Turing-pattern model of scaffolding proteins that establish spatial asymmetry during the cell cycle of Caulobacter crescentus

Summary The crescent-shaped bacterium Caulobacter crescentus divides asymmetrically into a sessile (stalked) cell and a motile (flagellated) cell. This dimorphic cell division cycle is driven by the asymmetric appearance of scaffolding proteins at the cell’s stalk and flagellum poles. The scaffolding proteins recruit enzyme complexes that phosphorylate and degrade a master transcription factor, CtrA, and the abundance and phosphorylation state of CtrA control the onset of DNA synthesis and the differentiation of stalked and flagellated cell types. In this study, we use a Turing-pattern mechanism to simulate the spatiotemporal dynamics of scaffolding proteins in Caulobacter and how they influence the abundance and intracellular distribution of CtrA∼P. Our mathematical model captures crucial features of wild-type and mutant strains and predicts the distributions of CtrA∼P and signaling proteins in mutant strains. Our model accounts for Caulobacter polar morphogenesis and shows how spatial localization and phosphosignaling cooperate to establish asymmetry during the cell cycle.


INTRODUCTION
In prokaryotic cells, asymmetric protein distributions contribute to diverse physiological processes including morphogenesis, stress response, and signal transduction. 1,2 A model organism for studying bacterial cell asymmetry is the oligotrophic aquatic bacterium, Caulobacter crescentus, in which at least 10% of proteins are non-uniformly distributed across a cell. 2,3 Caulobacter cells undergo a dimorphic division cycle, regulated by asymmetrically distributed proteins such as CtrA, a potent inhibitor of DNA replication. 4,5 Cell division produces two distinct progeny: a motile swarmer cell with high levels of phosphorylated CtrA (CtrA$P) and a sessile stalked cell with low levels of CtrA$P 6 ( Figure 1). As phosphorylated CtrA inhibits the initiation of DNA replication by binding to the chromosome origin of replication (Cori), the swarmer cell is blocked from DNA replication and cell division. To reproduce, the swarmer cell must eliminate CtrA$P, by degradation and/or dephosphorylation, a process that occurs during the swarmer-to-stalked (sw-to-st) transition. During this transition, the swarmer cell sheds its flagellum and makes a stalk. The stalk cell commences DNA replication and begins to generate a flagellum at the opposite pole. The predivisional cell, with a stalk at the ''old'' pole, a flagellum at the ''new'' pole, and a partially or fully replicated chromosome in the middle, exhibits dynamic localization of key proteins at the old and new poles (Figure 1). At cell division, the motile swarmer cell separates from the sessile stalk cell. The stalked cell recommences the cell division cycle immediately after cell division. [7][8][9] In addition to the changing asymmetric distributions of signaling proteins, cell cycle progression in Caulobacter is also controlled by temporal patterns of gene expression through the actions of three transcription factors (CtrA, GcrA, and DnaA) and a DNA methylase (CcrM). 15,16 GcrA and DnaA (unlike CtrA) promote the initiation of DNA replication, which converts fully methylated DNA sequences to hemi-methylated conditions. Later in the cycle, these sequences are returned to the fully methylated state by CcrM. The methylation state of genes affects their transcriptional activities.
The temporal dynamics of this complex network of interacting transcription factors and DNA methylation has been modeled mathematically by Li et al. 17,18 and Shen et al., 19 who have reproduced in silico the time courses of key regulatory proteins in wild-type (WT) and mutant cells and predicted the phenotypes of novel mutants. Murray et al. 20 studied a simpler model involving just GcrA, CtrA, and CcrM. Using a Figure 1. Dynamic localization of key proteins over the cell cycle of C. crescentus CtrA$P develops an asymmetric spatial distribution during the cell cycle (gray). The three scaffolding proteins, PopZ, PodJ, and SpmX, interact with each other and recruit (directly or indirectly) client proteins at specific poles, including DivJ, 10 PleC, 11 DivK, 10 CckA, 12 DivL, 12 and CpdR. 13 CckA (fluorescent yellow) is uniformly distributed throughout the cell in the swarmer cell and localized at the new pole during the predivisional stage. Some Caulobacter predivisional cells may also have old polar localized CckA (not shown). 14 ll OPEN ACCESS 2 iScience 26, 106513, April 21, 2023 iScience Article from the new pole after cytokinesis 30 (Figure 1, red). Interestingly, DivJ and PleC are localized at opposite poles (at the stalk and flagellum, respectively) during the cell cycle of Caulobacter (Figure 1, orange and purple). 31,10 The catalytic activity and polar localization of DivJ and PleC regulate the spatial phosphorylation and dephosphorylation of DivK. 2 Consequently, DivK is phosphorylated at the old pole and dephosphorylated at the new pole, and after cytokinesis, DivK is unphosphorylated in the newly formed swarmer cell.
Chen et al. 3 and Tropini and Huang 2 have constructed mathematical models to simulate the development of an asymmetrical distribution of CtrA$P in predivisional cells. Their models account for the phenotypes of relevant mutant cells and suggest that Caulobacter establishes robust asymmetry before cytokinesis. Subramanian et al. 22 Although these published spatiotemporal models provide an initial understanding of the mechanisms of spatial regulation and pattern formation in Caulobacter, they assume an initial non-uniform localization of DivJ and PleC, so the source of asymmetry remains elusive. The initial asymmetry is created by scaffolding proteins found at the old and new poles at the time of cytokinesis. 34 The key regulators of CtrA in Caulobacter respond to three scaffolding proteins-PodJ, PopZ, and SpmX. 10, 34 PodJ is localized at the flagellum pole, SpmX at the stalk pole, and PopZ at both poles during the Caulobacter cell cycle (Figure 1 Several hypotheses have been proposed for the initial polar localization of scaffolding proteins. 34 The nucleoid-occlusion hypothesis suggests that the poles, because they are devoid of chromosomes, provide

