How many sequences will two samples share?

The question: if two samples are drawn from the same population, how many sequences will they have in common by chance alone?

Overlap between repertoires is often read as evidence of a shared response. Before reading it that way, it helps to know how much overlap independent samples of one population produce on their own. BEAM computes it directly from the model of the population, without simulation.

Two samples

sharing(d1, d2) is the expected number of sequences found in both of two independent samples of depths d1 and d2.

sharing.py

import BEAM
from BEAM.analytics.exact_occupancy import ExactOccupancy

model = BEAM.load_pretrained("human_tcrb_cdr3_aa_foundation")
occupancy = ExactOccupancy(model)

print("  depth A    depth B   shared sequences   share of A's distinct sequences")
for a, b in [(1e5, 1e5), (1e5, 1e6), (1e6, 1e6), (1e7, 1e7)]:
    shared = occupancy.sharing(a, b).value
    distinct_a = occupancy.richness(a).value
    print(f"{a:9,.0f}  {b:9,.0f}  {shared:17,.0f}  {shared / distinct_a:32.1%}")

Output

  depth A    depth B   shared sequences   share of A's distinct sequences
  100,000    100,000              3,678                              3.8%
  100,000  1,000,000             14,633                             15.1%
1,000,000  1,000,000             82,670                              9.4%
10,000,000  10,000,000          1,090,052                             15.1%

Full program, including the set-up hidden here

Expected overlap between two independent samples of equal depth from the reference population.

Two samples of 100,000 cells share about 3,700 sequences by chance, 3.8% of either sample’s distinct sequences. The overlap grows with depth: at ten million cells each, 15% of one sample’s sequences are expected in the other. A deeper second sample also raises the overlap: 100,000 cells against a million share 15% of the smaller sample’s sequences.

Private sequences in a cohort

private(depths) is the expected number of sequences found in exactly one sample of a cohort.

private.py

import BEAM
from BEAM.analytics.exact_occupancy import ExactOccupancy

model = BEAM.load_pretrained("human_tcrb_cdr3_aa_foundation")
occupancy = ExactOccupancy(model)

# A cohort of 10 people, each sampled to 100,000 cells
depths = [100_000] * 10
private = occupancy.private(depths).value
all_distinct = occupancy.richness(sum(depths)).value

print(f"distinct sequences across the cohort: {all_distinct:,.0f}")
print(f"seen in exactly one person:           {private:,.0f} ({private / all_distinct:.1%})")

Output

distinct sequences across the cohort: 879,776
seen in exactly one person:           828,289 (94.1%)

Full program, including the set-up hidden here

Across ten people sampled to 100,000 cells each, 94% of all distinct sequences are expected to be private to one person. Shared sequences are the exception, even by chance.

The same calculation predicts how many distinct sequences a cohort accumulates as it is pooled. In the paper, without any adjustment, the reference model built without the 768 Emerson donors predicted 98.9% of their observed pooled distinct-sequence count.

Pitfalls

• This is the chance baseline. It assumes both samples are independent draws from one population. Real overlaps above it point to something shared: an HLA type, an exposure, or the same person sampled twice. Overlaps below it are possible too, when two populations differ.
• Depths are numbers of cells, not numbers of distinct sequences.
• Use the population the samples come from. The reference model describes a broad human aggregate. For overlaps within a specific group, such as one tissue or one disease cohort, a model of that group gives the relevant baseline.