Interaction of cis- and trans-RuCl2(DMSO)4 with the nucleotides GpA, d(GpA), ApG, d(ApG) and d(CCTGGTCC): high-field NMR characterization of the reaction products

 Both cis- and trans-RuCl₂(DMSO)₄ (cis-Ru and trans-Ru) react with ApG, GpA, d(ApG) and d(GpA) to yield products with bifunctional metal coordination of the bases. For each dinucleotide one major product and several minor species are formed. This is in contrast to previous results on analogous reactions between trans-Ru and d(GpG) where a substantial amount of an intermediate species was found. The rates of reaction between dinucleotides and cis-Ru are approximately 20-fold slower than for trans-Ru. The compounds formed with the two isomers exhibit identical proton NMR spectra, suggesting the same coordination mode for ruthenium in the final product. The two purine bases are coordinated to ruthenium through N7 in a head-to-head conformation with the glycosidic angles being in the anti range. Coupling constants indicate a relatively pure 3′-endo conformation for the 5′-sugar and mainly 2′-endo for the 3′-sugar. The similar bifunctional binding mode of cis- and trans-Ru(II) with dinucleotides as evident from the NMR spectra are in contrast to the different mode of interaction suggested earlier for cis- and trans-Ru complexes with DNA. trans-Ru interacts with the deoxyoctanucleotide d(CCTGGTCC), giving two main products during the first 2 h of incubation time. Four H8 guanine resonances are shifted downfield, characteristic of N7 metal coordination. The products are not analyzed in detail, but it is suggested that the structures may be described as two chiral G(N7/N7) chelates. Received: 20 August 1998 / Accepted: 20 January 1999     

Sugar interaction with biologically active cis-PtCl2(NH3)2 (anticancer) and its inactive trans isomer

The interaction of D-glucuronic and D-gluconic acids with cis- and trans-PtCl2(NH3)2 (cisplatin and transplatin) has been investigated in aqueous solution and solid complexes of the type cis-[PtL(NH3)2]L.H2O and trans-[PtL2(NH3)2]L.H2O, where L = D-glucuronate or D-gluconate anions, are isolated and characterized by means of Fourier transform-infrared and 1H-NMR spectroscopy, and molar conductivity and X-ray powder diffraction measurements. Spectroscopic and other evidence indicated that the sugar anions bind monodentately in trans-[PtL2(NH3)2].H2O and bidentately in cis-[PtL(NH3)2]L.H2O complexes through the carboxylate oxygen atoms and other sugar donor groups. The strong sugar intermolecular hydrogen-bonding network is altered to that of the sugar-OH...NH3(H2O)...OH-sugar, upon platinum-ammine interaction. The D-glucuronate anion has the beta-anomer configuration both in the free salt and in these platinum-sugar complexes.     

Specific platinum chelation by the guanines of the deoxyhexanucleotide d(T-G-G-C-C-A) upon reaction with cis-[Pt(NH3)2(H2O)2](NO3)2

The stoichiometric reaction between d-TpGpGpCpCpA (d(T-G-G-C-C-A)) and $\text{cis}$-[Pt(NH₃)₂(H₂O)₂](NO₃)₂ (8.4 × 10^?6 to 1.3 × 10^?4M in water at pH 5.5–6) gives a single complex. High pressure gel permeation chromatography and pH-dependent ¹H NMR analyses of the nonexchangeable base protons, show that it is a platinum chelate with the $\text{cis}$-Pt^II(NH₃)₂ moiety bound to the two N7 atoms of the adjacent guanines. A 3 × 10^?3M reaction gives the same platinum chelate, via the formation of intermediate complexes, together with unsoluble adducts.     

Synthesis and characterization of a d(ApG) platinated nonanucleotide duplex.

The nonamer 5'd(CTCAGCCTC) 3' 1 has been reacted with cis-diamminediaquaplatinum(II) in water at pH 4.2. The major reaction product was shown by enzymatic digestion and 1H NMR to be the d(ApG)cis-Pt(NH3)2 chelate [cis-Pt(NH3)2[d(CTCAGCCTC)-N7(4),N7(5)]] 1-Pt. When mixed with its complementary strand 2, 1-Pt forms a B DNA type duplex 3-Pt with a Tm of 35 degrees C (versus 58 degrees C for the unplatinated duplex). The NMR study of the exchangeable protons of 3-Pt revealed that the helix distortion is localized on the CA*G*-CTG moiety (the asterisks indicating the platinum chelation sites) with a strong perturbation of the A*(4)T(15) base pair related to a large tilt of A*(4).     

