Figure 1

Elastic rod model of DNA. (a) The elastic rod fitted into an all-atom structure of DNA. (b) Parametrization of the elastic rod: shown are the centerline $\mathbf{r}\left(s\right)$ and the intrinsic local frame $({\mathbf{d}}_1,{\mathbf{d}}_2,{\mathbf{d}}_3)$. (c) The principal normal $\mathbf{n}\left(s\right)=\ddot{\mathbf{r}}∕\mid \ddot{\mathbf{r}}\mid$ and the binormal $\mathbf{b}\left(s\right)={\mathbf{d}}_3\times \mathbf{n}$ vectors, together with ${\mathbf{d}}_3$, form the natural (Frenet-Serret) local frame for the 3D curve $\mathbf{r}\left(s\right)$. (d) A coordinate frame associated with a DNA base pair. The vectors are defined according to the general convention stated in [44] except that ${\mathbf{d}}_1$ and ${\mathbf{d}}_2$ are swapped.Reuse & Permissions

Figure 2

(a) The expressed lac operon. The biomolecules involved are (1) lac repressor (shown deactivated by four bound lactose molecules), (2) the DNA of E. coli, (3) RNA polymerase (shown first bound to the promoter and then transcribing the lac operon genes), and (4) mRNA (shown as being transcribed by the RNA polymerase). The flag shows the “$+1$” base pair of the DNA where the genes of the lac operon begin. The operator sites, marked as ${\mathrm{O}}_{1-3}$, are shown as shaded rectangles; the position of each operator’s central base pair is shown in brackets. (b) The repressed lac operon. The lac repressor is shown binding two operator sites, either ${\mathrm{O}}_1$ and ${\mathrm{O}}_3$ or ${\mathrm{O}}_1$ and ${\mathrm{O}}_2$, and the RNA polymerase is shown released from the operon. The end-to-end length of the DNA loop formed in each case is indicated. (c) The crystal structure of the lac repressor [37]. The DNA loop, missing in the crystal structure and shown here in light color, corresponds to an all-atom model fitted to one of the two elastic rod structures predicted for the 76 bp loop (see [18] and Sec. 3B). Small pieces of the crystal structure are omitted from the figure for clarity. (d) Same as (c) for the 385 bp loop; the all-atom DNA model is fitted to one of the four possible structures of the loop (see Sec. 3C and Fig. 6).Reuse & Permissions

Figure 3

Evolution of the elastic rod structure during the solution of the BVP for the short loop. (a) The initial solution: a closed circular loop. (b) The solution after the first iteration cycle. (c), (d) The solutions after the second iteration cycle, for the clockwise (c) and counter-clockwise (d) rotation of the $s=1$ end. (e), (f) The solutions after the third iteration cycle; the previous solutions are shown in light color; the views from the top include the forces that the DNA loop exerts on the DNA segments bound to the lac repressor. The protein-bound DNA segments from the lac repressor crystal structure are shown for reference only, as they played no role during the iteration cycles except for providing the boundary conditions.Reuse & Permissions

Figure 4

Extraneous solutions to the BVP obtained for a different orientation ${\psi }_{°}$ of the initial circular loop. The initial loop from Fig. 3 (a) is shown in panel (a) in light color.Reuse & Permissions

Figure 5

Distribution of curvature, twist, and energy in the elastic rod BVP solutions (U—top panel, O—bottom panel) for the 76 bp DNA loop. In each panel, left column: the isotropic solution with $\alpha =\beta =2∕3$; center column: the anisotropic solution with $\alpha =4∕15$, $\beta =16∕15$; right column: the anisotropic solution with electrostatics, for an ionic strength of $10\mathrm{mM}$.Reuse & Permissions

Figure 6

Four solutions of the BVP problem for the 385 bp long DNA loop. Underwound solutions are marked by the letter “U,” overwound ones by “O.”Reuse & Permissions

Figure 7

Changes in the predicted structure of the elastic loop due to the bending anisotropy. (a) The underwound (U) solution. (b) The overwound (O) solution. Structures obtained for $\alpha =\beta =2∕3$ are shown in light color; structures obtained for $\alpha =4∕15$, $\beta =16∕15$ are shown in dark color.Reuse & Permissions

Figure 8

Energy and geometry of the 76 bp loops in the broad range of elastic moduli $\alpha$ and $\beta$. The plots show the dependencies using coordinates $\mu =\alpha ∕\beta$ and $A∕C=2\alpha \beta ∕(\alpha +\beta )$ (30). (a) 3D plots for the elastic energy $U$ of the U loop, the average unwinding $\omega$ of the loop, and the rmsd of the loop centerline from that in the isotropic case $\mu =1$. The rmsd values are measured in DNA helical steps $h=3.4\mathrm{\AA }$. To present the best view, the plots for rmsd and $\omega$ are shown from a different viewpoint than that for $U$. The inset shows the contour line for $U=20\mathrm{kT}$, which is the DNA looping energy estimated from experiment [61]. (b) Cross sections of the maps in (a) for the fixed values of $A∕C=2∕3$ (top row) and $\mu =1$ (bottom row). The plots illustrate the behavior of the loop structure in response to changing the rigidity $(A∕C)$ and bending anisotropy $\left(\mu \right)$ of the loop. (c) Snapshots of the loop structures for the stated values of $A∕C$. The darkly colored loops correspond to $\mu =0.01$, and the lightly colored loops to $\mu =100$. For the U solution, the two loop structures practically overlap for all $A∕C$ values. (d)–(f) Same as (a)–(c) for the O solution. Note the more pronounced dependence of the O loop structure and energy on the bending anisotropy $\mu$.Reuse & Permissions

Figure 9

(a) Dependence of the elastic energy difference $\Delta U=U_{\mathrm{O}}-U_{\mathrm{U}}$ between O and U solutions, plotted in coordinates $\mu =\alpha ∕\beta$ and $⟨\alpha ⟩=(\alpha +\beta )∕2$. (b) Contours of the plot in (a) for $\Delta U$ values of $1,2,\dots ,7\mathrm{kT}$.Reuse & Permissions

Figure 10

Elastic energy difference between U and (a) O, (b) ${\mathrm{U}}^{\prime }$, and (c) ${\mathrm{O}}^{\prime }$ solutions for the 385 bp long DNA loop. 3D plots of the energy difference in the coordinates $\mu =\alpha ∕\beta$ and $⟨\alpha ⟩=(\alpha +\beta )∕2$ are shown on the left, and contour maps of the 3D plots for the $\Delta U$ values of 0.5, 1.0, 1.5, 2.0, 2.5, and $3.0\mathrm{kT}$ are shown on the right.Reuse & Permissions

Figure 11

Electrostatic interactions in the elastic rod problem. The electric field $\mathbf{E}\left(s\right)$ is computed at each point $s$ of the rod as the sum of the “external” field ${\mathbf{E}}^{\mathit{ext}}$, produced by the charges $q_i$ not associated with the elastic rod, and the “internal” field ${\mathbf{E}}^{\mathit{int}}$, produced by the charges $q_j^{\mathit{int}}=2e\chi$ placed in the maxima of the charge density $\sigma \left(s\right)$ of the elastic rod (32). $\Delta {\mathbf{r}}_i\left(s\right)=\mathbf{r}\left(s\right)-{\mathbf{R}}_i$ and $\Delta {\mathbf{r}}_j\left(s\right)=\mathbf{r}\left(s\right)-\mathbf{r}\left(s_j\right)$ [cf. Eq. (33)]. The part of the rod that lies within the cutoff $\delta s$ does not contribute to $\mathbf{E}\left(s\right)$.Reuse & Permissions

Figure 12

Changes in structure and energy of the 76 bp DNA loops after electrostatic interactions are taken into account. Left column: U solution; right column: O solution. (a), (b) Elastic (E), electrostatic (Q), and total (T) energy of the modeled loop versus the electrostatic weight $w_E$. (c), (d) Uncharged ($w_E=0$, in light color) and completely charged ($w_E=1$, in dark color) structures of the loops. The bottom views are rotated by 70° around the vertical axis relative to the top views. (e), (f) The rmsd between the charged and uncharged loop structures. The data correspond to $\alpha =4∕15$, $\beta =16∕15$, ionic strength $c_s=10\mathrm{mM}$, and the electrostatics exclusion radius $\delta s=H$.Reuse & Permissions

Figure 13

The effect of salt on the structure and energy of the 76 bp DNA loops. Left column: U solution; right column: O solution. (a), (b) Elastic (E), electrostatic (Q), and total (T) energy of the elastic loop versus the ionic strength $c_s$. Shown are only the plots for the exclusion radius of $H$; the energy plots for the other exclusion radii are almost indistinguishable. (c), (d) Snapshots of the elastic loop structures for $10\mathrm{mM}$ (dark color), $25\mathrm{mM}$ (medium color), and $100\mathrm{mM}$ (light color). Points where the snapshots were taken are shown as dots of the corresponding colors on the axes of panels (a), (b), (e), (f). The bottom views are rotated by 70° around the vertical axis relative to the top views. (e), (f) The rmsd of the loop structures from those computed without electrostatics (equivalent to infinitely high salt concentration). The lines, from top to bottom, correspond to the exclusion radii of $H$, $1.5H$, and $2H$.Reuse & Permissions

Figure 14

The effect of ionic strength on structure and energy of the 385 bp DNA loops. (a) Elastic (E), electrostatic (Q), and total (T) energy of the U, O, ${\mathrm{U}}^{\prime }$, and ${\mathrm{O}}^{\prime }$ loops plotted versus the ionic strength $c_s$. (b) The rmsd of the loops from their uncharged structures. The missing points in the ${\mathrm{U}}^{\prime }∕{\mathrm{O}}^{\prime }$ plot indicate solutions, missing due to nonconvergence of the iterative procedure (see text). The data correspond to $\delta s=3H$.Reuse & Permissions