# 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.

## Simulating trees¶

Running simulations is very straightforward in msprime:

>>> import msprime
>>> tree_sequence = msprime.simulate(sample_size=5, Ne=1000)
>>> tree = next(tree_sequence.trees())
>>> print(tree)
{0: 5, 1: 7, 2: 5, 3: 7, 4: 6, 5: 6, 6: 8, 7: 8, 8: -1}


Here, we simulate the coalescent for a sample of size 5 with an effective population size of 1000, and then print out a summary of the resulting tree. The simulate() function returns a 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 call next(tree_sequence.trees()).

Trees are represented within 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 msprime, however, nodes are integers: the leaves (i.e., our sample) are the integers $$0$$ to $$n - 1$$, and every internal node is some positive integer $$\geq n$$. The result of printing the tree is a summary of how these nodes relate to each other in terms of their parents. For example, we can see that the parent of nodes 1 and 3 is node 7.

This relationship can be seen more clearly in a picture:

This image shows the same tree as in the example but drawn out in a more familiar format (images like this can be drawn for any tree using the draw() method). We can see that the leaves of the tree are labelled with 0 to 4, and all the internal nodes of the tree are also integers with the root of the tree being 8. Also shown here are the times for each internal node in generations. (The time for all leaves is 0, and so we don’t show this information to avoid clutter.)

Knowing that our leaves are 0 to 4, we can easily trace our path back to the root for a particular sample using the get_parent() method:

>>> u = 0
>>> while u != msprime.NULL_NODE:
>>>     print("node {}: time = {}".format(u, tree.get_time(u)))
>>>     u = tree.get_parent(u)
node 0: time = 0.0
node 5: time = 107.921165302
node 6: time = 1006.74711128
node 8: time = 1785.36352521


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 the root if its parent is msprime.NULL_NODE, 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 get_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 get_branch_length() method:

>>> print(tree.get_branch_length(6))
778.616413923


The branch length for node 6 is 778.6 generations as the time for node 6 is 1006.7 and the time of its parent is 1785.4. 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.get_total_branch_length())
>>> 5932.15093686


## 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 in bases, and may be a floating point number. If length is not supplied, it is assumed to be 1. The recombination_rate parameter specifies the rate of crossing over per base 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 an effective population size of 1000:

>>> tree_sequence = msprime.simulate(
...    sample_size=5, Ne=1000, length=1e4, recombination_rate=2e-8)
>>> for tree in tree_sequence.trees():
...    print(tree.get_interval(), str(tree), sep="\t")
(0.0, 4701.4225005874)      {0: 6, 1: 5, 2: 6, 3: 9, 4: 5, 5: 7, 6: 7, 7: 9, 9: -1}
(4701.4225005874, 10000.0)  {0: 6, 1: 5, 2: 6, 3: 8, 4: 5, 5: 8, 6: 9, 8: 9, 9: -1}


In this example, we use the trees() method to iterate over the trees in the sequence. For each tree we print out the interval the tree covers (i.e., the genomic coordinates which all share precisely this tree) using the get_interval() method. Thus, the first tree covers the first 4.7kb of sequence and the second tree covers the remaining 5.3kb. We also print out the summary of each tree in terms of the parent values for each tree. Again, these differences are best illustrated by some images:

(We have suppressed the node time labels here for clarity.) 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 msprime uses the same instance of SparseTree 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, to add some mutations to our example above, we can use:

>>> tree_sequence = msprime.simulate(
...    sample_size=5, Ne=1000, length=1e4, recombination_rate=2e-8, mutation_rate=2e-8)
>>> print("Total mutations = ", tree_sequence.get_num_mutations())
>>> for tree in tree_sequence.trees():
>>>     print(tree.get_interval(), list(tree.mutations()), sep="\t")
Total mutations =  1
(0.0, 4701.4225005874)  []
(4701.4225005874, 10000.0)      [Mutation(position=5461.212369738915, node=6, index=0)]


