Asymmetry of 13C labeled 3-pyruvate affords improved site specific labeling of RNA for NMR spectroscopy
Selective isotopic labeling provides an unparalleled window within which to study the structure and dynamics of RNAs by high resolution NMR spectroscopy. Unlike commonly used carbon sources, the asymmetry of 13C-labeled pyruvate provides selective labeling in both the ribose and base moieties of nucleotides using Escherichia coli variants, that until now were not feasible. Here we show that an E. coli mutant strain that lacks succinate and malate dehydrogenases (DL323) and grown on [3-13C]-pyruvate affords ribonucleotides with site specific labeling at C5′ (~95%) and C1′ (~42%) and minimal enrichment elsewhere in the ribose ring. Enrichment is also achieved at purine C2 and C8 (~95%) and pyrimidine C5 (~100%) positions with minimal labeling at pyrimidine C6 and purine C5 positions. These labeling patterns contrast with those obtained with DL323 E. coli grown on [1, 3-13C]-glycerol for which the ribose ring is labeled in all but the C4′ carbon position, leading to multiplet splitting of the C1′, C2′ and C3′ carbon atoms. The usefulness of these labeling patterns is demonstrated with a 27-nt RNA fragment derived from the 30S ribosomal subunit. Removal of the strong magnetic coupling within the ribose and base leads to increased sensitivity, substantial simplification of NMR spectra, and more precise and accurate dynamic parameters derived from NMR relaxation measurements. Thus these new labels offer valuable probes for characterizing the structure and dynamics of RNA that were previously limited by the constraint of uniformly labeled nucleotides.
KeywordsSite selective labeling DL323 Pyruvate-3 Ribose and nucleobase RNA Structure and dynamics
Glucose-6-phosphate dehydrogenase mutant
Non-oxidative pentose phosphate pathway
Oxidative pentose phosphate pathway
RNA is increasingly viewed not only as an intermediary between DNA and protein, but also as a key regulator of gene expression and catalysis (Boisvert et al. 2007; Korostelev et al. 2008; Steitz 2008; Breaker 2009; Wahl et al. 2009; Newman and Nagai 2010). Methods to obtain very high-resolution structures and site specific dynamics of RNA molecules are therefore critical. Nuclear magnetic resonance (NMR) spectroscopy has emerged as an effective method for solving RNA 3D structures and for probing their motions. However, RNA molecules that are bigger than 40 nucleotides are difficult to characterize using NMR spectroscopy because of severe chemical shift overlap and rapid signal decay. Selective labeling techniques, first introduced to extend the utility of NMR spectroscopy for protein analysis, are increasingly being applied to RNA as well (Dayie 2008; Lu et al. 2010).
Of the three primary methods for incorporating selective isotopic labels into nucleic acids, biomass production using E. coli variants (Johnson et al. 2006; Johnson and Hoogstraten 2008; Dayie and Thakur 2010; Thakur et al. 2010a, b) appears to be more general and cost-effective than chemical synthesis (Milecki 2002) or de novo biosynthesis (Schultheisz et al. 2008, 2011). Selective introduction of 13C isotopes into the ribose ring using chemical synthesis entails difficult regio- and stereo-selective synthesis with low yields (Milecki 2002). The de novo biosynthetic method requires a large number of enzymes of which only a few are commercially available (Schultheisz et al. 2008, 2011). Additionally, the de novo method requires prohibitively expensive precursors that are limited in the labeling pattern that can be achieved with the biomass methods.
