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Aspects of the distribution of HbS in the United States.

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AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 52341-349
1980)
Aspects of the Distribution of HbS in the United States
BRAXTON M. ALFRED
Department of Anthropology and Sociology, and Laboratoq of Physical
Anthropolog,y, University of British Columbia, Vancouuer,
British Columbia, V 6 T 1W5
KEY WORDS
HbS, Age, Sex, Stable populations, Fitness,
Log-likelihood, Fertility
ABSTRACT
Frequencies of HbS obtained by several screening clinics a r e
analyzed for age, sex, and location effects. All seem to be present in some form,
though age and sex effects may be conditional on location. An attempt is made to
elaborate the common observation of increasing frequency with age. This is shown
to be the result of differences in fertility favoring the normal. A simulation which
includes 25% admixture was done. The results indicate a genetically relevant New
World experience for the population to be about 9- 12 generations with the heterozygote having fitness of 0.9GO.99.
Classical genetic theory predicts a decline
over time in the frequency of a n allele which is
maintained soley by heterozygote advantage
when the selective agent is no longer present.
The evidence for P. falciparum malaria being
central in maintaining the sickle cell polymorphism in West Africa is taken as convincing
(Serjeant, '74). There the normal, AA, genotype
has estimated fitness 0.89 relative to the heterozygote, AS (Cavalli-Sforzaand Bodmer, '7 1).
In the sequel the odds for AS, defined as the
ratio of the observed frequency of AS to that of
AA, will be analyzed. The reasons for using
odds rather t h a n relative frequencies a r e that a
simpler description and statistical analysis result, and that this ratio is conceptually close to
a common definition of relative fitness. In West
Africa the odds for AS are estimated as 54821
25374 = 0.2160 (Cavalli-Sforza and Bodmer,
'71). The odds for AS in the United States a r e
currently estimated as 2710131037 = 0.0873.
The south-east coastal areas of the U.S. a r e
sub-tropical becoming temperate inland. So P.
vivax would be expected to predominate but P.
falciparum would not be uncommon on the
coast (Russell, '52). Removal of West Africans
to this area should alter the relative fitness of
AA such that i t i s at least as great as that of AS
for all but possibly the coastal areas. I n the
West African context the superiority of AS is
usually attributed to its ameliorating effect on
mortality by protecting children a g a i n s t
malaria until natural immunity is acquired. I t
0002-948Y18015203-0341$02 00
0 1980 ALAN R I.ISS, INC
has been recognized for some time t h a t this
effect alone may not be sufficient to account for
contemporary observations of the frequency of
AS; and now some evidence suggests a n effect
on fertility as well (Eaton and Mucha, '71 ). The
magnitude of this effect, however, is thought to
be small (Cavalli-Sforza and Bodmer, '71), and,
I add, its direction unclear.
By 1970 there were about :% million Negroes
in the U.S. of whom about 34 were native born
(Thompson and Lewis, '65) Assuming t h e 1968
estimated g e n e r a t i o n t i m e of 25.3 y e a r s
(Keyfitzand Flieger, '68)to have been constant,
then the gene pool of U S . born Negroes has
experienced at least seven generations in a n
environment relatively free offalciparum. It is
commonly assumed t h a t this experience is of
the order of 1@12 generations.
Reed ('69) estimates the amount of white admixture currently present in U.S. Negroes to be
about 25% . Assuming a constant rate of admixture, y , the total amount at generation h is
proportional to ( y t l ) h= 1 + 0.25 which may
be solved for y = exp[(l/k)ln 1.251 - 1. For
example, when k = 10, y = 0.0226 i s t h e
amount of admixture per generation, which
over ten generations results in 25%. This effect
is largely independent of fitness and the result
is to accelerate the decrease in odds for AS.
Considering the sickling allele as a lethal
recessive leads to the expectation for its freReceived J u l y :il. 1978: accepted J u n e 20, 1979
34 1
342
BRAXTON M. ALFRED
quency p, = p,/l + tp,, at time t where p,) is the
starting frequency (Crow and Kimura, '70). So
the genotypic odds for type AS at time t a r e
given by
~~
~~~
1
2PU
+ p,,it -
1)
Taking p,, = 0.09 and t = 10 generations results
in odds 0.10. However, when the 25% admixture is included the expected odds a r e near 0.05.
