# Effects of G-Quadruplex Topology on Electronic Transfer Integrals

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Choice of Relevant G4 Structures, Spanning Viable Topologies

#### 2.2. Strength of Electronic Coupling

_{IF}, which was computed through the Marcus-Hush two-state model as implemented in Gaussian09. Technical details are included in Section 4. For a clean data set, we computed only intra-strand transfer integrals. Against comparison with a higher level of theory [25], we find that the two-state model is a good approximation, at least in G-G stacked couples.

#### 2.3. Correlation between Helix Shape Parameters and Electronic Coupling Parameters

_{IF}—and four quadruplex shape parameters—rise, twist, roll and shift—are illustrated in the bottom part of Figure 2. HL is the energy gap between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) of the +1 charged G-G couple, which is relevant to hole transfer. V

_{IF}is the computed transfer integral. The helix shape parameters are those illustrated in Figure 2. We can see big variations for all the parameters, distributed among all the structures. The roll of most conformations for all the structures is around the mid-range value. The twist of all conformations for structure 2JPZ is around the minimum value of 20 degrees; also for structure 186D the twist mostly assumes small values in all the 21 conformations of G-G couples. The other quantities are more uniformly spread. Considering all the 395 G-G couples together, the ranges are: V

_{IF}= 0.000 − 1.193 eV, HL = 0.264 − 0.276 eV, rise = 2.90 − 4.28 Å, twist = 14.27 − 41.37 degrees, roll = −25.58 − 26.76 degrees, shift = −1.43 − 1.81 Å.

_{IF}and each of the helix shape parameters shown in Figure 2—P(V

_{IF}-rise), P(V

_{IF}-twist), etc.—as well as between V

_{IF}and HL–P(V

_{IF}-HL). The V

_{IF}-HL Pearson’s correlation coefficients are reported in Table 1. We find a negative correlation between V

_{IF}and HL, which is consistent with the definition of the transfer integral in terms of the energy splitting at the reaction coordinate (Section 4). For the structures of group I and II the absolute value is larger than 0.7, with the exception of 1XAV; for the structures of group III smaller values occur. The V

_{IF}-helix Pearson’s correlation coefficients are reported in detail in the Supplementary Materials (Table S1). Here we discuss the salient features. The Pearson’s correlation coefficient does not reveal any remarkable correlation between V

_{IF}and the inter-guanine helix shape parameters: they are scattered between 0.07 and 0.69 in absolute values; some of them even vary between positive and negative correlation. Intuitively, one would think that the stacking distance (rise shape parameter) is a particularly crucial factor in determining the value of V

_{IF}: the larger the rise, the smaller the electronic coupling, with a negative correlation. However, we find a large spread of P(V

_{IF}-rise), from −0.63 to 0.17, with positive coefficients for structures with PDB ID’s 1XAV and 2HY9. Then we have searched for a homogeneous linear combination of the shape parameters that maximizes the Pearson’s correlation coefficient for each structure (Section 4.3). The values are much more meaningful, ranging between 0.49 to 0.85 for individual structures, while they are 0.59 and 0.44 for group I and II, respectively. The fact that cumulatively the correlation coefficients are smaller means that different linear combinations apply to different structures of the same group. Indeed, we note that in each group there are structures that conform to the intuition of a high negative correlation between V

_{IF}and rise and other structures that do not. We have thus looked more closely at the origin of the NMR studies, in order to make a selection. In group I we note that the quadruplex with PDB ID 2O3M [12] has a reasonable P(V

_{IF}-rise) value and that also P(V

_{IF}-twist) is negative, while these coefficients are oddly both positive in the quadruplex with PDB ID 1XAV [11]. Having these structures a parallel topology, one would expect a very regular behavior and similar to each other. They were resolved by different experimental groups, with PDB ID 1XAV being prior. Among the structures of group II, quadruplexes with PDB ID’s 2GKU [20] and 2JPZ [15] have the largest negative correlation coefficients of V

_{IF}with rise and twist: PDB ID 2GKU quadruplex [20] was resolved by the same group that solved PDB ID 2O3M; PDB ID 2JPZ quadruplex [22] was solved by the same group that solved PDB ID 1XAV. PDB ID 186D [17] is very old from 1994. PDB ID 2KZD quadruplex [16] was also resolved by Phan and coworkers (as PDB ID’s 2O3M and 2GKU), while PDB ID 2HY9 [14] was resolved by Yang and coworkers (as PDB ID’s 1XAV and 2JPZ). We have tried a restricted analysis of structure-function correlations using a limited set of structures resolved after 2006 and for which P(V

_{IF}-rise) is negative with large absolute value: We take PDB ID 2O3M as representative of group I, PDB ID’s 2GKU and 2JPZ as representative of group II. We find that the Pearson’s correlation coefficient between V

