Conformation Model of Back-Folding and Looping of a Single DNA Molecule Confined Inside a Nanochannel

Currently, no theory is available to describe the conformation of DNA confined in a channel when the nanochannel diameter is around the persistence length. Backfolded hairpins in the undulating wormlike chain conformation result in the formation of loops, which reduces the stretch of the molecule in the longitudinal direction of the channel. A cooperativity model is used to quantify the frequency and size of the loop domains. The predictions agree with results from the Monte Carlo simulation. A in nanofabrication have made it possible to fabricate quasi one-dimensional devices with a crosssectional diameter on the order of tens to hundreds of nanometers. These nanochannels serve as a platform for studying, among others, single DNA molecules. Furthermore, confinement in a nanospace results in significant modification of certain important biophysical phenomena, such as the knotting probability of circular DNA and the effect of macromolecular crowding on the conformation and folding of DNA. The conformation and dynamics of DNA in confinement have also been investigated with computer simulations. All of these results are invaluable in, for example, the development of a better genome analysis platform and our understanding of biological processes such as DNA packaging in viruses or segregation of DNA in bacteria. The conformation of a wormlike chain in a nanochannel is determined by the persistence length P, width w, unconfined radius of gyration Rg, and the cross-sectional diameter of the channel D. We have investigated the relative extension L∥/L, that is, the stretch of the molecule along the direction of the channel divided by its contour length, of DNA with Monte Carlo simulation (details will be presented below). A typical result is shown in Figure 1. Due to the interplay of the various length scales, regimes are established along the curve of L∥/L as a function of D/P. Odijk’s deflection and Daoud and de Gennes’s blob regimes are the two extreme regimes originally proposed. In the deflection regime (i.e., D < P), the chain is undulating since it is deflected by the walls. As a result of the undulation with deflection length λ ∼ DP, L∥/L is reduced with respect to its fully stretched value of unity according to = − L L c D P / 1 ( / ) d 2/3 (1) with c = 0.1701 for a circular cross-section. In the range of P/ w < D < Rg, the chain is in the blob regime. 21 In this regime, the chain statistics is described as a linearly packed array of subcoils (blobs). The relative extension is given by the scaling law ∝ − L L D P / ( / ) b 0.701 (2) Received: June 24, 2012 Accepted: July 31, 2012 Published: August 2, 2012 Figure 1. Monte Carlo results of the relative extension versus channel diameter divided by persistence length (50 nm). The chain width w = 10 nm and contour length L = 8 μm. The curves represent deflection (blue), deflection with S-loops (green), deflection with Sand C-loops (red), and blob (magenta) theories. The inset shows the results for w = 5 (○) and 7.5 (□) nm. Letter pubs.acs.org/macroletters © 2012 American Chemical Society 1046 dx.doi.org/10.1021/mz300323a | ACS Macro Lett. 2012, 1, 1046−1050 In the transition regime, the situation is less clear. Several attempts have been reported to bridge the gap. For the chain to crossover from the blob regime, the blob size reduces such that the volume interaction energy per blob becomes less than thermal energy kT. This results in modified or the same scaling of L∥/L with D, depending on the presumed chain statistics within the blob. On the other hand, for the crossover from the deflection regime, it has been proposed that the chain performs a one-dimensional random walk through the formation of back-folded hairpin conformations. Here, we seek the conformation of the chain in the range 1 < D/P < 2. In this range, the confined chain is neither in the full deflection conformation nor a series of blobs representation. However, it cannot resist thermal fluctuation that leads to hairpins along the chain. Indeed, folded structures of length 150−250 nm exist for DNA confined in channels with a crosssectional diameter of ∼100 nm. As shown by the snapshot of an equilibrated Monte Carlo conformation in Figure 2, there are two types of back-folded chain configurations. The C-loop results from the formation of a single hairpin and occurs at the end of the chain. An S-loop comes from a pair of hairpins somewhere in the middle. Both types of loops affect L∥/L, but the effect of the C-loops becomes vanishingly small for very long chains. Accordingly, we primarily focus on S-loop formation, but eventually, we will also include C-loops in our prediction for the stretch. Note that for 1 < D/P < 2, more than three parallel chain segments inside the channel is improbable, and a one-dimensional random walk is not established. The reduction in L∥/L by S-loop formation is determined by the average contour length Ls stored inside an S-loop and the number of S-loops fs per unit contour length. In the derivation of L∥/L, we make two assumptions. (i) We assume that the extension of each chain segment in an S-loop is not affected by the presence of the other two segments. (ii) The contour length of a hairpin chord is assumed to be πD/2. The relative extension then reads π = − − L L L L f L Df / / (1 2 /3 /3) ds d s s s (3) where the final term takes into account the fact that the hairpin chords do not contribute to the overall stretch. Expressions for Ls and fs are derived from a Bragg−Zimm type cooperativity model. Loop formation can be viewed as a dynamic process under constant thermal fluctuation. A pair of hairpins is initially created followed by the growth of an S-loop domain. Nucleation is hence determined by the free energy cost of hairpin formation, which is predominantly bending energy. Domain growth is mainly controlled by excluded volume interaction among parallel chain segments inside the loop. To implement the cooperativity model, the worm-like chain must be discretized into a sequence of units. Each unit can either be in a deflection or S-loop state. We define the basic unit as a chain segment with a contour length of πD, as being the length stored in two hairpin chords. The smallest S-loop consists hence of a single unit. For a chain with contour length L, the total number of units N = L/(πD). The free energy of the chain with a certain sequence of units can be written as = + F NF N F 2 conf s s d u (4) with Ns and Nd being the number of units in the S-loop state and the number of S-loops, respectively. The excess in free energy of a unit in the S-loop state with respect to the deflection state is denoted by Fs (the deflection state has been chosen to be the reference state). Note that each S-loop is sided by two hairpin junctions. The free energy of nucleation of an Sloop is hence 2Fu, with Fu being the required free energy to create a hairpin. The cooperativity model can be solved by using the transfer matrix method. In the ground state dominance, the thermally averaged free energy of the chain reads = − + + − + ⎡ ⎣⎢ ⎤ ⎦⎥ F kT N s s su / ln 1 ((1 ) 4 )