Modeling Phylogenetic Comparative Methods with Hybridization

Transcription

1 Modeling Phylogenetic Comparative Methods with Hybridization Tony Jhwueng NIMBioS Interdisciplinary Seminar Jan

2 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.

3 Phylogenetic Comparative Methods

4 A Phylogenetic Tree Human (Akha) Chimpanzee Gorilla 4.7 million years ago 7.2 million years ago (Takahata et al., 1995)

5 Comparative Data Examples: Body mass (adult male)(kg) Brain mass (adult male)(gram) X body = x 1 x 2 x 3 Data from Jerison (1973) = , Y brain = y 1 y 2 y 3 =

6 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.

7 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?

8 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.

9 Brownian Motion process: dy t = σdb t σ (rate parameter) measures the intensity of the random fluctuations in the evolutionary process. BM: σ=1 BM: σ=3 Trait Value y(t) Trait Value y(t) Time t Time t

10 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.

11 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). y 1 y 2 y 3 Human Chimp. Gorilla G 3 = y 1 y 2 y 3 y y y

12 FIC (Felsenstein, 1985): trait has evolved under the Brownian motion The variation of rate of evolution of the trait value is proportional to time. y 1 y 2 y 3 MVN µ µ µ, σ y 1 y 2 y 3 Human Chimp. Gorilla

13 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)

14 Some useful textbooks and softwares for PCMs: The Comparative Method in Evolutionary Biology by Harvey and Pagel, Inferring Phylogenies by Joseph Felsenstein, 2004, Ch 26. Analysis of Phylogenetics and Evolution with R by Emmanuel Paradis, Softwares: PHYLIP (Joseph Felsenstein) COMPARE (Emília Martins) BROWNIE (Brian O Meara) Various R packages at

15 Modeling PCMs with Hybridization

16 Hybridization is common in nature: Sunflower Wild Sunflowers in a Field (photo by Erin Silversmith)

17 Hybridization is common in nature: Sunflower Helianthus annuus. Helianthus petiolaris.

18 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.

19 Hybridization is common in nature: Cichild Lake Tanganyika (African Great Lake: the second largest freshwater lake in the world). (picture from Wikipedia)

20 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 Koblmüller et al. 2007)

21 Hybridization is common in nature: Cichlid Hybrid cichilds from Lake Tanganyika Lamprologus meleagris. (photo by Hagblom, Fredrik ) Lamprologus speciosus.(photo by Slaboch, Roman )

22 Hybridization is common in nature: Cichlid Hybrid cichilds from Lake Tanganyika Neolamprologus fasciatus. (photo by Jensen, Johnny) Neolamprologus (photo by Gagliardi, Flavio ) multifasciatus.

23 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

24 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?

25 Modeling PCMs with Hybridization If evolution involves ancient hybridizations (reticulate evolutionary events), instead of the phylogenetic tree, incorporate the phylogenetic network into comparative analysis.

26 Linder and Rieseberg 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

27 A 3 taxa network A nucleotide inherited from the A parent (species 1 at t 1 ) of hybrid B (hybrid at t 1 ) will be part of the subtree in which species 1 and hybrid are sister taxa Species 1 hybrid Species 2 t 2 A B C t 1 O

28 A 3 taxa network A nucleotide inherited from the C parent (species 2 at t 1 ) of hybrid B (hybrid at t 1 ) will be part of the subtree in which species 2 and hybrid are sister taxa. Species 1 hybrid Species 2 t 2 A B C t 1 O

29 A 3 taxa network Species 1 hybrid Species 2 t 2 A B C t 1 O

30 The affinity between the hybrid and other species Denote r as the trait of hybrid. Denote x 1 and x 2 as the trait of species 1 and 2, respectively. Define r = µ + β(x 1 + x 2 2µ). ( ) where β is called the hybrid parameter. Then cov(r, z) = cov(β(x 1 + x 2 ), z), z = x 1, r, x 2. ( ) The new PCMs are associate with the hybrid parameter β.

