Information

The Information module of MIToS defines types and functions useful to calculate information measures (e.g. Mutual Information (MI) and Entropy) over a Multiple Sequence Alignment (MSA). This module was designed to count Residues (defined in the MSA module) in special contingency tables (as fast as possible) and to derive probabilities from this counts. Also, includes methods for applying corrections to that tables, e.g. pseudocounts and pseudo frequencies. Finally, Information allows to use this probabilities and counts to estimate information measures and other frequency based values.

Features

• Estimate multi dimensional frequencies and probabilities tables from sequences, MSAs, etc...
• Correction for small number of observations
• Correction for data redundancy on a MSA
• Estimate information measures
• Calculate corrected mutual information between residues

using MIToS.Information

Additive Smoothing or fixed pseudocount λ for ResidueCount (in order to estimate probabilities when the number of samples is low).

Common values of λ are:

• 0 : No cell frequency prior, gives you the maximum likelihood estimator.
• 0.05 is the optimum value for λ found in Buslje et. al. 2009, similar results was obtained for λ in the range [0.025, 0.075].
• 1 / p : Perks prior (Perks, 1947) where p the number of parameters (i.e. residues, pairs of residues) to estimate. If p is the number of residues (20 without counting gaps), this gives you 0.05.
• sqrt(n) / p : Minimax prior (Trybula, 1958) where n is the number of samples and p the number of parameters to estimate. If the number of samples n is 400 (minimum number of sequence clusters for achieve good performance in Buslje et. al. 2009) for estimating 400 parameters (pairs of residues without counting gaps) this gives you 0.05.
• 0.5 : Jeffreys prior (Jeffreys, 1946).
• 1 : Bayes-Laplace uniform prior, aka. Laplace smoothing.

References

• Buslje, Cristina Marino, et al. "Correction for phylogeny, small number of observations and data redundancy improves the identification of coevolving amino acid pairs using mutual information." Bioinformatics 25.9 (2009): 1125-1131.

BLOSUM_Pseudofrequencies type. It takes to arguments/fields:

• α : Usually the number of sequences or sequence clusters in the MSA.
• β : The weight of the pseudofrequencies, a value close to 8.512 when α is the number of sequence clusters.

A ContingencyTable is a multidimensional array. It stores the contingency matrix, its marginal values and total. The type also has an internal and private temporal array and an alphabet object. It's a parametric type, taking three ordered parameters:

• T : The element type of the multidimensional array.
• N : It's the dimension of the array and should be an Int.
• A : This should be a type, subtype of ResidueAlphabet, i.e.: UngappedAlphabet, GappedAlphabet or ReducedAlphabet.

A ContingencyTable can be created from an alphabet if all the parameters are given. Otherwise, you need to give a type, a number (Val) and an alphabet. You can also create a ContingencyTable using a matrix and a alphabet. For example:

ContingencyTable{Float64,2,UngappedAlphabet}(UngappedAlphabet())
ContingencyTable(Float64, Val{2}, UngappedAlphabet())
ContingencyTable(zeros(Float64, 20, 20), UngappedAlphabet())

A Frequencies object wraps a ContingencyTable storing counts/frequencies.

You can use NoPseudocount() to avoid pseudocount corrections where a Pseudocount type is needed.

You can use NoPseudofrequencies() to avoid pseudocount corrections where a Pseudofrequencies type is needed.

A Probabilities object wraps a ContingencyTable storing probabilities. It doesn't perform any check. If the total isn't one, you must use normalize or normalize!on the ContingencyTable before wrapping it to make the sum of the probabilities equal to one.

Parametric abstract type to define pseudocount types

Parametric abstract type to define pseudofrequencies types

BLOSUM62 probabilities P(aa) for each residue on the UngappedAlphabet. SUM: 0.9987

Table with conditional probabilities of residues based on BLOSUM62. The normalization is done row based. The firts row contains the P(aa|A) and so one.

