Dalmatian Club of America
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 P_{h} to stand for the probability of the occurrence
"heads." Similarly, we use the notations P_{affected} or P_{carrier} 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.
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 malefemale pairs
represent matings and vertical descending lines represent the produce from
the matings. Unfilled symbols represent dogs that are normal or of unknown
status, blackfilled 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

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.

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.

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:
Snark:
P_{affected }= 0
P_{carrier }= 1
Vorpal:
P_{affected }= 0
P_{carrier }= 1
Boris:
P_{affected }= 0
P_{carrier }= 0

Now, applying formula
1 to each of these individuals:
For Snark
(sire of Jabber):
P_{s}
= P_{affected} + 1/2 x P_{carrier }= 0 + 1/2 x 1 = 1/2
For Vorpal (dam
of both Jabber and Gyre):
P_{d}
= 0 + 1/2 x 1 = 1/2
For Boris (sire
of Gyre):
P_{s}=
0 + 1/2 x 0 = 0

Moving to the next
generation and using formula 2:
For Jabber:
P_{affected}
= Ps x Pd = 1/2 x 1/2 = 1/4
For Gyre:
P_{affected} = 0 x 1/2 = 0

Now, applying formula
3:
For Jabber:
P_{carrier} = (1 Ps) x Pd + Ps x (1  Pd) = (1  1/2) x 1/2 +
1/2 x (1  1/2) = 1/2
For Gyre:
P_{carrier} = (1 0) x 1/2 + 0 x (1  1/2) = 1/2

And finally we
arrive at the puppy, Cornie:
Sire, Jabber:
Ps = P_{affected} + 1/2 x P_{carrier }= 1/4 + 1/2 x 1/2
= 1/2
Dam, Gyre:
Pd = 0 + 1/2 x 1/2 = 1/4
Puppy, Cornie:
P_{affected} = Ps x Pd = 1/2 x 1/4 = 1/8
P_{carrier} = (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 theory^{2,3} 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 = P_{affected}
+ 1/2 x P_{carrier}, where P_{affected} and P_{carrier}
refer to the sire's probability of being affected or carrier, respectively.
Pd = P_{affected}
+ 1/2 x P_{carrier}, where P_{affected} and P_{carrier}
refer to the dam's probability of being affected or carrier, respectively.
2. Affected
formula (the probability that the offspring is affected):
P_{affected}
= Ps x Pd, where P_{affected} refers here to the offspring's probability
of being affected.
3. Carrier formula
(the probability that the offspring is a carrier):
P_{carrier}
= (1 Ps) x Pd + Ps x (1  Pd), where P_{carrier} 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: P_{affected }and P_{carrier }as before,
but make the following replacements:
Set the new
P_{carrier }= original P_{carrier }/ (1 _{ }P_{affected})
Then set P_{affected
}= 0
References:

"The Value of Pedigree
Analysis," J. Charles Garvin, M.D., in the Spotter, vol. 21, no. 3, Spring
1991.

Schaum's Outline
of Probability and Statistics, 2d Ed., Murray Spiegel, McGrawHill

Schaum's Outline
of Genetics, 4th Ed., Susan Elrod, et al., McGrawHill.

Control of Canine
Genetic Diseases, George A. Padgett, DVM, Howell Book House, 1998.

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PEDIGREE Software Program
