Quantum Mechanical Investigation of the G-Quadruplex Systems of Human Telomere.

The three G-quadruplexes involved in the human telomere have been studied with an accurate quantum mechanical approach, and the possibility of reducing them to a simpler model has been tested. The similarities and the differences of these three systems are shown and discussed. Each system has been analyzed through different properties and compared to the others. In particular, we have considered: (1) the shape of the cavity and the atomic charges around it; (2) the electric field in and out of the cavity; (3) the stabilization energy due to the stacking of G-tetrads, to the H-bonds and to the ion interactions; and, finally, (4) to study the mechanism of the process of the ion inclusion in the cavity, the curves of potential energy due to the movement of the Na+ and K+ ions toward the cavity. The results suggest that a detailed study is essential in order to obtain the quantitative properties of these complex systems, but also that some qualitative behaviors can be schematized. Our study makes it clear that the entry of an ion in the cavity of these systems is a complex process, where it is possible to find stable structures with the ion out and in the cavity. Moreover, it is possible that more than one diabatic state is involved in this process.

Introduction

Nowadays, the interest in G-quadruplex systems is extremely high with thousands of studies on specific aspects of their structure or function and about research fields ranging from structural biology to medicinal chemistry and from supramolecular chemistry to nanotechnology. These systems, in fact, have been implicated in a variety of biological[1,2] and nonbiological areas.[3,4]

Several are the aspects of this kind of system treated in the literature. In particular, these systems were shown to be involved in various cellular processes, which include DNA replication, RNA transcription, and genome recombination.[5,6] Moreover, these systems can also be the object of the epigenetic phenomenon.[7] Today, the most prominent role of G-quadruplex systems in the biological area is associated with telomere structure and function. Telomeric ends of chromosomes, with noncoding repeated sequences of guanine-rich DNA, are fundamental in protecting the cell from recombination and degradation. In telomeric DNA sequences are present these four-stranded quadruplex structures that are involved in the structure of telomere ends.[8] The importance of this kind of structure is now firmly established and well documented in the literature.[9−11]

G-quadruplex is formed by consecutive G-quartets and connected loops, with the participation of metal ions, which dwell between G-quartets and are coordinated by several carbonyl oxygen atoms. G-quadruplexes can be stabilized by many species of cations, such as K+, Na+, NH4+, and divalent cations (Sr2+, Zn2+, Pb2+, etc.), among which the most physiologically relevant cations K+ and Na+ are widely noted.[12−17] Monovalent cations (especially K+), placed between G-quartet layers, have been accepted as one of the most important factor in the stability of G-quadruplex, but little is known about its process of binding in the human telomeric G-quadruplex. Also, the role of water molecules in K+ binding in human telomeres has not completely been clarified. These aspects are recently studied with molecular dynamics (MD).[18] In particular, these authors show that metal ions regulate topologies of G-quadruplexes through mechanisms that depend on the sizes of metal, cavities, and hydration states; the authors of ref (19) also found that the binding of central K+ with the parallel G-quadruplex is a coordinated step-wise process directed by water.

Several experimental techniques have been utilized for studying G-quadruplex: dynamic light scattering, single-crystal X-ray, electrospray mass spectrometry, circular dichroism (CD), and solid-state NMR spectroscopic studies.[20−25] In general, CD has been applied to chromophores with the main goal of assigning the absolute configuration of chiral molecules. Because of its sensitivity to stereochemical variations, CD has emerged to be an important technique for studying subtle conformational changes and supramolecular interactions in G-quadruplex systems. At present, these systems represent biological structures that are widely investigated with CD[8,22,26,27] with an extremely high number of experimental studies of this type. Also, the comparison[28] between duplex and G-quadruplex DNA has been characterized by CD spectroscopy when these two kinds of systems are oxidized and the sites of oxidation and the oxidation products are determined for these systems. The results indicate that the nature of the secondary structure of folded DNA greatly alters both the reactivity of these systems toward oxidative stress as well as the product outcome. This suggests that recognition of damage in telomeric sequences by repair enzymes may be profoundly different in G-quadruplex from that of B-form duplex DNA. Also, the NMR technique is an invaluable tool for the study of nucleic acid structures and many NMR studies have been carried out on G-quadruplex structures.[20,21,29] Qualitative NMR studies also have revealed that the structures formed by telomeric DNAs are differed according to the presence of Na+ or K+ as a counterion and the stability of these structures depended on the monovalent cation present,[30,31] too.