Unfortunately a biomass method based on E. coli metabolism, until recently, suffered from the disadvantage of isotopic scrambling in both the ribose and nucleobase atoms. This scrambling is caused by the admixture of the metabolic flux from both the oxidative and non-oxidative pentose phosphate pathways (noPPP) and a very efficient TCA cycle (Johnson et al. 2006; Johnson and Hoogstraten 2008). Hoffman and Holland earlier showed that wild-type E. coli grown using selectively labeled acetate enables the incorporation of 13C-labels at either the protonated or non-protonated carbon sites in the RNA ribose and nucleobases. LeMaster and coworkers also showed that a modified bacterial strain, in which the TCA cycle enzymes succinate dehydrogenase and malate dehydrogenase were disabled (DL323), could be grown on selectively labeled glycerol to provide alternative site labeling that removed the 13C–13C coupling in protein NMR spectra (LeMaster and Richards 1982; LeMaster and Kushlan 1996). Hoogstraten and coworkers then showed that the DL323 strain grown on selectively labeled glycerol also enabled site selective labeling within the ribose ring (Johnson et al. 2006; Johnson and Hoogstraten 2008). Earlier, Pardi and colleagues introduced labeling using 13C-formate in a background of 12C-glucose which site specifically labels purine C8 position exclusively; results were not presented for purine C2 position (Latham et al. 2005). Building on these previous ideas, we showed that an isolated two spin system could be achieved for both base and ribose depending on the strain and complementation of the growth media with formate (Dayie and Thakur 2010; Thakur et al. 2010a, b). Even under these newer conditions, the purine C2 positions were only labeled to ~26%, indicating a general limitation of the formate supplementation methodology (Dayie and Thakur 2010; Thakur et al. 2010a, b).
However until now, most biomass production of nucleotides had focused on using symmetric carbon sources such as glycerol, glucose, and acetate. For instance if glucose is the carbon source, then at the onset of the TCA cycle this input carbon source is already diluted to 50% by glycolysis and similarly the purine precursor 3-phosphoglycerate (3PG) is only 50% labeled. Similarly if [1, 3-13C]-glycerol is the carbon source, selectivity in the ribose and base regions are again reduced because of the symmetric nature of glycerol such that the purine C5 and all the ribose C1′ have substantial residual coupling that can render relaxation measurements inaccurate (Johnson et al. 2006; Dayie and Thakur 2010). This residual coupling arises from the C-1 carbon of three carbon precursors such as pyruvate and glycerol, and thus [1-13C]-pyruvate will similarly introduce coupling between C1′–C2′–C3′ and C2′–C3′. We, therefore, reasoned that the asymmetry of 13C-labeled pyruvate would provide selective labeling in both the ribose and base moieties of nucleotides that would be impossible with symmetric carbon sources.
Unlike the previous result with the use of [1, 3-13C]-glycerol (Johnson et al. 2006; Thakur et al. 2010a, b), the use of [3-13C]-pyruvate affords selective 13C enrichment of the ribose C1′ and C5′, of the purine C2 and C8, and of the pyrimidine C5 without any of the residual couplings described above. NMR T1 relaxation experiments and 13C-methylene TROSY or HSQC on a site selectively 13C-labeled 27-nt A-site RNA demonstrate that accurate T1 relaxation parameters and higher resolution spectra can be obtained. Thus, the technology to incorporate site specific labels into both the base and ribose moieties of purine and pyrimidine nucleotides promises to expand the scope of NMR studies to regions hitherto inaccessible.
Materials and methods
The mutant strains DL323 (CGSC # 7538, F-, sdh-1, &lambda-, mdh-2, rph-1) (Hansen and Juni 1979; LeMaster and Kushlan 1996) and K10-15-16 (CGSC # 4858 Hfr fhuA22, zwf-2, relA1, T2R, pfk-10; Fraenkel 1968; referred to here as K10zwf) and the wild-type K12 strain (Clowes 1968; Soupene et al. 2003; CGSC # 4401:F+) used in this work were obtained from the Yale Coli Genetic Stock Center (CGSC). Dr. Paliy kindly provided the wild-type K12 NCM3722 (Soupene et al. 2003).
Isotopic labeled compounds were purchased from Cambridge Isotope Laboratory (Andover, MA) and Isotec-Sigma-Aldrich (Miamisburg, OH): [3-13C]-pyruvate (99%), 15N-(NH4)2SO4 (99%).
Media for E. coli growth
Luria–Bertani (LB) media was prepared as described (Sambrook and Russell 2001) and LeMaster-Richards (LMR) minimal media was prepared as described (LeMaster and Richards 1982; Dayie and Thakur 2010; Thakur et al. 2010a, b). The LMR media contains 176 mM KH2PO4, 25 mM NaOH, 10 μl H2SO4, 12.6 mM (NH4)2SO4, 2 mM MgSO4, 10 μM FeSO4 and 0.2% trace metals, supplemented with the [3-13C]-pyruvate and 15N–(NH4)2SO4 enriched nitrogen source.
