# Bio.SubsMat package¶

## Module contents¶

Substitution matrices, log odds matrices, and operations on them.

### General:¶

This module provides a class and a few routines for generating substitution matrices, similar ot BLOSUM or PAM matrices, but based on user-provided data. The class used for these matrices is SeqMat

Matrices are implemented as a dictionary. Each index contains a 2-tuple, which are the two residue/nucleotide types replaced. The value differs according to the matrix’s purpose: e.g in a log-odds frequency matrix, the value would be log(Pij/(Pi*Pj)) where: Pij: frequency of substitution of letter (residue/nucleotide) i by j Pi, Pj: expected frequencies of i and j, respectively.

### Usage:¶

The following section is laid out in the order by which most people wish to generate a log-odds matrix. Of course, interim matrices can be generated and investigated. Most people just want a log-odds matrix, that’s all.

### Generating an Accepted Replacement Matrix:¶

Initially, you should generate an accepted replacement matrix (ARM) from your data. The values in ARM are the _counted_ number of replacements according to your data. The data could be a set of pairs or multiple alignments. So for instance if Alanine was replaced by Cysteine 10 times, and Cysteine by Alanine 12 times, the corresponding ARM entries would be: [‘A’,’C’]: 10, [‘C’,’A’] 12 As order doesn’t matter, user can already provide only one entry: [‘A’,’C’]: 22 A SeqMat instance may be initialized with either a full (first method of counting: 10, 12) or half (the latter method, 22) matrix. A Full protein alphabet matrix would be of the size 20x20 = 400. A Half matrix of that alphabet would be 20x20/2 + 20/2 = 210. That is because same-letter entries don’t change. (The matrix diagonal). Given an alphabet size of N: Full matrix size:N*N Half matrix size: N(N+1)/2

If you provide a full matrix, the constructor will create a half-matrix automatically. If you provide a half-matrix, make sure of a (low, high) sorted order in the keys: there should only be a (‘A’,’C’) not a (‘C’,’A’).

Internal functions:

### Generating the observed frequency matrix (OFM):¶

Use: OFM = _build_obs_freq_mat(ARM) The OFM is generated from the ARM, only instead of replacement counts, it contains replacement frequencies.

### Generating an expected frequency matrix (EFM):¶

Use: EFM = _build_exp_freq_mat(OFM,exp_freq_table) exp_freq_table: should be a freqTableC instantiation. See freqTable.py for detailed information. Briefly, the expected frequency table has the frequencies of appearance for each member of the alphabet

### Generating a substitution frequency matrix (SFM):¶

Use: SFM = _build_subs_mat(OFM,EFM) Accepts an OFM, EFM. Provides the division product of the corresponding values.

### Generating a log-odds matrix (LOM):¶

Use: LOM=_build_log_odds_mat(SFM[,logbase=10,factor=10.0,roundit=1]) Accepts an SFM. logbase: base of the logarithm used to generate the log-odds values. factor: factor used to multiply the log-odds values. roundit: default - true. Whether to round the values. Each entry is generated by log(LOM[key])*factor And rounded if required.

### External:¶

In most cases, users will want to generate a log-odds matrix only, without explicitly calling the OFM –> EFM –> SFM stages. The function build_log_odds_matrix does that. User provides an ARM and an expected frequency table. The function returns the log-odds matrix.

### Methods for subtraction, addition and multiplication of matrices:¶

• Generation of an expected frequency table from an observed frequency matrix.

• Calculation of linear correlation coefficient between two matrices.

• Calculation of relative entropy is now done using the _make_relative_entropy method and is stored in the member self.relative_entropy

• Calculation of entropy is now done using the _make_entropy method and is stored in the member self.entropy.

• Jensen-Shannon distance between the distributions from which the matrices are derived. This is a distance function based on the distribution’s entropies.

class Bio.SubsMat.SeqMat(data=None, alphabet=None, mat_name='', build_later=0)

Bases: `dict`

A Generic sequence matrix class.

The key is a 2-tuple containing the letter indices of the matrix. Those should be sorted in the tuple (low, high). Because each matrix is dealt with as a half-matrix.

__init__(self, data=None, alphabet=None, mat_name='', build_later=0)

Initialize.

User may supply:

• data: matrix itself

• mat_name: its name. See below.

• alphabet: an iterable over the letters allowed as indices into the matrix. If not supplied, constructor builds its own from that matrix.

• build_later: skip the matrix size assertion. User will build the matrix after creating the instance. Constructor builds a half matrix filled with zeroes.

make_entropy(self)

Calculate and set the entropy attribute.

sum(self)

Return sum of the results.

format(self, fmt='%4d', letterfmt='%4s', alphabet=None, non_sym=None, full=False)

Create a string with the bottom-half (default) or a full matrix.

User may pass own alphabet, which should contain all letters in the alphabet of the matrix, but may be in a different order. This order will be the order of the letters on the axes.

__str__(self)

Print a nice half-matrix.

__sub__(self, other)

Return integer subtraction product of the two matrices.

__mul__(self, other)

Element-wise matrix multiplication.

Returns a new matrix created by multiplying each element by other (if other is scalar), or by performing element-wise multiplication of the two matrices (if other is a matrix of the same size).

__rmul__(self, other)

Element-wise matrix multiplication.

Returns a new matrix created by multiplying each element by other (if other is scalar), or by performing element-wise multiplication of the two matrices (if other is a matrix of the same size).

class Bio.SubsMat.SubstitutionMatrix(data=None, alphabet=None, mat_name='', build_later=0)

Substitution matrix.

calculate_relative_entropy(self, obs_freq_mat)

Calculate and return relative entropy w.r.t. observed frequency matrix.

class Bio.SubsMat.LogOddsMatrix(data=None, alphabet=None, mat_name='', build_later=0)

Log odds matrix.

calculate_relative_entropy(self, obs_freq_mat)

Calculate and return relative entropy w.r.t. observed frequency matrix.

Bio.SubsMat.make_log_odds_matrix(acc_rep_mat, exp_freq_table=None, logbase=2, factor=1.0, round_digit=9, keep_nd=0)

Make log-odds matrix.

Bio.SubsMat.observed_frequency_to_substitution_matrix(obs_freq_mat)

Convert observed frequency table into substitution matrix.

Read a matrix from a text file.

Bio.SubsMat.two_mat_relative_entropy(mat_1, mat_2, logbase=2, diag=diagALL)

Return relative entropy of two matrices.

Bio.SubsMat.two_mat_correlation(mat_1, mat_2)

Return linear correlation coefficient between two matrices.

Bio.SubsMat.two_mat_DJS(mat_1, mat_2, pi_1=0.5, pi_2=0.5)

Return Jensen-Shannon Distance between two observed frequence matrices.