31P Chemical Shift Tensors for Canonical and Non-canonical

Feb 26, 2008 - Calculating the Response of NMR Shielding Tensor σ(P) andJ(P,C) Coupling Constants in Nucleic Acid Phosphate to Coordination of the Mg...
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3470

J. Phys. Chem. B 2008, 112, 3470-3478

31P

Chemical Shift Tensors for Canonical and Non-canonical Conformations of Nucleic Acids: A DFT Study and NMR Implications Jana Prˇ ecechteˇ lova´ , Petr Padrta, Marke´ ta L. Munzarova´ , and Vladimı´r Sklena´ rˇ * National Centre for Biomolecular Research, Faculty of Science, Masaryk UniVersity, Kotla´ rˇska´ 2, CZ-611 37 Brno, Czech Republic ReceiVed: July 31, 2007; In Final Form: December 19, 2007

31P

chemical shift anisotropy (CSA) tensors have been calculated for a set of selected DNA and RNA backbone conformations using density functional theory. The set includes canonical A-RNA, A-DNA, BI-DNA, BIIDNA, ZI-DNA, and ZII-DNA as well as four A-RNA-type, seven non-A-RNA-type, and three non-canonical DNA conformations. Hexahydrated dimethyl phosphate has been employed as a model. The 31P chemical shift tensors obtained are discussed in terms of similarities in the behavior observed for gauche-gauche (gg) and gauche-trans (gt) conformations around the P-O bonds. We show that torsion angles R and ζ are major determinants of the isotropic chemical shift δiso and of the δCSA 11 component of the traceless chemical shift tensor, which is revealed in separate ranges of both δiso and δCSA 11 for gg- and gt-conformers, respectively. A clear distinction between the two conformation types has not been found for the δCSA and δCSA 22 33 31 components, which is attributed to their different directional properties. The P CSA tensors exhibit CSA CSA considerable variations resulting in large spans of ∼16 ppm for δCSA 11 and ∼22 ppm for δ22 and δ33 . We examine the consequences of the CSA variations for predicting the chemical shift changes upon partial alignment δcsa and for the values of CSA order parameters extracted from the analysis of 31P NMR relaxation data. The theoretical 31P CSA tensors as well as the experimental 31P CSA tensor of barium diethyl phosphate (BDEP) are used to calculate δcsa for two eclipsed orientations of the CSA and molecular alignment tensors. Percentage differences between the CSA order parameters obtained using the theoretical 31P CSA tensors and the experimental 31P CSA tensor of BDEP, respectively, are also determined.

Introduction NMR structure determination of a nucleic acid backbone is complicated by a small number of experimental restraints. Low proton density, long distances among the protons from the 3′and 5′-ends, poor resolution in the ribose and deoxyribose spectral regions, and a complicated network of spin-spin interactions between the sugar protons limit the amount of reliable nuclear Overhauser effect (NOE) data. Phosphorus-31, the key element of the phosphodiester linkage, is a spin 1/2 nucleus with 100% natural abundance. Potentially, its NMR parameters could supply a wealth of structural information. 31P displays significant spin-spin scalar couplings to 1H and 13C nuclei along the nucleic acid backbone. A number of Karplus equations are available to restrain the backbone torsion angles.1 Unfortunately, large asymmetry of electron density in the vicinity of 31P nucleus and associated chemical shifts anisotropy (CSA) lead to fast, CSA driven relaxation resulting in a sizable line broadening. In nucleic acids, the variations of 31P isotropic chemical shifts are small, spanning typically less than 2 ppm, which, together with a broad linewidth of 31P resonances, substantially lowers the attainable resolution. As a result, the amount of available experimental data is significantly reduced, especially for larger, biologically relevant RNA and DNA molecules. The chemical shifts of nuclei having a substantial chemical shift anisotropy change when measured in partially aligned * Corresponding author. E-mail: [email protected]. Phone: +420549 49 7022. Fax: +420-549 49 2556.

media.2 The alignment induced changes δcsa can be used as conformational restraints for NMR structure determination of biomacromolecules. For sugar-phosphate backbone in nucleic acids, the induced chemical shifts of 31P provide valuable information about its conformation. Bax et al. employed the alignment induced changes of 31P chemical shifts as restrains in the molecular dynamics refinement.3,4 The usage of δcsa requires the knowledge of individual 31P chemical shift tensors and their orientations relative to the molecular reference frame for each nucleotide in the polynucleotide chain. Such data can be obtained exclusively by solid-state NMR measurements on single crystals, which are currently not applicable in the case of RNA and DNA oligonucleotides. Until now, the only available experimental values of the 31P chemical shift tensor for the phosphodiester group have been reported for barium diethyl phosphate.5 Since the isotropic 31P chemical shifts in nucleic acids span only a narrow range of several ppm, it has been suggested that the 31P CSA tensor uniform for all backbone phosphates in oligonucleotides and equal to the 31P CSA tensor of barium diethyl phosphate represents a justifiable simplification.3 Obviously, such an approach, employing the same CSA tensor for all phosphate groups in an oligonucleotide regardless of the phosphodiester conformation, is a potential source of errors. The lack of experimental information on 31P chemical shift tensors has considerable implications also for the analysis and interpretation of relaxation data, which provide information on the phosphorus order parameter, correlation time, and consequently on the dynamic behavior of nucleic acids. CSA

10.1021/jp076073n CCC: $40.75 © 2008 American Chemical Society Published on Web 02/26/2008

31P

Chemical Shift Tensors

J. Phys. Chem. B, Vol. 112, No. 11, 2008 3471 TABLE 1: Summary of the Conformations Studieda conf. class/PDB ID 8 10 17 15 1 3 11

Figure 1. DNA backbone torsion angles.

