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Magnesium-induced linear self-association of the FtsZ bacterial cell division protein monomer. The primary steps for FtsZ assembly.

The bacterial cell division protein FtsZ from Escherichia coli has been purified with a new calcium precipitation method. The protein contains one GDP and one Mg(2+) bound, it shows GTPase activity, and requires GTP and Mg(2+) to polymerize into long thin filaments at pH 6.5. FtsZ, with moderate ionic strength and low Mg(2+) concentrations, at pH 7.5, is a compact and globular monomer. Mg(2+) induces FtsZ self-association into oligomers, which has been studied by sedimentation equilibrium over a wide range of Mg(2+) and FtsZ concentrations. The oligomer formation mechanism is best described as an indefinite self-association, with binding of an additional Mg(2+) for each FtsZ monomer added to the growing oligomer, and a slight gradual decrease of the affinity of addition of a protomer with increasing oligomer size. The sedimentation velocity of FtsZ oligomer populations is compatible with a linear single-stranded arrangement of FtsZ monomers and a spacing of 4 nm. It is proposed that these FtsZ oligomers and the polymers formed under assembly conditions share a similar axial interaction between monomers (like in the case of tubulin, the eukaryotic homolog of FtsZ). Similar mechanisms may apply to FtsZ assembly in vivo, but additional factors, such as macromolecular crowding, nucleoid occlusion, or specific interactions with other cellular components active in septation have to be invoked to explain FtsZ assembly into a division ring.

The FtsZ protein, which is found in all known eubacteria and archaea as well as in chloroplasts with the exception of the Chlamydiae (1, 2), plays a central role in bacterial cell division (3)(4)(5). During the life of a cell, FtsZ molecules remain in the cytoplasm until the time previous to division when they localize at the midcell point, forming a ring at the inner face of the cytoplasmic membrane where the septum will form (6,7). The interactions of FtsZ with other septator proteins that are functionally relevant, such as PBP3 and FtsA, are being discovered (reviewed in Refs. 4, 5, 8, and 9). Several regulators of FtsZ ring formation have been found that either prevent/inhibit polymer formation (like the SOS response protein SulA (10), the MinCDE complex (11), and more recently, the ErzA protein from Bacillus subtilis (12)) or stabilize the FtsZ polymer (like the membrane protein ZipA (13,14)). Despite this information, FtsZ ring structure and assembly are still poorly understood.
FtsZ binds guanine nucleotides and has GTPase activity (15)(16)(17), which is dependent on protein concentration and probably is linked to self-association equilibria (15,16,18). Multiple sequence alignment and secondary structure predictions indicate that bacterial FtsZ is a homolog of eukaryotic tubulins (19,20). The x-ray crystal structure of the FtsZ from the archaeon Methanococcus jannaschii (21) and the electron crystallographic model structure of the ␣␤-tubulin dimer (22) have similar folds.
The nature of the polymers that FtsZ forms in vivo is unknown. FtsZ self-assembles in vitro to form filaments and rings (23)(24)(25)(26)(27)(28). Electron microscopy studies of FtsZ filaments show that the longitudinal spacing in the polymers is close to the 4-nm spacing found in microtubules (23,24). Similarly to microtubules (29) FtsZ assembly depends on divalent cations and guanine nucleotides and involves GTP hydrolysis (26). These physiological ligands have an important role in the control of the structure, associations, and assembly of tubulin. A high affinity Mg 2ϩ bound to the non-hydrolyzable GTP of ␣-tubulin regulates the structural stability of the ␣␤ heterodimer (30). The hydrolyzable GTP bound to ␤-tubulin regulates tubulin assembly and microtubule dynamic instability, where several lower affinity Mg 2ϩ -binding sites are also implicated (31)(32)(33)(34)(35)(36). Tubulin exists into two conformational states as follows: the active (straight) conformation with GTP bound in the ␤-subunit (GTP-tubulin) which favors self-assembly into the physiological microtubule polymers (37)(38)(39), and the inactive (curved) GDPbound state (36) which favors tubulin assembly into double rings over microtubule formation (33, 40 -43).
The aim of the present work was to study FtsZ self-association in solution and the specific role of divalent cations. The processes of association and assembly seem to be relevant in the biological function of FtsZ. We have addressed the question whether the smallest association state of FtsZ in solution is a monomer or an oligomer. We have quantitatively characterized the effect of Mg 2ϩ and Ca 2ϩ , ionic strength, and guanine nucleotides on the self-association of Escherichia coli FtsZ by means of analytical centrifugation (sedimentation equilibrium and velocity), and we have investigated the nature of the oligomeric species.

Chemicals
CaCl 2 , MgCl 2 , EDTA, Tris, and glycerol, all of analytical grade, were from Merck. Guanine nucleotides, GDP and GTP, were from Sigma and Roche Molecular Biochemicals, respectively. N-Acetyltyrosinamide was from Serva; guanidine hydrochloride was from Calbiochem, and fluorescein 5Ј-isothiocyanate was from Molecular Probes Inc. Other analyt-ical grade chemicals were from Merck or Sigma, except as otherwise indicated.

