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HYB: Punnett Square Question

It's been some time since this topic has been discussed on the list, so I'm
going to backtrack a bit. 

The Punnett Square is normally applied to a single locus, and used to
identify the combinations of alleles obtainable from two given parents and
the ratios in which they occur.  As a simple mathematical tool, however, it
can certainly be adapted for use in the manner Donald Eaves has suggested.

For the benefit of newcomers, though, I'm going to start by answering the
unasked question:  why bother?

Ten years ago, the Aril Society switched from using the Quantum System to
describe aril content to a modified Chromosome Set System.  The old system
was simple, but not always an accurate reflection of reality.  The new
system takes more genetics into account, but is much harder to understand. 
The average grower is little affected by it all, but hybridizers are still
adapting to the change and adjusting to its ramifications. 

For those unfamiliar with the Quantum System, it was not only extremely
easy to use but understanding it was almost intuitive.  Just add the
quantum content of the pod parent to that of the pollen parent, then divide
by two, and you get the quantum content of their offspring.  A few
examples:  two 1/2-breds produce more 1/2-breds; a 1/2-bred and a TB
produce 1/4-breds; an 1/8th-bred and a TB produce 1/16th-breds.    When the
minimum aril content was established as one-quarter, the possibilities were
further limited.  [This was the action that disenrolled Donald's example of

Simple.  Neat.  But, unfortunately, not an accurate reflection of what is
going on at a genetic level.  

Today's Modified Chromosome Set System does a much better job of depicting
that because, in the real world, chromosomes have a tendency toward being
passed on as sets.   It's more difficult to apply, though, and far from

Enter Donald's use of the Punnett Square.....

For a tetraploid, the Punnett Square includes four alleles for the locus: 
A1, A2, A3, and A4.  As each gamete contains only two alleles, the possible
combinations are:  A1A2, A1A3, A1A4, A2A3, A2A4, and A3A4.  That's six
possible combinations from each parent, so the Square itself is drawn with
six rows and six columns.

The example Donald cited is probably a bit confusing because it actually
requires a set of such squares to depict the range of possibilities using
MARY McCLELLAN instead of just one.  To simplify matters I'm going to use a
generic cross between a 1/4-bred and a TB as my example, and construct the
squares for entire chromosome sets rather than a single locus.  

The conventional shorthand for chromosome sets is A=Aril and B=Bearded. 
The composition of the quarterbred parent is thus ABBB and the combinations
of gametes it forms are: AB, AB, AB, BB, BB, BB.  The TB parent can form
only BB gametes, however, so BB will be used for all six cases.

Drawing a picture in e-mail so that it is received accurately by all
programs is beyond my ability, so you may want to resort to those archaic
tools known as pencil & paper.

1.      Draw a six-row, six-column square in the middle of the page.  This
is the Punnett Square.

2       Put the pod parent on the left-hand side and one of the possible
combinations on each row.  First, we'll assume the 1/4-bred is the pod
parent so the rows will be labeled:  AB, AB, AB, BB, BB, BB.

3.      Put the pollen parent at the top and one of the possible
combinations above each column.  In this case, that means each column is
just "BB".

4.      For each cell, add the row label and the column label to get its
combination.  Row 1, Column 1 is thus ABBB.  As, you'll find, are all other
cells in the first three rows.  Row 4, Column 1 is BBBB -- as are all of
the other cells in the last three rows.

Now do the same for a cross in which the TB is the pod parent and the
1/4-bred is the pollen parent.  This time, you should find that all cells
in the first three columns are ABBB and all cells in the last three columns
are BBBB.

So....   If you look at just the ratios of gametic formation and
recombination, you'd expect half of the offspring to be ABBB and half to be
BBBB in each case.  IF, however, you cross one of these offsping without
being sure which group it belongs to you have to draw two squares for it. 
And add more squares to the set with each succeeding generation to
completely depict the possibilities.

A set of generic Punnett Squares can simplify the process, though. With a
"best guess" as to the type of seedling [based on both genetic
possibilities and exhibited traits & fertility] it's possible to select the
most appropriate Punnett Square in the set and use it as a predictor
without having to recompute and redraw one for the specific set.  As long,
of course, as you've already drawn enough of them to be able to understand
and take advantage of the short cut.....

