Multiscale modeling of double-helical DNA and RNA: a unification through Lie groups

Abstract

Several different mechanical models of double-helical nucleic-acid structures that have been presented in the literature are reviewed here together with a new analysis method that provides a reconciliation between these disparate models. In all cases, terminology and basic results from the theory of Lie groups are used to describe rigid-body motions in a coordinate-free way, and when necessary, coordinates are introduced in a way in which simple equations result. We consider double-helical DNAs and RNAs which, in their unstressed referential state, have backbones that are either straight, slightly precurved, or bent by the action of a protein or other bound molecule. At the coarsest level, we consider worm-like chains with anisotropic bending stiffness. Then, we show how bi-rod models converge to this for sufficiently long filament lengths. At a finer level, we examine elastic networks of rigid bases and show how these relate to the coarser models. Finally, we show how results from molecular dynamics simulation at full atomic resolution (which is the finest scale considered here) and AFM experimental measurements (which is at the coarsest scale) relate to these models.

Figure 1

An overview of the models explored and some of the connections made between them.

Figure 2

Two examples of how helical RNA can be used to create nanostructures.

Figure 3

A parametrized set of reference frames defining an elastic filament.

Figure 4

Five examples of DNA bending. Images generated from PDB data.^–

Figure 5

Two ensembles of DNA conformations as measured using an AFM: (left) cisplatinated DNA; (right) naked DNA.

Figure 6

Two simulated ensembles of stochastic trajectories: (left) with a 36° bend; (right) without a bend.

Figure 7

Distribution of the distal end position of the model relative to the proximal end: (left) with a 36° bend; (right) without a bend.

Figure 8

The relationship between reference frames in the bi-rod model.

Figure 9

Examples of how the effective stiffness matrix, K(s), varies for the values associated with rotation. Results are shown for two different sets of parameters: (left) c₁ = 10, c₂ = 2, c₃ = 2, c₄ = 1; (right) c₁ = 10, c₂ = 4, c₃ = 5, c₃ = ¹/₂.

Figure 10

Examples of how the effective stiffness of the bi-rod model, K(s), differs from its steady-state value, K_ss, for various values of W_l/r and W. Here the Frobenius norm is used.

Figure 11

GCGC autocorrelation plot for the singular values of the cross-covariance matrix.

Figure 12

GCCG autocorrelation plot for the singular values of the cross-covariance matrix.

Figure 13

Probability density function with fit for the GCGC sequence for x₁.

Figure 14

Probability density function with fit for the GCGC sequence for x₄.

Figure 15

Chi-squared plot for the GCGC sequence parameters.