It populates a ContingencyTable (first argument) using the frequencies in the sequences (last positional arguments). The dimension of the table must match the number of sequences and all the sequences must have the same length. You must indicate the used weights and pseudocounts as second and third positional arguments respectively. You can use NoPseudofrequencies() and NoClustering() to avoid the use of sequence weighting and pseudocounts, respectively.

DEPRECATED: Use frequencies! instead. Note that frequencies! defines the weigths and pseudocounts using keyword arguments instead of positional arguments.

It returns a ContingencyTable wrapped in a Frequencies type with the frequencies of residues in the sequences that takes as arguments. The dimension of the table is equal to the number of sequences. You can use the keyword arguments alphabet, weights and pseudocounts to indicate the alphabet of the table (default to UngappedAlphabet()), a clustering result (default to NoClustering()) and the pseudocounts (default to NoPseudocount()) to be used during the estimation of the frequencies.

DEPRECATED: Use frequencies instead. Note that frequencies defines the alphabet, weigths and pseudocounts using keyword arguments instead of positional arguments.

normalize! makes the sum of the frequencies to be one, in place.

normalize returns another table where the sum of the frequencies is one.

APC

References

• Dunn, Stanley D., Lindi M. Wahl, and Gregory B. Gloor. "Mutual information without the influence of phylogeny or entropy dramatically improves residue contact prediction." Bioinformatics 24.3 (2008): 333-340.

BLMI takes an MSA and calculates a Z score (ZBLMI) and a corrected MI/MIp as described on Busjle et. al. 2009 but using using BLOSUM62 pseudo frequencies instead of a fixed pseudocount.

Keyword argument, type, default value and descriptions:

  - beta        Float64   8.512   β for BLOSUM62 pseudo frequencies
  - lambda      Float64   0.0     Low count value
  - threshold             62      Percent identity threshold for sequence clustering (Hobohm I)
  - maxgap      Float64   0.5     Maximum fraction of gaps in positions included in calculation
  - apc         Bool      true    Use APC correction (MIp)
  - samples     Int       50      Number of samples for Z-score
  - fixedgaps   Bool      true    Fix gaps positions for the random samples

This function returns:

  - Z score (ZBLMI)
  - MI or MIp using BLOSUM62 pseudo frequencies (BLMI/BLMIp)

It adds the pseudocount value to the table cells.

apply_pseudofrequencies!{T}(Pab::ContingencyTable{T,2,UngappedAlphabet}, pseudofrequencies::BLOSUM_Pseudofrequencies)

When a BLOSUM_Pseudofrequencies(α,β) is used, this function applies pseudofrequencies Gab over Pab, as a weighted mean of both. It uses the conditional probability matrix BLOSUM62_Pij and the real frequencies/probabilities Pab to estimate the pseudofrequencies Gab. α is the weight of the real frequencies Pab and β the weight of the pseudofrequencies.

Gab = Σcd Pcd ⋅ BLOSUM62( a | c ) ⋅ BLOSUM62( b | d ) Pab = (α ⋅ Pab + β ⋅ Gab )/(α + β)

buslje09 takes a MSA and calculates a Z score and a corrected MI/MIp as described on Busjle et. al. 2009.

keyword argument, type, default value and descriptions:

  - lambda      Float64   0.05    Low count value
  - clustering  Bool      true    Sequence clustering (Hobohm I)
  - threshold             62      Percent identity threshold for clustering
  - maxgap      Float64   0.5     Maximum fraction of gaps in positions included in calculation
  - apc         Bool      true    Use APC correction (MIp)
  - samples     Int       100     Number of samples for Z-score
  - fixedgaps   Bool      true    Fix gaps positions for the random samples
  - alphabet    ResidueAlphabet UngappedAlphabet()  Residue alphabet to be used

This function returns:

  - Z score
  - MI or MIp

cumulative allows to calculate cumulative scores (i.e. cMI) as defined in Marino Buslje et al. 2010:

"We calculated a cumulative mutual information score (cMI) for each residue as the sum of MI values above a certain threshold for every amino acid pair where the particular residue appears. This value defines to what degree a given amino acid takes part in a mutual information network."

