The thermodynamics of reversible cyclization in semiflexible polymers

Internal friction as a factor in the anomalous chain length dependence of DNA transcriptional dynamics

Recent experiments by Brückner et al. [Science 380, 1357 (2023)] have observed an anomalous chain length dependence of the time of near approach of widely separated pairs of genomic elements on transcriptionally active chromosomal DNA. In this paper, I suggest that the anomaly may have its roots in internal friction between neighboring segments on the DNA backbone. The basis for this proposal is a model of chain dynamics formulated in terms of a continuum scaled Brownian walk (sBw) of polymerization index N. The sBw is an extension of the simple Brownian walk model widely used in path integral calculations of polymer properties, differing from it in containing an additional parameter H (the Hurst index) that can be tuned to produce varying degrees of correlation between adjacent monomers. A calculation using the sBw of the mean time τc for chain closure predicts-under the Wilemski-Fixman approximation for diffusion-controlled reactions-that at early times, τc varies as the 2/3 power of N, in close agreement with the findings of the Brückner et al. study. Other scaling relations of that study, including those related to the probability of loop formation and the mean square displacements of terminal monomers, are also satisfactorily accounted for by the model.

Loop formation and translational diffusion of intrinsically disordered proteins

The conformational flexibility and dynamics of unfolded peptide chains is of major interest in the context of protein folding and protein functioning. The rate with which amino acids at different positions along the peptide chain meet sets an upper speed limit for protein folding. By using single-molecule photo-induced energy transfer spectroscopy, we have systematically measured end-to-end and end-to-internal site contact formation rates for several intrinsically disordered protein fragments (IDPs) (11 to 41 amino acids) and have also determined their hydrodynamic radius using dual-focus fluorescence correlation spectroscopy. For interpreting the measured values, we have developed a Brownian dynamics model (based on bead-rod chain dynamics in a thermal bath including hydrodynamic interactions) which quantitatively reproduces all measured data surprisingly well while requiring only two fit parameters. The model provides a complete picture of the peptides' dynamics and allows us to translate the experimental rates and radii into molecular properties of the peptides: We find a persistence length of l_{P}=5.2±1.9Å, a hydrodynamic radius of a=3.5±0.7Å per amino acid, and that excluded volume effects play an important role in the dynamics of IDPs.

The role of solvent quality and chain stiffness on the end-to-end contact kinetics of semiflexible polymers

Motivated by loop closure during protein folding and DNA packing, we systemically studied the effects of the solvent quality and chain stiffness on the thermodynamics and kinetics of the end-to-end contact formation for semiflexible polymer chains with reactive ends by Langevin dynamics simulations. In thermodynamics, a rich variety of products of the end-to-end contact have been discovered, such as loop, hairpin, toroid, and rodlike bundle, the populations of which are dependent on the solvent quality and chain stiffness. In kinetics, the overall pathways to form the end-to-end contact have been identified. The change of solvent quality and chain stiffness can tune the roughness of energy landscape and modulate the kinetic partitioning of the end-to-end contact formation pathways, leading to differing kinetic behaviors. In good or poor solvents, the first end-to-end contact rate k _c decreases with increasing the strength of bending stiffness k _θ monotonically. In very poor solvents, however, the dependence of the logarithm of the first end-to-end contact rate ln k _c on k _θ exhibits erratic behavior, which stems from more rugged energy landscape due to the polymer chain getting trapped into the intermediate state composed of the rodlike bundles with two ends in separation. For semiflexible chains, with increasing chain length N, the rate k _c increases initially and then decreases: in good solvents, the rate k _c exhibits a power-law relationship to chain length N with an exponent of ∼-1.50 in the region of long chains, which is in good agreement with the value derived from the experiment in the asymptotic limit of large N; and in poor solvents, the rate k _c exhibits a significantly stronger chain length dependence than those observed in good solvents in the region of long chains due to frustration to form the end-to-end contact along a specific path, especially the scaling exponent between the rate k _c and chain length N is ∼-3.62 for the case of polymer chains with k _θ = 4 at the solvent quality ε _ij = 1, in accord with the value obtained from the experiments.

