Mendelian Genetics (Part I) Due: Friday, September 25 th, 2013

Transcription

1 STAT 307 Fall 2015, Project 1 Mendelian Genetics (Part I) Due: Friday, September 25 th, 2013 The context: So, it turns out that in 1865 a man named Gregor Mendel published an article that provided a scientific explanation for heredity. This paper (and theory) subsequently caused nothing short of a revolution in biology. 1 Why we are doing this project: -Mendel s theory of genetics is great science -This theory shows the power of simple chance models in action The scoop: All of Mendel s experiments were done using simple house garden peas and I will provide a brief account of one of his experiments for us to statistically analyze. Pea seeds are either yellow or green. (*note: the seed color is a property of the seed itself and not of the parental plant; indeed, often one parent has seeds of both colors.) Mendel bred a pure yellow strain, that is, a strain in which every plant in every generation had only yellow seeds. Separately, he bred a pure green strain. He then proceeded to cross the plants of the pure yellow strain with plants of the pure green strain: for instance, he used pollen from the yellows to fertilize ovules on plants of the green strain. (The alternative, using pollen from the green strain with ovules from the yellow, produced exactly the same results.) Now, the seeds that result from a yellowgreen cross, and the plants into which they grow, are called firstgeneration hybrids. First-generation hybrid seeds are all yellow, indistinguishable from the seeds of the pure yellow strain. The green seems to have disappeared completely. 1 I.M. Lerner, Heredity, Evolution, and Society (W.H. Freeman, 1968). Friday, September 18 th, 2015 Page 1 of 6

2 STAT 307 Fall 2015, Project 1 These first-generation seeds grew into first-generation hybrid plants, which Mendel crossed with themselves, producing secondgeneration hybrid seeds. Some of these second-generation seeds were yellow, but some were green. So, the green disappeared for one generation, but reappeared in the second. Even more surprising, the green reappeared in a definite simple proportion: of the secondgeneration hybrid seeds, about 75% were yellow and 25% were green. So what is behind this regularity? To explain it, Mendel postulated the existence of the entities which we now call genes. 2 According to Mendel s theory, there were two different variants of a gene which paired up to control seed color. They will be denoted hereby y (for yellow) and g (for green). It is the gene-pair in the seednot the parent- which determines what color the seed will be, and all the cells making up a seed contain the same gene-pair. There are four different gene-pairs: y/y, y/g, g/y, and g/g. Gene-pairs control seed color by the rule: - y/y, y/g, and g/y make yellow - g/g makes green. As geneticists say, y is dominant and g is recessive. This completes the first part of the model. 2 Allele is perhaps the better term for a specific segment of DNA occupying a coding region. Presumably, several proteins are needed to determine seed color and, if so, the pure yellow and pure green strains would have many of the corresponding alleles in common, but would differ on one pair-the y/y and g/g in the text. Mendel himself referred to entities which controlled phenotypes. Friday, September 18 th, 2015 Page 2 of 6

3 Now, the seed grows up and becomes a plant; all cells in this plant also carry the seed s color gene-pair with one exception. Sex cells, either sperm or eggs, contain only one gene of the pair. For instance, a plant whose ordinary cells contain the gene-pair y/y will produce sperm cells each containing the gene y; similarly, it will produce egg cells each containing the gene y. On the other hand, a plant whose ordinary cells contain the gene-pair y/g will produce some sperm cells containing the gene y, and some sperm cells containing the gene g. In fact, half its sperm cells will contain y, and the other half will contain g; similarly, half its eggs will contain y, and the other half will contain g. This model accounts for the experimental results. Plants of the pure yellow strain have the color gene-pair y/y, so the sperm and eggs all just contain the gene y. Similarly, plants of the pure green strain have the gen-pair g/g, so their pollen and ovules just contain the gene g. Crossing a pure yellow with a pure green amounts for instance to fertilizing a g-egg by a y-sperm, producing a fertilized cell having the gene-pair y/g. This cell reproduces itself and eventually becomes a seed, in which all the cells have the gene-pair y/g and are yellow in color. The model has explained why all first-generation hybrid seeds are yellow, and none are green. Friday, September 18 th, 2015 Page 3 of 6

4 What about the second generation? A first-generation hybrid seed grows into a first-generation hybrid plant, with gene-pair y/g. This plant produces sperm cells, of which half will contain the gene y and the other half will contain the gene g; it also produces eggs, of which half will contain y and the other half will contain g. When two firstgeneration hybrids are crossed, each resulting second-generation hybrid seed gets one gene at random from each parent because it is formed by a random combination of a sperm cell and an egg. From the point of view of the seed, it s as if one ticket was chosen at random from each of two boxes: in each box, half the tickets are marked y, and the other half are marked g. The tickets are the genes, and there is one box for each parent. Friday, September 18 th, 2015 Page 4 of 6

5 As shown on the previous page, the seed has a 25% chance to get a gene-pair with two g s and be green; it has a 75% chance to get a genepair with one or two y s and be yellow. The number of seeds is small by comparison with the number of pollen grains, so the selections for the various seeds are essentially independent. The conclusion: the color of second-generation hybrid seeds will be determined as if by a sequence of draws with replacement from the following box: And that is how the model accounts for the reappearance of green in the second generation, for about 25% of the seeds. Mendel made a bold leap from his experimental evidence to his theoretical conclusions. His reconstruction of the chain of heredity was based entirely on statistical evidence of the kind discussed here. And he was right. Modern research in genetics and molecular biology is uncovering the chemical basis of heredity, and has provided ample direct proof for the existence of Mendel s hypothetical entities. Essentially, the same mechanism of heredity operates in all forms of life, from dolphins to fruit flies. So the genetic model proposed by Mendel unlocks one of the great mysteries of life. How is it that a pea-seed always produces a pea, and never a tomato or a whale? Furthermore, the answer turns out to involve chance in a crucial way. Friday, September 18 th, 2015 Page 5 of 6

6 Exercise Set I: 1) In some experiments, a first-generation hybrid pea is backcrossed with one parent. If a y/g plant is crossed with a g/g, about what percentage of the seeds will be yellow? Of 1,600 such seeds, what is the chance that over 850 will be yellow? (hint: the expected number of yellows is 1,600 x ½ = 800, with an SE of sqrt(1600) x ½ =20. Now, use the normal approximation.) 2) Flower color in snapdragons is controlled by one gene-pair. There are two variants of the gene, r (for red) and w (for white). The rules are: -r/r makes red flowers, -r/w and w/r make pink flowers, -w/w make white flowers. So neither r nor w is dominant. Their effects are additive, like mixing red paint with white paint. (a) (b) Work out the expected percentages of red-, white- and pink-flowered plants resulting from the following crosses: white x red, white x pink, pink x pink. With 400 plants from pink x pink crosses, what is the chance that between 190 and 210 will be pink-flowered? (hint: use the SE formula from above. Your SE should be 10.) Friday, September 18 th, 2015 Page 6 of 6