Substitution Models

---

layout: userdoc title: "Substitution Models" author: AUTHOR date: DATE docid: 10 icon: book doctype: manual tags:

• manual description: All common substitution models and usages. sections:
• name: DNA models url: dna-models
• name: Protein models url: protein-models
• name: Codon models url: codon-models
• name: Binary, morphological models url: binary-and-morphological-models
• name: Ascertainment bias correction url: ascertainment-bias-correction
• name: Rate heterogeneity across sites url: rate-heterogeneity-across-sites

---

Substitution models

All common substitution models and usages.

IQ-TREE supports a wide range of substitution models, including advanced partition and mixture models. This guide gives a detailed information of all available models.

TIP: If you do not know which model to use, simply run IQ-TREE with the standard model selection (-m TEST option) or the new ModelFinder (-m MFP). It automatically determines best-fit model for your data. {: .tip}

DNA models

Base substitution rates

IQ-TREE includes all common DNA models (ordered by complexity):

Model	df	Explanation	Code
JC or JC69	0	Equal substitution rates and equal base frequencies (Jukes and Cantor, 1969).	000000
F81	3	Equal rates but unequal base freq. (Felsenstein, 1981).	000000
K80 or K2P	1	Unequal transition/transversion rates and equal base freq. (Kimura, 1980).	010010
HKY or HKY85	4	Unequal transition/transversion rates and unequal base freq. (Hasegawa, Kishino and Yano, 1985).	010010
TN or TN93	5	Like HKY but unequal purine/pyrimidine rates (Tamura and Nei, 1993).	010020
TNe	2	Like TN but equal base freq.	010020
K81 or K3P	2	Three substitution types model and equal base freq. (Kimura, 1981).	012210
K81u	5	Like K81 but unequal base freq.	012210
TPM2	2	AC=AT, AG=CT, CG=GT and equal base freq.	010212
TPM2u	5	Like TPM2 but unequal base freq.	010212
TPM3	2	AC=CG, AG=CT, AT=GT and equal base freq.	012012
TPM3u	5	Like TPM3 but unequal base freq.	012012
TIM	6	Transition model, AC=GT, AT=CG and unequal base freq.	012230
TIMe	3	Like TIM but equal base freq.	012230
TIM2	6	AC=AT, CG=GT and unequal base freq.	010232
TIM2e	3	Like TIM2 but equal base freq.	010232
TIM3	6	AC=CG, AT=GT and unequal base freq.	012032
TIM3e	3	Like TIM3 but equal base freq.	012032
TVM	7	Transversion model, AG=CT and unequal base freq.	012314
TVMe	4	Like TVM but equal base freq.	012314
SYM	5	Symmetric model with unequal rates but equal base freq. (Zharkikh, 1994).	012345
GTR	8	General time reversible model with unequal rates and unequal base freq. (Tavare, 1986).	012345

The last column Code is a 6-digit code defining the equality constraints for 6 relative substitution rates: A-C, A-G, A-T, C-G, C-T and G-T. 010010 means that A-G rate is equal to C-T rate (corresponding to 1 in the code) and the remaining four substitution rates are equal (corresponding to 0 in the code). Thus, 010010 is equivalent to K80 or HKY model (depending on whether base frequencies are equal or not). 012345 is equivalent to GTR or SYM model as there is no restriction defined by such 6-digit code.

Moreover, IQ-TREE supports arbitrarily restricted DNA model via a 6-digit code, e.g. with option -m 012012+G.

NOTE: The digits in the codes do not necessarily have to have the same order as above. That means '101101' describes the same matrix as '010010'. The last rate, which corresponds to G-T, (and all rates with the same digit) is always set equal to 1.0 for convenience because the rates are relative.

If users want to fix model parameters, append the model name with a curly bracket {, followed by the comma-separated rate parameters, and a closing curly bracket }. For example, GTR{1.0,2.0,1.5,3.7,2.8} specifies 6 substitution rates A-C=1.0, A-G=2.0, A-T=1.5, C-G=3.7, C-T=2.8 and G-T=1.0.

Another example is for model TIM2 that has the 6-digit code 010232. Thus, TIM2{4.39,5.30,12.1} means that A-C=A-T=4.39 (coded 0), A-G=5.30 (coded 1), C-T=12.1 (coded 3) and C-G=G-T=1.0 (coded 2). This is, in turn, equivalent to specifying GTR{4.39,5.30,4.39,1.0,12.1}.

