Effect of the Min System on Timing of Cell Division in Escherichia coli

In Escherichia coli the Min protein system plays an important role in positioning the division site. We show that this system also has an effect on timing of cell division. We do this in a quantitative way by measuring the cell division waiting time (defined as time difference between appearance of a division site and the division event) and the Z-ring existence time. Both quantities are found to be different in WT and cells without functional Min system. We develop a series of theoretical models whose predictions are compared with the experimental findings. Continuous improvement leads to a final model that is able to explain all relevant experimental observations. In particular, it shows that the chromosome segregation defect caused by the absence of Min proteins has an important influence on timing of cell division. Our results indicate that the Min system affects the septum formation rate. In the absence of the Min proteins this rate is reduced, leading to the observed strongly randomized cell division events and the longer division waiting times.     

Single cell studies of the cell cycle and some models

Analysis of growth and division often involves measurements made on cell populations, which tend to average data. The value of single cell analysis needs to be appreciated, and models based on findings from single cells should be taken into greater consideration in our understanding of the way in which cell size and division are co-ordinated. Examples are given of some single cell analyses in mammalian cells, yeast and other microorganisms. There is also a short discussion on how far the results are in accord with simple models.     

Sloppy size control of the cell division cycle

In an asynchronous, exponentially proliferating cell culture there is a great deal of variability among individual cells in size at birth, size at division and generation time (= age at division). To account for this variability we assume that individual cells grow according to some given growth law and that, after reaching a minimum size, they divide with a certain probability (per unit time) which increases with increasing cell size. This model is called sloppy size control because cell division is assumed to be a random process with size-dependent probability. We derive general equations for the distribution of cell size at division, the distribution of generation time, and the correlations between generation times of closely related cells. Our theoretical results are compared in detail with experimental results (obtained by Miyata and coworkers) for cell division in fission yeast, Schizosaccharomyces pombe. The agreement between theory and experiment is superior to that found for any other simple models of the coordination of cell growth and division.     

Linking cell division to cell growth in a spatiotemporal model of the cell cycle

Cell division must be tightly coupled to cell growth in order to maintain cell size, yet the mechanisms linking these two processes are unclear. It is known that almost all proteins involved in cell division shuttle between cytoplasm and nucleus during the cell cycle; however, the implications of this process for cell cycle dynamics and its coupling to cell growth remains to be elucidated. We developed mathematical models of the cell cycle which incorporate protein translocation between cytoplasm and nucleus. We show that protein translocation between cytoplasm and nucleus not only modulates temporal cell cycle dynamics, but also provides a natural mechanism coupling cell division to cell growth. This coupling is mediated by the effect of cytoplasmic-to-nuclear size ratio on the activation threshold of critical cell cycle proteins, leading to the size-sensing checkpoint (sizer) and the size-independent clock (timer) observed in many cell cycle experiments.     

Videomicroscopic extraction of specific information on cell proliferation and migration in vitro

In vitro cell imaging is a useful exploratory tool for cell behavior monitoring with a wide range of applications in cell biology and pharmacology. Combined with appropriate image analysis techniques, this approach has been shown to provide useful information on the detection and dynamic analysis of cell events. In this context, numerous efforts have been focused on cell migration analysis. In contrast, the cell division process has been the subject of fewer investigations. The present work focuses on this latter aspect and shows that, in complement to cell migration data, interesting information related to cell division can be extracted from phase-contrast time-lapse image series, in particular cell division duration, which is not provided by standard cell assays using endpoint analyses. We illustrate our approach by analyzing the effects induced by two sigma-1 receptor ligands (haloperidol and 4-IBP) on the behavior of two glioma cell lines using two in vitro cell models, i.e., the low-density individual cell model and the high-density scratch wound model. This illustration also shows that the data provided by our approach are suggestive as to the mechanism of action of compounds, and are thus capable of informing the appropriate selection of further time-consuming and more expensive biological evaluations required to elucidate a mechanism.     

