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.
Basic PWM with frequencies
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. A basic PWM using relative frequencies is constructed by counting the occurrences of each symbol at each position and then normalising at each position. Formally, given a set X of N aligned sequences of length l, the elements of the PWM M are calculated:
where i (1,...,N), j (1,...,l), k is the set of symbols in the alphabet and I(a=k) is an indicator function where I(a=k) is 1 if a=k and 0 otherwise.
For example, given the following DNA sequences:
the PWM using relative frequencies is:
PWMs assume statistical independence between positions in the pattern, as the probabilities for each position are calculated independently of other positions. From the definition above, it follows that the sum of values for a particular position (that is, summing over all symbols) is 1. Each column can therefore be regarded as an independent multinomial distribution. This makes it easy to calculate the probability of a sequence given a PWM, by multiplying the relevant probabilities at each position. For example, the probability of the sequence S = GAGGTAAAC given the above PWM M can be calculated:
Pseudocounts (or Laplace estimators) are often applied to PWMs when a small dataset is used, in order to avoid matrix entries having a value of 0. This is equivalent to multiplying each column of the PWM by a Dirichlet distribution and allows the probability to be calculated for new sequences (that is, sequences which were not part of the original dataset): in the example above, without pseudocounts, any sequence which did not have a G in the 4th position or a T in the 5th position would have a probability of 0, regardless of the other positions.
Using log likelihoods
Often the element in PWMs are calculated as log likelihoods. That is, the elements of the PWM are transformed using a background model so that:
The simplest background model assumes that each letter appears equally frequently in the dataset. That is, the value of for all symbols in the alphabet (0.25 for nucleotides and 0.05 for amino acids). Applying this transformation to the PWM M from above gives:
The entries in the matrix make clear the advantage of adding pseudocounts, especially when using small datasets to construct M. The background model need not have equal values for each symbol: for example, when studying organisms with a high GC-content, the values for C and G may be increased with a corresponding decrease for the A and T values.
When the PWM elements are calculated using log likelihoods, the score of a sequence given a particular PWM can be calculated be adding (rather than multiplying) the relevant values at each position in the PWM. The sequence score gives an indication of how different the sequence is from a random sequence. The score is 0 if the sequence has the same probability of being a functional site and of being a random site. The score is greater than 0 if it is more likely to be a functional site than a random site, and less than 0 if it is more likely to be a random site than a functional site. The sequence score can also be interpreted in a physical framework as the binding energy for that sequence.
Information content of a PWM
The self-information of observing a particular symbol at a particular position of the motif is:
The expected (average) self-information of a particular element in the PWM is then:
Finally, the IC of the PWM is then the sum of the expected self-information of every element:
Often, it is more useful to calculate the information content with the background letter frequencies of the sequences you are studying rather than assuming equal probabilities of each letter (e.g., the GC-content of DNA of thermophilic bacteria range from 65.3 to 70.8, thus a motif of ATAT would contain much more information than a motif of CCGG). The equation for information content thus becomes
where is the background frequency for that letter. This corresponds to the Kullback-Leibler divergence or relative entropy. However, it has been shown that when using PSSM to search genomic sequences (see below) this uniform correction can lead to overestimation of the importance of the different bases in a motif, due to the uneven distribution of n-mers in real genomes, leading to a significantly larger number of false positives.
There are various algorithms to scan for hits of PWMs in sequences. One example is the MATCH algorithm which has been implemented in the ModuleMaster. More sophisticated algorithms for fast database searching with nucleotide as well as amino acid PWMs/PSSMs are implemented in the possumsearch software and are described by Beckstette, et al. (2006).
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