significantly contributes to relaxation processes of 31P nuclei.6,7 Chemical shift anisotropy ∆σ and chemical shift asymmetry η are needed to extract the CSA order parameters S2c from relaxation measurements. A representative example is the study by Roberts et al.8 who estimated the values of S2c for phosphate moieties in the DNA octamer [d(GGAATTCC)]2 from the measurements of the field dependence of the phosphorus longitudinal relaxations rates R1 via high-resolution field cycling NMR. As outlined above, the prerequisite for successful utilization of 31P induced chemical shifts for structure refinement and relaxation measurements is accurate knowledge of the 31P chemical shift tensor. The missing experimental data could be complemented by the results of theoretical calculations. 31P NMR properties of biomolecules were intensively studied theoretically in 1970s and 1980s. Although rough and approximate, the early semiempirical and coupled Hartree-Fock calculations provided valuable qualitative insight. The main focus of the studies was on 31P isotropic chemical shifts. Several papers suggested that P-O ester torsion angles ζ and R considerably influence δiso.9-13 A systematic study was carried out by Gorenstein and Kar who calculated a 31P chemical shift torsion angle counter map of dimethyl phosphate monoanion from CNDO electron densities.10,12 Moreover, the difference between the isotropic chemical shifts of the gauche-gauche (gg) and those of the gauche-trans (gt) conformers has been investigated extensively.9-12 Giessner-Prettre et al.13 were the first to note that C-O torsion angles  and β have the same impact on δiso as torsion angles ζ and R. It was also shown that the qualitative behavior of δiso as a function of β or  is identical for the gg- and gt-conformers. Several attempts9,10,12,13 were made to establish a correlation between the O-P-O bond angle and δiso. Such a correlation has been observed empirically for phosphate esters.14 However, it was concluded that the effect of the O-P-O bond angle is not dominant, being mediated primarily by coupled changes in backbone torsion angles. Recently, we have studied 31P CSA tensors as a function of backbone torsion angles R and ζ for BI and BII-DNA15 using density functional theory (DFT). Calculations performed with a solvated model of dimethyl phosphate demonstrated that the CSA δCSA 22 and δ33 principal components of the traceless chemical shift tensor vary within as much as 30 ppm if R and ζ torsion angles are in the range 〈270°, 330°〉 and 〈240°, 300°〉, respectively. The influence of β and  on the traceless chemical shift tensor components has also been studied by calculating the for two different choices of β and . It was dependence of δCSA ii shown that the C-O torsion angles do not change the qualitative trends, yet quantitative differences of ∼2 ppm have been

20 (canonical RNA) 19 22 30 31 BI-DNA BII-DNA A-DNA (-gt) A-DNA (-g-g) ZII-DNA (CG) ZII-DNA (GC) ZI-DNA (GC) 1PQT (4A-5A step) 1C11 (6T-7T step) 1JRV (9T-10A step)

Ri+1

βi+1

non-A-type 270 83 237 65 256 245 225 211 208 53 226 206 221 137

68 66 63 60 165 296 293

175 119 171 201 149 154 167

A-RNA-type 211 291 194 292 265 295 223 285 218 282

296 156 290 176 278

175 194 196 175 113

265 174 289 289 75 52 301

298 298 185 293 71 168 201

176 146 174 174 183 166 225

non-canonical DNAs 210 102 203 73 198 249

77 96 122

210 188 117

i

DNAs 184 246 203 203 267 189 240

ζi

a The first column shows the number of an RNA class as published by Schneider et al.,26 the type of a canonical DNA, or a PDB ID; i, ζi, Ri+1, and βi+1 refer to nucleic acid backbone torsion angles (Figure 1); A and T are abbreviations for adenine and thymine, respectively.

observed. The study consisted exclusively in systematic mapping within the ranges of R and ζ found in X-ray of δiso and δCSA ii structures of Β-DΝΑ. Α question remains, what ranges of tensor components would be obtained for particular combinations of R, β, , and ζ torsion angles occurring in canonical as well as non-canonical structures of both DNA and RNA. In this study, we run a series of 31P CSA tensor calculations using hexahydrated dimethyl phosphate as a model of the nucleic acid backbone conformation15 (Figure 1). The set of conformations studied include seven non-A-RNA-type and five A-RNAtype classes, and seven phosphate group conformations found in canonical DNAs (-gt A-DNA, -g-g A-DNA, BI-DNA, BIIDNA, ZI-DNA, cytosine-guanine and guanine-cytosine steps in ZII-DNA) as well as three unusual phosphate group conformations present in structures with PDB (Protein Data Bank) IDs 1PQT (DNA hairpin), 1C11 (double hairpin), and 1JRV (DNA bulge). Classification and torsion angles for individual conformations are summarized in Table 1. To estimate the influence of the 31P chemical shift tensor variations, the calculated CSA tensor components were used to obtain the values of induced chemical shifts for two eclipsed orientations of the CSA and molecular alignment tensors when the induced chemical shifts changes sample minimal and maximal amplitudes. In addition, the chemical shift anisotropy and asymmetry of the theoretical chemical shift tensors were employed to estimate the error resulting from applying the 31P CSA tensor of barium diethyl phosphate to calculations of CSA order parameters from experimental relaxation data. The results obtained are discussed in terms of deviations stemming from a simplified assumption that the 31P CSA tensor is uniform and thus does not vary for different combinations of torsion angles in the families of RNA and DNA conformations studied.