FtsZ Purification
In order to optimize FtsZ purification, in terms of protein purity, yield, and nucleotide bound, two methods have been compared. The first of them (Method 1) is a modification of the standard FtsZ purification by precipitation of this protein with ammonium sulfate followed by ion exchange (17,44). The second one (Method 2), developed in this work, takes into account the dependence of FtsZ solubility on divalent cations (28) and the methods of tubulin isolation by cycles of Mg 2ϩ -induced association (45).
The steps of cell growth and lysis were essentially the same in both methods. Briefly, FtsZ was overproduced in E. coli BL21(DE3) transformed with the plasmid pMFV56, which contains the E. coli ftsZ gene cloned in a pET-28a vector (Novagen). Cells were grown at 37°C in 1 liter of LB culture media (46) with kanamycin, to an A 600 of 0.5-0.6. Expression was induced with 0.4 mM isopropyl-1-thio-␤-D-galactopyranoside during 90 min (in Method 2, 180 min). Cells were harvested by centrifugation (10,000 ϫ g, during 15 min at 4°C), and pellets were resuspended either in 50 ml of Tris glycerol buffer (Tris glycerol buffer, 50 mM Tris-HCl, 50 mM KCl, 1 mM EDTA, 10% glycerol, pH 8.0) (Method 1) or in 20 ml of PEM buffer (PEM buffer, 50 mM PIPES, 1 5 mM MgCl 2 , 1 mM EDTA, pH 6.5) (Method 2), and then the cells were lysed on ice using an MSE sonicator. The soluble fraction, containing the FtsZ protein, was separated from the cell debris by centrifugation (100,000 ϫ g, for 2 h at 4°C).
In Method 1, the protein from the soluble fraction was precipitated with 30% ammonium sulfate. The sample was centrifuged (50,000 ϫ g, for 35 min at 4°C), and the pellet was resuspended in 5 ml of Tris glycerol buffer. In Method 2, the protein was extracted by two cycles of Ca 2ϩ -induced precipitation. In the first cycle, 1 mM GTP and 20 mM CaCl 2 were added to the solution, and the sample was incubated during 15 min at 30°C and then centrifuged (11,000 ϫ g, for 15 min at 4°C). The pellet, which contains FtsZ, was resuspended in 20 ml of PEM buffer and centrifuged as in the previous step to eliminate any insoluble material. The supernatant was subjected to an identical second cycle of Ca 2ϩ -induced precipitation. The final FtsZ pellet was resuspended in 5 ml of PEM buffer.
The materials from the previous step, highly enriched in FtsZ protein but contaminated with nucleic acid, were further purified by anion exchange chromatography using a 5-ml Hi-Trap Q-Sepharose column (in an Amersham Pharmacia Biotech FPLC system, at a flow rate of 1 ml/min), equilibrated in Tris glycerol buffer either without adding MgCl 2 (Method 1) or with 5 mM MgCl 2 (Method 2). FtsZ, due to its negative charge, was retained in the column and was eluted with a 0 -100% gradient of 1 M KCl in the same buffer. In both cases the protein eluted around 500 -600 mM KCl. The purified FtsZ aliquots were dialyzed either in Tris-200KCl buffer with 1 mM EDTA and 10% glycerol (Method 1) or in Tris-250KCl buffer with 10% glycerol (Method 2) and stored at Ϫ80°C. We have chosen Method 2 to isolate FtsZ because of the following: (i) the purity in SDS-PAGE was Ͼ98%, whereas was only Ͼ95% for Method 1; (ii) the stoichiometry of nucleotide binding was also higher for FtsZ purified using Method 2 (1.1 Ϯ 0.1 mol of bound GDP/mol of FtsZ, whereas only 0.6 Ϯ 0.1 for FtsZ Method 1); and (iii) the protein yield (ϳ40 mg of protein/liter culture) was higher than that obtained with Method 1 (ϳ20 mg/liter). The GTPase activity of FtsZ purified by both methods was ϳ120 nmol P i ⅐min Ϫ1 ⅐mg Ϫ1 , which corresponds to 5 molecules of GTP hydrolyzed at 37°C per min and molecule of FtsZ (at 5 M protein).

Protein and Nucleotide Quantitation
The determination of FtsZ concentration is not straightforward because the E. coli protein has few aromatic residues (only three tyrosines and no tryptophans (47)), and the bound nucleotide dominates the UV absorption spectrum. The nucleotide content of the protein was determined, after extraction with perchloric acid, by HPLC with an internal standard of guanosine, using an extinction coefficient of guanine nucleotides at 254 nm of 12,400 M Ϫ1 cm Ϫ1 (31,48). Two methods were employed to measure accurately the concentration of FtsZ at a given wavelength in the desired working buffer. This is important, for exam-ple, to measure the protein concentration gradients in analytical centrifugation experiments.
UV Spectroscopy-FtsZ concentration was first measured by taking its UV spectra in 6 M GdnHCl. From these spectra the concentration of the protein (which has three Tyr) was calculated after subtracting the GDP contribution at 280 nm, using the following molar extinction coefficients: 1280 M Ϫ1 cm Ϫ1 for Tyr (49,50) and 8,100 M Ϫ1 cm Ϫ1 for GDP. 2 Comparison with the spectra in 50 mM Tris-HCl, 50 mM KCl, pH 7.5, buffer yielded practical molar extinction coefficients for the native protein-GDP complex at 280 (13,800 Ϯ 900 M Ϫ1 cm Ϫ1 ) and 254 nm (17,100 Ϯ 1,100 M Ϫ1 cm Ϫ1 ). Note that these coefficients are only valid for FtsZ which has 1.1 Ϯ 0.1 molecules of bound GDP.
Fluorescence Spectroscopy-The intrinsic fluorescence signal of the Tyr residues of FtsZ was used to quantify FtsZ at low concentrations, since the emission spectra of the protein showed no fluorescent contaminants. The spectrofluorimetric measurements of FtsZ samples (diluted to Ͻ5 M to avoid inner filter effects) in 6 M GdnHCl were performed at 20°C in a Shimadzu RF-540 fluorimeter ( ex ϭ 275 nm; em ϭ 305 nm), with 0.5 ϫ 1 cm cells, using N-acetyltyrosinamide (1-40 M) as standard. This simple procedure gave the same results as UV spectroscopy within experimental error.
Quantitative amino acid analysis (51) was used to confirm the FtsZ quantitation methods previously described. The amino acid content of the protein was measured after acid hydrolysis (6 N HCl with norleucine as internal standard, at 110°C, during 20 h) in a Beckman 630 automatic amino acid analyzer. The amino acid composition, in particular the number of phenylalanine and tyrosine residues, was in agreement with that calculated from the E. coli FtsZ gene sequence (47). By taking into account the total amino acid residues of FtsZ from sequence (as well as using the content of several reference residues, like alanine, glutamine ϩ glutamate, and asparagine ϩ aspartate), the number of moles of FtsZ in the original sample was calculated. This method understimated the FtsZ concentration by around 10% with respect to the spectrophotometric method. Finally, a colorimetric protein assay, using the Coomassie Brilliant Blue G-25 reagent (Bio-Rad) had, in our hands, a poor reproducibility and overestimated the FtsZ content around 20%.

