Transmission genetics is exemplified by Mendel’s rules, which describe how genetic information is transmitted from parents to offspring. Consider a monohybrid cross:
| \(A_1A_2\) \(\times\) \(A_1A_2\) | ||
| \(\downarrow\) | ||
| \(\frac{1}{4}A_1A_1\) | \(\frac{1}{2}A_1A_2\) | \(\frac{1}{4}A_2A_2\) |
Population genetics describes how genetic information is transmitted from a population of parents to a population of offspring. Consider, for example, the following data from the Est-3 locus of Zoarces viviparus:1
| Genotype of offspring | |||
| Maternal genotype | \(A_1A_1\) | \(A_1A_2\) | \(A_2A_2\) |
| \(A_1A_1\) | 305 | 516 | |
| \(A_1A_2\) | 459 | 1360 | 877 |
| \(A_2A_2\) | 877 | 1541 | |
This table describes, empirically, the relationship between the genotypes of mothers and the genotypes of their offspring. Using these data we can also make some inferences about the genotypes of the fathers in this population, even though we didn’t collect them.2
305 out of 821 male gametes that fertilized eggs from \(A_1A_1\) mothers carried the \(A_1\) allele (37%).
877 out of 2418 male gametes that fertilized eggs from \(A_2A_2\) mothers carried the \(A_1\) allele (36%).
How many of the 2,696 male gametes that fertilized eggs from \(A_1A_2\) mothers carried the \(A_1\) allele?
We don’t know the paternal genotypes or we wouldn’t be asking this question.
There is no way to tell which of the 1360 \(A_1A_2\) offspring received \(A_1\) from their mother and which from their father.
Regardless of what the genotype of the father is, half of the offspring of a heterozygous mother will be heterozygous.3
Heterozygous offspring of heterozygous mothers contain no information about the frequency of \(A_1\) among fathers, so we don’t bother to include them in our calculations.
How many of the 1336 homozygous progeny of heterozygous mothers received an \(A_1\) allele from their father?
459 out of 1336 (34%)
How many of the offspring where the paternal contribution can be identified received an \(A_1\) allele from their father?
(305 + 459 + 877) out of (305 + 459 + 877 + 516 + 877 + 1541) or 1641 out of 4575 (36%)
The above calculations tell us what’s happening for this particular data set, but those of you who know me know that there has to be a little math coming so that we can describe the situation more generally.4 Here’s the notation:
| Genotype | Number | Sex |
|---|---|---|
| \(A_1A_1\) | \(F_{11}\) | female |
| \(A_1A_2\) | \(F_{12}\) | female |
| \(A_2A_2\) | \(F_{22}\) | female |
| \(A_1A_1\) | \(M_{11}\) | male |
| \(A_1A_2\) | \(M_{12}\) | male |
| \(A_2A_2\) | \(M_{22}\) | male |
Using that notation,
\[\begin{array}{cc} p_f = \frac{2F_{11}+F_{12}}{2F_{11}+2F_{12}+2F_{22}} & q_f = \frac{2F_{22}+F_{12}}{2F_{11}+2F_{12}+2F_{22}} \\ & \\ p_m = \frac{2M_{11}+M_{12}}{2M_{11}+2M_{12}+2M_{22}} & q_m = \frac{2M_{22}+M_{12}}{2M_{11}+2M_{12}+2M_{22}} \quad , \end{array}\] where \(p_f\) is the frequency of \(A_1\) in mothers and \(p_m\) is the frequency of \(A_1\) in fathers.5
Since every individual in the population must have one father and one mother, the frequency of \(A_1\) among offspring is the same in both sexes, namely \[p = \frac{1}{2}(p_f + p_m) \quad ,\] assuming that all matings have the same average fecundity and that the locus we’re studying is autosomal.6
Question: Why do those assumptions matter?
Answer: If \(p_f = p_m\), then the allele frequency among offspring is equal to the allele frequency in their parents, i.e., the allele frequency doesn’t change from one generation to the next. This might be considered the First Law of Population Genetics: If no forces act to change allele frequencies between zygote formation and breeding, allele frequencies will not change.
This is an example of what philosophers call a zero force law. Zero force laws play a very important role in scientific theories, because we can’t begin to understand what a force does until we understand what would happen in the absence of any forces. Consider Newton’s famous dictum:
An object in motion tends to remain in motion in a straight line. An object at rest tends to remain at rest.
or (as you may remember from introductory physics)7 \[F = ma \quad.\] If we observe an object accelerating, we can immediately infer that a force is acting on it. Not only that, we can also infer something about the magnitude of the force. However, if an object is not accelerating we cannot conclude that no forces are acting. It might be that opposing forces act on the object in such a way that the resultant is no net force. Acceleration is a sufficient condition to infer that force is operating on an object, but it is not necessary.
What we might call the “First Law of Population Genetics” is analogous to Newton’s First Law of Motion:
If all genotypes at a particular locus have the same average fecundity and the same average chance of being included in the breeding population, allele frequencies in the population will remain constant from one generation to the next.
For the rest of the semester we’ll be learning about the processes that cause allele frequencies to change and learning how to infer the properties of those processes from the changes that they induce. But you must always remember that while we can infer that some evolutionary process is happening if allele frequencies change from one generation to the next, we cannot infer the absence of an evolutionary process from a lack of allele frequency change.8
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