Single-cell behavior and population heterogeneity: solving an inverse problem to compute the intrinsic physiological state functions

The dynamics of isogenic cell populations can be described by cell population balance models that account for phenotypic heterogeneity. To utilize the predictive power of these models, however, we must know the rates of single-cell reaction and division and the bivariate partition probability density function. These three intrinsic physiological state (IPS) functions can be obtained by solving an inverse problem that requires knowledge of the phenotypic distributions for the overall cell population, the dividing cell subpopulation and the newborn cell subpopulation. We present here a robust computational procedure that can accurately estimate the IPS functions for heterogeneous cell populations. A detailed parametric analysis shows how the accuracy of the inverse solution is affected by discretization parameters, the type of non-parametric estimators used, the qualitative characteristics of phenotypic distributions and the unknown partitioning probability density function. The effect of finite sampling and measurement errors on the accuracy of the recovered IPS functions is also assessed. Finally, we apply the procedure to estimate the IPS functions of an E. coli population carrying an IPTG-inducible genetic toggle network. This study completes the development of an integrated experimental and computational framework that can become a powerful tool for quantifying single-cell behavior using measurements from heterogeneous cell populations.     

Cell volume distributions reveal cell growth rates and division times

A population of cells in culture displays a range of phenotypic responses even when those cells are derived from a single cell and are exposed to a homogeneous environment. Phenotypic variability can have a number of sources including the variable rates at which individual cells within the population grow and divide. We have examined how such variations contribute to population responses by measuring cell volumes within genetically identical populations of cells where individual members of the population are continuously growing and dividing, and we have derived a function describing the stationary distribution of cell volumes that arises from these dynamics. The model includes stochastic parameters for the variability in cell cycle times and growth rates for individual cells in a proliferating cell line. We used the model to analyze the volume distributions obtained for two different cell lines and one cell line in the absence and presence of aphidicolin, a DNA polymerase inhibitor. The derivation and application of the model allows one to relate the stationary population distribution of cell volumes to extrinsic biological noise present in growing and dividing cell cultures.     

Measurements and models of synchronous growth of fission yeast induced by temperature oscillations

Pulsing of temperature in a fermentor at intervals coincident with cell generation time was used to induce synchrony in a population of the fission yeast Schizosaccharomyces pombe. Measurements of culture protein, RNA, and DNA during synchronous growth confirm continuous synthesis of protein and RNA and discontinuos synthesis of DNA as previously reported. Flow microfluorometry of populations at different times during the synchrony cycle was used to monitor the changes in single-cell protein. RNA, and DNA frequency functions. These measurements illustrate very clearly the degree of synchrony and patterns of macromolecular synthesis and also confirm previous estimates of the cellular protein contents characteristic of dividing cells. Additional insights into single-cell kinetics and division controls are provided by two-parameter flow microfluorometry measurements and by mathematical modeling of population dynamics. Such data are necessary foundations for robust population balance models of microbial processes.     

Identification of various testicular cell populations in pubertal and adult cockerels

Precise identification of the male germinal stem cell population is important for their practical use in programs dedicated to the integration of exogenous genetic material in testicular tissues. In the present study, our aim was to identify germinal cell populations in the testes of pubertal and adult cockerels based on the detection of the nuclear DNA content by fluorescence-activated cell sorting (FACS) and on the expression of the Dazl and Stra8 genes in single-cell suspensions of testicular tissues. Cells with a tetraploid DNA content (4c) represent a small and equal fraction of the total germinal cell population in both pubertal and adult males. In contrast, the diploid (2c) and haploid (c) subpopulations differ significantly between ages as a consequence of different degrees of sexual maturation. A specific subpopulation of testicular cells, the side-scatter subpopulation of cells, or side population (SP), was identified at the junction between the haploid and diploid cell populations. The percentage of this cell subpopulation differs significantly in pubertal and adult cockerels, accounting for 4.1% and 1.3% of the total cell population, respectively. These four testicular cell populations were also tested for the expression of Dazl and Stra8 genes known to be expressed in premeiotic cells including stem spermatogonia. Both genes were expressed in SP, whereas the expression of either Dazl or Stra8 genes was detected only in the 4c and in the 2c testicular cell subpopulations, respectively. The correlation between the cell ploidy and Dazl/Stra8 expression was the same at both male ages. We conclude that SP cells might represent a subpopulation of germinal cells enriched in stem spermatogonia, which can be of great importance for transgenesis in chicken.     

