Portal de Periódicos da CAPES

Polymerization of FtsZ, a Bacterial Homolog of Tubulin

2001; Elsevier BV; Volume: 276; Issue: 15 Linguagem: Inglês

10.1074/jbc.m009033200

1083-351X

Laura Romberg, Martha N. Simon, Harold Erickson,

Origins and Evolution of Life

FtsZ is a bacterial homolog of tubulin that is essential for prokaryotic cytokinesis. In vitro, GTP induces FtsZ to assemble into straight, 5-nm-wide polymers. Here we show that the polymerization of these FtsZ filaments most closely resembles noncooperative (or "isodesmic") assembly; the polymers are single-stranded and assemble with no evidence of a nucleation phase and without a critical concentration. We have developed a model for the isodesmic polymerization that includes GTP hydrolysis in the scheme. The model can account for the lengths of the FtsZ polymers and their maximum steady state nucleotide hydrolysis rates. It predicts that unlike microtubules, FtsZ protofilaments consist of GTP-bound FtsZ subunits that hydrolyze their nucleotide only slowly and are connected by high affinity longitudinal bonds with a nanomolarK D. FtsZ is a bacterial homolog of tubulin that is essential for prokaryotic cytokinesis. In vitro, GTP induces FtsZ to assemble into straight, 5-nm-wide polymers. Here we show that the polymerization of these FtsZ filaments most closely resembles noncooperative (or "isodesmic") assembly; the polymers are single-stranded and assemble with no evidence of a nucleation phase and without a critical concentration. We have developed a model for the isodesmic polymerization that includes GTP hydrolysis in the scheme. The model can account for the lengths of the FtsZ polymers and their maximum steady state nucleotide hydrolysis rates. It predicts that unlike microtubules, FtsZ protofilaments consist of GTP-bound FtsZ subunits that hydrolyze their nucleotide only slowly and are connected by high affinity longitudinal bonds with a nanomolarK D. FtsZ is a bacterial homolog of tubulin that is essential for prokaryotic cytokinesis (for review see Refs. 1Bramhill D. Annu. Rev. Cell Dev. Biol... 1997; 13: 395-424Google Scholar, 2Lutkenhaus J. Addinall S.G. Annu. Rev. Biochem... 1997; 66: 93-116Google Scholar, 3Erickson H.P. Trends Cell Biol... 1997; 7: 362-367Google Scholar). InEscherichia coli, FtsZ assembles into a ring at the center of the cell just inside the inner membrane. This Z ring assembles early in the cell cycle, after which other proteins necessary for cell division are recruited to it. During cytokinesis, the Z ring contracts and disassembles. If the structure, timing, or localization of Z ring assembly is disrupted, cell division can fail or become aberrant. It is not yet clear how Z ring assembly is controlled; some properties of ring formation may derive from the inherent self-assembly characteristics of FtsZ, whereas others may require regulation or modification by additional factors in the cell. FtsZ and tubulin are homologs that share identical folds (4Nogales E. Downing K.H. Amos L.A. Lowe J. Nat. Struct. Biol... 1998; 5: 451-458Google Scholar) and assemble into polymers with many of the same properties. Like tubulin, FtsZ polymerizes in the presence of GTP (5Mukherjee A. Lutkenhaus J. J. Bacteriol... 1994; 176: 2754-2758Google Scholar, 6Bramhill D. Thompson C.M. Proc. Natl. Acad. Sci. U. S. A... 1994; 91: 5813-5817Google Scholar) and can form straight protofilaments that are ∼5 nm wide, with subunits spaced 4 nm apart (7Erickson H.P. Taylor D.W. Taylor K.A. Bramhill D. Proc. Natl. Acad. Sci. U. S. A... 1996; 93: 519-523Google Scholar, 8Löwe J. Amos L.A. EMBO J... 1999; 18: 2364-2371Google Scholar). The longitudinal bonds that connect the subunits in a tubulin protofilament are understood at atomic resolution, and the bonds between subunits in an FtsZ protofilament are likely to be very similar (9Nogales E. Wolf S.G. Downing K.H. Nature.. 1998; 391: 199-203Google Scholar). Residues on both sides of the longitudinal protein interface are conserved, and the GTP binds to one side of the interface and is necessary for formation of a protofilament. In return, hydrolysis of the GTP occurs only after formation of the longitudinal bond, when residues from the adjoining subunit contact the nucleotide (4Nogales E. Downing K.H. Amos L.A. Lowe J. Nat. Struct. Biol... 1998; 5: 451-458Google Scholar). GTP hydrolysis causes both FtsZ and tubulin filaments to adopt a curved conformation (7Erickson H.P. Taylor D.W. Taylor K.A. Bramhill D. Proc. Natl. Acad. Sci. U. S. A... 1996; 93: 519-523Google Scholar, 10Lu C.L. Reedy M. Erickson H.P. J. Bacteriol... 