Low-Energy DNA Bubble Dynamics via the Quantum Coulomb Potential

TL;DR

This work addresses DNA bubble dynamics below the melting point by recasting the Fokker-Planck description as an imaginary-time Schrödinger problem with a Coulomb-like potential. A low-energy approximation leads to a closed-form probability density for bubble size expressed through the modified Bessel function $I_{\mu+1/2}$, along with analytical first-passage time density and correlation functions that align with known Fokker-Planck results and Gamma-model limits. The method provides a time-accurate description for any $t$ and temperature $T<T_m$, and it offers a practical way to fit experimental data to extract the loop entropy parameter $c$. The framework can be extended to various $\gamma(T)$ profiles, potentially improving understanding of DNA denaturation kinetics and bubble dynamics.

Abstract

We developed a low-energy model that can be used at any time to describe the dynamics of DNA bubbles at temperatures below the melting point. The Schrödinger equation associated with this problem is solved in imaginary time with a quantum Coulomb potential, and we obtain an approximate expression for its more general physical solution as a linear combination of the states whose energies are close to the lower bound energy. We can then determine the probability density, the first-passage time density, and the correlation functions in terms of Bessel functions. Our findings are consistent with results obtained directly from the Fokker-Planck equation. Comparisons with the Gamma and Diffusion models are discussed.

Figures (2)

• Figure 1: First-passage time density functions for the Bessel model in Eq. \ref{['E2.11']} at $37\, ^{\circ}$C and different values of the parameter $\mu$. The inset graph shows a log-log plot of these functions and two straight gray dashed lines with slopes $-1.5$ and $-2.38$, respectively.
• Figure 2: Comparative graph between the correlations of the Bessel model and the Gamma model, in contrast to the Diffusion model of Ref. AltanBonnet2003 at $37\, ^{\circ}$C. The Bessel model shows a similar behavior to the Diffusion model with the parameter values set $x_0=2,\ \mu=0$. The correlations are centered at time $t_{1/2}$, so that $C(t_{1/2})=1/2$.