Pyrophosphate (PPi), a byproduct of numerous biosynthetic reactions [1] , re-enters metabolic pathways after being converted to phosphate via the action of specific enzymes-pyrophosphatases (PPases; EC 3.6.1.1). Soluble PPases belong to three non-homologous families, I, II, and III [2] [3] [4] . Family I PPases are found in all kingdoms of life, whereas Family II PPases and the less common Family III PPases are found in prokaryotes. With rare exceptions [5] , PPases of Families I and III are single-domain proteins.
Family II PPases have a more variable domain organization and are divided into two subfamilies. In the "canonical" subfamily, which has been well characterized both structurally and mechanistically [6] , each subunit of 33-34 kDa is formed by two catalytic domains, DHH and DHHA2, connected by a flexible linker, and the active site is located between the domains [7, 8] (Fig. 1A) . These PPases form stable homodimers in physiological conditions [9] , via association of the DHH domains [7, 8] , and are Mn 2+ -or Co 2+ -metalloproteins that additionally bind several Mg 2+ ions less tightly [9] . The metal ions stabilize the oligomeric structure of PPases [9] and are the key to catalysis; in the latter function they neutralize and position the substrate PPi and convert the nucleophilic water molecule to a highly reactive hydroxide ion [10] .
A quarter of Family II PPases, belonging to another subfamily (named "CBS-PPases"), contain a regulatory insert of ~250 residues comprised of two β-cystathionine synthase (CBS) domains and, frequently, one DRTGG domain within the DHH domain (Fig. 1B,C) [11, 12] . CBS-PPases are unique among different PPases in being differentially regulated (activated or inhibited) by adenine mono-and dinucleotides that bind to the CBS domains [11, 13, 14] . This property is thought to enhance the control of PPi level under stress conditions and, hence, the viability of the bacterial host [14] . The regulating nucleotides bind to the enzyme in a positively co-operative manner [13] , a phenomenon associated with the oligomeric structure of CBS-PPase.
The three-dimensional structure of the full-size CBS-PPase remains to be determined. Tuominen et al. [12] have reported on the crystal structure of the isolated regulatory part (residues 66-306) of Clostridium perfringens CBS-PPase. The regulatory part is dimerized by forming CBS-CBSʹ and DRTGG-DRTGGʹ contacts (prime sign refers to the other subunit) and retains the ability to bind adenine nucleotides [12] , which means that its native structure is well preserved in the absence of the catalytic domains. Accordingly, deletion of the regulatory domains did not inactivate CBS-PPase but abolished nucleotide effects on activity [13] . The catalytic domains of CBS-PPases should have a fold very similar to that of the canonical PPases, based on high sequence identity. Based on this information, a model of full-size CBS-PPase with a complete set of domains was suggested, which assumed its dimeric organization [12] .
Here we provide experimental evidence that, contrary to the above prediction, bacterial CBS-PPases form homotetramers. Most of the experiments described in this manuscript were performed with Desulfitobacterium hafniense CBS-PPase (dhPPase), which contains both CBS and DRTGG domains in its regulatory part [13] . However, critical controls run with four other CBS-PPases, including those lacking DRTGG domain, indicated that they are also tetrameric proteins. We have also determined the contributions of the catalytic and regulatory parts to tetramer stability and its dependence on substrate and adenine nucleotide binding to the catalytic and regulatory sites, respectively.
D. hafniense CBS-PPase (dhPPase), its separate catalytic part (dhPPaseΔCDC), and the CBS-PPases of Clostridium perfringens (cpPPase), Clostridium novyi (cnPPase), Eggerthella lenta CBS-PPase (elPPase), and Ethanoligenens harbinense CBS-PPase (ehPPase) were produced and purified as described elsewhere [12, 13, 15] . The last two enzymes (elPPase and ehPPase) lack the DRTGG domain in their structures. The gene for a separate regulatory part of dhPPase (dhCDC, residues 63-306 of dhPPase) was produced using overlap-extension PCR with Phusion DNA polymerase and the primers listed in Table S1 . The mutant gene was ligated into the NdeI and XhoI restriction sites of the pET-42b plasmid (Novagen) and transformed into Escherichia coli DH5α cells. The construct was confirmed by sequencing.
