A new mathematical approach predicts individual cell growth behavior using bacterial population information

A theoretical methodology has been developed for studying the growth kinetics of bacterial cells. It utilizes the steady-state cell length distribution in a bacterial population to predict the dependency of growth and division rates on cell length and age. The mathematical model has been applied to the analysis of two bacterial populations, a wild-type strain of Bacillus subtilis, and a minicell-producing strain that carries the divIVB1 mutation. The results show that our model describes the wild-type population very well and that the assumptions typically used in traditional methods are unrealistic. In the case of the minicell-producing mutant we find evidence that the rate of cell division must be a function not only of cell size but also of cell age.     

Cell division suppression in the Bacillus subtilis div IC-A1 minicell-producing mutant.

Growth and division patterns of Bacillus subtilis wild-type (div IV-A1+) and minicell-producing mutant (div IV-A1) clones were studied after spore germination during microcolony development in chambers that facilitate continuous observation with a phase contrast microscope. Data obtained from 13 div IV-A1+ clones were used to derive the equation DE equals [(mum minus 17.6)/8.8], which expresses the relationship of cell divisions present in clones of various lengths. This equation was used to determine the number of divisions expected in div IV-A1 clones if the mutant clones were able to divide as often as wild-type clones. The observed number of divisions present in mutant clones was found to be only 25.27% of the number expected on the basis of this equation. Although individual div IV-A1 clones varied in the percentage of division equivalents expressed, there appeared to be no correlation between the overall clone growth rate and the number of divisions expressed. Culturing div IV-A1+ and div IV-A1 clones together in the same growth chamber revealed that there were no diffusible interactions influencing the division phenotypes of either mutant or wild-type cells. At later stages of growth, mixed microcolonies containing cells of both genotypes were formed. A length analysis of individual cells in these populations indicated that the relative division suppression of mutant compared with wild-type cells characteristic of the initial stages of clone development was maintained. It is likely, therefore, that the excessive length of minicell-producing cells (div IV-A1) is a reflection primarily of division suppression in the mutant and not simply of mislocation of division along cell length.     

Altered membrane proteins in a minicell-producing mutant of Bacillus subtilis.

The Bacillus subtilis minicell-producing mutant divIV-B1 has a membrane protein profile that is strikingly different from that of the other minicell-producing mutant, divIV-A1, or that of wild-type strain CU403.     

Bacterial persistence: a model of survival in changing environments

The persistence phenotype is an epigenetic trait exhibited by a subpopulation of bacteria, characterized by slow growth coupled with an ability to survive antibiotic treatment. The phenotype is acquired via a spontaneous, reversible switch between normal and persister cells. These observations suggest that clonal bacterial populations may use persister cells, whose slow division rate under growth conditions leads to lower population fitness, as an "insurance policy" against antibiotic encounters. We present a model of Escherichia coli persistence, and using experimentally derived parameters for both wild type and a mutant strain (hipQ) with markedly different switching rates, we show how fitness loss due to slow persister growth pays off as a risk-reducing strategy. We demonstrate that wild-type persistence is suited for environments in which antibiotic stress is a rare event. The optimal rate of switching between normal and persister cells is found to depend strongly on the frequency of environmental changes and only weakly on the selective pressures of any given environment. In contrast to typical examples of adaptations to features of a single environment, persistence appears to constitute an adaptation that is tuned to the distribution of environmental change.     