Determination of FtsZ-bound Mg 2ϩ and Ca 2ϩ
The divalent cation content of FtsZ was measured in aliquots of 2 ml (25-50 M) either directly from the stock solution stored in Tris glycerol buffer or after equilibrium dialysis in Tris-50KCl buffer (Tris-50KCl buffer, 50 mM Tris-HCl, 50 mM KCl, pH 7.5) at 20°C. Measurements were performed by atomic absorption spectroscopy (52) and by inductively coupled plasma optical emission spectroscopy (53). Standard solutions of Mg 2ϩ were purchased from Sigma, and Ca 2ϩ stock solutions were prepared from a known amount of CaCO 3 . Only plasticware containers were employed, and all the solutions were prepared with fresh MilliQ grade water (Millipore Inc.).

Specific GTPase Activity
The GTPase activity of FtsZ (5 M in Tris-100KCl buffer with 5 mM MgCl 2 and 0.5 mM GTP) was measured at 37°C by quantitation of the inorganic phosphate with the malachite green-molybdate reagent (54,55).

FtsZ Polymerization
The formation of large FtsZ polymers, after incubation of the protein (ϳ12.5 M, in MES buffer (MES buffer, 50 mM MES-KOH, 50 mM KCl, pH 6.5)) with 1 mM GTP and 10 mM MgCl 2 at 30°C, was followed by 90°angle light scattering monitored with a High-Tech Scientific SS-51 stopped-flow device with a fluorescence detection system, employing 350 nm incident light and a filter with a cut-off below 330 nm for the scattered light. The dead time of the instrument was 2.5 ms, calibrated with the reaction of N-bromosuccinimide with N-acetyltryptophanamide (56). Light scattering at 90°was also monitored, using a Shimadzu RF-540 fluorimeter. FtsZ polymers were adsorbed to carbon-coated grids, negatively stained with 2% uranyl acetate, and inspected with a Jeol 1200 EX electron microscope.

Analytical Centrifugation
Sedimentation equilibrium experiments were performed at 20°C in an Optima XL-A (Beckman-Coulter) analytical ultracentrifuge equipped with UV-visible optics, using An50Ti and An60Ti rotors, with either 12-mm double sector or six-channel centerpieces of Epon charcoal. FtsZ (loading concentration between 5 and 100 M) was equilibrated in Tris-50KCl buffer, with 20 M GDP and the desired concentration of divalent cations and KCl, except otherwise indicated (i.e. experiments at pH 6.5 in PIPES or MES buffers). Omission of GDP in the buffer resulted in the partial dissociation of the GDP bound to the protein within the experiment time interval. Short column (50 -60 l), low speed sedimentation equilibrium was performed at three successive speeds (5,8, and 15,000 rpm), and the equilibrium scans were taken (typically after 10 h) at the most appropriate wavelength (240, 280, or 290 nm), depending upon the FtsZ concentration. Attention was given to the conservation of protein mass in soluble form during the experiment, and control sedimentation velocity experiments showed no aggregation within a 12-h time interval. In all cases, the base-line signals were measured after high speed centrifugation (40,000 rpm). Whole-cell apparent weight average molecular weights of FtsZ were obtained using the program EQASSOC (57). The partial specific volume of FtsZ was 0.738 ml/g, calculated from the amino acid composition with the program SEDNTERP (retrieved from the RASMB server (58)). In order to quantitatively characterize the effect of Mg 2ϩ on FtsZ self-association, the dependence of the weight average molecular weight (M w ) of FtsZ on protein concentration was calculated 3 from the local slopes of the transformed linearized data (ln c versus r 2 ) at defined radial intervals (57). Data from different loading protein concentrations and equilibrium speeds were superimposed and averaged over an interval of Ϯ0.1 log concentration units (59).
Sedimentation velocity experiments were performed at 20°C on parallel samples at 30,000, 40,000, and 50,000 rpm. Sedimentation coefficients were calculated either from the rate of movement of the solute boundaries, using the program XLAVEL (supplied by Beckman), or by globally fitting of multiple sedimentation velocity profiles with the program SVEDBERG (retrieved from the RASMB server (60)). From both methods we obtained similar apparent sedimentation coefficient values, which correspond to the maxima of the distribution of the apparent sedimentation coefficients, g(s*), calculated with the program DCDT (61), also taken from the RASMB server. The average sedimentation coefficients were calculated by integration as shown in Equation 1.

