Formation of the Z-ring is the first step in bacterial cytokinesis. Once the Z-ring has formed it becomes a scaffold for over 30 other proteins that form a divisome complex (Liu et al., 2015). The divisome carries out septal envelope synthesis that leads to the pinching off of one daughter cell from the other. In wild-type *E. coli*, pinching off occurs very accurately in the middle of the mother cell between two separated daughter nucleoids (Trueba, 1982; Den Blaauwen et al., 1999; Männik et al., 2012). In mutant cells, the inaccurate placement of the Z-ring relative to nucleoids can lead to cells lacking chromosomal DNA completely, i.e., minicells (Adler et al., 1967) or to cells that have an incomplete set of genetic material, i.e., have guillotined nucleoids (Niki et al., 1991;

**FIGURE 1 | Four molecular mechanisms that have been identified in spatial coordination between the chromosome and divisome in** *E. coli.* **(A)** SlmA-mediated nucleoid occlusion is a nucleoid driven mechanism which negatively regulates Z-ring formation in the vicinity of the chromosome except at the replication terminus region. **(B)** The Min system is independent of the nucleoid but it has been implicated in segregating and separating nucleoids. The Min system negatively regulates Z-ring formation at cell poles. **(C)** The Ter linkage is involved in determining the location of the Z-ring by a positive regulatory mechanism. The linkage is also involved in holding the Ter region fixed relative to the divisome. **(D)** FtsK translocase pumps DNA across the divisome in a directed manner leading to repositioning of chromosomes.

Cook and Rothfield, 1999; Hendricks et al., 2000). *E. coli* cells have developed a number of molecular systems to prevent this outcome. These systems include nucleoid occlusion (NO), the Min system, the Ter linkage, and FtsK translocase (**Figure 1**). There is evidence that other mechanisms may also be involved. Here, we will review the aforementioned known molecular systems and discuss some hypothetical ones that have been implicated in spatially coordinating the divisome and chromosome. We will limit our discussion to *E. coli*; coordination systems in other bacterial species show divergent molecular origins (Adams et al., 2014; Monahan et al., 2014). We will expand our scope in the last part of this review where we will discuss the minimal requirements for the coordination between cell division proteins and chromosomes that are necessary for the survival of any cell.

### **Nucleoid Occlusion**

The early discussion of coordination between cell division and chromosome replication/segregation centered on the idea of NO (Hussain et al., 1987; Mulder and Woldringh, 1989; Woldringh et al., 1990; Wu and Errington, 2011). The idea of NO is based on observations that constrictions in dividing cells were excluded from the regions occupied by the nucleoids. Woldringh and others proposed that the inhibitory effect of the nucleoid is mediated by short range interactions stemming from the nucleoid which are related to transcription and translation (Woldringh et al., 1991). Zaritsky and Woldringh (2003) further refined the idea proposing that molecular crowding of the inner surface of the plasma membrane in the vicinity of the nucleoid was responsible for this inhibitory effect. These authors hypothesized that crowding results from the transertion process. In transertion, nascent membrane proteins are synthesized concurrently with yet ongoing transcription of their mRNA and are inserted into the cell plasma membrane while their translation occurs (Norris, 1995; Woldringh, 2002). Altogether, this process links DNA to the inner membrane by a molecular chain that includes RNA polymerase, nascent mRNA, ribosome, and nascent protein. The nascent protein is hypothesized to insert itself into the plasma membrane by its N-terminal domain while still being translated. Taking that approximately 1/3 of all proteins in *E. coli* are membrane proteins, the number of transertional linkages should be considerable at any given point of the cell cycle and could lead to significant membrane crowding (Woldringh, 2002). Zaritsky and Woldringh (2003) hypothesized that crowding could be lower in membrane areas adjacent to the replication terminus region because of the smaller density of highly expressed membrane targeted genes. This assumption suggests that Z-ring assembly is least inhibited in membrane regions adjacent to the replication terminus in accord with experimental observations.

The effect of transertion-related crowding on Z-ring formation was experimentally studied (Sun and Margolin, 2004). These authors observed that severing transertional linkages by blocking transcription with rifampicin treatment indeed allowed Z-rings to form over the nucleoids. Conversely, severing transertional linkages by blocking translation with chloramphenicol caused nucleoid compaction and exclusion of Z-rings from the membrane regions in the proximity of nucleoids. Furthermore, in SecA(ts) cells where protein insertion was thermally inactivated, the NO effect was still present even though the nucleoids did not segregate properly. The two latter observations therefore were not in accord with the transertion-based crowding ideas. To explain their data, the authors concluded that the nucleoid structure and packing density most likely plays a role in excluding the Zrings from the regions adjacent to the nucleoid. Z-rings could localize "over" the expanded nucleoids but not over the compact ones. This idea has been supported by other measurements where nucleoid packing was altered (Sun and Margolin, 1998). So far, it remains unclear how the nucleoid packing density can influence Z-ring positioning especially during chloramphenicol treatment when the compacted nucleoid resides far from the cell membrane. Indirect effects of chloramphenicol and rifampicin treatment in these experiments could not be ruled out. In that light, more experiments are warranted to explore further the role of membrane crowding on Z-ring positioning.

### **SlmA Mediated Nucleoid Occlusion**

A major development in the understanding of NO came with the discovery of the SlmA protein from a synthetically lethal screen (Bernhardt and de Boer, 2005). Cells that lacked both *slmA* and *minCDE* were not able to divide in rich medium and gave rise to filamentous cells, indicating a defect in septation. The same authors also observed that some Z-rings could localize over unsegregated nucleoids in *slmA minCDE* cells suggesting SlmA played a role in Z-ring positioning. This role was also visible from studies of *slmA dnaA* cells in rich medium where a closing septum appeared in the middle of the nucleoid mass presumably because of the lack of a functional NO system (Bernhardt and de Boer, 2005). Spatial positioning of the Z-ring by SlmA is thought to result from a specific binding pattern of SlmA on *E. coli* chromosomal DNA. SlmA specific binding sites are distributed over the chromosome with a notable exception at the replication terminus region (Cho et al., 2011; Tonthat et al., 2011). Such a distribution, together with the inhibitory action of SlmA, can generate the NO effect, i.e., Zrings do not assemble over chromosomal DNA except near the replication terminus region. Note that the replication terminus region, which positions itself at mid-cell in the later part of cell cycle, is in the vicinity of the membrane region where the Z-ring typically assembles (**Figure 1A**).

Two possible molecular mechanisms by which SlmA inhibits the formation of the Z-ring have been proposed. One model posits that SlmA causes depolymerization of FtsZ protofilaments (Cho et al., 2011; Cho and Bernhardt, 2013; Du and Lutkenhaus, 2014). *In vitro* assays show that depolymerization of FtsZ only occurs at a significant rate when SlmA is bound to DNA in its specific binding sites (Cho et al., 2011). Consequently, depolymerization of protofilaments takes place everywhere in the nucleoid except at the replication terminus region. Further characterization of the depolymerization process shows that it occurs in two steps (Du and Lutkenhaus, 2014). In the first step, DNA-bound SlmA attaches to the highly conserved C-terminal tail of FtsZ. In this binding, SlmA competes with other FtsZ regulators (MinC, ClpX) and interaction partners (ZipA, FtsA, ZapD). In the second step, further interactions occur that lead to protofilament breakage. The resulting breakage appears to minimally affect the GTPase activity of FtsZ (Cho et al., 2011).

In an alternative model, it was proposed that DNA-bound SlmA does not depolymerize protofilaments but instead captures them and renders them incapable of Z-ring formation (Tonthat et al., 2011, 2013). The authors showed that SlmA binds to its specific binding sites as a dimer of dimers. Once the initial nucleation has occurred, the dimers can cooperatively spread on DNA. Tonthat et al. (2011, 2013) proposed that these higher order SlmA structures capture FtsZ filaments. However, so far microscopy of FtsZ fusion proteins has not confirmed any co-localization between FtsZ protofilaments and extended SlmA structures within the cell.

While characterization of SlmA at the molecular level has been extensive, understanding its role and function at the cellular level is still limited. How could SlmA that is bound to chromosomal DNA inhibit Z-ring formation at the cell membrane? Models described in (Du and Lutkenhaus, 2014) and (Tonthat et al., 2013), although different in their interaction mechanism between SlmA and FtsZ, both assume that DNA-bound SlmA comes into proximity of the cell membrane to influence the localization of the Z-ring. However, DNA-bound SlmA within the nucleoid makes only limited contacts with the membrane and therefore would interact infrequently with membrane-bound FtsZ. Transertional linkages may help to facilitate these contacts (Tonthat et al., 2013) but the existing microscopy data indicates that SlmA is localized within the nucleoid rather than in the vicinity of the cell surface (Bernhardt and de Boer, 2005). Alternatively, Bernhardt and de Boer (2005) proposed that depolymerization of protofilaments can happen within the nucleoid. The latter mechanism would lead only to a weak site-selective effect of SlmA on Z-ring localization. Existing experimental data (Männik et al., 2012) indeed shows that SlmA does not convey a strong site-selective effect on Z-ring location in fast growth conditions. How effectively SlmA inhibits Z-ring formation in various growth conditions remains yet to be characterized.

Although SlmA mediated NO has received the most attention recently, there is strong evidence that additional mechanisms beyond SlmA can lead to a NO effect in *E. coli*. It was observed that cell division proteins are positioned in accordance with NO in cells lacking SlmA. The NO effect was distinctly present even when the shapes and sizes of these cells were strongly perturbed (Männik et al., 2012). It was also observed that a replication-inhibited and unsegregated nucleoid at mid-cell blocks Z-ring formation independent of the SlmA and SOS response (Cambridge et al., 2014). How these inhibitory effects of nucleoids are mediated at a molecular level is currently not known, but from these data it is clear that the SlmA-related mechanism is not the only one that realizes NO in *E. coli* cells.

## **The Min System**

The three Min proteins, MinC, MinD, and MinE, form a wellunderstood geometric positioning system for the Z-ring in *E. coli* that defines the cell's geometric middle and prevents polar septations (Lutkenhaus, 2007, 2012; Shapiro et al., 2009; Moseley and Nurse, 2010). Fluorescent tagging of Min proteins has shown that MinC, MinD, and MinE exhibit remarkable oscillatory behavior in *E. coli* cells, moving back and forth between the two poles with a typical oscillatory period from 30 s to 1 min (Raskin and de Boer, 1999). Of these three proteins, only MinD and MinE are necessary to set up the oscillations while MinC, which follows and binds to MinD, acts as an inhibitor for Z-ring formation (Lutkenhaus, 2007). MinC binding to membrane-attached MinD activates its inhibitory function (Lutkenhaus, 2012). Due to the oscillations, the destabilizing effect of MinC on Z-ring formation is the strongest at the cell poles, where the time-averaged concentration of MinD-bound MinC is the highest. This negative regulation prevents minicelling at the poles. However, the Min system appears also to play a role in the precise localization of the Z-ring at midcell (Guberman et al., 2008).

In its inhibitory action, MinC resembles SlmA. MinC also binds to the conserved C-terminal domain of FtsZ and its subsequent interactions lead to breakage of the FtsZ polymer. GTPase activity of FtsZ is required for the MinC mediated breakage but at the same time MinC does not lead to increased GTPase activity of FtsZ (Dajkovic et al., 2008; Shen and Lutkenhaus, 2010). Measurements using *in vivo* reconstituted assays show how these two perhaps contradictory findings can be reconciled (Arumugam et al., 2014). Arumugam et al. (2014) propose that MinC dimers cap FtsZ filament ends. They also observe that FtsZ filaments lose monomers throughout its length. Such loss leads to breaks and gaps in the filament. In the absence of MinC these breaks could be annealed by the addition of new monomers but in the presence of MinC capping this would not occur and filaments become destabilized. The same authors also propose that MinC binding to FtsZ hinders protofilament bundling, which leads to further weakening of the Z-ring. Because of the several observed similarities between SlmA and MinC, it is tempting to speculate that a similar scenario also can be realized when SlmA depolymerizes FtsZ filaments.

Extensive modeling has been carried out to capture the oscillatory behavior of the Min system based on continuum models and stochastic simulations. For a cross-section of this work spanning from early to current models see (Meinhardt and de Boer, 2001; Kruse, 2002; Huang et al., 2003; Fange and Elf, 2006; Kerr et al., 2006; Halatek and Frey, 2012; Bonny et al., 2013). The oscillatory movement of MinD and MinE in the cell emerges in all of these models due to the ATP and MinE-modulated attachment of MinD to the plasma membrane. The computational models are able to semi-qualitatively reproduce experimentally measured oscillation patterns and oscillation periods. Although different models introduce slight variations in reactions occurring between MinD and MinE, they all can be categorized in mathematical terms as reaction-diffusion systems that exhibit Turing instability (Turing, 1952). Arguably the Min system in *E. coli* is the best studied example where the Turning instability mechanism leads to the formation of a dynamic pattern in a living organism.

The Min system functions autonomously from the nucleoid as shown convincingly in *in vitro* reconstituted assays (Loose et al., 2008; Ivanov and Mizuuchi, 2010; Schweizer et al., 2012). The same conclusion also can be drawn based on experiments with cells that lack nucleoids but have a functioning Min system. Remarkably, in these cells the Z-ring also can be placed relatively accurately in the middle of the cell (Sun and Margolin, 1998; Yu and Margolin, 1999; Pazos et al., 2014). Taking that the two daughter chromosomes separate from each other approximately at mid-cell in normal growth conditions, this system alone is perhaps sufficient to coordinate chromosomes and cell division proteins in *E. coli*. However, if the cell shape becomes aberrant (Männik et al., 2012) then the Min system and NO can define different locations for the cell division plane. In conflicting cases, it appears that the NO mechanism dominates over the Min system (Männik et al., 2012).

