We tested for evidence of non-reversibility of the nucleotide substitution process in 141 virus sequence datasets (33 ssDNA virus datasets, 30 dsDNA virus datasets, 31 dsRNA virus datasets, and 47 ssRNA virus datasets), all consisting of either full genome sequences (for unsegmented viruses) or complete genome component sequences (for viruses with segmented genomes). Specifically, for each dataset, we compared the goodness-of-fit of the GTR+G, NREV6+G, and NREV12+G models (where G represents gamma-distributed nucleotide substitution rates among sites; Yang, 1994).

Given that dsDNA viruses such as adenoviruses, papillomaviruses, and herpesviruses have both their DNA strands in existence for similar amounts of time before DNA-dependent-DNA polymerase enzymes copy both their + and – DNA strands during replication (Hanson, 2009), we had anticipated that the best fitting substitution model for sequence datasets of these viruses would be NREV6. Using weighted small sample corrected Akaike information criterion (AIC-c) scores to reveal trends of model support (Figure 1), it is surprising that NREV12 was overall the best-supported model (illustrated by the redder hues around the top corner of the dsDNA plot in Figure 2). Out of the 30 dsDNA datasets considered, we found that NREV6 provided the best fit to five datasets (HPV18, HPV45, HPV16, BPV, and SV40) and GTR provided the best fit to five (Alphapapillomavirus 6, JC polyomavirus, DPV, RTBV, and DBAV). NREV12 was the best fitting model for the remaining 20 datasets (Table 1). Further, likelihood ratio tests (LRTs) revealed strong overall support for NREV12, with this model providing a significantly better fit (p<0.05) than NREV6 for 25/30 of the dsDNA datasets and a significantly better fit than GTR for 24/30 of the datasets.

As NREV6 was not the best fitting model for most of the dsDNA virus datasets, we infer that, in most dsDNA virus species, strand-specific substitution biases are not ignorable. Further, the datasets where NREV6 was not the best fit are from species in families containing other species where NREV6 was the best fit, indicating that such strand-specific substitution biases are unlikely to be a consequence of some broadly conserved feature of viral life cycles in these families (such as ssDNA replicative intermediates). It is instead plausible that these differences may relate to:

Similarly, and equally surprising, we found that NREV12 was overall the best supported model for dsRNA viruses (illustrated by the redder hues around the top corner of the dsRNA plot in Figure 1).

NREV6 fit only 2 of the 31 dsRNA datasets better than both NREV12 and GTR (Human rotavirus A set H and Fiji virus). NREV12 was found to be the best fitting model for 21/31 datasets and GTR was the best fitting of 8/31 (Table 2). In all three Birnaviridae family datasets (which contain virus species with two genome segments) and in 17/22 of Reoviridae family datasets (which contain virus species with 10–12 genome segments), the NREV12 model provided the best fit. Based on the LRTs, strong overall support for NREV12 was found, with this model providing a significantly better fit (p<0.05) in 27/31 dsRNA virus datasets relative to NREV6 and 23/31 datasets relative to GTR.

We anticipated that NREV12 might fit many of these dsRNA datasets better than NREV6 simply because, during their infection cycles, the coding +strand of dsRNA viruses (the one from which protein translation occurs) tends to exist for longer periods within an infected cell than the non-coding –strand. Specifically, there are two main steps during double-stranded RNA virus replication (Wickner, 1993). Firstly, synthesis of the viral +strands from a dsRNA template occurs in the cytoplasm within viral particles. These +strands exist within the cell for prolonged periods in the absence of complementary –strands and are used as templates for translation of viral proteins. In the second step, the +strands remaining after translation act as templates for –strand synthesis, resulting in the formation of new dsRNA molecules. The +strands of dsRNA viruses are therefore likely more impacted by mutational processes, which in turn could explain the pervasive strand-specific substitution biases seen in this group of viruses.

For the ssRNA and ssDNA viruses where one genome strand exists during the virus life cycle for far longer periods of time than the other such that complementary substitutions would not be expected to occur at similar rates, we anticipated that NREV12 should provide a better fit than both NREV6 and GTR. Indeed, for ssRNA viruses, NREV12 was a better fit than NREV6 and GTR for 40/47 of the ssRNA datasets and 24/33 of the ssDNA virus datasets (Figure 1). Of the nine ssDNA virus datasets where NREV12 was not the best fitting model, GTR fit 7/9 better and NREV6 fit 2/9 better (Table 3). Of the seven ssRNA datasets where NREV12 was not the best fitting model, GTR fit 6/7 better and NREV6 fit 1/7 better.