31 Phylogenetic Tree and Phylogenetic Network in BM model y 1 y 2 y 3 Human Chimp. Gorilla x 1 r x 2 H. a. H. an. H. p y 1 y 2 y 3 y y y x 1 r x 2 x β 0 r 0.4β β 2 0.4β x β 1

32 Investigating the hybrid effect Test of assumption: Question 1: Using trait data with evolutionary evidence (network), does the trait of hybrid at origin fall between its parents with statistical significance? r = µ + β(x 1 + x 2 2µ) Hypothesis Testing: H 0 : β = 1 2

33 A simple approach for 3 taxa case Input trait data: (3,7,11)

34 A simple approach for 3 taxa case Statistical Model MVN µ µ µ, σ β 0 0.4β β 2 0.4β 0 0.4β 1

35 A simple approach for 3 taxa case Parameters are estimated by maximum likelihood analysis and confidence intervals are obtained from simulation. ˆµ = 6.78 (5.94, 8.32) ˆσ = 4.12 (2.97, 5.88) ˆβ = 0.47 (0.31, 0.65) Answer for Question 1 (H 0 : β = 0.5): Since 0.5 falls in the confidence interval (0.31, 0.65), the null hypothesis is not rejected. The trait of the hybrid in this data (3,7,11) to be more extreme than its parents is insignificant.

36 Investigating the hybrid effect Question 2: Does hybridization increase the rate of evolution? x 1 r x 2 H. a. H. an. H. p. x 1 r x 2 H. a. H. an. H. p

37 Investigating the hybrid effect Test different rate of evolution. (McPeek, 1995; O Meara et al. 2006) σ β 0 0.4β 0.8β β 0 0.4βt 1 1 σ β β 0.8β 2 0.4β + η βt

38 Investigating the hybrid effect Statistical models Y MVN(µ1, σ 2 G β ) (same rate) v.s. Y MVN(µ1, σ 2 (G β J) + η 2 J) (diff. rate) Hypothesis Testing: H 0 : σ 2 = η 2

39 A simple approach for 3 taxa case Answer for Question 2 (H 0 : σ 2 = η 2 ): Y MVN(µ1, σ 2 G β ) vs Y MVN(µ1, σ 2 (G β J) + η 2 J) By likelihood ratio test(lrt). Since 2{log L ˆβ/ log L ˆβ,ˆη } = 2.69 < χ 2 df =1 = 3.84 (P value = 0.101), we fail to reject the null hypothesis. Hence the heterogeneous rate of evolution between hybrid and other species is insignificant.

40 Phylogenetic Network 1 2 X X X

41 Algorithm for general construction of similarity matrix. 1 Input: phylogenetic network in enewick format. 2 Construct the similarity matrix through following recursive formula G β,k = K k 1 G β,k 1 K t k 1 + t k 1I k (BM) where K depends on hybridization (K h ) or speciation (K s ).

42 Quick look for algorithm of 4 taxa t t 2 9 t 1

43 [9] [5, 8]: Speciation t t t 1 9

44 [9] [5, 8]: Speciation G 2 = ( 5 8 ) 5 t t 1

45 [5, 8] [5, 7, 4]: Speciation t t 2 8 t 1 9

46 [5, 8] [5, 7, 4]: Speciation G 3 = t 1 + t t 1 + t 2 t t 1 t 1 + t 2

47 [5, 7, 4] [5, 6, 7, 4]: Hybridization t t 2 8 t 1 9

48 [5, 7, 4] [5, 6, 7, 4]: Hybridization G β,4 = t 1 + t 2 β(t 1 + t 2 ) β(t 1 + t 2 ) 2β 2 (t 1 + t 2 ) β(t 1 + t 2 ) βt β(t 1 + t 2 ) t 1 + t 2 t βt 1 t 1 t 1 + t 2

49 [5, 6, 7, 4] [1, 2, 3, 4]: Elongation t t 2 8 t 1 9

50 [5, 6, 7, 4] [1, 2, 3, 4]: Elongation G β,4 = t 1 + t 2 + t 3 β(t 1 + t 2 ) β(t 1 + t 2 ) t 3 + 2β 2 (t 1 + t 2 ) β(t 1 + t 2 ) βt β(t 1 + t 2 ) t 1 + t 2 + t 3 t βt 1 t 1 t 1 + t 2 + t 3

51 Statistical Model and Parameters Estimation Statistical model Y MVN(µ1, σ 2 G β ) Negative log likelihood function l(µ, σ 2, β Y) = n 2 log 2π + n 2 log σ log G β + 1 2σ 2 (Y µ1)t G 1 β (Y µ1) MLE estimation through derivative free method: golden section method.

