DNA as a programmable viscoelastic nanoelement

Abstract

The two strands of a DNA molecule with a repetitive sequence can pair into many different basepairing patterns. For perfectly periodic sequences, early bulk experiments of Pörschke indicate the existence of a sliding process, permitting the rapid transition between different relative strand positions. Here, we use a detailed theoretical model to study the basepairing dynamics of periodic and nearly periodic DNA. As suggested by Pörschke, DNA sliding is mediated by basepairing defects (bulge loops), which can diffuse along the DNA. Moreover, a shear force f on opposite ends of the two strands yields a characteristic dynamic response: An outward average sliding velocity v approximately 1/N is induced in a double strand of length N, provided f is larger than a threshold fc. Conversely, if the strands are initially misaligned, they realign even against an external force f < fc. These dynamics effectively result in a viscoelastic behavior of DNA under shear forces, with properties that are programmable through the choice of the DNA sequence. We find that a small number of mutations in periodic sequences does not prevent DNA sliding, but introduces a time delay in the dynamic response. We clarify the mechanism for the time delay and describe it quantitatively within a phenomenological model. Based on our findings, we suggest new dynamical roles for DNA in artificial nanoscale devices. The basepairing dynamics described here is also relevant for the extension of repetitive sequences inside genomic DNA.

FIGURE 1

DNA under a shear force. (a) A nonperiodic sequence unravels from both ends, driven by the length gain from converting stacked bases into longer single strands. (b) A periodic DNA sequence can open by sliding, mediated by bulge loops that are created at the ends and diffuse freely along the DNA. When a bulge loop reaches the opposite end, the two strands have effectively slipped against each other by a distance equal to the loop size.

FIGURE 2

DNA free-energy model. The free energy E(S) of a basepairing pattern S contains three separate contributions: first, a negative binding energy for basepairing. For simplicity, we assign the same binding energy for every basepair of type k, regardless of the neighboring bases. Second, a positive free energy cost for internal and bulge loops. We assign the same cost ɛ_ℓ for every loop, regardless of its length and base sequence, since the detailed choice of the loop cost function does not affect our main findings. Third, a stretching energy. For a given pattern S, the stretching energy can be written in the form −f L(S), with an effective length L(S), which is obtained from force-dependent base-to-base distances and for double and single strands, respectively. Note that L(S) does not correspond to the physical length of the DNA molecule (see main text).

FIGURE 3

Viscoelastic response of periodic DNA. (a) The shear force on an 80-bp dsDNA (with two 20-bp ssDNA linkers) is switched periodically between f_min = 11.4 pN and f_max = 19 pN (upper panel). The center and bottom panels show the extension-time-trace for heterogeneous and periodic DNA, respectively (energy parameters: ɛ_b = 1.11 k_BT and ɛ_ℓ = 2.8 k_BT, roughly corresponding to AT basepairs at 50°C (18)). The time units are Monte Carlo steps, the real-time equivalent of which is discussed in Kinetic Rates (see article). The heterogeneous DNA responds only elastically to the force jumps, mostly due to stretching of the linkers. The length of the periodic DNA shows a similar elastic strain, but in addition, the molecule elongates at a finite speed due to sliding, since f_max > f_c = 16.3 pN. When the molecule is relaxed, we find an elastic response and inward sliding, since f_min < f_c. The length of the periodic DNA fluctuates strongly due to loop formation and annihilation. (b) The viscoelastic behavior displayed by a periodic DNA molecule can be described by a generalized Zener model, where harmonic springs are substituted by anharmonic elastic elements describing polymer elasticity and the restoring force f_c. The ideal dashpot (with viscosity η) creates the viscous behavior of periodic dsDNA. (c) The response of the above idealized model to the same periodic force resembles the average response of periodic DNA.

FIGURE 4

(Top) Extension-time-trace for a perfectly periodic DNA of N = 120 basepairs and the same DNA with seven mutations, both under a shear force of f = 12.7 pN (energy parameters as in Fig. 3). Whereas the molecule without mutations starts sliding almost immediately, the molecule with mutations fluctuates about its initial length for some time τ_w before sliding starts. (Bottom) The time-trace of the binding state (open/closed) for the seven mutated basepairs in the sequence. Each mutated basepair (–7) is unbound wherever the corresponding thick horizontal line is broken, and bound where the line is shown. Note that the mutated basepairs do not open/close independently from each other. Instead, a mutated basepair opens only once all mutated basepairs to the left or right are already open. The black envelope curves emphasize the positions of the outermost bound mutation on each side. Their dynamics resembles a (biased) random walk. Sliding begins when all mutations are open.

FIGURE 5

Waiting time distributions. (Main panel) The histogram of waiting times τ_w of a 120-bp-long DNA sequence with M = 7 equidistant mutations subject to a force is well described by the distribution of collision times (dashed line) of the two-random-walker model (see main text and Fig. 6). The solid line shows the parameter-free asymptotic distribution of Eq. 5 for comparison. (Insets) Distribution of τ_w for forces above and below (f = 15.2 pN and f = 11.4 pN, respectively). The dashed lines are fits using the RW model with directional bias (see main text).

FIGURE 6

On a coarse-grained level, the dynamics of mutation opening/closing can be described by a model of two particles hopping on a one-dimensional lattice, with inward/outward hopping rates k_in, k_out. Their positions represent the two outermost closed mutations. When the particles collide, all mutations have opened. Equivalently, one can consider a single particle hopping on a triangular two-dimensional lattice. The first collision time then corresponds to the time to reach the diagonal absorbing boundary.

FIGURE 7

(a) The mean waiting time 〈τ_w〉 as a function of the system size (the mutation density of ν = 1/15 is kept fixed as the number of evenly spaced mutations M is increased). At low forces the scaling is exponential (circles, data for f = 11.4 pN; solid line, exponential fit), while we find power-law behavior at the force threshold (= 12.9 pN, squares) and above (f = 15.2 pN, diamonds). (b) The mean waiting time as a function of the applied force for a sequence of N = 240 bp with 5, 9, and 15 mutations. The dashed lines indicate the threshold force for each case. Below the threshold, 〈τ_w〉 rises sharply. (c) Different regimes of the DNA dynamics in the parameter space (f, ν). The Kramers regime (DNA rupture becomes exponentially slow with increasing system size) is separated from the (delayed) sliding regime by the line where the inward and outward hopping rates are equal, k_in = k_out (circles, data; solid line, interpolation). At forces larger than f*, the molecule dissociates by unraveling from both ends.