DNA Bubble Dynamics as a Quantum Coulomb Problem

We study the dynamics of denaturation bubbles in double-stranded DNA. Demonstrating that the associated Fokker-Planck equation is equivalent to a Coulomb problem, we derive expressions for the bubble survival distribution $W(t)$. Below $T_m$, $W(t)$ is associated with the continuum of scattering states of the repulsive Coulomb potential. At $T_m$, the Coulomb potential vanishes and $W(t)$ assumes a power-law tail with nontrivial dynamic exponents: the critical exponent of the entropy loss factor may cause a finite mean lifetime. Above $T_m$ (attractive potential), the long-time dynamics is controlled by the lowest bound state. Correlations and finite size effects are discussed.

Figure 1

Schematic of the potential $V(x)-{ϵ}^2/2$ of Eq. (4). (a) $T<T_m$: repulsive potential with continuous spectrum. The bubble fluctuations correspond to a Brownian walk in bubble size $x$ before collapse at $x=0$. (b) $T>T_m$: attractive potential that can trap a series of bound states. At long times the lowest bound state indicated in the figure dominates. The bubbles increase in size leading to denaturation.

Figure 2

Bubble lifetime distribution $W(t)$, Eq. (10), for $T_m=2T_r$ (full line) and $T_m=10T_r$ (dashed line). Inset: $\log \mathrm{\text{−}}\log $ plot at long $t$, with slopes $-2$, $-1.6$ indicated by the straight lines.

Figure 3

Drift-diffusion model and experimental data from Ref. 6 compared to the $\Gamma$ model, Eq. (12), for various parameters. The curve for $\mu =0$ and $ϵ=1/\sqrt{2}$ exactly matches the long-time behavior from Ref. 6.