# Introduction

This document illustrates some of the *phangorn* (Schliep 2011) specialized features which are
useful but maybe not as well-known or just not (yet) described
elsewhere. This is mainly interesting for someone who wants to explore
different models or set up some simulation studies. We show how to
construct data objects for different character states other than
nucleotides or amino acids or how to set up different models to estimate
transition rate.

The vignette *Trees* describes in detail how to estimate
phylogenies from nucleotide or amino acids.

# User defined data formats

To better understand how to define our own data type it is useful to
know a bit more about the internal representation of `phyDat`

objects. The internal representation of `phyDat`

object is
very similar to `factor`

objects.

As an example we will show here several possibilities to define nucleotide data with gaps defined as a fifth state. Ignoring gaps or coding them as ambiguous sites - as it is done in most programs, also in phangorn as default - may be misleading (see (Warnow 2012)). When the number of gaps is low and the gaps are missing at random coding gaps as separate state may be not important.

Let assume we have given a matrix where each row contains a character vector of a taxonomic unit:

`library(phangorn)`

`## Lade nötiges Paket: ape`

```
<- matrix(c("r","a","y","g","g","a","c","-","c","t","c","g",
data "a","a","t","g","g","a","t","-","c","t","c","a",
"a","a","t","-","g","a","c","c","c","t","?","g"),
dimnames = list(c("t1", "t2", "t3"),NULL), nrow=3, byrow=TRUE)
data
```

```
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
## t1 "r" "a" "y" "g" "g" "a" "c" "-" "c" "t" "c" "g"
## t2 "a" "a" "t" "g" "g" "a" "t" "-" "c" "t" "c" "a"
## t3 "a" "a" "t" "-" "g" "a" "c" "c" "c" "t" "?" "g"
```

Normally we would transform this matrix into an phyDat object and gaps are handled as ambiguous character like “?”.

```
<- phyDat(data)
gapsdata1 gapsdata1
```

```
## 3 sequences with 12 character and 11 different site patterns.
## The states are a c g t
```

Now we will define a “USER” defined object and have to supply a vector levels of the character states for the new data, in our case the for nucleotide states and the gap. Additional we can define ambiguous states which can be any of the states.

```
<- phyDat(data, type="USER", levels=c("a","c","g","t","-"),
gapsdata2 ambiguity = c("?", "n"))
```

```
## Warning in phyDat.default(data, levels = levels, return.index = return.index, :
## Found unknown characters (not supplied in levels). Deleted sites with unknown
## states.
```

` gapsdata2`

```
## 3 sequences with 10 character and 9 different site patterns.
## The states are a c g t -
```

This is not yet what we wanted as two sites of our alignment, which contain the ambiguous characters “r” and “y”, got deleted. To define ambiguous characters like “r” and “y” explicitly we have to supply a contrast matrix similar to contrasts for factors.

```
<- matrix(data = c(1,0,0,0,0,
contrast 0,1,0,0,0,
0,0,1,0,0,
0,0,0,1,0,
1,0,1,0,0,
0,1,0,1,0,
0,0,0,0,1,
1,1,1,1,0,
1,1,1,1,1),
ncol = 5, byrow = TRUE)
dimnames(contrast) <- list(c("a","c","g","t","r","y","-","n","?"),
c("a", "c", "g", "t", "-"))
contrast
```

```
## a c g t -
## a 1 0 0 0 0
## c 0 1 0 0 0
## g 0 0 1 0 0
## t 0 0 0 1 0
## r 1 0 1 0 0
## y 0 1 0 1 0
## - 0 0 0 0 1
## n 1 1 1 1 0
## ? 1 1 1 1 1
```

```
<- phyDat(data, type="USER", contrast=contrast)
gapsdata3 gapsdata3
```

```
## 3 sequences with 12 character and 11 different site patterns.
## The states are a c g t -
```

Here we defined “n” as a state which can be any nucleotide but not a gap “-” and “?” can be any state including a gap.

These data can be used in all functions available in
*phangorn* to compute distance matrices or perform parsimony and
maximum likelihood analysis.

# Markov models of character evolution

To model nucleotide substitutions across the edges of a tree T we can assign a transition matrix. In the case of nucleotides, with four character states, each 4 \(\times\) 4 transition matrix has, at most, 12 free parameters.

