We have analyzed π-π stacking interactions using different systems that were taken mainly from reference data present in the literature. We start with the prototypical system for studying π-π stacking, namely, the benzene dimer. The second set of reference data was taken from the supporting information of a paper by Mignon and co-workers [50], who studied hydrogen bonding and stacking in complexes of substituted benzene C6H5X with pyridine at MP2/6-31G*(0.25) with substituents X = H, F, NH2, Cl, CH3, OH, CN, COOH, CHO and NO2. All substitutions of H by another X in benzene were done at the para-position with respect to the nitrogen of pyridine (see Scheme 2). The third set of reference data was taken from a paper by Jurecka and co-workers [21], who investigated hydrogen bonding and stacking interactions in the cytosine dimer with MP2 and CCSD(T) methods, using several orientations (see Scheme 3) to investigate the potential energy surfaces. The fourth set of reference data was taken from a paper by Wu and Yang [14] (based on previous work by Hobza, Sponer and co-workers [17, 19, 22]) who studied the dependence of the stacking energy of DNA bases on the twist angle (see Scheme 4) and vertical separation using MP2/6-31G*(0.25) calculations.
Scheme 2Stack of substituted benzene and pyridine
Scheme 3Orientations of the stacked cytosine dimer from Jurecka and co-workers ([21]). The upper base is drawn in bold, the center of mass is indicated by a black dot, and the carbonylic oxygen and the hydrogen replacing the glycosidic bond are indicated for clarity
Scheme 4Geometries of stacked base dimers with a twist angle of 0 and 120 degrees. The upper base is drawn in bold, the center of mass is indicated by a black dot, and the hydrogens replacing the glycosidic bond are indicated for clarity. Note that for the homo dimers at zero twist angle the lower base is hidden behind the upper base
As mentioned already, the prototypical system for studying π-π stacking interactions is posed by the (parallel sandwich) benzene dimer. Because of its high symmetry (D6h), CCSD(T) energies using large basis sets (aug-cc-pVQZ) are available [51] to compare with (see Table 1). The minimum is found at 3.9 Å for this system with a stacking energy of −1.7 kcal mol−1. The standard density functionals are unable to give a proper description of this system, i.e., they either predict a purely repulsive energy surface (e.g., BP86; see Table 1), or give a shallow well (e.g., OLYP). Interestingly, the KT1 functional predicts a minimum that is very close (3.8–3.9 Å) to that of the CCSD(T) method, and has a well depth that is equally close (−1.6 kcal mol−1; see Table 1). The related KT2 and, surprisingly, the LDA functionals have a minimum at the same distance as CCSD(T) and KT1, but with a somewhat smaller well depth (−1.3 kcal mol−1). Also, the BHandH functional performs reasonably well with a similar equilibrium distance, but a reduced well depth (−0.9 kcal mol−1; see Table 1).
Table 1 Stacking energy (in kcal mol−1) of the benzene-benzene complex as a function of the vertical separation Rvert (in Å), computed with various density functionalsa and CCSD(T)/aug-cc-pVQZ*b
Reference data from Mignon and co-workers
Scheme 2 shows the structure of the π-stacked complexes of pyridine with (substituted) benzene, i.e. with benzene, fluorobenzene, aminobenzene, chlorobenzene, toluene, phenol, cyanobenzene, benzoic acid, benzaldehyde and nitrobenzene. For all of these complexes the coordinates were taken from the supporting information of the original paper by Mignon and co-workers [50]. The geometry of each dimer had been fully optimized and therefore the vertical separation is different for each complex; it is found in the range of 3.2–3.4 Å.
The interaction energies for several typical density functionals are compared to the original MP2 data in Table 2. Consistent with previous reports, the mean absolute deviations (MAD) with respect to the reference MP2 data are much larger for density functionals containing the Becke88 [43] exchange functional (e.g., BP86, MAD value 7.68 kcal mol−1) than for functionals based on PW91 [52] or PBE [53] exchange (e.g., PW91, MAD value 5.33 kcal mol−1). However, these standard density functionals all show repulsive interactions, in contrast to the MP2 results that indicate bound systems (see Table 2). Much better agreement with the MP2 data is observed for the KT1, KT2 and, surprisingly, LDA functionals, with MAD values of 0.25 kcal mol−1 (see Table 2). The BHandH functional, recently [12] shown to give good agreement with MP2/CCSD(T) energies, shows a larger MAD value of 0.4 kcal mol−1 (see Table 2). Because of the uncertainty of ca. ± 0.5 kcal mol−1 in the reference MP2 data with respect to the high-level CCSD(T) data (see above), it is uncertain which of the four density functionals (LDA, KT1 , KT2 or BHandH) performs best. However, the improvement over standard density functionals is obvious and promising.
