In a very large population, random genetic drift due to number regulation plays a negligible role in the lifecycle. The frequency of the disease-causing allele can then be treated as behaving deterministically, and as far as the frequency is concerned, the number of adults in the population is effectively infinite. In what follows, we shall use the shorthand infinite population to describe a population of effectively infinite size.
The function F x has the interpretation as the deterministic evolutionary force that acts on the frequency of the disease-causing allele the a allele in a very large population, when the frequency has the value x.
If, in a given generation, this force is non-zero then the frequency will be different in the following generation. We give the full form of F x in Equation 5 , and later the full form of the corresponding equilibrium frequency in Equation 9 , since these are quite sensitive to the precise values of the parameters h and u. A particular example of Equations 4 and 5 is for a recessive lethal allele, when mutation is neglected. However, when considering the evolution of lethal mutations we need to modify the above Wright-Fisher model so it incorporates strong selection.
This results in a modified Wright-Fisher model. The model we shall present is designed to be appropriate for a modern, post-industrial human population, where fertility is approximately two offspring per couple Hamilton et al. Details of the model are given in Part B of the Supplementary Material. The conventional Wright-Fisher model is based on the strong assumption that all randomness arises solely in the non-selective thinning of the population to the census population size 1.
This means, in particular, that selection is treated as a deterministic process, amounting to the population being effectively infinite during the time that selection occurs within the lifecycle the zygotic stage. For humans in modern post-industrial populations, the number of offspring produced is typically little more than that required to replace the population Hamilton et al. Thus, the number of zygotes produced is similar in number to the number of adults i.
To transparently avoid any possible consequences of an effectively infinite number of zygotes, we have used an explicitly probabilistic treatment of selection, where we have strongly limited the number of zygotes produced. However, as we show in Part B of the Supplementary Material , for most practical purposes there is a negligible difference between such a model, and the model where selection is treated as acting deterministically.
The main difference that arises because selection is not weak, but strong because of lethality of one genotype , is that selection cannot be directly approximated as acting at the level of alleles which is possible when selection is weak.
Rather, we find see Part B of the Supplementary Material for details that the frequency of the lethal genotype obeys the stochastic equation. We shall often refer to Equation 7 as the Wright-Fisher model describing a lethal genotype. A comparison of the standard weak-selection Wright-Fisher model Equation 6 , and the Wright-Fisher model describing a lethal genotype Equation 7 , indicates differences in the placement of factors of 2 in both arguments of the binomial random numbers present [the Bin n, p ].
Thus, the quantity F X appearing in Equation 7 continues to have the interpretation as the deterministic evolutionary force acting in an infinite population. We wish to point out one general implication of lethality within the context of a biallelic locus. This is that lethality of one homozygote generally constrains the frequency of the lethal allele in adults, such that independent of any model used, and independent of the size of the population, the frequency of the lethal allele can never exceed 1 2.
To see this we use a simple gene counting argument, as follows. We note that the lethal a allele only appears in heterozygote adults, but the wild type A allele appears in both viable homozygote adults and heterozygote adults. A particular implication of Equation 8 is that irrespective of the fitness of the heterozygote, it is impossible for there to be a selective sweep of the lethal a allele to fixation, and at most the frequency of this allele can only reach 1 2.
We note that the Wright-Fisher model for a lethal genotype Equation 7 , involves a binomial random number of the form Bin N, p , corresponding to the random number of successes on N independent trials. The N independence of this result indicates that independent of the population size finite or infinite the constraint of Equation 8 , on the frequency of the lethal allele in adults, will apply.
It is necessary and reassuring to see this constraint directly manifested in the Wright-Fisher model that was constructed for the problem at hand. Generally, any valid model describing a lethal allele must exhibit such a constraint on the frequency of the lethal allele. Equation 4 describes an infinite population.
In Part A of the Supplementary Material we give exact and approximate results for the equilibrium frequency following from Equation 4. In particular the equilibrium frequency has the exact form.
