# 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"
# Read in the recombination map using the read_hapmap method,
recomb_map = msprime.RecombinationMap.read_hapmap(infile)
# 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):
tables.nodes.add_row(time=T, flags=msprime.NODE_IS_SAMPLE)
t = T
while t > 0:
t -= 1
Pp = P.copy()
for j in range(N):
u = tables.nodes.add_row(time=t, flags=0)
Pp[j] = u
a = random.randint(0, N - 1)
b = random.randint(0, N - 1)
x = random.randint(1, L - 1)
tables.edges.add_row(0, x, P[a], u)
tables.edges.add_row(x, L, P[b], u)
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
# not allowing direct access to memory, which will change soon.
# 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.populations.add_row()
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:

- 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.
- 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.