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Effects of pleiotropy and drift on adaptation in static and changing environments

Final Report Summary - PLEIOCHANGE (Effects of pleiotropy and drift on adaptation in static and changing environments)

Quantitative traits are continuously varying biological characteristics and, as such, are a highly relevant part of the description and characterisation of many types of living organism. Typical examples of quantitative traits include morphological attributes, such as height or weight of an individual, physiological properties, such as fat content of muscles or blood pressure, as well as ecological traits, such as stress tolerance and disease resistance. There is long standing evidence that many of such traits are subject to stabilising selection, where the trait value associated with maximum fitness (the optimal trait value) lies at an intermediate value of the trait. A well-known example is the wing length of birds of a given species. After a storm, it is typically found that birds suffer increased mortality if their wing length is larger than the mean wing length. However it is also typically found that birds whose wing length is smaller than the mean wing length also suffer increased mortality. Thus the mean wing length is the optimal value and a deviation from this in either direction results in increased mortality, or equivalently, decreased fitness. This is stabilising selection. It represents a key aspect of the interaction between individuals and their environment (both living and physical), and much theoretical and experimental work has been carried out to understand the implications of this form of selection for the genetic variation displayed by such traits. In a very large (effectively infinite) population, where individuals are described by a single trait, the continuous value of both the traits and the effects of mutations would appear to suggest that any particular trait value, and hence also the associated genetic sequence that produces the trait, would occur at in an effectively infinitesimal proportion of the population. However, when individuals have more than one trait, and a single mutation results in simultaneous changes in more than one trait (such mutations are termed pleiotropic) we would similarly expect that particular values of the different traits in the problem would also occur in infinitesimal proportion of the population. However, in this case, a qualitatively new theoretical phenomenon is known to occur in an infinite asexual population where mutations are pleiotropic and affect three or more quantitative traits that are under stabilising selection.

It has been found that a single optimal genetic sequence, and hence particular trait values, may become predominant in the population. In particular, the sequence occurs at an appreciable (i.e. non-infinitesimal) proportion. This contrasts sharply with the results where only one or two-traits are affected by a mutation, and all genetic sequences at equilibrium are rare. Thus the phenomenon associated with pleiotropic mutations that affect three or more traits, which can be considered to be the generic case, leads to a non-smooth distribution of genotypic (i.e. purely genetic) trait effects, in contrast to what is usually assumed in the literature. Because standing genetic variation is a key factor in the adaptation of a population, such a qualitative change in the genotypic trait distribution is likely to have significant impacts on the response of a population to a changing environment.

Under the assumptions that mutations act pleiotropically, and affect more than two traits, and that selection is stabilising, the primary objective of the research project was to identify the highly nontrivial equilibrium distribution of genetic trait values that occurs when the known infinite population size results are extended to populations of finite size. This is a substantial generalisation from a deterministic theory to an intrinsically stochastic one. We adopted a dual approach to this problem, by using theoretical tools and performing individually based Monte Carlo simulations on the computer.

We proceeded by first modelling the dynamics of an asexual haploid population with a single quantitative trait. This was essential to make the appropriate generalisation to multiple quantitative traits. As is known in the literature, the infinite population case is described by an integral-differential equation and this, under an approximation for two extreme cases, leads to two distinct regimes: (a) the Gaussian (high mutation) regime and (b) the House of Cards (low mutation) regime. We made the plausible assumption that the number of offspring, from ll types of individual in the population, is sufficiently large that: (i) we can replace the number of mutations in a population of a given type by its expected value, and (ii) we can replace the number of individuals (of a given type) that survive viability selection by its expected value. In other words, we assumed we could neglect stochasticity in both mutation and selection. This allowed us to derive a 'semi-stochastic' approximation of trait evolution which is firmly in the tradition adopted for the Wright Fisher model of genetic drift. We made the approximation of discretising the possible trait values (putting them in 'bins'). The full finite population size treatment of this problem is new, and is far more complicated than the infinite population treatment, because the approach adopted (diffusion analysis) leads to a partial differential equation of a distribution which depends on time and a set of other variables representing the frequencies of the different possible trait values in the problem. The resulting multi-dimensional diffusion equation allowed us to show that at mutation-selection-drift equilibrium, the parameter space is effectively two dimensional in character. With u = trait mutation rate, m = standard deviation of mutation effects on the trait, s = strength of selection and N = population size, the variance of the single trait was predicted to be of the form Var(G) = m2x, where = u / (sm2) and = Nsm2, are the two composite parameters. Thus and very simply describe the key features of the problem. These results were tested by simulations, and providing the trait values were 'binned' i.e. were discretised, on a suitably fine mesh, the predictions of the theory were generally found to hold to with good validity and, indeed, to converge towards the well-established results Var(G) = m2x in the House of Card regime (<< 1) and Var(G) = m2x (/2) in the Gaussian regime (>> 1), when population sizes were made large (>> 1).

We then checked the validity of this 'semi-stochastic' diffusion approximation and the more general predictions it led to, namely that the parameter space is effectively two dimensional, by considering various combinations of parameters, s, u and m2 that lead to the same values of the composite parameters and , and, in principle, to the same scaled variance Var(G) / m2 for the single trait. We numerically identified the shape of the function. The behaviour of genetic variances found covered well the analytical approximations under the Gaussian and House of Card regimes.

With the establishment of results for the single trait case, we established the ingredients/methodology to tackle the multiple trait problem. We have developed a general formulation of the problem with multiple traits, in the context of a diffusion analysis. Again, the complexity of the problem is substantially higher than the single trait problem, since a direct numerical analysis, if it were possible (which it is not) would suffer from an extreme form of the 'curse of dimensionality' since the multiple trait problem involves a very large number of variables. While we have not fully completed the analysis and simulations associated with the multiple trait problem, we have carried out simulations and established preliminary results for the case of pleiotropic mutations, where two or three traits are simultaneously affected by a mutation. As in the one-trait situation, the variance seen in simulations converges appropriately in the Gaussian (high-mutation) regime, when the population size is large. The genetic variance does not seem to be changed when traits are subject to pleiotropic mutations. This is consistent with the theory. Simulations in the low mutation (House of Cards regime) are ongoing and take substantially longer to carry out, since one has to wait many generations before sufficient mutations have occurred (because the mutation rate is small) and equilibrium attained.

To summarise, we have broadly improved our understanding of the role that pleiotropy and drift have on the standing genetic variation in finite populations. The analysis has required considerations of deterministic and stochastic processes, and analytical approximations and simulations. We have made substantial progress, and laid the foundations from which all further work can proceed. We will continue to work on this, to put all of the details together, and provide a relatively complete understanding of how pleiotropy, the nature of the mating system, random genetic drift and selection all combine to determine the genotypic distribution at selection-mutation-drift equilibrium in a static environment. These results will give us the essential knowledge required to analyse the response of sexual and diploid organisms to environmental changes. We will determine the adaptive response of populations to environmental changes. We plan to have some outputs published on this work in the near future and will continue the work of the project. This work should lead to important insights into the highly topical question of how various species, with finite population sizes, can cope with ongoing environmental change.