An effective mesoscopic model of double-stranded DNA

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.

Introduction

In 1953, Watson and Crick demonstrated that DNA, the nucleic acid containing genetic material, is a double helix composed of two single-stranded phosphate backbone chains paired by complementary bases. In a cell, this double helical DNA, called B-form DNA, can be tightly bent, looped, and sometimes locally unwound for biological functions such as DNA packing, transcription, gene regulation, etc [1, 2]. Recent progress of single molecule techniques has enabled intensive studies on the elastic responses and structural changes of DNA under tension [3]. The mechanical property of B-DNA on length scales longer than several hundreds nanometers [4, 5] can be explained by a continuum elastic model that treats it as a twistable wormlike chain (WLC). The estimated value of its bending persistence length as large as ≈ 50 nm implies the double-stranded (ds) DNA barely bends over the length shorter than that. Recently, however, there have been a series of experimental evidences that, on such short length scales, sharp bendings occur much more readily than predicted by the WLC model [6, 7]. Experiments have shown that under strong tension the B-form DNA can be extended and unwound beyond its natural length, and furthermore at a force plateau of about 65 pN undergoes an overstretching transition. While the microscopic structure of the overstretched DNA is still in debate, WLC-based and two-state models have explained that the plateau can occur due to transition between the helical state and ladder-like state, dubbed as S-DNA [8–13].

The WLC model lacks the microscopic details such as double helical structures that can be manifested at nanoscales. Most fundamentally, DNA molecules are described at the atomistic level. Quantum chemical calculations have produced the structural properties and the associated energy parameters [14], establishing that the stacking interactions between two base planes induce a tight helical structure with a large bending stiffness [15]. However, these atomistic calculations and even molecular level simulations [16] are analytically unfeasible as well as computationally demanding to describe the properties from nanometer to micron scales that are relevant to recent single DNA experiments. What is urgently needed is an effective mesoscopic model to fill this huge gap. For magnetic systems, the effective model, like the Heisenberg or Ising Hamiltonian, has served the central role in describing collective transitions and dynamic behaviors emerging beyond the atomistic details. In this paper we propose such a model of DNA, relatively easy to implement analytically and simulationally, where it is treated as two strands of interacting coarse-grained beads. This mesoscopic model can describe the salient features of the double helical structures and also explain a variety of related experiments with predictive power. For the small deformations, the model is reduced to a generalized elastic chain model, which, in the continuum limit, becomes the usual twistable WLC model with the large bending and twist moduli, both of which are determined in terms of microscopic parameters such as the DNA geometry and stacking interaction strengths.

Energy model

We consider a mesoscopic model of a single dsDNA molecule, where the basic degrees of freedoms are the nucleotides (monomers denoted by beads in Fig. 1) governed by an energy function

1

For simplicity, we neglect the effect of base sequence heterogeneity in this study. The U_S is the intra-strand stretching energy for two single strands of N + 1 monomers, where is the 3-D position vector of nth monomer in strand i (i = 1,2). U_B arises mostly from the pairing interaction between complementary bases due to hydrogen bondings [see Fig. 1a], given by . The last term, U_D, stands for the interaction including mostly the inter-strand stacking interaction, which are not incorporated in U_S and U_B. The stacking interactions are complicated noncovalent interactions arising mainly from the Van der Waals, hydrophobic, and electrostatic interactions among subunits of nucleotides [17]. While the stacking interactions are known to play important roles in the helical structure, stability, and rigidity of the dsDNA molecule [18], they are hard to implement analytically within the mesoscopic DNA models in which structural details of the nucleotides are neglected. Because of this, in some DNA models they were implicitly considered as parameters associated with the helicity, stability, and stiffness of dsDNA [19–25]. From the fact that the stacking interactions mostly occur between neighboring base-pair planes, here we make an assumption that they are effectively described by the sum of the interactions between diagonally opposed monomers, called “diagonal interactions”

