# Tutorial¶

This is the tutorial for the Python interface to the msprime library. Detailed API Documentation is also available for this library. An ms compatible command line interface is also available if you wish to use msprime directly within an existing work flow. Please see the tskit documentation for more information on how to use the tskit Python API to analyse simulation results.

## Simulating trees¶

Running simulations is very straightforward in msprime:

>>> import msprime
>>> tree_sequence = msprime.simulate(sample_size=6, Ne=1000)
>>> tree = tree_sequence.first()
>>> print(tree.draw(format="unicode"))

10
┏━━┻━┓
┃    9
┃  ┏━┻━┓
8  ┃   ┃
┏┻┓ ┃   ┃
┃ ┃ ┃   7
┃ ┃ ┃ ┏━┻┓
┃ ┃ ┃ ┃  6
┃ ┃ ┃ ┃ ┏┻┓
3 5 0 4 1 2


Here, we simulate the coalescent for a sample of size six with an effective population size of 1000 diploids, and then print out a depiction of the resulting tree. The msprime library uses tskit to represent simulation results and the simulate() function returns a tskit.TreeSequence object, which provides a very efficient way to access the correlated trees in simulations involving recombination. In this example we know that there can only be one tree because we have not provided a value for recombination_rate, and it defaults to zero. Therefore, we access the only tree in the sequence using the first() method. Finally, we draw a simple depiction of the tree to the terminal using the tskit.Tree.draw() method.

Genealogical trees record the lines of descent along which genomes have been inherited. Since diploids have two copies of each autosomal chromosome, diploid individuals contain two such lines of descent: the simulation above provides the genealogical history of only three diploids.

Trees are represented within tskit (and therefore msprime) in a slightly unusual way. In the majority of libraries dealing with trees, each node is represented as an object in memory and the relationship between nodes as pointers between these objects. In tskit, however, nodes are integers. In the tree above, we can see that the leaves of the tree are labelled with 0 to 5, and all the internal nodes of the tree are also integers with the root of the tree being 10.

We can easily trace our path back to the root for a particular sample using the parent() method:

>>> u = 2
>>> while u != tskit.NULL:
>>>     print("node {}: time = {}".format(u, tree.time(u)))
>>>     u = tree.parent(u)
node 2: time = 0.0
node 6: time = 11.59282234272971
node 7: time = 129.57841077196494
node 9: time = 1959.4591339636365
node 10: time = 5379.737460469677


In this code chunk we iterate up the tree starting at node 0 and stop when we get to the root. We know that a node is a root if its parent is tskit.NULL, which is a special reserved node. (The value of the null node is -1, but we recommend using the symbolic constant to make code more readable.) We also use the time() method to get the time for each node, which corresponds to the time in generations at which the coalescence event happened during the simulation. We can also obtain the length of a branch joining a node to its parent using the branch_length() method:

>>> print(tree.branch_length(6))
117.98558842923524


The branch length for node 6 is about 118 generations, since the birth times of node 6 was 11 generations ago, and the birth time of its parent, node 7, was around 129 generations ago. It is also often useful to obtain the total branch length of the tree, i.e., the sum of the lengths of all branches:

>>> print(tree.total_branch_length)
13238.125493096279


## Recombination¶

Simulating the history of a single locus is a very useful, but we are most often interesting in simulating the history of our sample across large genomic regions under the influence of recombination. The msprime API is specifically designed to make this common requirement both easy and efficient. To model genomic sequences under the influence of recombination we have two parameters to the simulate() function. The length parameter specifies the length of the simulated sequence, and is a floating point number, so recombination (and mutation) can occur at any location along the sequence (the units are arbitrary). If length is not supplied, it is assumed to be 1.0. The recombination_rate parameter specifies the rate of crossing over per unit of length per generation, and is zero by default. See the API Documentation for a discussion of the precise recombination model used.