While the Min system is not directly involved in coordinating the Z-ring and chromosomes, an indirect involvement is possible (**Figure 1B**). Several authors have pointed out that deletion of the Min system leads to a small defect in chromosome segregation and in the separation of daughter nucleoids in *E. coli* (Mulder et al., 1990; Akerlund et al., 2002; Di Ventura et al., 2013; Jia et al., 2014). Di Ventura et al. (2013) have proposed that MinD, which is a homolog of ParA, binds to DNA. These authors propose that MinD oscillation and DNA binding provides a Brownian-ratchet mechanism for DNA segregation and separation (Di Ventura et al., 2013). Further experimental support to this interesting idea is still warranted.

### **The Ter Linkage**

The Min system and NO are negative regulators for cell division proteins, i.e., they inhibit Z-ring formation in certain locations of the cell. Recent research strongly suggests that there exists also a positive regulation mechanism in *E. coli*, which guides cell division proteins toward the replication terminus region of the chromosome (Bailey et al., 2014; **Figure 1C**). The presence of a positive regulation mechanism became evident in studies of cells that lacked both the Min system and SlmA-mediated NO. In these cells, the Z-ring positioned itself over the centers of segregating nucleoids instead of localizing at the cell poles or gaps between the nucleoids. The effect was particularly striking in cephalexin-treated cells that show many well separated nucleoids. Time-lapse measurements of *slmA min* cells showed that formation of the Z-ring commenced shortly after the arrival of the replication terminus region to the nucleoid center, even though transient associations of the Z-ring and the replication terminus region could be seen before the replication terminus centralized.

A different piece of evidence shows that the replication terminus region is anchored to the Z-ring (Espeli et al., 2012). The replication terminus region of *E. coli* chromosome forms a compact entity, termed the Ter macrodomain, which is organized by MatP proteins (Mercier et al., 2008; Dupaigne et al., 2012). MatP, like SlmA, is a DNA-binding protein. Interestingly, binding sites of these two proteins in *E. coli* chromosomal DNA are completely complementary—MatP specific binding sites, 23 total, can be found only in an 800 kb stretch around the *dif* sequence (located at 32.24*′* ) in the replication terminus region while SlmA binding sites are located, essentially, everywhere else. Work by Espeli et al. (2012) has shown that the anchoring of the Ter region to the Z-ring occurs due to the MatP C-terminal interaction with the Z-ring associated protein ZapB (Espeli et al., 2012), which indirectly interacts with FtsZ through ZapA (Galli and Gerdes, 2010). The anchor, which we refer as the Ter linkage, connects the replication terminus region to the Zring through a chain of DNA-MatP-ZapB-ZapA-FtsZ. Only the nearest components in the Ter linkage are thought to interact (**Figure 2A**). The function of the linkage is not fully established but it appears to guarantee that the positioning of the chromosomes relative to the divisome does not change after the Z-ring has formed.

There are thus two sequential processes occurring that coordinate the Z-ring and the replication terminus region (**Figure 2B**). First, a signal from the replication terminus region promotes assembly of the Z-ring in its immediate vicinity. Second, after the Z-ring has formed the replication terminus region becomes linked to the Z-ring by a connection involving MatP, ZapB, and ZapA. Interestingly, ZapA and ZapB, and to lesser degree MatP, are also needed in the first step (Bailey et al., 2014). How exactly MatP, ZapB, and ZapA promote Z-ring formation in the vicinity of the Ter macrodomain is not known. It is possible that positive regulation itself is not a direct consequence of MatP, ZapA, and ZapB but results from some other molecular system that associates with the replication terminus region. MatP, ZapA, and ZapB may help to reinforce this signal by linking the source of the signal to the nascent Z-ring and strengthening its effect. At this point, the hypothesis has not been tested and the mechanistic details of the positive regulation are yet to be established.

**FIGURE 2 | The Ter linkage coordinates localization of both the Z-ring and the Ter macrodomain region of the chromosome. (A)** A putative connection between the DNA in the Ter region and the Z-ring via the Ter linkage. Schematics from top to bottom show progressive zoom-ins of the Ter region (light blue ellipse) and the Z-ring (green circles). ZapB in the linkage could form filaments or even filament networks. **(B)** Cell-cycle dependent processes involving the Ter linkage. Top: in the beginning of cell cycle the Ter macrodomain region is localized at the nucleoid periphery close to the new pole of the cell. Positive regulation from the Ter region promotes transient Z-rings to form in its vicinity. Middle: as the cell cycle progresses, the Ter region moves to the center of the nucleoid. It has been hypothesized that the motion is related to replication of the Ter region (Espeli et al., 2012). The positive signal from the Ter region continues to promote Z-ring formation in its vicinity. Bottom: as the Z-ring fully forms the Ter region becomes attached to the Z-ring through a link involving MatP, ZapB, and ZapA (indicated by arrows).

### **Effect of the Divisome on the Nucleoid—the FtsK Translocase**

So far the discussion has focused on mechanisms where the nucleoid directly or indirectly determines the localization of cell division proteins. The opposite processes, in which the divisome affects positioning of the nucleoid, or at least part of it, are also present in *E. coli.* The Ter linkage, discussed in previous section, is one example where the divisome exerts its effect on the nucleoid. However, the Ter linkage appears to maintain rather than to actively re-arrange the chromosomal organization and positioning. Contrarily, DNA translocase FtsK (Begg et al., 1995) allows the divisome to actively re-arrange the *E. coli* chromosome in an ATP-dependent manner (**Figure 1D**). As is the case with the Ter linkage, the activity of FtsK seems to affect the Ter region of the chromosome (Deghorain et al., 2011; Stouf et al., 2013). Since the Ter region comprises about 20% of the total *E. coli* chromosome its re-arrangements have global implications to chromosome organization.

FtsK translocation activity leads to positioning of chromosomal *dif* site in the divisome during late stages of the cell cycle. The *dif* site is a 28 bp sequence in the replication terminus region where chromosome dimers are resolved by the XerCD recombinase. FtsK is capable of pumping DNA on both chromosome arms toward the *dif* site. The directionality in pumping is due to KOPS (FtsK orienting/polarizing sequence), which are oriented in opposite directions on the left and right arms of the *E. coli* chromosome. FtsK loads onto DNA at KOPS sites in a specific orientation. Unidirectional pumping by FtsK then leads to the movement of the *dif* site toward the divisome irrespective of which chromosome arm FtsK was loaded upon (Sherratt et al., 2010). Loading and translocation can only occur *in vivo* when FtsK is localized in the divisome where it forms hexameric units (Bisicchia et al., 2013). The translocation through the barrel in the FtsK hexameric assembly takes place at a rate of about 5 kb/s and stops when FtsK reaches the *dif* site (Sherratt et al., 2010). FtsK translocation activity releases most DNA-binding proteins from the translocated region of the chromosome (Lee et al., 2014). Among others, MatP proteins are also released. The latter leads to dissociation of the Ter linkage at late stages of cytokinesis (Stouf et al., 2013). It is likely that the Ter linkage facilities the activity of FtsK by maintaining this chromosomal region physically in the proximity of the divisome.

FtsK activity on DNA is not limited to translocation. At *dif* sites, FtsK is responsible for activation of the XerCD system that resolves chromosome dimers. FtsK has also been implicated in activating Topo IV (parC), which removes chromosome catenates (Espeli et al., 2003).

## **Minimal Requirements for Coordination**

Interestingly, none of the four molecular systems that have been discussed in this review are strictly essential in *E. coli*. One can delete *slmA* and *minCDE* together with either *matP*, *zapA*, or *zapB* from *E. coli* but the cells still remain viable in slow growth conditions (Bailey et al., 2014). Moreover, even though FtsK is essential, its DNA translocating domain (C-terminal domain) is not (Sherratt et al., 2010). Is it possible that *E. coli* harbors some additional molecular system that coordinates its divisome and chromosome, and this yet to be discovered system is indispensable? As discussed above, there is strong evidence that the NO effect can occur without SlmA. Although the molecular bases of this mechanism remains unknown it could be an essential mechanism. Alternatively, it is possible that there are no indispensable mechanisms that coordinate cell division and chromosome segregation in *E. coli*. One can ask what minimal coordination is needed between cell division proteins and chromosomes for any cell (not only *E. coli*) to propagate. For propagation of a sizeable cell population, the ultimate limit appears to be that on average more than half of the nucleoids need to survive cell division undamaged and emerge in newborn daughter cells (note that small cell populations can go extinct even when more than half of the nucleoids survive cell division). One way to fulfill this requirement in cells that lack any dedicated coordination mechanism between cell division and chromosomes is to increase cell size. If the division plane is placed randomly in the cell then the probability to produce viable daughter cells increases as the cell size increases (**Figure 3**). Multinucleoid cells are more likely to produce two viable daughters upon random placement of the division plane at the expense of losing some genetic material. However, rod-shaped bacterial cells with two nucleoids can also give rise to a viable population when their sizes are sufficiently large. In the latter case, it is assumed

that the physical size of the nucleoid does not depend on cell size and nucleoids are randomly placed in the cell before division. It remains to be proven if viable populations also can emerge in other cell geometries under these assumptions. Interestingly, as the systems depicted in **Figure 1** are progressively deleted from *E. coli*, the cells become larger (longer) but remain viable in slow growth conditions (Bailey et al., 2014) in accordance with this hypothesis.

their division plane placement is random.

Although large cells can cope with random placement of division planes, their fitness is very low because these cells lose a substantial amount of resources when they produce unviable cells or guillotine nucleoids. Mechanisms that coordinate cell division proteins and chromosomes are thus essential for cellular fitness. These mechanisms are highly efficient in modern bacteria. The probability that wild type *E. coli* produces minicells has been estimated to be less than 0.03% (Niki et al., 1991). It is, however, plausible that early protocells did not have any dedicated

### **References**

coordination systems. This argument is supported by findings that different bacterial species have evolved very different molecular mechanisms that coordinate divisomes and chromosomes (Monahan et al., 2014). As a corollary to this discussion, the lack of these coordination systems would imply that early protocells were perhaps larger than present day bacteria. Alternatively, early protocells might have had mechanisms that provided some coordination between division planes and chromosomes but which were not specifically dedicated for the task. For example, in a rodshaped bacterial cell, membrane mechanics dictates that divisions which partition a mother cell into two equal halves are energetically more favorable than asymmetric divisions (Shlomovitz and Gov, 2009). Also, it is likely that chromosomes could be pushed mechanically away as the division septum closes preventing them from being guillotined or that chromosomes could provide enough mechanical hindrance to prevent the septum from closing in cells that lack strong cell wall. These mechanisms could still be present in modern *E. coli* even though their influence is overridden by the more efficient molecular systems such as Min, SlmA, Ter linkage, and FtsK.

### **Concluding Remarks**

As this review emphasizes, there are several modular pathways coordinating chromosomal positioning with cell division in *E. coli*, which are redundant, at least, in slow growth conditions. Moreover, there is evidence that, in addition to the mechanisms known so far, there are other types of coordination which have not been described yet. All the mechanisms lead to increased cellular fitness but are not essential for cell viability. Perhaps surprisingly, one of the conclusions of this review is that the *E. coli* cell can cope with very limited coordination between cell division proteins and chromosomes.

### **Acknowledgments**

The authors thank Anna Jennings, Jaana Männik, and Conrad Woldringh for valuable comments on the manuscript. The work was supported in part by University of Tennessee start-up funds and NSF research grant MCB-1252890.

and SlmA in *Escherichia coli*. *PLoS Genet.* 10:e1004504. doi: 10.1371/journal. pgen.1004504

and cell division independently of SOS and the SlmA nucleoid occlusion protein in *Escherichia coli*. *J. Bacteriol.* 196, 36–49. doi: 10.1128/JB.01230-12

**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Copyright © 2015 Männik and Bailey. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

# **The Min system and other nucleoid-independent regulators of Z ring positioning**

*Veronica W. Rowlett and William Margolin\**

*Department of Microbiology and Molecular Genetics, University of Texas Medical School at Houston, Houston, TX, USA*

Rod-shaped bacteria such as *E. coli* have mechanisms to position their cell division plane at the precise center of the cell, to ensure that the daughter cells are equal in size. The two main mechanisms are the Min system and nucleoid occlusion (NO), both of which work by inhibiting assembly of FtsZ, the tubulin-like scaffold that forms the cytokinetic Z ring. Whereas NO prevents Z rings from constricting over unsegregated nucleoids, the Min system is nucleoid-independent and even functions in cells lacking nucleoids and thus NO. The Min proteins of *E. coli* and *B. subtilis* form bipolar gradients that inhibit Z ring formation most at the cell poles and least at the nascent division plane. This article will outline the molecular mechanisms behind Min function in *E. coli* and *B. subtilis*, and discuss distinct Z ring positioning systems in other bacterial species.

#### **Keywords: bacterial cell division, divisome, Min system, FtsZ, Z-ring positioning**

### **Introduction**

In bacteria, several proteins assemble at the cell center (midcell) to form the divisome, or cell division machine (Martos et al., 2012). At the start of divisome formation, FtsZ, a homolog of eukaryotic tubulin, polymerizes and forms a ring-like structure, the Z ring, at midcell (Bi and Lutkenhaus, 1991). The Z ring functions as a scaffold for cell division proteins and is an obvious target for regulating the site of cytokinesis. Decades of research in *E. coli* and *B. subtilis* have led to insights into the molecular mechanisms of NO and the Min system, which negatively regulate Z ring formation by preventing Z rings from forming over nucleoids or at cell poles, respectively. However, it is likely that other regulators contribute to Z ring positioning in these bacteria (Rodrigues and Harry, 2012). Recently, unique negative and positive regulators of Z ring formation have been identified in bacteria that lack NO and/or Min systems, highlighting the diversity of division site selection mechanisms.