Several theoretical studies on these systems have been performed but, because of the large number of atoms, a lot of these papers investigate the structures of these systems with MD simulations[32] including recent research.[33−37] Recently,[18] MD study has been applied to two of the most powerful stabilizers RHPS4 and BRACO-19 of G-quadruplex systems and reveals that the central K+ has little influence on the binding conformations of the bound stabilizers, but without the central K+, either RHPS4 or BRACO-19, cannot stabilize the structure of G-quadruplex. In contrast to MD, quantum mechanics (QM) approach can offer an inherently more accurate description of G-quadruplex structures, because it can consider the polarization[38−40] and, as known, it is more accurate in the treatment of hydrogen-bonding[41,42] and stacking interactions.[43,44] Only a few papers that use a QM method can be found in the literature. The oldest ones are applied to a reduced part of G-quadruplex[45−50] and use Hartree–Fock (HF) or density functional theory (DFT) with the B3LYP (or BLYP) functional and with 6-31G (or 6-311G) basis set. Only in ref (51), for the first time, as underlined by the authors, accurate QM computations (DFT-D3 with large atomic orbital basis sets) have been applied to these systems. Recently, QM calculations on dimers of G-quartet systems have been utilized to verify and parameterize the MD simulations,[52] to study the stabilization with different cations,[53] and the DFT-D3 method is applied to three-stack models[54] cutting from 2O4F and JPQ in PDB (Protein Data Bank). The authors of this study find stronger hydrogen-bonding and stacking interactions in the stem of the parallel G-quadruplex (2O4F), compared to those in the antiparallel system (JPQ). In this last paper, in any case, the optimization has been performed only on the positions of the hydrogen atoms, whereas all heavier atoms have been restrained to their experimental positions. This restriction in the optimization appears to be too strong in our view and it has been partially removed in this paper (see later).

In this paper, we will use a reliable quantum mechanical approach in order to compute some properties of the three different G-quadruplexes of the human telomere. In this context, “reliable” means that our approach is able to consider with the same accuracy both the hydrogen-bond interaction between the bases and the π–π interactions between the G-quartets. In this study, the fundamental properties of atomic charges, electric field, molecular structures, and energies of these systems are computed and analyzed. Two novelties of this paper must be underlined. First of all, for the first time, three complete G-quadruplex structures have been studied with a reliable quantum mechanical approach, while until now only had been studied parts of these systems; second, when we have utilized reduced systems we have tested their applicability to the specific properties considered. Moreover, in this paper, also the influence of water solvent and the optimization of the experimental structures are taken into consideration and discussed.

Systems Studied and Computational Method

In this paper, we are studying three different G-quadruplex structures of the human telomere: (a) the hybrid 1[55] (2HY9 code in PDB, model 1), (b) the hybrid 2[56] (2JPZ code, model 1), and (c) the basket-type[57] (2KF8 code), where, in the last case, only two G-quartet layers are present. Recently, the two polymorphic hybrid 1 and hybrid 2 conformers have been discriminated by a fluorescence study.[58] From now on, we will be using the notation Form 1, Form 2, and Form 3 for these three G-quadruplex systems as in ref (57). The Form 1 and Form 2 systems have 843 atoms (these two hybrid structures are different in the order of their loop orientations), while the Form 3 structure has 715 atoms. Figure shows schematically these systems while Figure SI_1–3 in the Supporting Information shows them in wireframe modality. These G-quadruplex systems have been identified experimentally. In particular, in KCl solution, experiments have before identified two similar folding motifs referred to as hybrid 1 and hybrid 2[59−62] and after the Form 3 that, despite the presence of only two G-tetrads in the core, it is more stable than the three-G-tetrad G-quadruplexes previously observed for human telomeric sequences in K+ solution. Recently,[63] quadruplex structures of a human telomere, stabilized by sodium and potassium ions, have been studied by using surface-enhanced Raman scattering (SERS) spectroscopy. In the SERS spectra, the bands typical of the basket-type and the mixed hybrid structures were identified and assigned. Moreover, a recent thermodynamic study[19] revealed that Na+ stabilized basket-like G-quadruplex. This paper revealed that metal ions selectively stabilize G-quadruplex topologies with a specific cavity of a certain size and also studied the role of the hydration states in this stabilization. The polymorphism present in the G-quadruplexes cannot be avoided in this study, because also in the experiments on the human telomeres the coexistence of these conformers has been found.[56,57,60,62,64−66]

Figure 1

Schematic representation of the three G-quadruplex systems.

