Revisiting roaming trajectories in ketene isomerization at higher dimensionality

Roaming dynamics have been observed in a three-dimensional model of the ketene isomerization reaction. The roaming trajectories sample the region between the outer potential barriers closest to the respective ketene isomers and involve turning points along the reaction coordinate in a polar representation. These roaming trajectories avoid the intrinsic reaction coordinate and the intermediates to which it is associated. Thus, one-dimensional transition state theory (TST) is generally insufficient as has been confirmed through an analysis of the reactive flux along the dividing surface (DS). A global representation of the DS, however, leads to accurate TST rate constants. The exact and TST microcanonical rates of isomerization have been obtained for the three-dimensional model and compare well to experiment. The global DS is therefore particularly important for obtaining rates in reactions that exhibit roaming. This work thus confirms the findings of our previous two-dimensional treatment of ketene isomerization (Ulusoy et al. in J. Phys. Chem. A 117:7553–7560, 2013).     

One-dimensional reaction coordinates for diffusive activated rate processes in many dimensions

For multidimensional activated rate processes controlled by diffusive crossing of a saddle point region, we show that a one-dimensional reaction coordinate can be constructed even when the diffusion anisotropy is arbitrary. The rate constant, found using the potential of mean force along this coordinate, is identical to that predicted by the multidimensional Kramers-Langer theory. This reaction coordinate minimizes the one-dimensional rate constant obtained using a trial reaction coordinate and is orthogonal to the stochastic separatrix, the transition state that separates reactants from products.     

Transition state barriers in multidimensional Marcus theory

Multidimensional Marcus theory is the extension of traditional Marcus theory to systems in which multiple particles are transferred. Rather than the intersecting parabolas of Marcus theory, multidimensional Marcus theory involves the intersection of paraboloids. In this paper, we examine the conditions under which a full multidimensional treatment of these paraboloids is necessary and when it is possible to use a simpler one-dimensional formalism. In particular, we examine transition state barrier energies, which are essential parameters in many reaction rate equations, and which depend on the formalism used. We find, based on both analytic calculations and numerical simulation, that the reduced one-dimensional treatment yields excellent agreement with the exact, multidimensional results over a wide variety of conditions for one particular choice of the single collective reaction coordinate. We also outline a procedure for calculating accurate multidimensional transition state barrier energies and apply it to a two-dimensional model of proton-coupled electron transfer.     

A direct transition state theory based analysis of the branching in NH2 + NO

A combination of high-level quantum-chemical simulations and sophisticated transition state theory analyses is employed in a study of the temperature dependence of the N2H + OH-->HNNOH recombination reaction. The implications for the branching between N2H + OH and N2 + H2O in the NH2 + NO reaction are also explored. The transition state partition function for the N2H + OH recombination reaction is evaluated with a direct implementation of variable reaction coordinate (VRC) transition state theory (TST). The orientation dependent interaction energies are directly determined at the CAS + 1 + 2/cc-pvdz level. Corrections for basis set limitations are obtained via calculations along the cis and trans minimum energy paths employing an approximately aug-pvtz basis set. The calculated rate constant for the N2H + OH-->HNNOH recombination is found to decrease significantly with increasing temperature, in agreement with the predictions of our earlier theoretical study. Conventional transition state theory analyses, employing new coupled cluster estimates for the vibrational frequencies and energies at the saddlepoints along the NH2 + NO reaction pathway, are coupled with the VRC-TST analyses for the N2H + OH channels to provide estimates for the branching in the NH2 + NO reaction. Modest variations in the exothermicity of the reaction (1-2 kcal mol-1), and in a few of the saddlepoint energies (2-4 kcal mol-1), yield TST based predictions for the branching fraction that are in satisfactory agreement with related experimental results. The unmodified results are in reasonable agreement for higher temperatures, but predict too low a branching ratio near room temperature, as well as too steep an initial rise.     

Polymer translocation through a nanopore under an applied external field

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.     

