The structure of DNA fragments: Quantum-chemical modelling

Highlights

• •(dA)5·(dT)5 anion in vacuum forms a DNA duplex with a conformation intermediate between a helix and a ladder.
• •The structure of the helical conformations of (dA)5·(dT)5 and (dG)5·(dC)5 is close to that of the A- and B-DNA forms.
• •B-DNA mini-helix is stabilized by the compensative Na+ counterions or explicit micro-hydration of minor and major groves.
• •Bulk water plays a minor role in stabilizing the structure.

Abstract

In this review, we analyze and systematize our computational studies of the nucleic acid duplex formations and thermodynamic stability under the different factors of investigation. The proposed structural models of mini-helix contains N nucleobase pairs (N = 3-5); QM structural data suggest that the helical conformations of mini-helix adopt geometrical parameters comparable to those of natural A- and B-DNA forms under specific conditions as micro hydration and charge compensation. The gas-phase models adopt non regular conformations between the helical form and a ladder form.. The natural helical shape of DNA mini-helix is stabilized by the presence of counterions or by explicit micro-hydration of the major and minor groves. The presence of aqueous solution is shown as a minor factor for the helical shape formation. The studies are performed at the level of density functional theory.

Graphical abstract

1. Introduction

An exhaustive investigation of DNA structure is explained by its role in maintaining the life of living organisms. Suggesting a famous double helix shape of DNA macromolecules, Watson and Crick made the significant first step in this direction. Since that, a DNA structure has been the subject of an enormous amount of experimental investigations that are mostly performed by an application of X-ray diffraction (see ref. in the book and review [1,2]) and NMR [2,3] techniques. As a result, the names of different DNA conformations take up almost all the letters of the Latin alphabet [4].

It is no doubt that such famous achievements in the understanding of a DNA structure would not be possible without the application of state-of-the-art computational algorithms that allow us to resolve and simulate a DNA structure. Those simulations started in parallel at the static and dynamics level using semi-empirical quantum-chemical methods (QM) and classical molecular dynamics (MD) methods. In spite that a historical survey is not the purpose of this review (see perfect description of these aspects in [5]), we would like to mention that the foundations of classical MD studies of single and double-stranded DNA and RNA have been established by such outstanding scientists as De Voe and Tinoco [6], Bradley [7], Pullman [8], Poltev (see [5] and the references therein), Kitaygorodski [9], Kollman [10] and many others. Those calculations use atomic charges among the initial parameters to start MD calculations. Therefore, QM calculations of nucleo bases and base pairs were an indispensable part of MD simulations at the very beginning.

In contrast to force field MD the ab initio and DFT QM provide more reliable data about the electronic structure of the molecule. Gradually, with the increasing power of supercomputers the comprehensive study of electronic structure and thermodynamics of molecular systems of 100 and more atoms become feasible. However, until approximately the very end of the 20th century, many quantum-chemical calculations have included just the single or the paired nucleobases. They served as the simplest structural models of the DNA chain to study the molecular basis of stability of the DNA duplex. An enormous number of calculations use these oversimplified models. The names of the following scientists should be mentioned in this regard: Pulman [8], Kwiatkowski [11], Hobza [12] Leszczynski [13], Hovorun [14]) and many others.

Nevertheless, there is a strong need to extend this investigation by studying the fragments that model double stranded DNA macromolecule more realistically. This means it is necessary to investigate DNA fragments containing all DNA structural components: nucleosides, counter-ions, and sugar-phosphate backbone. Also, those studies should include an influence of physiologically relevant conditions like water bulk, surrounding enzymes, etc. Only recently, it has been possible to perform such computations at QM level that provides accuracy close to experimental results. Nevertheless, even in the most recent publications, there are investigations of just dinucleotides step or single-strand DNA oligonucleotides. To the best of our knowledge, there are just a few publications where the authors investigated DNA fragments larger than two nucleotide steps by quantum-chemical calculations.

Below we describe our efforts to go beyond the “just two DNA bases” paradigm.

