Divalent Ion-Mediated DNA-DNA Interactions: A Comparative Study of Triplex and Duplex

Abstract

Ion-mediated interaction between DNAs is essential for DNA condensation, and it is generally believed that monovalent and nonspecifically binding divalent cations cannot induce the aggregation of double-stranded (ds) DNAs. Interestingly, recent experiments found that alkaline earth metal ions such as Mg²⁺ can induce the aggregation of triple-stranded (ts) DNAs, although there is still a lack of deep understanding of the surprising findings at the microscopic level. In this work, we employed all-atom dynamic simulations to directly calculate the potentials of mean force (PMFs) between tsDNAs, between dsDNAs, and between tsDNA and dsDNA in Mg²⁺ solutions. Our calculations show that the PMF between tsDNAs is apparently attractive and becomes more strongly attractive at higher [Mg²⁺], although the PMF between dsDNAs cannot become apparently attractive even at high [Mg²⁺]. Our analyses show that Mg²⁺ internally binds into grooves and externally binds to phosphate groups for both tsDNA and dsDNA, whereas the external binding of Mg²⁺ is much stronger for tsDNA. Such stronger external binding of Mg²⁺ for tsDNA favors more apparent ion-bridging between helices than for dsDNA. Furthermore, our analyses illustrate that bridging ions, as a special part of external binding ions, are tightly and positively coupled to ion-mediated attraction between DNAs.

Introduction

Nucleic acids are highly charged polyanionic biopolymers, and their collapse into compact state would experience very strong Coulomb repulsion (1). By interaction with polycations such as cationic protein and polyamines, DNAs can be tightly packaged inside cell nucleus, bacterial cytoplasm, and viral capsids (2, 3). In solutions, cations can bind to nucleic acids to reduce and modulate the repulsion between nucleic acid segments. Therefore, cations play an important role in the condensation and assembly of nucleic acids (4, 5, 6, 7, 8, 9). Extensive experiments have shown that cations with valence ≥3 can effectively condense double-stranded (ds) DNAs at millimolar ion concentrations (4, 5, 8), whereas monovalent ions and nonspecifically binding divalent ions such as Mg²⁺ generally cannot condense dsDNAs even at molar ion concentrations (2, 5, 9). The multivalent ion-mediated effective attraction has been proposed to be responsible for the condensation of DNAs and other like-charged polyelectrolytes (10, 11, 12, 13, 14, 15, 16, 17, 18). However, recent experiments have shown that the multivalent ion-mediated condensation of nucleic acids is far beyond what has been known (19, 20). First, recent UV absorption and small-angle x-ray scattering experiments have shown that cobalt hexamines can induce the aggregation of B-form dsDNAs although they cannot cause the aggregation of A-form dsRNAs (19). Second, recent small-angle x-ray scattering and osmotic stress experiments have shown that alkaline earth divalent ions cannot lead to the aggregation of dsDNAs, although they can cause the aggregation of triple-stranded (ts) DNAs (20).

Some polyelectrolyte theories and models have been developed and employed to understand or predict ion-mediated interactions between DNA helices (4, 5, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32). The counterion condensation theory is a successful double-limiting law, although it always predicts an effective attraction between dsDNAs even at a low monovalent salt (21, 22). The Poisson-Boltzmann (PB) theory is rather successful for predicting electrostatic properties of biopolymers in aqueous/monovalent salt solutions (23, 24, 25, 26, 27), although the theory always predicts an effective repulsion between dsDNAs due to the ignorance of ion-ion correlations. The tightly bound ion model can predict an ion-mediated effective attraction between dsDNAs, but this model ignores the realistic molecular charge distribution on atoms and may not make reliable predictions about detailed ion-binding near the DNA surface (28, 29, 30). To understand the recent experiments on the aggregation of dsDNAs and dsRNAs in cobalt hexamine solutions (19) and on the aggregation of dsDNAs and tsDNAs in alkaline earth divalent ion solutions (20), the electrostatic zipper model was extended. This model makes predictions in qualitative accordance with the experiments (31, 32), whereas the partition of binding ions into the minor/major grooves in the model is from the preset parameters (32). Besides the theoretical models, computer simulations can be considered as an effective tool to probe structural dynamics and ion effects for biomolecules (33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44). Very recently, the puzzling lack of dsRNA condensation was explained in Tolokh et al. (45), based on all-atom molecular dynamics (MD) simulations, and the different ion-binding modes (called external binding and internal binding) to dsDNAs and dsRNAs were proposed to be responsible for the different condensation behaviors of dsDNAs and dsRNAs in cobalt hexamine solutions (45). Furthermore, all-atom MD simulations were employed to directly calculate the cobalt hexamine-mediated interactions between two dsDNAs and between two dsRNAs (46). The external binding of multivalent ions (e.g., cobalt hexamine or spermine) to dsDNA favors the ion bridge between adjacent helices and an effective attraction between dsDNAs, whereas multivalent ions internally bind to dsRNA and cannot form apparent ion bridges at millimolar concentrations (45, 46, 47, 48).

