Reductive cleavage mechanism of methylcobalamin: elementary steps of Co-C bond breaking

Density functional theory has been applied to the investigation of the reductive cleavage mechanism of methylcobalamin (MeCbl). In the reductive cleavage of MeCbl, the Co-C bond is cleaved homolytically, and formation of the anion radical ([MeCbl]*-) reduces the dissociation energy by approximately 50%. Such dissociation energy lowering in [MeCbl]*- arises from the involvement of two electronic states: the initial state, which is formed upon electron addition, has dominant pi*corrin character, but when the Co-C bond is stretched the unpaired electron moves to the sigma*Co-C state, and the final cleavage involves the three-electron (sigma)2(sigma*)1 bond. The pi*corrin-sigma*Co-C states crossing does not take place at the equilibrium geometry of [MeCbl]*- but only when the Co-C bond is stretched to 2.3 A. In contrast to the neutral cofactor, the most energetically efficient cleavage of the Co-C bond is from the base-off form. The analysis of thermodynamic and kinetic data provides a rationale as to why Co-C cleavage in reduced form requires prior departure of the axial base. Finally, the possible connection of present work to B12 enzymatic catalysis and the involvement of anion-radical-like [MeCbl]*- species in relevant methyl transfer reactions is discussed.     

Photolysis of methylcobalamin: identification of the relevant excited states involved in Co-C bond scission

The relevant excited states involved in the photolysis of methylcobalamin (MeCbl) have been examined by means of time-dependent density functional theory (TD-DFT). The low-lying singlet and triplet excited states have been calculated along the Co-C bond at the TD-DFT/BP86/6-31g(d) level of theory in order to investigate the dissociation process of MeCbl. These calculations have shown that the photodissociation is mediated by the repulsive 3(sigmaCo-C --> sigma*Co-C) triplet state. The key metastable photoproduct involved in Co-C bond photolysis was identified as an S1 state having predominantly dCo --> pi*corrin metal-ligand charge transfer (MLCT) character.     

Carbon-cobalt bond distance and bond cleavage in one-electron reduced methylcobalamin: a failure of the conventional DFT method

Geometry optimizations at the HF, B3LYP, and CASSCF levels of electronic structure theory have been performed for methylcobalamin (MeCbl) model compounds in both the Co(III) (MeCbl(III)) and Co(II) (MeCbl(II)) formal oxidation states. Since the HOMO-LUMO and C-Co sigma-sigma MO gaps are significantly smaller in the MeCbl(II) compounds compared with MeCbl(III), a pseudo-Jahn Teller effect is possible. CASSCF calculations show that there is strong coupling between C-Co sigma-sigma MOs for the MeCbl(II) models leading to strong state mixing with significant total charge density transfer (approximately 0.4 e-), mainly from the C-Co sigma MO to C-Co sigma MO (approximately 0.3 e-). CASSCF(9:7) calculations show that the strong state mixing leads to an increase in the C-Co bond length for MeCbl(II) model compounds from 1.969 A (DFT and HF calculations) to 2.164 A in the base-on MeCbl(II) model and from 1.938 A to 2.144 A in the base-off MeCbl(II) model. Concomitantly, the Co-N axial bond length increases from 2.121 A (DFT) to 2.344 A in the CASSCF calculation. This coupling interaction between states can be used to explain the much lower Co-C bond dissociation enthalpy and much faster bond cleavage rate for the one-electron reduced methylcobalamin radical anion compared to MeCbl(III). It may also be important for axial bond distances in other Co(II) compounds.     

Spectroscopic and computational studies on the adenosylcobalamin-dependent methylmalonyl-CoA mutase: evaluation of enzymatic contributions to Co-C bond activation in the Co3+ ground state

