Buckling, fluctuations, and collapse in semiflexible polyelectrolytes

Dynamics of certain Euler-Bernoulli rods and rings from a minimal coupling quantum isomorphism

In some parameter and solution regimes, a minimally coupled nonrelativistic quantum particle in one dimension is isomorphic to a much heavier, vibrating, very thin Euler-Bernoulli rod in three dimensions with ratio of bending modulus to linear density (ℏ/2m)^{2}. For m=m_{e}, this quantity is comparable to that of a microtubule. Axial forces and torques applied to the rod play the role of scalar and vector potentials, respectively, and rod inextensibility plays the role of normalization. We show how an uncertainty principle ΔxΔp_{x}≳ℏ governs transverse deformations propagating down the inextensible, force and torque-free rod, and how orbital angular momentum quantized in units of ℏ or ℏ/2 (depending on calculation method) emerges when the force and torque-free inextensible rod is formed into a ring. For torqued rings with large wave numbers, a "twist quantum" appears that is somewhat analogous to the magnetic flux quantum. These and other results are obtained from a purely classical treatment of the rod, i.e., without quantizing any classical fields.

Compression-induced buckling of a semiflexible filament in two and three dimensions

The ability of biomolecules to exert forces on their surroundings or resist compression from the environment is essential in a variety of biologically relevant contexts. For filaments in the low-temperature limit and under a constant compressive force, Euler buckling theory predicts a sudden transition from a compressed to a bent state in these slender rods. In this paper, we use a mean-field theory to show that if a semiflexible chain is compressed at a finite temperature with a fixed end-to-end distance (permitting fluctuations in the compressive forces), it exhibits a continuous phase transition to a buckled state at a critical level of compression. We determine a quantitatively accurate prediction of the transverse position distribution function of the midpoint of the chain that indicates this transition. We find the mean compressive forces are non-monotonic as the extension of the filament varies, consistent with the observation that strongly buckled filaments are less able to bear an external load. We also find that for the fixed extension (isometric) ensemble, the buckling transition does not coincide with the local minimum of the mean force (in contrast to Euler buckling). We also show the theory is highly sensitive to fluctuations in length in two dimensions, and that the buckling transition can still be accurately recovered by accounting for those fluctuations. These predictions may be useful in understanding the behavior of filamentous biomolecules compressed by fluctuating forces, relevant in a variety of biological contexts.

Effect of magnesium ions on the structure of DNA thin films: an infrared spectroscopy study

Utilizing Fourier transform infrared spectroscopy we have investigated the vibrational spectrum of thin dsDNA films in order to track the structural changes upon addition of magnesium ions. In the range of low magnesium concentration ([magnesium]/[phosphate] = [Mg]/[P] < 0.5), both the red shift and the intensity of asymmetric PO2 stretching band decrease, indicating an increase of magnesium-phosphate binding in the backbone region. Vibration characteristics of the A conformation of the dsDNA vanish, whereas those characterizing the B conformation become fully stabilized. In the crossover range with comparable Mg and intrinsic Na DNA ions ([Mg]/[P] ≈ 1) B conformation remains stable; vibrational spectra show moderate intensity changes and a prominent blue shift, indicating a reinforcement of the bonds and binding in both the phosphate and the base regions. The obtained results reflect the modified screening and local charge neutralization of the dsDNA backbone charge, thus consistently demonstrating that the added Mg ions interact with DNA via long-range electrostatic forces. At high Mg contents ([Mg]/[P] > 10), the vibrational spectra broaden and show a striking intensity rise, while the base stacking remains unaffected. We argue that at these extreme conditions, where a charge compensation by vicinal counterions reaches 92-94%, DNA may undergo a structural transition into a more compact form.

Phase diagram of the ground states of DNA condensates

The phase diagram of the ground states of DNA in a bad solvent is studied for a semiflexible polymer model with a generalized local elastic bending potential characterized by a nonlinearity parameter x and effective self-attraction promoting compaction. x=1 corresponds to the wormlike chain model. Surprisingly, the phase diagram as well as the transition lines between the ground states are found to be a function of x. The model provides a simple explanation for the results of prior experimental and computational studies and makes predictions for the specific geometries of the ground states. The results underscore the impact of the form of the microscopic bending energy at macroscopic observable scales.

Finite-temperature buckling of an extensible rod

Thermal fluctuations can play an important role in the buckling of elastic objects at small scales, such as polymers or nanotubes. In this paper, we study the finite-temperature buckling transition of an extensible rod by analyzing fluctuation corrections to the elasticity of the rod. We find that, in both two and three dimensions, thermal fluctuations delay the buckling transition, and near the transition, there is a critical regime in which fluctuations are prominent and make a contribution to the effective force that is of order √T. We verify our theoretical prediction of the phase diagram with Monte Carlo simulations.

