Question

Assuming that an amino acid sequence is 250 amino acids long, how many different molecules, each with a unique sequence, could be formed?

Short answer

Answer: Approximately 2.85 x 10^325 different molecules with unique sequences can be formed.

Step-by-step solution

Identify the type of permutation problem

In this exercise, we are dealing with a permutation problem with repetitions. This is because any of the 20 different amino acids can occupy each position in the sequence without any restrictions on repeating the same amino acid at different positions.

Apply the formula for permutations with repetitions

The formula for calculating the number of permutations with repetitions is given by: n^r, where n is the number of choices (in this case, 20 amino acids) and r is the number of positions (in this case, 250 positions).

Calculate the number of unique sequences

Using the formula from step 2, we can calculate the number of unique sequences by raising the number of amino acids (20) to the power of the number of positions (250): Number of unique sequences = 20 ^ 250

Simplify the result and express it in scientific notation

The value of 20^250 is large, so it is more convenient to express it in scientific notation. Number of unique sequences ≈ 2.85 x 10^325

Key concepts

Amino Acid Sequence Variation

Variation in amino acid sequences is a cornerstone of protein diversity in living organisms. A protein's function is determined by its unique sequence of amino acids, which are the building blocks of proteins. Each protein sequence can fold into a three-dimensional structure that defines its activity within the cell.

Imagine a string of beads where each bead can be one of 20 different colors, representing the 20 standard amino acids found in nature. Now, if we have a chain that's 250 beads long, the variation possibilities become astronomical. Just like the beads on a string, each position in a protein's amino acid sequence can be occupied by any of these 20 amino acids. Consequently, a protein 250 amino acids in length can give rise to an immense number of possible sequence combinations, allowing for the vast diversity of proteins found within living systems.

This variation is not just theoretical; it's the basis for variations in traits, adaptations to environments, and evolutionary processes. When mutations occur in a gene, they can change one amino acid for another, potentially altering the protein's function. This is how new variations and ultimately new traits and functions can arise in a population.

Permutation Problem with Repetitions

When dealing with permutations in genetics, we often encounter the 'permutation problem with repetitions.' This type of problem arises when the order of items matters, but the items can be repeated in various positions throughout the sequence.

In our exercise, the sequence of amino acids is akin to a lock combination where the same amino acid can repeat, just as a number can repeat in a combination lock. This repetitiveness must be taken into account when computing the total number of permutations. Unlike combinations without repetition where items are unique, repetitions allow for a far greater number of possible outcomes.

An important thing to consider is that with repetitions, the problem's complexity increases with the number of positions to be filled, as each additional position multiplies the number of possible permutations by the number of choices. This concept is neatly encapsulated in the formula for permutations with repetition: \( n^r \), where \( n \) is the number of items to choose from and \( r \) is the number of positions.

Calculating Permutations in Genetics

Genetics often requires the calculation of permutations to predict the outcome of genetic combinations. As shown in our textbook problem, to find the number of distinct sequences that 250 amino acids can form, we use the permutation formula with repetitions: \( n^r \). This formula is also used in genetics to determine the number of possible genotypes or phenotypes that can arise from a set of genes.

In our case, \( n \) equals 20 (the number of common amino acids), and \( r \) equals 250 (the length of the sequenced chain of amino acids). Upon applying the formula, the result is \( 20^{250} \), which represents a number so vast that it is typically expressed in scientific notation to make it more comprehensible, in this case, approximately \( 2.85 \times 10^{325} \).

This calculation's magnitude highlights the incredible diversity and complexity at the molecular level in biology. By understanding these permutations, we can appreciate not only how traits are passed on but also how the body can generate a seemingly infinite variety of proteins, each with specific functions. This mathematical approach is a fundamental part of modern genetics, bioinformatics, and the study of biological diversity.