## **Minicells and the Min System**

Several decades ago, Adler et al. (1967) identified small, nucleoid free *E. coli* cells described as minicells. These minicells do not divide, but remain metabolically active for hours. Minicells also form in *B. subtilis* and other bacteria (Reeve et al., 1973). Historically, purified minicells were used to produce radiochemically pure proteins from high-copy plasmids that partition into minicells and could be selectively labeled in purified minicell preparations (Roozen et al., 1971).

In *E. coli*, Min proteins localize to the cell poles and function to prevent Z rings from forming near those poles (**Figure 1**; Bramkamp and van Baarle, 2009). In *B. subtilis*, the Min system has a

#### *Edited by:*

*Arieh Zaritsky, Ben-Gurion University of the Negev, Israel*

#### *Reviewed by:*

*Marc Bramkamp, Ludwig-Maximilians-University Munich, Germany Imrich Barak, Institute of Molecular Biology, Slovak Academy of Sciences, Slovakia Liz Harry, University of Technology, Sydney, Australia Orietta Massidda, Università di Cagliari, Italy*

#### *\*Correspondence:*

*William Margolin, Department of Microbiology and Molecular Genetics, University of Texas Medical School at Houston, 6431 Fannin, Houston, TX 77030, USA<EMAIL_ADDRESS>
#### *Specialty section:*

*This article was submitted to Microbial Physiology and Metabolism, a section of the journal Frontiers in Microbiology*

> *Received: 27 March 2015 Accepted: 30 April 2015 Published: 13 May 2015*

#### *Citation:*

*Rowlett VW and Margolin W (2015) The Min system and other nucleoid-independent regulators of Z ring positioning. Front. Microbiol. 6:478. doi: 10.3389/fmicb.2015.00478*

**by the Min system.** Shown are Min oscillations (dark gradients), dynamics of FtsZ (orange spheres) and eventual assembly as a Z ring at the midpoint of a wild-type cell through the process of cell division and partitioning of the Min system to daughter cells. Nucleoids are depicted as blue ovals; passage of time in seconds is represented by an arrow, passage of time in minutes by a double arrow.

somewhat different role (see next section). When Min proteins are deleted or non-functional, Z rings are able to form both at correct midcell locations and at cell poles, resulting in minicells and cells of heterogeneous length, as not all Z rings are functional to complete division (de Boer et al., 1989; Yu and Margolin, 1999). One of the Min proteins, MinC, directly interacts with FtsZ and inhibits FtsZ polymerization (Hu et al., 1999; Dajkovic et al., 2008; Shen and Lutkenhaus, 2009, 2010). As described below, MinC also interacts with partner proteins that localize it to cell poles, creating a gradient of MinC that is highest at cell poles and lowest at midcell (Monahan and Harry, 2012).

### **Min Systems of** *B. subtilis* **and** *E. coli*

In *B. subtilis*, DivIVA localizes the Min proteins to the poles of the cell (Cha and Stewart, 1997; Edwards and Errington, 1997). DivIVA functions as a generic cell-pole targeting protein, as it senses negative membrane curvature in a wide variety of species, including fission yeast (Edwards et al., 2000; Lenarcic et al., 2009). As a result, DivIVA relocalizes from cell poles to the new septum because of its sharp curvature. MinJ acts as an adaptor protein between polarly-targeted DivIVA and MinD, efficiently recruiting MinD to cell poles, where it binds the cytoplasmic membrane via its C-terminal amphipathic helix (Szeto et al., 2002). MinD binds MinC directly, and by doing so recruits the Z ring inhibitor MinC to the cell poles (Bramkamp et al., 2008; Patrick and Kearns, 2008). The Min system of *B. subtilis* was originally believed to migrate to existing cell poles to form a static bipolar gradient, with polar MinC inhibiting assembly of Z rings only at cell poles. However, more recent evidence indicates that Min proteins are recruited to midcell prior to septation and formation of new cell poles (Gregory et al., 2008). It is thought that localization of Min proteins to midcell prevents more than one Z ring from forming there, and plays a role in establishing a new bipolar gradient in daughter cells (Gregory et al., 2008; van Baarle and Bramkamp, 2010). The Min system thus does not seem to identify midcell as much as promote efficient use of the midcell site (Rodrigues and Harry, 2012).

The Min system in *E. coli* differs from that of *B. subtilis* because *E. coli* lacks DivIVA and instead contains a third Min protein, MinE. Because they lack DivIVA, *E. coli* cells need to establish a MinC bipolar gradient without the benefit of a polar targeting protein. As in *B. subtilis*, *E. coli* MinD directly binds the membrane and directly binds MinC (de Boer et al., 1991; Hu and Lutkenhaus, 2003). The MinE protein is critical for targeting the MinCD complex to the cell poles (see below). MinE forms a ring and causes MinD to be removed from one pole and migrate to the opposite pole (Raskin and de Boer, 1997; Hu et al., 2002). The dynamics between MinD and MinE create an oscillating system, in which MinC is a passenger (**Figure 2**; Hu and Lutkenhaus, 1999; Raskin and de Boer, 1999a,b). As a result of the oscillation, where most of the time is spent at the cell poles with only a short transit (typically 10 s) between them, the average position of MinC over time is at the poles and not at midcell, which leaves midcell as the least inhibited location for FtsZ assembly (**Figure 1**).

Although it is fairly well understood how the Min systems of *E. coli* and *B. subtilis* position Z rings during vegetative growth, relatively little is known about the requirements for placement of Z rings near the cell poles in preparation for asymmetric septation in *B. subtilis* and other Gram positive endospore-forming bacilli such as clostridia. A clue comes from the reconstitution of *E. coli*'s oscillating Min system in *B. subtilis*. Fluorescently-tagged MinD and MinE from *E. coli* oscillate from cell pole to cell pole when produced in *B. subtilis* cells deleted for native *minD* (Jamroškovič et al., 2012), consistent with the self-contained nature of the MinDE oscillator (Ramirez-Arcos et al., 2002). Interestingly, this artificial oscillation inhibited sporulation,

likely by preventing the assembly of Z rings near cell poles that is a hallmark of endospore formation in *B. subtilis*. This suggests that a constitutive oscillatory Min regime, while optimal for medial cell division in cells like *E. coli* that only grow vegetatively, may be problematic in cells that need to switch to asymmetric division for endospore formation (Barák, 2013). The presence of both MinDE and MinDJ/DivIVA in spore-forming clostridia may point to an ancestral Min system that exploited the advantages of each system, perhaps for switching from a vegetative oscillatory mode to a sporulation-induced static mode.

### **Molecular Mechanism of Min Oscillation in** *E. coli* **Cells**

The MinC protein has two distinct domains that synergistically inhibit polymerization of FtsZ. The N-terminal half of the protein is sufficient to antagonize the longitudinal interactions of FtsZ subunits within a protofilament, whereas the C-terminal half inhibits the lateral interactions between FtsZ protofilaments (Hu and Lutkenhaus, 2000; Shiomi and Margolin, 2007; Shen and Lutkenhaus, 2009, 2010). The C-terminus of MinC binds to MinD, which is required for the full inhibition of FtsZ lateral interactions, and is important for MinC dimerization (Hu and Lutkenhaus, 2000, 2003). MinD is a ParA family ATPase that contains a C-terminal membrane targeting sequence (MTS) as mentioned above (de Boer et al., 1991; Hu and Lutkenhaus, 2003). When bound to ATP, MinD dimerizes and associates with the membrane (Szeto et al., 2002; Hu and Lutkenhaus, 2003). MinC and MinD form copolymers of alternating MinC and MinD dimers that are similar to eukaryotic septins (Ghosal et al., 2014; Conti et al., 2015). MinCD complexes are removed from the membrane by MinE (Raskin and de Boer, 1997).

In *E. coli* cells lacking MinE, MinCD binds to the membrane and inhibits Z ring formation throughout the cell, resulting in lethal filamentation (de Boer et al., 1989). Thus, MinE is essential for the MinCD complex to assume its role as a spatial regulator. MinE packs many activities into its compact 88 amino acid size. First, it harbors an amphipathic helix at its N-terminus that serves as an MTS, and direct membrane interaction is important for its function (Ma et al., 2003; Hsieh et al., 2010). Earlier mathematical models did not account for this membrane binding, but more recent models have (Schweizer et al., 2012; Bonny et al., 2013). Upon recognizing MinD, dimers of MinE change conformation, bind MinD, displace MinC, and stimulate the ATPase activity of MinD, causing monomerization of MinD and its removal from the membrane (**Figure 2**; Raskin and de Boer, 1997; Hu et al., 2002; Suefuji et al., 2002; Lackner et al., 2003; Loose et al., 2011; Park et al., 2011). Once dislodged from the membrane, MinD-ATP is regenerated from MinD-ADP by ADP-ATP exchange and immediately re-binds the membrane (Huang et al., 2003). However, it will tend to bind the membrane far from its most recent complex, because the MinE complex that recently removed it can remain on the membrane via its MTS (Loose et al., 2011; Park et al., 2011). This allows MinE to remove other bound complexes of MinCD at the original site, including those that recently re-bound, before changing back to its original conformation and migrating to the opposite pole (Park et al., 2011).

In support of this molecular mechanism, MinE always lags behind MinD. Because MinD-ATP binds to the membrane cooperatively, newly formed MinD-ATP will form a large complex as far away as possible from the original site, which happens to be the opposite pole of a normal sized *E. coli* cell. MinE, with no substrate remaining at the original site, diffuses through the cytoplasm, and possibly the membrane, until it binds to the edge of the newly assembled MinD complex at the opposite pole. *In vivo*, this is observed as a ring of MinE at the edge of the polar zone of MinD. Because the MinD polar zones are large and extend far from the cell pole, MinE rings are often observed near midcell, where they then follow the edge of the MinD polar zone as it shrinks toward the pole (Raskin and de Boer, 1997).

While Min proteins oscillate between cell poles, proteins that are recruited in the early stages of cell division, including FtsZ, ZipA, ZapA, and ZapB, also oscillate oppositely from the Min system with the same period as the Min system (Thanedar and Margolin, 2004; Bisicchia et al., 2013; Tonthat et al., 2013). ZipA and FtsA function to tether the Z ring to the inner membrane (Pichoff and Lutkenhaus, 2005), and ZipA has also been shown to bundle FtsZ protofilaments *in vitro* (Hale and de Boer, 1999; Hale et al., 2000). ZapA recruits ZapB to the Z ring and these proteins function to stimulate and stabilize Z ring formation (Gueiros-Filho and Losick, 2002; Galli and Gerdes, 2010; Buss et al., 2013). As ZipA and ZapA bind to FtsZ directly, and ZapB binds to ZapA, all three proteins are dependent on the Z ring for their localization. Therefore, their counter-oscillatory behavior is likely caused by the periodic assembly and disassembly of Z ring precursor complexes in response to oscillating waves of MinC (Thanedar and Margolin, 2004; Bisicchia et al., 2013).

## **Other Factors Influencing Min Oscillation**

The oscillation of Min proteins can vary in response to changes in growth conditions, protein levels, and the cell cycle. For example, increased temperatures will shorten the period of Min oscillation from about a minute to several seconds (Touhami et al., 2006). Higher levels of MinD relative to MinE reduce MinD ATPase activity and lengthen the oscillation period, whereas lower MinD:MinE ratios increase MinD ATPase activity and shorten the period (Raskin and de Boer, 1999b; Hu et al., 2002), which is consistent with MinE as the driver of MinD dynamics. During later stages of cell division when the division septum is closing, MinD begins to "pause" at midcell prior to the formation of two separate oscillating systems, one in each daughter cell (Juarez and Margolin, 2010). Intriguingly, MinD is often observed to localize at opposite sides of the developing septum prior to doubling. This phenomenon is likely a result of the doubling in pole-topole distance prior to division. The pausing of MinD at midcell is likely required for equal partitioning of MinD into daughter cells and has been simulated mathematically (Di Ventura and Sourjik, 2011). This behavior might also prevent new Z rings from forming adjacent to existing rings, similar to the septal localization observed for Min proteins in *B. subtilis* as described above (Gregory et al., 2008; van Baarle and Bramkamp, 2010). This idea is supported by evidence that MinC also exhibits similar septal pausing (Hu and Lutkenhaus, 1999).

In addition to the pausing phenomenon, Min oscillation is strongly influenced by cell geometry. *E. coli* cells of normal length have a pole-to-pole pattern of Min localization, which is constrained by the essentially one-dimensional cellular rod shape (Raskin and de Boer, 1999b). If cells become elongated, such as in division-defective mutants, oscillation no longer extends from one pole to the other because the pole-to-pole distance becomes too long. Instead, the pattern changes to multiple (2 or more) oscillating units, some of which form traveling waves, in which MinD alternates from traveling through the membrane to traveling through the cytoplasm between poles (Raskin and de Boer, 1999b; Hu and Lutkenhaus, 2001; Bonny et al., 2013). The result is that zones of MinCD appear and disappear throughout the filamentous cell with *∼*7 µm spacing (Raskin and de Boer, 1999b; Meinhardt and de Boer, 2001). It is likely that these MinCD zones help restrict where Z rings form distal to cell poles within these filaments.

