Crystallization of the activated ternary complex of ribulose-1,5-bisphosphate carboxylase-oxygenase isolated from Rhodospirillum rubrum and from an Escherichia coli clone

Ribulose-1,5-bisphosphate carboxylase-oxygenase was purified from the photosynthetic bacterium Rhodospirillum rubrum as well as from an Escherichia coli clone overproducing the enzyme. Although the latter enzyme contains 25 additional amino acid residues at the N terminus, both preparations yielded isomorphous tetragonal, bipyramidal crystals of the ternary complex of the enzyme with CO2 and Mg2+. Crystallization is sensitive to variation in pH and to the addition of the transition state analog, 2-carboxyarabinitol-1,5-bisphosphate. The systematic absences in the X-ray diffraction photographs suggest a tetragonal space group P4(3)2(1)2 or the enantiomorph P4(1)2(1)2 with cell dimensions a = b = 83 A, c = 290 A. There is one molecule per asymmetric unit. The resolution on still photographs is 3 A. The crystals are comparable to some of those already published but differ from others.     

Crystallization and preliminary x-ray studies of spinach ribulose 1,5-bisphosphate carboxylase/oxygenase complexed with activator and a transition state analogue

Ribulose 1,5-bisphosphate carboxylase/oxygenase has been purified from spinach and crystallized by equilibrium vapor diffusion with polyethylene glycol 6000 as a precipitant. Crystals suitable for x-ray studies were obtained from a binary complex with a transition state analogue, 2-C-carboxy-D-arabinitol 1,5-bisphosphate, and a quaternary complex with 2-C-carboxy-D-arabinitol 1,5-bisphosphate, Mg2+, and HCO-3. Two forms of crystals were obtained in the presence of 2-C-carboxy-D-arabinitol 1,5-bisphosphate. Form B crystals are plates which have orthorhombic space group P2(1)2(1)2 with unit cell dimensions a = 184 A, b = 218 A, and c = 119 A. Form C crystals are tetragonal needles with space group I422 and with cell dimensions a = b = 275 A and c = 178 A. In both forms, the asymmetric unit contains half a molecule.     

Preliminary X-ray diffraction study of ribulose-1,5-bisphosphate carboxylase from Rhodospirillum rubrum

Crystals from the dimeric enzyme ribulose-1,5-bisphosphate carboxylase of the photosynthetic bacterium Rhodospirillum rubrum have been obtained from the gene product expressed in Escherichia coli. The crystals are of the quarternary complex comprising enzyme: activator CO2 (as a carbamate): Mg2+: 2- carboxyarabinitol -1,5-bisphosphate (as a transition state analog). X-ray diffraction photographs show symmetry consistent with space group P4(1)2(1)2 or the corresponding enantiomorphic space group. Cell parameters are a = b = 82 A, c = 324 A with two subunits per asymmetric unit. The crystals diffract to at least 3 A resolution.     

New crystal forms of ribulose-1,5-bisphosphate carboxylase/oxygenase from Rhodospirillum rubrum

Three crystal forms of the dimeric form of the enzyme ribulose-1,5-bisphosphate carboxylase from the photosynthetic bacterium Rhodospirillum rubrum have been obtained from the gene product expressed in Escherichia coli. Form A crystals formed from the quaternary complex comprising enzyme-activator carbamate-Mg2+-2'-carboxyarabinitol-1,5-bisphosphate are shown here to be devoid of ligands. In contrast, crystals of the quaternary complex formed with the hexadecameric L8S8 enzyme from spinach contain both the activator carbamate and 2'-carboxyarabinitol-1,5-bisphosphate. Form B crystals of the R. rubrum enzyme are monoclinic, space group P2(1) with cell dimensions a = 65.5 A, b = 70.6 A, c = 104.1 A and beta = 92.1 degrees, with two subunits per asymmetric unit. Rotation function calculations show a non-crystallographic 2-fold axis perpendicular to the monoclinic b-axis. Form C crystals are orthorhombic (space group P2(1)2(1)2(1)) with cell dimensions a = 79.4 A, b = 100.1 A and c = 131.0 A. The monoclinic crystal form diffracts to at least 2.0 A resolution on a conventional X-ray source.     

