Softening of DNA near melting as disappearance of an emergent property

Near the melting transition the bending elastic constant $\kappa$, an emergent property of double-stranded DNA (dsDNA), is shown not to follow the rodlike scaling for small-length $N$. The reduction in $\kappa$ with temperature is determined by the denatured bubbles for a continuous transition, e.g., when the two strands are Gaussian, but by the broken bonds near the open end in a Y-like configuration for a first-order transition as for strands with excluded volume interactions. In the latter case, a lever rule is operational, implying a phase coexistence although dsDNA is known to be a single phase.

Figure 1

Schematic diagram of renormalization of the elastic behavior of dsDNA due to bubbles. For Gaussian chains, tangents ${\mathbf{t}}_i$ and ${\mathbf{t}}_j$ can be used to define a persistence length (see text). Can one define a persistence length for configurations with bubbles?

Figure 2

(a)–(c) Possible configurations for two successive bonds. (a) Two bound bonds with three contacts and angle $\theta =0$. (b) Opening of a fork. (c) Same as panel (a) but with a bend ($\theta =\pi /2$), costing energy. There are reverse steps for Model I. $E_c$ and $E_b$ represent total contact and bending energies. (d) Identifying bubbles and the Y-fork.

Figure 3

Log-log plot of $\overline{\kappa }/N^{2\nu }$ vs $N$. (a) Model $\text{I}$ for flexible $\eta =0$ and semiflexible chain $\eta =3$, and different $\mathrm{\Delta }T=T-T_c$. The curve $\varphi \left(x\right)$ is a fit to the data points for $\epsilon /T=10,\eta /T=3$ using Eq. (1) with $l_p$ as a parameter. (b) Model $\text{II}$ for flexible $\eta =0$ and semiflexible chain $\eta =3$, and different $\mathrm{\Delta }T=T-T_c$. $\varphi \left(x\right)$ is a straight line of slope $(2-2\nu )$ representing the rodlike scaling regime. No other data sets show the initial slope of $\varphi \left(x\right)$.

Figure 4

Bubble, Y, and $\kappa$ for $\eta =0$ and 3 for Model $\text{I}$ (continuous transition). (a) Bubble fraction $f_b$, Y-fraction $f_Y$, and total fraction $1-n_c$ vs $(T-T_c)$. See text for the values of $T_c$. (b) Rescaled elastic modulus $(\overline{\kappa }/N^{2\nu })$ vs $(T-T_c)$, with $\nu =1/2$. Equation (3) is shown by solid lines, ${\varphi }_1$ for $\eta =0$ and ${\varphi }_2$ for $\eta =3$, with $n_c$ from panel (a). Inset in panel (b): Plot of $(\overline{\kappa }/N)-2$ vs ${(1-n_c)}^{0.1}$ for $\eta =0$ and 3.

Figure 5

Same as in Fig. 4 but for Model $\text{II}$ with first-order transition and $\nu =0.588$. (a) $f_b,f_Y,1-n_c$ vs $(T-T_c)$. In panel (b) $\nu =0.588$ has been used, and Eq. (4) is shown by solid lines, ${\varphi }_1$ for $\eta =0$ and ${\varphi }_2$ for $\eta =3$, with $f_Y$ taken from panel (a). Inset in panel (b): Plot of $(\overline{\kappa }/N^{1.176})-2.4$ vs $f_Y$ for $\eta =0$ and 3.

Figure 6

Contact number fluctuation. (a) Model $\text{I}$. (b) Model $\text{II}$. Both results are for semiflexible models with bending energy constant $\eta =3$.

Figure 7

Bubble size distribution at the melting point. (a) Model $\text{I}$ for $\eta =0$ and 3 and $T\approx T_c=0.928$ and 1.33, respectively, for a chain length $N=1000$. The straight line $\varphi \left(x\right)\sim x^{-1.7}$ is a fit to the intermediate region of the $\eta =0$ data points. (b) Model $\text{II}$ for $\eta =0$ and 3 at $T\approx T_c=0.75$ and 1.54, respectively, for chain length $N=500$. The straight line $\varphi \left(x\right)\sim x^{-2.4}$ is a fit to the intermediate region of the $\eta =3$ data points.

Figure 8

The mean squared end-to-end distance $R^2$ vs $\overline{\eta }$, obtained both analytically [Eq. (C1)] and from simulation at temperature $T$ much below the $\eta =0$ transition point $T_c$, $\epsilon /T=10\gg \epsilon /T_c=1.077.$ This is for length $N=1000$.

Figure 9

Log-log plot of the scaled mean squared end-to-end distance $R^2/N^{2\nu }$ vs the chain length $N$ for model $\text{II}$, with Boltzmann factors $exp(\epsilon /T)=7\times {10}^4$ for contact and $exp(-\overline{\eta })=0.005$ for bends, for the dsDNA, and a single self-avoiding semiflexible chain with the same Boltzmann factor for bending $exp(-\overline{\eta })=0.005$.

Figure 10

Plot of $n_b=f_b/(1-f_Y)$ vs $T-T_c$ for $\eta =0$ and 3 and various $N$, using the data of Fig. 5. The data collapse corroborates the picture of phase coexistence. The deviation near $T=T_c$ is the usual finite-size effect at a phase transition.