Nonlinear Least Squares Modeling of Data
Different self-association models containing adjustable parameters (see under "Theory") were fitted to the weight average molecular weight versus concentration data via Marquardt-Levenberg minimization of 2 (63). Goodness of fit was evaluated from the statistical distribution of 2 for an identical number of degrees of freedom (see Chapter 14 of Ref.

Circular Dichroism
The far-UV circular dichroism spectra of FtsZ (5-50 M, equilibrated in Tris-50KCl buffer, with 20 M GDP and a known concentration of MgCl 2 ) were acquired at 20°C in a JASCO J720 dichrograph, using 0.1and 0.01-cm cells. Changes in FtsZ secondary structure were estimated by deconvolution of the CD spectra using Yang (64) and LIMCOMB (65) methods, as described (19).

Nucleotide Exchange
The nucleotide bound to FtsZ was exchanged by incubating GDP-FtsZ with a large molar excess of GTP and further removal of the unbound nucleotides by gel filtration, as described for tubulin (48). In the case of FtsZ, the gel filtration step was performed in Tris-50KCl buffer, with only 50 M GTP, which permits us to follow the protein with the absorption optics in an analytical centrifuge.

Other Methods
The purity of FtsZ was checked by SDS-polyacrylamide gel electrophoresis performed essentially as described (19). Molecular weight markers from Amersham Pharmacia Biotech were employed. Gels containing 10% polyacrylamide were stained with Coomassie Blue. Microdensitometric scans of the gels were obtained with an Scanjet 6100c/T scanner (Hewlett-Packard) and analyzed with the program Scion Image (National Institutes of Health, Bethesda).

Magnesium-dependent Indefinite Protein Self-association-
Consider a protein molecule (A) with molar mass M 1 , that self-associates according to the following series of stepwise indefinite self-association equilibria shown in Equation 2.
We may define the apparent stepwise association constant K i as shown in Equation 3, where i ϭ 2, 3, 4, . . . , and c i is the molar concentration of i-mer. According to linkage theory (67)(68)(69), the effect of Mg 2ϩ on the equilibrium constant is shown in Equation 4.
where K i * is the value of K i at a reference Mg 2ϩ concentration c Mg * (1 mM in the present treatment), and n i is the net uptake of Mg 2ϩ upon the addition of 1 mol of FtsZ to a mole of (i Ϫ 1)-mer, which remains constant within the Mg 2ϩ concentration range employed. It will be assumed in the present treatment that n i ϭ n for all i and subsequently shown (see "Results") that this assumption satisfactorily accommodates the data.
It follows from Equation 3 that the concentration of each species is related to the monomer concentration by Equation 5, where Z i is a cumulative association constant shown in Equation 6.
The standard state free energy change associated with the assembly of one i-mer from i protomers is given by Equation 7.
The total concentration of protein expressed in moles of protomer per liter is then given by Equation 8.
Theoretical consideration of the energetics of indefinite linear self-association (70) suggests that the value of K i should tend toward an asymptotic limit K inf at sufficiently large oligomer size. For convenience, we denote by F i the dimensionless ratio K i /K inf . Upon substitution of variables, Equation 8 may then be rewritten as Equation 9.
Given the values of c tot , K inf , and the F j , Equation 9 may be solved (either analytically or numerically) for c 1 and subsequently for all c i . The weight average molecular weight may then be obtained from Equation 10.
Empirical Models for the Dependence of F i (or K i ) on i-The following two functions provide a gradual variation of K i from K 2 toward an asymptotic limit of K inf in the limit of large i.
For an inverse decay model we get Equation 11.
For an exponential decay model we get Equation 12.
where J ϭ K 2 /K inf , and ␣ and ␤ are empirical parameters characterizing the decay in their respective models. It will be shown subsequently that the dependence of association constants on the size of the aggregate derived from analysis of the experimental data is insensitive to the particular empirical model employed, so long as that model accurately describes the observed dependence of M w /M 1 on total protein concentration.
Simulation of the Dependence of s w /s 1 on the Concentration of Protein-The weight average sedimentation coefficient is related to the sedimentation coefficients of individual sedimenting species, s i , by Equation 13.
Estimates of c i are obtained from the best fit of equilibrium self-association models to the dependence of M w /M 1 on protein and Mg 2ϩ concentration, as described above, on the assumption that chemical equilibrium is achieved rapidly with respect to net transport in the sedimenting system. Estimates of s i are obtained from structural models of oligomers together with simple calculations using Kirkwood's frictional theory (71,72). The relationship between the sedimentation coefficient and oligomer size for different aggregate geometries (single and double-stranded linear oligomers) for identical spherical subunits is given by Equation 14 (71,72).
where s i is the sedimentation coefficient for i associated subunits in the oligomer, s 1 is the sedimentation coefficient of the monomer, R S is the Stokes' radius of FtsZ monomer (2.3 nm) and R jk is the distance between subunits j and k, for all sub- were compared with those based upon the simple approximation that all oligomers are spherical (72).