2

As depicted in Fig. 1a, this next-nearest neighbor interactions cause adjacent base-pair planes to stack each other, which effectively prevents the destacking, unpairing, and destabilization of the duplex DNA. As we shall see below, in the presence of U_D, the stacked base pairs have a preferential twist angle. Such effects are naturally lost when two single strands are apart. A similar treatment for the stacking interactions was employed in a simulation-based coarse-grained DNA model [26]. Each of the intermonomer interactions u_m(r) (m = s, b, d), which includes the electrostatic interaction modulated by the ionic environment, is assumed to be a bound potential having the minimum with positive curvature at the separation r = r_m.

Fig. 1.

Schematic figures of a DNA duplex showing a a ladder-like structure and b a helical structure. A nucleotide (monomer) in a strand interacts with one in complementary strand via u _b, base-pairing interaction, and other via u _d, diagonal interaction. The ladder is twisted into a helical structure represented by three geometrical constants; rise h, diameter D, and twist Ω (inset)

Helix formation

Suppose that a DNA molecule assumes a uniform and rigid helical structure with three geometrical constants, namely, the diameter (D), the twist (twisting angle between intra-strand neighboring bases, Ω), and the distance between neighboring bases (h) along the helix axis called the ‘rise’ per base [see Fig. 1b], . Since each strand is allowed to stretch, h and Ω, along with D, constitute three major independent parameters, with other geometrical details such as tilt, slide, and roll [2] neglected in this initial study. The net energy is then simplified to

The equilibrium DNA structure minimizing the energy can be found by solving , leading to the relations: (i) (ii) (iii) u′_b(D) = 0 with u′_m(r) = du_m/dr. From these, the equilibrium geometrical parameters for the helix are uniquely determined by remarkably simple formulae:

3a

3b

3c

Note that the geometry of the helix is given in terms of distance r_m where potentials are minima. From (3c) we find the criteria for helix formation (Ω₀ > 0); the DNA duplex assembles into a helix, provided that r_d is less than , the diagonal of the ladder. Our energy model does not distinguish the right-handed (+) helix from the left-handed (−) one. This helical symmetry may be broken by considering extra geometrical parameters such as tilt of bp plane. Selecting the B-form (right-handed) DNA with the known geometry, e.g., Ω₀ = + 0.60 rad, h₀ = 0.34 nm, and D₀ = 2 nm [2], we deduce a reasonable set of length parameters of B-DNA potentials, r_s = 0.68 nm, r_b = 2.0 nm, and r_d = 1.94 nm. If r_d is larger than d, the ladder conformation (Ω = 0, h = r_s, D = r_b) is the equilibrium structure in the present model.

To explicitly study how the net energy depends upon Ω, we consider harmonic potential for the stretching, for the base-pairing, and an anharmonic potential for the stacking interaction. The associated potential parameters are chosen to meet the experimental data about the B-DNA stretching, which will be explained next. Omitting the base pairing energy, which is independent of Ω, Fig. 2 shows the profile of E vs. Ω for D = 2 nm and d ≅ 2.1 nm for both cases r_d = 1.94 nm and r_d = 2.14 nm where the h is set so as to minimize E for a given Ω. Indeed, the net energy for the case r_d = 2.14 nm > d has a stable minimum at Ω = 0 (the ladder conformation). On the other hand, the curve for r_d = 1.94 nm in Fig. 2 shows a profile of E with the unstable maximum at Ω = 0 [27] and the stable minimum at Ω = Ω₀. This means that the DNA duplex with r_d < d, if initially prepared in a ladder structure, spontaneously assembles into a helical structure which is stable against thermal fluctuation. When subject to a stretching force above a critical value, however, the helical structure gives into the ladder structure, as described below.

Fig. 2.