However, there is still a lack of deep understanding on why alkaline earth ions such as Mg²⁺ can condense tsDNAs although they cannot condense dsDNAs. Unlike dsDNA, a tsDNA is formed by a third strand binding into the major groove of dsDNA via Hoogsteen (or reverse Hoogsteen) basepairing (49), and its charge density and structure are different from dsDNA to dsRNA. Therefore, tsDNA is a good structure model to study the mechanism of DNA condensation. Moreover, it is known that tsDNA participates in diverse biological functions such as gene regulation and DNA repair (50, 51), and may play a role in modulating the chromosome conformation in eukaryotic genomes (52). Furthermore, there is also a need for direct illustration of the relationship between bridging ions and the effective force between DNAs.

In this work, we employed all-atom MD simulations to directly calculate the potentials of mean force (PMFs) between tsDNAs, between dsDNAs, and between dsDNA and tsDNA in Mg²⁺ solutions. The Mg²⁺ distributions were analyzed in detail and direct comparisons were made between our calculations and the corresponding experimental measurements (20, 53). This work illustrates why Mg²⁺ can condense tsDNAs although it cannot condense dsDNAs at the microscopic level, and provides a direct illustration of the correlation between bridging Mg²⁺ ions and the effective attraction between tsDNAs.

Materials and Methods

All-atom MD simulations

In this work, we calculated the PMFs between dsDNAs (dsDNA-dsDNA), between tsDNAs (tsDNA-tsDNA), and between tsDNA and dsDNA (tsDNA-dsDNA) by all-atom MD simulations. The dsDNA and tsDNA are modeled as a 16-bp homopolymeric poly(dA):poly(dT) duplex in B-form and a poly(dT):poly(dA):poly(dT) triplex, which are built with Nucleic Acid Builder (54). The sequence for tsDNA was chosen according to the recent osmotic stress and small angle x-ray diffraction experiments (20), and the helical structure of tsDNA of a homo-sequence like poly(dT):poly(dA):poly(dT) can be constructed more conveniently than those of hetero-sequence (55). Correspondingly, the sequence of dsDNA was also taken as poly(dA):poly(dT). Specifically, the built tsDNA has an axial rise of 3.26 Å per residue and 12 residues per turn, and the structural parameters are taken from x-ray fiber diffraction (55); see the Supporting Material for details of building tsDNA. Because tsDNAs and dsDNAs were built in their respective standard forms, our calculations should not be affected essentially by the use of poly(dT):poly(dA):poly(dT) and poly(dA):poly(dT) strands.

The two DNA helices with axes kept in the z direction were separated in the x direction and immersed in a cubic box containing explicit water and ions, as shown in Fig. 1. The DNAs were harmonically constrained with a force constant of 1000 kJ/(mol·nm²) in the z direction, and thus the DNAs were allowed to move in the x-y plane and kept in parallel with each other. The ions were added to the systems according to the reported concentrations (20 and 100 mM [Mg²⁺], respectively) (20, 53). We employed the Amber parmbsc0 force field for DNA and the TIP3P water model for the explicit solvent (56, 57). Simultaneously, we adopted the hydrated magnesium ion model (58) with ion-pair specific corrections to the LJ parameters (59), which has been shown to improve the predictions for the Mg²⁺-mediated DNA assembly (58, 60).

Figure 1.

An illustration for two parallel DNAs (tsDNA-tsDNA, dsDNA-dsDNA, and tsDNA-dsDNA) in MgCl₂ solutions. Mg²⁺ ions are shown as green spheres and Cl⁻ ions are shown as cyan spheres (only parts of the ions are shown). The pairs of short DNAs are restrained in the z direction, and a spring with a spring constant k, which connects the mass centers of the two helices has been added to calculate the potentials of mean force between DNAs. The DNA-DNA distance is defined as the distance between the mass centers of two DNAs, and the shown DNA-DNA distances are 30 (A), 29 (B), and 28 Å (C), respectively. To see this figure in color, go online.

All the MD simulations were carried out with Gromacs 4.5 (61). The equilibrated systems of two (ds or ts) DNAs in MgCl₂ solutions were obtained as follows: first, using our previously described protocol (46), an energy minimization of 5000 steps was performed with the steepest descent algorithm, and the systems were slowly heated to 298 K and equilibrated until 0.5 ns. Second, the subsequent NPT simulations of 2 ns (time-step 1 fs, P = 1 atm) were performed and the DNAs were fixed. Subsequently, the steered MD procedures were performed for umbrella sampling. Third, all our MD simulations were continued for another 100 ns in the isothermic-isobaric ensemble (time-step 2 fs, P = 1 atm, T = 298 K). Our MD simulations generally reached equilibrium after ∼10 ns, as shown in Fig. S2 for the number of ions around DNAs versus the MD running time. The MD trajectories in equilibrium are used to calculate the PMF between two DNAs, as described below.

