Contents

Introductory Lecture (~15 minute video, accessible only from the CD-ROM)

Parent-Offspring Gene Dropping

All Founder Alleles Unique

A Simple Pedigree

Where do we get Pedigrees?

Gene Drop Simulations (with a Coin Toss)

Computer Simulation of Parent-Offspring Gene Dropping

Pedigree Gene Dropping

Autozygosity and Inbreeding

Gene Dropping to a Pair of Individuals--Kinship

Gene Dropping to a Set of Individuals--Allele Loss

Gene Drop Software--Complex Pedigrees

The basic genetic link is between parents and offspring. As is customary for our analysis, we assume a sexually reproducing, diploid population of individuals. An offspring has two parents, a father and a mother. In pedigree diagrams, the father is represented as a square, the mother as a circle. The movement of genes is downward on the drawing. Lines between individuals represent the downward passage of alleles, i.e., the dropping of alleles. There are different ways graphically to represent links between parents and offspring. The three common patterns are indicated below.

We employ the top method, as it is the most straightforward to deal with in computer software. For complex pedigrees, however, the other two methods can often be easier to read. The middle method is the most common method utilized. The bottom method, called the marriage node graph has advantages when the number of offspring is large and when there are many different pairings between parents. Each mating (marriage) is represented by a black dot that links the two parents, and from which the offspring are graphed.

Each individual represented in a parent-offspring relationship or in a pedigree diagram has a genotype. In most of what follows, we limit attention to a single genetic locus. The genotype of each individual depends on the two alleles the individual carries. These alleles can be analyzed structurally or functionally. Mendel, who first carried out such analyses in the last third of the nineteenth century, had to work functionally. He hadn't a clue about molecular biology, and he had no way to read chromosomes. Thus, he had to work with the products of the genes as represented in the morphology, anatomy and behavior of individuals of the species. That is, he analyzed the phenotypic products of the genes. He had to infer the genotype of an individual by doing test crosses, test matings. For example, with his famous pea plants, he discovered that there were two different flower color alleles, red and white, denoted, respectively as r and w. With two allelic types, there can be three genotypes, rr, rw and ww. Mendel found that red flowers crossed with red flowers, produced red flowers, and that white crossed with white produced white. However, red crossed with white, always produced pink. But pink crossed with pink produced, on average, 25% white, 25% red and 50% pink. He therefore concluded that the pink flowers were produced by the heterozygotic genotype, rw. The rw genotype crossed with rw would produce the genotypes rr, rw, wr and ww with equal probability. This represents his Law of Assortment. The genotypes rw and wr are indistinguishable, both functionally and structurally, which implies that there will be twice as many pink phenotypes.

A number of different conventions are used to represent alleles. When there are only two alleles, a common representation is 'A' for one allele, 'a' for the alternative form. If a second locus is analyzed, 'B' and 'b' can be used. This makes it straightforward to represent two-loci genotypes, e.g., AaBB or AAbb. When there are multiple allelic types at a locus in a population, their representation is problematic. One technique is to label them A₁, A₂, A₃, ... This technique is fine for books and manuscripts. However, it is often difficult for computers to handle subscripts. So, computers might use A1, A2, A3, ... However, it is more compact to use a, b, c, ... for the alleles. Each of these techniques has different advantages and limitations. We shall use different representations and will strive to be clear about which we are using in any particular example.

Here is a simple case. Both the father and the mother are homozygous AA genotypes. Thus, all of the offspring are also homozygous AA.

A second simple case has each of the parents homozygous, but for different alleles. All of the offspring must be heterozygous for the Aa genotype, since it is certain (absent mutation) that the father will contribute an A allele and that the mother will contribute an a allele.

In our third example, each parent is heterozygous for the A and a alleles. In this case, three offspring genotypes are possible, and these have a distribution of 1:2:1, or 25%, 50% and 25%.

All of this should be review from general biology courses.

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All Founder Alleles Unique

As mentioned above, it could be the case that all of the founders of a pedigree possess unique alleles. It is impossible that every allele could have a different function, could produce a different gene product. However, there could be differences at the structural level of the gene, that is, they could have different DNA sequences. (And with sequence analysis, we could find these.) Another requirement is that the founders would have to be randomly selected from a very large population in which the average relatedness of the animals is essentially zero (more about this later).