Interplay of the main chain, chiral side chains, and solvent in conformational transitions: poly([(R)-3,7-dimethyloctyl]-[(S)-3-methylpentyl]silylene).

Light scattering, sedimentation equilibrium, viscosity, circular dichroism (CD), and UV absorption (UV) measurements were made on dilute solutions of poly([(R)-3,7-dimethyloctyl]-[(S)-3-methylpentyl]silylene)(PRS) as functions of molecular weight. From light scattering and viscosity data, PRS is found to be a very stiff polymer of persistence length q as large as 103 nm at 25 degrees C, essentially a 7(3) helix found in the solid state; q increases only gradually with lowering temperature between -15 and 25 degrees C. The CD data show that PRS undergoes a conformational transition around 3 degrees C in isooctane (transition temperature T(c)). The CD signal is largely positive at low temperatures, passes through zero at T(c), and becomes largely negative at higher temperatures; T(c) is independent of sample's chain length N. This is a highly cooperative helix (M)-to-helix (P) transition depending remarkably on N, as PRS is substantially rodlike. The CD data are converted to the fraction f(P) of P helix as a function of N and analyzed successfully by a statistical mechanical theory based on a helix reversal model, where a polymer chain consists of M and P helices intervened by helix reversals, with the result that the free energy difference DeltaG(h) between P and M shows a temperature dependence similar to that of 2f(P) - 1, whereas the helix reversal energy is substantially constant at 1.2 x 10(4) J mol(-1); the latter value means that the helix reversal occurs only once in 100 Si units or less. This DeltaG(h) change and solvent dependence of T(c) are explained by a double-well potential for the rotation about Si-Si bonds, which incorporates into DeltaG(h) the solvent interactions with the helical grooves of side chains surrounding the main chain. Detailed features of UV absorption spectra at different temperature and molecular weights are also presented.     

Understanding the kinetics and mechanism of thermal cheletropic elimination of N<Subscript>2</Subscript> from (2,5-dihydro-1<Emphasis Type="Italic">H</Emphasis>-pyrrol-1-ium-1-ylidene) amide using RRKM and ELF theories

The cheletropic elimination process of N₂ from (2,5-dihydro-1H-pyrrol-1-ium-1-ylidene) amide (C₄H₆N₂) has been studied computationally using density functional theory, along with the M06-2X/aug-cc-pVTZ level of theory. The calculated energy profile has been supplemented with calculations of kinetic rate constants using transition state theory (TST) and statistical Rice–Ramsperger–Kassel–Marcus (RRKM) theory. This elimination process takes place spontaneously with an activation energy around 33 kJ/mol. Pressure dependence of the rate constants revealed that the TST approximation breaks down and fall-off expression is necessary for the kinetic modeling. At temperatures ranging from 240 to 360 K and atmospheric pressure, the unimolecular rate constant is evaluated from RRKM theory as \(k_{{(240 - 360\,{\text{K}})}}^{{1.0{\text{atm}}}} = 1.0249 \times 10^{12} \times {\text{e}}^{{ - \frac{{33.11\;{\text{kJ}}/{\text{mol}}}}{RT}}} \,{\text{s}}^{ - 1}\). Bonding changes along the reaction coordinate have been studied using bonding evolution theory. Electron localization function topological analysis reveals that the cheletropic elimination is characterized topologically by four successive structural stability domains (SSDs). Breaking of C–N bonds (Rx = 0.1992 amu^1/2 Bohr) and the other selected points separating the SSDs along the reaction coordinate occur in the vicinity of the transition state.     