Epithelial topology

It is universally accepted that genetic control over basic aspects of cell and molecular biology is the primary organizing principle in development and homeostasis of living systems. However, instances do exist where important aspects of biological order arise without explicit genetic instruction, emerging instead from simple physical principles, stochastic processes, or the complex self-organizing interaction between random and seemingly unrelated parts. Being mostly resistant to direct genetic dissection, the analysis of such emergent processes falls into a grey area between mathematics, physics and molecular cell biology and therefore remains very poorly understood. We recently proposed a mathematical model predicting the emergence of a specific non-Gaussian distribution of polygonal cell shapes from the stochastic cell division process in epithelial cell sheets; this cell shape distribution appears to be conserved across a diverse set of animals and plants.1 The use of such topological models to study the process of cellular morphogenesis has a long history, starting almost a century ago, and many insights from those original works influence current experimental studies. Here we review current and past literature on this topic while exploring some new ideas on the origins and implications of topological order in proliferating epithelia.     

Pattern of unequal cell division and development in Caulobacter crescentus

The pattern of asymmetric division has been examined in Caulobacter crescentus (gram-negative aquatic bacteria) by determining the position of the “division site” on cells of different ages. Measurements of cell width and length at this site, which corresponds to the point of eventual cell separation, were made on electron micrographs of cells stained with phosphotungstic acid. The results show that (i) the division site is formed early in the cell cycle and it constitutes the first visible feature on the growing stalked cell to differentiate the incipient swarmer cell, (ii) the division site is located asymmetrically (closer to the swarmer pole than the stalked pole) on the dividing cell, (iii) its position relative to the stalked and swarmer poles does not change during the cell cycle, and (iv) division is consequently unequal, with the swarmer cell always smaller than the stalked cell. The implications of these findings for general models of unequal cell division and stem cell development are discussed.     

Effects of Chromosome Underreplication on Cell Division in Escherichia coli

The key processes of the bacterial cell cycle are controlled and coordinated to match cellular mass growth. We have studied the coordination between replication and cell division by using a temperature-controlled Escherichia coli intR1 strain. In this strain, the initiation time for chromosome replication can be displaced to later (underreplication) or earlier (overreplication) times in the cell cycle. We used underreplication conditions to study the response of cell division to a delayed initiation of replication. The bacteria were grown exponentially at 39°C (normal DNA/mass ratio) and shifted to 38 and 37°C. In the last two cases, new, stable, lower DNA/mass ratios were obtained. The rate of replication elongation was not affected under these conditions. At increasing degrees of underreplication, increasing proportions of the cells became elongated. Cell division took place in the middle in cells of normal size, whereas the longer cells divided at twice that size to produce one daughter cell of normal size and one three times as big. The elongated cells often produced one daughter cell lacking a chromosome; this was always the smallest daughter cells, and it was the size of a normal newborn cell. These results favor a model in which cell division takes place at only distinct cell sizes. Furthermore, the elongated cells had a lower probability of dividing than the cells of normal size, and they often contained more than two nucleoids. This suggests that for cell division to occur, not only must replication and nucleoid partitioning be completed, but also the DNA/mass ratio must be above a certain threshold value. Our data support the ideas that cell division has its own control system and that there is a checkpoint at which cell division may be abolished if previous key cell cycle processes have not run to completion.     

Aging and immortality in a cell proliferation model

We investigate a model of cell division in which the length of telomeres within a cell regulates its proliferative potential. At each division, telomeres undergo a systematic length decrease as well as a superimposed fluctuation due to exchange of telomere DNA between the two daughter cells. A cell becomes senescent when one or more of its telomeres become shorter than a critical length. We map this telomere dynamics onto a biased branching-diffusion process with an absorbing boundary condition whenever any telomere reaches the critical length. Using first-passage ideas, we find a phase transition between finite lifetime and immortality (infinite proliferation) of the cell population as a function of the influence of telomere shortening, fluctuations, and cell division.     