Cell Division in Escherichia coli: Analysis of Double Mutants for Filamentation and Abnormal Septation of Deoxyribonucleic Acid-Less Cells

The link between chromosome termination, initiation of cell division, and choice of division sites was studied in Escherichia coli by preparing double mutants. Hybrid mutants containing div52-ts, a cell division initiation mutation, and min, mutations which affect the choice of division sites resulting in the septation of minicells, were characterized. The mutants produced minicells and normal cells coordinately under all conditions studied, although the fraction of minicells is half that of the parental minicell strain. The mutant gradually stopped dividing at both the median and minicell septation sites when transferred from 30 to 41 C in rich medium. A synchronous cell division of filaments was induced 15 min after addition of chloramphenicol to the medium, even at 41 C. Divisions were observed at both normal and minicell sites. These results indicate that div52-ts and min functions share a common step in a cell division pathway. A double mutant containing div52-ts and div27-ts, a dnaB mutant which divides in the absence of DNA synthesis, was characterized. The mutant continues to divide after a shift to the high temperature, although at a reduced rate. The behavior of this hybrid mutant suggests a hypothesis that the chromosome termination signal and div52-ts division initiation signal act on a single membrane site which is altered in div27-ts strains.     

Clonal Analysis of Cell Division in the Bacillus subtilis div IV-B1 Minicell-Producing Mutant

Spores of the Bacillus subtilis minicell-producing mutant div IV-B1 were germinated and grown to microcolonies in chambers which facilitate continuous observation of the developing clones with a phase-contrast microscope. Time lapse photographs were taken of 46 clones, covering the period from the beginning of outgrowth until at least two rounds of cell division had been completed. Cell lineages were constructed from contour length measurements of the photographs. These data include cell lengths, division site locations, and cell numbers in clones of various ages. From these data we have determined that the probability of a minicell being produced at any division by the div IV-B1 mutant is 0.31. The location of the abnormal division site which generates the first minicell produced in the outgrowing clone appears to be random with respect to the existing cell poles. In contrast, the location of the second abnormal division site, and hence the second minicell, is not random but rather occurs preferentially in proximity to the first minicell. This clustering of abnormal events suggests that division site location is related to pole age (generations), although other influences on minicell clustering cannot be ruled out at present.     

The effect of heterogeneity on invasion in spatial epidemics: from theory to experimental evidence in a model system

Heterogeneity in host populations is an important factor affecting the ability of a pathogen to invade, yet the quantitative investigation of its effects on epidemic spread is still an open problem. In this paper, we test recent theoretical results, which extend the established "percolation paradigm" to the spread of a pathogen in discrete heterogeneous host populations. In particular, we test the hypothesis that the probability of epidemic invasion decreases when host heterogeneity is increased. We use replicated experimental microcosms, in which the ubiquitous pathogenic fungus Rhizoctonia solani grows through a population of discrete nutrient sites on a lattice, with nutrient sites representing hosts. The degree of host heterogeneity within different populations is adjusted by changing the proportion and the nutrient concentration of nutrient sites. The experimental data are analysed via Bayesian inference methods, estimating pathogen transmission parameters for each individual population. We find a significant, negative correlation between heterogeneity and the probability of pathogen invasion, thereby validating the theory. The value of the correlation is also in remarkably good agreement with the theoretical predictions. We briefly discuss how our results can be exploited in the design and implementation of disease control strategies.     

Mutations in cell division proteins FtsZ and FtsA inhibit φX174 protein-E-mediated lysis of Escherichia coli

Electron microscopic studies emphasized that the protein-E-specific transmembrane tunnel structure, which permeabilizes Escherichia coli, is not randomly distributed over the cell envelope but is restricted to areas of potential division sites. These sites were located predominantly in the middle of the cell, but approximately one-third of these structures are found at the polar sites. Therefore, E. coli mutant strains with defects in cell division components were tested for their sensitivity to protein-E-mediated lysis. The ftsZ84 and the ftsA12 cell division mutant strains of E. coli were tolerant to protein-E-mediated lysis, whereas the ftsA3 mutant strain was lysed by protein E under conditions nonpermissive for division. The protein-E-tolerant phenotype of ftsZ84 and ftsA12 and the lysis-sensitive phenotype of other components of the septosome (e.g., ftsA3, ftsQ, and ftsI) suggest that initiation of cell division – rather than specific functions of cell division – plays an essential role in protein-E-mediated lysis. SulA-overproducing cells had a lysis-positive phenotype, the ring structure – but not the GTPase function - of FtsZ was impaired. Received: 14 April 1998 / Accepted: 9 June 1998     

Effect of cis-platinum(II)diamminodichloride on cell division of Hyphomicrobium and Caulobacter.