Stochastic state transitions give rise to phenotypic equilibrium in populations of cancer cells

Cancer cells within individual tumors often exist in distinct phenotypic states that differ in functional attributes. While cancer cell populations typically display distinctive equilibria in the proportion of cells in various states, the mechanisms by which this occurs are poorly understood. Here, we study the dynamics of phenotypic proportions in human breast cancer cell lines. We show that subpopulations of cells purified for a given phenotypic state return towards equilibrium proportions over time. These observations can be explained by a Markov model in which cells transition stochastically between states. A prediction of this model is that, given certain conditions, any subpopulation of cells will return to equilibrium phenotypic proportions over time. A second prediction is that breast cancer stem-like cells arise de novo from non-stem-like cells. These findings contribute to our understanding of cancer heterogeneity and reveal how stochasticity in single-cell behaviors promotes phenotypic equilibrium in populations of cancer cells.     

A Millifluidic Study of Cell-to-Cell Heterogeneity in Growth-Rate and Cell-Division Capability in Populations of Isogenic Cells of Chlamydomonas reinhardtii

To address possible cell-to-cell heterogeneity in growth dynamics of isogenic cell populations of Chlamydomonas reinhardtii, we developed a millifluidic drop-based device that not only allows the analysis of populations grown from single cells over periods of a week, but is also able to sort and collect drops of interest, containing viable and healthy cells, which can be used for further experimentation. In this study, we used isogenic algal cells that were first synchronized in mixotrophic growth conditions. We show that these synchronized cells, when placed in droplets and kept in mixotrophic growth conditions, exhibit mostly homogeneous growth statistics, but with two distinct subpopulations: a major population with a short doubling-time (fast-growers) and a significant subpopulation of slowly dividing cells (slow-growers). These observations suggest that algal cells from an isogenic population may be present in either of two states, a state of restricted division and a state of active division. When isogenic cells were allowed to propagate for about 1000 generations on solid agar plates, they displayed an increased heterogeneity in their growth dynamics. Although we could still identify the original populations of slow- and fast-growers, drops inoculated with a single progenitor cell now displayed a wider diversity of doubling-times. Moreover, populations dividing with the same growth-rate often reached different cell numbers in stationary phase, suggesting that the progenitor cells differed in the number of cell divisions they could undertake. We discuss possible explanations for these cell-to-cell heterogeneities in growth dynamics, such as mutations, differential aging or stochastic variations in metabolites and macromolecules yielding molecular switches, in the light of single-cell heterogeneities that have been reported among isogenic populations of other eu- and prokaryotes.     

Cell population-based framework of genetic epidemiology in the single-cell omics era

Genetic epidemiology is a rapidly advancing field due to the recent availability of large amounts of omics data. In recent years, it has become possible to obtain omics information at the single-cell level, so genetic epidemiological models need to be updated to integrate with single-cell expression data. In this perspective paper, we propose a cell population-based framework for genetic epidemiology in the single-cell era. In this framework, genetic diversity influences phenotypic diversity through the diversity of cell population profiles, which are defined as high-dimensional probability distributions of the state spaces of biomolecules of each omics layer. We discuss how biomolecular experimental measurement data can capture the different properties of this distribution. In particular, single-cell data constitute a sample from this population distribution where only some coordinate values are observable. From a data analysis standpoint, we introduce methodology for feature extraction from cell population profiles. Finally, we discuss how this framework can be applied not only to genetic epidemiology but also to systems biology.     

