Question

How many unique amino acid sequences are possible for a tripeptide containing only the amino acids gly, ala, and cys, with each amino acid occurring only once in each molecule?

Short answer

There are a total of 6 unique amino acid sequences possible for the given tripeptide using gly, ala, and cys without any repetition in each sequence.

Step-by-step solution

Determine the attributes of the problem

There are 3 distinct amino acids being used in creating the tripeptide and each amino acid can be used only once in the sequence. This results in a 3-element arrangement.

Use the permutation formula to calculate arrangements

In order to find the number of unique amino acids sequences, we should calculate the number of permutations of these 3 amino acids. The formula for permutation when given elements and the same quantity of slots is: \[P(n, r) = n! / (n - r)!\] In this case, since we have 3 amino acids and 3 slots to fill, both n and r are equal to 3. Therefore, the permutation formula becomes: \[P(3, 3) = 3! / (3 - 3)!\]

Calculate factorials

Calculate the factorials for the values in the formula: \[3! = 3 × 2 × 1 = 6\] \[(3 - 3)! = 0! = 1\]

Solve for the number of permutations

Insert the factorials into the permutation formula and solve for the number of permutations: \[P(3, 3) = 6 / 1 = 6\] There are a total of 6 unique amino acid sequences possible for the given tripeptide using gly, ala, and cys without any repetition in each sequence.

Key concepts

Understanding Tripeptides

A tripeptide is a type of peptide that is composed of three distinct amino acids linked together in a specific sequence. In the context of amino acid sequences, each tripeptide could have a different function or property depending on the order of its amino acids.
Tripeptides are the simplest form in terms of sequence length, compared to longer chains like polypeptides and proteins.

• Each position in the tripeptide is filled by one of the available amino acids.
• The order in which these amino acids are arranged is crucial towards the function of the tripeptide.

In our problem, we're focusing on a specific tripeptide formed from glycine (Gly), alanine (Ala), and cysteine (Cys). We need to understand how their arrangement can form different unique sequences.

Exploring Permutations

Permutations refer to arrangements where the order of elements matters. For amino acid sequences, the sequence or order will make a different permutation of the tripeptide.
In permutations, changing the sequence essentially creates a new unique arrangement. This is why permutations are key in calculating how many distinct sequences can be made out of the amino acids:

• For three amino acids, we have a total of 3 positions that need filling.
• Each different arrangement of these amino acids leads to a distinct sequence.

Permutation calculations become essential when determining possible unique sequences. The number of distinct permutations will tell us how many unique arrangements are possible, considering each arrangement as unique due to the order of amino acids.

The Role of Factorial Calculation

Factorial calculations help in determining how many permutations exist for a set number of items, where each item is placed once in each possible position.
The factorial of a number, denoted by an exclamation mark (!), is the product of all positive integers up to that number:

• The factorial of 3 (3!) is calculated as: 3 × 2 × 1 = 6.
• Factorials provide us with the total number of ways to arrange the entire set of elements.

The role of factorial in permutation boils down to counting the total sequences possible when each amino acid in the tripeptide occupies one unique position. This is why calculating 3! helped us determine the six unique arrangements possible for our amino acids Gly, Ala, and Cys.