RESULTS AND DISCUSSION
A-SD Turing patterns accurately capture the spatiotemporal dynamics of scaffolding proteins The simulated spatial dynamics of the scaffolding proteins PopZ, SpmX, and PodJ in WT cells are shown as heatmaps in Figure 3. PopZ is localized at the old pole throughout the cell cycle. At approximately 50 min, a second focus of PopZ appears at the new pole ( Figure 3A), matching experimental data. 11 SpmX, recruited by PopZ, sharply accumulates at the old pole at 10-20 min in our simulation ( Figure 3B), which agrees with experimental observations. 10 Long-form PodJ (PodJL) polymerizes at the new pole during S phase (Figure 3C). There it is truncated by the protease PerP into the short form, PodJS (Figure 1, blue). 11, 45 PodJS remains at the flagellated pole until it is degraded during the sw-to-st transition of the next cell cycle (Figure 3D). Total PodJ (PodJL+PodJS) is evident at the new pole for most of the cell cycle (see Figure S1), as observed by Chen et al. 45 and Zhao et al. 11 The new cell synthesizes PodJL, which localizes to the new pole because old polar SpmX inhibits the polymerization of PodJL. These simulation results are consistent with experimental observations. iScience Article Asymmetrical distributions of CtrA$P and DivK$P are reproduced by our model The polar accumulation of scaffolding proteins induces spatial distributions of many other proteins in the signaling network. For example, the scaffolding proteins recruit DivJ and PleC to opposite poles of the cell, where they function, respectively, as a kinase and phosphatase of DivK, thereby creating an asymmetrical distribution of DivK$P across the cell. Subsequently, DivK$P binds to DivL to promote the switch of CckA from kinase to phosphatase. 46,47 The CckA balance between kinase and phosphatase regulates the phosphorylation of CtrA and CpdR, thereby controlling the phosphorylation state and level of CtrA in the cell. In addition, PopZ binds to DivL and CpdR, 12 and PodJ participates in the localization of DivL. 36,48 Because the consequences of scaffolding-protein localizations are multifaceted and difficult to comprehend with confidence by informal reasoning alone, we include the detailed molecular mechanism of signaling modules in this work (green and yellow boxes in Figure 2) and explore the relationship between signaling proteins and scaffolding proteins. Figure 4 shows the simulated dynamics of proteins involved in two phosphotransfer modules. Overall, the simulations match well with experimental observations. For example, DivK$P accumulates in the stalked compartment after cell division. DivK$P remains localized at the old pole (the stalk pole) over the entire cell cycle, 30 while also showing temporary accumulation at the new pole in the predivisional stage. 30 Our simulation captures the behavior of DivK$P, although the temporary new-polar accumulation of DivK$P is limited ( Figure 4C). The asymmetric distribution of DivK is caused by new-polar PleC and old-polar DivJ, which regulate both the phosphorylation state and spatial localization of DivK (Figures 2 and 4A and B).
Experiments show that DivL is located at the new pole during S phase and less frequently at the old pole ( Figure 1, olive). 49 Our model captures the new-polar accumulation of DivL, which is mainly recruited by PodJ ( Figure 4D). Similarly to DivL, CckA is dispersed in the nascent swarmer cell and shows strong and stable new-polar accumulation later, 14,24 which is reproduced by our model ( Figure 4E). A portion of Caulobacter cells ($30%) exhibits old polar accumulation of CckA after the sw-to-st transition, which is not requisite for WT cell cycle. The total level of CpdR does not change much, while the phosphorylated CpdR level starts to increase in the predivisional stage ( Figures S2G and S2H). Following cell division, unphosphorylated CpdR accumulates in the stalked compartment ( Figure 4F). Our simulations of CpdR species are consistent with experiments. 13 Phosphorylated CtrA accumulates in the swarmer compartment ( Figure 4H), while unphosphorylated CtrA accordingly goes to the stalked compartment and subsequently is degraded. (Figure S2I). Because unphosphorylated CpdR promotes the proteolysis of CtrA, the iScience Article asymmetric distribution of CpdR reinforces a reduced level of CtrA$P in the stalked compartment, allowing the stalk cell to re-enter the DNA replication cycle.
A few studies suggest that CtrA$P develops a spatial gradient in the predivisional stage by assuming CckA kinase and phosphatase are localized at the flagellated and stalked pole 3,22 ; however, there has been no direct observation of a spatial gradient of CtrA$P across a predivisional cell. Our model exhibits a different pattern, suggesting that a spatial gradient of CtrA$P is not necessary for the development of asymmetry. In our model, the spatiotemporal dynamics of PleC and DivJ are sufficient to create an asymmetrical distribution of CtrA$P between swarmer and stalked daughter cells. We show the dynamics of specific species defined in our model in Figure  Temporal dynamics of signaling proteins are also successfully reproduced by our model In Figure 5, we compare the temporal dynamics of PodJL, PodJS, SpmX, DivJ, PleC, DivK, and CtrA with experimental measurements. In general, our temporal simulations fit the data quite well. PodJL increases steadily during the cell cycle, until PerP is expressed in late S phase, converting PodJL into PodJS (Figure 5A). Subsequently, PodJS is degraded at the beginning of the next cell cycle. SpmX level increases slowly throughout the cycle. Most of SpmX is in the stalked compartment at the end of cycle, which explains the low level of SpmX at the birth of a swarmer cell ( Figure 5B). Similarly, because most DivJ is localized at the old pole, the nascent swarmer cell inherits less DivJ, which explains the lower level of DivJ at t = 0 (Figure 5C). In experiments, PleC drops during the swarmer stage, and then rises steadily in the stalk and predivisional cell. Simulated PleC level drops and rises as well, but the turning point occurs significantly later in the cell cycle ( Figure 5D). The decrease of CtrA$P ( Figure 5F) at the sw-to-st transition signals the initiation of DNA replication and methylation-regulated gene expression in this model (see model details, Table S2). In our simulation, DNA replication commences approximately 25 min after cell separation.
Interactions among scaffolding proteins and higher polar affinity are required for their proper spatial localization iScience Article accumulation ( Figure 6A ). In a preliminary simulation of the DpodJ strain, polar localization of PopZ is drastically impaired (not shown), resulting in ectopic midcell accumulation of PopZ. These simulations suggest that, in addition to the clear role played by PodJ in stabilizing the polar localization of PopZ in WT cells, there is some other mechanism, independent of PodJ, that biases PopZ to polymerize at poles rather than midcell in DpodJ cells. For instance, the curvature of polar cell walls may provide higher affinity for scaffolding proteins like PopZ. 34 Since the specific mechanism for polar accumulation of PopZ (independent of PodJ) is unclear, we simply assume a higher polymerization rate of PopZ at the poles. Specifically, the autocatalytic polymerization rate of PopZ (k aut, PopZ ) is 25% larger in polar compartments than in central compartments, and similarly for PodJ. With this assumption of higher polar affinities, our model reproduces the phenotype of DpodJ cells, namely, that PopZ localizes solely to the old pole and PleC is dispersed uniformly across the cell throughout the division cycle ( Figure 6B). In addition, SpmX and DivJ co-localize with PopZ in the DpodJ simulation.
In our simulation of DpopZ mutant cells (k s,PopZ = 0), the polar localization of SpmX, DivJ, CpdR, and DivK is severely impaired ( Figures 6C and S3), in agreement with experimental observations, 35,56 although the remaining localized SpmX in Figure 4B of Bowman et al. 35 is not reproduced. Furthermore, PodJ shows slight bipolar accumulation ( Figure 6C), consistent with observations of Zhao et al. 11 and Bowman et al. 35 The delocalization of DivJ is likely caused by the delocalization of SpmX. Moreover, the bipolar localization of PodJ is likely a result of delocalized SpmX, because SpmX is an inhibitor on PodJ localization. 11 In general, DpopZ cells, though viable, are severely impaired, exhibiting defects in cell division. 57 To further investigate the function of SpmX, we set k s,SpmX = 0 to simulate DspmX cells (Table S3). In this simulation, PodJ exhibits bipolar localization, while DivJ is dispersed ( Figure 6D). Observations of DspmX cells reveal an increased number of cells with bipolar PodJ and ectopic midcell PodJ. 11 Thus, our model captures certain properties of DspmX cells. As described in (STAR Methods/chromosome replication, methylation, and cell division), we do not model the process of Z-ring constriction; instead, we artificially introduce Z-ring constriction at 95 min after the initiation of chromosome replication ( Figure S4). Assuming that the Z-ring does not close in DspmX, we provide, in Figure 6E, a 750 min simulation of DspmX, which shows that, without Z-ring closure, PodJ accumulates at poles and midcell in an elongated cell, as observed by Zhao et al. 11 The agreement between Figure 6E and experiments suggests that Z-ring constriction is impaired in DspmX mutant cells.
Altogether, our A-SD Turing model reproduces most phenotypes of mutant strains deleted of scaffolding proteins, suggesting that our hypothesized interactions among PopZ, PodJ, and SpmX are crucial for their correct localization. Our analysis of these mutant strains suggests that PopZ and PodJ likely have a    iScience Article Table S3. In the case of mild overexpression (PopJ op1 , see Figure 7B), PodJL and PopZ exhibit early accumulation at the new pole, as observed. For greater overexpression (PopJ op2 , see Figure 7C), PodJ is localized at poles and midcell, which is consistent with experiments. However, the cell division defect of cells that strongly overexpress PodJ is not captured by our model, because we do not consider the effects that overexpressed PodJ may have on the Z-ring constriction process.
In addition, we have considered the phenotype of mutant cells overexpressing PopZ (Table S3). These simulated cells ( Figure 7D) display an expanded localization pattern for DivJ, DivK$P, and CpdR, as observed in cells overexpressing PopZ at 0.3% xylose induction level (see Figures 4 and 5 in the study by Bowman et al. 35 ).