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Theory Chemical Shift Tensor. The chemical shift is a second rank tensor. In the principal axis system (PAS), the tensor is diagonal and is given as a sum of its isotropic and anisotropic part:16,17

( )(

)(

δCSA 0 0 δ11 0 0 δiso 0 0 11 CSA δ δ δ22 0 0 ) 0 + 0 iso 0 22 0 0 0 δ33 0 0 δiso δCSA 0 0 33

)

(1)

where

δ11 g δ22 g δ33

(2)

1 δiso ) (δ11 + δ22 + δ33) 3

(3)

and the trace of the second matrix on the right side of eq 1 is equal to zero: CSA CSA δCSA 11 + δ22 + δ33 ) 0

(4)

From now on, we shall denote the anisotropic part of the full chemical shift tensor as a CSA tensor. Besides the isotropic chemical shift δiso, two more parameters are commonly used to report chemical shift tensors in PAS: the chemical shift anisotropy ∆σ and chemical shift asymmetry η. The chemical shift anisotropy is defined as

1 ∆σ ) δ33 - (δ11 + δ22) 2

(5)

or combining eqs 4 and 5 as

3 ∆δ ) δCSA 2 33

(6)

and chemical shift asymmetry η, 0 e η e 1 is defined as

η)

δ22 - δ11 ) δ33 - δiso

- δCSA 11 CSA δ33

δCSA 22

1

∑ ∑ Aii(cos2θij)δCSA jj

3 i)x,y,z j)x,y,z

(7)

(8)

where Aii and δCSA are the elements of the diagonalized jj traceless molecular alignment tensor A and the CSA tensor, respectively. The resonance offsets δcsa are the source of structural information because of the term (cos2θij)δCSA jj , where θij is the angle between the ii- and the jj- principal axes of the two tensors. The molecular alignment tensor A (order matrix, Saupe matrix) is used to describe the preferential orientational averaging of a molecule in the magnetic field upon partial alignment.23-25 A is a 3 × 3 symmetric (Aij ) Aji) and traceless matrix

Axx + Ayy + Azz ) 0

where |Axx| e |Ayy| e |Azz| by convention. Therefore, only five out of the original nine alignment tensor components are independent. Most often, they are expressed as the principal order parameter (order parameter for the most ordered axis) Azz, an asymmetry parameter η ) (Axx - Ayy/Azz), and three Euler angles defining the orientation of the alignment tensor with respect to the coordinate system of the molecule. The asymmetry parameter reflects the deviation of the ordering from axial symmetry. The molecular alignment tensor is axially symmetric if η ) 0, that is,

Axx ) Ayy ) -

Induced Chemical Shift. 31P nuclei are known to have a large chemical shift anisotropy, which can be exploited for a structure refinement.3,4,18-20 As a result of the anisotropy of electron distribution in the vicinity of the phosphorus atom, the chemical shift observed in aligned media δalign differs from δiso obtained in solution by an anisotropic contribution, the so-called induced chemical shift, given as20-22

δcsa ) δalign - δiso )

Figure 2. Five types of a phosphate group conformation. Due to the symmetry of dimethyl phosphate, +g-g is identical with -g+g and the same applies for +gt vs t+g, and -gt vs t-g.

(9)

Azz 2

(10)

and asymmetric if η ) 1, that is,

Axx ) 0, Ayy ) - Azz

(11)

It is convenient to transform A from a molecule-fixed axis system to the principal axis frame in which all but diagonal elements Axx, Ayy, and Azz of the order matrix vanish.22 The calculated induced chemical shifts δcsa (eq 8) depend on parameters of the molecular alignment and the CSA tensors and on their mutual orientation. Extremes are found when the two coordinate systems have axes oriented in parallel. In order to easily discuss δcsa calculations, it is convenient to order the principal components of both tensors in terms of absolute values as |Txx| e |Tyy| e |Tzz|, where Tii is a principal tensor component. For the molecular alignment tensor |Axx| e |Ayy| e |Azz| by convention, whereas the components of the complete chemical shift tensor are typically ordered as δ11 g δ22 g δ33. Reordering the components of the CSA tensor in absolute values gives CSA CSA |δCSA 22 | e |δ11 | e |δ33 |. From now on, we will use labels CSA CSA CSA CSA CSA δxx , δyy , and δzz for components δCSA 22 , δ11 , and δ33 , respectively, which gives the desired order of absolute values CSA CSA |δCSA xx | e |δyy | e |δzz |. In using this notation, the smallest, middle-large, and largest components of both tensors are always labeled xx, yy, and zz. CSA Order Parameter. The chemical shift anisotropy contribution to the longitudinal relaxation rate R1 is given by the equation8

31P

Chemical Shift Tensors

J. Phys. Chem. B, Vol. 112, No. 11, 2008 3473

R1(CSA) ) CLωP2J(ωP)

(12)

where ωP is the Larmor frequency of phosphorus ωP ) γPB0 and J(ωP) is the spectral density function. The parameter CL is given as

CL )

(

)

1 η2 1+ (∆σ)2Sc2 15 3

(13)

where η and ∆σ are the phosphorus chemical shift asymmetry 2 and chemical shift anisotropy. If Sc,BDEP and S2c are the CSA order parameters obtained for the same value of CL using ∆σ and η of BDEP (∆σBDEP and ηBDEP) and of the dmp-conformations studied (∆σ and η), respectively, then the percentage 2 deviation of S2c from Sc,BDEP is given as

Sc,BDEP - Sc

2

2

∆Sc2 ) 100

((

Sc,BDEP

1+

(

) 100 1 +

2

1

)

2

ηBDEP (∆σBDEP)2 3

)

2

ηBDEP (∆σBDEP)2 3

-

)

1 (14) η2 1+ (∆σ)2 3

(

)

