Barrier crossing in one and three dimensions by a long chain
15 Nov 2010-Journal of Statistical Mechanics: Theory and Experiment (IOP Publishing)-Vol. 2010, Iss: 11, pp 11024
TL;DR: In this article, the authors considered the Kramers problem for a long chain polymer trapped in a biased double-well potential and showed that the time of crossing is proportional to the chain length, which is in agreement with the results for the kink mechanism.
Abstract: We consider the Kramers problem for a long chain polymer trapped in a biased double-well potential Initially the polymer is in the less stable well and it can escape from this well to the other well by the motion of its N beads across the barrier to attain the configuration having lower free energy In one dimension we simulate the crossing and show that the results are in agreement with the kink mechanism suggested earlier In three dimensions, it has not been possible to get an analytical 'kink solution' for an arbitrary potential; however, one can assume the form of the solution of the nonlinear equation as a kink solution and then find a double-well potential in three dimensions To verify the kink mechanism, simulations of the dynamics of a discrete Rouse polymer model in a double well in three dimensions are carried out We find that the time of crossing is proportional to the chain length, which is in agreement with the results for the kink mechanism The shape of the kink solution is also in agreement with the analytical solution in both one and three dimensions
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TL;DR: There is a range of polymer lengths in which the system is approximately translationally invariant, and a coarse-grained description of this regime is developed, and general features of the distribution of times for the polymer to pass through the pore may be deduced.
Abstract: Motivated by experiments in which a polynucleotide is driven through a proteinaceous pore by an electric field, we study the diffusive motion of a polymer threaded through a narrow channel with which it may have strong interactions. We show that there is a range of polymer lengths in which the system is approximately translationally invariant, and we develop a coarse-grained description of this regime. From this description, general features of the distribution of times for the polymer to pass through the pore may be deduced. We also introduce a more microscopic model. This model provides a physically reasonable scenario in which, as in experiments, the polymer's speed depends sensitively on its chemical composition, and even on its orientation in the channel. Finally, we point out that the experimental distribution of times for the polymer to pass through the pore is much broader than expected from simple estimates, and speculate on why this might be.
329 citations
TL;DR: This work applies transition state theory to out-of-equilibrium transport through confined environments: the thermally activated translocation of single DNA molecules over an entropic barrier helped by an external force field.
Abstract: Transition state theory (TST) provides a simple interpretation of many thermally activated processes. It applies successfully on timescales and length scales that differ several orders of magnitude: to chemical reactions, breaking of chemical bonds, unfolding of proteins and RNA structures and polymers crossing entropic barriers. Here we apply TST to out-of-equilibrium transport through confined environments: the thermally activated translocation of single DNA molecules over an entropic barrier helped by an external force field. Reaction pathways are effectively one dimensional and so long that they are observable in a microscope. Reaction rates are so slow that transitions are recorded on video. We find sharp transition states that are independent of the applied force, similar to chemical bond rupture, as well as transition states that change location on the reaction pathway with the strength of the applied force. The states of equilibrium and transition are separated by micrometres as compared with angstroms/nanometres for chemical bonds.
13 citations
TL;DR: TST with dynamical corrections based on short time trajectories started at the transition state gives rate constant estimates that agree within a factor of two with the molecular dynamics simulations over a wide range of bead coupling constants and polymer lengths.
Abstract: The rate of escape of an ideal bead-spring polymer in a symmetric double-well potential is calculated using transition state theory (TST) and the results compared with direct dynamical simulations. The minimum energy path of the transitions becomes flat and the dynamics diffusive for long polymers making the Kramers-Langer estimate poor. However, TST with dynamical corrections based on short time trajectories started at the transition state gives rate constant estimates that agree within a factor of two with the molecular dynamics simulations over a wide range of bead coupling constants and polymer lengths. The computational effort required by the TST approach does not depend on the escape rate and is much smaller than that required by molecular dynamics simulations.