Method for E. coli growth optimization
The growth evaluation of the E. coli mutant (DL323) was performed as described previously (Thakur et al. 2010a, b). Briefly, a 5 ml starter culture in LeMaster media supplemented with unlabeled carbon sources was inoculated from a single colony of DL323 E. coli grown on LB agar plates at 37°C overnight (~16 h). The overnight culture was pelleted, cells were washed twice in 1× phosphate-buffered saline (PBS), and re-suspended in fresh 5 ml of LMR medium; 1 ml of this re-suspension was added to a 50 ml culture in LMR medium for another overnight growth at 37°C. The overnight culture was pelleted prior to complete saturation of these cells, the cells were washed twice in 1×PBS, resuspended in a 50 ml of LMR medium with no carbon supplements, and then 5 ml from this resuspension was added to a 500 ml LMR medium supplemented with [3-13C]-pyruvate and 15N–(NH4)2SO4, and the solution incubated at 37°C.
P1 nuclease digestion, nucleotide separation and purification
The ribonucleoside monophosphates (rNMPs) were isolated from E. coli (DL323) mutant culture as described earlier (Batey et al. 1992; Thakur et al. 2010a, b). The nucleic acid mixture was digested with P1 nuclease and separated into individual ribo- or deoxy-monophosphates using a cis-diol boronate affinity column chromatography as described (Batey et al. 1992; Thakur et al. 2010a, b). The site specific labeling pattern of rNMPs was verified by NMR prior to re-phosphorylation to rNTPs.
Enzymatic phosphorylation of rNMPs
The detailed procedure for enzymatic phosphorylation of rNMP was described previously (Nyholm et al. 1995; Thakur et al. 2010a, b). Briefly, the individual rAMP, rCMP, rGMP or rUMP and kinases were freshly prepared and pre-incubated at 37°C for 10 min prior to mixing. The enzymatic conversion of the labeled rNMPs to rNTP was complete in 5 h. The progression of the phosphorylation reaction was monitored on a TARGA C18 column using conditions similar to those in the purification step (Supplementary Figure S1). Collected rNTP fractions were pooled and dialyzed overnight and lyophilized.
Site specific enrichment of A-Site RNA and in vitro transcription
A-Site RNA (5′-GGC GUC ACA CCU UCG GGU GAA GUC GCC-3′) was synthesized using in vitro transcription (Milligan et al. 1987; Milligan and Uhlenbeck 1989), with a His-tagged mutant (P266L) T7 RNA polymerase (Guillerez et al. 2005) using a mixture of uniformly labeled guanine and cytosine triphosphates (rGTP and rCTP) (Sigma Aldrich) or site specific labeled rGTP and rCTP. Another set of samples were made using a mixture of uniformly labeled adenosine triphosphates (rATP) (Sigma Aldrich) or site specific labeled rATP. Two terminal 2′-O-methyl modifications in the template strand were introduced to substantially reduce the amount of transcripts with extra nucleotides at the 3′-end (Kao et al. 1999). The optimal transcription conditions were as follows: 10 mM total NTP and 15 mM Mg2+. The reactions were carried out in a transcription buffer A (40 mM Tris–HCl, pH 8.1, 1 mM spermidine, 5 mM dithiothreitol (DTT), 0.01% Triton X-100, 80 mg/ml PEG 8000), 300 nM each DNA strand, and 1.5 μl of 2.4 mg/ml T7 RNAP (optimized amount) per 40 μl of transcription volume. After 3 h of incubation at 310 K, the RNA from the transcription reactions were purified and dialyzed as described (Dayie 2005). After dialysis, the RNA was lyophilized, and resuspended into NMR buffer (100 mM KCl, 10 mM potassium phosphate pH 6.2, 8 or 100% D2O), with or without 3 mM MgCl2, and a trace of sodium azide). The concentration of each NMR sample in Shigemi tubes in 250 μl of buffer was ~0.1 mM.
All NMR experiments were run on a four channel Bruker Avance III 600 MHz spectrometer equipped with actively shielded z-axis gradient triple resonance probe. The experiments were conducted at temperatures ranging from 15° to 45°C. The NMR data sets were processed and the peak positions and intensities were analyzed with Bruker’s TOPSPIN 2.1 as described previously (Dayie and Thakur 2010; Thakur et al. 2010a, b). One dimensional (1D) 13C spectra and two-dimensional non-constant-time (1H, 13C) heteronuclear single quantum correlation (HSQC) spectra (Bodenhausen and Ruben 1980; Bax et al. 1990) were acquired to analyze the rNMP fractions extracted from each bacterial strain. The fractional 13C enrichment at each carbon site was quantified directly by 1D proton methods or indirectly using 2-bond (2JHN) HSQC as described previously (Dayie and Thakur 2010; Thakur et al. 2010a, b).
