Capturing a phylogenetic tree when the number of character states varies with the number of leaves

We show that for any two values $\alpha, \beta >0$ for which $\alpha+\beta>1$ then there is a value $N$ so that for all $n \geq N$ the following holds. For any binary phylogenetic tree $T$ on $n$ leaves there is a set of $\lfloor n^\alpha \rfloor$ characters that capture $T$, and for which each character takes at most $\lfloor n^\beta \rfloor$ distinct states. Here `capture' means that $T$ is the unique perfect phylogeny for these characters. Our short proof of this combinatorial result is based on the probabilistic method.Comment: 3 pages, 0 figure

Phylogenetic information complexity: Is testing a tree easier than finding it?

Phylogenetic trees describe the evolutionary history of a group of present-day species from a common ancestor. These trees are typically reconstructed from aligned DNA sequence data. In this paper we analytically address the following question: is the amount of sequence data required to accurately reconstruct a tree significantly more than the amount required to test whether or not a candidate tree was the `true' tree? By `significantly', we mean that the two quantities behave the same way as a function of the number of species being considered. We prove that, for a certain type of model, the amount of information required is not significantly different; while for another type of model, the information required to test a tree is independent of the number of leaves, while that required to reconstruct it grows with this number. Our results combine probabilistic and combinatorial arguments.Comment: 15 pages, 3 figure

Tracing evolutionary links between species

The idea that all life on earth traces back to a common beginning dates back at least to Charles Darwin's {\em Origin of Species}. Ever since, biologists have tried to piece together parts of this `tree of life' based on what we can observe today: fossils, and the evolutionary signal that is present in the genomes and phenotypes of different organisms. Mathematics has played a key role in helping transform genetic data into phylogenetic (evolutionary) trees and networks. Here, I will explain some of the central concepts and basic results in phylogenetics, which benefit from several branches of mathematics, including combinatorics, probability and algebra.Comment: 18 pages, 6 figures (Invited review paper (draft version) for AMM

Four characters suffice to convexly define a phylogenetic tree

It was recently shown that just five characters (functions on a finite set X) suffice to convexly define a trivalent tree with leaf set X. Here we show that four characters suffice which, since three characters is not enough in general, is the best possible

Miljöpåverkansbedömning vid tillverkning av etanol från cellulosabaserade råvaror : ekologisk gård självförsörjande med drivmedel

Fuel produced from renewable resources is of big interest. In this project the difference between a few different renewable materials are compared on the matter of environmental load when ethanol is produced. The renewable materials that are compared are Salix, straw and reed canary-grass (rörflen). The cellulose in the material is cut down to small pieces and then hydrolysed so that the sugar from the cellulose can be fermented into ethanol. Tha amount of ethanols that is studied in this work is the smount of ethanol needed to cultivate 1000 ha. The functional unit in the report is 1000 ha. All the regulas crops and the crops for ethanolproduction in the system are organically cultivated. The goal for this study is to compare the final environmental load for the whole system of ethanol production, cultivation and usage of the ethanol in the end. This is assessed by calculation of global warming potential (GWP), acidification potential (AP) and eutrophfication potential (EP). The impact of GWP is dependant of the amount of CO2, CH4 and N2=. The fuel that are used for cultivation is based on ethanol with addition of some ignition improver of fossil origin. Since ethanol is produced from renewable sources the impact of GWP is small despite the ignition improver. The emissions from soil have the largest impact on GWP. The two sources that have the largest impact on AP is emissions from soil and the emissions from the steamproduction. Steam is needed during the ethanol production process. The emissions from the soil have the largest impact on EP. More than 90 percent of the total EP comes from the soil emissions. A comparison of the three different systems shows that the system with straw has the lowest environmental load.Drivmedel som tillverkas av förnyelsebara råvaror istället för fossila råvaror är naturligtvis av stort intresse. I detta arbete studeras skillnader i miljöpåverkan vid omvandling av olika förnyelsebara råvaror till etanoldrivmedel. Systemet bygger på att biomassa från salix, halm och rörflen används vid produktion av etanol. Cellulosan i biomassan klipps sönder och genom hydrolys blir sockret i cellulosan tillgängligt för jäsning till etanol. Mängden etanoldrivmedel som behandlas i studien är den mängd drivmedel som går åt för att odla 1000 ha. Målsättningen för studien är att studera skillnader i emissioner till luft, vatten och miljöpåverkan för de valda systemen. Detta bedöms genom att beräkna växthuseffekt, försurning och övergödning. Påverkan av växthuseffekten beror i denna studie av utsläppen av CO2, CH4 och N2O. Drivmedlet som används vid odlingen är baserad på etanol men med vissa tillsatser av tändförbättrare med fossilt ursprung. Detta gör att det bara är en lite del av bränslet som bidrar till växthuseffekten, etanolen ses som biologisk och bidrar inte till någon ökad växthuseffekt. Den klart största bidragande faktorn till växthuseffekten är markemissionerna där N2O är den bidragande källan. De största bidragande källorna till försurning kommer från markemissionerna och ångframställningen. När det gäller övergödning är det markemissionerna som är den största bidragande faktorn. Över 90 procent av den totala övergödningen beror på markemissionerna. En jämförelse mellan de tre olika systemen visar att halmsystemet är klart bäst med avseende på växthuseffekt, försurning och övergödning. Inget av de andra systemen är i närheten av att kunna jämföra sig med halmen