Min proteins also respond to changes in cell shape, indicating that oscillation is not restricted to a symmetrical pattern. For example, in *E. coli* cells such as *rodA* mutants that grow and divide as spheres, MinD and MinE will migrate from one location to another on the cell periphery in a seemingly disorganized pattern, although there is a strong bias for migration down a long axis (Corbin et al., 2002). This behavior has been simulated by mathematical models (Huang and Wingreen, 2004). Such round cell mutants divide in alternating perpendicular planes, much like Staphylococci except only in two dimensions instead of three; if the Min system is inactivated in these mutant cells, their division becomes highly irregular (Begg and Donachie, 1998; Corbin et al., 2002). When round cells divide, they pinch and form a long axis, thus establishing Min oscillations parallel to the growing septum. This pattern will then define the midpoint of future perpendicular Z ring or arc (Pas et al., 2001), which explains how this alternating division pattern can occur (Corbin et al., 2002). Other evidence for geometric control of Min oscillation comes from branched *E. coli* cells, where Min proteins travel in a clockwise or counterclockwise direction between cell branches (Varma et al., 2008).

Membrane curvature and/or phospholipid composition likely serve as physical cues for localization of DivIVA and MinD (Barák and Muchová, 2013). The anionic phospholipid cardiolipin contributes to membrane curvature, and is enriched both at cell poles and the septum of dividing *E. coli* and *B. subtilis* cells (Mileykovskaya and Dowhan, 2000; Kawai et al., 2004). Phosphatidylglycerol is also enriched at *E. coli* cell poles and may contribute to the anionic nature of polar membranes, which seems to be important for binding of MinD and MinE (Mileykovskaya et al., 2003; Mazor et al., 2008; Renner and Weibel, 2012; Oliver et al., 2014; Vecchiarelli et al., 2014). In mutants lacking phosphatidylethanolamine (PE), MinD does not oscillate in organized zones but instead assembles and disassembles dynamically at peripheral foci throughout the cell, probably because the levels of anionic phospholipids are too high (Mileykovskaya et al., 2003). Cells lacking PE also divide poorly and exhibit spiral FtsZ patterns (Mileykovskaya et al., 1998), but is not yet known if these effects result from perturbations of the Min system or of other proteins that affect FtsZ assembly and function. In *B. subtilis*, on the other hand, specific phospholipids do not seem to be involved in the polar targeting by DivIVA (Lenarcic et al., 2009).

## **Cell-free Reconstitution of Min Protein Dynamics**

The Min system has been an attractive subject for generating mathematical models to better understand Min protein dynamics (Kruse et al., 2007; Bonny et al., 2013). Such models of the Min system have both confirmed *in vivo* data, and have been used to predict patterns later confirmed using *in vivo* experiments (Bonny et al., 2013). These models implied that MinD and MinE were

sufficient for the oscillatory behavior, and as proof of this, purified MinD and MinE were shown to self-organize into wave patterns on supported lipid bilayers (SLBs) in the presence of ATP (Loose et al., 2008). MinD first binds ATP and forms a homogenous layer on the bilayer, and when MinE is added, planar waves form, with MinD at the leading edge and MinE at the trailing edge (Loose et al., 2008). This nicely mimics the lagging behavior of MinE observed in *E. coli* cells (Hale et al., 2001). Reconstitution of FtsZ and MinC on SLBs uncovered the molecular mechanism of MinC inhibition of FtsZ by demonstrating that MinC can both bind FtsZ to prevent monomers from incorporating into polymers, and remove monomers from existing polymers (Arumugam et al., 2014). When FtsZ was reconstituted on the SLBs with MinDE, with and without MinC, MinC was shown to be required for spatial organization of FtsZ on MinDE waves (Arumugam et al., 2014). Most convincingly, MinCDE oscillation from pole to pole was reconstituted in rod-shaped compartments covered with membranes, as was FtsZ assembly at the midpoint of those compartments (**Figure 3C**; Zieske and Schwille, 2013, 2014).

### **The Min System Acts Independently of the Nucleoid but May Influence Chromosome Partitioning**

Certain chromosome partitioning mutants of *E. coli*, such as those inactivating Topoisomerase IV or the MukBEF condensing complex, result in a relatively high percentage of anucleate cells (Hiraga et al., 1991; Stewart and D'Ari, 1992; Hayama and Marians, 2010). Importantly, Z rings are positioned near the center of anucleate cells, indicating that chromosomes, though active for NO, are not essential for Z ring positioning (Sun et al., 1998). Indeed, mathematical models for Min-mediated centering of the Z ring assume that Min proteins are sufficient without any other macromolecules needed. Moreover, Min oscillations occur independently of nucleoid structural changes (Sun and Margolin, 2001), and Z rings can be centered by the Min system alone in the reconstitution experiments described above. Finally, *E. coli* maxicells, which are generated by UV-mediated destruction of the chromosome and are thus nucleoid-free, retain an oscillating Min system and also support Z ring formation at midcell. This supports the *in vitro* reconstitution data and suggests that the Min system alone can restrict Z rings to midcell in *E. coli* (**Figure 3B**; Pazos et al., 2014).

One question in the field is what advantages might the ATPburning Min oscillation confer upon *E. coli*, considering that the *B. subtilis* Min system does not oscillate. Several studies have suggested that *E. coli* cells lacking Min systems have defects in chromosome partitioning (Bernander et al., 1989; Åkerlund et al., 1992, 2002). As the driving forces for chromosome partitioning in *E. coli* are unknown, this brings up the possibility that the Min oscillation somehow aids this process. Recent evidence suggests that MinD can bind chromosomal DNA directly, and therefore tether the chromosome to the membrane (Di Ventura et al., 2013), which might influence chromosome partitioning (Schofield et al., 2010).

Even when it is not oscillating, the *B. subtilis* Min system is also linked to chromosome replication. Another ParA family ATPase called Soj binds to chromosomal DNA and the DnaA initiator protein to activate DNA replication. However, Soj is also present as an inactive form that binds to MinD at cell poles, which prevents Soj activation of DNA replication (Murray and Errington, 2008).

### **Additional Regulators of Z ring Positioning in** *E. coli* **and** *B. subtilis*

Recent evidence suggests that spatial regulators of Z ring positioning other than NO and Min exist in *E. coli* and *B. subtilis*. In *E. coli* cells, deletions of the Min system and SlmA, the mediator of NO, are synthetically lethal when cells are grown in rich medium (Wu and Errington, 2004; Bernhardt and de Boer, 2005). Removing both systems results in too many potential locations for Z rings to form **Figure 3A**, preventing assembly of a single coherent ring at midcell (Yu and Margolin, 1999; Bernhardt and de Boer, 2004). However, in minimal medium, cells that lack Min proteins and NO can survive and divide quite well, although the reason for the growth medium-dependence is not clear (Bernhardt and de Boer, 2005). Upon further investigation, cells lacking both NO and Min had more precise Z ring placement and produced fewer minicells than cells that only lacked Min (Bailey et al., 2014). These findings suggest that when both systems are not present, other factors contribute to Z ring positioning (Cambridge et al., 2014).

These factors may involve the nucleoid itself. For example, a positive regulatory system that involves the Ter macrodomain region of the chromosome has been implicated in Z ring positioning, as it occupies the center of the nucleoid during Z ring positioning (Bailey et al., 2014). In this system, MatP serves to connect the Ter macrodomain to the divisome through interaction with ZapB (Espéli et al., 2012), and this network is involved in regulating septal constriction (Buss et al., 2015). Cells that lack all known systems still exhibit a bias for Z rings localizing to midcell, although with lower precision and accuracy, indicating that other overlapping factors help to position Z rings (Bailey et al.,

2014). *B. subtilis* cells that lack NO and Min proteins also precisely position Z rings, but Z ring formation is delayed, indicating their importance for Z ring formation at the proper time of the cell cycle (Rodrigues and Harry, 2012). Positive markers for Z ring formation in *B. subtilis* cells are also hypothesized (Moriya et al., 2010). In their "midcell potentiation" model, Moriya et al. (2010) proposed that as chromosome replication progresses, it acts along with NO and other factors to license midcell for Z-ring assembly. Essential chromosomal DNA replication proteins such as DnaX of *E. coli* (Bates and Kleckner, 2005) and DnaB of *B. subtilis* (Imai et al., 2000) localize at or near midcell. What first attracts these proteins to midcell is a key unanswered question.

## **Negative Spatial Regulation of Z ring Formation in** *Caulobacter*

Like many other bacterial species, *Caulobacter crescentus* lacks both NO and Min systems, but contains a protein that restricts Z ring formation to midcell (Thanbichler and Shapiro, 2006). MipZ is a member of the ParA-like family of ATPases that includes MinD, and is conserved in α-proteobacteria (Thanbichler and Shapiro, 2006). In *C. crescentus,* MipZ binds to the chromosomal replication origin (*oriC*) and forms a complex with proteins that are involved in chromosome partitioning. Prior to replication and partitioning, *oriC* and bound MipZ are at one cell pole, and FtsZ is at the opposite pole, farthest from MipZ. Upon chromosome replication, the duplicated *oriC* migrates to the opposite cell pole, forming a bipolar gradient (Thanbichler and Shapiro, 2006). Unlike MinD, MipZ is able to directly interact with FtsZ and inhibit polymer formation, therefore the bipolar gradient of MipZ permits assembly of FtsZ at midcell, at the lowest MipZ concentration (Thanbichler and Shapiro, 2006; Kiekebusch et al., 2012). In contrast to *E. coli*, this seems to be a mechanism to position the Z ring in response to both spatial and cell cycle cues without the need for a separate NO system.

## **Positive Spatial Regulators of Z ring Formation**

Until a few years ago, the known spatial regulators of the Z ring all acted to prevent Z ring formation at undesirable sites such as the cell poles or over the nucleoid. This negative regulation was satisfying because it generally exploited cell poles, which are a defined part of the cell that were once division sites, and did not need to invoke a midcell marker. Recently, however, several proteins that positively regulate Z ring formation have been identified in bacterial species that lack NO and/or Min systems. These proteins locate to the midcell division site and promote Z ring formation there, the very opposite of the Min/NO mechanism. The first reported example of a positive regulation system is the SsgA-SsgB pair in *Streptomyces coelicolor* (Willemse et al., 2011). SsgA-like proteins are present only in actinomycetes and are involved in forming multiple sporulation septa (Traag and van Wezel, 2008). *S. coelicolor* is able to grow vegetatively as mycelia in the absence of cell division, which enabled this study to be done in the absence of FtsZ. Notably, SsgA and SsgB localize to division sites between nucleoids in an *ftsZ* null mutant, indicating

that their localization is independent of FtsZ (McCormick et al., 1994; Willemse et al., 2011). SsgA recruits SsgB to midcell, where SsgB likely promotes FtsZ polymerization into discrete Z rings and serves as a membrane tether for the rings (Willemse et al., 2011). During formation of aerial spores, multiple Z rings are recruited to SsgB foci, ultimately forming sporulation septa (**Figure 4C**).

In another example of what seems to be positive regulation, PomZ, a ParA family ATPase, positions Z rings in the *∂*-proteobacterial species known for its ability to swarm and produce fruiting bodies, *Myxococcus xanthus* (Treuner-Lange et al., 2013). *M. xanthus* cells lacking PomZ have classic defects in cell division that are hallmarks of aberrant Z ring positioning, including filamentation and the formation of minicells (Treuner-Lange et al., 2013). PomZ localizes to midcell prior to, and in the absence of, FtsZ **Figure 4A**). However, a direct interaction between PomZ and FtsZ has not yet been observed, indicating that other unidentified proteins are involved in bridging this connection and/or positioning the Z ring (Treuner-Lange et al., 2013).

Rod-shaped Actinobacteria have less precise Z ring placement when compared to *E. coli* and *B. subtilis* (reviewed in Donovan and Bramkamp, 2014). Although lacking Min and NO homologs, *Corynebacterium glutamicum* produces a homolog of ParA, PldP, which may play a role in localizing the Z ring (Donovan et al., 2010). PldP localizes to midcell early in the cell cycle, and inactivation of PldP results in an increased variation in cell length as well as minicell formation (Donovan et al., 2010). Further work is required to confirm the role of PldP in Z ring positioning.

Until recently, the only known spatial regulators of FtsZ were in naturally rod-shaped species, and it was unknown how new Z rings formed in cocci. However, a transmembrane protein in *Streptococcus pneumoniae*, called either LocZ (Localizing at midcell of FtsZ) or MapZ (Mid-cell-anchored protein Z) has been found to localize to midcell and contribute to Z ring positioning, cell shape, and division (Fleurie et al., 2014; Holečková et al., 2014). Cells lacking LocZ/MapZ are viable, but have cell shape defects and form minicells (Holečková et al., 2014). MapZ/LocZ localizes as rings at new division sites prior to the arrival of FtsZ (**Figure 4B**), and it was shown that MapZ interacts directly with FtsZ (Fleurie et al., 2014; Holečková et al., 2014). Once the midcell Z ring forms, MapZ/LocZ gradually move from midcell to the cell quarters, the sites of the next division (Fleurie et al., 2014; Holečková et al., 2014). Together, the evidence suggests that this regulator may stimulate FtsZ assembly at midcell. Homologs of LocZ/MapZ are present in streptococci, lactococci, and enterococci, suggesting that these species regulate Z ring positioning by a similar mechanism (Fleurie et al., 2014; Holečková et al., 2014). There has been no published report of FtsZ assembly *in vitro* from streptococci, perhaps because this FtsZ assembles poorly without a stimulatory factor.

