(What Should Have Happened, But Didn't)
[This is a drama. So first be sure you read your part aloud; second be sure to follow all of the stage directions designated in brackets like this statement; and third don't just read óact. Give the play your best.]
Yule and Hardy met at a scientific meeting they were both attending. Yule was excited because he knew of the famous mathematician by reputation, but neither had met the other in person. They agreed to join one another for dinner at a local restaurant. As the play opens, dinner is proceeding.
Yule: It certainly is an honor to be able to talk with you about scientific matters, Professor Hardy. Your mathematical abilities are frequently discussed at my university.
Hardy: [Who is just finishing chewing a piece of his meal.] Well, Yule, that's all very well, but wouldn't it be far better actually to discuss my mathematics. It's far more interesting than my abilities.
Yule: Both are actually pretty interesting! But, say, discussing mathematics is exactly why I wanted to meet with you, Professor Hardy. I have just recently made an interesting mathematical discovery, published it, in fact. It is of some importance to the understanding of heredity. [Leans toward Hardy with a look of significance on his face.] I believe I have shown that the genetic principles of Mendel cannot be very important in nature.
Hardy: I confess to knowing little about the genetic principles of anybody, Professor Yule. My education has spent little time on biological subjects. [Lays down his silverware, apparently giving up trying to eat.] But, my, my, if I understand it correctly, your discovery is really quite significant. It seems to me that it was only a few years ago that Correns and De Vries over on the Continent claimed to have 'rediscovered' a set of principles of inheritance originally formulated by this Mendel.
Yule: [Becoming more excited.] Yes, indeed, they did claim that, Professor Hardy. Eight years ago, actually, but time flies when you're having fun. You'll be interested to know that Mendel was something of a mathematician himself and that he based his ideas about heredity on the occurrence of certain ratios that he got in his crosses.
Hardy: [Politely trying to seem impressed.] Ratios! Well, those aren't exactly the essence of higher mathematics, but I suppose there are some rudimentary issues that could be understood in terms of ratios. Thought of himself as a mathematician, did he? Tell me, how did ratios allow him to draw conclusions about heredity?
Yule: Mendel performed a series of controlled crosses between pea plants with different traits and recorded the numbers of offspring with each trait. As a result of the ratios he recovered in these crosses, he proposed a series of formal rules about the behavior of hereditary factors contained in the plants.
For example, he claimed that each individual contained two copies of each hereditary factor and donated one of them to each of his or her offspring. The pairs of factors could be of alternate forms. So, in one instance, he supposed that there were two forms of an hereditary factor for height. His plants could be tall or short, depending on which factor they contained. They could even contain one tall and one short factor, but since the tall factor masked the expression of the short factor, such plants could be tall.
Hardy: Goodness, this is quite a theory. Has he put his finger on these 'hereditary factors' in some way? Can they be demonstrated aside from their usefulness in accounting for his results?
Yule: Not so far. In fact, if my discovery is correct, they might well never be demonstrated since they will be shown to be unimportant. They may not even exist. But if they do, they are so far removed from our technical abilities to see them that I doubt if the material basis of these hereditary factors will ever be identified.
Hardy: Fascinating. Hereditary factors as abstract as any numbers I ever worked with. Do go on. What would be an example of the connection between the factors and ratios?
Yule: Suppose there were two peas that each contained one tall and one short factor. Let's designate the tall factor as 'T' and the short factor as 't'. Remember that T masks the expression of t. Only when a pea plant has the makeup of tt will it be short. Both TT and Tt will be tall. Remember also that each plant donates one factor to the next generation by way of its gametes. So if we cross two plants that are Tt, 1/4 of the plants will be TT, 1/4 will be tt and 1/2 will be Tt.
Hardy: [Becoming interested.] Of course! A practical demonstration of the expansion of a binomial. How interesting, even if practical. So, if I understand you correctly, a cross of two of these plants that are Tt will result in a ratio among the offspring of 3 tall to 1 short. Did he actually get that?
Yule: Virtually every time! The results are really quite consistent with his hypothesis, I must say. But there lies the problem that I have discovered. The very existence of this 3:1 ratio banishes any thought of this theory being of any importance in nature!
Hardy: You will have to explain that to me. I'm afraid I don't quite follow you.
Yule: Any naturalist knows that the abundance of different forms in nature virtually never displays itself as a 3:1 ratio. Collect shells, birds, insects or plants and you will certainly find alternate traits in color, size or form. But it is ever so rare to see these traits in the ratio of 3 to 1. [The next statement is said with considerable pride.] For that reason, I conclude ó and that was the substance of my talk at this meeting ó that Mendelian genetics is of little importance in nature!
Hardy: [In an embarrassed manner.] Your talk? You've made this point out loudóin public?
Yule: [Hesitating and suddenly unsure of himself.] Certainly. After all, when one makes an important discovery in science, one is obligated to report it to the community. But why this startled look on your face, Professor Hardy?
Hardy: Professor Yule, I'm afraid you've overlooked an important aspect of the problem. I can't be positive, of course, since I have only just now heard you describe the issues. But it seems to me that the very binomial expansion that is at the root of the ratios Mendel got will show that you are quite wrong about what is expected in nature.