Functional Properties of Purified FtsZ-Overproduced E. coli
FtsZ was purified with a new method employing cycles of Ca 2ϩ -induced protein precipitation that was based on tubulin purification procedures. This method compares with advantage to conventional FtsZ purification protocols (see "Experimental Procedures"). Purified FtsZ (Fig. 1, panel A) has 1 mol of bound GDP (but no GTP or GMP) and 1 mol of bound Mg 2ϩ per mol of protein, negligible Ca 2ϩ bound, 4 and it has functionally normal GTPase activity. The unusual UV absorption spectrum of purified FtsZ (Fig. 1, panel B), corresponding to its chemical composition, is dominated by the contribution of bound GDP. The vibrational absorption bands of phenylalanine (251, 257, 263, and 267 nm (49)), the absorption maximum of tyrosine (275 nm), and the lack of Trp residues can be appreciated.
FtsZ assembly was induced by GTP and Mg 2ϩ in MES buffer at 30°C and was monitored by stopped-flow light scattering. The polymers formed in a few seconds, disassembled upon GTP consumption (Fig. 1, panel C), and did not form with 1 mM GDP to 10 mM MgCl 2 (not shown). The polymers observed with negative stain electron microscopy consisted of thin filaments of variable length and curvature. They were apparently single- 4 Our divalent cation-free buffers typically contain 0.9 Ϯ 0.2 and Ͻ10 Ϯ 2 M residual Mg 2ϩ and Ca 2ϩ concentrations, respectively. and double-stranded, ϳ5 and ϳ10 nm wide, respectively (Fig.  1, panel D). FtsZ polymer solutions had a very small turbidity (typically ⌬A 350 nm,1 cm Յ0.05), like in the case of 6-nm actin filaments (73,74). These observations are qualitatively compatible with a recent report on the assembly of ϳ7-nm wide FtsZ polymers (27) and thin filaments previously observed (75). However, significantly thicker FtsZ polymers, such as ϳ20 nm wide polymers (27), filaments induced by DEAE-dextran (23), or Ͼ50 nm wide Ca 2ϩ -induced protofilament bundles (28), if they had been abundant in our solution, would have given a larger turbidity signal, similarly to microtubules (76).
The FtsZ Monomer-The nature of the smallest association state of FtsZ was studied under different solution conditions. In Fig. 2, panel A, the sedimentation equilibrium gradient of 8 M FtsZ equilibrated in Tris-50KCl buffer (ϳ1 M residual free Mg 2ϩ ) is shown. The experimental data are best fit by a single exponential gradient with an apparent weight average molecular weight of 40,800 Ϯ 2,400, which corresponds to the sequence molecular weight of FtsZ monomer (40,300). In sedimentation velocity experiments the same FtsZ sample sediments as a single homogeneous species of 3.4 Ϯ 0.1 S (inset in Fig. 2, panel A), which is compatible with a globular FtsZ monomer of relative frictional ratio of 1.1 Ϯ 0.1. At 25 M FtsZ, the M w was slighty higher (51,000), and the residuals distribution of the best fit was typical of an associating system (not shown). This behavior was confirmed with experiments at higher protein concentration (see Fig. 3, panel A), and it is in agreement with the increase of FtsZ sedimentation coefficient with protein concentration (see below, Fig. 6, panel B). From these experiments we concluded that FtsZ in solution is a globular monomer with a tendency to self-associate.
The secondary structure of FtsZ was checked by circular dichroism (CD) spectroscopy (Fig. 2, panel B). The spectra, characteristic of an ␣/␤ protein, showed an small increase (ϳ10%) in the 210 -220 nm CD signal with 1-10 mM MgCl 2 with respect to ϳ1 M cation, which is compatible with an ϳ5-10% increase in the ␣-helical content induced by mM MgCl 2 . This CD change was independent on FtsZ concentration (5-50 M range, not shown), which suggests that it is linked to cation binding more than to protein association.
Mg 2ϩ -induced FtsZ Self-association-The effect of Mg 2ϩ on the self-association of FtsZ was studied by means of low speed sedimentation equilibrium over a broad range of protein (5-100 M) and Mg 2ϩ (ϳ1 M-10 mM) concentrations, in Tris-50KCl buffer at 20°C. Sedimentation equilibrium gradients of FtsZ at a single protein concentration (75 M) and multiple Mg 2ϩ concentrations are shown in Fig. 3. In order to show the trend of the data, the corresponding best fit gradients for a single sedimenting species are also plotted. It is apparent upon inspection of the graphs that the molecular weight of FtsZ is increasing with increasing Mg 2ϩ concentration. The protein gradient at 1 M Mg 2ϩ is very shallow, and the single species best fit M w is ϳ69,000, a value 1.7 times larger than the corresponding to the sequence monomer molecular weight. At the highest Mg 2ϩ concentration (10 mM) the gradient is steep, corresponding to a best fit M w of ϳ390,000, but with a poor fit that indicates that the sample is heterogeneous in size. The dependence of M w upon protein and Mg 2ϩ concentrations, calculated as described under "Experimental Procedures," from these and many other data sets (not shown), is plotted in Fig. 4 (panel A). At high protein and Mg 2ϩ concentrations, M w approaches ϳ10 times that of the FtsZ monomer, and there is no hint of any asymptotic upper limit to M w . Such behavior is characteristic of an indefinite self-association.
