Propagation of tension along a polymer chain

Effect of local active fluctuations on structure and dynamics of flexible biopolymers

Active fluctuations play a significant role in the structure and dynamics of biopolymers (e.g. chromatin and cytoskeletal proteins) that are instrumental in the functioning of living cells. For a large range of experimentally accessible length and time scales, these polymers can be represented as flexible chains that are subjected to spatially and temporally varying fluctuating forces. In this work, we introduce a mathematical framework that correlates the spatial and temporal patterns of the fluctuations to different observables that describe the dynamics and conformations of the polymer. We demonstrate the power of this approach by analyzing the case of a point fluctuation on the polymer with an exponential decay of correlation in time with a finite time constant. Specifically, we identify the length and time scale over which the behavior of the polymer exhibits a significant departure from the behavior of a Rouse chain and the range of impact of the fluctuation along the chain. Furthermore, we show that the conformation of the polymer retains the memory of the active fluctuation from earlier times. Altogether, this work sets the basis for understanding and interpreting the role of spatio-temporal patterns of fluctuations in the dynamics, conformation, and functionality of biopolymers in living cells.

Nonequilibrium Thermodynamics of DNA Nanopore Unzipping

Using theory and simulations, we carried out a first systematic characterization of DNA unzipping via nanopore translocation. Starting from partially unzipped states, we found three dynamical regimes depending on the applied force f: (i) heterogeneous DNA retraction and rezipping (f<17  pN), (ii) normal (17  pN<f<60  pN), and (iii) anomalous (f>60  pN) drift-diffusive behavior. We show that the normal drift-diffusion regime can be effectively modeled as a one-dimensional stochastic process in a tilted periodic potential. We use the theory of stochastic processes to recover the potential from nonequilibrium unzipping trajectories and show that it corresponds to the free-energy landscape for single-base-pair unzipping. Applying this general approach to other single-molecule systems with periodic potentials ought to yield detailed free-energy landscapes from out-of-equilibrium trajectories.

Preaveraging description of polymer nonequilibrium stretching

This article focuses on a preaveraging description of polymer nonequilibrium stretching, where a single polymer undergoes a transient process from equilibrium to nonequilibrium steady state by pulling one chain end. The preaveraging method combined with mode analysis reduces the original Langevin equation to a simplified form for both a stretched steady state and an equilibrium state, even in the presence of self-avoiding repulsive interactions spanning a long range. However, the transient stretching process exhibits evolution of a hierarchal regime structure, which means a qualitative temporal change in probabilistic distributions assumed in preaveraging. We investigate the preaveraging method for evolution of the regime structure with consideration of the nonequilibrium work relations and deviations from the fluctuation-dissipation relation.

Unfolding and Translocation of Knotted Proteins by Clp Biological Nanomachines: Synergistic Contribution of Primary Sequence and Topology Revealed by Molecular Dynamics Simulations

We use Langevin dynamics simulations to model, at an atomistic resolution, how various natively knotted proteins are unfolded in repeated allosteric translocating cycles of the ClpY ATPase. We consider proteins representative of different topologies, from the simplest knot (trefoil 3₁), to the three-twist 5₂ knot, to the most complex stevedore, 6₁, knot. We harness the atomistic detail of the simulations to address aspects that have so far remained largely unexplored, such as sequence-dependent effects on the ruggedness of the landscape traversed during knot sliding. Our simulations reveal the combined effect on translocation of the knotted protein structure, i.e., backbone topology and geometry, and primary sequence, i.e., side chain size and interactions, and show that the latter can dominate translocation hindrance. In addition, we observe that due to the interplay between the knotted topology and intramolecular contacts the transmission of tension along the polypeptide chain occurs very differently from that of homopolymers. Finally, by considering native and non-native interactions, we examine how the disruption or formation of such contacts can affect the translocation processivity and concomitantly create multiple unfolding pathways with very different activation barriers.