Structural characterizations of some adducts of silver(I) nitrate and perchlorate with some cyclic di- (or tri-) ene organic ligands

Single crystal X-ray structural characterizations of some adducts of silver(I) nitrate and perchlorate with assorted organic poly-ene ligands (nbd = norbornadiene, bicyclo[2.2.1]hepta-2,5-diene; cod = 1,5-cyclooctadiene; cdt ≡ trans,trans,cis-cyclododeca-1,5,9-triene) are reported, all being polymeric in form (with the exception of mononuclear ionic AgClO₄:cod (1:2)), with chains comprised of alternating silver and nitrate/perchlorate components substituted or linked by unsaturated donors which complete the coordination spheres of the silver atoms. In AgNO₃:nbd (2:1) (a redetermination), pairs of silver/nitrate strands are linked in a one-dimensional polymer by the nbd ligands. In AgNO₃:nbd (1:1)_∞, meandering silver/nitrate strands containing pairs of independent silver and nitrate units in a crystallographic mirror plane are linked to either side with parallel planes by nbd ligands. In AgNO₃:cod (1:1)_∞, the cod ligands ‘chelate’ to the silver atoms in a silver/nitrate chain. In AgNO₃:cdt (1:2)_∞, pairs of ‘unidentate’ cdt ligands are pendant from a silver/nitrate chain, while in the (1:1)_∞ adduct, the cdt ligands bridge pairs of silver atoms from an adjacent chain forming a two-dimensional web. A common form of the bridging nitrate group in the above is as an O,O′-NO₃-O′,O″ bis-chelate, the pair of the bis-oxygen chelates having a common oxygen atom.     

Synthesis and characterization of trans-[Pt(NH3)2Cl2] adducts of d(CCTCGAGTCTCC).d(GGAGACTCGAGG)

The reaction of trans-diamminedichloroplatinum(II) (trans-DDP), the inactive isomer of the anticancer drug cisplatin, with the single-stranded deoxydodecanucleotide d(CCTCGAGTCTCC) in aqueous solution at 37 degrees C was monitored by reversed-phase HPLC. Consumption of the dodecamer follows pseudo-first-order reaction kinetics with a rate constant of 1.25 (4) x 10(-4) s-1. Two intermediates, shown to be monofunctional adducts in which Pt is coordinated to the guanine N7 positions, were trapped with NH4(HCO3) and identified by enzymatic degradation analysis. These monofunctional adducts and a third, less abundant, one are rapidly removed from the DNA by thiourea under mild conditions. When allowed to react further, the monofunctional intermediates formed a single main product that was characterized by 1H NMR spectroscopy and enzymatic digestion as the bifunctional 1,3-intrastrand cross-link trans-[Pt(NH3)2[d(CCTCGAGTCTCC)-N7-G(5),N7-G(7]]). Binding of the trans-[Pt(NH3)2]2+ moiety to the guanosine N7 positions decreases the pKa at N1 and leads to destacking of the intervening A(6) base. The double-stranded trans-DDP-modified and unmodified DNAs were obtained by annealing the complementary strand to the corresponding single strands and then studied by 31P and 1H NMR and UV spectroscopy. trans-DDP binding does not induce large changes in the O-P-O bond or torsional angles of the phosphodiester linkages in the duplex, nor does it significantly alter the UV melting temperature. trans-DDP binding does, however, cause the imino protons of the platinated duplex to exchange rapidly with solvent by 50 degrees C, a phenomenon that occurs at 65 degrees C for the unmodified duplex. A structural model for the platinated double-stranded oligonucleotide was generated through molecular dynamics calculations. This model reveals that the trans-DDP bifunctional adduct can be accommodated within the double helix with minimal distortion of the O-P-O angles and only local disruption of base pairing and destacking of the platinated bases. The model also predicts hydrogen bond formation involving coordinated ammine ligands that bridge the two strands.     