Energies and pressures in viruses: contribution of nonspecific electrostatic interactions

We summarize some aspects of electrostatic interactions in the context of viruses. A simplified but, within well defined limitations, reliable approach is used to derive expressions for electrostatic energies and the corresponding osmotic pressures in single-stranded RNA viruses and double-stranded DNA bacteriophages. The two types of viruses differ crucially in the spatial distribution of their genome charge which leads to essential differences in their free energies, depending on the capsid size and total charge in a quite different fashion. Differences in the free energies are trailed by the corresponding characteristics and variations in the osmotic pressure between the inside of the virus and the external bathing solution.

Torque-induced deformations of charged elastic DNA rods: thin helices, loops, and precursors of DNA supercoiling

We study the deformations of charged elastic rods under applied end forces and torques. For neutral filaments, we analyze the energetics of initial helical deformations and loop formation. We supplement this elastic approach with electrostatic energies of bent filaments and find critical conditions for buckling depending on the ionic strength of the solution. We also study force-induced loop opening, for parameters relevant for DNA. Finally, some applications of this nano-mechanical DNA model to salt-dependent onset of the DNA supercoiling are discussed.

Electrostatics of strongly charged biological polymers: ion-mediated interactions and self-organization in nucleic acids and proteins

Charges on biological polymers in physiologically relevant solution conditions are strongly screened by water and salt solutions containing counter-ions. However, the entropy of these counterions can result in surprisingly strong interactions between charged objects in water despite short screening lengths, via coupling between osmotic and electrostatic interactions. Widespread work in theory, experiment, and computation has been carried out to gain a fundamental understanding of the rich, yet sometimes counterintuitive, behavior of these polyelectrolyte systems. Examples of polyelectrolyte association in biology include DNA packaging and RNA folding, as well as aggregation and self-organization phenomena in different disease states.

Langevin dynamics of conformational transformations induced by the charge-curvature interaction

The role of thermal fluctuations in the conformational dynamics of a single closed filament is studied. It is shown that, due to the interaction between charges and bending degrees of freedom, initially circular chains may undergo transformation to polygonal shape.

End-to-end distribution for a wormlike chain in arbitrary dimensions

We construct an efficient methodology for calculating wormlike chain statistics in arbitrary D dimensions over all chain rigidities, from fully rigid to completely flexible. The structure of our exact analytical solution for the end-to-end distribution function for a wormlike chain in arbitrary D dimensions in Fourier-Laplace space (i.e., Fourier-transformed end position and Laplace-transformed chain length) adopts the form of an infinite continued fraction, which is advantageous for its compact structure and stability for numerical implementation. We then proceed to present a step-by-step methodology for performing the Fourier-Laplace inversion in order to make full use of our results in general applications. Asymptotic methods for evaluating the Laplace inversion (power-law expansion and Rayleigh-Schrödinger perturbation theory) are employed in order to improve the accuracy of the numerical inversions of the end-to-end distribution function in real space. We adapt our results to the evaluation of the single-chain structure factor, rendering simple, closed-form expressions that facilitate comparison with scattering experiments. Using our techniques, the accuracy of the end-to-end distribution function is enhanced up to the limit of the machine precision. We demonstrate the utility of our methodology with realizations of the chain statistics, giving a general methodology that can be applied to a wide range of biophysical problems.

Confined nanorods: jamming due to helical buckling

We investigate a longitudinally loaded elastic nanorod inside a cylindrical channel and show within the context of classical elasticity theory that the Euler buckling instability leads to a helical postbuckling form of the rod within the channel. The local pitch of the confined helix changes along the channel and so does the longitudinal force transmitted along the rod, diminishing away from the loaded end. This creates a possibility of jamming of the nanorod within the channel.

Stretching of buckled filaments by thermal fluctuations

We study the buckling instability of filaments or elastic rods in two spatial dimensions in the presence of thermal fluctuations. We present an analytical solution based on a renormalization-like procedure where we integrate out short wavelength fluctuations in order to obtain an effective theory governing the buckling instability. We calculate the resulting shift of the critical force by fluctuation effects and the average projected filament length parallel to the force direction as a function of the applied force and of the contour length of the filament. We find that, in the buckled state, thermal fluctuations lead to an increase in the mean projected length of the filament in the force direction. As a function of the contour length, the mean projected length exhibits a cusp at the buckling instability, which becomes rounded by thermal fluctuations. Our analytic results are confirmed by Monte Carlo simulations.

