Stochastic exits from dormancy give rise to heavy‐tailed distributions of descendants in bacterial populations

Variance in reproductive success is a major determinant of the degree of genetic drift in a population. While many plants and animals exhibit high variance in their number of progeny, far less is known about these distributions for microorganisms. Here, we used a strain barcoding approach to quantify variability in offspring number among replicate bacterial populations and developed a Bayesian method to infer the distribution of descendants from this variability. We applied our approach to measure the offspring distributions for five strains of bacteria from the genus Streptomyces after germination and growth in a homogenous laboratory environment. The distributions of descendants were heavy‐tailed, with a few cells effectively ‘winning the jackpot’ to become a disproportionately large fraction of the population. This extreme variability in reproductive success largely traced back to initial populations of spores stochastically exiting dormancy, which provided early‐germinating spores with an exponential advantage. In simulations with multiple dormancy cycles, heavy‐tailed distributions of descendants decreased the effective population size by many orders of magnitude and led to allele dynamics differing substantially from classical population genetics models with matching effective population size. Collectively, these results demonstrate that extreme variability in reproductive success can occur even in growth conditions that are far more homogeneous than the natural environment. Thus, extreme variability in reproductive success might be an important factor shaping microbial population dynamics with implications for predicting the fate of beneficial mutations, interpreting sequence variability within populations and explaining variability in infection outcomes across patients.     

Relationship between fitness and heterogeneity in exponentially growing microbial populations

Despite major environmental and genetic differences, microbial metabolic networks are known to generate consistent physiological outcomes across vastly different organisms. This remarkable robustness suggests that, at least in bacteria, metabolic activity may be guided by universal principles. The constrained optimization of evolutionarily motivated objective functions, such as the growth rate, has emerged as the key theoretical assumption for the study of bacterial metabolism. While conceptually and practically useful in many situations, the idea that certain functions are optimized is hard to validate in data. Moreover, it is not always clear how optimality can be reconciled with the high degree of single-cell variability observed in experiments within microbial populations. To shed light on these issues, we develop an inverse modeling framework that connects the fitness of a population of cells (represented by the mean single-cell growth rate) to the underlying metabolic variability through the maximum entropy inference of the distribution of metabolic phenotypes from data. While no clear objective function emerges, we find that, as the medium gets richer, the fitness and inferred variability for Escherichia coli populations follow and slowly approach the theoretically optimal bound defined by minimal reduction of variability at given fitness. These results suggest that bacterial metabolism may be crucially shaped by a population-level trade-off between growth and heterogeneity.     

二维Lotka-Volterra竞争系统的β持续生存与β绝灭

利用极限理论与延拓方法研究了二维Lotka-Volterra竞争务统在有限时间内的持续生存与绝灭问题,即β持续生存与β绝灭问题.给出了种群β持续生存与β绝灭的一些充分条件.所得结论表明:种群的β持续生存和β绝灭与种群的初始数量有关.在一定条件下,只要控制种群的初始数量在一定范围内,即可保证两种群永远β持续生存.     

The propagation of a cultural or biological trait by neutral genetic drift in a subdivided population

We study fixation probabilities and times as a consequence of neutral genetic drift in subdivided populations, motivated by a model of the cultural evolutionary process of language change that is described by the same mathematics as the biological process. We focus on the growth of fixation times with the number of subpopulations, and variation of fixation probabilities and times with initial distributions of mutants. A general formula for the fixation probability for arbitrary initial condition is derived by extending a duality relation between forwards- and backwards-time properties of the model from a panmictic to a subdivided population. From this we obtain new formulae(formally exact in the limit of extremely weak migration) for the mean fixation time from an arbitrary initial condition for Wright's island model, presenting two cases as examples. For more general models of population subdivision, formulae are introduced for an arbitrary number of mutants that are randomly located, and a single mutant whose position is known. These formulae contain parameters that typically have to be obtained numerically, a procedure we follow for two contrasting clustered models. These data suggest that variation of fixation time with the initial condition is slight, but depends strongly on the nature of subdivision. In particular, we demonstrate conditions under which the fixation time remains finite even in the limit of an infinite number of demes. In many cases-except this last where fixation in a finite time is seen--the time to fixation is shown to be in precise agreement with predictions from formulae for the asymptotic effective population size.     