Yule: Oh dear. And I was so sure....
Hardy: [Now completely absorbed in what he is saying.] It all seems to me to depend on what you start out with. Mendel was doing controlled crosses I presume, and he always started with Tt individuals in the cross we have been discussing. But nature need not be like that at all. [He takes out a pen and begins to write on a napkin.]
What would be an example of a ratio in humans, for example, that is far from your expected 3:1?
Yule: Well, a good example would be brachydactyly, a condition of the digits, which is dominant yet quite rare. I imagine there might be 1 brachydactylous person in every 10,000.
Hardy: Quite. Suppose that A is brachydactyly and a normal, and that we start from a population of pure brachydactylous and pure normal persons in this ratio of 1:10,000. (I use the word 'pure' because this is 1908. In the future people will say 'homozygous.') What will be the frequency of AA, Aa and aa in such a population?
Yule: [Answers the question.]
Hardy: All right, now suppose that the numbers are fairly large, so that mating may be regarded as random, that the sexes are evenly distributed among the possible genotypes and that all are equally fertile. When mating happens, what will be the frequency of A and a among the gametes?
Yule: [Answers the question.]
Hardy: Just so. Now a little mathematics of the multiplication table type will show you what the next generation will be like. I should have expected the very simple point I am going to make to have been familiar to biologists. [Yule flinches and blushes at this remark.] Remember that if you want to know the probability of two independent events, you multiply their individual probabilities. Let's make a table on this napkin of all the possible crosses that could occur and the probability of each one. We'll use the numbers you deduced just a moment ago. [Napkins are attached to the script.]
Now... oh... give me another napkin. [Yule does so.] Here is a diagram of the kind of offspring that each cross will produce. And here...drat, I need another napkin. Get one off of that table over there, Yule. [Yule does so.]
Yule: This is embarrassing, Professor Hardy. The waiter is looking at us with unmixed contempt.
Hardy: Then perhaps the waiter understands the binomial expansion, Yule. Just ignore him and look at this napkin. We just write in these numbers like this and... sooo... for each 1 AA, there are 20,000 Aa and 100,000,000 aa in the next generation or 1 in every 100,020,001 individuals. That yields a total of 20,001 brachydactylous individuals for each 100,020,000. We've obeyed Mendel's principles as you explained them to me, and we don't have a 3:1 ratio. Nor will we ever as long as mating is at random. That is the most interesting thing. If you repeat this exercise for the next generation, you will again get the same ratio we got for this generation. Try it at home, Yule, when you can get your hands on some decent paper.
Yule: [Quietly, and obviously humbled.] Well, it makes sense to me. Is there a general rule here, or is this something that has to be worked out this way for each case?
Hardy: [Becoming gruffer and more irritated.] Yule, what do they teach people in the school you went to!!! As I mentioned rather early in this play, this is an example of the binomial expansion applied to a practical situation. If we let p = frequency of A and q = the frequency of a, the frequency of all the genes is, of course (p+q) = 1. (Am I going too fast for you, Yule?)
Yule: Come on, Professor Hardy, I'm not nearly so slow as you suggest!
Hardy: Sorry, Yule. I apologize for being snide. Just trying to keep up my image as an insensitive mathematician. Anyway, if these assumptions are made, and if we also assume a very large population with no differences in fertility, no migration in or out of the population, no mutation from A to a or from a to A, a population, in short, with random mating and no external forces acting on it, (quite a sentence, wouldn't you say, and not done yet) then... the proportion of genotypes generation after generation is (p + q)2 = 1.
The expanded form of this equation is p2+2pq+q2 = 1. The first term will be the frequency of AA, the second of Aa and the third of aa. I can prove this for you algebraically if you'd like.
Yule: Not now, Professor Hardy. It's getting late, and I need to get some rest so I can go around retracting my statement all day tomorrow at the meeting.
Hardy: [Rising from his chair.] Just as well, Professor Yule. I agreed to call my friend Weinberg in Germany this evening. Some sort of discovery he says he's made and wants to talk to me about. All this activity is making me begin to lose my equilibrium.
|Types of one parent|
|AA (1)||Aa (0)||aa (10,000)|
|Types of the other parent||AA (1)||1||0||10,000|
|Types of one parent|
|Types of the other parent||AA||all AA||0.5 AA||all Aa|
|Aa||0.5 AA||0.25 AA||0.5 Aa|
|0.5 Aa||0.5 Aa||0.5 aa|
|aa||all Aa||0.5 Aa||all aa|
|Types of one parent|
|Types of the other parent||AA||Aa||aa|
|AA||all AA (1)||0.5 AA (0)||all Aa|
|0.5 Aa (0)||(10,000)|
|Aa||0.5 AA (0)||0.25 AA (0)||0.5 Aa|
|0.5 Aa (0)||0.5 Aa (0)||0.5 aa (0)|
|0.25 aa (0)|
|0.5 Aa (0)||0.5 Aa (0)||0.5 aa (0)|
|aa||all Aa||0.5 Aa (0)||all aa|
|(10,000)||0.5 aa (0)||(100,000,000)|