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. 2026 Jul 27;94(4):726–735. doi: 10.1007/s00239-026-10334-3

Evaluation of Molecular Phylogenetic Trees by an Information-Theoretic Metric

Takuma Nishimaki 1, Keiko Sato 1,✉
PMCID: PMC13433539  PMID: 42507036

Abstract

Understanding phylogenetic relationships among species is fundamental to many biological studies. Constructing phylogenetic trees based on molecular data is important not only for considering the evolution and classification of organisms, but also for understanding how genes retain and transmit information, which forms the basis of life science. However, it is currently difficult to determine which phylogenetic analysis method, including evolutionary models, is most appropriate for the target sequences, and which of the resulting phylogenetic trees most appropriately represents the underlying evolutionary relationships. Here, we propose a method for inferring common ancestral sequences and introduce an information-theoretic metric to provide a unified framework for quantitatively comparing phylogenetic trees based on information transmission between molecular sequences. Our metric can evaluate which phylogenetic tree is optimal from an information-theoretic point of view by determining and comparing the amount of information in phylogenetic trees generated by different methods in terms of the transmission of information between molecular sequences along the evolutionary process. The metric we introduced will facilitate understanding not only the evolution and classification of organisms, but also the phenomena of life that are deeply related to information transmission.

Supplementary Information

The online version contains supplementary material available at 10.1007/s00239-026-10334-3.

Keywords: Phylogenetic tree, Information theory, Molecular evolution, Ancestral sequences

Introduction

Understanding the evolution and classification of living organisms is one of the central themes in the life sciences and forms the foundation of much biological research. Appropriate mathematical analyses of molecular sequence data, such as genes and proteins, not only provide insights into the evolution and classification of organisms but also contribute to a more comprehensive understanding of life itself. The construction of phylogenetic trees using molecular data is essential for inferring phylogenetic relationships among organisms. Today, an abundance of molecular data is available, and there are a variety of methods for constructing phylogenetic trees. Commonly used methods include neighbor-joining (NJ) (Saitou and Nei 1987), maximum likelihood (ML) (Felsenstein 1981), and Bayesian inference (BI) (Rannala and Yang 1996) methods. All of them use evolutionary models of nucleotide or amino acid change; the NJ method determines the genetic differences between sequences based on the selected evolutionary model and constructs a phylogenetic tree based on these differences; the ML and BI methods select the most statistically supported phylogenetic tree that best represents a given set of sequences based on likelihood and posterior probability, respectively, depending on parameters such as the selected evolutionary model and branch length. In ML inference, evolutionary model selection is generally carried out using the Akaike information criterion (AIC) (Akaike 1974), the Bayesian information criterion (BIC) (Schwarz 1978), or the likelihood ratio test (Goldman 1993). For comparing alternative ML trees, the Shimodaira–Hasegawa test (Shimodaira and Hasegawa 1999) or the approximately unbiased test (Shimodaira 2002) is widely used. In BI inference, evolutionary model selection is commonly performed using the deviance information criterion (DIC) (Spiegelhalter et al. 2002), the widely applicable information criterion (Watanabe 2010), or the Bayes factor (Kass and Raftery 1995). In addition, comparison among BI trees or testing of specific phylogenetic hypotheses is typically based on posterior probabilities or Bayes factors. Despite methodological advances, there is currently no clear consensus regarding which phylogenetic inference method, including evolutionary models, is most appropriate for a given dataset, nor which analytical approach produces the most reliable phylogenetic tree (Cao et al. 2016; Anisimova et al. 2013; Lartillot 2023; Spielman 2020). Importantly, existing criteria evaluate models or trees within a given inferential framework, but few approaches provide a unified metric that allows direct comparison across fundamentally different phylogenetic methods.

