Pedigrees and Genes -- Using Pedigrees to Track Inherited Disorders - 2

James E. Seltzer, Ph.D.

Proper pedigree analysis is one technique that can be used in order to breed better dogs. Its value is its ability to decrease the unpredictability inherent in the independent assortment of genes.1

In the first part of this series, we examined several of the ways in which hereditary traits are expressed and passed on from one generation to the next. We reviewed the probability that a puppy will inherit a trait from a parent, assuming we know the parent to be a carrier for the gene that determines the trait. In this part we will dig deeper into a pedigree, examining the risks for hereditary defects passed down from grandparents and other more remote ancestors.

First, a brief comment on notation: When we speak of the probability of a random event, we mean the fraction of times that the event is likely to occur if we can repeat the exercise a large number of times, e.g., the fraction of times that "heads" comes up when we flip a coin many times. We use the notation P(heads) or Ph to stand for the probability of the occurrence "heads." Similarly, we use the notations Paffected or Pcarrier to designate the probability that a dog is affected or a carrier, respectively, for some hereditary characteristic.

As an example we will consider the defect, overshot bite, that we used in the first part of this series. We assume that the defect is transmitted as an autosomal recessive trait. We are going to use the pedigree of the puppy, Burble's Unicorn, "Cornie," to derive the probability that as an adult he will have an overshot bite.

The conventional breeder's pedigree shown in figure 1 tells us that Hatter, who is Cornie's double great grandsire is overshot, but that is not the whole story. Look at the less familiar geneticist's pedigree shown in figure 2. Two additional pieces of information emerge. First, the mating between Hatter and Rattie produced an affected female puppy. This means Rattie must also be a carrier. Second, the mating between Snark and Vorpal produced an affected puppy. So both Snark and Vorpal must also be carriers. Take note that unless we collect information on the siblings of the dogs whose names appear in a pedigree, we will not have the best information and cannot do a thorough job of assessing the risk of transmitting a hereditary defect.

 

 

Parents

GrandParents

G-GrandParents

CH Quadrille's Jabberwock CD

"Jabber"
(Sire)

Quadrille's Snark

"Snark"

CH Quadrille's Gryphon

Slithy Tove
Of Mimsy

CH Vorpal Queen Of Hearts

"Vorpal"

CH Mad Hatter Of Mimsy CDX

"Hatter"

Mome Rath Of Mimsy

"Rattie"

CH Red Queen Of Gyre

"Gyre"
(Dam)

Quadrille's Borogove

"Boris"

March Hare Of Uffish

Manxome
Mock Turtle

CH Vorpal Queen Of Hearts

"Vorpal"

CH Mad Hatter Of Mimsy CDX

"Hatter"

Mome Rath Of Mimsy

"Rattie"

Figure 1. Breeder's pedigree of Burble's Unicorn, Cornie. Hatter, Cornie's double great grandsire, has an overshot bite.

 

 

Figure 2. Geneticist's pedigree of Cornie with three identified cases of overshot bite. Square symbols denote males; circles denote females. Lines tying male-female pairs represent matings and vertical descending lines represent the produce from the matings. Unfilled symbols represent dogs that are normal or of unknown status, black-filled symbols are individuals affected with this disorder.

 

We want to calculate the probability that Cornie, when he gets his adult dentition, is going to end up with an overshot bite. We also want to know the probability that Cornie is a carrier for this disorder. Since overshot bite is assumed to be a recessive trait, we know that to be affected Cornie must have inherited defective genes from both his sire and his dam. However, to be a carrier, Cornie must have received a normal gene from one of his parents and a defective gene from the other.

Tracing the Defect
  1. Using the information from figure 2, go to the pedigree, figure 1, and mark a "C" for carrier next to the names Snark, Vorpal and Rattie. Place an "A" for affected next to Hatter's name.
  2. Starting at the oldest generation on the right in figure 1, trace the paths to the left taking note of which of the dogs are marked either with a "C" or an "A." When we find a dog so designated we shall need to begin with that dog and compute the risk as it is transmitted through the succeeding generations. In this example we move immediately to the grandparental generation where we have marked Snark and Vorpal. Boris has no mark and will be assumed to be clear for overshot bite. Snark and Vorpal are the parents of Jabber, the puppy's sire. Boris and Vorpal are the parents of Gyre, the puppy's dam.
  3. We shall need to use formulas 1, 2 and 3 from the Appendix as we work our way through the pedigree, one generation at a time. We start by making the following designations:

    4. Snark:

       P    affected = 0

       Pcarrier = 1

    Vorpal:

       P    affected = 0

       Pcarrier = 1

    Boris:

       P    affected = 0

       Pcarrier = 0

  4. Now, applying formula 1 to each of these individuals:

    5. For Snark (sire of Jabber):

      Ps = Paffected + 1/2 x Pcarrier = 0 + 1/2 x 1 = 1/2

    For Vorpal (dam of both Jabber and Gyre):

      Pd = 0 + 1/2 x 1 = 1/2

    For Boris (sire of Gyre):

      Ps= 0 + 1/2 x 0 = 0

  5. Moving to the next generation and using formula 2:

    6. For Jabber:

      Paffected = Ps x Pd = 1/2 x 1/2 = 1/4

    For Gyre:

