Analysis of gene copy number changes in tumor phylogenetics Jun - - PowerPoint PPT Presentation
Analysis of gene copy number changes in tumor phylogenetics Jun Zhou, Yu Lin, Vaibhav Rajan, Willia Hoskins, Bing Feng and Jijun Tang Background u Cancer can be modeled by molecular evolutionary processes (specifically deletion, insertion,
Jun Zhou, Yu Lin, Vaibhav Rajan, Willia Hoskins, Bing Feng and Jijun Tang
u Cancer can be modeled by
molecular evolutionary processes (specifically deletion, insertion, etc.)
u A mutational phylogenetic tree
can be built with nodes as clones and subclones and directed edges as mutation processes.
Image from Davis A. and Navin N., 2016
u Genetic marker: Copy number of an array of genes
detected by Fluorescent In Situ Hybridization (FISH) at single-cell level
u Caused by insertion/deletion of genes u Caused by chromosomal aberrations u Data structure: u A clone is represented as a tuple of copy numbers u A patient is represented as a matrix of copy numbers ->
main (and only) input to the phylogenetic problem
u Distance-based Minimum Tree u NP-hard
u Input: A set of vertices 𝑊 and a 𝑊 × 𝑊 distance matrix OR
a metric system.
u Output: A 1-connected tree T = (𝑊, 𝐹) with minimum
weight (sum of distances of vertices connected by an edge)
u Prim’s/Kruskal’s greedy algorithm in polynomial time
u Steiner nodes: unobserved nodes (absent in the dataset) u Input: A set of vertices 𝑊 and a metric system. u Output: A 1-connected tree T = 𝑊), 𝐹 where 𝑊) ⊇ 𝑊 with
minimum weight (sum of distances of vertices connected by an edge)
u NP-hard u For the case of 𝑊 = 3, this reduces to the Median
problem.
u Sankoff’s algorithm in linear time
Image from Zhou J. et al., 2016
u Rectilinear (Manhattan) metric: sum of absolute
difference between corresponding positions from 2 tuples
u Input: A set of vertices 𝑊. u Output: A 1-connected tree T = 𝑊), 𝐹 where 𝑊) ⊇ 𝑊 with
minimum weight (sum of distances of vertices connected by an edge) under the rectilinear metric.
u Hanan’s Theorem for 2-D problem: There exists a RSMT
containing only Steiner points from the Hanan’s grid
u Solution space is bounded u Generalized for n-D u Exact (and naïve) algorithm would enumerate all possible
sets of Steiner nodes, compute Minimum Spanning Tree on the new tree and compute the weight
u Inspiration from the Median problem u Sankoff’s algorithm in linear time u Inspiration from Maximum parsimony problem u Maximum parsimony tree: heuristics of MP borrowed
from TNT package
Image from Zhou J. et al., 2016
Image from Zhou J. et al., 2016
u Minimizing number of Steiner nodes added by carefully
selecting which nodes to add first.
u Steiner count for an observed node A: the number of
triplets containing A that require a Steiner node to
u Inference score for Steiner nodes: sum of Steiner counts
in the triplet defining it
Image from Zhou J. et al., 2016
u MP heuristics (TNT package) to derive a tree whose leaves
contains the dataset
u Dynamic programming to assign states to internal nodes u Contract trivial edges (edge with weight 0 under
rectilinear metric)
Image from Zhou J. et al., 2016
Image from Zhou
al., 2016
Image from Zhou
al., 2016
Image from Zhou J. et al., 2016
u
QA
u Input: A set of vertices 𝑊. u Output: A 1-connected tree T = 𝑊), 𝐹 where 𝑊) ⊇ 𝑊 with
minimum weight (sum of distances of vertices connected by an edge) under a generalized metric to incorporate large scale duplication events.
Image from Zhou J. et al., 2016