# 14 Pedigrees

Pedigrees are visual representations of genetic relations. They are extremely important for estimating inbreeding and following traits or diseases in lineages. In this lecture, we will learn:

- Symbology used in pedigrees
- Estimation of inbreeding
*F*from a given pedigree - Approaches for plotting pedigrees

In the analysis of pedigrees, we often use the following terminology:

**Consanguineous mating**(*lit.*“Of the same blood”). The condition where individuals who are related produce offspring.**Biparental Inbreeding**(see consanguineous mating).

A pedigree is simply a graphical representation characterizing the relationship amongst ancestral individuals. This tool is very helpful for understanding the process of inbreeding when it occurs during a handful of generations.

Perhaps the most studied pedigree is that of the Hapsburg Dynasty, dating from 1516-1700, which controlled most of what we call modern Europe. Some of the European Royalty were interbred to maintain familial relationships and consolidate power. Examples include:

King | Queen | Consanguineous Marriage |
---|---|---|

Philip I (1478-1506) | Joanna I of Castile & Aragon | Third cousins |

Charles I (1500-1558) | Isabella of Portugal | First cousins |

Philip II (1527-1598) | Mary of Portugal | Double first cousins |

\(\;\) | Mary I of England | First cousins once removed |

\(\;\) | Anna of Austria | Uncle-Niece |

Philip III (1578-1621) | Margaret of Austria | First cousins once removed |

Philip IV (1605-1665) | Elizabeth of Bourbon | Third cousins |

\(\;\) | Mariana of Austria | Uncle-Niece |

Charles II (1661-1700) | Marie Louise d’Orleans | Second cousins |

In this lineage, there were two genetic disorders that became paramount:

- Pituitary hormone deficiency

- Distal renal tubular acidosis

Pedigree Symbology

- Each row is a generation.
- Lines within a generation represent mating events that result in offspring.
- Lines between generations represent descent.
- Individuals are labeled uniquely.
- Sex is indicated by shape (square=male, circle=female, diamond=unknown).
- Traits can be mapped onto the pedigree using additional colors & symbols

Extraneous individuals may be removed from the depiction. Why is it that **C** and **F** are not shown on the pedigree on the right side?

Estimating Inbreeding From Pedigrees

The inbreeding coefficient, *F*, of an **individual** is determined by the probability that a pair of alleles carried by gametes are IBD from a recent common ancestor.

**F**has same grandfather on both sides.

**D**and**E**are half-sibs sharing father**B**.

**B**passed one of his alleles, say \(A_1\), to both**D**and**B**.**D**passed allele \(A_1\) to**F**and**E**passed allele \(A_1\) to**F**, which means**F**has some non-zero probability of being autozygous

Expectations for \(F\) in a pedigree.

- Label alleles in
**B**as \(\alpha\) and \(\beta\).

- The alleles
**B**gives to**D**&**E**are \(\{\alpha, \alpha\}\), \(\{\alpha, \beta\}\), \(\{\beta, \alpha\}\), or \(\{\beta, \beta\}\)

- Each potential allele pair occurs at: \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\)

- If
**B**gives \(\{\alpha,\alpha\}\) or \(\{\beta, \beta\}\) then the two alleles in the children of**B**are autozygous (by definition).

- If
**B**gives \(\{\alpha,\beta\}\) or \(\{\beta, \alpha\}\) then the two alleles in the children of**B**are autozygous*only if*the alleles \(\alpha\) and \(\beta\) are autozygous.

- The probability of \(\alpha\) and \(\beta\) being autozygous in
**B**is given by the inbreeding coefficient \(F\) for**B**(denoted \(F_B\)).

## 14.1 Expectations for F in a pedigree.

Transition probabilities for all cases of alleles within the children of **B** are:

- \(P({\alpha,\alpha}|\mathbf{B}) = \frac{1}{4}\)

- \(P({\alpha,\beta}|\mathbf{B}) = \frac{1}{4}F_A\)

- \(P({\beta,\alpha}|\mathbf{B}) = \frac{1}{4}F_A\)

- \(P({\alpha,\alpha}|\mathbf{B}) = \frac{1}{4}\)

Total probability for **B** donating autozygous alleles to his offspring are then:

\[
= \frac{1}{4} + \frac{1}{4}F_A + \frac{1}{4}F_A + \frac{1}{4} \\
= \frac{1}{2} + \frac{1}{2}F_A \\
= \frac{1}{2}(1.0 + F_A) \\
\]

Expectations for \(F\) in a pedigree.

