Identification of age-structured models: cell cycle phase transitions

A methodology is developed that determines age-specific transition rates between cell cycle phases during balanced growth by utilizing age-structured population balance equations. Age-distributed models are the simplest way to account for varied behavior of individual cells. However, this simplicity is offset by difficulties in making observations of age distributions, so age-distributed models are difficult to fit to experimental data. Herein, the proposed methodology is implemented to identify an age-structured model for human leukemia cells (Jurkat) based only on measurements of the total number density after the addition of bromodeoxyuridine partitions the total cell population into two subpopulations. Each of the subpopulations will temporarily undergo a period of unbalanced growth, which provides sufficient information to extract age-dependent transition rates, while the total cell population remains in balanced growth. The stipulation of initial balanced growth permits the derivation of age densities based on only age-dependent transition rates. In fitting the experimental data, a flexible transition rate representation, utilizing a series of cubic spline nodes, finds a bimodal G(0)/G(1) transition age probability distribution best fits the experimental data. This resolution may be unnecessary as convex combinations of more restricted transition rates derived from normalized Gaussian, lognormal, or skewed lognormal transition-age probability distributions corroborate the spline predictions, but require fewer parameters. The fit of data with a single log normal distribution is somewhat inferior suggesting the bimodal result as more likely. Regardless of the choice of basis functions, this methodology can identify age distributions, age-specific transition rates, and transition-age distributions during balanced growth conditions.     

Times from Infection to Disease-Induced Death and their Influence on Final Population Sizes After Epidemic Outbreaks

For epidemic models, it is shown that fatal infectious diseases cannot drive the host population into extinction if the incidence function is upper density-dependent. This finding holds even if a latency period is included and the time from infection to disease-induced death has an arbitrary length distribution. However, if the incidence function is also lower density-dependent, very infectious diseases can lead to a drastic decline of the host population. Further, the final population size after an epidemic outbreak can possibly be substantially affected by the infection-age distribution of the initial infectives if the life expectations of infected individuals are an unbounded function of infection age (time since infection). This is the case for lognormal distributions, which fit data from infection experiments involving tiger salamander larvae and ranavirus better than gamma distributions and Weibull distributions.     

Reversibility and the age of an allele II. Two-allele models,with selection and mutation

A Markov process with absorbing boundaries may be made recurrent by returning the process to the interior whenever a boundary is reached. The age of such a process may be defined as the length of time since the last return event. Examples drawn from two-allele genetic models are discussed, in which reversibility of the return process means that the age of an allele, whose present frequency in the population is known, has the same probability distribution as its future extinction time. Some discrete models are not reversible, yet if approximated by diffusion processes, the (approximate) age distribution is the same as the future extinction time distribution. Various results in the literature are unified by this viewpoint.     

A cytokinetic model for heterogeneous mammalian cell populations. I. Cell growth and cell death

A general model is proposed for describing the growth behavior of mammalian cell populations, which features:(a) a cell cycle time distribution function with properties such that mean and variance increase with increasing population size; (b) maturation age and maturation rate functions which constrain the maturational pathways of individual cells; and (c) a death rate function, where cell death is construed as irreparable damage to a cell's reproductive apparatus. The biological implications of the model are discussed, and methods for relating the model to real cell systems by means of commonly used experimental techniques are described. The model is compared with earlier models.     

Evolution of dispersal in density regulated populations: A haploid model

The evolution of dispersal is explored in a density-dependent framework. Attention is restricted to haploid populations in which the genotypic fitnesses at a single diallelic locus are decreasing functions of the changing number of individuals in the population. It is shown that migration between two populations in which the genotypic response to density is reversed can maintain both alleles when the intermigration rates are constant or nondecreasing functions of the population densities. There is always a unique symmetric interior equilibrium with equal numbers but opposite gene frequencies in the two populations, provided the system is not degenerate. Numerical examples with exponential and hyperbolic fitnesses suggest that this is the only stable equilibrium state under constant positive migration rates (m) less than . Practically speaking, however, there is only convergence after a reasonable number of generations for relatively small migration rates ( ). A migration-modifying mutant at a second, neutral locus, can successfully enter two populations at a stable migration-selection balance if and only if it reduces the intermigration rates of its carriers at the original equilibrium population size. Moreover, migration modification will always result in a higher equilibrium population size, provided the system approaches another symmetric interior equilibrium. The new equilibrium migration rate will be lower than that at the original equilibrium, even when the modified migration rate is a nondecreasing function of the population sizes. Therefore, as in constant viability models, evolution will lead to reduced dispersal.     

