Question

How many different tetrapeptides can be made under the following conditions? (a) The tetrapeptide contains one unit each of Asp, Glu, Pro, and Phe. (b) All 20 amino acids can be used, but each only once.

Short answer

Answer: (a) 24 different tetrapeptides can be made using one unit each of Asp, Glu, Pro, and Phe. (b) 116,280 different tetrapeptides can be made using all 20 amino acids with each one used only once.

Step-by-step solution

Determine the number of permutations for part (a) given specific amino acids

Since we are given four specific amino acids, we need to find the number of different ways these amino acids can be combined into a tetrapeptide. To do this, we find the permutation of four objects: Number of permutations = 4! = 4 × 3 × 2 × 1 = 24

Calculate the answer for part (a)

The number of different tetrapeptides containing one unit each of Asp, Glu, Pro, and Phe is 24.

Determine the number of permutations for part (b) given a choice of 20 amino acids

For part (b), we have 20 different amino acids to choose from, and each can only be used once. The situation is similar to part (a), but now we need to use the counting principle for permutations. We will find the permutation for selecting four objects out of 20 without repetition. Number of permutations = P(20, 4) = 20! / (20-4)! = 20! / 16! = 20 × 19 × 18 × 17

Calculate the answer for part (b)

The number of different tetrapeptides that can be formed using all 20 amino acids but using each one only once is: 20 × 19 × 18 × 17 = 116,280 Therefore, the answers are: (a) 24 different tetrapeptides can be made with the given conditions. (b) 116,280 different tetrapeptides can be made with the given conditions.

Key concepts

Amino Acids

Amino acids are the building blocks of proteins, playing a crucial role in virtually all processes within organisms. There are twenty standard amino acids, each with a unique side chain, which determine their distinct properties. These amino acids combine in varying sequences to form polypeptides and proteins, dictating their structure and function.

For example, in the given exercise, Asp (Aspartic acid), Glu (Glutamic acid), Pro (Proline), and Phe (Phenylalanine) are specific amino acids, each contributing a unique characteristic to the tetrapeptide structure. Understanding each amino acid's properties is essential for grasping how they affect peptide structure and function.

Permutation Calculation

Permutation calculation is a mathematical concept used to determine the number of possible arrangements in a set when the order of arrangements matters. When calculating permutations, each element's position in an arrangement is crucial, as changing the order creates a new permutation.

To calculate permutations for a given number of items, you can use the factorial operation, denoted by an exclamation mark (!). For instance, as demonstrated in the solution, a tetrapeptide containing four specific amino acids has 4 factorial (\(4!\)), or 24, different permutations. When calculating permutations from a larger pool, such as with 20 amino acids, one must use the permutation formula: \( P(n, k) = \frac{n!}{(n-k)!} \), where 'n' is the total number of objects to choose from, and 'k' is the number of objects to arrange.

Protein Structure

The structure of proteins is complex and organized into several levels. Primarily, the sequence of amino acids determines the protein's primary structure, which then folds into secondary structures like alpha helices and beta sheets. These further assemble into a three-dimensional tertiary structure, sometimes pairing up to form a quaternary structure.

Even a simple tetrapeptide, which is a sequence of just four amino acids, contributes to a larger protein's properties and function. The specific sequence of amino acids determines how the peptide will interact with its environment and fold into its functional form. Thus, the different permutations of a tetrapeptide will each have unique physical properties, influencing the broader protein structure and its role within a biological system.