References

• Marino Buslje, Cristina, et al. "Networks of high mutual information define the structural proximity of catalytic sites: implications for catalytic residue identification." PLoS computational biology 6.11 (2010): e1000978.

delete_dimensions!(out::ContingencyTable, in::ContingencyTable, dimensions::Int...)

This function fills a ContingencyTable with the counts/probabilities on in after the deletion of dimensions. i.e. This is useful for getting Pxy from Pxyz.

delete_dimensions(in::ContingencyTable, dimensions::Int...)

This function creates a ContingencyTable with the counts/probabilities on in after the deletion of dimensions. i.e. This is useful for getting Pxy from Pxyz.

frequencies!(table, seqs...; weights::WeightTypes, pseudocounts::Pseudocount)

It populates a ContingencyTable or Frequencies table (first argument) using the frequencies in the given sequences (last positional arguments). The dimension of the table must match the number of sequences and all the sequences must have the same length. You must indicate the used weights and pseudocounts as keyword arguments. Those arguments default to NoClustering() and NoPseudocount() respectively, to avoid the use of sequence weighting and pseudocounts.

frequencies(seqs...; alphabet=UngappedAlphabet(), weights=NoClustering(), pseudocounts=NoPseudocount()

This function returns a Frequencies object wrapping a ContingencyTable with the frequencies of residues in the sequences that takes as arguments. The dimension of the table is equal to the number of sequences. You can use the keyword arguments alphabet, weights and pseudocounts to indicate the alphabet of the table, a clustering result and the pseudocounts to be used during the estimation of the frequencies.

It calculates the gap intersection as percentage from a table of Frequencies.

It calculates the gap union as percentage from a table of Frequencies.

Wrapper function to GaussDCA.gDCA. You need to install GaussDCA:

using Pkg

Pkg.add(PackageSpec(url = "https://github.com/carlobaldassi/GaussDCA.jl", rev = "master"))

Look into GaussDCA.jl README for further information. If you use this wrapper, please cite the GaussDCA publication and the package's doi.

It's possible to indicate the path to the julia binary where GaussDCA is installed. However, it's recommended to use the same version where MIToS is installed. That is because this function use serialize/deserialize to transfer data between the processes.

GaussDCA Publication: Baldassi, Carlo, Marco Zamparo, Christoph Feinauer, Andrea Procaccini, Riccardo Zecchina, Martin Weigt, and Andrea Pagnani. "Fast and accurate multivariate Gaussian modeling of protein families: predicting residue contacts and protein-interaction partners." PloS one 9, no. 3 (2014): e92721.

getalphabet allows to access the stored alphabet object.

getcontingencytable allows to access the wrapped ContingencyTable in a Probabilities or Frequencies object.

getmarginals allows to access the array with the marginal values (NamedArray).

getmarginalsarray allows to access the array with the marginal values (Array without names).

gettable allows to access the table (NamedArray).

gettablearray allows to access the table (Array without names).

gettotal allows to access the stored total value.

kullback_leibler(msa::AbstractArray{Residue}; background::Union{Array{T,N}, Probabilities{T,N,A}, ContingencyTable{T,N,A}}=BLOSUM62_Pi, base::Number=ℯ, kargs...)

It calculates the Kullback-Leibler (KL) divergence from a multiple sequence alignment (MSA). You can use the keyword argument background to set the background distribution. This argument can take an Array, Probabilities, or ContingencyTable object. The background distribution must have the same size and alphabet as the probabilities. The default is the BLOSUM62_Pi table. The default base for the log is ℯ (base=ℯ), so the result is in nats. You can use base = 2 to get the result in bits.

