Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Modeling Phylogenetic Comparative Methods with Hybridization Tony Jhwueng NIMBioS Interdisciplinary Seminar Jan 25 2011 Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Outline: 1 Introduction: Phylognetic Comparative Methods (PCMs). 2 Modeling PCMs with Hybridization. Develop possible comparative methods when there are ancient hybridization events in addition to the usual speciation events. 3 Some ongoing work. Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Phylogenetic Comparative Methods Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work A Phylogenetic Tree Human (Akha) Chimpanzee Gorilla 4.7 million years ago 7.2 million years ago (Takahata et al., 1995) Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Comparative Data Examples: Body mass (adult male)(kg) Brain mass (adult male)(gram) 0 1 0 1 0 1 0 1 x1 55:5 y1 1361 Xbody = @ x2 A = @ 56:7 A ; Ybrain = @ y2 A = @ 440 A : x3 172:4 y3 570 Data from Jerison (1973) Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Phylogenetic Comparative Methods (PCMs) Phylogenetic Comparative Methods (PCMs) are statistical methods that incorporating phylogenetic tree for analyzing comparative data in the ecology and evolution literature. Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Phylogenetic Comparative Methods (PCMs) Phylogenetic comparative methods are commonly applied to such questions as: 1 What is the slope of an allometric scaling relationship ? e.g. how does brain mass vary in relation to body mass ? 2 What was the ancestral state of a trait? (Schluter et al. 1997; Hardy 2006; Ronquist 2004.) e.g. where did endothermy(warm-blooded animals) evolve in the lineage that led to mammals? Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work A PCM developed from Evolutionary Perspective Rely directly on explicit assumptions regarding the evolutionary process. 1. FIC (Felsenstein 1985): derived directly from population genetic theory and requires an assumption that the traits of interest have evolved via the Brownian motion process. Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Brownian Motion process: dyt = σdBt σ (rate parameter) measures the intensity of the random fluctuations in the evolutionary process. BM: σ=1 BM: σ=3 Trait Value y(t) Value Trait y(t) Value Trait −400 −200 0 200 400 −400 −200 0 200 400 0 2000 4000 6000 8000 0 2000 4000 6000 8000 Time t Time t Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work The Comparative Data is NOT Statisitcally Independent Since the species are related by shared evolutionary history it may not be reasonable to view comparative data as independent, identically distributed realizations of the same stochastic process. the distribution of comparative data depends on the assumption of stochastic process for trait evolution. Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Representing Phylogenetic Tree by Similarity Matrix G Scale the phylogenetic tree so that the length from the root to each tip is 1 Relationship between the trait of paired species is measured by the shared branch length (time). y1 y2 y3 Human Chimp. Gorilla y1 y2 y3 0 1 0.4 0.4 y1 10 :60 G3 = y2 @ 0:610 A 1 y3 0 0 1 0.6 Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work FIC (Felsenstein, 1985): trait has evolved under the Brownian motion The variation of rate of evolution of the trait value is proportional to time. 0 1 y1 y2 y3 0 1 0 1 0 Human Chimp.1 Gorilla y1 B µ 10 :60 C B 2 C y2 ∼ MVN B µ ; σ 0:610 C @ A B@ A @ 0.4 0.4 AC B C y3 @ µ 0 0 1 A 1 0.6 Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Other PCMs Developed from evolutionary perspective 2. PGLS (Martins and Hansen 1997, Butler and King 2004, Hansen 2008): expands the assumptions of BM to allow for other evolutionary scenarios (OU process) (for stabilizing selection). 3. PMM (Lynch 1991; Housworth et al. 2004): derived from quantitative genetics, and partitions phenotypic variation into phylogenetically heritable and nonheritable components. Developed from statistical perspective 4. ARM: spatial autoregressive method (Cheverud et al. 1985, Jittleman and Kot 1990). 5. PVR: phylogenetic eigenvector regression. Diniz-Filho et al. (1998) Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Some useful textbooks and softwares for PCMs: The Comparative Method in Evolutionary Biology by Harvey and Pagel, 1991. Inferring Phylogenies by Joseph Felsenstein, 2004, Ch 26. Analysis of Phylogenetics and Evolution with R by Emmanuel Paradis, 2006. Softwares: PHYLIP (Joseph Felsenstein) COMPARE (Em´ıliaMartins) BROWNIE (Brian O'Meara) Various R packages at http://cran.r-project.org/web/views/Phylogenetics.html. Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Modeling PCMs with Hybridization Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Hybridization is common in nature: Sunflower Wild Sunflowers in a Field (photo by Erin Silversmith) Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Hybridization is common in nature: Sunflower Helianthus annuus. Helianthus petiolaris. Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Hybridization is common in nature: Sunflower L. H. Rieseberg (1991) provided evidence that H. anomalus is a hybrid species derived from H. annuus and H. petiolaris. Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Hybridization is common in nature: Cichild Lake Tanganyika (African Great Lake: the second largest freshwater lake in the world). (picture from Wikipedia) Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Hybridization is common in nature: Cichild A typical shell-nest constructed by large Lamprologus callipterus males. These aggregations attract different species of obligatory and facultative gastropod-shell-breeders, which consequently live and breed in closest vicinity. (photo from Koblm¨ulleret al. 2007) Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Hybridization is common in nature: Cichlid Hybrid cichilds from Lake Tanganyika Lamprologus meleagris. (photo by Hag- Lamprologus speciosus.(photo by blom, Fredrik ) Slaboch, Roman ) Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Hybridization is common in nature: Cichlid Hybrid cichilds from Lake Tanganyika Neolamprologus fasciatus. (photo by Neolamprologus multifasciatus. Jensen, Johnny) (photo by Gagliardi, Flavio ) Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Purpose of this project Hybrid species are known for sharing some common phenotypes from their parents. When studying the trait of a group of related species involving multiple hybrids, Question 1: are hybrids constrained to be between the parents, or hybridization allows them to break free from their constraints ? Hybrid trait Hybrid trait Hybrid trait ? ? ? Trait values Species 1 Species 2 Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Purpose of this project Some exotic species can evolve rapidly especially after hybridization (Barrett and Richardson 1982). Question 2: When studying the trait of a group of related species involving multiple hybrids, does hybridization increase the rate of evolution ? Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Modeling PCMs with Hybridization If evolution involves ancient hybridizations (reticulate evolutionary events), instead of the phylogenetic tree, incorporate the phylogenetic network into comparative analysis. Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Linder and Rieseberg 2004. Assume the nonhybrid taxa are normal diploid organisms, in which each chromosome consists of a pair of homologs. In a diploid hybridization event, the hybrid inherits one of two homologs from each chromosome from each of its two parents. Species 1 Hybrid Species 2 Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work A 3 taxa network A nucleotide inherited from the A parent (species 1 at t1) of hybrid B (hybrid at t1) will be part of the subtree in which species 1 and hybrid are sister taxa Species 1 hybrid Species 2 t2 A B C t1 O Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work A 3 taxa network A nucleotide inherited from the C parent (species 2 at t1) of hybrid B (hybrid at t1) will be part of the subtree in which species 2 and hybrid are sister taxa. Species 1 hybrid Species 2 t2 A B C t1 O Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work A 3 taxa network Species 1 hybrid Species 2 t2 A B C t1 O Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work The affinity between the hybrid and other species Denote r as the trait of hybrid. Denote x1 and x2 as the trait of species 1 and 2, respectively. Define r = µ + β(x1 + x2 − 2µ): (??) where β is called the hybrid parameter. Then cov(r; z) = cov(β(x1 + x2); z); z = x1; r; x2: (?) The new PCMs are associate with the hybrid parameter β. Introduction: PCMs Modeling PCMs with Hybridization Some ongoing work Phylogenetic Tree and Phylogenetic Network in BM model y1 y2 y3 x1 r x2 Human Chimp. Gorilla H. a. H. an. H. p. 0.6 0.6 0.6 0.6 0.6 1 0.4 0.4 0.4 y1 y2 y3 x1 r x2 0 1 0 1 y1 1 0:4 0 x1 1 0:4β 0 2 y2 @ 0:4 1 0 A r @ 0:4β 0:6 + 0:8β 0:4β A y3 0 0 1 x2 0 0:4β 1 Introduction: