Question

The probabilities of identical sequences of amino acids. You are comparing protein amino acid sequences for homology. You have a 20-letter alphabet (20 different amino acids). Each sequence is a string \(n\) letters in length. You have one test sequence and \(s\) different data base sequences. You may find any one of the 20 different amino acids at any position in the sequence, independent of what you find at any other position. Let \(p\) represent the probability that there will be a 'match' at a given position in the two sequences. (a) In terms of \(s, p\), and \(n\), how many of the \(s\) sequences will be perfect matches (identical residues at every position)? (b) How many of the \(s\) comparisons (of the test sequence against each database sequence) will have exactly one mismatch at any position in the sequences?

Short answer

(a) \(s \times p^n\). (b) \(s \times (1-p) \times p^{n-1} \times n\).

Step-by-step solution

- Calculate the probability of a perfect match

If two sequences are to be identical at every position, the probability of matching at a single position is given by \(p\). Therefore, the probability of a perfect match for a sequence of length \(n\) is \(p^n\).

- Determine the expected number of perfect matches

Given \(s\) sequences in the database, the expected number of sequences that perfectly match with the test sequence is the total number of sequences times the probability of a perfect match. This can be represented as \(E_{perfect} = s \times p^n\).

- Calculate the probability of one mismatch

The probability of a mismatch at any position is \(1 - p\). For exactly one mismatch in a sequence of length \(n\), we want one position to be a mismatch \(1 - p\) and the other \(n-1\) positions to match \(p\). There are \(n\) possible positions where this mismatch can occur, thus the total probability is \((1-p) \times p^{n-1} \times n\).

- Determine the expected number of one-mismatch comparisons

The number of sequences out of \(s\) that will have exactly one mismatch is the total number of sequences times the probability of having exactly one mismatch. This can be represented as \(E_{one_{mismatch}} = s \times (1-p) \times p^{n-1} \times n\).

Key concepts

Probability Calculation

When dealing with protein sequence homology, understanding probability is crucial. In this context, probability helps us determine the likelihood of amino acids matching across different positions in sequences.

For example, let's calculate the probability of perfect matches. If the probability of an amino acid match at a single position is denoted by p, and n is the sequence length, the chance that two sequences perfectly match at all positions is given by \( p^n \).

Furthermore, to find out how many of the database sequences would perfectly match a test sequence, multiply the number of sequences, s, by the probability of a perfect match, represented as \( s \times p^n \).

We can also calculate the likelihood of having exactly one mismatch in the sequences. The probability of a mismatch at any position is thus \(1 - p\). For one mismatch in a sequence, the rest of the positions should match, giving us \( (1 - p) \times p^{n-1} \times n \). Multiply this probability by the number of sequences, s, to find how many sequences are expected to have exactly one mismatch: \( s \times (1 - p) \times p^{n-1} \times n \).

By mastering these probability calculations, you gain insight into how often specific sequence patterns might appear in your protein homology studies.

Amino Acid Sequences

Amino acid sequences are the building blocks of proteins and are key to understanding protein structure and function. Each sequence is composed of a series of amino acids linked together in a specific order.

With 20 different amino acids, each position in the sequence can be one of these 20 amino acids. Therefore, when comparing sequences, you are essentially matching these letters position by position.

Understanding the probability of matching involves considering each amino acid independently. For instance, if you want to compare a test sequence against multiple database sequences, each position's match is calculated independently of the others. This means probabilities can be multiplied across positions to determine total probabilities for sequences.

Recognizing patterns in these sequences and knowing how frequent certain amino acid matches can be is essential for research in fields like bioinformatics, genetics, and molecular biology. Knowledge of amino acid sequences allows scientists to predict protein function and identify similarities between different proteins or species.

Sequence Alignment

Sequence alignment is a method used to arrange protein or nucleotide sequences to identify regions of similarity. These similarities could indicate functional, structural, or evolutionary relationships between the sequences.

In sequence alignment, each position in the sequence is compared, and matches or mismatches are identified. When calculating homology, sequence alignment helps in determining where matches and mismatches occur, as well as their frequency.

For example, to find perfect matches, each amino acid in the test sequence is aligned and compared with each amino acid in the database sequences. Alignments are evaluated based on scoring systems and may involve different algorithms to optimize matching.

Basic types of sequence alignment include global alignment (aligns sequences end-to-end) and local alignment (finds regions of high similarity). These methods help in understanding the degree of conservation and variation in protein sequences, which is crucial for evolutionary biology and understanding protein function.

Knowledge in sequence alignment enables accurate comparison of sequences, aiding in discovering new proteins, understanding genetic variations, and even developing treatments for diseases.