Thermal denaturation of fluctuating finite DNA chains: The role of bending rigidity in bubble nucleation

Statistical DNA models available in the literature are often effective models where the base-pair state only (unbroken or broken) is considered. Because of a decrease by a factor of 30 of the effective bending rigidity of a sequence of broken bonds, or bubble, compared to the double stranded state, the inclusion of the molecular conformational degrees of freedom in a more general mesoscopic model is needed. In this paper we do so by presenting a one-dimensional Ising model, which describes the internal base-pair states, coupled to a discrete wormlike chain model describing the chain configurations [J. Palmeri, M. Manghi, and N. Destainville, Phys. Rev. Lett. 99, 088103 (2007)]. This coupled model is exactly solved using a transfer matrix technique that presents an analogy with the path integral treatment of a quantum two-state diatomic molecule. When the chain fluctuations are integrated out, the denaturation transition temperature and width emerge naturally as an explicit function of the model parameters of a well defined Hamiltonian, revealing that the transition is driven by the difference in bending (entropy dominated) free energy between bubble and double-stranded segments. The calculated melting curve (fraction of open base pairs) is in good agreement with the experimental melting profile of poly(dA)-poly(dT) and, by inserting the experimentally known bending rigidities, leads to physically reasonable values for the bare Ising model parameters. Among the thermodynamical quantities explicitly calculated within this model are the internal, structural, and mechanical features of the DNA molecule, such as bubble correlation length and two distinct chain persistence lengths. The predicted variation of the mean-square radius as a function of temperature leads to a coherent explanation for the experimentally observed thermal viscosity transition. Finally, the influence of the DNA strand length is studied in detail, underlining the importance of finite size effects, even for DNA made of several thousand base pairs. Simple limiting formulas, useful for analyzing experiments, are given for the fraction of broken base pairs, Ising and chain correlation functions, effective persistence lengths, and chain mean-square radius, all as a function of temperature and DNA length.

Figure 1

(Color online) Illustration of the different Ising parameters appearing in the Hamiltonian $\beta H$. Open (closed) base pairs are coded by a spin $\sigma =+1(-1)$. The energies indicate the cost of opening base pairs with respect to the ground states where all $\sigma$ are set to $+1$. The first line shows the cost, $2J$, of a domain wall. The second line indicates the energy, $2\mu$, required to open a base pair. The third line gives the difference in stacking energy between a segment of dsDNA and a denaturated one: dark (light) blue cigars indicate stacked states in dsDNA (ssDNA) and the absence of a cigar indicates the destacking of adjacent base pairs, which is already taken into account by the $J$ contribution.

Figure 2

(Color online) “Energies” ${\epsilon }_{l,\pm }=-\ln \left({\lambda }_{l,\pm }\right)$ for $l=0,1$ in the quantum formalism (related to the Landau-Zener problem) vs temperature (for parameter values used in Sec. 7 to fit experimental melting data: $\tilde{\mu }=4.46\mathrm{kJ}∕\mathrm{mol}$, $\tilde{J}=9.13\mathrm{kJ}∕\mathrm{mol}$, and $\tilde{K}=0$). We observe that far from the two transition temperatures, the eigenvalues reduce to the limiting forms as shown in Eq. (47). The inset is a zoom close to $T_m^{\infty }$ showing the level repulsion between the branches $(0,\pm )$. A similar level repulsion occurs near $T_1^{\infty }$ between the branches $(1,\pm )$.