In this example (which has the same genealogies as our example above because we use the same random seed), we have one mutation which falls on the second tree. Mutations are represented as an object with three attributes: position is the location of the mutation in genomic coordinates, node is the node in the tree above which the mutation occurs, and index is the (zero-based) index of the mutation in the list. Positions are given as a floating point value as we are using the infinite sites model. Every mutation falls on exactly one tree and we obtain the mutations for a particular tree using the mutations() method. Mutations are always returned in increasing order of position. The mutation in this example is shown as a red box on the corresponding branch:

We can calculate the allele frequency of mutations easily and efficiently using the get_num_leaves() which returns the number of leaves underneath a particular node. For example,:

>>> for tree in tree_sequence.trees():
...    for position, node in tree.mutations():
...        print("Mutation @ position {} has frequency {}".format(
...            mutation.position,
...            tree.get_num_leaves(mutation.node) / tree.get_sample_size()))
Mutation @ position 5461.21236974 has frequency 0.4


Sometimes we are only interested in a subset of the mutations in a tree sequence. In these situations, it is useful (and efficient) to update the tree sequence to only include the mutations we are interested in using the TreeSequence.set_mutations() method. Here, for example, we simulate some data and then retain only the common variants where the allele frequency is greater than 0.5.

import msprime

def set_mutations_example():
tree_sequence = msprime.simulate(
sample_size=10000, Ne=1e4, length=1e7, recombination_rate=2e-8,
mutation_rate=2e-8)
print("Simulated ", tree_sequence.get_num_mutations(), "mutations")
common_mutations = []
for tree in tree_sequence.trees():
for mutation in tree.mutations():
p = tree.get_num_leaves(mutation.node) / tree.get_sample_size()
if p >= 0.5:
common_mutations.append(mutation)
tree_sequence.set_mutations(common_mutations)
print("Reduced to ", tree_sequence.get_num_mutations(), "common mutations")


Running this code, we get:

>>> set_mutations_example()
Simulated  78202 mutations
Reduced to  5571 common mutations


## 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 TreeSequence.variants() method, which returns an iterator over all the variant objects arising from the trees and mutations. Each variant contains all the information in a mutation object, but also has the observed sequences for each sample in the genotypes field.

import msprime

def variants_example():
tree_sequence = msprime.simulate(
sample_size=20, Ne=1e4, length=5e3, recombination_rate=2e-8,
mutation_rate=2e-8, random_seed=10)
print("Simulated ", tree_sequence.get_num_mutations(), "mutations")
for variant in tree_sequence.variants():
print(variant.index, variant.position, variant.genotypes, sep="\t")


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. (The default form for the genotypes array here is a numpy.ndarray; however, the output can also be a plain Python bytes object. See the TreeSequence.variants() documentation for details.) Running the code, we get:

>>> variants_example()
Simulated  7 mutations
0       2146.29801511   [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
1       2475.24314909   [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0]
2       3087.04505359   [0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0]
3       3628.35359621   [1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1]
4       4587.85827679   [0 0 0 0 0 1 1 0 1 0 0 1 0 1 0 0 0 0 0 0]
5       4593.29453791   [1 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1]
6       4784.26662856   [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0]


This way of working with the sequence data is quite efficient because we do not need to keep the entire variant matrix in memory at once.

import msprime
import numpy as np

def variant_matrix_example():
print("\nCreating full variant matrix")
tree_sequence = msprime.simulate(
sample_size=20, Ne=1e4, length=5e3, recombination_rate=2e-8,
mutation_rate=2e-8, random_seed=10)
shape = tree_sequence.get_num_mutations(), tree_sequence.get_sample_size()
A = np.empty(shape, dtype="u1")
for variant in tree_sequence.variants():
A[variant.index] = variant.genotypes
print(A)


