Distribution Functions, Loop Formation Probabilities, and Force-Extension Relations in a Model for Short Double-Stranded DNA Molecules

We obtain, using transfer-matrix methods, the distribution function $P(R)$ of the end-to-end distance, the loop formation probability, and force-extension relations in a model for short double-stranded DNA molecules. Accounting for the appearance of “bubbles,” localized regions of enhanced flexibility associated with the opening of a few base pairs of double-stranded DNA in thermal equilibrium, leads to dramatic changes in $P(R)$ and unusual force-extension curves. An analytic formula for the loop formation probability in the presence of bubbles is proposed. For short heterogeneous chains, we demonstrate a strong dependence of loop formation probabilities on sequence.

Figure 1

(a) $P(R)$ calculated for a 150 bp DNA fragment with $\beta J_d=50$ and $\beta J_s=1$ for (i) $\beta \mu =6$, (ii) $\beta \mu =7$, and (iii) $\beta \mu =9$. Curve (iv) is $P(R)$ from the WLC calculations ($\beta \mu =\infty$). (b) $P(0)$ from the transfer-matrix calculation (symbols) compared to the predictions of the approximate formula of Eq. (4) (lines). The curves from top to bottom are for $\beta \mu =10$, 12, 15, 18, and $\infty$.

Figure 2

$P(\mathbf{R})$ for homogeneous dsDNA of length $L=50$ computed with a single bubble placed at $L/2$ (top curve), $L/2\pm 1$ (middle curve), and $L/2\pm 3$ (bottom curve). The inset shows $P(0)$ as a function of bubble position, illustrating how this quantity peaks sharply when the bubble is placed at the center.

Figure 3

Force-extension relation for homogeneous dsDNA of length $L=50$ at parameter values $\beta J=50$ with $\beta \mu =6$ and $\beta \mu =9$. Curves a and b are computed in the fixed-extension ensemble, whereas curves c and d are computed in the fixed force ensemble.