Here, we simulate the trees across over a 10kb region with a recombination rate of $$2 \times 10^{-8}$$ per base per generation, with a diploid effective population size of 1000:

>>> tree_sequence = msprime.simulate(
...     sample_size=6, Ne=1000, length=1e4, recombination_rate=2e-8)
>>> for tree in tree_sequence.trees():
...     print("-" * 20)
...     print("tree {}: interval = {}".format(tree.index, tree.interval))
...     print(tree.draw(format="unicode"))
--------------------
tree 0: interval = (0.0,  6016.224463474058)
11
┏━━┻━━┓
┃     10
┃  ┏━━┻━┓
┃  ┃    9
┃  ┃  ┏━┻┓
┃  7  ┃  ┃
┃ ┏┻┓ ┃  ┃
┃ ┃ ┃ ┃  6
┃ ┃ ┃ ┃ ┏┻┓
3 0 1 2 4 5

--------------------
tree 1: interval = (6016.224463474058, 10000.0)
10
┏━━┻━━┓
9     ┃
┏━┻┓    ┃
┃  ┃    8
┃  ┃  ┏━┻┓
┃  ┃  ┃  7
┃  ┃  ┃ ┏┻┓
┃  6  ┃ ┃ ┃
┃ ┏┻┓ ┃ ┃ ┃
2 4 5 3 0 1


In this example, we use the tskit.TreeSequence.trees() method to iterate over the trees in the sequence. For each tree we print out its index (i.e., its position in the sequence) and the interval the tree covers (i.e., the genomic coordinates which all share precisely this tree) using the tskit.Tree.index and tskit.Tree.interval attributes. Thus, the first tree covers the first 6kb of sequence and the second tree covers the remaining 4kb. We can see that these trees share a great deal of their structure, but that there are also important differences between the trees.

Warning

Do not store the values returned from the trees() iterator in a list and operate on them afterwards! For efficiency reasons tskit uses the same instance of tskit.Tree for each tree in the sequence and updates the internal state for each new tree. Therefore, if you store the trees returned from the iterator in a list, they will all refer to the same tree.

## Mutations¶

Mutations are generated in msprime by throwing mutations down on the branches of trees at a particular rate. The mutations are generated under the infinite sites model, and so each mutation occurs at a unique (floating point) point position along the genomic interval occupied by a tree. The mutation rate for simulations is specified using the mutation_rate parameter of simulate(). For example, the following chunk simulates 50kb of nonrecombining sequence with a mutation rate of $$1 \times 10^{-8}$$ per base per generation:

>>> tree_sequence = msprime.simulate(
...    sample_size=6, Ne=1000, length=50e3, mutation_rate=1e-8, random_seed=30)
>>> tree = tree_sequence.first()
>>> for site in tree.sites():
...     for mutation in site.mutations:
...         print("Mutation @ position {:.2f} over node {}".format(
...             site.position, mutation.node))
Mutation @ position 1556.54 over node 9
Mutation @ position 4485.17 over node 6
Mutation @ position 9788.56 over node 6
Mutation @ position 11759.03 over node 6
Mutation @ position 11949.32 over node 6
Mutation @ position 14321.77 over node 9
Mutation @ position 31454.99 over node 6
Mutation @ position 45125.69 over node 9
Mutation @ position 49709.68 over node 6

>>> print(tree.draw(format="unicode"))
10
┏━━┻━━┓
┃     9
┃   ┏━┻━┓
┃   ┃   8
┃   ┃  ┏┻┓
┃   7  ┃ ┃
┃  ┏┻┓ ┃ ┃
6  ┃ ┃ ┃ ┃
┏┻┓ ┃ ┃ ┃ ┃
0 4 2 5 1 3


It is also possible to add mutations to an existing tree sequence using the msprime.mutate() function.

## Variants¶

We are often interesting in accessing the sequence data that results from simulations directly. The most efficient way to do this is by using the tskit.TreeSequence.variants() method, which returns an iterator over all the tskit.Variant objects arising from the trees and mutations. Each variant contains a reference to the site object, as well as the alleles and the observed sequences for each sample in the genotypes field:

>>> tree_sequence = msprime.simulate(
...     sample_size=20, Ne=1e4, length=5e3, recombination_rate=2e-8,
...     mutation_rate=2e-8, random_seed=10)
>>> for variant in tree_sequence.variants():
...     print(
...         variant.site.id, variant.site.position,
...         variant.alleles, variant.genotypes, sep="\t")
0       2432.768327416852       ('0', '1')      [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0]
1       2577.6939414924095      ('0', '1')      [1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1]
2       2844.682702049562       ('0', '1')      [0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0]
3       4784.266628557816       ('0', '1')      [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0]


In this example we simulate some data and then print out the observed sequences. We loop through each variant and print out the observed state of each sample as an array of zeros and ones, along with the index and position of the corresponding mutation. In this example, the alleles are always '0' (the ancestral state) and '1' (the derived state), because we are simulating with the infinite sites mutation model, in which each mutation occurs at a unique position in the genome. More complex models are possible, however.