2000; 182: 164-170Google Scholar) and become more labile (11Mukherjee A. Lutkenhaus J. EMBO J... 1998; 17: 462-469Google Scholar). Curved GDP-FtsZ filaments have been visualized when stabilized by various polycations and are half the diameter of GDP-tubulin rings, indicating that the angle of curvature between GDP-FtsZ subunits is the same as that between the αβ heterodimers in tubulin rings (12Erickson H.P. Stoffler D. J. Cell Biol... 1996; 135: 5-8Google Scholar). For both proteins, assembly with GDP requires magnesium and is relatively weak (K D = 20–50 μm (13Frigon R.P. Timasheff S.N. Biochemistry.. 1975; 14: 4559-4566Google Scholar, 14Rivas G. Lopez A. Mingorance J. Ferrandiz M.J. Zorrilla S. Minton A.P. Vicente M. Andreu J.M. J. Biol. Chem... 2000; 275: 11740-11749Google Scholar)). In contrast to the conserved longitudinal contacts within tubulin and FtsZ protofilaments, the lateral contacts between protofilaments in a microtubule wall involve protein surfaces that are not conserved in FtsZ (4Nogales E. Downing K.H. Amos L.A. Lowe J. Nat. Struct. Biol... 1998; 5: 451-458Google Scholar). Because FtsZ polymers have not yet been visualized with high resolution in vivo, it is not known how the protofilaments in the Z ring are associated. Under many conditions in vitro, GTP induces FtsZ to form individual 5-nm-wide polymers that are stable without additional lateral interactions (5Mukherjee A. Lutkenhaus J. J. Bacteriol... 1994; 176: 2754-2758Google Scholar, 10Lu C.L. Reedy M. Erickson H.P. J. Bacteriol... 2000; 182: 164-170Google Scholar, 11Mukherjee A. Lutkenhaus J. EMBO J... 1998; 17: 462-469Google Scholar, 15Mukherjee A. Lutkenhaus J. J. Bacteriol... 1999; 181: 823-832Google Scholar). Nonetheless, even these thin FtsZ polymers could consist of two protofilaments joined along the narrowest 3 nm axis of the protein, and there is a precedent for such a 6 nm double filament forming in the presence of Ca2+ (8Löwe J. Amos L.A. EMBO J... 1999; 18: 2364-2371Google Scholar). FtsZ can form several types of lateral bonds when exogenous cations are added, resulting in a variety of paired protofilaments, bundles, double layered sheets, and tubes (6Bramhill D. Thompson C.M. Proc. Natl. Acad. Sci. U. S. A... 1994; 91: 5813-5817Google Scholar, 7Erickson H.P. Taylor D.W. Taylor K.A. Bramhill D. Proc. Natl. Acad. Sci. U. S. A... 1996; 93: 519-523Google Scholar, 8Löwe J. Amos L.A. EMBO J... 1999; 18: 2364-2371Google Scholar, 10Lu C.L. Reedy M. Erickson H.P. J. Bacteriol... 2000; 182: 164-170Google Scholar, 16Yu X.C. Margolin W. EMBO J... 1997; 16: 5455-5463Google Scholar). For tubulin, aberrant lateral bonds can result in the formation of zinc sheets (17Baker T.S. Amos L.A. J. Mol. Biol... 1978; 123: 89-106Google Scholar) and hooked and S-shaped polymers (18Himes R.H. Burton P.R. Gaito J.M. J. Biol. Chem... 1977; 252: 6222-6228Google Scholar). It is unclear for FtsZ whether any of the lateral contacts that formin vitro are relevant in vivo. Polymers that are multistranded assemble cooperatively (19Oosawa F. Kasai M. J. Mol. Biol... 1962; 4: 10-21Google Scholar). Cooperative assembly has three distinctive characteristics (see Fig.1 A) as follows: there is a critical concentration for assembly; there are kinetic lags in polymerization at low protein concentrations; and at equilibrium, subunits are distributed into two distinct populations, monomers and very long polymers. These properties derive from the two different phases of assembly, an unfavorable nucleation phase followed by a more favorable growth phase. Nucleation is unfavorable because individual bonds between monomers are weak, and therefore initiation of the polymer is difficult. However, once a stable nucleus has assembled, each new subunit can form multiple bonds to the growing polymer. Subunit addition becomes favorable and polymer growth rapid. The critical concentration is the minimum protein concentration at which any polymer forms; at all concentrations above this, subunits assemble until the concentration of