Our model captures the localization of DivK in existing mutant strains and predicts the phenotypes of novel mutations
Previous study has shown that phosphorylated DivK preferentially localizes at cell poles. 1 Among the currently known binding partners of DivK, PleC and DivL show significantly higher affinity for DivK$P in vivo, whereas DivJ binds to both phosphorylated and unphosphorylated forms. 46 Therefore, we consider the following complexes of DivK in our model: PleC:DivK$P, DivL:DivK$P, DivJ:DivK$P, and DivJ:DivK (see Figure 2). With these assumptions, our model reproduces the phenotypes of DivK localization in DivJand PleC-mutated cells, as described below.
DivJ is necessary for the polar localization of DivK. DivJ not only binds to DivK but also is the major kinase phosphorylating DivK. 1 Without DivJ (see the DdivJ mutant, Table S3), the level of DivK$P drops dramatically and DivK is delocalized. 1,7,53 In the kinase-defective DivJ strain (divJ-H338A), DivK can localize at the iScience Article old pole but fails to localize at the new pole. 1 Our simulations of DdivJ and divJ-H338A ( Figures 8B and 8C) are consistent with these observations, which suggest that DivJ determines the old-polar localization of DivK and that DivJ kinase activity is required for the new-polar accumulation of DivK. On the other hand, the viability of DdivJ cells calls into question our simulation of uniformly high levels of CtrA$P across a dividing cell, which would block DNA replication in the progeny.
PleC is believed to determine the release of DivK from the new pole rather than its localization there, because DivK$P continues to occupy the new pole after Z-ring closure in the PleC-deficient (DpleC) and the catalytically inactive (pleC-H610A) mutant strains. 1,21,30 Our simulations of DpleC and pleC-H610A in Figures 8D and 8E show that DivK$P fails to release from the new pole after cell division. We have also simulated the kinase-defective pleC strain pleC-F778L, in which DivK shows WT-like dynamics ( Figure 8F), consistent with experiments. 30 These results suggest that the kinase activity of PleC is dispensable for the transient new-polar accumulation of DivK in the predivisional stage.
Based on these results, we speculate that either PleC or DivL (or both) functions as the physical binding partner to recruit DivK$P to the new pole, which also requires the kinase activity of DivJ. To sum up, our model agrees with experimental observations that DivJ kinase activity is required for newpolar accumulation of DivK, while PleC is required for the timely release of DivK from the new pole. 1 In addition, our mutant simulations suggest that PleC localization and kinase activity are not necessary for the localization of DivK.

