NATIONAL INSTITUTE OF GENETICS

YATA 1,111 MISIMA, SIZUOKA-KEN,
411 JAPAN

Cable address: Genetics, Misima

October 24, 1975

Professor L. L. Cavalli-Sforza
Department of Genetics
Stanford University

School of Medicine

Stanford, California 94305

U. S. A.

Dear Luca:

It was a pleasure that I could see you (rather unexpectedly)
in my recent visit to the United States. Many thanks to you
for your hospitality then.

I have read your new manuscript "The Theory of Continuous
Variation: A Direct Approach Through the Joint Distribution
of Genotypes and Phenotypes" a copy of which you gave me at
that time.

I think that it is an interesting and original contribution.
Although I have not checked the details of your calculations
I have little doubt that they are correct. I am very happy
that one of my previous papers turned out to be of some use
to you. Incidentally, I note that you are right in saying
that S = K/2 (at the bottom of page 8) should read S = 1/(2K).

Probably the most unexpected result in your paper is
the demonstration that the heritability after selection can
be greater than unity.

Since I thought that a great deal of selective elimination
would be required to realize it, I made a few calculations
to check this point. If I understand your paper correctly,
the mean fitness_is (assuming that the mean and the optimum
coincide, i. e. Ff = uw);

oo 2 2
J L_ gr(#-u)*/2B | .-(£-1) “/28

Wo= —_— df
o J/27 YE
oc _ Ss
Wo = fey s
Noting that W : = W(u) = 1, the amount of selective
optimum

elimination is
-

Prof. L. L. Cavalli-Sforza October 24, 1975
w.o- W
L = -E——— = 1 - WS/(E + 8)
Op

which is the load due to the centripetal selection involved.
Thus

2
s = -D" of.

l1- (1 - 13)?

Substituting this into your condition s?

we get

< M(E + 2S) on page 8,

1-(i-1)4

<
(1 - uy?

miCd

as a condition for the after-selection-heritability to be
larger than unity. oO

The left hand side is the total phenotypic variance
devided by the amount of variance added each generation by
mutation. For example, if this is 99, L turns out to be larger
than 0.68, or 68% mortality. This seems to me an intense
selection for a human population.

I do not know whether the above point is of any interest
to you, but I am communicating it any way to show that I
studed your paper.

With best wishes to you as always. Please covey my kind
regards to Alba, and also to Jim McGregor.

Sincerely,

SNetoo

Motoo Kimura

MK /ym