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

>>> variant_matrix_example()
Creating full variant matrix
[[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 1 0 0 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 0 0 1 1 0 1 0 0 1 0 1 0 0 0 0 0 0]
[1 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0]]


## 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)
]
tree_seq = msprime.simulate(samples=samples)
tree = next(tree_seq.trees())
for u in range(tree_seq.get_num_nodes()):
print(u, tree.get_parent(u), tree.get_time(u), sep="\t")


In this example we create three samples, two taken at the present time and one taken 1.0 generations in the past. 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()
0       3       0.0
1       3       0.0
2       4       1.0
3       4       0.502039955384
4       -1      4.5595966593


Because nodes 0 and 1 were sampled at time 0, their times in the tree are both 0. Node 2 was sampled at time 1.0, and so its time is 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(
sample_size=n,
mutation_rate=theta / 4,
num_replicates=num_replicates)
for j, tree_sequence in enumerate(replicates):
S[j] = tree_sequence.get_num_mutations()
# 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      14.12173              52.4695318071
Analytical    14.14484              52.63903


Note that in this example we did not provide a value for the Ne argument to simulate(). In this case the effective population size defaults to 1, which can be useful for theoretical work. However, it is essential to remember that all rates and times must still be scaled by 4 to convert into the coalescent time scale.

## Population structure¶

Population structure in msprime closely follows the model used in the ms simulator: we have $$N$$ demes with an $$N\times N$$ matrix describing the migration rates between these subpopulations. The sample sizes, population sizes and growth rates of all demes 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 demes in a symmetric island model, and compare this with the analytical expectation.

import msprime
import numpy as np

def migration_example():
# M is the overall symmetric migration rate, and d is the number
# of demes.
M = 0.2
d = 3
# We rescale m into per-generation values for msprime.
m = M / (4 * (d - 1))
# Allocate the initial sample. Because we are interested in the
# between deme coalescence times, we choose one sample each
# from the first two demes.
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 demes. 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.
num_replicates = 1e6
replicates = msprime.simulate(
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 = next(tree_sequence.trees())
# Convert the TMRCA to coalecent units.
T[i] = tree.get_time(tree.get_root()) / 4
# Finally, calculate the analytical expectation and print
# out the results
analytical = d / 2 + (d - 1) / (2 * M)
print("Observed  =", np.mean(T))
print("Predicted =", analytical)


Running this example we get:

>>> migration_example()
Observed  = 6.50638181614
Predicted = 6.5


## 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.

Todo

Add a diagram of the model for convenience.

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.
dp = msprime.DemographyDebugger(
Ne=N_A,
population_configurations=population_configurations,
migration_matrix=migration_matrix,
demographic_events=demographic_events)
dp.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 |5.41e+04 2.41e-07         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 |5.41e+04  0.00123         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 |5.41e+04     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 assume Ne=10^4
# and have a sample of 100 individuals
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")


## Calculating LD¶

The msprime API provides methods to efficiently calculate population genetics statistics. For example, the LdCalculator class allows us to compute pairwise linkage disequilibrium coefficients. Here we use the get_r2_matrix() method to easily make an LD plot using matplotlib. (Thanks to the excellent scikit-allel for the basic plotting code used here.)

import msprime
import matplotlib.pyplot as pyplot

def ld_matrix_example():
ts = msprime.simulate(100, recombination_rate=10, mutation_rate=20,
random_seed=1)
ld_calc = msprime.LdCalculator(ts)
A = ld_calc.get_r2_matrix()
# Now plot this matrix.
x = A.shape[0] / pyplot.rcParams['savefig.dpi']
x = max(x, pyplot.rcParams['figure.figsize'][0])
fig, ax = pyplot.subplots(figsize=(x, x))
im = ax.imshow(A, interpolation="none", vmin=0, vmax=1, cmap="Blues")
ax.set_xticks([])
ax.set_yticks([])
for s in 'top', 'bottom', 'left', 'right':
ax.spines[s].set_visible(False)
pyplot.savefig("ld.svg")


When performing large calculations it’s often useful to split the work over multiple processes or threads. The msprime API can be used without issues across multiple processes, and the Python multiprocessing module often provides a very effective way to work with many replicate simulations in parallel.

