Self-avoiding wormlike chain model for double-stranded-DNA loop formation

We compute the effects of excluded volume on the probability for double-stranded DNA to form a loop. We utilize a Monte Carlo algorithm for generation of large ensembles of self-avoiding wormlike chains, which are used to compute the J factor for varying length scales. In the entropic regime, we confirm the scaling-theory prediction of a power-law drop off of $-1.92$, which is significantly stronger than the $-1.5$ power law predicted by the non-self-avoiding wormlike chain model. In the elastic regime, we find that the angle-independent end-to-end chain distribution is highly anisotropic. This anisotropy, combined with the excluded volume constraints, leads to an increase in the J factor of the self-avoiding wormlike chain by about half an order of magnitude relative to its non-self-avoiding counterpart. This increase could partially explain the anomalous results of recent cyclization experiments, in which short dsDNA molecules were found to have an increased propensity to form a loop.

Figure 1

WLC and SAWLC chains. A discrete chain is described by the locations of its joints ${\mathbf{r}}_i$, and a local coordinate system defined by three orthonormal vectors $\widehat{u_i},{\widehat{v}}_i,{\widehat{t}}_i$ at each link. The vector ${\widehat{t}}_i$ points along the direction of the ith link. (a) A WLC chain with five links and link length $l=1$. (b) A SAWLC chain with three links and $l=w=1$. ${\widehat{u}}_i$ and ${\widehat{v}}_i$ are not shown. (c) Representation of an ideally modeled chain with finite width.

Figure 2

Illustration of bond formation criteria. The termini and their bond directions are denoted by ${\mathbf{r}}_0,{\mathbf{r}}_N$ and ${\widehat{t}}_1,{\widehat{t}}_N$ respectively. The minimal possible separation between the termini is denoted by $d_{\min }$. The cross section of the volume $\delta \mathbf{r}$ in which ${\mathbf{r}}_N$ must reside is shown as a green (dark gray) wedge. The possible directions for ${\widehat{t}}_N$ are shown as a blue (light gray) cone with its base at ${\mathbf{r}}_N$. The rest of the chain is shown schematically with blue (light gray) arcs.

Figure 3

Forbidden directions for ${\widehat{t}}_i$. Purple (light gray) sphere is the excluded volume around ${\mathbf{r}}_{i-1}$ [the endpoint of the $(i-1)\text{th}$ link]. The red (dark gray) sphere is a potentially colliding object. If ${\widehat{t}}_i$ assumes any direction inside the black (dark gray) cone this would result in a collision of the hard-wall sphere at ${\mathbf{r}}_i$ with the red sphere.

Figure 4

${\theta }_i$ ranges mapping. ${\theta }_i$ ranges that do not cause a collision for all values of ${\varphi }_i$ are marked in green (dark gray). ${\theta }_i$ ranges that cause a collision for some values of ${\varphi }_i$ are marked in yellow (light gray). ${\theta }_i$ ranges that cause a collision for all values of ${\varphi }_i$ are marked in red (medium gray). (a) If ${\theta }_i\in [{\theta }_{+},{\theta }_{-}]$, a collision occurs for some values of ${\varphi }_i$. (b) If ${\theta }_i\in [0,{\theta }_{+}]$, a collision occurs for all values of ${\varphi }_i$.

Figure 5

Comparison of the simulation results to the RG calculations and WLC theory. Power-law exponent $\nu$ of the end-to-end distance $\sqrt{\langle R^2\rangle }$ is plotted as a function of chain length $L/b$. We compare three cases (top to bottom): a freely jointed “thick” polymer (SAFJC) with $\frac{w}{b}=1$ and $l=w$ (solid blue line, RG; blue triangles, simulation), a “thick” polymer (SAWLC) with dsDNA-like parameters $\frac{w}{b}=\frac{4.6}{106}\ll 1$ and $l=w$ (dashed green line, RG; green squares, simulation), and a zero-thickness dsDNA (WLC) with $\frac{l}{b}=\frac{4.6}{106}$ (dotted red line, WLC; red diamonds, simulation). The excluded volume parameter in the RG calculations was rescaled to match the excluded volume in the simulations with ${\overline{u}}_{\mathrm{RG}}=1.5u$ following Tree et al. [12].