To address this issue, we propose an information-theoretic metric that provides a unified framework for evaluating the reliability of phylogenetic trees inferred using different methods for a given molecular sequence dataset. This requires quantifying differences in evolutionary complexity encoded in molecular sequence data, thereby capturing the biological diversity among species. Several studies have introduced information-theoretic approaches in phylogenetics. For instance, Miyazaki et al. (1996) proposed an information-theoretic measure called the entropy evolution rate to quantify differences between DNA sequences when inferring phylogenetic trees, demonstrating its effectiveness for lineages with high evolutionary rates. Martin et al. (2005) applied mutual information to identify co-evolving residues in multiple sequence alignments of various protein families. In addition, Cao et al. (2016) introduced a model-independent method for estimating sequence differences based on mutual information, departing from conventional approaches that rely on model-dependent genetic distance estimation. Zhou et al. (2015) inferred the states at each internal node of phylogenetic trees representing the evolution of numeral systems, quantified the uncertainty of these inferences using entropy, and thereby assessed ancestral state reconstruction. Our proposed metric quantifies the amount of information transmitted from parent to child nodes within a phylogenetic tree, focusing on how genes store and convey information in the process of creating life. We compare the amount of information transmitted in phylogenetic trees constructed by different methods, and interpret the tree with the highest information transmission as the tree that most effectively transmits sequence information from ancestral to descendant nodes under the proposed information-theoretic metric. We also conduct phylogenetic simulations to evaluate whether the tree with the highest information transmission is consistent with the simulated true tree.

We compare the amount of information transmitted in phylogenetic trees constructed by different methods, and identify the tree with the highest information transmission as the tree that most effectively transmits sequence information from ancestral to descendant nodes under the proposed information-theoretic measure.

Methods

Inference of Ancestral Sequences

To calculate the amount of information in a rooted tree for \:n molecular sequences constructed using a given phylogenetic tree construction method, it is necessary to infer the \:n-1 ancestral sequences corresponding to each internal node. The \:n leaf nodes of the tree are assigned \:n aligned sequences of equal length \:k. Inline graphic is defined as the length of the longest path between all the leaf nodes. Each sequence Inline graphic of internal nodes of the tree is inferred by minimizing the sum of the weighted Hamming distances between the sequence of the internal node \:i and the sequence of each leaf node Inline graphic. That is, each internal node sequence is determined by Inline graphic (i = 1, 2, ⋅ ⋅ ⋅, n – 1), where Inline graphic is Inline graphic minus Inline graphic, the length of the path from internal node \:i to leaf node \:j, and is a weight corresponding to its importance in inferring the internal node sequence. Inline graphic is the Hamming distance of Inline graphic and Inline graphic, corresponding to the internal node \:i and the leaf node \:j, respectively. When inferring Inline graphic, in the case of a nucleotide sequence, each site can contain either four nucleotides or a gap (\:*), so there are Inline graphic candidates, and in the case of an amino acid sequence, each site can contain either 20 amino acids or \:*, so there are Inline graphic candidates. Since the weighted Hamming distance can be computed independently for each site, the cost per site is Inline graphic, and therefore the total computational complexity per internal node is Inline graphic.

Amount of Information in a Tree

Consider a rooted phylogenetic tree with an aligned sequence of length \:k assigned to each of the \:n leaf nodes and an inferred sequence of length \:k assigned to each of the \:n-1 internal nodes. The amount of information in the tree is calculated from the amount of information transmitted from parent to child nodes. For each of the \:n-1 sequences assigned to internal nodes and \:n sequences assigned to leaf nodes, a random variable takes values corresponding to 5 events (4 nucleotides and \:*) or 21 events (20 amino acids and \:*) with probability distribution Inline graphic or Inline graphic, respectively. Let Inline graphic be a set consisting of all nucleotides (or amino acids) and *. Let \:X be the random variable of the parent node, which takes value Inline graphic with probability Inline graphic. Let \:Y be the random variable of its child node, which takes value Inline graphic with probability Inline graphic. Here Inline graphic is the joint probability of \:X=x and \:Y=y satisfying Inline graphic and Inline graphic. That is, \:p(x,y) is the probability of \:x and \:y occurring across the \:k sites. Then the Shannon entropy of \:X is defined by

graphic file with name d33e489.gif

The mutual information Inline graphic between \:X and \:Y is

graphic file with name d33e506.gif

Based on entropy and mutual information, the ratio Inline graphic is defined as the proportion of information about the parent state \:X that is retained in the child state \:Y. This quantity can be interpreted as a measure of information transmission along an evolutionary branch. This normalized quantity ranges from 0 to 1, with higher values indicating greater retention of ancestral information. Although this quantity is related to sequence conservation between parent and child nodes, it is not equivalent to a conventional similarity measure. Unlike simple sitewise similarity metrics, mutual information captures statistical dependence between states and therefore reflects both character-state distributions and uncertainty in ancestral-state reconstruction. Furthermore, the proposed metric is evaluated across all parent-child relationships in the phylogenetic tree, thereby integrating topology and ancestral-state inference into a unified information-theoretic framework. Using this measure, we calculate the normalized information transmission for Inline graphic parent-child pairs in a phylogenetic tree of \:n sequences (species). We define the average of these values as the amount of information contained in the tree and use this quantity as a metric.