In our paper, all these G-quadruplex systems are studied with and without hydration (in the polarizable continuum model (PCM) approach); what is more, the reduced dimeric models (where only the two up G-tetrads of the three G-tetrads are extracted from the 2HY9 and 2JPZ structures with the backbones between their, as in ref (67). Similarly, the two G-tetrads and their backbones are extracted from the 2KF8 structure) of these three systems have been considered after these models have been tested and have reproduced a selection of G-quadruplex properties. Also, in the dimer cases, the hydration has been considered. The dimers extracted from the G-quadruplex systems have 244 atoms and have been studied with and without the inclusion of ions (K+, but in a case also Na+). No atoms (hydrogens or cations) are added near the phosphate groups, as in ref (51).

In the study of dimer systems, the initial geometry extracted from the G-quadruplex of the PDB has been partially optimized with respect to our basis set. In particular, we have optimized only the two G-quartets of these systems and kept fixed the position of the backbones according to those of the experimental data. At this point, we would like to stress that the authors of ref (54) have optimized only the hydrogen atoms from the structures extracted from the PDB. In our opinion, this type of optimization is not sufficient because the changing of the quadruplex cavity is essential in the study of the ion inclusion. Two are the reasons for our partial optimization. First, because of the presence of 244 atoms, is that the convergence of the full optimization is not easy to reach, and second, is that, during the full optimization, these calculations may be easily trapped in local minima, as highlighted in the literature.[52]

We have already demonstrated in our previous paper[67] on the G-quadruplex that the results obtained with a partial optimization (where only the positions of the atoms of guanines are optimized) are very similar to that of the full optimization. Also, in the cases studied here, we have performed this test on these reduced models and we have reached the same conclusion (result not reported in the paper). In order to understand the approach of the ion (K+ or Na+) to the cavity of these systems, we have studied the potential energy surface (PES), as a function of the ion position, in one of these dimers, the one extracted from the Form 3. Also, the inclusion of a water molecule in the cavity and the presence of it when the ion arrives have been studied in the case with K+ ion.

As in our previous paper on G-quadruplex[67] and in some of our studies on the Watson–Crick base of DNA,[68−71] this theoretical investigation has been performed with the Gaussian package[72] using DFT with the M06-2X[73] functional of Thrular in the cc-pVDZ basis set. For the cations studied, this basis set is not present in the Gaussian program so that we have used the 6-31G(d,p) basis set. As already explained,[67] the choice of the M06-2X functional for the DFT approach is motivated by the hope of describing both the hydrogen-bond interaction between the bases and the π–π interactions between the G-quartets. It is well known[74,75] that an acceptable description of these two kinds (hydrogen bond and π–π) of nonbonding interactions needs a good calculation of the dispersion term[45,51] or the use of the M06-2X functional. This functional, in fact, is probably the most accurate dispersion-uncorrected functional that is able to give good results for the stacked structures[76,77] because it contains a reasonable description of dispersion in the medium range (low but finite density regions)[78] of distances.

In any case, we have also computed the empirical correction of the basis set due to the dispersion energy, for the systems studied. We found that this correction does not affect the optimized geometry and the atomic charges, but it is important in the calculation of the stacking and hydrogen-bond stabilization. Hence, we use this correction in the calculations of these interaction energies. We have also used the counterpoise correction of energy. The basis set superposition error (BSSE) is a consequence of the finite basis set and is due to the superposition of the basis sets of the different fragments of the system. The absolute value of this correction is quite large (about 20 kcal/mol), but practically irrelevant in the computation of potential energy curves, where the different values are due to a change of only the position of one atom. Also, this correction is used in the calculation of interaction energies.

