How many generations does it take for the average descendant not to be genetically related to the ancestor?

Question

Parent 1 and 2 have children. Assume infinite, randomly-mating population size. How many generations until the median descendant by lineage of parent 1 has 0 base pairs inherited from parent 1?

I tried calculating this, but it was difficult to model variability in cross-over rates, and my computer is too slow for the Monte-Carlo I made.

Here is my guess:

from math import ceil, log2
import random
import numpy as np

haploid_genome_size_bp = 3300 * 1e6
diploid_genome_size_bp = haploid_genome_size_bp * 2
chromosome_pairs = 23

# Recomb hotspots
hotspot_length = 1500  # Taking the average length of 1-2 kb, in bp
hotspot_interval = 75000  # Taking the average interval of 50-100 kb, in bp
hotspots_per_genome = 30000  # Number of hotspots in the human genome

# Average crossover frequency per hotspot
avg_crossover_per_hotspot = 1 / 1300  # One crossover per 1300 meioses

total_hotspot_length_bp = hotspots_per_genome * hotspot_length

initial_genome_fraction = 1  # Starting with 100% of the genome from Parent 1
generations = 0  # Counter for the number of generations

# Loop to calculate the number of generations required for the descendant to have less than 1 bp from Parent 1
while initial_genome_fraction * diploid_genome_size_bp > 1:
    # Introduce variance in the average number of crossovers per chromosome, for probably needless complication
    # Using a Gaussian distribution centered around the average, with a standard deviation of 0.5
    avg_crossover_per_chromosome = np.random.normal(1.25, 0.5) #this doesn't actually affect the result

    # Total length affected by crossovers in one meiosis event bc hotspots
    total_crossover_length_bp = total_hotspot_length_bp * avg_crossover_per_hotspot * avg_crossover_per_chromosome * chromosome_pairs

    # Calculate the fraction of the genome that is affected by crossovers and recombination in each generation
    fraction_swapped_each_generation = 2 * (total_crossover_length_bp / diploid_genome_size_bp)

    # Update the fraction of the genome from Parent 1 in the descendant
    initial_genome_fraction *= 0.5 * (1 - fraction_swapped_each_generation) + 0.5 * fraction_swapped_each_generation / 2

    generations += 1

print(generations)

Result: 33 generations.

But really, information is lost sooner, since (approximately) non-SNPs won't contribute to information, non-SNPs are shared between individuals. Let's say there's 600 million SNPs, randomly distributed throughout the genome. Let's assume that parent 1 and the entire rest of the population differ at, say, 70% of SNPs (this value doesn't matter much, it's always 29 or 30).

from math import ceil, log2
import random
import numpy as np

total_snps = 600e6
haploid_genome_size_bp = 3300 * 1e6
diploid_genome_size_bp = haploid_genome_size_bp * 2
chromosome_pairs = 23

hotspot_length = 1500
hotspot_interval = 75000
hotspots_per_genome = 30000

avg_crossover_per_hotspot = 1 / 1300

total_hotspot_length_bp = hotspots_per_genome * hotspot_length

initial_snps_fraction = 0.7
generations = 0

while initial_snps_fraction * total_snps > 1:
    avg_crossover_per_chromosome = np.random.normal(1.25, 0.1)
    total_crossover_length_bp = total_hotspot_length_bp * avg_crossover_per_hotspot * avg_crossover_per_chromosome * chromosome_pairs
    fraction_swapped_each_generation = 2 * (total_crossover_length_bp / diploid_genome_size_bp)
    initial_snps_fraction *= 0.5 * (1 - fraction_swapped_each_generation) + 0.5 * fraction_swapped_each_generation / 2
    generations += 1
print(generations)

Result: 29 generations.

It is sensitive to the number of SNPs in existence. If we use 1 million SNPs, we get 15 generations.

Is this right? Can we find a more accurate estimate? My code probably has some mistakes.

Answer 1

Graham Coop performed simulations related to this question, based on actual transmission data for each chromosome in humans. He found that after 10 or 11 generations there was >50% chance that any particular ancestor contributed no genetic information to a specific descendant. Fewer than 5% of a person's 14th-generation ancestors will have contributed genetic information to them.

This blog post is part of a series. The first provides some more information about the data and assumptions feeding these simulations.

Answer 2

Father/Mother, DNA = D
Child, $\frac{1}{2} \times D$
Grandchild, $\frac{1}{2}\times \frac{1}{2} \times D= \left(\frac{1}{2}\right)^2 \times D$
Great grandchild, $\left(\frac{1}{2}\right)^3 \times D$
One's $n^{th}$ descendant has only $\left(\frac{1}{2}\right)^n$ of one's DNA.

Which generation and later will have only $1\%$ or less of your DNA?

$\left(\frac{1}{2}\right)^n \leq \frac{1}{100}$

$2^n \geq 100$

$n \geq 7$

That is to say, after 7 generations, one's genes are diluted to near homeopathic concentrations.

EDIT

Note that this is a(n) (over)simplfication of how heredity via DNA works. The actual processes are quite complsx.