Low concentrations of the radiomimetic agent cis-platinum(II)diamminodichloride (PDD) inhibited cell division in Caulobacter crescentus (0.1 mug/ml) and Hyphomicrobium sp. strain B-522 (1.0 mug/ml) without altering the length of prosthecae. After exposure, cells of C. crescentus appeared as long filaments, whereas only the bud portion of Hyphomicrobium underwent elongation. PDD-treated cells of both species were multinucleated. After the removal of PDD by washing, filaments of C. crescentus fragmented unequally and then normal growth resumed. In Hyphomicrobium (where division involves release of swarmer cells that arise as buds on the distal ends of hyphae), potential septation sites formed in the presence of PDD remained inactive after washing. Reinitiation of cell division in this species was dependent upon the synthesis of new hyphae that could arise from either end of the elongated bud. This finding suggests that the PDD-induced lesion at a given septation site is irreversible and, upon removal of this compound, alternate sites must be synthesized for the subsequent occurrence of cell division.     

Simulating mouse mammary gland development: cell ageing and its relation to stem and progenitor activity

BACKGROUND: Somatic stem and progenitor cell division is likely to be an important determinant of tumor development. Each division is accompanied by a risk of fixing genetic mutations, and/or generating innately immortal cells that escape normal physiological controls. AIM: Using biological information, we aimed to devise a theoretical model for mammary gland development that described the effect of various stem/progenitor cells activities on the demographics of adult mammary epithelial cell populations. RESULTS: We found that mammary ductal trees should develop in juvenile mice despite widely variant levels of activity in the progenitor compartment. Sequestration (inactivation) of progenitor cells dramatically affected the aging-maturation of the population without affecting the total regenerative capacity of the gland. Our results showed that if stem and progenitor cells can be demonstrated in glands regenerated by serial transplantation, they originated in a canonical primary stem cell (providing a functional definition of mammary stem cells). Finally, when the probability of symmetric division of stem cells increased above a threshold, the mammary epithelial population overall was immortal during serial transplantation. CONCLUSIONS: This model provides, (1) a theoretical framework for testing whether the phenotypes of genetically modified mice (many of which are breast cancer models) derive from changes of stem and progenitor activity, and (2) a means to evaluate the resolving power of functional assays of regenerative capacity in mammary epithelial cell populations.     

A model for an early stage of tomato fruit development: cell multiplication and cessation of the cell proliferative activity

Changes in cell number during the early period of tomato fruit development were analysed by means of a deterministic model of cell multiplication. The period commenced at the seed stage with one theoretical cell undergoing intensive cell division, and ended when the cell number became nearly constant. The model takes into consideration the proliferative activity of the fruit cell population which, a few days before flower anthesis, begins to decrease progressively after each mitotic cycle. Model parameters, namely the time at which proliferative activity diminishes, its rate of decrease and the length of the cell cycle, were estimated by fitting the model to observed cell population dynamics in tomato fruits growing in three different positions on the truss. It is hypothesized that the molecular mechanism responsible for the cessation of mitosis in growing fruits is associated with shortening of telomeric ends of nuclear DNA, as suggested previously for other growing cell populations.     

