Spatial Fleming-Viot Models with Selection and Mutation

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Consequently, a new look at genetic monitoring and discovery of population structure is required.

Spatial Fleming-Viot Models with Selection and Mutation by Donald A. Dawson

Additionally, the model would directly quantify the full distribution of actual population or evolutionary processes of interest as best as possible, decoupling these parameters from the elements that define population structure Excoffier, In particular, this model would:.

The basic observations for a general analysis with this hypothetical model would be multilocus genotypes, multilocus sequences, or full genome sequences of individuals, their geographic locations, and information on covariates that might influence local density, movement, and selection. Preliminary analyses using the model might indicate that a given system fits comfortably onto either border, justifying the use of one or the other set of standard analytical regimes.

However, most empirical cases are more likely to lie in the interior, so the model could also give an indication of the appropriateness of measures deriving from one or the other of the main paradigms. As a result, the SLFV model permits the simultaneous, yet independent, estimation of local population density and local dispersal rates, two key parameters of population processes integral to genetic monitoring studies. In what follows, we introduce informally this simple model, then present the steps involved in a more mathematically rigorous form to illustrate explicitly how the restrictive assumptions can be relaxed to obtain a model with the desired characteristics outlined in the previous section.

All coalescent events will be limited to individuals within this neighborhood. Existing individuals within the neighborhood are then randomly selected to be descendants of the new ancestor. As outlined below, the individuals need not be haploid. Sexual reproduction can be accommodated by selecting more than a single ancestor. The steps in this process can be formalized to illustrate the generalizations that are possible.

Thus, each individual i can be represented by the quantities i , x i , and s i. Let C t be the set of individuals extant at time t ; this can change at discrete points in time as reproductive or movement events occur. Iterate through the following steps until C contains only a single individual, the ancestor of the entire sample.

Generate an event at a location, which will involve a mixture of reproduction and movement. To do so, sample a spatial probability distribution E x from a family of spatial distributions across the landscape L. Nonhomogeneity in the landscape can be incorporated with different families of E x , which might, for example, depend on the distribution of habitats, land use patterns, other environmental characteristics, or the state genetic or demographic of the individuals.

Select a set of individuals based upon the spatial distribution E x. For every individual j in C , select it with a probability of. This will yield a set containing zero or more individuals, randomly selected according to the spatial distribution associated with the event and their state. In the case of no mutation, all individuals in will have the same state, but this restriction is not necessary.

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Depending on the number of individuals in , this event either has no effect or involves a mixture of reproduction and movement. Construct a new event. Sample a set of individuals, which will replace those in , from the distribution. Some or all of these individuals may be ancestors of some of those in ; the remainder are individuals in that have simply moved.

Thus, the distribution determines the mixture of reproduction and movement that occurs in the event. However, more complex distributions can depend on the locations of individuals in , on environmental characteristics across L , or on individual states.

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If mutation is possible, sample the state of these replacement individuals from the distribution. Finally, remove all individuals in from C and replace them with the newly sampled ones. Clearly the SLFV model is very general. The suitable locations for individuals can be continuously distributed either uniformly or not across the landscape, can be patchily distributed, can be limited to discrete positions, or can be a complex mixture of these. The flexibility of the SLFV model enables the spatial structure to emerge from the analysis rather than be imposed a priori.

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Developing software that reflects the range of applicability of the SLFV model remains an open challenge that is crucial to the advancement of genetic monitoring as well as population genetics. The selectively neutral, spatially homogeneous SLFV model is dependent on several parameters, the two most important of which govern how , the spatial distribution of new ancestors and coalescent events, reflects local population density and local dispersal rate. This means that the SLFV model is directly based on biological processes of known importance to the genetic composition of populations, a feature critical for genetic monitoring and discovery of population structure.

Additionally, spatially or temporally heterogeneous versions of the model could use spatial or temporal covariates, such as habitat characteristics, to parameterize the local population density and dispersal parameters. Two applications of the SLFV model illustrate both its power and the importance of relaxing the assumptions incorporated into existing software.