Regardless of these considerations, it is conventionally assumed in gene drop simulations that the founder alleles are unique. This is basically a bookkeeping decision. It takes no more execution time to drop unique alleles down a pedigree than to drop initially indistinguishable alleles. But by dropping unique alleles the maximum resolution of what happens will be obtained. This will be obvious in the studies immediately to follow. The next figure is an animation of parent offspring gene dropping when all of the parental alleles are assumed to be unique. We start from the left and give the parents the alleles, a, b, c and d. This animation, which loops continuously, shows only one of the possible outcome.

With four alleles in a population, there could be 10 genotypes: aa, ab, ac, ad, bb, bc, bd, cc, cd and dd. However, a number of these combinations are impossible in the parent-offspring pedigree illustrated above. In fact, the only possible offspring genotypes are ac, ad, bc and bd.

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A Simple Pedigree

A pedigree is nothing more than a connected set of parent-offspring relationships. Consider the following relatively simple pedigree.

This pedigree contains 10 individuals. The lines between individuals are drop paths for a single allele. The individuals at the top end of a path are, relatively speaking, ancestors and the individuals at the bottom end of a path are descendants. The top three individuals do not have their parents represented. They are taken as the founders of the pedigree. The other 7 individuals have fully described parent-offspring relationships, as indicated in the next animation.

This pedigree is not too complex. It begins with three individuals, a male and two females. These three parents produce three offspring, a son and two daughters. These three individuals then produce three daughters. The son, individual 6, mates with one of his daughters, individual 7, to produce a daughter, individual 10. So, there are four generations in this pedigree. However, in general, it is not all that easy to sort the individuals of a pedigree into clearly defined generations.

The next figure assumes that there are two alleles in the population, A and a, which could represent different gene functions. The founders are arbitrarily assigned genotypes. An example gene drop is given. Note that the genotype of individual 6 will always be heterozygous Aa.

The next figure (animation) illustrates a gene drop in which all of the founders are assigned unique alleles.

Their alleles and genotypes of the three founders are assigned, in a left to right fashion, and are all unique. All other animals in the pedigree receive their alleles from the founders. It is assumed that alleles do not change (mutate) as they drop through pedigrees. They validity of this assumption depends on the depth of the pedigree (e.g., the amount of time it covers) and also on the type of analysis used to distinguish the alleles. The function of alleles mutates extremely slowly. Mutation probabilities are on the order of one in a million to one in a hundred million. However, at the structural level, genes can mutate more rapidly. Nonetheless, mutation is ignored in the analysis of pedigree carried out below.

Note that individuals 9 and 10 in the above figure both ending up homozygous for the b allele (which they both received from their fraternal grandfather--their dad's dad). We comment on this shortly.

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Where do we get Pedigrees?

Homo sapiens is the best species for obtaining pedigrees, since human beings usually know who their father is, and they customarily keep written records of this information. For example, many of us have family trees. Computer software that aids in the construction of family trees sells briskly. You are the focus (the root) of your family tree, and it branches backwards in time to include all of your known ancestors. It can be mildly amusing to know something about your family tree. 'Gilpin' is, for instance, a relatively uncommon surname. My uncle produced a Gilpin Family Tree going back in the United States to the Mayflower (the 1640s). More interestingly, I find that my fraternal grandmother is a Babbitt, and I calculate I share 1/64th of my genes with Bruce Babbitt, currently (in the year 2000) the United States Secretary of the Interior.

However, such family trees are not too interesting genetically, as they are not closed. That is, with each generation backwards in time, all of the members of the family tree are unique. More interesting are closed human pedigrees. These usually come from islands or from small religious groups that intermarry among themselves. The issue is this: an individual has 2 parents, 4 grandparents, 8 great grandparents, and so on. In a small closed population, this progression fairly quickly surpasses the total members of the population. Thus, some ancestors will have to have multiple locations in the family tree. For science, the main interest of such closed pedigrees involves genetic diseases. Many genetic diseases are based on recessive alleles, that is, the disease only shows up (is expressed) in individuals who are homozygous for the recessive allele. In small, closed populations, it is easier to find such alleles and to infer their genetic transmission. For example, the country of Iceland, which has a population of about a quarter of a million individuals, has recently bargained away its 1000-year pedigree to a molecular genetics company, so that this company, armed with our new knowledge of the human genome, can search for the exact locations of genes that cause diseases. The bargain (Faustian?) is that Icelanders will receive any cures or therapeutics that derive from this research for free. I am again amused by this, as my maternal grandparents emigrated from Iceland to become farmers in Minnesota. So, I have one allele at every single one of my gene loci that derives from Iceland. Fortunately, I am unlikely to have in homozygous form any bad alleles from Iceland.