To demonstrate the usefulness of site selective labeling 2D 1H–13CH2 TROSY (Meissner et al. 1998; Miclet et al. 2004) spectra were measured using non-constant time evolution in the carbon dimension. Methylene CH2-TROSY experiments (Miclet et al. 2004) were run with the following modifications that dispense with selective decoupling: the WURST-4 decoupling waveform was omitted during the carbon t 1 evolution period, and a non-selective 180° pulse replaced the selective pulse on the C5′ carbon. To compare the usefulness of the modified CH2-TROSY experiments, the normal CH2-methylene optimized HSQC (Schleucher et al. 1994; Sattler et al. 1995) were also run under identical conditions: for each data set, 16 scans and 1,024 complex points were collected in t 2 and 512 complex points were collected in t 1 using the Echo-Anti echo method (Palmer et al. 1991; Kay et al. 1992) for quadrature detection; proton and carbon carrier was placed at 4.7 and 64 ppm respectively as described (Thakur et al. 2010a, b). The time domain data sets were zero filled in t 1 and t 2 before Fourier transformation to give a final real matrix size of 2,048 × 1,024 points.
Longitudinal (R1) relaxation rates were measured for ribose C1′ and cytosine nucleobase C5 carbons using TROSY detected experiments (Hansen and Al-Hashimi 2007). To compare the usefulness of the selective labels relative to uniform labels, the R1 relaxation experiments were run under identical conditions: for each data set, 64 scans and 1,024 complex points were collected in t 2, and 128 complex points were collected in t 1 using the Echo-Anti echo method (Palmer et al. 1991; Kay et al. 1992) for quadrature detection. Proton and carbon carrier was placed at 4.7 and 93 ppm respectively. The time domain data sets were zero filled in t 1 and t 2 before Fourier transformation to give a final real matrix size of 2,048 × 1,024 points. For the 2D longitudinal (R1) experiments, relaxation delays of 21.1 (2x), 61.4, 141.1, 301.2, 381.3, 461.3 (2x), 781.6 ms were used. The longitudinal relaxation decay curves were fitted using the Levenberg–Marquardt algorithm by assuming monoexponential decay.
Incorporation of13C into ribose and base of nucleotides
13C enrichment levels at various carbon positions within ribonucleotides harvested from K12, DL323, and K10zwf E. coli strains grown on [3-13C]-pyruvate
Carbon position labeled
DL323 E. coli strain
K10zwf E. coli strain
K12 E. coli strain
99 ± 1
85 ± 4
95 ± 2
93 ± 1
57 ± 1
98 ± 2
98 ± 4
81 ± 4
91 ± 1
15 ± 1
60 ± 4
64 ± 1
42 ± 4
82 ± 2
83 ± 2
9 ± 1
21 ± 1
9 ± 1
30 ± 1
95 ± 5
95 ± 1
Similar to DL323, K10zwf E. coli grown on [3-13C]-pyruvate leads to exclusive labeling of the C1′ and C5′ carbon position without 13C–13C coupling evident (Fig. 2). Importantly, in this case the abundance of 13C label at C1′ is higher than that obtained with DL323 (Table 1). Again similar to DL323 and K10zwf E. coli, wildtype K12 grown on [3-13C]-pyruvate also leads to labeling of the C1′ and C5′ carbon positions, but unlike the other two strains, sufficient residual labeling of the C2′ and C3′ carbon atoms leads to 13C–13C coupling (Figs. 1, 2). This residual labeling presumably arises from PEP derived from the mixture of [1, 2, 3-13C]-oxaloacetate and [2, 3, 4-13C]-oxaloacetate intermediates (Fig. 1).