12 citations
Journal Article•
TL;DR: In this paper, the authors investigate the dynamics of polymer translocation through a nanopore under an externally applied field using the two-dimensional fluctuating bond model with single-segment Monte Carlo moves.
Abstract: We investigate the dynamics of polymer translocation through a nanopore under an externally applied field using the two-dimensional fluctuating bond model with single-segment Monte Carlo moves. We concentrate on the influence of the field strength E, length of the chain N, and length of the pore L on forced translocation. As our main result, we find a crossover scaling for the translocation time tau with the chain length from tau approximately N2nu for relatively short polymers to tau approximately N1+nu for longer chains, where nu is the Flory exponent. We demonstrate that this crossover is due to the change in the dependence of the translocation velocity v on the chain length. For relatively short chains v approximately N-nu, which crosses over to v approximately N(-1) for long polymers. The reason for this is that with increasing N there is a high density of segments near the exit of the pore, which slows down the translocation process due to slow relaxation of the chain. For the case of a long nanopore for which R parallel, the radius of gyration Rg along the pore, is smaller than the pore length, we find no clear scaling of the translocation time with the chain length. For large N, however, the asymptotic scaling tau approximately N1+nu is recovered. In this regime, tau is almost independent of L. We have previously found that for a polymer, which is initially placed in the middle of the pore, there is a minimum in the escape time for R parallel approximately L. We show here that this minimum persists for weak fields E such that EL is less than some critical value, but vanishes for large values of EL.
9 citations
TL;DR: In this paper, an efficient method for evaluating the recrossing correction factor by constructing a sequence of hyperplanes starting at the transition state and calculating the probability that the system advances from one hyperplane to another towards the product is presented.
Abstract: The recrossing correction to the transition state theory estimate of a thermal rate can be difficult to calculate when the energy barrier is flat. This problem arises, for example, in polymer escape if the polymer is long enough to stretch between the initial and final state energy wells while the polymer beads undergo diffusive motion back and forth over the barrier. We present an efficient method for evaluating the correction factor by constructing a sequence of hyperplanes starting at the transition state and calculating the probability that the system advances from one hyperplane to another towards the product. This is analogous to what is done in forward flux sampling except that there the hyperplane sequence starts at the initial state. The method is applied to the escape of polymers with up to 64 beads from a potential well. For high temperature, the results are compared with direct Langevin dynamics simulations as well as forward flux sampling and excellent agreement between the three rate estimates is found. The use of a sequence of hyperplanes in the evaluation of the recrossing correction speeds up the calculation by an order of magnitude as compared with the traditional approach. As the temperature is lowered, the direct Langevin dynamics simulations as well as the forward flux simulations become computationally too demanding, while the harmonic transition state theory estimate corrected for recrossings can be calculated without significant increase in the computational effort.
7 citations
References
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TL;DR: In this paper, the authors consider a polymer of length N translocating through a narrow pore in the absence of external fields and show that the polymer dynamics is anomalous up to the Rouse time τR~N1+2ν, with a mean square displacement through the pore consistent with t(1+ν)/(1 + 2ν), with ν≈ 0.588 the Flory exponent.
Abstract: We consider a polymer of length N translocating through a narrow pore in the absence of external fields. The characterization of its purportedly anomalous dynamics has so far remained incomplete. We show that the polymer dynamics is anomalous up to the Rouse time τR~N1+2ν, with a mean square displacement through the pore consistent with t(1+ν)/(1+2ν), with ν≈0.588 the Flory exponent. This is shown to be directly related to a decay over time of the excess monomer density near the pore as t−(1+ν)/(1+2ν)exp(−t/τR). Beyond the Rouse time, translocation becomes diffusive. In consequence of this, the dwell time τd, the time a translocating polymer typically spends within the pore, scales as N2+ν, in contrast to previous claims.
