Question

The probability of a sequence (given a composition). A scientist has constructed a secret peptide to carry a message. You know only the composition of the peptide, which is six amino acids long. It contains one serine \(\mathbf{S}\), one threonine \(\mathbf{T}\), one cysteine \(\mathbf{C}\), one arginine \(\mathbf{R}\), and two glutamates \(\mathbf{E}\). What is the probability that the sequence SECRET will occur by chance?

Short answer

\( \frac{1}{360} \)

Step-by-step solution

Determine Total Number of Arrangements

Calculate the total number of possible arrangements of the peptide sequence. The peptide has 6 amino acids where S, T, C, R each appear once and E appears twice. Use the formula for permutations of multiset: P = \frac{n!}{n_1!n_2!n_3!...} where n is the total number of items and each n_i is the factorial of the frequency of each distinct item. For this problem: \(P = \frac{6!}{1!1!1!1!2!}\).

Calculate the Factorials

Calculate the factorials needed: \(6! = 7201! = 1\) \(2! = 2\).

Divide the Factorials

Calculate the probability by dividing the total arrangements by the product of the factorials of each amino acid's frequency: \(P = \frac{720}{1 \cdot 1 \cdot 1 \cdot 1 \cdot 2} = \frac{720}{2} = 360\). So, there are 360 possible unique sequences.

Determine Probability of SECRET

Since each sequence is equally likely, the probability of any single arrangement being 'SECRET' is \(P_{SECRET} = \frac{1}{360}\).

Key concepts

Amino Acid Composition

Amino acids are the building blocks of proteins, and their specific order in a peptide sequence determines its properties and functions. In this problem, our peptide sequence is composed of six amino acids: one serine (S), one threonine (T), one cysteine (C), one arginine (R), and two glutamates (E). Understanding the composition is crucial because it sets the stage for figuring out all possible ways these can be arranged. Knowing the composition also helps identify the factorial constraints we'll use in calculations.

Permutations of Multiset

Permutations refer to all possible arrangements of a set of objects. However, when the set contains duplicates, we use multiset permutations, which account for these repeated items. The formula for calculating permutations of a multiset is:

\[ P = \frac{n!}{n_1!n_2!n_3!...} \]
Here, \(n\) is the total number of items, and each \(n_i\) is the factorial of the frequency of each distinct item. In our peptide problem, \( n = 6 \) (total amino acids) and the frequencies of (S, T, C, R, E, E) are respectively 1, 1, 1, 1, 2.

Factorial Calculation

Factorials are numbers multiplied by each of their positive integers less than themselves. For example, \(6!\) (6 factorial) is:
\[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \]
We also need \(1!\) and \(2!\):
\[ 1! = 1 \] \[ 2! = 2 \times 1 = 2 \] Combining these into our formula for permutations of the multiset from the peptide sequence:

\[ P = \frac{720}{1 \times 1 \times 1 \times 1 \times 2} = \frac{720}{2} = 360 \] There are 360 unique possible arrangements of the amino acids.

Peptide Probability

To find the probability of the sequence 'SECRET' occurring by chance, we consider that any of the 360 unique sequences is equally likely. The probability of 'SECRET' is therefore:
\[ P_{SECRET} = \frac{1}{360} \]

This fraction indicates that out of 360 possible sequences, only one will be 'SECRET'. Therefore, the probability that the peptide sequence SECRET will occur by chance is approximately 0.00278.