Calculating PMF between DNAs

In this work, we employed umbrella sampling (62) with the pseudo-spring method (46, 63) to calculate the PMF between two DNAs. In the method, a pseudo-spring with spring constant k is added to link the mass centers of two DNAs. Based on the MD trajectories in equilibrium, the PMF ΔG between two DNAs can be calculated by (46, 63)

$$F=k\Delta x,$$ (1)

$$\Delta G\left(x\right)=G\left(x\right)-G\left(x_{\text{ref}}\right)={\int }_x^{x_{\text{ref}}}F\left(x^{\prime }\right)d{x}^{\prime },$$ (2)

where F is the effective force between DNAs. Δx is the mean deviation of the spring length away from the original length x₀ in equilibrium, and x_ref is the outer reference distance. In practice, the spring constant k is generally taken as 1000 kJ/(mol·nm²) and x_ref is taken as 40 Å. For the cases where two helices interact very strongly, we also use a higher spring constant of k = 2000 kJ/(mol·nm²). Furthermore, the MD data were used for calculating the PMFs with the weighted histogram analysis method (WHAM) (64), and the PMFs from the pseudo-spring method are very close to those from the WHAM method; see Figs. S3 and S4 for the PMFs and the technique details of the WHAM method. In the following, we used the DNA-DNA distance to denote the distance between the mass centers of two DNAs.

Continuum electrostatic calculations

For detailed analyses of the Mg²⁺-mediated PMFs between (ds and ts) DNAs, we also calculated the electrostatic potentials for the dsDNA and tsDNA in 20 mM MgCl₂ solutions by solving the PB equation with the APBS software (65). The parameters employed for APBS are listed as follows: the grid spacing is 0.3 Å, the dielectric constants are two for DNAs and 78 for water, and the solvent probe radius is 2 Å. The surface electrostatic potentials are visualized with the VMD program (66).

Results and Discussion

In this work, we directly calculated the Mg²⁺-mediated PMFs between dsDNAs, between tsDNAs, and between dsDNA and tsDNA using all-atom MD simulations. We compared our calculations with the available experimental measurements (20, 53), and detailed analyses were carried out to reveal the microscopic mechanism for the distinctive difference in Mg²⁺-mediated PMFs between dsDNAs and tsDNAs. Furthermore, we provided a direct illustration of the tight and positive coupling between bridging ions and the ion-mediated attraction between DNAs.

PMFs between DNAs in Mg²⁺ solutions

PMFs between dsDNAs

As shown in Fig. 2, the PMF between dsDNAs appears flat and is weakly repulsive at low (∼20 mM) [Mg²⁺]. When [Mg²⁺] is increased to ∼100 mM, the PMF between dsDNAs changes from a weak repulsion to a weak attraction. Such attraction is very slight (∼−0.02 k_BT per bp at a DNA-DNA distance of ∼29 Å), and should be insufficient to induce the condensation of short dsDNAs. The weakly repulsive PMF between dsDNAs at ∼20 mM [Mg²⁺] is in accordance with our MD simulation for two unconstrained dsDNAs. As shown in Fig. 2 C, the two dsDNAs fluctuate across a wide DNA-DNA distance range rather than preferring to remain close to each other during the MD process. Our calculated PMFs between dsDNAs are also consistent with the experiments showing that alkaline earth metal ions cannot induce dsDNA condensation even at high ion concentrations (20, 67). The change in DNA sequence should not cause a strong attractive PMF between dsDNAs, because the A·T sequence used in our MD simulations has been shown to be most prone to multivalent ion-mediated attraction between dsDNAs (68). The role of Mg²⁺ in mediating effective interaction between dsDNAs has been attributed to its +2 charge, which is not high enough when bridging dsDNAs to cause a strong effective attraction (4, 5, 9, 69).

Figure 2.

The potentials of mean force between DNAs is given for three cases: tsDNA-tsDNA, dsDNA-dsDNA, and tsDNA-dsDNA in ∼20 (A) and in ∼100 mM Mg²⁺ solutions (B). (C) Shown here is the distance between the mass center of the DNAs versus the MD simulation time from the MD simulations for two tsDNAs (red) and two dsDNAs (blue) without the connection spring in a 20 mM Mg²⁺ solution. (D) The inter-DNA forces were calculated from our MD simulations and the experimental measurements on osmotic pressure. It should be noted here that the osmotic pressure experiment only gives the repulsive part of the forces between DNAs (20, 53). To see this figure in color, go online.

PMFs between tsDNAs

In contrast to dsDNAs in Mg²⁺ solutions, the PMF between tsDNAs is strongly attractive with the free energy minimum ∼−2.5 k_BT (∼−1.48 kcal/mol) at a DNA-DNA distance of ∼29 Å for ∼20 mM [Mg²⁺], as shown in Fig. 2, A and B. When [Mg²⁺] is increased to ∼100 mM, the attractive PMF becomes stronger and the free energy minimum decreases to ∼−4.2 k_BT (∼−2.45 kcal/mol). The strong attractive PMF between tsDNAs mediated by Mg²⁺ is confirmed by the MD simulation for two unconstrained tsDNAs. Fig. 2 C shows that two unconstrained tsDNAs prefer to fluctuate weakly around the DNA-DNA distance of ∼29 Å. The strong attractive PMF between tsDNAs implies that tsDNAs have apparently stronger condensation propensity in Mg²⁺ solutions than dsDNAs, which is in accordance with the recent experiments (20). In our simulations, the DNAs were constrained in parallel and not separated very far from each other. Actually, the condensation of DNAs would not only depend on the PMF between parallel DNAs, but also on DNA concentration (47), DNA length (47), and DNA end-to-end stacking interaction (70), which are related to translational entropy, rotational entropy, and end-to-end assembly for short DNAs (47, 70), respectively; see Tolokh et al. (47) for the details of evaluating condensation free energy of DNAs. Because our MD simulations allow the rotation of DNA around their axes, we can roughly examine the relative orientation of the DNAs around their axes at different DNA-DNA distances. At large DNA-DNA distance, there is no obvious orientational correlation between the two DNAs. When the DNA-DNA distance becomes small, the two DNAs are inclined to interlock in a groove-to-backbone conformation where bridging ions can efficiently bridge the adjacent phosphate strands in the two DNAs. A more detailed analysis on the orientational correlation between DNAs can be found in Yoo and Aksimentiev (60).

