Author: Nan Xiao <https://nanx.me>

# protr: R package for generating various numerical representation schemes of protein sequences

#### Nan Xiao <https://nanx.me>

## Introduction

The protr package offers a unique and comprehensive toolkit for generating various numerical representation schemes of protein sequences. The descriptors included are extensively utilized in bioinformatics and chemogenomics research.

The commonly used descriptors listed in protr include amino acid composition, autocorrelation, CTD, conjoint traid, quasi-sequence order, pseudo amino acid composition, and profile-based descriptors derived by Position-Specific Scoring Matrix (PSSM).

The descriptors for proteochemometric (PCM) modeling include the scales-based descriptors derived by principal components analysis, factor analysis, multidimensional scaling, amino acid properties (AAindex), 20+ classes of 2D and 3D molecular descriptors (for example, Topological, WHIM, VHSE, among others), and BLOSUM/PAM matrix-derived descriptors.

The protr package also implemented parallelized similarity computation derived by pairwise protein sequence alignment and Gene Ontology (GO) semantic similarity measures. ProtrWeb, the web application built on protr, can be accessed from http://protr.org.

If you find protr useful in your research, please feel free to cite our paper:

> Nan Xiao, Dong-Sheng Cao, Min-Feng Zhu, and Qing-Song Xu. (2015). protr/ProtrWeb: R package and web server for generating various numerical representation schemes of protein sequences. Bioinformatics 31 (11), 1857-1859.

BibTeX entry:

    @article{Xiao2015,
      author  = {Xiao, Nan and Cao, Dong-Sheng and Zhu, Min-Feng and Xu, Qing-Song.},
      title   = {protr/{ProtrWeb}: {R} package and web server for generating various numerical representation schemes of protein sequences},
      journal = {Bioinformatics},
      year    = {2015},
      volume  = {31},
      number  = {11},
      pages   = {1857--1859},
      doi     = {10.1093/bioinformatics/btv042}
    }

## An example predictive modeling workflow

Here, we use the subcellular localization dataset of human proteins presented in Chou and Shen (2008) to demonstrate the basic usage of protr.

The complete dataset includes 3,134 protein sequences (2,750 different proteins) classified into 14 human subcellular locations. We selected two classes of proteins as our benchmark dataset. Class 1 contains 325 extracell proteins and class 2 includes 307 mitochondrion proteins. We aim to build a random forest classification model to classify these two types of proteins.

First, we load the protr package, then read the protein sequences stored in two separated FASTA files with `readFASTA()`:

    library("protr")

    # Load FASTA files
    # Note that `system.file()` is for accessing example files
    # in the protr package. Replace it with your own file path.
    extracell <- readFASTA(
      system.file(
        "protseq/extracell.fasta",
        package = "protr"
      )
    )
    mitonchon <- readFASTA(
      system.file(
        "protseq/mitochondrion.fasta",
        package = "protr"
      )
    )

To read protein sequences stored in PDB format files, use `readPDB()` instead. The loaded sequences will be stored as two lists in R, and each component is a character string representing one protein sequence. In this case, there are 325 extracell protein sequences and 306 mitonchon protein sequences:

    length(extracell)

    ## [1] 325

    length(mitonchon)

    ## [1] 306

To ensure that the protein sequences only have the 20 standard amino acid types, which is usually required for the descriptor computation, we use the `protcheck()` function to do the amino acid type sanity check and remove the non-standard sequences:

    extracell <- extracell[(sapply(extracell, protcheck))]
    mitonchon <- mitonchon[(sapply(mitonchon, protcheck))]

    length(extracell)

    ## [1] 323

    length(mitonchon)

    ## [1] 304

Two protein sequences were removed from each class. For the remaining sequences, we calculate the Type II PseAAC descriptor, i.e., the amphiphilic pseudo amino acid composition (APseAAC) descriptor (Chou 2005) and make class labels for classification modeling.

    # Calculate APseAAC descriptors
    x1 <- t(sapply(extracell, extractAPAAC))
    x2 <- t(sapply(mitonchon, extractAPAAC))
    x <- rbind(x1, x2)

    # Make class labels
    labels <- as.factor(c(rep(0, length(extracell)), rep(1, length(mitonchon))))

In protr, the functions of commonly used descriptors for protein sequences and proteochemometric (PCM) modeling descriptors are named after `extract...()`.

Next, we will split the data into a 75% training set and a 25% test set.

    set.seed(1001)

    # Split training and test set
    tr.idx <- c(
      sample(1:nrow(x1), round(nrow(x1) * 0.75)),
      sample(nrow(x1) + 1:nrow(x2), round(nrow(x2) * 0.75))
    )
    te.idx <- setdiff(1:nrow(x), tr.idx)

    x.tr <- x[tr.idx, ]
    x.te <- x[te.idx, ]
    y.tr <- labels[tr.idx]
    y.te <- labels[te.idx]

We will train a random forest classification model on the training set with 5-fold cross-validation, using the `randomForest` package.

    library("randomForest")
    rf.fit <- randomForest(x.tr, y.tr, cv.fold = 5)
    rf.fit

The training result is:

    ## Call:
    ##  randomForest(x = x.tr, y = y.tr, cv.fold = 5)
    ##                Type of random forest: classification
    ##                      Number of trees: 500
    ## No. of variables tried at each split: 8
    ##
    ##         OOB estimate of  error rate: 25.11%
    ## Confusion matrix:
    ##     0   1 class.error
    ## 0 196  46   0.1900826
    ## 1  72 156   0.3157895

With the model trained on the training set, we predict on the test set and plot the ROC curve with the `pROC` package, as is shown in Figure 1.

    # Predict on test set
    rf.pred <- predict(rf.fit, newdata = x.te, type = "prob")[, 1]

    # Plot ROC curve
    library("pROC")
    plot.roc(y.te, rf.pred, grid = TRUE, print.auc = TRUE)

The area under the ROC curve (AUC) is:

    ## Call:
    ## plot.roc.default(x = y.te, predictor = rf.pred, col = "#0080ff",
    ##                  grid = TRUE, print.auc = TRUE)
    ##
    ## Data: rf.pred in 81 controls (y.te 0) > 76 cases (y.te 1).
    ## Area under the curve: 0.8697

Figure 1: ROC curve for the test set of protein subcellular localization data.

## Package overview

The protr package (Xiao et al. 2015) implemented most of the state-of-the-art protein sequence descriptors with R. Generally, each type of the descriptor (feature) can be calculated with a function named `extractX()` in the protr package, where `X` stands for the abbreviation of the descriptor name. The descriptors are:

- Amino acid composition
  - `extractAAC()` - Amino acid composition
  - `extractDC()` - Dipeptide composition
  - `extractTC()` - Tripeptide composition
- Autocorrelation
  - `extractMoreauBroto()` - Normalized Moreau-Broto autocorrelation
  - `extractMoran()` - Moran autocorrelation
  - `extractGeary()` - Geary autocorrelation
- CTD descriptors
  - `extractCTDC()` - Composition
  - `extractCTDT()` - Transition
  - `extractCTDD()` - Distribution
- Conjoint triad descriptors
  - `extractCTriad()` - Conjoint triad descriptors
- Quasi-sequence-order descriptors
  - `extractSOCN()` - Sequence-order-coupling number
  - `extractQSO()` - Quasi-sequence-order descriptors
- Pseudo-amino acid composition
  - `extractPAAC()` - Pseudo-amino acid composition (PseAAC)
  - `extractAPAAC()` - Amphiphilic pseudo-amino acid composition (APseAAC)
- Profile-based descriptors
  - `extractPSSM()`
  - `extractPSSMAcc()`
  - `extractPSSMFeature()`

The descriptors commonly used in Proteochemometric Modeling (PCM) implemented in protr include:

- `extractScales()`, `extractScalesGap()` - Scales-based descriptors derived by Principal Components Analysis
  - `extractProtFP()`, `extractProtFPGap()` - Scales-based descriptors derived by amino acid properties from AAindex (a.k.a. Protein Fingerprint)
  - `extractDescScales()` - Scales-based descriptors derived by 20+ classes of 2D and 3D molecular descriptors (Topological, WHIM, VHSE, etc.)
- `extractFAScales()` - Scales-based descriptors derived by Factor Analysis
- `extractMDSScales()` - Scales-based descriptors derived by Multidimensional Scaling
- `extractBLOSUM()` - BLOSUM and PAM matrix-derived descriptors

The protr package integrates the function of parallelized similarity score computation derived by local or global protein sequence alignment between a list of protein sequences; Biostrings and pwalign provide the sequence alignment computation, and the corresponding functions listed in the protr package include:

- `twoSeqSim()` - Similarity calculation derived by sequence alignment between two protein sequences
- `parSeqSim()` - Parallelized pairwise similarity calculation with a list of protein sequences (in-memory version)
- `parSeqSimDisk()` - Parallelized pairwise similarity calculation with a list of protein sequences (disk-based version)
- `crossSetSim()` - Parallelized pairwise similarity calculation between two sets of protein sequences (in-memory version)

protr also integrates the function for parallelized similarity score computation derived by Gene Ontology (GO) semantic similarity measures between a list of GO terms / Entrez Gene IDs, the GO similarity computation is provided by GOSemSim, the corresponding functions listed in the protr package include:

- `twoGOSim()` - Similarity calculation derived by GO-terms semantic similarity measures between two GO terms / Entrez Gene IDs;
- `parGOSim()` - Pairwise similarity calculation with a list of GO terms / Entrez Gene IDs.

To use the `parSeqSim()` function, we suggest the users to install the packages foreach and doParallel first in order to make the parallelized pairwise similarity computation available.

In the following sections, we will introduce the descriptors and function usage in this order.

## Commonly used descriptors

Note: Users need to evaluate the underlying details of the descriptors provided intelligently, instead of using protr with their data blindly, especially for the descriptor types that have more flexibility. It would be wise for the users to use some negative and positive control comparisons where relevant to help guide the interpretation of the results.

A protein or peptide sequence with \(N\) amino acid residues can be generally represented as \(\{\,R_1, R_2, \ldots, R_n\,\}\), where \(R_i\) represents the residue at the \(i\)-th position in the sequence. The labels \(i\) and \(j\) are used to index amino acid position in a sequence, and \(r\), \(s\), \(t\) are used to represent the amino acid type. The computed descriptors are roughly categorized into 4 groups according to their major applications.

A protein sequence can be partitioned equally into segments. The descriptors designed for the complete sequence can often be applied to each individual segment.

### Amino acid composition descriptor

The amino acid composition describes the fraction of each amino acid type within a protein sequence. The fractions of all 20 natural amino acids are calculated as follows:

\[ f(r) = \frac{N_r}{N} \quad r = 1, 2, \ldots, 20. \]

where \(N_r\) is the number of the amino acid type \(r\) and \(N\) is the length of the sequence.

As was described above, we can use the function `extractAAC()` to extract the descriptors (features) from protein sequences:

    library("protr")

    x <- readFASTA(system.file(
      "protseq/P00750.fasta",
      package = "protr"
    ))[[1]]

    extractAAC(x)
    #>          A          R          N          D          C          E          Q
    #> 0.06405694 0.07117438 0.03914591 0.05160142 0.06761566 0.04804270 0.04804270
    #>          G          H          I          L          K          M          F
    #> 0.08185053 0.03024911 0.03558719 0.07651246 0.03914591 0.01245552 0.03202847
    #>          P          S          T          W          Y          V
    #> 0.05338078 0.08896797 0.04448399 0.02313167 0.04270463 0.04982206

Here, using the function `readFASTA()`, we loaded a single protein sequence (P00750, Tissue-type plasminogen activator) from a FASTA format file. Then, we extracted the AAC descriptors with `extractAAC()`. The result returned is a named vector whose elements are tagged with the name of each amino acid.