Computational Details Selection of Conformations. The following notation is used to indicate the values of torsion angles: +g ) +60° ( 60°, -g ) -60° ( 60°, and t ) 180° ( 60°. In order to conveniently describe common features of the (g(g and (g-g conformations around the P-O bonds, a simplified notation gg is used (i.e., signs are omitted). Similarly, (gt and t(g are grouped as gt. If signs are used, the first and second letters refer to the ζ and Ri+1 torsion angles, respectively. RNA conformations for DFT calculations were chosen from the list of eighteen non-A-RNA-type and fourteen A-RNA-type conformation classes identified by Schneider et al.26 The selection was made based on the following procedure. Non-ARNA-type conformations were divided into five groups according to the values of ζi and Ri+1 torsion angles: +g+g, -g+g, t+g, +gt, and t-g (Figure 2). Then, one or two representative conformations from each group were used for the calculations. Apart from seven non-A-RNA-type conformation classes, four A-RNA-type conformations with one of the torsion angles i, ζi, Ri+1, βi+1 differing by more than 50° from the corresponding angle in canonical A-RNA were selected. Canonical A-RNA was included for comparison. Moreover, three DNA phosphate group conformations with at least one of the torsion angles i, ζi, Ri+1, βi+1 values lying out of the range typical for B-DNA (A4-A5 step of 1PQT, T6-T7 step of 1C11, and T9-A10 step of 1JRV) were added to the set inspected. Canonical B-DNA (BI-DNA and BII-DNA), -gt A-DNA, -g-g A-DNA, and Z-DNA (ZI-DNA, and cytosine-guanine and guanine-cytosine steps of ZII-DNA) were studied as well. All conformations employed are listed in Table 1. For purposes of the discussion, the conformations are divided into three groups according to ζ and R torsion angles: (1) both ζ and R are +g or -g, (2) ζ is -g and R is +g, and (3) one of the two torsion angles is (g and the other is t. If necessary, the groups are further subdivided to distinguish various combinations of the  and β torsion angles: (i) one of the torsion angels is +g and the other is t, (ii) one of the torsion angles is -g and other is t, and (iii) both  and β are t (Table 2). Geometry Optimization. In order to calculate the 31P chemical shift tensors for the set of conformations selected, a model of dimethyl phosphate with six explicit water molecules

Figure 3. Orientations of the molecular alignment (solid line) and the CSA tensor (dashed line) that produce (a) minimum and (b) maximum csa value of δcsa, δcsa min and δmax, respectively.

was used. Justification for the model selection has been provided in our previous study15 of 31P chemical shift tensors in B-DNA. We denote the model system as dmp-IC (dimethyl phosphate Interconnected Cones of hydration), as two cones, each formed by three water molecules, are clustered around each of the anionic oxygens and interconnected via hydrogen bonds. All geometries of dmp-IC were optimized in Gaussian0327 at the DFT (density functional theory) level of theory. The B3LYP functional28,29 along with the 6-31G(d) basis set30,31 were used. The torsion angles , ζ, R, and β were frozen to their corresponding values in the sets of DNA and RNA conformations (Table 1).26,32 Other structural parameters were optimized. As a phosphate group connects two residues in an oligonucleotide, the torsion angles , ζ and R, β in the dmp-IC model correspond to the ith and (i + 1)th residue, respectively (Figure 1). Therefore, the torsion angles are denoted i, ζi, Ri+1, and βi+1 if necessary. Chemical Shift Tensor Calculations. The 31P chemical shift tensors were obtained from sum-over-states density functional perturbation theory calculations with the IGLO choice of the gauge origin (SOS-DFPT-IGLO)33 as implemented in the deMon-NMR code.34-36 The density functional calculations employed Perdew and Wang’s generalized gradient approximation (GGA)37 for exchange and Perdew’s GGA for correlation (PWP86).38,39 The orbital basis set IGLO-III40 combined with the corresponding experimental auxiliary basis set was assigned to all atoms. The convergence criteria for energy and density were set to 1.0 × 10-7 Hartree and 1.0 × 10-6, respectively. The numerical grid RADI128/EXTRAFINE was applied. The calculated chemical shieldings were converted to chemical shifts using the equation

δ(X,calc) ) σ(PH3,calc) - σ(X,calc) - 266.1

(15)

suggested by van Wu¨llen,41 where X denotes the conformation for which we calculate the 31P chemical shift, and 266.1 ppm is the difference between the absolute experimental chemical shieldings of PH3 (594.5 ppm) and 85% H3PO4 (328.4 ppm) at 300 K.42 The chemical shielding of PH3 calculated using the same method and basis set as the systems studied is 569.6 ppm. Induced Chemical Shift Calculations. Induced chemical shifts of all conformations were calculated for two eclipsed orientations of the CSA and molecular alignment tensor such CSA CSA that (a) Axx, Ayy, and Azz are collinear with δCSA xx , δyy , and δzz CSA (Figure 3a) and (b) Axx, Ayy, and Azz are collinear with δxx , CSA (Figure 3b).43 The two orientations give the δCSA zz , and δyy largest negative and largest positive values of δcsa, δcsa min and δcsa max, respectively, over all possible orientations of the two csa tensors. The values of δcsa min and δmax thus define the boundaries within which δcsa values are found.