A crystal form of ribulose-1,5-bisphosphate carboxylase/oxygenase from Nicotiana tabacum in the activated state

A new crystal form of ribulose-1,5-bisphosphate carboxylase/oxygenase (EC 4.1.1.39) from Nicotiana tabacum has been obtained at alkaline pH with polyethylene glycol 8000 in the presence of a non-ionic detergent, beta-octyl glucoside. The crystals are grown at room temperature by the hanging-drop vapor diffusion technique from a protein solution containing enzyme complexed with CO2, Mg2+, and the transition state analog 2-C-carboxy-D-arabinitol-1,5-bisphosphate. The crystals belong to the the space group P3(1)21 (or P3(2)21) with the cell parameters a = 204.6 A, and c = 117.4 A (1 A = 0.1 nm). The asymmetric unit contains half (L4S4: L, large subunit, 53,000 Mr; S, small subunit, 15,000 Mr) of a hexadecameric molecule (L8S8, 540,000 Mr). The crystals diffract to at least 2.6 A Bragg spacing and are suitable for X-ray structure determination.     

The activation of ribulose-1,5-bisphosphate carboxylase by carbon dioxide and magnesium ions. Equilibria, kinetics, a suggested mechanism, and physiological implications.

Ribulose-1,5-bisphosphate carboxylase was activated by incubation with CO2 and Mg2++, and inactivated upon removal of CO2 and Mg2+ by gel filtration. The activation process involved CO2 rather than HCO3-. The activity of the enzyme was dependent upon the preincubation concentrations of CO2 and Mg2+ and upon the preincubation pH, indicating that activation involved the reversible formation of an equilibrium complex of enzyme-CO2-Mg. The initial rate of activation was linearly dependent upon the CO2 concentration but independent of the Mg2+ concentration. Kinetic analyses indicated that the enzyme reacted first with CO2 in a rate-determining and reversible step, followed by a rapid reaction with Mg2+ to form an active ternary complex (see eq 1 in text). The pseudo-first order rate constant, kobsd, for the activation process at constant pH was derived: kobsd=k1[CO2] + (k2k4/k3[Mg2+]). Experimentally, kobsd was shown to be linearly dependent upon the CO2 concentration and inversely dependent upon the Mg2+ concentration. The activity of the enzyme after preincubation to equilibrium at constant concentrations of CO2 and Mg2+ increased as the preincubation pH was raised, indicating that CO2 reacted with an enzyme group whose pK was distinctly alkaline. It is proposed that the activation of ribulose-1, 5-biphosphate carboxylane involves the formation of a carbamate.     

The purification and preliminary X-ray diffraction studies of recombinant Synechococcus ribulose-1,5-bisphosphate carboxylase/oxygenase from Escherichia coli

X-ray crystallographic diffraction data has been collected for recombinant hexadecameric ribulose-P2 carboxylase from the cyanobacterium Synechococcus PCC6301 expressed in Escherichia coli. The enzyme has been purified and then crystallized in a number of crystal forms from polyethylene glycol solutions. The best crystals were obtained with enzyme that was first activated with the cofactors CO2 and Mg2+ in the presence of the tight-binding intermediate analogue, 2'-carboxyarabinitol 1,5-bisphosphate. One crystal form with plate-like morphology diffracts beyond 2.5 A but has one axis greater than 350 A. A second crystal form that diffracts to similar resolution grows with space group P212121 and unit cell dimensions of a = 223.9 A, b = 111.9 A, and c = 199.7 A. The crystal forms used to collect the diffraction data have been redissolved to determine that the recombinant ribulose-P2 carboxylase L8S8 molecule is indeed composed of equal numbers of large and small subunits and also that a quaternary complex between activated ribulose-P2 carboxylase E.CO2.Mg2+, and the analogue was present in the crystals. Denaturation of the redissolved enzyme in the absence of thiol-reducing agents established that the L-subunits of the L8 core are substantially dimeric, cross-linked by a disulfide bridge. Crystals of spinach ribulose-P2 carboxylase were likewise analyzed to show that dimers of the L-subunit were also predominant. This report identifies a single cysteine residue in the L-subunit that forms a bridge between those L-monomers that compose the four putative functional dimers of the L8 core.     

