Most model-based molecular phylogenetic methods assume that the sequences 19 diverged on a tree under homogeneous conditions. If evolution occurred under these 20 conditions, then it is unlikely that the sequences would become compositionally 21 heterogeneous. Conversely, if the sequences are compositionally heterogeneous, then it is 22 unlikely that they have evolved under homogeneous conditions. We present methods to detect 23 and analyse heterogeneous evolution in aligned sequence data and to examine-visually and 24 numerically-its effect on phylogenetic estimates. The methods are implemented in three 25 programs, allowing users to better examine under what conditions their phylogenetic data 26 may have evolved. 27 1Most model-based molecular phylogenetic methods assume that the sequences of 30 nucleotides or amino acids have evolved along the edges of a single bifurcating tree. Often, the 31 methods also assume that the evolutionary processes operating at the variable sites of these 32 data (i.e., the sites that are free to evolve) can be approximated by independent and 33 identically-distributed (iid) Markovian processes. Furthermore, it is often assumed that the 34 evolutionary processes were stationary, reversible and homogeneous (SRH) (for details, see 35 Bryant et al. 2005; Jayaswal et al. 2005; Ababneh et al. 2006a,b; Jermiin et al. 2017), with the 36 term homogeneity implying time-homogeneity (i.e., a constant rate of change between two 37 points in time). 38 In practice, when DNA has evolved under these conditions, commonly-used 39 phylogenetic methods are likely to identify the correct topology (Huelsenbeck and Hillis 1993; 40 Hillis et al. 1994a,b). However, the same methods may not be capable of identifying the 41 correct topology when DNA has evolved under more complex conditions (Huelsenbeck and 42 Hillis 1993; Hillis et al. 1994a,b; Ho and Jermiin 2004; Jermiin et al. 2004). One reason for 43 this failure is that the strength of the historical signal (i.e., the signal in DNA that is due to 44 the order and time of divergence events) decays over time (Ho and Jermiin 2004) whereas the 45 strength of the non-historical signals (Grundy and Naylor 1999) may increase over time (Fig. 46 1). This may lead to situations, where the non-historical signals-individually or jointly-may 47 become stronger than the historical signal (Ho and Jermiin 2004). Unless phylogenetic 48 methods are able to distinguish historical signals from non-historical signals, the latter may be 49 misinterpreted as being part of the historical signal. This is because the non-historical signals 50 are also phylogenetic signals.
51The non-historical signal is a mixed bag of signals that may arise over time due to 52 temporal variations in site-and lineage-specific evolutionary processes. For example, when the 53 homologous sites in a pair of sequences evolve under different conditions, evolutionary 54 processes cannot be homogeneous, and compositional heterogeneity across the sequences may 55 arise. When this ...