Transition state theory: variational formulation, dynamical corrections, and error estimates

Transition state theory (TST) is revisited, as well as evolutions upon TST such as variational TST in which the TST dividing surface is optimized so as to minimize the rate of recrossing through this surface and methods which aim at computing dynamical corrections to the TST transition rate constant. The theory is discussed from an original viewpoint. It is shown how to compute exactly the mean frequency of transition between two predefined sets which either partition phase space (as in TST) or are taken to be well-separated metastable sets corresponding to long-lived conformation states (as necessary to obtain the actual transition rate constants between these states). Exact and approximate criterions for the optimal TST dividing surface with minimum recrossing rate are derived. Some issues about the definition and meaning of the free energy in the context of TST are also discussed. Finally precise error estimates for the numerical procedure to evaluate the transmission coefficient kappaS of the TST dividing surface are given, and it is shown that the relative error on kappaS scales as 1/square root(kappaS) when kappaS is small. This implies that dynamical corrections to the TST rate constant can be computed efficiently if and only if the TST dividing surface has a transmission coefficient kappaS which is not too small. In particular, the TST dividing surface must be optimized upon (for otherwise kappaS is generally very small), but this may not be sufficient to make the procedure numerically efficient (because the optimal dividing surface has maximum kappaS, but this coefficient may still be very small).     

Failure of one-dimensional Smoluchowski diffusion models to describe the duration of conformational rearrangements in floppy, diffusive molecular systems: a case study of polymer cyclization

Motivated by recent experimental efforts to measure the duration of individual folding∕unfolding transitions in proteins and RNA, here we use simulations to study the duration of a simple transition mimicking an elementary step in biopolymer folding: the closure of a loop in a long polymer chain. While the rate of such a transition is well approximated by a one-dimensional Smoluchowski model that views the end-to-end distance dynamics of a polymer chain as diffusion governed by the one-dimensional potential of mean force, the same model fails rather dramatically to describe the duration of such transitions. Instead, the latter timescale is well described by a model where the chain ends diffuse freely, uninfluenced by the average entropic force imposed by the polymer chain. The effective diffusion coefficient then depends on the length scale of the loop closure transition. Our findings suggest that simple one-dimensional models, when applied to estimate the duration of reactive events in complex molecular systems, should be used with caution.     

Polymer reversal rate calculated via locally scaled diffusion map

A recent study on the dynamics of polymer reversal inside a nanopore by Huang and Makarov [J. Chem. Phys. 128, 114903 (2008)] demonstrated that the reaction rate cannot be reproduced by projecting the dynamics onto a single empirical reaction coordinate, a result suggesting the dynamics of this system cannot be correctly described by using a single collective coordinate. To further investigate this possibility we have applied our recently developed multiscale framework, locally scaled diffusion map (LSDMap), to obtain collective reaction coordinates for this system. Using a single diffusion coordinate, we obtain a reversal rate via Kramers expression that is in good agreement with the exact rate obtained from the simulations. Our mathematically rigorous approach accounts for the local heterogeneity of molecular configuration space in constructing a diffusion map, from which collective coordinates emerge. We believe this approach can be applied in general to characterize complex macromolecular dynamics by providing an accurate definition of the collective coordinates associated with processes at different time scales.     

Polymer translocation through a nanopore. II. Excluded volume effect

Following our previous study of a Gaussian chain translocation, we have investigated the transport of a self-avoiding chain from one sphere to another sphere through a narrow pore, using the self-consistent field theory formalism. The free energy landscape for polymer translocation is significantly modified by excluded volume interactions among monomers. The free energy barrier for the placement of one of the chain ends at the pore depends on the chain length N nonmonotonically, in contrast to the N-independence for Gaussian chains. This results in a nonmonotonic dependence of the average arrival time [tau0] on N for self-avoiding chains. When the polymer chain is partitioned between the donor and recipient spheres, a local free energy minimum develops, depending on the strength w of the excluded volume interaction and the relative sizes of the donor and recipient spheres. If the sizes of spheres are comparable, the average translocation time tau (the average time taken by the polymer, after the arrival at the pore, to convert from the donor to the recipient) increases with an increase in w for a fixed N value. On the other hand, for the highly asymmetric sizes of the donor and recipient spheres, tau decreases with an increase in w. As in the case of Gaussian chains, tau depends nonmonotonically on the pore length.     