Our model captures key characteristics of phosphotransfer processes
In DdivJ cells, the level of DivK$P is reduced and CtrA-dependent transcriptions increase, 7,65 and these properties are successfully recapitulated by our model ( Figure 8B). Our simulation of the DpleC strain shows increased level of DivK$P and reduced level of CtrA$P ( Figure 8D), which is consistent with experiments as well. 25, 65 In Table 1, we compare simulated levels of DivK$P to corresponding experimental measurements in four mutant strains. Our simulations capture the key trends of DivK$P level in these mutant cases. DivK$P level dramatically decreases in DspmX and DdivJ because the kinase activity of DivJ is largely impaired or deleted (Table 1, Figures S3 and 8B). As PleC mainly functions to dephosphorylate DivK, DpleC mutation results in increased DivK$P (Table 1 and Figure 8D). In addition, the higher level of DivK$P in DpodJ suggests that PodJ likely inhibits rather than activates the kinase activity of PleC, which is a debatable issue (see the study by Kowallis 34 ).
By interacting with DivL, DivK$P inhibits phosphotransfer to CtrA$P; hence, mutations that impact the localization and/or abundance of DivK$P should affect the spatiotemporal dynamics of CtrA$P. In this regard, our simulations of CtrA spatial dynamics for some relevant mutant strains (Figures 8 and S3) are subject to experimental verification. DpleC and pleC-H610A mutant strains, which fail to release DivK$P from the new pole, exhibit reverse distributions of CtrA$P in simulations ( Figures 8D and 8E). Our model iScience Article predicts the asymmetrical distribution of CtrA$P almost disappears in DdivJ, divJ-H338A, DpopZ, and DspmX mutant strains. All other mutant simulations in this study show higher levels of CtrA$P in the swarmer compartment, whereas the ''delocalized PleC'' and DpodJ mutant simulations display smaller differences of CtrA$P distribution between swarmer and stalked compartment (Figures 8 and S3). These predictions suggest that both the activity and the distribution of regulators contribute to the establishment of asymmetry.

