Fluctuating elastic rings: statics and dynamics

Natural tristability of a confined helical filament with anisotropic bending rigidities

We find that whenc 0 R ∼ 0.5 andτ 0 R < 0.11 < c 0 R , confining a helical filament with anisotropic bending rigidities within a cylindrical tube of radius R can create a natural tristable status which is consisted of two low-pitch helices and one high-pitch helix, where a helical filament is referred to as a filament that has both an intrinsic curvature ( c 0 ) and an intrinsic twist rate ( τ 0 ). The formation of the tristable status also requires that the filament has a significant difference between two bending rigidities and a large twisting rigidity. The relative heights of two low-pitch helices in a tristable status are close to zero, and the smaller the intrinsic twisting angle, the smaller the difference between these two heights. Moreover, at a large intrinsic twisting angle, two low-pitch helices display a large energy difference, and the energy difference increases with decreasing τ 0 . Meanwhile, the relative height of the high-pitch helix is always close to that of a straight line. Finally, at some specific intrinsic parameters, the tristable status can include an isoenergic status with two helices of the same energy but distinct pitches.

Charged elastic rings: deformation and dynamics

We report the counter-intuitive instability of charged elastic rings, and the persistence of sinusoidal deformations in the lowest-energy configurations by the combination of high-precision numerical simulations and analytical perturbation calculation. We also study the dynamical evolution of the charged ring under random disturbance, and reveal the modulation of the dominant frequencies by the electrostatic force. The purely mechanical analysis of the classical ring system presented in this work yields insights into the subtlety of long-range forces in the organization and dynamics of matter.

Bistability of a helical filament confined on a cylinder

The natural configuration of an intrinsically curved and twisted filament is uniquely a helix so that it can be referred to as a helical filament. We find that confining a helical filament on a cylinder can create a bistable state. When c_{0}R=0.5, where c_{0} is the intrinsic curvature of filament and R is the radius of cylinder, the phase diagram for the stability of a helix contains three regimes. Regime I has a small intrinsic twisting rate (ITR) and exhibits a bistable state which consists of two isoenergic helices. In regime II, the filament has a moderate ITR and the bistable state consists of a metastable low-pitch helix and a stable nonhelix. In regime III, the helix is unstable, owing to a large ITR. A similar phenomenon occurs when c_{0}R∼0.5. Monte Carlo simulation confirms these conclusions and indicates further that there are bistable nonhelices in regime III. This bistable system offers a prospective green material since the wide range of parameters and distinctive configurations for bistable states favor its realization and application.

Bistability induced by a spontaneous twisting rate for a two-dimensional intrinsically curved filament

We find that a moderate intrinsic twisting rate (ITR) can induce a bistable state for a force-free two-dimensional intrinsically curved filament. There are two different configurations of equal energy in a bistable state so that the filament is clearly different from its three-dimensional counterpart. The smaller the ITR or the larger the intrinsic curvature (IC), the clearer the distinction between two isoenergetic configurations and the longer the filament. In bistable states, the relationship between length and ITR is approximately a hyperbola and relationship between IC and critical ITR is approximately linear. Thermal fluctuation can result in a shift between two isoenergetic configurations, but large bending and twisting rigidities can prevent the shift and maintain the filament in one of these two configurations. Moreover, a filament can have a metastable state and at a finite temperature such a filament has the similar property as that of a filament with bistable state.

Crucial role of the intrinsic twist rate for the size of an intrinsically curved semiflexible biopolymer

We study the effects of the intrinsic curvature (IC), intrinsic twist rate (ITR), anisotropic bending rigidities, sequence disorder, and temperature (T) on the persistence length (l_{p}) of a two- or three-dimensional semiflexible biopolymer. We develop some general expressions to evaluate exactly these effects. We find that a moderate IC alone reduces l_{p} considerably. Our results indicate that the centerline of the filament keeps as a helix in a rather large range of T when ITR is small. However, a large ITR can counterbalance the effect of IC and the result is insensitive to the twist rigidity. Moreover, a weak randomness in IC and ITR can result in an "overexpanded" state. Meanwhile, when ITR is small, l_{p} is not a monotonic function of T but can have either minimum or maximum at some T, and in the two-dimensional case the maximum is more obvious than that in the three-dimensional case. These results reveal that to obtain a proper size at a finite T for an intrinsically curved semiflexible biopolymer, proper values of bending rigidities and ITR are necessary but a large twist rigidity may be only a by-product. Our findings are instructive in controlling the size of a semiflexible biopolymer in organic synthesis since the mean end-to-end distance and radius of gyration of a long semiflexible biopolymer are proportional to l_{p}.

