Cyclization of Rouse chains at long- and short-time scales

Conformational Dynamics of Poly(T) Single-Stranded DNA at the Single-Molecule Level

The conformational dynamics of single-stranded DNA (ss-DNA) are implicated in the mechanisms of several key biological processes such as DNA replication and damage repair and have been modeled with those of semiflexible or flexible polymer. The high flexibility and customizability of ss-DNA also make it an excellent polymeric material for materials engineering. Polythymidine (poly(T)) is an excellent model ss-DNA as a flexible polymer since it does not form any secondary structure. However, only limited experimental results have been reported of poly(T) conformational dynamics with a very short length that is not relevant to the aforementioned processes and applications. Here, we provide the first experimental results of the conformational dynamics of poly(T) with lengths in the range of 130-170 nucleotides at the single-molecule level. Our experiments are based on single-molecule FRET and a DNA hairpin structure of which the folding kinetics are governed by the conformational dynamics of poly(T). We found that the folding kinetics deviate far from those of a flexible polymer model with a harmonic bending potential. To this end, we derived a simple model for the kinetics of DNA hairpin folding from the self-avoiding-walk (SAW). Our model describes the conformational dynamics of poly(T) very well and enables estimation of the conformational dimensionality. The estimated dimensionalities suggest that ss-DNA is completely flexible at 100 mM or a higher NaCl concentration, but not at 50 mM. These results will help understand the conformational dynamics of ss-DNA implicated in several key biological processes and maximize the utility of ss-DNA for materials engineering. Also, our system and model provide an excellent platform to investigate the conformational dynamics of ss-DNA.

Kinetics of loop formation in worm-like chain polymers

A common theoretical approach to calculating reaction kinetics is to approximate a high-dimensional conformational search with a one-dimensional diffusion along an effective reaction coordinate. We employed Brownian dynamics simulations to test the validity of this approximation for loop formation kinetics in the worm-like chain polymer model. This model is often used to describe polymers that exhibit backbone stiffness beyond the monomer length scale. We find that one-dimensional diffusion models overestimate the looping time and do not predict the quantitatively correct dependence of looping time on chain length or capture radius. Our findings highlight the difficulty of describing high-dimensional polymers with simple kinetic theories.

Two-point approximation to the Kramers problem with coloured noise

We present a method, founded on previous renewal approaches as the classical Wilemski-Fixman approximation, to describe the escape dynamics from a potential well of a particle subject to non-Markovian fluctuations. In particular, we show how to provide an approximated expression for the distribution of escape times if the system is governed by a generalized Langevin equation (GLE). While we show that the method could apply to any friction kernel in the GLE, we focus here on the case of power-law kernels, for which extensive literature has appeared in the last years. The method presented (termed as two-point approximation) is able to fit the distribution of escape times adequately for low potential barriers, even if conditions are far from Markovian. In addition, it confirms that non-exponential decays arise when a power-law friction kernel is considered (in agreement with related works published recently), which questions the existence of a characteristic reaction rate in such situations.

End-to-surface reaction dynamics of a single surface-attached DNA or polypeptide

The dynamics of surface-attached polymers play a key role in the operation of a number of biological sensors, yet its current understanding is rather limited. Here we use computer simulations to study the dynamics of a reaction between the free end of a polymer chain and a surface, to which its other end has been attached. We consider two limiting cases, the diffusion-controlled limit, where the reaction is accomplished whenever the free chain end diffuses to within a specified distance from the surface, and the reaction-controlled limit, where slow, intrinsic reaction kinetics rather than diffusion of the chain is rate limiting. In the diffusion-controlled limit, we find that the overall rate scales as N(-b), where N is the number of monomers in the chain and b approximately = 2.2 for excluded volume chains. This value of the scaling exponent b is close to that derived from a simple approximate theory treating the dynamics of the chain end relative to the surface as one-dimensional diffusion in an effective potential. In the reaction-controlled limit, the value of the scaling exponent b is close to 1. We compare our findings with those for the related (and better studied) problem of end-to-end reactions within an unconstrained polymer chain and discuss their implications for electrochemical DNA sensors.

