Towards understanding of the complex structure of growing yeast populations

In a growing Saccharomyces cerevisiae population, cell size is finely modulated according to both the chronological and genealogical ages. This generates the complex heterogeneous structure typical of budding yeast populations. In recent years, there has been a growing interest in developing mathematical models capable of faithfully describing population dynamics at the single cell level. A multistaged morphologically structured model has been lately proposed based on the population balance theory. The model was able to describe the dynamics of the generation of a heterogeneous growing yeast population starting from a sub-population of daughter unbudded cells. In this work, which aims at validating the model, the simulated experiment was performed by following the release of a homogeneous population of daughter unbudded cells. A biparametric flow cytometric approach allowed us to analyse the time course joint distribution of DNA and protein contents at the single cell level; this gave insights into the coupling between growth and cell cycle progression that generated the final population structure. The comparison between experimental and simulated size distributions revealed a strong agreement for some unexpected features as well. Therefore, the model can be considered as validated and extendable to more complex situations.     

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.     

Plasmid stability in budding yeast populations: Steady-state growth with selection pressure

Plasmid gene product accumulation in a cell population depends on the fraction of plasmid-containing cells and the distribution of single-cell plasmid content. These important population properties have been related to plasmid replication regulation and kinetics and to plasmid segregation rules at the single-cell level using population balance mathematical models. Budding yeast populations are considered in detail because of the practical potential of yeast host-vector systems and because of the model complications introduced by the asymmetric division pattern observed for Saccharomyces cerevisiae at all but the largest growth rates. Solutions are presented for several different reasonable models of plasmid replication and segregation. The results offer potential for identification of important qualitative features of yeast plasmid replication and of model parameter values from average and segregated experimental data on yeast populations.     

An explicit representation of the Luria–Delbrück distribution

The probability distribution of the number of mutant cells in a growing single-cell population is presented in explicit form. We use a discrete model for mutation and population growth which in the limit of large cell numbers and small mutation rates reduces to certain classical models of the Luria-Delbrück distribution. Our results hold for arbitrarily large values of the mutation rate and for cell populations of arbitrary size. We discuss the influence of cell death on fluctuation experiments and investigate a version of our model that accounts for the possibility that both daughter cells of a non-mutant cell might be mutants. An algorithm is presented for the quick calculation of the distribution. Then, we focus on the derivation of two essentially different limit laws, the first of which applies if the population size tends to infinity while the mutation rate tends to zero such that the product of mutation rate times population size converges. The second limit law emerges after a suitable rescaling of the distribution of non-mutant cells in the population and applies if the product of mutation rate times population size tends to infinity. We discuss the distribution of mutation events for arbitrary values of the mutation rate and cell populations of arbitrary size, and, finally, consider limit laws for this distribution with respect to the behavior of the product of mutation rate times population size. Thus, the present paper substantially extends results due to Lea and Coulson (1949), Bartlett (1955), Stewart et al. (1990), and others.     

Bayesian models for the analysis of genetic structure when populations are correlated

MOTIVATION: Population allele frequencies are correlated when populations have a shared history or when they exchange genes. Unfortunately, most models for allele frequency and inference about population structure ignore this correlation. Recent analytical results show that among populations, correlations can be very high, which could affect estimates of population genetic structure. In this study, we propose a mixture beta model to characterize the allele frequency distribution among populations. This formulation incorporates the correlation among populations as well as extending the model to data with different clusters of populations. RESULTS: Using simulated data, we show that in general, the mixture model provides a good approximation of the among-population allele frequency distribution and a good estimate of correlation among populations. Results from fitting the mixture model to a dataset of genotypes at 377 autosomal microsatellite loci from human populations indicate high correlation among populations, which may not be appropriate to neglect. Traditional measures of population structure tend to overestimate the amount of genetic differentiation when correlation is neglected. Inference is performed in a Bayesian framework. CONTACT: fur@ohsu.edu.     