Cell lineage tree models of neurogenesis

The production of neurons to form the mammalian cortex, known as embryonic cortical neurogenesis, is a complex developmental process. Insight into the process of cell division during neurogenesis is provided by murine cortical cell lineage trees, recorded through experimental observation. Recurring patterns within cell lineage trees may be indicative of predetermined cell behaviour. The application of mathematical modelling to this process requires careful consideration and identification of the key features to be incorporated into the model. A biologically plausible stochastic model of evolution of cell lineage trees is developed, based on the most important known features of neurogenesis. Tractable means of measuring lineage tree shape are discussed. Symmetry is identified as a significant feature of shape and is measured using Colless's Index of Imbalance. Distributions of tree size and imbalance for large tree sizes are computed and results compared to experimental data. Several refinements to the model are investigated, when the cell division probabilities are weighted according to cell generation. Two models involving generation-dependent cell division probabilities produce imbalance distributions which are the most consistent with the available experimental results. The results indicate that a stochastic cell division mechanism is a plausible basis of mammalian neurogenesis.     

Regulation and SOS induction of division inhibition in Escherichia coli K12

Summary When Escherichia coli is subjected to treatments that damage DNA or perturb DNA replication considerable cell filamentation occurs. It has been postulated that this phenomenon is associated with the presence of a division inhibitor induced coordinately with the SOS functions. The role of this induction would be to delay septation during DNA repair to prevent the formation of DNAless cells. In this communication, we present evidence for such a division inhibitor based on the properties of a division mutant which is hyperactive in the septation delay. Cells of this mutant filament extensively after a nutritional shift-up, have drastically reduced colony-forming abilities on a rich medium but not on a minimal medium following treatment with ultraviolet radiation and, are deficient in the lysogenization of phage lambda; phenotypes which are characteristic of but expressed to a much lower extent in another type of division mutant called lon. Cells harboring the division mutation plus either one of the lexA mutant alleles, spr-51 or tsl-1, are filamentous suggesting that they are permanently derepressed for division inhibition. These results are in agreement with models that assign the regulation of cell division to a division inhibitor which is regulated by the lexA repressor protein.     

Analysis and Simulation of Division- and Label-Structured Population Models

In most biological studies and processes, cell proliferation and population dynamics play an essential role. Due to this ubiquity, a multitude of mathematical models has been developed to describe these processes. While the simplest models only consider the size of the overall populations, others take division numbers and labeling of the cells into account. In this work, we present a modeling and computational framework for proliferating cell populations undergoing symmetric cell division, which incorporates both the discrete division number and continuous label dynamics. Thus, it allows for the consideration of division number-dependent parameters as well as the direct comparison of the model prediction with labeling experiments, e.g., performed with Carboxyfluorescein succinimidyl ester (CFSE), and can be shown to be a generalization of most existing models used to describe these data. We prove that under mild assumptions the resulting system of coupled partial differential equations (PDEs) can be decomposed into a system of ordinary differential equations (ODEs) and a set of decoupled PDEs, which drastically reduces the computational effort for simulating the model. Furthermore, the PDEs are solved analytically and the ODE system is truncated, which allows for the prediction of the label distribution of complex systems using a low-dimensional system of ODEs. In addition to modeling the label dynamics, we link the label-induced fluorescence to the measure fluorescence which includes autofluorescence. Furthermore, we provide an analytical approximation for the resulting numerically challenging convolution integral. This is illustrated by modeling and simulating a proliferating population with division number-dependent proliferation rate.     

Theodor Boveri and the origin of malignant tumours

As long ago as 1914, Theodor Boveri suggested that there is an inhibitory mechanism in every normal cell that prevents the process of cell division until the inhibition has been overcome by a special stimulus. From his work on abnormal mitoses in the eggs of echinoderms, Boveri also suggested that the inhibitor resided in the chromosomes. The relevance of Boveri's ideas to modern cancer research is discussed in this Retrospective article.     