The other keyword arguments are passed to the mapfreq function.

kullback_leibler(probabilities::Probabilities{T,N,A}, background::Union{
    AbstractArray{T,N}, Probabilities{T,N,A}, ContingencyTable{T,N,A}}=BLOSUM62_Pi, 
    base::Number=ℯ)

It calculates the Kullback-Leibler (KL) divergence from a table of Probabilities. You can use the keyword argument background to set the background distribution. This argument can take an Array, Probabilities, or ContingencyTable object. The background distribution must have the same size and alphabet as the probabilities. The default is the BLOSUM62_Pi table. The default base for the log is ℯ (base=ℯ), so the result is in nats. You can use base = 2 to get the result in bits.

It efficiently map a function (first argument) that takes a table of Frequencies or Probabilities (third argument). The table is filled in place with the counts or probabilities of each column from the msa (second argument).

• weights (default: NoClustering()): Weights to be used for table counting.
• pseudocounts (default: NoPseudocount()): Pseudocount object to be applied to table.
• pseudofrequencies (default: NoPseudofrequencies()): Pseudofrequencies to be applied to the normalized (probabilities) table.

It efficiently map a function (first argument) that takes a table of Frequencies or Probabilities (third argument). The table is filled in place with the counts or probabilities of each pair of columns from the msa (second argument).

• weights (default: NoClustering()): Weights to be used for table counting.

• pseudocounts (default: NoPseudocount()): Pseudocount object to be applied to table.

• pseudofrequencies (default: NoPseudofrequencies()): Pseudofrequencies to be applied to the normalized (probabilities) table.

• usediagonal (default: true): If true, the function will be also applied to the diagonal elements.

• diagonalvalue (default: zero): Value to fill diagonal elements if usediagonal is false.

mapfreq(f, msa; rank = 1, dims = 2, alphabet = UngappedAlphabet(), 
    weights = NoClustering(), pseudocounts = NoPseudocount(), 
    pseudofrequencies = NoPseudofrequencies(), probabilities = true, 
    usediagonal = false, diagonalvalue = NaN, kargs...)

It efficiently map a function (first argument) that takes a table of Frequencies or Probabilities (depending on the probabilities keyword argument) calculated on sequences (dims = 1) or columns (dims = 2, the default) of an msa (second argument). If rank = 1, the default, the function is applied to each sequence or column. If rank = 2, the function is applied to each pair of sequences or columns. In that case, we can set the usediagonal keyword argument to true to apply the function to pairs of the same sequence or column. The diagonalvalue keyword argument is used to set the value of the diagonal elements if usediagonal is false. By default, the function is not applied to the diagonal elements (i.e. usediagonal = false) and the diagonalvalue is set to NaN. The alphabet keyword argument can be used to set the alphabet used to construct the contingency table. The function also accepts the following keyword arguments:

• weights (default: NoClustering()): Weights to be used for table counting.
• pseudocounts (default: NoPseudocount()): Pseudocount object to be applied to table.
• pseudofrequencies (default: NoPseudofrequencies()): Pseudofrequencies to be applied to the normalized (probabilities) table.

Note that the pseudofrequencies argument is only valid if probabilities = true. All the other keyword arguments are passed to the function f.

julia> using Random, MIToS.MSA, MIToS.Information

julia> msa = rand(Random.MersenneTwister(1), Residue, 3, 6) # random MSA as an example
3×6 Matrix{Residue}:
 F  A  F  D  E  V
 T  R  R  G  F  I
 N  V  S  W  Q  T

julia> mapfreq(sum, msa) # default: rank=1, dims=2, probabilities=true
1×6 Named Matrix{Float64}
Function ╲ Col │   1    2    3    4    5    6
───────────────┼─────────────────────────────
sum            │ 1.0  1.0  1.0  1.0  1.0  1.0

julia> mapfreq(sum, msa, probabilities=false)
1×6 Named Matrix{Float64}
Function ╲ Col │   1    2    3    4    5    6
───────────────┼─────────────────────────────
sum            │ 3.0  3.0  3.0  3.0  3.0  3.0

julia> mapfreq(sum, msa, dims=1)
3×1 Named Matrix{Float64}
Seq ╲ Function │ sum
───────────────┼────
1              │ 1.0
2              │ 1.0
3              │ 1.0

It efficiently map a function (first argument) that takes a table of Frequencies or Probabilities (third argument). The table is filled in place with the counts or probabilities of each sequence from the msa (second argument).