- The transition probabilities for the alleles that
**D**&**E**received from**B**and passes on to**F**are \(\frac{1}{2}\) for each.

- The total probability that the two alleles in
**F**are autozygous (Identical By Decent; IBD) is:

\[ F = \frac{1}{2}(1.0 + F_A) \frac{1}{2} \frac{1}{2} \\ = \left( \frac{1}{2} \right)^3 (1.0 + F_A) \]

Generalized Formula for Inbreeding | Chain Counting Method

\[
F = \left( \frac{1}{2} \right)^i (1.0 + F_A)
\] where the exponent \(i\) is the length of the *ancestral loop* (e.g., the number of steps starting at **F** and going trough the common ancestor and back to **F**) which in this case is **D BE** (the common ancestor is underlined).

Several Ancestral Loops

If there is more than one ancestral loop, then the final inbreeding coefficient, *F*, is the sum of the *F* estimated across each loop (assuming there are \(K\) different loops in the pedigree). In this example you would use both **GD AEH** and

**GD**.

__B__EH\[ F = \sum_{k=1}^{K} \left[ \left( \frac{1}{2} \right)^{i_k} (1.0 + F_k) \right] \]

N.B. Each ancestor may have different \(F_k\) values.

Strategies for Pedigrees

Here is a basic strategy for solving pedigree questions.

- Draw the pedigree

- Identify common ancestor(s)

- Trace ancestral loop(s)

- Annotate all ancestral loops indicating the common ancestor

- The length of each ancestral loop is \(i\)

- Plug into the equation for inbreeding statistic
*F*for each loop.

- Final
*F*is sum of all estimated*F*from each loop.

Easy Example, F=?

```
library(kinship2)
id <- LETTERS[1:6]
dadid <- c(NA,NA,NA,"B","B","D")
momid <- c(NA,NA,NA,"A","C","E")
sex <- c("female","male","female","male","female","female")
ped <- pedigree( id, dadid, momid, sex)
ped
```

```
## Pedigree object with 6 subjects
## Bit size= 3
```

`plot(ped)`

- Ancestral loop:
**D**,__B__E - \(i = 3\),
- \(F_F = \left( \frac{1}{2} \right)^3(1+F_A)\).

Easy Example, F=?

```
library(kinship2)
id <- LETTERS[1:9]
momid <- c(NA,NA,NA,"B","B",NA,"C","E","H")
dadid <- c(NA,NA,NA,"A","A",NA,"D","F","G")
sex <- c("male","female","female","male","female","male","male","female","female")
ped <- pedigree( id, dadid, momid, sex)
plot(ped)
```

- Ancestral loop:
**GD**,__A__EH**GD**, \(F_I = \left[ \left( \frac{1}{2} \right)^5(1+F_A) \right] + \left[ \left( \frac{1}{2} \right)^5(1+F_B) \right]\).__B__EH

Medium Example, F=?

```
id <- LETTERS[1:8]
dadid <- c(NA,NA,"A","A",NA,"D","D","F")
momid <- c(NA,NA,"B","B",NA,"C","E","G")
sex <- c("male","female","female","male","female","male","female","unknown")
ped <- pedigree( id, dadid, momid, sex)
suppressWarnings(plot(ped))
```

- Ancestral loop:
**GD**,__A__EH**GD**, \(F_I = \left[ \left( \frac{1}{2} \right)^5(1+F_A) \right] + \left[ \left( \frac{1}{2} \right)^5(1+F_B) \right]\).__B__EH

