Statistics of loop formation along double helix DNAs

We compute relative position distributions of distant sites along discretized semiflexible polymers, focusing on encounter statistics for pairs of sites along a double-stranded DNA molecule (dsDNA), using a transfer-matrix approach. We generalize the usual semiflexible polymer, considering nonlinear elasticity effects arising from inhomogeneities which either appear at any position via thermal fluctuation, or which occur at specific “quenched” locations. We apply our theory to two problems associated with dsDNA looping. First, we discuss how local flexible defects in double-helix structure facilitate cyclization of short dsDNA molecules. Flexible defects greatly enhance cyclization rate, and strongly modify its dependence on the closure orientational boundary condition. This effect is relevant to free-solution cyclization experiments, and to loop formation in vivo. Second, we present calculations of force dependence of the probability of formation of loops along single dsDNAs which show how the probability of loop formation is suppressed by tension.

Figure 1

dsDNA synapsis and loop formation. (a) Synapsis between two dsDNA segment (gray curves), mediated by protein-DNA complexes (black circles). (b) Loop formation mediated by proteins bound at two specific positions along a dsDNA molecule; the boundary condition in this sketch has $a\approx 90°$ synapsis angle between the interacting sites. (c) Cyclization of a linear dsDNA molecule, by the action of the enzyme DNA ligase (black circle), which is thought to require a “parallel-end” boundary condition. (d) Role of localized sharp bends in facilitating dsDNA looping for parallel-end and antiparallel-end boundary condition.

Figure 2

Persistence lengths for fluctuating-defect models of dsDNA. In all cases, the segment length $b=1\mathrm{nm}$ and the unperturbed double helix bending constant $a=50$, corresponding to 50 nm persistence length. (a) Flexible-hinge model persistence length as a function of defect free energy $\mu$. The excited hinges have bending stiffness $a^{\prime }=1$. For $\mu >10$, the defects are so rare that they do not significantly perturb the net flexibility, but for $\mu <10$, the hinges reduce the net persistence length. (b) Hinge density as a function of $\mu$. For $\mu >10$, it is less than 1 hinge per 1000 nm of the molecule. (c) Spontaneous-bend “kink” model persistence length as a function of defect free energy $\mu$, for various preferred bend angles. The kink bend constant is fixed at $a^{\prime }=50$. In each case, large $\mu$ causes the persistence length to revert to the unperturbed 50 nm value; small $\mu$ greatly reduces the net persistence length. Results are shown for angles $\gamma =0$ (solid), $\pi ∕9$ (dash), $\pi ∕6$ (short dash), $\pi ∕4$ (dot), and $\pi ∕2$ (dash-dot). As the bending angle is increased, the net persistence length is gradually reduced. For a $\pi ∕2$ angle, the kink-dense model (low $\mu$) has essentially zero persistence length as defined by tangent vector correlations. (d) Spontaneous-bend model persistence length as a function of defect free energy $\mu$, at fixed preferred bend angle $\gamma =\pi ∕4$, for various kink flexibilities. Results are shown for $a^{\prime }=50$ (dash-dot-dot), 10 (dash-dot), 1 (dot), 0.1 (dash), and 0.01 (solid). As the kinks are made more flexible, the effect of the defects on persistence length becomes more pronounced.

Figure 3

Distribution of end-to-end radius for the simple semiflexible polymer model of dsDNA ($b=1\mathrm{nm}$, $a=50$, no hinge or kink defects), as a function of radius in units of molecule contour length $(R∕l_0)$), for the parallel-end boundary condition. Results are shown for molecular lengths 100 (rightmost peak), 170, 280, and 3000 (leftmost peak) nm, corresponding to sequence lengths of 300, 510, 840, and 9000 bp.

Figure 4

$J$ factors versus molecule length, for the simple semiflexible polymer with persistence length 50 nm. Results are shown for three different cyclization boundary conditions: parallel (lowest curve), antiparallel (middle curve), and free (highest curve).