Dielectric relaxation of DNA aqueous solutions

We report on a detailed characterization of complex dielectric response of Na-DNA aqueous solutions by means of low-frequency dielectric spectroscopy (40 Hz-110 MHz). Results reveal two broad relaxation modes of strength 20<Deltaepsilon(LF)<100 and 5<Deltaepsilon(HF)<20, centered at 0.5 kHz<nu(LF)<70 kHz and 0.1 MHz<nu(HF)<15 MHz. The characteristic length scale of the low-frequency (LF) process, 50<L(LF)<750 nm, scales with DNA concentration as c(DNA)(-0.29+/-0.04) and is independent of the ionic strength in the low added salt regime. Conversely, the measured length scale of the LF process does not vary with DNA concentration but depends on the ionic strength of the added salt as I(s)(-1) in the high added salt regime. On the other hand, the characteristic length scale of the high-frequency (HF) process, 3<L(HF)<50 nm, varies with DNA concentration as c(DNA)(-0.5) for intermediate and large DNA concentrations. At low DNA concentrations and in the low added salt limit the characteristic length scale of the HF process scales as c(DNA)(-0.33). We put these results in perspective regarding the integrity of the double stranded form of DNA at low salt conditions as well as regarding the role of different types of counterions in different regimes of dielectric dispersion. We argue that the free DNA counterions are primarily active in the HF relaxation, while the condensed counterions play a role only in the LF relaxation. We also suggest theoretical interpretations for all these length scales in the whole regime of DNA and salt concentrations and discuss their ramifications and limitations.

Conformational transformations induced by the charge-curvature interaction: Mean-field approach

A simple phenomenological model for describing the conformational dynamics of biological macromolecules via the nonlinearity-induced instabilities is proposed. It is shown that the interaction between charges and bending degrees of freedom of closed molecular aggregates may act as drivers giving impetus to conformational dynamics of biopolymers. It is demonstrated that initially circular aggregates may undergo transformation to polygonal shapes and possible application to aggregates of bacteriochlorophyl a molecules is considered.

Condensation of DNA-actin polyelectrolyte mixtures driven by ions of different valences

Multivalent ions can induce condensation of like-charged polyelectrolytes into compact states, a process that requires different ion valences for different polyelectrolyte species. In this work we examine the condensation behavior in binary anionic polyelectrolyte mixtures consisting of DNA coils and F-actin rods in the presence of monovalent, divalent, and trivalent ions. As expected, monovalent ions do not condense either component and divalent ions selectively condense F-actin rods out of the polyelectrolyte mixture. For trivalent ions, however, we observe a microphase separation between the two polyelectrolytes into coexisting finite-sized F-actin bundles and DNA toroids. Further, by increasing the DNA volume fraction in the mixture, condensed F-actin bundles can be completely destabilized, leading to only DNA condensation within the mixture. We examine a number of possible causes and propose a model based on polyelectrolyte competition for ions.

Direct observation of counterion organization in F-actin polyelectrolyte bundles

Attractions between like-charged polyelectrolytes have been observed in a variety of systems (W.M. Gelbart, R.F. Bruinsma, P.A. Pincus, V.A. Parsegian, Phys. Today 53, September issue, 38 (2000)). Recent biological examples include DNA, filamentous viruses, and F-actin. Theoretical investigations on idealized systems indicate that counterion correlations play a central role, but no experiments that specifically probe such correlations have been performed. Using synchrotron X-ray diffraction, we have directly observed the organization of multivalent ions on cytoskeletal filamentous actin (a well-defined biological polyelectrolyte) and found an unanticipated symmetry-breaking collective counterion mechanism for generating attractions. Surprisingly, the counterions do not form a lattice that simply follows actin's helical symmetry; rather, the counterions organize into "frozen" ripples parallel to the actin filaments and form structures reminiscent of charge density waves. Moreover, these 1D counterion charge density waves form a coupled mode with twist deformations of the oppositely charged actin filaments. This counterion organization is not sensitive to thermal fluctuations in temperature range accessible to protein-based polyelectrolyte systems. Moreover, the counterion density waves are "pinned" to the spatial periodicity of charges on the actin filament even if the global filament charge density is varied, indicating the importance of charge periodicity on the polyelectrolyte substrate.

Electrostatic contribution to the persistence length of a semiflexible dipolar chain

We investigate the electrostatic contribution to the persistence length of a semiflexible polymer chain whose segments interact via a screened Debye-Hückel dipolar interaction potential. We derive the expressions for the renormalized persistence length on the level of a 1/D-expansion method already successfully used in other contexts of polyelectrolye physics. We investigate different limiting forms of the renormalized persistence length of the dipolar chain and show that, in, general, it depends less strongly on the screening length than in the context of a monopolar chain. We show that for a dipolar chain the electrostatic persistence length in the same regime of the parameter phase space as the original Odijk-Skolnick-Fixman (OSF) form for a monopolar chain depends logarithmically on the screening length rather than quadratically. This can be understood solely on the basis of a swifter decay of the dipolar interactions with separation compared to the monopolar electrostatic interactions. We comment also on the general contribution of higher multipoles to the electrostatic renormalization of the bending rigidity.