52 Introduction: PCMs Modeling PCMs with Hybridization Application: Koblmu ller et al. (2007) Cichild in Lake Tanganyika: 5 hybrids (in bold) Some ongoing work

53 Phylogenetic Network: regraph from Koblmüller et al. (2007) 1 2 X X X

54 Input enewick format ((((((1 : 0.495, ((((2 : 0.215, (5 : 0.215)27 : 0)26 : 0.1, (14 : 0.315)34 : 0)33 : 0.055, (X1 : 0.1, (3 : 0.1)21 : 0)20 : 0.27)37 : 0.06, ((21 : 0, 4 : 0.1)22 : 0.26, (27 : 0, (X 2 : 0.17, (6 : 0.17)24 : 0)23 : 0.045)28 : 0.145)36 : 0.07)38 : 0.065)39 : 0.11, 7 : 0.605)41 : 0.045, ((8 : 0.575, 9 : 0.575)40 : 0.06, 10 : 0.635)42 : 0.015)43 : 0.125, ((((24 : 0, 11 : 0.17)25 : 0.075, 12 : 0.245)29 : 0.475, (13 : 0.68, (34 : 0, (15 : 0.275, (16 : 0.275)31 : 0)30 : 0.04)35 : 0.365)44 : 0.04)45 : 0.02, (31 : 0, X 3 : 0.275)32 : 0.465)46 : 0.035)47 : 0.11, (17 : 0.835, 18 : 0.835)48 : 0.05)49 : 0.115, 19 : 1)50 : 0; Trait data: total lengths of cichild (cm). (Froese and Pauly, 2010) Y = (y 1, y 2,, y 19 ) = (13.5, 12.4, 7, 5.8, 5.5, 16, 15, 25, 6.1, 6.5, 5.5, 5, 7.8, 15, 4.3, 4, 8.6, 10.3, 8.5)

55 Some result Model does not fit the raw data, log transform the data. MLE estimators ˆβ = 0.43, ˆµ = 2.06, ˆσ = Since ˆβ = 0.43 < 0.5, the trait values of those hybrid species are not more variable than those of the other species. As 0.5 falls in ( ˆβ 25, ˆβ 975 ) = ( , 1.05). The total length of those hybrids to be more extreme than their parents is insignificant.

56 Some ongoing work in this project 1 Contribute R packages bmhyd (BM model) and ouhyd (OU model). 2 Modeling with more parameters R i = β i (X i + Y i ), i = 1, 2,, d. 3 Improve these methods to allow adaptive radiations. 4 Develop sampling strategy to study performance (bias and power of β) of BM and OU model. 5 Model events such as horizontal gene transfers that affects traits but are biologically different from hybridization.

57 Acknowledgement: NIMBioS Elizabeth Housworth (Dept. of Mathematics, Indiana University Bloomington) Brian O Meara (EEB, Univ. of Tennessee, Knoxville) Emília Martin (Dept. of Biology, Indiana University Bloomington) Vasileios Maroulas (Dept. of Math, Univ. of Tennessee, Knoxville)

58 Thanks!

59 Bias and Power analysis for the hybrid parameter Access the bias and power of parameter β using Access 5, 9, 17, 33, 65 and 129 taxa phylogenetic network, each contains one ancient hybrid. The time where the ancient hybridization occurs. Set t 1 by t 1 = 0.1, 0.5, 0.9, separately.

60 A sample network for simulation t 4 t 3 t 2 17 taxa 1 hybrid t 1

61 Bias analysis of BM model Bias of β Red : t 1 = 0.1 Black : t 1 = 0.5 Blue : t 1 = β

62 Power analysis of BM model Power of β = Up : t 1 = 0.9 Middle : t 1 = 0.5 Down : t 1 = β