2. Structure of 2´-deoxyribonucleotides

Canonical 2-deoxyribonucleotide The simplest structural models that have all three chemical units that compose a DNA (a DNA base, a 2′-deoxyribose sugar, and a phosphate group) are the canonical 2´-deoxyribonucleotides (DNTs). Namely, these are deoxythymidilic (pdT), deoxycytidilic (pdC), deoxyadenylic (pdA), and deoxyguanylic (pdG) acids and their anions (Fig. 1).

Fig. 1.

Chemical structure of 2´-deoxyribonucleotides.

It is generally accepted that nucleotides are not rigid molecules [1,17]. The traditional source of the nonrigidity of nucleotides is the rotation of the base and sugar units around each other. The phenomenon of the pseudorotation of the furanose ring is another source of nonrigidity. However, the results of numerous investigations (see for example [18] and references therein) suggest that DNT molecules could have some more sources of flexibility than those mentioned above. This conclusion follows from the results of gas-phase calculations of isolated DNA bases. It was shown [19,20] that the amino group in cytosine, guanine, and adenine adopts a non-planar configuration with relatively small inversion barriers. The reason is the behavior of the hydrogen atoms of the amino group. They undergo the large amplitude motion even when they are involved in hydrogen bonding with a complementary DNA base. Another phenomenon of non-planarity relates to conformations of pyrimidine and purine rings. They also possess a non-planar geometry, [12,20,21].

Further we summarize the QM results for 2-deoxyribonucleotides [15,16,[22], [23], [24]]. It was found that the geometry of these biomolecules is very sensitive to the mutual orientation of the nucleobases and furanose rings, the intramolecular hydrogen bonds, the charge and conformation of sugar-phosphate backbone. It was also observed that incorporated into nucleotides, a nucleobase moiety remains nonplanar and nonrigid similarly to the isolated nucleobase.

New important properties are introduced by the interaction of the nucleobases with the furanose ring and the phosphate groups. This interaction leads to a considerable deformation of the pyrimidine rings and a partial pyramidality of the amino group. The monoanionic nucleobases which adopt the syn-orientation are deformed the most. The structural data of the nucleotide pdC northern confirmation (Fig. 2) suggest that the deformed conformation is stabilized by the N4-H...O-P hydrogen bond between the amino-phosphate groups. A significant distortion of the planarity of the pyrimidine ring and an increase in the pyramidality of the amino group is clearly present [21].

Fig. 2.

The local minimum structure of north/syn conformer of pdC monoanion with orthogonal orientation of base with respect to ribose (DFT/B3LYP/6-31G(d), vacuum). Dashed lines indicate intramolecular hydrogen bonds. The structure is used to visualise critical properties, such as the piramidality of the amino group, nonplanarity of the nucleobase rings, and the formation of hydrogen bonds.

The obtained QM structural data in aqueous solution and gas phase for isolated nucleotides could point on possible mechanism of anti-motif stabilization of 2´-deoxyribonucleotides in DNA double helix (Table 1S). When the charge of the 2′-deoxyribonucleotide anions is fully compensated by the protons, the syn-conformations become more favorable due to the stabilization by intermolecular hydrogen bonds. In contrast, the presence of negative charge resulting from the successive removal of protons from the phosphate group stabilizes the south/anti-conformations of the 2′-deoxyribonucleotide anions. Thus, the anti-conformations, which are completely dominant in natural DNA, become more favorable (with the exception of those of the 2′-deoxyguanosine phosphate). The energy difference between the syn- and anti-conformations is in the range of 1 kcal/mol. We believe that mechanism of nucleotide charge compensation has biological importance for the stabilization of the anti-motif in DNA strands.