Use of clones increases the power of physiological experiments on coastal Douglas-fir

The statistical benefit of reduced variability in experiments with clones was quantified with cold hardy Douglas-fir ( Pseudotsuga menziesii var. menziesii [Mirb.] Franco) ramets from six clones from a single full-sib family and a check group of open-pollinated seedlings from a bulk seed collection from the same geographic seed source. All groups were cold deacclimated under controlled conditions. The variability in response for physiological attributes (cold hardiness, root growth potential, days to bud-break) was compared within and among groups by using statistical power analyses. Sample sizes required to detect significant differences of varying magnitude between two hypothetical treatment means were calculated. Differences among clones in the average response exhibited for each physiological attribute were large, but within-clone variability was low, relative to the check group of seedlings. Selection of plant material for a hypothetical experiment from this population of several identifiable clones would have consistently resulted in an experiment with greater power than an experiment using these ramets without the clone identities. From a statistical perspective, the best approach to reduce the number of replicates needed to detect treatment differences was selection of experimental plant material from a single clone, especially with prior screening for the most homogeneous clone for the physiological attributes and time periods of interest. However, from a biological perspective, use of a single clone should be approached with caution because of the lack of representation of natural population variability and the possible inability to broadly apply experimental results.     

Comparative statics and stochastic dynamics of age-structured populations

Arguments from the comparative statics of populations with fixed vital rates are of limited use in studying age-structured populations subject to stochastically varying vital rates. In an age-structured population that experiences a sequence of independently and identically distributed Leslie matrices, the expectation of the Malthusian parameters of the Leslie matrices has no exact interpretation either as the ensemble average of the long-run rate of growth of each sample path of the population (Eq. (3)) or as the long-run rate of growth of the ensemble average of total population size (Eq. (4)). On the other hand, the Malthusian parameter of the expectation of a sequence of Leslie matrices is exactly the logarithm of the finite growth rate of the ensemble average of total population size when Leslie matrices are independently and identically distributed (though not in general when Leslie matrices are sequentially dependent). These observations appear to contradict the claims of a recent study using computer simulation of age-structured populations with stochastically varying vital rates.     

Simulating natural selection as a culling mechanism on finite populations with the hawk-dove game

The behaviors of individuals and species are often explained in terms of evolutionary stable strategies (ESSs). The analysis of ESSs determines which, if any, combinations of behaviors cannot be invaded by alternative strategies. Two assumptions required to generate an ESS (i.e., an infinite population and payoffs described only on the average) do not hold under natural conditions. Previous experiments indicated that under more realistic conditions of finite populations and stochastic payoffs, populations may evolve in trajectories that are unrelated to an ESS, even in very simple games. The simulations offered here extend earlier research by employing truncation selection with random parental selection in a hawk-dove game. Payoffs are determined in pairwise contests using either the expected outcome, or the result of a random variable. In each case, however, the mean fraction of hawks over many generations and across many independent trials does not conform to the expected ESS. Implications of these results and philosophical underpinnings of ESS theory are offered.     

Design and analysis of experiments on random drift and inbreeding depression

While the genetic consequences of inbreeding and small population size are of fundamental importance in many areas of biology, empirical research on these phenomena has proceeded in the absence of a well-developed statistical methodology. The usual approach is to compare observed means and variances with the expectations of Wright's neutral, additive genetic model for quantitative characters. If the observations deviate from the expectations more than can be accounted for by sampling variance of the parameter estimates, the null hypothesis is routinely rejected in favor of alternatives invoking evolutionary forces such as selection or nonadditive gene action. This is a biased procedure because it treats sequential samples from the same populations as independent, and because it ignores the fact that the expectations of the neutral additive genetic model will rarely be realized when only a finite number of lines are studied. Even when genes are perfectly additive and neutral, the variation among the properties of founder populations, the random development of linkage disequilibrium within lines, and the variance in inbreeding between lines reduce the likelihood that Wright's expectations will be realized in any particular set of lines. Under most experimental designs, these sources of variation are much too large to be ignored. Formulas are presented for the variance-covariance structure of the realized within- and between-line variance under the neutral additive genetic model. These results are then used to develop statistical tests for detecting the operation of selection and/or inbreeding depression in small populations. A number of recommendations are made for the optimal design of experiments on drift and inbreeding, and a method is suggested for the correction of data for general environmental effects. In general, it appears that we can best understand the response of populations to inbreeding and finite population size by studying a very large number (>100) of self-fertilizing or full-sib mated lines in parallel with one or more stable control populations.     