Table 2 Stacking energy (in kcal mol−1) of substituted benzene-pyridine complexes (see Scheme 2), computed with various density functionalsa and MP2/6-31G*(0.25)b
Reference data from Jurecka and co-workers
Scheme 3 shows the fourteen orientations of the stacked cytosine dimer that have been considered in the paper by Jurecka and co-workers [21] (with a vertical separation of 3.3 Å for the first orientation, and 3.4 Å for all others). The corresponding reference energies and our DFT interaction energies are collected in Table 3. The latter also contains the MAD values with respect to two sets of reference energies, i.e. the usual MP2/6-31G*(0.25) results (MAD1) as well as the MP2 energies obtained after complete-basis-set (CBS) extrapolation corrected for the difference between MP2 and CCSD(T) with the 6-31G*(0.25) basis set (MAD2). Similar to what was found for the benzene-pyridine systems (see above), the MAD values for standard density functionals are found between 5 and 14 kcal mol−1, resulting in many cases in erroneous repulsive stacking interactions between the cytosine fragments (see Table 3). Improved results are again obtained with LDA, KT1, KT2, and to a lesser extent BHandH, which have MAD1 values (i.e. deviations with respect to MP2/6-31G*(0.25)) of 0.9, 0.8, 0.6 and 1.5 kcal mol−1 respectively, and MAD2 values of 0.4, 0.5, 0.7 and 0.5 kcal mol−1 respectively with the CCSD(T) data (see Table 3).
Table 3 Stacking energies (in kcal mol−1) for several orientations of the cytosine dimer (see Scheme 3), computed with various density functionals and MP2a,b
Reference data from Wu and Yang
In the paper by Wu and Yang, [14] the potential energy surface (PES) was calculated as a function of the twist angle between two stacked bases (see Scheme 1 and methodological section for the definition of the twist angle) using MP2/6-31G*(0.25), which was chosen as a reliable reference method based on the paper by Hobza and co-workers [22]. Although thymine was not included by Wu and Yang, we report here the energy profiles for all combinations of the five RNA/DNA bases, i.e., thymine, adenine, guanine, cytosine, uracil (see Scheme 4 for the geometries). The vertical separation is 3.4 Å for the C-C, G-C and G-G stacked base dimers, and 3.3 Å for all other systems. This difference in vertical separations was chosen based on earlier papers by Sponer and co-workers [19].
From the tests with the benzene dimer (Table 1), the Mignon (Table 2) and Jurecka reference data (Table 3), it is already evident that standard density functionals do not give reliable interaction energies, which is now shown to be true also for the energy profiles as function of the twist angle. Figure 1 shows the PES for two stacked bases, namely C-C and G-U, which are representative for all possible base pair combinations (the complete figure for all combinations of stacked bases can be found in the Supporting Information). Interestingly, although the standard density functionals typically underestimate the interaction energy, the shape of the PES is in all cases highly similar to that based on the more accurate MP2 data. This suggests that, at least trends and qualitative features in the rotational energy profile can be correctly reproduced with DFT which otherwise shows a (functional and basis-set dependent) constant shift with respect to the more accurate MP2 rotational profile.
It is, however, more instructive to look at functionals that by themselves already give accurate interaction energies, such as LDA, KT1, KT2 and BHandH. Figure 2 shows the corresponding rotational-energy profiles, again for the stacked bases C-C and G-U. From Fig. 2 it is evident that the LDA, KT1, KT2, and to a lesser extent BHandH are visually (and virtually) indistinguishable from the reference MP2 data.