The reaction of Pt-antitumor drugs with selected nucleophiles. I. The reaction of cis-[Pt(NH3)2Cl2] with glycine

Various Pt(II)-glycine coordination compounds were characterized by 1H and 13C NMR spectroscopy, some of them also by electrophoretic and chromatographic behavior. The results were applied to the analysis of the reaction mixtures of cis-[Pt(NH3)2Cl2] and glycine obtained under various conditions. Cis-[Pt(NH3)2Cl2] reacts with glycine to give cis-diammine-(glycine-N,O)-Pt(II) and cis-diammine-bis(glycine-N-)Pt(II). Their ratio depends primarily on the pH of the reaction medium. Conformation of these compounds is discussed based on the observed Pt-C and Pt-H NMR coupling constants.     

Spectrophotometric studies on complexes formed between tri-, di- and monobutyltin(IV) chlorides and 3-hydroxyflavone in different solvents

The complex formation of tri-, di- and monobutyltin(IV) chloride with 3-hydroxyflavone was studied spectrophotometrically in methanol, chloroform and toluene (the last two after extraction with buffer pH 9.0) by the molar ratio method. The composition of the chelates formed was found to be solvent and molar ratio (C_M/C_L) dependent. The complex formation constants were calculated.     

A 1H and 31P NMR study of cis-Pt(NH3)2[d(CpGpG)-N7(2),N7(3)]. The influence of a 5'-terminal cytosine, on the structure of the cis-Pt(NH3)2[d(GpG)-N7,N7] intrastrand cross-link

Proton NMR studies at 300 MHz and 500 MHz have been carried out on the trinucleoside bisphosphate d(CpGpG) and on cis-Pt(NH3)2[d(CpGpG)-N7(2),N7(3)] [abbreviated as d(CpGpGp) . cisPt]. For the Pt adduct, 13C and 31P NMR was also used for characterizing the oligonucleotide. d(CpGpG) appears to revert to a B-DNA-type single helix at lower temperatures. The relatively small concentration dependence of the proton chemical shifts, in comparison with shifts due to intramolecular stacking effects, indicates that the compound is essentially single-stranded. In d(CpGpGp) . cisPt, the first nucleoside, C(1), stacks well on top of the second, G(2), despite the N conformation of the G(2) sugar ring. The platinated GpG part in this trimer adopts largely the same structure as in cis-Pt(NH3)2[d(GpGpG)-N7(1),N7(2)] [den Hartog, J. H. J., et al. (1982) Nucleic Acids Res. 10, 4715-4730]. Main differences however, are changes in H8 chemical shifts and a 0.6-ppm downfield shift of the third nucleotide phosphorus, P(3), in d(CpGpGp) . cisPt with respect to P(2) in d(GpG) . cisPt. The latter shift change is likely to be induced by a structural alteration, caused by stacking of C(1) on top of G(2). Also, the large chemical shift differences between the two H8 protons in d(NpGpG) . cisPt fragments is discussed; the deviation from a mirror symmetry of the two guanine bases seems to be the main origin of this effect. The chemical shift changes, observed in the proton and phosphorus NMR chemical shift temperature and chemical shift pH profiles have been explained in terms of stack-destack equilibria changes.     

Lanthanide iodides as promoters of acetonitrile amination. Molecular structure of MeC(NH)NHPr, MeC(NH)NHBu and {Dy[MeC(NH)NEt2]6}I3

Amination of acetonitrile by the amines MeNH₂, PrⁿNH₂, PrⁱNH₂, Bu^tNH₂, and Et₂NH is efficiently promoted by the lanthanide iodides LnI₂ (Ln = Nd, Dy, Tm), LnI₃ (Ln = Pr, Nd, Dy) and LnI₃(THF)₃ (Ln = Pr, Nd, Dy). The formed mono- and N,N′-disubstituted amidines MeC(NH)NHR (R = Prⁱ, Bu^t), MeC(NH)NEt₂, MeC(NR)NHR (R = Me, Prⁿ) were isolated mainly as the complexes with starting iodide of general composition LnI₂(amidine)_x (1) or LnI₃(amidine)_x (2) (x = 3-8). In the products 1, which evidently are the mixtures of LnI²⁺, and LnI₃ derivatives, the metal exists in trivalent state but one of the ligands actually is amidinate anion. A part of the generated amidines remains in the reaction solutions in free form. Heating of the 1 and 2 in vacuum at 150-200 °C affords corresponding amidine and the complexes with reduced amount of the amidine ligands LnI₂(amidine)_y (3) or LnI₃(amidine)_y (4) (y = 2-3). The products 3 and 4 displayed the same catalytic activity in the acetonitrile-amine cross-coupling as the initial iodides. SmI₂ and especially YbI₂ revealed lower activity. The structure of isopropylacetamidine (5), tert-butylacetamidine (6) and {Dy[MeC(NH)NEt₂]₆}I₃(MeCN) (7) were determined by X-ray diffraction analysis.     