Landscapes of E(Ω)/N for DNA helix (r _d = 1.94 nm) and ladder (r _d = 2.14 nm). The parameter values used are N = 1,000, D = r _b = 2.0 nm, r _s = 0.68 nm, k _s = 263 k _B T/nm², V = 2.5 k _B T, and a = 28.2 nm^− 1. See the Appendix for the form of the anharmonic potential used and the determination of parameter values

Force-induced helix deformation and helix-ladder transition

Consider a dsDNA pulled by the force f applied to one of strands or the forces f /2 applied to both strands at an end, with the other end held fixed. Just below the overstretching transition, the dsDNA is uniformly deformed with the energy . The modified helical structures h(f ), D(f ), and Ω(f ) are obtained by solving . As shown by single DNA molecule experiments, the stretched DNA undergoes small structural deformation relative to the B-form below the overstretching transition (see Fig. 3a, b). In this regime, the interaction potentials u_m(r) can be simplified to harmonic potentials, from which we can analytically derive the equations of helix deformation for h(f ), D(f ), and Ω(f ):

4a

4b

4c

Fig. 3.

a h vs. f obtained from experiment (circle, [28]) and theory (solid line). b Ω vs. f obtained from experiment (square, [29]) and theory (solid line). c E/N vs. h for DNA helix (Ω = Ω₀) and ladder (Ω = 0). The coexistence region of the two structures is shown by dotted line. h ₁ and h ₂ are the DNA rises at initiation and termination of the region, respectively. All potential parameters used in Fig. 2

The above self-consistent equations reproduce the force-zero relations (3a–3c) for h₀, Ω₀, and D₀ as f→0. Once h(f ) is solved from (4a) for given potential parameters and force, D(f ) and Ω(f ) are subsequently obtained from the other equations. Note that (4a) & (4c), respectively, describe the force-extension and stretch-twist relations of dsDNA that have been measured by single-molecule experiments. In the case that an anharmonic potential is used for the diagonal interaction in the regime near and above the overstretching transition, the above relation for Ω(f ) is replaced by (13) in Appendix. This equation is not analytically tractable, but can be numerically solved self-consistently.

According to the above formulae, the response of B-DNA against the force is such that the rise h increases and twist Ω decreases with the force f , consistent with corresponding experimental data [28, 29]. By fitting the theory with these data as shown in Fig. 3a, b, we estimate potential parameters: , V ≅ 2.5 k_BT, a ≅ 28.2 nm^− 1 [for ], and [for ]. See the Appendix for detail. It is noteworthy that, below the overstretching transition, the DNA is stretched with negligible unwinding, presumably due to the high energy cost of destacking.

When the DNA is highly extended, however, the ladder form can be energetically favorable compared to the helix form. This is shown in Fig. 3c, where the energy profiles of DNA helix (Ω = Ω₀) and ladder (Ω = 0) are depicted as a function of h using the potential parameters determined above. The condition of energy minimization dictates a coexistence of helix and ladder structures for the region between h₁ and h₂, represented by the dotted double-tangent,

5

where E₁ = E(Ω₀,h₁) and E₂ = E(0,h₂). The tangent is the critical force plateau responsible for the overstretching transition. We numerically solve the double-tangent relation (5) with the given potential parameters determined above. The critical force is estimated to be ≈ 60 pN, which is close to the experimental value 65 pN. It has been conjectured from experiments that underlying within overstretching transition are three different physical mechanisms (strand unpeeling, local DNA melting, and the strand unwinding with intact bp called S-DNA) [30–36]. Moreover, recent experiments showed that the DNA structure during the overstretching process is governed by several factors including the local bp stability and the topology of the stretched DNA [34–36]. While our study using a simple model is not capable of fully elucidating all possible modes of the overstretched DNA, it shows a possibility that a discontinuous structural transition in favor of S-DNA can occur by the helix-ladder transition in their coexistence, induced by a critical force. We find that in our model the (free) energy change during this transition is about 2.7 kcal/mol, which is comparable to the experimental value ≈ 2 kcal/mol measured by Zhang et al. [34, 35] for the nonhysteretic overstretching transition originating in S-DNA. However, our ladder model for S-DNA differs from a previously proposed model of S-DNA speculated by Léger et al. from experiment [11], which is the underwound right-handed double helix of pitch ~22 nm and helical repeat ~37.5 bp [37]. Finally, we emphasize that our approach contrasts with earlier models for the B-S transition in that it explicitly accounts for structural geometries for B- and S-forms from the model interactions, and the transition condition self-consistently without fitting free parameter.