This way of working with the sequence data is quite efficient because we do not need to keep the entire genotype matrix in memory at once. However, if we do want the full genotype matrix it is simple to obtain:

>>> A = tree_sequence.genotype_matrix()
>>> A
array([[0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
[0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=uint8)


In this example, we run the same simulation but this time store the entire variant matrix in a two-dimensional numpy array. This is useful for integrating with tools such as scikit allel.

## Historical samples¶

Simulating coalescent histories in which some of the samples are not from the present time is straightforward in msprime. By using the samples argument to msprime.simulate() we can specify the location and time at which all samples are made.

def historical_samples_example():
samples = [
msprime.Sample(population=0, time=0),
msprime.Sample(0, 0),  # Or, we can use positional arguments.
msprime.Sample(0, 1.0),
msprime.Sample(0, 1.0)
]
tree_seq = msprime.simulate(samples=samples)
tree = tree_seq.first()
for u in tree.nodes():
print(u, tree.parent(u), tree.time(u), sep="\t")
print(tree.draw(format="unicode"))


In this example we create four samples, two taken at the present time and two taken 1.0 generations in the past, as might represent one modern and one ancient diploid individual. There are a number of different ways in which we can describe the samples using the msprime.Sample object (samples can be provided as plain tuples also if more convenient). Running this example, we get:

>>> historical_samples_example()
6    -1    2.8240255501413247
4    6    0.0864109319103291
0    4    0.0
1    4    0.0
5    6    1.9249243960710336
2    5    1.0
3    5    1.0
6
┏━┻━┓
┃   5
┃  ┏┻┓
┃  2 3
┃
4
┏┻┓
0 1


Because nodes 0 and 1 were sampled at time 0, their times in the tree are both 0. Nodes 2 and 3 were sampled at time 1.0, and so their times are recorded as 1.0 in the tree.

## Replication¶

A common task for coalescent simulations is to check the accuracy of analytical approximations to statistics of interest. To do this, we require many independent replicates of a given simulation. msprime provides a simple and efficient API for replication: by providing the num_replicates argument to the simulate() function, we can iterate over the replicates in a straightforward manner. Here is an example where we compare the analytical results for the number of segregating sites with simulations:

import msprime
import numpy as np

def segregating_sites_example(n, theta, num_replicates):
S = np.zeros(num_replicates)
replicates = msprime.simulate(
Ne=0.5,
sample_size=n,
mutation_rate=theta / 2,
num_replicates=num_replicates)
for j, tree_sequence in enumerate(replicates):
S[j] = tree_sequence.num_sites
# Now, calculate the analytical predictions
S_mean_a = np.sum(1 / np.arange(1, n)) * theta
S_var_a = (
theta * np.sum(1 / np.arange(1, n)) +
theta**2 * np.sum(1 / np.arange(1, n)**2))
print("              mean              variance")
print("Observed      {}\t\t{}".format(np.mean(S), np.var(S)))
print("Analytical    {:.5f}\t\t{:.5f}".format(S_mean_a, S_var_a))


Running this code, we get:

>>> segregating_sites_example(10, 5, 100000)
mean              variance
Observed      14.17893          53.0746740551
Analytical    14.14484          52.63903


Note that in this example we set $$N_e = 0.5$$ and the mutation rate to $$\theta / 2$$ when calling simulate(). This works because msprime simulates Kingman’s coalescent, for which $$N_e$$ is only a time scaling; since $$N_e$$ is the diploid effective population size, setting $$N_e = 0.5$$ means that the mean time for two samples to coalesce is equal to one time unit in the resulting trees. This is helpful for converting the diploid per-generation time units of msprime into the haploid coalescent units used in many theoretical results. However, it is important to note that conventions vary widely, and great care is needed with such factor-of-two rescalings.

## Population structure¶

Population structure in msprime closely follows the model used in the ms simulator: we have $$N$$ subpopulations with an $$N\times N$$ matrix describing the migration rates between these subpopulations. The sample sizes, population sizes and growth rates of all subpopulations can be specified independently. Migration rates are specified using a migration matrix. Unlike ms however, all times and rates are specified in generations and all populations sizes are absolute (that is, not multiples of $$N_e$$).