Frequency of RNA-RNA interaction in a model of the RNA World

The RNA World model for prebiotic evolution posits the selection of catalytic/template RNAs from random populations. The mechanisms by which these random populations could be generated de novo are unclear. Non-enzymatic and RNA-catalyzed nucleic acid polymerizations are poorly processive, which means that the resulting short-chain RNA population could contain only limited diversity. Nonreciprocal recombination of smaller RNAs provides an alternative mechanism for the assembly of larger species with concomitantly greater structural diversity; however, the frequency of any specific recombination event in a random RNA population is limited by the low probability of an encounter between any two given molecules. This low probability could be overcome if the molecules capable of productive recombination were redundant, with many nonhomologous but functionally equivalent RNAs being present in a random population. Here we report fluctuation experiments to estimate the redundancy of the set of RNAs in a population of random sequences that are capable of non-Watson-Crick interaction with another RNA. Parallel SELEX experiments showed that at least one in 10(6) random 20-mers binds to the P5.1 stem-loop of Bacillus subtilis RNase P RNA with affinities equal to that of its naturally occurring partner. This high frequency predicts that a single RNA in an RNA World would encounter multiple interacting RNAs within its lifetime, supporting recombination as a plausible mechanism for prebiotic RNA evolution. The large number of equivalent species implies that the selection of any single interacting species in the RNA World would be a contingent event, i.e., one resulting from historical accident.     

A proposal for using the ensemble approach to understand genetic regulatory networks

Understanding the genetic regulatory network comprising genes, RNA, proteins and the network connections and dynamical control rules among them, is a major task of contemporary systems biology. I focus here on the use of the ensemble approach to find one or more well-defined ensembles of model networks whose statistical features match those of real cells and organisms. Such ensembles should help explain and predict features of real cells and organisms. More precisely, an ensemble of model networks is defined by constraints on the "wiring diagram" of regulatory interactions, and the "rules" governing the dynamical behavior of regulated components of the network. The ensemble consists of all networks consistent with those constraints. Here I discuss ensembles of random Boolean networks, scale free Boolean networks, "medusa" Boolean networks, continuous variable networks, and others. For each ensemble, M statistical features, such as the size distribution of avalanches in gene activity changes unleashed by transiently altering the activity of a single gene, the distribution in distances between gene activities on different cell types, and others, are measured. This creates an M-dimensional space, where each ensemble corresponds to a cluster of points or distributions. Using current and future experimental techniques, such as gene arrays, these M properties are to be measured for real cells and organisms, again yielding a cluster of points or distributions in the M-dimensional space. The procedure then finds ensembles close to those of real cells and organisms, and hill climbs to attempt to match the observed M features. Thus obtains one or more ensembles that should predict and explain many features of the regulatory networks in cells and organisms.     

The synergistic effects of stochasticity and dispersal on population densities

Using laboratory experiments, simulation models, and analytical techniques, we examined the impact of dispersal on the mean densities of patchily distributed populations. Even when dispersal leads to no net additions or removals of individuals from a population, it may nonetheless increase mean population densities if the net immigration rate is positive when populations are growing and negative when they are declining. As a model system for exploring this phenomenon, we used the yeastlike fungus Aureobasidium pullulans. In a laboratory experiment, we showed that dispersal can both ensure persistence and increase mean population densities even when dispersal among populations causes no direct addition or loss of fungal cells. From the laboratory data, we constructed a plausible model of A. pullulans dynamics among apple leaves within an orchard. This simulation model demonstrated that the effect of dispersal on mean densities is enhanced by three factors: weak density dependence of the dynamics within populations, high environmental variability affecting population growth rates, and lack of synchrony among the fluctuations of populations. Using an analytical model, we showed that the underlying mechanisms for this phenomenon are general, suggesting that a large effect of dispersal on mean population densities is possible in many natural systems.     

