Statistical physics of the melting of inhomogeneous DNA

We studied how the inhomogeneity of a sequence affects the phase transition that takes place at DNA melting. Unlike previous works, which considered thermodynamic quantities averaged over many different inhomogeneous sequences, we focused on precise sequences and investigated the succession of local openings that lead to their dissociation. For this purpose, we performed transfer-integral-type calculations with two different dynamical models: namely, the heterogeneous Dauxois-Peyrard-Bishop model and the model based on finite stacking enthalpies we recently proposed. It appears that, for both models, the essential effect of heterogeneity is to let different portions of the investigated sequences open at slightly different temperatures. Besides this macroscopic effect, the local aperture of each portion indeed turns out to be very similar to that of a homogeneous sequence with the same length. Rounding of each local opening transition is therefore merely a size effect. For the Dauxois-Peyrard-Bishop model, sequences with a few thousand base pairs are still far from the thermodynamic limit, so that it is inappropriate, for this model, to discuss the order of the transition associated with each local opening. In contrast, sequences with several hundred to a few thousand base pairs are pretty close to the thermodynamic limit for the model we proposed. The temperature interval where a power law holds is consequently broad enough to enable the estimation of critical exponents. On the basis of the few examples we investigated, it seems that, for our model, disorder does not necessarily induce a decrease of the order of the transition.

Figure 1

(Color online) Top: plot, for increasing temperatures, of $⟨y_n⟩$ as a function of the site number $n$ for the 1793-bp human $\beta$-actin cDNA sequence (NCB entry code NM_001101). These curves were obtained from TI calculations performed with the JB model. Bottom: plot, as a function of $n$, of the AT percentage averaged over 40 consecutive bp of the actin sequence.

Figure 2

(Color online) Top: plot, for increasing temperatures, of $⟨y_n⟩$ as a function of the site number $n$ for the 2399-bp inhibitor of the hepatocyte growth factor activator sequence [30]. These curves were obtained from TI calculations performed with the JB model. Bottom: plot, as a function of $n$, of the AT percentage averaged over 40 consecutive bp of the inhibitor sequence.

Figure 3

(Color online) Comparison of $⟨y_n⟩$ profiles for the 1793-bp actin sequence at 322 K (bottom plot) and 346 K (top plot) obtained from TI calculations (dashed lines) and MD simulations (solid lines) performed with the JB model. The main plots show the profile of the whole sequence, while the inserts zoom in on 300 base pairs.

Figure 4

(Color online) Comparison of $⟨y_n⟩$ profiles for the 1793-bp actin sequence at 350 K obtained from TI calculations (small crosses) and MD simulations (solid line) performed with the heterogeneous DPB model. The main plot shows the profile of the whole sequence, while the inset zooms in on 300 base pairs.

Figure 5

(Color online) Plot of the fraction of open base pairs as a function of temperature $T$ for the 1793-bp actin sequence, obtained from TI calculations performed with the JB model. The criterion for a base pair $n$ to be open is that $⟨y_n⟩$ be larger than the threshold of $10\text{Å}$.

Figure 6

(Color online) Plots of the specific heat per particle, $c_V$, as a function of temperature $T$ for the 1793-bp actin sequence (top plot) and the 2399-bp inhibitor sequence (bottom plot), obtained from TI calculations performed with the JB model. $c_V$ is expressed in units of the Boltzmann constant $k_B$.

Figure 7

(Color online) Log-log plots of the specific heat per particle, $c_V$, as a function of the reduced temperature $t$ for the 2399-bp inhibitor sequence (solid lines), a 2000-bp homogeneous sequence (dashed lines), and an infinitely long homogeneous sequence (dot-dashed lines), obtained from TI calculations performed with the JB model (bottom plot) and the DPB model (top plot). $c_V$ is expressed in units of the Boltzmann constant $k_B$.

Figure 8

(Color online) Log-log plots, as a function of the reduced temperature $t$, of the average depth $⟨y_n⟩$ of bubbles centered around $n=1300$, $n=1450$, and $n=1640$ (top plot) and $n=318$ and $n=441$ (bottom plot) for the 1793-bp actin sequence. These results were obtained from TI calculations performed with the JB model. The critical temperature of each portion of the sequence is indicated on the corresponding plot.

Figure 9

(Color online) Plots of $\text{ln}\left(C_{ij}\right)$ as a function of $\left|i-j\right|$ for a homogeneous sequence with 10 000 base pairs at several temperatures comprised between 340 K and 367.2 K. $i=1$ for all plots. These results were obtained from TI calculations performed with the homogeneous JB model. The critical temperature of this system is $T_c=367.47\text{K}$.

Figure 10

(Color online) Plots of $\text{ln}\left(C_{ij}\right)$ as a function of $\left|i-j\right|$ for the 1793-bp actin sequence at several temperatures regularly spaced between 340 K and 350 K. $i=180$ for the main plot and $i=1250$ for the smaller vignette. The horizontal and vertical scales are identical for both plots, but the vignette $\left(i=1250\right)$ was horizontally shifted so that identical values of $j$ are vertically aligned. These results were obtained from TI calculations performed with the JB model.