monomer left in solution has fallen to the critical concentration. Most biological polymers are multistranded and assemble cooperatively with the properties described above. Cooperative polymers such as actin and microtubules may be common because they form filaments that are strong in the middle but can easily be remodeled by removing subunits from their ends (20Erickson H.P. J. Mol. Biol... 1989; 206: 465-474Google Scholar). In addition, a cell can use cooperative assembly to control its spatial organization by localizing polymer nucleation sites and assembling long filaments that can span the cytoplasm. In contrast to multistranded polymers, single-stranded polymers can be assembled with an identical bond at each step of polymerization (21Adams Jr., E.T. Lewis M.S. Biochemistry.. 1968; 7: 1044-1053Google Scholar,22Reisler E. Pouyet J. Eisenberg H. Biochemistry.. 1970; 9: 3095-3102Google Scholar) (see Fig. 1 B). For such isodesmic polymers, there is no separate nucleation and growth phase, and so there is no lag in the assembly kinetics. Instead, assembly is rapid because every monomer in solution can act as a start site for polymer growth. There is also no critical concentration for assembly, and as the total protein concentration rises, both polymer and monomer populations can increase simultaneously. Finally, fragmentation of a single-stranded polymer in the middle requires breaking only a single bond. Disassembly thus occurs not only at the ends of the polymers but also along their length, and isodesmic polymers remain relatively short. Single-stranded, isodesmic polymers are less common in biology than cooperative, multistranded polymers, and none so far has been shown to be the major form of a protein in vivo. Isodesmic assembly was first analyzed by studying β-lactoglobulin (21Adams Jr., E.T. Lewis M.S. Biochemistry.. 1968; 7: 1044-1053Google Scholar) and glutamate dehydrogenase (22Reisler E. Pouyet J. Eisenberg H. Biochemistry.. 1970; 9: 3095-3102Google Scholar), but both these proteins function in vivoas monomers. The Drosophila septins can assemble in vitro both isodesmically, forming short, single-stranded polymers (23Field C.M. Al-Awar O. Rosenblatt J. Wong M.L. Alberts B. Mitchison T.J. J. Cell Biol... 1996; 133: 605-616Google Scholar), and cooperatively, forming multistranded filaments, 1M. Glotzer, personal communication.1M. Glotzer, personal communication. depending on which subunits are combined. Relevant to the present study, tubulin-GDP can assemble isodesmically into single, curved protofilaments that eventually close to form tubulin rings (13Frigon R.P. Timasheff S.N. Biochemistry.. 1975; 14: 4559-4566Google Scholar). Similarly, GDP-FtsZ assembly is noncooperative and deviates only slightly from isodesmic polymerization (14Rivas G. Lopez A. Mingorance J. Ferrandiz M.J. Zorrilla S. Minton A.P. Vicente M. Andreu J.M. J. Biol. Chem... 2000; 275: 11740-11749Google Scholar). The two basic mechanisms of assembly described above are often complicated by nucleotide hydrolysis or conformational changes within a subunit. In actin and tubulin, these complications lead to phenomena such as treadmilling and dynamic instability (24Erickson H.P. O'Brien E.T. Annu. Rev. Biophys. Struct. Biol... 1992; 21: 145-166Google Scholar, 25Desai A. Mitchison T.J. Annu. Rev. Cell Dev. Biol... 1997; 13: 83-117Google Scholar). Prion fiber assembly also shows characteristics that do not fall neatly into either mechanism described above. Prion assembly shows lag phases that are concentration-independent, and monomer, small oligomer, and polymer coexist in assembly reactions (26Serio T.R. Cashikar A.G. Kowal A.S. Sawicki G.J. Moslehi J.J. Serpell L. Arnsdorf M.F. Lindquist S.L. Science.. 