Cyclization kinetics of Gaussian semiflexible polymer chains

We consider the dynamics and the cyclization kinetics of Gaussian semiflexible chains, in which the interaction potential tends to align successive bonds. We provide asymptotic expressions for the cyclization time, for the eigenvalues and eigenfunctions, and for the mean square displacement at all time and length scales, with explicit dependence on the capture radius, on the positions of the reactive monomers in the chain, and on the finite number of beads. For the cyclization kinetics, we take into account non-Markovian effects by calculating the distribution of reactive conformations of the polymer, which are not taken into account in the classical Wilemski-Fixman theory. Comparison with numerical simulations confirms the accuracy of this non-Markovian theory.

Dynamics of the reaction between the free end of a tethered self-avoiding polymer and a flat penetrable surface: a renormalization group study

The average time τr for one end of a long, self-avoiding polymer to interact for the first time with a flat penetrable surface to which it is attached at the other end is shown here to scale essentially as the square of the chain's contour length N. This result is obtained within the framework of the Wilemski-Fixman approximation to diffusion-limited reactions, in which the reaction time is expressed as a time correlation function of a "sink" term. In the present work, this sink-sink correlation function is calculated using perturbation expansions in the excluded volume and the polymer-surface interactions, with renormalization group methods being used to resum the expansion into a power law form. The quadratic dependence of τr on N mirrors the behavior of the average time τc of a free random walk to cyclize, but contrasts with the cyclization time of a free self-avoiding walk (SAW), for which τr ∼ N(2.2). A simulation study by Cheng and Makarov [J. Phys. Chem. B 114, 3321 (2010)] of the chain-end reaction time of an SAW on a flat impenetrable surface leads to the same N(2.2) behavior, which is surprising given the reduced conformational space a tethered polymer has to explore in order to react.

Non-Markovian polymer reaction kinetics

Describing the kinetics of polymer reactions, such as the formation of loops and hairpins in nucleic acids or polypeptides, is complicated by the structural dynamics of their chains. Although both intramolecular reactions, such as cyclization, and intermolecular reactions have been studied extensively, both experimentally and theoretically, there is to date no exact explicit analytical treatment of transport-limited polymer reaction kinetics, even in the case of the simplest (Rouse) model of monomers connected by linear springs. We introduce a new analytical approach to calculate the mean reaction time of polymer reactions that encompasses the non-Markovian dynamics of monomer motion. This requires that the conformational statistics of the polymer at the very instant of reaction be determined, which provides, as a by-product, new information on the reaction path. We show that the typical reactive conformation of the polymer is more extended than the equilibrium conformation, which leads to reaction times significantly shorter than predicted by the existing classical Markovian theory.

Dynamics of loop formation in a semiflexible polymer

The dynamics of loop formation by linear polymer chains has been a topic of several theoretical and experimental studies. Formation of loops and their opening are key processes in many important biological processes. Loop formation in flexible chains has been extensively studied by many groups. However, in the more realistic case of semiflexible polymers, not much results are available. In a recent study [K. P. Santo and K. L. Sebastian, Phys. Rev. E 73, 031923 (2006)], we investigated opening dynamics of semiflexible loops in the short chain limit and presented results for opening rates as a function of the length of the chain. We presented an approximate model for a semiflexible polymer in the rod limit based on a semiclassical expansion of the bending energy of the chain. The model provided an easy way to describe the dynamics. In this paper, using this model, we investigate the reverse process, i.e., the loop formation dynamics of a semiflexible polymer chain by describing the process as a diffusion-controlled reaction. We make use of the "closure approximation" of Wilemski and Fixman [G. Wilemski and M. Fixman, J. Chem. Phys. 60, 878 (1974)], in which a sink function is used to represent the reaction. We perform a detailed multidimensional analysis of the problem and calculate closing times for a semiflexible chain. We show that for short chains, the loop formation time tau decreases with the contour length of the polymer. But for longer chains, it increases with length obeying a power law and so it has a minimum at an intermediate length. In terms of dimensionless variables, the closing time is found to be given by tau approximately Ln exp(const/L), where n=4.5-6. The minimum loop formation time occurs at a length Lm of about 2.2-2.4. These are, indeed, the results that are physically expected, but a multidimensional analysis leading to these results does not seem to exist in the literature so far.