Network-based analysis of stochastic SIR epidemic models with random and proportionate mixing

In this paper, we outline the theory of epidemic percolation networks and their use in the analysis of stochastic susceptible-infectious-removed (SIR) epidemic models on undirected contact networks. We then show how the same theory can be used to analyze stochastic SIR models with random and proportionate mixing. The epidemic percolation networks for these models are purely directed because undirected edges disappear in the limit of a large population. In a series of simulations, we show that epidemic percolation networks accurately predict the mean outbreak size and probability and final size of an epidemic for a variety of epidemic models in homogeneous and heterogeneous populations. Finally, we show that epidemic percolation networks can be used to re-derive classical results from several different areas of infectious disease epidemiology. In an Appendix, we show that an epidemic percolation network can be defined for any time-homogeneous stochastic SIR model in a closed population and prove that the distribution of outbreak sizes given the infection of any given node in the SIR model is identical to the distribution of its out-component sizes in the corresponding probability space of epidemic percolation networks. We conclude that the theory of percolation on semi-directed networks provides a very general framework for the analysis of stochastic SIR models in closed populations.     

A size-structured model of bacterial growth and reproduction

We consider a size-structured bacterial population model in which the rate of cell growth is both size- and time-dependent and the average per capita reproduction rate is specified as a model parameter. It is shown that the model admits classical solutions. The population-level and distribution-level behaviours of these solutions are then determined in terms of the model parameters. The distribution-level behaviour is found to be different from that found in similar models of bacterial population dynamics. Rather than convergence to a stable size distribution, we find that size distributions repeat in cycles. This phenomenon is observed in similar models only under special assumptions on the functional form of the size-dependent growth rate factor. Our main results are illustrated with examples, and we also provide an introductory study of the bacterial growth in a chemostat within the framework of our model.     

Cell Growth and Size Homeostasis in Silico

Cell growth in size is a complex process coordinated by intrinsic and environmental signals. In a research work performed by a different group, size distributions of an exponentially growing population of mammalian cells were used to infer cell-growth rate in size. The results suggested that cell growth was neither linear nor exponential, but subject to size-dependent regulation. To explain the observed growth pattern, we built a mathematical model in which growth rate was regulated by the relative amount of mRNA and ribosomes in a cell. Under the growth model and a stochastic division rule, we simulated the evolution of a population of cells. Both the sampled growth rate and size distribution from this in silico population agreed well with experimental data. To explore the model space, alternative growth models and division rules were studied. This work may serve as a starting point to understand the mechanisms behind cell growth and size regulation using predictive models.     

Cell Growth and Size Homeostasis in Silico

Cell growth in size is a complex process coordinated by intrinsic and environmental signals. In a research work performed by a different group, size distributions of an exponentially growing population of mammalian cells were used to infer cell-growth rate in size. The results suggested that cell growth was neither linear nor exponential, but subject to size-dependent regulation. To explain the observed growth pattern, we built a mathematical model in which growth rate was regulated by the relative amount of mRNA and ribosomes in a cell. Under the growth model and a stochastic division rule, we simulated the evolution of a population of cells. Both the sampled growth rate and size distribution from this in silico population agreed well with experimental data. To explore the model space, alternative growth models and division rules were studied. This work may serve as a starting point to understand the mechanisms behind cell growth and size regulation using predictive models.     

Phylodynamic Inference for Structured Epidemiological Models

Coalescent theory is routinely used to estimate past population dynamics and demographic parameters from genealogies. While early work in coalescent theory only considered simple demographic models, advances in theory have allowed for increasingly complex demographic scenarios to be considered. The success of this approach has lead to coalescent-based inference methods being applied to populations with rapidly changing population dynamics, including pathogens like RNA viruses. However, fitting epidemiological models to genealogies via coalescent models remains a challenging task, because pathogen populations often exhibit complex, nonlinear dynamics and are structured by multiple factors. Moreover, it often becomes necessary to consider stochastic variation in population dynamics when fitting such complex models to real data. Using recently developed structured coalescent models that accommodate complex population dynamics and population structure, we develop a statistical framework for fitting stochastic epidemiological models to genealogies. By combining particle filtering methods with Bayesian Markov chain Monte Carlo methods, we are able to fit a wide class of stochastic, nonlinear epidemiological models with different forms of population structure to genealogies. We demonstrate our framework using two structured epidemiological models: a model with disease progression between multiple stages of infection and a two-population model reflecting spatial structure. We apply the multi-stage model to HIV genealogies and show that the proposed method can be used to estimate the stage-specific transmission rates and prevalence of HIV. Finally, using the two-population model we explore how much information about population structure is contained in genealogies and what sample sizes are necessary to reliably infer parameters like migration rates.     