Preparative isolation of di-, tri- and tetrapyrimidine nucleotides from depurinated herring sperm DNA (author's transl)

The pyrimidine nucleotides p(dT)3, p(dT)2p, pdCp, p(dC)2p and p(dC)3 and the mixtures of sequence isomers p(dC, dT), p(dC, dT)p, p(dT2, dU), (dT2, dU)p, p(dC, dT2), p(dC, dT2)p, p(dC, dT3), p(dC, dT), p(dC2, dT)p, and p(dC2, dT2) are isolated on a preparative scale from depurinated hydrolysates of herring sperm DNA by the following procedure. The DNA hydrolysate is first separated into a low and a high molecular weight mixture of pyrimidine nucleotides by column chromatography on DEAE-cellulose. The nucleoside bisphosphates pdCp and pdTp, the dinucleotides p(dC)2 and p(dT)2 and the sequence isomers p(dC, dT) are largely separated out of the mixture of the low molecular weight pyrimidine nucleotides. The remaining mixture is rechromatographed at pH 3.5 on QAE-Sephadex. This separates the pyrimidine nucleotides containing a majority of cytidylic acid units from those containing more thymidylic acid units, which are then fractionated at pH 7.5 according to the number of bases in the chain. The pyrimidine nucleotides and mixtures of sequence isomers separated to 83-99% purity by column chromatography are further separated by paper chromatography and are obtained in chromatographically pure form after this step. The structures of the isolated DNA fragments are determined from chromatographic data, absorption and enzymatic degradation.     

Electronic structure of DNA by DV-X alpha cluster calculations. I. d(GG).d(CC), d(CG)2, d(GC)2 A and B conformations

The electronic structure of d(GG).d(CC), d(CG)2, d(GC)2 which are stacked base pairs in the DNA double helix, are elucidated for both A and B conformations in detail by DV-X alpha cluster calculations. These three DNA double helix fragments are constructed from the same bases, G and C, but the electronic structure of the fragments for A and B conformations differs from each other characteristically. In particular, the electronic states of the O2 and O3 in phosphates differ drastically from each other, and might play a crucial role as recognition sites in various reaction processes concerning DNA. These differences are caused by the delicate differences in the admixture of the orbital components and the intra- and inter-bases interactions. Contour maps of the wavefunction of the HOMO and LUMO are compared among the stacking isomers.     

DNA-platinum interactions in vitro with trans- and cis-Pt(NH3)2Cl2.

In the present study the nature and the hydrolysis of DNA-Pt complexes with the platinum compounds, [Pt(dien)Cl]Cl, trans- and cis-Pt(NH3)2Cl2, using potentiometric chloride determinations, have been investigated. The trans-Pt(NH3)2Cl2 and the [Pt(dien)Cl]Cl react with the GC planes at the N7(G) sites, while the cis-Pt(NH3)2Cl2 compound reacts with the GC planes and forms a chelate by using the N7(G) and O6(G) sites. The complex is a specific 1:1 Pt:DNA adduct. The platinum atom in cis-Pt(NH3)2Cl2 liberates both chlorine atoms on chelation. A mechanism for the in vivo antitumor activity of the cis-Pt(NH3)2Cl2 is proposed and the structure activity relationship is discussed.     

