# Source code for dendropy.model.coalescent

#! /usr/bin/env python

##############################################################################
##  DendroPy Phylogenetic Computing Library.
##
##  Copyright 2010-2015 Jeet Sukumaran and Mark T. Holder.
##
##  See "LICENSE.rst" for terms and conditions of usage.
##
##  If you use this work or any portion thereof in published work,
##
##     Sukumaran, J. and M. T. Holder. 2010. DendroPy: a Python library
##     for phylogenetic computing. Bioinformatics 26: 1569-1571.
##
##############################################################################

"""
Functions, classes, and methods for working with Kingman's n-coalescent
framework.
"""

import math
import dendropy
from dendropy.utility import GLOBAL_RNG
from dendropy.utility import constants
from dendropy.calculate import probability
from dendropy.calculate import combinatorics

###############################################################################
## Calculations and statistics

[docs]def discrete_time_to_coalescence(n_genes, pop_size=None, n_to_coalesce=2, rng=None): """ A random draw from the "Kingman distribution" (discrete time version): Time to go from n_genes genes to n_genes-1 genes in a discrete-time Wright-Fisher population of pop_size genes; i.e. waiting time until n-genes lineages coalesce in a population of pop_size genes. Parameters ---------- n_genes : integer The number of genes in the sample. pop_size : integer The effective *haploid* population size; i.e., number of genes in the population: 2 * N in a diploid population of N individuals, or N in a haploid population of N individuals. n_to_coalesce : integer The waiting time that will be returned will be the waiting time for this number of genes in the sample to coalesce. rng : Random object The random number generator instance. Returns ------- k : integer A randomly-generated waiting time (in discrete generations) for n_to_coalesce genes to coalesce out of a sample of n_genes in a population of pop_size genes. """ if not pop_size: time_units = 1.0 else: time_units = pop_size if rng is None: rng = GLOBAL_RNG p = pop_size / combinatorics.choose(n_genes, n_to_coalesce) tmrca = probability.geometric_rv(p) return tmrca * time_units
[docs]def time_to_coalescence(n_genes, pop_size=None, n_to_coalesce=2, rng=None): """ A random draw from the "Kingman distribution" (discrete time version): Time to go from n_genes genes to n_genes-1 genes in a continuous-time Wright-Fisher population of pop_size genes; i.e. waiting time until n-genes lineages coalesce in a population of pop_size genes. Given the number of gene lineages in a sample, n_genes, and a population size, pop_size, this function returns a random number from an exponential distribution with rate $\\choose(pop_size, 2)$. pop_size is the effective *haploid* population size; i.e., number of gene in the population: 2 * N in a diploid population of N individuals, or N in a haploid population of N individuals. If pop_size is 1 or 0 or None, then time is in haploid population units; i.e. where 1 unit of time equals 2N generations for a diploid population of size N, or N generations for a haploid population of size N. Otherwise time is in generations. The coalescence time, or the waiting time for the coalescence, of two gene lineages evolving in a population with haploid size $N$ is an exponentially-distributed random variable with rate of $N$ an expectation of $\\frac{1}{N}$). The waiting time for coalescence of *any* two gene lineages in a sample of $n$ gene lineages evolving in a population with haploid size $N$ is an exponentially-distributed random variable with rate of $\\choose{N, 2}$ and an expectation of $\\frac{1}{\choose{N, 2}}$. Parameters ---------- n_genes : integer The number of genes in the sample. pop_size : integer The effective *haploid* population size; i.e., number of genes in the population: 2 * N in a diploid population of N individuals, or N in a haploid population of N individuals. n_to_coalesce : integer The waiting time that will be returned will be the waiting time for this number of genes in the sample to coalesce. rng : Random object The random number generator instance to use. Returns ------- k : float A randomly-generated waiting time (in continuous time) for n_to_coalesce genes to coalesce out of a sample of n_genes in a population of pop_size genes. """ if rng is None: rng = GLOBAL_RNG if not pop_size: time_units = 1.0 else: time_units = pop_size rate = combinatorics.choose(n_genes, n_to_coalesce) tmrca = rng.expovariate(rate) return tmrca * time_units