### **Conclusion and Perspectives**

The wide varieties of Z ring positioning systems identified in recent years highlight the complexity and diversity of bacteria (Monahan et al., 2014). The two systems that are best understood are the negative regulators of Z ring placement, NO and Min. Further studies will be required to elucidate the mechanism of recently identified systems, and to identify factors that contribute to Z ring positioning that are currently unknown. A common theme among many known Z ring positioning systems is the involvement of ParA-like proteins, which are also involved in chromosome segregation (Lutkenhaus et al., 2012). Another important question for further study is how positive regulators of Z ring positioning are recruited to midcell, particularly as the midpoint of a rod shaped cell does not feature any known molecular marker. In the cases of PomZ and MapZ/LocZ, it remains possible that their positioning at midcell is indirectly orchestrated by negative regulatory factors that depend on cell poles as cues. On the other hand, there is evidence that round cells such as *Staphylococcus aureus* may directly use spatial cues derived from previous septation events. Prior division septa result in orthogonal belts of distinct cell wall material that encircle the cell, acting to mark those sites for several generations. It is thought that the intersection of these belts mark the site for new cell division events (Turner et al., 2010). NO also has a role in Z ring placement in *S. aureus*, as loss of NO perturbs Z ring placement (Veiga et al., 2011). It remains to be determined how widespread this type of mechanism is and which proteins help mediate assembly of Z rings in these systems.

### **Acknowledgments**

The authors are grateful for support from grant GM61074 from the National Institutes of Health and from the University of Texas Graduate School of Biomedical Sciences.

## **References**

septum placement in *Streptococcus pneumoniae*. *mBio* 6, e01700–e01714. doi: 10.1128/mBio.01700-14

**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Copyright © 2015 Rowlett and Margolin. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

# Simultaneous regulation of cell size and chromosome replication in bacteria

Po-Yi Ho and Ariel Amir\*

School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA

Bacteria are able to maintain a narrow distribution of cell sizes by regulating the timing of cell divisions. In rich nutrient conditions, cells divide much faster than their chromosomes replicate. This implies that cells maintain multiple rounds of chromosome replication per cell division by regulating the timing of chromosome replications. Here, we show that both cell size and chromosome replication may be simultaneously regulated by the long-standing initiator accumulation strategy. The strategy proposes that initiators are produced in proportion to the volume increase and is accumulated at each origin of replication, and chromosome replication is initiated when a critical amount per origin has accumulated. We show that this model maps to the incremental model of size control, which was previously shown to reproduce experimentally observed correlations between various events in the cell cycle and explains the exponential dependence of cell size on the growth rate of the cell. Furthermore, we show that this model also leads to the efficient regulation of the timing of initiation and the number of origins consistent with existing experimental results.

Keywords: replication initiation, cell cycle, cell size regulation, multiple forks, mathematical modeling

### 1. Introduction

Bacterial cells are extremely proficient in regulating and coordinating the different processes of the cell cycle. The Cooper-Helmstetter model proposes a molecular mechanism that couples two such processes, the replication of the chromosome and the division of the cell (Cooper and Helmstetter, 1968). In the model, cell division occurs a constant duration after the initiation of chromosome replication. The model implies a tight coordination between replication initiation and cell division such that in cells able to double faster than their chromosomes can replicate, multiple rounds of replications proceed simultaneously (Yoshikawa et al., 1964; Cooper and Helmstetter, 1968). To answer how cells regulate the timing of initiation, it was proposed that replication initiation factors accumulate to a critical amount per origin of replication to trigger the initiation of replication (Helmstetter et al., 1968). Since the conception of the above model, many experiments and models have attempted to capture the molecular mechanisms responsible for the initiation of multiple rounds of replication. However, no model has been completely satisfactory (Donachie and Blakely, 2003).

As a result of the coupling between replication and division, the average cell size per origin is approximately a constant independent of the growth rate of the cell (Donachie, 1968). Furthermore, it is now understood that a common size regulation strategy for organisms including bacteria and budding yeast is the incremental model in which division occurs upon the addition of a constant

#### Edited by:

Arieh Zaritsky, Ben-Gurion University of the Negev, Israel

#### Reviewed by:

Charles Edward Helmstetter, Florida Institute of Technology, USA Joao D. Xavier, Memorial Sloan Kettering Cancer Center, USA Avinoam Rabinovitch, Ben-Gurion University of the Negev, Israel

#### \*Correspondence:

Ariel Amir, School of Engineering and Applied Sciences, Harvard University, 29 Oxford Street, Cambridge, 02138 MA, USA<EMAIL_ADDRESS>
#### Specialty section:

This article was submitted to Microbial Physiology and Metabolism, a section of the journal Frontiers in Microbiology

> Received: 31 March 2015 Accepted: 17 June 2015 Published: 10 July 2015

#### Citation:

Ho P and Amir A (2015) Simultaneous regulation of cell size and chromosome replication in bacteria. Front. Microbiol. 6:662. doi: 10.3389/fmicb.2015.00662 size dependent on the growth rate of the cell (Amir, 2014; Campos et al., 2014; Soifer et al., 2014; Taheri-Araghi et al., 2015). However, the molecular mechanisms responsible for the incremental model of size control remain in question.

Our main result in this work is to show that the initiator accumulation strategy not only regulates size according to the incremental model, but also regulates simultaneously the timing of initiation and the number of origins of replication. The strategy says that replication initiates upon the accumulation of a critical amount of replication initiation factors per origin. We emphasize the importance of the partitioning of replication initiation factors amongst origins, which we show is essential in order for the multiple replication forks to be adequately regulated. We show, analytically and numerically, that this strategy robustly regulates both cell size and the number of origins. Agreement between existing experiments and predictions of the above model reveals essential features that must be captured in any molecular mechanisms coordinating replication initiation and cell division. Finally, we make distinct predictions regarding the distribution of cell sizes at initiation of replication.

### 2. Methods

#### 2.1. Multiple Origins Accumulation Model

We consider the regulation strategy in which replication initiates upon the accumulation of a critical amount of replication initiation factors, or "initiators," per origin of replication (Helmstetter et al., 1968). We assume that the initiators are expressed via an autorepressor model, as seen in **Figure 1** (Sompayrac and Maaloe, 1973). In this model, a protein is expressed such that its concentration c remains constant and independent of the growth rate of the cell, which is plausible to achieve through autorepression. Therefore, an increase in the volume of the cell corresponds to a proportional increase in the copy number of this autorepressing protein. A second protein is the initiator and is expressed under the same promoter as the first, but in contrast to the first protein, it is localized at the origins of replication. For simplicity, we assume that the initiators are equally partitioned amongst the origins. Initiation then occurs when a critical copy number per origin Ncritical of the localized initiators is reached, after which the initiators are assumed to degrade. Under these assumptions, the copy number of the initiator effectively measures the increase in volume since initiation.

More precisely, if a cell initiated a round of replication at volume v<sup>i</sup> into O number of origins, the amount of initiators Ninitiators immediately after initiation is zero. To initiate the next round of replication, the cell must accumulate ONcritical initiators, but because the initiator is expressed under the same promoter as the autorepressor, the cell must also accumulate ONcritical autorepressors. Because the concentration of the autorepressor is constant, this implies that the cell must accumulate a corresponding volume 1 = Ncritical/c per origin, independent of the growth rate, to trigger the next initiation.

Thus, on a phenomenological level, the above biophysical model maps to the following regulation strategy for initiation,

$$\nu\_i^{\text{tot,next}} \approx \nu\_i + O\Delta,\tag{1}$$

Equation (1) says that if a cell initiated a round of replication at cell volume v<sup>i</sup> into O number of origins, then the cell will attempt to initiate another round of replication at total volume v tot,next i , which is the sum of the volumes of all cells in the lineage since the initiation event at v<sup>i</sup> (typically two cells). This is not to be confused with the threshold model in which cells initiate upon reaching a threshold volume proportional to the number of origins, v next <sup>i</sup> ∝ O. For the rest of this manuscript, O will denote the number of origins after initiation at cell volume v<sup>i</sup> but before initiation at total cell volume v tot,next i .

We assume an exponential mode of growth for cell volume with a constant doubling time τ and a corresponding constant growth rate λ = ln 2/τ (Godin et al., 2010). From Equation (1) and the exponential mode of growth, durations between initiations are

$$t\_i = \frac{1}{\lambda} \ln \left( 1 + \frac{O \Delta}{\nu\_i} \right) + \xi,\tag{2}$$

where ξ represents some noise in the initiation process. An initiation event will trigger a division event after a constant duration C + D, where C and D are respectively, the constant duration required to replicate the chromosome and the constant duration between replication termination and division (Cooper and Helmstetter, 1968). We will refer to Equation (1) as the multiple origins accumulation model (i.e., initiators are accumulated per origin). **Figure 2** illustrates this regulation strategy. We note that the strategy described here is mathematically equivalent to the "replisome" model of Bleecken (1971) (not to be confused with the current use of the term replisome).

Finally, we will not take into account additional biological mechanisms that act at the level of the initiation of chromosome replication, such as oriC sequestration, Dam methylation, and the "eclipse" phenomenon (Bogan and Helmstetter, 1997; Zaritsky et al., 2007; Campbell and Kleckner, 2010). While these mechanisms are important to prevent rapid re-initiations, by themselves they are insufficient in ensuring an appropriately coordinated coupling between chromosome replication and cell division, which is the main focus of our work.

#### 2.2. Numerical Simulations

We can numerically simulate the multiple origins accumulation model given C + D, τ , and 1 as experimentally measurable parameters. First, we initialize a population of N cells with uniformly distributed cell ages. Durations between initiations are calculated as Equation (2) and the noise in the initiation process is assumed to be normally distributed with standard deviation σ<sup>τ</sup> , though the precise nature of the noise does not affect any of our conclusions. It is assumed that in an initiation event, the number of origins in a cell is doubled. The corresponding division event occurs after a constant time C+D. In a division event, the number of origins in a cell, along with the size of the cell, is halved, and two identical cells are generated. We neglect the stochasticity arising from asymmetric divisions, which do not significantly affect any of the results. There are no division events without the corresponding initiation events. Following this procedure, a population of cells will robustly reach stationarity regardless of initial conditions, as seen in **Figures 3**, **4**.

### 3. Results

### 3.1. Multiple Origins Accumulation Robustly and Efficiently Regulates the Number of Origins of Replication

An important measurable consequence of the tight coupling between replication initiation and cell division is the average

FIGURE 3 | ti , O, and vb approaching stationary distributions in numerical simulations of the multiple origins accumulation model. First, we initialize a population of N cells with uniformly distributed cell ages. Durations between initiations of replication are calculated as Equation (2) and the noise in the initiation process is assumed to be normally distributed. In an initiation event, the number of origins in a cell is doubled. The corresponding division event occurs after a constant time C + D. In a division event, the number of origins in a cell, along with the size of the cell, is halved, and two identical cells are generated. There are no division events without the corresponding initiation events. Following this procedure, a population of cells will robustly reach a stationary distribution of cell sizes and number of origins per cell regardless of initial conditions. The plots here track one lineage of cells. Solid lines are moving averages. Here, C + D = 70 mins, τ = 20 mins, and στ = 4 mins. These are biologically realistic choices. We set 1 = 1/2 (C+D)/τ so that vb ≈ 1.

number of origins of replication per cell. It has been theoretically shown that the average number of origins per cell is

$$
\langle \mathcal{O} \rangle = 2^{(\mathcal{C} + \mathcal{D})/\mathfrak{r}}.\tag{3}
$$

The derivations leading up to Equation (3), summarized in Appendix, hinge on assuming an efficient process: the population is growing exponentially and reaches a stationary distribution of cell ages, implying that there are no delays due to chromosome replication. These assumptions are independent of any molecular mechanisms for initiation and thus should be fulfilled by any efficient mechanism. We will show that the multiple origins accumulation model reproduces Equation (3).

The multiple origins accumulation model regulates initiation via negative feedback on the volume at initiation described by Equations (1) and (2). The feedback enables cells to maintain a stationary average volume at initiation and a stationary average duration between initiations despite noise in the initiation process. Specifically, if a cell initiated replication at volume v<sup>i</sup> = O1, then the duration to the next initiation event is

$$t\_i(O) \approx \frac{1}{\lambda} \ln \left( 1 + \frac{O \Delta}{O \Delta} \right) = \text{tr.} \tag{4}$$

But if a cell initiated replication at a slightly larger volume v<sup>i</sup> = O1 + δv, the duration to the next initiation event is

$$t\_i'(O) \approx \frac{1}{\lambda} \ln \left( 1 + \frac{O\Delta}{O\Delta + \delta\nu} \right) \lesssim \text{r.} \tag{5}$$

Equations (4) and (5) say that a cell that initiated at a slightly larger volume than average tend to initiate again faster than average so that its volume at next initiation is again near the average. Similar reasoning says that cells that initiated at slightly smaller volumes tend to initiate again slower than average. In this way, cells maintain a stationary average volume at initiation and a stationary average duration between initiations.

Furthermore, the feedback enables cells to maintain a balanced cell cycle, in which there is on average one and only one initiation event per cell cycle. In the case of negligible noise, a balanced cell cycle implies that cells will initiate at cell age (Bremmer and Dennis, 1996)

$$a\_l = 1 + \lfloor \frac{\mathcal{C} + D}{\pi} \rfloor - \frac{\mathcal{C} + D}{\pi},\tag{6}$$

where a = 0 represents cell birth, a = 1 represents cell division, and ⌊⌋ is the mathematical floor operator (largest integer smaller or equal to the argument). But in the case of realistic noise, a cell may initiate an extra round of replication if the noise is negative enough, ξ/τ . ⌊(C + D)/τ ⌋ − (C + D)/τ , which corresponds to an extra initiation at volume v ′ <sup>i</sup> = 2O1−δv. The multiple origins accumulation model is robust to these stochastic events because a cell that initiated an extra round of replication will initiate again after

$$t\_i(2O) \approx \frac{1}{\lambda} \ln \left( 1 + \frac{2O\Delta}{2O\Delta - \delta \nu} \right) \gtrsim \text{tr.}\tag{7}$$

In other words, cells with extra rounds of replication will initiate slower than those without so that the stationary average duration between initiations is maintained. The cell cycle following the extra initiation will typically not have any initiations, so that the initiation following the extra initiation will occur at approximately the average volume at initiation. A cell that missed a round of replication will return to a balanced initiation process in the analogous manner. In this way, the multiple origins accumulation model is able to efficiently maintain a balanced cell cycle in fast growth conditions. In contrast, the model simulated by Campos et al. (2014) is not robust to the noise in the initiation process, because in their model, the incremental volume needed to trigger initiation is not partitioned between origins. We will elaborate on this in Section 3.3.