All CBS-PPases and their deletion variants were produced in Escherichia coli BL21 cells. Collected cells with produced dhCDC were suspended in lysis buffer (25 mM Tris-HCl, 10 mM MgCl2, pH 7.3) and disrupted with 2-3 cycles of freezing in liquid nitrogen followed by thawing in a water bath at room temperature in the presence of 0.2 mg/ml lysozyme. Genomic DNA was destroyed by DNAse I (Roche, Germany). The dhCDC was isolated from the cell extract by ionexchange chromatography in the lysis buffer with a linear gradient of 0-0.3 M NaCl followed by gel filtration in 50 mM Hepes/KOH, pH 7.5, containing 2 mM MgCl2 and 25 mM KCl. dhCDC was identified in eluate by SDS-PAGE in the Laemmli system [16] . The dhCDC-containing fractions obtained after gel filtration were divided into aliquots, frozen, and stored at -50 °C. The protein was electrophoretically homogeneous.
Protein concentrations were determined with a Nanodrop spectrophotometer (Thermo Scientific) using subunit molecular mass/A 0. 
The initial rates of PPi hydrolysis were measured using a continuous Pi assay [17] at a sensitivity of 2-50 μM Pi per recorder scale. The assay medium contained 140 μM PPi (yielding 50 μM MgPPi complex), 0.1 M Tes-KOH, pH 7.2, 5 mM MgCl2 and 0.1 mM CoCl2, except where specified. The reaction was initiated by adding enzyme and continued for 2-3 min at 25 °C. The dead time of the phosphate analyzer (i.e., the time from feeding the sample into the analyzer to mixing it with the quenching acidic molybdate solution) was 10-20 s in the standard mode. To increase the precision of the initial velocity estimation for reactions yielding non-linear Pi production traces, the dead time was decreased to 1 s by an appropriate modification of the inlet system [18] . The rate values obtained in replicate measurements usually agreed within 5-10%.
Cross-linking of CBS-PPase with glutaraldehyde was performed for 15 min at 25 °C in 25 мМ HEPES-KOH buffer, pH 7.5. The reaction was terminated by adding 1/10 volume of 1 M Tris/HCl, pH 7.5. Cross-linked samples were diluted to 0.3 mg/ml protein and separated by electrophoresis in 4-12% gradient polyacrylamide gels, using identical protein loads (9 µg per lane), on a Mini Protein II apparatus (Bio-Rad) in the Laemmli system [16] . Prestained PAGE rulers (Thermo Fisher Scientific) were used as molecular mass standards. The gels were fixed for 20 min with 10% acetic acid/25% isopropanol, stained for 40 min with 0.15% Coomassie G-250 and 0.024% CuSO4 in 10% acetic acid/30% ethanol, and destained with 5% acetic acid/15% ethanol.
Analytical ultracentrifugation was performed at 25 °C in a Spinco E instrument (Beckman Instruments) equipped with a computerized data collection unit, with scanning at 280 nm. Samples containing 10-20 µM proteins in the isolation buffer were preincubated for 6 h at 25 °C, unless otherwise indicated. MgCl2 and CoCl2 were not added to the samples containing EDTA. The sedimentation velocity was measured at 60,000 rpm, and sedimentation coefficients (s20,w) and molecular masses were estimated with the program SedFit [19] .
Nonlinear least-squares fittings were performed using the program SCIENTIST (Micromath), which allows use of differential and implicit equations. Equations 1 and 2, derived from Scheme 1, describe time-courses of activity (A) resulting from tetramer (T) dissociation into dimers (D). At dT/dt = 0, these equations describe the equilibrium activity as a function of enzyme concentration. AT and AD are specific activities of tetramer and dimer, respectively, T is the fraction of enzyme in tetrameric form at time t, [E]t is total enzyme concentration, expressed in monomers, ka and kd are the apparent rate constants for tetramer formation and breakdown, respectively. 