Interaction of the antitumor agents cis,cis,trans-PtIV(NH3)2Cl2(OH)2 and cis,cis,trans-PtIV[(CH3)2CHNH2]2Cl2(OH)2 and their reduction products with PM2 DNA

The ability of two platinum(IV) antitumor agents, cis,cis,trans-PtIV[(CH3)2CHNH2]2Cl2(OH)2 (2) and cis,cis,trans-PtIV(NH3)2Cl2(OH)2 (4), to interact with PM2 DNA was examined. Analysis using gel electrophoresis showed that neither compound is able to alter the electrophoretic mobilities of the three forms of PM2 DNA in the gel. However, incubation of 2 and 4 with 2 equiv of Fe(ClO4)2 X 6H2O or 1 equiv of ascorbic acid results in reduction to yield the divalent complexes cis-PtII(NH3)2Cl2 (1) and cis-PtII-[(CH3)2CHNH2]2Cl2 (3). The structures of the reduction products were characterized by using elemental analysis as well as infrared and 195Pt NMR spectroscopies. Both 1 and 3 were found to bind to and unwind supercoiled form I PM2 DNA. The aforementioned observations support the suggestion that reduction is a means of activating the antitumor properties of 2 and 4.     

Electronic structure of DNA by DV-X alpha cluster calculations: II. d(GG).d(CC), d(CG)2, d(GC)2 A and B conformations. (Part 2) Sugars and bases

The electronic structure of d(GG).d(CC), d(CG)2, d(GC)2 which are stacked base pairs in the DNA double helix, are elucidated for both A and B conformations in detail by DV-X alpha cluster calculations. These three DNA double helix fragments are contracted from the same bases, G and C, but the electronic structures of the fragments for both A and B conformations are different from each other characteristically. There are some delicate differences in the admixture of the orbital components and the overlap populations of intra- and inter- strand stacked bases among the stacking isomers. On the other hand, the electronic states of sugars differ in the 5'-3' direction, but are not almost dependent on stacked base pairs.     

Molecular-Mechanical studies of the mismatched base analogs of d(CGCGAATTCGCG)2:d(CGTGAATTCGCG)2, d(CGAGAATTCGCG)2, d(CGCGAATTCACG)2, d(CGCGAATTCTCG)2, and d(CGCAGAATTCGCG)·d(CGCGAATTCGCG)

Molecular-mechanical simulations have been carried out on “mismatched base” analogs of the DNA double-helical structure d(CGCGAATTCGCG)₂, in which the base pairs CG at the 3 and 10 positions have been replaced by CA, AG, TC, and TG base pairs, as well as an insertion analog in which an extra adenine has been incorporated into one strand of the above structure between bases 3 and 4. The results of these simulations (calculated relative stabilities, structures, and nmr ring-current shifts) have been compared with calorimetric and nmr data. The calculated relative stabilities of the double-helical parent dodecamer and the various “wobble” base pairs qualitatively correlate with the experimental melting temperatures. The base-pairing structure for the GT wobble pair is in agreement with that previously determined from nmr experiments. For the GA base pair, the structure with both bases anti has a slightly more favorable energy from base pairing and stacking than a structure with non-Watson-Crick H-bonding with adenine syn, in agreement with nmr experiments. The CA wobble base is calculated to favor an adenine 6NH₂ …? cytosine N3 H-bond over cytosine 4NH₂ …? adenine N1, again, in agreement with nmr experiments. There is no definitive experimental data on the TC base pair, but the existence of (somewhat long and weak) H-bonds involving cytosine 4NH₂ …? thymine 4CO and cytosine N3 …? thymine HN3 seems reasonable. We find a structure in which the extra adenine base of the insertion analogs sits “inside” the double helix.     

Synthesis, X-ray crystal structure, anti-fungal and anti-cancer activity of [Ag2(NH3)2(salH)2] (salH2=salicylic acid)

[Ag(2)(NH(3))(2)(salH)(2)] (salH(2)=salicylic acid) was synthesised from salicylic acid and Ag(2)O in concentrated aqueous NH(3) and the dimeric Ag(I) complex was characterised using X-ray crystallography. The complex is centrosymmetric with each metal coordinated to a salicylate carboxylate oxygen and to an ammonia nitrogen atom in an almost linear fashion. The two [Ag(NH(3))(salH)] units in the complex are linked by an Ag-Ag bond. Whilst metal-free salH(2) did not prevent the growth of the fungal pathogen Candida albicans [Ag(2)(NH(3))(2)(salH)(2)], [Ag(2)(salH)(2)] and some simple Ag(I) salts greatly inhibited cell reproduction. SalH(2), [Ag(2)(NH(3))(2)(salH)(2)] [Ag(2)(salH)(2)] and AgClO(4) produced a dose-dependent cytotoxic response against the three human derived cancer cell lines, Cal-27, Hep-G2 and A-498, with the Ag(I)-containing reagents being the most effective.     