Results and Discussion

Three are the main properties studied in this paper. First of all, we have computed the atomic natural bond orbital (NBO,[79] where we remember that NBOs are generally localized 2-center orbitals that describe the Lewis-like molecular bonding pattern of electron pairs in optimally compact form) charges of the systems of Figure and of the dimers extracted from them. In this paper, we will show only the atomic charges of the atoms around the cavity and those of the atoms involved in the hydrogen bridge (H-bridge), in particular. The remaining atomic charges can be found in the Supporting Information. Second, we have investigated the electric field of these systems and investigated the entry of the ion in their cavity. In the cases studied, the role of hydration has been included and analyzed. Finally, for a dimer (that one extracted from the basket-like system), we have computed the potential energy curve as a function of the position of the K+ or Na+ ions and a water molecule respect to the center of the cavity in the system where the positions of the atoms of the G-quartets have been optimized.

In Figures –4 [case (a) G-quadruplex and (b) corresponding dimer], we are showing the shape of this cavity, the charges of the main atoms involved in the cage and the hydrogen bond distances for the Form 1 (Figure ), Form 2 (Figure ), and Form 3 (Figure ), respectively. In Figure SI_5-7, the same results are shown for the identical systems (no new optimization) of Figures –4 but in the presence of hydration in PCM model. Some considerations can be done from the analysis of Figures –4 and SI_5-7.

Figure 2

Shape of the cavity, NBO atomic charges, and H-bond lengths of (a) 2HY9 G-quadruplex and (b) corresponding dimer. The hydrogen bond distances (Å) are indicated. The atoms in yellow are in the up plane where the charges and distances are in bold characters.

Figure 4

As Figure , (a) 2KF8 G-quadruplex and (b) corresponding dimer.

Figure 3

As Figure , (a) 2JPZ G-quadruplex and (b) corresponding dimer.

Each system has a particular cavity shape that is not largely modified in the dimer after the partial optimization. We underline that in the cases of dimers we have performed an optimization of the geometry of the eight guanines with the constraint of experimental geometry of the backbones.

The H-bridges of both G-quadruplexes and dimers show a large negative charge on the heavy atoms and a positive charge on the hydrogen, with the larger charge on the N atom.

The hydration changes the NBO charges of the atoms involved in the H-bridges. In particular, the hydration generally makes the negative charge of the oxygen atom to increase and decrease that of N atom, while the positive charge of the H atom in some cases increases while in other cases decreases. In general, the hydration makes the global charges of the atoms involved in the H-bridges to increase, with a variation of the charge larger for the oxygen than for the H and N atoms.

The H-bond distances can vary greatly as a consequence of the optimization of the guanine, until 0.58 Å. Generally, there is a decrease of the H-bond distance in the optimized dimer and, hence, a stronger H-bond, but there are also some H-bond distances that are increased when the optimization of the guanines positions are performed.

After the optimization of the guanine positions in the dimer, the charges of the atoms involved in the H-bridges are larger. In the cases studied, the larger variation is that of the oxygen atom.

In the comparison among the different G-quadruplex structures, we would like to underline some similarities and differences.

There is a similar global charge in the cavity of the three G-quadruplex structures; in particular, the global negative and positive charges in the cavity of Form 1 and Form 2 are practically identical and those of Form 3 slightly larger (this is true for both negative and positive global charges).

The order of the average H-bond distance is Form 1 > Form 2 > Form 3 in the G-quadruplexes, but in the dimers this average distance is similar in the three cases.

In order to compare the G-quadruplex and the corresponding dimer of the three systems, in Table we show the order of the difference between the dimer and the corresponding G-quadruplex for some properties. In particular, we show the variations of the H-bond average distance and those of the atomic charges of the atoms involved in the H-bridges with and without hydration. Here, it is important to point out that the variations of the atomic charges around the cavity for the three hydrated systems follow the same order: Form 2 > Form 1 > Form 3.

Table 1

a

	large	medium	small
ΔH-b	form 1	form 2	form 3
ΔCO	form 1	form 2	form 3
ΔCH	form 2	form 1	form 3
ΔCN	form 3	form 2	form 1
ΔCO(h)	form 2	form 1	form 3
ΔCH(h)	form 2	form 1	form 3
ΔCN(h)	form 2	form 1	form 3

ΔH-b = variation of the length of the H-bond; ΔCX variation of the atomic charge of X; ΔCX(h) variation of the charge of X in the hydrated system.

