Question

How many unique amino acid sequences are possible for a tripeptide containing only the amino acids gly, ala, and \(c y s,\) with each amino acid occurring only once in each molecule?

Short answer

There are 6 unique amino acid sequences possible for a tripeptide containing only glycine (gly), alanine (ala), and cysteine (cys), with each amino acid occurring only once in each molecule. This is calculated using the permutation formula \(P(n,r) = \frac{n!}{(n-r)!}\), where \(n = 3\) and \(r = 3\).

Step-by-step solution

Identify the number of elements to arrange

First, we need to identify the number of elements we have to arrange. In this case, we have 3 unique amino acids: glycine (gly), alanine (ala), and cysteine (cys).

Apply the permutation formula

The permutation formula is given by: $$P(n, r) = \frac{n!}{(n-r)!}$$ where \(P(n, r)\) is the number of ways to arrange \(n\) objects in \(r\) positions without repetition. In this case, we have \(n = 3\) unique amino acids, and we want to arrange them in \(r = 3\) positions. Therefore, we can plug the values into the formula: $$P(3, 3) = \frac{3!}{(3-3)!}$$

Calculate the factorial values and solve

Now, we can calculate the factorial values of the numbers: $$3! = 3 \times 2 \times 1 = 6$$ $$(3 - 3)! = 0! = 1$$ Plug them back into the formula: $$P(3, 3) = \frac{6}{1}$$

Solve for the number of unique sequences

Now, we can solve the equation: $$P(3, 3) = \frac{6}{1} = 6$$ Therefore, there are 6 unique amino acid sequences possible for a tripeptide containing only the amino acids glycine (gly), alanine (ala), and cysteine (cys), with each amino acid occurring only once in each molecule.

Key concepts

Amino Acid Sequence

A sequence of amino acids refers to the specific order in which amino acids are linked together to form a protein or peptide. Each amino acid is like a building block, and when you put them in a certain order, they create a functional molecule. Proteins are typically long chains of these amino acids, while peptides are shorter chains. For instance, a tripeptide is a kind of peptide that has three amino acids in its chain. In this exercise, we're exploring sequences made only of three specific amino acids: glycine (gly), alanine (ala), and cysteine (cys).

Understanding amino acid sequences is crucial for studying protein functions and structures because their sequence determines how they fold and what roles they play in biological processes. Even though the sequence might appear straightforward, changing the order of amino acids can drastically alter a protein’s characteristics and functionality.

Tripeptide

A tripeptide is a short peptide chain made up of three amino acids linked together by peptide bonds. Think of it as a short protein sequence. The sequence in which the amino acids are arranged in a tripeptide affects its properties and characteristics. Given three amino acids, like the ones in this problem — glycine (gly), alanine (ala), and cysteine (cys) — there are several possible ways to arrange them.

In the context of biochemistry and molecular biology, tripeptides are significant because they can model how larger proteins might behave. Although a tripeptide is much simpler than a full protein, studying them gives insight into more complex systems. These sequences can be constructed and analyzed to understand their roles in metabolic pathways and how they might interact with other molecules.

Factorial Calculation

Factorial calculation is a fundamental concept in mathematics, particularly when it comes to permutations. The factorial of a number, represented as "n!", is the product of all whole numbers descending from "n" to 1. For instance, the factorial of 3 (written as 3!) is 3 x 2 x 1, which equals 6. This calculation helps determine the number of ways to arrange a set of items, which is a cornerstone in understanding permutations.

In our exercise, we're interested in determining the possible permutations of three distinct amino acids in a tripeptide. The process involves calculating the factorial of the total number of amino acids, which gives us the number of unique arrangements possible. Using the permutations formula,

• You identify the total items (in our case, the amino acids: gly, ala, cys).
• Apply the formula for permutations, which includes the factorial of these items: \(P(n, r) = \frac{n!}{(n-r)!}\).
• Calculate the factorial to find possible sequences.

By understanding how to compute these factorials, students can solve many combinatorial problems, applying this method to other scenarios in science and mathematics.