When we wish to work with a single very large dataset, however, threads can offer better resource usage because of the shared memory space. The Python threading library gives a very simple interface to lightweight CPU threads and allows us to perform several CPU intensive tasks in parallel. The msprime API is designed to allow multiple threads to work in parallel when CPU intensive tasks are being undertaken.

Note

In the CPython implementation the Global Interpreter Lock ensures that only one thread executes Python bytecode at one time. This means that Python code does not parallelise well across threads, but avoids a large number of nasty pitfalls associated with multiple threads updating data structures in parallel. Native C extensions like numpy and msprime release the GIL while expensive tasks are being performed, therefore allowing these calculations to proceed in parallel.

In the following example we wish to find all mutations that are in approximate LD ($$r^2 \geq 0.5$$) with a given set of mutations. We parallelise this by splitting the input array between a number of threads, and use the LdCalculator.get_r2_array() method to compute the $$r^2$$ value both up and downstream of each focal mutation, filter out those that exceed our threshold, and store the results in a dictionary. We also use the very cool tqdm module to give us a progress bar on this computation.

import threading
import numpy as np
import tqdm
import msprime

def find_ld_sites(
tree_sequence, focal_mutations, max_distance=1e6, r2_threshold=0.5,
results = {}
progress_bar = tqdm.tqdm(total=len(focal_mutations))

ld_calc = msprime.LdCalculator(tree_sequence)
for focal_mutation in focal_mutations[start: start + chunk_size]:
a = ld_calc.get_r2_array(
focal_mutation, max_distance=max_distance,
direction=msprime.REVERSE)
rev_indexes = focal_mutation - np.nonzero(a >= r2_threshold)[0] - 1
a = ld_calc.get_r2_array(
focal_mutation, max_distance=max_distance,
direction=msprime.FORWARD)
fwd_indexes = focal_mutation + np.nonzero(a >= r2_threshold)[0] + 1
indexes = np.concatenate((rev_indexes[::-1], fwd_indexes))
results[focal_mutation] = indexes
progress_bar.update()

t.start()
t.join()
progress_bar.close()
return results

ts = msprime.simulate(
sample_size=1000, Ne=1e4, length=1e7, recombination_rate=2e-8,
mutation_rate=2e-8)
counts = np.zeros(ts.get_num_mutations())
for t in ts.trees():
for mutation in t.mutations():
counts[mutation.index] = t.get_num_leaves(mutation.node)
doubletons = np.nonzero(counts == 2)[0]
print(
"Found LD sites for", len(results), "doubleton mutations out of",
ts.get_num_mutations())


In this example, we first simulate 1000 samples of 10 megabases and find all doubleton mutations in the resulting tree sequence. We then call the find_ld_sites() function to find all mutations that are within 1 megabase of these doubletons and have an $$r^2$$ statistic of greater than 0.5.

The find_ld_sites() function performs these calculations in parallel using 8 threads. The real work is done in the nested thread_worker() function, which is called once by each thread. In the thread worker, we first allocate an instance of the LdCalculator class. (It is critically important that each thread has its own instance of LdCalculator, as the threads will not work efficiently otherwise.) After this, each thread works out the slice of the input array that it is responsible for, and then iterates over each focal mutation in turn. After the $$r^2$$ values have been calculated, we then find the indexes of the mutations corresponding to values greater than 0.5 using numpy.nonzero(). Finally, the thread stores the resulting array of mutation indexes in the results dictionary, and moves on to the next focal mutation.

Running this example we get:

>>> threads_example()
100%|████████████████████████████████████████████████| 4045/4045 [00:09<00:00, 440.29it/s]
Found LD sites for 4045 doubleton mutations out of 60100