Complex Models

Partition and mixture models and usages.

Table of Contents

• Partition models
  • Partition file format
  • Partitioned analysis
• Mixture models
  • What is the difference between partition and mixture models?
  • Defining mixture models
  • Profile mixture models
  • NEXUS model file

This document gives detailed descriptions of complex maximum-likelihood models available in IQ-TREE. It is assumed that you know the basic substitution models already.

Partition models

Partition models are intended for phylogenomic (e.g., multi-gene) alignments, which allow each partition to have its own substitution models and evolutionary rates. IQ-TREE supports three types of partition models:

1. Edge-equal partition model with equal branch lengths: All partitions share the same set of branch lengths.
2. Edge-proportional partition model with proportional branch lengths: Like above but each partition has its own partition specific rate, that rescale all its branch lengths. This model accomodates different evolutionary rates between partitions (e.g. between 1st, 2nd, and 3rd codon positions).
3. Edge-unlinked partition model: each partition has its own set of branch lengths. This is the most parameter-rich partition model, that accounts for e.g., heterotachy (Lopez et al., 2002).

NOTICE: The edge-equal partition model is typically unrealistic as it does not account for different evolutionary speeds between partitions, whereas edge-unlinked partition model can be overfitting if there are many short partitions. Therefore, the edge-proportional partition model is recommended for a typical analysis.

Partition file format

To apply partition models users must first prepare a partition file in RAxML-style or NEXUS format. The RAxML-style is defined by the RAxML software, which may look like:

DNA, part1 = 1-100
DNA, part2 = 101-384

This means two DNA partitions of an alignment, where one groups aligment sites 1-100 into part1 and 101-384 into part2.

The NEXUS format is more complex but more powerful. For example, the above partition scheme may look like:

#nexus
begin sets;
    charset part1 = 1-100;
    charset part2 = 101-384;
    charpartition mine = HKY+G:part1, GTR+I+G:part2;
end;

The first line contains the keyword #nexus to indicate a NEXUS file. It has a sets block, which contains two character sets (charset command) named part1 and part2. Furthermore, with charpartition command we set the model HKY+G for part1 and GTR+I+G for part2. This is not possible with RAxML-style format (i.e., one cannot specify +G rate model for one partition and +I+G rate model for the other partition).

One can also specify non-consecutive sites of a partition, e.g. under RAxML-style format:

DNA, part1 = 1-100, 250-384
DNA, part2 = 101-249\3, 102-249\3
DNA, part3 = 103-249\3

or under NEXUS format:

#nexus
begin sets;
    charset part1 = 1-100 250-384;
    charset part2 = 101-249\3 102-249\3;
    charset part3 = 103-249\3;
end;

This means, part2 contains sites 101, 102, 104, 105, 107, ..., 246, 248, 249; whereas part3 contains sites 103, 106, ..., 247. This is useful to specify partitions corresponding to 1st, 2nd and 3rd codon positions.

Moreover, the NEXUS file allows each partition to come from separate alignment file (not possible under RAxML-style format) with e.g.:

#nexus
begin sets;
    charset part1 = aln1.phy: 1-100\3 201-300;
    charset part2 = aln1.phy: 101-200;
    charset part3 = aln2.phy: *;
    charpartition mine = HKY:part1, GTR+G:part2, WAG+I+G:part3;
end;

Here, part1 and part2 correspond to sub-alignments of aln1.phy file and part3 is the entire alignment file aln2.phy. Note that aln2.phy is a protein alignment in this example. In fact, IQ-TREE fully supports mixed data types between partitions!

If you want to specify codon model for a partition, use the CODON keyword (otherwise, the partition may be detected as DNA):

#nexus
begin sets;
    charset part1 = aln1.phy:CODON, 1-300;
    charset part2 = aln1.phy: 301-400;
    charset part3 = aln2.phy: *;
    charpartition mine = GY:part1, GTR+G:part2, WAG+I+G:part3;
end;

Note that this assumes part1 has standard genetic code. If not, append CODON with the right genetic code ID.

Partitioned analysis

Having prepared a partition file, one is ready to start a partitioned analysis with -q (edge-equal), -spp (edge-proportional) or -sp (edge-unlinked) option. See this tutorial for more details.

Mixture models

What is the difference between partition and mixture models?

Mixture models allow more than one substitution model along the sequences like partition models. However, while a partition model assigns each alignment site with a given specific model, mixture models do not have this information: each site has a probability of belonging to each of the mixture components (also called categories or classes). In other words, the site-to-model assignment is unknown.

For example, the discrete Gamma rate heterogeneity is the simplest type of mixture model, where there are several rate categories and each site belongs to a rate category with a probability. The likelihood of a site under a mixture model is computed as the weighted average of the site-likelihood under each mixture category.