For species other than us, obtaining pedigrees is far more difficult. The best studied species are those that have a history of economic (or sporting) interest to us. Breeders customarily keep pedigree information for such species as race horses, pure bred dogs, pigs, cattle, chickens and so forth. They keep such information to be able to avoid inbreeding in their lines of animals, and to be able to select for animals who have genetically based traits, e.g., high milk yield in cattle or superior running speed in horses, that they wish to emphasize.

Animal pedigrees are usually maintained in 'stud books.' Why this name? The biggest problem in recording pedigree information comes from the males. Who is the father? The mother is often obvious. For a variety of reasons, the father can be problematic. For domestic species, the sexually receptive female is often only provided a single partner, the stud.

Zoos also keep stud book information. Initially, this was done locally and was mainly done to prevent close inbreeding. However, in the late 1970s, the zoos found themselves playing an extremely important role in species conservation. For example, it was the case that the entire species of the Arabian oryx, eight animals, was suddenly found to reside in the London zoo. How did this happen? The wild population was rapidly and imprudently hunted to extinction (links to the story). Extirpated from the wild, the oryx's only hope for survival was in the captively bred population. The species had to be bred back to a larger population size and then reintroduced to the wild. Genetic management was important in this regrowth period. This story seems to have a happy ending. But this species and many others underscored the value of the Zoo Ark, and it warned zoo managers and zoo curators that they could again and again find themselves playing the savior of last resort. Today, zoos cooperatively manage the entire pedigree of about 100 animal species that are extirpated or endangered in the wild (links to AAZP and ISIS web pages)

There are many species that have similar stories to the Arabian oryx: California condors, the black-footed ferret, Speakes gazelle, the golden lion tamarin. We will examine many of their stories in the analyses to follow.

There are also cases of where we have been about to obtain pedigrees from wild populations. Usually these come from islands or small nature preserves. In Addo Elephant Nature Park in South Africa, the completely insular elephant population began (in 1938) with 12 animals and included only one mature male.

Sometimes, with small populations, an unknown pedigree can be inferred through techniques of paternity exclusion (link) based on knowledge of alleles at one or more gene loci. The power to perform such analyses is rapidly increasing.

Finally, we can obtain pedigrees from individual-based computer simulations. That is, a number of assumptions about the movement, demography and behavior of the individuals of a closed population can be programmed and then the consequences of all of these assumptions can be followed forward in time. As every animal in the population has a unique ID number, it is possible to output the resulting pedigree information and analyze it. While this might sound contrived, it can give valuable management insights. For example, one can run a baseline model and then one can stipulate that mature males be harvested (something which can often produce a revenue stream for the park). This harvest might allow more males to get the chance to reproduce, and the genetic consequences of this can be explored.

Thus, there can be many situations that provide pedigrees for conservation biology analysis.

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Gene Drop Simulations (with a Coin Toss)

An individual dropping alleles has two alleles to drop, but only one will travel the path to the descendent. How is this allele chosen? Basically, by a coin toss. It's a random 50:50 proposition.

One way to perform gene drops, such as illustrated in a couple of the figures above, it to do a Monte Carlo simulation using a coin. This is a stochastic simulation, which involves chance events. If the coin toss comes up 'heads' the allele at the left in the individual's genotype is chosen to be dropped, and if 'tails' the allele to the right.

The figure below shows how the alleles of individual 4 can be selected from its parents, individuals 1 and 2, using the coin toss simulation process. The 'heads' for the father-to-daughter drop selects the father's left allele, the a allele. The 'tails' for the mother-to-daughter drop selects the mother's right allele, the d allele.

The seven non-founder individuals in this pedigree have alleles dropped into them. This requires fourteen coin tosses. These fourteen tosses can be given an order:

• paternal allele for individual 3
• maternal allele for individual 3
• paternal allele for individual 4
• .
• .
• .
• maternal allele for individual 10

There are 2¹⁴ (=16384) different ways to toss a coin 14 times in succession. However, this does not mean that there can be sixteen thousand distinct gene drop patterns, as you will discover by simulating the gene dropping process on some of the pedigrees to be examined in computer software introduced below.

