Estimation of population growth or decline in genetically monitored populations

Mathematical Modeling of Biofilm Structures Using COMSTAT Data

Mathematical modeling holds great potential for quantitatively describing biofilm growth in presence or absence of chemical agents used to limit or promote biofilm growth. In this paper, we describe a general mathematical/statistical framework that allows for the characterization of complex data in terms of few parameters and the capability to (i) compare different experiments and exposures to different agents, (ii) test different hypotheses regarding biofilm growth and interaction with different agents, and (iii) simulate arbitrary administrations of agents. The mathematical framework is divided to submodels characterizing biofilm, including new models characterizing live biofilm growth and dead cell accumulation; the interaction with agents inhibiting or stimulating growth; the kinetics of the agents. The statistical framework can take into account measurement and interexperiment variation. We demonstrate the application of (some of) the models using confocal microscopy data obtained using the computer program COMSTAT.

Pseudo-marginal Metropolis–Hastings sampling using averages of unbiased estimators

We consider a pseudo-marginal Metropolis–Hastings kernel ${\mathbb{P}}_m$ that is constructed using an average of $m$ exchangeable random variables, and an analogous kernel ${\mathbb{P}}_s$ that averages $s<m$ of these same random variables. Using an embedding technique to facilitate comparisons, we provide a lower bound for the asymptotic variance of any ergodic average associated with ${\mathbb{P}}_m$ in terms of the asymptotic variance of the corresponding ergodic average associated with ${\mathbb{P}}_s$. We show that the bound is tight and disprove a conjecture that when the random variables to be averaged are independent, the asymptotic variance under ${\mathbb{P}}_m$ is never less than $s/m$ times the variance under ${\mathbb{P}}_s$. The conjecture does, however, hold for continuous-time Markov chains. These results imply that if the computational cost of the algorithm is proportional to $m$, it is often better to set $m=1$. We provide intuition as to why these findings differ so markedly from recent results for pseudo-marginal kernels employing particle filter approximations. Our results are exemplified through two simulation studies; in the first the computational cost is effectively proportional to $m$ and in the second there is a considerable start-up cost at each iteration.

Ecological and genetic impact of the 2011 Tohoku Earthquake Tsunami on intertidal mud snails

Natural disturbances often destroy local populations and can considerably affect the genetic properties of these populations. The 2011 Tohoku Earthquake Tsunami greatly damaged local populations of various coastal organisms, including the mud snail Batillaria attramentaria, which was an abundant macroinvertebrate on the tidal flats in the Tohoku region. To evaluate the impact of the tsunami on the ecology and population genetic properties of these snails, we monitored the density, shell size, and microsatellite DNA variation of B. attramentaria for more than ten years (2005–2015) throughout the disturbance event. We found that the density of snails declined immediately after the tsunami. Bayesian inference of the genetically effective population size (N_e) demonstrated that the N_e declined by 60–99% at the study sites exposed to the tsunami. However, we found that their genetic diversity was not significantly reduced after the tsunami. The maintenance of genetic diversity is essential for long-term survival of local populations, and thus, the observed genetic robustness could play a key role in the persistence of snail populations in this region which has been devastated by similar tsunamis every 500–800 years. Our findings have significant implications for understanding the sustainability of populations damaged by natural disturbances.

Prediction and estimation of effective population size

Effective population size (Ne) is a key parameter in population genetics. It has important applications in evolutionary biology, conservation genetics and plant and animal breeding, because it measures the rates of genetic drift and inbreeding and affects the efficacy of systematic evolutionary forces, such as mutation, selection and migration. We review the developments in predictive equations and estimation methodologies of effective size. In the prediction part, we focus on the equations for populations with different modes of reproduction, for populations under selection for unlinked or linked loci and for the specific applications to conservation genetics. In the estimation part, we focus on methods developed for estimating the current or recent effective size from molecular marker or sequence data. We discuss some underdeveloped areas in predicting and estimating Ne for future research.

