14
J. Wakeley
• The time during which there are i lineages ancestral to the sample, denoted T i ,
follows an exponential distribution with parameter i(i − 1)/2
• T i and T j are statistically independent for i = j
For reference, the gene genealogy in Fig. 1.1 is drawn with the lengths of coalescent intervals T 2 , T 3 , and T 4 equal to their expected values from the exponential
distribution, E[T i ] = 2/(i(i − 1)). In the case of parent-independent mutation, such
as in the infinite-sites model considered here, we may include a fourth property:
• Mutations occur with rate θ /2 along each branch of the coalescent tree
Then, conditional on the tree, the number of mutations over a length t of the tree
follows a Poisson distribution with parameter tθ /2. For example, the 11 mutations
on the gene genealogy in Fig. 1.1 are exactly the number expected if θ = 6 because
the total length of the gene genealogy in Fig. 1.1 is 11/3 (see E[T Total ] below).
1.4.1 The Size and Shape of a Gene Genealogy
Considering just the gene genealogy, without mutations, a great deal can be gleaned
from the first three properties listed above. Because lineages always coalesce in
pairs in the standard neutral coalescent, every gene genealogy includes exactly n − 1
coalescent events. Let T MRCA represent the time to the most recent common ancestor
of the sample and let T Total represent the total length of the gene genealogy, or the
sum of the lengths of all the branches in the tree. These two measures have been used
extensively to characterize the sampling properties of gene genealogies. Knowledge
of T MRCA may be of direct biological interest, while knowledge of T Total is important
because T Total quantifies the total opportunity for mutations to occur in the ancestry
of the sample.
From their definitions, T MRCA is simply the sum of the individual times T i , or
T MRCA =
n
i=2
T i
and T Total is obtained similarly, except that each time is weighted by the number of
lineages that existed during the interval, so that
T Total =
n
i=2
iT i
The n − 1 times, T n , T n − 2 , . . . , T 2 , are called coalescence intervals here to avoid
confusion with T MRCA , which is often referred to as the coalescence time.
Using the properties of the exponential distribution, namely, that E[T i ] = 2/
(i(i − 1)) and Var[T i ] = 4/(i(i − 1)) 2 and the fact that the n − 1 coalescent intervals
are statistically independent, so that Cov[T i , T j ] = 0 for i = j, one obtains the
J. Wakeley
• The time during which there are i lineages ancestral to the sample, denoted T i ,
follows an exponential distribution with parameter i(i − 1)/2
• T i and T j are statistically independent for i = j
For reference, the gene genealogy in Fig. 1.1 is drawn with the lengths of coalescent intervals T 2 , T 3 , and T 4 equal to their expected values from the exponential
distribution, E[T i ] = 2/(i(i − 1)). In the case of parent-independent mutation, such
as in the infinite-sites model considered here, we may include a fourth property:
• Mutations occur with rate θ /2 along each branch of the coalescent tree
Then, conditional on the tree, the number of mutations over a length t of the tree
follows a Poisson distribution with parameter tθ /2. For example, the 11 mutations
on the gene genealogy in Fig. 1.1 are exactly the number expected if θ = 6 because
the total length of the gene genealogy in Fig. 1.1 is 11/3 (see E[T Total ] below).
1.4.1 The Size and Shape of a Gene Genealogy
Considering just the gene genealogy, without mutations, a great deal can be gleaned
from the first three properties listed above. Because lineages always coalesce in
pairs in the standard neutral coalescent, every gene genealogy includes exactly n − 1
coalescent events. Let T MRCA represent the time to the most recent common ancestor
of the sample and let T Total represent the total length of the gene genealogy, or the
sum of the lengths of all the branches in the tree. These two measures have been used
extensively to characterize the sampling properties of gene genealogies. Knowledge
of T MRCA may be of direct biological interest, while knowledge of T Total is important
because T Total quantifies the total opportunity for mutations to occur in the ancestry
of the sample.
From their definitions, T MRCA is simply the sum of the individual times T i , or
T MRCA =
n
i=2
T i
and T Total is obtained similarly, except that each time is weighted by the number of
lineages that existed during the interval, so that
T Total =
n
i=2
iT i
The n − 1 times, T n , T n − 2 , . . . , T 2 , are called coalescence intervals here to avoid
confusion with T MRCA , which is often referred to as the coalescence time.
Using the properties of the exponential distribution, namely, that E[T i ] = 2/
(i(i − 1)) and Var[T i ] = 4/(i(i − 1)) 2 and the fact that the n − 1 coalescent intervals
are statistically independent, so that Cov[T i , T j ] = 0 for i = j, one obtains the