Effective population size of a population with stochastically varying size

For a Wright–Fisher model with mutation whose population size fluctuates stochastically from generation to generation, a heterozygosity effective population size is defined by means of the equilibrium average heterozygosity of the population. It is shown that this effective population size is equal to the harmonic mean of population size if and only if the stochastic changes of population size are uncorrelated. The effective population size is larger (resp. smaller) than the harmonic mean when the stochastic changes of population size are positively (resp. negatively) autocorrelated. These results and those obtained so far for other stochastic models with fluctuating population size suggest that the property that effective population sizes are always larger than the harmonic mean under the fluctuation of population size holds only for continuous time models such as diffusion and coalescent models, whereas effective population sizes can be equal to or smaller than the harmonic mean for discrete time models.     

Piecewise exponential survival curves with smooth transitions

Several models of a population survival curve composed of two piecewise exponential distributions are developed. In one formulation the hazard rate changes at a point that is an unobservable random variable that varies between individuals. The population hazard function may decrease with age even when all individuals' hazards are increasing. In a second formulation, the population hazard function is modeled directly. Several models are fit to the survival history of a cohort of 5751 highly inbred male Drosophila melanogaster and the British coal mining disaster data.     

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.     

Widespread conservation of genetic redundancy during a billion years of eukaryotic evolution

Genetic redundancy means that two genes can perform the same function. Using a comprehensive phylogenetic analysis, we show here in both Saccharomyces cerevisiae and Caenorhabditis elegans that genetic redundancy is not just a transient consequence of gene duplication, but is often an evolutionary stable state. In multiple examples, genes have retained redundant functions since the divergence of the animal, plant and fungi kingdoms over a billion years ago. The stable conservation of genetic redundancy contrasts with the more rapid evolution of genetic interactions between unrelated genes and can be explained by theoretical models including a 'piggyback' mechanism in which overlapping redundant functions are co-selected with nonredundant ones.     

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.     

Structured models of metapopulation dynamics

I develop models of metapopulation dynamics that describe changes in the numbers of individuals within patches. These models are analogous to structured population models, with patches playing the role of individuals. Single species models which do not include the effect of immigration on local population dynamics of occupied patches typically lead to a unique equilibrium. The models can be used to study the distributions of numbers of individuals among patches, showing that both metapopulations with local outbreaks and metapopulations without outbreaks can occur in systems with no underlying environmental variability. Distributions of local population sizes (in occupied patches) can vary independently of the total population size, so both patterns of distributions of local population sizes are compatible with either rare or common species. Models which include the effect of immigration on local population dynamics can lead to two positive equilibria, one stable and one unstable, the latter representing a threshold between regional extinction and persistence.     

Co-Occurrence of Preference Functions and Acceptance Thresholds in Female Choice: Mate Discrimination in the Lesser Wax Moth

Basic economic models adapted from foraging theory predict that decisions in mate choice may be determined either by ‘best‐of‐n’ preference functions or by sequential rules incorporating acceptance thresholds. However, in some species, more complex determinations that incorporate versions of both protocols are found. To understand the functions of co‐occurring protocols, we studied mating decisions in the lesser wax moth, Achroia grisella (Lepidoptera: Pyralidae), an acoustic species in which females prefer males, the advertisement songs of which are delivered at relatively high ‘pulse‐pair’ rates. In addition to this preference, A. grisella females avoid mating with a male, the song of which does not exceed a minimum pulse‐pair rate, and they hold to this criterion even when no other singing males are present and regardless of song amplitude. Thus, mating decisions are not simply based on acoustic power (pulse‐pair rate × amplitude). We recorded male songs and female responses in an A. grisella population and found that male pulse‐pair rates showed a median of 87/s and ranged from 50 to 115/s, while female acceptance thresholds for male song showed a median of 60/s and ranged from 30 to 105/s. The distributions of thresholds were approximately normal and were not significantly skewed toward the right. Male song rates declined slightly with age, but female thresholds remained stable over the adult lifespan. Both the male and female traits showed significant repeatability within individuals. Whereas phylogenetic inference indicates that hearing in pyralid moths originated as a means of avoiding predation by insectivorous bats, the specific distribution of female acceptance thresholds suggests that currently this protocol does not primarily function to preclude inappropriate, and potentially lethal, responses to bat echolocations: pulse rates in the searching‐phase echolocations used by either aerial‐hawking or substrate‐gleaning bats mostly range from 10 to 20/s, and the lack of positive skew in the distribution of thresholds indicates an absence of directional selection from the left. Rather, we infer that thresholds augment preference functions in A. grisella by precluding mating with males which are markedly inferior in a critical song character. In general, co‐occurring protocols may be important where population density fluctuates markedly, as preference functions may be ineffective in preventing mating with inferior males when density is low.     