Structural characterization and the determination of negative cooperativity in the tight binding of 2-carboxyarabinitol bisphosphate to higher plant ribulose bisphosphate carboxylase

When CO2/Mg2+-activated spinach leaf ribulose-1,5-bisphosphate carboxylase (EC 4.1.1.39) is incubated with the transition-state analog 2-carboxyarabinitol 1,5-bisphosphate, an essentially irreversible complex is formed. The extreme stability of this quaternary complex has allowed the use of native analytical isoelectric focusing, anion-exchange chromatography, and nondenaturing polyacrylamide gel electrophoresis to probe the mechanism of the binding process and the effects of ligand tight-binding on the structure of the protein molecule. Changes in the chromatographic and electrophoretic properties of the enzyme upon tight binding of the inhibitor reveal that the ligand induces a conformational reorganization which extends to the surface of the protein molecule and, at saturation, results in a 16% decrease in apparent molecular weight. Analysis of ligand binding by isoelectric focusing shows that (i) incubating the protein with a stoichiometric molar concentration of ligand (site basis) results in an apparently charge homogeneous enzyme population with an isoelectric point of 4.9, and (ii) substoichiometric levels of ligand produce differential effects on each of the charge microheterogeneous native enzyme forms. These stoichiometry-dependent changes in electrofocusing band patterns were employed as a probe of cooperativity in the ligand tight-binding process. The tight-binding reaction was shown to be negatively cooperative.     

Crystal structure of activated ribulose-1,5-bisphosphate carboxylase complexed with its substrate, ribulose-1,5-bisphosphate

The three-dimensional structure of the complex of ribulose-1,5-bisphosphate carboxylase from Rhodospirillum rubrum, CO2, Mg2+, and ribulose bisphosphate has been determined with x-ray crystallographic methods to 2.6-A resolution. Ribulose-1,5-bisphosphate binds across the active site with the two phosphate groups in the two phosphate binding sites of the beta/alpha barrel. The oxygen atoms of the carbamate and the side chain of Asp-193 provide the protein ligands to the bound Mg2+ ion. The C2 and the C3 or C4 oxygen atoms of the substrate are also within the first coordination sphere of the metal ion. At the present resolution of the electron density maps, two slightly different conformations of the substrate, with the C3 hydroxyl group "cis" or "trans" to the C2 oxygen, can be built into the observed electron density. The two different conformations suggest two different mechanisms of proton abstraction in the first step of catalysis, the enolization of the ribulose 1,5-bisphosphate. Two loop regions, which are disordered in the crystals of the nonactivated enzyme, could be built into their respective electron density. A comparison with the structure of the quaternary complex of the spinach enzyme shows that despite the different conformations of loop 6, the positions of the Mg2+ ion, and most atoms of the substrate are very similar when superimposed on each other. There are, however, some significant differences at the active site, especially in the metal coordination sphere.     

A site-specific mutation within the active site of ribulose-1,5-bisphosphate carboxylase of Rhodospirillum rubrum

In vitro mutagenic techniques have generated an asp→glu substitution at residue 198 adjacent to the carbamate-divalent metal ion binding site of Rhodospirillum rubrum ribulose 1,5-bisphosphate carboxylase. A single C→A nucleotide change in the coding strand created the mutant and introduced a new EcoRI restriction site on the expression plasmid pRR2119. Although the carboxylase:oxygenase ratio remained the same, the mutant enzyme had slightly altered kinetic properties. The e.p.r. spectra of the quaternary complexes enzyme.activator carbamate.Mn²⁺.2-carboxyarabinitol 1,5-bisphosphate and enzyme.activator carbamate.Mn²⁺.4-carboxyarabinitol 1,5-bisphosphate for mutant and wild-type enzymes were different, indicating that the metal ion was in a slightly altered environment. These findings are consistent with the hypothesis that, besides the carbamate at lys 201, the carboxyl group of asp 198 contributes to the formation of the divalent metal ion binding site.     

In situ assay of ribulose-1,5-bisphosphate carboxylase/oxygenase in Thiobacillus neapolitanus.

Cells permeabilized with chloroform yielded ribulose-1,5-bisphosphate carboxylase/oxygenase (RuBisCO) activities nearly equal to those of cell extracts, thus indicating that both cytoplasmic and carboxysomal RuBisCO are functional in situ. The carboxysomal and cytoplasmic RuBisCO both form the CO2-Mg2(+)-enzyme ternary complex, as evidenced by stabilization with 2-C-carboxy-D-arabinitol-1,5-bisphosphate (CABP), a potent competitive inhibitor of RuBisCO. The data are consistent with the hypothesis that the carboxysome is functional in carbon dioxide fixation.     