Polymer depletion profiles around nonspherical colloidal particles

We study the effect of chain self-avoidance on the polymer density profiles that are induced by a single colloidal particle of nonspherical shape such as an ellipsoid, a dumbbell, or a lens in a solution of nonadsorbing polymers. For colloid sizes sigma much smaller than the size R(x) of the polymers, we observe a pronounced difference between ideal and self-avoiding chains. In the case of ideal polymers, the surfaces of constant density always have the same character as the surface of the particle, e.g., are oblate for an oblate ellipsoid. In the self-avoiding case, however, the character changes with increasing distance r from the particle, and an oblate particle induces prolate surfaces of constant density if sigma相似文献   

Electron tunneling dynamics in anharmonic bath

Tunneling transition probability for a particle interacting with an anharmonic bath is found in a time-dependent Hartree approximation. The general expression is presented in terms of medium Keldysh functions that are assumed to be known. Furthermore, the transition probability is calculated in the noninteracting-blip approximation where the rate constant does not exhibit an activation dependence at high temperatures. The reorganization energy E(r) and the renormalized reaction heat epsilon are expressed in terms of the correlation matrix for a solvent and internal modes in both quantum and classical regimes. It is shown that E(r) and epsilon are temperature dependent.     

Time-dependent elongational viscosity of a solution of elastic dumbbells

The elongational viscosity of dilute polymer solutions must always be measured under nonsteady-state conditions. To predict the time dependence of this viscosity for a polymer solution in which a constant stretching rate is maintained, a simple model is considered in which the polymer molecules are represented by elastic dumbbells. The non-Hookean elastic force in the dumbbell is determined by the conformational entropy of the chain. Use is made of Peterlin's approximation which replaces the elastic force by a function of the root-mean-square end-to-end distance. Application to the transient state is straightforward; it can be extended to include the effect of “internal” chain viscosity by means of approximations that are similar to Peterlin's.     

Polymer translocation through a nanopore: a two-dimensional Monte Carlo study

We investigate the problem of polymer translocation through a nanopore in the absence of an external driving force. To this end, we use the two-dimensional fluctuating bond model with single-segment Monte Carlo moves. To overcome the entropic barrier without artificial restrictions, we consider a polymer which is initially placed in the middle of the pore and study the escape time tau required for the polymer to completely exit the pore on either end. We find numerically that tau scales with the chain length N as tau approximately N(1+2nu), where nu is the Flory exponent. This is the same scaling as predicted for the translocation time of a polymer which passes through the nanopore in one direction only. We examine the interplay between the pore length L and the radius of gyration R(g). For LR(g), we find tau approximately N. In addition, we numerically find the scaling function describing crossover between short and long pores. We also show that tau has a minimum as a function of L for longer chains when the radius of gyration along the pore direction R( parallel) approximately L. Finally, we demonstrate that the stiffness of the polymer does not change the scaling behavior of translocation dynamics for single-segment dynamics.     

Steady-state properties and dynamic behavior of polymer chains in dilute solution under elongational flow

The behavior of polymer chains in dilute solution under a steady, homogeneous elongational flow has been studied employing Brownian dynamics simulation. We first consider the dependence of polymer properties in steady state on the elongational rate, ϵ. When this rate exceeds some critical value, ϵ_c, the properties show a dramatic change from the values typical of the coil state to those of a stretched conformation. We describe the dependence of ϵ_c on chain length for different polymer/solvent conditions. Following the trajectories of individual molecules, we have characterized dynamic aspects of the coil-stretch transition. Each chain suffer the transition after some time, t_trans, has elapsed after the flow start-up. The values of t_trans vary remarkably from one molecule to another, and we have characterized the statistical distribution of this quantity. We also determine the kinetics of the coil-to-stretch process, which seems to follow a first-order kinetics after some induction time. The dependence of the statistical and kinetic parameters on chain length and elongational rate has been determined.     