In Figures –7 are shown the cavity shapes with the main atomic charges and distances of the dimers extracted from the three G-quadruplex systems, when a K+ ion (placed at the center of these systems) is included in the optimization (a) without and (b) with water solvent. The comparison between the dimers with and without a K+ ion shows that this ion lengthens, and hence weakens, the H-bonds of these systems, while the modification of the shape of cavity is not large. Otherwise, the hydration of these systems has specific effects on the charges of each structure; in particular:

Figure 5

Shape of the cavity, NBO atomic charges, and H-bond lengths of (a) 2HY9 + K+ dimer and (b) corresponding hydrated (water in PCM) dimer. The hydrogen bond distances (Å) are indicated. The atoms in yellow are in the up plane where the charges and distances are in bold characters.

Figure 7

As Figure , (a) 2KF8 + K+ dimer and (b) corresponding hydrated (water in PCM) dimer.

The Form 1 has an increase of the negative charge of the oxygen and a decrease of that of the nitrogen, while the hydrogen charge varies in non-uniform way. The effect on the oxygen is larger, in any case.

The Form 2 shows the larger variation of the charges compared with the other structures, but in this case both the oxygen and the nitrogen atoms can increase or decrease their negative charge.

The Form 3 has a similar variation of charge than Form 1, with the charge of the oxygen that can increase or decrease while that of the nitrogen and hydrogen has small variations.

As Figure , (a) 2JPZ + K+ dimer and (b) corresponding hydrated (water in PCM) dimer.

The charge of the K+ ion in the cavity of the dimers is 0.623, 0.619, and 0.648 in the Form 1, Form 2, and Form 3 (and 0.624, 0.622, and 0.653 in the hydrated systems), respectively, and it is smaller of the initial +1 of the ion in any case, with a fraction of electron (in the range 0.35–0.4) that moves from the dimer to the ion. Because even the atoms involved in the H-bonds are more negative in the presence of the K+ ion, the interaction of this cation with these dimer systems causes in all cases a movement of a fraction of electron toward the cavity. At our knowledge, this effect is never underlined in the literature. Finally, these dimers show eight different bonds (with a bond distance in the range 2.59–3.02 Å) between the central K+ ion and the oxygen atoms of the cavity. These K–O bonds are of different lengths and strengths, and also, this asymmetry has never been underlined in the literature before.

In Table , we show the energy stabilization of the dimers (with and without a K+ ion) due to the stacking interaction and to the H-bonds. All of these results have been obtained with the inclusion of the BSSE correction and the empirical D3 correction on the M062X functional.

Table 2

a

	stacking	H-bond	O–K+
form 1	–16.40	–15.37
form 1 + K+	–16.20	–18.99	–12.19
form 2	–19.09	–16.41
form 2 + K+	–13.94	–17.99	–12.83
form 3	–16.79	–11.03
form 3 + K+	–11.05	–17.67	–13.51

Stabilization energies in kcal/mol. Both the energies of H-bond and O–K+ interaction are average value. Remember that in the dimer system, there are 16 H-bonds and 8 O–K+ interactions.

From the analysis of Table , we can see that:

The stablest system is that of Form 1, while the other two systems have similar stabilization energy.

The stacking energy is smaller than the H-bonds energy (in the Table , we have shown the average energy of one H-bond, but in the dimeric system there are 16 bonds) and of the O–K+ interaction (eight interactions).

The addition of a K+ ion decreases the stacking energy (due to steric effect) but increases the H-bond stabilization (and forms eight interactions between the oxygen atom and the K+ ion).

In Figure it is shown the electric field in the cavity of the three G-quadruplexes and in that of the corresponding dimers with the K+ ion in the center. The Cartesian axes of these systems are shown in Figure SI_1–3. In Figure , the positive part of the field is plotted in black and the negative part in green. It is evident that the electric field in the cavity of the G-quadruplex is, in all cases, completely negative without the ion. When the K+ ion is added, instead, there are positive and negative parts of the electric field in the cavity and only the 2FK8 system has a cavity with an electric field completely positive.

