






Stephen J. Willson. Reconstruction of certain phylogenetic networks from their treeaverage distances. In BMB, Vol. 75(10):18401878, 2013. Keywords: explicit network, from distances, galled tree, normal network, phylogenetic network, phylogeny, unicyclic network. Note: http://www.public.iastate.edu/~swillson/TreeAverageReconPaper9.pdf.
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"Trees are commonly utilized to describe the evolutionary history of a collection of biological species, in which case the trees are called phylogenetic trees. Often these are reconstructed from data by making use of distances between extant species corresponding to the leaves of the tree. Because of increased recognition of the possibility of hybridization events, more attention is being given to the use of phylogenetic networks that are not necessarily trees. This paper describes the reconstruction of certain such networks from the treeaverage distances between the leaves. For a certain class of phylogenetic networks, a polynomialtime method is presented to reconstruct the network from the treeaverage distances. The method is proved to work if there is a single reticulation cycle. © 2013 Society for Mathematical Biology."






Stephen J. Willson. CSD Homomorphisms Between Phylogenetic Networks. In TCBB, Vol. 9(4), 2012. Keywords: explicit network, from network, from quartets, phylogenetic network. Note: http://www.public.iastate.edu/~swillson/Relationships11IEEE.pdf, preliminary version entitled Relationships Among Phylogenetic Networks.
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"Since Darwin, species trees have been used as a simplified description of the relationships which summarize the complicated network N of reality. Recent evidence of hybridization and lateral gene transfer, however, suggest that there are situations where trees are inadequate. Consequently it is important to determine properties that characterize networks closely related to N and possibly more complicated than trees but lacking the full complexity of N. A connected surjective digraph map (CSD) is a map f from one network N to another network M such that every arc is either collapsed to a single vertex or is taken to an arc, such that f is surjective, and such that the inverse image of a vertex is always connected. CSD maps are shown to behave well under composition. It is proved that if there is a CSD map from N to M, then there is a way to lift an undirected version of M into N, often with added resolution. A CSD map from N to M puts strong constraints on N. In general, it may be useful to study classes of networks such that, for any N, there exists a CSD map from N to some standard member of that class. © 2012 IEEE."



Stephen J. Willson. Treeaverage distances on certain phylogenetic networks have their weights uniquely determined. In ALMOB, Vol. 7(13), 2012. Keywords: from distances, from network, normal network, phylogenetic network, phylogeny, reconstruction, treechild network. Note: hhttp://www.public.iastate.edu/~swillson/TreeAverageDis10All.pdf.
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"A phylogenetic network N has vertices corresponding to species and arcs corresponding to direct genetic inheritance from the species at the tail to the species at the head. Measurements of DNA are often made on species in the leaf set, and one seeks to infer properties of the network, possibly including the graph itself. In the case of phylogenetic trees, distances between extant species are frequently used to infer the phylogenetic trees by methods such as neighborjoining.This paper proposes a treeaverage distance for networks more general than trees. The notion requires a weight on each arc measuring the genetic change along the arc. For each displayed tree the distance between two leaves is the sum of the weights along the path joining them. At a hybrid vertex, each character is inherited from one of its parents. We will assume that for each hybrid there is a probability that the inheritance of a character is from a specified parent. Assume that the inheritance events at different hybrids are independent. Then for each displayed tree there will be a probability that the inheritance of a given character follows the tree; this probability may be interpreted as the probability of the tree. The treeaverage distance between the leaves is defined to be the expected value of their distance in the displayed trees.For a class of rooted networks that includes rooted trees, it is shown that the weights and the probabilities at each hybrid vertex can be calculated given the network and the treeaverage distances between the leaves. Hence these weights and probabilities are uniquely determined. The hypotheses on the networks include that hybrid vertices have indegree exactly 2 and that vertices that are not leaves have a treechild. © 2012 Willson; licensee BioMed Central Ltd."






