Position weight matrix

A position weight matrix (PWM), also known as a position-specific weight matrix (PSWM) or position-specific scoring matrix (PSSM), is a commonly used representation of motifs (patterns) in biological sequences.

PWMs are often derived from a set of aligned sequences that are thought to be functionally related and have become an important part of many software tools for computational motif discovery.

The position weight matrix was introduced by American geneticist Gary Stormo and colleagues in 1982 as an alternative to consensus sequences. Consensus sequences had previously been used to represent patterns in biological sequences, but had difficulties in the prediction of new occurrences of these patterns. The first use of PWMs was in the discovery of RNA sites that function as translation initiation sites. The perceptron algorithm was suggested by Polish American mathematician Andrzej Ehrenfeucht in order to create a matrix of weights which could distinguish true binding sites from other non-functional sites with similar sequences. Training the perceptron on both sets of sites resulted in a matrix and a threshold to distinguish between the two sets. Using the matrix to scan new sequences not included in the training set showed that this method was both more sensitive and precise than the best consensus sequence.

The advantages of PWMs over consensus sequences have made PWMs a popular method for representing patterns in biological sequences and an essential component in modern algorithms for motif discovery.

A PWM has one row for each symbol of the alphabet: 4 rows for nucleotides in DNA sequences or 20 rows for amino acids in protein sequences. It also has one column for each position in the pattern. In the first step in constructing a PWM, a basic position frequency matrix (PFM) is created by counting the occurrences of each nucleotide at each position. From the PFM, a position probability matrix (PPM) can now be created by dividing that former nucleotide count at each position by the number of sequences, thereby normalising the values. Formally, given a set X of N aligned sequences of length l, the elements of the PPM M are calculated:

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