Figure 8

Electric field of the 2HY9, 2JPZ, and 2KF8 G-quadruplexes (from left to right) and of the corresponding dimers (from the top to bottom) with K+ ion in the center of the cavity. In green, the negative part of the field; in black, the positive one. The cavity has been explored in the range from −2 to 2 Å from the center in the directions between the G-quartets (x-axis) and perpendicular (y-axis) to them (see Figure SI_1–3 for the Cartesian axes).

As already pointed out, only in the example case of the Form 3, we have studied the potential energy curve due to the movement of the Na+ and K+ ions toward the cavity of the dimer extracted from this quadruplex. In these calculations, the geometry of the cavity has been optimized for each position of the ion because, without this optimization, there would have been a too large barrier for the entry of the ion in the cavity (see Figure 15a,b, in ref (67), for an example). In these cases, the ion approaches the cavity from the upper plane, as suggested by simulation studies.[80,81] We remember that the two G-quartet planes of the dimer are non-equal because this structure is extracted from the quadruplexes (Form 3) where it is evident that an upper plane is different from a lower one.

In Figure a, there is the PES, as a function of the position respect to the center of the cavity, of: Na+ (black curve), H2O case (red curve), and H2O and water solvent in the PCM method (blue curve). In Figure b are shown the PES of K+ (black curve), K+ with hydration (red curve), and K+ when an explicit molecule of water is present in the cavity and it is kicked out by the ion (blue curve). In Figure a it is evident that the Na+ ion comes into the cavity without a barrier because no other stable position exists out of the cavity in this case. The case with a molecule of water has been studied for analyzing the possibility that the cavity is not empty when the ion arrives, as considered in the Figure b. In this last case, it is evident that only the hydrated water molecule has a reasonable energy barrier for entering the cavity of this system. From the comparison between the Figure a,b, the preference of the 2KF8 G-quadruplex for the Na+ ion appears evident, as found in the experimental studies.[64] In all cases of Figure b, the existence of two kinds of minima is evident (and, hence, of stable structures), one with K+ in the cavity (the stablest one) and the other with K+ outside the cavity. In particular, in the case of K+ + H2O, there are two stable positions outside the cavity, and in the case of K+, a minimum outside and another one around the up plane of guanine (we want to remember that the two plane of guanine are around −1.5 and 1.5 Å). The barriers for the entry of the K+ ion in the three cases of Figure b are different, but only in the case where the K+ ion is alone (black curve), there is a practically null barrier.

Figure 9

Potential energy curves of the Form 3 as a function of the distance from the center of the cavity; X = Na+, H2O, and H2O hydrated in (a) and K+ ion in (b). In figure (b) K-H2O means that a molecule of water is initially present in the center of the cavity, but then its position is optimized when the K+ ion moves toward the cavity. In figure (b), different areas are highlighted (see text).

Different Areas Are Highlighted (See Text)

All curves of potential energy of Figure b can be seen as formed from more than one diabatic state. This behavior can be also highlighted by plotting the charges of the atoms involved in the H-bridges as a function of the different position of K+ ion.

In Figures –12, we are showing the variations of the average charge of the O (Figure ), H (Figure ), and N (Figure ) atoms involved in the H-bridges around the cavity. By comparing Figures b and –12, it appears evident that a correspondence can be found between the change in the behavior of the potential energy curves and those of these average atomic charges. We believe that this correspondence represents an evidence of the nature of the interaction between the atoms of the cavity and the K+ ion, and it can be rationalized with different diabatic states involved in this complex process. At your knowledge, it is the first time that the entry of a K+ ion into the cavity of a G-quadruplex has been schematized as a complex process that involved some different minima and a multiplicity of diabatic states.

Figure 10

NBO oxygen charges as a function of K+ position. The suffix “d” and “u” in Hbu and Hbd is related to the up and down (respect to the center of cavity) planes of guanines.

Figure 12

NBO nitrogen charge, as in Figure .

Figure 11

NBO hydrogen charge, as in Figure .

