Distinguishing Convergence on Two-Taxon and Three-Taxon Networks - PowerPoint PPT Presentation

References Distinguishing Convergence on Two-Taxon and Three-Taxon Networks Jonathan Mitchell Supervisors: Barbara Holland, Jeremy Sumner University of Tasmania November 6, 2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References Convergence δ · π π � M M pushing through δ − − − − − − − − − − → δ M ′ M ′′ M ′ M ′′ Action of the splitting operator on an edge. � • M plays the role of implementing correlated changes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References Convergence δ · π π � M M pushing through δ − − − − − − − − − − → δ M ′ M ′′ M ′ M ′′ Action of the splitting operator on an edge. � • M plays the role of implementing correlated changes. • Sumner et al. [2012] showed that the model can also be used for convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References Convergence • Examples of convergence are hybridisation, horizontal gene transfer or convergence of morphological traits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References Convergence • Examples of convergence are hybridisation, horizontal gene transfer or convergence of morphological traits. • Compare non-clock-like and clock-like trees to networks with convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References Convergence • Examples of convergence are hybridisation, horizontal gene transfer or convergence of morphological traits. • Compare non-clock-like and clock-like trees to networks with convergence. • Are our convergence-divergence networks identifiable ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References Convergence • Examples of convergence are hybridisation, horizontal gene transfer or convergence of morphological traits. • Compare non-clock-like and clock-like trees to networks with convergence. • Are our convergence-divergence networks identifiable ? • Can our convergence-divergence networks be distinguished from simpler trees? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References Convergence-Divergence Network Convergence of two taxa is as follows: 00 01 10 11 , Q on two taxa. with each character state transition having the same rate, λ , from the binary symmetrical model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References Convergence-Divergence Network τ 1 τ 2 τ 3 τ 4 ? A three-taxon convergence-divergence network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References Convergence-Divergence Network τ 1 τ 2 τ 3 τ 4 A three-taxon convergence-divergence network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References Process • Given a tree or network, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References Process • Given a tree or network, 1. Transform the basis of the rate matrix of the model, eg. Hadamard transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References Process • Given a tree or network, 1. Transform the basis of the rate matrix of the model, eg. Hadamard transformation. 2. Determine the probability distribution of the tree or network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References Process • Given a tree or network, 1. Transform the basis of the rate matrix of the model, eg. Hadamard transformation. 2. Determine the probability distribution of the tree or network. 3. Determine if the time parameters can be recovered from the probability distribution ( identifiability ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References Process • Given a tree or network, 1. Transform the basis of the rate matrix of the model, eg. Hadamard transformation. 2. Determine the probability distribution of the tree or network. 3. Determine if the time parameters can be recovered from the probability distribution ( identifiability ). 4. Determine the constraints on the probability distribution, i.e. the probability space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References Process • Given a tree or network, 1. Transform the basis of the rate matrix of the model, eg. Hadamard transformation. 2. Determine the probability distribution of the tree or network. 3. Determine if the time parameters can be recovered from the probability distribution ( identifiability ). 4. Determine the constraints on the probability distribution, i.e. the probability space. • From here we can compare the probability spaces of competing trees and networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References Process • Given a tree or network, 1. Transform the basis of the rate matrix of the model, eg. Hadamard transformation. 2. Determine the probability distribution of the tree or network. 3. Determine if the time parameters can be recovered from the probability distribution ( identifiability ). 4. Determine the constraints on the probability distribution, i.e. the probability space. • From here we can compare the probability spaces of competing trees and networks. • If two trees or networks have the same probability space they are said to not be distinguishable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References Three-Taxon Networks 1 τ 1 τ 1 τ 3 3 τ 2 τ 2 1 2 3 2 Network 2 Network 1 τ 1 τ 1 τ 2 τ 2 τ 3 τ 3 1 2 3 1 2 3 Network 3 Network 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References Three-Taxon Networks τ 1 τ 1 τ 2 τ 2 τ 3 τ 3 1 2 3 1 2 3 Network 5 Network 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References Three-Taxon Networks τ 1 τ 1 τ 2 τ 2 τ 3 τ 3 τ 4 τ 4 1 2 3 1 2 3 Network 7 Network 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References Three-Taxon Networks τ 1 τ 2 τ 3 τ 4 1 2 3 Network 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References An Example: Three-Taxon Clock-Like Tree τ 1 τ 2 1 2 3 Three-taxon clock-like tree. • In the regular basis, P , and the Hadamard basis, � P , the probability distribution is       p 000 q 000 1       0 p 001 q 001             0 p 010 q 010            e − 2 τ 2  p 011 q 011   �     P = , P = = .       0 p 100 q 100             e − 2( τ 1 + τ 2 ) p 101 q 101             e − 2( τ 1 + τ 2 ) p 110 q 110 p 111 q 111 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References An Example: Three-Taxon Clock-Like Tree • Make the substitutions, x i = e − τ i , to convert to polynomial functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References An Example: Three-Taxon Clock-Like Tree • Make the substitutions, x i = e − τ i , to convert to polynomial functions. • For the three-taxon clock-like tree, { q 011 = x 2 q 101 = x 2 1 x 2 q 110 = x 2 1 x 2 2 , 2 , 2 } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References An Example: Three-Taxon Clock-Like Tree • Make the substitutions, x i = e − τ i , to convert to polynomial functions. • For the three-taxon clock-like tree, { q 011 = x 2 q 101 = x 2 1 x 2 q 110 = x 2 1 x 2 2 , 2 , 2 } . • Constraints are, { q 101 = q 110 , q 011 ≥ q 101 } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .