidal rods (fd virus particles) within a part of the paranematic-nematic biphasic region. Banding occurs uniformly throughout the cell gap within a shear-rate interval (.gamma-, .gamma+) , which depends on the fd concentration. For shear rates below the lower-border shear rate .gamma- only shear elongation of inhomogeneities, which are formed due to paranematic-nematic phase separation, is observed. Within a small region just above the upper-border shear rate .gamma+ , banding occurs heterogeneously. An essential difference in the kinetics of vorticity banding is observed, depending on the morphology of inhomogeneities formed during the initial stages of the paranematic-nematic phase separation. Particle tracking and polarization experiments indicate that the vorticity bands are in a weak rolling flow, superimposed on the applied shear flow. We propose a mechanism for the origin of the banding instability and the transient stability of the banded states. This mechanism is related to the normal stresses generated by inhomogeneities formed due to the underlying paranematic-nematic phase transition.

Elasticity Due to Semiflexible Protein Assemblies near the Critical Gel Concentration and Beyond

We conclude that the elasticity of gelatin gels over a large range of gelatin concentrations can be described in a single model in which a percolation model for low helix concentrations has been combined with a model based on the deflection length (for high helix concentrations). The model gives a better view of the physics behind the gel strength of a gelatin gel over its entire concentration range and should be applicable to any other semi-flexible rod system over a large concentration range.

Semiflexible Rod in a Nematic Environment with Applications to Biological Polymers

We present the theory for a long polymer rod immersed in a nematic environment and find the effects of a nematic solvent on small fluctuations of the tangent vector perpendicular to the rod axis. We are thus able to calculate the physically interesting properties associated with our combined polymer and nematic system. These include tangent−tangent correlation functions and the scattering structure factor, which reproduces known qualitative experimental results rather well. As one biological application of our model, we analyze the possible effects of a weak nematic solvent on the mechanical properties of individual hemoglobin fibers. As another biological application, we model the effects of a strong nematic environment, provided by fd virus, on the elastic and conformational properties of wormlike micelles. The work presented here can be viewed as constituting a microscopic model derivation of previous, more phenomenologically inspired, theories that deal with similar polymer/nematic systems. Interestingly, we find that a simple boundary condition of the nematic order at the rod surface agrees well with available experimental data.

The force generated by biological membranes on a polymer rod and its response: statics and dynamics.

We propose a theory for the force exerted by a fluctuating membrane on a polymer rod tip. Using statistical mechanical methods, the expression for the generated force is written in terms of the distance of the rod tip from the membrane "frame." We apply the theory in calculating the stall force and membrane displacement required to cease the growth of a growing fiber induced by membrane fluctuations, as well as the membrane force and membrane displacement required for rod/fiber buckling. We also consider the dynamics of a growing fiber tip under the influence of a fluctuation-induced membrane force. We discuss the importance of our results in various biological contexts. Finally, we present a method to simultaneously extract both the rigidity of the semiflexible rod and the force applied by, e.g., the membrane from the measurements of the bending fluctuations of the rod. Such a measurement of the force would give information about the thermodynamics of the rod polymerization that involves the usual Brownian ratchet mechanism.

Isotropic-nematic phase transition in suspensions of filamentous virus and the neutral polymer Dextran.

We present an experimental study of the isotropic-nematic phase transition in an aqueous mixture of charged semiflexible rods ( fd virus) and neutral polymer (Dextran). A complete phase diagram is measured as a function of ionic strength and polymer molecular weight. At high ionic strength we find that adding polymer widens the isotropic-nematic coexistence region with polymers preferentially partitioning into the isotropic phase, while at low ionic strength the added polymer has no effect on the phase transition. The nematic order parameter is determined from birefringence measurements and is found to be independent of polymer concentration (or equivalently the strength of attraction). The experimental results are compared with the existing theoretical predictions for the isotropic-nematic transition in rods with attractive interactions.

Measuring the nematic order of suspensions of colloidal fd virus by x-ray diffraction and optical birefringence.