### Dipeptide composition descriptor

Dipeptide composition gives a 400-dimensional descriptor, defined as:

\[ f(r, s) = \frac{N_{rs}}{N - 1} \quad r, s = 1, 2, \ldots, 20. \]

where \(N_{rs}\) is the number of dipeptides represented by amino acid type \(r\) and type \(s\). Similar to `extractAAC()`, we use `extractDC()` to compute the descriptors:

    dc <- extractDC(x)
    head(dc, n = 30L)
    #>          AA          RA          NA          DA          CA          EA
    #> 0.003565062 0.003565062 0.000000000 0.007130125 0.003565062 0.003565062
    #>          QA          GA          HA          IA          LA          KA
    #> 0.007130125 0.007130125 0.001782531 0.003565062 0.001782531 0.001782531
    #>          MA          FA          PA          SA          TA          WA
    #> 0.000000000 0.005347594 0.003565062 0.007130125 0.003565062 0.000000000
    #>          YA          VA          AR          RR          NR          DR
    #> 0.000000000 0.000000000 0.003565062 0.007130125 0.005347594 0.001782531
    #>          CR          ER          QR          GR          HR          IR
    #> 0.005347594 0.005347594 0.000000000 0.007130125 0.001782531 0.003565062

Here, we only showed the first 30 elements of the result vector and omitted the rest of the result. The element names of the returned vector are self-explanatory, as before.

### Tripeptide composition descriptor

Tripeptide composition gives an 8000-dimensional descriptor, defined as:

\[ f(r, s, t) = \frac{N_{rst}}{N - 2} \quad r, s, t = 1, 2, \ldots, 20 \]

where \(N_{rst}\) is the number of tripeptides represented by amino acid type \(r\), \(s\), and \(t\). With the function `extractTC()`, we can easily obtain the length-8000 descriptor, to save some space, here we also omitted the long outputs:

    tc <- extractTC(x)
    head(tc, n = 36L)
    #>         AAA         RAA         NAA         DAA         CAA         EAA
    #> 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000
    #>         QAA         GAA         HAA         IAA         LAA         KAA
    #> 0.001785714 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000
    #>         MAA         FAA         PAA         SAA         TAA         WAA
    #> 0.000000000 0.000000000 0.000000000 0.001785714 0.000000000 0.000000000
    #>         YAA         VAA         ARA         RRA         NRA         DRA
    #> 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000
    #>         CRA         ERA         QRA         GRA         HRA         IRA
    #> 0.000000000 0.000000000 0.000000000 0.001785714 0.000000000 0.000000000
    #>         LRA         KRA         MRA         FRA         PRA         SRA
    #> 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000

### Autocorrelation descriptors

Autocorrelation descriptors are defined based on the distribution of amino acid properties along the sequence. The amino acid properties used here are various types of amino acids index (Retrieved from the AAindex database, see Kawashima, Ogata, and Kanehisa (1999), Kawashima and Kanehisa (2000), and Kawashima et al. (2008); see Figure 2 for an illustrated example). Three types of autocorrelation descriptors are defined here and described below.

All the amino acid indices are centralized and standardized before the calculation, i.e.

\[ P_r = \frac{P_r - \bar{P}}{\sigma} \]

where \(\bar{P}\) is the average of the property of the 20 amino acids:

\[ \bar{P} = \frac{\sum_{r=1}^{20} P_r}{20} \quad \textrm{and} \quad \sigma = \sqrt{\frac{1}{2} \sum_{r=1}^{20} (P_r - \bar{P})^2} \]

Figure 2: An illustrated example in the AAIndex database.

#### Normalized Moreau-Broto autocorrelation descriptors

For protein sequences, the Moreau-Broto autocorrelation descriptors can be defined as:

\[ AC(d) = \sum_{i=1}^{N-d} P_i P_{i + d} \quad d = 1, 2, \ldots, \textrm{nlag} \]

where \(d\) is called the lag of the autocorrelation; \(P_i\) and \(P_{i+d}\) are the properties of the amino acids at position \(i\) and \(i+d\); \(\textrm{nlag}\) is the maximum value of the lag.

The normalized Moreau-Broto autocorrelation descriptors are defined as:

\[ ATS(d) = \frac{AC(d)}{N-d} \quad d = 1, 2, \ldots, \textrm{nlag} \]

The corresponding function for this descriptor is `extractMoreauBroto()`. A typical call would be:

    moreau <- extractMoreauBroto(x)
    head(moreau, n = 36L)
    #>  CIDH920105.lag1  CIDH920105.lag2  CIDH920105.lag3  CIDH920105.lag4
    #>      0.081573213     -0.016064817     -0.015982990     -0.025739038
    #>  CIDH920105.lag5  CIDH920105.lag6  CIDH920105.lag7  CIDH920105.lag8
    #>      0.079058632     -0.042771564     -0.036320847      0.024087298
    #>  CIDH920105.lag9 CIDH920105.lag10 CIDH920105.lag11 CIDH920105.lag12
    #>     -0.005273958      0.052274763      0.082170073      0.005419919
    #> CIDH920105.lag13 CIDH920105.lag14 CIDH920105.lag15 CIDH920105.lag16
    #>      0.083292042      0.004810584      0.001872446     -0.001531495
    #> CIDH920105.lag17 CIDH920105.lag18 CIDH920105.lag19 CIDH920105.lag20
    #>     -0.011917230      0.071161551      0.033473197      0.026882737
    #> CIDH920105.lag21 CIDH920105.lag22 CIDH920105.lag23 CIDH920105.lag24
    #>      0.073075402      0.115272790      0.041517897     -0.027025993
    #> CIDH920105.lag25 CIDH920105.lag26 CIDH920105.lag27 CIDH920105.lag28
    #>      0.033477388     -0.003245255      0.078117010     -0.028177304
    #> CIDH920105.lag29 CIDH920105.lag30  BHAR880101.lag1  BHAR880101.lag2
    #>      0.046695832      0.020584423      0.052740185      0.030804784
    #>  BHAR880101.lag3  BHAR880101.lag4  BHAR880101.lag5  BHAR880101.lag6
    #>      0.037170476     -0.058993771      0.070641780     -0.089192490

The eight default properties used here are:

- AccNo. CIDH920105 - Normalized Average Hydrophobicity Scales
- AccNo. BHAR880101 - Average Flexibility Indices
- AccNo. CHAM820101 - Polarizability Parameter
- AccNo. CHAM820102 - Free Energy of Solution in Water, kcal/mole
- AccNo. CHOC760101 - Residue Accessible Surface Area in Tripeptide
- AccNo. BIGC670101 - Residue Volume
- AccNo. CHAM810101 - Steric Parameter
- AccNo. DAYM780201 - Relative Mutability

Users can change the property names of the AAindex database with the argument `props`. The AAindex data shipped with protr can be loaded by `data(AAindex)` which has detailed information on each property. With the arguments `customprops` and `nlag`, users can specify their own properties and lag value to calculate with. For example:

    # Define 3 custom properties
    myprops <- data.frame(
      AccNo = c("MyProp1", "MyProp2", "MyProp3"),
      A = c(0.62, -0.5, 15), R = c(-2.53, 3, 101),
      N = c(-0.78, 0.2, 58), D = c(-0.9, 3, 59),
      C = c(0.29, -1, 47), E = c(-0.74, 3, 73),
      Q = c(-0.85, 0.2, 72), G = c(0.48, 0, 1),
      H = c(-0.4, -0.5, 82), I = c(1.38, -1.8, 57),
      L = c(1.06, -1.8, 57), K = c(-1.5, 3, 73),
      M = c(0.64, -1.3, 75), F = c(1.19, -2.5, 91),
      P = c(0.12, 0, 42), S = c(-0.18, 0.3, 31),
      T = c(-0.05, -0.4, 45), W = c(0.81, -3.4, 130),
      Y = c(0.26, -2.3, 107), V = c(1.08, -1.5, 43)
    )

    # Use 4 properties in the AAindex database and 3 customized properties
    moreau2 <- extractMoreauBroto(
      x,
      customprops = myprops,
      props = c(
        "CIDH920105", "BHAR880101",
        "CHAM820101", "CHAM820102",
        "MyProp1", "MyProp2", "MyProp3"
      )
    )

    head(moreau2, n = 36L)
    #>  CIDH920105.lag1  CIDH920105.lag2  CIDH920105.lag3  CIDH920105.lag4
    #>      0.081573213     -0.016064817     -0.015982990     -0.025739038
    #>  CIDH920105.lag5  CIDH920105.lag6  CIDH920105.lag7  CIDH920105.lag8
    #>      0.079058632     -0.042771564     -0.036320847      0.024087298
    #>  CIDH920105.lag9 CIDH920105.lag10 CIDH920105.lag11 CIDH920105.lag12
    #>     -0.005273958      0.052274763      0.082170073      0.005419919
    #> CIDH920105.lag13 CIDH920105.lag14 CIDH920105.lag15 CIDH920105.lag16
    #>      0.083292042      0.004810584      0.001872446     -0.001531495
    #> CIDH920105.lag17 CIDH920105.lag18 CIDH920105.lag19 CIDH920105.lag20
    #>     -0.011917230      0.071161551      0.033473197      0.026882737
    #> CIDH920105.lag21 CIDH920105.lag22 CIDH920105.lag23 CIDH920105.lag24
    #>      0.073075402      0.115272790      0.041517897     -0.027025993
    #> CIDH920105.lag25 CIDH920105.lag26 CIDH920105.lag27 CIDH920105.lag28
    #>      0.033477388     -0.003245255      0.078117010     -0.028177304
    #> CIDH920105.lag29 CIDH920105.lag30  BHAR880101.lag1  BHAR880101.lag2
    #>      0.046695832      0.020584423      0.052740185      0.030804784
    #>  BHAR880101.lag3  BHAR880101.lag4  BHAR880101.lag5  BHAR880101.lag6
    #>      0.037170476     -0.058993771      0.070641780     -0.089192490

About the standard input format of `props` and `customprops`, see `?extractMoreauBroto` for details.

#### Moran autocorrelation descriptors

For protein sequences, the Moran autocorrelation descriptors can be defined as:

\[ I(d) = \frac{\frac{1}{N-d} \sum_{i=1}^{N-d} (P_i - \bar{P}') (P_{i+d} - \bar{P}')}{\frac{1}{N} \sum_{i=1}^{N} (P_i - \bar{P}')^2} \quad d = 1, 2, \ldots, 30 \]

where \(d\), \(P_i\), and \(P_{i+d}\) are defined in the same way as in the first place; \(\bar{P}'\) is the considered property \(P\) along the sequence, i.e.,

\[ \bar{P}' = \frac{\sum_{i=1}^N P_i}{N} \]

\(d\), \(P\), \(P_i\) and \(P_{i+d}\), \(\textrm{nlag}\) have the same meaning as above.