Thus there is a discrepancy between observed
and expected odds. One of the central concerns
of this paper is the resolution of this difference.
Before considering this, I address the logically prior topic of t h e fitness of the heterozygote. A common observation in the U.S. is that
the trait frequency increases with age. If the
major part of a fitness differential is due to
mortality the frequency should decline with
age.
The analysis of these features of the current
distribution leads to a new interpretation of the
history of the trait as well as advancing understanding of its role in fitness under relaxed
selection.
THE AGE EFFECT
In this section the data from screening projects are analyzed for age, and other effects. The
data from Florida are due to Wienker ('74);
those from New York are due to Janerich, et al.
('73);and those from Stockton a r e from a clinic
conducted by the University of California, San
Francisco (Alfred, et al., '78).Table 1 presents
the observed frequencies of AA and AS from
these three sources. The Tampa data were generated by the screening program of the Hillsborough County Health Department which
included both stationary and mobile clinics.
The New York data were produced by the New
York State Department of Health and exclude
only New York City. The Stockton observations
were made at a stationary clinic.
Attention is restricted to the two genotypes
AA and AS to reduce bias due to prior knowledge of genotype by t h e patient. T h e age
categories a r e those used by Janerich et al. ('73)
to effect comparability. The minimum age considered is one year.
In Figure 1 will he found the odds for AS by
age category, sex, and location. As may he observed, there is considerable apparent variability around t h e overall expected odds. It
should be noted t h a t the mid-point of age category 1is 5.5 years, of category 2 it is 15years, of
category 3 i t is 25 years. The mid-point of category 4 is greater than 45 years.
METHODS
For details and extensions of the techniques
used here refer to Goodman ('70, '71, ' 7 2 ) ,and
Bishop, et al. ('75).
The four way frequency distribution in Table
1can be represented by the general element f,,L,
where i = 1 , 2 , 3 , 4indexes the 4 age categories
(variable A ) ,j = 1, 2 indexes male and female
(S),
k = 1 , 2 , 3 indexes the 3 locations (L),and 1
= 1, 2 indexes normal or sickler genotype iGj.
The variable G will be considered the dependent variable. Various models, hypotheses, for
the observations may he constructed by differe n t combinations of the marginal frequencies.
For example, the hypothesis that genotype is
independent of age-sex-location jointly will be
represented by the notation "ASL,G" and asserts that the observations can he adequately
reproduced by knowing only the genotype frequencies and, separately, the joint frequencies
TABLE 1 . Ohserued frcqiiencies vf A A cmd AS nt Trirnpci, New Yor-k,
und Stockton by age nnd
Tampa
Location tL):
SEX
New York
Stockton
-~
AA
AS
AA
AS
AA
AS
Male
1 to
10yedrs
10 to 20 years
20 to 30year s
' 30 years
2714
1312
357
464
274
153
47
48
:3 129
3874
664
546
229
293
56
52
73
45
18
23
14
2
3
2
Female
1 to
10 years
10 to 20 years
20 to 30 year5
30 years
2915
2168
1190
1045
256
251
134
134
3339
4711
1002
1202
250
311
83
97
74
88
35
49
Genotype [GI:
Sex IS),Age ( A )
I
~
343
DISTRIBUTION OF HbS
0.200
0.175
0.1 50
x
0.125
c
0
c
al
-4
L
0
0.075
-0
0
0.050
0
0.025
0.000
FEMALE
MALE
1
I
I
2
3
1
4
A g e C ategor y
Fig. 1. Odds for AS by age category, sex, and location.
of ASL. The hypothesis that genotype and age
a r e related independently of ASL is represented by “ASL,GA”. The assertion now is that
one must know t h e genotype-age frequencies in
addition to the ASL frequencies. And so on. The
object of the procedure is to locate the simplest
combination of marginal frequencies required
to reproduce t h e data satisfactorily. The procedures that have been developed for treating
problems of this sort are directly analogous to
analysis of variance without the assumptions of
normality and homoscedasticity.
