Stochastic dynamics and denaturation of thermalized DNA

In the first part of the paper, the stochastic dynamics of the Peyrard-Bishop-Dauxois (PBD) DNA model is studied. A one-dimensional averaged Itô stochastic differential equation governing the total energy of the system and the associated Fokker-Planck equation governing the transition probability density function of the total energy are derived from the Langevin equations for the base-pair (bp) separation of the PBD DNA model by using the stochastic averaging method for quasinonintegrable Hamiltonian systems. The stationary probability density function of the average energy and the mean square of the bp separation are obtained by solving the reduced Fokker-Planck equation. In the second part of the paper, the local denaturation of the thermalized PBD DNA model is studied as a first-passage-time problem in the energy. A backward Kolmogorov equation and a Pontryagin equation are derived from the averaged Itô equation and solved to yield the waiting-time distribution and the mean bp opening time. All the analytical results are confirmed with those from Monte Carlo simulation. It is pointed out that the proposed method may yield a reasonable mean bp opening time if the friction coefficient is fixed using experimental results.

Figure 1

A schematic representation of the PBD DNA model. It consists of two backbones with side chains facing each other. The neighboring bps interact with each other with a PBD potential and each base interacts with its complementary base with a Morse potential.

Figure 2

Results obtained from Monte Carlo simulation of the thermalized PBD DNA model with 50 bps. $\gamma =1$ for $25°\text{C}$. The separation distance of the bps is shown by the gray scale. (a) In the case of constant excitation intensity $D_{\xi }=0.02$. The horizontal coordinate is time $t$ (in units of $t_0$). (b) In the case of a linear intensity $D_{\xi }$ ramp ranging from 0 to 0.04. The horizontal coordinate is time or temperature. (c) Average energy $E$ (in units of $k_B{T}_r$). The data for the two curves are taken from Figs. 3a, 3b.

Figure 3

Stationary probability density $p\left(E\right)$ of the average energy $E$ (in units of $k_B{T}_r$) for PBD DNA model with different numbers of bps. The theoretical results from Eq. (22) are shown using a solid line and the corresponding results from simulation are shown by using symbols $●$ $◆$. $\gamma =0.6$ for $37°\text{C}$, $\epsilon =0.1$, $N=6$.

Figure 4

Mean square $E\left[y_i^2\right]$ of bp separation distance $y_i$ as a function of excitation intensity $D_{\xi }$; $\epsilon =0.1$, $N=3$.

Figure 5

Mean time $\tau$ (in units of $t_0$) for opening one bp as function of the bp numbers $N$ in the fork.

Figure 6

Waiting-time distribution $W\left(t\right)$ for a fork with six pb’s, $\epsilon =0.01$, $D_{\xi }=\epsilon k_B{T}_r$.

Figure 7

Probability density $\rho \left(T\right)$ of first-passage time $T$ (in units of $t_0$) for a fork with six pb’s, $\epsilon =0.01$, $D_{\xi }=\epsilon k_B{T}_r$.

Figure 8

Mean time $\tau$ (in units of $t_0$) for opening one bp in a fork with six pb’s as a function of friction coefficient $\epsilon$ (in units of $m_b{t}_0^{-1}$). $D_{\xi }=\epsilon k_B{T}_r$.