The multiple origins accumulation model is therefore able to robustly regulate the timing of initiations in face of extra initiations. Extra initiations can occur not only because of noise in the initiation process, but also because of a shift in the growth rate of the cell such as that found in a shift-up experiment, in which a population of cells is abruptly switched from one nutrient condition to a richer nutrient condition allowing for faster growth. The increase in growth rate corresponds to a decrease in the duration between initiations. Therefore, cell in a shift-up experiment will initiate extra rounds of replication in the cycle immediately following the shift-up, but as we have seen, the multiple origins accumulation model is able to appropriately regulate the timing of initiations to reflect the new growth rate. Simulations of the multiple origins accumulation model reached stationary distributions of cell ages, durations between initiations, cell sizes, and number of origins per cell, regardless of initial conditions or the magnitude of the noise ξ in Equation (2), as seen in **Figures 3**, **4**. Simulations also show that the number of origins is regulated as in Equation (3), as seen in **Figure 5**. The above considerations show that the multiple origins accumulation model regulates the number of origins robustly and efficiently in face of noise in the initiation process.

FIGURE 5 | **<sup>h</sup>**O**<sup>i</sup>** as a function of <sup>τ</sup> . Simulations are the same as Figure 3, with a varying τ = 20–100 mins, a fixed C + D = 70 mins, and στ /τ = 0.2. Dashed line plots (Equation 3). Similarly, inset plots log O as a function of λ.

### 3.2. Multiple Origins Accumulation Robustly Regulates Cell Size

It was recently shown that the multiple origins accumulation model of replication initiation reduces to the incremental model of size regulation (Amir, 2014)

$$
\nu\_d \approx \nu\_b + \nu\_0. \tag{8}
$$

Equation (8) says that if a cell is born with volume v<sup>b</sup> , then the cell will attempt to divide at volume vd, where v<sup>0</sup> is the constant incremental volume from birth to division. In fact, v<sup>0</sup> can be expressed in terms of known parameters. First, if a cell initiated replication at volume v<sup>i</sup> , then the corresponding division event will occur at total volume v tot <sup>d</sup> <sup>=</sup> <sup>v</sup>i<sup>2</sup> (C+D)/τ . But there will have been log<sup>2</sup> O division events since initiation at v<sup>i</sup> , so that the corresponding volume at birth is

$$\nu\_b = \frac{\nu\_d^{\rm tot}}{O} = \frac{\nu\_i 2^{(C+D)/\tau}}{O}. \tag{9}$$

In a balanced cell cycle, the next initiation event will occur at total volume v tot,next <sup>i</sup> ≈ v<sup>i</sup> + O1 and the corresponding division event will occur at total volume v tot,next <sup>d</sup> <sup>=</sup> <sup>v</sup> tot,next i 2 (C+D)/τ . Similarly, there will have been log<sup>2</sup> O division events since initiation at v next i , so that the corresponding volume at division is

$$\nu\_d = \frac{\nu\_d^{\text{tot,next}}}{O} = \frac{\nu\_i 2^{(C+D)/\tau}}{O} + \Delta 2^{(C+D)/\tau}.\tag{10}$$

Therefore,

$$
\nu\_d - \nu\_b \approx \Delta 2^{(C+D)/\tau}.\tag{11}
$$

Within the multiple origins accumulation model, this derivation is valid for any C + D and τ (Amir, 2014).

The incremental model of size regulation predicts distributions, correlations, correlation coefficients, and scalings consistent with existing measurements (Amir, 2014; Campos et al., 2014; Soifer et al., 2014; Taheri-Araghi et al., 2015). In particular, the average cell volume at birth

$$
\langle \boldsymbol{\nu}\_b \rangle \approx \Delta 2^{(\mathbf{C} + \mathbf{D})/\mathbf{r}}.\tag{12}
$$

Equation (12) says that the average cell volume at birth is exponentially dependent on the growth rate, a well-known and well-tested result for E. coli and B. subtilis (Schaechter et al., 1958; Sharpe et al., 1998; Taheri-Araghi et al., 2015). Simulations of the multiple origins accumulation model also confirm this result, as seen in **Figure 7**. Thus, the multiple origins accumulation model robustly regulates cell size.

### 3.3. Master Accumulation Predictions are Inconsistent with Existing Experiments

Consider the regulation strategy

$$\nu\_i^{\text{tot,next}} \approx \nu\_i + \Delta \,. \tag{13}$$

Equation (13) says that if a cell initiated a round of chromosome replication at cell volume v<sup>i</sup> with O number of origins of replication, then the cell will attempt to initiate another round of replication at total volume v tot,next i , where 1 is a constant volume independent of the growth rate. This is an incremental model of size control applied at initiation. If we assume the same mode of initiator expression as before, then the regulation strategy described by Equation (13) corresponds to replication initiation upon accumulation of a critical amount of initiators without partitioning of initiators between origins. Instead, a plausible molecular picture is that of initiators accumulating at a "master" origin, whose initiation triggers the cascade initiation of other origins (Lobner-Olesen et al., 1994). We will therefore refer to Equation (13) as the master accumulation model. In contrast to the multiple origins accumulation model described above, here in the master accumulation model, the total volume at next initiation does not depend on the number of origins present in the cell.

As before, we assume an exponential mode of growth with a constant doubling time τ . The durations between initiations are therefore

$$t\_i(O) = \frac{1}{\lambda} \ln \left( 1 + \frac{\Delta}{\nu\_i} \right) + \xi,\tag{14}$$

where ξ represents some noise in the initiation process. Again as before, an initiation event will trigger a division event after a constant duration C+D. Equation (14) differs from Equation (2) by a missing factor of O. As we show below, the factor of O is essential in regulating appropriately the timing of initiations.

However, essential in regulating appropriately the timing of initiations and the master accumulation model does not reproduce the well-known exponential scaling of cell size with growth rate. The derivation follows and is similar to that of the multiple origins accumulation model. First, if a cell initiated replication at volume v<sup>i</sup> , then the corresponding division event will occur at total volume v tot <sup>d</sup> <sup>=</sup> <sup>v</sup>i<sup>2</sup> (C+D)/τ . But there will have been log<sup>2</sup> O division events since initiation at v<sup>i</sup> , so that the corresponding volume at birth is

$$\nu\_b = \frac{\nu\_d^{\rm tot}}{O} = \frac{\nu\_i 2^{(C+D)/\tau}}{O}. \tag{15}$$

In a balanced cell cycle, the next initiation event will occur at total volume v tot,next <sup>i</sup> ≈ v<sup>i</sup> + 1 and the corresponding division event will occur at total volume v tot,next <sup>d</sup> <sup>=</sup> <sup>v</sup> tot,next i 2 (C+D)/τ . Similarly, there will have been log<sup>2</sup> O division events since initiation at v next i , so that the corresponding volume at division is

$$\nu\_d = \frac{\nu\_d^{\text{tot, next}}}{O} = \frac{\nu\_i 2^{(C+D)/\tau}}{O} + \frac{\Delta 2^{(C+D)/\tau}}{O}.\tag{16}$$

Therefore,

$$\nu\_d - \nu\_b = \frac{\Delta 2^{(C+D)/\tau}}{O}.\tag{17}$$

This derivation is valid for any C + D and τ . However, from Equation (3), O should scale exponentially with the growth rate like 2(C+D)/τ so that

$$
\langle \boldsymbol{\nu}\_{b} \rangle \approx \boldsymbol{\nu}\_{d} - \boldsymbol{\nu}\_{b} \sim \Delta \,. \tag{18}
$$

Equation (18) says that the average cell size is a constant roughly independent of the growth rate, a prediction contradicting the well-tested exponential scaling with growth rate for the model organisms mentioned above.

Furthermore, the above reasoning assumes that the master accumulation model can maintain a balanced cell cycle. But the master accumulation model cannot robustly maintain a balanced cell cycle in face of noise as Equation (14) demonstrates. Specifically, if a cell initiated replication at volume v<sup>i</sup> = 1, then the duration to the next initiation event is

$$t\_i(O) \approx \frac{1}{\lambda} \ln \left( 1 + \frac{\Delta}{\Delta} \right) = \text{tr.} \tag{19}$$

But the cell may proceed to initiate an extra round of replication if the noise is negative enough, ξ/τ . ⌊(C + D)/τ ⌋ − (C + D)/τ , which corresponds to an extra round of initiation at volume v ′ <sup>i</sup> = 21 − δv. The next initiation will then occur after

$$t\_i(2O) \approx \frac{1}{\lambda} \ln\left(1 + \frac{\Delta}{2\Delta - \delta\nu}\right) \gtrsim \log\_2\left(\frac{3}{2}\right)\varepsilon. \tag{20}$$

Implying that:

$$\mathfrak{t}\_{\mathfrak{i}}(\mathfrak{L}\mathcal{O}) \lesssim \mathfrak{t}\_{\mathfrak{i}}(\mathcal{O}) \tag{21}$$

Equation (21) says that cells with more origins will initiate faster than those with less, giving rise to cells with average durations between birth and division not equal to τ . In other words, the master accumulation model does not robustly regulate the initiation process to maintain a balanced cell cycle. Indeed, simulations of the master accumulation model do not converge to a balanced cell cycle. Likewise, Campos et al. (2014) carried out simulations of the master accumulation model and obtained "widely abnormal cell size distributions." Given the above inconsistent predictions, the master accumulation model can be ruled out as a possible regulation strategy for replication initiation.

### 3.4. Multiple Origins Accumulation Suggests That Variations in C **+** D Are Small

In claiming that the master accumulation model gives incorrect correlations between growth rate dependent variables, Campos et al. (2014) simulated the master accumulation model and reported negative correlations between cell size at birth v<sup>b</sup> and cell size differences between birth and division 1v, whereas none is observed experimentally. In contrast to claims in Campos et al. (2014), the negative correlations do not provide evidence against the multiple origins accumulation model nor the master accumulation model. Instead, the negative correlations provide evidence that variability in the durations from initiation to division C + D should be small. Indeed, the multiple origins accumulation model, because of its reduction to the incremental model of size control, predicts no correlations between v<sup>b</sup> and 1v, given that variations in C + D are small compared to variations in τ . Simulations assuming that durations from initiation to the corresponding division are normally distributed with mean C + D and standard deviation σC+<sup>D</sup> show that the correlations between v<sup>b</sup> and 1v become increasingly negative as σC+D/σ<sup>τ</sup> increases, as seen in **Figure 6**. **Figure 6** shows that as long as σC+D/σ<sup>τ</sup> < 0.3, the correlations between v<sup>b</sup> and 1v will be close to zero. This is intuitive because when σC+<sup>D</sup> is small compared to σ<sup>τ</sup> , fluctuations in cell sizes at birth arise due to variations in cell sizes at initiation, but these variations are negatively fed back into the multiple origins accumulation model as explained in Section 3.1. On the other hand, when σC+<sup>D</sup> is comparable to σ<sup>τ</sup> , some fluctuations in cell sizes at birth arise due to variations in durations between initiation and division, but these variations are not accounted for by the multiple origins accumulation model. Variations of this nature give rise to the negative correlations between v<sup>b</sup> and 1v.

### 3.5. Multiple Origins Accumulation Predicts Proportionality between Cell Size and the Number of Origins Per Cell

The simultaneous regulation of cell size and the number of origins per cell in the multiple origins accumulation model gives rise to a strict relationship between the two variables. In the multiple origins accumulation model, the average cell volume at birth (Equation 12) is exponentially dependent on the growth rate, while the average number of origins per cell (Equation 3) also scales exponentially with the growth rate. Therefore, we have that

$$
\langle \boldsymbol{\nu}\_{\boldsymbol{b}} \rangle \approx \Delta \,\langle \boldsymbol{O} \rangle \tag{22}
$$

That is, the multiple origins accumulation model predicts that given a fixed volume increment per origin 1, the average volume at birth and the average number of origins per cell will scale appropriately with respect to a varying (C + D)/τ to give rise to the above approximate proportionality. The critical size regulation strategy proposed by Donachie assumed this proportionality (Donachie, 1968), but is inconsistent with measured correlations in E. coli because thresholding size at any point in the cell cycle washes away the memory of the initial conditions, and therefore leads to a vanishing correlation coefficient between size at birth and size at division - contrary to measurements; on the other hand, multiple origins accumulation predicts this proportionality, and is consistent with measured correlations (Amir, 2014; Campos et al., 2014; Soifer et al., 2014; Taheri-Araghi et al., 2015). Simulations of the multiple origins accumulation model confirm that cell size is indeed approximately proportional to the number of origins per cell, as seen in **Figure 7**. We emphasize that the approximate proportionality is a property predicted by the multiple origins accumulation model. In contrast, other strategies that do not regulate the number of origins, such as the master accumulation model, would not predict it.

