Theoretical investigation of finite size effects at DNA melting

We investigated how the finiteness of the length of a sequence affects the phase transition that takes place at the DNA melting temperature. For this purpose, we modified the transfer integral method to adapt it to the calculation of both extensive (partition function, entropy, specific heat, etc.) and nonextensive (order parameter and average separation between paired bases) thermodynamic quantities of finite sequences with open boundary conditions, and applied the modified procedure to two different dynamical models. We characterized in some detail the three effects that take place when the length of the sequence is decreased, namely, (i) the decrease of the critical temperature, (ii) the decrease of the peak values of all quantities that diverge at the thermodynamic limit but remain finite for finite sequences, like the specific heat and the correlation length, and (iii) the broadening of the temperature range over which the transition affects the dynamics of the system. We also performed a finite size scaling analysis of the two models and showed that the singular part of the free energy can indeed be expressed in terms of a homogeneous function. However, Josephson’s identity is satisfied for none of the investigated models, so that the derivation of the characteristic exponents which appear, for example, in the expression of the specific heat requires some care.

Figure 1

(Color online) Log-log plots, as a function of the sequence length $N$, of the reduced critical temperature shift $1-T_c\left(N\right)∕T_c\left(\infty \right)$ (top plot), the value of the correlation length $\xi$ at $T_c\left(N\right)$ (middle plot), and the value of the specific heat $c_V$ at $T_c\left(N\right)$ (bottom plot). $\xi$ is in units of the separation between two successive base pairs, and $c_V$ in units of $k_B$. Squares and circles show the results obtained with the TI method for the DPB and JB models, respectively. Open symbols denote results obtained with grids of 4201 $y$ values, while the few filled symbols denote results obtained with larger grids of 6201 $y$ values (see end of Sec. 4A). The solid and dash-dotted lines show the results of the adjustment of power laws against the calculated points.

Figure 2

(Color online) Entropy per site $s$ as a function of the rescaled temperature $t\left(N\right)$ for an infinitely long chain (circles) and a sequence with $N=100\text{base pairs}$ (squares), according to the DPB (top plot) and the JB models (bottom plot). $s$ is in units of $k_B$.

Figure 3

(Color online) Log-log plots of the specific heat per site $c_V$ as a function of the negative $-t\left(N\right)$ of the rescaled temperature for the DPB (top plot) and the JB models (bottom plot), and seven values of the sequence length $N$ ranging from 100 to $\infty$. $c_V$ is in units of $k_B$. Note that, at the thermodynamic limit of infinitely long chains, $c_V$ becomes infinite at the critical temperature, but numerical limitations of the TI method prevent the observation of such divergence.

Figure 4

(Color online) Log-log plots of the correlation length $\xi$ (bottom plot) and the average base pair separation $⟨y⟩$ (top plot) as a function of the negative $-t\left(N\right)$ of the rescaled temperature for the JB model and seven values of the sequence length $N$ ranging from 100 to $\infty$. $\xi$ is in units of the separation $a$ between two successive base pairs and $⟨y⟩$ in units of the inverse $1∕A_M$ of the Morse potential parameter. Note that, at the thermodynamic limit of infinitely long chains, $\xi$ and $⟨y⟩$ become infinite at the critical temperature, but numerical limitations of the TI method prevent the observation of such divergence.

Figure 5

(Color online) $c_V∕N^{\rho }$, for six values of the sequence length $N$ ranging from 100 to $10000$, as a function of $-t{N}^{\sigma }$ for the DPB model and values of $\rho$ and $\sigma$ obtained from Eq. (32) (top plot) or adjusted by hand (bottom plot).

Figure 6

(Color online) $c_V∕N^{\rho }$, for six values of the sequence length $N$ ranging from 100 to $10000$, as a function of $-t{N}^{\sigma }$ for the JB model and values of $\rho$ and $\sigma$ obtained from Eq. (32) (top plot) or adjusted by hand (bottom plot).