Demographic model
Let’s imagine the following demographic model of Neanderthal
introgression into the ancestors of all non-Africans (represented by
“EUR” and “PAP” populations, approximating European and Papuan people
living today), followed by Denisovan introgression into the ancestors of
Papuans:
anc_all <- population("ancestor_all", time = 700e3, N = 10000, remove = 640e3)
afr <- population("AFR", parent = anc_all, time = 650e3, N = 10000)
anc_arch <- population("ancestor_archaics", parent = anc_all, time = 650e3, N = 10000, remove = 390e3)
nea <- population("NEA", parent = anc_arch, time = 400e3, N = 2000, remove = 30e3)
den <- population("DEN", parent = anc_arch, time = 400e3, N = 2000, remove = 30e3)
nonafr <- population("nonAFR", parent = afr, time = 100e3, N = 3000, remove = 39e3)
eur <- population("EUR", parent = nonafr, time = 45e3, N = 5000)
pap <- population("PAP", parent = nonafr, time = 45e3, N = 5000)
gf <- list(
gene_flow(from = nea, to = nonafr, rate = 0.03, start = 55000, end = 50000),
gene_flow(from = den, to = pap, rate = 0.07, start = 35000, end = 30000)
)
model <- compile_model(
populations = list(anc_all, afr, anc_arch, nea, den, nonafr, eur, pap),
gene_flow = gf,
generation_time = 30,
serialize = FALSE
)
plot_model(
model, sizes = FALSE,
order = c("AFR", "EUR", "nonAFR", "PAP", "ancestor_all", "DEN", "ancestor_archaics", "NEA")
)
Tree sequence simulation
Let’s now simulate a 50Mb tree sequence from this model, recording 50
diploid individuals from EUR and PAP populations:
samples <- schedule_sampling(model, times = 0, list(eur, 50), list(pap, 50))
ts <- msprime(model, sequence_length = 100e6, recombination_rate = 1e-8, samples = samples, random_seed = 42)
Summaries of ancestral proportions
When we summarise the ancestry proportions in target EUR and PAP
populations, we see that the EUR population only carries about ~3% of
Neanderthal ancestry and that this is also true for the PAP population.
However, we also see that Papuans carry about 7% of Denisovan ancestry.
This is consistent with our model, but also with the expectation from empirical data.
summary <- tracts %>%
group_by(name, node_id, pop, source_pop) %>%
summarise(prop = sum(length) / 100e6)
#> `summarise()` has grouped output by 'name', 'node_id', 'pop'. You can override
#> using the `.groups` argument.
summary %>% group_by(pop, source_pop) %>% summarise(mean(prop)) %>% arrange(source_pop, pop)
#> `summarise()` has grouped output by 'pop'. You can override using the `.groups`
#> argument.
#> # A tibble: 3 × 3
#> # Groups: pop [2]
#> pop source_pop `mean(prop)`
#> <fct> <fct> <dbl>
#> 1 EUR NEA 0.0303
#> 2 PAP NEA 0.0348
#> 3 PAP DEN 0.0739
Let’s visualize these proportions at an individual level:
summary %>%
ggplot(aes(source_pop, prop, color = source_pop, fill = source_pop)) +
geom_jitter() +
coord_cartesian(ylim = c(0, 0.2)) +
geom_hline(yintercept = c(0.03, 0.08), linetype = 2) +
ylab("ancestry proportion") +
facet_wrap(~ pop) +
ggtitle("Ancestry proportions in each individual",
"(vertical lines represent 3% and 7% baseline expectations")
“Chromosome painting” of ancestry tracts
Because the tracts object contains the coordinates of
every single ancestry segment in each of the simulated individuals, we
can “paint” each chromosome with each of the two archaic human
ancestries:
tracts %>%
mutate(chrom = paste(name, " (node", node_id, ")")) %>%
ggplot(aes(x = left, xend = right, y = chrom, yend = chrom, color = source_pop)) +
geom_segment(linewidth = 3) +
theme_minimal() +
labs(x = "position [bp]", y = "haplotype") +
ggtitle("True ancestry tracts along each chromosome") +
theme(axis.text.y = element_blank(), panel.grid = element_blank()) +
facet_grid(pop ~ ., scales = "free_y")
By lining up NEA & DEN ancestry tracts in both EUR and PAP
populations, we can see how the common origin of Neanderthal ancestry in
both non-African populations manifests in a significant overlap of NEA
tracts between both populations.
Average tract lengths:
Let’s compute simple summaries of tract lengths in the simulated
data, and compare them to theoretical expectations.
tracts %>%
group_by(pop, source_pop) %>%
summarise(mean(length))
#> `summarise()` has grouped output by 'pop'. You can override using the `.groups`
#> argument.
#> # A tibble: 3 × 3
#> # Groups: pop [2]
#> pop source_pop `mean(length)`
#> <fct> <fct> <dbl>
#> 1 EUR NEA 65765.
#> 2 PAP NEA 69204.
#> 3 PAP DEN 100317.
Theoretical expectations (from Racimo and Slatkin 2015, Box 1)
m <- 0.03
t <- 52500 / 30
r <- 1e-8
mean_nea <- 1 / ((1 - m) * r * (t - 1))
mean_nea
#> [1] 58943.84
m <- 0.07
t <- 37500 / 30
r <- 1e-8
mean_den <- 1 / ((1 - m) * r * (t - 1))
mean_den
#> [1] 86090.38
As we can see, our simulations are not that far of from the
theoretical expectations, giving us confidence that our simulations (and
the ancestry tract extraction algorithm) are working as expected.