The age structure of populations of Saccharomyces cerevisiae

A discrete deterministic model is described for the growth of an age-structured population of yeast, Saccharomyces cerevisiae, incorporating recent information on the asymmetry of cell division and control of the cell cycle in this species. Solutions are obtained for the age structure of the population at equilibrium, and for the equilibrium distribution of relative frequency of cells through the cell cycle. The model is applied to experimental data on the changing age structure of nonequilibrium populations of yeast. The model predicts well both the transient behavior and the equilibrium structure of such populations. It is shown that the asymmetry of cell division explains (1) the excess of newly formed daughter cells in the population as compared to the frequency of older cells and (2) the damped oscillations in the frequencies of cells of different ages as demographic equilibrium is approached.     

Size control models of Saccharomyces cerevisiae cell proliferation.

By using time-lapse photomicroscopy, the individual cycle times and sizes at bud emergence were measured for a population of saccharomyces cerevisiae cells growing exponentially under balanced growth conditions in a specially constructed filming slide. There was extensive variability in both parameters for daughter and parent cells. The data on 162 pairs of siblings were analyzed for agreement with the predictions of the transition probability hypothesis and the critical-size hypothesis of yeast cell proliferation and also with a model incorporating both of these hypotheses in tandem. None of the models accounted for all of the experimental data, but two models did give good agreement to all of the data. The wobbly tandem model proposes that cells need to attain a critical size, which is very variable, enabling them to enter a start state from which they exit with first order kinetics. The sloppy size control model suggests that cells have an increasing probability per unit time of traversing start as they increase in size, reaching a high plateau value which is less than one. Both models predict that the kinetics of entry into the cell division sequence will strongly depend on variability in birth size and thus will be quite different for daughters and parents of the asymmetrically dividing yeast cells. Mechanisms underlying these models are discussed.     

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.     

Stochastic Models of Lymphocyte Proliferation and Death

Quantitative understanding of the kinetics of lymphocyte proliferation and death upon activation with an antigen is crucial for elucidating factors determining the magnitude, duration and efficiency of the immune response. Recent advances in quantitative experimental techniques, in particular intracellular labeling and multi-channel flow cytometry, allow one to measure the population structure of proliferating and dying lymphocytes for several generations with high precision. These new experimental techniques require novel quantitative methods of analysis. We review several recent mathematical approaches used to describe and analyze cell proliferation data. Using a rigorous mathematical framework, we show that two commonly used models that are based on the theories of age-structured cell populations and of branching processes, are mathematically identical. We provide several simple analytical solutions for a model in which the distribution of inter-division times follows a gamma distribution and show that this model can fit both simulated and experimental data. We also show that the estimates of some critical kinetic parameters, such as the average inter-division time, obtained by fitting models to data may depend on the assumed distribution of inter-division times, highlighting the challenges in quantitative understanding of cell kinetics.     

Population Dynamics of Metastable Growth-Rate Phenotypes

Neo-Darwinian evolution has presented a paradigm for population dynamics built on random mutations and selection with a clear separation of time-scales between single-cell mutation rates and the rate of reproduction. Laboratory experiments on evolving populations until now have concentrated on the fixation of beneficial mutations. Following the Darwinian paradigm, these experiments probed populations at low temporal resolution dictated by the rate of rare mutations, ignoring the intermediate evolving phenotypes. Selection however, works on phenotypes rather than genotypes. Research in recent years has uncovered the complexity of genotype-to-phenotype transformation and a wealth of intracellular processes including epigenetic inheritance, which operate on a wide range of time-scales. Here, by studying the adaptation dynamics of genetically rewired yeast cells, we show a novel type of population dynamics in which the intracellular processes intervene in shaping the population structure. Under constant environmental conditions, we measure a wide distribution of growth rates that coexist in the population for very long durations (>100 generations). Remarkably, the fastest growing cells do not take over the population on the time-scale dictated by the width of the growth-rate distributions and simple selection. Additionally, we measure significant fluctuations in the population distribution of various phenotypes: the fraction of exponentially-growing cells, the distributions of single-cell growth-rates and protein content. The observed fluctuations relax on time-scales of many generations and thus do not reflect noisy processes. Rather, our data show that the phenotypic state of the cells, including the growth-rate, for large populations in a constant environment is metastable and varies on time-scales that reflect the importance of long-term intracellular processes in shaping the population structure. This lack of time-scale separation between the intracellular and population processes calls for a new framework for population dynamics which is likely to be significant in a wide range of biological contexts, from evolution to cancer.     