### 3.6. Multiple Origins Accumulation Predicts Bimodal Cell Sizes at Initiation

In addition to the approximate proportionality between cell size and the number of origins per cell, the multiple origins accumulation model predicts that the distribution of cell sizes at initiation will be approximately bimodal because cells will

<sup>σ</sup>C**+**D/σ<sup>τ</sup> .

initiate extra rounds of replication when the noise is large enough. Simulations show that the distribution of cell sizes at initiation is indeed bimodal, with one large peak corresponding to a subpopulation whose cells initiated the expected number of rounds of replication and a smaller subpopulation whose cells initiated extra rounds of replication, as seen in **Figure 8**. Naively, because the distribution of cell sizes is lognormal in the multiple origins accumulation model, the distribution of cell sizes at initiation should be approximately the sum of two lognormal distributions with means O01 and 2O01, where O<sup>0</sup> = 2 ⌊(C+D)/τ ⌋ , and the ratio between the frequencies of the two peaks equal to the probability that ξ/τ . ⌊(C + D)/τ ⌋ − (C + D)/τ . However, the value O01 overestimates the average sizes at initiation of cells that initiated extra rounds of replication

because of correlations between the volumes at initiation and the probability for extra rounds of replication. The correlations arise from Equation (2), which says that a smaller volume at initiation correlates with a larger probability for extra rounds of replication during the current cell cycle. The bimodal distribution of cell sizes at initiation highlights how the multiple origins accumulation model, without invoking other mechanisms, can robustly maintain a balanced cell cycle despite noise in the initiation process and is an experimentally testable prediction of our model.

### 3.7. Multiple Origins Accumulation Predictions Are Consistent with Experiments on Mutants

Existing experiments on mutants of E. coli and B. subtilis produced results consistent with the predictions of the multiple origins accumulation model. Experiments have shown by manipulating the cell size of E. coli via mutations that a decrease in v is correlated with a decrease in C + D and that a decrease in v is also correlated with a decrease in hOi (Hill et al., 2012). In the language of the multiple origins accumulation model, v is controlled by 1 and (C + D)/τ . We assume that the mutations left unchanged the replication initiation mechanism in E. coli., so that 1 is a constant throughout. But τ remained approximately constant with changes in v, so a decrease in v must correspond to a decrease in C + D. One particular measurement reported for E. coli cells an average doubling time τ ≈ 25 mins for wildtype and mutant cells, C + D ≈ 40 mins + 20 mins for wildtype cells, and (C + D) ′ ≈ 30 mins + 20 mins for mutant cells (Hill et al., 2012). Given these values and Equation (12), the relative change in cell sizes corresponding to the reported difference in the durations from initiation to division should be 2((C+D) ′−(C+D))/τ <sup>≈</sup> <sup>0</sup>.75. That is, the mutant cells should be 0.75 times the size of the wildtype cells. This is in excellent agreement with the 25% decrease in volume reported. Moreover, the size at initiation did not change for these smaller mutants, consistent with our model which predicts the size at initiation to depend only on 1 and not on C + D. In contrast, in the case of B. subtilis, C + D in smaller mutant cells remained constant, suggesting that 1 is the quantity which was changed. Based on this interpretation, our model would predict that both size at initiation and at birth would change proportionally. Indeed, it was found that both the average mutant cell size and the mutant cell size at initiation both decreased by approximately 35% (Hill et al., 2012). The above experiments also observed the predicted approximate proportionality between cell size and the number of origins per cell in both E. coli and B. subtilis (Hill et al., 2012). It remains to be shown that cell sizes at initiation fall into an approximate bimodal distribution. The agreement between experimental results and the predictions made by the multiple origins accumulation model speaks to the importance of regulating simultaneously cell size and the number of origins.

### 4. Discussion

The multiple origins accumulation model proposes that replication initiates upon the accumulation of a critical amount of initiators per origin. If the initiators are expressed as in the autorepressor model, this strategy corresponds to Equation (1), which in turn reduces to the incremental model of size control, which predicts distributions, correlations, and scalings consistent with existing measurements. Specifically, the average cell size scales exponentially with the growth rate of the cell, Equation (12), as does the average number of origins per cell, Equation (3). The model robustly regulates both cell size and the number of origins per cell such that cell size is approximately proportional to the number of origins per cell, Equation (22). These predictions are consistent with existing experiments on E. coli and B. subtilis (Hill et al., 2012). A proportionality between ploidy and cell size has also been observed in other organisms, including yeast (Marshall et al., 2012). The multiple origins accumulation model is a general regulation strategy that may illuminate the source of the approximate proportionality between cell size and the number of origins across organisms.

An essential feature of the multiple origins accumulation model is the tight coupling between chromosome replication and cell division. The differences between the multiple origins accumulation model and the master accumulation model emphasizes this coupling and the importance of regulating the timing of initiation. By negatively regulating cell size in response to the number of origins via Equation (1), the multiple origins accumulation model is able to maintain a balanced cell cycle and achieve robustness in face of noise in the initiation process. However, the master accumulation model described by Equation (13) is a regulation strategy without such a feedback mechanism. The master accumulation model is unable to maintain a balanced cell cycle and does not predict the exponential scaling of cell size. This suggests that regulation strategies must account for the number of origins per cell in order to regulate appropriately the frequency of division.

The coupling between the number of origins to the division frequency could be demonstrated via a shift-up experiment. It was found for E. coli that cells maintain their rate of division for a duration of C + D after a shift-up, a phenomenon known as rate maintenance (Kjeldgaard et al., 1958; Cooper, 1969). The multiple origins accumulation model naturally accounts for rate maintenance, because division always occurs at time C + D after initiation (chromosome replication rate is independent of growth rate). Furthermore, the model offers a robust mechanism to regulate, after a transient, the number of origins per cell appropriately with the new growth rate via Equation (3). The existence of a rate maintenance period implies that division is coupled to replication initiation, and the incremental model applied at birth and division is a valid phenomenological description only at stationarity. Instead, it is the underlying molecular mechanism of replication initiation that dictates the frequency of division.

Although the multiple origins accumulation model captures many aspects of the coupling between replication and division, experiments with minichromosomes suggest that the molecular mechanism is more complicated. Minichromosomes are plasmids containing the oriC sequence coding for chromosomal origins. In general, minichromosomes initiate replications in coordination with chromosomes and do not affect the growth properties of the cell, such as the doubling time or the average cell size (Leonard and Helmstetter, 1986). However, if more than ∼ 40 minichromosomes are present in a cell, replication initiation is no longer synchronous, the doubling time increases, the average number of origins per cell decreases, and the average cell size decreases (Lobner-Olesen, 1999). Another experiment inserted a second origin into E. coli chromosome and observed again that the extra origin does not affect the growth properties of the cell (Wang et al., 2011). These result points to a more complicated molecular mechanism than accumulation of initiators per origin. Several mechanisms have been suggested, but none has been completely satisfactory. For example, the master accumulation mechanism discussed in Section 3.3 is ruled out for being unable to robustly regulate the number of origins. Another plausible regulation strategy is one in which replication initiates when a critical ratio of active to inactive initiators is reached (Donachie and Blakely, 2003). The validity of this strategy remains to be tested.

The molecular mechanism underlying the regulation of replication initiation is yet to be unraveled, but here we have given significant constraints regarding the potential mechanisms. Specifically, this work and previous works have shown that the molecular mechanism in question should satisfy both the incremental model of size control and the mathematical

### References

form of the multiple origins accumulation model described by Equation (1), so that the predicted distributions, correlations, and scalings remain intact and consistent with existing experiments.

### Author Contributions

PH and AA developed the theoretical model and conducted the mathematical analysis.

### Acknowledgments

The authors thank Lydia Robert and Nancy Kleckner for useful discussions.

**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2015 Ho and Amir. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

### Appendix

### Derivations of the Average Number of Origins per Cell

The average number of origins per cell has been calculated previously in two distinct derivations (Cooper and Helmstetter, 1968; Bremer and Churchward, 1977). The two derivations seemed to make different assumptions to arrive at the same conclusions, bringing into question the necessity of the underlying assumptions. Here, we reproduce the two derivations and show that both derivations in fact make the same assumptions.

The model of the cell cycle under consideration is due to Cooper and Helmstetter (1968). In this model, replication initiation occurs on average every doubling time τ . An initiation event then triggers a division event after a constant duration C + D, where C and D are respectively, the constant duration required to replicate the chromosome and the constant duration between replication termination and division. Given C, D, and τ , we want to find the average number of origins per cell. The average number of origins per cell is defined as hOi = hOtotal/Ni, where Ototal is the total number of origins in a population of cells, N is the number of cells in that population, and brackets denote the ensemble average.

First, we reproduce the derivation due to Cooper and Helmstetter (1968). To calculate the average number of origins per cell, we must first define the probability distribution underlying the ensemble average. In an asynchronous population of exponentially growing cells, the cells must be exponentially distributed in the cell cycle for ensemble averages to be stationary with respect to time (Powell, 1956). Defining cell age a = 0 at birth and a = 1 at division, the exponential distribution of cell ages is

$$\text{p }(a) = \left(\ln 2\right) 2^{1-a}.\tag{23}$$

We can now calculate the desired ensemble averages. For example, if 0 < C+D <sup>τ</sup> < 1, then a cell younger than τ − (C + D) /τ will not be replicating its chromosome and will have only one origin, whereas a cell older than τ − (C + D) /τ will be replicating its chromosome and will have two origins. We have assumed that the amount of time a cell spends with more than two origins is negligible, which is plausible for weak noise in the initiation process. The average number of origins per cell in an asynchronous, exponentially growing population is then

$$\begin{split} \langle O \rangle &= \ln 2 \left[ \left( \int\_0^{\frac{\mathfrak{r} - (C+D)}{\mathfrak{r}}} 2^{1-a} da \right) + 2 \left( \int\_{\frac{\mathfrak{r} - (C+D)}{\mathfrak{r}}}^1 2^{1-a} da \right) \right] \\ &= \left[ \left( 2 - 2^{(C+D)/\mathfrak{r}} \right) + 2 \left( 2^{(C+D)/\mathfrak{r}} - 1 \right) \right] \\ &= 2^{(C+D)/\mathfrak{r}}. \end{split} \tag{24}$$

Similarly if 1 < C+D <sup>τ</sup> < 2, then a cell must initiate replication not only for its daughter cells, but also for its granddaughter cells. In this case, a cell with cell age less than 2τ − (C + D) /τ will not be replicating its chromosome for its granddaughters and will have only two origins, whereas a cell with cell age more than 2τ − (C + D) /τ will be replicating its chromosome for its granddaughters and will have four origins. Again, we have assumed weak noise.

Thus, we see that

$$p\left(O=O\_0\right) = \ln 2 \int\_0^{\Delta T} 2^{1-a} da,\tag{25}$$

$$p\left(O=2O\_0\right) = \ln 2 \int\_{\Delta T}^{1} 2^{1-a} da,\tag{26}$$

where O<sup>0</sup> = 2 ⌊(C+D)/τ ⌋ , and 1T = ⌊(C + D)/τ ⌋ + 1 τ − (C + D). Simplification gives

$$
\langle O \rangle = 2^{(C+D)/\mathfrak{r}},\tag{27}
$$

which generalizes (Equation 24) and is valid for any C + D and τ . The two assumptions made in this derivation are that the population is growing exponentially and that the population has reached a stationary distribution of cell ages.

Next, we reproduce the derivation due to Bremer and Churchward (1977). Assuming exponential growth, the number of cells must grow exponentially as N ∝ 2 <sup>t</sup>/τ . Similarly, the total number of origins must grow at the same exponential rate so that Otot ∝ 2 <sup>t</sup>/τ . But an initiation event triggers a division event after a constant duration C + D, so the number of cells must on average lag behind the total number of origins by 2(C+D)/τ . The average number of origins per cell must then be hOi = hOtot/Ni = 2 (C+D)/τ . Although the distribution of cell ages was not explicitly involved in this derivation, the assumption of a stationary ensemble average under exponential growth is satisfied if and only if the distribution of cell ages is exponential (Powell, 1956).

The exponential distribution of cell ages is not always realized in experimental setups. For example, single-cell experiments that track a lineage of cells, such as those in Taheri-Araghi et al. (2015), will follow a different distribution, as discussed in Robert et al. (2014). Experiments that track a single cell will follow a uniformly distributed cell age. In that case, Equation (27) is replaced by

$$
\langle O \rangle = O\_0 \left( 1 + \frac{C+D}{\mathfrak{r}} - \lfloor \frac{C+D}{\mathfrak{r}} \rfloor \right). \tag{28}
$$

Simulations tracking a population of cells with uniformly distributed cell ages confirm this result. The differences between Equations (27) and (28) do not significantly change the predictions of the multiple origins accumulation model.

## Size sensors in bacteria, cell cycle control, and size control

#### Lydia Robert 1, 2, 3 \*

<sup>1</sup> UMR1319 Micalis, Institut National de la Recherche Agronomique, Jouy-en-Josas, France, <sup>2</sup> UMR Micalis, AgroParisTech, Jouy-en-Josas, France, <sup>3</sup> Laboratoire Jean Perrin (Université Pierre et Marie Curie-Centre National de la Recherche Scientifique UMR8237), Université Pierre et Marie Curie, Paris, France

Bacteria proliferate by repetitive cycles of cellular growth and division. The progression into the cell cycle is admitted to be under the control of cell size. However, the molecular basis of this regulation is still unclear. Here I will discuss which mechanisms could allow coupling growth and division by sensing size and transmitting this information to the division machinery. Size sensors could act at different stages of the cell cycle. During septum formation, mechanisms controlling the formation of the Z ring, such as MinCD inhibition or Nucleoid Occlusion (NO) could participate in the size-dependence of the division process. In addition or alternatively, the coupling of growth and division may occur indirectly through the control of DNA replication initiation. The relative importance of these different size-sensing mechanisms could depend on the environmental and genetic context. The recent demonstration of an incremental strategy of size control in bacteria, suggests that DnaA-dependent control of replication initiation could be the major size control mechanism limiting cell size variation.

