SL, PB, and JJT conceived, designed, and performed the research and wrote the paper. BS revised the manuscript for important intellectual content. PB, BS, and JJT obtained funding for this research.
The authors have declared that no competing interests exist.
Progression of a cell through the division cycle is tightly controlled at different steps to ensure the integrity of genome replication and partitioning to daughter cells. From published experimental evidence, we propose a molecular mechanism for control of the cell division cycle in
The cell cycle is the sequence of events by which a growing cell replicates all its components and divides them more or less evenly between two daughter cells. The timing and spatial organization of these events are controlled by gene–protein interaction networks of great complexity. A challenge for computational biology is to build realistic, accurate, predictive mathematical models of these control systems in a variety of organisms, both eukaryotes and prokaryotes. To this end, we present a model of a portion of the molecular network controlling DNA synthesis, cell cycle–related gene expression, DNA methylation, and cell division in stalked cells of the α-proteobacterium
The events of the cell division cycle must be carried out in a coordinated fashion. Coordination is maintained by underlying molecular regulatory systems of great complexity. Intensive studies of these protein interaction networks by mathematical modeling have assisted our understanding of cell cycle regulation in yeasts [
Although progress in understanding cell cycle regulation in bacteria has lagged behind eukaryotes, the recent discovery of master regulatory proteins, CtrA and GcrA, in
At this stage of the model, the regulation of CtrA proteolysis has been incorporated in a simplistic way, concentrating on the phosphorylation of DivK in the stalked cell compartment at cell division and ignoring (for now) the roles of other proteins, such as RcdA, CpdR, and ClpXP [
Three cell cycle phases can be distinguished in a swarmer cell: a growth and differentiation (G1) phase that lasts approximately 30 min, a DNA synthesis (S) phase that takes approximately 90 min, and a cell division (G2/M) phase, lasting approximately 30 min, that culminates in the separation of mother (stalked) and daughter (swarmer) cells. The stalked cell cycle lacks G1 phase. The color scheme denotes protein variations through the cell division cycle: GcrA (blue), CtrA (red), DnaA (green). The θ-like structure denotes replicating DNA. The ring in the middle of the cell indicates Z-ring formation and constriction, leading to cell separation (cytokinesis). PD, predivisional.
Regulation of genes by CtrA is shown in red, by GcrA in blue, by DnaA in green, and by CcrM in cyan. The cyan stars indicate those genes whose transcription is regulated by DNA methylation. The CtrA-driven up-regulation of the
CtrA is present at a high level in swarmer cells, whereas in stalked cells, it changes from low to high level during the cell cycle [
Expression of
Proteolysis of CtrA (and CtrA∼P) is significantly accelerated by the phosphorylated form of DivK protein, DivK∼P, via the ClpXP protease pathway [
CtrA is active when phosphorylated [
GcrA is an activator of components of the replisome and of the segregation machinery [
DNA replication proceeds in three phases: initiation, elongation, and termination. The origin of DNA replication (Cori) in
Several cell cycle–related genes (
In
The multicomponent Z-ring organelle, which forms and constricts at the mid-cell plane, plays an important role in compartmentation of the predivisional cell and its subsequent division [
The Fts proteins (FtsZ, FtsQ, FtsA, and FtsW) have been identified as crucial elements of the Z-ring.
Using
The double-stranded closed curve at the top represents DNA. Cori is the origin of DNA replication and
Equations of the Model
To simulate the molecular regulation of a wild-type stalked-cell division cycle, we solve the equations in
Basal Parameter Values for the Wild-Type Stalked-Cell Division Cycle
Initial Values of Model Variables, for a Newborn, Wild-Type Stalked Cell
Here and in subsequent figures, the simulation begins when the initiation of DNA replication has completed. Three cell cycles are presented.
(A) Here and in subsequent figures, the scale for [Ini] is on the left and the scales for [Elong] and [DNA] are on the right. DNA replication (green curve) takes 90 min, as observed [
(B) The methylation states of Cori and of three genes. As replication starts, Cori(red) is hemimethylated (
(C) Early in the cycle, GcrA (blue) is increasing and triggers production of CtrA (red). When CtrA is high, it represses synthesis of GcrA and activates its own degradation by up-regulating DivK∼P (green).