110 citations
TL;DR: In this paper, three-dimensional dynamic Monte Carlo simulations of polymer translocation through a cylindrical hole in a planar slab under the influence of an external driving force are performed.
Abstract: Three-dimensional dynamic Monte Carlo simulations of polymer translocation through a cylindrical hole in a planar slab under the influence of an external driving force are performed. The driving force is intended to emulate the effect of a static electric field applied in an electrolytic solution containing charged monomer particles, as is relevant to the translocation of certain biopolymers through protein channel pores embedded in cell membranes. The time evolution of the probability distribution of the translocation coordinate (the number of monomers that have passed through the pore) is extracted from three-dimensional (3-D) simulations over a range of polymer chain lengths. These distributions are compared to the predictions of a 1-D Smoluchowski equation model of the translocation coordinate dynamics. Good agreement is found, with the effective diffusion constant for the 1-D Smoluchowski model being nearly independent of chain length.
110 citations
TL;DR: It is shown that for a polymer, which is initially placed in the middle of the pore, there is a minimum in the escape time for R parallel approximately L such that EL is less than some critical value, but vanishes for large values of EL.
Abstract: We investigate the dynamics of polymer translocation through a nanopore under an externally applied field using the two-dimensional fluctuating bond model with single-segment Monte Carlo moves. We concentrate on the influence of the field strength E, length of the chain N, and length of the pore L on forced translocation. As our main result, we find a crossover scaling for the translocation time τ with the chain length from τ∼N2ν for relatively short polymers to τ∼N1+ν for longer chains, where ν is the Flory exponent. We demonstrate that this crossover is due to the change in the dependence of the translocation velocity v on the chain length. For relatively short chains v∼N−ν, which crosses over to v∼N−1 for long polymers. The reason for this is that with increasing N there is a high density of segments near the exit of the pore, which slows down the translocation process due to slow relaxation of the chain. For the case of a long nanopore for which R‖, the radius of gyration Rg along the pore, is smaller...
106 citations
TL;DR: A comparative discussion of the underlying mechanisms of DNA transfer in mesophilic and extremely thermophilic bacteria is placed on, highlighting conserved and distinctive features of these transformation machineries.
Abstract: Horizontal gene flow is a driving force for bacterial adaptation. Among the three distinct mechanisms of gene transfer in bacteria, conjugation, transduction, and transformation, the latter, which includes competence induction, DNA binding, and DNA uptake, is perhaps the most versatile mechanism and allows the incorporation of free DNA from diverse bacterial species. Here we review DNA transport machineries mediating uptake of naked DNA in gram-positive and gram-negative bacteria. Different putative models of transformation machineries comprising components similar to proteins of type IV pili are presented. Emphasis is placed on a comparative discussion of the underlying mechanisms of DNA transfer in mesophilic and extremely thermophilic bacteria, highlighting conserved and distinctive features of these transformation machineries.
104 citations
TL;DR: The result suggests that the polymers partition into the lumen of the pore according to the simple scaling law of Daoud and de Gennes, cpore/csolution = exp(−N(a/D)5/3), which might be applied to determine the approximate dimensions of cavities within other similar proteins.
Abstract: The dependence of the rate on polymer mass was examined for the reaction of four sulfhydryl-directed poly(ethylene glycol) reagents with cysteine residues located in the lumen of the staphylococcal α-hemolysin pore. The logarithms of the apparent rate constants for a particular site in the lumen were proportional to N, the number of repeat units in a polymer chain. The proportionality constant was −(a/D)5/3, where a is the persistence length of the polymer (≈3.5Å) and D is the diameter of the pore. Despite some incongruencies with the assumptions of the derivation, the result suggests that the polymers partition into the lumen of the pore according to the simple scaling law of Daoud and de Gennes, cpore/csolution = exp(−N(a/D)5/3). Therefore, the measured reaction rates yield an estimate of the diameter of the pore and might be applied to determine the approximate dimensions of cavities within other similar proteins.
103 citations