PMFs between dsDNA and tsDNA

It would be interesting to examine the PMF between dsDNA and tsDNA because the PMFs between dsDNAs and between tsDNAs are distinctively different. As shown in Fig. 2, the PMF between dsDNA and tsDNA is visibly attractive, and such effective attraction becomes stronger at higher [Mg²⁺]. The free energy minimum is ∼−1.5 k_BT (∼−0.9 kcal/mol) at ∼20 mM [Mg²⁺] and becomes ∼−2.3 k_BT (∼−1.3 kcal/mol) at ∼100 mM [Mg²⁺], suggesting that the attractive PMF between dsDNA and tsDNA is weaker (stronger) than that between tsDNAs (dsDNAs). The attractive PMF between tsDNA and dsDNA implies that dsDNAs-tsDNAs have a stronger condensation propensity than dsDNAs in Mg²⁺ solutions. Certainly, the obvious dsDNA-tsDNA condensation in Mg²⁺ solutions is still required to be measured experimentally to validate our findings here.

In the following, we made quantitative comparisons with the experimental measurements (20, 53). Based on the assumption that inter-DNA forces in a DNA aggregate are additive (60), the inter-DNA force for dsDNAs and tsDNAs can be approximately calculated from the experimental osmotic pressures Π (29, 60),

$$F_{\text{inter}\text{-}\text{DNA}}=dh\Pi /\sqrt{3},$$ (3)

where d is the mean distance between two nearest neighbors in a DNA array, and h is the length of 16-bp dsDNA or tsDNA. As shown in Fig. 2 D, the calculated inter-DNA force between dsDNAs is weakly repulsive at ∼20 mM [Mg²⁺] and that between tsDNAs is strongly attractive, which agrees well with the experimental data (20, 53). Alternatively, we can also calculate the osmotic pressure for dsDNA and tsDNA arrays based on the additive assumption (29, 60), and make direct comparisons with the experimental measurements on osmotic pressures (20, 53). As shown in Fig. S5, the calculated osmotic pressures are in good agreement with the experimental data for both tsDNAs and dsDNAs. The slight deviation between the calculations (∼29 Å) and experiments (∼30 Å) may be attributed to the slight nonadditive effect at high divalent salts (10, 71), and it remains difficult in general to completely achieve a quantitative agreement between the calculations and experiments.

As described above, the experimental findings that Mg²⁺ cannot condense dsDNAs whereas it can condense tsDNAs is attributed to the Mg²⁺-mediated forces between dsDNAs and between tsDNAs (20, 72). The PMF between dsDNAs in a Mg²⁺ solution is repulsive (or very weakly attractive at high [Mg²⁺]), whereas the PMF between tsDNAs is strongly attractive. The question that follows is: why can Mg²⁺ ions cause an effective attraction for tsDNAs, although they cannot for dsDNAs?

Mg²⁺ distribution around DNA: triplex versus duplex

To understand the distinctively different roles of Mg²⁺ in mediating effective interactions between dsDNAs and between tsDNAs, we analyzed the Mg²⁺ distributions around dsDNA and around tsDNA, particularly given the different ion-binding modes between dsDNA and dsRNA (46, 47, 48). The cylindrical distributions c(r) values of Mg²⁺ around dsDNA and around tsDNA are calculated according to Wu et al. (46), where r is the axial distance from the helical axis. As shown in Fig. 3, A and B, Mg²⁺ ions begin to bind at an axial distance of r ∼5 Å and prefer to accumulate in two distribution regions separated by an apparent local minimum for both dsDNA and tsDNA. The inner Mg²⁺-binding regions are in the axial distance ranges of 5 < r < 10 Å for dsDNA and 6 < r < 12 Å for tsDNA, which correspond to the ion binding into a groove. The outer Mg²⁺-binding regions are in the ranges of 10 < r < 14 Å for dsDNA and 12 < r < 17 Å for tsDNA, which correspond to the ion binding to phosphate groups. The ranges of the inner region and outer region for Mg²⁺-binding are slightly different for dsDNA and tsDNA, which is attributed to the larger helical radius of tsDNA than that of dsDNA (∼12.5 Å for tsDNA and ∼11 Å for dsDNA) (20). As suggested in the literature (46, 47, 48), the Mg²⁺-binding in the inner region is called the “internal binding Mg²⁺”, whereas that in the outer region is called the “external binding Mg²⁺”. Because the total number of ions binding to a DNA (per phosphate) is almost constant, the number of internal binding ions and the number of external binding ions are negatively coupled: the fewer (more) internal binding ions would correspond to the more (fewer) external binding ions. The internal binding ions of one DNA are generally separated from the other DNA by a large distance, and consequently make much less contribution to modulating the inter-DNA attractive force than an ion positioned between two DNAs. Thus a large ratio of external binding ions is generally more conducive to an effective attraction between two DNAs.