With `extractMoran()` (which has the identical parameters as `extractMoreauBroto()`), we can compute the Moran autocorrelation descriptors (only print out the first 36 elements):

    # Use the 3 custom properties defined before
    # and 4 properties in the AAindex database
    moran <- extractMoran(
      x,
      customprops = myprops,
      props = c(
        "CIDH920105", "BHAR880101",
        "CHAM820101", "CHAM820102",
        "MyProp1", "MyProp2", "MyProp3"
      )
    )

    head(moran, n = 36L)
    #>  CIDH920105.lag1  CIDH920105.lag2  CIDH920105.lag3  CIDH920105.lag4
    #>      0.062895724     -0.044827681     -0.045065117     -0.055955678
    #>  CIDH920105.lag5  CIDH920105.lag6  CIDH920105.lag7  CIDH920105.lag8
    #>      0.060586377     -0.074128412     -0.067308852     -0.001293384
    #>  CIDH920105.lag9 CIDH920105.lag10 CIDH920105.lag11 CIDH920105.lag12
    #>     -0.033747588      0.029392193      0.061789800     -0.023368437
    #> CIDH920105.lag13 CIDH920105.lag14 CIDH920105.lag15 CIDH920105.lag16
    #>      0.062769417     -0.024912264     -0.028298043     -0.031584063
    #> CIDH920105.lag17 CIDH920105.lag18 CIDH920105.lag19 CIDH920105.lag20
    #>     -0.043466730      0.047830694      0.005883901     -0.001769769
    #> CIDH920105.lag21 CIDH920105.lag22 CIDH920105.lag23 CIDH920105.lag24
    #>      0.049334048      0.096427969      0.015147594     -0.060092509
    #> CIDH920105.lag25 CIDH920105.lag26 CIDH920105.lag27 CIDH920105.lag28
    #>      0.007549152     -0.033987885      0.056307675     -0.061844453
    #> CIDH920105.lag29 CIDH920105.lag30  BHAR880101.lag1  BHAR880101.lag2
    #>      0.021484780     -0.008461776      0.014229951     -0.009142419
    #>  BHAR880101.lag3  BHAR880101.lag4  BHAR880101.lag5  BHAR880101.lag6
    #>     -0.003272262     -0.109613332      0.033346233     -0.141538598

#### Geary autocorrelation descriptors

Geary autocorrelation descriptors for protein sequences can be defined as:

\[ C(d) = \frac{\frac{1}{2(N-d)} \sum_{i=1}^{N-d} (P_i - P_{i+d})^2}{\frac{1}{N-1} \sum_{i=1}^{N} (P_i - \bar{P}')^2} \quad d = 1, 2, \ldots, 30 \]

where \(d\), \(P\), \(P_i\), \(P_{i+d}\), and \(\textrm{nlag}\) have the same meaning as above.

For each amino acid index, there will be \(3 \times \textrm{nlag}\) autocorrelation descriptors. The usage of `extractGeary()` is exactly the same as `extractMoreauBroto()` and `extractMoran()`:

    # Use the 3 custom properties defined before
    # and 4 properties in the AAindex database
    geary <- extractGeary(
      x,
      customprops = myprops,
      props = c(
        "CIDH920105", "BHAR880101",
        "CHAM820101", "CHAM820102",
        "MyProp1", "MyProp2", "MyProp3"
      )
    )

    head(geary, n = 36L)
    #>  CIDH920105.lag1  CIDH920105.lag2  CIDH920105.lag3  CIDH920105.lag4
    #>        0.9361830        1.0442920        1.0452843        1.0563467
    #>  CIDH920105.lag5  CIDH920105.lag6  CIDH920105.lag7  CIDH920105.lag8
    #>        0.9406031        1.0765517        1.0675786        0.9991363
    #>  CIDH920105.lag9 CIDH920105.lag10 CIDH920105.lag11 CIDH920105.lag12
    #>        1.0316555        0.9684585        0.9353130        1.0201990
    #> CIDH920105.lag13 CIDH920105.lag14 CIDH920105.lag15 CIDH920105.lag16
    #>        0.9340933        1.0207373        1.0251486        1.0290464
    #> CIDH920105.lag17 CIDH920105.lag18 CIDH920105.lag19 CIDH920105.lag20
    #>        1.0414375        0.9494403        0.9905987        0.9987183
    #> CIDH920105.lag21 CIDH920105.lag22 CIDH920105.lag23 CIDH920105.lag24
    #>        0.9472542        0.9010009        0.9828848        1.0574098
    #> CIDH920105.lag25 CIDH920105.lag26 CIDH920105.lag27 CIDH920105.lag28
    #>        0.9897955        1.0290018        0.9400066        1.0584150
    #> CIDH920105.lag29 CIDH920105.lag30  BHAR880101.lag1  BHAR880101.lag2
    #>        0.9762904        1.0029734        0.9818711        1.0051730
    #>  BHAR880101.lag3  BHAR880101.lag4  BHAR880101.lag5  BHAR880101.lag6
    #>        0.9967069        1.1012905        0.9595859        1.1337056

### Composition/Transition/Distribution

The CTD descriptors are developed by Dubchak et al. (1995) and Dubchak et al. (1999).

Figure 3: The sequence of a hypothetic protein indicating the construction of composition, transition, and distribution descriptors of a protein. The sequence index indicates the position of an amino acid in the sequence. The index for each type of amino acid in the sequence (`1`, `2` or `3`) indicates the position of the first, second, third, … of that type of amino acid. 1/2 transition indicates the position of `12` or `21` pairs in the sequence (1/3 and 2/3 are defined similarly).

Step 1: Sequence Encoding

The amino acids are categorized into three classes according to their attributes, and each amino acid is encoded by one of the indices 1, 2, 3 according to which class it belongs. The attributes used here include hydrophobicity, normalized van der Waals volume, polarity, and polarizability. The corresponding classification for each attribute is listed in Table 1.

Table 1: Amino acid attributes and the three-group classification of the 20 amino acids by each attribute
| Group 1 | Group 2 | Group 3
--- | --- | --- | ---
Hydrophobicity | Polar | Neutral | Hydrophobicity
| R, K, E, D, Q, N | G, A, S, T, P, H, Y | C, L, V, I, M, F, W
Normalized van der Waals Volume | 0-2.78 | 2.95-4.0 | 4.03-8.08
| G, A, S, T, P, D, C | N, V, E, Q, I, L | M, H, K, F, R, Y, W
Polarity | 4.9-6.2 | 8.0-9.2 | 10.4-13.0
| L, I, F, W, C, M, V, Y | P, A, T, G, S | H, Q, R, K, N, E, D
Polarizability | 0-1.08 | 0.128-0.186 | 0.219-0.409
| G, A, S, D, T | C, P, N, V, E, Q, I, L | K, M, H, F, R, Y, W
Charge | Positive | Neutral | Negative
| K, R | A, N, C, Q, G, H, I, L, M, F, P, S, T, W, Y, V | D, E
Secondary Structure | Helix | Strand | Coil
| E, A, L, M, Q, K, R, H | V, I, Y, C, W, F, T | G, N, P, S, D
Solvent Accessibility | Buried | Exposed | Intermediate
| A, L, F, C, G, I, V, W | R, K, Q, E, N, D | M, S, P, T, H, Y

For example, for a given sequence `MTEITAAMVKELRESTGAGA`, it will be encoded as `32132223311311222222` according to its hydrophobicity.

Step 2: Compute Composition, Transition, and Distribution Descriptors

Three types of descriptors, Composition (C), Transition (T), and Distribution (D) can be calculated for a given attribute as follows.

#### Composition

Composition is defined as the global percentage for each encoded class in the protein sequence. In the above example, using the hydrophobicity classification, the numbers for encoded classes `1`, `2`, `3` are 5, 10, and 5, so that the compositions for them will be \(5/20=25\%\), \(10/20=10\%\), and \(5/20=25\%\), where 20 is the length of the protein sequence. The composition descriptor can be expressed as

\[ C_r = \frac{n_r}{n} \quad r = 1, 2, 3 \]

where \(n_r\) is the number of amino acid type \(r\) in the encoded sequence; \(N\) is the length of the sequence. An example for `extractCTDC()`:

    extractCTDC(x)
    #>  hydrophobicity.Group1  hydrophobicity.Group2  hydrophobicity.Group3
    #>             0.29715302             0.40569395             0.29715302
    #> normwaalsvolume.Group1 normwaalsvolume.Group2 normwaalsvolume.Group3
    #>             0.45195730             0.29715302             0.25088968
    #>        polarity.Group1        polarity.Group2        polarity.Group3
    #>             0.33985765             0.33274021             0.32740214
    #>  polarizability.Group1  polarizability.Group2  polarizability.Group3
    #>             0.33096085             0.41814947             0.25088968
    #>          charge.Group1          charge.Group2          charge.Group3
    #>             0.11032028             0.79003559             0.09964413
    #> secondarystruct.Group1 secondarystruct.Group2 secondarystruct.Group3
    #>             0.38967972             0.29537367             0.31494662
    #>   solventaccess.Group1   solventaccess.Group2   solventaccess.Group3
    #>             0.43060498             0.29715302             0.27224199

The result shows the elements that are named as `PropertyNumber.GroupNumber` in the returned vector.

#### Transition

A transition from class 1 to 2 is the percent frequency with which 1 is followed by 2 or 2 is followed by 1 in the encoded sequences. The transition descriptor can be computed as

\[ T_{rs} = \frac{n_{rs} + n_{sr}}{N - 1} \quad rs = \text{12}, \text{13}, \text{23} \]

where \(n_{rs}\), \(n_{sr}\) are the numbers of dipeptide encoded as `rs` and `sr` in the sequence; \(N\) is the length of the sequence. An example for `extractCTDT()`:

    extractCTDT(x)
    #> prop1.Tr1221 prop1.Tr1331 prop1.Tr2332 prop2.Tr1221 prop2.Tr1331 prop2.Tr2332
    #>   0.27094474   0.16042781   0.23351159   0.26737968   0.22638146   0.17112299
    #> prop3.Tr1221 prop3.Tr1331 prop3.Tr2332 prop4.Tr1221 prop4.Tr1331 prop4.Tr2332
    #>   0.21033868   0.20499109   0.23707665   0.27272727   0.15151515   0.24598930
    #> prop5.Tr1221 prop5.Tr1331 prop5.Tr2332 prop6.Tr1221 prop6.Tr1331 prop6.Tr2332
    #>   0.18181818   0.02139037   0.15686275   0.21925134   0.22816399   0.15864528
    #> prop7.Tr1221 prop7.Tr1331 prop7.Tr2332
    #>   0.25133690   0.21568627   0.18003565

#### Distribution

The distribution descriptor describes the distribution of each attribute in the sequence.

There are five “distribution” descriptors for each attribute, and they are the position percent in the whole sequence for the first residue, 25% residues, 50% residues, 75% residues, and 100% residues for a certain encoded class. For example, there are 10 residues encoded as `2` in the above example, the positions for the first residue `2`, the 2nd residue `2` (25% * 10 = 2), the 5th `2` residue (50% * 10 = 5), the 7th `2` (75% * 10 = 7) and the 10th residue `2` (100% * 10) in the encoded sequence are 2, 5, 15, 17, 20, so that the distribution descriptors for `2` are: 10.0 (2/20 * 100), 25.0 (5/20 * 100), 75.0 (15/20 * 100), 85.0 (17/20 * 100), 100.0 (20/20 * 100).

To compute the distribution descriptor, use `extractCTDD()`:

    extractCTDD(x)
    #>   prop1.G1.residue0  prop1.G1.residue25  prop1.G1.residue50  prop1.G1.residue75
    #>           0.3558719          23.1316726          50.1779359          73.8434164
    #> prop1.G1.residue100   prop1.G2.residue0  prop1.G2.residue25  prop1.G2.residue50
    #>          99.8220641           0.5338078          27.4021352          47.3309609
    #>  prop1.G2.residue75 prop1.G2.residue100   prop1.G3.residue0  prop1.G3.residue25
    #>          75.2669039         100.0000000           0.1779359          19.5729537
    #>  prop1.G3.residue50  prop1.G3.residue75 prop1.G3.residue100   prop2.G1.residue0
    #>          51.7793594          75.6227758          99.6441281           0.3558719
    #>  prop2.G1.residue25  prop2.G1.residue50  prop2.G1.residue75 prop2.G1.residue100
    #>          25.6227758          48.0427046          75.4448399         100.0000000
    #>   prop2.G2.residue0  prop2.G2.residue25  prop2.G2.residue50  prop2.G2.residue75
    #>           1.4234875          23.3096085          54.4483986          76.3345196
    #> prop2.G2.residue100   prop2.G3.residue0  prop2.G3.residue25  prop2.G3.residue50
    #>          99.4661922           0.1779359          22.7758007          48.9323843
    #>  prop2.G3.residue75 prop2.G3.residue100   prop3.G1.residue0  prop3.G1.residue25
    #>          69.5729537          99.8220641           0.1779359          20.9964413
    #>  prop3.G1.residue50  prop3.G1.residue75 prop3.G1.residue100   prop3.G2.residue0
    #>          50.8896797          74.5551601          99.6441281           0.5338078
    #>  prop3.G2.residue25  prop3.G2.residue50  prop3.G2.residue75 prop3.G2.residue100
    #>          26.5124555          46.2633452          75.4448399         100.0000000
    #>   prop3.G3.residue0  prop3.G3.residue25  prop3.G3.residue50  prop3.G3.residue75
    #>           0.3558719          24.1992883          50.5338078          73.8434164
    #> prop3.G3.residue100   prop4.G1.residue0  prop4.G1.residue25  prop4.G1.residue50
    #>          99.8220641           0.3558719          26.5124555          48.3985765
    #>  prop4.G1.residue75 prop4.G1.residue100   prop4.G2.residue0  prop4.G2.residue25
    #>          76.1565836          99.2882562           1.4234875          21.5302491
    #>  prop4.G2.residue50  prop4.G2.residue75 prop4.G2.residue100   prop4.G3.residue0
    #>          51.4234875          75.8007117         100.0000000           0.1779359
    #>  prop4.G3.residue25  prop4.G3.residue50  prop4.G3.residue75 prop4.G3.residue100
    #>          22.7758007          48.9323843          69.5729537          99.8220641
    #>   prop5.G1.residue0  prop5.G1.residue25  prop5.G1.residue50  prop5.G1.residue75
    #>           0.8896797          20.8185053          48.9323843          69.5729537
    #> prop5.G1.residue100   prop5.G2.residue0  prop5.G2.residue25  prop5.G2.residue50
    #>          99.8220641           0.1779359          24.9110320          49.1103203
    #>  prop5.G2.residue75 prop5.G2.residue100   prop5.G3.residue0  prop5.G3.residue25
    #>          75.2669039         100.0000000           0.3558719          26.1565836
    #>  prop5.G3.residue50  prop5.G3.residue75 prop5.G3.residue100   prop6.G1.residue0
    #>          64.2348754          77.4021352          99.2882562           0.1779359
    #>  prop6.G1.residue25  prop6.G1.residue50  prop6.G1.residue75 prop6.G1.residue100
    #>          22.9537367          50.8896797          74.3772242          99.8220641
    #>   prop6.G2.residue0  prop6.G2.residue25  prop6.G2.residue50  prop6.G2.residue75
    #>           1.6014235          21.5302491          49.2882562          70.8185053
    #> prop6.G2.residue100   prop6.G3.residue0  prop6.G3.residue25  prop6.G3.residue50
    #>          98.9323843           0.3558719          29.0035587          48.2206406
    #>  prop6.G3.residue75 prop6.G3.residue100   prop7.G1.residue0  prop7.G1.residue25
    #>          77.4021352         100.0000000           0.5338078          23.4875445
    #>  prop7.G1.residue50  prop7.G1.residue75 prop7.G1.residue100   prop7.G2.residue0
    #>          50.0000000          74.5551601          98.9323843           0.3558719
    #>  prop7.G2.residue25  prop7.G2.residue50  prop7.G2.residue75 prop7.G2.residue100
    #>          23.1316726          50.1779359          73.8434164          99.8220641
    #>   prop7.G3.residue0  prop7.G3.residue25  prop7.G3.residue50  prop7.G3.residue75
    #>           0.1779359          27.2241993          48.0427046          75.4448399
    #> prop7.G3.residue100
    #>         100.0000000

### Conjoint triad descriptors

Conjoint triad descriptors are proposed by Shen et al. (2007). The conjoint triad descriptors were used to model protein-protein interactions based on the classification of amino acids. In this approach, each protein sequence is represented by a vector space consisting of descriptors of amino acids. To reduce the dimensions of vector space, the 20 amino acids were clustered into several classes according to their dipoles and volumes of the side chains. The conjoint triad descriptors are calculated as follows:

Step 1: Classification of Amino Acids

Electrostatic and hydrophobic interactions dominate protein-protein interactions. These two kinds of interactions may be reflected by the dipoles and volumes of the side chains of amino acids, respectively. Accordingly, these two parameters were calculated using the density-functional theory method B3LYP/6-31G and the molecular modeling approach. Based on the dipoles and volumes of the side chains, the 20 amino acids can be clustered into seven classes (See Table 2). Amino acids within the same class likely involve synonymous mutations because of their similar characteristics.

Table 2: Classification of amino acids based on dipoles and volumes of the side chains
No. | Dipole Scale\(^1\) | Volume Scale\(^2\) | Class
--- | --- | --- | ---
1 | \(-\) | \(-\) | Ala, Gly, Val
2 | \(-\) | \(+\) | Ile, Leu, Phe, Pro
3 | \(+\) | \(+\) | Tyr, Met, Thr, Ser
4 | \(++\) | \(+\) | His, Asn, Gln, Tpr
5 | \(+++\) | \(+\) | Arg, Lys
6 | \(+'+'+'\) | \(+\) | Asp, Glu
7 | \(+^3\) | \(+\) | Cys

1 Dipole Scale (Debye): \(-\), Dipole < 1.0; \(+\), 1.0 < Dipole < 2.0; \(++\), 2.0 < Dipole < 3.0; \(+++\), Dipole > 3.0; \(+'+'+'\), Dipole > 3.0 with opposite orientation.

2 Volume Scale (\(\overset{\lower.5em\circ}{\mathrm{A}}\lower.01em^3\)): \(-\), Volume < 50; \(+\), Volume > 50.

3 Cys is separated from class 3 because of its ability to form disulfide bonds.

Step 2: Conjoint Triad Calculation

The conjoint triad descriptors considered the properties of one amino acid and its vicinal amino acids and regarded any three continuous amino acids as a unit. Thus, the triads can be differentiated according to the classes of amino acids, i.e., triads composed of three amino acids belonging to the same classes, such as ART and VKS, can be treated identically. To conveniently represent a protein, we first use a binary space \((\mathbf{V}, \mathbf{F})\) to represent a protein sequence. Here, \(\mathbf{V}\) is the vector space of the sequence features, and each feature \(v_i\) represents a sort of triad type; \(\mathbf{F}\) is the frequency vector corresponding to \(\mathbf{V}\), and the value of the \(i\)-th dimension of \(\mathbf{F} (f_i)\) is the frequency of type \(v_i\) appearing in the protein sequence. For the amino acids that have been categorized into seven classes, the size of \(\mathbf{V}\) should be \(7 \times 7 \times 7\); thus \(i = 1, 2, \ldots, 343\). The detailed description for (\(\mathbf{V}\), \(\mathbf{F}\)) is illustrated in Figure 4.

Figure 4: Schematic diagram for constructing the vector space (\(\mathbf{V}\), \(\mathbf{F}\)) of protein sequences. \(\mathbf{V}\) is the vector space of the sequence features; each feature (\(v_i\)) represents a triad composed of three consecutive amino acids; \(\mathbf{F}\) is the frequency vector corresponding to \(\mathbf{V}\), and the value of the \(i\)-th dimension of \(\mathbf{F} (f_i)\) is the frequency that \(v_i\) triad appeared in the protein sequence.

Clearly, each protein correlates to the length (number of amino acids) of protein. In general, a long protein would have a large value of \(f_i\), which complicates the comparison between two heterogeneous proteins. Thus, we defined a new parameter, \(d_i\), by normalizing \(f_i\) with the following equation:

\[ d_i = \frac{f_i - \min\{\,f_1, f_2 , \ldots, f_{343}\,\}}{\max\{\,f_1, f_2, \ldots, f_{343}\,\}} \]

The numerical value of \(d_i\) of each protein ranges from 0 to 1, enabling the comparison between proteins. Accordingly, we obtain another vector space (designated \(\mathbf{D}\)) consisting of \(d_i\) to represent protein.

To compute conjoint triads of protein sequences, we can use:

    ctriad <- extractCTriad(x)
    head(ctriad, n = 65L)
    #> VS111 VS211 VS311 VS411 VS511 VS611 VS711 VS121 VS221 VS321 VS421 VS521 VS621
    #>   0.1   0.3   0.6   0.2   0.4   0.0   0.3   1.0   0.6   0.5   0.0   0.2   0.3
    #> VS721 VS131 VS231 VS331 VS431 VS531 VS631 VS731 VS141 VS241 VS341 VS441 VS541
    #>   0.0   0.2   0.4   0.5   0.2   0.3   0.3   0.1   0.3   0.3   0.2   0.2   0.0
    #> VS641 VS741 VS151 VS251 VS351 VS451 VS551 VS651 VS751 VS161 VS261 VS361 VS461
    #>   0.1   0.2   0.2   0.2   0.5   0.1   0.2   0.0   0.0   0.1   0.4   0.2   0.3
    #> VS561 VS661 VS761 VS171 VS271 VS371 VS471 VS571 VS671 VS771 VS112 VS212 VS312
    #>   0.2   0.0   0.1   0.1   0.3   0.1   0.0   0.1   0.0   0.1   0.8   0.4   0.4
    #> VS412 VS512 VS612 VS712 VS122 VS222 VS322 VS422 VS522 VS622 VS722 VS132 VS232
    #>   0.6   0.1   0.5   0.2   0.8   0.5   0.2   0.3   0.2   0.0   0.2   0.1   0.3

by which we only outputted the first 65 of 343 dimensions to save space.

### Quasi-sequence-order descriptors

The quasi-sequence-order descriptors are proposed by Chou (2000). They are derived from the distance matrix between the 20 amino acids.

#### Sequence-order-coupling number

The \(d\)-th rank sequence-order-coupling number is defined as:

\[ \tau_d = \sum_{i=1}^{N-d} (d_{i, i+d})^2 \quad d = 1, 2, \ldots, \textrm{maxlag} \]

where \(d_{i, i+d}\) is the distance between the two amino acids at position \(i\) and \(i+d\).

Note: maxlag is the maximum lag and the length of the protein must be not less than \(\textrm{maxlag}\).

