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: It is shown that an electric field can drive single-stranded RNA and DNA molecules through a 2.6-nm diameter ion channel in a lipid bilayer membrane, which could in principle provide direct, high-speed detection of the sequence of bases in single molecules of DNA or RNA.
Abstract: We show that an electric field can drive single-stranded RNA and DNA molecules through a 2.6-nm diameter ion channel in a lipid bilayer membrane. Because the channel diameter can accommodate only a single strand of RNA or DNA, each polymer traverses the membrane as an extended chain that partially blocks the channel. The passage of each molecule is detected as a transient decrease of ionic current whose duration is proportional to polymer length. Channel blockades can therefore be used to measure polynucleotide length. With further improvements, the method could in principle provide direct, high-speed detection of the sequence of bases in single molecules of DNA or RNA.
3,251 citations
TL;DR: It is demonstrated that this nanopore behaves as a detector that can rapidly discriminate between pyrimidine and purine segments along an RNA molecule.
1,044 citations
TL;DR: The force–velocity relationship of the motor is established and it is found that the rate-limiting step of the machine's cycle is force dependent even at low loads, suggesting that this force may be available for initiating the ejection of the DNA from the capsid during infection.
Abstract: As part of the viral infection cycle, viruses must package their newly replicated genomes for delivery to other host cells. Bacteriophage straight phi29 packages its 6.6-microm long, double-stranded DNA into a 42 x 54 nm capsid by means of a portal complex that hydrolyses ATP. This process is remarkable because entropic, electrostatic and bending energies of the DNA must be overcome to package the DNA to near-crystalline density. Here we use optical tweezers to pull on single DNA molecules as they are packaged, thus demonstrating that the portal complex is a force-generating motor. This motor can work against loads of up to 57 pN on average, making it one of the strongest molecular motors reported to date. Movements of over 5 microm are observed, indicating high processivity. Pauses and slips also occur, particularly at higher forces. We establish the force-velocity relationship of the motor and find that the rate-limiting step of the motor's cycle is force dependent even at low loads. Notably, the packaging rate decreases as the prohead is filled, indicating that an internal force builds up to approximately 50 pN owing to DNA confinement. Our data suggest that this force may be available for initiating the ejection of the DNA from the capsid during infection.
1,022 citations
TL;DR: Current blockade and time distributions for single-stranded DNA polymers during voltage-driven translocations through a single alpha-hemolysin pore imply that, while polymers longer than the pore are translocated at a constant speed, the velocity of shorter polymers increases with decreasing length.
Abstract: We measure current blockade and time distributions for single-stranded DNA polymers during voltage-driven translocations through a single alpha-hemolysin pore. We use these data to determine the velocity of the polymers in the pore. Our measurements imply that, while polymers longer than the pore are translocated at a constant speed, the velocity of shorter polymers increases with decreasing length. This velocity is nonlinear with the applied field. Based on this data, we estimate the effective diffusion coefficient and the energy penalty for extending a molecule into the pore.
923 citations
TL;DR: It is demonstrated that DNA translocation speeds can be reduced by an order of magnitude over previous results by controlling the electrolyte temperature, salt concentration, viscosity, and the electrical bias voltage across the nanopore.
Abstract: Reducing a DNA molecule's translocation speed in a solid-state nanopore is a key step toward rapid single molecule identification. Here we demonstrate that DNA translocation speeds can be reduced by an order of magnitude over previous results. By controlling the electrolyte temperature, salt concentration, viscosity, and the electrical bias voltage across the nanopore, we obtain a 3 base/μs translocation speed for 3 kbp double-stranded DNA in a 4−8 nm diameter silicon nitride pore. Our results also indicate that the ionic conductivity inside such a nanopore is smaller than it is in bulk.
572 citations