The time-courses of PPi hydrolysis by dhPPaseΔCDC, accompanied by first-order inactivation (dissociation) of the enzyme, were fit to Equation 3, where [P] is phosphate concentration, ki is the rate constant for enzyme inactivation, v0 is the initial rate of product formation, t is time, and a is the background signal of the instrument.
The full-length cpPPase model was constructed using cpPPase amino acid sequence and the structures of canonical Family II PPase of Bacillus subtilis and AMP-complexed cpCBS (PDB IDs: 2HAW and 3L31) [12] as templates. These dimeric structures were manually arranged, using UCSF Chimera v. 1.11.2 [20] , into a preliminary tetrameric form that was used as the input for Modeller v. 9.23 [21] operated in "automodel" mode with restrained known contacts within dimers.
Cross-linking experiments provided strong evidence for the tetrameric structure of CBS-PPase. Full-size dhPPase was reacted with glutaraldehyde, using a reaction time and cross-linker concentration that allowed only partial modification, in order to minimize intermolecular cross-linking. To estimate its extent, the reaction was performed at various protein concentrations, keeping in mind that only the intermolecular but not intramolecular reaction should depend on the concentration. The reaction products were separated by SDS-PAGE ( Fig. 2A ). Dimer and tetramer were the major cross-linked species, but for an unknown reason each migrated as two-three close bands. The intensity of the bands attributable to tetramer did not depend on protein concentration, ruling out intermolecular cross-linking.
Both deletion variants of dhPPase produced only dimers upon cross-linking ( Fig. 2B ,C). Crosslinked dhCDC, but not dhPPaseΔCDC dimer also migrated as two close bands in SDS-PAGE (Fig.  2C) , indicating that the regulatory part was responsible for the unusual behavior of full-size dhPPase in SDS-PAGE ( Fig. 2A) . Cross-linking of an equimolar mixture of dhCDC and dhPPaseΔCDC (20 µM each) yielded only homodimers of each separate protein, but not their heterodimer (Fig. S1 ).
Similar results indicating tetrameric structure were obtained with cnPPase and cpPPase (Fig.  S2 ). With ehPPase and elPPase, which lack DRTGG domain in their structures, reaction with glutaraldehyde under a variety of conditions (reaction time and glutaraldehyde concentration) yielded high-mass aggregates that did not enter polyacrylamide gel. Dimethyl suberimidate, at a concentration of up to 20 mM (2 h reaction time), generated a small amount of dimers with ehPPase ( Fig. S3) and no cross-linked products with elPPase (data not shown).
Sedimentation velocity measurements were performed with all full-size CBS-PPases and two deletion variants without regulatory or catalytic parts. Figure 3 shows representative data for the full-size dhPPase, and Table 1 summarizes the final results for three dhPPase forms and all other CBS-PPases, as generated by the SedFit program [19] . Sedimentograms revealed tetramers as the major species in all full-size CBS-PPases. The identification of the major species with s20,w of 9.3−10.2 S in dhPPase, cnPPase, and cpPPase as tetramers was based on the results of the crosslinking experiments and was consistent with the s20,w values reported for proteins of known oligomeric composition [22] . In contrast, all deletion variants, representing the catalytic or regulatory part of CBS-PPases, were predominantly dimeric. Species larger than tetramers were observed in minor amounts in the full-size PPases but were absent in their deletion variants.
Preincubation of protein samples for 0.2 h with 1 mM EDTA before sedimentation induced partial dissociation of tetrameric dhPPase into dimers and monomers (Fig. 3D ). The c(s) distribution was characteristic of a mixture of rapidly interconverting tetramers, dimers, and monomers -the tetramer peak was shifted to lower s values, and all peaks were asymmetrical, wide and not well separated. A longer preincubation with EDTA increased the fraction of monomers, as indicated by a further shift of the peaks to lower s values and appearance of a clear low-mass shoulder and stimulated, in addition, conversion of dimeric dhPPase into larger enzyme species (Fig. 3E) . Dimeric dhPPaseΔCDC was nearly completely dissociated into monomers in the presence of EDTA (Table 1) , whereas EDTA-treated dhCDC retained its dimeric structure ( Table 1 ). The allosteric inhibitor AMP and activator 5',5-P1,P4-diadenosine tetraphosphate (Ap4A) exhibited minor effects, if any, on the quaternary structure of dhPPase (Table 1) .
