Assumptions of the model
A number of simplifying assumptions have been made in formulating the model.
1. We propose to model, at this stage, only the average behavior of cells and do not address naturally occurring fluctuations in cell cycle progression.
2. The rise of DivK~P in stalked compartments after constriction of the Z-ring is a necessary but not sufficient condition for CtrA degradation. In our coarse-grained model of CtrA proteolysis, we use DivK~P as a signal for starting rapid degradation of CtrA. In other words, DivK~P determines 'when' the degradation of CtrA is turned on, but the 'how' (the machinery that degrades CtrA, involving RcdA, CpdR, and ClpXP) is assumed to be there when needed and is not modeled at present.
3. CtrA is activated by phosphorylation (by kinases, CckA and DivL), and a complete model of the Caulobacter cell cycle should take this into account. Unfortunately, little is known about the phosphorylation and dephosphorylation of CtrA and how these processes are temporally regulated. During the division cycle of wild-type cells, the levels of CtrA and CtrA~P rise and fall together (Ausmees and Jacobs-Wagner, 2003; Jacobs et al., 2003), so we need not distinguish between the two forms. Therefore, in the current model, we keep track of CtrA synthesis and degradation only, assuming that CtrA~P is a fixed fraction of total CtrA. This assumption, though a great oversimplification, is harmless enough for most of the mutants we consider in this paper. But it seems to cause serious problems for exactly those mutants (ctrAop, ctrAΔ3, ctrAD51E and ctrAD51EΔ3 in wild-type background) that interfere with normal synthesis, degradation or activation of CtrA (Domian et al., 1997). Later versions of the model will have to include CtrA~P as a variable, when we have a better of idea of the mechanisms controlling CtrA phosphorylation.
It is known that DivK~P promotes the proteolysis of both CckA and CtrA~P. Hence, DivK~P works to eliminate CtrA~P activity both by shutting down its supply and by destroying the existing protein. Our coarse-grained model lumps these two effects together as a pathway for removing active CtrA.
4. The dnaA locus is very close to the origin site (Cori) (Nierman et al., 2001). Within its promoter, potential CtrA and DnaA boxes and methylation sites exist for regulating its expression (Marczynski and Shapiro, 2002; Zweiger et al., 1994; Zweiger and Shapiro, 1994). GcrA is a repressor for dnaA expression (Holtzendorff et al., 2004), and CtrA seems to be an activator (Laub et al., 2002). However, DnaA protein concentration varies very little during the Caulobacter cell cycle (Zweiger and Shapiro, 1994). Although we include the regulatory signals in the model, they do not affect much the dynamics of a stalked cell because DnaA level is nearly constant throughout the cell cycle due to DnaA's long half-life.
5. Initiation of DNA replication is triggered by the combined conditions of low CtrA, high DnaA, and fully methylated DNA origin site. In addition, initiation requires sufficient replication machinery, which is correlated to a high level of GcrA. We combine these regulatory effects into a single term. We assume that once initiation of DNA replication is successful, DNA elongation starts immediately. Elongation of new DNA strands is linear in time until it finishes, based on experimental data indicating that the speed of DNA replication in C. crescentus is almost constant (Dingwall and Shapiro, 1989).
6. Full constriction of the Z-ring requires accumulation and activation of a number of proteins, including FtsZ, FtsQ, FtsA and FtsW, some of which are stimulated by CtrA. To simplify the model, we use Fts as a combined component to relay the signal from CtrA to Z-ring constriction. The transition from Z-ring open (= 1) to fully constricted (= 0) is modeled as a Goldbeter-Koshland ultrasensitive switch (Goldbeter and Koshland, 1981).
7. We include the effects of DNA methylation on gene expression in our model because these effects mediate important feedback loops between DNA synthesis and the master regulatory proteins, and because DNA methylation can be a useful target for new drug development. In our model, the genes ccrM, dnaA, ctrA and fts as well as the origin of DNA replication are regulated by methylation.
Methylation plays a minor role in the regulation of GcrA production (Collier et al., 2006), so we disregard it in our model. We allow a modest contribution of DNA methylation to regulating the production of DnaA. ccrM gene expression is significantly affected by its methylation state (Stephens et al., 1995; Reisenauer and Shapiro, 2002). The activity of ctrA-P1 is known to depend on hemimethylation (Reisenauer and Shapiro, 2002), and the activity of ctrA-P2 seems to depend in some other way on DNA replication (Wortinger et al., 2000). For simplicity, we assume that both ctrA promoters are turned on by hemimethylation of the gene.
Among fts genes, the ftsZ promoter has a methylation site (Reisenauer et al., 1999; Reisenauer and Shapiro, 2002) but the ftsQ promoter does not (Wortinger et al., 2000). Scanning the ftsQ gene for the consensus sequence GANTC using the Regulatory Sequence Analysis Tools (http://rsat.ulb.ac.be/rsat/), we found a GAGTC segment in the coding sequence, suggesting that the ftsQ gene might also be affected by methylation. Since our 'Fts' variable is a combination of Fts proteins, we conclude that our fts gene should be regulated by methylation.
The effects of methylation on gene promoters and Cori are described by probabilities to be methylated or hemimethylated during the cell cycle. The probabilities (h.. variables) are in turn controlled by the progression of DNA replication and by the activity of CcrM (Zweiger et al., 1994; Reisenauer et al., 1999).
8. ccrM transcription is tightly regulated by CtrA protein, but accumulation of CcrM protein shows a noticeable delay from the transcriptional activation of its gene (Grunenfelder et al., 2001), resulting in delayed activation of DNA methylation (Stephens et al., 1995). This delay is mimicked in our model by an intermediate variable I in the CtrA-to-CcrM pathway.
9. We recognize the importance of spatial controls in the Caulobacter cell cycle. However, at this stage, we are trying to model the stalked cell cycle as far as possible without explicitly tracking the spatial localization of regulatory proteins. That would require a more sophisticated mathematical framework and is planned for the next stage of the model. As the result of this simplification, our model makes no distinction between the stalked and swarmer parts of the predivisional cell. Right after compartmentation and before cytokinesis, we keep track of proteins in the stalked cell compartment only. At this stage, the distinction between swarmer and stalked cells is made by the phosphorylation state of DivK (being completely phosphorylated in stalked compartment).
10. We assume cells grow steadily in time, with a mass-doubling time of about 120 min and with the accumulated material shed at each division in the swarmer cell. In the present model there is no coupling between cell growth and division, as in our models of eukaryotic cell proliferation (Csikasz-Nagy et al., 2006). Hence, there is no need for us to keep track of cell size, except to notice that, if cell division is delayed or blocked, then the stalked cell will grow longer than normal and eventually be described as a filamentous morphology.
webmaster