[docs]def expected_tmrca(n_genes, pop_size=None, n_to_coalesce=2): """ Expected (mean) value for the Time to the Most Recent Common Ancestor of n_to_coalesce genes in a sample of n_genes drawn from a population of pop_size genes. Parameters ---------- n_genes : integer The number of genes in the sample. pop_size : integer The effective *haploid* population size; i.e., number of genes in the population: 2 * N in a diploid population of N individuals, or N in a haploid population of N individuals. n_to_coalesce : integer The waiting time that will be returned will be the waiting time for this number of genes in the sample to coalesce. rng : Random object The random number generator instance. Returns ------- k : float The expected waiting time (in continuous time) for n_to_coalesce genes to coalesce out of a sample of n_genes in a population of pop_size genes. """ nc2 = combinatorics.choose(n_genes, n_to_coalesce) tmrca = (float(1)/nc2) if pop_size is not None: return tmrca * pop_size else: return tmrca
[docs]def coalesce_nodes(nodes, pop_size=None, period=None, rng=None, use_expected_tmrca=False): """ Returns a list of nodes that have not yet coalesced once period is exhausted. This function will a draw a coalescence time, t, from an exponential distribution with a rate of choose(k, 2), where k is the number of nodes. If period is given and if this time is less than period, or if period is not given, then two nodes are selected at random from nodes, and coalesced: a new node is created, and the two nodes are added as child_nodes to this node with an edge length such the the total length from tip to the ancestral node is equal to the depth of the deepest child + t. The two nodes are removed from the list of nodes, and the new node is added to it. t is then deducted from period, and the process repeats. The function ends and returns the list of nodes once period is exhausted or if any draw of t exceeds period, if period is given or when there is only one node left. As each coalescent event occurs, *all* nodes have their edges extended to the point of the coalescent event. In the case of constrained coalescence, all uncoalesced nodes have their edges extended to the end of the period (coalesced nodes have the edges fixed by the coalescent event in their ancestor). Thus multiple calls to this method with the same set of nodes will gradually 'grow' the edges, until all the the nodes coalesce. The edge lengths of the nodes passed to this method thus should not be modified or reset until the process is complete. Parameters ---------- nodes : iterable[|Node|] An interable of |Node| objects representing a sample of neutral genes (some, all, or none of these nodes may have descendent nodes). pop_size : integer The effective *haploid* population size; i.e., number of genes in the population: 2 * N in a diploid population of N individuals, or N in a haploid population of N individuals. period : numeric The time that the genes have to coalesce. If pop_size is 1 or 0 or None, then time is in haploid population units; i.e. where 1 unit of time equals 2N generations for a diploid population of size N, or N generations for a haploid population of size N. Otherwise time is in generations. rng : Random object The random number generator instance to use. If not specified, the default RNG will be used. use_expected_tmrca : bool If |True|, then instead of random times, the *expected* times will be used. Returns ------- nodes : iterable[|Node|] A list of nodes once period is exhausted or if any draw of t exceeds period, if period is given or when there is only one node left. """ # idiot-check, because I can be an idiot if not nodes: return [] # set the random number generator if rng is None: rng = GLOBAL_RNG # define the function needed to create new coalescence nodes new_node = nodes[0].