The conformation of single nucleobases within DNA double helix plays not less vital role than the conformation of sugar-phosphate backbone. Not without reason, Watson and Crick emphasized the importance of the tautomerism of nucleobases for DNA structures saying that only canonical tautomers of nucleobases can form hydrogen-bonded complexes with their natural counterparts. Thus, tautomeric modifications of nucleobases can lead to errors in replication and transcription – the origin of point mutations in DNA [[25], [26]]. Here, we investigate the factors contributing to the stability of the canonical nucleobase's tautomers by tautomeric equilibrium theoretical calculations. We consider the tautomeric interconversion as a chemical reaction of the type C⇌R, where C represents the ground state of the canonical form of nucleobase (Fig. 3, 1a-4a) and R represents the ground state of corresponding “rare” form (Fig. 3, 1b-4b). Only prototropic tautomerism has been considered for C/R reactions. The thermodynamics calculations have been performed for tautomeric interconversions C⇌R in the gas phase. The tautomerization Gibbs free energy difference between tautomers is defined as:

$$\Delta G_{C/R}^{gas}=\Delta E_{C/R}^{el}+\Delta E\left(ZPE\right)+\Delta E^t-T\Delta S,$$ (1)

where the difference in total energy from the electronic structure calculation $\Delta E_{C/R}^{el}$ is obtained at the same level method as zero-temperature vibrational energy ΔE(ZPE), thermal corrections $\Delta E^t$ and entropy contributions $T\Delta S$ are from the corresponding frequency calculations.

Fig. 3.

The most stable tautomers of the nucleobases are considered for further modelling (variations in hydrogen atoms are shown in red).

The gas phase calculations prove the evidence (Table 1S, 2S) that the tautomeric properties of isolated DNA bases and anti-conformers of 2′-deoxyribonucleotides are practically identical. They do not depend on the amount of uncompensated negative charge. However, the tautomeric properties of isolated adenine, thymine, guanine and cytosine differ considerably. Namely, the canonical form of adenine and thymine is much more stable than the canonical form of guanine and cytosine. QM calculations predict (which is consistent with experimental data, see e.g. [27,28]) that about 1% of a gas phase of guanine and cytosine at equilibrium consists of their tautomeric (”rare”) forms. This is more than enough to provide the observable frequency of point mutations. In contrast, the calculated values of the equilibrium constants in the case of adenine and thymine indicate a negligible contribution of the “rare” forms to the composition of the gas phase state of these compounds and accordingly to the frequency of point mutations. Therefore, if the gas phase data correctly describe the situation in a replisome, only 2′-deoxyribonucleotides possessing south/anti- and north/anti-conformations and containing guanine and cytosine could make a significant contribution to the rate of spontaneous point mutations due to the formation of biologically relevant amounts of "rare" tautomers. However, we found a strong influence of the uncompensated negative charge of the 2′-deoxyribonucleotides, which increases the concentration of syn-conformations. We have also found that a spontaneous conformational change associated with tautomeric proton transfer could lead to additional stabilization of the "rare" isomers for DNA nucleobases such as cytosine and thymine.

Table 1.

Relative Gibbs free energies and equilibrium constants for tautomeric equilibrium in gas phase. [22].

DNA Base	MP2				DFT
	6-31G(d)		6-311++G(d,p)a		6-31G(d)		6-311++G(d,p)b
	ΔG298 K		ΔG298 K		ΔG298 K		ΔG298 K
Adenine	13.0	3.1×10−10	13.2	2.2×10−10	12.6	6.0×10−10	12.3	9.2×10−10
Cytosine	0.8	2.6×10−1	1.5	8.0×10−2	1.9	4.1×10−2	2.0	4.4×10−2
Guanine	2.4	1.8×10−2	0.5	4.3×10−1	2.3	2.1×10−2	1.4	9.5×10−2
Thymine	14.5	2.4×10−11	11.9	1.9×10−9	14.2	4.0×10−11	13.1	2.6×10−10

3. Structure of mini-helixes

As mentioned in the previous section, 2-deoxyribonucleotides are the simplest structural models consisting of all DNA units. This already allows to distinguish the properties of fragments that characterize a DNA structure (e.g. anti- vs. syn- and north- vs. south- conformers). The next minimal DNA fragment capable of simulating the geometric structure of different DNA conformations should consist of at least three fused nucleotide pairs, e.g. d(A)₃-d(T)₃ or d(G)₃-d(C)₃. The advantage of such a model is illustrated in Fig. 4. The system under consideration is the smallest possible fragment of a DNA helix, 2/3 of which belongs to the terminal DNA fragments.

Fig. 4.