Based on the LRTs, strong overall support for NREV12 was found with this model providing a significantly better fit (p<0.05) than NREV6 for 45/47 of the ssRNA virus datasets (Table 4) and 31/33 of the ssDNA virus datasets (Table 3). Similarly, based on LRTs, NREV12 provided a significantly better fit than GTR for 27/33 of the ssDNA virus datasets (Table 3) and 40/47 of the ssRNA virus datasets (Table 4).

We found that the degree of non-reversibility (DNR) estimates alone did not cleanly differentiate between datasets for which NREV12 was or was not best supported (Tables 1–4). For the 107 nucleotide sequence datasets with a model preference of NREV12, 10 had estimated DNRs that were greater than 0.5, 13 had DNRs between 0.25 and 0.5, and 84 had DNRs between 0.0225 and 0.25. For the 10 nucleotide sequence datasets with a model preference of NREV6, one had an estimated DNR greater than 0.5, four had estimated DNRs between 0.25 and 0.5, and five had estimated DNRs between 0.064 and 0.25 (Figure 1). For the 24 nucleotide sequence datasets with a model preference of GTR, none had estimated DNRs greater than 0.5, one had an estimated DNR between 0.25 and 0.5, and the remainder had estimated DNRs between 0.0335 and 0.25.

The dsDNA virus dataset with the highest DNR was Human mastadenovirus D (DNR = 0.646), the dsRNA virus dataset with the highest estimated DNR was Porcine_rotavirus_B (0.351), the ssRNA virus dataset with the highest DNR was SARS-CoV-2 (DNR = 1.536), and the ssDNA virus dataset with the highest DNR was Torque teno sus virus (DNR = 1.56).

Therefore, while NREV12 appears to be generally more appropriate than either NREV6 or GTR for describing mutational processes in ssRNA, ssDNA, dsDNA, and dsRNA viruses, this might only be particularly relevant from a practical perspective when datasets of these viruses yield DNR estimates that are greater than 0.25. For such datasets, NREV12 (and possibly NREV 6 in some instances) might be especially useful for both determining the direction of evolution across phylogenetic trees (i.e. it could potentially be used to root these trees) and for quantifying genomic strand-specific nucleotide substitution biases (Harkins et al., 2009).

To determine whether it is worthwhile to use NREV12 rather than GTR for phylogenetic inference when NREV12 is the best fitting nucleotide substitution model, we used simulated datasets to compare the accuracy of phylogenetic trees inferred using these models.

We found that, regardless of dataset diversity and the nucleotide substitution model used, phylogenetic inference tended to become less accurate (i.e. weighted Robinson-Foulds [wRF] scores increased) as DNR increased (Figure 2). This tendency was, however, more pronounced when using a (mis-specified) GTR model than when using a (correctly specified) NREV12 model with, for any given dataset having DNR>0, the use of NREV12 tending to yield more accurate phylogenetic trees than when GTR was used. There were, however, only statistically significant improvements (p<0.05, paired t-test) in the accuracy of phylogenetic trees inferred using NREV12 relative to those inferred using GTR in lower diversity datasets (i.e. those with average pairwise nucleotide sequence identities (APIs) of 85%, 90%, and 95%) and then only for DNR>8. It is noteworthy that the highest estimated DNR in any of the empirical datasets that we analysed was 1.56 more than fourfold lower than the point where statistically significant differences in phylogenetic inference accuracies became apparent in the simulated datasets.

We obtained viral nucleotide sequences from the National Centre for Biotechnology Information Taxonomy database (http://www.ncbi.nlm.nih.gov/taxonomy) and the Los Alamos National Laboratory HIV sequence database (https://www.hiv.lanl.gov/content/index). These included gene and whole-genome sequences for viruses with ssRNA, ssDNA, dsRNA, and dsDNA genomes (datasets are summarised in Supplementary file 1). An outgroup sequence from a closely related virus species was added to each dataset to help root phylogenetic trees. The sequences in each of the datasets were aligned using MUSCLE (Edgar, 2004) implemented in Aliview (Larsson, 2014)⁠, and maximum likelihood phylogenetic trees were constructed from each alignment using RAxML v8.2 (Stamatakis, 2016)⁠.