Mechanical property of the helical configuration for a twisted intrinsically straight biopolymer

We explore the effects of two typical torques on the mechanical property of the helical configuration for an intrinsically straight filament or biopolymer either in three-dimensional space or on a cylinder. One torque is parallel to the direction of a uniaxial applied force, and is coupled to the cross section of the filament. We obtain some algebraic equations for the helical configuration and find that the boundary conditions are crucial. In three-dimensional space, we show that the extension is always a monotonic function of the applied force. On the other hand, for a filament confined on a cylinder, the twisting rigidity and torque coupled to the cross section are irrelevant in forming a helix if the filament is isotropic and under free boundary condition. However, the twisting rigidity and the torque coupled to the cross section become crucial when the Euler angle at two ends of the filament are fixed. Particularly, the extension of a helix can subject to a first-order transition so that in such a condition a biopolymer can act as a switch or sensor in some biological processes. We also present several phase diagrams to provide the conditions to form a helix.

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.

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.

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.

Entropic elasticity of dilated and contorted idealized circular chains

Thermal energy provides random motion to particles that leads to the well-known entropic force, which favors the clumping of linear and circular molecules. We evaluate the entropic force which resists the radial dilation and subsequent twisting out of plane of circular polymers by developing mechanical models and performing molecular dynamics simulations. We find that dilating a circular chain is analogous to stretching its linear counterpart. We also find that the torque applied to an already dilated ring and the resulting twist out of plane are related by a linear relationship for a wide range of deformed configurations and, using this result, we are able to predict the angular fluctuations of such a macromolecule.

Coupling of twist and writhe in short DNA loops

While bending and twist can be treated as independent degrees of freedom for linear DNA molecules, the loop closure constraint introduces a coupling between these variables in circular DNA. We performed Monte Carlo simulations of wormlike rods with both bending and twist rigidity in order to study the coupling between the writhe and twist distributions for various DNA lengths. We find that for sufficiently short DNA, the writhe distribution differs from that of a model with bending energy only. We show that the factorization approximation introduced by previous researchers coincides, within numerical accuracy, with our simulation results, and conclude that the closure constraint is fully accounted for by the White-Fuller relation. Experimental tests of our results for short DNA plasmids are proposed.

Disordered, stretched, and semiflexible biopolymers in two dimensions

We study the effects of intrinsic sequence-dependent curvature for a two-dimensional semiflexible biopolymer with short-range correlation in intrinsic curvatures. We show exactly that when not subjected to any external force, such a system is equivalent to a system with a well-defined intrinsic curvature and a proper renormalized persistence length. We find the exact expression for the distribution function of the equivalent system. However, we show that such an equivalent system does not always exist for the polymer subjected to an external force. We find that under an external force, the effect of sequence disorder depends upon the averaging order, the degree of disorder, and the experimental conditions, such as the boundary conditions. Furthermore, a short to moderate length biopolymer may be much softer or has a smaller apparent persistent length than what would be expected from the "equivalent system." Moreover, under a strong stretching force and for a long biopolymer, the sequence disorder is immaterial for elasticity. Finally, the effect of sequence disorder may depend upon the quantity considered.

Collapse and folding of pressurized rings in two dimensions

Hydrostatically pressurized circular rings confined to two dimensions (or cylinders constrained to have only z -independent deformations) undergo Euler-type buckling when the outside pressure exceeds a critical value. We perform a stability analysis of rings with arclength-dependent bending moduli and determine how weakened bending modulus segments affect the buckling critical pressure. Rings with a fourfold symmetric modulation are particularly susceptible to collapse. In addition we study the initial postbuckling stages of the pressurized rings to determine possible ring folding patterns.

A semiflexible polymer ring acting as a nano-propeller

The dynamics of a rotating elastic nano-ring driven in a viscous fluid by an externally applied torque about a specific axis is studied using elasto-hydrodynamic simulations. We show that a helical deformation of the ring filament is excited, and that this leads to directed propulsion which is independent of the direction of rotation. It is found that the propulsive force and efficiency initially increase as the torque is increased, and then decrease discontinuously at a buckling transition at a critical torque. This unique propulsive behavior at the shape transition arises due to its specific geometry, i.e., circularity of an elastic filament. The implications of the behavior for artificial microscopic devices are discussed.