Schematic diagram of a duplex composed of three stacked base pairs and a sugar-phosphate backbone (the distributions of the marked tortion angles are further analysed in Fig. 5).

This fact has prevented us from making reliable investigations of the geometrical parameters only for this central fragment. Nevertheless, in many cases we have obtained reasonable agreement between experimental and MD data and the results of the current investigations.

3.1. B-forms

The first comprehensive density functional study (DFT) of the Watson-Crick trideoxyribonucleoside diphosphate homopolymers d(A)₃d(T)₃ and d(G)₃d(C)₃ was carried out in [29,30]. The geometric properties were determined at theory levels M06-2x [29] and B97-D3 [30] in the gas phase and in the dielectric continuum type of the solvent (water). The analysis was performed for the following models: Model 1 represents the duplex in the anionic form in vacuum. Model 2 corresponds to Model 1, except that the negative charges of the phosphate groups are neutralized by Na+ cations. Originally, the sodium cations were equidistant (about 2.4 Å) from the two phosphate oxygen atoms along the backbone. To form Model 3, the structure of the first model was immersed in a continuum-type dielectric medium that simulates water. However, this is not simply a hydrated anionic form of the mini-helixes under consideration. Indeed, this model includes the average influence of compensated ions (see the explanations in [29,30]). To form Model 4, Model 2 was hydrated in the same way as Model 3.

The main difference between certain DNAs and the others is the shape of their helices. These parameters are determined by the specific orientation of the nucleobase pairs, by the conformation of the sugars and conformation of the sugar-phosphate torsions. However, according to the "nucleobase centered" hypothesis [31], the main role is played by the interaction of the base pairs, which even in the isolated state can form stacking complexes that have almost the same twisting parameters as the A and B forms [32,33]. The key geometrical parameters that characterize A- and B- forms are collected in the Table 3S.

The examination of the parameters shown in Table 3S indicates that, with a few exceptions, the outcomes of the two approximations B97-D3/def2- SV (P) and M06-2x/6-31G(d,p) are quantitatively similar. We found that the helical conformation adopts geometrical characteristics that are similar to the shape of a B-DNA in both duplexes under consideration. Of course, when the mini-helix is hydrated, the agreement with the ideal B-DNA form and the averaged data of the classical MD is considerably better (see results for Model 3 and especially for Model 4)

We have reconstructed the conformational wheels for the B-DNA duplex in Fig 5. so that it is clear how the backbone behaves differently in different Models, which depicts the regions of torsions [34]. As depicted in Fig. 5, the B-DNA-like duplexes' torsion angle distributions are extremely broad but essentially conform to the experimental ranges. It is evident that PCM Model 4 provides the best fit for the torsions. Models 2 and 4 show that the direct addition of sodium cations results in almost identical α-, β-, γ- and δ-torsions in both strands. Interestingly, especially in PCM Models 3 and 4, the torsion angles of the modelled B-DNA mini-helix are within the ranges of the two conformational classes BI and BII.

Fig. 5.

Conformation wheels for the B-DNA duplex type derived from experimental data of Schneider [34], showing the ranges in blue with average values in grey. The average values for BI-DNA-form are shown in yellow and the average values for DNA BII-form are shown in red. The distribution of the calculated torsion angles is rather broad, yet consistent with the experimentally obtained margins (blue).

3.2. A-forms

The general difference in the structure between A- and B-forms of DNA is presented in Fig. 6. Similar to the results presented in the previous section, the study of A-type mini-helixes was conducted at the B97-D3 level of DFT theory. Models 2, 3 and 4 presented above were used. It was found that the conditions of all three models maintain the obtained conformations of the mini-helixes as A-ones since all of them have negative sliding orientation of the bases, more rolling than in the B-form and less twisting than in the B-form. The comparison of the geometrical parameters of d(A)₃d(T)₃ and d(G)₃d(C)₃ mini-helixes belonging to the A- and B-forms was carried out in [35]. As expected, the structural parameters of hydrated model correspond much better to the geometry of the ideal A-form than those describing the geometry of the mini-helixes in vacuum. This is especially true for the (dG:dC)₃ duplexes.

Fig. 6.

The A- and B-form of DNA macromolecule (side and top view) [35].