When a term such as “ G A S appears in a
model, the hierarchical principle implies t h a t
all the following terms are present: GAS, GA,
GS, AS, G, A, S.The descriptive notation for the
models will include only t h e highest order
terms, e.g., GAS, required. To further simplify
t h e notation t h e term “ASL” will be dropped
from model descriptions even though i t is
present in all. The reason for holding ASL constant is to remove from further consideration
the peculiarities due to the sampling frame.
That is, the uniqueness of the age-sex distribution at each location is of interest only with
regard to its effect on G so that source of variability is statistically removed and subsequent
tests are performed on the residual variability.
The test statistic used is the log-likelihood
ratio chi-square (LLR) which has considerable
advantages over the Pearson version. In Bishop
e t al. (’75)the statistic is named G’, which, due
to the obvious conflict in notation, will not be
used.
RESULTS
An hypothesis is specified by dropping one or
more of t h e possible terms. The parameter
which weights t h e term is set to zero. The
hypothesis is then tested by determining the
BKAXTON M. ALFRED
344
log-likelihood ratio chi-square for goodness of
fit. When a given term is assumed present the
marginal frequencies specified by the descriptive notation are fixed. In Table 2, in the column headed “Model”are found only the highest
order terms which are assumed to be present by
hypothesis. A model, e g . , no. 6 in Table 2,
specified as GA, GL (with ASL understood) may
be read as the 3-way interaction of age-sex-location plus the 2-way interaction of genotypeage plus the 2-way interaction of genotype-location. Model no. 1 is the hypothesis of the
independence of genotype and all other variablesjointly; clearly it does not fit the data. And
model no. 18 tests the effect of dropping the
highest order interaction, i.e., GASL; as the
probability of the fit without it is greater than
0.10, it is assumed to be unnecessary.
In a routine search of a table such as this,
beginning with the simplest model and adding
“one step’’terms to improve the fit, i.e., reduce
the chi-square value, is called the forward procedure; beginning with the most complex and
dropping terms which are unnecessary in order
to simplify the model, is called the backward
procedure.
Working forward through Table 2 it is observed that adding the GL interaction results
in a significant improvement in fit over the
independence model-chi-square has been reduced by 119.16 36.99 = 82.17 which with 23
- 21 = 2 degrees of freedom is highly significant. So, while the GL interaction alone may
not be suficient (chi-square = 36.99 with 21
~
degrees of freedom and probability 0.017), it
should be included in some form in the final
model. Adding the GA term reduces chi-square
by 12.42 which, with 3 degrees of freedom, is
significant a t the 0.01 level. If now the GS term
should be included, chi-square is reduced by
2.72 which, with 1 degree of freedom, is just
significant a t the 0.10 level. It is questionable
whether the GS term should be included but the
forward procedure terminates either a t model
no. 6 or a t model no. t%i.e., there is no other
term that will effect a significant reduction in
chi-square.
Working backward from model no. 18, if the
GSL term is dropped, model no. 15 results and
chi-square is not significantly changed. Dropping A from the GAS term produces model no.
12 and again no significant increase. And dropping the GAL interaction from the GAL term
(i.e., to model no. 8 ) causes a change of 9.33 in
chi-square which, with 6 degrees of freedom, is
not significant. And once more the problem of
whether the GS term should be included or
dropped ti.e., to model no. 6) arises. Simplicity
would require t h a t i t be dropped. And the
backward procedure, then, terminates a t either
model no. 8 or a t model no. G-i.e., no other
term can be dropped which will not significantly increase the value of chi-square. ( I t
should be mentioned that the backward path is
not unique. The branching from model no. 15
could have gone to model no. 11 and thence to
model no. 6 with equal validity, thereby obviating the need to consider the inclusion of the GS
TABLE 2 . Tests of irll possible hypotheses regarding
the odds f i r the A S genotype
Log-1ikeli hood
Model
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
ia
G
GA
GS
GI,
GA,GS
GA,GL
GS,GL
GA,GS,GL
GAS
GAS, G1.