Conclusion
To study the establishment of cell polarity and asymmetric division in the alpha-proteobacterium C. crescentus, we integrate a Turing-type model of spontaneous pattern formation in reaction-diffusion equations with a protein signaling network linking scaffolding proteins to phosphotransfer pathways. Symmetry breaking derives from an A-SD mechanism, where ''substrate'' is monomeric protein subunits and ''activator'' is polymeric protein, whose branched structure supports autocatalytic growth of the polymer. Quite naturally, the substrate molecules diffuse much more rapidly than the polymeric material, which is the primary requirement for developing distinct foci of polymerization (i.e., Turing patterns). In our model, the three scaffolding proteins (PodJ, PopZ, and SpmX) are polymerized based on the A-SD mechanism and interact with each other. Constructed in this way, our model accounts for the observed patterns of polymerization of scaffolding proteins at the two poles of a growing Caulobacter cell. Subsequently, the spatial distribution of the scaffolding proteins spatially influences the two phosphotransfer modules that ultimately control the accumulation and phosphorylation of the master transcription factor CtrA in the growing cell. Phosphorylated CtrA (CtrA$P) is abundant in a newborn, motile swarmer cell. At the transition from the swarmer (flagellated) morphology to the sessile (stalked) morphology, CtrA activity is cleared by dephosphorylation and degradation. The flagellum is dropped and a stalk develops in its place at the ''old'' pole. The stalk cell initiates DNA replication and eventually generates a flagellum at the opposite end (the ''new'' pole) of the cell. In the predivisional cell, CtrA is resynthesized and phosphorylated by CckA-kinase activity at the new pole. After the Z-ring closes, CtrA$P is found only in the nascent swarmer cell (with kinase activity). In the nascent stalk cell, CtrA is degraded by a CpdR-dependent protease. Eventually, the cell divides asymmetrically into a sessile stalk cell (mostly devoid of CtrA) and a motile swarmer cell (with abundant CtrA$P).
The network of biochemical reactions that controls these processes of spatial differentiation and temporal development, as it has been worked out by molecular cell biologists over past decades, is exceedingly complex. Even a simplified, partial description of the network consists of dozens of proteins in different phosphorylation states and in complex with different partners (see Figure 2). By informal reasoning alone, it is impossible to know how successful this conception of the control system might be in explaining known properties of Caulobacter cell replication, in accounting for phenotypes of many regulatory mutations, and in predicting the behavior of cells under novel circumstances. To answer these questions requires the precision of a mathematical model based on well-known properties of the control system, calibrated against quantitative experimental observations, tested by its predictions of known mutant phenotypes, and (ultimately) pushed to make novel predictions that can be tested experimentally.
A mathematical model of such detail is necessarily complex, consisting of 39 differential equations (see Methods S2) involving 110 parameters (Table S4), of which 41 had to be estimated from experimental observations of protein time courses and spatial distributions. These parameters were estimated from data on a few key species in wild-type cells. Then the model (with the estimated parameter values) was verified by comparing simulations with the whole set of available data on protein abundances and distributions in wild-type cells and a variety of well-studied mutant strains. The surprisingly good match between model simulations and experimental data demonstrates the strength of the model.