Rapid kinetics of lysis in human natural cell-mediated cytotoxicity: Some implications

The entire lytic process of natural cell-mediated cytotoxicity against sensitive target cells can occur rapidly, within minutes. This was demonstrated by ⁵¹chromium release and in single-cell assays. At the cellular level, most of the target cell lysis occurred within 15–30 min after binding to effector cells. The enriched natural killer cell subpopulation of lymphocytes obtained by Percoll density gradient centrifugation (containing >70% large granular lymphocytes (LGL)) was the most rapidly lytic population by ⁵¹chromium release. However, in the single-cell assay, the rate of lysis of bound target cells was quite similar for the LGL-enriched effector subpopulation and the higher density subpopulation of effector cells recognized previously. Both the light and dense effector cells contained similar numbers of target binding cells. Therefore, that the light subpopulation effected lysis more rapidly and to a greater extent than the dense subpopulation suggested that the low-density effector cells probably recycled more rapidly than those of higher density. This was corroborated by the finding that when conjugates were formed at 29 °C for the single-cell assay, a significant number of dead unconjugated targets could be observed only on the slides made with the LGL-enriched effector cells but not on those made with dense effector cell. Lysis continued to increase in the chromium-release assay probably because of recycling, recruitment, and/or heterogeneity of the effector cells, and/or because of heterogeneity or delayed death of the target cells.     

Revealing non-genetic adhesive variations in clonal populations by comparative single-cell force spectroscopy

Cell populations often display heterogeneous behavior, including cell-to-cell variations in morphology, adhesion and spreading. However, better understanding the significance of such cell variations for the function of the population as a whole requires quantitative single-cell assays. To investigate adhesion variability in a CHO cell population in detail, we measured integrin-mediated adhesion to laminin and collagen, two ubiquitous ECM components, by AFM-based single-cell force spectroscopy (SCFS). CHO cells generally adhered more strongly to laminin than collagen but population adhesion force distributions to both ECM components were broad and partially overlapped. To determine the levels of laminin and collagen binding in individual cells directly, we alternatingly measured single cells on adjacent microstripes of collagen and laminin arrayed on the same adhesion substrate. In repeated measurements (≥60) individual cells showed a stable and ECM type-specific adhesion response. All tested cells bound laminin more strongly, but the scale of laminin over collagen binding varied between cells. Together, this demonstrates that adhesion levels to different ECM components are tightly yet differently set in each cell of the population. Adhesion variability to laminin was non-genetic and cell cycle-independent but scaled with the range of α6 integrin expression on the cell surface. Adhesive cell-to-cell variations due to varying receptor expression levels thus appear to be an inherent feature of cell populations and should to be considered when fully characterizing population adhesion. In this approach, SCFS performed on multifunctional adhesion substrates can provide quantitative single-cell information not obtainable from population-averaging measurements on homogeneous adhesion substrates.     

Slit scanning of Saccharomyces cerevisiae cells: quantification of asymmetric cell division and cell cycle progression in asynchronous culture.

Slit scanning flow cytometry has been applied to the analysis of the cell cycle and cell-cycle-dependent events in Saccharomyces cerevisiae, yielding information on the low-resolution spatial distribution of cellular components in single cells of unperturbed cell populations. Because this process is rapid, large numbers of cells can be analyzed to give distributions of parameters in a given population. To study asymmetric cell division and cell cycle progression, forward-angle light scattering (FALS) signals together with fluorescence signals from acriflavine-stained nuclei have been measured in cells from exponentially growing yeast populations. An algorithm has been developed that assigns the position of the bud neck in the FALS signals so that both FALS and DNA signals can be analyzed in terms of the contributions from the mother cell and the cell bud. The data indicate that mother cell FALS, on average, remains constant while FALS due to the cell bud increases as a cell progresses through the cell cycle. By identifying mitotic cells and measuring their properties, we have found that the coefficient of variation for the distribution of FALS is smallest within the dividing cell population and largest within the newborn cell population, in accordance with the critical size control mechanism of yeast cell growth. The use of this experimental approach to provide data for statistical population models is discussed.     

A quantitative study of the development of type II pneumocytes in fetal lung

Cell populations dissociated from fetal rabbit lungs were analyzed by laser flow cytometry for the presence of type II pneumocytes. These cells are distinguishable by the staining of their lamellar bodies with the fluorescent lipophilic dye, phosphine-3R and by their intensity of low-angle light scatter. Lung cells were obtained by enzymatic dissociation from fetal rabbits at gestational ages of 24 d, 27 d, and from 2-d newborn rabbits. Flow cytometric analysis was sufficiently sensitive to discriminate between fetuses. Quantitative analysis of type II pneumocytes showed that newborn rabbits had a distinct cell subpopulation in a region of low-angle light scatter and phosphine-3R fluorescence intensity similar to that previously reported on type II cells from adult rabbits. By contrast, 24-d gestation rabbits had a negligible type II cell subpopulation. Fetuses of 27 and 30 d gestation showed a slow but progressive increase in the numbers of cells in the type II region. Mathematical analyses of light scatter and fluorescence intensity distributions were used to define statistically significant (P less than .05) boundaries that characterize the development of the type II cell subpopulation in fetal rabbit lung. The methods employed offer new possibilities for quantification of developing lung cell subpopulations of particular interest to the problem of respiratory distress syndrome in human neonates.     

