Force generation by a dynamic Zring in Escherichia coli cell division
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Edited by J. Richard McIntosh, University of Colorado, Boulder, CO, and approved November 14, 2008 (received for review September 2, 2008)
Abstract
FtsZ, a bacterial homologue of tubulin, plays a central role in bacterial cell division. It is the first of many proteins recruited to the division site to form the Zring, a dynamic structure that recycles on the time scale of seconds and is required for division to proceed. FtsZ has been recently shown to form rings inside tubular liposomes and to constrict the liposome membrane without the presence of other proteins, particularly molecular motors that appear to be absent from the bacterial proteome. Here, we propose a mathematical model for the dynamic turnover of the Zring and for its ability to generate a constriction force. Force generation is assumed to derive from GTP hydrolysis, which is known to induce curvature in FtsZ filaments. We find that this transition to a curved state is capable of generating a sufficient force to drive cellwall invagination in vivo and can also explain the constriction seen in the in vitro liposome experiments. Our observations resolve the question of how FtsZ might accomplish cell division despite the highly dynamic nature of the Zring and the lack of molecular motors.
FtsZ is a fundamental bacterial division protein that is conserved across nearly all bacterial and archaeal species, and has been found to play a role in chloroplast division in plants as well as in mitochondrial division in algae (1, 2). It is a cytoskeletal protein, homologous to tubulin, that assembles into short filaments (3) that are the building blocks of the bacterial division ring referred to as the Zring. The Zring forms at midcell (4, 5) under the regulating influence of a nucleoidassociated signal (6, 7) and the Min system (8).
By analogy with eukaryotic cell division, one might expect bacterial division to proceed by a filament sliding motion similar to that of actin and myosin. However, bacteria have no known cytoskeletal molecular motors to generate the sliding force. In addition, FtsZ filaments sometimes form “Zspirals” that are capable of invaginating despite not forming a closed loop structure around the cell (9). An alternate hypothesis (10) is that the Zring constricts by progressively bending into a more highly curved structure. This idea is supported by the observation that FtsZ has a GTP hydrolysis cycle, similar to that of tubulin, that allows it to take on 2 preferred states. When bound to GTP, filaments prefer to be straight but, upon hydrolysis, the preferred conformation is curved with a radius of curvature of ≈12–13 nm (10, 11). The fact that FtsZ alone is capable of generating a constriction force was clearly demonstrated by Osawa et al. (12). In their in vitro liposomeFtsZ assay, FtsZ rings formed on the inside of tubular liposomes and proceeded to constrict the liposomes in a GTPdependent manner, in the absence of any other protein. The viability of this hydrolyzeandbend hypothesis, in either the in vivo or in vitro context, has not been rigorously tested yet in the sense of quantitative modeling. In vivo, the Zring forms early in the cell cycle and is maintained at a constant size for tens of minutes until constriction begins (13). During this time, it contains ≈30% of the total FtsZ available in the cell (14, 15). However, throughout this period, either FtsZ subunits or short FtsZ filaments are constantly being incorporated into and removed from the ring, resulting in a turnover halftime on the order of tens of seconds, as indicated by FRAP measurements (14, 15). This rapid turnover is coordinated with nucleotide state in that GDPbound subunits tend to disassemble from the ring more rapidly than GTPbound subunits (16). The halftime is consistent with in vitro measurement of FtsZ polymerization kinetics, in particular, hydrolysis and disassembly rates that are reported to be ≈0.1 s^{−1} and 3 s^{−1}, respectively (14, 17, 18). However, given that the filaments themselves are allegedly generating the constriction force, it is not clear that significant force can be generated when disassembly follows hydrolysis so rapidly. There seems to be a narrow window of forcegenerating opportunity.
Although much progress has been made in understanding FtsZ biochemistry, the details of the molecular structure of the in vivo Zring and its maintenance are still largely unknown. For example, an important question is whether the dominant pathway for Zring maintenance is growth by direct subunit addition or incorporation of preformed filaments. One recent electron cryotomographic study of fixed cells reported that Zrings appeared to consist of a scattering of short filaments loosely connected by lateral contacts (19). Like tubulin, FtsZ forms filaments with a strong longitudinal bond (20) and can also associate laterally (10). Lateral bonds are weak (0.1 − 0.3 k_{B}T per subunit) (21) compared with longitudinal bonds and with lateral bonds in microtubules (22, 23). However, lateral bonding of longer filaments would have proportionally higher energy and could lead to a greater net contribution to Zring maintenance.
The notion that the Zring is built by lateral association of cytosolic filaments to filaments already incorporated into the ring is bolstered by the finding that MinC, an inhibitor of in vivo Zring assembly, inhibits lateral association of filaments in vitro (24).