In the following example, we calculate the mean coalescence time for a pair of lineages sampled in different subpopulations in a symmetric island model, and compare this with the analytical expectation.

import msprime
import numpy as np

def migration_example(num_replicates=10**4):
# M is the overall symmetric migration rate, and d is the number
# of subpopulations.
M = 0.2
d = 3
m = M / (2 * (d - 1))
# Allocate the initial sample. Because we are interested in the
# between-subpopulation coalescence times, we choose one sample each
# from the first two subpopulations.
population_configurations = [
msprime.PopulationConfiguration(sample_size=1),
msprime.PopulationConfiguration(sample_size=1),
msprime.PopulationConfiguration(sample_size=0)]
# Now we set up the migration matrix. Since this is a symmetric
# island model, we have the same rate of migration between all
# pairs of subpopulations. Diagonal elements must be zero.
migration_matrix = [
[0, m, m],
[m, 0, m],
[m, m, 0]]
# We pass these values to the simulate function, and ask it
# to run the required number of replicates.
replicates = msprime.simulate(Ne=0.5,
population_configurations=population_configurations,
migration_matrix=migration_matrix,
num_replicates=num_replicates)
# And then iterate over these replicates
T = np.zeros(num_replicates)
for i, tree_sequence in enumerate(replicates):
tree = tree_sequence.first()
T[i] = tree.time(tree.root) / 4
# Finally, calculate the analytical expectation and print
# out the results
analytical = d / 4 + (d - 1) / (4 * M)
print("Observed  =", np.mean(T))
print("Predicted =", analytical)


Again, we set $$N_e = 0.5$$ to agree with convention in theoretical results, where usually one coalescent time unit is, in generations, the effective number of haploid individuals. Running this example we get:

>>> migration_example()
Observed  = 3.254904176088153
Predicted = 3.25


## Demography¶

Msprime provides a flexible and simple way to model past demographic events in arbitrary combinations. Here is an example describing the Gutenkunst et al. out-of-Africa model. See Figure 2B for a schematic of this model, and Table 1 for the values used. Coalescent simulation moves from the present back into the past, so times are in units of generations ago, and we build the model with most recent events first.

Todo

Add a diagram of the model for convenience.

import math
def out_of_africa():
# First we set out the maximum likelihood values of the various parameters
# given in Table 1.
N_A = 7300
N_B = 2100
N_AF = 12300
N_EU0 = 1000
N_AS0 = 510
# Times are provided in years, so we convert into generations.
generation_time = 25
T_AF = 220e3 / generation_time
T_B = 140e3 / generation_time
T_EU_AS = 21.2e3 / generation_time
# We need to work out the starting (diploid) population sizes based on
# the growth rates provided for these two populations
r_EU = 0.004
r_AS = 0.0055
N_EU = N_EU0 / math.exp(-r_EU * T_EU_AS)
N_AS = N_AS0 / math.exp(-r_AS * T_EU_AS)
# Migration rates during the various epochs.
m_AF_B = 25e-5
m_AF_EU = 3e-5
m_AF_AS = 1.9e-5
m_EU_AS = 9.6e-5
# Population IDs correspond to their indexes in the population
# configuration array. Therefore, we have 0=YRI, 1=CEU and 2=CHB
# initially.
population_configurations = [
msprime.PopulationConfiguration(
sample_size=0, initial_size=N_AF),
msprime.PopulationConfiguration(
sample_size=1, initial_size=N_EU, growth_rate=r_EU),
msprime.PopulationConfiguration(
sample_size=1, initial_size=N_AS, growth_rate=r_AS)
]
migration_matrix = [
[      0, m_AF_EU, m_AF_AS],
[m_AF_EU,       0, m_EU_AS],
[m_AF_AS, m_EU_AS,       0],
]
demographic_events = [
# CEU and CHB merge into B with rate changes at T_EU_AS
msprime.MassMigration(
time=T_EU_AS, source=2, destination=1, proportion=1.0),
msprime.MigrationRateChange(time=T_EU_AS, rate=0),
msprime.MigrationRateChange(
time=T_EU_AS, rate=m_AF_B, matrix_index=(0, 1)),
msprime.MigrationRateChange(
time=T_EU_AS, rate=m_AF_B, matrix_index=(1, 0)),
msprime.PopulationParametersChange(
time=T_EU_AS, initial_size=N_B, growth_rate=0, population_id=1),
# Population B merges into YRI at T_B
msprime.MassMigration(
time=T_B, source=1, destination=0, proportion=1.0),
# Size changes to N_A at T_AF
msprime.PopulationParametersChange(
time=T_AF, initial_size=N_A, population_id=0)
]
# Use the demography debugger to print out the demographic history
# that we have just described.
dd = msprime.DemographyDebugger(
population_configurations=population_configurations,
migration_matrix=migration_matrix,
demographic_events=demographic_events)
dd.print_history()