RNA Pore Translocation with Static and Periodic Forces: Effect of Secondary and Tertiary Elements on Process Activation and Duration

We use MD simulations to study the pore translocation properties of a pseudoknotted viral RNA. We consider the 71-nucleotide-long xrRNA from the Zika virus and establish how it responds when driven through a narrow pore by static or periodic forces applied to either of the two termini. Unlike the case of fluctuating homopolymers, the onset of translocation is significantly delayed with respect to the application of static driving forces. Because of the peculiar xrRNA architecture, activation times can differ by orders of magnitude at the two ends. Instead, translocation duration is much smaller than activation times and occurs on time scales comparable at the two ends. Periodic forces amplify significantly the differences at the two ends, for both activation times and translocation duration. Finally, we use a waiting-times analysis to examine the systematic slowing downs in xrRNA translocations and associate them to the hindrance of specific secondary and tertiary elements of xrRNA. The findings provide a useful reference to interpret and design future theoretical and experimental studies of RNA translocation.

How a local active force modifies the structural properties of polymers

We study the dynamics of a polymer, described as a variant of a Rouse chain, driven by an active terminal monomer (head). The local active force induces a transition from a globule-like to an elongated state, as revealed by the study of the end-to-end distance, the variance of which is analytically predicted under suitable approximations. The change in the relaxation times of the Rouse-modes produced by the local self-propulsion is consistent with the transition from globule to elongated conformations. Moreover, also the bond-bond spatial correlation for the chain head are affected by the self-propulsion and a gradient of over-stretched bonds along the chain is observed. We compare our numerical results both with the phenomenological stiff-polymer theory and several analytical predictions in the Rouse-chain approximation.

Translocation of Charged Polymers through a Nanopore in Monovalent and Divalent Salt Solutions: A Scaling Study Exploring over the Entire Driving Force Regimes

Langevin dynamics simulations are performed to study polyelectrolytes driven through a nanopore in monovalent and divalent salt solutions. The driving electric field E is applied inside the pore, and the strength is varied to cover the four characteristic force regimes depicted by a rederived scaling theory, namely the unbiased (UB) regime, the weakly-driven (WD) regime, the strongly-driven trumpet (SD(T)) regime and the strongly-driven isoflux (SD(I)) regime. By changing the chain length N, the mean translocation time is studied under the scaling form 〈 τ 〉 ∼ N α E - δ . The exponents α and δ are calculated in each force regime for the two studied salt cases. Both of them are found to vary with E and N and, hence, are not universal in the parameter's space. We further investigate the diffusion behavior of translocation. The subdiffusion exponent γ p is extracted. The three essential exponents ν s , q, z p are then obtained from the simulations. Together with γ p , the validness of the scaling theory is verified. Through a comparison with experiments, the location of a usual experimental condition on the scaling plot is pinpointed.

Reducing the variance in the translocation times by prestretching the polymer

Langevin dynamics simulations of polymer translocation are performed where the polymer is stretched via two opposing forces applied on the first and last monomer before and during translocation. In this setup, polymer translocation is achieved by imposing a bias between the two pulling forces such that there is net displacement towards the trans side. Under the influence of stretching forces, the elongated polymer ensemble contains less variations in conformations compared to an unstretched ensemble. Simulations demonstrate that this reduced spread in initial conformations yields a reduced variation in translocation times relative to the mean translocation time. This effect is explored for different ratios of the amplitude of thermal fluctuations to driving forces to control for the relative influence of the thermal path sampled by the polymer. Since the variance in translocation times is due to contributions coming from sampling both thermal noise and initial conformations, our simulations offer independent control over the two main sources of noise and allow us to shed light on how they both contribute to translocation dynamics. Simulation parameter space corresponding to experimentally relevant conditions is highlighted and shown to correspond to a significant decrease in the spread of translocation times, thus indicating that stretching DNA prior to translocation could assist nanopore-based sequencing and sizing applications.