In the last few years, much theoretical work has been published on the kinetics and mechanics of FtsZ. Several models of FtsZ in vitro kinetics have been proposed (20, 21, 25–29). Surovtsev et al. (29) also studied in vivo kinetics of the Zring but did not address the question of how energy is transduced into a mechanical constriction force.
Using a formalism derived from elasticity theory, Andrews and Arkin (30) and Horger et al. (31) recently provided a comprehensive numerical characterization of the shapes that a linear polymer can take on when constrained to a cylindrical geometry. This provided an appealing explanation for the helical and ring shapes formed by many bacterial polymers. Their work has implications for the mechanics of FtsZ constriction but was focused on the shapes of static polymers rather than the mechanical interaction between dynamic polymers and the cell membrane/wall.
By modeling the process of cellwall growth, Lan et al. (32) derived estimates of the force required to drive cellwall invagination. By treating the Zring force as a free parameter in their detailed cellwall model, they concluded that the Zring must generate at least 8 pN of force.
Through these studies, much has been elucidated about the kinetics and mechanics of the Zring. However, no one has yet addressed the question of whether any particular putative forcegenerating mechanism is capable of generating sufficient force to explain the observed constriction phenomena in vivo or in vitro. Furthermore, the issue of how the Zring can generate this constriction force over a period of minutes, despite the fact that its component parts are turning over on the scale of seconds, has also not been resolved.
In this article, we propose a simple mathematical model that describes the mechanism originally suggested by Erickson et al. (10) for Zring constriction. We use this model to examine in vivo Zring maintenance and force generation in the context of rapid turnover. We also consider the observed phenotype of the ftsZ84 mutant, whose FtsZ GTPase activity is reduced. Next, we apply the model to the in vitro Zring reconstitution experiments of Osawa et al. (12), taking into consideration the balance of forces between Zring and membrane including the GTP dependence of the observed constriction phenomenon. Our results quantitatively support the hydrolyzeandbend mechanism.
Model
Our model consists of 2 components: (i) a kinetic description of Zring maintenance including the incorporation of new filaments into the ring, hydrolysis of GTPbound subunits in the ring, and disassembly of GDPbound subunits from the ring and (ii) a mechanical characterization of the Zring used for analysis of both in vivo and in vitro force generation. Our formal description of the ring treats it as a collection of filaments with weak lateral bonds that maintain their individual identities as far as kinetics are concerned but cohere as a single mechanical structure. An illustration of the relevant structures is provided in Fig. 1. The kinetic model tracks 2 primary quantities: the length distribution of filaments in the ring, p(l,t), where l is measured in numbers of subunits, and the number of hydrolyzed subunits, S_{D}(t), in the ring. From these 2 primary quantities, several secondary quantities can be calculated: the total number of FtsZ subunits in the ring S = ∫_{0}^{∞}lp(l,t)dl; the total number of filaments, and hence the total number of filament tips, in the ring, F = ∫_{0}^{∞}p(l,t)dl; the cytosolic concentration of FtsZ, Z_{T} − S/(N_{C}V), where Z_{T} is the total concentration of FtsZ, and the factor N_{C}V converts numbers of molecules to concentration in micromoles.
Filaments of all lengths are assumed to be randomly distributed throughout the ring. Changes in the length distribution p(l,t) arise through incorporation of FtsZ filaments from the cytosol and dissociation of hydrolyzed subunits from filament tips.
We assume a quasisteady exponential length distribution for FtsZ filaments in the cytosol, which is consistent with polymer kinetics, independent of the issue of cooperative versus isodesmic assembly or the presence of fragmentation and annealing (25). The assumption of a cytosolic filament distribution is justified by the fact that the cytosolic concentration is well above the critical concentration (15). Additionally, FRAP experiments of cytosolic FtsZ give a halftime approximately twice that of monomeric GFP, consistent with predictions for an exponential length distribution [see supporting information (SI) Appendix]. We use a mean length λ = 30 subunits (17) and an amplitude determined by the current cytosolic concentration. The incorporation rate of filaments of length l is proportional to the cytosolic concentration of filaments of that length. We assume that cytosolic filaments consist entirely of GTPbound subunits. Although incorporated filaments certainly contain some fraction of hydrolyzed subunits, explicitly including this in the model is equivalent to scaling the hydrolysis rate (see SI Appendix).
