Weak convergence of discrete time non-markovian processes related to selection models in population genetics

We consider discrete time stochastic processes defined by solutions to some non-linear difference equations whose coefficients are autocorrelated random sequences. It is proved that these processes converge weakly in D[0, T] to diffusion processes, under the assumption that the random sequences satisfy some mixing condition. Diffusion approximation for stochastic selection models in population genetics is discussed, as the application of this limit theorem.     

Weak selection and protein evolution

The "nearly neutral" theory of molecular evolution proposes that many features of genomes arise from the interaction of three weak evolutionary forces: mutation, genetic drift, and natural selection acting at its limit of efficacy. Such forces generally have little impact on allele frequencies within populations from generation to generation but can have substantial effects on long-term evolution. The evolutionary dynamics of weakly selected mutations are highly sensitive to population size, and near neutrality was initially proposed as an adjustment to the neutral theory to account for general patterns in available protein and DNA variation data. Here, we review the motivation for the nearly neutral theory, discuss the structure of the model and its predictions, and evaluate current empirical support for interactions among weak evolutionary forces in protein evolution. Near neutrality may be a prevalent mode of evolution across a range of functional categories of mutations and taxa. However, multiple evolutionary mechanisms (including adaptive evolution, linked selection, changes in fitness-effect distributions, and weak selection) can often explain the same patterns of genome variation. Strong parameter sensitivity remains a limitation of the nearly neutral model, and we discuss concave fitness functions as a plausible underlying basis for weak selection.     

Population dynamics in variable environments. IV. Weak ergodicity in the Lotka equation

The Hilbert projective metric is applied to the continuous-time Lotka equation in demography to establish weak ergodicity: populations with the same time-varying fecundity and mortality schedules ultimately have the same age composition. The analysis displays clearly the dynamic content of Lotka's equation and identifies a contraction operator which forces convergence of birth sequences over time. The relationship between primitivity in the discrete (Leslie) and continuous (Lotka) demographic models is made clear.     

Directed evolution of biocatalytic processes

The benefits of applying biocatalysts to organic synthesis, such as their high chemo-, regio-, and enantio-specificity and selectivity, must be seriously considered, especially where chemical routes are unavailable, complex or prohibitively expensive. In cases where a potential biocatalytic route is not yet efficient enough to compete with chemical synthesis, directed evolution, and/or process engineering could be implemented for improvements. While directed evolution has demonstrated great potential to enhance enzyme properties, there will always be some aspects of biocatalytic processes that it does not address. Even where it can be successfully applied, the resources required for its implementation must currently be weighed against the feasibility of, and resources available for developing a chemical synthesis route. Here, we review the potential of combining directed evolution with process engineering, and recent developments to improve their implementation. Favourable targets for the directed evolution of new biocatalysts are the syntheses of highly complex molecules, especially where chemistry, metabolic engineering or recombineering provide a partial solution. We also review some of the recent advances in the application of these approaches alongside the directed evolution of biocatalysts.     

Neuronal selectivity,population sparseness,and ergodicity in the inferior temporal visual cortex

The sparseness of the encoding of stimuli by single neurons and by populations of neurons is fundamental to understanding the efficiency and capacity of representations in the brain, and was addressed as follows. The selectivity and sparseness of firing to visual stimuli of single neurons in the primate inferior temporal visual cortex were measured to a set of 20 visual stimuli including objects and faces in macaques performing a visual fixation task. Neurons were analysed with significantly different responses to the stimuli. The firing rate distribution of 36% of the neurons was exponential. Twenty-nine percent of the neurons had too few low rates to be fitted by an exponential distribution, and were fitted by a gamma distribution. Interestingly, the raw firing rate distribution taken across all neurons fitted an exponential distribution very closely. The sparseness a ^s or selectivity of the representation of the set of 20 stimuli provided by each of these neurons (which takes a maximal value of 1.0) had an average across all neurons of 0.77, indicating a rather distributed representation. The sparseness of the representation of a given stimulus by the whole population of neurons, the population sparseness a ^p, also had an average value of 0.77. The similarity of the average single neuron selectivity a ^s and population sparseness for any one stimulus taken at any one time a ^p shows that the representation is weakly ergodic. For this to occur, the different neurons must have uncorrelated tuning profiles to the set of stimuli.     

Polynucleotide evolution and branching processes

The theory of multitype branching processes is applied to the kinetics of polynucleotide replication. The results obtained are compared with the solutions of the deterministic differential equations of conventional chemical kinetics.     

The evolution of the Uruguayan population

Uruguay is usually considered an immigrants' country that, unlike other South American countries, has no native population and only a slight African contribution. The perception of its identity frequently refers to this characteristic: there are no regional differences and population variability is minimal. The object of this paper is to analyze the processes that shaped the present population and the regional diversity in their historical context. For this purpose parish and civil archives of three Departments were reviewed for the period 1840–1990: Cerro Largo and Tacuarembo, in the Northeast and Montevideo, in the south. The origin of the population, individual migration distance, consanguinity and marriage age were taken into account. With the exception of age, differences were found between the capital, Montevideo, and the Northeast for the selected variables during most of the period analyzed. On the basis of these results, the importance of Montevideo as the point of entry of both immigrants and slaves, the regional homogeneity and the concept of national identity are discussed.     

