Thermodynamics of DNA Loops with Long-Range Correlated Structural Disorder

We study the influence of a structural disorder on the thermodynamical properties of 2D-elastic chains submitted to mechanical/topological constraint as loops. The disorder is introduced via a spontaneous curvature whose distribution along the chain presents either no correlation or long-range correlations (LRC). The equilibrium properties of the one-loop system are derived numerically and analytically for weak disorder. LRC are shown to favor the formation of small loop, larger the LRC, smaller the loop size. We use the mean first passage time formalism to show that the typical short time loop dynamics is superdiffusive in the presence of LRC. Potential biological implications on nucleosome positioning and dynamics in eukaryotic chromatin are discussed.

Figure 1

Left: spontaneous trajectory for two 2D semiflexible chains with uncorrelated ($H=0.5$, black) and LRC ($H=0.8$, gray) structural disorder. Right: winding constraint (top) and cyclization constraint (bottom).

Figure 2

Single loop of size $l$ in a chain of length $L={10}^5$ under the winding constraint and for $A=200$, ${\sigma }_0=0.01$, and $\alpha =2\pi$. (a) Energy landscapes for a $l=200$ loop along chains with uncorrelated ($H=0.5$, black) and LRC ($H=0.8$, gray) disorders. (b) Free energy rms fluctuations $\Lambda (l)$ vs $l$; the symbols correspond to numerical estimate for different disorders: $H=0.5$ (○), 0.6 (□), 0.7 (☆), 0.8 (△), and 0.9 (◇); the solid lines correspond to Eq. (4). (c) Reduced correlation function $C(y,200)/\Lambda (200)$ vs $y$; the symbols have the same meaning as in (b); the solid curves correspond to Eq. (5).

Figure 3

Free energy of the single loop system ${\mathcal{F}}_H(l,L)$ vs $l$ under the cyclization constraint and for $L=15000$, $A=200$, $\alpha =2\pi$, and ${\sigma }_0=0.01$ (a) and 0.05 (b). The symbols correspond to a single chain’s free energy ${\mathcal{F}}_H(l,L)$ obtained from the (exact) numerical computation of $f(s,l)$ for $H=0.5$ (□, ■), 0.6 (□, ■), 0.7 (☆, ★), 0.8 (△, ▴) and 0.9 (◇, ◆); the (×) correspond to the pure case without disorder. The continuous curves stand for the corresponding quenched free energies ${\overline{\mathcal{F}}}_H(l,L)$ averaged over 100 “typical” single chain free energies computed using Eq. (3) for the enthalpic part (i.e., the winding energy) and $c_c=7/2$ for the entropy cost; the dotted curve corresponds to the exact expression for the pure case. (c) ${\mathcal{F}}_H(l)-{\mathcal{F}}_{1/2}(l)$ vs $H$ between LRC and uncorrelated chains, for loop size $l=200$; the dashed curves correspond to perturbative approximation. (d) Optimal loop length $l^{*}(H)$ vs $H$; the dashed curves correspond to the perturbative expression (10); the horizontal line indicates the optimal loop length $l^{*}=1128$ for the pure system. In (c) and (d) the symbols and the continuous curves have the same meaning as in (a) and (b).

Figure 4

MFPT $\tau (N)$ (measured in number of steps) for $A=l=200$, $\alpha =2\pi$, $L=15000$, and ${\sigma }_0=0.01$. (a) PDF of $\tau (N)$ calculated for 100 000 uncorrelated ($H=0.5$, black) and LRC ($H=0.8$, gray) chains for $N=100$ (thin) and 200 (thick). (b) Most probable MFPT vs $N$ for $H=0.5$ (○) and 0.8 (△); mean MFPT vs $N$ for $H=0.6$ (■) and 0.8 (▴); the dashed curves correspond to the analytical quenched average [Eq. (10)]; the dotted curves correspond to the small displacement approximation [Eq. (11)]; the continuous line corresponds to the pure diffusion case $\tau (N)=N^2$.