Dissociation of subunits is governed by 2 factors: position in a filament and nucleotide state. Because interior subunits are stabilized by the lattice, disassembly is biased to the tips of filaments, whose flux is proportional to k_{off}p. Because GDPbound subunits disassemble more rapidly than GTPbound ones (16), this tip expression is multiplied by a prefactor S_{D}/S, the fraction of subunits that are GDPbound. In a simpler model described in SI Appendix, we assume that disassembly is uniform throughout the lattice instead of restricted to filament tips, leading to force estimates that are insufficient to explain constriction. The equation for p(l,t) is The 4πR accounts for the number of available binding sites for new filaments that we assume can only attach along the outer edges of the barrelshaped Zring.
The number of GDPbound subunits in the ring, S_{D}, evolves according to The first term accounts for hydrolysis, which is independent of subunit position in a filament, and the second term accounts for disassembly, found by integrating the disassembly term in the p equation. Parameter values are given in Table 1.
For the mechanical properties of the Zring, we assume that the ring always forms a stable and smooth circle of radius R. The spring energy of an individual hydrolyzed subunit E =
An expression, ℋ, for the mechanical energy stored in a Zring is derived in SI Appendix. Taking the derivative of this expression with respect to R gives the radial force generated by the ring, where B is the bending modulus of a filament, δ is the size of a subunit, f_{D} = S_{D}/S is the fraction of hydrolyzed subunits in the ring, and κ_{D} is the preferred curvature of a hydrolyzed subunit. We use this expression to understand in vivo force generation by the Zring, a quantity that we compare with the 8pN estimate of Lan et al. (32) as a test of the viability of the hydrolyzeandbend mechanism.
When considering the in vitro experiments of Osawa et al. (12), we must include the membrane's resistance to constriction in a forcebalance calculation. We treat the membrane force as linear in the deviation from the preferred radius of the tubular liposome (R_{0} ≈ 1 μm). The resulting force balance equation, which we take to be in mechanical equilibrium, can be written as where F_{M} is the membrane force scale. This approximation is valid for relatively small deformations observed by Osawa et al. (12). For larger deformations, the nonlinear elastic energy should be considered as should the possibility of membrane rupture (see SI Appendix).
Results
In Vivo ZRing Constriction Forces.
The force generated by the Zring has been hypothesized to be the primary driving force behind constriction during cell division (10). The mechanics of the invagination process are complicated by the fact that the cell wall is relatively stiff and must be remodeled rather than simply pulled inward. A model for this process was recently proposed by Lan et al. (32), through which the authors derive the aforementioned estimate of 8 pN required for constriction to proceed. To determine whether our Zring model has such a forcegenerating capacity, we consider the expression for the radial force (Eq. 3) given in Model above.
This expression provides a map of the forcegenerating capacity of the Zring as a function of parameters. For a ring consisting of S = 4,000 subunits [≈30% of the total FtsZ (14)], the force of constriction is shown in Fig. 2A as a family of level curves over the plane of possible values of ring radius R and fraction of subunits hydrolyzed f_{D}. To the right of the rightmost curve, the force is >80 pN. To the left of and below the leftmost curve, it is <8 pN; this corresponds to a small fraction of FtsZ–GDP subunits and/or a small ring radius. Note that the absolute minimal radius for which this constriction mechanism can generate an inward force is the intrinsic radius of curvature of FtsZ–GDP, κ_{D}^{−1} = 12.5 nm (11). For radii R > 60 nm, the force is always >8 pN, provided f_{D} > 0.2. Thus, we predict that ≈20% or more of the FtsZ in the ring must be hydrolyzed.
This purely mechanical analysis indicates that FtsZ filaments with a small preferred radius of curvature are capable of generating the required force under the conditions described. However, the kinetics of FtsZ turnover in the ring are surprisingly fast, with halftimes as small as 9 s (14, 15). With a disassembly rate of hydrolyzed subunits significantly faster than the hydrolysis rate (see Table 1), it is not obvious that the ring can maintain a sufficient fraction of hydrolyzed subunits (>20%) or even a sufficient total number of subunits.
Assuming R changes slowly according to a mechanism such as the one described by Lan et al. (32), we calculate the quasisteady state of the kinetic Eqs. 1 and 2 for fixed R and use the resulting values of S and S_{D} in Eq. 3 to calculate the force generated by rings of various sizes. For the in vivo parameters shown in Table 1, we find that for a cell starting with a radius of 400 nm, the dynamic ring is capable of generating the requisite 8 pN of constriction force all the way down to a radius of ≈25 nm. This force–radius relationship is depicted in Fig. 2B.
It is important to note that for the in vivo rate of filament incorporation k_{in}, no estimate could be found in the literature. However, Stricker et al. (14) and Anderson et al. (15) both observed that ≈30% of the total FtsZ in a cell is found in the Zring. This observation provides a phenomenological means of determining k_{in}, as described in SI Appendix. No further fitting of parameters was required.
