Length-scale-dependent elasticity in DNA from coarse-grained and all-atom models

We investigate the influence of nonlocal couplings on the torsional and bending elasticities of DNA. Such couplings have been observed in the past by several simulation studies. Here, we use a description of DNA conformations based on the variables tilt, roll, and twist. Our analysis of both coarse-grained (oxDNA) and all-atom models indicates that these share strikingly similar features: there are strong off-site couplings for tilt-tilt and twist-twist, while they are much weaker in the roll-roll case. By developing an analytical framework to estimate bending and torsional persistence lengths in nonlocal DNA models, we show how off-site interactions generate a length-scale-dependent elasticity. Based on the simulation-generated elasticity data, the theory predicts a significant length-scale-dependent effect on torsional fluctuations but only a modest effect on bending fluctuations. These results are in agreement with experiments probing DNA mechanics from single base pair to kilobase pair scales.

Figure 1

(a) Toy model of length-scale-dependent elasticity consisting of a linear chain with neighbors and next-neighbor springs with stiffnesses $K$ and $K^{\prime }$, respectively [Eq. (1)]. (b) Momentum space stiffness of the model (7) for $K=1$ and $K^{\prime }=3$. The one-step $K_1$, two-step $K_2$, and asymptotic stiffnesses $K_{\infty }={\tilde{K}}_0$ [Eqs. (12, 13, 14)] are shown. In the case shown here ($K^{\prime }>0$), the system is softer at short scales: $K_1<K_2<...<K_{\infty }$.

Figure 2

Schematic illustration of the effect of a local perturbation at site $n=0$ resulting in an exponentially decaying stretching profile $\langle u_m\rangle$, see Eq. (18). This depiction represents the case $K^{\prime }<0$, where the stretching decays monotonically (for the sake of clarity, we do not show next-neighbor springs).

Figure 3

Mapping of a DNA configuration into a rigid base-pair representation [23] that consists of a series of triads each attached to a single basepair, capturing the local geometry of the molecule. These triads are constructed from a set of three mutually orthogonal unit vectors ${\widehat{\mathcal{T}}}_n=\left({\widehat{\mathbf{f}}}_n{\widehat{\mathbf{v}}}_n{\widehat{\mathbf{u}}}_n\right)$, where ${\widehat{\mathbf{u}}}_n$ is the local tangent, ${\widehat{\mathbf{v}}}_n$ connects the two backbones and ${\widehat{\mathbf{f}}}_n$ points toward the major groove. Deformations of the chain are parametrized by the rotation vectors ${\mathrm{\Theta }}_n$ rotating the triads ${\widehat{\mathcal{T}}}_n$ into their sequentially adjacent triads ${\widehat{\mathcal{T}}}_{n+1}$.

Figure 4

(a) Red dots: Simulation data reporting the entries of the stiffness matrix in momentum space ${\tilde{M}}_q$ for oxDNA2 as obtained from Eq. (27) for a sequence of length 150. In the analysis, two nucleotides at the two ends were eliminated, which gives 146 triads and thus $N=145$ deformation vectors ${\mathrm{\Delta }}_m$. The units are in nm. The entry ${\tilde{A}}_q^{tr}$ has been multiplied by a factor of 10 to facilitate its visibility. The stiffness matrix has the structure given in (26). All its entries are symmetric in $q$, except for the tilt-roll term ${\tilde{A}}_q^{tr}$ which is antisymmetric. Blue dashed lines: Fits of the data to Eqs. (42) and (43), with fitting parameters given in Table 1. (b) Plots of $l_{\text{B}}$ and $l_{\text{T}}/2$ vs $m$ the relative distance in numbers of base-pair steps between the considered segments. Green lines are obtained from the stiffness matrix data using Eqs. (31) and (35). The red line is the approximation (C19). In this case, the difference between the two approximations for $l_{\text{B}}$ is very small. Black dashed lines are obtained by direct calculations of correlation functions from simulations. The oscillatory behavior of the bending persistence length stems from a light helicity of the traced contour.

Figure 5

Red dots and solid squares: Elements of the stiffness matrix ${\tilde{M}}_q$ as obtained from all-atom data for sequences of length (a) $N=20$ (average of 9 seq.) and (b) $N=32$ (average of three seq.). Dashed lines: Fits of the forms (42) and (43).

Figure 6

(a) Estimated length-scale dependence of the persistence lengths as obtained from the analysis of the all-atom data in Table 2. Assuming that these data are representatives for the behavior of very long sequences, we used Eqs. (35) and (31) to calculate $l_{\text{B}}$ and $l_{\text{T}}$ (green lines). The red line is the approximation (C19) for $l_{\text{B}}$. (b) Calculation of the propagation of perturbations induced by generalized forces acting on the site $m=0$. This data is calculated with Eq. (41) using the data in Table 2. Results are given in degrees (the quantities plotted are $180a\tau /\pi , 180a\rho /\pi$, and $180a\mathrm{\Omega }/\pi$).

Figure 7

Integration contours in the complex $y$ plane used for the evaluation of the integral (B1). The two cases correspond to (a) $K^{\prime }>0$ and (b) $-K/4<K^{\prime }<0$.

Figure 8

Monte Carlo simulations with positive (left) and negative (right) off-diagonal couplings. In both cases, couplings between step parameters up to two steps displaced were included. The black lines show the bending persistence length as deduced directly from the tangent-tangent correlation function [Eq. (34)]. Indicated in red is the expression derived for infinitesimal rotations [Eq. (C19)] and in green the improved expression [Eq. (C33)].

Figure 9

Entries of the momentum space coupling matrices ${\tilde{M}}_q$ for the full spectrum of rescaled momenta for all individual simulations. Results of the 21-mer ($N=15$) and 32-mer ($N=27$) simulations are plotted in black and red, respectively.