Limitations of the study and future directions
Although our model reproduces many key characteristics of experimental observations, there are some notable discrepancies between the model and some mutant phenotypes. For example, our model cannot explain the initiation of DNA replication in DdivJ cells 31 or the deviant division patterns of DpopZ cells. 57 Reconciling some of these discrepancies may lead to a more accurate and predictive mathematical model in the future. Some differences may stem from the fact that our model does not explicitly track the regulation of DNA synthesis, of chromosome methylation, of Z-ring closure, or of cell separation. Other limitations derive from the current parameterization procedure, which is not highly efficient for simulating mutant ll OPEN ACCESS 1. Integrate this spatial model with previous temporal models of the CtrA-DnaA-GcrA-CcrM regulatory network to investigate the comprehensive control of DNA replication and chromosome methylation.
2. Supplement the model with interactions among PopZ, ParA, ParB, and FtsZ to account for the spatial regulations of chromosome segregation and cell division.
3. Extend the parameter estimation-verification procedure to derive a representative distribution of parameter sets that all provide ''reasonably'' good fits to the constraining datasets, in order to assess the robustness of the model and to estimate the reliability of predictions made by the model across the range of parameters sets.
4. Convert the deterministic model into a stochastic version to capture the inherent variability of bacterial cell development.

STAR+METHODS
Detailed methods are provided in the online version of this paper and include the following:

ACKNOWLEDGMENTS
This work was partially supported by the National Science Foundation (USA) under awards MCB-1613741 and CCF-1909122. The funding sources played no role in the design of the study, in the collection, analysis, and interpretation of data, and in writing the manuscript.

AUTHOR CONTRIBUTIONS
C.X. conducted this research. All authors contributed to the conceptualization of this work. Y.C. supervised this work. C.X. and J.J.T. wrote this manuscript, and all authors approve the final manuscript.

DECLARATION OF INTERESTS
The authors declare no competing interests.  All the results presented in the main text were calculated from a ten-compartment version of the PDEs in Methods S2 using the estimated parameter values in Table S4 for WT cells. Simulation time increases quadratically with compartment number ( Figure S6). Ten-compartment is a suitable compromise between accuracy and efficiency of simulating the model.
To simulate mutant cells, we make appropriate changes to some of the parameters, as specified in Table S3. To wipe out any biases introduced by the initial conditions, the simulation is run for a few cell cycles to reach a stable, repetitive sequence of cell cycle events. For a WT cell or a mutant cell that initiates DNA replication within 300 min in our simulation, we plot results of the fifth cycle. For mutant cells that do not initiate DNA replication by 300 min, we plot simulated results of 0-300 min.
We qualitatively summarize the regulatory network of our model in Figure 2, the key assumptions in Methods S1, and the roles of individual regulators in Table S5.

