# DiscML: an R package for estimating evolutionary rates of discrete characters using maximum likelihood

- Tane Kim
^{1, 2}and - Weilong Hao
^{1}Email author

**15**:320

https://doi.org/10.1186/1471-2105-15-320

© Kim and Hao; licensee BioMed Central Ltd. 2014

**Received: **26 July 2014

**Accepted: **25 September 2014

**Published: **27 September 2014

## Abstract

### Background

The study of discrete characters is crucial for the understanding of evolutionary processes. Even though great advances have been made in the analysis of nucleotide sequences, computer programs for non-DNA discrete characters are often dedicated to specific analyses and lack flexibility. Discrete characters often have different transition rate matrices, variable rates among sites and sometimes contain unobservable states. To obtain the ability to accurately estimate a variety of discrete characters, programs with sophisticated methodologies and flexible settings are desired.

### Results

DiscML performs maximum likelihood estimation for evolutionary rates of discrete characters on a provided phylogeny with the options that correct for unobservable data, rate variations, and unknown prior root probabilities from the empirical data. It gives users options to customize the instantaneous transition rate matrices, or to choose pre-determined matrices from models such as birth-and-death (BD), birth-death-and-innovation (BDI), equal rates (ER), symmetric (SYM), general time-reversible (GTR) and all rates different (ARD). Moreover, we show application examples of DiscML on gene family data and on intron presence/absence data.

### Conclusion

DiscML was developed as a unified R program for estimating evolutionary rates of discrete characters with no restriction on the number of character states, and with flexibility to use different transition models. DiscML is ideal for the analyses of binary (1s/0s) patterns, multi-gene families, and multistate discrete morphological characteristics.

## Keywords

## Background

Many evolutionary processes involve transitions among different discrete characteristic states, including changes in morphological characteristics [1], sequence gain and loss [2, 3], gene family expansion and contraction [4], gain and loss of mobile promoters [5] and epigenetic characteristics such as methylation [6]. Evolutionary rates of discrete characters have been estimated using programs primarily developed for constructing ancestral character states such as the ACE function of the APE package [7] in R, standalone programs BayesTraits [8] and Mesquite [9]. Recently, great efforts have been made to estimate gene family turnover rates. The GLOOME program maps gain and loss rates using binary characters (or 1s/0s) [10], while Count [11], BEGFE [12], BadiRate [13], and CAFE3 [14] employ birth-and-death (BD) models to study gene family expansion and contraction.

Some of these programs have advanced (or realistic) features that are not implemented in other programs. For instance, the BayesTraits program implements a *Γ*-distribution for rate variation [8]. The GLOOME program allows the estimation of prior root probabilities of the character states [10, 15]. The BadiRate program allows variable birth rates and death rates, and corrects for unobservable data [13]. Furthermore, many multistate characters do not necessarily evolve in a BD manner [16], and should therefore be modeled using transition rate matrices other than BD.

In order to perform accurate rate estimation on a variety of discrete characters, we have developed a unified program DiscML by implementing the advanced features mentioned above as well as flexible options for transition rate matrices.

## Implementation

DiscML estimates the evolutionary rates of discrete characters by fitting the distribution of all character states (the data) on a given phylogeny. The data need to be in a matrix format (vector format for a single site) as required in many other phylogenetic programs in R (see examples in Additional file 1). The provided phylogeny is required to have branch lengths, as branch lengths will be used as a relative time scale in the analysis. The evolutionary rates, transition rate matrices, and additional parameters discussed below will be optimized to maximize the likelihood of the data. The optimization is achieved using the PORT routines [17] implemented in the nlminb function in R.

### Implementation of rate variation in the analysis

Rate variation among the character sites has long been recognized and implemented in DNA analyses [18], but has been missing from most analyses of non-DNA discrete characters (but see [8]). DiscML considers rate variation among the character sites by implementing a discrete *Γ* distribution (with the option of alpha=TRUE).

### Estimation of prior root probabilities

Most programs for the analysis of discrete characters assume only uniformly distributed prior root probabilities, e.g., ${\pi}_{1}={\pi}_{2}=\mathrm{..}={\pi}_{a}=\frac{1}{a}$, (*a* is the total number of character states). DiscML allows the estimation of prior root probabilities ( *π*_{
a
}) for different character states (with the option of rootprobability=TRUE).

### Flexibility on both the transition model and the number of character states

*Q*), which can be customized by users. This option could open the door for novel evolutionary analyses on different discrete characters. Several transition rate matrices are pre-determined in DiscML: model=~ER~ (equal rates, i.e., all entries in equation 1 are equal), model=~SYM~ (symmetric, i.e.,

*α*

_{1}=

*α*

_{2},

*β*

_{1}=

*β*

_{2},

*γ*

_{1}=

*γ*

_{2}, ..), and model=~ARD~ (all rates different, i.e., all entries are free to vary). ER and SYM are reversible matrices, while ARD matrices are irreversible.