Methylmalonyl-CoA mutase (MMCM) is an enzyme that utilizes the adenosylcobalamin (AdoCbl) cofactor to catalyze the rearrangement of methylmalonyl-CoA to succinyl-CoA. Despite many years of dedicated research, the mechanism by which MMCM and related AdoCbl-dependent enzymes accelerate the rate for homolytic cleavage of the cofactor's Co-C bond by approximately 12 orders of magnitude while avoiding potentially harmful side reactions remains one of the greatest subjects of debate among B(12) researchers. In this study, we have employed electronic absorption (Abs) and magnetic circular dichroism (MCD) spectroscopic techniques to probe cofactor/enzyme active site interactions in the Co(3+)Cbl "ground" state for MMCM reconstituted with both the native cofactor AdoCbl and its derivative methylcobalamin (MeCbl). In both cases, Abs and MCD spectra of the free and enzyme-bound cofactor are very similar, indicating that replacement of the intramolecular base 5,6-dimethylbenzimidazole (DMB) by a histidine residue from the enzyme active site has insignificant effects on the cofactor's electronic properties. Likewise, spectral perturbations associated with substrate (analogue) binding to holo-MMCM are minor, arguing against substrate-induced enzymatic Co-C bond activation. As compared to the AdoCbl data, however, Abs and MCD spectral changes for the sterically less constrained MeCbl cofactor upon binding to MMCM and treatment of holoenzyme with substrate (analogues) are much more substantial. Analysis of these changes within the framework of time-dependent density functional theory calculations provides uniquely detailed insight into the structural distortions imposed on the cofactor as the enzyme progresses through the reaction cycle. Together, our results indicate that, although the enzyme may serve to activate the cofactor in its Co(3+)Cbl ground state to a small degree, the dominant contribution to the enzymatic Co-C bond activation presumably comes through stabilization of the Co(2+)Cbl/Ado. post-homolysis products.     

Accurate redetermination of the X-ray structure and electronic bonding in adenosylcobalamin

The electronic structure of adenosylcobalamin (B12 coenzyme, AdoCbl) has been calculated by a density functional method, using the orthogonalized linear combination of the atomic orbital method (OLCAO). Since a fixed accurately determined geometry was needed in such calculations, the crystal structure of adenosylcobalamin has been redone and refined to R = 0.065, using synchrotron diffraction data. Comparison with the recently reported electronic structures of cyano- (CNCbl) and methylcobalamin (MeCbl) shows that the net charges and bond orders vary only on the axial donors. The values in the three cobalamins suggest that the Co-C bond in MeCbl has a strength similar to that in AdoCbl, but it is significantly weaker that that in CNCbl. Present results are compared with those previously reported for the analogous corrin derivatives; i.e., simplified cobalamins with the side chains a-f replaced by H atoms. Despite a qualitative agreement, a discrepancy in the calculated HOMO-LUMO gap is found.     

Which DFT functional performs well in the calculation of methylcobalamin? Comparison of the B3LYP and BP86 functionals and evaluation of the impact of empirical dispersion correction

Controversy remains regarding the suitable density functionals for the calculation of vitamin B(12) systems that contain cobalt. To identify the optimum functionals, geometry optimization calculations were performed on a full-size model of methylcobalamin (MeCbl) using the B3LYP, B3LYP-D, BP86, and BP86-D methods in conjunction with the 6-31G* basis set. Single-point energy evaluations were also performed with the 6-311+G(2d,p) basis set. Consistent with previous studies, the BP86-optimized geometry showed fairly good agreement with the experimental geometry. Various factors that may influence the homolytic bond dissociation energy (BDE) of the Co-C bond of MeCbl were systematically evaluated with these methods. Our analysis demonstrated that dispersion was the largest correction term that influenced the magnitude of BDE. Previous studies have shown that B3LYP significantly underestimates BDE, whereas BP86 gives BDE values that are fairly close to the experimental values (36-37 kcal/mol). The same trend in the relative magnitudes of the BDEs was observed in the present calculations. However, BP86 underestimated the BDE for a full model of MeCbl. When the amount of Hartree-Fock exchange in the B3LYP functional was reduced to 15% and the dispersion correction was made (i.e., B3LYP*-D), the calculated BDE was in good accord with experimental values. B3P86-D also performed well. A detailed analysis was undertaken to determine which atoms in cobalamin have large dispersion interactions with a methyl fragment of MeCbl.     

