ВЫСОКОМОЛЕКУЛЯРНЫЕ СОЕДИНЕНИЯ, Серия C, 2013, том 55, № 7, с. 866-875
УДК 541.64:539.199
TRANSLOCATION DYNAMICS OF A SEMI-FLEXIBLE CHAIN UNDER A BIAS: COMPARISON WITH TENSION PROPAGATION THEORY
© 2013 г. Aniket Bhattacharya
Department of Physics, University of Central Florida, Orlando, Florida 32816-2385, USA
e-mail: aniket@physics.ucf.edu
Abstract — We study translocation dynamics of a semi-flexible polymer through a nanoscopic pore in two dimensions (2D) using Langevin dynamics simulation in presence of an external force inside the pore. We observe that for a given chain length N the mean first passage time (MFPT) (т) increases for a stiffer chain. By repeating the calculation for various chain lengths N and bending rigidity parameter кь we calculate the
translocation exponent а ((т) ~ Na). For chain lengths N and bending rigidity Kb considered in this paper we find that the translocation exponent a satisfies the inequality a < 1 + v, where v is the equilibrium Flory exponent for a given chain stiffness, as previously observed in various simulation studies for fully flexible chains. We observe that the peak position of the residence time W(s) as a function of the monomer index s shifts at a lower s-value with increasing chain stiffness кb. We also monitor segmental gyration (Rg(s}) both at the cis and trans side during the translocation process and find that for кь ^ 0 the late time cis conformations are nearly identical to the early time trans conformations, and this overlap continues to increase for stiffer chains. Finally, we try to rationalize dependence of various quantities on chain stiffness кь using Sakaue's tension propagation (TP) theory [Phys. Rev. E 76, 021803 (2007)] and Brownian Dynamics Tension Propagation (BDTP) theory due to Ikonen et al. [Phys. Rev. E 85, 051803 (2012); J. Chem. Phys. 137, 085101 (2012)] originally developed for a fully flexible chain to a semi-flexible chain.
DOI: 10.7868/S0507547513070015
INTRODUCTION
Translocation of polymer chains threading through a narrow channel has remained an active field of research for more than a decade [1, 2]. The phenomenon is of particular interest in the context of biopolymers as translocation is an important ubiquitous process in molecular biology. Translocation of DNA and RNA across nuclear pores, protein transport through membrane channels, and virus injection are examples of such processes [3]. In a series of pioneering experiments using single stranded as well as double stranded DNA translocating through a -hemolysin protein pore and synthetic nanopores [4—9], where the histogram of the mean first passage time (MFPT) was obtained by measuring the fluctuation in the channel current, it was demonstrated that a nanopore can be used to determine sequences of a heteropolymer. Further experimental and theoretical studies revealed that the interaction of the individual nucleotides with the nanopore has remarkable manifestation not only in the characteristic MFPT but also in the residence time distribution of the individual nucleotides transported across the pore [8—12]. Therefore, it is no surprise that recently "nano-pore" based techniques have been commercialized and are being used to detect sequences [13].
These exciting experiments have provided enough enthusiasm to develop a proper theoretical framework for polymer translocation through a nanopore. Sung and Park [14] and Muthukumar [15] considered translocation as a one dimensional barrier crossing problem and derived expression for the translocation exponent
a ((x) ~ Na) using free energy expression for a threaded polymer through the pore (Fig. 1).
These initial predictions were followed by many others [16—31] using back of the envelope estimates and dynamical scaling arguments [17, 18], analyzing folds of the chains [19, 20], incorporation of memory effects [21—24], mass and energy conservations [25— 28], and tension propagation (TP) along the chain backbone [25, 29—31]. These experimental and theoretical developments have been supplemented by a large number of simulation studies which played crucial role in the theoretical developments in the field [10-12, 30-56].
Almost all of the aforementioned theoretical and simulation studies have been addressed in the context of a fully flexible chain. For example, a recent important theoretical development in terms of a TP model [25, 26], its subsequent modification [29], and adoption to Brownian dynamics (BDTP) [30, 31] considers fully flexible chains. But when one considers impor-
tant biopolymers, e.g., DNA, RNA, Actin filaments, all of them exhibit finite bending rigidity under various physiological conditions. Therefore, it is rather demanding to extend the TP formalism for the semiflexible chains and wonder how the chain stiffness will affect the tension propagation and hence the dynamics of a translocating chain.