#### Edited by:

Conrad L. Woldringh, University of Amsterdam, Netherlands

#### Reviewed by:

Piet A. De Boer, Case Western Reserve University, USA Manuel Loic Campos, Howard Hughes Medical Institute at Yale, USA

#### \*Correspondence:

Lydia Robert, Laboratoire Jean Perrin (Université Pierre et Marie Curie-Centre National de la Recherche Scientifique UMR8237), UPMC, 75005 Paris, France<EMAIL_ADDRESS>
#### Specialty section:

This article was submitted to Microbial Physiology and Metabolism, a section of the journal Frontiers in Microbiology

> Received: 27 February 2015 Accepted: 09 May 2015 Published: 29 May 2015

#### Citation:

Robert L (2015) Size sensors in bacteria, cell cycle control, and size control. Front. Microbiol. 6:515. doi: 10.3389/fmicb.2015.00515

Keywords: size control, bacterial cell cycle, replication initiation, incremental model, DnaA, FtsZ, MinCD

## 1. Introduction

All dividing cells have to coordinate different steps of the cell cycle, such as DNA replication, chromosome segregation, and cytokinesis. In eukaryotes, cell cycle control has been widely studied and the overall logic is well-understood (Murray and Hunt, 1993). The orderly progression in the eukaryotic cycle relies on a biochemical engine composed of cyclins and cyclin-dependendent kinases (CDKs). The periodic activity of cyclin-CDK complexes regulates the cell cycle transitions such as the initiation of DNA replication or the entry into mitosis (Murray and Hunt, 1993). In addition, specific control mechanisms, often called checkpoints, ensure that late events do not occur before the completion of earlier events (Hartwell and Weinert, 1989). In contrast, the logic of the bacterial cell cycle, even though intensively studied for more than half a century, remains unclear. Bacteria do not possess any cyclins or CDKs. Numerous regulatory mechanisms have been discovered and characterized in details, such as the control of DNA replication initiation (Katayama et al., 2010) or the SOS response, which inhibits division in case of DNA damage or replication arrest (Simmons et al., 2008). However, how these different mechanisms are organized and connected to ensure an orderly progression into the cycle remains unclear (Haeusser and Levin, 2008).

In many organisms, progression through the cycle is coupled with cellular growth. In the budding yeast Saccharomyces cerevisiae, the duration of the G1 phase depends on the cell size at birth, smaller cells spending a longer time in G1 (Johnston et al., 1977; Di Talia et al., 2007; Turner et al., 2012). Likewise, in the fission yeast Schizosaccharomyces pombe, mitotic entry is delayed in smaller cells (Fantes, 1977; Sveiczer et al., 1996). In bacteria, cell cycle progression was as well-assumed to be under size control. In 1968, building on the seminal physiological studies of Schaechter et al. (1958) and Cooper and Helmstetter (1968) on Salmonella typhimurium and Escherichia coli, Donachie showed that at the population level, the average initiation mass, i.e., the ratio of cell mass to the number of replication origins is constant regardless of the growth rate and culture medium (Donachie, 1968). This lead him to propose a size control mechanism at the single cell level, in which the cell initiates DNA replication when it reaches a critical size, and divides a constant time after initiation. Although widely accepted for decades, this model became controversial in recent years (Voorn et al., 1993; Wold et al., 1994; Boye et al., 1996; Bates and Kleckner, 2005; Chien et al., 2012; Hill et al., 2012). Nevertheless, definitive evidence for size control in bacteria was provided by recent studies using quantitative data on single cell growth and division in both E. coli, Bacillus subtilis, and Caulobacter crescentus (Campos et al., 2014; Osella et al., 2014; Robert et al., 2014; Soifer et al., 2014; Taheri-Araghi et al., 2015).

However, no size-sensing module has been identified so far, and the molecular basis of size control stays unclear. Cells could sense their length,volume or mass. Here I use cell size as a catch-all descriptor and describe how cell size could be sensed through different pathways involved in the control of cytokinesis, chromosome segregation or replication initiation. Importantly, the potential size-sensing mechanisms are not mutually exclusive and size control could be implemented redundantly at several cell cycle stages. Such redundancy has been evidenced in the fission yeast S. pombe. In addition to the major size control acting at mitosis entry (Fantes and Nurse, 1977), a second size control mechanism acts at the G1/S transition (Fantes and Nurse, 1978). This second size-control is usually invisible since the G2/M control produces cells whose size at birth already exceeds the requirement of the G1/S control. Nevertheless, this second mechanism can be revealed when the primary G2/M control is perturbed, such as in the wee1 mutant (Fantes and Nurse, 1978). S. cerevisiae has also been proposed to exhibit an usually invisible size-control, acting at the G2/M transition (Murray and Hunt, 1993; Turner et al., 2012). Several size checkpoints could as well-exist in bacteria, where the cytokinesis, the chromosome segregation and the initiation of replication might all depend on cell size. As in yeast, the most stringent of these size control mechanisms would be responsible for the limitation of the variations of the cell size in bacteria. Interestingly, recent experimental results on size control in bacteria argue against the current critical size paradigm and suggest an incremental strategy: division does not occur at a critical size but rather when a constant size has been added to the size at birth (Campos et al., 2014; Soifer et al., 2014; Taheri-Araghi et al., 2015). Such phenomenological description sheds a new light on the mechanism limiting cell size variations. Therefore, after a description of all the potential size-sensing mechanisms involved in the control of cytokinesis, chromosome segregation and replication initiation, I will discuss which of these mechanisms could be responsible for the limitation of cell size variations, in light of this incremental principle.

### 2. The Min System: a Geometric Size-Sensor Controlling Cytokinesis?

A critical event in the bacterial division process is the polymerization of the tubulin-like protein FtsZ into an annular structure called the Z ring, which locates the division site and recruits the numerous proteins required to carry out cytokinesis (Bi and Lutkenhaus, 1991; Addinall and Holland, 2002; Margolin, 2005; Harry et al., 2006). In E. coli and B. subtilis, the positioning of the Z ring has been widely studied. It is generally assumed to rely on two inhibitory systems: Nucleoid Occlusion (NO) and the Min system (Lutkenhaus, 2007; Wu and Errington, 2011). The latter is based on the protein complex MinCD, an inhibitor of FtsZ polymerization that concentrates at the cell ends, and is therefore essential to prevent polar divisions (Adler et al., 1967; De Boer et al., 1989; Lutkenhaus, 2007). In E. coli, the localization pattern of MinCD emerges from remarkable oscillations of the complex from pole to pole, which create over time a gradient of concentration showing a minimum around mid-cell (Hu and Lutkenhaus, 1999; Raskin and de Boer, 1999; Hale et al., 2001). In contrast, in B. subtilis MinCD does not oscillate and the gradient is static (Lutkenhaus, 2007). In both organisms, the concentration of the MinCD complex along the cell axis may vary with cell length and MinCD has therefore been proposed to serve as a size ruler (Raskin and de Boer, 1999). Nevertheless, there is no experimental evidence yet allowing confirming or invalidating this hypothesis.

The function and localization of MinCD is very similar to those of the Pom1 kinase in the fission yeast S.pombe (Padte et al., 2006; Huang et al., 2007). This kinase exhibits a gradient of concentration in the cell, with maxima at cell ends and a minimum around mid-cell. This localization allows Pom1 to prevent the assembly of the division septum at the cell ends. It was also recently suggested to serve as a size sensing device responsible for size control at mitotic entry. The work of two different teams published in 2009 established that Pom1 is a dose-dependent inhibitor of mitosis, whose concentration at the cell middle decreases when the cells elongate (Martin and Berthelot-Grosjean, 2009; Moseley et al., 2009). Pom1 was shown to regulate the G2/M transition by inhibiting mitotic activators localized at the middle of the cells. The following model of size-dependent G2/M transition was proposed: when the cells are short, the high concentration of Pom1 at mid-cell where it can interact with mitotic regulators inhibits mitotic entry. When the cell grows, this inhibition progressively weakens as the concentration of Pom1 at mid-cell decreases. Pom1 was therefore proposed to be a dedicated size sensor involved in cell cycle control. This discovery appeared promising for understanding the molecular basis of size control. Nevertheless, a few years later, one of the team involved in this discovery presented evidence that Pom1 is in fact not the size sensor but rather a modulator that affects the link between the measured size and the division probability, thus changing the size threshold at mitosis (Novák, 2013; Wood and Nurse, 2013). This conclusion was supported by the behavior of the pom11 mutant, which is shorter but exhibits a wild type correction for cell size fluctuations, compressing or extending the duration of the

cycle according to the cell's size at birth (Wood and Nurse, 2013).

Bacteria possess some geometric sensors similar to Pom1. In E. coli and B. subtilis, the potential Pom1-like geometric sensor is the MinCD complex. In C. crescentus, which has no MinCD orthologs, a gradient of the inhibitor of FtsZ polymerization MipZ emanates from the cell poles after the segregation of the replication origins (Goley et al., 2007). The mechanism that was proposed for size-sensing through Pom1 could in principle apply to its bacterial counterparts: cell elongation could lead to a decrease of MinCD/MipZ concentration in the cell middle, thus triggering the formation of the Z ring. Alternatively, these regulators could be, like Pom1, modulators of cell size at division.

### 3. FtsZ, a Size Sensor?

In the fission yeast, very recent results suggest that the cdr2 mitotic activator regulated by Pom1 could be the size sensor controlling mitotic entry (Pan et al., 2014). Cdr2 is localized in a band of cortical nodes at mid-cell, where it accumulates when the cell elongates. Pan et al. therefore proposed that a critical cdr2 concentration is attained when the cell reaches a critical size and triggers mitotic entry. The bacterial FtsZ protein, the target of MinCD regulation, has also been proposed to act as a size sensor (Chien et al., 2012). In this model, division would occur when the amount of FtsZ inside the cell reaches a threshold. In support of this hypothesis, FtsZ levels are known to modulate the size at division in E. coli (Chien et al., 2012). In C. crescentus, FtsZ expression and degradation are regulated during the cell cycle. As a consequence, the concentration oscillates during the cycle with a maximum at the onset of constriction, after which FtsZ is degraded (Quardokus et al., 1996; Kelly et al., 1998). This dynamics is in agreement with division being triggered by a critical level of FtsZ. In contrast, in E. coli and B. subtilis, FtsZ is constitutively expressed and its concentration is constant throughout the cycle. Its total amount is therefore proportional to cell size.

The capacity of the yeast protein Cdr2 to serve as a size-sensor mainly relies on its localization, so that the local concentration at mid-cell increases when cells elongate. Similarly, specific localization of FtsZ could be responsible for a size-dependent local concentration at mid-cell. In addition to ring structures, FtsZ can assemble into helices in E. coli and B. subtilis (Ben-Yehuda and Losick, 2002; Thanedar and Margolin, 2004; Harry et al., 2006). FtsZ has also been shown to oscillate from one pole to the other, out of phase with the Min system (Thanedar and Margolin, 2004; Bisicchia et al., 2013). The Z ring may therefore form from the reorganization of moving helices. However, helical structures of FtsZ have still to be confirmed. In particular, such helical structures can be artifacts of fluorescent fusions, as demonstrated for the MreB protein (Domínguez-Escobar et al., 2011; Van Teeffelen et al., 2011; Swulius and Jensen, 2012). Therefore, the localization of FtsZ is still unclear, but some sizedependence of the local concentration at mid-cell is possible, leading to a critical size for Z ring formation. Importantly, in E. coli and B. subtilis the ring is known to form well before constriction (Den Blaauwen et al., 1999). In addition, recent data suggests that at the single cell level, the timing of its formation is independent of the timing of constriction and the cell length at birth (Tsukanov et al., 2011). It is therefore possible that once the ring is formed, the amount of FtsZ it contains increases up to a threshold where the ring becomes able to recruit all the downstream components of the divisome and constrict. In this case, division timing could depend on cell size through FtsZ accumulation inside the ring.

### 4. Chromosome Segregation: a Size-Dependent Event?

Chromosome segregation occurs in three steps: separation of the newly replicated origins, bulk chromosome segregation and separation of the replicated termini (Wang et al., 2013). Bulk chromosome segregation appears as an important transition, leading to the appearance of a DNA free space at mid-cell and relief of the division inhibition mediated by Nucleoid Occlusion (NO). NO prevents the formation of the Z ring in the vicinity of the chromosome (Mulder and Woldringh, 1989; Wu and Errington, 2011). The mechanism of NO is not fully elucidated yet but some molecular bases have been provided by the discovery of the Noc protein of B. subtilis and SlmA protein of E. coli, which associate with DNA and inhibit FtsZ polymerization (Wu and Errington, 2004; Bernhardt and De Boer, 2005). Interestingly, these NO proteins bind to specific DNA sequences that decorate a large portion of the chromosome but are absent from the Ter region (Wu and Errington, 2011). Therefore, NO proteins are largely removed from mid-cell following bulk chromosome segregation.

The mechanism underlying bulk chromosome segregation are still unclear but entropic forces have been proposed to either drive or facilitate segregation (Jun and Wright, 2010; Di Ventura et al., 2013). Such forces can in theory segregate mixed chains of polymers in a situation of confinement. A single E. coli chromosome is more than 1 mm in length and has to be compacted more than 1000-fold to fit inside the cell (Wang et al., 2013). Replicated chromosomes are thus confined inside the cell, whose shape and size will determine the intensity of the entropic forces. Importantly, if segregation is entropy-driven, it should be easier in longer cells. During cell elongation, the intensity of the entropic forces increases, which could lead to size-dependent segregation. In agreement with this hypothesis, experimental results on cell cycle progression in single cells suggest that the timing of nucleoid splitting depends on cell growth rather than on replication progression (Bates and Kleckner, 2005). Therefore, if driven by entropic forces, the very act of segregation could be a size-dependent signal, transmitted to the division machinery through the relief of NO.