Why Encode Inheritance?
In a pedigree, relatives can share alleles because they descend from common ancestors. To describe that sharing precisely, it is useful to record which parental homolog each person inherited at a particular locus. An inheritance vector encodes those meiosis-specific choices as bits. Once founder alleles are labelled, the vector lets us trace every pedigree member’s alleles back to their ancestral copies and determine which relatives share alleles identical by descent (IBD).
The focus here is the relationship between a vector state and IBD sharing. A vector represents transmissions, not affection status: marking people as affected identifies the relatives of interest, but does not itself specify which alleles they inherited. In nonparametric linkage analysis, the relevant question is whether the observed sharing pattern among affected relatives is unusual under a null model with no linkage between the marker and the disease.
From Bits to Founder Alleles
Fix an ordering of the nonfounders, and write the two bits for person \(i\) as \((v_{i,p},v_{i,m})\), with the paternal-transmission bit first and the maternal-transmission bit second. A bit of 0 means the transmitting parent passed on that parent’s homolog 0; a bit of 1 means homolog 1. The bits identify homologs in the parent, not in the child.
For example, label a founder father’s two allele copies \(F_0,F_1\) and a founder mother’s copies \(M_0,M_1\). A child with vector entry \((0,1)\) inherits \(F_0\) from the father and \(M_1\) from the mother. If that child later has children, the child’s own homologs become the parental homologs selected by those later transmissions. Thus tracing through a pedigree requires applying each bit to the transmitting parent’s already-traced homologs.
Assume initially that founder allele copies are unrelated and not IBD with one another. Each founder copy then has its own label. A person’s two allele labels follow deterministically from the inheritance vector: founders start with their own labels, and each nonfounder receives the labels selected by their two bits. If founders are related or inbred, their allele copies may already be IBD; the appropriate known founder relationships must then be included rather than treating every founder copy as distinct.
For an outbred pedigree with unrelated founders, paternal-side labels and maternal-side labels are distinct. In that setting, the IBD count for two relatives can be found by comparing their traced paternal labels and their traced maternal labels. The more general matching definition also works when founder relatedness means an IBD match could occur across those sides.
Worked Examples
Worked Example: One Affected Sib Pair
Suppose two full siblings, A and B, are both affected. Their parents are unrelated, noninbred founders with paternal allele labels \(F_0,F_1\) and maternal labels \(M_0,M_1\). Use the vector entries \((v_{i,p},v_{i,m})\), and suppose A has \((0,0)\) while B has \((1,0)\).
A inherits the paternal homolog selected by bit 0 and the maternal homolog selected by bit 0: \(A=(F_0,M_0)\). B inherits \(F_1\) paternally and \(M_0\) maternally: \(B=(F_1,M_0)\).
The paternal labels differ, but the maternal labels match. Therefore the siblings share one allele IBD, \(M_0\), and their IBD state is 1. Both are affected, but that status is not used to calculate this particular vector’s sharing state; the transmissions and founder labels determine it.
Worked Example: Sharing Among Three Affected Siblings
Consider three affected full siblings A, B, and C, with the same unrelated, noninbred founder parents. Suppose their paternal and maternal bits are \(A=(0,0)\), \(B=(0,1)\), and \(C=(1,0)\). The founder labels remain \(F_0,F_1\) and \(M_0,M_1\).
Tracing the bits gives \(A=(F_0,M_0)\), \(B=(F_0,M_1)\), and \(C=(F_1,M_0)\). For A and B, the paternal alleles match at \(F_0\), while the maternal alleles differ, so their IBD state is 1. For A and C, the paternal alleles differ, while the maternal alleles match at \(M_0\), so their IBD state is also 1. For B and C, \(F_0\ne F_1\) and \(M_1\ne M_0\), so their IBD state is 0.
This example shows why a vector for a larger pedigree is useful: one joint set of transmissions determines the sharing states for several relative pairs. Those pairwise states are not independent observations; they arise from the same underlying inheritance pattern.