Base frequencies

Users can specify three different kinds of base frequencies:

FreqType	Explanation
+F	Empirical base frequencies. This is the default if the model has unequal base freq.
+FQ	Equal base frequencies.
+FO	Optimized base frequencies by maximum-likelihood.

For example, GTR+FO optimizes base frequencies by ML whereas GTR+F (default) counts base frequencies directly from the alignment.

Finally, users can fix base frequencies with e.g. GTR+F{0.1,0.2,0.3,0.4} to fix the corresponding frequencies of A, C, G and T (must sum up to 1.0).

Lie Markov models

Starting with version 1.6, IQ-TREE supports a series of Lie Markov models (Woodhams et al., 2015), many of which are non-reversible models. Lie Markov models have a consistent property, which is lacking in other common models such as GTR. The following table shows the list of all Lie Markov models (the number before . in the name shows the number of parameters of the model):

Model	Rev?	Freq	Note
1.1	Yes	0	equiv. to JC
2.2b	Yes	0	equiv. to K2P
3.3a	Yes	0	equiv. to K3P
3.3b	No	0
3.3c	Yes	0	equiv. to TNe
3.4	Yes	1
4.4a	Yes	3	equiv. to F81
4.4b	Yes	1
4.5a	No	1
4.5b	No	1
5.6a	No	0
5.6b	No	3
5.7a	No	2
5.7b	No	0
5.7c	No	0
5.11a	No	2
5.11b	No	0
5.11c	No	0
5.16	No	1
6.6	No	1	equiv. to STRSYM (strand symmetric model)
6.7a	No	3	F81+K3P
6.7b	No	3
6.8a	No	3
6.8b	No	1
6.17a	No	1
6.17b	No	1
8.8	No	3
8.10a	No	3
8.10b	No	1
8.16	No	3
8.17	No	3
8.18	No	3
9.20a	No	2
9.20b	No	0	Doubly stochastic
10.12	No	3
10.34	No	3
12.12	No	3	equiv. to UNREST (unrestricted model)

Column Rev? shows whether the model is reversible or not. Column Freq shows the number of free base frequencies. 0 means equal base frequency; 1 means f(A)=f(G) and f(C)=f(T); 2 means f(A)+f(G)=0.5=f(C)+f(T); 3 means unconstrained frequencies.

All Lie Markov models can have one of the following prefices:

Prefix	Meaning
RY	purine-pyrimidine pairing (default)
WS	weak-strong pairing
MK	aMino-Keto pairing

Protein models

Amino-acid exchange rate matrices

IQ-TREE supports all common empirical amino-acid exchange rate matrices (alphabetical order):

Model	Region	Explanation
Blosum62	nuclear	BLOcks SUbstitution Matrix (Henikoff and Henikoff, 1992). Note that BLOSUM62 is not recommended for phylogenetic analysis as it was designed mainly for sequence alignments.
cpREV	chloroplast	chloroplast matrix (Adachi et al., 2000).
Dayhoff	nuclear	General matrix (Dayhoff et al., 1978).
DCMut	nuclear	Revised Dayhoff matrix (Kosiol and Goldman, 2005).
FLU	viral	Influenza virus (Dang et al., 2010).
HIVb	viral	HIV between-patient matrix HIV-Bm (Nickle et al., 2007).
HIVw	viral	HIV within-patient matrix HIV-Wm (Nickle et al., 2007).
JTT	nuclear	General matrix (Jones et al., 1992).
JTTDCMut	nuclear	Revised JTT matrix (Kosiol and Goldman, 2005).
LG	nuclear	General matrix (Le and Gascuel, 2008).
mtART	mitochondrial	Mitochondrial Arthropoda (Abascal et al., 2007).
mtMAM	mitochondrial	Mitochondrial Mammalia (Yang et al., 1998).
mtREV	mitochondrial	Mitochondrial Vertebrate (Adachi and Hasegawa, 1996).
mtZOA	mitochondrial	Mitochondrial Metazoa (Animals) (Rota-Stabelli et al., 2009).
mtMet	mitochondrial	Mitochondrial Metazoa (Vinh et al., 2017).
mtVer	mitochondrial	Mitochondrial Vertebrate (Vinh et al., 2017).
mtInv	mitochondrial	Mitochondrial Inverterbrate (Vinh et al., 2017).
Poisson	none	Equal amino-acid exchange rates and frequencies.
PMB	nuclear	Probability Matrix from Blocks, revised BLOSUM matrix (Veerassamy et al., 2004).
rtREV	viral	Retrovirus (Dimmic et al., 2002).
VT	nuclear	General 'Variable Time' matrix (Mueller and Vingron, 2000).
WAG	nuclear	General matrix (Whelan and Goldman, 2001).
GTR20	general	General time reversible models with 190 rate parameters. WARNING: Be careful when using this parameter-rich model as parameter estimates might not be stable, especially when not having enough phylogenetic information (e.g. not long enough alignments).