The overall picture becomes clear when the roll-slide correlation is examined for mini-helixes in A- and B-shapes (see Fig. 7). Analysis of the roll-slip correlation shown in Fig. 7 shows that the roll and slip values of the (dG:dC)₃ mini-helixes considered belong to the range that characterizes the A-DNA form. As can be seen, the situation is more complex for similar parameters of the (dA:dT)₃ mini-helix, as these parameters belong to the range that lies at the boundary between the A and B forms. This could be explained by at least two different arguments. It is known that the (dA:dT)₃ sequence strongly prefers the B-type conformation, in contrast to the (dG:dC)₃ sequence. [4,5] For this reason, structural data of AA /TT step in the A-form could lie at the boundary between A- and B-forms, indicating a tendency towards such a preference.

Fig. 7.

Plots of roll versus slide for two base-pair steps of duplex were discussed. The numbers indicate the Model. The dashed line from Roll, Slide =-10°,-1 Å to +20°, -0.2 Å, represents the break between A- and B-type DNA geometries, which lie to the left and right, respectively, of the line (Calladine & Drew[3]).

We found some unexpected results when we looked at the energies of the A- and B-mini-helixes. In all models, the A-configuration poses the greatest stability, as shown by the data in Table 4S. We refer to Beveridge and colleagues' published findings [36] to explain these data. The authors discovered that hydration energy is the most critical factor in the dominance of a B-form of DNA in water solution using data from classical molecular dynamic simulations. As a result, we speculate that an explicit, more accurate modeling of the hydration is required to observe the B-form's dominance.

The following was found through the examination of the interaction and binding energies:

• (i)It is anticipated that the mini-helixes will remain stable in both vacuum and water solution. This result is true both for A- and B-forms of mini-helix.
• (ii)The strand specificity is clearly visible in the energies of oligonucleotide relaxation.
• (iii)Another intriguing hypothesis that can be evaluated based on the calculation that has been presented is that the DNA most likely chose a particular base sequence (AT or GC) in order to form an A- or B-conformation through the helix.

4. Hydrated model of mini-helix

The mini-helixes that consist of five hydrogen-bonded pairs (see Fig. 8, Fig. 9, Fig. 10) of condensed deoxyribonucleotides (c.a. d(A)₅d(T)₅ and d(G)₅d(C)₅) have a size of half B-DNA helix turn. They also represent the category of mini-helixes, where the central base pair is screened by two base pairs placed up and down correspondently. As a result, the structures under consideration (see Figs. 8 and 9) now resemble small DNA pieces. The following shortcuts have been introduced in order to evaluate certain pieces.AT5 – B-DNA type mini-helix which has five AT base pairs and sugar-phosphate backbone, uncompensated by Na⁺ ions (total electric charge equal to -8).

Fig. 8.

Geometry of AT5 mini-helix (A-main, B-top, C-left, and D-right view) in vacuum [37]. Visual analysis of the C- and D- orientations suggests that the structure is an intermediate between a helix and a ladder.

Fig. 9.

Structure of AT5wW mini-helix (main, top, left, and right view) in vacuum (water molecules are omitted) [37]. Visual analysis suggests close correspondence to the structure of B-DNA (Fig. 6).

Fig. 10.

Structure of (dA)5•(dT)5 mini-helixes in water solution (main view) [37]. All presented structures visually are very similar to those of the B-DNA. Consequently, the explicit micro-hydration and counterion compensation appear to play central role in maintaining the helical structure of B-DNA.

AT5Na – B-DNA type mini-helix which has five AT base pairs and sugar-phosphate backbone, compensated by Na⁺ ions (total electric charge equal to zero).

AT5w – B-DNA type mini-helix which has five AT base pairs, sugar-phosphate backbone, uncompensated by Na⁺ ions (total electric charge equal to -8). It has explicit water molecules in a minor groove.

AT5Naw – B-DNA type mini-helix which has five AT base pairs, sugar-phosphate backbone, compensated by Na⁺ ions (total electric charge equal to zero). It has explicit water molecules in a minor groove.