__class__ # make a shallow copy of the node list nodes = list(nodes) # start tracking the time remaining time_remaining = period # If there is no time constraint, we want to continue coalescing # until there is only one gene left in the pool. If there is a # time constraint, we continue as long as there is time remaining, # but we do not control for that here: it is automatically taken # care of when the time drawn for the next coalescent event # exceeds the time remaining, and triggers a break from the loop while len(nodes) > 1: if use_expected_tmrca: tmrca = expected_tmrca(len(nodes), pop_size=pop_size) else: # draw a time to coalesce: this will be an exponential random # variable with parameter (rate) of BINOMIAL[n_genes 2] # multiplied pop_size tmrca = time_to_coalescence(len(nodes), pop_size=pop_size, rng=rng) # if no time_remaining is given (i.e, we want to coalesce till # there is only one gene left) or, if we are working under the # constrained coalescent, if the time to the next coalescence # event is not longer than the time_remaining if time_remaining is None or tmrca <= time_remaining: # stretch out the edges of all the nodes to this time for node in nodes: if node.edge.length is None: node.edge.length = 0.0 node.edge.length = node.edge.length + tmrca # pick two nodes to coalesce at random to_coalesce = rng.sample(nodes, 2) # create the new ancestor of these nodes new_ancestor = new_node() # add the nodes as child nodes of the new node, their # common ancestor, and set the ancestor's edge length new_ancestor.add_child(to_coalesce[0]) new_ancestor.add_child(to_coalesce[1]) new_ancestor.edge.length = 0.0 # remove the nodes that have coalesced from the pool of # nodes nodes.remove(to_coalesce[0]) nodes.remove(to_coalesce[1]) # add the ancestor to the pool of nodes nodes.append(new_ancestor) # adjust the time_remaining left to coalesce if time_remaining is not None: time_remaining = time_remaining - tmrca else: # the next coalescent event takes place after the period constraint break # adjust the edge lengths of all the nodes, so they are at the # correct height, with the edges 'lining up' at the end of # coalescent period if time_remaining is not None and time_remaining > 0: for node in nodes: if node.edge.length is None: node.edge.length = 0.0 node.edge.length = node.edge.length + time_remaining # return the list of nodes that have not coalesced return nodes
[docs]def node_waiting_time_pairs(tree, ultrametricity_precision=constants.DEFAULT_ULTRAMETRICITY_PRECISION): """ Returns a list of tuples of (nodes, coalescent interval time) on the tree. That is, each element in the list is tuple pair consisting of where: the first element of the pair is an internal node representing a coalescent event on the tree, and the second element of the pair is the time between this coalescence event and the earlier (more recent) one. Parameters ---------- tree : |Tree| A tree instance. ultrametricity_precision : float When calculating the node ages, an error will be raised if the tree is not ultrametric. This error may be due to floating-point or numerical imprecision. You can set the precision of the ultrametricity validation by setting the ultrametricity_precision parameter. E.g., use ultrametricity_precision=0.01 for a more relaxed precision, down to 2 decimal places. Use ultrametricity_precision=False to disable checking of ultrametricity. Returns ------- x : list of tuples (node, coalescent interval) Returns list of tuples of (node, coalescent interval [= time between last coalescent event and current node age]) """ tree.calc_node_ages(ultrametricity_precision=ultrametricity_precision) ages = [(n, n.age) for n in tree.internal_nodes()] ages.sort(key=lambda x: x[1]) intervals = [] intervals.append(ages[0]) for i, d in enumerate(ages[1:]): nd = d[0] prev_nd = ages[i][0] intervals.append( (nd, nd.age - prev_nd.age) ) return intervals