Figure 3

(Color online) (a) Variation of the transition probabilities ${⟨1,\pm \mid 0,+⟩}^2$ with temperature (red solid line for + and dashed blue line for −). For $T<T_1^{\infty }$, ${⟨1,+\mid 0,+⟩}^2\simeq {\phi }_{U,\infty }$ and ${⟨1,-\mid 0,+⟩}^2\simeq {\phi }_{B,\infty }$. One observes that for $T_m^{\infty }<T<T_1^{\infty }$, ${⟨1,-\mid 0,+⟩}^2=1$, which underlines the relevance of ${\xi }_{1,-}^p$ in this temperature range (with parameter values $\tilde{\mu }=4.46\mathrm{kJ}∕\mathrm{mol}$, $\tilde{J}=9.13\mathrm{kJ}∕\mathrm{mol}$, and $\tilde{K}=0$; see Sec. 7). (b) Tangent-tangent correlation function given by Eq. (100) $(N,i\to \infty )$ for three different temperatures: just before the transition (dotted blue line), controlled by ${\xi }_{1,+}^p\simeq {[1∕{\xi }_U^p+1∕{\xi }_I]}^{-1}\simeq {\xi }_U^p$; slightly above $T_m^{\infty }$ (solid red line), where the correlation length ${\xi }_{1,-}^p\simeq {\xi }_B^p$ is dominant for $r<r^{*}\simeq 20$; and after the transition $(T_m^{\infty }<T<T_1^{\infty })$ where the correlation length ${\xi }_{1,+}^p\simeq {\xi }_I$ disappears in favor of ${\xi }_{1,-}^p\simeq {\xi }_B^p$ (dashed green line).

Figure 4

(Color online) Variation with temperature of the various correlation lengths appearing in the model results: the Ising correlation length ${\xi }_I$ (solid red line); persistence lengths of the coupled system, ${\xi }_{\mathrm{eff}}^p$ (appearing for long chains) and ${\xi }_{1,\pm }^p$ (dashed-dotted blue and green lines, respectively); and of the pure chains, ${\xi }_{U,B}^p$ (dashed purple lines) (in units of $a$). At $T_m^{\infty }$, the Ising correlation length is peaked but finite, which is related to the point of closest approach of the two (0,±) branches (zoom of Fig. 2), and the effective persistence length ${\xi }_{\mathrm{eff}}^p$ rapidly crosses over from ${\xi }_U^p$ to ${\xi }_B^p$ (parameter values $\tilde{\mu }=4.46\mathrm{kJ}∕\mathrm{mol}$, $\tilde{J}=9.13\mathrm{kJ}∕\mathrm{mol}$, and $\tilde{K}=0$).

Figure 5

Fraction of broken base pairs for a poly(dA)-poly(dT) vs temperature (solution of 0.1 SSC, $N=1815$). The solid line represents the theoretical law for $\tilde{\mu }=1.64k_B{T}_m$ and $\tilde{J}=3.35k_B{T}_m$, where $T_m=326.4\mathrm{K}$. The case $N\to \infty$ for the same parameter values corresponds to the broken line.

Figure 6

(Color online) Linear-log plot of melting curves for $N=100$ (dashed-dotted green line), 500 (dotted purple line), 1815 (solid blue line), and $\infty$ (dashed red line) in decreasing order at low temperature, $T<326\mathrm{K}$ (parameter values $\tilde{\mu }=4.46\mathrm{kJ}∕\mathrm{mol}$, $\tilde{J}=9.13\mathrm{kJ}∕\mathrm{mol}$, and $\tilde{K}=0$). We note that all the curves intersect at $T^{*}$, as discussed in the text. $T_m$ is defined by ${\phi }_B=0.5$. Inset: Log-log plot of model results for the shift in transition width $\Delta T_m-\Delta T_m^{\infty }$ vs polymer length. Dots correspond to the model results and the solid line is a law in $1∕N$.

Figure 7

Average melting maps for different temperatures for $N=1815$ and the same parameter values used in Fig. 5: plot of the fraction of broken bases, ${\phi }_{B,i}$ as a function of the base position $i$ for, in increasing order, $310\mathrm{K}$, $0.99T^{*}=322.87\mathrm{K}$, $T^{*}=326.13\mathrm{K}$, $T_m^{\infty }=326.24\mathrm{K}$, $T_m=326.4\mathrm{K}$, and $1.01T_m=329.66\mathrm{K}$.

Figure 8

Zoom of the melting map shown in Fig. 7 for $T_m^{\infty }$ (lower curve) and $T_m$ (upper curve).