Unlike fully flexible polymers, one requires another length scale to describe a semi-flexible polymer in the bulk, namely its persistence length lp [57, 58]. In coarse-grained models (section MODEL) this is introduced through a bending rigidity parameter kb (for simplicity we consider a 2D case). The details of the molecular architecture and interactions of the individual building blocks are masked in this parameter. To understand the bulk properties and for length scale larger than lp, it is often useful to think a semi-flexible polymer as a flexible polymer of basic building block of length ~l [60, 61]. However, in case of polymers threading through a pore, it is not clear if such coarse-graining will be appropriate when lp greater than the pore diameter a pore.
Let us now consider some aspects of theoretical and simulation studies of driven polymer translocation through a nanopore which will serve as the motivating factors for this article. Simulation studies for a fully flexible chain have revealed that for driven polymer translocation the chain is out of equilibrium and the configurations at the cis side are different than those at the trans side [51—53, 55]. It has also been observed in MD simulation using bead spring model of polymer that for chain length N ~ 500 the translocation exponent a < 1 + v [48]. On the theoretical front two developments are worth mentioning. First, this out of equilibrium aspects have been nicely captured in Sakaue's TP model [25, 26]. Secondly, by including pore friction into Sakaue's TP theory Ikonen et al. have demonstrated that the physical basis of the inequality a < 1 + v is the chain length dependent pore friction and the finite chain length effect persist up to very long
chains N ~ 106 [30, 31].
An important aspect of the BDTP model is that by extending Sakaue's theory [25, 26] for a finite chain length N [30,31], it introduces a new time scale ttp when the tension front reaches the last monomer on the cis side (Fig. 2d). As will be discussed in more detail in section-III, a monomer inside the pore translocating during t < ttp experiences progressively increasing viscous drag due to increased number of monomers at the cis side responding to the tension front (Figs. 2b—2c). For t > tp, the viscous drag at the monomer inside the pore decreases due to the decreased number of monomers at the cis side as they translocate to the trans side [30,31]. This results is a nonmonotonic dependence of the residence time of the individual monomers at the pore and thus explains
N - s
cis
trans
Fig. 1. Minimalist view of a polymer chain translocating through an ideal pore in a thin wall from cis to trans side in terms of the translocation (s) coordinate. The picture shows an instant of translocation when s segments are at the trans side with remaining N - s segments at the cis side of a chain of length N.
the peak in the residence time distribution W(s) ~ s observed in various simulations studies of polymer translocation [10, 30, 31, 34, 35].
Evidently, this TP idea that a segment of a chain responding to a propagating disturbance need to be restricted to a fully flexible chain. The purpose of this paper is to demonstrate that it is possible to extend the ideas of the TP theory for semi-flexible chains. By carrying out Langevin dynamics simulation for several chain lengths and of different stiffness first we demonstrate that both the MFPT (x) as well as the end-to-end distance {Rn) for a semi-flexible chain increase as a function of chain stiffness parameter k b. We then borrow ideas from the BDTP model and demonstrate that many of our findings, e.g., residence time of the individual monomers, segmental gyration radii at the cis and trans sides for the semi-flexible chains can be explained in terms of the ratio ttp /(x), which we find from simulation results is a decreasing function of the chain stiffness k b. The format of the rest of the paper is as follows. In the next section we introduce the model. Results are presented in Section-III where we calculate several physical quantities, e.g., the MFPT (x), the residence time W(s) ~ s, segmental gyration radii (Rg(s)), end-to-end distance RN and show this connection explicitly. We conclude in section-IV with some remarks, although somewhat speculative, how the TP idea can be used to unravel the dynamics of real biopolymers.
THE MODEL
We have used a bead spring model of a polymer chain with excluded volume, spring and bending potentials as follows [59]. The excluded volume interac-
Fig. 2. Depiction of tension propagation (TP) along the chain backbone; (a) at t = 0 the whole chain is at the cis side and relaxed; the position of a monomer M is shown with a solid circle; (b) at t = t' tension front (dashed arc) is at the location of the monomer M but having s' monomers at the trans side, M - s' monomers (dashed-dot) under tension and N - M monomers are still relaxed. (c) same as (b) but at a later time when the tension front has reached another monomer M' > M , and (d) at t = ttp when the tension front reaches the last monomer hav
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