Modelling multi-pulse population dynamics from ultrafast spectroscopy

Current advanced laser, optics and electronics technology allows sensitive recording of molecular dynamics, from single resonance to multi-colour and multi-pulse experiments. Extracting the occurring (bio-) physical relevant pathways via global analysis of experimental data requires a systematic investigation of connectivity schemes. Here we present a Matlab-based toolbox for this purpose. The toolbox has a graphical user interface which facilitates the application of different reaction models to the data to generate the coupled differential equations. Any time-dependent dataset can be analysed to extract time-independent correlations of the observables by using gradient or direct search methods. Specific capabilities (i.e. chirp and instrument response function) for the analysis of ultrafast pump-probe spectroscopic data are included. The inclusion of an extra pulse that interacts with a transient phase can help to disentangle complex interdependent pathways. The modelling of pathways is therefore extended by new theory (which is included in the toolbox) that describes the finite bleach (orientation) effect of single and multiple intense polarised femtosecond pulses on an ensemble of randomly oriented particles in the presence of population decay. For instance, the generally assumed flat-top multimode beam profile is adapted to a more realistic Gaussian shape, exposing the need for several corrections for accurate anisotropy measurements. In addition, the (selective) excitation (photoselection) and anisotropy of populations that interact with single or multiple intense polarised laser pulses is demonstrated as function of power density and beam profile. Using example values of real world experiments it is calculated to what extent this effectively orients the ensemble of particles. Finally, the implementation includes the interaction with multiple pulses in addition to depth averaging in optically dense samples. In summary, we show that mathematical modelling is essential to model and resolve the details of physical behaviour of populations in ultrafast spectroscopy such as pump-probe, pump-dump-probe and pump-repump-probe experiments.     

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.     

Contingency and statistical laws in replicate microbial closed ecosystems

Contingency, the persistent influence of past random events, pervades biology. To what extent, then, is each course of ecological or evolutionary dynamics unique, and to what extent are these dynamics subject to a common statistical structure? Addressing this question requires replicate measurements to search for emergent statistical laws. We establish a readily replicated microbial closed ecosystem (CES), sustaining its three species for years. We precisely measure the local population density of each species in many CES replicates, started from the same initial conditions and kept under constant light and temperature. The covariation among replicates of the three species densities acquires a stable structure, which could be decomposed into discrete eigenvectors, or "ecomodes." The largest ecomode dominates population density fluctuations around the replicate-average dynamics. These fluctuations follow simple power laws consistent with a geometric random walk. Thus, variability in ecological dynamics can be studied with CES replicates and described by simple statistical laws.     

Influence of molecular noise on the growth of single cells and bacterial populations

During the last decades experimental studies have revealed that single cells of a growing bacterial population are significantly exposed to molecular noise. Important sources for noise are low levels of metabolites and enzymes that cause significant statistical variations in the outcome of biochemical reactions. In this way molecular noise affects biological processes such as nutrient uptake, chemotactic tumbling behavior, or gene expression of genetically identical cells. These processes give rise to significant cell-to-cell variations of many directly observable quantities such as protein levels, cell sizes or individual doubling times. In this study we theoretically explore if there are evolutionary benefits of noise for a growing population of bacteria. We analyze different situations where noise is either suppressed or where it affects single cell behavior. We consider two specific examples that have been experimentally observed in wild-type Escherichia coli cells: (i) the precision of division site placement (at which molecular noise is highly suppressed) and (ii) the occurrence of noise-induced phenotypic variations in fluctuating environments. Surprisingly, our analysis reveals that in these specific situations both regulatory schemes [i.e. suppression of noise in example (i) and allowance of noise in example (ii)] do not lead to an increased growth rate of the population. Assuming that the observed regulatory schemes are indeed caused by the presence of noise our findings indicate that the evolutionary benefits of noise are more subtle than a simple growth advantage for a bacterial population in nutrient rich conditions.     