X-ray structures and homolysis of some alkylcobalt(III) phthalocyanine complexes

The first crystallographic data for sigma-bonded alkylcobalt(III) phthalocyanine complexes are reported. A single-crystal X-ray structure of CH(3)CH(2)Co(III)Pc (Pc = dianion of phthalocyanine) reveals that the solid consists of centrosymmetric face-to-face dimers in which the CH(3)CH(2)Co(III)Pc units retain their square pyramidal geometry. The structure appears to be the first one reported for a five-coordinate RCo(III)(chelate) complex with an electron-deficient equatorial system. The Co-C bond in CH(3)CH(2)Co(III)Pc (2.031(5) A) is the longest found in five-coordinate RCo(III)(chel) complexes (R = simple primary alkyl group). Another X-ray study demonstrates that CH(3)Co(III)Pc(py) has a distorted octahedral geometry with axial bonds of very similar length to those in methylcobalamin. The axial bonds are shorter than those in its octaethylporphyrin analogue, in accordance with a weaker trans axial influence in six-coordinate complexes containing an electron-deficient phthalocyanine equatorial ligand. A different trend has been observed for five-coordinate RCo(III)(chel) complexes: electron-rich equatorial systems seem to make the Co-C axial bond shorter. Kinetic data for the homolysis of RCo(III)Pc complexes (R = Me, Et) in dimethylacetamide are also reported. Homolysis of ethyl derivatives is faster. The Co-C bond dissociation energies (BDEs) for the pyridine adducts of the methyl and the ethyl derivative are 30 +/- 1 and 29 +/- 1 kcal/mol, respectively. The BDE for CH(3)CoPc(py) is considerably lower than that for MeCbl despite the very similar lengths of the axial bonds in the two complexes. The results of this work do not support any correlation between the Co-C bond length and the bond strength as defined by BDE.     

The enzymatic activation of coenzyme B12

The enzymatic "activation" of coenzyme B12 (5'-deoxyadenosylcobalamin, AdoCbl), in which homolysis of the carbon-cobalt bond of the coenzyme is catalyzed by some 10(9)- to 10(14)-fold, remains one of the outstanding problems in bioinorganic chemistry. Mechanisms which feature the enzymatic manipulation of the axial Co-N bond length have been investigated by theoretical and experimental methods. Classical mechanochemical triggering, in which steric compression of the long axial Co-N bond leads to increased upward folding of the corrin ring and stretching of the Co-C bond is found to be feasible by molecular modeling, but the strain induced in the Co-C bond seems to be too small to account for the observed catalytic power. The modeling study shows that the effect is a steric one which depends on the size of the axial nucleotide base, as substitution of imidazole (Im) for the normal 5,6-dimethylbenzimidazole (Bzm) axial base decreases the Co-C bond labilization considerably. An experimental test was thus devised using the coenzyme analog with Im in place of Bzm (Ado(Im)Cbl). Studies of the enzymatic activation of this analog by the B12-dependent ribonucleoside triphosphate reductase from Lactobacillus leichmannii coupled with studies of the non-enzymatic homolytic lability of the Co-C bond of Ado(Im)Cbl show that the enzyme is only slightly less efficient (3.8-fold, 0.8 kcal mol(-1)) at activating Ado(Im)Cbl than at activating AdoCbl itself. This suggests, in agreement with the modeling study, that mechanochemical triggering can make only a small contribution to the enzymatic activation of AdoCbl. Another possibility, electronic stabilization of the Co(II) homolysis product by compression of the axial Co-N bond, requires that enzymatic activation be sensitive to the basicity of the axial nucleotide. Preliminary studies of the enzymatic activation of a coenzyme analog with a 5-fluoroimidazole axial nucleotide suggest that the catalysis of Co-C bond homolysis may indeed be significantly slowed by the decrease in basicity.     

A combined density functional theory and molecular mechanics study of the relationship between the structure of coenzyme B12 and its binding to methylmalonyl-CoA mutase

A combined density functional theory (DFT) and molecular mechanics (MM) approach was applied to investigate the relationship between the structure of a free coenzyme B12, and bound to methylmalonyl-CoA mutase. It was found that, upon coenzyme binding to apoenzyme, the Co-C bond remains intact, while the C-Naxial bond becomes slightly elongated and labilized. The labilization of the Co-Naxial bond that takes place in coenzyme B12-dependent enzymes is most likely necessary for fine-tuning of the cobalt-nitrogen (axial base) distance. The controlling of this distance is important to inhibit abiological site reaction involving heterolysis of the Co-C bond but is not important for biologically relevant Co-C bond homolysis.     