Loops in DNA: an overview of experimental and theoretical approaches

DNA loop formation plays a central role in many cellular processes. The aim of this paper is to present the state of the art and open problems regarding the experimental and theoretical approaches to DNA looping. A particular attention is devoted to the effects of the protein bridge size and of protein induced sharp DNA bending on DNA loop formation enhancement.

Opening of a weak link in a semiflexible ring polymer

The dynamics of contact formation between different parts of a long chain molecule is of considerable interest in biology. The related processes of opening of a loop or closing to form a loop also are of considerable interest and have attracted the attention of experimentalists/theorists. For closing, results are available in the completely flexible limit. However, this limit is not realized in many cases. Recently, there have been investigations for the semiflexible case too. We develop an approach, which leads to an easy description of the dynamics, incorporating semiflexibility rigorously into account. With this approach, the dynamics of a semiflexible polymer ring formed by a weak bond between the two ends can be modeled as the escape of a particle over a barrier in a multidimensional potential energy surface. We then calculate the rate of opening using a multidimensional transition state theory. Effects of friction on the rate are also taken into account using the standard coupling to a bath of harmonic oscillators. We find that for shorter chains (i.e., semiflexible), the rate of opening is strongly length dependent and is well described by the equation A(L/lp)v exp(Blp/L), with L as the length, and lp as the persistence length, A, B as the constants, and v approximately 1.2.

Multiple time scale dynamics of distance fluctuations in a semiflexible polymer: A one-dimensional generalized Langevin equation treatment

Time-dependent fluctuations in the distance x(t) between two segments along a polymer are one measure of its overall conformational dynamics. The dynamics of x(t), modeled as the coordinate of a particle moving in a one-dimensional potential well in thermal contact with a reservoir, is treated with a generalized Langevin equation whose memory kernel K(t) can be calculated from the time-correlation function of distance fluctuations C(t) identical with x(0)x(t). We compute C(t) for a semiflexible continuum model of the polymer and use it to determine K(t) via the GLE. The calculations demonstrate that C(t) is well approximated by a Mittag-Leffler function and K(t) by a power-law decay on time scales of several decades. Both functions depend on a number of parameters characterizing the polymer, including chain length, degree of stiffness, and the number of intervening residues between the two segments. The calculations are compared with the recent observation of a nonexponential C(t) and a power law K(t) in the conformational dynamics within single molecule proteins [Min et al., Phys. Rev. Lett. 94, 198302 (2005)].

Dynamics of chain closure: Approximate treatment of nonlocal interactions

The Wilemski-Fixman model of diffusion controlled-reactions [J. Chem. Phys. 58, 4009 (1973)] is combined with a generalized random walk description of chain conformations to predict the dependence of the closure time tau on the chain length N of polymers with reactive end groups and nonlocal interactions. The nonlocal interactions are modeled by a modification to the connectivity term in the Edwards continuum representation of the polymer. The modification involves a parameter h lying between 0 and 1 that is a measure of the extent of correlation between adjacent monomers on the chain backbone. Different choices of h correspond to chain conformations of different average radial dimensions. In particular, the values 1/3, 1/2 and 3/5 provide approximations to the statistics of polymers in poor, theta and good solvents, respectively. The closure time tau of such chains is calculated analytically for different N. In all cases, tau is found to vary as a power law in N, Nb, with b a function of h. For the special case h = 1/3, which models collapsed polymers and globular proteins, b is about 1.6-1.7.