_{IF}and the helix shape can be maximized by taking a unique homogeneous linear combination of the helix shape parameters. Specifically, the linear combination of the opposite of the rise, 1/2 of the roll and −5/3 of the twist maximizes the shape-V

_{IF}correlation. A slight further improvement is achieved by including also −1/3 of the shift. Scatter plots obtained for group I and group II using this combination of the helix shape parameters are shown in Figure 3 and reveal a significant linear correlation: the linear fits have a R

^{2}of 0.62 and 0.65 for group I and group II, respectively. For the restricted set of conformations represented in Figure 3, spanning parallel and hybrid topologies of the human telomeric sequence, the global shape variable can be tuned to design efficient molecular wires based on G-quadruplex.

## 3. Discussion

## 4. Materials and Methods

#### 4.1. Structural Analysis

#### 4.2. Computational Approach—Electronic Structure Calculations

_{2}. These fragments, deprived of the backbone, were subjected to quantum mechanical (QM) calculations in the framework of density functional theory.

_{2}dimer is related to the energy splitting between HOMO and HOMO-1 of the dimer, which in turn are determined by the HOMO of the unit guanine. Thus, we computed intra-strand and inter-strand transfer integrals between guanines in consecutive tetrads. When two guanine molecules approach each other, their interaction induces an energy splitting between the two highest occupied molecular orbitals in the dimer. The transfer integral V

_{IF}can be expressed in terms of the energy difference between the HOMO and HOMO-1 in the interacting guanine dimer [46,47]: ${V}_{IF}=\sqrt{{\left({E}_{HOMO}-{E}_{HOMO-1}\right)}^{2}-{\left({\epsilon}_{1}-{\epsilon}_{2}\right)}^{2}}$, where ${\epsilon}_{1}$ and ${\epsilon}_{2}$ are the energy levels of the HOMOs of the two interacting guanines. If the two interacting guanines were identical we would have ${\epsilon}_{1}={\epsilon}_{2}$. In our work, however, the geometries of two stacked guanines are never identical: they are not ideal structure formulas but come from real NMR data and have not been further optimized. Consequently, ${\epsilon}_{1}\ne {\epsilon}_{2}$ in the formula for ${V}_{IF}$.

_{IF}introduces errors in the estimated quantity, which can in principle be bypassed by more accurate theories [25,37,40]. We chose in this work a simple method that can routinely be applied to several fairly large structures, which are computationally prohibitive for more sophisticated approaches. We verified that this method gives satisfactory accuracy for the transfer integral of two stacked guanines. We used the standard G-G stacking geometry as an example: the transfer integral computed by us with the energy splitting approach and the technical ingredients specified above is 0.083 eV. This is a good approximation of the value 0.075 eV determined by Migliore and co-workers using a refined formula for ${V}_{IF}$, a more complete basis set for the electronic wave functions and diabatic states for the dimer [25]. To our purposes of analyzing the topological dependence of ${V}_{IF}$ and revealing possible structure-${V}_{IF}$ correlations this accuracy is sufficient and allows us to sample about a thousand relevant geometries. Furthermore, the similar fragment charge difference approach was successfully employed for the investigation of several G-quadruplex conformations [26]: the reliability of this and other semi-empirical methods is also discussed in the existing literature [47,48,49].

#### 4.3. Statistical Analysis—Maximizing the Pearson’s Correlation Coefficient with the Transfer Integral for a Linear Combination of Helix Shape Parameters

_{1}= 7.4 Å and d

_{2}= 4.0 Å being the size of the long and short side of guanine, respectively. z is a homogeneous linear combination of the local inter-base helix parameters, with coefficients ${\alpha}_{n}$ that we wish to optimize to maximize the Pearson’s correlation coefficient between z and the transfer integral V

_{IF}–P(z,V

_{IF}), or between z and the HOMO-LUMO gap HL–p(z,HL).

_{n}, z, V

_{IF}and HL we have several instances, because we have considered different NMR structures, each of them with a number of models. So we use another index i that labels the G-G stacked couples, as a superscript. N is the total number of such couples in each considered group. A group could be Group I, Group II or Group III, or any single pdb (e.g., 2O3M), or any other classification that we want to consider.

_{IF}or HL) is:

_{i}with its definition from Equation (1):

- compute the covariance of each inter-base helix parameter x
_{n}with V_{IF}and with all the other inter-base helix parameters x_{m}; the former is vector $\overrightarrow{b}$ (this vector can also be obtained for the electronic quantity t = HL), the latter is matrix A; - diagonalize matrix A;
- use the eigenvalues and eigenvectors of A to determine the coefficients ${a}_{n}$ that maximize the shape-electronic Pearson’s correlation coefficient;
- calculate the Pearson’s correlation coefficient of this combination, which is an effective helix parameter, with V
_{IF}(and HL), reported in Table S1.