[docs]def extract_coalescent_frames(tree, ultrametricity_precision=constants.DEFAULT_ULTRAMETRICITY_PRECISION): """ Returns a list of tuples describing the coalescent frames on the tree. That is, each element in the list is tuple pair consisting of where: the first element of the pair is the number of separate lineages remaining on the tree at coalescence event, and the second element of the pair is the time between this coalescence event and the earlier (more recent) one. Parameters ---------- tree : |Tree| A tree instance. ultrametricity_precision : float When calculating the node ages, an error will be raised if the tree is not ultrametric. This error may be due to floating-point or numerical imprecision. You can set the precision of the ultrametricity validation by setting the ultrametricity_precision parameter. E.g., use ultrametricity_precision=0.01 for a more relaxed precision, down to 2 decimal places. Use ultrametricity_precision=False to disable checking of ultrametricity. Returns ------- x : dict Returns dictionary, with key = number of alleles, and values = waiting time for coalescent for the given tree """ nwti = node_waiting_time_pairs(tree, ultrametricity_precision=ultrametricity_precision) # num_genes = len(tree.taxon_namespace) num_genes = len(tree.leaf_nodes()) num_genes_wt = {} for n in nwti: num_genes_wt[num_genes] = n[1] num_genes = num_genes - len(n[0].child_nodes()) + 1 # num_alleles_list = sorted(num_genes_wt.keys(), reverse=True) return num_genes_wt
[docs]def log_probability_of_coalescent_frames(coalescent_frames, haploid_pop_size): """ Under the classical neutral coalescent \citep{Kingman1982, Kingman1982b}, the waiting times between coalescent events in a sample of $k$ alleles segregating in a population of (haploid) size $N_e$ is distributed exponentially with a rate parameter of :math\\frac{{k \choose 2}}{N_e}:: .. math:: \\Pr(T) = \\frac{{k \\choose 2}}{N_e} \\e{- \\frac{{k \\choose 2}}{N_e} T}, where $T$ is the length of (chronological) time in which there are $k$ alleles in the sample (i.e., for $k$ alleles to coalesce into $k-1$ alleles). """ lp = 0.0 for k, t in coalescent_frames.items(): k2N = (float(k * (k-1)) / 2) / haploid_pop_size # k2N = float(combinatorics.choose(k, 2)) / haploid_pop_size lp = lp + math.log(k2N) - (k2N * t) return lp
[docs]def log_probability_of_coalescent_tree(tree, haploid_pop_size, ultrametricity_precision=constants.DEFAULT_ULTRAMETRICITY_PRECISION): """ Wraps up extraction of coalescent frames and reporting of probability. """ return log_probability_of_coalescent_frames(extract_coalescent_frames(tree), haploid_pop_size)
############################################################################### ## Tree Simulations
[docs]def contained_coalescent_tree(containing_tree, gene_to_containing_taxon_map, edge_pop_size_attr="pop_size", default_pop_size=1, rng=None): """ Returns a gene tree simulated under the coalescent contained within a population or species tree. containing_tree The population or species tree. If edge_pop_size_map is not None, and population sizes given are non-trivial (i.e., >1), then edge lengths on this tree are in units of generations. Otherwise edge lengths are in population units; i.e. 2N generations for diploid populations of size N, or N generations for diploid populations of size N. gene_to_containing_taxon_map A TaxonNamespaceMapping object mapping Taxon objects in the containing_tree TaxonNamespace to corresponding Taxon objects in the resulting gene tree. edge_pop_size_attr Name of attribute of edges that specify population size. By default this is "pop_size". If this attribute does not exist, default_pop_size will be used. The value for this attribute should be the haploid population size or the number of genes; i.e. 2N for a diploid population of N individuals, or N for a haploid population of N individuals. This value determines how branch length units are interpreted in the input tree, containing_tree. If a biologically-meaningful value, then branch lengths on the containing_tree are properly read as generations. If not (e.g. 1 or 0), then they are in population units, i.e. where 1 unit of time equals 2N generations for a diploid population of size N, or N generations for a haploid population of size N. Otherwise time is in generations. If this argument is None, then population sizes default to default_pop_size. default_pop_size Population size to use if edge_pop_size_attr is None or if an edge does not have the attribute. Defaults to 1. The returned gene tree will