Distribution of ancestry tract lengths
Finally, let’s plot the distributions of lengths of each of the
ancestry tracts. The case of archaic human introgression is very well
studied so it’s perhaps not that exciting to look at these figures. That
said, in less well studied species, it might be interesting to use these
kinds of simulations for inference of introgression times and
proportions via Approximate Bayesian Computation or by another
method:
expectation_df <- data.frame(
pop = c("EUR", "PAP", "PAP"),
source_pop = c("NEA", "NEA", "DEN"),
length = c(mean_nea, mean_nea, mean_den)
)
p_densities <- tracts %>%
ggplot(aes(length, color = source_pop)) +
geom_density() +
geom_vline(data = expectation_df, aes(xintercept = length, color = source_pop),
linetype = 2) +
facet_wrap(~ pop) +
ggtitle("Distribution of tract lengths per different ancestries")
cowplot::plot_grid(p_densities, p_densities + scale_x_log10(), nrow = 2)
Pure msprime tree sequence
Finally, as a sanity check, let’s use the pure msprime
simulation example from the official
tspop documentation to test that ts_tracts()
behaves as expected even on a standard msprime tree-sequence
object.
First, let’s run the simulation code exactly as it is:
import msprime
pop_size = 500
sequence_length = 1e7
seed = 98765
rho = 3e-8
# Make the Demography object.
demography = msprime.Demography()
demography.add_population(name="RED", initial_size=pop_size)
#> Population(initial_size=500, growth_rate=0, name='RED', description='', extra_metadata={}, default_sampling_time=None, initially_active=None, id=0)
demography.add_population(name="BLUE", initial_size=pop_size)
#> Population(initial_size=500, growth_rate=0, name='BLUE', description='', extra_metadata={}, default_sampling_time=None, initially_active=None, id=1)
demography.add_population(name="ADMIX", initial_size=pop_size)
#> Population(initial_size=500, growth_rate=0, name='ADMIX', description='', extra_metadata={}, default_sampling_time=None, initially_active=None, id=2)
demography.add_population(name="ANC", initial_size=pop_size)
#> Population(initial_size=500, growth_rate=0, name='ANC', description='', extra_metadata={}, default_sampling_time=None, initially_active=None, id=3)
demography.add_admixture(
time=100, derived="ADMIX", ancestral=["RED", "BLUE"], proportions=[0.5, 0.5]
)
#> Admixture(time=100, derived='ADMIX', ancestral=['RED', 'BLUE'], proportions=[0.5, 0.5])
demography.add_census(time=100.01) # Census is here!
#> CensusEvent(time=100.01)
demography.add_population_split(
time=1000, derived=["RED", "BLUE"], ancestral="ANC"
)
#> PopulationSplit(time=1000, derived=['RED', 'BLUE'], ancestral='ANC')
# Simulate.
ts = msprime.sim_ancestry(
samples={"RED": 0, "BLUE": 0, "ADMIX" : 2},
demography=demography,
random_seed=seed,
sequence_length=sequence_length,
recombination_rate=rho
)
Let’s save the msprime tree sequence to disk so that we can
load it into R (i.e., approximating what you might want to do should you
want to use ts_tracts() without running a slendr
simulation first):
import tempfile
path = tempfile.NamedTemporaryFile(suffix=".trees").name
ts.dump(path)
Now let’s move to R again, load the tree sequence into
slendr and extract ancestry tracts from it using
ts_tracts():
sim_ts <- ts_read(reticulate::py$path)
squashed_tracts <- ts_tracts(sim_ts, census = 100.01, squashed = TRUE)
#>
#> PopAncestry summary
#> Number of ancestral populations: 2
#> Number of sample chromosomes: 4
#> Number of ancestors: 118
#> Total length of genomes: 40000000.000000
#> Ancestral coverage: 40000000.000000
head(squashed_tracts)
#> # A tibble: 6 × 4
#> sample left right population
#> <dbl> <dbl> <dbl> <dbl>
#> 1 0 0 419848 0
#> 2 0 419848 483009 1
#> 3 0 483009 1475765 0
#> 4 0 1475765 2427904 1
#> 5 0 2427904 3635390 0
#> 6 0 3635390 4606954 1
tail(squashed_tracts)
#> # A tibble: 6 × 4
#> sample left right population
#> <dbl> <dbl> <dbl> <dbl>
#> 1 3 7134130 7362300 1
#> 2 3 7362300 7369409 0
#> 3 3 7369409 7596783 1
#> 4 3 7596783 8289015 0
#> 5 3 8289015 8918727 1
#> 6 3 8918727 10000000 0
By setting squashed = FALSE, we get the full,
un-squashed ancestry segments, each with its appropriate ancestral node
ID:
full_tracts <- ts_tracts(sim_ts, census = 100.01, squashed = FALSE)
#>
#> PopAncestry summary
#> Number of ancestral populations: 2
#> Number of sample chromosomes: 4
#> Number of ancestors: 118
#> Total length of genomes: 40000000.000000
#> Ancestral coverage: 40000000.000000
head(full_tracts)
#> # A tibble: 6 × 5
#> sample left right ancestor population
#> <int> <dbl> <dbl> <int> <int>
#> 1 0 0 33027 74 0
#> 2 0 33027 155453 33 0
#> 3 0 155453 290542 46 0
#> 4 0 290542 419848 18 0
#> 5 0 419848 483009 83 1
#> 6 0 483009 1475765 28 0
tail(full_tracts)
#> # A tibble: 6 × 5
#> sample left right ancestor population
#> <int> <dbl> <dbl> <int> <int>
#> 1 3 8477625 8672850 94 1
#> 2 3 8672850 8849756 95 1
#> 3 3 8849756 8918727 131 1
#> 4 3 8918727 9165035 44 0
#> 5 3 9165035 9176562 47 0
#> 6 3 9176562 10000000 58 0
By comparing the two tables above to the pandas data frames
in the tspop
documentation, we can see that we obtained the same results.