Near-optimal protocols in complex nonequilibrium transformations

The development of sophisticated experimental means to control nanoscale systems has motivated efforts to design driving protocols that minimize the energy dissipated to the environment. Computational models are a crucial tool in this practical challenge. We describe a general method for sampling an ensemble of finite-time, nonequilibrium protocols biased toward a low average dissipation. We show that this scheme can be carried out very efficiently in several limiting cases. As an application, we sample the ensemble of low-dissipation protocols that invert the magnetization of a 2D Ising model and explore how the diversity of the protocols varies in response to constraints on the average dissipation. In this example, we find that there is a large set of protocols with average dissipation close to the optimal value, which we argue is a general phenomenon.

Assessing current genetic status of the Hainan gibbon using historical and demographic baselines: implications for conservation management of species of extreme rarity

Evidence-based conservation planning is crucial for informing management decisions for species of extreme rarity, but collection of robust data on genetic status or other parameters can be extremely challenging for such species. The Hainan gibbon, possibly the world's rarest mammal, consists of a single population of ~25 individuals restricted to one protected area on Hainan Island, China, and has persisted for over 30 years at exceptionally low population size. Analysis of genotypes at 11 microsatellite loci from faecal samples for 36% of the current global population and tissue samples from 62% of existing historical museum specimens demonstrates limited current genetic diversity (Na = 2.27, Ar = 2.24, He  = 0.43); diversity has declined since the 19th century and even further within the last 30 years, representing declines of ~30% from historical levels (Na = 3.36, Ar = 3.29, He  = 0.63). Significant differentiation is seen between current and historical samples (FST  = 0.156, P = 0.0315), and the current population exhibits extremely small Ne (current Ne  = 2.16). There is evidence for both a recent population bottleneck and an earlier bottleneck, with population size already reasonably low by the late 19th century (historical Ne  = 1162.96). Individuals in the current population are related at the level of half- to full-siblings between social groups, and full-siblings or parent-offspring within a social group, suggesting that inbreeding is likely to increase in the future. The species' current reduced genetic diversity must be considered during conservation planning, particularly for expectations of likely population recovery, indicating that intensive, carefully planned management is essential.

Moderate genetic drift is driven by extreme recruitment events in the invasive mollusk Crepidula fornicata

Effective population size (Ne) is a measure of genetic drift and is thus a central parameter in evolution, conservation genetics and invasion biology. Interestingly, in native marine species, Ne is typically several orders of magnitude lower than the census size. This pattern has often been explained by high fecundity, variation in reproductive success and pronounced early mortality, resulting in genetic drift across generations. Data documenting genetic drift and/or Ne in marine invasive species are, however, still scarce. We examined the importance of genetic drift in the invasive species Crepidula fornicata by genotyping 681 juveniles sampled during each annual recruitment peak over nine consecutive years in the Bay of Morlaix (Brittany, France). Observed variations in genetic diversity were partially explained by variation in recruitment intensity. In addition, substantial temporal genetic differentiation was documented (that is, genetic drift), and was attributed to nonrandom variance in the reproductive success of different breeding groups across years in the study species. Using a set of single-sample and temporal estimators for Ne, we estimated Ne to be three or four orders of magnitude smaller than the census size (Nc). On one hand, this reduction in Ne relative to Nc appeared congruent with, although slight higher than, values commonly observed in native marine species. Particular life-history traits of this invasive species may play an important role in buffering genetic drift. On the other hand, Ne still remained far below Nc, hence, possibly reducing the efficiency of selection effects.

Unbiased Bayesian inference for population Markov jump processes via random truncations

We consider continuous time Markovian processes where populations of individual agents interact stochastically according to kinetic rules. Despite the increasing prominence of such models in fields ranging from biology to smart cities, Bayesian inference for such systems remains challenging, as these are continuous time, discrete state systems with potentially infinite state-space. Here we propose a novel efficient algorithm for joint state/parameter posterior sampling in population Markov Jump processes. We introduce a class of pseudo-marginal sampling algorithms based on a random truncation method which enables a principled treatment of infinite state spaces. Extensive evaluation on a number of benchmark models shows that this approach achieves considerable savings compared to state of the art methods, retaining accuracy and fast convergence. We also present results on a synthetic biology data set showing the potential for practical usefulness of our work.