The model also provides an estimate for the fraction of the subunits in the ring that are GDP bound, f_{D} = 0.53 (see SI Appendix). Currently, there is no reported value to which this can be compared, even in vitro, so this stands as a prediction of the model. Interestingly, a simpler model for Zring kinetics that does not account for any spatial structure, with subunit dissociation occurring anywhere from within the ring, predicts a value of f_{D} = 0.036 (see SI Appendix). As shown in Fig. 2A, this is clearly too small to generate sufficient forces for constriction. The model presented here introduces enough structure to account for constriction forces without adding details that are not resolved experimentally.
Reduced GTPase Mutant and FRAP Recovery.
The mutant ftsZ84 hydrolyzes GTP at a rate one tenth that of wildtype cells (14, 15). In vivo, this reduced GTPase activity was found to increase the fraction of FtsZ in the ring to 65% (14). By FRAP analysis Anderson et al. (15) found a halftime for the mutant Zring of 30 ± 10 s, a slowing of the subunit turnover by a factor of 3 compared with the wildtype halftime of 9 ± 3 s. An earlier study under different conditions found slower rates of 248 and 32 s for mutant and wild type, respectively (14).
To simulate this mutant phenotype, we used an appropriately reduced hydrolysis rate k_{hyd}^{84} and calculated the steadystate value of S for a radius of R = 400 nm, with all other parameters unchanged. The fraction of FtsZ in the ring increased to 69%, fairly close to the observed 65%.
To explain the halftimes in both wildtype and mutant cells, we consider an estimate for the average time a subunit resides in the ring: The first term is the time required for subunit hydrolysis, and the second is the average time required for an arbitrary subunit to be released by disassembly. This gives an estimated halflife of 8.3 s for wild type and 56 s for the mutant.
To support these estimates, we simulated the FRAP experiment numerically by holding the cell radius fixed and integrating the system of Eqs. 1 and 2 starting from an initially empty ring until a steady state was reached. We found that wild type recovers with a halftime of 5.2 s, whereas the mutant halftime is 15.7 s. Thus, the model accurately reproduces the 3fold increase observed by Anderson et al. (15). The halftimes themselves are faster than those reported by Anderson et al. (15). This difference could be due to variability in k_{hyd} and k_{off} in vivo, where the FRAP assay was carried out, as compared with the in vitro kinetic measurements from which we took our parameter values.
The model predicts a hydrolyzed fraction of subunits in the Zring of f_{D} = 0.1 for the mutant cells. With such a reduction in the hydrolyzed fraction, it seemed possible that the constriction force might fall below the required 8 pN. This would be inconsistent with the observation that the mutant cells still manage to divide (14). We find that force generated by the mutant's Zring is sufficient by the criterion of Lan et al. (32) down to a radius of ≈135 nm, in contrast with the 25nm result for wildtype cells.
ZRings in Liposomes.
In the in vitro experiments of Osawa et al. (12), the physical problem is simpler than that described by Lan et al. (32) in that the Zring is only competing against the elastic resistance of the tubular vesicle wall. Thus, no model for cellwall production is required. To model this system, we use the same kinetic Eqs. 1 and 2 as in the in vivo context with changes only in the values of those parameters listed under “in vitro” in Table 1. At every moment, the dynamic values of S and S_{D} determine the mechanical equilibrium value of R, which solves Eq. 4.
Can the hydrolyzeandbend mechanism generate constrictions that are consistent with experiment? Osawa et al. (12) report that Zrings colocalize with membrane constrictions that vary in their extent. Our own measurements from their published images and movies indicate that Zringassociated constriction radii range from ≈60% of the liposome radius to a complete lack of perceptible constriction for faint rings. In addition, Osawa et al. (12) report that thinnerwalled liposomes could not be successfully produced.
In the physiological regime, the steadystate ring radius given by our model does not depend on the values of F_{M} and k_{in} independently but rather on their ratio (see SI Appendix); a stiffer membrane can be balanced by a thicker Zring. Fig. 3 illustrates the 2 sides of the forcebalance equation (Eq. 4) for the wildtype value of k_{in} and an estimate of the membrane force scale for liposomes with 2, 3, and 4 bilayers (see SI Appendix), but the analysis applies equally well to other parameter combinations with the same ratios. The membrane force curves (dashed lines) fade out below 400 nm, where the linear approximation is no longer valid and where we hypothesize membrane rupture might occur (see SI Appendix). The solid curve corresponds to the force exerted by the Zring. For the middle dashed curve (two bilayers), the steady constriction radius is 765 nm. The shaded region represents variation of k_{in} by 20%, which is an ad hoc means of representing stochastic variation in S (S = 4,000 ± 800). This predicts a range of constriction radii from ≈690–820 nm.