Figure 5

Effect of thermally activated flexible hinges on cyclization of short dsDNAs. Hinge flexibility is $a^{\prime }=1$ (3 base persistence length); unperturbed double helix flexibility is $a=50$ (50 nm persistence length). (a) $J$ factor versus molecule length for parallel-end boundary condition. Results (from bottom to top) for $\mu =\infty$ (no hinge), 12, 11, and 10. Open circles show data of Widom and Cloutier [27]. (b) Influence of closure boundary condition on cyclization in the hinge model ($a=50$, $a^{\prime }=1$, and $\mu =11$). $J$-factor versus molecule length is plotted for parallel (squares), antiparallel (stars), and free (filled circles) closure boundary conditions. (c) Number of hinge defects on molecule in cyclized state, versus molecule length, for $\mu =11$. For molecules shorter than 300 bp, cyclization becomes affected by molecules with hinge excitations. Results are shown for the parallel boundary condition $(\text{solid}+\text{squares})$, and also for the antiparallel $(\text{dashed}+\text{filled circles})$ and free $(\text{dotted}+\text{stars})$ boundary conditions. (d) Nonlinear bending elasticity versus angle for hinge model, for defect energies $\mu =1$ (solid), 10 (dashed), and 20 (dotted). The elasticity shows a “crossover” behavior from the small-angle, large-scale persistence length $(a=50)$ of 50 nm, to a much smaller rigidity $(a^{\prime }=1)$, at a critical bend angle controlled by $\mu$.

Figure 6

Enhancement of $J$ factor of dsDNA $(a=50)$ by a single preferred-angle kink located at the midpoint of the molecule. The kink angle is $\gamma =\pi ∕2$ and is stiff $(a^{\prime }=50)$. $J$ factors are shown for parallel (filled circles) and free (filled squares) boundary conditions, in the absence of any other defects (i.e., no thermally excited hinges). $J$ factors are further affected by combining the fluctuating hinges and the permanent kink together (cross-in-circles for parallel boundary condition, and cross-in-squares for free boundary condition). $J$ factors of defect-free dsDNA are also shown for parallel (open circles) and free (open squares) boundary conditions.

Figure 7

Effect of thermally activated flexible hinges for $\mu =11$ on force-extension curve is negligible. (a) Force-extension curves for defect-free dsDNA (open circles) and for dsDNA with hinge fluctuation (filled circles); the two results coincide. (b) and (c) Persistence lengths for unperturbed dsDNA and dsDNA with hinge fluctuation are extracted from the force-extension curve at high force: 48.9 nm for unperturbed dsDNA, and 48.5 nm for dsDNA with hinge fluctuation.

Figure 8

End-to-end distributions of a $10\mu \mathrm{m}$ (30 kb) dsDNA, in longitudinal (force) and transverse directions. (a) Open circles are the longitudinal distribution, $\rho \left(z\right)$, at $f=0.1\mathrm{pN}$. Solid line is a Gaussian distribution with its peak matched to that of $\rho \left(z\right)$. (b) Open circles are transverse distribution, $\rho \left(x\right)$, at $f=0.1\mathrm{pN}$. It can be fit closely by a Gaussian distribution (solid line). (c) The variance of the transverse fluctuation is equal to the end-to-end distance divided by force in $k_{B}T$ units ($(k_{B}T⟨z⟩∕f)$, for all forces.

Figure 9

The general end-to-end distance distribution of a 334 nm (1000 bp) dsDNA at a force $f=0.04\mathrm{pN}$. Here $z$ is the coordinate of the force direction, and $x$ is the coordinates of a direction transverse to the force direction. The probability density is in units of ${\mathrm{nm}}^{-3}$.

Figure 10

Effect of force on $J$ factors of dsDNA with different lengths. Results are shown for a 2 kb unperturbed dsDNA (filled squares), an 1 kb unperturbed dsDNA (stars), 0.5 kb without fluctuating hinge defects (filled circles), and 0.5 kb with fluctuating hinge defects ($\mu =11$, $a^{\prime }=1$, open circles).