# Shear Unzipping of DNA

Original entry: Naveen Sinha, APPHY 226, Spring 2009

Second Entry: Nick Chisholm, AP 225, Fall 2009 (In Progress)

## Contents

### Shear Unzipping of DNA

Buddhapriya Chakrabarti, David R. Nelson

The Journal of Physical Chemistry B 113 (12), 3831-3836 (2009).

## Soft matter keywords:

deGennes, polymer, shear

## Summary

Biologists have already investigated the helical unzipping of DNA, which occurs during DNA replication and other biological processes (see Fig. 1b). However, the problem of *shear*-induced unzipping in such systems (see Fig. 1a) has yet to be investigated. Although not as prevalent in biological systems, it is directly relevant to artificially constructed nanostructures that are bound together with DNA duplexes. The shear situation is different from the helical one because the stress is distributed throughout the length of the DNA, rather than at just the bond nearest the end.

## Soft Matter Aspects

When studying an analogous system, deGennes used the model of a ladder connected by harmonic springs. This led to the prediction of a linear increase in the minimal force needed to unzip the DNA as a function of overlap length (<math>\beta \gamma</math> and <math>\delta \epsilon</math> in Fig. 1). The model also predicted a saturation in this force above a critical DNA length. A physical version of this system has been realized by the Prentiss group, in which a double-stranded DNA is trapped between a glass capillary and a magnetic bead, which can be pulled away from the surface to stretch the DNA (i.e. the force, F, would be applied at <math>\alpha</math> and <math>\omega</math> in Fig. 1). To more accurately model this situation, the authors look at both complementary base-pair interactions on opposite strands of the DNA and next-nearest neighbor interactions, also on opposite strands. In addition, the authors look at the case of a heteropolymer, with both A-T and G-C pairs.

The authors find that most of the bond breakages occur near the endpoints, with a frequency that occurs with a characteristic length scale. The unzipping process advances in jumps and plateaus, rather than uniformly. In future work, the model could be modified to take into account the helical twist of the DNA. It would also be interesting to investigate how these findings could be applied to other heteropolymers.

*written by: Naveen N. Sinha*

## General Information

**Authors**: B. Chakrabarti, and D. R. Nelson

**Publication**: The Journal of Physical Chemistry B **113** (12), 3831-3836 (2009)

## Soft Matter Keywords

DNA, Homopolymer Polymer, Shear

## Summary

This paper is a theoretical study of shear denaturation (unzipping) of DNA. Using a microscopic Hamiltonian, the authors derive a nonlinear generalization of a ladder model of shear unzipping proposed by deGennes. Previously studied is the physics behind single molecule DNA unzipping, which is the helicase-mediated unzipping that occurs during DNA replication (see Figure 1(b)). Although the case of shear unzipping (see Figure 1(a)) is less biologically relevant, its study leads to interesting materials science applications; the authors note that these include determining the strength of DNA or gold nanoparticle assemblies.

More specifically, the authors model the sugar phosphate backbone of the DNA by Hookean springs and expand in the displacements of the nearest neighbor interactions of complementary base pairs on opposite rungs of the ladder, and the next nearest neighbor interactions on different rungs of the ladder (this is a simple model of stacking interactions). Both are expanded to quadratic order, and then the eigenmode corresponding to shear deformation is projected out, thus leading to a nonlinear Hamiltonian.

## Soft Matter Discussion

Instead of delving into the mathematics in great detail (which, at times, to the unfamiliar spectator, this could seem overwhelming), I will present a more conceptual approach to this paper with some rather mild mathematics for support.

Briefly examining Figure 1(b), one notices that in the tensile unzipping that is occurring, the bases are sequentially stretched by forces that are exerted on the same end of the duplex. However, for shear unzipping (as in Figure 1(a)), one notices that the forces act on opposite ends of the duplex, and the stretching that occurs is spread out over many base pairs.

In order to account for these parameters, the authors introduce a Hamiltonian which describes the interactions between nearest neighbors on the same strand, along with the interactions between the nearest and next nearest-neighbor nucleotides of opposite strands. The interactions between nearest neighbors on the same strand are modeled by Hookean springs with spring constant <math>Q</math>, and <math>a</math> being the equilibrium unstretched separation between adjacent nucleotides. The interactions between nearest and next-nearest neighbor nucleotides of opposite strands are modeled by a Lennard-Jones potential (which, traditionally, models the interaction between a pair of neutral atoms or molecules). Mathematically, this is shown below (see Figure 1(c) for clarification on location of the vectors and forces).

<math> H = \frac{1}{2} Q \sum_{n=1}^L([|\vec{r_{n+1}} - \vec{r_{n}}| - a]^{2} + [|\vec{s_{n+1}} - \vec{s_{n}}| - a]^{2}) + \sum_{n=1}^L[V_{LJ}(|\vec{r_{n}} - \vec{s_{n}}|) + V_{LJ}'(|\vec{r_{n+1}} - \vec{s_{n}}|) + V_{LJ}'(|\vec{s_{n+1}} - \vec{r_{n}}|)] </math>

The direct nucleotide pairing is decribed by the Lennard-Jones pair potential:

<math> V_{LJ}(|\vec{r_{n}} - \vec{s_{n}}|) = 4\epsilon [(\frac{\sigma}{|\vec{r_{n}} - \vec{s_{n}}|})^{12} - (\frac{\sigma}{|\vec{r_{n}} - \vec{s_{n}}|})^6] </math>

whereas the interactions between the next nearest neighbors of opposite strands is given by the Lennard-Jones potential:

<math>V_{LJ}'(|\vec{r_{n}} - \vec{s_{n}}|) = 4\epsilon' [(\frac{\sigma'}{|\vec{r_{n}} - \vec{s_{n}}|})^{12} - (\frac{\sigma'}{|\vec{r_{n}} - \vec{s_{n}}|})^6</math> where <math>\epsilon \approx 1.6k_{B}T</math> and <math>\sigma' = 2^{\frac{-1}{6}} \sqrt{2^{\frac{1}{3}}\sigma^{2} + a^{2}}</math>.

Adding in the force terms (see Figure 1(a)), we find the total energy to be:

<math> H' = H - \vec{F} \cdot (\vec{r_{1}} - \vec{s_{L}}) </math>.

Of particular interest is the calculation of the rupture force (<math>F_{c}</math>); for small segments, they find that <math>F_{c} = f_{0} L</math> (where L is the hybridized length, or the number of nucleotides on each strand), and for long segments, they find that <math>F_{c} \approx \frac{2f_{0}}{\kappa a}</math> (where <math>\kappa</math> is the elastic screening length, i.e. <math>\kappa^{-1}</math> is the distance over which the strain relaxes over). This can be explained by the fact that for short segments, each bond is stressed during the shear unzipping (i.e. one adds all the spring constants of each nucleotide), whereas for long segments only those nucleotides within <math>\kappa^{-1}</math> (to first order) are included (this turns out to be approximately <math>\frac{1}{\kappa a}</math> nucleotides).

Figure 2 shows the extension curves (normalized by the separation of each adjacent base pair, a) vs. the force (normalized by the rupture force of a single bond, <math>f_{0}</math>) for different hybridized lengths L. Rupture is indicated by a sharp upturn in <math>\Delta x</math>. The inset shows the strain profiles <math>\delta_{n}</math> for a <math>L = 80</math> system.

The most interesting part of this article, in my opinion, was seeing the connection between solid-state physics and soft matter physics.

## Reference

[1] B. Chakrabarti, and D. R. Nelson, "Shear Unzipping of DNA," The Journal of Physical Chemistry B **113** (12), 3831-3836 (2009).