As expected, all of the C2 and C8 atoms of purine are substantially labeled to >90% for nucleotides derived from DL323 and K12 cultures, but those from K10zwf have these labels reduced to 85 and 57% respectively partly because the scrambling from the TCA cycle apparently feeds more efficiently into the purine biosynthetic pathway for K10zwf than K12 (Figs. 2, 3; Supplementary Figure S2). The origin of these differences is not clear. Nonetheless, some of the scrambled malate or oxaloacetate from the TCA cycle is likely fed back into PEP such that purine C4, purine C5, and pyrimidine C6 positions are labeled in K10zwf and K12 but not DL323 (Fig. 3). As a result, K10zwf and K12 provide 13C–13C spin pairs in the purine and pyrimidine base moieties that can potentially hinder accurate extraction of relaxation parameters (shown below).
Applications of site selective labels for 13C NMR study of nucleic acids
Use of the site selectively labeled NTPs removes the crowding due to the splitting of signals from J-coupling and eliminates the C2′, C3′, and C4′ resonances while preserving only the C1′ and C5′ signals (Fig. 4). In addition to improving resolution, sensitivity is also greatly enhanced. For instance, cytosine C5 has a large chemical shielding anisotropy (CSA) and so decays rapidly, and as expected most of the C5 resonances are absent in the spectra of the uniformly labeled sample. These signals are preserved in the spectra of the site selectively labeled sample (Fig. 4).
In the C8 purine and C6 pyrimidine regions, the selective labels once again improve the resolution and sensitivity (Fig. 5). By eliminating the C6 pyrimidine signals, the guanosine C8 signals can be readily identified without concerns about potential overlap with cytosine C6, especially for a sample that is labeled with all four nucleotides. These observations underscore the difference selective labeling makes on sensitivity and resolution.
Site selective labeling affords more accurate relaxation rate measurements using non-constant time non-selective pulse experiments
Here X = C4 or C6. The physical constants γH and γC are the gyromagnetic ratios of 1H and 13C respectively (γC = 6.7283 × 107 rad (Ts)−1, γH = 26.75 × 107 rad (Ts)−1), rHC is the length of the internuclear 1H–13C bond vector, rCC is the length of the internuclear 13C–13C bond vector, and \( \hbar \) is Planck’s constant divided by 2π (1.054592 × 10−27). An asymmetric chemical shielding tensor can be decomposed into an isotropic and two axially symmetric anisotropic terms with the symmetry axes arbitrarily chosen as the two principal axes along X and Y, as discussed previously for carbonyl backbone motional analysis (Dayie and Wagner 1997). More general expressions for anisotropic motion can be found in the literature (Werbelow 1994). Because each term J(ωX − ωC) in (3) and (5) is proportional to J(0) which scales linearly with molecular weight, it is clear that the dipole–dipole contribution cannot be neglected for uniformly labeled RNA samples.
Previous labeling technologies using wild type E. coli strains and symmetric carbon sources spurred the development of new solution NMR methods to characterize the structure and dynamics of small to medium sized uniformly labeled RNA molecules (Batey et al. 1995; Hall 1995; Fürtig et al. 2003; Latham et al. 2005; Dayie 2008; Lu et al. 2010). A number of alternative labeling approaches have been proposed since then, and it is important therefore to discuss the advantages and limitations of these different methods. First, uniform labeling introduces direct one-bond and residual dipolar couplings that prevents accurate measurement of 13C relaxation rates (Johnson et al. 2006; Dayie and Thakur 2010; Thakur et al. 2010a, b) and decreases the resolution and sensitivity of NMR experiments. For instance, growth of wildtype E. coli strain K12 on [1-13C]-acetate yields 13C label at the ribose C1′, C2′, and C3′ atomic sites but very little label at C4′ and C5′ (Hoffman and Holland 1995). As a result, C2′ and C3′ have substantial multiplet structure and even C1′ has residual multiplet structure (Supplementary Figure S3). In addition, the protonated aromatic carbons (purine C2 and C8 and pyrimidine C5 and C6) are not labeled, making this particular labeling pattern less useful for routine NMR applications. In contrast, growth of wildtype E. coli strain K12 on [2-13C]-acetate yields 13C label at all five ribose carbon positions of C1′, C2′, C3′, C4′ and C5′ (Hoffman and Holland 1995). Since each site is highly enriched, all five positions have substantial multiplet structure (Supplementary Figure S3). Also for the aromatic carbons, the protonated carbon C5 and C6 of pyrimidines and C2 and C8 of purines are labeled at very high level (Hoffman and Holland 1995; Supplementary Figure S3). Similarly, growth of wildtype E. coli strain K12 on [2-13C]-glucose yields 13C label only at C1′, C2′, and C4′. No label is observed at C3′ and C5′. For the aromatic carbons, the protonated carbon C5 and C6 of pyrimidine and C2 and C8 of purine are also labeled at very high level (Hoffman and Holland 1995). These second set of labeling patterns are therefore very useful for NMR applications that require substantial coupling for transfer of coherences, but problematic for applications that require suppression of these coupling networks (Supplementary Figure S3 E).
