Problem 89 - Calculus for Biology and Medicine

Question

The number of amino acid substitutions in a given sequence is Poisson distributed with mean 3 . What is the probability of at least two substitutions?

Step-by-Step Solution

Answer

The probability of at least two substitutions is approximately 0.8008.

Step 1: Understand the Poisson Distribution

In this problem, the number of amino acid substitutions follows a Poisson distribution. The Poisson distribution is used for counting events that occur randomly within a given interval, here with a mean (or lambda, \( \lambda \)) of 3.

Step 2: Define the Probability that We Need to Calculate

We need to find the probability of having at least two substitutions, which translates into finding \( P(X \geq 2) \). Since it is easier to calculate probabilities for fewer events, we'll use the complement rule: \( P(X \geq 2) = 1 - P(X < 2) \).

Step 3: Calculate \( P(X < 2) \)

To find \( P(X < 2) \), we calculate the sum of the probabilities for 0 and 1 substitutions: \( P(X=0) + P(X=1) \). The formulas for these in a Poisson distribution are given by \( P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!} \).

Step 4: Compute \( P(X=0) \)

Substitute \( \lambda = 3 \) and \( k = 0 \) into the Poisson formula: \( P(X=0) = \frac{e^{-3} \cdot 3^0}{0!} = e^{-3} \).

Step 5: Compute \( P(X=1) \)

Substitute \( \lambda = 3 \) and \( k = 1 \) into the Poisson formula: \( P(X=1) = \frac{e^{-3} \cdot 3^1}{1!} = 3e^{-3} \).

Step 6: Sum the Probabilities and Calculate \( P(X \geq 2) \)

Calculate \( P(X < 2) = P(X=0) + P(X=1) = e^{-3} + 3e^{-3} = 4e^{-3} \). Then, \( P(X \geq 2) = 1 - P(X < 2) = 1 - 4e^{-3} \).

Step 7: Calculate Numerical Values

Plug in the numerical value of \( e^{-3} \approx 0.0498 \): \( P(X < 2) = 4 \times 0.0498 = 0.1992 \). Therefore, \( P(X \geq 2) = 1 - 0.1992 = 0.8008 \).

Key Concepts

amino acid substitutions

Amino acid substitutions refer to the process where one amino acid in a protein sequence is replaced by another. This is a natural process that occurs due to mutations in the DNA coding for the protein. While it's a simple swap at the molecular level, its impact on the protein's structure and function can be significant.

In the context of genetics and protein studies, these substitutions can lead to slight changes or completely alter a protein's function. The study of these substitutions can help in understanding diseases, evolution, and protein engineering.

• In nature, mutations might be neutral; however, some can be harmful or beneficial, influencing an organism's fitness.
• Researchers might look at the frequency of substitutions to understand patterns and evolutionary pressures.

For instance, if amino acids that are crucial for a protein's function are frequently substituted, it could indicate an adaptation or a naturally occurring resistance to certain conditions.

probability calculation

Probability calculation in this context revolves around determining the likelihood of observing a certain number of events—in this case, amino acid substitutions—using Poisson distribution. The Poisson distribution is ideal for modeling the number of events happening in a fixed interval of time or space when these events are rare and occur independently of each other.

To calculate the probability of a specific number of substitutions, you use the formula:
\[ P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!} \]
where:

• \( \lambda \) is the average number of substitutions, and in our scenario, it's given as 3.
• \( k \) is the actual number of events (substitutions) you want to find the probability for.
• \( e \) is the base of natural logarithms, approximately equal to 2.718.

The probability calculation becomes incredibly useful in fields like bioinformatics and genetics, assisting in predicting protein changes due to mutations.

complement rule

The complement rule in probability is a simple yet powerful concept. It helps in calculating the probability of an event by using its opposite. Instead of directly calculating the probability of the desired event, one calculates the probability of the opposite event and subtracts it from 1. This often simplifies the problem.

In the context of Poisson distributions and our problem, we wish to calculate the probability of having at least two substitutions, denoted as \( P(X \geq 2) \). Direct calculation here could get cumbersome, so the complement rule is applied:

• The event \( X \geq 2 \) is the complement of \( X < 2 \).
• The calculation thus becomes \( P(X \geq 2) = 1 - P(X < 2) \).

This way, you only need to compute \( P(X=0) \) and \( P(X=1) \), then subtract these probabilities from 1. It saves effort and reduces the chance of calculation errors, making it a practical tool in statistical analyses.