The function `extractSOCN()` is used for computing the sequence-order-coupling numbers:

    extractSOCN(x)
    #>  Schneider.lag1  Schneider.lag2  Schneider.lag3  Schneider.lag4  Schneider.lag5
    #>        204.2036        199.8708        206.8102        197.4828        193.3366
    #>  Schneider.lag6  Schneider.lag7  Schneider.lag8  Schneider.lag9 Schneider.lag10
    #>        208.1936        195.5476        200.9789        196.7110        193.9931
    #> Schneider.lag11 Schneider.lag12 Schneider.lag13 Schneider.lag14 Schneider.lag15
    #>        199.7031        204.9389        187.0140        198.4702        205.4526
    #> Schneider.lag16 Schneider.lag17 Schneider.lag18 Schneider.lag19 Schneider.lag20
    #>        193.1274        187.3529        190.4949        202.8853        198.5299
    #> Schneider.lag21 Schneider.lag22 Schneider.lag23 Schneider.lag24 Schneider.lag25
    #>        191.1013        185.0074        189.9857        202.7113        201.6267
    #> Schneider.lag26 Schneider.lag27 Schneider.lag28 Schneider.lag29 Schneider.lag30
    #>        194.5770        185.9939        204.1297        191.1629        183.9073
    #>   Grantham.lag1   Grantham.lag2   Grantham.lag3   Grantham.lag4   Grantham.lag5
    #>    6674686.0000    6761609.0000    7138892.0000    6748261.0000    6291229.0000
    #>   Grantham.lag6   Grantham.lag7   Grantham.lag8   Grantham.lag9  Grantham.lag10
    #>    6839853.0000    6594164.0000    6556148.0000    6620183.0000    6770614.0000
    #>  Grantham.lag11  Grantham.lag12  Grantham.lag13  Grantham.lag14  Grantham.lag15
    #>    6495689.0000    6865537.0000    6297267.0000    6498247.0000    6615566.0000
    #>  Grantham.lag16  Grantham.lag17  Grantham.lag18  Grantham.lag19  Grantham.lag20
    #>    6572680.0000    6569081.0000    6173947.0000    6570829.0000    6471308.0000
    #>  Grantham.lag21  Grantham.lag22  Grantham.lag23  Grantham.lag24  Grantham.lag25
    #>    6461649.0000    5939432.0000    6532121.0000    6652472.0000    6480660.0000
    #>  Grantham.lag26  Grantham.lag27  Grantham.lag28  Grantham.lag29  Grantham.lag30
    #>    6382281.0000    6276521.0000    6537634.0000    6442991.0000    6350157.0000

Users can also specify the maximum lag value with the `nlag` argument.

Note: Besides the Schneider-Wrede physicochemical distance matrix (Schneider and Wrede 1994) used by Kuo-Chen Chou, another chemical distance matrix by Grantham (1974) is also used here. So the descriptors dimension will be `nlag * 2`. The quasi-sequence-order descriptors described next also utilized the two matrices.

#### Quasi-sequence-order descriptors

For each amino acid type, a quasi-sequence-order descriptor can be defined as:

\[ X_r = \frac{f_r}{\sum_{r=1}^{20} f_r + w \sum_{d=1}^{\textrm{maxlag}} \tau_d} \quad r = 1, 2, \ldots, 20 \]

where \(f_r\) is the normalized occurrence for amino acid type \(i\) and \(w\) is a weighting factor (\(w=0.1\)). These are the first 20 quasi-sequence-order descriptors. The other 30 quasi-sequence-order are defined as:

\[ X_d = \frac{w \tau_{d-20}}{\sum_{r=1}^{20} f_r + w \sum_{d=1}^{\textrm{maxlag}} \tau_d} \quad d = 21, 22, \ldots, 20 + \textrm{maxlag} \]

Figure 5: A schematic drawing to show (a) the 1st-rank, (b) the 2nd-rank, and (3) the 3rd-rank sequence-order-coupling mode along a protein sequence. (a) Reflects the coupling mode between all the most contiguous residues, (b) that between all the 2nd most contiguous residues, and (c) that between all the 3rd most contiguous residues. This figure is from Chou (2000).

A minimal example for `extractQSO()`:

    extractQSO(x)
    #>  Schneider.Xr.A  Schneider.Xr.R  Schneider.Xr.N  Schneider.Xr.D  Schneider.Xr.C
    #>    6.096218e-02    6.773576e-02    3.725467e-02    4.910842e-02    6.434897e-02
    #>  Schneider.Xr.E  Schneider.Xr.Q  Schneider.Xr.G  Schneider.Xr.H  Schneider.Xr.I
    #>    4.572164e-02    4.572164e-02    7.789612e-02    2.878770e-02    3.386788e-02
    #>  Schneider.Xr.L  Schneider.Xr.K  Schneider.Xr.M  Schneider.Xr.F  Schneider.Xr.P
    #>    7.281594e-02    3.725467e-02    1.185376e-02    3.048109e-02    5.080182e-02
    #>  Schneider.Xr.S  Schneider.Xr.T  Schneider.Xr.W  Schneider.Xr.Y  Schneider.Xr.V
    #>    8.466970e-02    4.233485e-02    2.201412e-02    4.064145e-02    4.741503e-02
    #>   Grantham.Xr.A   Grantham.Xr.R   Grantham.Xr.N   Grantham.Xr.D   Grantham.Xr.C
    #>    1.835033e-06    2.038926e-06    1.121409e-06    1.478221e-06    1.936980e-06
    #>   Grantham.Xr.E   Grantham.Xr.Q   Grantham.Xr.G   Grantham.Xr.H   Grantham.Xr.I
    #>    1.376275e-06    1.376275e-06    2.344765e-06    8.665435e-07    1.019463e-06
    #>   Grantham.Xr.L   Grantham.Xr.K   Grantham.Xr.M   Grantham.Xr.F   Grantham.Xr.P
    #>    2.191845e-06    1.121409e-06    3.568120e-07    9.175167e-07    1.529194e-06
    #>   Grantham.Xr.S   Grantham.Xr.T   Grantham.Xr.W   Grantham.Xr.Y   Grantham.Xr.V
    #>    2.548657e-06    1.274329e-06    6.626509e-07    1.223356e-06    1.427248e-06
    #>  Schneider.Xd.1  Schneider.Xd.2  Schneider.Xd.3  Schneider.Xd.4  Schneider.Xd.5
    #>    3.457972e-02    3.384600e-02    3.502111e-02    3.344162e-02    3.273951e-02
    #>  Schneider.Xd.6  Schneider.Xd.7  Schneider.Xd.8  Schneider.Xd.9 Schneider.Xd.10
    #>    3.525537e-02    3.311390e-02    3.403364e-02    3.331093e-02    3.285068e-02
    #> Schneider.Xd.11 Schneider.Xd.12 Schneider.Xd.13 Schneider.Xd.14 Schneider.Xd.15
    #>    3.381760e-02    3.470422e-02    3.166883e-02    3.360882e-02    3.479121e-02
    #> Schneider.Xd.16 Schneider.Xd.17 Schneider.Xd.18 Schneider.Xd.19 Schneider.Xd.20
    #>    3.270408e-02    3.172623e-02    3.225829e-02    3.435647e-02    3.361893e-02
    #> Schneider.Xd.21 Schneider.Xd.22 Schneider.Xd.23 Schneider.Xd.24 Schneider.Xd.25
    #>    3.236099e-02    3.132904e-02    3.217206e-02    3.432701e-02    3.414334e-02
    #> Schneider.Xd.26 Schneider.Xd.27 Schneider.Xd.28 Schneider.Xd.29 Schneider.Xd.30
    #>    3.294954e-02    3.149609e-02    3.456720e-02    3.237140e-02    3.114275e-02
    #>   Grantham.Xd.1   Grantham.Xd.2   Grantham.Xd.3   Grantham.Xd.4   Grantham.Xd.5
    #>    3.402298e-02    3.446605e-02    3.638918e-02    3.439801e-02    3.206838e-02
    #>   Grantham.Xd.6   Grantham.Xd.7   Grantham.Xd.8   Grantham.Xd.9  Grantham.Xd.10
    #>    3.486488e-02    3.361253e-02    3.341875e-02    3.374516e-02    3.451195e-02
    #>  Grantham.Xd.11  Grantham.Xd.12  Grantham.Xd.13  Grantham.Xd.14  Grantham.Xd.15
    #>    3.311057e-02    3.499580e-02    3.209915e-02    3.312361e-02    3.372162e-02
    #>  Grantham.Xd.16  Grantham.Xd.17  Grantham.Xd.18  Grantham.Xd.19  Grantham.Xd.20
    #>    3.350302e-02    3.348467e-02    3.147055e-02    3.349358e-02    3.298629e-02
    #>  Grantham.Xd.21  Grantham.Xd.22  Grantham.Xd.23  Grantham.Xd.24  Grantham.Xd.25
    #>    3.293706e-02    3.027516e-02    3.329628e-02    3.390974e-02    3.303396e-02
    #>  Grantham.Xd.26  Grantham.Xd.27  Grantham.Xd.28  Grantham.Xd.29  Grantham.Xd.30
    #>    3.253250e-02    3.199340e-02    3.332438e-02    3.284195e-02    3.236875e-02

where users can also specify the maximum lag with the argument `nlag` and the weighting factor with the argument `w`.

### Pseudo-amino acid composition (PseAAC)

This group of descriptors is proposed by Chou (2001). PseAAC descriptors are also named as the type 1 pseudo-amino acid composition. Let \(H_1^o (i)\), \(H_2^o (i)\), \(M^o (i)\) (\(i=1, 2, 3, \ldots, 20\)) be the original hydrophobicity values, the original hydrophilicity values and the original side chain masses of the 20 natural amino acids, respectively. They are converted to the following qualities by a standard conversion:

\[ H_1 (i) = \frac{H_1^o (i) - \frac{1}{20} \sum_{i=1}^{20} H_1^o (i)}{\sqrt{\frac{\sum_{i=1}^{20} [H_1^o (i) - \frac{1}{20} \sum_{i=1}^{20} H_1^o (i) ]^2}{20}}} \]

\(H_2^o (i)\) and \(M^o (i)\) are normalized as \(H_2 (i)\) and \(M (i)\) in the same way.

Figure 6: A schematic drawing to show (a) the first-tier, (b) the second-tier, and (3) the third-tier sequence order correlation mode along a protein sequence. Panel (a) reflects the correlation mode between all the most contiguous residues, panel (b) between all the second-most contiguous residues, and panel (c) between all the third-most contiguous residues. This figure is from Chou (2001).

Then, a correlation function can be defined as

\[ \Theta (R_i, R_j) = \frac{1}{3} \bigg\{ [ H_1 (R_i) - H_1 (R_j) ]^2 + [ H_2 (R_i) - H_2 (R_j) ]^2 + [ M (R_i) - M (R_j) ]^2 \bigg\} \]

This correlation function is an average value for the three amino acid properties: hydrophobicity, hydrophilicity, and side chain mass. Therefore, we can extend this definition of correlation functions for one amino acid property or for a set of \(n\) amino acid properties.

For one amino acid property, the correlation can be defined as:

\[ \Theta (R_i, R_j) = [H_1 (R_i) - H_1(R_j)]^2 \]

where \(H (R_i)\) is the amino acid property of amino acid \(R_i\) after standardization.

For a set of n amino acid properties, it can be defined as: where \(H_k (R_i)\) is the \(k\)-th property in the amino acid property set for amino acid \(R_i\).

\[ \Theta (R_i, R_j) = \frac{1}{n} \sum_{k=1}^{n} [H_k (R_i) - H_k (R_j)]^2 \]

where \(H_k(R_i)\) is the \(k\)-th property in the amino acid property set for amino acid \(R_i\).