PREDICTIVE MODELING OF CHOLERA OUTBREAKS IN BANGLADESH

Despite seasonal cholera outbreaks in Bangladesh, little is known about the relationship between environmental conditions and cholera cases. We seek to develop a predictive model for cholera outbreaks in Bangladesh based on environmental predictors. To do this, we estimate the contribution of environmental variables, such as water depth and water temperature, to cholera outbreaks in the context of a disease transmission model. We implement a method which simultaneously accounts for disease dynamics and environmental variables in a Susceptible-Infected-Recovered-Susceptible (SIRS) model. The entire system is treated as a continuous-time hidden Markov model, where the hidden Markov states are the numbers of people who are susceptible, infected, or recovered at each time point, and the observed states are the numbers of cholera cases reported. We use a Bayesian framework to fit this hidden SIRS model, implementing particle Markov chain Monte Carlo methods to sample from the posterior distribution of the environmental and transmission parameters given the observed data. We test this method using both simulation and data from Mathbaria, Bangladesh. Parameter estimates are used to make short-term predictions that capture the formation and decline of epidemic peaks. We demonstrate that our model can successfully predict an increase in the number of infected individuals in the population weeks before the observed number of cholera cases increases, which could allow for early notification of an epidemic and timely allocation of resources.

Coalescent Inference Using Serially Sampled, High-Throughput Sequencing Data from Intrahost HIV Infection

Human immunodeficiency virus (HIV) is a rapidly evolving pathogen that causes chronic infections, so genetic diversity within a single infection can be very high. High-throughput "deep" sequencing can now measure this diversity in unprecedented detail, particularly since it can be performed at different time points during an infection, and this offers a potentially powerful way to infer the evolutionary dynamics of the intrahost viral population. However, population genomic inference from HIV sequence data is challenging because of high rates of mutation and recombination, rapid demographic changes, and ongoing selective pressures. In this article we develop a new method for inference using HIV deep sequencing data, using an approach based on importance sampling of ancestral recombination graphs under a multilocus coalescent model. The approach further extends recent progress in the approximation of so-called conditional sampling distributions, a quantity of key interest when approximating coalescent likelihoods. The chief novelties of our method are that it is able to infer rates of recombination and mutation, as well as the effective population size, while handling sampling over different time points and missing data without extra computational difficulty. We apply our method to a data set of HIV-1, in which several hundred sequences were obtained from an infected individual at seven time points over 2 years. We find mutation rate and effective population size estimates to be comparable to those produced by the software BEAST. Additionally, our method is able to produce local recombination rate estimates. The software underlying our method, Coalescenator, is freely available.

Simulation-based Bayesian Analysis of Complex Data

Our ability to collect large datasets is growing rapidly. Such richness of data offers great promise in terms of addressing detailed scientific questions in great depth. However, this benefit is not without scientific difficulty: many traditional analysis methods become computationally intractable for very large datasets. However, one can frequently still simulate data from scientific models for which direct calculation is no longer possible. In this paper we propose a Bayesian perspective for such analyses, and argue for the advantage of a simulation-based approximate Bayesian method that remains tractable when tractability of other methods is lost. This method, which is known as "approximate Bayesian computation" [ABC], has now been used in a variety of contexts, such as the analysis of tumor data (a tumor being a complex population of cells), and the analysis of human genetic variation data (which arise from a population of individual people). We review a number of ABC methods, with specific attention to the use of ABC in agent-based models, and give pointers to software that allows straightforward implementation of the ABC approach. In this way we demonstrate the utility of simulation-based analyses of large datasets within a rigorous statistical framework.