Evidence for the existence of discrete activator and substrate sites for CO2 on ribulose-1,5-bisphosphate carboxylase.

When incubated with CO2 and Mg2+, ribulose-1,5-bis-phosphate carboxylase forms a ternary complex of enzyme . CO2 . Mg. This complex was prepared using high specific activity [14C]O2 and injected into a solution containing a large (50- to 112-fold) molar excess of [12C]O2 and sufficient ribulose 1,5-bisphosphate to permit the catalytic site to turn over several times. The enzyme was then rapidly separated from the other components by gel filtration and its radiospecific activity was determined to be 30 to 60 times that of the medium. If the CO2 activator and the CO2 substrate sites were one and the same, then, following turnover, the enzyme should have been in isotopic equilibrium with the medium. The finding that this was not the case, by a factor of about 40, indicates that the CO2 activator site is physically distinct from the CO2 substrate site.     

Modulation of the tight binding of carboxyarabinitol 1,5-bisphosphate to the large subunit of ribulose 1,5-bisphosphate carboxylase/oxygenase

The large subunit (L) of ribulose 1,5-bisphosphate carboxylase/oxygenase (rubisco) from Synechococcus PCC 6301 was expressed in Escherichia coli, purified as the octamer L8, and analyzed for its ability to tightly bind the transition state analog, 2-carboxyarabinitol 1,5-bisphosphate (CABP). [14C]CABP remained tightly bound to L8 after challenging with [12C]CABP and gel filtration, indicating that L8 alone without the small subunit (S) could tightly bind CABP. Binding of CABP to L8 induced a shift in the gel filtration profile due to apparent aggregation of L8. Aggregation did not occur with the L8S8-CABP complex nor with L8-CABP in the presence of 150 mM MgCl2. If ionic strength was increased with either KCl or MgCl2 during or after the binding of [14C]CABP to L8, [14C]CABP in the complex exchanged with [12C]CABP and was lost from the protein. Ionic strength strongly affected the rate constant (k4) for [14C]CABP dissociation from the L8-[14C]CABP complex, but had little effect on k4 for the L8S8-CABP complex. The differences in CABP binding characteristics between the L8-CABP and L8S8-CABP complexes demonstrate that S is intimately involved in maintaining the stability of the tight binding of CABP to the active site. These are the same interactions stabilizing the intermediate, 3-keto-2-carboxyarabinitol 1,5-bisphosphate, to native rubisco during CO2 fixation.     

Preparation of crystals of chicken cytosolic aspartate amino- transferase, suitable for high resolution x-ray structural analysis

The crystals of cytosolic chicken aspartate aminotransferase were grown from polyethylene glycol solutions. Two of the four crystal modifications obtained diffract to 1.8 A resolution. The crystals of the free holoenzyme belong to space group P2(1)2(1)2(1) with unit cell dimensions of a = 56.9, b = 126.9, c = 124.6 A. The crystals of the enzyme-maleate complex belong to the same space group with slightly different unit cell dimensions of a = 56.5, b = 126.1, c = 124.6 A. The influence of ions of several divalent metals, dioxane and non-ionic detergent beta-octylglucoside on crystallization have been investigated. The best crystals were obtained in the presence of Mg2+ ions. These crystals were used for data collection on the diffractometer.     

Evidence supporting lysine 166 of Rhodospirillum rubrum ribulosebisphosphate carboxylase as the essential base which initiates catalysis

