Question

Suppose that a population consists of a fixed number, say, \(m\), of genes in any generation. Each gene is one of two possible genetic types. If exactly \(i\) (of the \(m\) ) genes of any generation are of type 1 , then the next generation will have \(j\) type 1 (and \(m-j\) type 2 ) genes with probability $$ \left(\begin{array}{c} m \\ j \end{array}\right)\left(\frac{i}{m}\right)^{j}\left(\frac{m-i}{m}\right)^{m-j}, \quad j=0,1, \ldots, m $$ Let \(X_{n}\) denote the number of type 1 genes in the \(n\) th generation, and assume that \(X_{0}=i\) (a) Find \(E\left[X_{n}\right]\). (b) What is the probability that eventually all the genes will be type \(1 ?\)

Short answer

(a) The expected number of type 1 genes in the nth generation is the same as the initial number of type 1 genes \(i\), so \(E[X_n] = i\). (b) Since the expected number of type 1 genes in each generation remains the same, the probability that eventually all the genes will be type 1 is 0.

Step-by-step solution

We are given that, if there are \(i\) genes of type 1 in a generation, the probability of having \(j\) genes of type 1 in the next generation is given by the following formula: \[ \left(\begin{array}{c} m \\\ j \end{array}\right)\left(\frac{i}{m}\right)^{j}\left(\frac{m-i}{m}\right)^{m-j}, \quad j=0,1, \ldots, m \] This is a binomial probability formula, which is reasonable as each gene can be either type 1 or type 2 with associated probabilities. #Step 2: Express the expected value in the nth generation#

The expected value of \(X_n\), denoted by \(E[X_n]\), is given by the weighted sum of the number of genes of type 1 in the nth generation, multiplied by their respective probabilities: \[ E[X_n] = \sum_{j = 0}^{m} j \cdot \left(\begin{array}{c} m \\\ j \end{array}\right)\left(\frac{i}{m}\right)^{j}\left(\frac{m-i}{m}\right)^{m-j} \] #Step 3: Simplify the expected value expression#

The expression can be simplified using the identity \(E[X_n] = m \cdot \frac{i}{m} = i\). This means that the expected number of type 1 genes in the nth generation is the same as the initial number of type 1 genes \(i\). Therefore, the answer to part (a) is: \(E[X_n] = i\) #Step 4: Examine the long-term behavior#

In order to determine the probability that eventually all genes will be type 1, we need to consider the long-term behavior of the system. Since the expected number of type 1 genes in each generation remains the same, the population does not tend to have more or fewer type 1 genes over time. Consequently, the probability that eventually all genes will be type 1 is 0. Answer: (a) \(E[X_n] = i\) (b) The probability that eventually all the genes will be type 1 is 0.

Key concepts

Genetic Drift

Genetic drift is a fundamental concept in population genetics which refers to random changes in the frequency of alleles (versions of a gene) within a population. This phenomenon is especially pronounced in small populations where the randomness of allele transmission can have a more significant impact. Unlike natural selection, which is a directional force pushing for traits that confer a survival advantage, genetic drift does not necessarily favor any particular allele. Instead, alleles can increase or decrease in frequency due solely to chance.

For example, consider a population with only two alleles, A and B. If allele A is passed down more frequently just by luck (say, in a small isolated group), it might become more common in the next generation, regardless of whether it provides any evolutionary advantage. Over time, these random shifts can lead to major changes in the genetic makeup of a population and can even result in the fixation of certain alleles, where one allele is present in all members of the population, thereby reducing genetic diversity.

Moreover, in scenarios where a population undergoes a bottleneck event—a dramatic reduction in size—it's possible for genetic drift to cause alleles to disappear entirely, potentially impacting the population's long-term survival and adaptability.

Expected Value of Binomial Distribution

The expected value of a binomial distribution is a measure of the central tendency or 'average' outcome we might expect from a binomial random process. A binomial distribution arises when we perform a series of independent trials, each with two possible outcomes (success or failure) and a consistent probability of success. In our genetic example, each gene being of type 1 or type 2 can be thought of as such a trial.

Mathematically, the expected value of a binomial distribution with number of trials, or sample size, denoted by 'n' and the probability of success 'p', is calculated as the product of these two factors, expressed as \( E[X] = n \cdot p \). In the context of our exercise, where 'i' type 1 genes are present initially, the expected number of type 1 genes in future generations remains constant at 'i' since each gene from the initial generation has a \(\frac{i}{m}\) chance of being type 1 in the next generation. This reflects the transmission of genetic traits from parents to offspring, where the average amount of genetic material inherited remains stable over time if no other evolutionary forces play a role.

Long-term Behavior in Probability Models

Analyzing long-term behavior in probability models is key to understanding the future state of a system, especially in processes influenced by random events like genetic drift. In binomial probability models, such as the one presented in the exercise for genetic types, we often assess whether the distribution of outcomes converges to a particular state or behavior over an extended number of trials or generations.

In our genetic drift scenario, even though the expected number of type 1 genes remains the same across generations, the actual count can fluctuate randomly around this expectation. However, probability models suggest that as the number of trials (or generations, in this context) grows indefinitely, the outcome tends to stabilize due to the Law of Large Numbers. This law states that the long-term average of results from repeated independent trials should be close to the expected value.

However, the possibility of all genes becoming type 1 involves fixation, an event that has a probability of 0 in an infinite model like ours. This is because, without a force pushing towards fixation, random fluctuations across generations don't typically result in every gene converging to the same type. Thus, unless influenced by other factors such as natural selection, small mutation rates, or non-random mating, the probability model predicts a maintenance of genetic variability without ever reaching fixation.