Stability of noisy Metropolis-Hastings

Pseudo-marginal Markov chain Monte Carlo methods for sampling from intractable distributions have gained recent interest and have been theoretically studied in considerable depth. Their main appeal is that they are exact, in the sense that they target marginally the correct invariant distribution. However, the pseudo-marginal Markov chain can exhibit poor mixing and slow convergence towards its target. As an alternative, a subtly different Markov chain can be simulated, where better mixing is possible but the exactness property is sacrificed. This is the noisy algorithm, initially conceptualised as Monte Carlo within Metropolis, which has also been studied but to a lesser extent. The present article provides a further characterisation of the noisy algorithm, with a focus on fundamental stability properties like positive recurrence and geometric ergodicity. Sufficient conditions for inheriting geometric ergodicity from a standard Metropolis-Hastings chain are given, as well as convergence of the invariant distribution towards the true target distribution.

Exact likelihood-free Markov chain Monte Carlo for elliptically contoured distributions

Recent results in Markov chain Monte Carlo (MCMC) show that a chain based on an unbiased estimator of the likelihood can have a stationary distribution identical to that of a chain based on exact likelihood calculations. In this paper we develop such an estimator for elliptically contoured distributions, a large family of distributions that includes and generalizes the multivariate normal. We then show how this estimator, combined with pseudorandom realizations of an elliptically contoured distribution, can be used to run MCMC in a way that replicates the stationary distribution of a likelihood based chain, but does not require explicit likelihood calculations. Because many elliptically contoured distributions do not have closed form densities, our simulation based approach enables exact MCMC based inference in a range of cases where previously it was impossible.

Generalized correlation integral vectors: A distance concept for chaotic dynamical systems

Several concepts of fractal dimension have been developed to characterise properties of attractors of chaotic dynamical systems. Numerical approximations of them must be calculated by finite samples of simulated trajectories. In principle, the quantities should not depend on the choice of the trajectory, as long as it provides properly distributed samples of the underlying attractor. In practice, however, the trajectories are sensitive with respect to varying initial values, small changes of the model parameters, to the choice of a solver, numeric tolerances, etc. The purpose of this paper is to present a statistically sound approach to quantify this variability. We modify the concept of correlation integral to produce a vector that summarises the variability at all selected scales. The distribution of this stochastic vector can be estimated, and it provides a statistical distance concept between trajectories. Here, we demonstrate the use of the distance for the purpose of estimating model parameters of a chaotic dynamic model. The methodology is illustrated using computational examples for the Lorenz 63 and Lorenz 95 systems, together with a framework for Markov chain Monte Carlo sampling to produce posterior distributions of model parameters.

Statistical inference on genetic data reveals the complex demographic history of human populations in central Asia

The demographic history of modern humans constitutes a combination of expansions, colonizations, contractions, and remigrations. The advent of large scale genetic data combined with statistically refined methods facilitates inference of this complex history. Here we study the demographic history of two genetically admixed ethnic groups in Central Asia, an area characterized by high levels of genetic diversity and a history of recurrent immigration. Using Approximate Bayesian Computation, we infer that the timing of admixture markedly differs between the two groups. Admixture in the traditionally agricultural Tajiks could be dated back to the onset of the Neolithic transition in the region, whereas admixture in Kyrgyz is more recent, and may have involved the westward movement of Turkic peoples. These results are confirmed by a coalescent method that fits an isolation-with-migration model to the genetic data, with both Central Asian groups having received gene flow from the extremities of Eurasia. Interestingly, our analyses also uncover signatures of gene flow from Eastern to Western Eurasia during Paleolithic times. In conclusion, the high genetic diversity currently observed in these two Central Asian peoples most likely reflects the effects of recurrent immigration that likely started before historical times. Conversely, conquests during historical times may have had a relatively limited genetic impact. These results emphasize the need for a better understanding of the genetic consequences of transmission of culture and technological innovations, as well as those of invasions and conquests.