Worked Example: Tracing IBD Between First Cousins
Suppose P1 and P2 are siblings whose parents are founders. They each have a child, C1 and C2, respectively; C1 and C2 are first cousins. The siblings’ founder father has alleles \(G_0,G_1\), and their founder mother has \(H_0,H_1\). Their other parents (the cousins’ unrelated spouses) are founders with distinct allele labels.
Use entries in the order P1, P2, C1, C2. Suppose the vector entries are P1 \((0,0)\), P2 \((0,1)\), C1 \((0,1)\), and C2 \((0,0)\). P1 therefore inherits \(G_0,H_0\); P2 inherits \(G_0,H_1\). The siblings share \(G_0\) IBD.
C1’s paternal bit 0 selects P1’s paternal homolog, which carries \(G_0\). C2’s paternal bit 0 selects P2’s paternal homolog, which also carries \(G_0\). Their maternal bits select alleles from their unrelated spouses, so those labels do not match under the stated assumptions. The cousins therefore share one allele IBD, \(G_0\), and have IBD state 1.
The important step is tracing a child’s transmission through the parent’s inherited homologs. The bit for C1 does not select \(G_0\) directly; it selects P1’s paternal homolog, whose label was determined by P1’s own earlier transmission.
Why the Vector Determines the IBD State
The correspondence can be justified by tracing the pedigree from its founders forward. Founder allele labels are given. For each nonfounder, the paternal bit selects one of the father’s two allele labels, and the maternal bit selects one of the mother’s two labels. Repeating this step assigns both labels to every person. Comparing those labels for a relative pair then identifies their shared ancestral allele copies. No additional information about affection status is needed to carry out this tracing.
Assign a distinct ancestral-copy label to each founder allele, unless specified founder relationships imply IBD.
For each nonfounder, copy the selected paternal homolog’s label and the selected maternal homolog’s label into that individual.
Count the maximum number of matching ancestral-copy labels across their two alleles to obtain IBD state 0, 1, or 2.
Under Mendelian transmission at a locus, each parent transmits either homolog with probability \(1/2\), absent segregation distortion or another violation of that assumption. Conditional on parental genotypes, a particular vector state for \(m\) nonfounders has probability \(2^{-2m}\) if the \(2m\) meioses are independent. This describes the transmission mechanism; it is not a claim that every vector remains equally probable after conditioning on arbitrary observed data or ascertainment.
Assumptions and Common Pitfalls
The vector-to-IBD mapping is exact once the pedigree, the vector, and the founder-origin information are specified. Interpreting probabilities for vector states requires additional assumptions. At a single locus, the standard Mendelian model assigns probability \(1/2\) to each transmitted parental homolog. To treat different meioses as independent in the simple probability calculation above, assume ordinary Mendelian segregation and no transmission dependence beyond the specified pedigree structure. When testing affected-family sharing for linkage, the null model also concerns the relationship between the marker and disease; it is not enough merely to call a transmission vector “random.”
A common mistake is to read an inheritance-vector bit as an allele label. For instance, bit 0 for a child does not mean the child has founder allele \(F_0\). It means the child received homolog 0 from that child’s parent. That homolog might carry a different founder-origin label, depending on the parent’s own transmissions. The correction is to trace homolog labels generation by generation before comparing relatives.
Another pitfall is to infer IBD just because two relatives have the same observed allele value. IBD concerns descent from the same ancestral allele copy, not merely matching allele states; two copies can match in state without sharing descent. Conversely, the vector identifies descent only relative to the pedigree and founder-origin assumptions being used. Marker genotype information can help infer which vector states are compatible with observations, but the vector-to-IBD tracing itself is a separate step.
Check Your Understanding
Use the definitions and assumptions in this tutorial to answer the following questions.
- For a nonfounder with vector entry \((1,0)\), which homolog does the person receive from each parent?
- Two full siblings have entries \((0,1)\) and \((0,0)\), with unrelated noninbred founder parents. What is their IBD state, and which founder allele is shared?
- Why must a child’s transmission bit be interpreted using the allele labels of that child’s parent?
- In a pedigree with three affected siblings, why can the three pairwise IBD states not be treated as if they came from three unrelated transmission vectors?
- What extra information is needed before assuming that two distinct founder allele labels cannot be IBD?