## 5. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

GBA | glycosydic bond angle |

DFT | density functional theory |

Nr. | number |

HOMO | highest occupied molecular orbital |

LUMO | lowest unoccupied molecular orbital |

HL | HOMO-LUMO gap |

BHH | Becke half-and-half functional |

NMR | nuclear magnetic resonance |

PDB | protein data bank |

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**Figure 2.**Top: illustration of the inter-base helix parameters between stacking guanines. Bottom: heat mapss of the structural and electronic parameters on which correlations have been evaluated. The 7 structures of group I and group II are arranged on the vertical axis, from bottom to top in each panel: PDB ID 1XAV, 2O3M, 2GKU, 2HY9, 2JPZ, 2KZD, 186D. The horizontal axis represents the different representative conformations (or NMR model) deposited for each structure in the nucleic acid database (details in Table 3). The scale is shown on the right vertical axis: HL and V

_{IF}are in eV, rise and shift are in Å, twist and roll are in degrees.

**Figure 3.**Scatter plots to illustrate the structural-electronic correlation for topology groups I and II. The insets illustrate the representative structures for group I (2O3M) and group II (2GKU top left, 2JPZ bottom right).

**Figure 4.**Exemplifying structure of a (G4)

_{2}dimer. This is taken from the G-quadruplex 1XAV, which contains three tetrads. The tetrads shown here included guanines 4, 8, 13, 17, 5, 9, 14, 18, as labeled in the PDB file.

Group | PDB Code | Nr. G4 ^{§} | P(V_{IF}-HL) ^{†} |
---|---|---|---|

I | 1XAV | 3 | −0.49 |

2O3M | 3 | −0.73 | |

II | 2GKU | 3 | −0.71 |

2HY9 | 3 | −0.87 | |

2JPZ | 3 | −0.81 | |

2KZD | 3 | −0.82 | |

186D | 3 | −0.78 | |

III | 2KOW | 2 | −0.92 |

201D | 4 | −0.60 | |

143D | 3 | −0.57 | |

2KKA | 2 | −0.41 | |

2KM3 | 2 | −0.94 | |

148D | 2 | −0.87 | |

2KF8 | 2 | −0.94 |

^{§}Nr. G4 is the number of G-tetrads.

^{†}P(V

_{IF}-HL) is the Pearson’s correlation coefficient between the electronic structure parameters V

_{IF}(transfer integral) and HL (energy gap), which are defined and commented later.

PDB Code | Nr. Models ^{†} | Nr. Tetrads ^{‡} | Nr. Guanines | Strand Directions | Parent Sequence | |
---|---|---|---|---|---|---|

Group I | 1XAV [11] | 20 | 3 | 22 | ++++ | 5’-TGAGGGTGGGTAGGGTGGGTAA-3’ |

2O3M [12] | 11 | 3 | 22 | ++++ | 5’-AGGGAGGGCGCTGGGAGGAGGG-3’ | |

Group II | 2GKU [13] | 12 | 3 | 24 | ++−+ | 5’-TTGGGTTAGGGTTAGGGTTAGGGA-3’ |

2HY9 [14] | 10 | 3 | 26 | ++−+ | 5’-AAAGGGTTAGGGTTAGGGTTAGGGAA-3’ | |

2JPZ [15] | 10 | 3 | 26 | +−++ | 5’-TTAGGGTTAGGGTTAGGGTTAGGGTT-3’ | |

2KZD [16] | 10 | 3 | 20 | +−++ | 5’-AGGGIAGGGGCTGGGAGGGC-3’ | |

186D [17] | 7 | 3 | 24 | +−++ | 5’-TTGGGGTTGGGGTTGGGGTTGGGG-3’ | |

Group III | 2KOW [18] | 10 | 2 | 20 | +−+− | 5’-TAGGGTAGGGTAGGGTAIGG-3’ |

201D [19] | 6 | 4 | 28 | +−+− | 5’-GGGGTTTTGGGGTTTTGGGGTTTTGGGG-3’ | |

143D [20] | 6 | 3 | 22 | +−+− | 5’-AGGGTTAGGGTTAGGGTTAGGG-3’ | |

2KKA [21] | 8 | 2 | 23 | +−+− | 5’-AGGGTTAGGGTTAIGGTTAGGGT-3’ | |

2KM3 [22] | 10 | 2 | 22 | +−+− | 5’-AGGGCTAGGGCTAGGGCTAGGG-3’ | |

148D [23] | 12 | 2 | 15 | +−+− | 5’-GGTTGGTGTGGTTGG-3’ | |

2KF8 [24] | 10 | 2 | 22 | +−+− | 5’-GGGTTAGGGTTAGGGTTAGGGT-3’ |

^{†}The number of models in the NMR structure from the protein databank.