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TABLE 2: Calculated 31P Chemical Shift Anisotropy Tensorsa i

ζi

Ri+1

βi+1

conf. class/PDB IDb

δCSA 22

δCSA 33

∆σ

φx

φy

φz

21.9 21.0 22.9 26.9 19.4 26.0 16.8 21.6 14.2 15.4

-110.1 -105.7 -111.3 -116.5 -106.9 -112.4 -101.8 -105.8 -105.6 -106.3

-165.2 -158.6 -167.0 -174.8 -160.4 -168.6 -152.7 -158.7 -158.4 -159.5

2.5 0.6 2.4 0.6 2.6 1.9 2.3 1.1 4.6 3.9

0.6 0.4 1.3 0.1 1.9 2.4 0.8 0.9 1.2 1.5

2.6 0.5 2.0 0.6 3.2 1.5 2.2 0.6 4.6 3.9

ζ,Ri+1 ) -g+g 5.0 85.1 7.4 82.9

22.2 12.0

-107.4 -95.1

-161.1 -142.7

3.0 1.6

7.9 8.8

8.5 8.7

ζ,Ri+1 ) (gt or t(g 8.4 77.9 10.8 77.7 7.0 81.8 6.3 80.2 6.3 80.1 10.5 76.7 9.5 76.0 7.2 80.0 10.8 77.1 10.8 75.7

27.9 23.5 21.2 33.5 26.9 23.5 24.1 33.9 29.5 29.6

-105.7 -101.1 -103.0 -113.6 -106.9 -100.2 -100.1 -113.9 -106.5 -105.4

-158.6 -151.7 -154.5 -170.4 -160.4 -150.3 -150.2 -170.9 -159.8 -158.1

4.0 4.8 4.5 6.4 4.8 4.9 4.0 3.6 7.4 5.7

5.6 4.0 8.8 6.2 8.2 7.1 5.6 3.5 7.2 7.6

5.6 3.6 8.9 1.9 7.6 5.1 4.0 1.3 1.6 5.1

δiso

δCSA 11

-g -g -g t/-g t t t t t t

+g -g +g +g -g -g -g -g +g +g

+g -g +g +g -g -g -g -g +g +g

t t t +g/t +g/t t t t t t

ζ,Ri+1 ) +g+g or -g-g 8 4.7 88.3 22 7.3 84.7 ZII-DNA (CG) 5.5 88.5 10 7.0 89.7 31 5.7 87.5 20 (canonical RNA) 6.1 86.3 BI-DNA 6.4 84.9 A-DNA (-g-g) 5.8 84.3 1PQT 7.1 91.5 1C11 6.6 90.7

-g t

-g/t -g/t

+g t/+g

t +g/t

17 1JRV

t t t t t t t t -g/t -g/t

t t t +g -g -g -g +g -g t

+g -g -g t t t t t t -g

t t t t t t t t t t

15 3 11 1 19 30 A-DNA (-gt) ZII-DNA (GC) ZI-DNA (GC) BII-DNA

a , ζ, R i i i+1, and βi+1 refer to nucleic acid backbone torsion angles: +g ) +60° ( 60°, -g ) -60° ( 60°, and t ) 180° ( 60°, -g/t and g/t indicate that the value of the corresponding torsion angle lies within 10° from the boundary of the (g and t region; b is the number of an RNA class CSA CSA as published by Schneider et al.,26 the type of a canonical DNA, or a PDB ID; δiso refers to the isotropic chemical shift; δCSA 11 , δ22 , and δ33 are the CSA components of the traceless chemical shift tensor; the chemical shift anisotropy ∆σ ) (3/2) δ33 ; φx, φy, and φz are the deviations of the principal axes 11, 22, and 33 from the Cartesian axes x, y, and z of the molecular coordinate system (Figure 4), respectively.

The calculations of the induced chemical shifts were carried out for an example of an almost axially symmetric (Axx ) -7.594 × 10-4, Ayy ) -7.874 × 10-4, Azz ) 1.547 × 10-3) and almost fully asymmetric molecular alignment tensor (Axx ) -5.734 × 10-6, Ayy ) -9.498 × 10-4, Azz ) 9.556 × 10-4) obtained for Dickerson’s dodecamer (1NAJ)4 and DNA hairpin (1KR8),44 respectively, via fitting experimental data to the respective structures. CSA Order Parameter. In order to assess the effect of the 31P CSA tensor on the CSA order parameters S2 derived from c relaxation measurements, we have calculated the percentage error ∆S2c according to the eq 14. For BDEP, the chemical shift anisotropy ∆σBDEP ) -150 ppm and chemical shift asymmetry CSA CSA ηBDEP ) 0.6 were used (δCSA 11 ) 80 ppm, δ22 ) 20 ppm, δ33 ) 5 -100 ppm). Results and Discussion Isotropic Chemical Shifts. Results of our calculations, summarized in Table 2, reveal that backbone torsion angles ζ and R clearly govern isotropic chemical shifts, while the C-O torsion angles  and β introduce only minor changes. The same behavior has also been found for the δCSA 11 principal component (cf. below). As a result, the set of nucleic acid backbone conformations studied can be clearly separated into two groups based on the value of the isotropic chemical shift obtained. Whereas (g(g and -g(g combinations of the ζ and Ri+1 torsion angles exhibit isotropic chemical shifts between 4.7 and 7.4 ppm, (gt and t(g conformations around the P-Ο bonds tend to give larger values of δiso ranging from 6.3 to 10.8 ppm. Maximum values of δiso amount to 9.5-10.8 ppm and were obtained for five -gt and t-g conformers including both canonical (B-DNA, A-DNA, Z-DNA, and A-RNA) and noncanonical (non-A-RNA type, class 3) nucleic acids. These