Chemistry of constrained dioxocyclam ligands with Co(III): unusual examples of C-H and C-N bond cleavage

The reactions between H(2)dc3 and Co(acac)(3) have been studied in the presence and absence of base. In the presence of base, a complex with an intramolecular Co-C bond, Co(dc3-C-(8))(H(2)O), 1, is formed, presumably through heterolytic C-H bond activation. An X-ray crystallographic study demonstrates the presence of a Co-C bond and shows that the diazacyclooctane (daco) subunit adopts the chair-boat conformation with respect to the metal. The cobalt-carbon bond induces strain in the macrocycle as demonstrated by bond angles significantly deviating from tetrahedral. The (13)C NMR resonance of the carbon atom bound to cobalt (-10.5 ppm) suggests significant ionic character in the cobalt-carbon bond. However, we were unable to cleave this bond in the presence of strong acid. In the absence of base, the reaction of Co(acac)(3) with H(2)dc3 resulted in C-N cleavage of the ligand and the formation of a complex of dioxocyclam, Co(dc)(acac), 2. This complex has subsequently been prepared in high yield by the reaction of Co(acac)(3) with dioxocyclam. An X-ray crystallographic study demonstrates that dioxocyclam adopts the heretofore unreported cis configuration, having folded along a N-Co-N axis that is perpendicular to the Co-acac plane.     

DFT study of Co-C bond cleavage in the neutral and one-electron-reduced alkyl-cobalt(III) phthalocyanines

Density functional theory (DFT) has been applied to the analysis of the structural and electronic properties of the alkyl-cobalt(III) phthalocyanine complexes, [CoIIIPc]-R (Pc = phthalocyanine, R = Me or Et), and their pyridine adducts. The BP86/6-31G(d) level of theory shows good reliability for the optimized axial bond lengths and bond dissociation energies (BDEs). The mechanism of the reductive cleavage was probed for the [CoIIIPc]-Me complex which is known as a highly effective methyl group donor. In the present analysis, which follows a recent study on the reductive Co-C bond cleavage in methylcobalamin (J. Phys. Chem. B 2007, 111, 7638-7645), it is demonstrated that addition of an electron and formation of the pi-anion radical [CoIII(Pc*)]-Me- significantly lowers the energetic barrier required for homolytic Co-C bond dissociation. Such BDE lowering in [CoIII(Pc*)]-Me- arises from the involvement of two electronic states: upon electron addition, a quasi-degenerate pi*Pc state is initially formed, but when the cobalt-carbon bond is stretched, the unpaired electron moves to a sigma*Co-C state and the final cleavage involves the three-electron (sigma)2(sigma*)1 bond. As in corrin complexes, the pi*Pc-sigma*Co-C states crossing does not take place at the equilibrium geometry of [CoIII(Pc*)]-Me- but only when the Co-C bond is stretched to approximately 2.3 A. The DFT computed Co-C BDE of 23.3 kcal/mol in the one-electron-reduced phthalocyanine species, [CoIII(Pc*)]-Me-, is lowered by approximately 37% compared to the neutral Py-[CoIIIPc]-Me complex where BDE = 36.8 kcal/mol. A similar comparison for the corrin-containing complexes shows that a DFT computed BDE of 20.4 kcal/mol for [CoIII(corrin*)]-Me leads to approximately 45% bond strength reduction, in comparison to 37.0 kcal/mol for Im-[CoIII(corrin)]-Me+. These results suggest some preference by the alkylcorrinoids for the reductive cleavage mechanism.     

Characterization of Co-C bonding in dichlorovinylcobaloxime complexes

This study combines theory and experiment in an examination of Co-C bonding and reductive Co-C cleavage in cobalt dichlorovinyl complexes. It is motivated by the role of dichlorovinyl complexes as intermediates in the dechlorination of trichloroethylene by cobalamin and cobalamin model complexes. A series of seven cis-1,2-dichlorovinyl(L)cobaloxime complexes were prepared (L = m- and p-substituted pyridines; cobaloxime = bis(dimethylglyoximato)cobalt). The complexes were characterized using 1H NMR, 13C NMR, cyclic voltammetry, and X-ray crystallography. Examination of the metrical parameters of the Co-C=C unit across the series shows very little change in the C=C bond length and a slight increase in the Co-C bond length with increasing electron-donating ability of the pyridine ligand. These structural changes along with electronic structure calculations indicate that Co-C pi-bonding is not important in these complexes. The stronger Co-C bonds of vinylcobaloximes compared to those of alkylcobaloximes are best explained by the higher s character at C. Changes in the reduction potential across the series indicate that the pyridine-bound form is the primary electrochemically active species. Theoretical examination of the Co-C cleavage following reduction supports the direct formation of the cis-1,2-dichlorovinyl anion and not the cis-1,2-dichlorovinyl radical.     

