Master equation approach to DNA breathing in heteropolymer DNA

After crossing an initial barrier to break the first base-pair (bp) in double-stranded DNA, the disruption of further bps is characterized by free energies up to a few $k_{B}T$. Thermal motion within the DNA double strand therefore causes the opening of intermittent single-stranded denaturation zones, the DNA bubbles. The unzipping and zipping dynamics of bps at the two zipper forks of a bubble, where the single strand of the denatured zone joins the still intact double strand, can be monitored by single molecule fluorescence or NMR methods. We here establish a dynamic description of this DNA breathing in a heteropolymer DNA with given sequence in terms of a master equation that governs the time evolution of the joint probability distribution for the bubble size and position along the sequence. The transfer coefficients are based on the Poland-Scheraga free energy model. We derive the autocorrelation function for the bubble dynamics and the associated relaxation time spectrum. In particular, we show how one can obtain the probability densities of individual bubble lifetimes and of the waiting times between successive bubble events from the master equation. A comparison to results of a stochastic Gillespie simulation shows excellent agreement.

Figure 1

(Color online) Clamped DNA domain with internal bps $x=1$ to $M$, statistical weights $u_{\mathrm{hb}}\left(x\right)$, $u_{\mathrm{st}}\left(x\right)$, and tag position $x_T$. The DNA sequence enters through the statistical weights $u_{\mathrm{st}}\left(x\right)$ and $u_{\mathrm{hb}}\left(x\right)$ for disrupting stacking and hydrogen bonds, respectively. The bubble breathing process consists of the initiation of a bubble and the subsequent motion of the forks at positions $x_L$ and $x_R$, see also Fig. 2.

Figure 2

Possible bubble (un)zipping transitions: for $m⩾2$, the four transfer rates ${\mathsf{t}}_{L∕R}^{\pm }(x_L,m)$ completely determine the transitions out of this state; the coefficients ${\mathsf{t}}_L^{\pm }(x_L\mp 1,m\pm 1)$ and ${\mathsf{t}}_R^{\pm }(x_L,m\mp 1)$ specify the possible jumps into this state. Jumps between state $m=1$ and ground state $m=0$ are described by ${\mathsf{t}}_G^{+}\left(x_L\right)$ and ${\mathsf{t}}_G^{-}\left(x_L\right)$.

Figure 3

Schematic representation of the $(x_L,m)$-lattice on which the DNA-bubble jump process takes place, with the permitted transitions; compare to Figs. 1, 2. The boundary conditions Eq. (3) are also indicated. For $m⩾1$ jumps are determined by the rates ${\mathsf{t}}_{L∕R}^{\pm }$. A jump between the ground state $m=0$ and one of the $m=1$ states occurs with rates ${\mathsf{t}}_G^{\pm }$. Note that the ground state $m=0$ is unique; from the ground state, a bubble can be initiated at any point in the row $m=1$, with the appropriate weight ${\mathsf{t}}_G^{+}\left(x_L\right)$. Note that all allowed jumps correspond to a change in bubble size $m$, i.e., there are no allowed “horizontal” transitions in the $(x_L,m)$-lattice.

Figure 4

Enumeration scheme for the numerical analysis: The two-dimensional grid points $(x_L,m)$ are replaced by a one-dimensional running variable $s$. See text for details.

Figure 5

Schematic of the $(x_L,m)$-points, region $\mathbb{R}1$ $\left(\mathbb{R}0\right)$, for which the stochastic variable takes the value $I=1$ $(I=0)$. The boundary points regions $\mathbb{R}{1}^{\prime }$ and $\mathbb{R}{0}^{\prime }$ are also indicated. The illustration is for the case $M=5$ and $x_T=3$, with $\Delta =0$.

Figure 6

(Color online) Top: Fluorescence time series $I\left(t\right)$ for the T7 promoter sequence, with tag positions $x_T=41$ (first panel) and $x_T=38$ (second panel), and $\Delta =0$. Bottom: Waiting time $\left[\psi \left(\tau \right)\right]$ and fluorescence survival time $\left[\varphi \left(\tau \right)\right]$ densities, in units of $k$. The data points are results from the Gillespie algorithm, while the lines represent the numeric results obtained from the master equation (ME). All results are for $T=37°\mathrm{C}$ and $100\mathrm{mM}$ NaCl with DNA parameters from [11].

Figure 7

(Color online) Autocorrelation $A\left(t\right)$ and spectral density $f\left(\tau \right)$ for a tagged bp in a homopolymer region: $u=u_{\mathrm{hb}}{u}_{\mathrm{st}}$. Top: Close-to-end-tagging far from $T_m$ ($x_T=2$, $u=0.6$). Middle: Center-tagging far from $T_m$ ($x_T=20$, $u=0.6$). Bottom: Center-tagging close to $T_m$ ($x_T=20$, $u=0.9$). In the $A\left(t\right)$ plots the full lines (blue online) are the exact result. The dashed (green online) curves are approximated from Eqs. (52, 54). In the spectral density plot the data were collected into ten bins. The light (green online) bars are the approximate one-variable results, Eq. (55), and the dark (blue online) bars are the exact result. The length of the DNA segment was $M=40$. The approximate expression only works well for internal tagging and below $T_m$.

Figure 8

(Color online) Top: Mean correlation time ${\tau }_{\mathrm{corr}}={\int }_0^{\infty }d{t}^{\prime }A\left(t^{\prime }\right)∕A\left(0\right)$ vs NaCl concentration for various temperatures $T$ for the AT9 construct of Ref. 16, showing a critical slowing down at the melting concentration (compare lower panel). The triangles denote the melting concentration for infinitely long random AT and GC stretches. The curve denoted by ${\tau }_{\max 1\mathrm{D}}$ is the result given in Eq. (60), and ${\tau }_{\max 2\mathrm{D}}=\max \left\{{\tau }_p\right\}$, $p=1,\dots ,S$ is the maximum relaxation time of the full problem specified in Sec. 3. Bottom: Opening probability of bp $x_T$.