Inferring epidemiological dynamics with Bayesian coalescent inference: the merits of deterministic and stochastic models

Estimation of epidemiological and population parameters from molecular sequence data has become central to the understanding of infectious disease dynamics. Various models have been proposed to infer details of the dynamics that describe epidemic progression. These include inference approaches derived from Kingman’s coalescent theory. Here, we use recently described coalescent theory for epidemic dynamics to develop stochastic and deterministic coalescent susceptible–infected–removed (SIR) tree priors. We implement these in a Bayesian phylogenetic inference framework to permit joint estimation of SIR epidemic parameters and the sample genealogy. We assess the performance of the two coalescent models and also juxtapose results obtained with a recently published birth–death-sampling model for epidemic inference. Comparisons are made by analyzing sets of genealogies simulated under precisely known epidemiological parameters. Additionally, we analyze influenza A (H1N1) sequence data sampled in the Canterbury region of New Zealand and HIV-1 sequence data obtained from known United Kingdom infection clusters. We show that both coalescent SIR models are effective at estimating epidemiological parameters from data with large fundamental reproductive number R0 and large population size S0. Furthermore, we find that the stochastic variant generally outperforms its deterministic counterpart in terms of error, bias, and highest posterior density coverage, particularly for smaller R0 and S0. However, each of these inference models is shown to have undesirable properties in certain circumstances, especially for epidemic outbreaks with R0 close to one or with small effective susceptible populations.

Pseudo-Marginal Bayesian Inference for Gaussian Processes

The main challenges that arise when adopting Gaussian process priors in probabilistic modeling are how to carry out exact Bayesian inference and how to account for uncertainty on model parameters when making model-based predictions on out-of-sample data. Using probit regression as an illustrative working example, this paper presents a general and effective methodology based on the pseudo-marginal approach to Markov chain Monte Carlo that efficiently addresses both of these issues. The results presented in this paper show improvements over existing sampling methods to simulate from the posterior distribution over the parameters defining the covariance function of the Gaussian Process prior. This is particularly important as it offers a powerful tool to carry out full Bayesian inference of Gaussian Process based hierarchic statistical models in general. The results also demonstrate that Monte Carlo based integration of all model parameters is actually feasible in this class of models providing a superior quantification of uncertainty in predictions. Extensive comparisons with respect to state-of-the-art probabilistic classifiers confirm this assertion.

AABC: approximate approximate Bayesian computation for inference in population-genetic models

Approximate Bayesian computation (ABC) methods perform inference on model-specific parameters of mechanistically motivated parametric models when evaluating likelihoods is difficult. Central to the success of ABC methods, which have been used frequently in biology, is computationally inexpensive simulation of data sets from the parametric model of interest. However, when simulating data sets from a model is so computationally expensive that the posterior distribution of parameters cannot be adequately sampled by ABC, inference is not straightforward. We present "approximate approximate Bayesian computation" (AABC), a class of computationally fast inference methods that extends ABC to models in which simulating data is expensive. In AABC, we first simulate a number of data sets small enough to be computationally feasible to simulate from the parametric model. Conditional on these data sets, we use a statistical model that approximates the correct parametric model and enables efficient simulation of a large number of data sets. We show that under mild assumptions, the posterior distribution obtained by AABC converges to the posterior distribution obtained by ABC, as the number of data sets simulated from the parametric model and the sample size of the observed data set increase. We demonstrate the performance of AABC on a population-genetic model of natural selection, as well as on a model of the admixture history of hybrid populations. This latter example illustrates how, in population genetics, AABC is of particular utility in scenarios that rely on conceptually straightforward but potentially slow forward-in-time simulations.

Population genetics inference for longitudinally-sampled mutants under strong selection

Longitudinal allele frequency data are becoming increasingly prevalent. Such samples permit statistical inference of the population genetics parameters that influence the fate of mutant variants. To infer these parameters by maximum likelihood, the mutant frequency is often assumed to evolve according to the Wright–Fisher model. For computational reasons, this discrete model is commonly approximated by a diffusion process that requires the assumption that the forces of natural selection and mutation are weak. This assumption is not always appropriate. For example, mutations that impart drug resistance in pathogens may evolve under strong selective pressure. Here, we present an alternative approximation to the mutant-frequency distribution that does not make any assumptions about the magnitude of selection or mutation and is much more computationally efficient than the standard diffusion approximation. Simulation studies are used to compare the performance of our method to that of the Wright–Fisher and Gaussian diffusion approximations. For large populations, our method is found to provide a much better approximation to the mutant-frequency distribution when selection is strong, while all three methods perform comparably when selection is weak. Importantly, maximum-likelihood estimates of the selection coefficient are severely attenuated when selection is strong under the two diffusion models, but not when our method is used. This is further demonstrated with an application to mutant-frequency data from an experimental study of bacteriophage evolution. We therefore recommend our method for estimating the selection coefficient when the effective population size is too large to utilize the discrete Wright–Fisher model.