Changes in the oligomeric state of the catalytically active dhPPase forms could be conveniently monitored by activity measurements, because activities of the associated and dissociated forms were different and they converted into each other slowly on the time scale of the enzyme assay, like with canonical Family II PPases [9] . Dilution of dhPPase into the same medium induced its time-dependent inactivation until a constant non-zero activity value was attained (Fig. 4A) . Omission of Mg 2+ from the medium did not change the inactivation time-course (data not shown). EDTA, which depleted the enzyme from both Co 2+ and Mg 2+ , stimulated the inactivation and decreased the final activity level to nearly zero. The dhPPaseΔCDC variant was inactivated much faster upon dilution, and the inactivation was only slightly stimulated by EDTA (Fig. 4B) . The final level of activity attained after equilibration of the diluted enzyme increased with increasing enzyme concentration in the absence of EDTA but remained at a nearly zero level in the EDTA-containing medium (Fig. 5) .
The time-courses in Fig. 4 could be analyzed in terms of a model assuming reversible dissociation of CBS-PPase into two equal part (dimers with dhPPase and monomers with dhPPaseΔCDC). The data in Fig. 4 were fit to Eqns 1 and 2 with dD/dt set to zero, yielding parameter values summarized in Table 2 . Lowering the temperature to 0 °C slightly decreased kd and ka for both dhPPase and dhPPaseΔCDC with no or minor effect on Kd ( Table 2 ).
The following experiment indicated that the dilution-induced inactivation of dhPPase in the presence of EDTA was completely reversible. EDTA-treated dhPPase with a nearly zero activity (Fig. 4A ) was stripped of EDTA and reconstituted with Co 2+ by three cycles of a 10-fold concentration on a Centricon centrifugal filter and dilution with the buffer containing 100 μM CoCl2 and 2 mM MgCl2 instead of EDTA. This procedure reactivated dhPPase by 54% (i. e., to the level observed in Fig. 4A ) after prolonged incubation in the absence of ligands. The easy reversibility of dhPPase inactivation is consistent with EDTA-induced Co 2+ depletion and/or enzyme dissociation into inactive parts upon dilution.
The dilution-induced inactivation (tetramer dissociation) of dhPPase was less pronounced in the presence of AMP and, especially, ATP or Ap4A (Fig. 4A) . Accordingly, the dependences of the specific activity on enzyme concentration in the equilibrated system were shifted to lower concentrations in the presence of the adenosine phosphates (Fig. 5A) . Furthermore, the activity of dhPPase half-inactivated by incubation in the diluted state in the absence of ligands was completely restored upon a 70-min incubation with 5 µM Ap4A (Fig. 4A ). Of note, the nucleotides were ineffective with the full-size enzyme in the presence of EDTA or with dhPPaseΔCDC (data not shown). In terms of Scheme 1, nucleotide effects on Kd, at least for AMP and ADP, resulted mainly from a change in kd and increased with the size of the nucleotide ( Table 2) .
The effect of PPi on oligomer stability could not be tested in the same way because PPi is rapidly consumed by CBS-PPase. To overcome this difficulty, we estimated rates of enzyme dissociation from the time-courses of dhPPaseΔCDC-catalyzed PPi hydrolysis at different levels of enzyme saturation with substrate (MgPPi complex). Two MgPPi concentrations were used -5 and 100 µM, corresponding to 56 and 95% saturation, respectively, based on the Km value of 4 µM [13] . The choice of dhPPaseΔCDC for this experiment is explained by its propensity to dissociate on the time scale of the enzyme assay compared with the full-size enzyme. At the low concentration of the enzyme used in the activity assay (10 -9 M), dimer dissociation and, hence, enzyme inactivation are expected to proceed to near completion, according to Fig. 5 .