Conclusions

In this paper we have studied with a reliable quantum mechanical approach the three G-quadruplexes involved in the human telomere. It is a first time that a complete theoretical analysis of this very important biological system has been performed at this level of calculation. The general conclusion that can be drawn from this study is that these three systems are similar according to some characteristics and different in others. The first statement explains the similar behavior of these systems and the possibility of a transformation of each other when the conditions of the environment change. This analysis has also highlighted some differences in the properties of these systems: the shape of cavity, the atomic charges, the global electric field, the H-bonds, and the preference for an ion or another. Hence, the results of these calculations provide also an answer to the specificity of each structure.

Finally, we would like to highlight some general consideration that can be obtained from this paper.

The global atomic charges of the atoms involved in the H-bonds around the cavity are substantially equal in the three G-quadruplexes. Nevertheless, the strength of the H-bonds is quite different in these three G-quadruplexes. This different behavior of the charge and the strength of the H-bonds points out that the H-bond has certainly an important ionic component, but this is flanked from other quantum mechanical aspects. The interaction of the cavity of the G-quadruplexes with the ion and the water, and the environment in general, can be understood with the help of these atomic charges, but also the variation of the strength of the H-bonds must be considered for this respect.

Generally speaking, the hydration makes the global charges of the atoms involved in the H-bridges to increase with a larger variation of the charge of the oxygen then for the H and N atoms. The introduction of the water solvent, also in a simple model such as the PCM one, is important for moving the system studied from the ideal case to the real one present in the cell.

By the analysis of the results of this paper, we can state that it is possible to reduce the G-quadruplex systems to a simpler model, as a dimer, mainly for the properties localized in the cavity or around the cavity. This approximation is largely applied in the literature, but it has been tested only by us, in ref (67) and in this paper. In the dimer models, in any case, the optimization of the G-quartet parts is essential and sufficient. This optimization has as a consequence the increase of the atomic charges of the atoms involved in the H-bridges around the cavity and of the H-bonds strength. This optimization is not used in general in the literature, as demonstrated in ref (54).

The addition of a K+ ion to the dimers changes some of their properties: (1) weakens the bonds H around the cavity, (2) generates a small modification of the shape of the cavity, (3) changes the atomic charges, and (4) forms a strong interaction between the ion and the oxygen atoms of the cavity. In any case, the addition of the ion stabilizes the quadruplex systems because the weakening of the H-bonds is compensated by the formation of the bonds between the ion and the oxygen atoms of the cavity. In fact, these systems shows eight bonds between the central K+ ion and the oxygen atoms of the cavity, different from each other, but this difference has never been underlined and considered in the literature. Important to emphasize is also the change of the atomic charges around the cavity. In all cases studied, a fraction of about 0.4 electron moves from the dimer to the ion and, because even the atoms involved in the H-bridges become more negative in the presence of the K+ ion, there is also an additional movement of a fraction of electron toward the cavity. Otherwise, the hydration of these systems with the ion has different effects on the charges of each specific structure.

The electric field in the cavity of the G-quadruplex is in all cases studied negative without the ion. When the ion is added, positive and negative electric field in different parts of the cavity can be found and only the 2FK8 system has a cavity with an electric field practically completely positive.

The stabilization energy of these systems is due primary to the H-bonds network, second to the ion interaction with the oxygen atoms and, finally, to the stacking between the G-quartets.

The entry of an ion into the cavity can be a complex process where it is possible to find stable structures with the ion outside or inside the cavity. In some case, it is possible that more than one diabatic state is involved in this process. We have studied the potential energy curves due to the movement of the Na+ and K+ ions and also of a water molecule toward the cavity of the dimer extracted from Form 3 of quadruplex. These curves are able to explain the preference of the basket-like G-quadruplex (2KF8) for the Na+ ion compared with the K+ one and also show the presence of a not-too-large barrier for the entry of K+ even if in the cavity there is already a water molecule. In the case of K+ ion, the curve suggests that more diabatic states are involved in the process.

As a conclusion, we underline that the results of this paper can help in the understanding of the behavior of the quadruplex systems of the human telomere, but this aim can only be obtained by accurate theoretical calculations. Of course, a QM approach to these complex systems can work only on a modelization of the real condition in the cell, and hence, the theoretical results must be integrated with experimental results in order to operate in the biological context.