Persistence length of a strongly charged rodlike polyelectrolyte in the presence of salt

The persistence length of a single, intrinsically rigid polyelectrolyte chain, above the Manning condensation threshold is investigated theoretically in the presence of added salt. Using a loop expansion method, the partition function is consistently calculated, taking into account corrections to mean-field theory. Within a mean-field approximation, the well-known results of Odijk, Skolnick, and Fixman are reproduced. Beyond mean field, it is found that density correlations between counterions and thermal fluctuations reduce the stiffness of the chain, indicating an effective attraction between monomers for highly charged chains and multivalent counterions. This attraction results in a possible mechanical instability (collapse), alluding to the phenomenon of DNA condensation. In addition, we find that more counterions condense on slightly bent conformations of the chain than predicted by the Manning model for the case of an infinite cylinder. Finally, our results are compared with previous models and experiments.

Conformational instability of rodlike polyelectrolytes due to counterion fluctuations

The effective elasticity of highly charged stiff polyelectrolytes is studied in the presence of counterions, with and without added salt. The rigid polymer conformations may become unstable due to an effective attraction induced by counterion density fluctuations. Instabilities at the longest, or intermediate length scales, may signal collapse to globule, or necklace states, respectively. In the presence of added salt, a generalized electrostatic persistence length is obtained, which has a nontrivial dependence on the Debye screening length. It is also found that the onset of conformational instability is a reentrant phenomenon as a function of polyelectrolyte length for the unscreened case, and the Debye length or salt concentration for the screened case. This may be relevant in understanding the experimentally observed reentrant condensation of DNA.

Free expansion of elastic filaments

The dynamics of an elastic polymer filament undergoing contour length expansion is studied using computer simulation. The expansion occurs by development of transverse buckling waves that grow through a coarsening process. The growing buckles locally organize into a helical structure with a characteristic persistence length. The helical domain boundaries are eliminated from the relaxing structure by unwinding through the ends of the rod. The growth of the helical domains results in self-propulsive motion of the expanding rod, as one large helix spanning the entire chain relaxes during the late stages of the dynamics. Stability analyses and scaling arguments are provided to explain the simulation results.

Buckling and nonlocal elasticity of charged membranes

The elastic behavior of an interacting, and, in specific, of a charged flexible membrane is considered. In the first part of this paper the effective nonlocal elastic energy of a membrane due to a pairwise and arbitrary intra-membrane interaction is derived. Nonlocal elasticity is included to all orders, this description, therefore, corresponds to an infinite resummation of the standard gradient expansion. In the second part, the pair interaction between segments of an (on average) neutral membrane consisting of mobile positive and negative charges is derived both field theoretically in the Gaussian approximation and using a simple ion-pairing approximation. This model might also apply to strongly charged membranes with strongly condensed counter ions. The resulting contribution to the elastic energy is negative and thus favors undulations of the membrane. The bending modulus is extracted from the large-scale or small-momentum behavior of the elastic kernel and found to be comparable to k(B)T for the case where ion pairing is dominant. The large-momentum elastic response exhibits a markedly different scaling than the small-momentum regime and sensitively depends on the small-distance cutoff and thus on molecular details.

Elastically driven linker aggregation between two semiflexible polyelectrolytes

The behavior of mobile linkers connecting two semiflexible charged polymers, such as polyvalent counterions connecting DNA or F-actin chains, is studied theoretically. The chain bending rigidity induces an effective repulsion between linkers at large distances while the interchain electrostatic repulsion leads to an effective short-range interlinker attraction. We find a rounded phase transition from a dilute linker gas where the chains form large loops between linkers to a dense disordered linker fluid connecting parallel chains. The onset of chain pairing occurs within the rounded transition.

Binding of molecules to DNA and other semiflexible polymers

A theory is presented for the binding of small molecules such as surfactants to semiflexible polymers. The persistence length is assumed to be large compared to the monomer size but much smaller than the total chain length. Such polymers (e.g., DNA) represent an intermediate case between flexible polymers and stiff, rodlike ones, whose association with small molecules was previously studied. The chains are not flexible enough to actively participate in the self-assembly, yet their fluctuations induce long-range attractive interactions between bound molecules. In cases where the binding significantly affects the local chain stiffness, those interactions lead to a very sharp, cooperative association. This scenario is of relevance to the association of DNA with surfactants and compact proteins such as RecA. External tension exerted on the chain is found to significantly modify the binding by suppressing the fluctuation-induced interaction.