The measured time-courses (Fig. 6) were nearly linear at high and markedly nonlinear at low substrate concentrations, indicating that substrate stabilized dimer in the course of the enzymatic reaction. Substrate consumption was quite low by the end of the recording time (0.7% and 4.3%, respectively), ruling out substrate depletion as an explanation of the nonlinearity. Also, full-size dhPPase, which dissociates much slower (Table 2) , generated linear Pi accumulation curves under identical conditions (Fig. 6 ). Fig. 6 yielded inactivation rate constant (ki) of 1.53 and 0.28 min -1 for the dhPPaseΔCDC-catalyzed reaction at 5 and 100 µM MgPPi, respectively. The former ki value is consistent with the kd value for dhPPaseΔCDC dissociation estimated in the absence of substrate (144 h -1 , i.e. 2.4 min -1 ; Table 2 ), keeping in mind that 5 µM substrate partially protected the tetramer against dissociation, i.e., decreased ki. Based on the above values of the rate constant for dimer dissociation, one can conclude that bound substrate decelerated the reaction by approximately tenfold.
The results reported above indicate that CBS-PPases of five bacterial species are active as homotetramers. Three of these PPases contain the full set of domains (DHH, DHHA2, DRTGG, and two copies of CBS), and the remaining two lack the DRTGG domain. Both CBS-PPase types are regulated by monoadenosine phosphates (AMP, ADP, and ATP) that bind to CBS domains, but only the former type is activated by diadenosine polyphosphates [14] , important cellular alarmons [23] . Most other reported PPases are homohexamers of approximately 20-kDa subunits (prokaryotic Family I), homodimers of 32-70-kDa subunits (eukaryotic Family I, prokaryotic canonical Family II, and cation-transporting membrane PPases), or monomers (Family III). Formation of tetramers of 21-25-kDa subunits has been claimed in reports of a preliminary nature on Family I PPases from thermophilic and acidophilic bacteria [24] [25] [26] . Conclusive evidence for homotetramer formation has been only reported for a unique Family I vacuolar PPase from Trypanosoma brucei, which has an additional EF hand domain in its 47-kDa subunit [27] . CBS-PPase is thus unique among different PPases by the type and size of its oligomer structure.
The first evidence for the tetrameric structure of CBS-PPases came from cross-linking experiments, which revealed tetrameric species after reaction of three CBS-PPases with glutaraldehyde. Interestingly, the cross-linking reaction using glutaraldehyde produced spurious results with the DRTGG-lacking CBS-PPases (ehPPase and elPPase), and another cross-linker, dimethylsuberimidate, produced only dimers. Notably in this regard, the efficiency of a crosslinking reaction is determined by the existence of a pair of suitably located amino groups across the subunit interface in a dimeric protein and at least two such pairs at different interfaces in a tetramer. The lack of such amino group pairs in ehPPase and elPPase seems to be the most likely explanation of the low yield of their cross-linked tetramers. This conclusion is supported by a comparison of the lysine content and distribution in different CBS-PPases (Table S3 ). The total number of lysines per subunit is, on the average, 3.4 times larger in the DRTGG domain-containing CBS-PPases than in their DRTGG domain-lacking counterparts. This difference is mainly due to the low lysine content in the CBS domains of the latter CBS-PPases.
Sedimentation experiments were therefore crucial to confirm tetrameric organization of the CBS-PPases, including those that could not be cross-linked. The critical point in this type of analysis was conversion of the sedimentation coefficient into molecular mass. The samples containing EDTA-treated CBS-PPase (Fig. 3D) were most informative in this regard as they contained a dynamic mixture of all three oligomeric forms -tetramer, dimer, and monomer. An additional reference point was provided by the canonical sgPPase that mimics the catalytic part of CBS-PPase, dhPPaseΔCDC, because its dimeric structure was confirmed by X-ray crystallography [8] .
Dilution of CBS-PPase solution caused reversible dissociation and inactivation of the tetrameric enzyme. A question to answer is whether the dissociation lead to dimers or monomers. We used the simplest model assuming dimer formation, but, given the relatively high point scatter in Fig. 5A , the models involving tetramer  monomer or tetramer  dimer  monomer equilibrium could not be ruled out. The identification of the oligomeric species generated upon dilution of the full-size enzyme thus requires further investigations.
Reversible dissociation inactivates CBS-PPase and might therefore contribute to regulation of its activity in cells. However, this regulatory mechanism is operable only if enzyme concentration is sufficiently low to allow oligomer dissociation in changing conditions. CBS-PPase concentration in a cell can be estimated based on the rate of PPi production in bacteria. The upper limit for this rate was estimated to be 82.5 mM/min [1] , which can be counterbalanced by 3 µM PPase with a turnover number of 450 s -1 , characteristic of CBS-PPase activated by ATP. At this concentration, CBS-PPase is dissociated by approximately 10% (Fig. 5) and this percentage increases with cellular PPi concentration because PPi stabilizes the tetramer (Fig. 6) . However, this mechanism of activity regulation would be far less efficient compared to the mechanism based on adenine nucleotide binding [11, 13] .
As the published model of cpPPase dimer [12] cannot explain tetramer formation, we have produced alternative models using a Modeller program. The following considerations were taken into account when choosing the principle of tetramer organization. The N-and C-termini of both monomers are located on the same side of the dimeric regulatory insert [12] , whereas the two loops within which it is to be inserted are on the opposite sides of the dimeric bsPPase [10] , which mimics the catalytic part of CBS-PPase. Hence, two dimers of the catalytic part should be arranged as a tetramer in such a way to allow close proximity of two loops belonging to different dimers for fusing with the regulatory domains. Furthermore, two dimers of the catalytic parts should not form a tight contact with each other, and the same refers to the regulatory parts, as none of these parts forms a tetramer separately. In other words, strong contacts that maintain tetramer stability should be formed by catalytic and regulatory parts belonging to different pairs of monomers.
Two modelled structures that best satisfied these requirements were selected. The model that we prefer (Model I) is shown in Fig. 7 , and the other one (Model II) is shown in Fig. S4 . Both models can be described as a dimer of dimers. Model I positions the regulatory part nearly perpendicular to the plane going through the contact zone of two DHH domains that interact by forming an extended β-sheet both in the modelled structures and in bsPPase dimer [10] . The DHH domains of monomers 1 and 2 strongly interact with each other but not with monomers 3 and 4, and vice versa (Fig. 7 , left top panel). In contrast, the regulatory parts of monomers 1 and 2 interact crosswise -only with the regulatory parts of monomers 3 and 4, respectively. Accordingly, deletion of either catalytic or regulatory part should prevent tetramer formation. In Model II, each pair of the catalytic domains is rotated by 180° around the axes crossing the tetramer in its largest dimension and by 30° around a perpendicular axis. This allows additional contact of the DHH domains through the α-helices formed by residues 396−409. DHHA2 domains do not participate in oligomeric contacts in Model II, but may form weak contacts in the closed conformation of the catalytic domains in Model I.
Both models are consistent with known positive co-operativity of CBS-PPase catalysis and regulation. Kinetic co-operativity (decreased Km value for interaction with the second substrate molecule) [13] apparently results from interactions in pairs of monomers 1-2 and 3-4, as only this type of interaction occurs in co-operative dimeric Family II PPases devoid of the regulatory CBS domains [28] . AMP, ADP, and ATP are bound by CBS-PPase co-operatively with a stoichiometry of one per monomer [14] . This co-operativity clearly results from interactions within the pairs of the regulatory parts of monomers 1-3 and 2-4, but not between these pairs because they lie far apart in CBS-PPase tetramer. This interpretation is consistent with the non-cooperative binding of adenosine polyphosphates that demonstrated a stoichiometry of one per two monomers [14] because each molecule of this ligand occupies both regulatory sites in each of the two distant monomer pairs [12] . Dilution-induced dissociation of CBS-PPase apparently yields dimers 1-3 and 2-4, which are inactive because the DHH domains cannot form the catalytically competent association via extended β-sheet.