A general modeling framework for describing spatially structured population dynamics

Variation in movement across time and space fundamentally shapes the abundance and distribution of populations. Although a variety of approaches model structured population dynamics, they are limited to specific types of spatially structured populations and lack a unifying framework. Here, we propose a unified network‐based framework sufficiently novel in its flexibility to capture a wide variety of spatiotemporal processes including metapopulations and a range of migratory patterns. It can accommodate different kinds of age structures, forms of population growth, dispersal, nomadism and migration, and alternative life‐history strategies. Our objective was to link three general elements common to all spatially structured populations (space, time and movement) under a single mathematical framework. To do this, we adopt a network modeling approach. The spatial structure of a population is represented by a weighted and directed network. Each node and each edge has a set of attributes which vary through time. The dynamics of our network‐based population is modeled with discrete time steps. Using both theoretical and real‐world examples, we show how common elements recur across species with disparate movement strategies and how they can be combined under a unified mathematical framework. We illustrate how metapopulations, various migratory patterns, and nomadism can be represented with this modeling approach. We also apply our network‐based framework to four organisms spanning a wide range of life histories, movement patterns, and carrying capacities. General computer code to implement our framework is provided, which can be applied to almost any spatially structured population. This framework contributes to our theoretical understanding of population dynamics and has practical management applications, including understanding the impact of perturbations on population size, distribution, and movement patterns. By working within a common framework, there is less chance that comparative analyses are colored by model details rather than general principles.     

Approximation of a physiologically structured population model with seasonal reproduction by a stage-structured biomass model

Seasonal reproduction causes, due to the periodic inflow of young small individuals in the population, seasonal fluctuations in population size distributions. Seasonal reproduction furthermore implies that the energetic body condition of reproducing individuals varies over time. Through these mechanisms, seasonal reproduction likely affects population and community dynamics. While seasonal reproduction is often incorporated in population models using discrete time equations, these are not suitable for size-structured populations in which individuals grow continuously between reproductive events. Size-structured population models that consider seasonal reproduction, an explicit growing season and individual-level energetic processes exist in the form of physiologically structured population models. However, modeling large species ensembles with these models is virtually impossible. In this study, we therefore develop a simpler model framework by approximating a cohort-based size-structured population model with seasonal reproduction to a stage-structured biomass model of four ODEs. The model translates individual-level assumptions about food ingestion, bioenergetics, growth, investment in reproduction, storage of reproductive energy, and seasonal reproduction in stage-based processes at the population level. Numerical analysis of the two models shows similar values for the average biomass of juveniles, adults, and resource unless large-amplitude cycles with a single cohort dominating the population occur. The model framework can be extended by adding species or multiple juvenile and/or adult stages. This opens up possibilities to investigate population dynamics of interacting species while incorporating ontogenetic development and complex life histories in combination with seasonal reproduction.     

An Age-of-Allele Test of Neutrality for Transposable Element Insertions

How natural selection acts to limit the proliferation of transposable elements (TEs) in genomes has been of interest to evolutionary biologists for many years. To describe TE dynamics in populations, previous studies have used models of transposition–selection equilibrium that assume a constant rate of transposition. However, since TE invasions are known to happen in bursts through time, this assumption may not be reasonable. Here we propose a test of neutrality for TE insertions that does not rely on the assumption of a constant transposition rate. We consider the case of TE insertions that have been ascertained from a single haploid reference genome sequence. By conditioning on the age of an individual TE insertion allele (inferred by the number of unique substitutions that have occurred within the particular TE sequence since insertion), we determine the probability distribution of the insertion allele frequency in a population sample under neutrality. Taking models of varying population size into account, we then evaluate predictions of our model against allele frequency data from 190 retrotransposon insertions sampled from North American and African populations of Drosophila melanogaster. Using this nonequilibrium neutral model, we are able to explain ∼80% of the variance in TE insertion allele frequencies based on age alone. Controlling for both nonequilibrium dynamics of transposition and host demography, we provide evidence for negative selection acting against most TEs as well as for positive selection acting on a small subset of TEs. Our work establishes a new framework for the analysis of the evolutionary forces governing large insertion mutations like TEs, gene duplications, or other copy number variants.     

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.