Role of non-equilibrium conformations on driven polymer translocation

One of the major theoretical methods in understanding polymer translocation through a nanopore is the Fokker-Planck formalism based on the assumption of quasi-equilibrium of polymer conformations. The criterion for applicability of the quasi-equilibrium approximation for polymer translocation is that the average translocation time per Kuhn segment, ⟨τ⟩/NK, is longer than the relaxation time τ0 of the polymer. Toward an understanding of conditions that would satisfy this criterion, we have performed coarse-grained three dimensional Langevin dynamics and multi-particle collision dynamics simulations. We have studied the role of initial conformations of a polyelectrolyte chain (which were artificially generated with a flow field) on the kinetics of its translocation across a nanopore under the action of an externally applied transmembrane voltage V (in the absence of the initial flow field). Stretched (out-of-equilibrium) polyelectrolyte chain conformations are deliberately and systematically generated and used as initial conformations in translocation simulations. Independent simulations are performed to study the relaxation behavior of these stretched chains, and a comparison is made between the relaxation time scale and the mean translocation time (⟨τ⟩). For such artificially stretched initial states, ⟨τ⟩/NK < τ0, demonstrating the inapplicability of the quasi-equilibrium approximation. Nevertheless, we observe a scaling of ⟨τ⟩ ∼ 1/V over the entire range of chain stretching studied, in agreement with the predictions of the Fokker-Planck model. On the other hand, for realistic situations where the initial artificially imposed flow field is absent, a comparison of experimental data reported in the literature with the theory of polyelectrolyte dynamics reveals that the Zimm relaxation time (τZimm) is shorter than the mean translocation time for several polymers including single stranded DNA (ssDNA), double stranded DNA (dsDNA), and synthetic polymers. Even when these data are rescaled assuming a constant effective velocity of translocation, it is found that for flexible (ssDNA and synthetic) polymers with NK Kuhn segments, the condition ⟨τ⟩/NK < τZimm is satisfied. We predict that for flexible polymers such as ssDNA, a crossover from quasi-equilibrium to non-equilibrium behavior would occur at NK ∼ O(1000).

Non-Markovian dynamics of reaction coordinate in polymer folding

We develop a theoretical description of the critical zipping dynamics of a self-folding polymer. We use tension propagation theory and the formalism of the generalized Langevin equation applied to a polymer that contains two complementary parts which can bind to each other. At the critical temperature, the (un)zipping is unbiased and the two strands open and close as a zipper. The number of broken base pairs n(t) displays a subdiffusive motion characterized by a variance growing as 〈Δn²(t)〉 ∼ t^α with α < 1 at long times. Our theory provides an estimate of both the asymptotic anomalous exponent α and of the subleading correction term, which are both in excellent agreement with numerical simulations. The results indicate that the tension propagation theory captures the relevant features of the dynamics and shed some new insights on related polymer problems characterized by anomalous dynamical behavior.

Active diffusion of model chromosomal loci driven by athermal noise

Active diffusion, i.e., fluctuating dynamics driven by athermal noise, is found in various out-of-equilibrium systems. Here we discuss the nature of the active diffusion of tagged monomers in a flexible polymer. A scaling argument based on the notion of tension propagation clarifies how the polymeric effect is reflected in the anomalous diffusion exponent, which may be of relevance to the dynamics of chromosomal loci in living cells.

Driven anomalous diffusion: An example from polymer stretching

The way tension propagates along a chain is a key to govern many anomalous dynamics in macromolecular systems. After introducing the weak and the strong force regimes of the tension propagation, we focus on the latter, in which the dynamical fluctuations of a segment in a long polymer during its stretching process is investigated. We show that the response, i.e., average drift, is anomalous, which is characterized by the nonlinear memory kernel, and its relation to the fluctuation is nontrivial. These features are discussed on the basis of the generalized Langevin equation, in which the role of the temporal change in spring constant due to the stress hardening is pinpointed. We carried out the molecular dynamics simulation, which supports our theory.

Mechanical systems biology of C. elegans touch sensation

The sense of touch informs us of the physical properties of our surroundings and is a critical aspect of communication. Before touches are perceived, mechanical signals are transmitted quickly and reliably from the skin's surface to mechano-electrical transduction channels embedded within specialized sensory neurons. We are just beginning to understand how soft tissues participate in force transmission and how they are deformed. Here, we review empirical and theoretical studies of single molecules and molecular ensembles thought to be involved in mechanotransmission and apply the concepts emerging from this work to the sense of touch. We focus on the nematode Caenorhabditis elegans as a well-studied model for touch sensation in which mechanics can be studied on the molecular, cellular, and systems level. Finally, we conclude that force transmission is an emergent property of macromolecular cellular structures that mutually stabilize one another.