2000; 289: 1317-1321Google Scholar). A conformational change is thought to be necessary to activate the prion protein before it can assemble into multistranded fibers, and preformed fibers may accelerate the rate of this conformational change. For FtsZ, cycles of GTP hydrolysis and conformational changes in the subunit may also introduce complexities into its polymerization mechanism. The straight 5-nm-wide FtsZ polymers that form in GTP are probably the building blocks of any larger structure that forms in vivo. In the cell, controlled nucleation followed by favorable growth would be an economic explanation of the FtsZ's tight localization and its ability to span the 3-μm circumference of an E. coli cell. However, if FtsZ polymers have the structure of an isolated tubulin protofilament, they would be single-stranded and would be expected to show isodesmic assembly. Our goal was to determine whether the assembly of these protofilaments occurs via isodesmic or cooperative assembly. Therefore, we determined whether these filaments are in fact single-stranded, whether there is a concentration-dependent lag phase during assembly indicative of nucleation, and whether there is a critical concentration for assembly. We found that unlike microtubules, the assembly of these FtsZ-GTP protofilaments appears isodesmic. E. coli FtsZ was overexpressed from a pET11b vector (Novagen) and purified largely as in Lu et al. (27Lu C. Stricker J. Erickson H.P. Cell Motil. Cytoskeleton.. 1998; 40: 71-86Google Scholar) with a few modifications. Cells were grown in 500 ml of LB at 37 °C to A 600 = 1, induced with 0.5 mmisopropyl-1-thio-β-d-galactopyranoside, and grown for an additional 2 h. Cells were sedimented and resuspended in 10 ml of 50 mm Tris, pH 8, 100 mm NaCl, 1 mmEDTA, 1 mm 4-(2-aminoethyl)benzenesulfonyl fluoride and frozen at −80 °C. Cells were thawed, and lysozyme was added to 0.1 mg/ml and incubated for 10 min, and magnesium acetate was added to 10 mm. Cells were sonicated, and DNase was added to 10 μg/ml, and the extract was incubated for 1 h. Lysate was spun for 30 min at 220,000 × g at 4 °C. The supernatant was mixed with 0.25 volume saturated (room temperature) ammonium sulfate, incubated on ice for 20 min and spun at 80,000 ×g for 10 min at 4 °C. The supernatant was removed, and 1/12 volume ammonium sulfate was added. This was spun again at 80,000 × g for 10 min at 4 °C. The pellet was resuspended in 5 ml of polymerization buffer (50 mmNaMES,2 pH 6.5, 2.5 mm magnesium acetate, 1 mm EGTA) and spun again at 160,000 × g for 10 min at 4 °C. The final supernatant was brought to 10% glycerol and 50 μm GDP, aliquoted, snap-frozen in liquid nitrogen, and stored at −80 °C. Protein concentrations were measured using the BCA assay (Pierce) with bovine serum albumin as a standard, calibrated for the 0.75 color ratio of FtsZ/bovine serum albumin (27Lu C. Stricker J. Erickson H.P. Cell Motil. Cytoskeleton.. 1998; 40: 71-86Google Scholar). All assembly reactions were carried out in the above polymerization buffer (50 mm NaMES, pH 6.5, 2.5 mm magnesium acetate, 1 mm EGTA). Reactions were supplemented with magnesium acetate so that the total magnesium concentration was 2 mmin excess of any nucleotide. Assembly reactions were performed at room temperature unless otherwise indicated. On the day the protein was to be assayed, the FtsZ was put through a cycle of calcium-aided assembly and disassembly (based on Ref. 16Yu X.C. Margolin W. EMBO J... 1997; 16: 5455-5463Google Scholar) to select for active, nonaggregated protein. Thawed protein was prespun in a TLA100 rotor for 15 min at 350,000 ×g at 4 °C. The protein was then diluted 5-fold in polymerization buffer to lower the glycerol concentration (final protein concentration ∼1 mg/ml) and brought to 10 mmCaCl2 and 2 mm GTP. After incubation for 3 min at room temperature, polymer was sedimented for 15 min at 350,000 × g at 20 °C. The pellet was resuspended well in polymerization buffer and incubated on ice for 30 min. A final spin (15 min, 350,000 × g, 4 °C) removed any remaining aggregated protein. Negative stain electron microscopy was used to visualize FtsZ filaments. Carbon-coated copper grids (400 mesh, Electron Microscopy Sciences) were glow discharged for 5 s before use. For some assays, before applying the FtsZ a drop of 0.2 mg/ml cytochrome c was pipetted onto the carbon, incubated for 30 s, and then blotted with filter paper. This procedure aided good staining at low FtsZ concentrations but did not visibly affect filament number or length. Cycled protein was diluted to the desired concentration in polymerization buffer and nucleotide was added, and the reaction was incubated for several minutes at room temperature. A drop of FtsZ solution was then applied to the carbon and incubated for 10 s before the excess was blotted. The grid was immediately rinsed with 3–4 drops 2% uranyl