CoTPP电化学催化还原溴代环己烷的机理

以电化学循环伏安、现场ESR电化学以及现场薄层电化学方法研究了电生Co(Ⅰ)TPP与溴代环己烷的反应机制.在DMF中,Co(Ⅱ)/Co(Ⅰ)的氧化还原有明显的催化溴代环己烷还原的特征,反应现场有自由基生成.反应产物之一是Co-C键化合物,可以在-1.30V(SCE)一电子还原.当存在CH₂=CHCN时,生成另一种Co-C键化合物,该化合物在-1.10V(SCE)处一电子还原.证明溴代环己烷与Co(Ⅰ)TPP反应主要是经过形成烷基自由基的机制进行的.     

Cis-influence of hydroporphyrin macrocycles on axial ligation equilibria and alkyl exchange reactions of alkylcobalt(III) porphyrin complexes

Stability constants are reported for the coordination of pyridine and substituted pyridines to the alkylcobalt(III) complexes of octaethylporphyrin (OEP), t-octaethylchlorin (OEC), and ttt-octaethylisobacteriochlorin (OEiBC) in toluene solution. The stability constants correlate with the base strength of the nitrogenous ligand. A cis-influence of the macrocycle saturation level on the stability constants is observed. Stability constants for coordination of a given pyridine ligand to an alkylcobalt(III) complex are roughly 10 times smaller than the stability constants for the corresponding cobalt(II) complex. Analysis of a thermodynamic cycle demonstrates that this leads to decreased stability of the complex with respect to Co-C bond homolysis upon ligand coordination, a "base-on" effect. Alkyl exchange occurs between cobalt complexes of different tetrapyrroles. Equilibrium data establish that the exchange is nonstatistical and that the Co-C bond is stabilized by increasing the saturation of the tetrapyrrole macrocycle.     

How the Co-C bond is cleaved in coenzyme B12 enzymes: a theoretical study

The homolytic cleavage of the organometallic Co-C bond in vitamin B12-dependent enzymes is accelerated by a factor of approximately 10(12) in the protein compared to that of the isolated cofactor in aqueous solution. To understand this much debated effect, we have studied the Co-C bond cleavage in the enzyme glutamate mutase with combined quantum and molecular mechanics methods. We show that the calculated bond dissociation energy (BDE) of the Co-C bond in adenosyl cobalamin is reduced by 135 kJ/mol in the enzyme. This catalytic effect can be divided into four terms. First, the adenosine radical is kept within 4.2 angstroms of the Co ion in the enzyme, which decreases the BDE by 20 kJ/mol. Second, the surrounding enzyme stabilizes the dissociated state by 42 kJ/mol using electrostatic and van der Waals interactions. Third, the protein itself is stabilized by 11 kJ/mol in the dissociated state. Finally, the coenzyme is geometrically distorted by the protein, and this distortion is 61 kJ/mol larger in the Co(III) state. This deformation of the coenzyme is caused mainly by steric interactions, and it is especially the ribose moiety and the Co-C5'-C4' angle that are distorted. Without the polar ribose group, the catalytic effect is much smaller, e.g. only 42 kJ/mol for methyl cobalamin. The deformation of the coenzyme is caused mainly by the substrate, a side chain of the coenzyme itself, and a few residues around the adenosine part of the coenzyme.     

Spectroscopic and computational studies of Co2+corrinoids: spectral and electronic properties of the biologically relevant base-on and base-off forms of Co2+cobalamin