Exercise 22.1: Perform a coin flipping simulation for the ten-animal pedigrees shown above. (hint)

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Computer Simulation of Parent-Offspring Gene Dropping

The coin flipping approach to simulating gene dropping is slow and prone to error. As we have seen in the programming course, it is straightforward to simulate coin flipping on a computer. A True Basic program for performing this coin-flipping simulating is

IF RND < 0.5 THEN
   PRINT "heads"
ELSE
   PRINT "tails"
END IF
END

In this program, RND is an intrinsic function of the True Basic programming language that produces a (pseudo) random number distributed uniformly between 0.0 and 1.0. This number will be below 0.5 fifty percent of the time, so, after a large number of executions, this program would print 'tails' half the time and 'heads' half the time. Thus, it might not be too difficult to write a program to perform gene drops on pedigrees utilizing a digital computer. In fact, we wrote the basic core of such a program in the earlier programming course. Here is a slightly abbreviated version of that True Basic program that performs the parent-offspring gene drop that we need to have.

RANDOMIZE                        ! required to make executions independent
   
DEF FN_RND_INDEX = INT(2*RND+1)  ! either returns 1 or 2,
                                 ! which is equivalent to heads or tails
   
DIM mom$(2), dad$(2)             ! each of dad and mom have two alleles
DIM offspring$(2)                ! the offspring will receive two alleles
   
LET mom$(1) = "a"    ! these are the structurally unique alleles
LET mom$(2) = "b" 
LET dad$(1) = "c"
LET dad$(2) = "d"
   
! this ends the initialization, now the simulation begins
   
LET offspring$(1) = mom$(FN_RND_INDEX) ! randomly selects the first offspring
                                       ! allele from the mother
LET offspring$(2) = dad$(FN_RND_INDEX) ! selects the second offspring allele
   
PRINT offspring$(1)&offspring$(2)    ! prints the genotype of the offspring
END

This program, if executed repeatedly, will print 'ac', 'ad', 'bc' and 'bd' with equal probability. The secret to this program is the user defined function FN_RND_INDEX, which returns either a 1 or a 2 with equal probability. This function is used as a random index to a parent's two alleles to select one or the other of them with equal probability. This short program is the core of a full gene drop program.

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Pedigree Gene Dropping

Presented below is a longer program that extends the logic of the parent-offspring gene drop program to perform the full gene drop simulation for the ten-individual pedigree we have been examining above. This program fills up the two allelic positions for each of the 7 offspring animals, based specifically on who their mother and father are. It assumes that each of the alleles of the three founder parents are structurally unique. The offspring are filled from top to bottom, as it is impossible to drop alleles to, say, individual 8 before alleles are dropped into individuals 4, 5 and 6. This program performs a single drop.

RANDOMIZE
DEF FN_RND_INDEX = INT(2*RND+1) ! either returns 1 or 2,
                                ! which is equivalent to heads and tails
DIM animal1$(2)                 ! dimension two alleles for
DIM animal2$(2)                 ! each of the animals
DIM animal3$(2)
DIM animal4$(2)
DIM animal5$(2)
DIM animal6$(2)
DIM animal7$(2)
DIM animal8$(2)
DIM animal9$(2)
DIM animal10$(2)
   
LET animal1$(1) = "a"          ! give animals 1, 2 and 3 unique alleles
LET animal1$(2) = "b"          ! as these are the three founders
LET animal2$(1) = "c" 
LET animal2$(2) = "d"   
LET animal3$(1) = "e"
LET animal3$(2) = "f"
   
LET animal4$(1) = animal2$(FN_RND_INDEX) ! select individual 4 alleles
LET animal4$(2) = animal1$(FN_RND_INDEX) ! from mother 2 and father 1
   
LET animal5$(1) = animal2$(FN_RND_INDEX) ! select individual 5 alleles
LET animal5$(2) = animal1$(FN_RND_INDEX) ! from mother 2 and father 1
   
LET animal6$(1) = animal3$(FN_RND_INDEX) ! and so forth...
LET animal6$(2) = animal1$(FN_RND_INDEX)
   