A set of descriptors named sequence order-correlated factors are defined as:

\[\begin{align*} \theta_1 & = \frac{1}{N-1} \sum_{i=1}^{N-1} \Theta (R_i, R_{i+1})\\ \theta_2 & = \frac{1}{N-2} \sum_{i=1}^{N-2} \Theta (R_i, R_{i+2})\\ \theta_3 & = \frac{1}{N-3} \sum_{i=1}^{N-3} \Theta (R_i, R_{i+3})\\ & \ldots \\ \theta_\lambda & = \frac{1}{N-\lambda} \sum_{i=1}^{N-\lambda} \Theta (R_i, R_{i+\lambda}) \end{align*}\]

\(\lambda\) (\(\lambda < L\)) is a parameter to be specified. Let \(f_i\) be the normalized occurrence frequency of the 20 amino acids in the protein sequence, a set of \(20 + \lambda\) descriptors called the pseudo-amino acid composition for a protein sequence can be defined as:

\[ X_c = \frac{f_c}{\sum_{r=1}^{20} f_r + w \sum_{j=1}^{\lambda} \theta_j} \quad (1 < c < 20) \]

\[ X_c = \frac{w \theta_{c-20}}{\sum_{r=1}^{20} f_r + w \sum_{j=1}^{\lambda} \theta_j} \quad (21 < c < 20 + \lambda) \]

where \(w\) is the weighting factor for the sequence-order effect and is set to \(w = 0.05\) in protr as suggested by Kuo-Chen Chou.

With `extractPAAC()`, we can compute the PseAAC descriptors directly:

    extractPAAC(x)
    #>         Xc1.A         Xc1.R         Xc1.N         Xc1.D         Xc1.C
    #>    9.07025432   10.07806035    5.54293319    7.30659376    9.57415734
    #>         Xc1.E         Xc1.Q         Xc1.G         Xc1.H         Xc1.I
    #>    6.80269074    6.80269074   11.58976941    4.28317565    5.03903018
    #>         Xc1.L         Xc1.K         Xc1.M         Xc1.F         Xc1.P
    #>   10.83391488    5.54293319    1.76366056    4.53512716    7.55854527
    #>         Xc1.S         Xc1.T         Xc1.W         Xc1.Y         Xc1.V
    #>   12.59757544    6.29878772    3.27536961    6.04683621    7.05464225
    #>  Xc2.lambda.1  Xc2.lambda.2  Xc2.lambda.3  Xc2.lambda.4  Xc2.lambda.5
    #>    0.02514092    0.02500357    0.02527773    0.02553159    0.02445265
    #>  Xc2.lambda.6  Xc2.lambda.7  Xc2.lambda.8  Xc2.lambda.9 Xc2.lambda.10
    #>    0.02561910    0.02486131    0.02506656    0.02553952    0.02437663
    #> Xc2.lambda.11 Xc2.lambda.12 Xc2.lambda.13 Xc2.lambda.14 Xc2.lambda.15
    #>    0.02491262    0.02533803    0.02351915    0.02479912    0.02548431
    #> Xc2.lambda.16 Xc2.lambda.17 Xc2.lambda.18 Xc2.lambda.19 Xc2.lambda.20
    #>    0.02478210    0.02513770    0.02457224    0.02543046    0.02500889
    #> Xc2.lambda.21 Xc2.lambda.22 Xc2.lambda.23 Xc2.lambda.24 Xc2.lambda.25
    #>    0.02476967    0.02342389    0.02431684    0.02610300    0.02626722
    #> Xc2.lambda.26 Xc2.lambda.27 Xc2.lambda.28 Xc2.lambda.29 Xc2.lambda.30
    #>    0.02457082    0.02343049    0.02588823    0.02490463    0.02451951

The `extractPAAC()` function also provides the additional arguments `props` and `customprops`, which are similar to those arguments for Moreau-Broto/Moran/Geary autocorrelation descriptors. For their minor differences, please see `?extracPAAC`. Users can specify the lambda parameter and the weighting factor with the arguments `lambda` and `w`.

Note: In the work of Kuo-Chen Chou, the definition for “normalized occurrence frequency” was not given. In this work, we define it as the occurrence frequency of amino acids in the sequence normalized to 100%, and hence, our calculated values are not the same as their values.

### Amphiphilic pseudo-amino acid composition (APseAAC)

Amphiphilic Pseudo-Amino Acid Composition (APseAAC) was proposed in Chou (2001). APseAAC is also recognized as the type 2 pseudo-amino acid composition. The definitions of these qualities are similar to the PAAC descriptors. From \(H_1 (i)\) and \(H_2 (j)\) defined before, the hydrophobicity and hydrophilicity correlation functions are defined as:

\[\begin{align*} H_{i, j}^1 & = H_1 (i) H_1 (j)\\ H_{i, j}^2 & = H_2 (i) H_2 (j) \end{align*}\]

From these qualities, sequence order factors can be defined as:

\[\begin{align*} \tau_1 & = \frac{1}{N-1} \sum_{i=1}^{N-1} H_{i, i+1}^1\\ \tau_2 & = \frac{1}{N-1} \sum_{i=1}^{N-1} H_{i, i+1}^2\\ \tau_3 & = \frac{1}{N-2} \sum_{i=1}^{N-2} H_{i, i+2}^1\\ \tau_4 & = \frac{1}{N-2} \sum_{i=1}^{N-2} H_{i, i+2}^2\\ & \ldots \\ \tau_{2 \lambda - 1} & = \frac{1}{N-\lambda} \sum_{i=1}^{N-\lambda} H_{i, i+\lambda}^1\\ \tau_{2 \lambda} & = \frac{1}{N-\lambda} \sum_{i=1}^{N-\lambda} H_{i, i+\lambda}^2 \end{align*}\]

Figure 7: A schematic diagram to show (a1/a2) the first-rank, (b1/b2) the second-rank and (c1/c2) the third-rank sequence-order-coupling mode along a protein sequence through a hydrophobicity/hydrophilicity correlation function, where \(H_{i, j}^1\) and \(H_{i, j}^2\) are given by Equation (3). Panel (a1/a2) reflects the coupling mode between all the most contiguous residues, panel (b1/b2) between all the second-most contiguous residues, and panel (c1/c2) between all the third-most contiguous residues. This figure is from Chou (2005).

Then a set of descriptors called Amphiphilic Pseudo-Amino Acid Composition (APseAAC) are defined as:

\[ P_c = \frac{f_c}{\sum_{r=1}^{20} f_r + w \sum_{j=1}^{2 \lambda} \tau_j} \quad (1 < c < 20) \]

\[ P_c = \frac{w \tau_u}{\sum_{r=1}^{20} f_r + w \sum_{j=1}^{2 \lambda} \tau_j} \quad (21 < u < 20 + 2 \lambda) \]

where \(w\) is the weighting factor, its default value is set to \(w = 0.5\) in protr.

A minimal example for `extracAPAAC()` is:

    extractAPAAC(x)
    #>                 Pc1.A                 Pc1.R                 Pc1.N
    #>          3.537412e+01          3.930458e+01          2.161752e+01
    #>                 Pc1.D                 Pc1.C                 Pc1.E
    #>          2.849582e+01          3.733935e+01          2.653059e+01
    #>                 Pc1.Q                 Pc1.G                 Pc1.H
    #>          2.653059e+01          4.520027e+01          1.670445e+01
    #>                 Pc1.I                 Pc1.L                 Pc1.K
    #>          1.965229e+01          4.225242e+01          2.161752e+01
    #>                 Pc1.M                 Pc1.F                 Pc1.P
    #>          6.878302e+00          1.768706e+01          2.947844e+01
    #>                 Pc1.S                 Pc1.T                 Pc1.W
    #>          4.913073e+01          2.456536e+01          1.277399e+01
    #>                 Pc1.Y                 Pc1.V  Pc2.Hydrophobicity.1
    #>          2.358275e+01          2.751321e+01          2.196320e-04
    #>  Pc2.Hydrophilicity.1  Pc2.Hydrophobicity.2  Pc2.Hydrophilicity.2
    #>          1.025766e-03         -3.088876e-04         -1.834385e-04
    #>  Pc2.Hydrophobicity.3  Pc2.Hydrophilicity.3  Pc2.Hydrophobicity.4
    #>          1.174146e-03          7.400156e-04         -1.105715e-03
    #>  Pc2.Hydrophilicity.4  Pc2.Hydrophobicity.5  Pc2.Hydrophilicity.5
    #>         -4.493680e-04          1.766358e-03          1.471212e-03
    #>  Pc2.Hydrophobicity.6  Pc2.Hydrophilicity.6  Pc2.Hydrophobicity.7
    #>         -1.441572e-03         -4.913600e-03         -1.678053e-05
    #>  Pc2.Hydrophilicity.7  Pc2.Hydrophobicity.8  Pc2.Hydrophilicity.8
    #>          7.312356e-04         -1.885399e-03         -1.928708e-03
    #>  Pc2.Hydrophobicity.9  Pc2.Hydrophilicity.9 Pc2.Hydrophobicity.10
    #>         -2.931177e-03         -1.555660e-03          2.916597e-03
    #> Pc2.Hydrophilicity.10 Pc2.Hydrophobicity.11 Pc2.Hydrophilicity.11
    #>          3.602591e-03          1.055082e-04          8.697920e-04
    #> Pc2.Hydrophobicity.12 Pc2.Hydrophilicity.12 Pc2.Hydrophobicity.13
    #>         -9.276413e-04         -2.001384e-03          1.705044e-03
    #> Pc2.Hydrophilicity.13 Pc2.Hydrophobicity.14 Pc2.Hydrophilicity.14
    #>          4.364007e-03          7.883453e-04         -9.441693e-04
    #> Pc2.Hydrophobicity.15 Pc2.Hydrophilicity.15 Pc2.Hydrophobicity.16
    #>         -3.133437e-04         -3.599332e-03          3.689079e-05
    #> Pc2.Hydrophilicity.16 Pc2.Hydrophobicity.17 Pc2.Hydrophilicity.17
    #>          2.483867e-03          4.832798e-04          2.465788e-03
    #> Pc2.Hydrophobicity.18 Pc2.Hydrophilicity.18 Pc2.Hydrophobicity.19
    #>         -3.142728e-04          2.021961e-03          6.421283e-05
    #> Pc2.Hydrophilicity.19 Pc2.Hydrophobicity.20 Pc2.Hydrophilicity.20
    #>         -8.896690e-04         -2.986886e-04          9.304039e-04
    #> Pc2.Hydrophobicity.21 Pc2.Hydrophilicity.21 Pc2.Hydrophobicity.22
    #>         -6.777458e-04          1.646818e-03          3.193506e-03
    #> Pc2.Hydrophilicity.22 Pc2.Hydrophobicity.23 Pc2.Hydrophilicity.23
    #>          3.270656e-03          2.533569e-03          2.478252e-03
    #> Pc2.Hydrophobicity.24 Pc2.Hydrophilicity.24 Pc2.Hydrophobicity.25
    #>         -2.489106e-03         -1.031008e-03         -3.992322e-03
    #> Pc2.Hydrophilicity.25 Pc2.Hydrophobicity.26 Pc2.Hydrophilicity.26
    #>         -2.596060e-03          8.690771e-04         -1.221378e-03
    #> Pc2.Hydrophobicity.27 Pc2.Hydrophilicity.27 Pc2.Hydrophobicity.28
    #>          5.208649e-03          4.617400e-03         -1.088584e-03
    #> Pc2.Hydrophilicity.28 Pc2.Hydrophobicity.29 Pc2.Hydrophilicity.29
    #>         -2.512263e-03          1.387641e-03          2.060890e-03
    #> Pc2.Hydrophobicity.30 Pc2.Hydrophilicity.30
    #>          3.177340e-04          1.451909e-03

This function has the same arguments as `extractPAAC()`.

### Profile-based descriptors

The profile-based descriptors for protein sequences are available in the protr package. The feature vectors of profile-based methods were based on the PSSM by running PSI-BLAST and often show good performance. See Ye, Wang, and Altschul (2011) and Rangwala and Karypis (2005) for details. The functions `extractPSSM()`, `extractPSSMAcc()`, and `extractPSSMFeature()` are used to generate these descriptors. Users need to install the NCBI-BLAST+ software package first to make the functions fully functional.