*n*to state

*n*+1, while the death processes correspond to transitions from state

*n*to state

*n*−1. The BD transitions can be represented as matrices containing non-zero entries only between the neighboring states (equation 2). Several pre-determined BD transition rate matrices are available: BDER (equal rates), BDSYM (symmetric, i.e.,

*α*

_{1}=

*α*

_{2},

*β*

_{1}=

*β*

_{2},

*γ*

_{1}=

*γ*

_{2}, ..), BDISYM (symmetric, all entries except

*α*are equal, i.e.,

*α*

_{1}=

*α*

_{2},

*β*

_{1}=

*β*

_{2}=

*γ*

_{1}=

*γ*

_{2}=..), and BDARD (all rates different).

*Q*s) are calibrated [19], i.e., each

*Q*satisfies

so that the evolutionary rate parameter ( *μ*) is the average number of transition events per site per evolutionary time unit [20].

### Forced reversibility and flexible irreversible options

*π*) for different character states are estimated, reversible transition matrices will no longer necessarily result in reversible evolutionary processes (because of potentially different probabilities of character states). Since it is sometimes of biological interest to assume reversibility (i.e., the expected

*x*→

*y*changes equal to the

*y*→

*x*changes), DiscML can allow forced reversibility by setting reversible=TRUE. In practice, reversibility is obtained by multiplying the corresponding root probabilities (equation 4) to the entries in reversible transition matrices, e.g., ER and SYM. Such a practice is conceptually the same with the general time-reversible (GTR) DNA substitution model [21]. In DiscML, model=~GTR~ is equivalent to the combination of model=~SYM~ and reversible=TRUE.

In DiscML, the default setting is reversible=FALSE and users have the flexibility to conduct analysis by assuming irreversible evolutionary processes. Unlike in reversible processes, the root position can greatly affect the maximum likelihood calculation in irreversible cases [22, 23]. Therefore, it is only meaningful to perform irreversible analysis on a rooted tree. If the provided phylogenetic tree is unrooted, DiscML will first reroot the tree by midpoint rooting, and perform analysis on the midpoint rooted tree.

### Correction for unobservable data

*L*

_{−}is the likelihood of unobservable patterns. The correction for unobservable data (shown as ‘+0’ in Table 1) is essential for systems such as gene family data due to the complete loss of some ancient genes, but not suitable for single-site analyses and for systems in which all character states are observable (e.g., nucleotide bases).

**DiscML estimates from the gene family data in the Bacillaceae (B1, B2, B3) clades**

Models | Parameters | B1 | B2 | B3 |
---|---|---|---|---|

ER |
| 3.073 | 0.677 | 0.540 |

(1s/0s only) | Ln | -15150 | -16467 | -22229 |

ER+0 |
| 1.887 | 0.463 | 0.388 |

(1s/0s only) | Ln | -13682 | -15268 | -21207 |

BDER |
| 2.490 | 0.590 | 0.485 |

Ln | -20901 | -22196 | -29127 | |

BDISYM |
| 2.669 | 0.556 | 0.438 |

Ln | -19684 | -20973 | -27811 | |

BDARD |
| 5.746 | 1.369 | 1.450 |

Ln | -18254 | -20073 | -26578 | |

ER |
| 2.940 | 0.638 | 0.459 |

Ln | -21411 | -23273 | -31405 | |

SYM |
| 2.635 | 0.546 | 0.427 |

Ln | -19615 | -20947 | -27801 | |

ARD |
| 5.601 | 1.345 | 1.314 |

Ln | -18143 | -19678 | -26239 | |

GTR |
| 3.731 | 0.739 | 0.632 |

(SYM+ | Ln | -17753 | -19337 | -25381 |

ER+0 |
| 2.339 | 0.531 | 0.395 |

Ln | -20595 | -22586 | -30753 | |

ER+ |
| 2.935 | 0.624 | 0.454 |

Ln | -20070 | -21783 | -28771 | |

ER+ |
| 3.205 | 0.638 | 0.459 |

Ln | -21398 | -23273 | -31405 | |

ER+0+ |
| 1.358 | 0.236 | 0.240 |

Ln | -18719 | -19960 | -26712 | |

ER+0+ |
| 3.630 | 0.379 | 0.387 |

Ln | -16839 | -17960 | -23398 |

### Site and branch specific estimations

Even though the default setting of DiscML is to perform rate estimation by fitting the distribution pattern of all character sites on a phylogeny, there is an option to perform rate estimation on individual sites (ind=TRUE). Individual rates can be graphically displayed using plotmu=TRUE. Furthermore, DiscML allows branch specific rate estimation, which can be specified using ‘$’ on branches in the provided tree file. For instance, (((taxon1$1: 0.01, taxon2$1: 0.01)$3: 0.01, taxon3$2: 0.02)$3: 0.01, taxon4$2: 0.03); specifies three rates, one for the branches leading to taxon1 and taxon2 ($1), one for the branches leading to taxon3 and taxon4 ($2), and one for the remaining branches ($3). The modified tree files are no longer in the conventional Newick format, we have developed a function read.tree2 in DiscML to read such modified tree files.