• weights (default: NoClustering()): Weights to be used for table counting.
• pseudocounts (default: NoPseudocount()): Pseudocount object to be applied to table.
• pseudofrequencies (default: NoPseudofrequencies()): Pseudofrequencies to be applied to the normalized (probabilities) table.

It efficiently map a function (first argument) that takes a table of Frequencies or Probabilities (third argument). The table is filled in place with the counts or probabilities of each pair of sequences from the msa (second argument).

• weights (default: NoClustering()): Weights to be used for table counting.

• pseudocounts (default: NoPseudocount()): Pseudocount object to be applied to table.

• pseudofrequencies (default: NoPseudofrequencies()): Pseudofrequencies to be applied to the normalized (probabilities) table.

• usediagonal (default: true): If true, the function will be also applied to the diagonal elements.

• diagonalvalue (default: zero): Value to fill diagonal elements if usediagonal is false.

marginal_entropy(table::Union{Frequencies{T,N,A},Probabilities{T,N,A}}; margin::Int=1, 
    base::Number=ℯ)

It calculates marginal entropy (H) from a table of Frequencies or Probabilities. It takes two keyword arguments: margin and base. The first one is used to indicate the margin used to calculate the entropy, e.g. it estimates the entropy H(X) if margin is 1, H(Y) for 2, etc. The default value of margin is 1. The second keyword argument is used to change the base of the log. The default base for the log is ℯ (base=ℯ), so the result is in nats. You can use base = 2 to get the result in bits.

mutual_information(msa::AbstractArray{Residue}; base::Number=ℯ, kargs...)

It calculates Mutual Information (MI) from a multiple sequence alignment (MSA). The default base for the log is ℯ (base=ℯ), so the result is in nats. You can use base = 2 to get the result in bits. The minimum value for rank is 2 (the default value). By defualt, it uses counts/frequencies to calculate the MI, as it's faster. You can use the keyword argument probabilities = true to calculate the MI from probabilities.

```jldoctest julia> using Random, MIToS.MSA, MIToS.Information

julia> msa = rand(Random.MersenneTwister(37), Residue, 3, 4) 3×4 Matrix{Residue}: T R F K S H C I G G R V

julia> mutual_information(msa) 4×4 Named PairwiseListMatrices.PairwiseListMatrix{Float64, false, Vector{Float64}} Col1 ╲ Col2 │ 1 2 3 4 ────────────┼─────────────────────────────────── 1 │ NaN 1.09861 1.09861 1.09861 2 │ 1.09861 NaN 1.09861 1.09861 3 │ 1.09861 1.09861 NaN 1.09861 4 │ 1.09861 1.09861 1.09861 NaN

````

mutual_information(table::Union{Frequencies{T,2,A},Probabilities{T,2,A}}; base::Number=ℯ)

It calculates Mutual Information (MI) from a table of Frequencies or Probabilities. The default base for the log is ℯ (base=ℯ), so the result is in nats. You can use base = 2 to get the result in bits. Note that calculating MI from Frequencies is faster than from Probabilities.

mutual_information(table::Union{Frequencies{T,3,A},Probabilities{T,3,A}}; base::Number=ℯ)

It calculates Mutual Information (MI) from a table of Frequencies or Probabilities with three dimensions. The default base for the log is ℯ (base=ℯ), so the result is in nats. You can use base = 2 to get the result in bits.

julia> using Random, MIToS.MSA, MIToS.Information

julia> msa = rand(Random.MersenneTwister(37), Residue, 3, 4)
3×4 Matrix{Residue}:
 T  R  F  K
 S  H  C  I
 G  G  R  V

julia> Nxyz = frequencies(msa[:, 1], msa[:, 2], msa[:, 3]);

julia> mutual_information(Nxyz)
1.0986122886681093

normalized_mutual_information(msa::AbstractArray{Residue}; kargs...)