The orientational distribution function of the nematic phase of suspensions of the semiflexible rodlike virus fd is measured by x-ray diffraction as a function of concentration and ionic strength. X-ray diffraction from a single-domain nematic phase of fd is influenced by interparticle correlations at low angle, while only intraparticle scatter contributes at high angle. Consequently, the angular distribution of the scattered intensity arises from only the single-particle orientational distribution function at high angle but it also includes spatial and orientational correlations at low angle. Experimental measurements of the orientational distribution function from both the interparticle (structure factor) and intraparticle (form factor) scattering were made to test whether the correlations present in interparticle scatter influence the measurement of the single-particle orientational distribution function. It was found that the two types of scatter yield consistent values for the nematic order parameter. It was also found that x-ray diffraction is insensitive to the orientational distribution function's precise form, and the measured angular intensity distribution is described equally well by both Onsager's trial function and a Gaussian. At high ionic strength, the order parameter S of the nematic phase coexisting with the isotropic phase approaches theoretical predictions for long semiflexible rods S=0.55, but deviations from theory increase with decreasing ionic strength. The concentration dependence of the nematic order parameter also better agrees with theoretical predictions at high ionic strength indicating that electrostatic interactions have a measurable effect on the nematic order parameter. The x-ray order parameters are shown to be proportional to the measured birefringence, and the saturation birefringence of fd is determined enabling a simple, inexpensive way to measure the order parameter. Additionally, the spatial ordering of nematic fd was probed. Measurements of the nematic structure factor revealed a single large peak in contrast to nematics of rigid rods.

Theory of linear viscoelasticity of semiflexible rods in dilute solution

We present a theory of the linear viscoelasticity of dilute solutions of freely draining, inextensible, semiflexible rods. The theory is developed expanding the polymer contour about a rigid rod reference state, in a manner that respects the inextensibility of the chain, and is asymptotically exact in the rodlike limit where the polymer length L is much less than its persistence length Lp. In this limit, the relaxation modulus G(t) exhibits three time regimes: At very early times, less than a time τ∥∝L8/Lp5 required for the end-to-end length of a chain to relax significantly after a deformation, the average tension induced in each chain and G(t) both decay as t−3/4. Over a broad range of intermediate times, τ∥≪t≪τ⊥, where τ⊥∝L4/Lp is the longest relaxation time for the transverse bending modes, the end-to-end length decays as t−1/4, while the residual tension required to drive this relaxation and G(t) both decay as t−5/4. As later times, the stress is dominated by an entropic orientational stress, giving ...

Entropic interactions in suspensions of semiflexible rods: short-range effects of flexibility.

We compute the entropic interactions between two colloidal spheres immersed in a dilute suspension of semiflexible rods. Our model treats the semiflexible rod as a bent rod at fixed angle, set by the rod contour and persistence lengths. The entropic forces arising from this additional rotational degree of freedom are captured quantitatively by the model, and account for observations at short range in a recent experiment. Global fits to the interaction potential data suggest the persistence length of the fd virus is about two to three times smaller than the commonly used value of 2.2 microm.

An exactly solvable Ogston model of gel electrophoresis. VI. Towards a theory for macromolecules.

In this article, we present a generalized version of our lattice model of low-field gel electrophoresis that allows us to treat the case of macromolecules such as short linear or circular oligomers and semi-flexible rods. We show that free-solution electrophoresis problems can be seen as random walks in the conformational space of the analyte. For sufficiently small molecules, our mathematical approach provides exact mobilities. In a quenched gel-like environment, however, both conformational and positional degrees of freedom must be used, but exact solutions can also be obtained. As an example, we then investigate several two-dimensional model gels, as well as a simple channel system where we see evidence of entropic effects that cannot be captured by the traditional Ogston concept of free volume.

Concentration-dependent sedimentation of colloidal rods

In this paper we first develop an approximate theory for the leading order concentration dependence of the sedimentation coefficient for rodlike colloids/polymers/macromolecules. To first order in volume fraction φ of rods, the sedimentation coefficient is written as 1+αφ. For large aspect ratios L/D (L is the rod length, D its thickness) α is found to vary like ∝(L/D)2/ln(L/D). This theoretical prediction is compared to experimental results. Then, experiments on fd virus are described, both in the isotropic and nematic phase. First-order in concentration results for this very long and thin (semiflexible) rod are in agreement with the above-mentioned theoretical prediction. Sedimentation profiles for the nematic phase show two sedimentation fronts. This result indicates that the nematic phase becomes unstable with the respect to isotropic phase during sedimentation.

Self-Diffusion of Rodlike Polymers in Isotropic Solutions

The self-diffusion coefficient, D s , of fluorescently tagged poly(γ-benzyl α,L-glutamate) has been measured throughout the isotropic regime in pyridine. Intrinsic viscosity, phase boundary studies, epifluorescence microscopy, computer modeling, and the diffusion rates themselves show that the labeling does not appreciably perturb the semiflexible rod structure of the polymer. It is also demonstrated that pyridine is a good solvent for poly(γ-benzyl α,L-glutamate), comparable to N,N'-dimethylformamide. As concentration increases, two or three regimes are found depending on the rod length. Dilute behavior holds to number densities, ν, well exceeding L -3 , where L is the rod length