## Proteochemometric modeling descriptors

Proteochemometric (PCM) modeling utilizes statistical modeling to model the ligand-target interaction space. The below descriptors implemented in protr are extensively used in proteochemometric modeling.

- Scales-based descriptors derived by Principal Components Analysis

  - Scales-based descriptors derived by Amino Acid Properties from AAindex (Protein Fingerprint)
  - Scales-based descriptors derived by 20+ classes of 2D and 3D molecular descriptors (Topological, WHIM, VHSE, etc.)

- Scales-based descriptors derived by Factor Analysis

- Scales-based descriptors derived by Multidimensional Scaling

- BLOSUM and PAM matrix-derived descriptors

Note that every scales-based descriptor function can freely combine more than 20 classes of 2D and 3D molecular descriptors to construct highly customized scales-based descriptors. Of course, these functions are designed to be flexible enough that users can provide totally self-defined property matrices to construct scales-based descriptors.

For example, to compute the “topological scales” derived by PCA (using the first 5 principal components), one can use `extractDescScales()`:

    x <- readFASTA(system.file(
      "protseq/P00750.fasta",
      package = "protr"
    ))[[1]]

    descscales <- extractDescScales(
      x,
      propmat = "AATopo",
      index = c(37:41, 43:47),
      pc = 5, lag = 7, silent = FALSE
    )
    #> Summary of the first 5 principal components:
    #>                             PC1      PC2       PC3       PC4        PC5
    #> Standard deviation     2.581537 1.754133 0.4621854 0.1918666 0.08972087
    #> Proportion of Variance 0.666430 0.307700 0.0213600 0.0036800 0.00080000
    #> Cumulative Proportion  0.666430 0.974130 0.9954900 0.9991700 0.99998000

The `propmat` argument invokes the `AATopo` dataset shipped with the protr package. The argument `index` selects the 37 to 41 and the 43 to 47 columns (molecular descriptors) in the `AATopo` dataset to use. The parameter `lag` was set for the Auto Cross Covariance (ACC) to generate scales-based descriptors of the same length. At last, we printed the summary of the first 5 principal components (standard deviation, proportion of variance, cumulative proportion of variance).

The result is a length 175 named vector, which is consistent with the descriptors before:

    length(descscales)
    #> [1] 175
    head(descscales, 15)
    #>     scl1.lag1     scl2.lag1     scl3.lag1     scl4.lag1     scl5.lag1
    #> -2.645644e-01 -1.717847e-02  1.975438e-02 -7.930659e-05 -3.710597e-05
    #>     scl1.lag2     scl2.lag2     scl3.lag2     scl4.lag2     scl5.lag2
    #>  3.548612e-01  1.343712e-01  5.699395e-03 -5.489472e-04 -6.364577e-05
    #>     scl1.lag3     scl2.lag3     scl3.lag3     scl4.lag3     scl5.lag3
    #>  2.011431e-02 -9.211136e-02 -1.461755e-03  6.747801e-04  2.386782e-04

For another example, to compute the descriptors derived by the BLOSUM62 matrix and use the first 5 scales, one can use:

    x <- readFASTA(system.file(
      "protseq/P00750.fasta",
      package = "protr"
    ))[[1]]

    blosum <- extractBLOSUM(
      x,
      submat = "AABLOSUM62",
      k = 5, lag = 7, scale = TRUE, silent = FALSE
    )
    #> Relative importance of all the possible 20 scales:
    #>  [1]  1.204960e+01  7.982007e+00  6.254364e+00  4.533706e+00  4.326286e+00
    #>  [6]  3.850579e+00  3.752197e+00  3.538207e+00  3.139155e+00  2.546405e+00
    #> [11]  2.373286e+00  1.666259e+00  1.553126e+00  1.263685e+00  1.024699e+00
    #> [16]  9.630187e-01  9.225759e-01  7.221636e-01  1.020085e-01 -6.314726e-16

The result is a length 175 named vector:

    length(blosum)
    #> [1] 175
    head(blosum, 15)
    #>     scl1.lag1     scl2.lag1     scl3.lag1     scl4.lag1     scl5.lag1
    #>  0.0042370555 -0.0021502057  0.0005993291  0.0006456375  0.0014849592
    #>     scl1.lag2     scl2.lag2     scl3.lag2     scl4.lag2     scl5.lag2
    #> -0.0014919096  0.0032873726  0.0011734162 -0.0021758536 -0.0018127568
    #>     scl1.lag3     scl2.lag3     scl3.lag3     scl4.lag3     scl5.lag3
    #> -0.0029413528  0.0001494193  0.0003298806 -0.0017877430 -0.0051044133

Dealing with gaps. In proteochemometrics (sequence alignment), gaps can be very useful since a gap in a certain position contains information. The protr package has built-in support for such gaps. We deal with the gaps by using a dummy descriptor to code for the 21st type of amino acid. The function `extractScalesGap()` and `extractProtFPGap()` can be used to deal with such gaps. See `?extractScalesGap` and `?extractProtFPGap` for details.

## Similarity calculation by sequence alignment

Similarity computation derived by local or global protein sequence alignment between a list of protein sequences is of great need in protein research. However, this type of pairwise similarity computation is often computationally intensive, especially when many protein sequences exist. Luckily, this process is also highly parallelizable. The protr package integrates the function of parallelized similarity computation derived by local or global protein sequence alignment between a list of protein sequences.

The function `twoSeqSim()` calculates the alignment result between two protein sequences. The function `parSeqSim()` calculates the pairwise similarity with a list of protein sequences in parallel:

    s1 <- readFASTA(system.file("protseq/P00750.fasta", package = "protr"))[[1]]
    s2 <- readFASTA(system.file("protseq/P08218.fasta", package = "protr"))[[1]]
    s3 <- readFASTA(system.file("protseq/P10323.fasta", package = "protr"))[[1]]
    s4 <- readFASTA(system.file("protseq/P20160.fasta", package = "protr"))[[1]]
    s5 <- readFASTA(system.file("protseq/Q9NZP8.fasta", package = "protr"))[[1]]
    plist <- list(s1, s2, s3, s4, s5)
    psimmat <- parSeqSim(plist, cores = 4, type = "local", submat = "BLOSUM62")
    psimmat

    ##            [,1]       [,2]       [,3]       [,4]       [,5]
    ## [1,] 1.00000000 0.11825938 0.10236985 0.04921696 0.03943488
    ## [2,] 0.11825938 1.00000000 0.18858241 0.12124217 0.06391103
    ## [3,] 0.10236985 0.18858241 1.00000000 0.05819984 0.06175942
    ## [4,] 0.04921696 0.12124217 0.05819984 1.00000000 0.05714638
    ## [5,] 0.03943488 0.06391103 0.06175942 0.05714638 1.00000000

Similarly, the function `crossSetSim()` calculates the pairwise similarity between two sets of protein sequences in parallel.

Note that for a small number of proteins, calculating their pairwise similarity scores derived by sequence alignment in parallel may not significantly reduce the overall computation time since each task only requires a relatively short time to finish. Thus, computational overheads may exist and affect the performance. In our benchmarks, we used about 1,000 protein sequences on 64 CPU cores and observed significant performance improvement compared to the sequential computation.

### Scaling up for a large number of sequences

`batches`. A useful argument for reducing memory usage is `batches`. It splits the similarity computations into smaller batches. A larger batch number reduces the memory need for each parallel process but also adds the overhead of forking new processes. This is a trade-off between time (CPU) and space (memory). Also, use `verbose = TRUE` to print the computation progress.

`parSeqSimDisk()`. For similarity computations over an even larger number of protein sequences, please try `parSeqSimDisk()` and set `batches` to a large number. Unlike the in-memory version, this disk-based function caches the partial results in each batch to the hard drive and merges all the results together in the end, which further reduces memory usage.

Example. Let’s assume we have 20,000 sequences. By setting `cores = 64` and `batches = 10000` in `parSeqSimDisk()`, the workload will be around `20000^2/2/64/10000 = 312.5` alignments per batch per core. This could give you relatively quick feedback about whether the memory size is going to be sufficient, and how long the computation will take in total after the first few batches.

## Similarity calculation by GO semantic similarity measures

The protr package also integrates the function for similarity score computation derived by Gene Ontology (GO) semantic similarity measures between a list of GO terms or Entrez Gene IDs.

The function `twoGOSim()` calculates the similarity derived by GO terms semantic similarity measures between two GO terms / Entrez Gene IDs, and the function `parGOSim()` calculates the pairwise similarity with a list of GO terms / Entrez Gene IDs:

    # By GO Terms
    go1 <- c(
      "GO:0005215", "GO:0005488", "GO:0005515",
      "GO:0005625", "GO:0005802", "GO:0005905"
    ) # AP4B1
    go2 <- c(
      "GO:0005515", "GO:0005634", "GO:0005681",
      "GO:0008380", "GO:0031202"
    ) # BCAS2
    go3 <- c(
      "GO:0003735", "GO:0005622", "GO:0005840",
      "GO:0006412"
    ) # PDE4DIP
    golist <- list(go1, go2, go3)
    parGOSim(golist, type = "go", ont = "CC", measure = "Wang")

    ##       [,1]  [,2] [,3]
    ## [1,] 1.000 0.344 0.17
    ## [2,] 0.344 1.000 0.24
    ## [3,] 0.170 0.240 1.00

    # By Entrez gene id
    genelist <- list(c("150", "151", "152", "1814", "1815", "1816"))
    parGOSim(genelist, type = "gene", ont = "BP", measure = "Wang")

    ##        150   151   152  1814  1815  1816
    ## 150  1.000 0.702 0.725 0.496 0.570 0.455
    ## 151  0.702 1.000 0.980 0.456 0.507 0.551
    ## 152  0.725 0.980 1.000 0.460 0.511 0.538
    ## 1814 0.496 0.456 0.460 1.000 0.687 0.473
    ## 1815 0.570 0.507 0.511 0.687 1.000 0.505
    ## 1816 0.455 0.551 0.538 0.473 0.505 1.000

## Miscellaneous tools

In this section, we will briefly introduce some useful tools the protr package provides.