The DemographyDebugger provides a method to debug the history that you have described so that you can be sure that the migration rates, population sizes and growth rates are all as you intend during each epoch:

=============================
Epoch: 0 -- 848.0 generations
=============================
start     end      growth_rate |     0        1        2
-------- --------       -------- | -------- -------- --------
0 |1.23e+04 1.23e+04              0 |     0      3e-05   1.9e-05
1 |2.97e+04   1e+03           0.004 |   3e-05      0     9.6e-05
2 |5.41e+04    510           0.0055 |  1.9e-05  9.6e-05     0

Events @ generation 848.0
- Mass migration: lineages move from 2 to 1 with probability 1.0
- Migration rate change to 0 everywhere
- Migration rate change for (0, 1) to 0.00025
- Migration rate change for (1, 0) to 0.00025
- Population parameter change for 1: initial_size -> 2100 growth_rate -> 0
==================================
Epoch: 848.0 -- 5600.0 generations
==================================
start     end      growth_rate |     0        1        2
-------- --------       -------- | -------- -------- --------
0 |1.23e+04 1.23e+04              0 |     0     0.00025     0
1 | 2.1e+03  2.1e+03              0 |  0.00025     0        0
2 |   510   2.27e-09         0.0055 |     0        0        0

Events @ generation 5600.0
- Mass migration: lineages move from 1 to 0 with probability 1.0
===================================
Epoch: 5600.0 -- 8800.0 generations
===================================
start     end      growth_rate |     0        1        2
-------- --------       -------- | -------- -------- --------
0 |1.23e+04 1.23e+04              0 |     0     0.00025     0
1 | 2.1e+03  2.1e+03              0 |  0.00025     0        0
2 |2.27e-09 5.17e-17         0.0055 |     0        0        0

Events @ generation 8800.0
- Population parameter change for 0: initial_size -> 7300
================================
Epoch: 8800.0 -- inf generations
================================
start     end      growth_rate |     0        1        2
-------- --------       -------- | -------- -------- --------
0 | 7.3e+03  7.3e+03              0 |     0     0.00025     0
1 | 2.1e+03  2.1e+03              0 |  0.00025     0        0
2 |5.17e-17     0            0.0055 |     0        0        0


Warning

The output of the DemographyDebugger.print_history() method is intended only for debugging purposes, and is not meant to be machine readable. The format is also preliminary; if there is other information that you think would be useful, please open an issue on GitHub

Once you are satisfied that the demographic history that you have built is correct, it can then be simulated by calling the simulate() function.

## Recombination maps¶

The msprime API allows us to quickly and easily simulate data from an arbitrary recombination map. In this example we read a recombination map for human chromosome 22, and simulate a single replicate. After the simulation is completed, we plot histograms of the recombination rates and the simulated breakpoints. These show that density of breakpoints follows the recombination rate closely.

import numpy as np
import scipy.stats
import matplotlib.pyplot as pyplot

def variable_recomb_example():
infile = "hapmap/genetic_map_GRCh37_chr22.txt"

# Now we get the positions and rates from the recombination
# map and plot these using 500 bins.
positions = np.array(recomb_map.get_positions()[1:])
rates = np.array(recomb_map.get_rates()[1:])
num_bins = 500
v, bin_edges, _ = scipy.stats.binned_statistic(
positions, rates, bins=num_bins)
x = bin_edges[:-1][np.logical_not(np.isnan(v))]
y = v[np.logical_not(np.isnan(v))]
fig, ax1 = pyplot.subplots(figsize=(16, 6))
ax1.plot(x, y, color="blue")
ax1.set_ylabel("Recombination rate")
ax1.set_xlabel("Chromosome position")

# Now we run the simulation for this map. We simulate
# 50 diploids (100 sampled genomes) in a population with Ne=10^4.
tree_sequence = msprime.simulate(
sample_size=100,
Ne=10**4,
recombination_map=recomb_map)
# Now plot the density of breakpoints along the chromosome
breakpoints = np.array(list(tree_sequence.breakpoints()))
ax2 = ax1.twinx()
v, bin_edges = np.histogram(breakpoints, num_bins, density=True)
ax2.plot(bin_edges[:-1], v, color="green")
ax2.set_ylabel("Breakpoint density")
ax2.set_xlim(1.5e7, 5.3e7)
fig.savefig("hapmap_chr22.svg")