The epsilon-amino group of Lys-166 of Rhodospirillum rubrum ribulosebisphosphate carboxylase/oxygenase was postulated as the essential base which initiates catalysis by abstracting the proton at C-3 of ribulose 1,5-bisphosphate (Hartman, F. C., Soper, T. S., Niyogi, S. K., Mural, R. J., Foote, R. S., Mitra, S., Lee, E. H., Machanoff, R., and Larimer, F. W. (1987) J. Biol. Chem. 262, 3496-3501). To scrutinize this possibility, the site-directed Gly-166 mutant, totally devoid of ribulosebisphosphate carboxylase activity, was examined for its ability to catalyze each of three partial reactions. When carbamylated at Lys-191 (i.e. activated with CO2 and Mg2+), wild-type enzyme catalyzed the hydrolysis of 2-carboxy-3-keto-D-arabinitol 1,5-bisphosphate, the six-carbon reaction intermediate of the carboxylase reaction (Pierce, J., Andrews, T. J., and Lorimer, G. H. (1986a) J. Biol. Chem. 261, 10248-10256). Likewise, when carbamylated at Lys-191, the Gly-166 mutant also catalyzed the hydrolysis of this reaction intermediate. The carbamylated wild type catalyzed the enolization of ribulose 1,5-bisphosphate as indicated by the transfer of 3H radioactivity from [3-3H]ribulose, 1,5-bisphosphate to the medium. However, even when carbamylated at Lys-191, the mutant protein did not catalyze the enolization of ribulose 1,5-bisphosphate. Additionally, unlike the decarbamylated wild-type enzyme, which catalyzed the decarboxylation of 2-carboxy-3-keto-D-arabinitol 1,5-bisphosphate in the absence of Mg2+, the mutant protein was inactive in this partial reaction. These properties exclude the epsilon-amino group of Lys-166 as an obligatory participant in the hydrolysis of 2-carboxy-3-keto-D-arabinitol 1,5-bisphosphate. In contrast, these properties are consistent with the epsilon-amino group of Lys-166 functioning as an acid-base catalyst in the enolization of ribulose 1,5-bisphosphate (when the enzyme is carbamylated) and in the decarboxylation of 2-carboxy-3-keto-D-arabinitol 1,5-bisphosphate (when the enzyme is decarbamylated). Alternatively, Lys-166 may stabilize the transition states of these two partial reactions.     

Ribulose 1,5-bisphosphate carboxylase/oxygenase from Pseudomonas oxalacticus.

Ribulose 1,5-bisphosphate carboxylase/oxygenase was purified by a rapid, facile procedure from formate-grown Pseudomonas oxalaticus. The electrophoretically homogeneous enzyme had specific activities of 1.9 mumol of CO2 fixed per min per mg of protein and 0.15 mumol of O2 consumed per min per mg of protein. The amino acid composition was similar to that of other bacterial sources of the enzyme. The molecular weights determined by sedimentation equilibrium and by gel filtration were 421,000 and 450,000, respectively. Upon sodium dodecyl sulfate electrophoresis of enzyme purified under conditions which would limit proteolysis, two types of large (L) subunits and two types of small (S) subunits were observed with apparent molecular weights of 57,000, 55,000, 17,000 and 15,000. By densitometric scans at two different protein concentrations the stoichiometry of the total large to total small subunits was 1:1, implying an L6S6 structure. Electron micrographs of the enzyme revealed an unusual structure that was inconsistent with a cubical structure. The enzyme had an unusually high Km for ribulose 1,5-bisphosphate (220 microM) and was strongly inhibited by 6-phosphogluconate in the ribulose 1,5-bisphosphate carboxylase assay (Ki = 270 microM). One, 5, and 12 days after purification the enzyme was half-maximally activated at 0.13 microM, 0.23 mM, and 0.70 mM CO2, respectively, at saturating Mg2+. At saturating CO2, enzyme 1 day afer purification responded sigmoidally to Mg2+ and was half-maximally activated by 0.85 mM Mg2+ in the absence of 6-phosphogluconate (Hill coefficient, h = 2.0) and by 0.19 mM Mg2+ in the presence of mM 6-phosphogluconate (h = 1.7).     

Calcium supports loop closure but not catalysis in Rubisco

Rubisco (ribulose-1,5-bisphosphate carboxylase/oxygenase) catalyses CO(2) assimilation in biology. A prerequisite for catalysis is an activation process, whereby an active site lysine is selectively carbamylated. The carbamyl group is then stablised by a metal ion, which in vivo is Mg(2+). Other divalent metal ions can replace Mg(2+) as activators in vitro, but the nature of the metal ion strongly influences the catalytic activity of the enzyme and has a differential effect on the ratio of the carboxylation reaction and the competing oxygenation reaction. Biochemical studies show that calcium promotes carbamylation but not catalysis. To investigate the role of the metal in catalysis, we have determined two structures of the enzyme complexed with Ca(2+) and the transition state analogue 2-carboxy-D-arbinitol-1,5-bisphosphate (2CABP). One of the complexes was prepared by soaking 2CABP into crystals of the enzyme-Ca(2+)-product complex, while the other was obtained by cocrystallising the enzyme with calcium and 2CABP under activating conditions. The two crystals belong to different space groups, and one was merohedrally twinned. Both complexes show very similar three-dimensional features. The enzyme is carbamylated at Lys201, and requisite loops close over the bound ligands in the active site, shielding them from the solvent in a manner similar to the corresponding complex with Mg(2+). However, there are subtle differences that could explain the particular role of Ca(2+) in these processes. The larger radius of the calcium ion and its reduced Lewis-acid character causes a significant increase in the required proton hop distance between the C3 proton and the carbamate on Lys201 in the calcium complex. This alone could explain the inability of calcium to sustain catalysis in Rubisco. Similar effects are also expected on subsequent proton transfer steps in the catalytic cycle. Here we also discuss the effect of metal substitution on the dynamics of the ligands around the metal ion.     