Dynamic compression of single nanochannel confined DNA via a nanodozer assay

We show that a single DNA molecule confined and extended in a nanochannel can be dynamically compressed by sliding a permeable gasket at a fixed velocity relative to the stationary polymer. The gasket is realized experimentally by optically trapping a nanosphere inside a nanochannel. The trapped bead acts like a "nanodozer," directly applying compressive forces to the molecule without requirement of chemical attachment. Remarkably, these strongly nonequilibrium measurements can be quantified via a simple nonlinear convective-diffusion formalism and yield insights into the local blob statistics, allowing us to conclude that the compressed nanochannel-confined chain exhibits mean-field behavior.

Transient behaviour of a polymer dragged through a viscoelastic medium

We study the dynamics of a polymer that is pulled by a constant force through a viscoelastic medium. This is a model for a polymer being pulled through a cell by an external force, or for an active biopolymer moving due to a self-generated force. Using the Rouse model with a memory dependent drag force, we find that the center of mass of the polymer follows a subballistic motion. We determine the time evolution of the length and the shape of the polymer. Through an analysis of the velocity of the monomers, we investigate how the tension propagates through the polymer. We discuss how polymers can be used to probe the properties of a viscoelastic medium.

Anomalous dynamics of DNA hairpin folding

By means of computer simulations of a coarse-grained DNA model we show that the DNA hairpin zippering dynamics is anomalous; i.e., the characteristic time τ scales nonlinearly with N, the hairpin length, τ ∼ N(α) with α>1. This is in sharp contrast to the prediction of the zipper model for which τ ∼ N. We show that the anomalous dynamics originates from an increase in the friction during zippering due to the tension built in the closing strands. From a simple polymer model we get α = 1+ν ≈ 1.59 with ν being the Flory exponent, a result which is in agreement with the simulations. We discuss transition path times data where such effects should be detected.

Relationships between the winding angle, the characteristic radius, and the torque for a long polymer chain wound around a cylinder: implications for RNA winding around DNA during transcription

Long polymer chains are ubiquitous in biological systems and their mechanical properties have significant impact upon biological processes. Of particular interest is the situation in which polymer chains are wound around each other or around other objects. We have analyzed the parameters of a long Gaussian polymer chain wound around a cylinder as a function of the torque applied to the ends of the chain. We have shown that for sufficiently long polymer chains, an average winding angle and a characteristic radius of the chain can be determined from a modified Bessel function of purely imaginary order, in which the value of the order is equivalent to the applied torque, normalized to the product of the absolute temperature and the Boltzmann constant. The obtained results are consistent with a simplified interpretation in terms of "torsional blobs," and this could be extended to nonideal chains with excluded volumes. We have also extended our results to the case of a polymer chain rotating in viscous medium. Our results could be used to estimate the mechanical strains that appear in DNA and RNA during transcription, as these might initiate formation of unusual DNA structures, invasion of RNA into the DNA duplex (R-loop formation), and modulation of the interactions of DNA and RNA with proteins.

Cis-trans dynamical asymmetry in driven polymer translocation

During polymer translocation driven by, e.g., voltage drop across a nanopore, the segments in the cis side are incessantly pulled into the pore, which are then pushed out of it into the trans side. This pulling and pushing of polymer segments are described in the continuum level by nonlinear transport processes known, respectively, as fast and slow diffusions. By matching solutions of both sides through the mass conservation across the pore, we provide a physical basis for the cis and trans dynamical asymmetry, a feature repeatedly reported in recent numerical simulations. We then predict how the total driving force is dynamically allocated between cis (pulling) and trans (pushing) sides, demonstrating that the trans-side event adds a weak finite-chain length effect to the dynamical scaling.

Memory effect and fluctuating anomalous dynamics of a tagged monomer

We analyze the anomalous dynamics of a tagged monomer under external navigation. The memory effect causing the anomaly is elucidated, which depends on the magnitude of the force. In particular, the nonlinear and nonequilibrium memory effect under strong force is characterized by the force-dependent self-affine process for the tension transmission along the connectivity. Utilizing such knowledge, a generalized Langevin equation approach is proposed to quantify the fluctuating dynamics of driven anomalous walkers.