Sequence-dependent effects on the properties of semiflexible biopolymers

Using a path integral technique, we show exactly that for a semiflexible biopolymer in constant extension ensemble, no matter how long the polymer and how large the external force, the effects of short-range correlations in the sequence-dependent spontaneous curvatures and torsions can be incorporated into a model with well-defined mean spontaneous curvature and torsion as well as a renormalized persistence length. Moreover, for a long biopolymer with large mean persistence length, the sequence-dependent persistence lengths can be replaced by their mean. However, for a short biopolymer or for a biopolymer with small persistence lengths, inhomogeneity in persistence lengths tends to make physical observables very sensitive to details and therefore less predictable.

Internal dynamics of superhelical DNA

We present the first data on the temporal kinetics of monomer mean square displacements in DNA circles with defined degrees of superhelicity. The segmental dynamics of specifically labeled DNA plasmids with superhelical densities between 0 and -0.016 was assessed by fluorescence correlation spectroscopy. Introduction of superhelicity leads to progressively faster dynamics in the long time regime corresponding to the coil diffusion as observed previously by Langowski et al. [Biopolymers 34, 639 (1994)10.1002/bip.360340506], but also in the short time range corresponding to the segmental motion within the DNA coil.

Elasticity of two-dimensional filaments with constant spontaneous curvature

We study the mechanical property of a two-dimensional filament with constant spontaneous curvature and under uniaxial applied force. We derive the equation that governs the stable shape of the filament and obtain analytical solutions for the equation. We find that for a long filament with positive initial azimuth angle (the azimuth angle is the angle between x axis and the tangent of the filament) and under large stretching force, the azimuth angle is a two-valued function of the arclength, decreases first, and then increases with increasing arclength. Otherwise, the azimuth angle is a monotonic function of arclength. At finite temperature, we derive the differential equation that governs the partition function and find exact solution of the partition function for a filament free of force. We obtain closed-form expressions on the force-extension relation for a filament under low force and for a long filament under strong stretching force. Our results show that for a biopolymer with moderate length and not too small spontaneous curvature, the effect of the spontaneous curvature cannot be replaced by a simple renormalization of the persistence length in the wormlike chain model.

Fluctuating semiflexible polymer ribbon constrained to a ring

Twist stiffness and an asymmetric bending stiffness of a polymer or a polymer bundle is captured by the elastic ribbon model. We investigate the effects a ring geometry induces to a thermally fluctuating ribbon, finding bend-bend coupling in addition to twist-bend coupling. Furthermore, due to the geometric constraint the polymer's effective bending stiffness increases. A new parameter for experimental investigations of polymer bundles is proposed: the mean square diameter of a ribbonlike ring, which is determined analytically in the semiflexible limit. Monte Carlo simulations are performed which affirm the model's prediction up to high flexibility.

Distribution functions for filaments under tension

We develop a biased Monte Carlo simulation technique to measure the distribution functions of the extension and the end-to-end distance of fluctuating filaments stretched by external force. The method is applicable for arbitrary ratio of the persistence length to the contour length and for arbitrary forces, and also for the case of steric constraints, such as an external wall. The fundamental idea underlying the algorithm is to account explicitly for the length-scale dependence of the effective elastic moduli. We find that orientational fluctuations and wall effects produce non-Gaussian distributions for nearly rigid filaments in the small to intermediate force regime. The simulation results are tested against analytic expressions for the force-extension curves, both in the semiflexible and nearly stiff limits.

Elasticity and stability of a helical filament

We derive the general shape equations in terms of Euler angles for a uniform elastic rod with spontaneous torsion and curvatures and subjected to external force and torque. Our results based on an analytic formalism show that the extension of a helical rod may undergo a one-step discontinuous transition with increasing stretching force. This agrees quantitatively with experimental observations for a helix in a chemically defined lipid concentrate. The larger the twisting rigidity, the larger the jump in the extension. The effect of torque on the jump is, however, dependent on the value of the spontaneous torsion. In contrast, increasing the spontaneous torsion encourages the continuous variation of the extension. An "over-collapse" behavior is observed for the rod with asymmetric bending rigidity, and an intrinsic asymmetric elasticity under twisting force is found.