Figure 6

The J-factor power law. (a) Comparison of J-factor power-law exponents $\chi$ for the WLC ($\frac{w}{b}=0$) (red diamonds), SAWLCs with $\frac{w}{b}=0.046$ (green squares) and $\frac{w}{b}=0.2$ (magenta circles), and the SAFJC ($\frac{w}{b}=1$) (blue triangles), using $d_{\min }=w$ and $\frac{\varepsilon }{w}=4$. The blue (bottom) and red (top) dashed lines correspond to the SAFJC and WLC power laws of $-1.92$ and $-1.5$, respectively. (b)–(d) Effect of $\varepsilon$ on the J-factor power-law exponent. (b) J-factor power-law exponent for the SAFJC with various $\frac{\varepsilon }{w}$. (c) J-factor power-law exponent for the SAWLC with $\frac{\omega }{b}=\frac{4.6}{106}$ and with various $\frac{\varepsilon }{w}$. (d) J-factor power-law exponent for the WLC ($\frac{w}{b}=0$) with $b=106$ nm and with various $\frac{\varepsilon }{w_{\mathrm{dsDNA}}}$, where $w_{\mathrm{dsDNA}}$ is taken to be equal to that of dsDNA.

Figure 7

WLC and SAWLC ($d_{\min }=0$) comparison in the elastic regime. The insets show a cross section of the data in the xz plane. ${\mathbf{r}}_0$ is in the origin and ${\widehat{t}}_1=\widehat{z}$. (a) ${log}_{10}({\mathbf{C}}_{\mathrm{WLC}}\left(\mathbf{r}\right)/N_A)$ in the xz plane for WLC with $L=162$ bp. (b) ${log}_{10}({\mathbf{C}}_{\mathrm{SAWLC}-0}\left(\mathbf{r}\right)/N_A)$ in the xz plane for SAWLC with $L=162$ bp, $\frac{w}{b}=\frac{4.6}{106}$ and $d_{\min }=0$. (c) ${log}_{10}\left(\frac{{\mathbf{C}}_{\mathrm{WLC}}\left(\mathbf{r}\right)}{{\mathbf{C}}_{\mathrm{SAWLC}-0}\left(\mathbf{r}\right)}\right)$. (d) A blowup of the rectangular section marked in (c). Note, shorter chains were also examined, however, the accumulated statistical data around the origin in those chains was insufficient, and precluded a comprehensive analysis. Qualitatively, however, the shorter chains exhibited the same behavior as shown below in the data ranges that contained significant statistical data.

Figure 8

WLC and SAWLC ($d_{\min }=w$) comparison in the elastic regime. The insets show a cross section of the data in the xz plane. ${\mathbf{r}}_0$ is at the origin and ${\widehat{t}}_1=\widehat{z}$. (a) ${log}_{10}({\mathbf{C}}_{\mathrm{SAWLC}-\mathrm{w}}\left(\mathbf{r}\right)/N_A)$ in the xz plane for SAWLC with $L=162$ bp, $\frac{w}{b}=\frac{4.6}{106}$, and $d_{\min }=w$. (b) ${log}_{10}\left(\frac{{\mathbf{C}}_{\mathrm{WLC}}\left(\mathbf{r}\right)}{{\mathbf{C}}_{\mathrm{SAWLC}-\mathrm{w}}\left(\mathbf{r}\right)}\right)$.

Figure 9

Effect of $d_{\min }$ on the J factor in the elastic regime. (a) The volume enclosed in the light-green outline is inaccessible to a SAWLC chain. (b) J factor for the WLC and SAWLC, varying $d_{\min }$, constant $d_{\min }+\varepsilon =4.6$ nm. Recent experimental data [7] are shown with black squares. (c) J factor for the WLC and SAWLC, varying $d_{\min }$, constant $\varepsilon =1$ nm. Recent experimental data [7] are shown with black squares. Note the particular choice of ${\mathit{d}}_{\min }+\varepsilon$ is based on accepted physical values for the effective chain thickness in standard saline conditions. Even though the effects are slightly different in both cases, the magnitude of the effects is less than half an order of magnitude for these values.