Figure 3.

(A and B) Shown here are the Mg²⁺ concentration distributions around DNA and the corresponding net charge distribution Q(r) per unit charge on the DNA molecules as functions of the radial distance r from the central axis of DNA. To avoid the end effect, the central full helix turn in the middle of the DNA is selected for the analyses. To see this figure in color, go online.

For dsDNA, the peak of c(r) for external binding Mg²⁺ has a similar height and width to that for internal binding. However, for tsDNA, the peak of c(r) for external binding Mg²⁺ is significantly higher and wider than that of internal binding. Our calculations further show that at ∼20 mM [Mg²⁺], the number of internal binding and external binding Mg²⁺ ions is 2.5 and 4.2 for the central full turn of dsDNA, and 2.6 and 8.1 for the central full turn of tsDNA, respectively. This indicates that the distinctive difference in Mg²⁺-binding between dsDNA and tsDNA is attributed to the preference for ion binding modes: for dsDNA, the internal binding of Mg²⁺ ions into the groove is only slightly weaker than the external binding to phosphate groups; however, for tsDNA, the external binding to phosphate groups is dominant over the internal binding into the groove. The strong external binding of ions would favor strong ion-bridging between DNAs to cause an apparently effective attraction, as suggested in recent studies (46, 47, 48). Therefore, the strong external binding of Mg²⁺ is responsible for the strong attraction between tsDNAs, whereas for dsDNA, the external binding of Mg²⁺ is relatively weak and consequently cannot induce an apparent attraction between dsDNAs. In addition, as shown in Fig. 3, A and B, when [Mg²⁺] increases from ∼20 to ∼100 mM, the external binding of Mg²⁺ becomes stronger for both tsDNAs and dsDNAs due to the lower binding penalty of ions. As a result, the PMF between tsDNAs becomes more attractive, and the PMF between dsDNAs can become weakly attractive at high [Mg²⁺], as described above.

Why does Mg²⁺ bind to dsDNA and tsDNA in different ways? To understand the distinctively different ways that of Mg²⁺ binds to dsDNA and tsDNA, we examined the different structures of dsDNA and tsDNA as well as the electrostatic potential around them. As shown in Fig. 1 A, B-form dsDNA has a narrow minor groove (width ∼6 Å) and a wide major groove (width ∼12 Å), whereas for tsDNA, the binding of the third nucleotide strand into the major groove of the DNA duplex causes tsDNA to have three grooves: two of them are very narrow (∼3 and ∼4 Å) and one is slightly wide (∼9 Å). Here, the groove widths were calculated as the smallest separation distances between the P atoms in two strands minus 5.8 Å (approximately the sum of the van der Waals radii of two phosphate groups) (73). As a result, the accessible volume of ions in the grooves for tsDNA would be smaller than that for dsDNA, as shown in Fig. 4 C. The smaller ion-accessible volume in the grooves would cause Mg²⁺ binding to tsDNA to be slightly more external, than to dsDNA. More importantly, more strands in tsDNA can cause stronger electrostatic potential in the vicinity and different binding sites compared with dsDNA. We have calculated the electrostatic potential on the Mg²⁺-accessible surface for both tsDNA and dsDNA with the PB solver of APBS (65), although the PB theory may not capture some details in the vicinity of DNA surface (74). As shown in Fig. 4, A and B, a dsDNA has two kinds of preferential sites for ion binding: N7 atoms in the wide major groove and phosphate groups at the narrow minor groove. In contrast, for tsDNA, the most preferential binding sites for Mg²⁺ ions are phosphate groups at the two very narrow grooves, and the binding into the slightly wide groove is the next preferential binding site. Therefore, for dsDNA, the internal binding (into the major groove) and external binding to phosphate groups (at the minor groove) are both favorable, which is in accordance with the cylindrical distribution of Mg²⁺ around dsDNA. However, for tsDNA, the external binding (to phosphate groups at two narrow grooves) is dominant over internal binding (into the slightly wide groove), corresponding to the Mg²⁺ distribution around tsDNA in Fig. 3. Furthermore, Fig. 4 shows that the surface electrostatic potential of tsDNA is apparently stronger than that of dsDNA, which would cause stronger and tighter binding of Mg²⁺; see also Fig. 4, D–F.

Figure 4.

(A and B) Given here is the electrostatic potential at the hydrated Mg²⁺ accessible surface of tsDNA and dsDNA. Shown is the electrostatic potential computed 3 Å away (radius of hydrated Mg²⁺ is ∼3Å) from the DNA surface. (C) The ion-inaccessible volume per unit length (Å) is given for tsDNA and dsDNA. (D) Given here are the radial distribution functions g(r) of Mg²⁺ with respect to the phosphate atoms of tsDNA and dsDNA. (E and F) Shown here is the distribution of residence time for the Mg²⁺ within 5.4 Å of the phosphate atoms for tsDNA and dsDNA, where 5.4 Å corresponds to the peak location in (D). To see this figure in color, go online.