An elastic model of DNA bending and twisting

It is striking that the tight helical structure and large bending stiffness of DNA are attained by complementary binding of very flexible ssDNAs with persistence length as small as ~1 nm. Understandably, this feature is mainly due to stacking interactions between two adjacent base-pair planes, which are known to be a complex non-covalent interaction arising from the Van der Waals, hydrophobic, electrostatic interactions, etc [2]. Here we show how the stacking interaction within our mesoscopic model gives rise to very large bending and twisting moduli. At the physiological temperature, the DNA duplex undergoes thermal deformation about the uniform and rigid B-form structure we have considered in Sect. 3. We evaluate the energy of the thermally induced bending and twisting of the helix from our helical bead-spring model and thereby construct an elastic model that is reduced to the wormlike chain in the continuum limit. As long as only thermal deformation is concerned without any external mechanical forces, the study in this limit is validated by the fact that thermally induced bp opening is a rare occasion with a probability ~10^− 5 − 10^− 7 due to the energy barrier for initiating a denatured open structure [38–40], which is much larger than thermal energy. With this background, let us now consider a DNA double helix in physiological environment, with its all base pairs intact while their bp distances are in principle allowed to fluctuate around the equilibrium value D₀. Suppose that a kink occurs at n-th bp plane, with the unit vectors chosen to be along the base-pairing direction and along the helical axis [see Fig. 4].

Fig. 4.

Schematic picture of local coordinates and a kink formed at n-th and (n + 1)-th bp planes. The kink is represented by the angles θ and ϕ, which satisfy the relation:

In the absence of any local deformation, z-axes of all base coordinates coincide with their (x-y) planes rotated by Ω₀ about the z-axis relative to adjacent planes. To describe the dislocation of monomers due to the kink, as shown in Fig. 4, we introduce three Euler-like angles, θ, ϕ, and ψ: The first two angles describe how (n + 1)-th bp plane is bent with respect to n-th bp plane, representing the magnitude and direction of bending angle while ψ represents the angle of rotation between adjacent bp planes, which is Ω₀ for the unperturbed helix. We choose this angle representation in order to incorporate the ϕ-dependence (anisotropy) on the bending energy characteristic of the diagonal interaction. The bending operation mentioned above is obtained by rotating (n + 1)-th coordinate by θ clockwise about the axis [see Fig. 4]. The transformed position of (n + 1)-th monomers then is twisted by ψ counterclockwise about the bent axis . Its final position of is obtained as

6

and with D replaced by − D.

The kink increases the energy by an amount , where , , and . In this study, we neglect the fluctuation of diameter for simplicity (See Appendix B). We expand ΔE(θ,ϕ,ψ) upto second order in θ and ω = ψ − Ω₀ as , where the derivatives are evaluated at the condition of unperturbed B-form helix, θ = ω = 0, h = h₀, and D = D₀. The first, second, and third terms are, respectively, the stacking-induced bending, twisting, and twist-bending coupling energies. Evaluation of the elastic moduli shows that the twist-bending coupling term becomes zero in our symmetric helical model of B-DNA. Thus, the free energy is written as

7

where . Strikingly, the DNA bending due to a single kink is not isotropic, as the induced bending energy depends on the bending direction ϕ. For ϕ = π/2, i.e., for bending perpendicular to the base-pairing direction, the bending energy and thus the induced modulus are zero because the diagonal distances (d_i) remain invariant as r_d. In terms of stacking interactions among base pairs in double-stranded DNA, this result may be viewed such that the energy cost for DNA bending is, at the microscopic level, highly dependent on the degree of destacking induced by the bending. Microscopic bending towards the direction minimizing destacking can easily occur, presumably inducing anisotropic and sharp bending of DNA molecules studied by experiment [6, 7].