Modeling the impact of single-cell stochasticity and size control on the population growth rate in asymmetrically dividing cells

Microbial populations show striking diversity in cell growth morphology and lifecycle; however, our understanding of how these factors influence the growth rate of cell populations remains limited. We use theory and simulations to predict the impact of asymmetric cell division, cell size regulation and single-cell stochasticity on the population growth rate. Our model predicts that coarse-grained noise in the single-cell growth rate λ decreases the population growth rate, as previously seen for symmetrically dividing cells. However, for a given noise in λ we find that dividing asymmetrically can enhance the population growth rate for cells with strong size control (between a “sizer” and an “adder”). To reconcile this finding with the abundance of symmetrically dividing organisms in nature, we propose that additional constraints on cell growth and division must be present which are not included in our model, and we explore the effects of selected extensions thereof. Further, we find that within our model, epigenetically inherited generation times may arise due to size control in asymmetrically dividing cells, providing a possible explanation for recent experimental observations in budding yeast. Taken together, our findings provide insight into the complex effects generated by non-canonical growth morphologies.     

Changes in cell diameter during the division cycle of Escherichia coli

Extensive measurements of steady-state populations of several Escherichia coli strains have consistently indicated that cell diameter decreases with increasing cell length. This was observed both after electron microscopy of air-dried cells and after phase-contrast microscopy of living cells. The analysis was made by considering separately the unconstricted cells and three classes (slight, medium, and deep) of constricted cells in the population. During slow growth, cells with the average newborn length were up to 8% thicker than unconstricted cells twice as long. This decrease in diameter is less at higher growth rates. Despite the small changes and the large variation of the diameter in any particular length class, significant negative correlations between diameter and length were obtained. Cell diameter increases again at the end of the cell cycle as indicated by an increase of average diameter in the three consecutive classes of constriction.     

Mapping the environmental fitness landscape of a synthetic gene circuit

Gene expression actualizes the organismal phenotypes encoded within the genome in an environment-dependent manner. Among all encoded phenotypes, cell population growth rate (fitness) is perhaps the most important, since it determines how well-adapted a genotype is in various environments. Traditional biological measurement techniques have revealed the connection between the environment and fitness based on the gene expression mean. Yet, recently it became clear that cells with identical genomes exposed to the same environment can differ dramatically from the population average in their gene expression and division rate (individual fitness). For cell populations with bimodal gene expression, this difference is particularly pronounced, and may involve stochastic transitions between two cellular states that form distinct sub-populations. Currently it remains unclear how a cell population's growth rate and its subpopulation fractions emerge from the molecular-level kinetics of gene networks and the division rates of single cells. To address this question we developed and quantitatively characterized an inducible, bistable synthetic gene circuit controlling the expression of a bifunctional antibiotic resistance gene in Saccharomyces cerevisiae. Following fitness and fluorescence measurements in two distinct environments (inducer alone and antibiotic alone), we applied a computational approach to predict cell population fitness and subpopulation fractions in the combination of these environments based on stochastic cellular movement in gene expression space and fitness space. We found that knowing the fitness and nongenetic (cellular) memory associated with specific gene expression states were necessary for predicting the overall fitness of cell populations in combined environments. We validated these predictions experimentally and identified environmental conditions that defined a "sweet spot" of drug resistance. These findings may provide a roadmap for connecting the molecular-level kinetics of gene networks to cell population fitness in well-defined environments, and may have important implications for phenotypic variability of drug resistance in natural settings.     