Reaction-diffusion equations and compartment-based simulation
To model the spatiotemporal dynamics of a generic protein, we propose the reaction-diffusion equation where C(x,t) is the concentration of the protein and D is its diffusion coefficient. Chemical Reaction Rates = rate of synthesis -rate of degradation -rate of dilution G molecular interaction rates. By dividing the spatial domain into N compartments of length l = L/N, we convert this partial differential equation (PDE) into a set of N ordinary differential equations (ODEs) for C i (t), i = 1, ..., N. For a ten-compartment model,

8
> > > > > > > > < > > > > > > > > : The first compartment represents the new pole of a Caulobacter cell and the tenth compartment is the old pole. The number of compartments N influences the computational complexity of the model. For initially exploring the model and searching its parameter space, we used a simpler four-compartment model, described in Methods S3. After the model was initially verified and the parameters estimated, we extended the four-compartment scheme to ten compartments to explore the spatiotemporal dynamics of proteins and provide more accurate simulations, as presented in the main text.
To take into account the fact that a Caulobacter cell is growing as a result of new cell wall materials being added uniformly along the long axis, 68  Localization of scaffolding proteins based on an activator-substrate depletion mechanism for turing patterns In 1952, Alan Turing proposed a chemical reaction-diffusion model for the spontaneous generation of spatial patterns in an otherwise homogeneous reaction vessel. 71 Since then, many authors have investigated the criteria for pattern formation in activator-inhibitor (AI) and activator-substrate depletion (A-SD) mechanisms. [72][73][74] In short, the 'activator' must be produced autocatalytically, and the 'inhibitor' (or the 'substrate') must diffuse much faster than the activator. Furthermore, for the A-SD mechanism, the rate of the conversion of substrate into activator must be proportional to [

Chromosome replication, methylation, and cell division
In addition to CtrA, there are other proteins controlling the initiation of chromosome replication. One key regulator is DnaA, which binds to Cori to initiate replication. 15 During replication, a fully-methylated chromosome becomes a pair of hemi-methylated chromosomes. CcrM, a DNA methyltransferase that is activated as replication is completed, remethylates promoters at specific methylation sites. 79 In this study, we do not explicitly model the control of replication by GcrA and DnaA as well as CtrA and the methylation of chromosomes by CcrM because these events are not closely coupled to spatial regulations, although they are vital to temporal checkpoints during the cell cycle. We assume that DNA replication is initiated (time = T ini ) when CtrA$P drops below a threshold, Q 20 (Table S2). The replication period (S phase) of WT cells is approximately 90 min, 80 so we set the termination time T term = T ini + 90 min. For promoters with methylation sites (ctrA, pleC, perP, and podJ), we use the factor ((1Àe)$S podJ + e) to model the effect of methylation (see yellow rectangles with 'Meth' in Figure 2): S = 0: fully-methylated promoters with lower rate of transcription. S = 1: hemi-methylated promoters.
where e is a small number indicating the suppressed expression of genes when fully methylated (see Methods S2). Because bacterial genes are replicated in the linear order in which they are located on the chromosome, 81 we set S = 1 at the time when the replication fork passes a gene, based on its genome coordinates. 82,66 When replication terminates, S is set back to 0. The switching parameters for these events are listed in Table S2.
Z-ring closure is not modelled in this study. Instead, we assume that compartmentalization is completed (Z-ring closes) 5 min after DNA replication terminates. The swarmer and stalked cells separate (i.e., the simulated cell cycle completes) about 25 min after Z-ring closure. The cycle timeline, shown in Figure S4, is estimated from experimental measurements for wild-type Caulobacter strain (CB15N) grown in M2G or PYE at 28 C. 80,83 For the case of mutant cells, where the level of CtrA$P is either too low or too high to trigger DNA replication (Table S2), we enforce the following operations: 1. if the average [CtrA$P] is lower than the threshold Q at t = 15 min, then set T ini = 15 min; 2. if the average [CtrA$P] has never been lower than Q by t = 300 min, then terminate the simulation at 300 min.
ll OPEN ACCESS calculates a spatial fitting cost (for WT simulation) that includes penalties related to the spatial dynamics of PopZ and DivK. SP j is the spatial penalty of protein j, and Weight j denotes the contribution of SP j to the score of the second objective function (Weight PopZ = 100; Weight DivK = 1). The spatial penalties are defined as: