Spatial Moran models, II: cancer initiation in spatially structured tissue.

We study the accumulation and spread of advantageous mutations in a spatial stochastic model of cancer initiation on a lattice. The parameters of this general model can be tuned to study a variety of cancer types and genetic progression pathways. This investigation contributes to an understanding of how the selective advantage of cancer cells together with the rates of mutations driving cancer, impact the process and timing of carcinogenesis. These results can be used to give insights into tumor heterogeneity and the "cancer field effect," the observation that a malignancy is often surrounded by cells that have undergone premalignant transformation.

Exponential distance statistics to detect the effects of population subdivision.

Statistical tests are needed to determine whether spatial structure has had a significant effect on the genetic differentiation of subpopulations. Here we introduce a new family of statistics based on a sum of an exponential function of the distances between individuals, which can be used with any genetic distance (e.g., nucleotide differences, number of nonshared alleles, or separation on a phylogenetic tree). The power of the tests to detect genetic differentiation in Wright-Fisher island models and stepping stone models was calculated for various sample sizes, rates of migration and mutation, and definitions of spatial neighborhoods. We found that our new test was in some cases more powerful than the Ks* statistic of Hudson et al. (Mol. Biol. Evol. 9, 138-151, 1992), but in all cases was slightly less powerful than both a traditional chi2 test without lumping of rare haplotypes and the S(nn) test of Hudson (Genetics 155, 2011-2014, 2000). However, when we applied our new tests to three data sets, we found in some cases highly significant results that were missed by the other tests.

Dynamics of microsatellite divergence under stepwise mutation and proportional slippage/point mutation models.

Recently Kruglyak, Durrett, Schug, and Aquadro showed that microsatellite equilibrium distributions can result from a balance between polymerase slippage and point mutations. Here, we introduce an elaboration of their model that keeps track of all parts of a perfect repeat and a simplification that ignores point mutations. We develop a detailed mathematical theory for these models that exhibits properties of microsatellite distributions, such as positive skewness of allele lengths, that are consistent with data but are inconsistent with the predictions of the stepwise mutation model. We use our theoretical results to analyze the successes and failures of the genetic distances (delta(mu))(2) and D(SW) when used to date four divergences: African vs. non-African human populations, humans vs. chimpanzees, Drosophila melanogaster vs. D. simulans, and sheep vs. cattle. The influence of point mutations explains some of the problems with the last two examples, as does the fact that these genetic distances have large stochastic variance. However, we find that these two features are not enough to explain the problems of dating the human-chimpanzee split. One possible explanation of this phenomenon is that long microsatellites have a mutational bias that favors contractions over expansions.

Distribution and abundance of microsatellites in the yeast genome can Be explained by a balance between slippage events and point mutations.

We fit a Markov chain model of microsatellite evolution introduced by Kruglyak et al. to data on all di-, tri-, and tetranucleotide repeats in the yeast genome. Our results suggest that many features of the distribution of abundance and length of microsatellites can be explained by this simple model, which incorporates a competition between slippage events and base pair substitutions, with no need to invoke selection or constraints on the lengths. Our results provide some new information on slippage rates for individual repeat motifs, which suggest that AT-rich trinucleotide repeats have higher slippage rates. As our model predicts, we found that many repeats were adjacent to shorter repeats of the same motif. However, we also found a significant tendency of microsatellites of different motifs to cluster.

Lessons on pattern formation from planet WATOR.

It is well known that if reacting species experience unequal diffusion rates, then dynamics that lead to a constant steady state in a "well-mixed" environment can in a spatial setting lead to interesting patterns. In this paper, we focus on complementary pattern formation mechanisms that operate even when the diffusion rates are equal. In particular, we can say that when the mean-field ODE has an attracting periodic orbit then the stochastic spatial model will have large-scale spatial structures in equilibrium. We explore this mechanism in depth through the dynamics of the simulator WATOR.

Selective mapping: a strategy for optimizing the construction of high-density linkage maps.

Historically, linkage mapping populations have consisted of large, randomly selected samples of progeny from a given pedigree or cell lines from a panel of radiation hybrids. We demonstrate that, to construct a map with high genome-wide marker density, it is neither necessary nor desirable to genotype all markers in every individual of a large mapping population. Instead, a reduced sample of individuals bearing complementary recombinational or radiation-induced breakpoints may be selected for genotyping subsequent markers from a large, but sparsely genotyped, mapping population. Choosing such a sample can be reduced to a discrete stochastic optimization problem for which the goal is a sample with breakpoints spaced evenly throughout the genome. We have developed several different methods for selecting such samples and have evaluated their performance on simulated and actual mapping populations, including the Lister and Dean Arabidopsis thaliana recombinant inbred population and the GeneBridge 4 human radiation hybrid panel. Our methods quickly and consistently find much-reduced samples with map resolution approaching that of the larger populations from which they are derived. This approach, which we have termed selective mapping, can facilitate the production of high-quality, high-density genome-wide linkage maps.

Spatial models for hybrid zones.