### Retrieve protein sequences from UniProt

This function `getUniProt()` gets protein sequences from uniprot.org by protein ID(s). The input `ID` is a character vector specifying the protein ID(s). The returned sequences are stored in a list:

    ids <- c("P00750", "P00751", "P00752")
    prots <- getUniProt(ids)
    prots

    ## [[1]]
    ## [1] "MDAMKRGLCCVLLLCGAVFVSPSQEIHARFRRGARSYQVICRDEKTQMIYQQHQSWLRPVLRSNRVEYCWCN
    ## SGRAQCHSVPVKSCSEPRCFNGGTCQQALYFSDFVCQCPEGFAGKCCEIDTRATCYEDQGISYRGTWSTAESGAECT
    ## NWNSSALAQKPYSGRRPDAIRLGLGNHNYCRNPDRDSKPWCYVFKAGKYSSEFCSTPACSEGNSDCYFGNGSAYRGT
    ## HSLTESGASCLPWNSMILIGKVYTAQNPSAQALGLGKHNYCRNPDGDAKPWCHVLKNRRLTWEYCDVPSCSTCGLRQ
    ## YSQPQFRIKGGLFADIASHPWQAAIFAKHRRSPGERFLCGGILISSCWILSAAHCFQERFPPHHLTVILGRTYRVVP
    ## GEEEQKFEVEKYIVHKEFDDDTYDNDIALLQLKSDSSRCAQESSVVRTVCLPPADLQLPDWTECELSGYGKHEALSP
    ## FYSERLKEAHVRLYPSSRCTSQHLLNRTVTDNMLCAGDTRSGGPQANLHDACQGDSGGPLVCLNDGRMTLVGIISWG
    ## LGCGQKDVPGVYTKVTNYLDWIRDNMRP"
    ##
    ## [[2]]
    ## [1] "MGSNLSPQLCLMPFILGLLSGGVTTTPWSLARPQGSCSLEGVEIKGGSFRLLQEGQALEYVCPSGFYPYPVQ
    ## TRTCRSTGSWSTLKTQDQKTVRKAECRAIHCPRPHDFENGEYWPRSPYYNVSDEISFHCYDGYTLRGSANRTCQVNG
    ## RWSGQTAICDNGAGYCSNPGIPIGTRKVGSQYRLEDSVTYHCSRGLTLRGSQRRTCQEGGSWSGTEPSCQDSFMYDT
    ## PQEVAEAFLSSLTETIEGVDAEDGHGPGEQQKRKIVLDPSGSMNIYLVLDGSDSIGASNFTGAKKCLVNLIEKVASY
    ## GVKPRYGLVTYATYPKIWVKVSEADSSNADWVTKQLNEINYEDHKLKSGTNTKKALQAVYSMMSWPDDVPPEGWNRT
    ## RHVIILMTDGLHNMGGDPITVIDEIRDLLYIGKDRKNPREDYLDVYVFGVGPLVNQVNINALASKKDNEQHVFKVKD
    ## MENLEDVFYQMIDESQSLSLCGMVWEHRKGTDYHKQPWQAKISVIRPSKGHESCMGAVVSEYFVLTAAHCFTVDDKE
    ## HSIKVSVGGEKRDLEIEVVLFHPNYNINGKKEAGIPEFYDYDVALIKLKNKLKYGQTIRPICLPCTEGTTRALRLPP
    ## TTTCQQQKEELLPAQDIKALFVSEEEKKLTRKEVYIKNGDKKGSCERDAQYAPGYDKVKDISEVVTPRFLCTGGVSP
    ## YADPNTCRGDSGGPLIVHKRSRFIQVGVISWGVVDVCKNQKRQKQVPAHARDFHINLFQVLPWLKEKLQDEDLGFL"
    ##
    ## [[3]]
    ## [1] "APPIQSRIIGGRECEKNSHPWQVAIYHYSSFQCGGVLVNPKWVLTAAHCKNDNYEVWLGRHNLFENENTAQF
    ## FGVTADFPHPGFNLSLLKXHTKADGKDYSHDLMLLRLQSPAKITDAVKVLELPTQEPELGSTCEASGWGSIEPGPDB
    ## FEFPDEIQCVQLTLLQNTFCABAHPBKVTESMLCAGYLPGGKDTCMGDSGGPLICNGMWQGITSWGHTPCGSANKPS
    ## IYTKLIFYLDWINDTITENP"

### Read FASTA files

The function `readFASTA()` provides a convenient way to read protein sequences stored in FASTA format files. See `?readFASTA` for details. The returned sequences are stored in a named list, whose components are named with the protein sequences’ names.

### Read PDB files

The Protein Data Bank (PDB) file format is a text file format that describes the three-dimensional structures of proteins. The `readPDB()` function allows you to read protein sequences stored in PDB format files. See `?readPDB` for details.

### Sanity check for amino acid types

The `protcheck()` function checks if the protein sequence’s amino acid types are in the 20 default types, which returns `TRUE` if all the amino acids in the sequence belong to the 20 default types:

    x <- readFASTA(system.file("protseq/P00750.fasta", package = "protr"))[[1]]
    # A real sequence
    protcheck(x)
    #> [1] TRUE
    # An artificial sequence
    protcheck(paste(x, "Z", sep = ""))
    #> [1] FALSE

### Remove gaps from sequences

The `removeGaps()` function allows us to remove/replace gaps (`-`) or any “irregular” characters from amino acid sequences to prepare them for feature extraction or sequence alignment-based similarity computation.

### Protein sequence partition

The `protseg()` function partitions the protein sequences to create sliding windows. This is usually required when creating feature vectors for machine learning tasks. Users can specify a sequence `x`, a character `aa` (one of the 20 amino acid types), and a positive integer `k`, which controls the window size (half of the window).

This function returns a named list. Each component contains one of the segmentations (a character string). Names of the list components are the positions of the specified amino acid in the sequence.

    protseg(x, aa = "M", k = 5)
    #> $`48`
    #> [1] "DEKTQMIYQQH"
    #>
    #> $`242`
    #> [1] "LPWNSMILIGK"
    #>
    #> $`490`
    #> [1] "TVTDNMLCAGD"
    #>
    #> $`525`
    #> [1] "LNDGRMTLVGI"

### Auto cross covariance computation

Auto Cross Covariance (ACC) is extensively used in the scales-based descriptors computation, this approach calculates the auto covariance and auto cross covariance for generating scale-based descriptors of the same length. Users can write their own scales-based descriptor functions with the help of `acc()` function in the protr package.

### Pre-computed 2D and 3D descriptor sets for the 20 amino acids

The protr package ships with more than 20 pre-computed 2D and 3D descriptor sets for the 20 amino acids to use with the scales-based descriptors. Please use `data(package = "protr")` to list all the datasets included in the protr package.

### BLOSUM and PAM matrices for the 20 amino acids

The BLOSUM and PAM matrices for the 20 amino acids can be used to calculate BLOSUM and PAM matrix-derived descriptors with function `extractBLOSUM()`, the datasets are named in `AABLOSUM45`, `AABLOSUM50`, `AABLOSUM62`, `AABLOSUM80`, `AABLOSUM100`, `AAPAM30`, `AAPAM40`, `AAPAM70`, `AAPAM120`, and `AAPAM250`.

### Metadata of the 20 amino acids

As the reference, the `AAMetaInfo` dataset contains metadata of the 20 amino acids used for the 2D and 3D descriptor calculation in the protr package. This dataset includes each amino acid’s name, one-letter representation, three-letter representation, SMILE representation, PubChem CID, and PubChem link. See `data(AAMetaInfo)` for details.

## ProtrWeb

The web application built on protr, namely ProtrWeb, is available at http://protr.org.

ProtrWeb does not require the user to have any programming knowledge. It is a user-friendly web application that computes the protein sequence descriptors in the protr package.

Figure 8: A screenshot of the web application ProtrWeb.

Source code repository for this Shiny web application: https://github.com/nanxstats/protrweb.

## Summary

The summary of the descriptors in the protr package is listed in Table 3.

Table 3: List of commonly used descriptors in protr
Descriptor Group | Descriptor Name | Descriptor Dimension | Function Name
--- | --- | --- | ---
Amino Acid Composition | Amino Acid Composition | 20 | extractAAC()
| Dipeptide Composition | 400 | extractDC()
| Tripeptide Composition | 8000 | extractTC()
Autocorrelation | Normalized Moreau-Broto Autocorrelation | 240\(^1\) | extractMoreauBroto()
| Moran Autocorrelation | 240\(^1\) | extractMoran()
| Geary Autocorrelation | 240\(^1\) | extractGeary()
CTD | Composition | 21 | extractCTDC(), extractCTDCClass()
| Transition | 21 | extractCTDT(), extractCTDTClass()
| Distribution | 105 | extractCTDD(), extractCTDDClass()
Conjoint Triad | Conjoint Triad | 343 | extractCTriad(), extractCTriadClass()
Quasi-Sequence-Order | Sequence-Order-Coupling Number | 60\(^2\) | extractSOCN()
| Quasi-Sequence-Order Descriptors | 100\(^2\) | extractQSO()
Pseudo-Amino Acid Composition | Pseudo-Amino Acid Composition | 50\(^3\) | extractPAAC()
| Amphiphilic Pseudo-Amino Acid Composition | 80\(^4\) | extractAPAAC()

1 The number depends on the choice of the number of properties of amino acids and the choice of the maximum values of the `lag`. The default is to use 8 types of properties and `lag = 30`.

2 The number depends on the maximum value of `lag`. By default, `lag = 30`. Two distance matrices are used, so the descriptor dimension is \(30 \times 2 = 60\) and \((20 + 30) \times 2 = 100\).

3 The number depends on the choice of the number of the set of amino acid properties and the choice of the \(\lambda\) value. The default is to use 3 types of properties proposed by Kuo-Chen Chou and \(\lambda = 30\).

4 The number depends on the choice of the \(\lambda\) vlaue. The default is \(\lambda = 30\).

The scales-based PCM descriptors in the protr package are listed in Table 4.

Table 4: List of PCM descriptors in protr
Derived by | Descriptor Class | Function Name |
--- | --- | --- | ---
Principal Components Analysis | Scales-based descriptors derived by Principal Components Analysis | extractScales(), extractScalesGap() |
| Scales-based descriptors derived by amino acid properties from AAindex (a.k.a. Protein Fingerprint) | extractProtFP(), extractProtFPGap() |
| Scales-based descriptors derived by 2D and 3D molecular descriptors (Topological, WHIM, VHSE, etc.) | extractDescScales() |
Factor Analysis | Scales-based descriptors derived by Factor Analysis | extractFAScales() |
Multidimensional Scaling | Scales-based descriptors derived by Multidimensional Scaling | extractMDSScales() |
Substitution Matrix | BLOSUM and PAM matrix-derived descriptors | extractBLOSUM() |

The amino acid descriptor sets used by scales-based descriptors provided by the protr package are listed in Table 5. Note that the non-informative descriptors (like the descriptors have only one value across all the 20 amino acids) in these datasets have already been filtered out.

Table 5: List of the pre-calculated descriptor sets of the 20 amino acids in protr
Dataset Name | Descriptor Set Name | Dimensionality | Calculated by
--- | --- | --- | ---
AA2DACOR | 2D Autocorrelations Descriptors | 92 | Dragon
AA3DMoRSE | 3D-MoRSE Descriptors | 160 | Dragon
AAACF | Atom-Centred Fragments Descriptors | 6 | Dragon
AABurden | Burden Eigenvalues Descriptors | 62 | Dragon
AAConn | Connectivity Indices Descriptors | 33 | Dragon
AAConst | Constitutional Descriptors | 23 | Dragon
AAEdgeAdj | Edge Adjacency Indices Descriptors | 97 | Dragon
AAEigIdx | Eigenvalue-Based Indices Descriptors | 44 | Dragon
AAFGC | Functional Group Counts Descriptors | 5 | Dragon
AAGeom | Geometrical Descriptors | 41 | Dragon
AAGETAWAY | GETAWAY Descriptors | 194 | Dragon
AAInfo | Information Indices Descriptors | 47 | Dragon
AAMolProp | Molecular Properties Descriptors | 12 | Dragon
AARandic | Randic Molecular Profiles Descriptors | 41 | Dragon
AARDF | RDF Descriptors | 82 | Dragon
AATopo | Topological Descriptors | 78 | Dragon
AATopoChg | Topological Charge Indices Descriptors | 15 | Dragon
AAWalk | Walk and Path Counts Descriptors | 40 | Dragon
AAWHIM | WHIM Descriptors | 99 | Dragon
AACPSA | CPSA Descriptors | 41 | Accelrys Discovery Studio
AADescAll | All the 2D Descriptors Calculated by Dragon | 1171 | Dragon
AAMOE2D | All the 2D Descriptors Calculated by MOE | 148 | MOE
AAMOE3D | All the 3D Descriptors Calculated by MOE | 143 | MOE

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