LET animal7$(1) = animal4$(FN_RND_INDEX)
LET animal7$(2) = animal6$(FN_RND_INDEX)
   
LET animal8$(1) = animal4$(FN_RND_INDEX)
LET animal8$(2) = animal6$(FN_RND_INDEX)
   
LET animal9$(1) = animal5$(FN_RND_INDEX)
LET animal9$(2) = animal6$(FN_RND_INDEX)
   
LET animal10$(1) = animal7$(FN_RND_INDEX)
LET animal10$(2) = animal6$(FN_RND_INDEX)
   
PRINT animal10$(1)&animal10$(2)     ! print the genotype of animal 10
   
END

Show in the figure below is the pedigree under consideration and a small Java program for performing this gene drop and displaying the genotype of individual 10. Press the Go button a number of times to see different simulations of this dropping process.

Exercise: 22.2 Perform 100 gene drops (press the Go button 100 time) and keep track of the number of aa, bb, cc, dd, ee and ff homozygous genotypes that occur. Beware that sometimes after pressing the Go button, there is seemingly no change. What this is is just a drop that produces the same result, and it does this so fast that there is no flicker of the computer screen. So, record the result after every pressing of the Go button.

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Autozygosity and Inbreeding

We have just seen that individual 10 in the above pedigree can sometimes get two copies of the same allele. It's not simply the case that these two alleles will have the same function or behavior. Rather, they will be two identical copies of a unique allele in one of the founders. They are Identical by Descent -- IBD. They are also said to be autozygous. For example, the genotype of individual 10 produced by random gene dropping might be bb. The allele b is one of the two alleles of the male founder of this pedigree, individual 1. The only way an individual can receive two identical copies of a founder allele is if it receives one through its mother and one through its father. Thus, there needs to be two gene drop paths, one through the mother and one through the father, that lead back to the common ancestor from whom the allele is received. For the case of bb autozygosity, this is shown in the next figure.

In fact, this figure shows how either the a or the b allele can drop through the pedigree (following the bold line paths) to produce an autozygous aa or bb individual 10. There are two additional configurations of paths that lead back to a common founder and that, therefore, provide independent paths to autozygosity. These two additional configurations are shown below.

Notice that there are no such paths that lead back to founder 2. Thus, it is impossible to be autozygous for founder alleles c or d. You should have observed this in the Exercise 22.2, above.

The inbreeding coefficient of an individual is the probability that it will be autozygous, that it has received two identical copies of an allele from one of its ancestors. It doesn't matter which ancestor it receives them from. Thus, in the example just given, the probability of inbreeding is the sum of the probabilities of being autozygous aa, bb, ee and ff. The simulations of gene dropping we have been exploring do not allow the calculation of exact probabilities. Rather, they allow for an estimation of this probability. The larger the number of trials, the closer the estimation.

Programming Problem 22.1. Modify the True Basic program listed above to perform 1000 gene drops thereby to estimate the inbreeding coefficient of individual 10. (answer)

Programming Problem 22.2. Modify the code of the previous program to calculate the inbreeding coefficient of individual 10 in the pedigree shown below, which is a five generation example of sib-sib mating. That is, the brother and daughter mate, and the grandson and granddaughter mate, and so on. (hint)

Here is a Java Applet that executes basically the same steps to estimate the inbreeding coefficient of individual 10 in the pedigree we have been exploring. This is a slightly more general program which allows the user to input the number of replications used in the simulation. This number can be inputted into the text field at the top right of the applet.

Exercise 22.3. Run this applet with 100 replications 10 times; record and average the results. Run the applet with 1000 replications 10 times; record and average the results. Run the applet with 10000 replications 10 times; record and average the results. Run the applet with 100000 replications 10 times; record and average the results. Comment on the pattern that you observe.

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Gene Dropping to a Pair of Individuals--Kinship

The gene dropping patterns explored in the foregoing involve outcomes in a single individual. What fraction of the drops result in the designated individual having an autozygous genotype? This fraction is an estimation of that target individual's inbreeding coefficient. It is possible to do something similar with pairs of individuals in a pedigree. For example, we could analyze the pair of individuals 6 and 9 (indicated by the red circles) and individuals 6 and 7 (indicated by green circles) in the pedigree below.

To introduce this technique, look at a particular gene drop, as shown below.

