Abstract
Watson and Crick’s epochal presentation of the double helix structure in 1953 has paved the way to intense exploration of DNA’s vital functions in cells. Also, recent advances of single molecule techniques have made it possible to probe structures and mechanics of constrained DNA at length scales ranging from nanometers to microns. There have been a number of atomistic scale quantum chemical calculations or molecular level simulations, but they are too computationally demanding or analytically unfeasible to describe the DNA conformation and mechanics at mesoscopic levels. At micron scales, on the other hand, the wormlike chain model has been very instrumental in describing analytically the DNA mechanics but lacks certain molecular details that are essential in describing the hybridization, nano-scale confinement, and local denaturation. To fill this fundamental gap, we present a workable and predictive mesoscopic model of double-stranded DNA where the nucleotides beads constitute the basic degrees of freedom. With the inter-strand stacking given by an interaction between diagonally opposed monomers, the model explains with analytical simplicity the helix formation and produces a generalized wormlike chain model with the concomitant large bending modulus given in terms of the helical structure and stiffness. It also explains how the helical conformation undergoes overstretch transition to the ladder-like conformation at a force plateau, in agreement with the experiment.
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Acknowledgements
We thank F. Ree, R. Netz, and R. Metzler for encouraging discussions and valuable comments. This research was supported by Korea Research Foundation (NRF-2013R1A12008900).
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Appendices
Appendix A Determination of the intermonomer interactions
Here we describe the detail of how the potential parameters of intermonomer interactions are determined with associated experimental data from the formulae of force-dependent geometrical parameters (4).
1.1 A.1 The case of harmonic intermonomer potential
1.1.1 A.1.1 Determination of k s
To find k s , we choose experimental force-extension data in the range where the DNA is fully stretched (h > h 0) but not overstretched (h < h 1 ≅ 0.58 nm). Because the change of twist and diameter of the DNA is known to be small in this range of force, we here assume constant diameter D 0 and twist Ω0. This corresponds to the limiting case that k b and k d approach infinity in (4). Then we find k s that minimizes the square error between the experiment \(h^{exp}(f_{\,\,i})\) and the theory h(f ) from (4a):
We used recent experimental data of λ-DNA at 150 mM NaCl and pH 7.4 by C. Danilowicz et al. [28]. The optimized value for k s was 263 \(k_BT/\mathrm{nm}^2\). The fitted theoretical curve is compared to the experimental one in Fig. 3a. Note that the fitted value of k s is close to the value inferred from the stretch modulus of ssDNA S ss via the relation \(k_s=S_{ss}/r_s=284~k_BT/\mathrm{nm}^2\) with S ss = 800 pN [9]. This means that the k s is a major contribution to elasticity of each single strand in our model. At the best fit, \(\mathcal{E}\) was 0.000009.
1.1.2 A.1.2 Determination of k d
For the estimation of k d we used the twist-stretch experiment by Gore et al. [29]; the authors measured the twist of a DNA double helix as a function of applied force f . The twist deviation per bp from Ω0 can be estimated from the twist numbers. According to the study, the response of the DNA against the pulling is that Ω mildly increases until the force reaches ≈ 30 pN and beyond this value Ω decreases with increasing force. We used data points belonging to the latter regime. Inserting \(k_s=263{k_BT}/\mathrm{nm}^2\) and assuming D = D 0 in (4), we found the optimized value of k d that minimizes the square error
The best fit value was \(k_d\cong1673 {k_BT}/\mathrm{nm}^2\), at which \(\mathcal{E}_2=0.000008\). Note that the obtained value for k d is large, as the experimentally measured twist deviation is in fact very small below the overstretching transition. The large value justifies the use of an harmonic potential in deriving (4).
1.1.3 A.1.3 Determination of k b
One can roughly estimate k b from the base-pairing potential used in the Peyrard-Bishop-Daixios DNA denaturation model [44], with an approximation \(u_b(r)=D(e^{-b(r-r_b)}-1)^2\approx b^2D(r-r_b)^2\). From the well-established parameter values of the model used for simulating the thermal denaturation of the DNA double helix [44], the spring constants are estimated to \(k_b=7056~{k_BT}/\mathrm{nm}^2\) for AT parameters and \(k_b=28566~{k_BT}/\mathrm{nm}^2\) for GC parameters. Similarly, the parameters used in the Breathing Wormlike Chain model suggest that \(k_b=2240~{k_BT}/\mathrm{nm}^2\) from [41] and \(k_b=10530~{k_BT}/\mathrm{nm}^2\) from [45]. Although the estimated value fluctuates widely depending on the parameter values chosen, it is clear that k b is a very large value compared to the other two k s and k d , which is consistent with our assumption of D ≅ D 0 within the force range of our interest. In fact, correct estimation of k b is not necessary in our study, as it is irrelevant for predicting overstretching force of helix-ladder transition and elastic moduli of B-DNA.