In the case of a fully recessive a allele i. This last result can be determined from Equation 3. The above results indicate that for some degree of recessiveness i. Figure 2. This is a feature that is apparent in Figure 2. Given a mutation rate of e.
Let us now consider a very large effectively infinite population with mutation rate u. This elevated fitness value persists until generation t f. This assumption allows us to use some of the approximate results we have presented above, and thereby gain some analytical insights.
Figure 3 illustrates that a relatively small discontinuous change in the heterozygote fitness in the figure from 1 to 1. Figure 3. This figure contains plots of the logarithm of the mean frequency of the disease-causing allele, log 10 E [ X t ] , against the time, t.
For a large effectively infinite population, there are negligible deviations of the frequency from its expected value and E [ X t ] then coincides with the frequency itself, X t. The infinite population results are given by the black curves.
Finite population results are given by colored curves. The figure illustrates transient behavior that the frequency can exhibit in populations with different mutation rates, u , and different population sizes, N when the following are assumed. The figure was obtained using Equation 4 , for an effectively infinite population, and from the Wright-Fisher model describing a lethal genotype, based on Equation 7.
With regard to Drosophila eye color, when the P 1 male expresses the white-eye phenotype and the female is homozygous red-eyed, all members of the F 1 generation exhibit red eyes. Now, consider a cross between a homozygous white-eyed female and a male with red eyes.
Punnett square analysis of Drosophila eye color : Punnett square analysis is used to determine the ratio of offspring from a cross between a red-eyed male fruit fly X W Y and a white-eyed female fruit fly X w X w. Sex-linkage studies provided the fundamentals for understanding X-linked recessive disorders in humans, which include red-green color blindness and Types A and B hemophilia. Because human males need to inherit only one recessive mutant X allele to be affected, X-linked disorders are disproportionately observed in males.
Females must inherit recessive X-linked alleles from both of their parents in order to express the trait. Color perception in different types of color blindness : In this chart you can see what people with different types of color blindness can see versus the normal color vision line at top. When they inherit one recessive X-linked mutant allele and one dominant X-linked wild-type allele, they are carriers of the trait and are typically unaffected. Carrier females can manifest mild forms of the trait due to the inactivation of the dominant allele located on one of the X chromosomes.
However, female carriers can contribute the trait to their sons, resulting in the son exhibiting the trait, or they can contribute the recessive allele to their daughters, resulting in the daughters being carriers of the trait. Although some Y-linked recessive disorders exist, typically they are associated with infertility in males and are, therefore, not transmitted to subsequent generations.
Inheritance of a recessive X-linked disorder : The son of a woman who is a carrier of a recessive X-linked disorder will have a 50 percent chance of being affected.
A daughter will not be affected, but she will have a 50 percent chance of being a carrier like her mother. Privacy Policy. Skip to main content.
Reproduction, Chromosomes, and Meiosis. Search for:. Patterns of Inheritance. When they do, they have a 1 in 2 chance of passing on the allele to the next generation, keeping the allele in the gene pool achondroplasiacs are never homozygous: two alleles is a fatal combination.
And as a result, this allele persists in the population. If you feel that you understand the concepts in this tutorial, take this brief quiz. If not, carefully re-read the material above, and then take the quiz. Which square represents the genotype of an individual who is a carrier for Tay-Sachs? Which square shows the genotype of an individual who would have Tay-Sachs disease?
Which of the following individuals MUST be a carrier for the lethal allele? Which of the following individuals inherited two copies of the lethal allele?
The prevalence of the disease will then reflect the rate at which such pathogenic mutations occur together with the rate at which individuals carrying them are eliminated before they have off-spring. In this case, the allele will be subject to positive selection, that is, it will increase in frequency.
This increase will continue until the number of individuals carrying the allele reaches a point where the number of offspring with two copies of the mutant pathogenic allele becomes significant. These homozygous individuals and the alleles they carry then become subject to strong negative selection.
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