### Additional features

DiscML allows binary (1s/0s) analysis on data with more than two character states by converting all non-zero characters to 1s with simplify=TRUE.

## Results and discussion

*M*

_{00}model in [20]. The optimization in [20] was achieved using the Nelder-Mead simplex method [25], while the optimization in Table 1 was achieved using the PORT routines [17]. Importantly, the DiscML estimates are identical to the previous estimates for all three clades. As expected, the parameter-rich models consistently outperformed the nested simplistic models (e.g., Ln

*L*BDARD > Ln

*L*BDISYM > Ln

*L*BDER; Ln

*L*ARD > Ln

*L*SYM > Ln

*L*ER). Consistent with previous studies [3, 20, 26], rate estimates in closely related clades tend to be higher than those in distantly related clades due to the transient nature of many acquired genes (Table 1). Tested on an Intel Core i7 (3.4 Ghz) 16 GB RAM Dell desktop, the computation using DiscML is fast (Table 2). For instance, the ER (1s/0s only) analysis took 49 seconds (0 m 49 s) for B1 (5453 gene families), 60 seconds (1 m 00 s) for B2 (5614 gene families), and 86 seconds (1 m 26 s) for B3 (6813 gene families). Computational time increases with the complexity of transition rate matrices and the addition of estimated parameters. For instance, the ER+0+

*π*+

*Γ*analysis took 82 m 22 s for B1, 81 m 20 s for B2, and 178 m27 s for B3 (Table 2).

**Computational time on an Intel Core i7 (3.4 Ghz) 16 GB RAM Dell desktop to generate the results in Table**
1

Models | B1(5453) | B2(5614) | B3(6813) |
---|---|---|---|

ER (1s/0s only) | 0 m 49 s | 1 m 00 s | 1 m 26 s |

ER+0 (1s/0s only) | 1 m 39 s | 2 m 01 s | 3 m 03 s |

BDER | 0 m 48 s | 1 m 06 s | 1 m 36 s |

BDISYM | 1 m 58 s | 2 m 20 s | 3 m 01 s |

BDARD | 7 m 54 s | 6 m 58 s | 8 m 28 s |

ER | 1 m 04 s | 1 m 15 s | 1 m 17 s |

SYM | 3 m 14 s | 4 m 47 s | 5 m 31 s |

ARD | 9 m 53 s | 9 m 12 s | 16 m 59 s |

GTR(SYM+ | 9 m 04 s | 9 m 54 s | 11 m 44 s |

ER+0 | 1 m 36 s | 2 m 34 s | 2 m 21 s |

ER+ | 2 m 41 s | 3 m 13 s | 4 m 40 s |

ER+ | 12 m 00 s | 39 m 01 s | 45 m 23 s |

ER+0+ | 82 m 22 s | 81 m 20 s | 178 m 27 s |

ER+0+ | 80 m 13 s | 67 m 33 s | 91 m 42 s |

*μ*

_{1}) and internal branches (

*μ*

_{2}) as illustrated in Figure 1B. Our results in Table 3 support the previous findings of higher gene turnover rates on external branches than those on internal branches [26, 30].

**Separate rates on branches estimated from the gene family data in the Bacillaceae (B1, B2, B3) clades**

Models | Parameters | B1 | B2 | B3 |
---|---|---|---|---|

( |
| 2.940 | 0.638 | 0.459 |

Ln | -21411 | -23273 | -31405 | |

( |
| 4.430 | 0.674 | 0.477 |

| 0.306 | 0.526 | 0.344 | |

Ln | -21045 | -23267 | -31395 | |

2 | 732 | 14 | 20 |

## Conclusion

We illustrated the versatility of DiscML on different types of data and analyses. With a great flexibility and fast computational speed, we are confident that DiscML can be used in a variety of studies on different discrete characters.

## Availability and requirements

**Project name:** DiscML**Project home page:**http://cran.r-project.org/web/packages/DiscML/index.html**Operating system(s):** Platform independent.**Programming language:** R.**Other requirements:** R (2.14 or newer); R-package: ape from CRAN.**License:** GNU GPL

## Declarations

### Acknowledgements

The authors would like to thank two anonymous reviewers for their helpful comments, Dr. Edward Golenberg for critical reading of the manuscript, Dr. Brian Golding for suggestions during the development of DiscML, Dr. Baojun Wu for assistance in collecting yeast mitochondrial data used in Figures 2 and 3. The work was supported by funds from Wayne State University to WH.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.