The evolution of population stability as a by-product of life-history evolution

Proposed mechanisms for the evolution of population stability include group selection through longterm persistence, individual selection acting directly on stability determining the demographic parameters, and the evolution of stability as a by-product of life-history evolution. None of these hypotheses currently has clear empirical support. Using two sets of Drosophila melanogaster populations, we provide experimental evidence of stability evolving as a correlated response to selection on traits not directly related to demography. Four populations (FEJs) were selected for faster development and early reproduction for 125 generations, and the other four (JBs) were ancestral controls. All FEJ and JB populations have been maintained on discrete generations at moderate density, thus eliminating differential selection on stability determining demographic parameters. We derived eight small populations from each FEJ and JB population, and subjected four small populations each to either stabilizing or destabilizing food regimes. Census data on these 64 small populations over 20 generations clearly showed that the FEJ populations have significantly less temporal fluctuations in their numbers in both food regimes compared to their controls. This greater stability of the FEJ populations is probably a by-product of the evolution of reduced fecundity and pre-adult survivorship, as a correlated response to selection for rapid development.     

Connecting geographical distributions with population processes

The geographical distribution of a species is determined by a large number of complex processes operating over spatial scales spanning 10 orders of magnitude. Patterns in population processes have been described at numerous scales. We show that two patterns, measured at different scales, jointly allow us to infer heretofore unknown patterns in the distribution of demographic patterns across the geographical range of a species. The resulting model describes three fundamentally different modes of geographical variation in vital rates of populations. One mode is characterized by a positive nonlinear relationship between the maximum rate of population growth and the intensity of intraspecific competition across a geographical range. That is, populations that grow rapidly are also those where individuals experience the greatest per capita negative effect of the presence of other individuals. The second mode of behaviour is described by a negative nonlinear relationship between maximum growth rate and density dependence. Under this scenario, populations with low capacity to grow rapidly have highest intensities of intraspecific competitive effects. A third mode of behaviour is characterized by a weak positive relationship between growth rate and intraspecific competition, with very little geographical variation in maximum growth rate. A survey of studies relating temporal means and variances in population abundance for a variety of species indicate that the second mode of geographical variation in population dynamics across species ranges is the most common, though a few species appear to be characterized by the third mode.     

Conditional diffusion processes in population genetics

Modern population genetics is involved largely with molecular processes and in particular with the observed outcome of an evolutionary molecular process. The investigation of data of this form requires a mathematical apparatus of conditional diffusion processes. The relevant theory is given in this paper. Many results can be obtained more or less directly: however, the complete theory requires formation of the conditional diffusion equation. The form of this equation, together with several consequences flowing from it, are given.     

Ancestral processes in population genetics-the coalescent

A special stochastic process, called the coalescent, is of fundamental interest in population genetics. For a large class of population models this process is the appropriate tool to analyse the ancestral structure of a sample of n individuals or genes, if the total number of individuals in the population is sufficiently large. A corresponding convergence theorem was first proved by Kingman in 1982 for the Wright-Fisher model and the Moran model. Generalizations to a large class of exchangeable population models and to models with overlying mutation processes followed shortly later. One speaks of the "robustness of the coalescent, as this process appears in many models as the total population size tends to infinity. This publication can be considered as an introduction to the theory of the coalescent as well as a review of the most important "convergence-to-the-coalescent-theorems. Convergence theorems are not only presented for the classical exchangeable haploid case but also for larger classes of population models, for example for diploid, two-sex or non-exchangeable models. A review-like summary of further examples and applications of convergence to the coalescent is given including the most important biological forces like mutation, recombination and selection. The general coalescent process allows for simultaneous multiple mergers of ancestral lines.     

Markov population processes as models of primate social and population dynamics

Thanks to recently developed theory of Markov population processes, models of how an individual primate migrates from one casual social group to another or from one breeding troop to another can now deal exactly with transition rates which depend nonlinearly on the sizes of both the group (or troop) left and the group (or troop) entered. Examples of such models presented here are consistent with existing observations of primate social and population dynamics and are more plausible as explanations of these data than previous linear models.     

Developmental processes and the evolution of plant clonality

A common type of clonal plant grows limited ramets that are connected by modified stems. Three major developmental processes are required for this clonality: the differentiation of specialized plagiotropic stems which act as spacers, the localized formation of adventitious roots, and a limitation of vertical development which is coupled with repeated rejuvenation. Intermediate forms, in which one of these processes occurs without the others, are readily found. Studies of apical initiation and differentiation in a-clonal plants suggest mechanisms for these processes, based on modified hormonal correlations and maturation processes within apices. The consideration of developmental processes, which has been relatively neglected, can therefore be a key to understanding the possibilities and limitations of clonal evolution. For example, comparative developmental studies point to convergent or parallel evolution, in which a similar outcome has been based on different developmental mechanisms. A consideration of developmental processes also throws new light on clonal organization and raises concrete questions that are amenable to experimental work.     

Some models of population structure and evolution

Models of population structure analyzed include: the uniform model (k populations all exchanging at a constant rate) and the multiuniform model (s clusters of k populations exchanging at different rates within and between clusters). Analogies of the latter model with trees of descent have been shown. The algorithm used allows exact solutions for the equilibrium variances and covariances in a variety of cases, including the circular stepping-stone model.Effects of multiple sources of stabilizing pressures have been shown in the multiuniform and the star model. The latter is of interest in situations of population radiation and of centripetal migration.