Las expresiones de la indignación

eters which it is claimed quantify the differentiation between subpopulations are defined by a number of authors.

Indeed the param eter 0 = {9j\j = 1 , . . . , M) of the model defined in Chapter 3 can be seen as a subpopulation differentiation parameter. If reasonable models are to be developed, it is very useful to be able to relate the param eters of such models to meaningful ‘real world’ quantities. W hat does 0 really mean? Which, if any, of the population genetics differentiation parameters is it equivalent to? These are interesting and im portant questions which are not always answered satisfactorily.

As will be seen in this chapter, the degree of difference between param eters which have in some instances been regarded as equivalent is as great as th a t between sample and population variances.

To dem onstrate this, the subpopulation differentiation param eters of Nei [Nei,1987] and Weir and Cockerham [Weir and Cockerham, 1984] are compared in the context of the hierarchical model described in Chapter 3. Papers employ­ ing models of population substructure often make no distinction between the two, classing them as equivalent.

To simplify m atters, the two allele (0 and 1) single binary locus case is considered here, where P r(%*6 = 1|G) = Gi for each band (6 = 1,2) within each

individual i in subpopulation Vi.

We define n (= « (/);/ = 1 , . . . , 77) as the collection of prior probabilities of a

randomly selected individual i being in subpopulation Vi'. K,{1) = Fr{Ii = I),

5.1

N ei

Nei considers the fixation index F to be a function of the param eters defining the allele probabilities {Gi) of the 77 actual subpopulations. Using only these

present generation parameters, no assumption is required about pedigrees of individuals, selection and migration in the past.

We define

9 = /=!

the average of the subpopulation allele ‘1’ probabilities, weighted by k.

If the current population is in Hardy-Weinberg equilibrium we have the case where 77 = 1 and g = Gi'.

P r(X , = (0 ,0 )|^,G ) = ( l - g f - ,

P r(X i = (0, l ) | p , G) = 2 g { l - g ) ;

P r(X , = ( l,l ) |p ,G ) = g^.

Following the thinking of Wright [Wright, 1951], any departure from these Hardy-Weinberg probabilities can be measured by the fixation index F so th a t

P r(X , = (0 ,0 )|^ ,G ) = { l - F ) { l - g f F F ( l - g ) - (33)

P r(X , = (0 ,l)|p ,G ) = 2 ( l - F ) g { l - g ) ; (34)

P r(X i = (l , l )| ff , G) = ( 1 - F ) g ^ + Fg. (35)

There are a number of possible causes of departure from Hardy-Weinberg pro­ portions, including inbreeding, assortative m ating and selection. At this point, however, we consider only the effect upon overall proportions of a population be­ ing split into 77 randomly mating subpopulations. The overall pair probabilities

are then given by

V

/=!

— (1 ~ ^)^ + ^ ( 0 [(1 ~ ^ i ) “ (1 “ g)Ÿ

1=1

P r(X , = ( 0 ,l) |s ,G ) = 2 Y ^ K { l ) G , ( \ - G i ) - = 2 g { l - g ) - 2 a \ (37)

/ = 1

P r(X i = ( l , l ) |5, G) = j ^K ( l )G ] = g'^ + a'^-, (38)

Z=1

where <7^ = YJi=i the ‘sample’ variance of the gene ‘1’ proportion

across subpopulations. The homozygotic frequencies are increased above the Hardy-Weinberg level by an amount cr^ with an appropriate reduction in the heterozygotic frequency. Comparison of equations (33)-(35) to (36)-(38) shows th a t the fixation index in the case of 2 alleles is then given by

F =

g { ^ - 9)'

5.2

W eir and Cockerham

In terms of the hierarchical model, Weir and Cockerham work at the level above Nei. Weir and Cockerham define a coancestry coefficient 6 th a t does not depend upon the number of subpopulations observed and governs the level of variability of allele probability (Gi) across subpopulations. It is a param eter related to the ancestral population from which the currently observed subpopulations have developed. To Weir and Cockerham, the observed subpopulations are just a sample of those th at could have evolved from the ancestral population under similar conditions. This is directly comparable to the hierarchical model in which the 77 subpopulations are a sample generated given the param eters of the

level above.

The subpopulations (P/; I = 1, . . . ,77) are assumed to have descended sepa­

rately from the single ancestral population.

Random m ating is assumed within subpopulations, and we define 7 as the

mean of the process generating subpopulation probabilities (Q ), i.e.

E[G/|7] = 7,

independently for all 1. Weir and Cockerham then define their subpopulation differentiation param eter 9 by

Referring to the DAG of Figure 5, the probabilities {Gi) are defined at the level of the subpopulations. We assume th at these probabilities are generated with a mean 7 and variance V. The im portant point here is th a t 7 and V are

param eters at the level above (Gi) in the hierarchy. In the previous section, g

and are defined at the same level as (G/).

Pr(%, = ( l , l ) | 7 , n = E[Pr(X, = ( l , l ) | 7 , R ) = E[Pr(%, = ( l , l ) | G , 7 , n =

E[/{(/)G^|7,y)

= + = ^ + f (40)

1=1

By comparison of equations (39) and (40), we see th a t 6 = a measure of subpopulation differentiation at the level above Nei’s param eter F.

5.3

H ow are th e m easures o f su b p op u lation differentia­

tio n related?

Firstly it is interesting to compare directly the definitions of Nei’s param eter F,

and Weir and Cockerham’s 6.

6 =

7(1 - 7) ’

Presented in this way, one can see the justification behind the earlier statem ent th a t the comparison is similar to th at between a sample variance and a pop­ ulation variance. The differences in definition between g = i^{l)Gi and

7 = ElGil'y] should also be noted.

It is also helpful to consider the expectation of the ‘sample variance’ (= — p)^), given 7 and the prior variance V and 7.

E [a^|y,7] = E [ ' £ < l ) { G , - g Y \ V , j ] I

= e E k ( ; ) g ? - s ^ | v , 7 ]

= E «(OEfG^lF,7] - V ax(s|F,7) - E [g |y ,7]^

l

= V ar(g |y ,7) + E[g\V,7]' - V a r ( ^ i^{l)G,\V,7) - Ÿ

l = V + - Y ^ K { i f v l I E[(7^|y,7] = y E 4 0 ( i - 4 0 ) I

In the special case where k,{1) = ^ for / = 1, . . . , 77, this simplifies to 9|TT -1 ^ “ 1

meaning th at

E [ F s ( l -5)10,7] = ^ 0 7(1 - 7), (41)

Equation (41) clarifies the role of the subpopulation differentiation parameters,