GAL
GAL, GS
GSL
CSI,, GA
GAS, GAL
GAS,GSL
GAL, GSL
GAS. GAL, GSL
Degrees of
freedom
rat10
chi-square
Probabi I i ty
23
20
22
21
19
18
20
17
16
14
I2
11
18
15
119.16
102.93
118.65
36.99
101.43
24.57
35.66
21.85
100.72
21.04
15.50
12.52
33.85
20.17
11.70
19.34
11.36
10.49
0.000
0.000
0.000
0.017
0.000
0.137
0.017
0.191
0.000
0.101
0.215
0.326
0.013
0.166
0.165
0.081
0.252
0.106
8
12
9
6
345
DISTRIBUTION OF HbS
TABLE 3. Tests of all possible hypotheses regurriing the odds for the AS genotype ot
each locution separately
Tampa
Stock ton
Model
1.
2.
3.
4.
G
GA
GS
GA, GS
New York
-~
Degrees of
freedom
LLR
chi-square
P
LLR
chi-square
P
chi-square
P
7
4
8.31
7.35
6.40
5.72
0.306
0.118
0.380
0.126
18.93
4.59
18.85
3.68
0.008
0.331
0.005
0.296
9.76
3.56
8.62
1.96
0.203
0.469
0.196
0.581
6
3
term.) So, to this point in the analysis, the parsimonious model is ASL, GA, GL. Note that a n
age effect is evident even a t this level. The GS
term, potentially of considerable interest, will
not be retained at this level, but will reappear
a t lower levels a s will be seen shortly.
If model no. 6, which is a “non-elementary”
hypothesis (Goodman, ’70) in the sense that it
cannot be expressed in simple conditional
probabilities, is accepted then we should consider each of the locations separately. Table 3
contains tests of all possible hypotheses a t each
of the locations. (When considering the data by
location, the age-sex, AS, term is included in all
models.) Only in the Tampa data is there reason to accept an hypothesis other than the independence of genotype and all other variables
jointly though it is noted that the GA term
improves the fit significantly for New York and
GS slightly improves the fit at Stockton. Thus
the Tampa and New York data appear to include a n age-genotype interaction. Consequently we may conclude that the presence of
a n age-genotype interaction in Table 1is due to
those sources, and the possible sex-genotype
interaction is due mainly to the Stockton data.
Fortunately Dr. Wienker’s data include information on birthplace. These frequencies are
presented in Table 4, I. None of the possible
unsaturated models provides a n adequate fit of
the observed odds for the AS genotype; thus the
highest order interaction is required for these
data. However, considering each of the three
birthplaces separately (Table 4, 111, note that
for those born in Florida but not in Tampa, the
is 6.49
model of independence fit-hi-square
with 7 degrees of freedom and probability
0.484. For those born outside Florida the model
which includes the age-genotype interaction is
required+hi-square 3.74 with 4 degrees of
freedom and probability 0.442. Only for those
born in Tampa is an age-sex-genotype interaction required.
1,LR
Now we may conclude that the age-genotype
interaction included in model no. 6 (Table 2) is
due primarily to the “Tampa born,” those “born
outside of Florida” and New York subsets of the
total; and the possible sex-genotype interaction
in model no. 8 is due to the Tampa born and
Stockton populations. It should be pointed out
that the age effect is present in the total data
set. Detailed analysis of the recognizable subsets indicates the possible source of the effect,
but in no way do these results invalidate concluding that the effect characterizes the U.S.
distribution.
The sex independent odds for the population
born outside Florida are, for each of the four age
categories: 0.10,0.13,0.12,0.11.For New York
the same odds are: 0.07,0.07,0.08,0.09.For the
Tampa born males the odds are: 0.09,0.08,0.14,
0.05; and for females: 0.07, 0.11, 0.08, 0.20.
(Herethere seems to be a tendency for the male
odds to decrease and the female odds to increase
with age.) Overall, the odds are 0.08,0.08,0.10,
0.10. It is clear by inspection that the effect is
not a monotonic decline with age.