When the in vitro assays were carried out at concentrations of GTP that would allow for complete GTP depletion Osawa et al. (12) found that the Zring relaxed over a period of ≈20 s,, and the liposome returned to its original radius. We replicated this experiment by first allowing the system to find its steady state (90 s) and then imposed Z_{T} = 0 (see SI Appendix) using the in vivo value of k_{in} and the 3bilayer liposome force scale. The resulting time course is shown in Fig. 4. Association of new polymers ceased and S and S_{D} decreased, whereas R increased to its original radius. The constriction and release had halftimes of 6.4 s and 8.8 s, respectively.
Discussion
We have developed a model of the bacterial Zring based on the hydrolyzeandbend mechanism first proposed by Erickson et al. (10). Our model explains how FtsZ can generate the required force for cell division and sustain the force to almost total constriction (25 nm in wild type), even while the individual subunits are turning over rapidly.
The model provides estimates of the time scale for subunit turnover that are consistent with both wildtype and ftsZ84 mutant FRAP measurements. It also makes predictions for the relative abundance of GTP and GDP in the Zring. These results depend on the assumption that disassociation from the ring is from filament tips only. A Zring that allows dissociation of subunits from within filaments loses its hydrolyzed subunits too quickly and cannot generate the required force.
The model also reproduces the in vitro experiments of Osawa et al. (12) where the Zring was reconstituted inside liposomes. Constriction of the liposomes is slight (down to ≈60% of the original radius) and depends on the ratio of membrane resistance F_{M} and the polymer incorporation rate k_{in}, but not on either parameter individually. Our results are therefore robust to inaccuracy in parameter estimates provided the ratio is correct. Estimates from data of the number of lipid bilayers and of the number of subunits in the ring would provide a good test of the model.
An alternative hypothesis for force generation in the absence of motors is a Hill sleevetype mechanism, where polymers slide relative to one another, reducing free energy by increasing the number of lateral bonds (35). However, the mutants described by Addinall and Lutkenhaus (9) (ftsZ26 and rodA_{sui}) were capable of forming “Zspirals” and “Zarcs” that did not form closed loops but were nonetheless capable of generating invaginations. Although a sliding mechanism, which requires tension along the length of the structure, could still accomplish this, it would require selective anchoring of the FtsZ structure at its open ends. In addition, the lateral bonding not only provides an energy gradient and, hence, force but also comes with an energy barrier through the necessity of breaking bonds. Our estimates indicate that the force generated is significant but, relative to the thermal energy scale, so is the barrier (see SI Appendix).
The Zring has been observed to maintain 30% of the FtsZ in a cell even during constriction (14). Our model does not reproduce this result but instead predicts that the ring shrinks as it constricts. This could be reconciled by the upregulation of k_{in} or downregulation of k_{hyd} or k_{off} coincident with the onset of constriction, which is certainly possible given the multiple FtsZ regulatory pathways (36).
Osawa et al. (12) were not able to produce tubular liposomes with thinner walls. One possible reason is suggested by Fig. 3. By accounting for only small deformations in the membrane force calculation, we implicitly assumed that membrane stretch was minimal. In the case of large constrictions, our estimate of the stretch (see SI Appendix) indicates that for constriction radii <0.7 R_{0}, membrane rupture becomes a possibility. For thickwalled liposomes, the upper solution prevents the system from entering this regime, but thinwalled liposomes have no such protection. It may be that thinwalled liposomes initially tubulate but are ruptured as soon as Zrings form. This could also explain why constriction radii <0.6R_{0} were never seen.
A more detailed mechanical model, similar to the work of VanBuren et al. (22, 37) on microtubule dynamic instability, would be a valuable step forward from the model presented here. However, such a mechanically detailed treatment requires a better understanding of the details of in vivo kinetics, including the still elusive ultrastructure of and kinetic rates within the Zring.
Acknowledgments
We thank B. Marshall, D. Fagnan, R. Wong, and A. Mogilner for useful discussion and the Pacific Institute for Mathematical Sciences International Graduate Training Centre Summer School in Mathematical Biology. This work was supported by the National Sciences and Engineering Research Council of Canada.
Footnotes
 ^{1}To whom correspondence should be addressed. Email: cytryn{at}math.ubc.ca

Author contributions: J.F.A. and E.N.C. designed research, performed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/cgi/content/full/0808657106/DCSupplemental.
 © 2008 by The National Academy of Sciences of the USA
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