Identity by Descent

Identity by descent (IBD) is a fundamental concept in population genetics and statistical genomics describing two homologous alleles—or two continuous homologous chromosomal segments—that are identical because both were inherited by replication from the exact same ancestral chromosome in a shared common ancestor without intervening mutation. Introduced in probabilistic form by French mathematician Gustave Malécot in Les Mathématiques de l'Hérédité (1948) as an extension of Sewall Wright's (1921, 1922) path coefficients, identity by descent is strictly distinguished from identity by state (IBS) (also called allozigosity), in which two alleles share the same nucleotide or amino acid state regardless of whether they coalesce in a recent common ancestor or arose independently via recurrent mutation or ancient standing polymorphism.

Whereas two alleles that are identical by descent (barring fresh de novo mutation) are necessarily identical by state, two alleles that are identical by state—such as two common adenosine ('A') nucleotides at a single-nucleotide polymorphism (SNP) with population allele frequency p = 0.60—need not be identical by descent relative to a defined reference pedigree or historical time horizon. Because every pair of homologous chromosomes in a finite species ultimately coalesces at some point in the deep past under the Wright–Fisher model, classical IBD is always defined relative to a specified founder reference generation (in pedigree analysis) or a coalescent time threshold T (in population genomics).

In modern high-density SNP array and whole-genome sequencing analyses, IBD is extended from single point loci to continuous IBD chromosomal tracts bounded by meiotic genetic recombination breakpoints. The length and abundance of these shared genomic tracts enable precise quantification of individual inbreeding via runs of homozygosity (F_ROH), reconstruction of recent demographic history (N_e[t]), gene mapping, and forensic genealogical kinship identification.

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Autozygosity vs. Allozygosity

At any single diploid locus within an individual I, the two homologous alleles (one maternal, one paternal) fall into one of two mutually exclusive ancestral states relative to the founder generation:

• Autozygous (Identical by Descent): Both alleles are physical replicates of a single ancestral copy carried by a common ancestor of the individual's mother and father. The probability that two homologous alleles within individual I are autozygous defines the inbreeding coefficient (F_I). Because an autozygous locus is guaranteed to be homozygous (AA or aa), an individual with inbreeding coefficient F exhibits homozygous genotype frequencies of:

f(AA) = p²(1 − F) + pF

f(Aa) = 2*pq*(1 − F)

f(aa) = q²(1 − F) + qF

• Allozygous (Not Identical by Descent): The two alleles are drawn independently from the ancestral founder gene pool with probability 1 − F, yielding standard Hardy–Weinberg homozygote proportions (p² and q²) and heterozygosity 2*pq*.

Malécot's Coefficient of Kinship (Coancestry) and Wright's Path Equation

For any two individuals X and Y, Malécot (1948) defined the coefficient of kinship (or coefficient of coancestry, denoted Θ_XY or f_XY) as the probability that a single allele sampled at random from locus L in individual X and a single allele sampled at random from the same homologous locus in individual Y are identical by descent.

Because a diploid offspring I inherits one random allele from parent X and one random allele from parent Y, the inbreeding coefficient F_I of the offspring is identically equal to the kinship coefficient of its parents:

F_I = Θ_XY

If X and Y share one or more common ancestors A in a pedigree, where n₁ is the number of generational meioses (steps) connecting X to ancestor A and n₂ is the number of meioses connecting Y to ancestor A, Sewall Wright's path-counting equation gives:

F_I = Θ_XY = Σ_{all paths A} (½)^{n₁ + n₂ + 1} (1 + F_A)

where F_A is the inbreeding coefficient of common ancestor A itself. The exponent n₁ + n₂ + 1 reflects the probability (½)^n₁+*n*₂ that a specific allele in A is transmitted down every meiotic step to both X and Y, summed over the two alleles carried by A (2 × [½]^{n₁+*n*₂+2} = [½]^{n₁+*n*₂+1}), plus the probability ½*F*_A that X and Y inherit the two different alleles of A which are themselves already IBD.

Furthermore, for any two non-inbred diploid relatives X and Y, Jacquard's (1974) condensed identity states simplify to three Cotterman IBD sharing probabilities (k₀, k₁, k₂)—the probabilities that X and Y share exactly 0, 1, or 2 alleles IBD at a single autosomal locus (where k₀ + k₁ + k₂ = 1):

Θ_XY = ¼ k₁ + ½ k₂

R_XY = 2 Θ_XY = ½ k₁ + k₂

where R_XY is Wright's coefficient of relatedness (the expected proportion of the genome shared IBD). Note that while parent–offspring pairs and full siblings both have a kinship coefficient of Θ = ¼ (R = ½), their Cotterman distributions are completely distinct: parent–offspring pairs share exactly 1 allele IBD at every autosomal locus (k₀ = 0, k₁ = 1, k₂ = 0), whereas full siblings share 0 alleles IBD at 25% of loci (k₀ = ¼), 1 allele IBD at 50% of loci (k₁ = ½), and both alleles IBD at 25% of loci (k₂ = ¼).

