We present the first comprehensive analysis of a diploid human genome that combines single-molecule sequencing with single-molecule genome maps. Our hybrid assembly markedly improves upon the contiguity observed from traditional shotgun sequencing approaches, with scaffold N50 values approaching 30 Mb, and we identified complex structural variants (SVs) missed by other high-throughput approaches. Furthermore, by combining Illumina short-read data with long reads, we phased both single-nucleotide variants and SVs, generating haplotypes with over 99% consistency with previous trio-based studies. Our work shows that it is now possible to integrate single-molecule and high-throughput sequence data to generate de novo assembled genomes that approach reference quality.

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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 f lies, and yeast. The best fit results lead to slippage rate estimates that are highest in mice, followed by humans, then yeast, and then fruit f lies. 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.

Let W 1 .... , W N be N nonnegative random variables and let 9J~ be the class of all probability measures on [0, m). Define a transformation T on 9J~ by letting T# be the distribution of WaX 1 +... + WNXN, where the X i are independent random variables with distribution /~, which are independent of W1, ..., W N as well. In earlier work, first Kahane and Peyriere, and then Holley and Liggett, obtained necessary and sufficient conditions for T to have a nontrivial fixed point of finite mean in the special cases that the W~ are independent and identically distributed, or are fixed multiples of one random variable. In this paper we study the transformation in general. Assuming only that for some 7 > 1, EWES< oe for all i, we determine exactly when T has a nontrivial fixed point (of finite or infinite mean). When it does, we find all fixed points and prove a convergence result. In particular, it turns out that in the previously considered cases, T always has a nontrivial fixed point. Our results were motivated by a number of open problems in infinite particle systems. The basic question is: in those cases in which an infinite particle system has no invariant measures of finite mean, does it have invariant measures of infinite mean? Our results suggest possible answers to this question for the generalized potlatch and smoothing processes studied by Holley and Liggett.

Spatial pattern, how it arises and how it is maintained, are central foci for ecological theory. In recent years, some attention has shifted from continuum models to spatially discrete analogues, which allow easy treatment of local stochastic effects and of non-local spatial influences. Many of these fall within the area of mathematics known as `interacting particle systems', which provides a body of results that facilitate the interpretation of the suite of simulation models that have been considered, and point towards future analyses. In this paper we review the basic mathematical literature. Three influential examples from the ecological literature are considered and placed within the general framework, which is shown to be a powerful one for the study of spatial ecological interactions.

We consider a simplified model of a social network in which individuals have one of two opinions (called 0 and 1) and their opinions and the network connections coevolve. Edges are picked at random. If the two connected individuals hold different opinions then, with probability 1 − α, one imitates the opinion of the other; otherwise (i.e., with probability α), the link between them is broken and one of them makes a new connection to an individual chosen at random (i) from those with the same opinion or (ii) from the network as a whole. The evolution of the system stops when there are no longer any discordant edges connecting individuals with different opinions. Letting ρ be the fraction of voters holding the minority opinion after the evolution stops, we are interested in how ρ depends on α and the initial fraction u of voters with opinion 1. In case (i), there is a critical value α c which does not depend on u, with ρ ≈ u for α > α c and ρ ≈ 0 for α < α c . In case (ii), the transition point α c ðuÞ depends on the initial density u. For α > α c ðuÞ, ρ ≈ u, but for α < α c ðuÞ, we have ρðα,uÞ ¼ ρðα,1∕2Þ. Using simulations and approximate calculations, we explain why these two nearly identical models have such dramatically different phase transitions.I n recent years, a variety of research efforts from different disciplines have combined with established studies in social network analysis and random graph models to fundamentally change the way we think about networks. Significant attention has focused on the implications of dynamics in establishing network structure, including preferential attachment, rewiring, and other mechanisms (1-5). At the same time, the impact of structural properties on dynamics on those networks has been studied, (6), including the spread of epidemics (7-10), opinions (11-13), information cascades (14-16), and evolutionary games (17,18). Of course, in many real-world networks the evolution of the edges in the network is tied to the states of the vertices and vice versa. Networks that exhibit such a feedback are called adaptive or coevolutionary networks (19,20). As in the case of static networks, significant attention has been paid to evolutionary games (21)(22)(23)(24) and to the spread of epidemics (25-29) and opinions (30-35), including the polarization of a network of opinions into two groups (36,37). In this paper, we examine two closely related variants of a simple, abstract model for coevolution of a network and the opinions of its members. Holme-Newman ModelOur starting point is the model of Holme and Newman (38-41). They begin with a network of N vertices and M edges, where each vertex x has an opinion ξðxÞ from a set of G possible opinions and the number of people per opinion γ N ¼ N∕G stays bounded as N gets large. On each step of the process, a vertex x is picked at random. If its degree dðxÞ ¼ 0, nothing happens. For dðxÞ > 0, (i) with probability α an edge attached to vertex x is selected and the other end of that edge is moved to a vertex chosen at random from those with o...

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