Modeling the mechanical properties of DNA nanostructures

We discuss generalizations of a previously published coarse-grained description [Mergell et al., Phys. Rev. E 68, 021911 (2003)] of double stranded DNA (dsDNA). The model is defined at the base-pair level and includes the electrostatic repulsion between neighbor helices. We show that the model reproduces mechanical and elastic properties of several DNA nanostructures (DNA origamis). We also show that electrostatic interactions are necessary to reproduce atomic force microscopy measurements on planar DNA origamis.

Figure 1

(a) AFM imaging of a rectangular origami, (b) section of the AFM picture, (c) schematic representation of the dsDNA strands as measured by AFM.

Figure 2

Representation of the different levels of description for a small origami, made of three single strands of DNA, noted respectively $B_1$, $B_2$, and $B_3$. (a) Scheme showing the connectivity of the ensemble: both $B_1$ and $B_2$ bind to $B_3$ (the scaffold), $B_1$ binds to two noncontiguous subsets of $B_3$. (b) SOP representation of this construction. Ellipses (in gray) representing base pairs are connected by springs (corresponding to the sugar-phosphate backbone, represented by cylinders). (c) Atomistic (ball and stick) representation. A ribbon has been superimposed to the sugar-phosphate backbone. (d) Zoom of the crossover region.

Figure 3

Illustration of the global Monte Carlo moves used in this paper. The structure before the move is made of two parallel straight helices linked by nine crossovers. (a) The structure after a pivot move. Panels (b) and (c) illustrate the structure after application of a crank-shaft move to the structure in (a). For this move, two randomly selected bases (noted $\alpha$ and ${\alpha }^{\prime }$) define an axis of rotation. (b) Side view, (c) top view.

Figure 4

Structure of the (a) 2-oDNA(298) and (c) 3-oDNA(298) structures. Panels (b) and (d) are alternative schematic representations of respectively the 2-oDNA(298) and 3-oDNA(298) structures that highlight the topology of the crossovers. Each color corresponds to a different staple.

Figure 5

Structure of a 4-oDNA(298) after 10${}^6$ MC steps (Sim 1).

Figure 6

(a) Vectors ${\mathbf{T}}_1$ and ${\mathbf{T}}_5$ used to define the global twist of the structure, made here with $n=3$ strands. (b) Evolution of the global twist ${\text{Tw}}_g$ for a 2-oDNA (298) structure in the three simulations Sim 1, Sim 2, and Sim 3. (c) Same data for a 3-oDNA(298) structure. The sampling ratio is ${210}^7$ MC steps.

Figure 7

In the absence of additional bonds, the combined effect of electrostatic repulsion and stacking attraction leads to unstacking of base pairs (a). The zoom in (b) represents the electrostatic interactions as red (dashed) arrows, the stacking interactions as green (continuous) arrows. (c) Extra liaisons added at the crossover are in green and red and framed by two circles.

Figure 8

Distance between two neighbor strands as a function of the position along the strand. A typical origami structure is displayed below.

Figure 9

Distance between two neighbor strands as a function of the position along the strand. Comparison for three values of the Debye length.

Figure 10

The correlation between the tangent vectors of a same strand as a function of the distance along the strand. For each structure, the symbols represent the measured correlation, the continuous lines represent the fitted data.

Figure 11

Four structures with 141 base pairs each and different distribution of the junctions, respectively (a) 4 crossovers, (b) 6, (c) 8, (d) 14. Notice that here we extend somewhat the notion of crossover to include the junctions at the ends.

Figure 12

Four different structures with 141 base pairs each and different distribution of the junctions, respectively, (a) 4 crossovers, (b) 6, (c) 8, (d) 14. Typical configurations observed at the end of the MC simulations.

Figure 13

Typical configurations obtained after MC equilibration for three different structures with 298 base pairs each, with, respectively (a) 2 rows, (b) 3 rows, and (c) 4 rows.

Figure 14

Structure of the simulated DAE structures. For each structure, the total number of bases is indicated at the bottom.

Figure 15

Umbrella sampling simulations of 2-DNA${}_p\left(141\right)$ and 5-DNA${}_p\left(141\right)$ structures. The end-to-end distance is varied from 7 to 45 nm, with overlapping intervals of length 1 nm. The data from the umbrella sampling are fitted by an analytic WLC model (continuous lines). Left panel: energy. Right panel: force.

Figure 16

Structure of an oDNA similar to the four-helix bundle of Ref. [28]. Top: side view of the structure. Bottom: shattered view of the same structure. The circles in the bottom panel show how the links between the two 2-oDNA${}_p$ structures are determined.