(D) When a cell enters the predivisional phase, high CtrA activates the expression of
The main physiological events of the division cycle can be traced back to characteristic signatures of protein expression, as described in the Introduction. The division cycle starts with initiation of DNA replication (
As DNA synthesis progresses, certain genetic loci become hemimethylated in order along the chromosome (
High CtrA down-regulates
In
(A) The simulated total DNA (green), Elongation (blue), and Initiation (red) variables. The experimental data for total DNA are taken from
(B) Curves are simulated probabilities of hemimethylated states of Cori and three genes. The appearance of hemimethylated gene sites in our simulation reflects the nearly linear growth of overall DNA-hemimethylation observed experimentally (data points [black] from
(C) Time courses of GcrA (blue) and CtrA (red) are compared against experimental data from
(D) The generalized Fts protein time course (blue) is compared to the measured profile of FtsQ from
(E) The relative activities of
The only serious objection that might be raised is to our simulation of DivK∼P (
The phenotypes of mutant cells provide crucial hints for deciphering the biochemical circuitry underlying any aspect of cell physiology. A mathematical model must be consistent with known phenotypes of relevant mutants. To make this test, we simulate cell cycle mutants of
Altered Parameter Values for Mutant Simulations.
This mutant is obtained by deleting the normal
Experiments show that the CtrA depletion strain,
When the temperature-sensitive strain,
A number of different mutations can cause increased levels of CtrA in cells: by constitutive expression of the gene, by producing a poorly degraded form of CtrA, by producing a constitutively active form of CtrA (not needing to be phosphorylated), or by combinations of these mutational strategies [
Overproduction of CtrA does not interfere with normal cell cycling: the
When the genomic copy of
The vertical column of open circles here and on subsequent figures indicates the time at which the mutation is introduced.
In the GcrA depletion strain,
When the
In the
Similarly, when
Deletion of
The model predicts that cells depleted of DivK may undergo four to five division cycles before dying, because (in the model) DivK protein is only slowly degraded, and its only job is to trigger degradation of CtrA. To the extent that other processes may render DivK nonfunctional and that DivK may have other essential roles in the cell [
A point mutation (D90G) of
If the mutation is introduced in our simulation before DNA replication starts, its start will be delayed because it takes longer for CtrA to be reduced (by the background degradation rate alone) to the replication-permissible level (
If
In this family of
In experiments, a
In [
In a CcrM-depleted strain (Δ
In the CcrM overproduction mutant, the wild-type
Lon protease is required for CcrM degradation in living cells [
Elongation of DNA during S phase ceases in cells depleted of DnaC [
Based on our simulations (
No mutant overproducing GcrA has been reported in the literature. Our model predicts that small overexpression of
Phenotypes of
We propose (
These assumptions are formulated as a mathematical model (
Our present model is based heavily on an earlier conjecture [
Finally, most of division-control proteins (such as CtrA, DivK, CcrM, FtsZ, and FtsQ) are conserved among α-proteobacteria [
To understand the molecular logic of cell cycle regulation in
Our model includes:
Seven proteins: DnaA, GcrA, CtrA, CcrM, DivK (inactive), and DivK∼P (phosphorylated, active form), and a “representative” Fts protein.
Two phenomenological variables,
The progression of DNA replication (including initiation, elongation, and termination) and its methylation (including probabilities of hemimethylation of
Accordingly, our mathematical model consists of 16 nonlinear differential equations presented in
A common trend in developing complex models in molecular cell biology is to start from a simple coarse-grained (“phenomenological”) model and then refine and expand it step by step (as data become available) into an increasingly more comprehensive model. (A good example is the progression of models of the budding yeast cell cycle [
First, we propose to model, at this stage, only the average behavior of cells and do not address naturally occurring fluctuations in cell cycle progression.
Second, 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.
Third, CtrA is activated by phosphorylation (by kinases CckA and DivL), and a complete model of the
It is known that DivK∼P promotes the proteolysis of CtrA∼P [
Fourth, the
Fifth, 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
Sixth, 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 [
Seventh, 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
Methylation plays a minor role in the regulation of GcrA production [
Among
The effects of methylation on gene promoters and Cori are described by probabilities to be methylated or hemimethylated during the cell cycle. The probabilities (
Eighth,
Ninth, we recognize the importance of spatial controls in the
Tenth, 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 [
Parameter values for our model (
The phenotypes of relevant mutants were collected from the literature. To simulate each mutant, we use exactly the same equations (
Equations of the model were solved numerically with Matlab 2006a (The MathWorks). Machine-readable files for reproducing our simulations are made available in Text S1 and on our Web site (
The vertical column of open circles here and on subsequent figures indicates the time at which the mutation is introduced. WT, wild-type.
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The authors thank our many colleagues in the Tyson and Sobral research groups for valuable discussions of the model, and especially Harley McAdams of Stanford University for bringing this problem to our attention and educating us on many aspects of