It was of interest to compare CBS-PPase models with published structures of other CBS domain-containing proteins. Most of them form homodimers, with the CBS domains participating in subunit interactions [29] , but higher order structures have been also reported. Human cystathionine β-synthase (CBS) contains two catalytic and two CBS domains and appears to be the closest structural analog of CBS-PPase. CBS is homotetrameric in solution. Its structure predicted from determined structure of a dimeric mutant form [30] is a dimer of dimers, wherein strong interactions between catalytic domains in dimer are buttressed by CBS domain-mediated interactions within and between dimers. Like with CBS-PPase, removal of the CBS domains renders CBS dimeric [31] . In homotetrameric putative acetoin dehydrogenase TTHA0829 from Thermus thermophilus, the constituting domains have opposite roles, compared to the CBS-PPase models. Its core domain (aspartate-kinase chorismate-mutase tyrA), together with CBS2 domain, participates only in dimer formation, like the CBS and DRTGG domains of CBS-PPase model, whereas two dimers interact only through CBS1 domains that have -helical and loop extensions [32] . Homooctameric inosine monophosphate dehydrogenase is a dimer of plane tetramers and exists in two forms with different contacts between the tetramers -either predominantly between the catalytic domains or predominantly between the regulatory CBS domains [33] . In the latter case, pairs of the CBS domains (Bateman modules) belonging to different tetramers stack in an unusual antiparallel mode. In tetrameric inosine monophosphate dehydrogenase, the CBS domains do not participate in oligomerization [34] . These structures emphasize variable roles of CBS domain in protein oligomerization.
Further structural studies of CBS-PPase are clearly needed to test the validity of our structural models and provide insight into its regulatory mechanism, which apparently depends on interactions between domains and subunits. Crystallization attempts with CBS-PPase have met with serious difficulties, presumably associated with enormous flexibility of its molecule, the problem common to all CBS domain-containing proteins. Perhaps, different approaches should be accessed in future investigations towards the three-dimensional structure of CBS-PPase. Table 1 Sedimentation velocity data. Values separated by slash refer to different oligomeric forms. Samples contained 10 µM full-length CBS-PPases or 20 µM their truncated variants. AMP (200 µM) and Ap4A (50 µM) were added to protein samples immediately before the sedimentation run; 1 mM EDTA was added immediately before the run (the time of incubation with EDTA during rotor acceleration was roughly estimated as 0.2 h), 3 h prior to the run, or 6 h prior to the run.
Ligand/additive s20,w a , S Fraction distribution a , % [20] . The theoretical masses of the closest oligomeric forms (tetramer, dimer or monomer) are shown in parentheses and marked as T, D or M, respectively. c From Parfenyev et al. [9] . . Aliquots were withdrawn in time, and activity was measured in the standard assay medium. The lines represent the best fits for Eqns 1 and 2 and were created using parameter values found in Table 2 . Activity extrapolated to zero time (typically 290 s -1 ) was taken as 100 % for each inactivation curve. The dashed lines ending with symbols show reactivation achieved in 60-65 min upon addition of 5 µM Ap4A to the sample incubated for 60 min without additions (---♦; the final activity value was corrected for activation due to Ap4A carryover to the activity assay medium), or upon EDTA removal by centrifugal ultrafiltration from the sample incubated for 60 min with EDTA (---•). The metal salts were omitted in the incubations with EDTA. After a 2-h equilibration, aliquots were withdrawn and enzymatic activity was measured in the standard assay medium. The lines represent the best fits of Eqns 1 and 2 with dαD/dt = 0, using the parameter values found in Table 2 . Activity extrapolated to infinite enzyme concentration was taken as 100 % for each fitted curve. Fig. 6 . Time-courses of Pi production by dhPPaseΔCDC and dhPPase measured at two substrate concentrations. Total PPi concentration was 280 μM or 14 μM, corresponding to 100 and 5 μM MgPPi, as indicated below the curves. Final enzyme concentrations were 0.5 nM (leftmost curve) or 1 nM (two other curves). Stock enzyme solution contained 5.8 µM dhPPaseΔCDC or 1 µM dhPPase in the isolation buffer; the enzymes were by 37 and 79 % dimeric or tetrameric, respectively, in the stock solution, according to Fig. 5 . The phosphate analyzer was operated in a "short dead-time" mode (see Materials and methods) at a sensitivity of 5 μM Pi per recorder scale. 