## Completing forwards simulations¶

The msprime simulator generates tree sequences using the backwards in time coalescent model. But it is also possible to output tree sequences from forwards-time simulators such as SLiM. There are many advantages to using forward-time simulators, but they are usually quite slow compared to similar coalescent simulations. In this section we show how to combine the best of both approaches by simulating the recent past using a forwards-time simulator and then complete the simulation of the ancient past using msprime. (We sometimes refer to this “recapitation”, as we can think of it as adding a “head” onto a tree sequence.)

First, we define a simple Wright-Fisher simulator which returns a tree sequence with the properties that we require (please see the API section for a formal description of these properties):

import random
import numpy as np

def wright_fisher(N, T, L=100, random_seed=None):
"""
Simulate a Wright-Fisher population of N haploid individuals with L
discrete loci for T generations. Based on Algorithm W from
https://www.biorxiv.org/content/biorxiv/early/2018/01/16/248500.full.pdf
"""
random.seed(random_seed)
tables = msprime.TableCollection(L)
P = np.arange(N, dtype=int)
# Mark the initial generation as samples so that we remember these nodes.
for j in range(N):
t = T
while t > 0:
t -= 1
Pp = P.copy()
for j in range(N):
Pp[j] = u
a = random.randint(0, N - 1)
b = random.randint(0, N - 1)
x = random.randint(1, L - 1)
P = Pp

# Now do some table manipulations to ensure that the tree sequence
# that we output has the form that msprime needs to finish the
# simulation. Much of the complexity here is caused by the tables API

# Mark the extant population as samples also
flags = tables.nodes.flags
flags[P] = msprime.NODE_IS_SAMPLE
tables.nodes.set_columns(flags=flags, time=tables.nodes.time)
tables.sort()
# Simplify with respect to the current generation, but ensuring we keep the
# ancient nodes from the initial population.
tables.simplify()
# Unmark the initial generation as samples
flags = tables.nodes.flags
time = tables.nodes.time
flags[:] = 0
flags[time == 0] = msprime.NODE_IS_SAMPLE
# The final tables must also have at least one population which
# the samples are assigned to
tables.nodes.set_columns(
flags=flags, time=time,
population=np.zeros_like(tables.nodes.population))
return tables.tree_sequence()


We then run a tiny forward simulation of 10 two-locus individuals for 5 generations, and print out the resulting trees:

num_loci = 2
N = 10
wf_ts = wright_fisher(N, 5, L=num_loci, random_seed=3)
for tree in wf_ts.trees():
print("interval = ", tree.interval)
print(tree.draw(format="unicode"))


We get:

interval =  (0.0, 1.0)
0                 7
┃                 ┃
25                ┃
┏━━━━┻━━━━┓            ┃
23        24           ┃
┏━┻━┓    ┏━━╋━━━┓        ┃
┃   21   ┃  ┃   22       20
┃  ┏┻━┓  ┃  ┃  ┏┻━┓   ┏━━╋━━┓
10 14 19 11 18 15 17  12 13 16

interval =  (1.0, 2.0)
0          8    4     7
┃          ┃   ┏┻━┓   ┃
21         ┃   ┃  ┃   ┃
┏━━┳━━┳━┻┳━━┳━━┓   ┃   ┃  ┃   ┃
14 19 10 13 16 18  11  15 17  12


Because our Wright Fisher simulation ran for only 5 generations, there has not been enough time for the trees to fully coalesce. Therefore, instead of having one root, the trees have several — the first tree has 2 and the second 4. Nodes 0 to 9 in this simulation represent the initial population of the simulation, and so we can see that all samples in the first tree trace back to one of two individuals from the initial generation. These unary branches joining samples and coalesced subtrees to the nodes in the initial generation are essential as they allow use to correctly assemble the various fragments of ancestral material into chromosomes when creating the initial conditions for the coalescent simulation. (Please see the API section for more details on the required properties of input tree sequences.)