We evaluated the fit of NREV12, NREV6, and GTR to the 141 individual sequence datasets using a previously published model test (Harkins et al., 2009) implemented as a HyPhy package module (Pond and Muse, 2005). This script (https://github.com/veg/hyphy-analyses/tree/master/NucleotideNonREV; Delport and Kosakovsky Pond, 2023) is also available as a module in the Datamonkey web server (Weaver et al., 2018), which takes as input a rooted maximum likelihood phylogenetic tree (minus the rooting sequence) and its corresponding nucleotide sequence alignment. The three models described above: GTR, NREV6, and NREV12 are then fitted to the data using maximum likelihood (ML). The equilibrium frequencies (EF) of the GTR model match those empirically observed in the alignment, while EFs for NREV6 and NREV12 are inferred by the model, satisfying the condition πQ=0\begin{document}$\boldsymbol {\pi Q=0}$\end{document}. An additional model NREV12 + F is estimated, where the distribution of nucleotides at the root of the tree is estimated by maximum likelihood, instead of being set to the empirical frequencies. Nested models were compared by the LRT with significance evaluated using the χd2\begin{document}$\chi _d^2$\end{document} distribution with d=difference in degrees of freedom. For all models, we also computed the small AIC-c score.

We further defined the DNR as the absolute difference between the relative rate differences of two nucleotide pairs; i.e., for two nucleotides, x and y, there exists a relative rate of x to y substitutions that we will refer to as m, and a relative rate of y to x substitutions that we will refer to as n. Under the NREV12 model, the DNR between x and y is defined simply as the absolute difference between m and n: (|m-n|) and will hereby be referred to as ijDNR\begin{document}$ij_{DNR}$\end{document}, where i\begin{document}$i$\end{document} and j\begin{document}$j$\end{document} are two nucleotides. We use DNR as a mathematical representation of the degree to which the rates of all pairs of reverse substitutions differ from one another. For each of the 140 individual viral alignments, we calculated the average DNR of the six ijDNR\begin{document}$ij_{DNR}$\end{document} estimates inferred using the NREV12 model.

We tested the accuracy of phylogenetic tree inference under reversible and non-reversible models using simulated datasets with varying APIs evolved under the NREV12 model with different DNRs. The goal of these tests was not to exhaustively evaluate model misspecification issues during phylogenetic tree inference, but rather to check, in instances where viral taxa are known to be evolving in a detectably non-reversible manner (i.e. where NREV12 or NREV6 fits the data better than GTR), whether not accounting for this might decrease the accuracy of phylogenetic inference. Using IQ-TREE, a phylogenetic inference program that has the option to apply an NREV12-like model (referred to in IQ-TREE as the UNREST model), a phylogenetic tree was inferred from an alignment of real sequences (Avian Leukosis virus) (Figure 3) with an API of ~90%. The branch lengths on this tree were then scaled to create four other phylogenetic trees representing sequences with approximately 95%, 85%, 80%, and 75% API. These five trees are hereafter referred to as ‘true’ trees, and each individual tree was used as the starting point of a different set of simulations.

Phylogenetic trees were inferred from these 5500 simulated datasets and compared to the phylogenetic trees used to simulate the datasets (i.e. the true trees) using wRF distances to assess the impact of varying DNR on the accuracy of phylogenetic inference. We further tested whether the accuracy of phylogenetic inference could be improved for sequences that had evolved under DNR >0 by using NREV12 instead of GTR. Specifically, for every simulated dataset, a phylogenetic tree was inferred using GTR, and another using NREV12 and the wRF distances of each of these trees to the true tree was determined. For each of the analysed DNRs, a paired t-test was then used to compare the wRF scores of trees inferred using GTR and NREV12. We were particularly interested in determining whether trees inferred using a mis-specified model (i.e. GTR in this case) would be less accurate than trees inferred with a correctly specified model (i.e. NREV12).