Model Analysis of FtsZ Association-The empirical models for indefinite self-association described above under "Theory" (Equations 4 and 9 -12) provide a satisfactory description, to within experimental precision, of the combined experimental data (M w /M 1 versus [FtsZ] at four Mg 2ϩ concentrations: 1, 2, 5, and 10 mM) with four adjustable parameters. Two of the parameters, the dimensionless ratio J (ϵK 2 /K inf ) and the decay parameter (either ␣ or ␤), describe a gradual reduction of the stepwise association constant K i toward an asymptotic value K inf in the limit of sufficiently large oligomer size. The parameter K inf * is an offset whose value depends upon the definition of the standard reference state, and the parameter n describes the linkage between protein association and Mg 2ϩ uptake (Equation 4). The dependences of M w /M 1 on protein and Mg 2ϩ concentrations, calculated from the best least square fits of both indefinite self-association models to the combined data (which are visually indistinguishable) is plotted together with the data in Fig. 4 (panel A), and the best fit parameter values are given in the figure legend. Four salient points emerge from our results.
1. The parameter n is constrained to a narrow set of values around 1 (n ϭ 0.93 Ϯ 0.2, 95% confidence limits), indicating unambiguously that 1 additional mole of Mg 2ϩ is bound for each protomer of FtsZ added to a growing oligomer. 5 Thus, for a given oligomer size i, the stepwise association constant K i increases in direct proportion to the concentration of Mg 2ϩ (Fig.  5, panel A); K i increases an order of magnitude between 0.1 (curve 1) and 1 mM (curve 2) Mg 2ϩ and another order upon raising [Mg 2ϩ ] to 10 mM (curve 3).
2. The data are inconsistent with an isodesmic model (J ϭ 1) at a 95% level of confidence, as judged by the best fit value of 2 (63) (see dotted line in Fig. 4, panel B). This model assumes that the stepwise association constant K i is independent of i. Therefore, according to our analysis, the free energy of monomer addition becomes more positive with increasing oligomer size, although by an amount that is significantly smaller than that calculated by Chatelier (70) on the basis of a simple rigid body statistical-thermodynamic model. The data are also inconsistent, within experimental precision, with defined oligomer models, such as a monomer-dimer-n-mer model (see dotted line in Fig. 4, panel B).
3. Although the values of the individual parameters K inf * , J, and ␣ or ␤ are not individually well determined by the data, the correlations between them required to achieve a satisfactory fit are such that the derived dependence of the stepwise association constants K i on Mg 2ϩ concentration and oligomer size is, to a very good approximation, independent of individual parameter values and even independent of the particular model employed, so long as the fit to the data is within experimental uncertainty. This is illustrated in Fig. 5, panel A, which shows the maximum divergence between K i calculated from acceptable fits to the data with the two different models. For any value of i between 2 and 25, the indicated uncertainty in the absolute value of K i amounts to less than 25% or, equivalently, an uncertainty of ϳ0.1 RT (60 cal/mol at room temperature) in the stepwise free energy of association.
4. The size dependence of the cumulative free energy change of i-mer assembly, ⌬G 1i (Fig. 5, panel B) is well determined. This result is not entirely unexpected, as the cumulative association constants Z i (and free energy changes) are more directly linked to the calculation of the species distribution, plotted in Sedimentation Velocity Shape of FtsZ Oligomers-In order to confirm the Mg 2ϩ -induced FtsZ self-association and to obtain information about the shape of the oligomers, sedimentation velocity experiments were performed. In Fig. 6, panel A (curves  a-d) From the combined analytical centrifugation data, and taking into account the proposed FtsZ association scheme, the shape of the Mg 2ϩ -induced FtsZ was analyzed. The dependence of s w (considering oligomer sizes up to 50-mer) with protein concentration for different structural models of FtsZ oligomers was calculated using Equations 13 and 14 in the hydrodynamic treatment described under "Theory." For a linear FtsZ oligomer with an axial spacing of 4.3 nm (23, 24, 77), this dependence is reasonably compatible with the experimental data (solid lines 5 Subsequent modeling of the data was carried out with n constrained to a value of unity. The solid line corresponds to the best fit to the indefinite self-association model. The corresponding best fits to an isodesmic (dashed line; J ϭ 1, log K inf * ϭ 5.4) and to a monomer-dimer-dodecamer (dotted line; log K 2 ϭ 7.9, log K 12 ϭ 27.7) association models, both inconsistent at a 95% level of confidence, are also plotted.