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Exponential Decay of IBD Tract Lengths

In a real genome composed of finite chromosomes rather than unlinked point loci, meiotic crossing-over breaks ancestral chromosomes at a rate of 1 Morgan (100 centiMorgans, cM) per meiosis. Consider two present-day haplotypes that coalesce in a single common ancestor g generations ago (separated by 2*g* total meioses: g steps up from haplotype 1 to the ancestor and g steps down to haplotype 2). Along the chromosome, recombination crossovers occur as a spatial Poisson process with rate λ = 2*g* crossovers per Morgan.

Consequently, the genetic length L (in Morgans) of an unbroken IBD chromosomal segment inherited from a common ancestor g generations in the past follows an exponential probability distribution with rate constant 2*g*:

f(L) = 2*g* exp(−2*gL*)

The expected mean length *E*[L] of an IBD tract originating g generations ago is therefore the reciprocal of the total meiotic path length:

*E*[L] = 1 / (2*g*) Morgans = 100 / (2*g*) cM = 50 / g cM

For example:

• First cousins (g = 2 generations to shared grandparents, 4 meioses per parental path): Shared IBD segments have an expected mean length of 100 / 4 = 25.0 cM.
• Ancestors g = 10 generations ago (~250–300 years in humans, 20 meioses): Shared IBD segments average 50 / 10 = 5.0 cM.
• Ancestors g = 50 generations ago (~1,250–1,500 years, 100 meioses): Shared IBD segments average 50 / 50 = 1.0 cM.

Because short IBD segments (1–3 cM) trace back to common ancestors 20 to 100 generations ago, whereas long IBD segments (>10 cM) trace back to ancestors within the last 1 to 5 generations, the length spectrum of shared IBD tracts in a population Sample allows algorithms such as IBDNe (Browning & Browning, 2015) to reconstruct historical trajectories of effective population size N_e(t) over the past 5 to 200 generations.

Genomic Inbreeding: Runs of Homozygosity (F_ROH)

When an individual's maternal and paternal chromosomes share an IBD segment from a common ancestor, the overlapping chromosomal region appears in the individual's diploid genome as a continuous Run of Homozygosity (ROH) devoid of heterozygous SNPs. McQuillan et al. (2008) defined the genomic inbreeding coefficient F_ROH as the total physical or genetic length of all autosomal ROH segments above a threshold length (typically L ≥ 1.0 or 1.5 Mb) divided by the total mappable autosomal genome length (L_auto ≈ 2,880 Mb in humans):

F_ROH = ( Σ L_ROH ) / L_auto

Unlike classical pedigree F_ped—which measures only the expected average autozygosity and misses unlisted historical consanguinity—F_ROH measures the realized proportion of the genome that is actually autozygous after the stochastic variance of Mendelian segregation and recombination, making F_ROH the gold-standard predictor of inbreeding depression in wild and clinical populations (Kardos et al., 2016).

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What is the difference between Identity by Descent (IBD) and Identity by State (IBS)?

Two alleles are Identical by State (IBS) if they have the same chemical nucleotide or amino acid sequence, regardless of their evolutionary origin. Two alleles are Identical by Descent (IBD) only if they are physical copies replicated from the exact same ancestral chromosome in a shared common ancestor within a defined reference pedigree or timeframe without intervening mutation. All unmutated IBD alleles are IBS, but common alleles can be IBS purely by chance without being recently IBD.

How can genomics distinguish parent–offspring pairs from full siblings when both share 50% of their DNA?

Although both parent–offspring pairs and full siblings have a relatedness coefficient of R = 0.50 and a kinship coefficient of Θ = 0.25, their Cotterman IBD distributions differ completely. A parent transmits exactly one copy of every autosome to their child, so parent–offspring pairs share 1 allele IBD across 100% of the genome (k₁ = 1, k₂ = 0). Full siblings inherit independently from both parents, sharing 0 alleles IBD across 25% of the genome, 1 allele IBD across 50%, and 2 alleles IBD (k₂ = 0.25) across 25% of the genome.

Why does the length of shared IBD segments decrease with each generation?

During meiosis in each generation, homologous chromosomes undergo crossing-over at a rate of approximately one crossover per Morgan (100 centiMorgans) per gamete. If two individuals share a common ancestor g generations ago, their chromosomes are separated by 2*g* independent meioses, breaking ancestral blocks according to an exponential probability density f(L) = 2*g* exp(−2*gL*) with rate constant 2*g* and reducing the expected mean IBD segment length to *E*[L] = 100 / (2*g*) = 50 / g centiMorgans.

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1. Malécot, G. (1948). Les Mathématiques de l'Hérédité. Masson et Cie. (English translation: The Mathematics of Heredity, W. H. Freeman, 1969). https://doi.org/10.1002/bimj.19710130313
2. Wright, S. (1922). "Coefficients of inbreeding and relationship." The American Naturalist, 56(645), 330–338. https://doi.org/10.1086/279872
3. Thompson, E. A. (2013). "Identity by descent: variation in meiosis, across genomes, and in populations." Genetics, 194(2), 301–326. https://doi.org/10.1534/genetics.112.148825
4. Browning, S. R., & Browning, B. L. (2012). "Identity by descent between distant relatives: detection and applications." Annual Review of Genetics, 46(1), 617–633. https://doi.org/10.1146/annurev-genet-110711-155534