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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Journal Article•
212 citations
TL;DR: In this paper, the effect of the length of the pore and diameter of the nanopore on the velocity of polymer translocation is investigated, and it is shown that the size and geometry of pore are important factors in polymer dynamics.
Abstract: Polymer translocation through a nanopore in a membrane is investigated theoretically. Recent experiments on voltage-driven DNA and RNA translocations through a nanopore indicate that the size and geometry of the pore are important factors in polymer dynamics. A theoretical approach is presented which explicitly takes into account the effect of the nanopore length and diameter for polymer motion across the membrane. It is shown that the length of the pore is crucial for polymer translocation dynamics. The present model predicts that for realistic conditions (long nanopores and large external fields) there are two regimes of translocation depending on polymer size: for polymer chains larger than the pore length, the velocity of translocation is nearly constant, while for polymer chains smaller than the pore length the velocity increases with decreasing polymer size. These results agree with experimental data.
200 citations
TL;DR: In this paper, the scaling analysis of the effect of entropic barriers on self-avoiding polymer chains in an infinite periodic array of cubic cavities separated by short bottlenecks was analyzed by Monte Carlo simulations and analyzed by scaling arguments.
Abstract: Dynamic and static properties of a self-avoiding polymer chain in an infinite periodic array of cubic cavities separated by short bottlenecks have been investigated by Monte Carlo simulations and analyzed by scaling arguments. At the locations of the bottlenecks, entropic barriers are set up due to the reduction of the number of possible chain configurations in these positions. Consequently the chain diffusion coefficient D is smaller than the Rouse diffusion coefficient D,. The scaling analysis of the effect of entropic barriers shows that D/Do decays exponentially with chain length N if the cross section of the bottleneck C is large and that it is independent of N for small C but is now a function of C, according to N-' In (DIDo) = A - sN-' where s depends inversely on C and A depends inversely on the size of the cavity and is negative. The simulation results are in agreement with the scaling analysis demonstrating that the chain diffusion in this problem is dominantly controlled by the entropic barriers.
177 citations
TL;DR: In this article, the authors derived an expression for the free energy as a function of the number of polymers passing through a narrow pore in a membrane using the Smoluchowski equation.
Abstract: We study the process of charged polymer translocation, driven by an external electric potential, through a narrow pore in a membrane We assume that the number of polymer segments, m, having passed the entrance pore mouth, is a slow variable governing the translocation process Outside the pore the probability that there is an end segment at the entrance pore mouth, is taken as the relevant parameter In particular we derive an expression for the free energy as a function of m, F(m) F(m) is used in the Smoluchowski equation in order to obtain the flux of polymers through the pore In the low voltage regime we find a thresholdlike behavior and exponential dependence on voltage Above this regime the flux depends linearly on the applied voltage At very high voltages the process is diffusion limited and the flux saturates to a constant value The model accounts for all features of the recent experiments by Henrickson et al [Phys Rev Lett 85, 3057 (2000)] for the flux of DNA molecules through an α-hemolysin pore as a function of applied voltage
148 citations
TL;DR: The nuclear uptake of extended linear DNA molecules is demonstrated by a combination of fluorescence microscopy and single-molecule manipulation techniques, and it is found that uptake of DNA is independent of ATP or GTP hydrolysis, but is blocked by wheat germ agglutinin.
Abstract: Gene transfer to eukaryotic cells requires the uptake of exogenous DNA into the cell nucleus. Except during mitosis, molecular access to the nuclear interior is limited to passage through the nuclear pores. Here we demonstrate the nuclear uptake of extended linear DNA molecules by a combination of fluorescence microscopy and single-molecule manipulation techniques, using the latter to follow uptake kinetics of individual molecules in real time. The assays were carried out on nuclei reconstituted in vitro from extracts of Xenopus eggs, which provide both a complete complement of biochemical factors involved in nuclear protein import, and unobstructed access to the nuclear pores. We find that uptake of DNA is independent of ATP or GTP hydrolysis, but is blocked by wheat germ agglutinin. The kinetics are much slower than would be expected from hydrodynamic considerations. A fit of the data to a simple model suggests femto-Newton forces and a large friction relevant to the uptake process.
146 citations