This function calculates the Normalized Mutual Information (nMI) from a multiple sequence alignment using the mapfreq function—all the keyword arguments are passed to mapfreq. The mutual information score is normalized by the joint entropy of the two variables: $nMI(X, Y) = MI(X, Y) / H(X, Y)$ By default, it uses counts/frequencies to estimate the nMI, as it's faster than using probabilities.

It calculates a Normalized Mutual Information (nMI) from a table of Frequencies or Probabilities. The mutual information score is normalized by the joint entropy of the two variables: $nMI(X, Y) = MI(X, Y) / H(X, Y)$

It takes a MSA or a file and a FileFormat as first arguments. It calculates the percentage of gaps on columns pairs (union and intersection) using sequence clustering (Hobohm I).

Argument, type, default value and descriptions:

    - clustering  Bool      true    Sequence clustering (Hobohm I)
    - threshold             62      Percent identity threshold for sequence clustering (Hobohm I)

This function returns:

    - pairwise gap union as percentage
    - pairwise gap intersection as percentage

It populates a ContingencyTable (first argument) using the probabilities in the sequences (last positional arguments). The dimension of the table must match the number of sequences and all the sequences must have the same length. You must indicate the used weights, pseudocounts and pseudofrequencies as second, third and fourth positional arguments respectively. You can use NoClustering(), NoPseudocount() and NoPseudofrequencies() to avoid the use of sequence weighting, pseudocounts and pseudofrequencies, respectively.

It returns a ContingencyTable wrapped in a Probabilities type with the probabilities of residues in the sequences that takes as arguments. The dimension of the table is equal to the number of sequences. You can use the keyword arguments alphabet, weights, pseudocounts and pseudofrequencies to indicate the alphabet of the table (default to UngappedAlphabet()), a clustering result (default to NoClustering()), the pseudocounts (default to NoPseudocount()) and the pseudofrequencies (default to NoPseudofrequencies()) to be used during the estimation of the probabilities.

shannon_entropy(msa::AbstractArray{Residue}; base::Number=ℯ, 
    probabilities::Bool=false, usediagonal::Bool=true, kargs...)

It calculates the Shannon entropy (H) on a MSA. You can use the keyword argument base to change the base of the log. The default base for the log is ℯ (base=ℯ), so the result is in nats. You can use base = 2 to get the result in bits. It uses mapfreq under the hood, so it takes the same keyword arguments. By default, it measures the entropy of each column in the MSA. You can use dims = 1 to measure the entropy of each sequence. You can also set rank = 2to measure the joint entropy of each pair of sequences or columns. This function sets by default the probabilities keyword argument to false because it's faster to calculate the entropy from counts/frequencies. It also sets usediagonal = true to also calculate the entropy of the individual variables (sequences or columns).

```jldoctest julia> using MIToS.MSA, MIToS.Information

julia> msa = Residue['C' 'G'; 'C' 'L'; 'C' 'I'] 3×2 Matrix{Residue}: C G C L C I

julia> shannonentropy(msa) 1×2 Named Matrix{Float64} Function ╲ Col │ 1 2 ────────────────┼───────────────── shannonentropy │ 0.0 1.09861

shannon_entropy(table::Union{Frequencies{T,N,A},Probabilities{T,N,A}}; base::Number=ℯ)

It calculates the Shannon entropy (H) from a table of Frequencies or Probabilities. Use last and optional positional argument to change the base of the log. The default base for the log is ℯ (base=ℯ), so the result is in nats. You can use base = 2 to get the result in bits.