Simple, but Not Too Simple: Modeling the Dynamics of DNA and RNA Buckling

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Imagine a straight elastic rod subject to an axial compressing force. If the force is less than a certain critical value, the rod remains straight; however, if the critical force is exceeded, the straight shape becomes unstable, and a large bending of the rod occurs. Similarly, a straight rod subject to torsion undergoes a lateral deformation for torques above a critical level. These are just two examples of the phenomenon of elastic instability, first studied by Leonhard Euler (1). In Euler’s time, the main construction materials were wood and stone, requiring robust structural elements for which the problem of elastic stability is less important. It was only with the advent of modern bridges, ships, and aircraft that his theory found extensive practical applications and has been largely developed (2).

In the age of biology, can we make use of these developments to study elastic phenomena at the subcellular level? There are no metallic struts and shells there. Rather, we are confronted with macromolecular structures consisting of intricately assembled building blocks. To what extent these structures can be approximated by simple elements such as elastic rods with uniform material properties is not evident. But there is yet another aspect not relevant in the macroscopic world: because all the objects inside the cell are so small, thermal fluctuations of their shapes cannot be neglected. Everything moves there, and the exact onset of instability may be less important than the range of conformations accessible to a particular structure.

A prominent elastic object inside a cell is its DNA, a long, double-helical macromolecule storing genetic information in all cellular life. During fundamental cellular processes such as replication and transcription, DNA is subjected to torsional strains. The torsional state of DNA can be defined by the value of the linking number, Lk, the number of crosses one strand makes across the other. In the relaxed state, Lk is equal to Lk₀, which increases by 1 every 10.5 basepairs, the helical repeat of the B-form DNA. A nonzero excess linking number Lk − Lk₀ indicates a supercoiled state. In the cell, DNA supercoiling plays a role in genome packaging and in transcription regulation. When torsionally strained, a straight piece of DNA will eventually undergo a buckling transition and form an intertwined structure, the plectoneme (Fig. 1), partly releasing the strain. An analogous phenomenon occurs in double-helical RNA, a molecule present in RNA viruses and involved in gene regulation by RNA interference. Quantitative understanding of the process of DNA and RNA buckling has been a challenge for experiment, theory, and simulation. Rather than taking up a single shape as in the case of a macroscopic object, the torsionally strained DNA or RNA may constantly switch between the prebuckling and postbuckling state because of its thermal agitation. The process can be observed experimentally using the magnetic tweezer setup, in which one end of a kilobase-long DNA or RNA is fixed to a surface and the other end to a micrometer-sized magnetic particle (3). The particle can be manipulated by an external magnet to impose a defined linking number, and its vertical position can be monitored. At a certain excess linking number, the system reaches the buckling point, in which the probabilities (or, equivalently, the mean dwell times) of both states are equal (Fig. 1).

Figure 1.

Buckling transitions induced by torsional stress. At the buckling point, the mean dwell times for the prebuckling and postbuckling states are equal. The switching between the two states is much slower for RNA than for DNA.

What properties of the DNA or RNA molecule govern the buckling, and what are the structural details of the transition? It is up to theory and simulation to answer these questions. The work by Ott et al. (4), in this issue of Biophysical Journal, makes an important contribution in this direction.

Ott et al. (4) simulate the buckling process using the worm-like chain model of DNA and RNA. In their model, the molecule is represented by a straight elastic rod with uniform, isotropic bending and torsional stiffness. Electrostatic interactions and excluded volume terms are added, and the model is discretized into a chain of beads. Thermal agitation is mimicked by stochastic forces in the equations of motion, which are integrated using a Brownian dynamics numerical scheme. Hydrodynamic interactions between different parts of the molecule are neglected. The magnetic particle and the surface are modeled by planar excluded volume potential barriers. For computational efficiency, the simulated molecule may be shorter than the one in the experiment. This difference and the missing viscous drag from the magnetic particle are circumvented by an elegant rescaling scheme to achieve model similarity. The chain is kept stretched by a force in the piconewton range.

The model is first verified for DNA by comparing observables such as the linking number and the jump size Δz at the buckling point (Fig. 1) to experimental values (5,6). The agreement is quantitative. Thus, this simple model that neglects many aspects of DNA structure and elasticity already faithfully captures equilibrium properties of DNA buckling. This adds confidence to the detailed structural changes at the buckling transition reported by the authors. Furthermore, they found that although the mean dwell time at the buckling point (the buckling time) exponentially increases with the bending rigidity, it is nearly insensitive to the torsional rigidity of the molecule. Because RNA has higher bending stiffness than DNA, it is expected that the RNA buckling time will be much longer. The model passes also this critical test, at least qualitatively; although the experimental ratio of RNA and DNA buckling times is 40 (at the 2-pN stretching force), the model predicts a factor of 7.

How significant is this discrepancy? Assuming an Arrhenius kinetics at the buckling point, the buckling time τ_b is given by

$${\tau }_b={\tau }_0\text{exp}\left(\Delta G^{\#}/k_{B}T\right),$$

where ΔG^# is the free-energy barrier between the (isoenergetic) prebuckling and postbuckling states at the buckling point. Inserting the ratios of the buckling times into the equation, we find that the barrier for RNA is higher than that for DNA by 3.7 k_BT, or roughly 2.2 kcal/mol, for the experiment and 1.9 k_BT (1.2 kcal/mol) for the simulation.

The 1-kcal/mol discrepancy in the barrier height may occur for many reasons, but three of them stand out. First, the DNA and RNA in the model differ essentially just by the higher RNA bending stiffness. However, the real double-stranded RNA adopts the A-form, exhibiting groove dimensions and the sugar-phosphate backbone conformation that is quite different from DNA. These structural details may influence the initial formation (nucleation) of the plectonemes. Furthermore, the DNA and RNA helices are not uniform but exhibit sequence-dependent variations of their structures and stiffness. At the kilobase scale, these variations may enter in the form of a structural disorder known to affect the buckling energy barrier (7). Third, the hydrodynamic interactions are neglected in the model to speed up the computations; here, a recently proposed scheme, scaling as O(N) instead of O(N³) with the number of particles (8), may be a promising option. Before introducing these refinements, it would be extremely interesting to see how the simple model behaves with respect to other experimental results; for instance, whether it can reproduce the dependence of the jump size on the stretching force that decreases linearly for RNA (3).

The study of Ott et al. (4) exposes the power and the limits of the worm-like chain model, probably the simplest possible physical description of flexible DNA and RNA helices, to capture the buckling phenomenon. Its success is astonishing, its limits are revealing, and the work is a pleasure to read.

Editor: Jason Kahn.