cis-Platinum induced distortions in DNA. Conformational analysis of d(GpCpG) and cis-pt(NH3)2[d(GpCpG)], studied by 500-MHz NMR

Proton NMR studies at 500 MHz in aqueous solution were carried out on the G-G chelated deoxytrinucleosidediphosphate platinum complex cis-Pt(NH3)2[d(GpCpG], on the uncoordinated trinucleotide d(GpCpG) and on the constituent monomers cis-Pt(NH3)2[d(Gp)]2, cis-Pt(NH3)2[d(pG)]2, d(Gp), d(pCp) and d(pG). Complete NMR spectral assignments are given and chemical shifts and coupling constants are analysed to obtain an impression of the detailed structure of d(GpCpG) and the distortion of the structure due to chelation with [cis-Pt(NH3)2]2+. Platination of the guanosine monophosphates affects the sugar conformational equilibrium to favour the N conformation of the deoxyribose ring. This feature is also apparent in ribose mononucleotides and is possibly caused by an increased anomeric effect. In cis-Pt(NH3)2[d(pG)]2 the phase angle of pseudorotation of the S-type sugar ring is 20 degrees higher than in 'free' d(pG) which might be an indication for an ionic interaction between the positive platinum and the negatively charged phosphate. It appears that d(GpCpG) reverts from a predominantly random coil to a normal right-handed B-DNA-like single-helical structure at lower temperatures, whereas the conformational features of cis-Pt(NH3)2[d(GpCpG)] are largely temperature-independent. In the latter compound much conformational freedom along the backbone angles is seen. The cytosine protons and deoxyribose protons exhibit almost no shielding effect as should normally be exerted by the guanine bases in stacking positions. This is interpreted in terms of a 'turning away' of the cytosine residue from both chelating guanines. Conformational features of cis-Pt(NH3)2[d(GpCpG)[ are compared with the 'bulge-out' of the ribose-trinucleotide m6(2)ApUpm6(2)A.     

Characterization of di-, tri-, and tetranucleotide microsatellite markers with perfect repeats for Trypanosoma brucei and related species

Trypanosoma brucei, unicellular parasites causing human sleeping sickness and animal nagana, have a great impact on the socio-economic environment of sub-Saharan Africa. The dynamics of the parasite are still poorly understood. We have characterized 14 polymorphic di-, tri-, and tetranucleotide microsatellite loci with perfect repeats (only one motif) exhibiting between 5 and 16 alleles in T. brucei isolates from all over Africa and from all described subspecies. The microsatellites will be useful in addressing population genetic questions in T. brucei to better understand the population structure and spread of this important parasite.     

Electrochemical investigations of platinum-methyluracil-blue and the related binuclear complex [(NH3)2Pt(MeU)2Pt(NH3)2](NO3)2

The redox behavior of the head-to-head bis(μ- (1-methyluracilato-N³,O²)-bis(cis-diammine platinum(II)) dinitrate, PtMeU, and platinum 1-methyluracil blue, PtMeUB, was studied by cyclic voltammetry (CV), rotating disk voltammetry (RDV), and controlled-potential coulometry (CPC). Redox titrimetry, electrochemistry/electron paramagnetic resonance spectroscopy (EPR), and liquid chromatography (LC) served as complementary techniques. The former reactant exhibits two-step electro-oxidation, consistent with the formation of a mixed-valence Pt(II, III) state en route to Pt(III, III). The latter also appears to oxidize to a uniform Pt(III) state. Although the oxidative-reductive electrochemistry of both reactants exhibits chemical reversibility, the heterogeneous electron-transfer kinetics are notably sluggish. The latter appears to be associated with the formation of an inhibiting film on the electrode surface. A slow conversion of PtMeU to a PtMeUB-like state was revealed by CV and LC. The complex, oligomeric nature of PtMeUB was revealed by means of gradient LC examination. Comparing oxidative and reductive electrolysis curves for PtMeUB yielded an average platinum oxidation state of 2.08. All observed behavior for PtMeUB, as well as for PtMeU, is accounted for by invoking +2 and +3 oxidation states for platinum; redox titrimetry using Ce(IV) revealed inconsequential oxidation of both of these systems beyond the III state. An estimate of molecular weight for the platinum blue was made by employing RDV in conjunction with the Einstein-Stokes equation.