Discrete and statistical approaches to genetics

This thesis presents a number of major innovations in related but different areas of research. The contributions range along a continuum from mathematical phylogenetics, to development of statistical methodology for detecting recombination and finally to the application of statistical techniques to understand Feline Immunodeficiency Virus (FIV) an important pathogen. An underlying theme is the application of combinatorial and statistical ideas to problems in evolutionary biology and genetics.Chapter 2 and Chapter 3 give a number of results relevant to mathematical phylogenetics, in particular maximum parsimony. Chapter 2 presents a new formulation of maximum parsimony in terms of character subdivision, providing a direct link with the character compatibility problem, also known as the perfect phylogeny problem. Specialization of this result to two characters gives a simple formula based on the intersection graph for calculating the parsimony score for a, pair of characters. Chapter 3 further explores maximum parsimony. In particular, it is shown that a maximum parsimony tree for a sequence of characters minimizes a subtree-prune and regraft (SPR) distance to the sets of trees on which each character is convex. Similar connections are also drawn between the Robinson-Foulds distance and a new variant of Dollo parsimony.Chapter 4 presents an application of the work in Chapters 2 and 3 to develop a statistical test for detecting recombination. An extensive coalescent based simulation study shows that this new test is both robust and powerful in a variety of different circumstances compared to a number of current methods. In fact, a simple model of mutation rate correlation is shown to mislead a number of competing tests, causing recombination to be falsely inferred. Analysis of empirical data sets confirm that the new test is one of the best approaches to distinguish recurrent mutation from recombination.Finally, Chapter 5 uses the test developed in Chapter 4 to localize recombinant breakpoints in 14 genomic strains of FIV taken from a wild population of cougars. Based on the technique, three recombinant strains of FIV are identified. Previous studies have focused on the epidemiology and population structure of the virus and this study shows that recombination has also played an important role in the evolution of FIV

Four characters suffice to convexly define a phylogenetic tree

It was recently shown that just five characters (functions on a finite set X) suffice to convexly define a trivalent tree with leaf set X. Here we show that four characters suffice which, since three characters are not enough in general, is the best possible

A phylogenetic investigation of Begonia L. section knesebeckia (Klotzsch) a.dc

This thesis uses morphological and molecular data to address the phylogeny of Begonia and in particular section Knesebeckia. Begonia is a species rich genus containing an estimated 1400 species distributed throughout the tropics and subtropics although none are present in Australia and New Zealand. Despite its horticultural importance few studies of the genus have been made and there is no published phylogeny for Begonia. Begonia section Knesebeckia is the only section from the c. 80 sections in Begonia to be currently recognised as containing species from more than one continent. There are 77 species (33 American and 44 Asian) currently attributed to Knesebeckia. With the prospect of an interesting biogeographic study, the aim of this research was to test the monophyly of Knesebeckia. A cladistic analysis of the morphology of Begonia was conducted using a sample of 86 taxa representing the morphological and geographical variation in Begonia. This analysis included all the available American Knesebeckia and 11 Asian Knesebeckia. A total of 66 macro- and micro-morphological characters was included in the analysis and resulted in 5340 trees contained in two islands of equally most parsimonious trees. From these results several conclusions were able to be made: 1) Knesebeckia is not monophyletic; 2) B. dichroa and B. olbia are not closely affiliated to other American Knesebeckia and are contained in a clade which is sister to a clade containing species with fleshy indehiscent fruit, mainly from Africa; 3) groupings of species suggested for the remainder of the American Knesebeckia include some species which may better be represented in section Begonia-, a clade centred around B. incarnata, the type of Knesebeckia; and a clade centred around B. boissieri for which the section Quadriperigonia should be reinstated; 4) several species were suggested as being closely related to section Gireoudia, e.g. B. peltata, and representatives from sections with a chromosome number of 2n = 28 (e.g. Gireoudia and Weilbachia) are recommended for inclusion in any future study of American Knesebeckia', 5) Asian Knesebeckia are shown to be paraphyletic, with representatives from sections Alicida, Diploclinium, Lauchea, Monophyllon, Parvibegonia, Putzeysia and Trilobaria nested in the clade of Asian Knesebeckia; 6) section Diploclinium is shown to be polyphyletic; species groups centred around B. putii and B. rubella are recommended for consideration in the context of any future analysis of Asian Knesebeckia. A cladistic analysis of sequence data for 30 species of Begonia from the noncoding chloroplast DNA region trnC - trnD was compared to an analysis of the morphology for the same 30 species. The results were not congruent and it was argued that the molecular phylogeny is more reliable than the morphology phylogeny for these 30 species and that combining the data sets for a "total evidence" view is not conducive to the production of a reliable and conservative estimate of the phylogeny of Begonia. The molecular phylogeny corroborates the conclusion in the larger morphological study that Knesebeckia is not monophyletic. The implications of the molecular phylogeny for the phylogeny of Begonia are discussed. The properties and evolution of this region of noncoding chloroplast DNA are investigated and compared to previous studies of other noncoding regions. The region is found to be AT rich with many insertions and deletions (indels). These indels are shown to be phylogenetically informative, exhibiting little homoplasy. A review of the literature relating to Knesebeckia indicate that B. grandis, an Asian species included in Knesebeckia from its inception, is the type of section Diploclinium. Eleven Asian species are provided with updated descriptions and admitted to section Diploclinium. Some existing species are placed as synonyms of these species and one new combination is made