To test whether failure to account for non-reversibility might decrease the accuracy of phylogenetic inference, we simulated the evolution of 5500 nucleotide sequence alignments evolved non-reversibly under varying DNR along the five true phylogenetic trees: 100 datasets per true tree per simulated DNR. Specifically, simulations were done using HyPhy (Pond and Muse, 2005), with relative rates ranging from a completely reversible matrix (Equation 4)(4)Q={qij}=(−0.16610.140.166−0.1311.10110.131−0.1880.141.1010.188−)\begin{document}$$\displaystyle \boldsymbol{\it Q=\left\{q_{i j}\right\}=\left(\begin{array}{cccc} - & 0.166 & 1 & 0.14 \\ 0.166 & - & 0.131 & 1.101 \\ 1 & 0.131 & - & 0.188 \\ 0.14 & 1.101 & 0.188 & - \end{array}\right)}$$\end{document}

representing DNR = 0 – through matrices with DNR = 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20 (Table 5). These baselines-simulated substitution rates are reflective of those seen in empirical viral nucleotide sequence datasets.

At each DNR, the relative rates used conformed to standard measures of non-reversibility under the Kolmogorov conditions according to which non-reversibly evolving sequence datasets should yield three irreversibility indices (IRI1, IRI2, and IRI3) that are all non-zero (Squartini and Arndt, 2008). It should be noted that all simulations under NREV12 were performed under the stationarity criterion: πeQt=π\begin{document}$\rm {\pi e^{Qt}=\pi}$\end{document} (where Q is the rate matrix, π is the nucleotide frequency distribution, and t≥0).

We used the wRF (implemented in the R phangorn package; Schliep, 2011⁠) to quantify differences between the true trees used to simulate datasets and the trees inferred from these datasets using the GTR or NREV12 models. wRF considers differences between both the topology and branch lengths of actual and inferred trees (Kuhner and Yamato, 2015; Robinson and Foulds, 1981).

The non-reversible nucleotide substitution model, NREV12, provides a substantially better fit to most virus nucleotide sequence datasets than does the widely used reversible substitution model, GTR. NREV12 also provides a better fit to most virus nucleotide sequence datasets than does NREV6; a non-reversible model that would be expected to best describe the evolution of double-stranded genome sequences that display no strand-specific nucleotide substitution biases. This suggests that, contrary to our expectations, substantial strand-specific nucleotide substitution biases (i.e. estimated DNRs>0.25) are common during viral evolution irrespective of genome type. Such biases should be expected for any viruses where one genome strand either is in existence for substantially longer periods of time than the other, or is more exposed to mutagenic processes than the other during transmission, replication, or gene expression.

We had anticipated that, given evidence of sequences evolving both non-reversibly and with strand-specific substitution biases, inferring trees using a model such as NREV12 that appropriately accounts for this might: (1) minimise the impact of increasing DNR on the accuracy of phylogenetic inference (i.e. wRF scores presented in blue in Figure 2 might have been expected to not increase with increasing DNR) and (2) yield significantly more accurate phylogenetic inferences than when using GTR for all datasets where NREV12 was the most appropriate model and DNRs were greater than zero. However, increasing DNR clearly decreased the accuracy of phylogenetic inference even when using NREV12, and, for datasets where DNRs were greater than zero, using GTR did not consistently yield significantly less accurate phylogenetic inferences than those attained using NREV12. From a practical perspective, choosing a non-reversible nucleotide substitution model to construct phylogenetic trees from virus genome sequences that display strand-specific nucleotide substitution biases is not guaranteed to yield more accurate phylogenetic trees. Nevertheless, in instances where strand-specific substitution biases are higher than ~0.5 (such as are found in our SARS-CoV-2, Torque teno sus virus, and Banana bunchy top virus datasets), it may be prudent to select a model such as NREV12 (such as is implemented in programs like IQ-TREE) over GTR as the better of two suboptimal choices.

The lack of available data regarding the proportions of viral life cycles during which genomes exist in single- and double-stranded states makes it difficult to rationally predict the situations where the use of models such as GTR, NREV6, and NREV12 might be most justified: particularly in light of the poor overall performance of NREV6 and GTR relative to NREV12 with respect to describing mutational processes in viral genome sequence datasets. We therefore recommend case-by-case assessments of NREV12 vs NREV6 vs GTR model fit when deciding whether it is appropriate to consider the application of non-reversible models for phylogenetic inference and/or phylogenetic model-based analyses such as those intended to test for evidence of natural selection or the existence of molecular clocks.