The process of completing this tree sequence using a coalescent simulation begins by first examining the root segments on the input trees. We get the following segments:

[(0, 2, 0), (0, 2, 7), (1, 2, 8), (1, 2, 4)]


where each segment is a (left, right, node) tuple. As nodes 0 and 7 are present in both trees, they have segments spanning both loci. Nodes 8 and 4 are present only in the second tree, and so they have ancestral segments only for the second locus. Note that this means that we do not simulate the ancestry of the entire initial generation of the simulation, but rather the exact minimum that we need in order to complete the ancestry of the current generation. For instance, root 8 has not coalesced over the interval from 1.0 to 2.0, while root 0 has not coalesced over the entire segment from 0.0 to 2.0.

We run the coalescent simulation to complete this tree sequence using the from_ts argument to simulate(). Because we have simulated a two locus system with a recombination rate of 1 / num_loci per generation in the Wright-Fisher model, we want to use the same system in the coalescent simulation. To do this we create recombination map using the RecombinationMap.uniform_map() class method to easily create a discrete map with the required number of loci. (Please see the API section for more details on the restrictions on recombination maps when completing an existing simulation.) We also use a Ne value of N / 2 since the Wright-Fisher simulation was haploid and msprime is diploid.

recomb_map = msprime.RecombinationMap.uniform_map(num_loci, 1 / num_loci, num_loci)
coalesced_ts = msprime.simulate(
Ne=N / 2, from_ts=wf_ts, recombination_map=recomb_map, random_seed=5)


After running this simulation we get the following trees:

interval =  (0.0, 1.0)
26
┏━━━━━━━━┻━━━━━━━┓
0                7
┃                ┃
25               ┃
┏━━━━┻━━━━┓           ┃
23        24          ┃
┏━┻━┓    ┏━━╋━━━┓       ┃
┃   21   ┃  ┃   22      20
┃  ┏┻━┓  ┃  ┃  ┏┻━┓  ┏━━╋━━┓
10 14 19 11 18 15 17 12 13 16

interval =  (1.0, 2.0)
28
┏━━━━┻━━━━━┓
┃          27
┃        ┏━┻━━┓
26       ┃    ┃
┏━━━━┻━━━━┓   ┃    ┃
0         7   4    8
┃         ┃  ┏┻━┓  ┃
21        ┃  ┃  ┃  ┃
┏━━┳━━┳━┻┳━━┳━━┓  ┃  ┃  ┃  ┃
14 19 10 13 16 18 12 15 17 11


The trees have fully coalesced and we’ve successfully combined a forwards-time Wright-Fisher simulation with a coalescent simulation: hooray!

### Why record the initial generation?¶

We can now see why it is essential that the forwards simulator records the initial generation in a tree sequence that will later be used as a from_ts argument to simulate(). In the example above, if node 7 was not in the tree sequence, we would not know that the segment that node 20 inherits from on [0.0, 1.0) and the segment that node 12 inherits from on [1.0, 2.0) both exist in the same node (here, node 7).

However, note that although the intial generation (above, nodes 0, 4, 7, and 8) must be in the tree sequence, they do not have to be samples. The easiest way to do this is to (a) retain the initial generation as samples throughout the forwards simulation (so they persist through simplify()), but then (b) before we output the final tree sequence, we remove the flags that mark them as samples, so that simulate() does not simulate their entire history as well. This is the approach taken in the toy simulator provided above (although we skip the periodic simplify() steps which are essential in any practical simulation for simplicity).

### Topology gotchas¶

The trees that we output from this combined forwards and backwards simulation process have some slightly odd properties that are important to be aware of. In the example above, we can see that the old roots are still present in both trees, even through they have only one child and are clearly redundant. This is because the tables of from_ts have been retained, without modification, at the top of the tables of the output tree sequence. While this redundancy is not important for many tasks, there are some cases where they may cause problems:

1. When computing statistics on the number of nodes, edges or trees in a tree sequence, having these unary edges and redundant nodes will slightly inflate the values.
2. If you are computing the overall tree “height” by taking the time of the root node, you may overestimate the height because there is a unary edge above the “real” root (this would happen if one of the trees had already coalesced in the forwards-time simulation).