Stage duration distributions in matrix population models

Matrix population models are a standard tool for studying stage‐structured populations, but they are not flexible in describing stage duration distributions. This study describes a method for modeling various such distributions in matrix models. The method uses a mixture of two negative binomial distributions (parametrized using a maximum likelihood method) to approximate a target (true) distribution. To examine the performance of the method, populations consisting of two life stages (juvenile and adult) were considered. The juvenile duration distribution followed a gamma distribution, lognormal distribution, or zero‐truncated (over‐dispersed) Poisson distribution, each of which represents a target distribution to be approximated by a mixture distribution. The true population growth rate based on a target distribution was obtained using an individual‐based model, and the extent to which matrix models can approximate the target dynamics was examined. The results show that the method generally works well for the examined target distributions, but is prone to biased predictions under some conditions. In addition, the method works uniformly better than an existing method whose performance was also examined for comparison. Other details regarding parameter estimation and model development are also discussed.     

On the Gurtin-MacCamy conjecture concerning the stability of a simple nonlinear age-dependent population model

The conjecture that an age-dependent population model, involving a survival function dependent only on the population and a fertility function dependent on population and exponentially on age, could not entail Hopf bifurcation into stable orbits is shown to be incorrect.     

Age distribution and fertility of populations of the arctic-alpine species Oxyria digyna

Age distributions and fruit production of populations of Oxyria digyna (L) Hill were studied in Norway and Greenland. The observed age distributions were described by the negative exponential function and by the power function. Three kinds of distributions were observed: steep, flat and bell-shaped with the corresponding hint production: 33–917, 8–108 and 76–303 fruits per individual. The population with the steep age distributions and high fruit productions were assumed to be increasing. They were found in sites with almost no vegetation (≤10% cover). The populations with the flat age distributions and the low fruit production were assumed to be stable. They were found in sites with a vegetational cover of 50–75%. The populations with the bell-shaped age distributions and the intermediate fruit productions were assumed to be decreasing. They were found in sites with a closed vegetation (cover ≥100%).     

Density-dependent selection migration model with non-monotone fitness functions

This paper studies the classical single locus, diallelic selection model with diffusion for a continuously reproducing population. The phase variables are population density and allele frequency (or allele density). The genotype fitness depend only on population density but include one-hump functions of the density variable. With mild assumptions on genotype fitnesses, we study the geometry of the nullclines and the asymptotic behavior of solutions of the selection model without diffusion. For the diffusion model with zero Neumann boundary conditions, we use this geometric information to show that if the initial data satisfy certain conditions then the corresponding solution to the reaction-diffusion equation converges to the spatially constant stable equilibrium which is closest to the initial data.Research partially supported by NSF grant DMS-8920597Research supported by funds provided by the USDA-Forest Service, Southeastern Forest Experiment Station, Pioneering (Population Genetics of Forest Trees) Research Unit, Raleigh, North Carolina     

Using radius frequency distribution functions as a metric for quantifying root systems

Root radius frequency distributions have been measured to quantify the effect of plant type, environment and methodology on root systems, however, to date the results of such studies have not been synthesised. We propose that cumulative frequency distribution functions can be used as a metric to describe root systems because (1) statistical properties of the frequency distribution can be determined, (2) the parameters for these can be used as a means of comparison, and (3) the analytical expressions can be easily incorporated into models that are dependent upon root geometry. We collated a database of 96 root radii frequency distributions and botanical and methodology traits for each distribution. To determine if there was a frequency distribution function that was best suited to root radii measurements we fitted the exponential, Rayleigh, normal, log-normal, logistic and Weibull cumulative distribution functions to each distribution in our database. We found that the log-normal function provided the best fit to these distributions and that none of the distribution functions was better or worse suited to particular shapes. We derived analytical expressions for root surface and volume and found that they are a valid, and simpler method for incorporating root architectural traits into analytical models. We also found that growth habit and growth media had a significant effect on mean root radius.     

