Electrostatic contribution to twist rigidity of DNA

The electrostatic contribution to the twist rigidity of DNA is studied, and it is shown that the Coulomb self-energy of the double-helical sugar-phosphate backbone makes a considerable contribution—the electrostatic twist rigidity of DNA is found to be $C_{\text{elec}}\approx 5\text{nm}$, which makes up about $7\%$ of its total twist rigidity $(C_{\text{DNA}}\approx 75\text{nm})$. The electrostatic twist rigidity is found, however, to depend only weakly on the salt concentration, because of a competition between two different screening mechanisms: (1) Debye screening by the salt ions in the bulk, and (2) structural screening by the periodic charge distribution along the backbone of the helical polyelectrolyte. It is found that, depending on the parameters, the electrostatic contribution to the twist rigidity could stabilize or destabilize the structure of a helical polyelectrolyte.

Figure 1

A schematic picture of double-helical $B$-DNA with the negative charges lying on the sugar-phosphate backbone in a periodic manner.

Figure 2

The diagram delineating the different regimes in the parameter space of a helical polyelectrolyte, where ${\omega }_0=2\pi ∕P$, and $a$ is the radius of the helix. The line separating the two screening regimes has slope 1, whereas the slope of the boundary denoting the onset of instability in the Debye screening regime is set by the inverse of a cutoff number $n_{\mathrm{c}}$ (see below). A stable (unstable) state corresponds to positive (negative) electrostatic twist rigidity only. The overall stability of the macromolecule, however, is determined by the total twist rigidity which consists of both mechanical and electrostatic contributions (see Sec. 4).

Figure 3

A schematic picture of the surface charge distribution of $B$-DNA. The geometrical parameters of DNA are shown in the picture: $b$ is the vertical distance between two successive charges on each strand, $\zeta$ is the distance between the two strands along the $z$ axis (given by the width of the minor groove in $B$-DNA), and $P$ is the pitch of the helix.

Figure 4

$C_{\text{elec}}∕C_0$ as a function of $a{\omega }_0$. This plot corresponds to $\kappa a=1.2$ and $\zeta {\omega }_0=2.1$.

Figure 5

$C_{\text{elec}}∕C_0$ as a function of $a{\omega }_0$. This plot corresponds to $\kappa a=0.2$ and $\zeta {\omega }_0=2.1$.

Figure 6

$C_{\text{elec}}∕C_0$ as a function of $\zeta {\omega }_0∕\pi$. This plot corresponds to $\kappa a=1.2$ and $a{\omega }_0=2.2$.

Figure 7

$C_{\text{elec}}∕C_0$ as a function of $\kappa a$. This plot corresponds to $\zeta {\omega }_0=2.1$ and $a{\omega }_0=2.2$.

Figure 8

$W_{\text{elec}}∕W_0$ as a function of $a{\omega }_0$. This plot corresponds to $\kappa a=1.2$ and $\zeta {\omega }_0=2.1$.

Figure 9

$W_{\text{elec}}∕W_0$ as a function of $a{\omega }_0$. This plot corresponds to $\kappa a=0.2$ and $\zeta {\omega }_0=2.1$.

Figure 10

$W_{\text{elec}}∕W_0$ as a function of $\zeta {\omega }_0$. This plot corresponds to $\kappa a=1.2$ and $a{\omega }_0=2.2$.

Figure 11

$W_{\text{elec}}∕W_0$ as a function of $\kappa a$. This plot corresponds to $\zeta {\omega }_0=2.1$ and $a{\omega }_0=2.2$.

Figure 12

Difference between the asymptotic form of $I_n\left(nx\right)K_n\left(nx\right)$ and its exact value as a function of $n$ for several values of $x$. The solid line corresponds to $x=0.8$, the dashed line corresponds to $x=1.8$, and the dotted line corresponds to $x=2.8$. The difference is less than $0.25\%$.

Figure 13

Difference between the asymptotic form of $I_n\left(nx\right)K_n\left(nx\right)$ and its exact value as a function of $x$ for different values of $n$. The solid line corresponds to $n=2$, the dashed line corresponds to $n=3$, and the dotted line corresponds to $n=5$. The difference is less than $0.3\%$.

Figure 14

The auxiliary functions $f_0\left(x\right)$, $f_2\left(x\right)$, and $f_4\left(x\right)$.

Figure 15

The auxiliary functions $g_0\left(x\right)$, $g_2\left(x\right)$, and $g_4\left(x\right)$.