Crystal structure of the ternary complex of ribulose-1,5-bisphosphate carboxylase, Mg(II), and activator CO2 at 2.3-A resolution

The activated ternary complex, enzyme-CO2-Mg(II), of the dimeric ribulose-1,5-bisphosphate carboxylase/oxygenase from Rhodospirillum rubrum can be prepared in the same crystal form that was used for the crystallographic structure determination of the native nonactivated enzyme (Schneider, G., Br?nden, C.-I., & Lorimer, G. (1986) J. Mol. Biol. 187, 141-143). The three-dimensional structure of the activated enzyme has been determined to a nominal resolution of 2.3 A by protein crystallographic methods. The activator CO2 forms a carbamate with Lys191, located at the bottom of the funnel-shaped active site. In both subunits, this labile adduct is stabilized by a Mg(II) ion, bound to the carbamate and the side chains of Asp193 and Glu194. One solvent molecule was found within the first coordination sphere of the metal ion. The metal-binding site in ribulose-1,5-bisphosphate carboxylase consists thus of at least three protein ligands, all located on loop 2 of the beta/alpha barrel. One additional metal ligand, the side chain of the conserved Asn111, was observed close to the Mg(II) ion in the B-subunit. Other structural differences at the active site between the activated and nonactivated enzyme are limited to side-chain positions. Nevertheless, it is obvious that the hydrogen-bonding pattern in the vicinity of the activator site is completely altered.     

Crystallization and preliminary X-ray diffraction study of the bacterially expressed Fv from the monoclonal anti-lysozyme antibody D1.3 and of its complex with the antigen, lysozyme

The associated heavy (VH) and light (VL) chain variable domains (Fv) of the monoclonal anti-lysozyme antibody D1.3, secreted from Escherichia coli, have been crystallized in their antigen-bound and free forms. FvD1.3 gives tetragonal crystals, space group P4(1)2(1)2 (or P4(3)2(1)2), with a = 90.6 A, c = 56.4 A. The FvD1.3-lysozyme complex crystallizes in space group C2, with a = 129.2 A, b = 60.8 A, c = 56.9 A and beta = 119.3 degrees. The crystals contain one molecule of Fv or of the Fv-lysozyme complex in their asymmetric units and diffract X-rays to high resolution, making them suitable for X-ray crystallographic studies.     

Partition kinetics of ribulose-1,5-bisphosphate carboxylase from Rhodospirillum rubrum

When the enzymatically generated intermediate 2-carboxy-3-keto-D-arabinitol-1,5-bisphosphate (II) was used as a substrate with fresh enzyme, 70% reacted to produce 3-phosphoglycerate (3PGA). When a reaction mixture of enzyme plus [1-32P]ribulose 1,5-bisphosphate (RuBP) was quenched in the steady state with the tightly bound inhibitor 2-carboxyarabinitol-1,5-bisphosphate, 30% of the enzyme-bound species was released as 3PGA and 70% as RuBP. The major source for this partition was the ternary substrates Michaelis complex. The level of carboxylated intermediate in the steady state was determined to be 8% of active sites under the conditions of substrate saturation. No burst was seen in the appearance of product when 6.5 eq of [1-32P]RuBP was mixed with enzyme plus saturating CO2 and the reaction followed in the steady state. From these data plus the steady-state Vmax and Km of RuBP it is possible to derive the five bulk rate constants represented in the scheme ECO2 + RuBP in equilibrium ERuBPCO2 in equilibrium E X II----E + 2(3PGA).