have the following extra attributes: pop_node_genes A dictionary with nodes of containing_tree as keys and a list of gene tree nodes that are uncoalesced as values. Note that this function does very much the same thing as constrained_kingman(), but provides a very different API. """ if rng is None: rng = GLOBAL_RNG gene_tree_taxon_namespace = gene_to_containing_taxon_map.domain_taxon_namespace if gene_tree_taxon_namespace is None: gene_tree_taxon_namespace = dendropy.TaxonNamespace() for gene_taxa in pop_gene_taxa_map: for taxon in gene_taxa: gene_tree_taxon_namespace.add(taxon) gene_tree = dendropy.Tree(taxon_namespace=gene_tree_taxon_namespace) gene_tree.is_rooted = True pop_node_genes = {} pop_gene_taxa = gene_to_containing_taxon_map.reverse for nd in containing_tree.postorder_node_iter(): if nd.taxon and nd.taxon in pop_gene_taxa: pop_node_genes[nd] = [] gene_taxa = pop_gene_taxa[nd.taxon] for gene_taxon in gene_taxa: gene_node = dendropy.Node() gene_node.taxon = gene_taxon pop_node_genes[nd].append(gene_node) #gene_nodes = [dendropy.Node() for i in range(len(gene_taxa))] #for gidx, gene_node in enumerate(gene_nodes): # gene_node.taxon = gene_taxa[gidx] # pop_node_genes[nd].append(gene_node) for edge in containing_tree.postorder_edge_iter(): if edge_pop_size_attr and hasattr(edge, edge_pop_size_attr): pop_size = getattr(edge, edge_pop_size_attr) else: pop_size = default_pop_size if edge.head_node.parent_node is None: if len(pop_node_genes[edge.head_node]) > 1: final = coalesce_nodes(nodes=pop_node_genes[edge.head_node], pop_size=default_pop_size, period=None, rng=rng) else: final = pop_node_genes[edge.head_node] gene_tree.seed_node = final[0] else: uncoal = coalesce_nodes(nodes=pop_node_genes[edge.head_node], pop_size=pop_size, period=edge.length, rng=rng) if edge.tail_node not in pop_node_genes: pop_node_genes[edge.tail_node] = [] pop_node_genes[edge.tail_node].extend(uncoal) gene_tree.pop_node_genes = pop_node_genes return gene_tree
[docs]def pure_kingman_tree(taxon_namespace, pop_size=1, rng=None): """ Generates a tree under the unconstrained Kingman's coalescent process. Parameters ---------- taxon_namespace: |TaxonNamespace| instance A pre-populated |TaxonNamespace| where the contained |Taxon| instances represent the genes or individuals sampled from the population. pop_size : numeric The size of the population from the which the coalescent process is sampled. Returns ------- t : |Tree| A tree sampled from the Kingman's neutral coalescent. """ if rng is None: rng = GLOBAL_RNG # use the global rng by default nodes = [dendropy.Node(taxon=t) for t in taxon_namespace] seed_node = coalesce_nodes(nodes=nodes, pop_size=pop_size, period=None, rng=rng, use_expected_tmrca=False)[0] tree = dendropy.Tree(taxon_namespace=taxon_namespace, seed_node=seed_node) return tree
[docs]def pure_kingman_tree_shape(num_leaves, pop_size=1, rng=None): """ Like :func:dendropy.model.pure_kingman_tree, but does not assign taxa to tips. Parameters ---------- num_leaves : int Number of individuals/genes sampled. pop_size : numeric The size of the population from the which the coalescent process is sampled. Returns ------- t : |Tree| A tree sampled from the Kingman's neutral coalescent. """ if rng is None: rng = GLOBAL_RNG # use the global rng by default nodes = [dendropy.Node() for t in range(num_leaves)] seed_node = coalesce_nodes(nodes=nodes, pop_size=pop_size, period=None, rng=rng, use_expected_tmrca=False)[0] tree = dendropy.Tree(seed_node=seed_node) return tree
[docs]def mean_kingman_tree(taxon_namespace, pop_size=1, rng=None): """ Returns a tree with coalescent intervals given by the expected times under Kingman's neutral coalescent. """ if rng is None: rng = GLOBAL_RNG # use the global rng by default nodes = [dendropy.Node(taxon=t) for t in taxon_namespace] seed_node = coalesce_nodes(nodes=nodes, pop_size=pop_size, period=None, rng=rng, use_expected_tmrca=True)[0] tree = dendropy.Tree(taxon_namespace=taxon_namespace, seed_node=seed_node) return tree