DISCUSSION
The locales producing a statistically significant age trend are New York, the “Tampa born”
and “born outside Florida” subsets. In none of
the data sets is there a suggestion of a decline
other than the sex-conditional trend for Tampa
born males. This will be discussed further.
Historically the black migration from the
South to the Northeast was concentrated in the
period before World War 11. On the strength of
the similarity of odds between “Tampa born”
and New York, this migration would appear to
have been from urban areas. The main westward movement occurred during and after the
war. The similarity of odds between “Florida
but not Tampa” and Stockton suggests that this
was a movement of predominantly rural popu-
346
HFiAXTON M. AI,FRED
TABLE 4 . Frequencies, I ,
of
A A and A S ohserued, and tests of hypotheses, 11, at Tornpa
hy age, sex, and hrrthplace
I
Birthplace tL):
Tampa
Genotype 1 G 1:
Sex tS) Age tAl
Florida
Other
AA
AS
AA
AS
AA
AS
717
65
45
17
4
89
86
36
47
8
12
7
9
1908
65 1
209
201
96
343
35
770
61
87
236
125
19
25
72
137
94
121
5
12
11
13
2006
1261
860
799
190
152
104
96
Male
1t o . 10 years
10 to c 20 years
20 t o . 30 years
3 30 years
575
118
74
23
Female
1t o
10 years
10 to
20 years
20 to <. 30 yeari
’30 years
8.37
II
Birthplace:
Tampa
Measure:
Ll,R
chi-square
Florida
P
Other
1,I.R
chi-square
P
LLR
chi-square
P
6.49
X98
4.05
0.26
0.484
0.408
0.670
0.967
13.81
:3.74
13.52
2.47
0.054
0 442
0.035
0.480
Model’
-~
1.
2.
3.
4.
G
GA
GS
GA, GS
’ Fikwrrs tn p;trt.nthe\ei
17)
(4)
t6J
13)
22.42
14.30
21.76
13.94
0.002
0.006
0.002
n.oo:j
:ire degrees of frccdom
lations. Thus i t appears unnecessary to invoke
micro-environmental explanations for the observed differences among locales, except for the
rural-urban difference in the Southeast. This
difference has been adequately discussed and
documented by many (Pollitzer, e t al., ’70).
All locales are characterized by apparent sex
effects but only in the “Tampa born” does this
reach statistically detectable levels. There the
marginal odds are 0.088 for males and 0.098 for
females. If this were a n X-chromosome mediated effect, one might expect the difference to
be constant over all ages. It is not. So either the
effect is more complex or it is a sampling artifact. If the effect were real and t h e same
among locales one would expect similar results
from all sites. This is not observed. Given the
magnitude of the Tampa project one must invent complex sampling scenarios to reproduce
the observations by sampling bias. These data
cannot elaborate further. Janerich was able to
exclude bias due to factors related to hematocrit level.
An observation which is not fully understood
but commonly made in the U S . involves a n
increase in the frequency of type AS with age.
This topic will now be considered.
The regression of In odds on age, In (odds) = a
+ b(age), was done and the results are below:
a
b
r:!
parameter
-2.57
0.01
0.71
In a demographically stable population, the
proportion of people in the age range rr to n +
d o , c(a),is given by c ( a )= b exp(- a r ) p ( a ) where
h is the birthrate, I‘ the intrinsic rate of increase
andp/rz) the probability of surviving from birth
to age ci (Keyfitz, ’68);r is defined by r = b - d
where d is the death rate. Writing c,(a) and
c,(a) for the proportions of sicklers and normals
respectively in the age range, the odds for AS,
w,(a), are
w,!a)
~
~
c,ia)
c,ln)
~
h, expi --ar,)p,ia)
-~
b, expi -ar,)p,!:i)
11)
DISTRIBUTION OF HbS
Taking natural logs and rearranging
(In p,ia)
ar,)
As a crude first approximation, assume t h a t
mortality, q, is a constant pressure acting
equally a t all ages. Then survivorship can be
q)". The
represented in t h e form pia) = (1
terms involving p ( a ) above can now be written
In p ( a )= a In(l - q) then substituting b q = r
into (2) the expression for In odds becomes
-
-
13)
q,) + q , )
tlntl
~
ib,
b,) I
~
-
which is linear i n a . Note that terms of the form
In(1 q) + q are asymptotically zero as 0 si q s 1
goes to zero. For q < O.Ol,ln(l q) + q
5x
lo-' and, for present purposes, may be set to
zero. Keyfitz and Flieger ('68)estimate the U.S.