Therefore, the stronger external binding to tsDNA than to dsDNA is attributed to the enhancement of the electrostatic potential and smaller ion-accessible volume in the grooves caused by the third nucleotide strand, which leads to stronger external binding of Mg²⁺ to bridge adjacent helices to induce an effective attraction. We also note that a larger fraction of spermine in the external shell for tsRNA of poly(A·U·U) than that for dsRNA of poly(A·U) was proposed to understand the condensation of poly(A·U) in spermine solutions, because the tsRNA of poly(A·U·U) can possibly be formed in the presence of spermine (48). Additionally, the recent MD simulations for RNA triplex observed that multivalent cations of spermines have a preference to bind more externally to a RNA triplex than to a RNA duplex (48), which is in analogy to our observation that the external binding of Mg²⁺ to a tsDNA is much more pronounced than to a dsDNA.

Bridging ions between DNAs: triplex versus duplex

Going beyond a single helix, to understand the Mg²⁺-mediated PMFs between (ts and ds) DNAs, we can illustrate Mg²⁺ concentration distributions around two tsDNAs, two dsDNAs, and a tsDNA and a dsDNA. As shown in Fig. 5, there is a strong ion-bridging configuration between tsDNAs, whereas that between dsDNAs appears obviously weaker, which corresponds to the strongly attractive PMF between tsDNAs in Mg²⁺ solutions. For tsDNA-dsDNA, the ion-bridging is slightly weaker than for tsDNA-tsDNA and corresponds to that the Mg²⁺-mediated attraction between tsDNA and dsDNA is slightly weaker than that between tsDNAs. Furthermore, for a quantitative view of Mg²⁺ distributions, we calculated the 2D distributions for tsDNA-tsDNA, dsDNA-dsDNA, and tsDNA-dsDNA. As shown in Fig. 5, for tsDNA-tsDNA at ∼20 mM [Mg²⁺], the local concentration of Mg²⁺ in the interfacial region is very high (∼6 M), but for dsDNA-dsDNA, the Mg²⁺ concentration (∼4 M) is visibly lower than that for tsDNA-tsDNA. When DNAs approach each other, Mg²⁺ ions accumulate in the interfacial region to form ion-bridges between two DNAs, as shown in Fig. S6. Such an ion-bridge has been proposed to be responsible for the multivalent ion-mediated attraction between DNAs and between other polyelectrolytes (46, 47, 60, 68, 75, 76). As suggested above, the external binding of Mg²⁺ favors the ion-bridge and the effective attraction between DNAs.

Figure 5.

(Top panels) Given here are the illustrations for the region of high Mg²⁺ density (solid area >10 M, and transparent area >5 M) around two 16-bp DNAs in 20 mM Mg²⁺ solutions, respectively. (Bottom panels) Given here are the distributions of Mg²⁺ for the three cases corresponding to the top panels averaged over the MD trajectory and over the DNAs in the z direction. The DNA-DNA distance is 30 Å for tsDNA-tsDNA (A), 28 Å for dsDNA-dsDNA (B), and 29 Å for tsDNA-dsDNA (C). To see this figure in color, go online.

Because there is no available physical definition for bridging ions, we would define bridging ions as those residing within a short cutoff distance r_c of 9 Å from the surface atoms of both DNAs. This definition is based on the spatial positions of ions relative to both the two DNAs, and the slight change in cutoff distance r_c around 9 Å does not qualitatively influence our analyses (see below). As shown in Fig. 6, the number of bridging ions increases as the two DNAs become close, and such increase in the number of bridging Mg²⁺ is much more pronounced for tsDNAs, indicating a stronger ion-bridging effect for tsDNAs and consequently a stronger effective attraction between tsDNAs. It is not strange that the number of bridging ions is not very sensitive to the increase of bulk [Mg²⁺] (from ∼20 to ∼100 mM), because multivalent ion-binding near the DNA surface is more dependent on the electrostatic field nearby than on bulk ion concentration (21, 72). However, the increase of bulk [Mg²⁺] would reduce the entropy penalty for ion-binding and consequently could cause stronger effective attraction between DNAs (63, 72). When the DNA-DNA distance becomes less than a certain value (e.g., ∼29 Å for tsDNAs), the increase in the number of bridging ions slows down and the rapid increase in Coulomb repulsion between the negative charges on the two DNAs results in a strong short-range repulsion between DNAs. Moreover, as shown in Fig. S7, the bridging Mg²⁺ ions between DNAs are dynamic (with a residence time of no more than a nanosecond) and the residence time of bridging Mg²⁺ for tsDNAs is longer than that for dsDNAs, indicating a more pronounced bridging effect for tsDNAs.

Figure 6.

(A and B) Shown here are the numbers of bridging Mg²⁺ ions and internal binding Mg²⁺ ions (in insets) per phosphate group as functions of the DNA-DNA distance. Here, it should be noted that the radius of tsDNA is larger than that of dsDNA. In general, the number of bridging ions for tsDNA-tsDNA is apparently more than that for dsDNA-dsDNA, whereas the number of internal binding ions for tsDNA-tsDNA is visibly less than that for dsDNA-dsDNA. To see this figure in color, go online.