1.2 A.2 The case of anharmonic intermononer potential
1.2.1 A.2.1 Determination of V and a
As shown above, the diagonal interaction u d is well explained by a harmonic potential in the regime well below the overstretching transition where the double helix maintains its B-form. In order to explain the overstretching transition in the regime of high forces, however, u d should be described by a nonlinear short-ranged attractive potential. This is because the harmonic potential implicitly assumes that the B-form is only one stable structure. Since the diagonal interactions represent stacking interactions between successive bp planes, it is reasonably described by a short-ranged potential such as the Morse potential:
The parameters V and a can be determined by matching the corresponding theoretical Ω(f ) with the twist-stretch experiment [29].
From the energy minimization condition, \(\frac{{\partial} \overline{E}}{{\partial} \Omega}=\frac{{\partial} \overline{E}}{{\partial} h}=\frac{{\partial} \overline{E}}{{\partial} D}=0\), we find the new relation of Ω(f ) for the above Morse potential:
This formula replaces (4c) of Ω(f ) in the relations of structural deformation. We numerically solved this equation to find Ω(f ) with the use of h(f ) in (4a) with the fitted value of \(k_s=263k_BT/\mathrm{nm}^2\) and D(f ) = D 0. We found the optimized values of V and a that minimize the square error
The best-fit values were V = 2.5 k B T and a = 28.2 nm − 1, where \(\mathcal{E}_3=0.000006\)(\(<\mathcal{E}_2\)). Figure 3b shows the comparison of experimental and theoretical curves of Ω(f ).
In contrast to the nonlinear description of u d for the overstretching transition, [46] reports that u s is described well by a harmonic potential up to f ~65 pN from fitting their ssDNA elasticity model with the force-extension experiment; the anharmonic quartic term may need to be considered at higher forces.
Appendix B Thermal deformation of helix diameter
In the Appendix, we discuss three elastic terms associated with thermal deformation of the helix diameter that were omitted in the study of Sect. 5. The variation of the helix diameter D by the thermal kink of energy ΔE shown in Fig. 4 is described, up to second order, by the elastic energies \(\frac{1}{2}\left(\frac{{\partial}^2 \Delta E}{{\partial} D^2}\right)_0 (D-D_0)^2+\left(\frac{{\partial}^2 \Delta E}{{\partial} D{\partial}\theta}\right)_0\theta(D-D_0)+\left(\frac{{\partial}^2 \Delta E}{{\partial} D{\partial}\omega}\right)_0\omega(D-D_0) \). Here the first, second, and third terms account for respectively the elastic energy of local diameter fluctuation, the bending-diameter coupling, and the twist-diameter coupling. As in the study of Sect. 5, we evaluate the corresponding elastic moduli at around the B-form structure (θ = ω = 0, h = h 0, and D = D 0), finding that (1) \(\left(\frac{{\partial}^2 \Delta E}{{\partial} D^2}\right)_0=0.07k_s+0.1k_d+k_b\), (2) \(\left(\frac{{\partial}^2 \Delta E}{{\partial} D{\partial}\theta}\right)_0=0\), and (3) \(\left(\frac{{\partial}^2\Delta E}{{\partial} D{\partial}\omega}\right)_0=0.5k_s-0.07k_d\). These results have several implications: First, the elastic modulus (1) shows that the thermal deformation of the helix diameter is very small compared to thermal bending and twisting, since k b is a much larger value than k d and k s based on their determined values (see A.1.3). Also note that the elastic modulus (1) has positive contributions from k s and k d because the increased helix diameter leads to an increase in the stretching and diagonal interactions. Second, there is no bending-diameter coupling in the small perturbation regime considered in our study. In this regime, the variation of intra-strand and diagonal distances of strand 1 by helix bending of angle θ turns out to be opposite that of strand 2 while the change of helix diameter gives the same effect to both strands. This eventually makes the net effect of the bending-diameter coupling zero. Third, the twist-diameter coupling is determined by the competition of stretching and diagonal interaction. This behavior mainly arises from the fact that twisting a helix increases the intra-distance while decreasing the diagonal distance. From the chosen values of k s and k d obtained from the experiment, it is found that the twist-diameter coupling modulus is positive (albeit small). This means that the expansion of the helix diameter results in the un-twist of a helix (equivalently, the increase of the helix twist leads to the shrink of the helix diameter). However, for almost perfectly bound base pairs, this effect may not be significant because the variation of the helix diameter is very small.
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Jeon, JH., Sung, W. An effective mesoscopic model of double-stranded DNA. J Biol Phys 40, 1–14 (2014). https://doi.org/10.1007/s10867-013-9333-9
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DOI: https://doi.org/10.1007/s10867-013-9333-9