The pipeline was used to validate the estimation of neighborhood size from simulated data and subsequently to estimate both neighborhood size and dispersal radius from empirical data on Berkheya cuneata Asteraceae from South Africa. In their model, dispersal radius R was the maximum distance individuals could disperse, and neighborhood size was the number of individuals within the area of an event of radius R. For validation, , datasets were generated for eight individuals sampled at 10 unlinked loci.

The same pipeline implementing the selectively neutral, spatially homogeneous SLFV model was used to generate , datasets; rejection ABC was used to sample from the posterior distributions of both neighborhood size and dispersal distance.

The median estimates of neighborhood size and dispersal distance were This study illustrates several important points regarding practical use of the SLFV model. First, the two most biologically important parameters, neighborhood size and dispersal distance, are identifiable; that is, they can be estimated separately using the SLFV model. Second, it is possible to obtain useful estimates even from relatively small datasets composed of no more than dozens of individuals or handfuls of loci. Third, there is room for improved computational efficiency to accommodate larger datasets.

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A second example using the selectively neutral, spatially homogeneous SLFV model reinforces these points and illustrates additional ones. Effective population density d and dispersal intensity Wright, were estimated using the SLFV model based upon a sample of 50 individuals sampled at either two or ten different sites. Additionally, parameter estimates were obtained using the structured coalescent Hudson, ; Notohara, under the assumption of either two or ten discrete populations.

Estimates from the structured coalescent were upwardly biased to a large degree, though much less so for ten than for two populations. Estimates from the SLFV model were much better, although the precision declined with larger values of dispersal intensity.

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Each dataset yielded an estimate of the posterior distributions for neighborhood size and dispersal radius Wright, Comparison of the five distributions for these two parameters revealed that the two biological replicates yielded similar distributions, an indication of consistency despite moderate sample size. Further, the — flu season was different from the other four; it was characterized by a smaller neighborhood size and a larger dispersal radius.

This observation indicates limited infection rates and broader climatic tolerance, which is consistent with the known history longer duration and milder incidence of that epidemic. This study reinforces the point that neighborhood size and dispersal rates can be estimated separately using the SLFV model. Distinguishing between them is important, especially in the case of genetic monitoring where either or both might shift as they did with influenza through time.

Detecting those shifts may in fact be a major reason for undertaking a monitoring program.

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It also reinforces the point that useful estimates can be obtained for typical samples using a reasonable amount of computation. Thus, the SLFV model can be developed into a practical approach to genetic monitoring. It may also serve the task much better than other methods, such as those based upon F ST or the structured coalescent, that impose a priori assumptions upon the spatial structure of the populations under study.

For example, given information on the spatial layout of distinct habitat types, one could estimate different densities or dispersal rates for each habitat. It would also reveal valuable information on the basic biology of the species under study.

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  7. Importantly, differences among habitats or other spatially defined factors would emerge naturally from the analysis if they exist rather than be imposed at the outset by selection of the analysis framework. Of course, as with landscape genetics models, SLFV models with too many parameters will be impossible to estimate. How many and which parameters can be estimated remains an open question, and software implementations of more complex, and possibly biologically realistic, models are required to investigate this.

    Inefficiencies of implementation or limited sets of MCMC operators might also be shortcomings leading to analyses taking longer to complete or being limited in scope. These are purely technical limitations related to the early stage of development of the SLFV model, and can be overcome by improvements in software design coupled with additional investigation of model performance.

    A feature of the SLFV model as currently implemented is that no distinction, other than location, is made among individuals with respect to their likelihood of birth; in the backward in time version of the model described above, the probability distribution E x that selects individuals influenced by an event depends only on location.

    Spatial Fleming-Viot Models with Selection and Mutation

    Greater biological realism could be incorporated into the model by allowing E x to depend on, for example, the demographic state of individuals or their genotype. These states need not even be static; they could be projected through time from one event to the next much as phylogenetic analysis projects state change along lineages.

    Further, these projections could incorporate structured population models Caswell, in a natural way. Like the Moran model, the SLFV model applies to overlapping generations, as reproductive events are not synchronized across the population in any way other than by the geographic scale of each event. Interestingly, this feature contrasts with most other models, which have the opposite limitation of applying to nonoverlapping generations.