Time-reversible Markov models are used to describe how characters change over time, and use fewer parameters. Time-reversible means that these models need not be directed in time, and the Markov property states that these models depend only on the current state. These models are used in analysis of phylogenies using maximum likelihood and MCMC, computing pairwise distances, as well in simulating sequence evolution.

We will now describe the General Time-Reversible (GTR) model (Tavaré 1986). The parameters of the GTR model are the equilibrium frequencies \(\pi = (\pi_A ,\pi_C ,\pi_G ,\pi_T)\) and a rate matrix \(Q\) which has the form \[\begin{equation} Q = \begin{pmatrix} \ast & \alpha\pi_C & \beta\pi_G & \gamma\pi_T \\ \alpha\pi_A & \ast & \delta\pi_G & \epsilon\pi_T \\ \beta\pi_A & \delta\pi_C & \ast & \eta\pi_T \\ \gamma\pi_A & \epsilon\pi_C & \eta\pi_G & \ast \\ \end{pmatrix} (1) \end{equation}\]

where we need to assign 6 parameters \(\alpha, \dots, \eta\). The elements on the
diagonal are chosen so that the rows sum to zero. The Jukes-Cantor (JC)
(Jukes and Cantor 1969) model can be
derived as special case from the GTR model, for equal equilibrium
frequencies \(\pi_A = \pi_C = \pi_G = \pi_T =
0.25\) and equal rates set to \(\alpha
= \beta = \gamma = \delta = \eta\). Table @ref(tab:models) lists
all built-in nucleotide models in *phangorn*. The transition
probabilities which describe the probabilities of change from character
\(i\) to \(j\) in time \(t\), are given by the corresponding entries
of the matrix exponential \[
P(t) = (p_{ij}(t)) = e^{Qt}, \qquad \sum_j p_{ij}=1
\] where \(P(t)\) is the
transition matrix spanning a period of time \(t\).

# Estimation of non-standard transition rate matrices

In the last section @ref(user) we described how to set up user defined data formats. Now we describe how to estimate transition matrices with pml.

Again for nucleotide data the most common models can be called
directly in the `optim.pml`

function (e.g. “JC69”, “HKY”,
“GTR” to name a few). Table 2 lists all the available nucleotide models,
which can estimated directly in `optim.pml`

. For amino acids
several transition matrices are available (“WAG”, “JTT”, “LG”,
“Dayhoff”, “cpREV”, “mtmam”, “mtArt”, “MtZoa”, “mtREV24”, “VT”,“RtREV”,
“HIVw”, “HIVb”, “FLU”, “Blosum62”, “Dayhoff_DCMut” and “JTT-DCMut”) or
can be estimated with `optim.pml`

. For example Mathews et
al. (2010) (Mathews, Clements, and Beilstein
2010) used this function to estimate a phytochrome amino acid
transition matrix.

We will now show how to estimate a rate matrix with different transition (\(\alpha\)) and transversion ratio (\(\beta\)) and a fixed rate to the gap state (\(\gamma\)) - a kind of Kimura two-parameter model (K81) for nucleotide data with gaps as fifth state (see table 1).

a | c | g | t | - | |
---|---|---|---|---|---|

a | |||||

c | \(\beta\) | ||||

g | \(\alpha\) | \(\beta\) | |||

t | \(\beta\) | \(\alpha\) | \(\beta\) | ||

- | \(\gamma\) | \(\gamma\) | \(\gamma\) | \(\gamma\) |

If we want to fit a non-standard transition rate matrices, we have to
tell `optim.pml`

, which transitions in Q get the same rate.
The parameter vector subs accepts a vector of consecutive integers and
at least one element has to be zero (these gets the reference rate of
1). Negative values indicate that there is no direct transition is
possible and the rate is set to zero.

```
library(ape)
<- unroot(rtree(3))
tree <- pml(tree, gapsdata3)
fit <- optim.pml(fit, optQ=TRUE, subs=c(1,0,1,2,1,0,2,1,2,2),
fit control=pml.control(trace=0))
fit
```

```
## model: Mk
## loglikelihood: -33.01
## unconstrained loglikelihood: -28.43
##
## Rate matrix:
## a c g t -
## a 0.000e+00 2.584e-06 1.000e+00 2.584e-06 0.6912
## c 2.584e-06 0.000e+00 2.584e-06 1.000e+00 0.6912
## g 1.000e+00 2.584e-06 0.000e+00 2.584e-06 0.6912
## t 2.584e-06 1.000e+00 2.584e-06 0.000e+00 0.6912
## - 6.912e-01 6.912e-01 6.912e-01 6.912e-01 0.0000
##
## Base frequencies:
## a c g t -
## 0.2 0.2 0.2 0.2 0.2
```