in Fig. 6, panels B and C). The behavior for a double-stranded linear oligomer model (using for simplicity 4.3 nm as the lateral inter-protofilament spacing) deviates from the sedimentation velocity data (dashed-dotted line in Fig. 6, panel C). It is clear that a spherical FtsZ oligomer is not compatible with the experimental data (dotted line in Fig. 6, panel C). In fact, any departure from a linear oligomerization consisting of a more compact structure or a markedly flexible/curved oligomer shape would result in an increase of the s w model values and a poorer fit to the data. Only a significant increase of the axial intermonomer spacing (up to 6 nm) could lower the average sedimentation coefficient values of the double-stranded oligomer model near the data (short dashed line in Fig. 6, panel  C). However, this increasing in axial spacing is neither supported by the data available from electron microscopy (23,24,77) nor by the available crystal dimensions of FtsZ molecules (21). Therefore, within the accuracy of the present data and model, a single-stranded linear oligomeric structure is favored.
Qualitative Effects of Solution Variables (Ionic Strength, pH, and Ca 2ϩ ) on Mg 2ϩ -induced FtsZ Association-The role of ionic strength on the Mg 2ϩ -induced self-association of FtsZ was studied by sedimentation equilibrium. The results obtained are summarized in Table I. At 25 M protein, the degree of association of FtsZ, equilibrated in Tris-500KCl buffer (Tris-500KCl buffer, 50 mM Tris-HCl, 500 mM KCl, pH 7.5) with 20 M GDP and 10 mM MgCl 2 , is not significantly larger than 2, while in the corresponding FtsZ sample at 50 mM KCl this value was around 6. Therefore, the data show that increasing the ionic strength of the solution (250 -500 mM KCl) diminishes the tendency of FtsZ to self-associate, even at high Mg 2ϩ concentration (5-10 mM). This effect has been confirmed by parallel sedimentation velocity experiments (Table I and  Preliminary analytical centrifugation experiments performed in PIPES and MES buffers (pH 6.5) indicate a favored FtsZ oligomer formation. The sedimentation coefficient of FtsZ (ϳ20 M) in MES buffer with 1 mM Mg 2ϩ is 6.8 Ϯ 0.2 S, whereas it was only 5.4 Ϯ 0.1 S for the equivalent sample in Tris-50KCl buffer. The presence of a minor (Ͻ10%) and faster species (ϳ 10 S) not observed in Tris buffer would be an indication that the scheme of FtsZ association might be different than that proposed with pH 7.5 buffer. A loss of mass observed in the sedimentation equilibrium experiments 6 suggests that these experimental conditions also affect the solubility of FtsZ, which can be compatible with the pH dependence of FtsZ polymer formation in vitro ( Fig. 1 (44, 78)). Finally, preliminary sedimentation velocity data showed that Ca 2ϩ induces FtsZ oligomer formation. For example, the sedimentation coefficient of 25 M FtsZ increases from 3.5 Ϯ 0.1 S (no Ca 2ϩ added 4 ) to 6.6 Ϯ 0.2 S (ϩ 5 mM Ca 2ϩ ) in Tris-50KCl buffer 4 (open circles and dashed line in Fig. 6, panel A, respectively).
Effect of Bound Nucleotide on Mg 2ϩ -induced FtsZ Association-After characterizing the Mg 2ϩ -induced association of FtsZ in its GDP-bound form (the reference ground state), the effects of GTP binding were explored, because the ␥-phosphate of the nucleotide switches FtsZ into the active state for assembly (Fig. 1). The nucleotide exchange method (see "Experimental Procedures") renders FtsZ with ϳ80 -90% GTP bound when done in 0.1 mM GTP and in the absence of Mg 2ϩ (ϳ1 M), and the protein sediments as a 3.5 Ϯ 0.1 S species (not shown), the same as the GDP-FtsZ monomer (3.4 Ϯ 0.1 S). However, when the divalent cation concentration was increased to 1-10 mM only GDP-FtsZ was obtained, due to GTP hydrolysis, which is compatible with the GTPase activity of FtsZ found (this work and see Refs. 15-17 and 75). 6 J. Ferná ndez and G. Rivas, unpublished data. To overcome these problems which precluded getting FtsZ with GTP bound in the presence of 1-10 mM Mg 2ϩ at low GTP concentrations (experimental condition required to properly measure the protein with the absorption optics of the XL-A ultracentrifuge), a second approach was taken, using an FtsZ that was gently labeled with an extrinsic chromophore (FITC). The fluorescein labeling apparently did not alter the association properties of FtsZ when compared with the unlabeled protein as checked by sedimentation velocity (see legend of Fig.  7). Therefore, this FITC-FtsZ can be used to study the effect of GTP on the physical state of the protein at high GTP concentration (2 mM) by monitoring the protein in the visible region. Under these conditions a significant concentration of GTP with respect to GDP is present in the solution during the time scale of the experiment (1-2 h), as measured by HPLC and GTPase assays performed under identical conditions (not shown). The data shown in Fig. 7 indicate that GTP seems to reduce the FtsZ self-association induced by Mg 2ϩ . It can be assumed that a large fraction of FtsZ is present in its GTP-bound state. If the association scheme remains the same, this would be compatible with a weakening of the monomer-monomer interaction upon GTP binding.