For these reasons it is usually better to remove this redundancy from your computed tree sequence which is easily done using the tskit.TreeSequence.simplify() method:

final_ts = coalesced_ts.simplify()

for tree in final_ts.trees():
print("interval = ", tree.interval)
print(tree.draw(format="unicode"))


giving us:

interval =  (0.0, 1.0)
17
┏━━━┻━━━━┓
┃        15
┃     ┏━━┻━━┓
┃     13    14
┃   ┏━┻┓  ┏━╋━━┓
10  ┃  11 ┃ ┃  12
┏━╋━┓ ┃ ┏┻┓ ┃ ┃ ┏┻┓
2 3 6 0 4 9 1 8 5 7

interval =  (1.0, 2.0)
19
┏━━━━━┻━━━━━┓
┃           18
┃         ┏━┻┓
17        ┃  ┃
┏━━━┻━━┓      ┃  ┃
┃      ┃      ┃  16
┃      ┃      ┃ ┏┻┓
┃      11     ┃ ┃ ┃
┃ ┏━┳━┳┻┳━┳━┓ ┃ ┃ ┃
2 4 9 0 3 6 8 1 5 7


This final tree sequence is topologically identical to the original tree sequence, but has the redundant nodes and edges removed. Note also that he node IDs have been reassigned so that the samples are 0 to 9 — if you need the IDs from the original tree sequence, please set map_nodes=True when calling simplify to get a mapping between the two sets of IDs.

## Recording the full ARG¶

In msprime we usually want to simulate the coalescent with recombination and represent the output as efficiently as possible. As a result, we don’t store individual recombination events, but rather their effects on the output tree sequence. We also do not explicitly store common ancestor events that do not result in marginal coalescences. For some purposes, however, we want to get information on the full history of the simulation, not just the minimal representation of its outcome. The record_full_arg option to simulate() provides this functionality, as illustrated in the following example:

def full_arg_example():
ts = msprime.simulate(
sample_size=5, recombination_rate=0.1, record_full_arg=True, random_seed=42)
print(ts.tables.nodes)
print()
for tree in ts.trees():
print("interval:", tree.interval)
print(tree.draw(format="unicode"))


Running this code we get:

id      flags   population      individual      time    metadata
0       1       0       -1      0.00000000000000
1       1       0       -1      0.00000000000000
2       1       0       -1      0.00000000000000
3       1       0       -1      0.00000000000000
4       1       0       -1      0.00000000000000
5       0       0       -1      0.31846010419674
6       0       0       -1      0.82270149120229
7       0       0       -1      1.21622732856555
8       131072  0       -1      1.51542116580501
9       131072  0       -1      1.51542116580501
10      262144  0       -1      2.12814260094490
11      0       0       -1      2.16974122606933

interval: (0.0, 0.7323522972251177)
11
┏━━┻━┓
┃    10
┃    ┃
┃    8
┃    ┃
7    ┃
┏━┻━┓  ┃
┃   6  ┃
┃  ┏┻┓ ┃
5  ┃ ┃ ┃
┏┻┓ ┃ ┃ ┃
0 4 2 3 1

interval: (0.7323522972251177, 1.0)
11
┏━━┻━┓
┃    10
┃    ┃
┃    9
┃    ┃
7    ┃
┏━┻━┓  ┃
┃   6  ┃
┃  ┏┻┓ ┃
5  ┃ ┃ ┃
┏┻┓ ┃ ┃ ┃
0 4 2 3 1


After running the simulation we first print out the node table, which contains information on all the nodes in the tree sequence. Note that flags column contains several different values: all of the sample nodes (at time 0) have a flag value of 1 (tskit.NODE_IS_SAMPLE). Other internal nodes have a flag value of 0, which is the standard for internal nodes in a coalescent simulations.

Nodes 8 and 9 have flags equal to 131072 (NODE_IS_RE_EVENT), which tells us that they correspond to a recombination event in the ARG. A recombination event results in two extra nodes being recorded, one identifying the individual providing the genetic material to the left of the breakpoint and the other identifying the individuals providing the genetic material to the right. The effect of this extra node can be seen in the trees: node 8 is present as a ‘unary’ node in the left hand tree and node 9 in the right.

Node 10 has a flags value of 262144 (NODE_IS_CA_EVENT), which tells us that it is an ARG common ancestor event that did not result in marginal coalescence. This class of event also results in unary nodes in the trees, which we can see in the example.

If we wish to reduce these trees down to the minimal representation, we can use tskit.TreeSequence.simplify(). The resulting tree sequence will have all of these unary nodes removed and will be equivalent to (but not identical, due to stochastic effects) calling simulate() without the record_full_arg argument.

Migrations nodes are also recording in the ARG using the NODE_IS_MIG_EVENT flag. See the Node flags section for more details.