nonwhite stable population from the 1960 observed population. There the death rate is estimated as 6.84 per thousand; with this estimate the terms under consideration become
ln(l 0.00684) t 0.00684 - -2 x 10 This
means that any slope in the In w may be considered to bedue to differences in birth rates; i.e.,
~
~
-
-I.
-
So when b, -,b,, i.e., when the birth rate of
normals is greater than that for heterozygotes,
the slope of the odds regressed on age is positive, which is somewhat counterintuitive. The
parameters estimated earlier may now be displayed as
or bJb, = 0.08, and by - b, = 0.01.
Keyfitz and Flieger ('68) estimate the total
birth rate about 0.035. Now there is a system of
three equations i n two unknowns:
0.08b,
h,
b,
I
11\ + b,
b,
-
=
=
0
0.01
0.035
An approximate solution of this system exists:
b, = 0.023 and b, = 0.009. The implication is
t h a t the birth r a t e among normals is about 2.5
times higher t h a n among heterozygotes. This
should not be taken as a n exact estimate, how-
347
ever, due to the sources of error which were
included as a result of simplification at three
places. First was the survivorship function. It
seems quite reasonable to assume a n exponential decay function to represent the force of
mortality. Reality is not so simple unfortunately. The magnitude of the error introduced
by this assumption is unknown but should not
be unacceptably large relative to other sources.
Second was the assumption of demographic
stability. The U.S. nonwhite population has not
achieved its stable distribution yet. But again,
making the assumption allows analysis to proceed without, i t is hoped, distorting t h e outcome too greatly. And third was the assumption
that mortality is insignificant relative to fertility. This is probably the least serious. Consider
the quantity in brackets in equation 3. Each of
the first two terms is certainly less t h a n 10- '
and their difference less than lo-". When combined with the last term, birth rate difference
in the neighborhood of
or greater, it is seen
that the effect of mortality can legitimately be
assumed negligible.
The results obtained here strongly indicate
t h a t compensatory fertility f a v o r i n g AS
(Cavalli-Sforza and Bodmer, '71) is not occurring; quite the contrary.
In the next section I shall investigate the
historical dynamic of the change in frequency
of the heterozygote.
THE: EVOLUTIONARY DYNAMIC WITH ADMIXTURE
AND DIFFERENTIAL FERTILITY
In this section I will consider the effect of
heterozygote fitness on the odds in the presence
of admixture. A set of experiments with a deterministic simulator using discrete time (generations) was performed. The logic will be discussed below. The following constraints apply:
1)initial odds a r e 0.216 and terminal odds
0.087; 2) admixture occurs at a constant rate
such that the total is 25%; 3)only the time
periods beginning 7-12 generations ago are
studied; 4) the homozygote for S has fitness 0.1.
The object is to determine the functional relationship between heterozygote fitness and
time within these bounds.
The assumption t h a t initial odds for this
process are identical to current odds in Africa is
acknowledged to be questionable. The arguments against i t are cogent but not compelling.
Some starting point is required, however, and
this particular one can at least be rationalized.
Admixture was assumed to be structured as
one-way selective gene flow by equally increasing the frequencies of the matings AA x AA,
348
BRAXTON M. ALFRED
AA x AS, AA x SS over that expected under a
random mating regimen in accordance with the
expression obtained earlier. I t h a s been
suggested that temporal variability in admixture would noticeably modify predictions. Consider the extreme case of all admixture occurring in generation 1. The equilibrium odds for
AS change from 2q/p to [q(2 + ap)J/ p[ 1 + ap(1
+ q)l where q is the frequency of the allele for
type S, initially estimated as 0.09, p = 1 - q,
and n is the admixture rate structured as described above and estimated as 1.25. Numerically the odds change from 0.20 to 0.14, about
30%. This result serves as a caution in interpreting predictions based on the assumption of
a constant admixture rate. Likewise spatial
variability in allele frequency with low interdemic flow, and selective admixture could generate equally severe disturbances.