maximum values are obtained only when one of the torsion angles R, ζ falls within the -gauche range of (285°, 300°) and the other is found in the trans range between 174° and 206°. This is in agreement with our previous results for an idealized -gt conformation,15 for which δiso amounts to 11.5 ppm if the dmp-IC model is used and (R, ζ, β, ) are set to (-60°, 180°, 150°, 250°). Traceless Chemical Shift Tensor Components. An inspection of Table 2 shows that for different conformations the CSA components δCSA 22 and δ33 vary within ∼22 ppm, while changes CSA of the component δ11 are slightly less pronounced, altering within ∼16 ppm. Similarly to isotropic chemical shifts, the torsion angles ζ and R are the major determinants of the δCSA 11 principal component. The δCSA values obtained from our 11 calculations clearly split into two ranges, each of which refers to a different conformation type around the P-O bonds. Whereas δCSA 11 of gg-conformers amounts to ∼83-92 ppm, gtconformers display values between 76 and 82 ppm. Changes in β and  only modulate δCSA 11 within the ranges established. Separated ranges of tensor component values corresponding to the gg- and gt-conformation types can also be traced for δCSA 22 , yet those ranges are only approximate and not all conformations of a given type fall within the corresponding of gg-conformations varies boundaries. For example, δCSA 22 between 12 and 23 ppm, while δCSA of gt-conformations 22 amounts to 24-34 ppm. Exceptions to the rule are conformation classes 10, 11, and 20. Similarly, the δCSA 33 component ranges approximately from -117 to -106 ppm for the gg-conformation and from -106 to -100 ppm for the gt-conformation. The two intervals partially overlap as δCSA 33 of about -106 ( 1 ppm has been obtained for several gg- as well as gt-conformations. that lie out of the Moreover, there are four values of δCSA 33

31P

Chemical Shift Tensors

J. Phys. Chem. B, Vol. 112, No. 11, 2008 3475 TABLE 3: Induced Chemical Shifts Calculated Using the Almost Axially Symmetric Molecular Alignment Tensor of 1NAJa

Figure 4. Directions of the principal components of the 31P CSA tensor with respect to the Cartesian coordinate system. The y axis bisects the O1-P-O2 angle; the z axis lies in the O1-P-O2 plane and is perpendicular to y; and the x axis is orthogonal to both y and z.

range established for the corresponding conformation type (δCSA of BI-DNA, 1JRV, class 1, and ZII-DNA (GC)). The 33 for 1JRV is not surprising as both anomalous value of δCSA 33 C-O torsion angles of the 9T-10A phosphodiester group adopt transition geometries. Other irregularities in the behavior of CSA δCSA 22 and δ33 suggest that the two components result from a coupled effect of ζ and R with the more remote torsion angles β and . The different behavior of δCSA compared with δCSA and 11 22 CSA δ33 is most likely due to the directional properties of δCSA 11 . Whereas δCSA points from phosphorus along the backbone, 11 and δCSA are perpendicular to the backbone (Figure 4). δCSA 22 33 Consequently, the paramagnetic contributions to δCSA will 11 arise from orbital couplings in the O1-P-Ο2 plane. This plane is structurally separated from the conformational variations along the backbone; hence, we expect only second-order effects on δCSA 11 , which explains its lower sensitivity to β and . Potentially, the analysis of the full chemical shift tensor components in terms of dia- and paramagnetic contributions45 could provide a support for our claim. However, substantial variations in δpara among the studied conformations do not allow a straightforward comparison and thus do not afford a clear explanation. CSA Parameter. As a result of direct proportionality (eq 6), the dependence of ∆σ on the conformation of the phosphate groups reflects the properties of the δCSA 33 component. Except for the conformations BI-DNA, 1JRV, class 1, and ZII-DNA (GC), the gg-conformers generally display values of -175 ppm to -158 ppm, while ∆σ of gt-conformers range from -160 to -150 ppm. That gives a total span of ∼32 ppm, which has important implications for the interpretation of experimental 31P NMR relaxation data as discussed below. Angles between the Standard Coordinate System and the Principal Axes. The standard coordinate system is defined as follows: the y axis bisects the O1-P-O2 angle, the z axis lies in the O1-P-O2 plane and is perpendicular to y, and the x axis is orthogonal to both y and z (Figure 4). The angular angles between the 11-, 22-, and 33-principal axes and the Cartesian axes x, y, and z of the standard coordinate system are denoted as φx, φy, and φz. A survey of Table 2 reveals the following: in the overwhelming majority of cases, two of the angles (φx, φy, φz) have similar values but deviate substantially from the third angle. The third angle is mostly smaller than the other two angles. The “2 + 1” scheme for (φx, φy, φz) can be understood on the

conf. class

δcsa min/ppb

∆csa min/%

δcsa max/ppb

δcsa max/%

BDEP 8 10 17 15 1 3 11 20 (canonical RNA) 19 22 30 31 BI-DNA BII-DNA A-DNA (-gt) A-DNA (-g-g) ZII-DNA (CG) ZII-DNA (GC) ZI-DNA (GC) 1PQT 1C11 1JRV

-77.6 -85.5 -90.4 -83.3 -82.0 -88.1 -78.5 -80.0 -87.2 -83.0 -82.1 -77.8 -83.0 -79.0 -81.7 -77.7 -82.2 -86.4 -88.3 -82.6 -82.1 -82.5 -73.8

10.2 16.5 7.4 5.7 13.5 1.2 3.1 12.4 7.0 5.8 0.3 7.0 1.8 5.3 0.1 5.9 11.3 13.8 6.4 5.8 6.3 -4.9

62.4 68.9 70.0 66.5 60.9 62.7 60.7 63.9 67.4 62.6 66.1 59.9 68.3 66.3 59.2 59.4 65.8 69.1 62.6 60.2 71.3 70.8 64.7

10.4 12.2 6.6 -2.4 0.5 -2.7 2.4 8.0 0.3 5.9 -4.0 9.5 6.3 -5.1 -4.8 5.5 10.7 0.3 -3.5 14.3 13.5 3.7

a csa δmin and δcsa max represent the induced chemical shifts calculated using the molecular alignment tensor of 1NAJ according to eqs 16 and 17, respectively, for the two orientations of the CSA and molecular csa alignment tnesor shown in Figure 3. ∆csa min and ∆max are the deviations csa csa of δmin and δmax calculated for a particular phosphate group conforcsa mation from δcsa min and δmax obtained using BDEP, respectively.