^{‡}Number of G-tetrads that compose the G-quadruplex. Strand directions in the quadruplex stem: + means parallel, - means antiparallel. The first strand is + starting from the 5’ end of the parent strand. The parent sequence indicates the DNA G4-motif.

PDB Code | Nr. G-G Couples for V_{IF} * Statistics (Figure 1) | Nr. G-G Couples for Conformation Fluctuations and Structure-Electronic Correlations (Figure 2 and Figure 3) | |
---|---|---|---|

Group I | 1XAV | 160 | 160 |

2O3M | 88 | 88 | |

total group I | 248 | 248 | |

Group II | 2GKU | 96 | 36 |

2HY9 | 80 | 30 | |

2JPZ | 80 | 30 | |

2KZD | 80 | 30 | |

186D | 56 | 21 | |

total group II | 392 | 147 | |

Group III | 2KOW | 40 | 0 |

201D | 72 | 0 | |

143D | 48 | 0 | |

2KKA | 32 | 0 | |

2KM3 | 40 | 0 | |

148D | 48 | 0 | |

2KF8 | 40 | 0 | |

total group III | 320 | 0 |

_{IF}is the transfer integral, computed according to the details described in the Computational approach below.

PDB Code | Strand 1 | Strand 2 | Strand 3 | Strand 4 | Strand Directions | |
---|---|---|---|---|---|---|

Group I | 1XAV | 4(a) 5(a) 6(a) | 8(a) 9(a) 10(a) | 13(a) 14(a) 15(a) | 17(a) 18(a) 19(a) | ++++ |

2O3M | 2(a) 3(a) 4(a) | 6(a) 7(a) 8(a) | 13(a) 14(a) 15(a) | 10(a) 21(a) 22(a) | ++++ | |

Group II | 2GKU | 3(s) 4(a) 5(a) | 9(s) 10(a) 11(a) | 17(a) 16(s) 15(s) | 21(s) 22(a) 23(a) | ++−+ |

2HY9 | 4(s) 5(a) 6(a) | 10(s) 11(a) 12(a) | 18(a) 17(s) 16(s) | 22(s) 23(a) 24(a) | ++−+ | |

2JPZ | 4(s) 5(a) 6(a) | 12(a) 11(s) 10(s) | 16(s) 17(a) 18(a) | 22(s) 23(a) 24(a) | +−++ | |

2KZD | 2(s) 3(a) 4(a) | 10(a) 9(s) 8(s) | 13(s) 14(a) 15(a) | 17(s) 18(a) 19(a) | +−++ | |

186D | 3(s) 4(a) 5(a) | 12(a) 11(s) 10(s) | 16(s) 17(a) 18(a) | 21(s) 22(a) 23(a) | +−++ | |

Group III | 2KOW | 3(s) 4(a) | 9(a) 8(s) | 14(s) 15(a) | 19(a) 20(s) | +−+− |

201D | 1(s) 2(a) 3(s) 4(a) | 12(a) 11(s) 10(a) 9(s) | 17(s) 18(a) 19(s) 20(a) | 28(a) 27(s) 26(a) 25(s) | +−+− | |

143D | 2(a) 3(s) 4(a) | 10(s) 9(a) 8(s) | 14(a) 15(s) 16(a) | 22(s) 21(a) 20(s) | +−+− | |

2KKA | 2(s) 3(a) | 9(a) 8(s) | 15(s) 16(a) | 21(a) 20(s) | +−+− | |

2KM3 | 3(s) 4(a) | 10(a) 9(s) | 15(s) 16(a) | 22(a) 21(s) | +−+− | |

148D | 1(s) 2(a) | 6(a) 5(s) | 10(s) 11(a) | 15(a) 14(s) | +−+− | |

2KF8 | 1(s) 2(a) | 8(a) 7(s) | 14(s) 15(a) | 20(a) 19(s) | +−+− |

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**MDPI and ACS Style**

Sun, W.; Varsano, D.; Di Felice, R. Effects of G-Quadruplex Topology on Electronic Transfer Integrals. *Nanomaterials* **2016**, *6*, 184.
https://doi.org/10.3390/nano6100184

**AMA Style**

Sun W, Varsano D, Di Felice R. Effects of G-Quadruplex Topology on Electronic Transfer Integrals. *Nanomaterials*. 2016; 6(10):184.
https://doi.org/10.3390/nano6100184

**Chicago/Turabian Style**

Sun, Wenming, Daniele Varsano, and Rosa Di Felice. 2016. "Effects of G-Quadruplex Topology on Electronic Transfer Integrals" *Nanomaterials* 6, no. 10: 184.
https://doi.org/10.3390/nano6100184