DISCUSSION
The role of the FtsZ protein in E. coli division has been described as forming part of the initial steps in the formation of the septum (4). Its assembly into the cell center forming a so-called FtsZ division ring seems to precede any other visible signs of septum formation and takes place shortly after completion of DNA replication (7). In this work we have shown that purified FtsZ associates from a monomer into Mg 2ϩ -induced linear oligomers. 7 In the following discussion we propose that these interactions are the primary steps of FtsZ assembly.
The smallest association state of FtsZ in solution is a compact and globular monomer, a species whose presence is favored by moderate ionic strength and low Mg 2ϩ concentrations. As in the case of tubulin, the eukaryotic homolog of FtsZ, magnesium determines the association properties of FtsZ in solution. Our data are best described by an indefinite association scheme in which the affinity of addition of a protomer gradually decreases with increasing oligomer size. In addition, protein association is linked to the binding of an additional Mg 2ϩ per FtsZ association step (Fig. 4, panel A). The shape of the Mg 2ϩ -induced FtsZ oligomers which best explains the combined analysis of the sedimentation velocity and equilibrium data is a linear single-stranded oligomeric structure. This would be compatible with the 5-10 subunits long straight protofilaments found in rotary-shadowed electron micrographs of FtsZ under nonassembly conditions (75,77). Under assembly solution conditions single (and double-stranded) straight filaments can be visualized after negative stain electron microscopy (Fig. 1), similarly to previous observations (23,78). Minirings and curved filament ends have been previously observed by electron microscopy of FtsZ polymers adsorbed onto cationic lipid, and the ring curvature has been related to either GDP or GTP being bound at the FtsZ nucleotide-binding site (23,77,78,79). In our solution study we were unable to detect FtsZ mini-rings by sedimentation velocity and electron microscopy. 7 After submission of this paper, Sossong and co-workers (92) have reported evidence suggesting that the activation of FtsZ GTPase is due to Mg 2ϩ -induced oligomerization, forming dimers, trimers, and tetramers. Their sedimentation equilibrium analysis at a single speed (18,000 rpm), using a narrower range of protein concentration, in the absence and in the presence of 5 mM MgCl 2 , is qualitatively compatible with the fact that Mg 2ϩ promotes higher order linear FtsZ oligomer formation (this study). A similar Mg 2ϩ -induced CD change (Fig. 2B) has been observed. This could be explained if the FtsZ oligomers were less flexible than in the case of tubulin, impeding the curvature required for the end-closure reaction in solution.
The endwise association of ␣␤-tubulin dimers induced by Mg 2ϩ proceeds by an indefinite association mechanism (34,35). Linear oligomers equivalent to fragments of microtubule protofilaments are formed. Therefore, the energetics of the linear oligomerization of FtsZ in solution provides a strong counterpart to the tubulin-like structural model of FtsZ protofilaments (23,24,78,80). FtsZ oligomer formation is promoted by reducing the ionic strength. This is also the case in tubulin, where microtubule assembly is favored at low salt conditions and is inhibited by salt (KCl or NaCl) at concentrations in excess of 100 -150 mM (39,81). Similarly to tubulin (41), the self-association of FtsZ is apparently stronger with GDP than with GTP bound at the nucleotide site. In the case of FtsZ this nucleotide effect seems to be a weakening of the monomer-monomer association, whereas for tubulin it was suggested to be due to the conformation of the GDP-liganded protein being more favorable for the closure reaction (41). The effect is intriguing, since in both cases the GTP-liganded protein is the active species forming larger polymers. FtsZ oligomers and polymers, observed in this work under different solution conditions of temperature and buffer composition, might be alternative association pathways as in the case of tubulin (33, 41) (see Scheme 1).
We have found that oligomerization depends to a large extent on FtsZ concentration and Mg 2ϩ ions, and it is favored by low ionic strength. The estimates for the number of FtsZ copies present in the cell (from 5,000 to 20,000 (6, 75, 82)) would indicate that FtsZ is at a concentration of ϳ10 M; the physiological free concentration of Mg 2ϩ is ϳ1-2 mM (83); the intra-cellular pH is around 7.6 -7.8 (84), and the total intracellular K ϩ concentration ranges about 0.1 to 1 M (85). By taking into account these physiological concentrations of protein and ions inside the cell, our results summarized in Fig. 4, panel A, would suggest that under physiological micro-solute conditions isolated FtsZ is reluctant to associate. This would be consistent with earlier data that indicated that the majority of the FtsZ molecules are found in the soluble fraction of the cell (82), and with a previous report by Jones and Holland (86) that an extended sonic treatment of the samples at 10 mM Mg 2ϩ was needed for the association of FtsZ to the membrane fraction of maxicells.
However, there are other important factors that might regulate the state of association and assembly of FtsZ in vivo. (i) Excluded volume effects due to crowding (87,88) at the high macromolecular concentrations estimated for E. coli cytoplasm (several hundreds of mg/ml; Refs. 85 and 89) would favor that FtsZ exists in an aggregated/oligomeric state inside the cell. Zimmerman and Trach (89) have discussed the effects of such a high volume occupancy inside the cell upon the association equilibria of a protein of the size of FtsZ monomer (40,000) and calculated that if the protein is in a mixture of monomers and dimers (or tetramers) at equilibrium the association constant would be shifted 10 -40-fold (or 10 3 to 10 5 ) toward dimerization (or tetramer formation). (ii) A large volume of the cell interior is occupied by the nucleoid, what has led to conjecture, following the nucleoid occlusion model of Woldringh (90), that septation cannot occur in regions occupied by the nucleoid. Migration of FtsZ (due to electrostatic repulsions) toward nucleoidfree regions of the cell (namely the poles and after segregation has taken place, the cell center) would result in a substantial increase in local concentration of FtsZ which also could facilitate FtsZ oligomer formation. Activation of the MinCD division inhibitor would prevent division at the poles, whereas such a mechanism would not be operating at mid-cell. However, recent data indicating that the FtsZ division ring is visible prior to the estimated time of nucleoid segregation would make this SCHEME 1. A model scheme that summarizes the results and implications of this work. The reaction studied is the indefinite Mg 2ϩ -induced association of FtsZ into linear oligomers (reaction 1). It is proposed that this is a predominant interaction in the cooperative nucleated condensation polymerization of FtsZ (simply represented as the formation of a double-stranded filament in reaction 2, which is based on previous results (Refs. 23,75,77,78), and this work, and hypothetical reaction 3), and that cellular modulation of these interactions (pathway 4) would result in the correct formation of the Z-ring during bacterial division (see text). assumption unlikely (7). (iii) The FtsZ oligomers involved in the formation of the ring might be clustered by specific binding to another protein or structure, presently unknown, that can be further stabilized by FtsA and Zip A, given their proven interaction with FtsZ (13,91). A simple way to summarize the implications of the results presented in this work is shown in Scheme 1. It is proposed that the primary interaction for FtsZ assembly in vivo is the relatively weak endwise association of FtsZ monomers, which is induced by Mg 2ϩ , and leads as well to oligomer formation. Binding of GTP and changes in solution conditions (temperature, lowering pH) promote additional FtsZ-FtsZ interactions that would result in a cooperative bidimensional polymer assembly. Modulation of the energetics of self-association by additional factors, such as molecular crowding, nucleoid occlusion, or specific molecular interactions with other cellular components active in septation have to be invoked to explain FtsZ assembly into a division ring.