Position weight matrix

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) and 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:

${\displaystyle M_{k,j}={\frac {1}{N}}\sum _{i=1}^{N}I(X_{i,j}=k),}$

where i ${\displaystyle \in }$ (1,...,N), j ${\displaystyle \in }$ (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:

GAGGTAAAC TCCGTAAGT CAGGTTGGA ACAGTCAGT TAGGTCATT TAGGTACTG ATGGTAACT CAGGTATAC TGTGTGAGT AAGGTAAGT

The corresponding PFM is:

${\displaystyle M={\begin{matrix}A\\C\\G\\T\end{matrix}}{\begin{bmatrix}3&6&1&0&0&6&7&2&1\\2&2&1&0&0&2&1&1&2\\1&1&7&10&0&1&1&5&1\\4&1&1&0&10&1&1&2&6\end{bmatrix}}.}$

Therefore, the resulting PPM is:[1]

${\displaystyle M={\begin{matrix}A\\C\\G\\T\end{matrix}}{\begin{bmatrix}0.3&0.6&0.1&0.0&0.0&0.6&0.7&0.2&0.1\\0.2&0.2&0.1&0.0&0.0&0.2&0.1&0.1&0.2\\0.1&0.1&0.7&1.0&0.0&0.1&0.1&0.5&0.1\\0.4&0.1&0.1&0.0&1.0&0.1&0.1&0.2&0.6\end{bmatrix}}.}$

Both PPMs and 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 PPM, by multiplying the relevant probabilities at each position. For example, the probability of the sequence S = GAGGTAAAC given the above PPM M can be calculated:

${\displaystyle p(S\vert M)=0.1\times 0.6\times 0.7\times 1.0\times 1.0\times 0.6\times 0.7\times 0.2\times 0.2=0.0007056.}$

Pseudocounts (or Laplace estimators) are often applied when calculating PPMs if based on a small dataset, in order to avoid matrix entries having a value of 0.[2] This is equivalent to multiplying each column of the PPM 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.

Most often the elements in PWMs are calculated as log odds. That is, the elements of a PPM are transformed using a background model ${\displaystyle b}$ so that:

${\displaystyle M_{k,j}=\mathrm {log_{2}} \;(M_{k,j}/b_{k}).}$

describes how an element in the PWM (left), ${\displaystyle M_{k,j}}$, can be calculated. The simplest background model assumes that each letter appears equally frequently in the dataset. That is, the value of ${\displaystyle b_{k}=1/\vert k\vert }$ for all symbols in the alphabet (0.25 for nucleotides and 0.05 for amino acids). Applying this transformation to the PPM M from above (with no pseudocounts added) gives:

${\displaystyle M={\begin{matrix}A\\C\\G\\T\end{matrix}}{\begin{bmatrix}0.26&1.26&-1.32&-\infty &-\infty &1.26&1.49&-0.32&-1.32\\-0.32&-0.32&-1.32&-\infty &-\infty &-0.32&-1.32&-1.32&-0.32\\-1.32&-1.32&1.49&2.0&-\infty &-1.32&-1.32&1.0&-1.32\\0.68&-1.32&-1.32&-\infty &2.0&-1.32&-1.32&-0.32&1.26\end{bmatrix}}.}$

The ${\displaystyle -\infty }$ 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 can be calculated by 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.[1] The sequence score can also be interpreted in a physical framework as the binding energy for that sequence.