Cell population heterogeneity in expression of a gene-switching network with fluorescent markers of different half-lives

We studied the distribution of expression levels amongst the cells of an Escherichia coli population carrying a gene-switching network, known as the genetic toggle. We employed two green fluorescent protein (GFP) reporter proteins with different half-lives and characterized the effect of isopropyl-beta-D-thiogalactopyranoside (IPTG) inducer concentration on fluorescence distribution characteristics. Our flow cytometric measurements indicated that there is a spread of fluorescence phenotypes of one to three orders of magnitude, due to the highly heterogeneous nature of the cell populations under investigation. Moreover, the shape of the distribution at a specific quasi-time-invariant reference state, defined for comparison purposes, strongly depended on inducer concentration. For very low and very high inducer concentrations, the distributions at the reference state are unimodal. On the contrary, for intermediate IPTG concentrations, two distinct subpopulations were formed below and above a single-cell threshold, resulting in distributions with a bimodal shape. The region of inducer concentrations where bimodality is observed is the same and independent of GFP half-life. Bimodal number density functions are not only obtained at the reference state. Transient studies revealed that even in cases where the distribution at the reference state is unimodal, the distribution becomes bimodal for a period of time required for the population to pass through the single-cell induction threshold. However, this feature was only captured by the system with the reduced half-life GFP. A simple single-cell model was used to shed light into the effect of inducer concentration and GFP half-life on the shape of the experimentally measured number density functions. The wide range of fluorescent phenotypes and the inability of the average population properties to fully characterize network behavior, indicate the importance of taking into account cell population heterogeneity when designing such a gene-switching network for biotechnological and biomedical applications.     

Toward a self-similar theory of microbial populations

A fresh quest is made of segregated cell models of microbial populations with a view to determine whether the multivarite distribution of physiological states, during transient growth, can attain self-similar forms (i.e., become time invariant) when each physiological state variable is scaled with respect to its population average. Such self-similar growth situations are believed to be more general than those of balanced growth. The conditions under which self-similarity is possible are investigated. Thus conditions are stipulated on the synthesis rates of different physiological entities, cell division rate, and the partitioning of the parent cell's components among the daughter cells (assuming binary division) in order for self-similar growth to be attained. Subject to the attainment of self-similar growth, it is shown that cytometric data can be analyzed systematically to determine how the rates of syntheses of various biochemical entities and cell division rates vary with the physiological entities that are measured. Inverse problems, represented by algebraic systems, are identified which will potentially allow flow cytometric data to be inverted to yield quantitative information on the absolute rates of cellular growth and reproductory processes as a function of the cell states chosen for measurement. It is suggested that the methods become more effective when cytometry can be used to make direct observations on dividing cells so that the number of unknowns in the inverse problem can be reduced, thus facilitating its more complete solution. Preliminary analysis of cytometric data obtained in the literature show promise of self-similarity and thus the possibility of application of the methods discussed here. (c) 1994 John Wiley & Sons, Inc.     

RELATIONSHIP BETWEEN THE NUMBER OF DIVIDING AND NONDIVIDING CELLS OF CYANOBACTERIA IN NORTH ATLANTIC PICOPLANKTON1

In the North Atlantic over a wide geographic region that includes various oceanic regimes and a temperature range from 10 to 22° C, an increase in the number of nondividing Synechococcus cells (X) was generally accompanied by a greater-than-proportional increase in the number of dividing cells (Y). As a result, the fraction of dividing cells (FDC = Y · (Y + X)^?1) was positively related to population size (Y + X). Recognizing that FDC is generally greater in a rapidly growing population than in a slowly growing one, our empirical finding implies a positive correlation between specific growth rate and standing stock for Synechococcus. One notable exception occurred during winter (T < 5°C) in a eutrophic coastal embayment when a decrease in cell abundance was not matched by a decrease in FDC.     

Asynchrony in the growth and motility responses to environmental changes by individual bacterial cells

Knowing how individual cells respond to environmental changes helps one understand phenotypic diversity in a bacterial cell population, so we simultaneously monitored the growth and motility of isolated motile Escherichia coli cells over several generations by using a method called on-chip single-cell cultivation. Starved cells quickly stopped growing but remained motile for several hours before gradually becoming immotile. When nutrients were restored the cells soon resumed their growth and proliferation but remained immotile for up to six generations. A flagella visualization assay suggested that deflagellation underlies the observed loss of motility. This set of results demonstrates that single-cell transgenerational study under well-characterized environmental conditions can provide information that will help us understand distinct functions within individual cells.