DNA denaturation bubbles at criticality

The equilibrium statistical properties of DNA denaturation bubbles are examined in detail within the framework of the Peyrard-Bishop-Dauxois model. Bubble formation in homogeneous DNA is found to depend crucially on the presence of nonlinear base-stacking interactions. Small bubbles extending over fewer than ten base pairs are associated with much larger free energies of formation per site than larger bubbles. As the critical temperature is approached, the free energy associated with further bubble growth becomes vanishingly small. An analysis of average displacement profiles of bubbles of varying sizes at different temperatures reveals almost identical scaled shapes in the absence of nonlinear stacking; nonlinear stacking leads to distinct scaled shapes of large and small bubbles.

Figure 1

(Color online) Bubble statistics in the case of vanishing nonlinear base stacking, $\rho =0$. Upper panel: probability of a bubble extending to $n$ sites for a variety of temperatures. Note that the decay becomes weaker as the critical temperature is approached. Also shown are, for the two lowest temperatures, fits to a stretched exponential function. Inset: For comparison, I show the probability of a cluster of $n$ successive sites with bound base pairs (pure exponential). Lower panel: ratios of successive probabilities vs inverse bubble size for a range of temperatures.

Figure 2

(Color online) Summary of critical results in the case $\rho =0$: as extracted from the numerical TI solution and the asymptotic behavior of the curves shown in the lower panel of Fig. 1: (i) spectral gap $\Delta ϵ$ (open squares) vs temperature; (ii) activation free energy $\Delta f_b$ associated with bubble growth (open circles); (iii) the square root $\Delta {ϵ}^{1/2}$ (solid squares) is known to depend linearly on the temperature; rounding is due to the finiteness of the matrix used; (iv) $\Delta f_b^{1/2}$ (solid circles) is found to depend linearly on the temperature; remarkably, it exhibits no rounding; (v) the fraction of bound sites $p$ (diamonds), as obtained from (6). The dotted lines represent linear fits to the data (cf. text for discussion). Inset: the exponent $c$ vs temperature.

Figure 3

(Color online) Summary of critical results in the case $\rho =1$: (i) spectral gap $\Delta ϵ$ (diamonds) and (ii) fraction of bound sites $p$ (stars) vs temperature, as obtained from the TI numerical solution; (iii) activation free energy $\Delta f_b$ associated with bubble growth (solid squares) and (iv) exponent $c$ of the asymptotic form (10) (triangles, inset), as extracted from the $n\to \infty$ asymptotics of Fig. 4. The dotted lines represent linear fits to the data (i) and (ii), both yielding, respectively, $T_c=0.80$ within numerical accuracy.

Figure 4

(Color online) Bubble statistics in the case of nonvanishing nonlinear base stacking, $\rho =1$. Upper panel: probability of a bubble extending to $n$ sites at $T=0.76$ and $T=0.79$; also shown (solid lines) are fits to the function (9); cf. text for a discussion of the fitting parameters. Inset: for comparison I show the probability of a cluster of $n$ bound sites (pure exponential). Lower panel: ratios of successive probabilities vs inverse bubble size for a range of temperatures. Inset: a close-up of the asymptotic region for the two highest temperatures; dotted lines represent quadratic extrapolations from the last three points.

Figure 5

(Color online) The functions $\mathcal{R}\left(x\right)$ (open symbols, left $y$ axis) and $\beta \left(x\right)$ (solid symbols, right $y$ axis). Upper panel, $\rho =0$: $T=0.85$ (squares), $T=1.20$ (circles). The density of eigenvalues can be fitted with a power law $\zeta =0.51$, regardless of temperature; the function $\beta \left(x\right)$ also exhibits a power-law behavior as $x\to 0$; at the highest temperature a value $\eta =0.81$ is obtained. Lower panel: $\rho =1$: $T=0.63$ (squares), $T=0.79$ (circles). The density of eigenvalues exhibits a slight anomaly around $x=.1$; in the region $x\to 0$ it follows a power law $\zeta =0.41$ (slope of dotted line, guide to the eye). The function $\beta \left(x\right)$ exhibits a strong anomaly around $x=0.1$ and does not seem to settle to a pure power law behavior; however, the effective slope is quite high (dashed line, guide to the eye has a slope of 2.5).

Figure 6

(Color online) Bubble shapes and sizes in the case $\rho =0$. Upper panel: relative average displacements of sites in a bubble vs relative site coordinate. Results from a variety of sizes and temperatures collapse on a single curve. Lower panel: the maximal displacement of an $n$-site bubble for various temperatures.

Figure 7

(Color online) Bubble shapes and sizes in the case $\rho =1$. Upper panel: relative average displacements of sites in a bubble vs relative site coordinate. Results from different temperatures and sizes collapse on different curves for large (open symbols) and small (solid symbols) bubbles. Lower panel: the maximal displacement of an $n$-site bubble for various temperatures. Note the crossover which occurs around $n=10-20$ at temperatures both near and far from $T_c$.

Figure 8

(Color online) Comparison of $\Delta f_b=-T\text{ln}{\mu }_0$ obtained in the context of the associated eigenvalue problem with the TI spectral gap $\Delta ϵ$ in the cases $\rho =1$ (open symbols) $\rho =0$ (solid symbols); in the latter case, the square root is plotted.