Investigating population history using temporal genetic differentiation

The rapid advance of sequencing technology, coupled with improvements in molecular methods for obtaining genetic data from ancient sources, holds the promise of producing a wealth of genomic data from time-separated individuals. However, the population-genetic properties of time-structured samples have not been extensively explored. Here, we consider the implications of temporal sampling for analyses of genetic differentiation and use a temporal coalescent framework to show that complex historical events such as size reductions, population replacements, and transient genetic barriers between populations leave a footprint of genetic differentiation that can be traced through history using temporal samples. Our results emphasize explicit consideration of the temporal structure when making inferences and indicate that genomic data from ancient individuals will greatly increase our ability to reconstruct population history.

Simultaneous reconstruction of evolutionary history and epidemiological dynamics from viral sequences with the birth-death SIR model

The evolution of RNA viruses, such as human immunodeficiency virus (HIV), hepatitis C virus and influenza virus, occurs so rapidly that the viruses' genomes contain information on past ecological dynamics. Hence, we develop a phylodynamic method that enables the joint estimation of epidemiological parameters and phylogenetic history. Based on a compartmental susceptible–infected–removed (SIR) model, this method provides separate information on incidence and prevalence of infections. Detailed information on the interaction of host population dynamics and evolutionary history can inform decisions on how to contain or entirely avoid disease outbreaks. We apply our birth–death SIR method to two viral datasets. First, five HIV type 1 clusters sampled in the UK between 1999 and 2003 are analysed. The estimated basic reproduction ratios range from 1.9 to 3.2 among the clusters. All clusters show a decline in the growth rate of the local epidemic in the middle or end of the 1990s. The analysis of a hepatitis C virus genotype 2c dataset shows that the local epidemic in the Córdoban city Cruz del Eje originated around 1906 (median), coinciding with an immigration wave from Europe to central Argentina that dates from 1880 to 1920. The estimated time of epidemic peak is around 1970.

Effective population size, genetic variation, and their relevance for conservation: the bighorn sheep in Tiburon Island and comparisons with managed artiodactyls

The amount of genetic diversity in a finite biological population mostly depends on the interactions among evolutionary forces and the effective population size (N(e)) as well as the time since population establishment. Because the N(e) estimation helps to explore population demographic history, and allows one to predict the behavior of genetic diversity through time, N(e) is a key parameter for the genetic management of small and isolated populations. Here, we explored an N(e)-based approach using a bighorn sheep population on Tiburon Island, Mexico (TI) as a model. We estimated the current (N(crnt)) and ancestral stable (N(stbl)) inbreeding effective population sizes as well as summary statistics to assess genetic diversity and the demographic scenarios that could explain such diversity. Then, we evaluated the feasibility of using TI as a source population for reintroduction programs. We also included data from other bighorn sheep and artiodactyl populations in the analysis to compare their inbreeding effective size estimates. The TI population showed high levels of genetic diversity with respect to other managed populations. However, our analysis suggested that TI has been under a genetic bottleneck, indicating that using individuals from this population as the only source for reintroduction could lead to a severe genetic diversity reduction. Analyses of the published data did not show a strict correlation between H(E) and N(crnt) estimates. Moreover, we detected that ancient anthropogenic and climatic pressures affected all studied populations. We conclude that the estimation of N(crnt) and N(stbl) are informative genetic diversity estimators and should be used in addition to summary statistics for conservation and population management planning.