A model for the length of tracts of identity by descent in finite random mating populations

Linkage disequilibrium (LD) reflects coinheritance of an ancestral segment by chromosomes in a population. To begin to understand the effects of population history on the extent of LD, we model the length of a tract of identity-by-descent (IBD) between two chromosomes in a finite, random mating population. The variance of an IBD tract is large: a model described by (Genet. Res. Cambridge 35 (1980) 131) underestimates this variance. Using Fisher's concept of junctions, we predict the mean length of an IBD tract, given the age of the population and the population sizes over time. We derive results also for subdivided populations, given times of subdivision events and sizes of the resulting subpopulations. The model demonstrates that population growth and subdivision strongly affect the expected length of an IBD tract in small populations. These effects are less dramatic in large populations.     

Label-Free Detection of Neuronal Differentiation in Cell Populations Using High-Throughput Live-Cell Imaging of PC12 Cells

Detection of neuronal cell differentiation is essential to study cell fate decisions under various stimuli and/or environmental conditions. Many tools exist that quantify differentiation by neurite length measurements of single cells. However, quantification of differentiation in whole cell populations remains elusive so far. Because such populations can consist of both proliferating and differentiating cells, the task to assess the overall differentiation status is not trivial and requires a high-throughput, fully automated approach to analyze sufficient data for a statistically significant discrimination to determine cell differentiation. We address the problem of detecting differentiation in a mixed population of proliferating and differentiating cells over time by supervised classification. Using nerve growth factor induced differentiation of PC12 cells, we monitor the changes in cell morphology over days by phase-contrast live-cell imaging. For general applicability, the classification procedure starts out with many features to identify those that maximize discrimination of differentiated and undifferentiated cells and to eliminate features sensitive to systematic measurement artifacts. The resulting image analysis determines the optimal post treatment day for training and achieves a near perfect classification of differentiation, which we confirmed in technically and biologically independent as well as differently designed experiments. Our approach allows to monitor neuronal cell populations repeatedly over days without any interference. It requires only an initial calibration and training step and is thereafter capable to discriminate further experiments. In conclusion, this enables long-term, large-scale studies of cell populations with minimized costs and efforts for detecting effects of external manipulation of neuronal cell differentiation.     

Population density-dependent metal tolerance: One possible basis and its ecological implications

A population density-dependent copper (Cu) resistance mechanism in a gram-negative soil bacterium, strain TDCd1, was shown to be inducible and was accompanied by changes in the protein composition of the outer membrane of the cell envelope. Characteristically, following inoculation of TDCd1 into Cu-supplemented growth media, there was a period of growth inhibition during which the number of individuals tolerant to Cu gradually increased, even though microcultural experiments indicated that some cells died during this period. We concluded that the population density dependency of the resistance mechanism resulted from the interactions between the rate of cell death, the time taken for Cu resistance to develop, and the size of the initial population. Therefore, the ability of relatively large populations of microorganisms to grow in metal-supplemented media under laboratory conditions may have little ecological significance for sparse populations in natural environments.     

A quantitative study of growth variability of tumour cell clones in vitro

Abstract.   Objectives : In this study, we quantify growth variability of tumour cell clones from a human leukaemia cell line. Materials and methods : We have used microplate spectrophotometry to measure growth kinetics of hundreds of individual cell clones from the Molt3 cell line. Growth rate of each clonal population has been estimated by fitting experimental data with the logistic equation. Results : Growth rates were observed to vary between different clones. Up to six clones with growth rates above or below mean growth rate of the parent population were further cloned and growth rates of their offspring were measured. Distribution of growth rates of the subclones did not significantly differ from that of the parent population, thus suggesting that growth variability has an epigenetic origin. To explain observed distributions of clonal growth rates, we have developed a probabilistic model, assuming that fluctuation in the number of mitochondria through successive cell cycles is the leading cause of growth variability. For fitting purposes, we have estimated experimentally by flow cytometry the average maximum number of mitochondria in Molt3 cells. The model fits nicely observed distributions in growth rates; however, cells in which mitochondria were rendered non-functional (ρ⁰ cells) showed only 30% reduction in clonal growth variability with respect to normal cells. Conclusions : A tumour cell population is a dynamic ensemble of clones with highly variable growth rates. At least part of this variability is due to fluctuations in the initial number of mitochondria in daughter cells.