Here are some conventions how the models are estimated:

If a model is supplied the base frequencies bf and rate matrix Q are optimized according to the model (nucleotides) or the adequate rate matrix and frequencies are chosen (for amino acids). If optQ=TRUE and neither a model or subs are supplied than a symmetric (optBf=FALSE) or reversible model (optBf=TRUE, i.e. the GTR for nucleotides) is estimated. This can be slow if the there are many character states, e.g. for amino acids. Table 2 shows how parameters are optimized and number of parameters to estimate. The elements of the vector subs correspond to \(\alpha, \dots, \eta\) in equation (1)

model | optQ | optBf | subs | df |
---|---|---|---|---|

JC | FALSE | FALSE | \(c(0, 0, 0, 0, 0, 0)\) | 0 |

F81 | FALSE | TRUE | \(c(0, 0, 0, 0, 0, 0)\) | 3 |

K80 | TRUE | FALSE | \(c(0, 1, 0, 0, 1, 0)\) | 1 |

HKY | TRUE | TRUE | \(c(0, 1, 0, 0, 1, 0)\) | 4 |

TrNe | TRUE | FALSE | \(c(0, 1, 0, 0, 2, 0)\) | 2 |

TrN | TRUE | TRUE | \(c(0, 1, 0, 0, 2, 0)\) | 5 |

TPM1 | TRUE | FALSE | \(c(0, 1, 2, 2, 1, 0)\) | 2 |

K81 | TRUE | FALSE | \(c(0, 1, 2, 2, 1, 0)\) | 2 |

TPM1u | TRUE | TRUE | \(c(0, 1, 2, 2, 1, 0)\) | 5 |

TPM2 | TRUE | FALSE | \(c(1, 2, 1, 0, 2, 0)\) | 2 |

TPM2u | TRUE | TRUE | \(c(1, 2, 1, 0, 2, 0)\) | 5 |

TPM3 | TRUE | FALSE | \(c(1, 2, 0, 1, 2, 0)\) | 2 |

TPM3u | TRUE | TRUE | \(c(1, 2, 0, 1, 2, 0)\) | 5 |

TIM1e | TRUE | FALSE | \(c(0, 1, 2, 2, 3, 0)\) | 3 |

TIM1 | TRUE | TRUE | \(c(0, 1, 2, 2, 3, 0)\) | 6 |

TIM2e | TRUE | FALSE | \(c(1, 2, 1, 0, 3, 0)\) | 3 |

TIM2 | TRUE | TRUE | \(c(1, 2, 1, 0, 3, 0)\) | 6 |

TIM3e | TRUE | FALSE | \(c(1, 2, 0, 1, 3, 0)\) | 3 |

TIM3 | TRUE | TRUE | \(c(1, 2, 0, 1, 3, 0)\) | 6 |

TVMe | TRUE | FALSE | \(c(1, 2, 3, 4, 2, 0)\) | 4 |

TVM | TRUE | TRUE | \(c(1, 2, 3, 4, 2, 0)\) | 7 |

SYM | TRUE | FALSE | \(c(1, 2, 3, 4, 5, 0)\) | 5 |

GTR | TRUE | TRUE | \(c(1, 2, 3, 4, 5, 0)\) | 8 |

## Predefined models for user defined data

So far there are 4 models which are just a generalization from nucleotide models allowing different number of states. In many cases only the equal rates (ER) model will be apropriate.