basis of the orthogonality of the axes. The two of the angles with similar values indicate the type of rotation leading from the standard coordinate system to the principal axes system. For example, if φy ≈ φz, the rotation can be composed of (1) rotation of y, z around x and (2) rotation of x, y, z around the axis of the two angles confined by y, z. Table 2 also shows that maximum angles of 6° - 9° were obtained for the 22- and the 33- principal axes in -g+g and gt conformations. Second-largest deviations of 4°-7° were found for the 11- and 22-principal axes in gt-conformers. On the contrary, the smallest, yet still substantial values (2°-5°) of the angles between the principal axes and the axes of the Cartesian coordinate system were observed for +g+g and -g-g conformations. Moreover, there are three examples of φx, φy, φz about 1° or smaller in Table 2, again for +g+g and -g-g. Considering the fact that both examples of -g+g conformers in Table 2 are transition geometries in terms of R and/or ζ, we conclude that the largest deviations of tensor components from the Cartesian axes were obtained for gt-conformations. In many cases, the deviations exceed 5°, which was the maximum value found in our study of the 31P CSA tensors in B-DNA.15 In the previous paper, the deviations were determined for CSA tensors calculated in vacuo. A comparison of angles (φx, φy, φz) for solvated and unsolvated dmp-conformations shows that the slightly higher values of (φx, φy, φz) presented here can be attributed to the solvent effect. Induced Chemical Shifts. General expressions for the smallest and largest value of the chemical shift measured in a csa partially oriented media, δcsa min and δmax, respectively, can be easily derived from eq 8. In the case of the eclipsed orientations, six terms in eq 8 vanish and using the symmetry properties of A we obtain

3476 J. Phys. Chem. B, Vol. 112, No. 11, 2008

δcsa min )

δcsa max )

1 CSA (A δCSA + AyyδCSA yy + Azzδzz ) ) 3 xx xx δCSA δCSA 1 xx yy 3 1 ) Azz δCSA (16) + δCSA zz 3 2 2 3 2 zz

(

)

1 + AzzδCSA (A δCSA + AyyδCSA zz yy ) ) 3 xx xx δCSA δCSA zz xx 3 1 1 + δCSA ) Azz δCSA (17) yy 3 2 2 3 2 yy

(

)

for the axially symmetric molecular alignment tensor and

δcsa min )

1 CSA (A δCSA + AyyδCSA yy + Azzδzz ) ) 3 xx xx 1 CSA - Azz(δCSA yy - δzz ) (18) 3

δcsa max )

1 + AzzδCSA (A δCSA + AyyδCSA zz yy ) ) 3 xx xx 1 CSA + Azz(δCSA yy - δzz ) (19) 3

for the asymmetric molecular alignment tensor. In all four equations, two components of the alignment tensor vanish and only the Azz component remains. By contrast, the CSA tensor reduces to a single component only for the symmetric A. In that case, the lower limit of the induced chemical shift δcsa min for a given phosphorus atom depends on the largest principal component of the traceless chemical shift csa CSA tensor δCSA zz . Similarly, δmax is directly proportional to δyy (eq 17). For the asymmetric molecular alignment tensor, δcsa min and δcsa depend on the complete CSA tensor in addition to A zz. For max CSA and δ to express eqs 18 the sake of clarity, we choose δCSA yy zz and 19 but we note that eq 4 can be used to recast all eqs 1619 with different components of the CSA tensor. csa Table 3 shows δcsa min and δmax calculated using the symmetrical molecular alignment tensor of the Dickerson’s dodecamer. Whereas δcsa min ranges from -90.4 to -73.8 ppb, giving a span of ∼17 ppb, δcsa min falls within the range of ∼12 ppb bracketed by the values 59.2 and 71.3 ppb, respectively. Besides, csa the percentage deviations ∆csa min and ∆max of induced chemical csa csa shifts δmin and δmax obtained using the calculated CSA tensors for particular conformations and the experimental CSA tensor of barium diethyl phosphate, respectively, are provided. Alcsa though ∆csa min and ∆max are negligible in many cases, they are larger than 10% for six and five conformation classes, respectively. The maximum deviations amount to ∆csa min ) 16.5% and ) 14.3%. ∆csa max If the asymmetric molecular alignment tensor of 1KR8 is csa employed, δcsa min and δmax vary within a range of ∼9.5 ppb, the boundary values being -65.6 and -56.0 ppb for δcsa min and 55.9 csa csa (Table 4). ∆ and ∆ are above 10% and 65.4 ppb for δcsa max min max for four conformation classes with the maximum error of 14.5%. CSA Order Parameter. Table 5 shows the percentage error ∆S2c resulting from the uniform usage of the 31P CSA tensor of BDEP instead of the CSA tensors calculated for a given conformation. Except for three cases, the CSA order parameter decreases upon employing the values of ∆σ and η of the calculated CSA tensors. In the set of conformation classes inspected, the largest error obtained amounts to ∼25%. Apart from the maximum error obtained for the non-A-RNA-type conformation class 10, significant errors of ∼10-18% were found for eleven conformation classes.