Protein mixture models

IQ-TREE also supports a series of protein mixture models:

Model	Explanation
C10 to C60	10, 20, 30, 40, 50, 60-profile mixture models (Le et al., 2008a) as variants of the CAT model (Lartillot and Philippe, 2004) for ML. Note that these models assume Poisson AA replacement and implicitly include a Gamma rate heterogeneity among sites.
EX2	Two-matrix model for exposed/buried AA sites (Le et al., 2008b).
EX3	Three-matrix model for highly exposed/intermediate/buried AA sites (Le et al., 2008b).
EHO	Three-matrix model for extended/helix/other sites (Le et al., 2008b).
UL2, UL3	Unsupervised-learning variants of EX2 and EX3, respectively.
EX_EHO	Six-matrix model combining EX2 and EHO (Le and Gascuel, 2010).
LG4M	Four-matrix model fused with Gamma rate heterogeneity (Le et al., 2012).
LG4X	Four-matrix model fused with FreeRate heterogeneity (Le et al., 2012).
CF4	Five-profile mixture model (Wang et al., 2008).

One can even combine a protein matrix with a profile mixture model like:

• LG+C20: Applying LG matrix instead of Poisson for all 20 classes of AA profiles and a Gamma rate heterogeneity.
• LG+C20+F: Applying LG matrix for 20 classes plus the 21th class of empirical AA profile (counted from the current data) and Gamma rate heterogeneity.
• JTT+CF4+G: Applying JTT matrix for all 5 classes of AA profiles and Gamma rate heteorogeneity.

Moreover, one can override the Gamma rate by FreeRate heterogeneity:

• LG+C20+R4: Like LG+C20 but replace Gamma by FreeRate heterogeneity.

User-defined empirical protein models

If the matrix name does not match any of the above listed models, IQ-TREE assumes that it is a file containing AA exchange rates and frequencies in PAML format. It contains the lower diagonal part of the matrix and 20 AA frequencies, e.g.:

0.425093 
0.276818 0.751878 
0.395144 0.123954 5.076149 
2.489084 0.534551 0.528768 0.062556 
0.969894 2.807908 1.695752 0.523386 0.084808 
1.038545 0.363970 0.541712 5.243870 0.003499 4.128591 
2.066040 0.390192 1.437645 0.844926 0.569265 0.267959 0.348847 
0.358858 2.426601 4.509238 0.927114 0.640543 4.813505 0.423881 0.311484 
0.149830 0.126991 0.191503 0.010690 0.320627 0.072854 0.044265 0.008705 0.108882 
0.395337 0.301848 0.068427 0.015076 0.594007 0.582457 0.069673 0.044261 0.366317 4.145067 
0.536518 6.326067 2.145078 0.282959 0.013266 3.234294 1.807177 0.296636 0.697264 0.159069 0.137500 
1.124035 0.484133 0.371004 0.025548 0.893680 1.672569 0.173735 0.139538 0.442472 4.273607 6.312358 0.656604 
0.253701 0.052722 0.089525 0.017416 1.105251 0.035855 0.018811 0.089586 0.682139 1.112727 2.592692 0.023918 1.798853 
1.177651 0.332533 0.161787 0.394456 0.075382 0.624294 0.419409 0.196961 0.508851 0.078281 0.249060 0.390322 0.099849 0.094464 
4.727182 0.858151 4.008358 1.240275 2.784478 1.223828 0.611973 1.739990 0.990012 0.064105 0.182287 0.748683 0.346960 0.361819 1.338132 
2.139501 0.578987 2.000679 0.425860 1.143480 1.080136 0.604545 0.129836 0.584262 1.033739 0.302936 1.136863 2.020366 0.165001 0.571468 6.472279 
0.180717 0.593607 0.045376 0.029890 0.670128 0.236199 0.077852 0.268491 0.597054 0.111660 0.619632 0.049906 0.696175 2.457121 0.095131 0.248862 0.140825 
0.218959 0.314440 0.612025 0.135107 1.165532 0.257336 0.120037 0.054679 5.306834 0.232523 0.299648 0.131932 0.481306 7.803902 0.089613 0.400547 0.245841 3.151815 
2.547870 0.170887 0.083688 0.037967 1.959291 0.210332 0.245034 0.076701 0.119013 10.649107 1.702745 0.185202 1.898718 0.654683 0.296501 0.098369 2.188158 0.189510 0.249313 