Mathematical Determination of Cell Population Doubling Times for Multiple Cell Lines

Cell cycle times are vital parameters in cancer research, and short cell cycle times are often related to poor survival of cancer patients. A method for experimental estimation of cell cycle times, or doubling times of cultured cancer cell populations, based on addition of paclitaxel (an inhibitor of cell division) has been proposed in literature. We use a mathematical model to investigate relationships between essential parameters of the cell division cycle following inhibition of cell division. The reduction in the number of cells engaged in DNA replication reaches a plateau as the concentration of paclitaxel is increased; this can be determined experimentally. From our model we have derived a plateau log reduction formula for proliferating cells and established that there are linear relationships between the plateau log reduction values and the reciprocal of doubling times (i.e. growth rates of the populations). We have therefore provided theoretical justification of an important experimental technique to determine cell doubling times. Furthermore, we have applied Monte Carlo experiments to justify the suggested linear relationships used to estimate doubling time from 5-day cell culture assays. We show that our results are applicable to cancer cell populations with cell loss present.     

FtsZ regulates frequency of cell division in Escherichia coli.

Cell division is regulated so that it occurs only once per cell cycle. In Escherichia coli, a rod-shaped bacterium, division normally takes place at the center of the long axis of the cell; however, in the minicell mutant, division can also take place at the cell pole. Such divisions take place at the expense of normal divisions, resulting in an overall increase in nucleated cell length. We report here that increasing the level of FtsZ can completely suppress the cell length of the minicell mutant by increasing the frequency at which cell division events take place. This result suggests that the level of FtsZ controls the frequency of cell division in E. coli.     

Mutation in the structural gene for release factor 1 (RF-1) of Salmonella typhimurium inhibits cell division.

A temperature-sensitive mutant of Salmonella typhimurium LT2 was isolated. At the nonpermissive temperature cell division stopped and multinucleated filaments were formed. DNA, RNA, or protein synthesis was not affected until after about two generations. Different physiological conditions, such as anaerobiosis and different growth media, suppress the division deficiency at high temperatures. Certain mutations causing a reduced polypeptide chain elongation rate also suppress the division deficiency. The mutation is recessive and shown to be in the structural gene for release factor I (prfA). DNA sequencing of both the wild-type (prfA+) and mutant (prfA101) allele revealed a GC-to-AT transition in codon 168. Like other known prfA mutants, prfA101 can suppress amber mutations. The division defect in the prfA101 mutant strain could not be suppressed by overexpression of the ftsQAZ operon. Moreover, at the nonpermissive temperature the mutant shows a normal heat shock and SOS response and has a normal ppGpp level. We conclude that the prfA101-mediated defect in cell division is not directed through any of these metabolic pathways, which are all known to affect cell division. We speculate that the altered release factor I induces aberrant synthesis of an unidentified protein(s) involved in the elaborate process of septation.     

Exploring intracellular space: function of the Min system in round-shaped Escherichia coli

The MinCDE proteins help to select cell division sites in normal cylindrical Escherichia coli by oscillating along the long axis, preventing unwanted polar divisions. To determine how the Min system might function in cells with multiple potential division planes, we investigated its role in a round-cell rodA mutant. Round cells lacking MinCDE were viable, but growth, morphology and positioning of cell division sites were abnormal relative to Min+ cells. In round cells with a long axis, such as those undergoing cell division, green fluorescent protein (GFP) fusions to MinD almost always oscillated parallel to the long axis. However, perfect spheres or irregularly shaped cells exhibited MinD movement to and from multiple sites on the cell surface. A MinE-GFP fusion exhibited similar behavior. These results indicate that the Min proteins can potentially localize anywhere in the cell but tend to move a certain maximum distance from their previous assembly site, thus favoring movement along the cell's long axis. A new model for the spatial control of division planes by the Min system in round cells is proposed.     

Steady-State growth of budding yeast populations in well-mixed continous-flow microbial reactors

A population-balance mathematical model of microbial growth in a flow reactor is formulated which incorporates an asymmetric-division, budding-cycle model of coordinated cell and nuclear division cycles for the budding yeast Saccharomyces cerevisiae. Analytical solutions are obtained for limiting nutrient and cell-number concentrations in the reactor as functions of basic cell cycle parameters. Frequency functions for cell mass and DNA content in the resident yeast population are also derived under different assumptions concerning cell mass and DNA synthesis and bud scar accumulation. These results, which correspond to experimentally observable medium and population variables, provide new bases for evaluating budding-yeast-cell cycle models and for deducing kinetics of mass and DNA synthesis in single cells growing in steady-state, asynchronous populations.     