acetate, blotted, and air-dried. Filaments were visualized and photographed using a Phillips 301 electron microscope at × 50,000 magnification. For length determinations, 0.75–3.5 μm FtsZ was assembled with either 2 mm GTP or 0.5 mm GMPCPP and visualized by negative stain. Image quality improved significantly at the higher protein concentrations, although overlapping filaments made lengths more difficult to measure above 2 μm FtsZ. Three fields on a grid were photographed from 10 different reactions with each nucleotide. The negatives were scanned into a computer at 600 dpi. Filaments were traced using NIH image, and the length distributions for a given grid were pooled and plotted. The polymer length distributions were largely exponential. However, because the shortest polymers were more likely to be obscured in the noise of an image or overlooked during measurement, the frequency of observing these polymers was lower than expected for an exponential distribution. The peak polymer length correlated with image quality, with the best images showing exponential length distributions down to 8 subunits (32 nm). To avoid biasing the average filament length because of incomplete counting of short filaments, the average filament length for each grid was mathematically calculated from an exponential fitting to the data, and only data from filaments that were longer than 25 subunits were included in the fittings. Changing the cut-off from 25 to 50 subunits did not alter the best fit curve. Because the average length did not change significantly between the FtsZ concentrations tested, the data from all experiments were pooled to determine the final average length reported for each nucleotide. A malachite green-sodium molybdate assay was used to measure production of inorganic phosphate (28Geladopoulos T.P. Sotiroudis T.G. Evangelopoulos A.E. Anal. Biochem... 1991; 192: 112-116Google Scholar). Protein was diluted to the desired concentration in polymerization buffer at room temperature, and GTP was added to initiate the reaction. At four time intervals, reactions were stopped by addition of a 1× volume of cold 0.6 m perchloric acid. The samples were stored on ice until all time points were collected, at which time a 2× volume of filtered malachite green solution (0.15 g of malachite green, 1 g of sodium molybdate, 0.25 g of Triton X-100 in 0.7 m HCl) was added. The samples were then incubated at room temperature for 30 min, and the A 650 was measured. NaPO4buffer was used to create a standard curve, and the reaction was normalized by including a control without FtsZ. FtsZ was assembled in polymerization buffer plus either 2 mm GTP/5 mm EDTA or 0.5 mm GMPCPP. A 3-μl sample of the reaction mix was injected into a drop of buffer on a thin (2–3 nm) carbon film supported by a thick holey film over a titanium grid to which tobacco mosaic virus had previously been applied as an internal standard. The grid was washed first 5 times with injection buffer, then 10 times with 20 mm ammonium acetate, blotted to a thin layer, plunged into liquid nitrogen slush, and freeze-dried overnight under vacuum before being transferred to the microscope. The grids were visualized using a custom built microscope (STEM1) at the Brookhaven National Laboratory. The microscope was operated in a dark field mode in which annular detectors collect nearly all the scattered electrons. Because the FtsZ filaments were radiation-sensitive, it was necessary to keep the electron dose low to have meaningful measurements. With a high dose, the mass/length measurements could be lower by 50% (data not shown). However, at lower doses, results became reproducible. Areas with relatively clean backgrounds and an adequate number of filaments were chosen for analysis. A digital image was saved consisting of 512 × 512 pixels, each of which shows the number of scattered electrons, which is directly proportional to the mass thickness in that pixel. Relatively straight, short (usually ∼30 nm) segments were chosen along a filament for mass measurements. Because there were significant numbers of unpolymerized molecules in the background, it was necessary to mask these molecules before the background was computed and subtracted from the intensity summed over the filaments. The microscope calibration factor