DNA | USER |
---|---|

JC | ER |

F81 | FREQ |

SYM | SYM |

GTR | GTR |

There is an additional model ORDERED, which assumes ordered characters and only allows to switch between neighboring states. Table 3 show the corresponding rate matrix.

a | b | c | d | e | |
---|---|---|---|---|---|

a | |||||

b | 1 | ||||

c | 0 | 1 | |||

d | 0 | 0 | 1 | ||

e | 0 | 0 | 0 | 1 |

# Codon substitution models

A special case of the transition rates are codon models.
*phangorn* now offers the possibility to estimate the \(d_N/d_S\) ratio (sometimes called ka/ks),
for an overview see (Yang 2014). These
functions extend the option to estimates the \(d_N/d_S\) ratio for pairwise sequence
comparison as it is available through the function `kaks`

in
*seqinr*. The transition rate between between codon \(i\) and \(j\) is defined as follows: \[\begin{eqnarray}
q_{ij}=\left\{
\begin{array}{l@{\quad}l}
0 & \textrm{if i and j differ in more than one position} \\
\pi_j & \textrm{for synonymous transversion} \\
\pi_j\kappa & \textrm{for synonymous transition} \\
\pi_j\omega & \textrm{for non-synonymous transversion} \\
\pi_j\omega\kappa & \textrm{for non synonymous transition}
\end{array}
\right. \nonumber
\end{eqnarray}\]

where \(\omega\) is the \(d_N/d_S\) ratio, \(\kappa\) the transition transversion ratio and \(\pi_j\) is the the equilibrium frequencies of codon \(j\). For \(\omega\sim1\) the an amino acid change is neutral, for \(\omega < 1\) purifying selection and \(\omega > 1\) positive selection.

Here we use data set from and follow loosely the example in (Bielawski and Yang 2005). We first read in an
alignment and phylogenetic tree for 45 sequences of the nef Gene in the
Human HIV-2 Genome using `read.phyDat`

function.

```
<- system.file("extdata/trees", package = "phangorn")
fdir <- read.phyDat(file.path(fdir, "seqfile.txt"), format="sequential")
hiv_2_nef <- read.tree(file.path(fdir, "tree.txt")) tree
```

With the tree and data set we can estimate currently 3 different site models:

- The M0 model with a constant \(\omega\), where \(\omega\) estimates the average over all
sites of the alignment. M0 does not allow for distinct \(\omega\) and identify classes, therefore we
will not retrieve any information regards positive selection.

- The M1a or nearly neutral model estimates two different \(\omega\) value classes (\(\omega=1\) & \(\omega<1\)).
- The M2a or positive selection model estimates three different classes of \(\omega\) (negative selection \(\omega<1\), neutral selection \(\omega=1\), positive selection \(\omega>1\)). One can use a likelihood ratio test to compare the M1a and M2a to for positive selection.

```
<- codonTest(tree, hiv_2_nef)
cdn cdn
```

```
## model Frequencies estimate logLik df AIC BIC dnds_0 dnds_1 dnds_2
## 1 M0 F3x4 empirical -9773 98 19741 20087 0.50486 NA NA
## 2 M1a F3x4 empirical -9313 99 18824 19168 0.06282 1 NA
## 3 M2a F3x4 empirical -9244 101 18689 19040 0.05551 1 2.469
## p_0 p_1 p_2 tstv
## 1 1.0000 NA NA 4.418
## 2 0.5564 0.4436 NA 4.364
## 3 0.5227 0.3617 0.1156 4.849
```

Currently the choice of site models is limited to the three models mentioned above and are no branch models implemented so far.

We can identify sites under positive selection using the Na"ive empirical Bayes (NEB) method of (Nielsen and Yang 1998): \[ P(\omega|x_h) = \frac{P(X_h|\omega_i)p_i}{P(X_h)} = \frac{P(X_h|\omega_i)p_i}{\sum_j P(X_h|\omega_j)p_j} \]

`plot(cdn, "M1a")`

`plot(cdn, "M2a")`

A lot if implementations differ in the way the codon frequencies are
derived. The M0 model can be also estimated using `pml`

and
`optim.pml`

functions. There are several ways to estimate the
codon frequencies \(\pi_j\). The
simplest model is to assume they have equal frequencies (=1/61). A
second is to use the empirical codon frequencies, either computed using
`baseFreq`

or using the argument `bf="empirical"`

in `pml`

. This is usually not really good as some codon are
rare and have a high variance. One can estimate the frequencies from
nucleotide frequencies with the F1x4 model. Last but not least the
frequencies can be derived from the base frequencies at each codon
position, the F3x4 model is set by the argument
`bf="F3x4"`