       Paffected = 0 x 1/2 = 0

  6. Now, applying formula 3:

    7. For Jabber:

       Pcarrier = (1- Ps) x Pd + Ps x (1 - Pd) = (1 - 1/2) x 1/2 + 1/2 x (1 - 1/2) = 1/2

    For Gyre:

       Pcarrier = (1- 0) x 1/2 + 0 x (1 - 1/2) = 1/2

  7. And finally we arrive at the puppy, Cornie:

Sire, Jabber:

   Ps = Paffected + 1/2 x Pcarrier = 1/4 + 1/2 x 1/2 = 1/2

Dam, Gyre:

   Pd = 0 + 1/2 x 1/2 = 1/4

Puppy, Cornie:

   Paffected = Ps x Pd = 1/2 x 1/4 = 1/8

   Pcarrier = (1- Ps) x Pd + Ps x (1 - Pd) = (1 - 1/2 ) x 1/4 + 1/2 x (1 - 1/4 ) = 1/2

So we see that Cornie has a 12.5% chance of having an overshot bite as an adult. His chance of being a carrier for the defect is 50%. The chance that he is clear (neither affected or carrier) is only 37.5%.

Summary

To do a risk analysis for the transmission of an autosomal recessive characteristic using a pedigree we need to work through the pedigree from the oldest ancestors for which we have data to the youngest generations using three formulas successively, generation following generation. The steps outlined here for tracing risk probabilities through a pedigree might seem onerous at first, and, for long and detailed pedigrees you will probably need at least a pocket calculator. However, the routine is not complicated and can be broken down to only a few steps that are applied repeatedly:

  • Lay out the pedigree and mark each of the ancestors for which you have information as affected if they have the disorder or as carrier if they produced a puppy with the disorder.
  • Starting with each ancestor at the right (oldest) end of the pedigree, trace the descendents path to the left until you encounter the most recent, i.e., youngest, ancestor that you have marked as affected or as a carrier. Compute the probability, Ps or Pd, that the dog or bitch so designated has transmitted a defective gene to its progeny. Since you already have information about this dog or bitch, you can disregard all of the ancestors from earlier generations that lie behind this individual.
  • After this initial calculation, move left to the next generation and compute the probabilities that the animals in this generation are carriers or affected. Take care that the Ps and Pd used are the numbers that were calculated for the animal's parents. Continue this procedure until you reach the final generation or the expected litter from a proposed breeding.

You may have data on some dogs that are definitely known to be unaffected. However, lacking DNA testing, you will not know that these ancestors are clear, i.e., neither carriers nor affecteds. For these individuals you can simply set the affected probability to zero (but not the carrier probability) to take cognizance of this data. Actually, should you choose to be mathematically rigorous, a further tweaking of the probabilities can be made (see the Unaffected correction in the Appendix for the adjustment).

Want to explore further the use of pedigrees to trace inherited disorders? You can find a lengthy discussion on this topic in the book, Control of Canine Genetic Diseases, by George A. Padgett, DVM.4

Not very nimble with your pocket calculator? There are several easy to use computer programs for pedigree management that include calculations for risks of genetic disorders.5 A freeware program for PCs that includes this option can be downloaded from http://www.geocities.com/willowind_dals/pedigree.html

Appendix

Three useful formulas from probability theory2,3 are the joint probability for two independent random events, the probability of one or the other of two mutually exclusive random events, and the probability of a random event not occurring. To work through a pedigree from oldest generation to youngest, the following specific formulas are essential. We shall not take the time to derive these formulas, but it is not difficult to do so from general equations.

1. Transmission formula (the probability, Ps for sire and Pd for dam, that a parent transmits a defective gene to the offspring):

Ps = Paffected + 1/2 x Pcarrier, where Paffected and Pcarrier refer to the sire's probability of being affected or carrier, respectively.

Pd = Paffected + 1/2 x Pcarrier, where Paffected and Pcarrier refer to the dam's probability of being affected or carrier, respectively.

2. Affected formula (the probability that the offspring is affected):

Paffected = Ps x Pd, where Paffected refers here to the offspring's probability of being affected.

3. Carrier formula (the probability that the offspring is a carrier):

Pcarrier = (1- Ps) x Pd + Ps x (1 - Pd), where Pcarrier refers here to the offspring's probability of being a carrier.

4. Unaffected correction: If an ancestor is known to be unaffected, then calculate the probabilities: Paffected and Pcarrier as before, but make the following replacements:

Set the new Pcarrier = original Pcarrier / (1 - Paffected)

Then set Paffected = 0

 

References:

  1. "The Value of Pedigree Analysis," J. Charles Garvin, M.D., in the Spotter, vol. 21, no. 3, Spring 1991.
  2. Schaum's Outline of Probability and Statistics, 2d Ed., Murray Spiegel, McGraw-Hill
  3. Schaum's Outline of Genetics, 4th Ed., Susan Elrod, et al., McGraw-Hill.
  4. Control of Canine Genetic Diseases, George A. Padgett, DVM, Howell Book House, 1998.
  5. WinCanis ©, PEDIGREE Software Program