was checked against that of tobacco mosaic virus, and the summed intensities (minus the background) multiplied by the calibration factor gave the mass values for the specimen. Cycled FtsZ was diluted to the desired concentration in polymerization buffer and introduced into a quartz cuvette in a FluoroMax fluorimeter with a custom-made temperature-controlled cuvette holder. The cuvette was illuminated with 310 nm light, and the 90° light scattering was detected. The slit widths were set at 0.5 mm and the signal was integrated once a second. A base line of scattering for the unpolymerized protein was established and then polymerization was initiated by the addition of 1 mm GTP or GDP or 0.25 mm GMPCPP. The nucleotide was introduced into the solution on the end of a custom-made plunger that fit snugly into the cuvette with a 1-mm clearance around all sides. The plunger was inserted into the cuvette and rapidly withdrawn to mix the contents. Typical dead times were 2–3 s. The noise in the signal increased when using this plunger due to small bubbles coming out of solution as the plunger was rapidly withdrawn. Pre-cycled protein was diluted to the desired concentrations with polymerization buffer. GTP was added to a final concentration of 5–10 mm, GDP to 2 mm, or GMPCPP to 0.5 mm, along with additional magnesium acetate so that there was a 2 mm excess of magnesium over nucleotide. 75 μl of each reaction was added to a polycarbonate tube and centrifuged in the TLA100 rotor in a TL100 ultracentrifuge (Beckman) for 15 min at 350,000 × g at 20 °C. Removing the entire supernatant and assaying the supernatant and pellet gave poorly reproducible results, so we removed and assayed only the upper two-thirds of the supernatant, leaving the pellet undisturbed. The upper 50 μl of supernatant was immediately removed from the tube, and the concentration of protein in the original sample and the supernatant were measured. To limit GTP depletion and diffusion of the pellet from the tube walls after sedimentation, no more than seven reactions were sedimented simultaneously to reduce the handling time. Results were identical using 5 or 10 mm GTP, confirming that the nucleotide is not being significantly depleted during the 25 min required to complete an assay. Each protein concentration was tested 2–4 times per experiment, and 2–4 separate experiments were combined and averaged to obtain accurate data. To calibrate the assay, seven proteins with known S values between 4.5 and 30 were sedimented in the same manner. The resulting curve was linear up to 19 S, at which point 94% of the protein sedimented. The resulting standard curve indicated that the percent protein sedimented = 3.6 × 10−3 + 4.9 × 10−2·S (R = 0.97). FtsZ monomers are 3.4 S (14Rivas G. Lopez A. Mingorance J. Ferrandiz M.J. Zorrilla S. Minton A.P. Vicente M. Andreu J.M. J. Biol. Chem... 2000; 275: 11740-11749Google Scholar), and the expected S values of single-stranded FtsZ polymers of a given length were calculated from the theory of Kirkwood (29Kirkwood J.G. J. Polymer Sci... 1954; 12: 1-14Google Scholar). The above calculations predict that 85% of a 3.4 S FtsZ monomer should remain in the supernatant after centrifugation; 5-mers, which should be 8 S, would be 40% sedimented, and 40-subunit filaments, which should be 14.5 S, would be 70% sedimented. The theory of isodesmic polymer assembly at equilibrium can be used to predict an apparent association constant for assembly of FtsZ into protofilaments. Here we have extended this theory to include a step of GTP hydrolysis followed by rapid subunit dissociation. The modified theory fits the observed maximum hydrolysis rates and length distributions of FtsZ and allows the prediction of a true association constant for the addition of GTP subunits. The average length of an isodesmic filament is determined by the affinity constant for assembly and the total protein concentration. To use length measurements to estimate an apparent affinity constant for GTP- and GMPCPP-FtsZ polymerization, we derived the following Equations Equation 1, Equation 2, Equation 3, Equation 4, based on previous models for isodesmic assembly (19Oosawa F. Kasai M. J. Mol. Biol... 1962; 4: 10-21Google Scholar, 22Reisler E. Pouyet J. Eisenberg H. Biochemistry.. 