Continuous cultivation of fission yeast: Analysis of single-cell protein synthesis kinetics

A fundamental problem in microbial reactor analysis is identification of the relationship between environment and individual cell metabolic activity. Population balance equations provide a link between experimental measurements of composition frequency functions in microbial populations on the one hand and macromolecular synthesis kinetics and cell division control parameters for single cells on the other. Flow microfluorometry measurements of frequency functions for single-cell protein content in Schizosaccharomyces pombe in balanced exponential growth have been analyzed by two different methods. One approach utilizes the integrated form of the population balance equation known as the Collins-Richmond equation, and the other method involves optimization of parameters in assumed kinetic and cell division functional forms in order to best fit measured frequency functions with corresponding model solutions. Both data interpretation techniques indicate that rates of protein synthesis increase most in small protein content cells as the population specific growth rate increases, leading to parabolic single-cell protein synthesis kinetics at large specific growth rates. Utilization of frequency function data for an asynchronous population is shown in this case to be a far more sensitive method for determination of single-cell kinetics than is monitoring the metabolic dynamics of a single cell or, equivalently, synchronous culture analyses.     

Cell growth and division in Escherichia coli: a common genetic control involved in cell division and minicell formation

Escherichia coli Div 124(ts) is a conditional-lethal cell division mutant formed from a cross between a mutant that produces polar anucleated minicells and a temperature-sensitive cell division mutant affected in a stage of cross-wall synthesis. Under permissive growth temperature (30 C), Div 124(ts) grows and produces normal progeny cells and anucleated minicells from its polar ends. When transferred to nonpermissive growth temperature (42 C), growth and macromolecular synthesis continue, but cell division and minicell formation are inhibited. Growth at 42 C results in formation of filamentous cells showing some constrictions along the length of the filaments. Return of the filaments from 42 to 30 C results in cell division and minicell formation in association with the constrictions and other areas along the length of the filaments. This gives rise to a "necklace-type" array of cells and minicells. Recovery of cell division is observed after a lag and is followed by a burst in cell division and finally by a return to the normal growth characteristic of 30 C cultures. Recovery of cell division takes place in the presence of chloramphenicol or nalidixic acid when these are added at the time of shift from 42 to 30 C, and indicates that a division potential for filament fragmentation is accumulated while the cells are at 42 C. This division potential is used for the production of both minicells and cells of normal length. The conditional-lethal temperature sensitive mutation controls a step(s) in cross-wall synthesis common to cell division and minicell formation.     

A single-locus model of speciation

The crucial phase of speciation is argued to be the evolution of mating cross-incompatibility (prezygotic incompatibility) between the genotypes distinguishing the prospective species populations. Based on this idea, a single-locus model of speciation is presented, which is shown to be biologically plausible and may help to settle the controversy as to the biological significance of single-locus modes of speciation. The model involves three alleles, two of which characterize in homozygous state the prospective species populations and in heterozygous state their hybrids. The third allele represents a mutant which is equivalent to one of the first two alleles with the exception that it inhibits mating with carriers of the third allele. This third allele is fixed in one population and immigrates into a second population which contains the mutant inhibiting matings with members of the former population. Migration in the reverse direction does not occur. Proceeding from a widely applicable concept of fitness and mating preference it is shown that postzygotic incompatibility (hybrid or heterozygote disadvantage) alone suffices to trigger evolutionary replacement of the extant mating relations in the population receiving immigrants by any arbitrary degree of prezygotic incompatibility. This corroborates Wallace's hypothesis and emphasizes the potential biological relevance of speciation by reinforcement (parapatric speciation) at single gene loci.