The rate of intramolecular loop formation in DNA and polypeptides: the absence of the diffusion-controlled limit and fractional power-law viscosity dependence

The problem of determining the rate of end-to-end collisions for polymer chains has attracted the attention of theorists and experimentalists for more than three decades. The typical theoretical approach to this problem has focused on the case where a collision is defined as any instantaneous fluctuation that brings the chain ends to within a specific capture distance. In this paper, we study the more experimentally relevant case, where the end-to-end collision dynamics are probed by measuring the excited state lifetime of a fluorophore (or other lumiphore) attached to one chain end and quenched by a quencher group attached to the other end. Under this regime, a "contact" is defined not by the chain ends approach to within some sharp cutoff but, instead, typically by an exponentially distance-dependent process. Previous theoretical models predict that, if quenching is sufficiently rapid, a diffusion-controlled limit is attained, where such measurements report on the probe-independent, intrinsic end-to-end collision rate. In contrast, our theoretical considerations, simulations, and an analysis of experimental measurements of loop closure rates in single-stranded DNA molecules all indicate that no such limit exists, and that the measured effective collision rate has a nontrivial, fractional power-law dependence on both the intrinsic quenching rate of the fluorophore and the solvent viscosity. We propose a simple scaling formula describing the effective loop closure rate and its dependence on the viscosity, chain length, and properties of the probes. Previous theoretical results are limiting cases of this more general formula.

Excluded volume effects on the intrachain reaction kinetics

On the basis of the recently developed optimized Rouse-Zimm theory of chain polymers with excluded volume interactions, we calculate the long-time first-order rate constant k(1) for end-to-end cyclization of linear chain polymers. We first find that the optimized Rouse-Zimm theory provides the longest chain relaxation times tau(1) of excluded volume chains that are in excellent agreement with the available Brownian dynamics simulation results. In the free-draining limit, the cyclization rate is diffusion-controlled and k(1) is inversely proportional to tau(1), and the k(1) values calculated using the Wilemski-Fixman rate theory are in good agreement with Brownian dynamics simulation results. However, when hydrodynamic interactions are included, noticeable deviations are found. The main sources of errors are fluctuating hydrodynamic interaction and correlation hole effects as well as the non-Markovian reaction dynamic effect. The physical natures of these factors are discussed, and estimates for the magnitudes of required corrections are given. When the corrections are included, the present theory allows the prediction of accurate k(1) values for the cyclization of finite-length chains in good solvents as well as the correct scaling exponent in the long-chain limit.

Renormalization group analysis of polymer cyclization with non-equilibrium initial conditions

We develop a renormalization group approach for cyclizing polymers for the case when chain ends are initially close together (ring initial conditions). We analyze the behavior at times much shorter than the longest polymer relaxation time. In agreement with our previous work (Europhys. Lett. 73, 621 (2006)) we find that the leading time dependence of the reaction rate k(t) for ring initial conditions and equilibrium initial conditions are related, namely k (ring)(t) proportional, variant t (-delta) and k (eq)(t) proportional, variant t (1-delta) for times less than the longest polymer relaxation time. Here delta is an effective exponent which approaches delta = 5/4 for very long Rouse chains. Our present analysis also suggests a "sub-leading" term proportional to (ln t)/t which should be particularly significant for smaller values of the renormalized reaction rate and early times. For Zimm dynamics, our RG analysis indicates that the leading time dependence for the reaction rate is k(t) approximately 1/t for very long chains. The leading term is again consistent with the expected relation between ring and equilibrium initial conditions. We also find a logarithmic correction term which we "exponentiate" to a logarithmic form with a Landau pole. The presence of the logarithm is particularly important for smaller chains and, in the Zimm case, large values of the reaction rate.