Individual-based modeling of phytoplankton: evaluating approaches for applying the cell quota model

Present phytoplankton models typically use a population-level (lumped) modeling (PLM) approach that assumes average properties of a population within a control volume. For modern biogeochemical models that formulate growth as a nonlinear function of the internal nutrient (e.g. Droop kinetics), this averaging assumption can introduce a significant error. Individual-based (agent-based) modeling (IBM) does not make the assumption of average properties and therefore constitutes a promising alternative for biogeochemical modeling. This paper explores the hypothesis that the cell quota (Droop) model, which predicts the population-average specific growth or cell division rate, based on the population-average nutrient cell quota, can be applied to individual algal cells and produce the same population-level results. Three models that translate the growth rate calculated using the cell quota model into discrete cell division events are evaluated, including a stochastic model based on the probability of cell division, a deterministic model based on the maturation velocity and fraction of the cell cycle completed (maturity fraction), and a deterministic model based on biomass (carbon) growth and cell size. The division models are integrated into an IBM framework (iAlgae), which combines a lumped system representation of a nutrient with an individual representation of algae. The IBM models are evaluated against a conventional PLM (because that is the traditional approach) and data from a number of steady and unsteady continuous (chemostat) and batch culture laboratory experiments. The stochastic IBM model fails the steady chemostat culture test, because it produces excessive numerical randomness. The deterministic cell cycle IBM model fails the batch culture test, because it has an abrupt drop in cell quota at division, which allows the cell quota to fall below the subsistence quota. The deterministic cell size IBM model reproduces the data and PLM results for all experiments and the model parameters (e.g. maximum specific growth rate, subsistence quota) are the same as those for the PLM. In addition, the model-predicted cell age, size (carbon) and volume distributions are consistent with those derived analytically and compare well to observations. The paper discusses and illustrates scenarios where intra-population variability in natural systems leads to differences between the IBM and PLM models.     

Topological solution for cell proliferation in intestinal crypt. I. Elastic growth without cell loss

The cells of an intestinal crypt are tightly packed and, consequently, cell renewal must proceed in accordance with topological laws implicit in the hexagonal cell patterns. The division wave is proposed as the simplest way of proliferation, satisfying topological requirements in steady state. Six pentagonal cells, persisting by topological necessity in the crypt bottom, are the sources of division waves for the whole crypt. The positions of the six pentagonal cells specify the order of cell division. The division, reciprocally, changes the positions of the pentagons which, in turn, specify the order of division in the new cells, and so on. The resulting order of cell division accounts for maintenance of the crypt structure, cell movement toward the villus and cessation of division. Since the pattern of elastic growth is dictated entirely by topological considerations, it does not depend on the genetic constitution of the organism. This model is different from conventional models in which the crypt is assumed to be composed of fixed longitudinal cell columns, the cells of the bottom contributing collectively to the proliferative potential of the whole crypt.     

Nucleoid-independent identification of cell division sites in Escherichia coli

The mechanism used by Escherichia coli to determine the correct site for cell division is unknown. In this report, we have attempted to distinguish between a model in which septal position is determined by the position of the nucleoids and a model in which septal position is predetermined by a mechanism that does not involve nucleoid position. To do this, filaments with extended nucleoid-free regions adjacent to the cell poles were produced by simultaneous inactivation of cell division and DNA replication. The positions of septa that formed within the nucleoid-free zones after division was allowed to resume were then analyzed. The results showed that septa were formed at a uniform distance from cell poles when division was restored, with no relation to the distance from the nearest nucleoid. In some cells, septa were formed directly over nucleoids. These results are inconsistent with models that invoke nucleoid positioning as the mechanism for determining the site of division site formation.     

Cell cycle modelling

Models able to describe the events of cellular growth and division and the dynamics of cell populations are useful for the understanding of functional control mechanisms and for the theoretical support for automated analysis of flow cytometric data and of cell volume distributions. This paper reports on models that we have developed with this aim for different kinds of cells. The models are composed by two subsystems: one describes the growth dynamics of RNA and protein, and the second accounts for DNA replication and cell division, and describe in a rather unitary frame the cell cycle of eukaryotic cells, like mammalian cells and yeast, and of prokaryotic cells. The model is also used to study the effects of various sources of variability on the statistical properties of cell populations, and we find that in microbial cells the main source of variability appears to be an inaccuracy of the molecular mechanism that monitors cell size. In normal mammalian cells another source of variability, that depends upon the interaction with growth factors which give competence, is apparent. An extended version of the model, which comprises also this additional variability, is presented and used to describe the properties of mammalian cell growth.