The only selective force presumed to be
operating is differential fertility. Thus the
product of the fitness of the partners in a mnting is taken as the fertility of the pair. The
random mating frequencies are obtained and
modified by admixture. These matings then
produce the offspring for the next generation.
The results of the experiments are presented
here as ordered pairs, the first member of
which,g, is the number of generations, and the
second, f,is the fitness of the heterozygote: ( 7 ,
0.925),18,0.945),19,0.96),i10,0.97),(11,0.98),
(12,0.986).These pairs are to be read as “if the
number of generations was g the heterozygote
must have fitness f in order to change the odds
from 0.216 to 0.087.”
There is no way to select among these alternatives with confidence a t this time. Intuitively, however, motivated by previous results,
a fitness for the heterozygote less than 0.925
would almost certainly have been recognized.
So the conclusion is reached that the genetically relevant New World experience of this
gene pool is about 9-12 generations with heterozygote fitness about 0.96 - 0.99.
A second set of experiments was conducted to
investigate the effect of “compensatory fertility” (Cavilli-Sforza and Bodmer, ’71). This factor, when included, was assumed to increase
the fertility of AS only and so affected the fertility of matings involving this type by 4%. (Double heterozygote mating fertility was also increased by 4% .)
For the period being investigated, the effect
of admixture on the predicted odds was minor
overall, being of the approximate magnitude of
0.004 to 0.001 lower than with no admixture.
The effect of compensatory fertility with no
admixture was about a n order of magnitude
larger, i.e., about 0.03 to 0.01 higher than without the differential. The combined effect of admixture and compensatory fertility was intermediate but expectably nearer t h e value
obtained with the fertility differential alone.
A third series of model experiments was performed with admixture rate assumed to be proportional to genotypic fitness such that the
total was 25%. The rate of decay of predicted
odds was only slightly greater than when admixture was assumed to be independent of fitness.
And finally a series similar to the first was
done using the assumption that the fitness of
the normal changed linearly from 0.9 to 1.0
over the specified number of generations. The
results are: ( 7 , 0.88), ( 8 , 0.90), (9, 0.9151, (10,
0.925),(11,0.935),(12,0.94).Thebasicassumption implies that the fertility of the normal is
depressed in the malarial environment and recovers linearly. It will be noted that heterozygote fitnesses seem too low and consequently
too much time is required.
The qualitative results of the simulation experiments are consistent with those obtained
by analyzing the age distribution. Specifically,
both indicate that the normal has greater fitness than the heterozygote. The difference is
due to fertility a n d is apparently small,
probably no larger than a 4% advantage for the
normal.
CONCLUSIONS
The results obtained here were surprising to
me initially. Clearly the reason is a bias with
regard to the concept of fitness. Specifically
when thinking of fitness as survival of individuals, attention is naturally directed to
mortality. One then assumes equal fertility
combined with selective mortality as the evolutionary dynamic. There is, however, little, if
any, evidence for greater mortality among
sickle cell heterozygotes than among normals.
And, since the frequency apparently increases
with age it follows that normals have higher
mortality. But if that is so then the overall
frequency should be increasing. And it is not.
The apparent paradox is an artifact of the initial bias. The sickle cell trait observations in
the U.S. are characterized by decaying overall
frequency combined with increasing frequency
with age. There is no mortality force that will
produce these conditions in the presence of
equal fertility. One is forced to the conclusion
DISTRIBUTION OF HhS
that the evolutionary dynamic is one involving
a fertility differential favoring t h e normal
combined with (effectively) equal mortality.
ACKNOWLEDGMENTS
I a m deeply indebted to Dr. Malcolm Greig,
U.B.C. Computing Centre for assistance with
the statistical parts; and t0U.B.C. for sufficient
computing time to survive many false starts.
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