0.079066 0.055941 0.041977 0.053052 0.012937 0.040767 0.071586 0.057337 0.022355 0.062157 0.099081 0.064600 0.022951 0.042302 0.044040 0.061197 0.053287 0.012066 0.034155 0.069147 

(This is an example of an LG matrix taken from PAML package). Note that the amino-acid order in this file is:

 A   R   N   D   C   Q   E   G   H   I   L   K   M   F   P   S   T   W   Y   V
Ala Arg Asn Asp Cys Gln Glu Gly His Ile Leu Lys Met Phe Pro Ser Thr Trp Tyr Val

Amino-acid frequencies

By default, AA frequencies are given by the model. Users can change this with:

FreqType	Explanation
+F	empirical AA frequencies from the data.
+FO	ML optimized AA frequencies from the data.
+FQ	Equal AA frequencies.

Users can also specify AA frequencies with, e.g.:

+F{0.079066,0.055941,0.041977,0.053052,0.012937,0.040767,0.071586,0.057337,0.022355,0.062157,0.099081,0.064600,0.022951,0.042302,0.044040,0.061197,0.053287,0.012066,0.034155,0.069147}

(Example corresponds to the AA frequencies of the LG matrix).

Codon models

To apply a codon model one should use the option -st CODON to tell IQ-TREE that the alignment contains protein coding sequences (otherwise, IQ-TREE thinks that it contains DNA sequences and will apply DNA models). This implicitly applies the standard genetic code. You can change to an other genetic code by appending the appropriate ID to the CODON keyword:

Code	Genetic code meaning
CODON1	The Standard Code (same as -st CODON)
CODON2	The Vertebrate Mitochondrial Code
CODON3	The Yeast Mitochondrial Code
CODON4	The Mold, Protozoan, and Coelenterate Mitochondrial Code and the Mycoplasma/Spiroplasma Code
CODON5	The Invertebrate Mitochondrial Code
CODON6	The Ciliate, Dasycladacean and Hexamita Nuclear Code
CODON9	The Echinoderm and Flatworm Mitochondrial Code
CODON10	The Euplotid Nuclear Code
CODON11	The Bacterial, Archaeal and Plant Plastid Code
CODON12	The Alternative Yeast Nuclear Code
CODON13	The Ascidian Mitochondrial Code
CODON14	The Alternative Flatworm Mitochondrial Code
CODON16	Chlorophycean Mitochondrial Code
CODON21	Trematode Mitochondrial Code
CODON22	Scenedesmus obliquus Mitochondrial Code
CODON23	Thraustochytrium Mitochondrial Code
CODON24	Pterobranchia Mitochondrial Code
CODON25	Candidate Division SR1 and Gracilibacteria Code

(The IDs follow the specification at http://www.ncbi.nlm.nih.gov/Taxonomy/Utils/wprintgc.cgi).

Codon substitution rates

IQ-TREE supports several codon models:

Model	Explanation
MG	Nonsynonymous/synonymous (dn/ds) rate ratio (Muse and Gaut, 1994).
MGK	Like MG with additional transition/transversion (ts/tv) rate ratio.
MG1KTS or MGKAP2	Like MG with a transition rate (Kosiol et al., 2007).
MG1KTV or MGKAP3	Like MG with a transversion rate (Kosiol et al., 2007).
MG2K or MGKAP4	Like MG with a transition rate and a transversion rate (Kosiol et al., 2007).
GY	Nonsynonymous/synonymous and transition/transversion rate ratios (Goldman and Yang, 1994).
GY1KTS or GYKAP2	Like GY with a transition rate (Kosiol et al., 2007).
GY1KTV or GYKAP3	Like GY with a transversion rate (Kosiol et al., 2007).
GY2K or GYKAP4	Like GY with a transition rate and a transversion rate (Kosiol et al., 2007).
ECMK07 or KOSI07	Empirical codon model (Kosiol et al., 2007).
ECMrest	Restricted version of ECMK07 that allows only one nucleotide exchange.
ECMS05 or SCHN05	Empirical codon model (Schneider et al., 2005).