## 14.2 Drawing Pedigrees

Example Pedigree

```
library(kinship2)
id <- LETTERS[1:5]
dadid <- c(NA,NA,"A","A","A")
momid <- c(NA,NA,"B","B","B")
sex <- c("male","female","male","female","female")
data <- data.frame( id, dadid,momid,sex)
data
```

```
## id dadid momid sex
## 1 A <NA> <NA> male
## 2 B <NA> <NA> female
## 3 C A B male
## 4 D A B female
## 5 E A B female
```

Example Pedigree

```
ped <- pedigree(data$id, data$dadid, data$momid, data$sex)
ped
```

```
## Pedigree object with 5 subjects
## Bit size= 4
```

`summary(ped)`

```
## Length Class Mode
## id 5 -none- character
## findex 5 -none- numeric
## mindex 5 -none- numeric
## sex 5 factor numeric
```

Example Pedigree

`plot.pedigree(ped)`

Decay of \(F\), The effects of size in the ancestral loop

```
library(ggplot2)
df <- data.frame( i=seq(4,20,by=2) )
df$F <- 0.5^df$i
ggplot( df, aes(x=i,y=F) ) + geom_line(color="red")
```

## 14.3 Skills

This lecture covered the creation and analysis of pedigree data. At the end of this lecture you should be able to:

- first skill
- second skill

```
id <- LETTERS[1:9]
momid <- c(NA,NA,NA,"B","B",NA,"C","E","H")
dadid <- c(NA,NA,NA,"A","A",NA,"D","F","G")
sex <- c("male","female","female","male","female","male","male","female","female")
brown_eyes <- c(1,0,0,1,1,NA,1,1,0)
college <- c(0,1,0,0,0,0,1,1,1)
likes_asparagus <- c(0,1,0,0,1,1,0,1,1)
traits <- cbind( brown_eyes, college, likes_asparagus)
ped <- pedigree( id, dadid, momid, sex, affected = traits)
plot(ped)
pedigree.legend( ped, location="bottomleft",radius=.2)
```

```
library(gstudio)
p1 <- c("Ai","Aj")
p2 <- c("Ak","Al")
offs <- c( locus( c(p1[1],p2[1]) ),
locus( c(p1[2],p2[1]) ),
locus( c(p1[1],p2[2]) ),
locus( c(p1[2],p2[2]) ) )
offs
```

`## [1] "Ai:Ak" "Aj:Ak" "Ai:Al" "Aj:Al"`

Relationship | \(k_0\) | \(k_1\) | \(k_2\) | \(r\) |
---|---|---|---|---|

Identical Twins | 0 | 0 | 1 | 1 |

Full Sibs | 0.25 | 0.5 | 0.25 | 0.5 |

Parent Offspring | 0 | 1 | 0 | 0.5 |

Half Sib | 0.5 | 0.5 | 0 | 0.25 |

Aunt-nephew | 0.5 | 0.5 | 0 | 0.25 |

First Cousin | 0.75 | 0.25 | 0 | 0.125 |

Unrelated | 1 | 0 | 0 | 0 |

sdf

```
ped <- pedigree( id, dadid, momid, sex, affected = traits)
plot(ped)
pedigree.legend( ped, location="bottomleft",radius=.2)
```

`kinship(ped)`

```
## A B C D E F G H I
## A 0.500 0.000 0.000 0.2500 0.2500 0.000 0.12500 0.12500 0.12500
## B 0.000 0.500 0.000 0.2500 0.2500 0.000 0.12500 0.12500 0.12500
## C 0.000 0.000 0.500 0.0000 0.0000 0.000 0.25000 0.00000 0.12500
## D 0.250 0.250 0.000 0.5000 0.2500 0.000 0.25000 0.12500 0.18750
## E 0.250 0.250 0.000 0.2500 0.5000 0.000 0.12500 0.25000 0.18750
## F 0.000 0.000 0.000 0.0000 0.0000 0.500 0.00000 0.25000 0.12500
## G 0.125 0.125 0.250 0.2500 0.1250 0.000 0.50000 0.06250 0.28125
## H 0.125 0.125 0.000 0.1250 0.2500 0.250 0.06250 0.50000 0.28125
## I 0.125 0.125 0.125 0.1875 0.1875 0.125 0.28125 0.28125 0.53125
```