July 5, 1947.

Dear Dr. Rhoades-

How is everything?

You may recall that I mentioned in my seminar on coli linkage
that for most of my map, I could give only relative, not absolute
distances. For some time now, I have been trying to develop a theory
to estimate the xkbai absolute distance involved by means of the

efreauency of the triple-exchange types. Briefly the situation is

this: AbCa » and call the re; ioss between, ané b, a, etc.
ape | -

the only types recovered (as prototrophs) are:
ABcD 27.3%  AbcD 46.3% ADCL 24.3% , which are exchanges
BG ; >
in repions a,b and c respectively, and ABCI 2.1% which is a triple-
exchange tyne.
Lhe theory relating the frequencg of *sBCT to the absolute distances,
on a two-stranc theory, is very simple, and barring interiemce, cs
map of 72 morgans between A and D is obtained. I am still trying to
develop a rigoroug theory for a similar estimate on the four-strand
theory, with only beginning success, bec&use of the complexity of
the problem when dealing with 3, 4, and 5 random crossovers per tetrad.
My first apyroximate solution is sbout 80 morgens, which is surprisingly
close to the two-stranc figure. Po you know of sny theoretical treat-
ment of multiple crossing over that would fécilitate my lubors?
Tn the course of this armchair experinentation (which is a good
deal more laborious than lab.-desk work!), I developed an "Operator"
notation anc method of dealing: with maltiple-crossing over, concerning
which I shoulé like to heer your opininn reletive to «.) its ude macy
b) its usefulness in general and c) its novelty. The method is a good

deal easier to use than to explain; it concerns the problem of cal-
culating the nature of all the digierent tetrads which result from
combinations of various crossovers:

Write:

A b Cc d

 

 

POP re

 

“a 5 G D

Each crossover is written with the region involved, and the strands
exchanged as subsoripts, e.g. 8,3-, boye, ete.The crossovers in a given
tetrad are written tn sequence, And may be unlimited in number.

Each such symbol is regarded as an operator, which affects only that
strand bearing the same subscript as it does (i.e. of the same rank).
The effect of a c.-o. operation tm, going from right to left is a) to
substitute certain allels by their alternative, depending on the
region of the crossover, i.e.

Axy subtitutes a/A and A/a, written (a), operating on strands of

rank x and y.
byy subtitutes a/A; b/B and vice versa, .... (ab)

Cyy substitutes a/A;b/B and c/C, ana vice versa, .....(abe).

A second effect of each operation, e.g. a on s, is to change its
rank from x to y with respect to any Prther operators.

This,is, of course, merely a restatement of the interaction of cross-
overs, but the symbolic formulation is useful, since one can write
down certain combinations at once:

ab =(a).(ab) = (b) abe = a.be = (a)(ac)= (c).
a,c =[a).(abe)= (bc)
bec = (b).(abe)= (ac)

Hol

E.G., 1f s is ABCD, °
ABoD

ab.s = akeg@hx= AbCD abe.s =

ac.s = AbcD a,S = aBCcT

be.s = aBel bd.S = abcD
aa.s =bb.s=cc.s= ABCD  O,.«S= abcd.

Obviously the repétition of an operator restores the original configu-
ration,

To break down a sequence of crossovers into its component operations

on each strand, one starts from right to left, and writes the appropriate
operation, using the subscript to indicate the new rank, until the c.o,.8s
are exhausted. E.G., 15°83 4° Donel, is:

On 8, the right handmost operator hs a,4. One writes AgeS,+ Or abcd

On So, We see Coq on the righthand end, and write ¢,.s,,. Looking for
adaifional operat ors of rank 4,now, we seef @,,, an write A, C4-S2-
Finally, 8,3 is seen to be an operator of rank‘i, so we write
aza104-82. ° this is,of course, aBcD. For sz, we see bg3, and then
-3-

no acditional rank 2 operators. bo-sz is AbeD. Finally, For Sq we

2
write(from right to left): ¢» then, ba; then Ayes 1.e. ajbzep -S4
or ABCD. the tetrad is then:
abCd
aBca
AbeD
ABCD
For most ptirposes, the final subscripts can be neglected, except that
1,2 and 3,4 are the centromere markers, at either end you choose.
ihe extension of this system to polysomgic segregations is,I think,
evident. After 5 minutes practice, the tetrads can be written down
by inspection, instead of having to construct involved diagrams of
the chromatids, particularly with tri- amd quadrivaleht essociations.
I am busy now trying to work out derived rules to see if the manipulation
can be cispensed with altogether, Whether or not this system can be
adapted to loop- and ring- formations, in aberra.ion heterozygotes
is another possibility I have not yet looked into.
Your opinkmion on all this would be greatly appreciated. It is hard
to believe that noone else has done this sort of thing before, but
I have not yet encountered it.

Best regards,

Yours sincerely,

Joshua Lederberg