Co(2+)cobalmain (Co(2+)Cbl) is implicated in the catalytic cycles of all adenosylcobalamin (AdoCbl)-dependent enzymes, as in each case catalysis is initiated through homolytic cleavage of the cofactor's Co-C bond. The rate of Co-C bond homolysis, while slow for the free cofactor, is accelerated by 12 orders of magnitude when AdoCbl is bound to the protein active site, possibly through enzyme-mediated stabilization of the post-homolysis products. As an essential step toward the elucidation of the mechanism of enzymatic Co-C bond activation, we employed electronic absorption (Abs), magnetic circular dichroism (MCD), and resonance Raman spectroscopies to characterize the electronic excited states of Co(2+)Cbl and Co(2+)cobinamide (Co(2+)Cbi(+), a cobalamin derivative that lacks the nucleotide loop and 5,6-dimethylbenzimazole (DMB) base and instead binds a water molecule in the lower axial position). Although relatively modest differences exist between the Abs spectra of these two Co(2+)corrinoid species, MCD data reveal that substitution of the lower axial ligand gives rise to dramatic changes in the low-energy region where Co(2+)-centered ligand field transitions are expected to occur. Our quantitative analysis of these spectral changes within the framework of time-dependent density functional theory (TD-DFT) calculations indicates that corrin-based pi --> pi transitions, which dominate the Co(2+)corrinoid Abs spectra, are essentially insulated from perturbations of the lower ligand environment. Contrastingly, the Co(2+)-centered ligand field transitions, which are observed here for the first time using MCD spectroscopy, are extremely sensitive to alterations in the Co(2+) ligand environment and thus may serve as excellent reporters of enzyme-induced perturbations of the Co(2+) state. The power of this combined spectroscopic/computational methodology for studying Co(2+)corrinoid/enzyme active site interactions is demonstrated by the dramatic changes in the MCD spectrum as Co(2+)Cbi(+) binds to the adenosyltransferase CobA.     

Characterization of protein contributions to cobalt-carbon bond cleavage catalysis in adenosylcobalamin-dependent ethanolamine ammonia-lyase by using photolysis in the ternary complex

Protein contributions to the substrate-triggered cleavage of the cobalt-carbon (Co-C) bond and formation of the cob(II)alamin-5'-deoxyadenosyl radical pair in the adenosylcobalamin (AdoCbl)-dependent ethanolamine ammonia-lyase (EAL) from Salmonella typhimurium have been studied by using pulsed-laser photolysis of AdoCbl in the EAL-AdoCbl-substrate ternary complex, and time-resolved probing of the photoproduct dynamics by using ultraviolet-visible absorption spectroscopy on the 10(-7)-10(-1) s time scale. Experiments were performed in a fluid dimethylsulfoxide/water cryosolvent system at 240 K, under conditions of kinetic competence for thermal cleavage of the Co-C bond in the ternary complex. The static ultraviolet-visible absorption spectra of holo-EAL and ternary complex are comparable, indicating that the binding of substrate does not labilize the cofactor cobalt-carbon (Co-C) bond by significantly distorting the equilibrium AdoCbl structure. Photolysis of AdoCbl in EAL at 240 K leads to cob(II)alamin-5'-deoxyadenosyl radical pair quantum yields of <0.01 at 10(-6) s in both holo-EAL and ternary complex. Three photoproduct states are populated following a saturating laser pulse, and labeled, P(f), P(s), and P(c). The relative amplitudes and first-order recombination rate constants of P(f) (0.4-0.6; 40-50 s(-1)), P(s) (0.3-0.4; 4 s(-1)), and P(c) (0.1-0.2; 0) are comparable in holo-EAL and in the ternary complex. Time-resolved, full-spectrum electron paramagnetic resonance (EPR) spectroscopy shows that visible irradiation alters neither the kinetics of thermal cob(II)alamin-substrate radical pair formation, nor the equilibrium between ternary complex and cob(II)alamin-substrate radical pair, at 246 K. The results indicate that substrate binding to holo-EAL does not "switch" the protein to a new structural state, which promptly stabilizes the cob(II)alamin-5'-deoxyadenosyl radical pair photoproduct, either through an increased barrier to recombination, a decreased barrier to further radical pair separation, or lowering of the radical pair state free energy, or a combination of these effects. Therefore, we conclude that such a change in protein structure, which is independent of changes in the AdoCbl structure, and specifically the Co-C bond length, is not a basis of Co-C bond cleavage catalysis. The results suggest that, following the substrate trigger, the protein interacts with the cofactor to contiguously guide the cleavage of the Co-C bond, at every step along the cleavage coordinate, starting from the equilibrium configuration of the ternary complex. The cleavage is thus represented by a diagonal trajectory across a free energy surface, that is defined by chemical (Co-C separation) and protein configuration coordinates.