The pair indicated by the green circles share a copy of the b allele. The pair indicated by the red circles share no allele in common. We would like to develop a statistic that represents the likelihood that two individuals in a pedigree share alleles identical by descent. We can't call this sharing autozygosity, as the shared alleles are not in the same genotype. We call it kinship.

We define kinship as the probability that alleles randomly selected from two individuals are identical by descent. Consider the two individuals enclosed by the green circles in the pedigree above. There is a 0.5 probability of selecting the b allele from individual 6 and a 0.5 probability of selecting the b allele from individual 7. Thus, there is a 0.25 probability of jointly selecting the b allele in both individuals. Note that this is not the kinship between the two individuals, as this value depends on a particular gene drop. As with estimating the inbreeding coefficient, the kinship coefficient must be obtained from thousands of gene drops.

Exercise 22.4. Compute the probability of selecting alleles that are identical by descent for the two pairs of individuals indicated by the colored circles in the particular instance of a gene dropping shown below.

A computer program to estimate kinship does the gene dropping exactly as before. The difference comes with how it does the computation of the kinship statistic. With inbreeding, the comparison was between the first and the second alleles of the target individual. The exact statement was:

IF animal10$(1) = animal10$(2) THEN ...

To do the kinship comparison (say, for animals 6 and 7) the following statement will do what is required:

IF animal6$(FN_RND_INDEX) = animal7$(FN_RND_INDEX) THEN ...

Recall that the user defined function FN_RND_INDEX yields either 1 or 2 with equal probability. With the execution of this statement, one of the two alleles from individual 7 is selected, one of the two alleles from individual 6 is selected, and the two are compared to see whether they are identical. The Java Applet performs kinship coefficient estimates for specified pairs of individuals in the pedigree with which we have been working. The two individuals are specified in the two text fields at the top and then the Go button is pressed.

Kinship can even be computed between an individual and itself. If the individual is not inbred, the kinship must be 0.50, as one allele is drawn randomly and then a second is drawn randomly. Half the time the two draws will be of the same allele. However, with inbreeding there is the possibility that the alleles will be the same even though the selection is of different alleles.

Exercise 22.5. Use the Kinship Applet to compute the entire ten-by-ten kinship matrix. (Note that the kinship between 6 and 7 is the same as the kinship between 7 and 6, thus the matrix is symmetrical about the main diagonal.) (answer)

The utility of kinship can be understood by comparing the kinship of individuals 6 and 7 to the inbreeding coefficient of individual 10. You've estimated both of these. How do the two compare?

They should both be very close to 0.3125. Why? Is there a reason these two should be the same? Notice that individual 10 is the offspring of individual's 6 and 7.

Basically, the random drawing of single alleles from two individuals for kinship comparison is exactly the same as drawing two alleles (gametes) from two individuals for the sexual reproduction to a new offspring. That is, the kinship coefficient between potential parents, actual or potential, precomputes the inbreeding coefficient that their offspring will have. As such, the kinship matrix can provide good guidance for the genetic management of populations.

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Gene Dropping to a Set of Individuals--Allele Loss

We have used the random dropping of genes through pedigrees to estimate inbreeding and kinship coefficients. As we shall see in Module 25, there are more accurate and elegant ways to do this, but perhaps not so intuitive. This is not true for the next statistic of conservation importance.

If a pedigree contains n founders, it contains 2n founder alleles, assuming all are unique. Imagine a pedigree covering multiple generations, with only a set of the animals at the bottom of the pedigree still alive. We can ask the following question: what fraction of the initial founder alleles will be represented in this set of individuals?

Consider the 10 animal pedigree for brother-sister mating (sib mating) that we introduced earlier. It starts with 4 unique alleles. How many of the founder alleles will be present in the fifth generation, i.e., in animals 9 and 10? And what kind of an answer do we want from this question? Perhaps we might obtain the answer: 2.2 alleles. This tells us something. Roughly, have the initial alleles get through the pedigree. Or, we might obtain a fuller answer: 2.2 +/- 0.4 alleles. This second form of the answer tells us that there would only be about a 2% probability that 3 alleles would pass through this pedigree, as 3.0 is 2.2 plus two of the standard deviations. That all 4 alleles would pass down to the fifth generation is extremely unlikely. Put another way, it is quite likely that one founder allele will be lost, and very likely that two will be lost.