The last three models (ECMK07, ECMrest or ECMS05) are called empirical codon models, whereas the others are called mechanistic codon models.

Moreover, IQ-TREE supports combined empirical-mechanistic codon models using an underscore separator (_). For example:

• ECMK07_GY2K: The combined ECMK07 and GY2K model, with the rate entries being multiplication of the two corresponding rate matrices.

Thus, there can be many such combinations.

Starting with version 1.5.6: If the model name does not match any of the above listed models, IQ-TREE assumes that it is a file containing lower diagonal part of non-stop codon exchange rate matrix, non-stop codon frequencies and a list of non-stop codons. The rest of the file will be ignored. Example files (ECMrest.dat and ECMunrest.dat) can be downloaded from the supplementary material (Kosiol et al., 2007).

NOTE: Branch lengths under codon models are interpreted as number of nucleotide substitutions per codon site. Thus, they are typically 3 times longer than under DNA models.

Codon frequencies

IQ-TREE supports the following codon frequencies:

FreqType	Explanation
+F	Empirical codon frequencies counted from the data.
+FQ	Equal codon frequencies.
+F1X4	Unequal nucleotide frequencies but equal nt frequencies over three codon positions.
+F3X4	Unequal nucleotide frequencies and unequal nt frequencies over three codon positions.

If not specified, the default codon frequency will be +F3X4 for MG-type models, +F for GY-type models and given by the model for empirical codon models.

Binary and morphological models

The binary alignments should contain state 0 and 1, whereas for morphological data, the valid states are 0 to 9 and A to Z.

Model	Explanation
JC2	Jukes-Cantor type model for binary data.
GTR2	General time reversible model for binary data.
MK	Jukes-Cantor type model for morphological data.
ORDERED	Allowing exchange of neighboring states only.

Except for GTR2 that has unequal state frequencies, all other models have equal state frequencies.

TIP: If morphological alignments do not contain constant sites (typically the case), then an ascertainment bias correction model (+ASC) should be applied to correct the branch lengths for the absence of constant sites. {: .tip}

Ascertainment bias correction

An ascertainment bias correction (+ASC) model (Lewis, 2001) should be applied if the alignment does not contain constant sites (such as morphological or SNPs data). For example:

• MK+ASC: For morphological data.
• GTR+ASC: For SNPs data.

+ASC will correct the likelihood conditioned on variable sites. Without +ASC, the branch lengths might be overestimated.

Rate heterogeneity across sites

IQ-TREE supports all common rate heterogeneity across sites models:

RateType	Explanation
+I	allowing for a proportion of invariable sites.
+G	discrete Gamma model (Yang, 1994) with default 4 rate categories. The number of categories can be changed with e.g. +G8.
+I+G	invariable site plus discrete Gamma model (Gu et al., 1995).
+R	FreeRate model (Yang, 1995; Soubrier et al., 2012) that generalizes the +G model by relaxing the assumption of Gamma-distributed rates. The number of categories can be specified with e.g. +R6 (default 4 categories if not specified). The FreeRate model typically fits data better than the +G model and is recommended for analysis of large data sets.
+I+R	invariable site plus FreeRate model.

TIP: The new ModelFinder (-m MFP option) tests the FreeRate model, whereas the standard procedure (-m TEST) does not. {: .tip}

Users can fix the parameters of the model. For example, +I{0.2} will fix the proportion of invariable sites (pinvar) to 0.2; +G{0.9} will fix the Gamma shape parameter (alpha) to 0.9; +I{0.2}+G{0.9} will fix both pinvar and alpha. To fix the FreeRate model parameters, use the syntax +Rk{w1,r1,...,wk,rk} (replacing k with the number of categories). Here, w1, ..., wk are the weights and r1, ..., rk the rates for each category.

NOTE: For the +G model IQ-TREE implements the mean approximation approach (Yang, 1994). The same is done in RAxML and PhyML. However, some programs like TREE-PUZZLE implement the median approximation approach, which makes the resulting log-likelihood not comparable. IQ-TREE can change to this approach via the -gmedian option.