Prˇecechteˇlova´ et al. TABLE 4: Induced Chemical Shifts Calculated Using the Almost Fully Asymmetric Molecular Alignment Tensor of 1KR8a conf. class

δcsa min/ ppb

∆csa min/ %

δcsa max/ ppb

∆csa max/ %

BDEP 8 10 17 15 1 3 11 20 (canonical RNA) 19 22 30 31 BI-DNA BII-DNA A-DNA (-gt) A-DNA (-g-g) ZII-DNA (CG) ZII-DNA (GC) ZI-DNA (GC) 1PQT 1C11 1JRV

-57.2 -63.1 -65.6 -61.2 -58.4 -61.6 -56.9 -58.8 -63.2 -59.5 -60.5 -56.3 -61.8 -59.3 -57.6 -56.0 -60.4 -63.5 -61.7 -58.4 -62.6 -62.6 -56.6

10.3 14.7 7.0 2.1 7.7 -0.5 2.8 10.5 4.0 5.8 -1.6 8.0 3.7 0.7 -2.1 5.6 11.0 7.9 2.1 9.4 9.4 -1.1

57.1 62.9 65.4 61.1 58.2 61.5 56.7 58.6 63.0 59.3 60.4 56.1 61.7 59.2 57.4 55.9 60.3 63.4 61.5 58.2 62.6 62.5 56.5

10.2 14.5 7.0 1.9 7.7 -0.7 2.6 10.3 3.9 5.8 -1.8 8.1 3.7 0.5 -2.1 5.6 11.0 7.7 1.9 9.6 9.5 -1.1

a δcsa and δcsa represent the induced chemical shifts calculated min max using the molecular alignment tensor of 1KR8 according to eqs 18 and 19, respectively, for the two orientations of the CSA and molecular csa alignment tensor shown in Figure 3. ∆csa min and ∆max are the deviations csa csa of δmin and δmax calculated for a particular phosphate group conforcsa mation from δcsa min and δmax obtained using BDEP, respectively.

TABLE 5: Percentage Differences between the CSA Order Parameters Obtained Using the Theoretical 31P CSA Tensors and the Experimental 31P CSA Tensor of Barium Diethyl Phosphate, Respectivelya

a∆S2 c

conf. class

∆S2c /%

8 10 17 15 1 3 11 20 (canonical RNA) 19 22 30 31 BI-DNA BII-DNA A-DNA (-gt) A-DNA (-g-g) ZII-DNA(CG) ZII-DNA (GC) ZI-DNA (GC) 1PQT 1C11 1JRV

17.6 24.8 12.9 6.7 17.8 0.0 5.4 19.1 9.5 10.6 -2.0 13.7 6.0 5.2 -2.6 10.4 19.0 18.1 7.4 14.8 15.1 -4.5

2 2 ) 100(Sc,BDEP - S2c )/Sc,BDEP . (eq 14).

Conclusions Within the set of studied conformations, our results for the chemical shift anisotropy tensors reveal an interesting and useful distinction in the sensitivity of δiso and the individual CSA components to the variations of torsion angles. The isotropic chemical shifts and the anisotropic component along the nucleic acid backbone are mainly influenced by the torsion angles R

31P

Chemical Shift Tensors

and ζ. Consequently, different ranges of δiso and δCSA 11 values were found for the gg- and gt-conformers, respectively. Increased shielding (a decreased chemical shift) in the ggconformation is in accord with the largest, experimentally observed difference of the chemical shifts in the Z-DNA46 where the 31P resonance in the CG step (+g+g) is shifted by -2.06 ppm with respect to the GC step (+gt). Similar trends have been traced also in other structures for which the 31P chemical shifts were deposited in the BioMagResBank (http://www.bmrb.wisc.edu). For most conformations, the anisotropic components perpendicular to the backbone also fall to distinct ranges corresponding to gg- and gt-conformational classes. However, CSA there are several conformations, for which δCSA 22 or δ33 value does not fit the interval of the corresponding conformation type. The exceptions are presumably caused by the larger sensitivity of the perpendicular components to the more distant torsion angles β and . The total span of CSA tensor component values CSA CSA is larger for δCSA 22 and δ33 (21 and 22 ppm) than for δ11 (16 ppm). The isotropic chemical shift and the chemical shift anisotropy range within ∼6 and ∼32 ppm, respectively. For the choice of the Cartesian coordinate system such that the origin is on phosphorus, y axis bisects the O1-P-O2 angle, z axis lies in the O1-P-O2 plane, and x axis is perpendicular to y and z; the angles φx, φy, and φz between the Cartesian and principal axes range between 0° and 9°. The angles larger than 5° were found for gt-conformers. In case that the molecular alignment tensor is either axially symmetric or fully asymmetric, the boundary values of the δcsa interval depend only on the Azz component of the molecular and δCSA (i.e., δCSA and alignment tensor and on the δCSA yy zz 11 CSA δ33 ) components of the traceless chemical shift tensor. Thus, and δCSA for a given structure (i.e., a given Azz), the δCSA yy zz csa components determine the maximum span of δ values. To assess the effect of using a uniform CSA tensor, we have calculated the smallest and largest possible values of the induced csa chemical shift, δcsa min and δmax, for the set of conformation classes as well as for barium diethyl phosphate. The calculations were carried out for an example of an almost axially symmetric and almost fully asymmetric molecular alignment tensor. It was csa csa shown that maximum deviations of δcsa min and δmax from δmin csa (BDEP) and δmax (BDEP), respectively, range between 10 and 16.5%. The chemical shift asymmetry η and chemical shift anisotropy ∆σ of the calculated 31P CSA tensors, and of the experimental 31P CSA tensor for barium diethyl phosphate were used to calculate the percentage differences ∆S2c between the CSA 2 , respectively. The maximum order parameters S2c and Sc,BDEP 2 value of ∆Sc was as large as 25% and nonnegligible errors of 10-18% were also obtained for half of the conformations studied. The results of our study show that the variations of the 31P CSA tensor component values for a wide range of arrangements of the sugar-phosphate backbone in DNA and RNA oligonucleotides are large enough to influence the structure refinement using data obtained in partially aligned media. In addition, the variations considerably affect the CSA order parameters derived from experimental 31P relaxation data and thus have an impact on the analysis of backbone dynamics. Acknowledgment. This work was supported by Grants MSM0021622413 to M.M. and LC06030 to V.S. from the Ministry of Education, Youth and Sports of the Czech Republic, and by Grant 204/03/H016 from the Science Foundation of the Czech Republic to J.P. and by FSG-V-RNA project of the Sixth

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