To remove these coupling networks, alternate 13C–12C labeling schemes have been shown to be important for tackling these difficulties in RNA molecules (Johnson et al. 2006; Johnson and Hoogstraten 2008; Dayie and Thakur 2010; Thakur et al. 2010a, b). However some of the earlier schemes had their unique problems. For instance, the labeling pattern of rNTPs derived from DL323 grown on [1,3-13C]-glycerol is not attractive for relaxation studies because the ribose C2′ position is doublet, the C1′ retains some residual doublet arising from 13C2′–13C1′ isotopomers, and the C4′ has no label (Thakur et al. 2010a, b). In the base region the pyrimidine C5 site has multiplet structure, again precluding its use for accurate relaxation measurements (Thakur et al. 2010a, b). The use of G6PDH mutant strains rather than DL323 using 2-13C-glycerol was also recommended in previous work (Johnson et al. 2006).
Recently de novo biosynthetic methods have been proposed that potentially resolve these problems (Schultheisz et al. 2008, 2011). However, the new method requires 29 enzymes for making ATP and GTP and 18 enzymes for making CTP and UTP. Of these enzymes only a few are commercially available, which makes this method excellent for adaptation by companies but not by individual laboratories. Additionally, the de novo method requires selectively labeled glucose and serine which are not readily available commercially, or prohibitively expensive, or both. For instance the de novo method of labeling ribose C1′ and C5′ in the context of purine C2, C5, and C8, indicated above by our method, requires 13C-2,6-glucose that is not commercially available and 13C-3-serine that costs ~$4,000/g; the biomass method is able to make these labels readily.
Of greater interest the main advantage of using the E. coli strain DL323 grown on a non-symmetric carbon source such as [3-13C]-pyruvate is that it provides isolated two-spin systems that are ideal for relaxation measurements as well as improving resolution and sensitivity. For example in the ribose region, the C1′ and C5′ labels are completely isolated from C2′ and C4′ spins respectively. Similarly in the base region, the purine C2 and C8 and pyrimidine C5 spins are isolated from the network of carbon spins within the aromatic ring. For instance in a uniformly labeled sample, adenine C2 has substantial 2JC2C5 coupling of 11 Hz to the C5 carbon, and the purine C8 carbons have 2JC4C8 coupling of ~9 Hz to the C5 carbon (Wijmenga and van Buuren 1998). In addition, the carbon atoms C4 (149–154 ppm)/C5 (116–120 ppm)/C6 (156–161 ppm)/C8 (131–142 ppm) resonate within a chemical shift range that precludes effectively decoupling C8 from the C4, C5, or C6 carbon atoms. These complications again prevent accurate relaxation measurements using CPMG experiments. Formate supplementation has been proposed as a potential cost-effective solution (Latham et al. 2005; Dayie and Thakur 2010; Thakur et al. 2010a, b). However, the C8 is labeled to ~90% and C2 to only ~25%, limiting the general usefulness of formate labeling. Thus the confounding effects of scalar coupling from adjacent labeled sites are eliminated using the current method proposed in this work, unlike the case for previous labels.
Any RNA sequence can be site specifically labeled with site alternate 13C-enriched isotopes efficiently prepared from site-specifically labeled rNTPs using in vitro transcription by T7 RNA polymerase. We have presented an efficient method for preparing these ribonucleotides containing site specific distributions of 13C enrichment that has significant advantages in terms of cost, flexibility, resolution and sensitivity over previously reported procedures. These labels will be particularly beneficial to investigators using heteronuclear NMR spectroscopy to study the structure and dynamics of RNA and RNA complexes of increasingly large sizes, currently an important and active area of research.
This work was supported in part by the University of Maryland Nano-Biotechnology Award and the National Institutes of Health grant GM077326 to T.K.D. We are grateful to Professors Jon Dinman and Steve Rokita for helpful discussions.
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