The DNA bending at a mesoscopic scale can be viewed as many consecutive kinks with small bending angles. Denoting θ_n, ϕ_n, and ω_n for bending and twisting angles defined by (n + 1)-th and n-th bp planes, we obtain a generalized twistable wormlike chain model as The ϕ_n is integrated out to yield a renormalized bending energy ΔE_B at a mesoscopic scale , where I₀(x) is the modified Bessel function of zeroth order. For very small θ_n, , so that , which remarkably shows that the anisotropic bending behavior predicted at microscopic scale vanishes. The energy of deformation can then be written as

8

where ΔL_b and L_t are respectively stacking-induced bending and twist persistence lengths given by

9a

9b

which are evidently determined by the helix geometry and the stiffness parameter Λ characteristic of stacking strength. In particular, the stacking-induced increase of the bending persistence length is given by ΔL_b = L_b − 2L_bs, where the L_b and L_bs are persistence lengths of dsDNA and ssDNA, respectively [41]. The potential and geometry parameters we have found yield ΔL_b~42 nm and L_t~230 nm. In view of uncertainty and scarcity of the available parameters we used, it is striking that the estimated values are in reasonable agreement with the corresponding values expected from the measured values L_b~50±5 nm [42], L_bs~1 − 4 nm [9], and L_t~75 − 120 nm [43]. This points to consistent validity of the model in describing the double helix structure and the emerging mechanical properties at mesoscopic scales.

Summary and discussion

We have introduced a simple mechanical model accounting for DNA structure and elasticity. Using this model, we studied how a DNA duplex self-assembles into the helix structure due to the stacking interactions, and also how the helix is deformed against the stretching force in comparison with related single-molecule experiments. We have found that an overstretching transition with the plateau, as shown in typical force-extension experiments, can be induced by the coexistence of a helix and ladder structure at a critical force close to the experimental value. We have also shown analytically how a wormlike-chain-like elastic model, frequently used in DNA mechanics, can be derived from our model. The bending and twist stiffnesses are explained in terms of basic interactions and DNA geometrical constants, and their estimated values from the model are in reasonable agreement with the corresponding experimental values.

The basic model we explored here can be improved by further tuning of parameters, and certainly by incorporating features neglected in this initial study. Further analytical calculations and numerical simulations in particular offer new possibilities to study a variety of a single DNA phenomena from nano- to micron-length scales. Some examples are the effects of sequence heterogeneity, ionic solutions, and torsional constraints on mechanics and local denaturation, protein–DNA interaction and the associated enhanced looping flexibility, which may be implemented by incorporating appropriate potentials in the model.

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).

A.1 The case of harmonic intermonomer potential

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 > h₀) but not overstretched (h < h₁ ≅ 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 D₀ 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 and the theory h(f ) from (4a):

10

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 . 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 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, was 0.000009.

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 and assuming D = D₀ in (4), we found the optimized value of k_d that minimizes the square error

11

The best fit value was , at which . 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).

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 . 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 for AT parameters and for GC parameters. Similarly, the parameters used in the Breathing Wormlike Chain model suggest that from [41] and 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 ≅ D₀ 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.

A.2 The case of anharmonic intermononer potential

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:

12

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, , we find the new relation of Ω(f ) for the above Morse potential:

13

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 and D(f ) = D₀. We found the optimized values of V and a that minimize the square error

14

The best-fit values were V = 2.5 k_BT and a = 28.2 nm^− 1, where (). 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 . 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 = h₀, and D = D₀), finding that (1) , (2) , and (3) . 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.