Figure 3 shows the interaction energy as a function of the vertical separation for both the standard (upper diagrams) and promising (lower diagrams) density functionals, together with MP2 reference curve in each of the four diagrams. Not surprisingly, the density functionals that gave larger errors for the rotational energy profile, also give larger errors for the vertical separation profiles: almost all standard density functionals show potential energy surfaces that are repulsive, or at best weakly attractive with a very shallow minimum at elongated distances. In contrast, the promising functionals (i.e., KT1, KT2, LDA and BHandH) do provide well-defined minima with equilibrium distances (3.2–3.3 Å) close to the reference MP2 data. Similar to the uncertainty in the reference energy data of some 0.5–1.0 kcal mol−1 (see above), we may expect a similar uncertainty for the vertical separation, which we very roughly estimate at 0.1–0.2 Å. The differences between the equilibrium distance of the promising density functionals and that of the reference MP2 falls well within this estimated uncertainty. Thus, KT1, KT2, LDA and BHandH perform well not only for rotational energy profiles but also for the PES as a function of the vertical separation.
Decomposition of interaction energy
Now that we have established the reliability of the KT1, KT2, LDA and BHandH functionals for the interaction energy of stacked DNA bases, it is interesting to study the actual interaction into greater detail. This is done through a quantitative decomposition of the interaction energy ΔE
int into electrostatic attraction ΔV
elstat, Pauli repulsion ΔE
Pauli and orbital interactions ΔE
oi (see Methods). Table 4 provides the results of the energy decomposition at KT1/TZ2P for the five π-stacked homo dimers, for each one at a twist angle of 0° and 180° (which is the lowest-energy conformation) and at two different vertical separations. Note that the lowest-energy conformation is always achieved at a twist angle of 180° but that the optimal stacking distance (vertical separation) varies by 0.1 – 0.2 Å between the different π-stacked homo dimers. The energy changes by −6.8 (A-A), −11.8 (C-C), −10.2 (G–G), −11.2 (T-T) and −7.9 (U-U) kcal mol−1, respectively, when going from a twist angle of 0° to 180° (the lowest-energy conformation). This change in energy is almost completely resulting from a change in electrostatics that change by −5.4 (A-A), −13.1 (C-C), −10.3 (G-G), −6.6 (T-T) and −8.3 (U-U) kcal mol−1 respectively. The sum of Pauli repulsion and orbital interactions changes much less, by ca. 1.4 kcal mol−1 or less, except for T-T where the sum of Pauli repulsion and orbital interactions changes by 4.6 kcal mol−1 due to the influence of the peripheral methyl groups. But also the changes in each term individually, i.e., either ΔE
Pauli or ΔE
oi, are much smaller (typically 2 to 5 times, but in some cases even more) than the changes in ΔV
elstat (again, only in the case of T-T, the change in ΔE
Pauli is nearly as large as that in ΔV
elstat). Here we note that the same energy decomposition analyses for the five π-stacked homo dimers at LDA/TZ2P instead of KT1/TZ2P yield values that differ by only 1 kcal mol−1 or less (not shown in Table 4).
Table 4 Decomposition of interaction energiesa (in kcal mol−1) between stacked DNA bases in different geometries, computed at KT1/TZ2P
Additivity of interaction energies of stacked base pair dimers
With the good performance of KT1 for stacking interactions kept in mind, we went one step further, and looked also at the interactions within stacked dimers
of hydrogen-bonded base pairs (see Scheme 5) [54]. The geometry was obtained by taking the geometry of one hydrogen-bonded base pair and putting another one at 3.4 Å on top of the former. The base pair dimer is stabilized by hydrogen bonding between the bases within one layer, and stacking interactions between the bases in different layers of the stack. In this case, because there are two bases per layer, there are also interactions present between bases that are, in a sense diagonally, on opposite sides and in different layers [55], the so-called cross terms. For example for the AT-AT system as shown in Scheme 5, this refers to the interaction between the thymine of the top layer with the adenine of the bottom layer.
Scheme 5Structure of stacked DNA base pair dimer AT-AT with twist angle of 36°. The dot represents the B-DNA helical axis of rotation.[54]
In order to investigate in more detail the importance of the various energy terms, in particular, the cross-terms, we approximate the total stacking interaction between the base pairs in the stacked base pair dimer by the sum of four pairwise interactions between two bases, i.e., we approximate the total (stacking) interaction energy ΔE
int between the layers AB and CD in Scheme 6 by the sum ΔE
add of interactions A-C, B-D, A-D and B-C as shown in Equations 4a,b:
$$\Delta E_{{{\text{add}}}} = \Delta E_{{{\text{AC}}}} + \Delta E_{{{\text{BD}}}} + \Delta E_{{{\text{AD}}}} + \Delta E_{{{\text{BD}}}} $$
(4a)
$$\Delta E_{{\operatorname{int} }} = \Delta E_{{{\text{add}}}} + \Delta E_{{{\text{coop}}}} $$
(4b)
Scheme 6Additivity approximation for the π-π interaction between two stacked Watson-Crick base pairs in terms of pairwise interactions between individual bases
The remaining term ΔE
coop that, added to the approximate ΔE
add yields again the real stacking interaction ΔE
int, consists of cooperativity effects between the different interactions.
