The magazine of the Institute & Faculty of Actuaries
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# Genetics: Keep it in the family

All children take after their parents. This follows from the fact that every person possesses a set of individually identifiable ‘genes’ that, between them, determine his or her whole genetic make-up. Every individual inherits on average 50% of each parent’s genes.

The pool of genes in the whole human population is virtually infinite. But when two individuals with a common ancestor marry, it is possible that the same gene is possessed by both of them. In itself that may be no bad thing but, if the gene happens to be of a certain — albeit rare — type, the consequences can be disastrous, with the possibility that any child of that union will be at high risk of severe illness or even deformity. The marriage of close, and even not so close, relatives will therefore increase the chance of the two individuals having genes in common — and thus make it more likely that both will have the same rogue gene, which could put any child that they produce at risk.

Consanguinity
It is desirable therefore to be able to measure the consanguinity, or closeness, of any particular pair of family members. This is done by calculating what is called their Coefficient of Relatedness (CR), which — given everyone has the same number of genes — is the proportion of their genes that each has in common with the other.

The method of calculating this coefficient depends on the exact family relationship between M and N, the two individuals concerned. M and N do not have to be of opposite sex, since consanguinity does not just determine the potential for genetic mishaps; it also results in the sharing of family traits as a whole.

Single families
The simplest relationship is when M and N each descends from one and only one common union. This makes them either siblings or, say, uncle and niece, or cousins — possibly distant, possibly one or more times removed.

First of all the number of generations, or ‘steps’, necessary to connect each individual to their common progenitors must be counted — call these m and n respectively. Their CR is then calculated as 2 x (½)m x (½)n , the factor 2 reflecting the fact that there are two such progenitors.

Figure 1 displays the two progenitors as A and B; m=3 and n=2; and the CR of M and N (first cousins once removed) is 2 x (½)5 = 1/16 or 6.25%. The CR of double cousins — the children of, say, two brothers and two sisters — is thus 25%. This is the same as the lowest CR of any of the relationships of those who, with its opening salvo, “A man may not marry his grandmother”, the Table in the Anglican Prayer Book decrees are not permitted to marry on the grounds of kindredship (as opposed to those of mere affinity). Whether this means that in consequence double cousins are not legally allowed to marry, I am not certain.

Marriages within a family
A complication arises when cousins within one (or even both) branches of a family marry each other, as seen in Figure 2. Here, there are more ways than one of tracing each individual M and N’s connection back to A and B. Each such connection is termed a ‘path’ and here there are two M-paths (each of three steps) and two N-paths (one of four steps, the other of three). For either progenitor, any M-path can be combined with any N-path to form what is termed a ‘route’, a way of connecting M and N via that common ancestor. For either A or B in this example there are thus four routes connecting M and N, resulting in eight routes altogether.

Each such route contributes to the CR of M and N by an amount (½)m+n where m and n, as before, are the number of steps in the M-path and N-path respectively. The total CR in this example is therefore 2 x 2 x (½)3 x [(½)4 + (½)3” = 9.4-.

When families inter-marry
Complications really set in when members of the two different branches inter-marry. To calculate the CR of M and N, a chart must be drawn up that includes only those who are antecedents of both M and N (see Figure 3). From this chart, it is first necessary to identify all those who are common ancestors of both M and N: A, B, C and D in this example. Spouses are, in the absence of any second marriages, included by definition, which is why D’s spouse is included whereas that of his sibling E is not. These members form the ‘core’ of the family tree. The following steps must then be followed:
1) Identify all unions between members of the ‘core’, ignoring any union that has only one immediate offspring;
2) Select the union that is closest to M and N; regard the two parties to the union as progenitors; and considering them and their descendants alone, calculate the CR of M and N as described above. This will be one component of their total CR;
3) Repeat this process successively for each of the other unions identified in 1), omitting all routes where the M-path includes an individual that also appears in the N-path. The total CR for M and N will be the sum total of all the component CRs thus calculated.

Applying these steps to Figure 3, the two relevant unions are A=B and C=D. Starting with C=D, the contribution to the CR is 2 x (½)2 x (½)4 = 1/32 = 3.12- Continuing with A=B, it is first necessary to enumerate all M-paths and N-paths thus:
>> One M-path (M, F, D, A=B)
>> Three N-paths (N, K, J, G, D, A=B), (N, K, J, H, E, A=B) and (N, L, I, E, A=B).

One must then combine each M-path with every N-path in turn and ascertain if they have any elements in common (apart from their common progenitors). Here, that of (M, F, D, A=B) combined with (N, K, J, G, D, A=B) falls into that category — both paths ‘pass through’ D, so any contribution they may make to the CR is already included in the above C=D calculation. That combination can thus be ignored.

The one M-path can, however, combine successfully with each of the other two N-paths to produce an additional contribution to the CR of 2 x (½)3 x [(½)5 + (½)4” = 3/128 = 2.34%. The total CR for M and N is thus 3.12 + 2.34 = 5.5%.

A royal example
Inter-marriages occur quite frequently within royal families, a notable example being the Bourbon kings of Spain in the 18th and 19th centuries. Of King Charles IV’s family, four of the princes — Anton Pascal, Charles, Ferdinand and the illegitimate Francisco — all ended up marrying their nieces. Indeed, to complicate things still further, Ferdinand (later to become King Ferdinand VII) went one better — he married, in turn, daughters of two of his sisters, one of whom, Isabella, had married her cousin.

This consanguinity was later compounded by the fact that another Isabella and another Francisco — both great-grandchildren of Charles IV — themselves married in 1846 to reign jointly as Queen Isabella II and King King-consort Francisco of Spain. It is instructive therefore to calculate the CR of these two persons (Figure 4).

Here, the four ‘core’ unions — Isabella and Francis I; Charles IV and Maria; Maria on her own; and Charles III and Mary Amalia — contribute CR components of 1/8, 1/16, 3/32 and 1/32 respectively. The total CR of the union of Isabella and Francisco, based on Figure 4, is therefore 10/32 or 31.25%. However, that is not quite all. Charles IV and Maria were themselves first cousins, both being grandchildren of Philip V, the founder of the dynasty and his queen. This contributes a further 1/32 to the CR, bringing it up to 11/32 or 34.4%.

Since, however, there was circumstantial evidence at the time that the product of this union (which included the future Alfonso XII) was not fathered by Francisco, it would be idle to claim that this was the most consanguineous Western royal union of modern times.

Nevertheless, it is certainly possible to claim that Isabella II and Francisco were the most consanguineous queen and king-consort, since their 34.4 % CR certainly far exceeds that of the other famous contenders, William III and Mary II of Great Britain; who, being first cousins and nothing more, had a CR of a mere 12.5%.

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John MacLeod retired in 1992. His interests include history, writing and bridge