In addition, we calculated the numbers of internal binding Mg²⁺ as functions of DNA-DNA distance for the tsDNA-tsDNA, dsDNA-dsDNA, and tsDNA-dsDNA. Fig. 6 shows that the numbers of internal binding Mg²⁺ ions remain almost invariable as two DNAs approach each other, indicating that the internal ion binding does not respond to the approaching DNAs and consequently makes no contribution to the effective attraction between the tsDNAs. This is reasonable because internal binding ions generally bind deeply into the grooves of a DNA and the approach of another DNA can only have a negligible effect on such ion binding.

Correlation between bridging ions and inter-DNA force

To reveal the role of bridging ions in modulating the effective attraction between DNAs, we further analyzed the bridging Mg²⁺ and instantaneous inter-DNA forces for tsDNAs in a 20 mM Mg²⁺ solution. Additional MD simulations were performed, in which tsDNAs were strongly restrained to keep them immobile, and the instantaneous force between the tsDNAs was calculated by (60, 77, 78)

$$F\left(d\right)=-\frac{\partial U\left({\mathit{r}}^N\right)}{\partial d},$$ (4)

where U(r^N) is the total potential energy of the system, and d is the reaction coordinate (the distance between the mass centers of two DNAs). As the pair of DNAs is aligned along the z direction and separated in the x direction, the force only relates to the x component of the force acted on each atom of the DNAs. For this case, the inter-DNA force can be calculated by

$$F=-\frac{1}{2}\left(F_{1x}-F_{2x}\right),$$ (5)

where F_1x and F_2x are the total force in the x direction acting on the left DNA and right DNA, respectively; see the Supporting Material for more details.

Fig. S9 shows the instantaneous force between tsDNAs and the instantaneous number of bridging Mg²⁺ as functions of MD running time, where water molecules are removed in the analyses due to the strong fluctuations. It is shown that the two curves fluctuate versus the MD running time and there is a visible correlation between the two curves: the fluctuation of the number of bridging Mg²⁺ to a high value would correspond to a stronger attraction force. Such a correlation coefficient is ∼−0.42, suggesting that bridging Mg²⁺ does contribute to the effective attraction between tsDNAs. Moreover, we examined the effect of the cutoff distance r_c on the correlation between the instantaneous inter-DNA force and the number of bridging Mg²⁺. As shown in Fig. S8, with the different values for the cutoff distance r_c for defining the bridging Mg²⁺, the instantaneous number of bridging ions also has a visible correlation with the instantaneous inter-DNA force, suggesting that the above analyses are robust. Fig. S9 also shows the instantaneous numbers of internal binding Mg²⁺ and external binding Mg²⁺. The correlation coefficients between the instantaneous inter-DNA force and the numbers of internal binding Mg²⁺ and external binding Mg²⁺ are ∼0.08 and ∼−0.14, respectively. This may suggest that the effective force between tsDNAs is positively correlated with the internal binding ions and negatively correlated with the external binding ions in a slight manner. The correlations between the inter-DNA force and the number of binding Mg²⁺ have been shown more directly in Fig. 7. In an analogy to the correlation coefficients calculated above, the correlation between inter-DNA force and the numbers of internal binding Mg²⁺ and external binding Mg²⁺ are weakly positive and negative, respectively. Furthermore, the correlation between the inter-DNA force and the number of bridging Mg²⁺ is strongly negative, i.e., the large and small numbers of bridging Mg²⁺ correspond to the attractive (negative) and repulsive (positive) inter-DNA forces for tsDNAs, respectively. Therefore, the above analyses show that the bridging Mg²⁺ ions as a special part of external binding ions are strongly coupled to the effective attraction between tsDNAs.

Figure 7.

Shown here is the coupling between inter-DNA force and internal binding Mg²⁺ (A), external binding Mg²⁺ (B), bridging Mg²⁺ (C), and bridging Mg²⁺ with the contribution of water molecules (D). Here, positive (negative) force indicates the effective repulsion (attraction). The inter-DNA forces corresponding to same number of bridging (or internal binding or external binding) Mg²⁺ ions are averaged and the error bars are obtained as standard errors of the averages. The distance between two tsDNAs is 32 Å. To see this figure in color, go online.

Furthermore, although the inclusion of water may bring a strong fluctuation, we examined the correlation between the number of bridging Mg²⁺ and the instantaneous inter-DNA force with the contribution of water molecules. We extracted ∼8000 conformations from the MD trajectory and performed energy minimization to remove all kinetic energy from the system to reduce the thermal noise. We then calculated the instantaneous inter-DNA force and number of bridging Mg²⁺ ions. Fig. 7 D shows that, as an analogy to Fig. 7 C, the negative correlation between the inter-DNA force and number of bridging Mg²⁺ still exists, except at a much lower (absolute) correlation slope. This illustrates that the bridging ions are critically important for the effective attraction between tsDNAs, and the number of bridging ions can emerge as an “order parameter” that correlates well with the ion-mediated attraction between DNAs.