Overall, the Modified Chromosome-Set model provides a more accurate
depiction of reality than the "1/8th" assigned to all offspring by the
Quantum System, but there are still other real-world factors that is
doesn't consider so this seems like the appropriate point at which to
address Jeff's comment..... 

I have often said that NO model depicts the real world with complete
accuracy.  The purpose of a model is to provide an overview, to help people
understand something without having to delve into the underlying theory. 
The quantum one is very easy to use, but doesn't even attempt to describe
what's going at at the genetic level.  The chromosome-set one is harder to
apply, but at least includes the most fundamental genetic concepts although
it falls far short of accurately representing everything that goes on at
the genetic level.   

The average person who grows arilbreds really need NOT be concerned about
all this.  Those who show arilbreds need only to understand the
classification codes and their usage.  Pollen daubers can have a lot of fun
without studying models -- merrily crossing the fertile halfbreds to
produce gobs of nice seedlings.  Beginning hybridizers, who want to advance
to more complex crosses, can use the models as they learn genetics. 
Advanced hybridizers can use them to identify avenues of further
exploration by asking what factors not included by accepted models could
explain what is happening in seedling  beds.  

That said, let's get back to Donald's example.  Here are some things I've
noted that the Modified Chromosome Set model does not take into account:

Environmental factors.  Even if seeds are produced in the expected ratios,
if one type is favored by local growing conditions then more of that type
can be expected to survive. 

Selective Fertilization.  Much has been written about the natural advantage
afforded like gametes in the fertilization process.   When the 1/4-bred is
the pod parent this does not come into play, but when the TB is the pod
parent it means that the 1/4-bred's BB gametes would be favored over its AB
gametes so that the actual mix of offspring can be expected to include more
BBBB types than ABBB.  Left alone, this means that a population consisting
of a mixture of BBBBs and ABBBs would have fewer and fewer ABBBs down
through the generations.

Seedling Selection.  The itself and the Punnett Square that depicts the
possible combinations describe only the POSSIBILITIES. Once a hybridizer
has entered the picture and made a selection, it's a whole new world.  With
hybridizer selection, that same mixed population of BBBBs and ABBBs would
soon be dominated by the ABBBs.  I don't know how many seedlings he bloomed
from the cross that produced MARY McCLELLAN, but Tom Craig was selecting
for aril traits so that, genetically, MARY McCLELLAN is mostly likely an
ABBB-type arilbred regardless of how many generations may have elapsed from
is species ancestry. 

Homologous Chromosomes.  Although chromosomes TEND to be passed on as sets,
when species share homologous chromosomes the resultant sets are often
mixed.  Were it not so, we would not have our modern TBs.  The affinity of
aril and bearded chromosomes is less than that of species within the
bearded family itself, but the relative fertility between the two families
is evidence that it does exist.  Chromosome studies have found aril markers
in cultivars counted as having 48 chromosomes, which on count alone would
be classified as a tetraploid TB. 

Yes, assessing progeny from a wide cross can be frustrating and
time-consuming -- but, most of all, it is fascinating!

Sharon McAllister

Message text written by Donald Eaves:
MARY McCLELLAN is a disenrolled arilbred, being listed as 1/8 onco.
The aril ancestry derives from the pod parent BLUE OX which was a
cross of ACROPOLIS (I assume a TB?) x C.G. White oncobred (I
assume a 1/4 bred?)  In order to put this in a Punnett square to
calculate the odds of seedling resulting from a cross of MMc x 1/4 bred,
would the following be correct?

AA AA AA Aa if 'A' is TB and 'a' arilbred?  Also, is there a standard
method of putting the pod/pollen parents on a specific axis of the

For iris progeny from a cross, it may prove to be frustrating and time
consuming to prove, however.<

and Jeff Walters replied:

>I don't think you can represent MMc this way, since it is a tetraploid,
an octoploid. MMc would be either AAAA or AAAa for a given tetrad of
chromosomes. Probabilistically half the chromosome tetrads would be of one
kind and half of the other. However, Sharon McAllister has stated that this
sort of quantum model does not very accurately represent the way in which
genetic material is passed down in interspecific crosses, if I understand
what she has tried to explain.

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