AT5W – B-DNA type mini-helix which has five AT base pairs, sugar-phosphate backbone uncompensated by Na⁺ ions (total electric charge equal to -8). It has explicit water molecules in a major groove.

AT5NaW – B-DNA type mini-helix which has five AT base pairs, sugar-phosphate backbone, compensated by Na⁺ ions (total electric charge equal to zero). It has explicit water molecules in a major groove.

AT5wW – B-DNA type mini-helix which has five AT base pairs, sugar-phosphate backbone, uncompensated by Na⁺ ions (total electric charge equal to -8). It has explicit water molecules in minor and major grooves.

AT5NawW – B-DNA type mini-helix which has five AT base pairs, sugar-phosphate backbone, compensated by Na⁺ ions (total electric charge equal to zero). It has explicit water molecules in minor and major grooves.

The abbreviations GC5, GC5Na, GC5wW, and GC5NawW should be read similarly while replacing ‘A’ with ‘G’ and ‘T’ with ‘C’.

Visual inspection (see Fig. 8) demonstrates that a (dA)₅(dT)₅ mini-helix that is not compensated by counterions placed in a vacuum poses an intermediate structure between a helix and a ladder. One also may note (see Fig. 10) that the explicit micro-hydration of both minor and major grooves or the counterion compensation of the backbone charge are the minimal external factors that do stabilize DNA helixes of the B-type. Both these factors simply screen a large value of electrostatic repulsion of phosphate groups. As the result (dA)_5•(dT)₅ duplex forms a mini-helix structure. The distribution of basic structural parameters of the middle AT deoxyribonucleotide as in water solution is drawn in Fig. 11.

Fig. 11.

The distribution of the following basic structural parameters of the middle AT deoxyribonucleotide in water solution [37]. The following common features can be identified: i) the slide and roll parameters deviate significantly; ii) a relatively short value of h-rise that means that all (dA)₅•(dT)5 mini-helixes are predicted to be flatter relatively to an ideal form; iii) h-twist, χ, and δ have the values are quite close to the ones characterizing an ideal B-DNA.

The (dG)₅(dC)₅ duplex (Fig. 12) exhibits a different kind of behavior. The duplex maintains the B-type DNA geometry due to the presence of an additional hydrogen bond. However, gas phase geometry is the most distorted among all considered cases. Micro hydration of minor and major grooves and Na+ counterions compensating for the backbone charge shift the fundamental structural parameters to a range close to that of the ideal B-type of DNA. As a result, the interaction with the microenvironment is again crucial in this case.

Fig. 12.

Structure of (dG)5•(dC)5 mini-helixes in vacuum (main view), indicating the similar role of the microenvironment as described in Fig. 10.

Finalizing, we would like to mention numerical ESI-MS evidence that show that completely dehydrated and deionized DNA duplexes do not dissociate during the experiment as part of the experimental and computational support for findings described above [38], [39], [40]. Also, classical MD simulations show that B-DNA dissolved in water without counterions maintains its helix-like shape throughout a simulation (up to a ms timescale) [41,42,43].

6. Final remarks

To summarize the current status of QM calculations of structural and energetic properties of the DNA fragments, we would like to highlight the following.

The most investigated area, however still not comprehensive, is the studies of isolated and hydrated DNA bases. We have already mentioned the reviews [11], [12], [13], [14] and the references therein. In spite that the area of the calculations of nucleosides is not covered in this review, they are also investigated quite carefully but not so detail as DNA bases (see for example, [44], [45], [46] and references therein). There is much more room for investigations in the case of 2′- deoxyribonucleotides. In our opinion, the priority studies must be directed to the investigation of protonation and deprotonation properties of nucleotides in an aqueous solution. Those properties could be directly connected with their hydrogen-bonding ability during Watson-Crick bonding. There is no data on the rigidity of 2′-deoxyribonucleotides in water solution. And, finally, the area of QM studies of DNA mini-helixes is completely open for investigation. Here for the first time, one may model QM different DNA conformations and study numerous properties which are known from the experimental studies, and go beyond the experiment using a specific arsenal of computational methods and approximations. The first examples of such studies are presented above in this review.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

• No data was used for the research described in the article.