.

```
<- cdn$estimates[["M0"]]$tree # tree with edge lengths
treeM0 <- pml(treeM0, dna2codon(hiv_2_nef), bf="F3x4")
M0 <- optim.pml(M0, model="codon1", control=pml.control(trace=0))
M0 M0
```

```
## model: codon1
## loglikelihood: -9773
## unconstrained loglikelihood: -1372
## dn/ds: 0.5049
## ts/tv: 4.418
## Freq: F3x4
```

For the F3x4 model can optimize the codon frequencies setting the
option to `optBf=TRUE`

in `optim.pml`

.

```
<- optim.pml(M0, model="codon1", optBf=TRUE, control=pml.control(trace=0))
M0_opt M0_opt
```

```
## model: codon1
## loglikelihood: -9668
## unconstrained loglikelihood: -1372
## dn/ds: 0.51
## ts/tv: 4.581
## Freq: F3x4
```

# Generating trees

*phangorn* has several functions to generate tree topologies,
which may are interesting for simulation studies. `allTrees`

computes all possible bifurcating tree topologies either rooted or
unrooted for up to 10 taxa. One has to keep in mind that the number of
trees is growing exponentially, use `howmanytrees`

from
*ape* as a reminder.

```
<- allTrees(5)
trees par(mfrow=c(3,5), mar=rep(0,4))
for(i in 1:15)plot(trees[[i]], cex=1, type="u")
```

`nni`

returns a list of all trees which are one nearest
neighbor interchange away.

`nni(trees[[1]])`

`## 4 phylogenetic trees`

`rNNI`

and `rSPR`

generate trees which are a
defined number of NNI (nearest neighbor interchange) or SPR (subtree
pruning and regrafting) away.

# Session info

```
## R version 4.2.1 (2022-06-23)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 22.04.1 LTS
##
## Matrix products: default
## BLAS: /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
## LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.20.so
##
## locale:
## [1] LC_CTYPE=de_AT.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=de_AT.UTF-8 LC_COLLATE=C
## [5] LC_MONETARY=de_AT.UTF-8 LC_MESSAGES=de_AT.UTF-8
## [7] LC_PAPER=de_AT.UTF-8 LC_NAME=C
## [9] LC_ADDRESS=C LC_TELEPHONE=C
## [11] LC_MEASUREMENT=de_AT.UTF-8 LC_IDENTIFICATION=C
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] phangorn_2.10.0 ape_5.6-3
##
## loaded via a namespace (and not attached):
## [1] igraph_1.3.4 Rcpp_1.0.9 rstudioapi_0.14 knitr_1.40
## [5] magrittr_2.0.3 lattice_0.20-45 R6_2.5.1 quadprog_1.5-8
## [9] rlang_1.0.5 fastmatch_1.1-3 fastmap_1.1.0 highr_0.9
## [13] stringr_1.4.1 tools_4.2.1 parallel_4.2.1 grid_4.2.1
## [17] nlme_3.1-159 xfun_0.33 cli_3.4.0 jquerylib_0.1.4
## [21] htmltools_0.5.3 yaml_2.3.5 digest_0.6.29 Matrix_1.5-1
## [25] codetools_0.2-18 sass_0.4.2 prettydoc_0.4.1 cachem_1.0.6
## [29] evaluate_0.16 rmarkdown_2.16 stringi_1.7.8 compiler_4.2.1
## [33] bslib_0.4.0 generics_0.1.3 jsonlite_1.8.0 pkgconfig_2.0.3
```

# References

*Statistical Methods in Molecular Evolution*, 103–24. New York, NY: Springer New York. https://doi.org/10.1007/0-387-27733-1_5.

*Mammalian Protein Metabolism*, edited by H. N. Munro, 21–132. Academic Press.

*Phil. Trans. R. Soc. B*365: 383–95.

*Bioinformatics*27 (4): 592–93. https://doi.org/10.1093/bioinformatics/btq706.

*Lectures on Mathematics in the Life Sciences*, no. 17: 57–86.

*PLOS Currents Tree of Life*.

*Molecular Evolution: A Statistical Approach*. Oxford: Oxford University Press.