We introduce a spatially explicit model of natural hybrid zones that allows us to consider how patterns of allele frequencies and linkage disequilibria change over time. We examine the influence of hybrid zone origins on patterns of variation at two loci, a locus under selection in a two-patch environment, and a linked neutral locus. We consider several possible starting conditions that represent explicit realizations of two alternative scenarios for hybrid zone origins: primary intergradation and secondary contact. Our results indicate that in some circumstances, differences in hybrid zone origins will result in substantially different patterns of variation that may persist for thousands of generations. Our conclusions are generally similar to those previously derived from partial differential equations, but there are also some important differences.

Quantification of homoplasy for nucleotide transitions and transversions and a reexamination of assumptions in weighted phylogenetic analysis.

Nucleotide transitions are frequently down-weighted relative to transversions in phylogenetic analysis. This is based on the assumption that transitions, by virtue of their greater evolutionary rate, exhibit relatively more homoplasy and are therefore less reliable phylogenetic characters. Relative amounts of homoplastic and consistent transition and transversion changes in mitochondrial protein coding genes were determined from character-state reconstructions on a highly corroborated phylogeny of mammals. We found that although homoplasy was related to evolutionary rates and was greater for transitions, the absolute number of consistent transitions greatly exceeded the number of consistent transversions. Consequently, transitions provided substantially more useful phylogenetic information than transversions. These results suggest that down-weighting transitions may be unwarranted in many cases. This conclusion was supported by the fact that a range of transition: transversion weighting schemes applied to various mitochondrial genes and genomic partitions rarely provided improvement in phylogenetic estimates relative to equal weighting, and in some cases weighting transitions more heavily than transversions was most effective.

Local frequency dependence and global coexistence.

In sessile organisms such as plants, interactions occur locally so that important ecological aspects like frequency dependence are manifest within local neighborhoods. Using probabilistic cellular automata models, we investigated how local frequency-dependent competition influenced whether two species could coexist. Individuals of the two species were randomly placed on a grid and allowed to interact according to local frequency-dependent rules. For four different frequency-dependent scenarios, the results indicated that over a broad parameter range the two species could coexist. Comparisons between explicit spatial simulations and the mean-field approximation indicate that coexistence occurs over a broader region in the explicit spatial simulation.

Equilibrium distributions of microsatellite repeat length resulting from a balance between slippage events and point mutations.

We describe and test a Markov chain model of microsatellite evolution that can explain the different distributions of microsatellite lengths across different organisms and repeat motifs. Two key features of this model are the dependence of mutation rates on microsatellite length and a mutation process that includes both strand slippage and point mutation events. We compute the stationary distribution of allele lengths under this model and use it to fit DNA data for di-, tri-, and tetranucleotide repeats in humans, mice, fruit flies, and yeast. The best fit results lead to slippage rate estimates that are highest in mice, followed by humans, then yeast, and then fruit flies. Within each organism, the estimates are highest in di-, then tri-, and then tetranucleotide repeats. Our estimates are consistent with experimentally determined mutation rates from other studies. The results suggest that the different length distributions among organisms and repeat motifs can be explained by a simple difference in slippage rates and that selective constraints on length need not be imposed.

Spatial aspects of interspecific competition.

Using several variants of a stochastic spatial model introduced by Silvertown et al., we investigate the effect of spatial distribution of individuals on the outcome of competition. First, we prove rigorously that if one species has a competitive advantage over each of the others, then eventually it takes over all the sites in the system. Second, we examine tradeoffs between competition and dispersal distance in a two-species system. Third, we consider a cyclic competitive relationship between three types. In this case, a nonspatial treatment leads to densities that follow neutrally stable cycles or even unstable spiral solutions, while a spatial model yields a stationary distribution with an interesting spatial structure.

Allelopathy in Spatially Distributed Populations

In a homogeneously mixing population of E. coli, colicin-producing and colicin-sensitive strategies both may be evolutionarily stable for certain parameter ranges, with the outcome of competition determined by initial conditions. In contrast, in a spatially-structured population, there is a unique ESS for any given set of parameters; the outcome is determined by how effective allelopathy is in relation to its costs. Furthermore, in a spatially-structured environment, a dynamic equilibrium may be sustained among a colicin-sensitive type, a high colicin-producing type, and a "cheater" that expends less on colicin production but is resistant. Copyright 1997 Academic Press Limited

From individuals to epidemics.

Heterogeneous mixing fundamentally changes the dynamics of infectious diseases; finding ways to incorporate it into models represents a critical challenge. Phenomenological approaches are deficient in their lack of attention to underlying process; individual-based models, on the other hand, may obscure the essential interactions in a sea of detail. The challenge then is to find ways to bridge these levels of description, starting from individual-based models and deriving macroscopic descriptions from them that retain essential detail, and filter out the rest. In this paper, attempts to achieve this transformation are described for a class of models where nonrandom mixing arises from the spatial localization of interactions. In general, the epidemic threshold is found to be larger owing to spatial localization than for a homogeneously mixing population. An improved estimate of the dynamics is developed by the use of moment equations, and a simple estimate of the threshold in terms of a 'dyad heuristic'. For more general models in which local infection is not described by mass action, the connection with related partial differential equations is investigated.