Defining mixture models

IQ-TREE supports a number of predefined protein mixture models. Here, we give more details how to define new mixture models in IQ-TREE. To start with, the following command:

iqtree -s example.phy -m "MIX{JC,HKY}"

is a valid analysis. Here, we specify a mixture model (via MIX keyword in the model string) with two components (JC and HKY model) given in curly bracket and comma separator. IQ-TREE will then estimate the parameters of both mixture components as well as their weights: the proportion of sites belonging to each component.

NOTICE: Do not forget the double-quotes around model string! Otherwise, the Terminal might not recognize the model string properly.

Mixture models can be combined with rate heterogeneity, e.g.:

iqtree -s example.phy -m "MIX{JC,HKY}+G4"

Here, we specify two models and four Gamma rate categories. Effectively it means that there are 8 mixture components! Each site has a probability belonging to either JC or HKY and to one of the four rate categories.

Profile mixture models

Sometimes one only wants to model the changes in nucleotide or amino-acid frequencies along the sequences while keeping the substitution rate matrix the same. This can be specified in IQ-TREE via FMIX{...} model syntax. For convenience the mixture components can be defined in a NEXUS file like this (example corresponds to the CF4 model of (Wang et al., 2008)):

#nexus
begin models;
    frequency Fclass1 = 0.02549352 0.01296012 0.005545202 0.006005566 0.01002193 0.01112289 0.008811948 0.001796161 0.004312188 0.2108274 0.2730413 0.01335451 0.07862202 0.03859909 0.005058205 0.008209453 0.03210019 0.002668138 0.01379098 0.2376598;
    frequency Fclass2 = 0.09596966 0.008786096 0.02805857 0.01880183 0.005026264 0.006454635 0.01582725 0.7215719 0.003379354 0.002257725 0.003013483 0.01343441 0.001511657 0.002107865 0.006751404 0.04798539 0.01141559 0.000523736 0.002188483 0.004934972;
    frequency Fclass3 = 0.01726065 0.005467988 0.01092937 0.3627871 0.001046402 0.01984758 0.5149206 0.004145081 0.002563289 0.002955213 0.005286931 0.01558693 0.002693098 0.002075771 0.003006167 0.01263069 0.01082144 0.000253451 0.001144787 0.004573568;
    frequency Fclass4 = 0.1263139 0.09564027 0.07050061 0.03316681 0.02095119 0.05473468 0.02790523 0.009007538 0.03441334 0.005855319 0.008061884 0.1078084 0.009019514 0.05018693 0.07948 0.09447839 0.09258897 0.01390669 0.05367769 0.01230413;

    frequency CF4model = FMIX{empirical,Fclass1,Fclass2,Fclass3,Fclass4};
end;

Here, the NEXUS file contains a models block to define new models. More explicitly, we define four AA profiles Fclass1 to Fclass4 (each containing 20 AA frequencies). Then, the frequency mixture was defined with

FMIX{empirical,Fclass1,Fclass2,Fclass3,Fclass4}

That means, we have five components: the first corresponds to empirical AA frequencies to be inferred from the data and the remaining four components specified in this NEXUS file. Please save this to a file, say, mymodels.nex. One can now start the analysis with:

iqtree -s some_protein.aln -mdef mymodels.nex -m JTT+CF4model+G

-mdef option is to specify the NEXUS file containing user-defined models. Here, the JTT matrix is applied for all alignment sites and one varies the AA profiles along the alignment. One can use the NEXUS syntax to define all other profile mixture models such as C10 to C60.

NEXUS model file

In fact, IQ-TREE uses this NEXUS model file internally to define all protein mixture models. In addition to define state frequencies, one can specify the entire model with rate matrix and state frequencies together. For example, the LG4M model (Le et al., 2012) was defined by:

#nexus
begin models;
    model LG4M1 =
        0.269343
        0.254612 0.150988
        0.236821 0.031863 0.659648
        ....;
    ....
    model LG4M4 = ....;
    
    model LG4M = MIX{LG4M1,LG4M2,LG4M3,LG4M4}*G4;
end;

Here, we first defined the four matrices LG4M1, LG4M2, LG4M3 and LG4M4 in PAML format (see protein models). Then LG4M was defined as mixture model with these four components fused with Gamma rate heterogeneity (via *G4 syntax instead of +G4). That means, we have totally 4 mixture components instead of 16. The first component LG4M1 is rescaled by the rate of the lowest Gamma rate category. The fourth component LG4M4 corresponds to the highest rate.

Note that both frequency and model commands can be embedded into a single model file.