A three-state model with loop entropy for the overstretching transition of DNA

Abstract

We introduce a three-state model for a single DNA chain under tension that distinguishes among B-DNA, S-DNA, and M (molten or denatured) segments and at the same time correctly accounts for the entropy of molten loops, characterized by the exponent c in the asymptotic expression S approximately -c ln n for the entropy of a loop of length n. Force extension curves are derived exactly by employing a generalized Poland-Scheraga approach and then compared to experimental data. Simultaneous fitting to force-extension data at room temperature and to the denaturation phase transition at zero force is possible and allows us to establish a global phase diagram in the force-temperature plane. Under a stretching force, the effects of the stacking energy (entering as a domain-wall energy between paired and unpaired bases) and the loop entropy are separated. Therefore, we can estimate the loop exponent c independently from the precise value of the stacking energy. The fitted value for c is small, suggesting that nicks dominate the experimental force extension traces of natural DNA.

Figure 1

Comparison of force extension curves obtained by different methods for c = 0. The curve obtained via the exact transfer matrix calculation Eq. 20 is already for N = 2 accurately reproduced by the approximate Legendre transformation Eq. 13. The dominating singularity method equation, i.e., Eq. 16 (or, equivalently, Eq. 21), is strictly valid in the thermodynamic limit but agrees with the Legendre transform already for a modest value of N = 10. The units of the abscissa is in extension per basepair. Parameters for λ-DNA in the absence of DDP are used (see Section E in the Supporting Material).

Figure 2

(Bottom panel) Force extension curve of double-stranded λ-DNA with and without DDP. (Symbols) Experimental data (20); (lines) fits with the three-state model for c = 0. The main difference between the two curves is the lack of cooperativity in the BS-transition in the presence of DDP which we take into account by choosing vanishing interaction energies V_ij = 0, i, j = B, S, M. (Top panel) Fraction N_i/N of segments in the different states, as follows from Eq. 24 in the absence of DDP.

Figure 3

Various force-extension curves of the three-state model with fit parameters for λ-DNA without DDP. (a) (Lower panel) Force extension curves for different values of the loop exponent c, showing no phase transition (c ≤ 1), a continuous (1 < c ≤ 2), or a first-order phase transition (c > 2). (Open circles) Critical forces. (Inset) Magnification of the region around the transition. (Upper panel) Fraction of bases in the three states for c = 3/2. The critical transition, above which all bases are in the molten M-state, is discerned as a kink in the curves. (b) Comparison of experimental data (circles) and theory for c ≠ 0. The curve for c = 0 and V_SM = 0, already shown in Fig. 2, is obtained by fitting l_S, g⁰_s, and g⁰_M to the experimental data, the values of which are kept fixed for all curves shown. The curve for V_SM = 0 and c = 0.6 results by fitting c and slightly improves the fit quality. The curve c = 0 and V_SM = 1.1 × 10⁻²¹ J is obtained by fitting V_SM. The curve for V_SM = 1.1 × 10⁻²¹ J and c = 0.3 is obtained by fitting c and keeping V_SM fixed. (Inset) First derivative of x(F), illustrating that increasing c leads to a growing asymmetry around the transition region. (c) Temperature dependence of the force extension curves. Increasing temperature leads to a decrease of the BS-plateau force. In the presence of a true denaturing transition, i.e., for c > 1, the critical force F_c decreases with increasing temperature, and for F > F_c, the force extension curve follows a pure WLC behavior.

Figure 4

(Solid line) Critical force F_c for c = 3/2, at which a singularity occurs according to Eq. 28. Phase boundaries for c = 0 (thick lines) and c = 3/2 (thin lines) are defined by N_M/N = 0.5 (dotted) and N_B/N = 0.5 (dashed). (Dot) The melting temperature T_c. (Insets) Behavior of the phase boundaries near the melting temperature, $F\propto \sqrt{T_{\text{c}}-T}$. Parameters for λ-DNA without DDP are used.

Figure 5

Relative fraction of segments in the B-state, N_B/N, as a function of temperature for different loop exponents c = 0, 1.5, and 2.1 and for finite BS interfacial energy V_BS = V_BM = 1.2 × 10⁻²⁰ J (bold lines) and for V_BS = V_BM = 0 (thin lines). (Circles) Positions of the phase transition. For all curves, parameters for λ-DNA without DDP have been used, g⁰_M (T) = 1.5 × 10⁻¹⁹ J – T × 4.2 × 10⁻²² J/K for c = 0, and g⁰_M (T) = 1.6 × 10⁻¹⁹ J – T × 4.6 × 10⁻²² J/K for c > 0.