1970; 9: 3095-3102Google Scholar, 30Thusius D. Dessen P. Jallon J.M. J. Mol. Biol... 1975; 92: 413-432Google Scholar): Ci=Ci−1·C1·2KA=(2KA)(i−1)·C1iEquation 1 Ct=C1/(1−2KAC1)2Equation 2 where C i is the concentration of polymers of length i; C 1 is the monomer concentration; C t is the total concentration of subunits in all polymers; and K A is the association constant for adding a subunit. These equations are the same as those found in Oosawa and Kasai (19Oosawa F. Kasai M. J. Mol. Biol... 1962; 4: 10-21Google Scholar), except that K A is multiplied by a factor of 2 because the i + 1 subunit can be added to either end of a growing polymer.C1=4KACt+1−(8KACt+1)1/28KA2CtEquation 3 Equation 3 is the quadratic solution of Equation 2,Ln=ΣiCiΣCi=4KACt−1+8KACf+1Equation 4 Equation 4 is the number average length of the polymers (L n). At total FtsZ concentrations of ∼2 μm, the average lengths of GTP and GMPCPP filaments were 23 and 38 subunits, respectively. According to Eq. 4, the apparent affinities needed to produce filaments of such lengths areK A, app = 1.25 × 108and 3.3 × 108m−1, corresponding to dissociation constants of 8 and 3 nm. These apparent affinity constants underestimate the actual affinity at a GTP-FtsZ interface because in addition to dissociation of GTP-bound FtsZ subunits, filaments fragment due to GTP hydrolysis, as described below. The following polymerization Model 1 accounts for the on-off reactions of subunits in both the GTP and GDP states, and the rate of GTP hydrolysis (see also Fig. 7, which will be addressed under "Discussion"): FT = FtsZ bound to GTP; FD = FtsZ bound to GDP; and F = another subunit of any nucleotide state. The rates are assumed to be identical whether monomers or the ends of long polymers are interacting. In reality, the entropy and diffusion of long polymers will be different from that of monomers, but these deviations from isodesmic assembly are likely to be less than 2-fold, as was found by Rivas et al. (14Rivas G. Lopez A. Mingorance J. Ferrandiz M.J. Zorrilla S. Minton A.P. Vicente M. Andreu J.M. J. Biol. Chem... 2000; 275: 11740-11749Google Scholar). The association of protein subunits was assumed to be diffusion-limited (k on ≈ 5 × 106m−1 s−1) (31Northrup S.H. Erickson H.P. Proc. Natl. Acad. Sci. U. S. A... 1992; 89: 3338-3342Google Scholar) for both GTP and GDP-FtsZ. Nucleotide exchange is likely to be very rapid (k nucleotide ≈ 35/s), based on the 7 μm K D of GDP for FtsZ monomer (32Mukherjee A. Dai K. Lutkenhaus J. Proc. Natl. Acad. Sci. U. S. A... 1993; 90: 1053-1057Google Scholar) and a diffusion-limited on rate for nucleotide binding (≈5 × 106). Dissociation of GDP-FtsZ is also likely to be very rapid; with a diffusion-limited k on(GDP-FtsZ) and a K D = 20 μm (14Rivas G. Lopez A. Mingorance J. Ferrandiz M.J. Zorrilla S. Minton A.P. Vicente M. Andreu J.M. J. Biol. Chem... 2000; 275: 11740-11749Google Scholar),k off(GDP-FtsZ) would be 100/s. In contrast to the rapid dissociation at a GDP-FtsZ interface, the dissociation at a GTP-FtsZ interface would be slow, giving rise to the stability of the protofilaments. The 3 nm apparent affinity constant calculated for GMPCPP filaments can be used as a first approximation for the affinity at GTP-FtsZ interfaces. A diffusion-limited on rate then predictsk off(GTP-FtsZ) = 0.014/s. The GTP hydrolysis rate can be shown to be on the same order as this dissociation rate of GTP-FtsZ. The overall nucleotide turnover rates are known from experimental measurements (in the present study 1.5/min or 0.025/s for GTP, 0.23/min or 0.0038/s for GMPCPP). The model assumes that hydrolysis occurs only after subunit association, and that all subunits in the polymer hydrolyze GTP at a rate characterized by the first order rate constant k hydrolysis. At high protein concentrations, these overall turnover rates are valid estimates of k hydrolysis. This is because subunit association is no longer rate-limiting, and there is only a small fraction protein that is either monomer or at a polymer end and so not contributing to the turnover rate. We postulate that the subunits in the protofilaments are largely bound to GTP and that the K A, app estimated above is determined by three parameters.k on(GTP-FtsZ) gives the rate of addition of GTP subunits (or annealing of polymers with GTP at their ends). The off ra