A computer program to do this is very similar to the ones we have written above. It has to be recoded to account for the different structure of the pedigree. Next, we establish a vector histogram, dimension it to hold 4 elements, indexed from 1 to 4, and set all of these equal to zero:

DIM histogram(4)
MAT histogram = ZER

Inside the gene drop loop, we have to count the different alleles that reach individuals 9 and 10. A straightforward way to do this is to go through the four alleles in individuals 9 and 10 four times, counting the number of a alleles, then a b alleles, then a c alleles and finally a d alleles.

Some True Basic code to do this is

LET a_count = 0
LET b_count = 0
LET c_count = 0
LET d_count = 0
IF animal$9(1) = "a" THEN LET a_count = a_count+1
IF animal$9(1) = "b" THEN LET b_count = b_count+1
IF animal$9(1) = "c" THEN LET c_count = c_count+1 
IF animal$9(1) = "d" THEN LET d_count = d_count+1
IF animal$9(2) = "a" THEN LET a_count = a_count+1
IF animal$9(2) = "b" THEN LET b_count = b_count+1
IF animal$9(2) = "c" THEN LET c_count = c_count+1 
IF animal$9(2) = "d" THEN LET d_count = d_count+1
IF animal$10(1) = "a" THEN LET a_count = a_count+1
IF animal$10(1) = "b" THEN LET b_count = b_count+1
IF animal$10(1) = "c" THEN LET c_count = c_count+1 
IF animal$10(1) = "d" THEN LET d_count = d_count+1
IF animal$10(2) = "a" THEN LET a_count = a_count+1
IF animal$10(2) = "b" THEN LET b_count = b_count+1
IF animal$10(2) = "c" THEN LET c_count = c_count+1 
IF animal$10(2) = "d" THEN LET d_count = d_count+1

This code could be written more efficiently, but it does the job just fine. Next, from these counts of alleles of the four founder types, the count of allele diversity must be obtained. This can be done with the code:

LET diversity_count = 0
IF a_count > 0 THEN LET diversity_count = diversity_count + 1
IF b_count > 0 THEN LET diversity_count = diversity_count + 1
IF c_count > 0 THEN LET diversity_count = diversity_count + 1
IF d_count > 0 THEN LET diversity_count = diversity_count + 1

So, at the end of the loop, the variable diversity_ count is the count of unique founder alleles that have passed through the pedigree. From this we can estimate the probabilities for the passage of 1, 2, 3 or 4 alleles. Here is a link to the True Basic code for this program (source code). The Java Applet below performs the same calculation.

Note the high fraction of the time that 3 of the initial 4 alleles are lost.

Exercise 22.6. Describe what you think will happen with the pedigree illustrated below.

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Gene Drop Software--Complex Pedigrees

Some of the techniques explored above are applied to more complex pedigrees in the Java Applets below. There are two applets, each contain the same ten pedigrees. These pedigrees are selected by clicking on the choice box at the top right of the applet screen and highlighting the choice you desire. The chararacter and histrory of some of these pedigrees will be discussed in the next module. The screen is initially blank. After selecting a pedigree there are buttons to pressed and moust actions that can be taken.

The first applet does single trial gene dropping, which you trigger by pressing the Gene Drop button. It also does a thousand gene drops and calculates the fraction of autozygosity, which is an estimate of the inbreeding coefficient. You can press this button multiple times to get a better approximation of these values.

Exercise 22.7. Having selected the Back Cross pedigree, estimate the inbreeding coefficients and plot them against the generation level of the succession of females. What asymptote is being reach? How can you explain this?

 The next Java Applet deals with the same ten pedigrees, but it treats the issue of allele loss. This applet allows mouse actions. You can click on an animal to select it, and it will turn green. To deselect it, click on it again. With one or fewer animals selected, a normal gene drop is performed and displayed, just as in the applet immediately above. However, when two or more animals are selected, one thousand gene drops are performed and for each drop the number of unique alleles reaching the set of selected animals is counted. A frequency histogram of the number of alleles reaching the selected animals is then drawn for the 1000 trials. Click on the button multiple times to see the stability of this estimate.

Exercise 22.8. For the sib mating pedigrees, plot the modal number of alleles that reach the two animals at a generation level versus the generation level.

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