Stephen J. Willson. Restricted trees: simplifying networks with bottlenecks. In BMB, Vol. 73(10):23222338, 2011. Keywords: from network, phylogenetic network. Note: http://arxiv.org/abs/1005.4956.
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"Suppose N is a phylogenetic network indicating a complicated relationship among individuals and taxa. Often of interest is a much simpler network, for example, a species tree T, that summarizes the most fundamental relationships. The meaning of a species tree is made more complicated by the recent discovery of the importance of hybridizations and lateral gene transfers. Hence, it is desirable to describe uniform welldefined procedures that yield a tree given a network N. A useful tool toward this end is a connected surjective digraph (CSD) map φ:N→N′ where N′ is generally a much simpler network than N. A set W of vertices in N is "restricted" if there is at most one vertex u∉W from which there is an arc into W, thus yielding a bottleneck in N. A CSD map φ:N→N′ is "restricted" if the inverse image of each vertex in N′ is restricted in N. This paper describes a uniform procedure that, given a network N, yields a welldefined tree called the "restricted tree" of N. There is a restricted CSD map from N to the restricted tree. Many relationships in the tree can be proved to appear also in N. © 2011 The Author(s)."






Stephen J. Willson. Regular Networks Can Be Uniquely Constructed from Their Trees. In TCBB, Vol. 8(3):785796, 2010. Keywords: explicit network, from rooted trees, phylogenetic network, phylogeny, reconstruction, regular network. Note: http://www.public.iastate.edu/~swillson/RegularNetsFromTrees5.pdf.
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"A rooted acyclic digraph N with labeled leaves displays a tree T when there exists a way to select a unique parent of each hybrid vertex resulting in the tree T. Let Tr(N) denote the set of all trees displayed by the network N. In general, there may be many other networks M, such that Tr(M) = Tr(N). A network is regular if it is isomorphic with its cover digraph. If N is regular and D is a collection of trees displayed by N, this paper studies some procedures to try to reconstruct N given D. If the input is D=Tr(N), one procedure is described, which will reconstruct N. Hence, if N and M are regular networks and Tr(N) = Tr(M), it follows that N = M, proving that a regular network is uniquely determined by its displayed trees. If D is a (usually very much smaller) collection of displayed trees that satisfies certain hypotheses, modifications of the procedure will still reconstruct N given D. © 2011 IEEE."



Stephen J. Willson. Properties of normal phylogenetic networks. In BMB, Vol. 72(2):340358, 2010. Keywords: normal network, phylogenetic network, phylogeny, regular network. Note: http://www.public.iastate.edu/~swillson/RestrictionsOnNetworkspap9.pdf, slides available at http://www.newton.cam.ac.uk/webseminars/pg+ws/2007/plg/plgw01/0904/willson/.
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"A phylogenetic network is a rooted acyclic digraph with vertices corresponding to taxa. Let X denote a set of vertices containing the root, the leaves, and all vertices of outdegree 1. Regard X as the set of vertices on which measurements such as DNA can be made. A vertex is called normal if it has one parent, and hybrid if it has more than one parent. The network is called normal if it has no redundant arcs and also from every vertex there is a directed path to a member of X such that all vertices after the first are normal. This paper studies properties of normal networks. Under a simple model of inheritance that allows homoplasies only at hybrid vertices, there is essentially unique determination of the genomes at all vertices by the genomes at members of X if and only if the network is normal. This model is a limiting case of more standard models of inheritance when the substitution rate is sufficiently low. Various mathematical properties of normal networks are described. These properties include that the number of vertices grows at most quadratically with the number of leaves and that the number of hybrid vertices grows at most linearly with the number of leaves. © 2009 Society for Mathematical Biology."






Stephen J. Willson. Reconstruction of certain phylogenetic networks from the genomes at their leaves. In JTB, Vol. 252(2):185376, 2008. Keywords: labeling, polynomial. Note: http://www.public.iastate.edu/~swillson/ReconstructNormalHomopap6.pdf.
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"A network N is a rooted acyclic digraph. A baseset X for N is a subset of vertices including the root (or outgroup), all leaves, and all vertices of outdegree 1. A simple model of evolution is considered in which all characters are binary and in which backmutations occur only at hybrid vertices. It is assumed that the genome is known for each member of the baseset X. If the network is known and is assumed to be "normal," then it is proved that the genome of every vertex is uniquely determined and can be explicitly reconstructed. Under additional hypotheses involving timeconsistency and separation of the hybrid vertices, the network itself can also be reconstructed from the genomes of all members of X. An explicit polynomialtime procedure is described for performing the reconstruction. © 2008 Elsevier Ltd. All rights reserved."