In Table 5, we report the energy values resulting from the additivity scheme for the stacked AT-AT, GC-GC and AU-AU base pair dimers at three different values of the twist angle (0, 36 and 180°). Importantly, the additive approximation ΔE
add of the real stacking interaction ΔE
int largely accounts for the latter. ΔE
coop has in most cases a positive sign and is small compared to either ΔE
add or ΔE
int. Furthermore, the dependence of the ΔE
coop term on the twist angle is much smaller than that of the interaction energy ΔE
int. For example, ΔE
coop varies from 1.0 to 1.6 kcal mol−1 whereas ΔE
int varies from −1.4 to −10.7 kcal mol−1 for the stacked AT-AT base pair dimer (see Table 5). Larger cooperativity effects are found in the case of the stacked GC-GC base pair dimer, where ΔE
coop values are found between 2.0 and 4.4 kcal mol−1. Note however that the changes in ΔE
coop are again small compared to the changes in ΔE
int.
Table 5 Interaction energy (kcal mol−1) between stacked DNA base pairs in terms of the additivity scheme, computed at KT1/TZ2Pa,b
The cross-terms in the additivity scheme are as important as the non-cross terms, especially when rotating the base pair dimer from a twist angle of 0 degrees to finally a twist angle of 180 degrees. At the end point of 180°, the A and B base have switched, and the cross terms are no longer AD and BC, but AC and BD; this switch occurs after a twist angle of 90°. For instance for the AT-AT base pair dimer at 0°, the cross terms are stabilizing (−2.5 kcal mol−1) and dominate the ΔEadd of −2.2 kcal mol−1. At 180°, the non-cross terms dominate (−10.9 kcal mol−1) with still a substantial contribution from the cross terms (−1.4 kcal mol−1). Note also the dominance of the cross terms (−10.2 kcal mol−1) over the non-cross terms (−3.0 kcal mol−1) for the GC-GC base pair dimer at 180°. This is mainly due to favorable hydrogen-bonding interactions between the cross G-G pair.
When rotating base pair dimers from a twist angle of 0° to 36°, the cross terms in the additivity scheme reduce slightly. However, this is dominated by the increase of favorable interactions between the stacked bases. For instance for AT-AT, the cross terms drop from −2.5 to −2.0 kcal mol−1, but the non-cross terms increase from +0.0 to −9.2 kcal mol−1. The same trend is observed for the AU-AU and GC-GC base pair dimers.
Again we note that the same energy decomposition analyses for stacked DNA base pairs at LDA/TZ2P instead of KT1/TZ2P yield values that differ by only 1 kcal mol−1 or less (not shown in Table 5).
Significance of the observed trends for the structure of DNA
The stabilizing interactions for the structure of DNA mainly originate from hydrogen-bonding interactions between bases in one plane and the π-π stacking interactions between stacked bases. However, we have shown here that in addition to these terms there is also an important contribution of the cross terms that arise between bases that are neither hydrogen-bonded within one layer, nor stacked above each other. All these interaction energies can now be understood and quantified with density functional theory using appropriate functionals. For the hydrogen-bonding interactions, the BP86 and PW91 functionals seem to perform best, [3] while for π-stacking interactions KT1, KT2 and (surprisingly) LDA work well.
The experimentally observed value for the twist angle (36°) is retrieved in our study on base pair dimers as the orientation with optimal stacking interactions. We have shown that this is predominantly determined by the classical electrostatic component of the stacking energy. The rotation to a twist angle of 36° destroys part of the favorable cross terms (see above), which is however completely overcome by the increase in favorable stacking interactions between the stacked bases. The twist angle and vertical separation that are experimentally observed are of course influenced by both the sugar-phosphate backbone and the presence of solvent and/or counter ions. These factors will be tackled in forthcoming work.