A Monte Carlo algorithm to study polymer translocation through nanopores. I. Theory and numerical approach

The process during which a polymer translocates through a nanopore depends on many physical parameters and fundamental mechanisms. We propose a new one-dimensional lattice Monte Carlo algorithm that integrates various effects such as the entropic forces acting on the subchains that are outside the channel, the external forces that are pulling the polymer through the pore, and the frictional effects that involve the chain and its environment. Our novel approach allows us to study the polymer as a single Brownian particle diffusing while subjected to a position-dependent force that includes both the external driving forces and the internal entropic bias. Frictional effects outside and inside the pore are also considered. This Monte Carlo method is much more efficient than other simulation methods, and it can be used to obtain scaling laws for various polymer translocation regimes. In this first part, we derive the model and describe a subtle numerical approach that gives exact results for both the escape probability and the mean translocation time (and higher moments of its distribution). The scaling laws obtained from this model will be presented and discussed in the second part of this series.     

Stochastic transition states: reaction geometry amidst noise

Classical transition state theory (TST) is the cornerstone of reaction-rate theory. It postulates a partition of phase space into reactant and product regions, which are separated by a dividing surface that reactive trajectories must cross. In order not to overestimate the reaction rate, the dynamics must be free of recrossings of the dividing surface. This no-recrossing rule is difficult (and sometimes impossible) to enforce, however, when a chemical reaction takes place in a fluctuating environment such as a liquid. High-accuracy approximations to the rate are well known when the solvent forces are treated using stochastic representations, though again, exact no-recrossing surfaces have not been available. To generalize the exact limit of TST to reactive systems driven by noise, we introduce a time-dependent dividing surface that is stochastically moving in phase space, such that it is crossed once and only once by each transition path.     

An exact formulation of hyperdynamics simulations

We introduce a new formula for the acceleration weight factor in the hyperdynamics simulation method, the use of which correctly provides an exact simulation of the true dynamics of a system. This new form of hyperdynamics is valid and applicable where the transition state theory (TST) is applicable and also where the TST is not applicable. To illustrate this new formulation, we perform hyperdynamics simulations for four systems ranging from one degree of freedom to 591 degrees of freedom: (1) We first analyze free diffusion having one degree of freedom. This system does not have a transition state. The TST and the original form of hyperdynamics are not applicable. Using the new form of hyperdynamics, we compute mean square displacement for a range of time. The results obtained agree perfectly with the analytical formula. (2) Then we examine the classical Kramers escape rate problem. The rate computed is in perfect agreement with the Kramers formula over a broad range of temperature. (3) We also study another classical problem: Computing the rate of effusion out of a cubic box through a tiny hole. This problem does not involve an energy barrier. Thus, the original form of hyperdynamics excludes the possibility of using a nonzero bias and is inappropriate. However, with the new weight factor formula, our new form of hyperdynamics can be easily implemented and it produces the exact results. (4) To illustrate applicability to systems of many degrees of freedom, we analyze diffusion of an atom adsorbed on the (001) surface of an fcc crystal. The system is modeled by an atom on top of a slab of six atomic layers. Each layer has 49 atoms. With the bottom two layers of atoms fixed, this system has 591 degrees of freedom. With very modest computing effort, we are able to characterize its diffusion pathways in the exchange-with-the-substrate and hop-over-the-bridge mechanisms.     

First passage time distribution of chaperone driven polymer translocation through a nanopore: homopolymer and heteropolymer cases

Combining the advection-diffusion equation approach with Monte Carlo simulations we study chaperone driven polymer translocation of a stiff polymer through a nanopore. We demonstrate that the probability density function of first passage times across the pore depends solely on the Pe?clet number, a dimensionless parameter comparing drift strength and diffusivity. Moreover it is shown that the characteristic exponent in the power-law dependence of the translocation time on the chain length, a function of the chaperone-polymer binding energy, the chaperone concentration, and the chain length, is also effectively determined by the Pe?clet number. We investigate the effect of the chaperone size on the translocation process. In particular, for large chaperone size, the translocation progress and the mean waiting time as function of the reaction coordinate exhibit pronounced sawtooth-shapes. The effects of a heterogeneous polymer sequence on the translocation dynamics is studied in terms of the translocation velocity, the probability distribution for the translocation progress, and the monomer waiting times.