On the number of segregating sites in genetical models without recombination.

The distribution is obtained for the number of segregating sites observed in a sample from a population which is subject to recurring, new, mutations but not subject to recombination. After allowance is made for the different effective population sizes, the results apply approximately to three population models, due to Wright, Burrows and Cockerham, and Moran. Included as extreme special cases are the distributions of the number of segregating sites in the whole population and of the number of heterozygous sites in a diploid individual. Some results of Fisher, Haldane, Kimura, and Ewens concerning the means of the distributions for different models are confirmed, but the variances, and the distributions themselves, are new.     

No‐analog climates and shifting realized niches during the late quaternary: implications for 21st‐century predictions by species distribution models

Empirically derived species distributions models (SDMs) are increasingly relied upon to forecast species vulnerabilities to future climate change. However, many of the assumptions of SDMs may be violated when they are used to project species distributions across significant climate change events. In particular, SDM's in theory assume stable fundamental niches, but in practice, they assume stable realized niches. The assumption of a fixed realized niche relative to climate variables remains unlikely for various reasons, particularly if novel future climates open up currently unavailable portions of species’ fundamental niches. To demonstrate this effect, we compare the climate distributions for fossil‐pollen data from 21 to 15 ka bp (relying on paleoclimate simulations) when communities and climates with no modern analog were common across North America to observed modern pollen assemblages. We test how well SDMs are able to project 20th century pollen‐based taxon distributions with models calibrated using data from 21 to 15 ka. We find that taxa which were abundant in areas with no‐analog late glacial climates, such as Fraxinus, Ostrya/Carpinus and Ulmus, substantially shifted their realized niches from the late glacial period to present. SDMs for these taxa had low predictive accuracy when projected to modern climates despite demonstrating high predictive accuracy for late glacial pollen distributions. For other taxa, e.g. Quercus, Picea, Pinus strobus, had relatively stable realized niches and models for these taxa tended to have higher predictive accuracy when projected to present. Our findings reinforce the point that a realized niche at any one time often represents only a subset of the climate conditions in which a taxon can persist. Projections from SDMs into future climate conditions that are based solely on contemporary realized distributions are potentially misleading for assessing the vulnerability of species to future climate change.     

A field test of simple dispersal models as predictors of movement in a cohort of lake-dwelling brook charr

1. Dispersal can be a major determinant of the distribution and abundance of animals, as well as a key mechanism linking behaviour to population dynamics, but progress in understanding dispersal has been hampered by the lack of a general framework for modelling dispersal. 2. This study tested the capacity of simple models to summarize and predict the lake-wide dispersal of an emerging cohort of young-of-the-year brook charr Salvelinus fontinalis, over 12 surveys conducted during a 2-month period. 3. The models are based on two types of dispersal kernel, the normal distribution from a simple diffusion process, and a Laplace distribution depicting exponential decay of the frequency of dispersers away from the point of origin. In all, four models were assessed: one-group diffusion (D1S) and exponential (E1S) models assuming homogeneous dispersal behaviour within the cohort, and two-group diffusion (D2S) and exponential (E2S) models accounting for intrapopulation differences in dispersal between sedentary and mobile individuals. 4. A rigorous cross-validation, based on calibrating the models to the distributions from the first two surveys only and then validating them on the remaining 10 distributions, was used to compare model predictions with observed values for five properties of the dispersal distributions: counts in individual shoreline sections; mean lateral displacement, variance and kurtosis of displacements; and the percentage of long-distance dispersers. 5. Substantial intrapopulation heterogeneity in dispersal behaviour was apparent: 83% of all individuals were estimated to be sedentary and the remainder mobile. Remarkably, the two-group exponential model E2S - calibrated to data from only two surveys conducted 3.5 and 8.5 days after the beginning of emergence - predicted reasonably well all properties of the spatial distribution of the cohort until the end of the study, 7 weeks later. 6. Standardized measures of mobility derived from simple models may lead to better understanding of population dynamics and improved management. Specifically, the ability to accurately predict long-distance dispersal may be critical to assessing population persistence and cohort strength whenever key habitats, such as refugia or productive areas supporting a large proportion of the cohort, are sparsely distributed or distant from the point of origin.