In addition, the comparison between Fig. 7, C and D, shows that water molecules would provide a global screening for the inter-DNA force, i.e., the attractive (repulsive) force at large (small) numbers of bridging Mg²⁺ would become apparently weaker with the contribution of water molecules; see also Fig. S10. Such an effect is attributed to the nature of the strong electric dipole of water molecules. The rotation and translation of water molecules would significantly reduce the DNA-DNA Coulomb repulsion, which is also reflected by the high dielectric constant (∼78 at 25°C) of water in light of continuous dielectric model (79). However, the above analyses are only from the viewpoint of a simple exclusion/inclusion of water. In fact, the bridging ions and water molecules are strongly coupled, and the bridging ions can serve as an important modulator for water structure between two like-charged DNAs, as there would be orientational frustration for the water molecules between like-charged DNAs without the bridging (multivalent) ions. The bridging Mg²⁺ ions between tsDNAs can induce the structuring of water molecules with the O-atom toward the Mg²⁺ and the H-atom toward the phosphate groups, and such an effect is very apparent at 32 Å, where there are visible hydrogen-bonds between the H-atom and phosphate groups in the x direction and the attraction between tsDNAs is the strongest, as shown in Fig. S11. Therefore, the bridging ions contribute substantially to the effective attraction between tsDNAs, and water molecules may also make a contribution to such effective attraction through the structuring and hydrogen bonding with phosphate groups mediated by (multivalent) bridging ions (80, 81, 82).

Conclusions

This work shows that the Mg²⁺-mediated interaction between tsDNAs is apparently attractive, and such attraction becomes stronger at higher [Mg²⁺], whereas the effective interaction between dsDNAs mediated by Mg²⁺ cannot become apparently attractive even at high [Mg²⁺]. Our analyses show that such distinctive differences in the PMFs for tsDNAs and dsDNAs are attributed to the different ion-binding modes: Mg²⁺ will internally bind to groove and externally bind to the phosphate groups for both tsDNA and dsDNA, whereas the external binding to tsDNA is significantly stronger than that to dsDNA. The stronger external binding of Mg²⁺ to tsDNA (than to dsDNA) is attributed to the enhancement of electrostatic potential and less ion-accessible volume in the grooves caused by the third strand, and would favor more apparent ion-bridging between adjacent tsDNAs. Our additional calculations show that the PMF between tsDNA and dsDNA is attractive and the effective attraction is visibly weaker than that between tsDNAs. Furthermore, our microscopic analyses illustrate a tight coupling between bridging Mg²⁺ and the inter-DNA force. Our calculations are in accordance with existing experiments (20, 53). First, the calculated PMF between tsDNAs mediated by Mg²⁺ is strongly attractive whereas that between dsDNAs is not apparently attractive even at high [Mg²⁺], which implies that tsDNAs would have the apparently greater condensation tendency than dsDNAs in Mg²⁺ solutions. This is consistent with the experiments where Mg²⁺ can condense tsDNAs but it cannot condense dsDNAs (20). Second, the calculated inter-DNA forces from our MD simulations are close to those from the osmotic stress measurements (20, 53) for both tsDNAs and dsDNAs based on the additivity assumption (29, 60).

However, this work also involved some approximations and simplifications. First, in the MD simulations, the dsDNA and tsDNA are treated as rigid helical structures and the effect of DNA flexibility was ignored. Because the persistence length of DNA is large (≥500 Å) (38, 39, 83, 84), the ignorance of DNA flexibility would be a reasonable simplification. Second, in our MD simulations, we adopted the hydrated Mg²⁺ model proposed by Yoo and Aksimentiev (58), because the model can achieve a quantitative agreement with experiments on DNA assembly (58, 60). Although the ion model ignores the dehydration effect of Mg²⁺, previous studies have shown that Mg²⁺ generally keeps its hydration shell in solutions and seldom becomes dehydrated when interacting with DNA/RNA helices (85, 86). An accurate description of divalent ions in MD simulations requires realistic treatment of the polarizability and kinetics of the surrounding water (87), and developing a general model for ions, especially for Mg²⁺, remains a challenge. Third, the two DNAs are constrained in parallel in our simulations and thus the nonparallel configurations of DNAs were ignored in our work. Some previous studies have shown that the parallel configuration would be most favorable for large DNA-DNA distances (>∼28 Å) (88), whereas for small DNA-DNA distances, the parallel configuration will be unstable due to the strong short-range repulsion, and two DNAs would favor a typical x-shape configuration (89). Moreover, the nonparallel (rotational) configurations of DNAs could make unfavorable entropy contributions to DNA condensation free energy. The contribution of nonparallel configurations would become less important for long DNAs. This is beyond the scope of this work, and deserves to be studied separately. Nevertheless, this work will be very helpful for understanding the distinctively different condensation behaviors of tsDNAs and dsDNAs in Mg²⁺ solutions, as well as the multivalent ion-mediated attraction between DNAs.

Author Contributions

Z.-J.T., J.-P.S., and Z.-L.Z. designed the research. Z.-L.Z., Y.-Y.W., and K.X. performed the calculations. Z.-J.T., Z.-L.Z., and J.-P.S. analyzed the data. Z.-L.Z. and Z.-J.T. wrote the manuscript. All authors discussed the results and reviewed the manuscript.

Editor: Chris Chipot.