Using punnett squares to express sex/gene linkage and recombination frequency

Question

I have been using punnett squares to visualize mating experiments. I am looking at a pairing which is known to have a crossover event occur. I know from empirically collected data the recombination frequency is ~3%. Can punnett squares be used to express the results bearing the recombination frequency in mind or should I consider a different visualisation technique?

The end result of what I want to achieve is the expected genotypic/phenotypic outcomes taking recombination into account.

In addition the pairing has a sex-linked mutation which further complicates things e.g. aB/Ab x Ab/Y

Here is an example punnett square:

 a-B/A-b x A-b/Y
 ------------------------- 
 | x   | A-b     | Y     |
 ------------------------- 
 | a-B | A-b/a-B | a-B/Y |
 ------------------------- 
 | A-b | A-b/A-b | A-b/Y |
 ------------------------- 

If a want to represent a recombination event in the punnett square (e.g. to produce a-b/A-b and a-b/Y does this mean I would need to add a-b as an additional trait to the punnett square like so?

a-B/A-b/a-b x A-b/Y
 ------------------------- 
 | x   | A-b     | Y     |
 ------------------------- 
 | a-B | A-b/a-B | a-B/Y |
 ------------------------- 
 | A-b | A-b/A-b | A-b/Y |
 ------------------------- 
 | a-b | A-b/a-b | a-b/Y |
 ------------------------- 

If that is the case, that would give me a frequency of 16.666666% for each.

Answer 1

Punnett squares show the possible gametes from each parent. I suppose you could write out all the possible recombined gametes with their frequency in the same format, explicitly stating the percentage of each (usually 50% or 25% is assumed but not written). So if you have the female at the top of the table (sorry, I tried some html tags but they're not taking; I should RTFM)

........ AB - Ab - aB - ab
....... 48.5 1.5 1.5 48.5 %
ab 50%: 24.25 0.75 0.75 24.25 %
Y 50%: 24.25 0.75 0.75 24.25 %

Per your last sentence, the male can only pass on its X genotype, which I've assumed as ab for a backcross, or Y - no crossover - so this would be a 4x2 Punnett square. The odds of each box in the square (not considering lethality) are still found by multiplying the odds of each allele as ratios, then converting to percentage. You have to add up all the percentages that match a phenotype as you score it to find phenotypic ratios.

This page explains this idea with better formatting, but not for a sex-linked trait.

Answer 2

The Punnett squares are a general way to visualize the product of any two polynomials; for example, you know that the result of (a + b)(c + d) is the sum of the four terms ab, ac, bc, bd. You may put it this way:

  | a   b
----------
c | ac  bc
d | ad  bd

Suppose that the female sex gametes are of two types, X1 and X2, which are transmitted with probabilities 0.9 and 0.1; suppose that males form three types of gametes, X1, X2, and Y, with probabilities 0.45, 0.05 and 0.5. Then, you may visualize the outcome of a general population mating this way:

    |Fem   X1   X2
    |      .9  .1
--------------------
Male|
X1  |     X1X1  X1X2
.45 |     .405  .045
    |
X2  |     X1X2  X2X2
.05 |     .045  .005
    |
Y   |     X1 Y  X2 Y
.5  |     .45   .05

(You may note that this situation roughly depicts the population genetics of color blindness)