[docs]def constrained_kingman_tree(pop_tree, gene_tree_list=None, rng=None, gene_node_label_fn=None, num_genes_attr='num_genes', pop_size_attr='pop_size', decorate_original_tree=False): """ Given a population tree, pop_tree this will return a *pair of trees*: a gene tree simulated on this population tree based on Kingman's n-coalescent, and population tree with the additional attribute 'gene_nodes' on each node, which is a list of uncoalesced nodes from the gene tree associated with the given node from the population tree. pop_tree should be a DendroPy Tree object or an object of a class derived from this with the following attribute num_genes -- the number of gene samples from each population in the present. Each edge on the tree should also have the attribute pop_size_attr is the attribute name of the edges of pop_tree that specify the population size. By default it is pop_size. The should specify the effective *haploid* population size; i.e., number of gene in the population: 2 * N in a diploid population of N individuals, or N in a haploid population of N individuals. If pop_size is 1 or 0 or None, then the edge lengths of pop_tree is taken to be in haploid population units; i.e. where 1 unit equals 2N generations for a diploid population of size N, or N generations for a haploid population of size N. Otherwise the edge lengths of pop_tree is taken to be in generations. If gene_tree_list is given, then the gene tree is added to the tree block, and the tree block's taxa block will be used to manage the gene tree's taxa. gene_node_label_fn is a function that takes two arguments (a string and an integer, respectively, where the string is the containing species taxon label and the integer is the gene index) and returns a label for the corresponding the gene node. if decorate_original_tree is True, then the list of uncoalesced nodes at each node of the population tree is added to the original (input) population tree instead of a copy. Note that this function does very much the same thing as contained_coalescent(), but provides a very different API. """ # get our random number generator if rng is None: rng = GLOBAL_RNG # use the global rng by default if gene_tree_list is not None: gtaxa = gene_tree_list.taxon_namespace else: gtaxa = dendropy.TaxonNamespace() if gene_node_label_fn is None: gene_node_label_fn = lambda x, y: "%s_%02d" % (x, y) # we create a set of gene nodes for each leaf node on the population # tree, and associate those gene nodes to the leaf by assignment # of 'taxon'. for leaf_count, leaf in enumerate(pop_tree.leaf_node_iter()): gene_nodes = [] for gene_count in range(getattr(leaf, num_genes_attr)): gene_node = dendropy.Node() gene_node.taxon = gtaxa.require_taxon(label=gene_node_label_fn(leaf.taxon.label, gene_count+1)) gene_nodes.append(gene_node) leaf.gene_nodes = gene_nodes # We iterate through the edges of the population tree in post-order, # i.e., visiting child edges before we visit parent edges. For # each edge visited, we take the genes found in the child nodes, # and run the coalescent simulation on them attacheded by the length # of the edge. Any genes that have not yet coalesced at the end of # this period are added to the genes of the tail (parent) node of # the edge. if decorate_original_tree: working_poptree = pop_tree else: # start with a new (deep) copy of the population tree so as to not # to change the original tree working_poptree = dendropy.Tree(pop_tree) # start with a new tree gene_tree = dendropy.Tree() gene_tree.taxon_namespace = gtaxa for edge in working_poptree.postorder_edge_iter(): # if mrca root, run unconstrained coalescent if edge.head_node.parent_node is None: if len(edge.head_node.gene_nodes) > 1: final = coalesce_nodes(nodes=edge.head_node.gene_nodes, pop_size=pop_size, period=None, rng=rng) else: final = edge.head_node.gene_nodes gene_tree.seed_node = final[0] else: if hasattr(edge, pop_size_attr): pop_size = getattr(edge, pop_size_attr) else: # this means all our time will be in population units pop_size = 1 uncoal = coalesce_nodes(nodes=edge.head_node.gene_nodes, pop_size=pop_size, period=edge.length, rng=rng) if not hasattr(edge.tail_node, 'gene_nodes'): edge.tail_node.gene_nodes = [] edge.tail_node.gene_nodes.extend(uncoal) gene_tree.is_rooted = True if gene_tree_list is not None: gene_tree_list.append(gene_tree) return gene_tree, working_poptree else: return gene_tree, working_poptree