Comparison of the Amount of Information in Phylogenetic Trees

Rooted phylogenetic trees were constructed from real sequence data using the NJ, ML, and BI methods, and the amount of information in each tree was compared. Three datasets were used: the cytochrome b (CYTB) gene nucleotide sequences, the encoded amino acid sequences, and the 16S rRNA nucleotide sequences from 34 species of domestic animals used in the phylogenetic analysis of Zhang et al. (2020). We also used two other datasets: the spike (S) gene nucleotide sequences and the encoded amino acid sequences of 121 human coronavirus strains (HCoV-OC43, 36; HCoV-229E, 37; HCoV-NL63, 48) collected by Tang et al. (2023). CYTB and 16S rRNA are well-established molecular markers in animals and microorganisms, respectively. In addition, we used the S gene and S protein of coronaviruses as examples of rapidly evolving sequences to evaluate our metric under different evolutionary rates. The minimum and maximum sequence lengths for each dataset are listed in Table 1. The multiple sequence alignment was performed using the default parameters of MAFFT (version 7.526) (Katoh et al. 2019). P*R*O*P (Nishimaki and Sato 2021) was used for phylogenetic tree construction by the NJ method. Kimura two-parameter (K2P) (Kimura 1980) for nucleotide sequences and Jukes-Cantor (JC) (Jukes and Cantor 1969) for amino acid sequences were selected as evolutionary models. The treatment of gaps was performed using two patterns: pairwise deletion of gap sites and + Gap (Nishimaki and Sato 2019, 2021). RAxML-NG (version 1.1.2) (Kozlov et al. 2019) was used for phylogenetic tree construction using the ML method. From the options available in the tool, a relatively simple evolutionary model and a relatively complex evolutionary model were selected. For nucleotide sequences, the K2P model and the general time reversible plus gamma (GTR + G) model (Tavaré 1986; Yang 1993) were used. For amino acid sequences, the Dayhoff model (Dayhoff et al. 1978) and the Jones-Taylor-Thornton plus gamma (JTT + G) model (Jones et al. 1992; Yang 1993) were used. All other parameters were defaults. We also checked the AIC and BIC values obtained using RAxML-NG. MrBayes (version 3.2.7a) (Ronquist et al. 2012) was used for phylogenetic tree construction with the BI method. As with the use of RAxML-NG, a relatively simple evolutionary model and a relatively complex evolutionary model were selected from the options in MrBayes. The Hasegawa-Kishino-Yano (HKY) (Hasegawa et al. 1985) and GTR + G models were applied to nucleotide sequences, and the Poisson (Zuckerkandl and Pauling 1965) and JTT + G models were applied to amino acid sequences. All other parameters were defaults. Note that we calculated the DIC approximations given by Gelman et al. (2014) based on the log-likelihoods written in the MrBayes .p files. For the unrooted trees of P*R*O*P NJ, RAxML ML, and MrBayes BI, we rooted the tree at the midpoint of the longest leaf-to-leaf length in the unrooted tree. For each rooted phylogenetic tree generated by the above phylogenetic analysis methods, we assigned the aligned sequences to the leaf nodes, inferred the sequences of the internal nodes, calculated the amount of information, and evaluated which phylogenetic analysis method could construct phylogenetic trees with higher amounts of information. In addition, for the CYTB, 16S rRNA, and S gene datasets, and for the CYTB protein and S glycoprotein datasets, 100 random phylogenetic trees each were generated using msprime (Kelleher et al. 2016). These trees were generated to test whether their amount of information was less than that of the phylogenetic trees generated by NJ, ML, and BI. First, each random phylogenetic tree was generated using only the number of sequences, and the aligned sequences were randomly assigned to the leaves of the tree. Then, the internal sequences of the random phylogenetic tree were inferred and the amount of information in the tree was calculated.

Table 1.

The minimum and maximum sequence lengths for each dataset

Dataset Sequence length
Min. Max.
CYTB gene 1140 nt 1144 nt
16S rRNA gene 1564 nt 1623 nt
S gene 3513 nt 4101 nt
CYTB protein 379 aa 381 aa
S glycoprotein 1170 aa 1366 aa
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In addition to the primary analysis using our weighted Hamming distance method on midpoint-rooted trees, we performed a supplementary ancestral sequence reconstruction using a maximum likelihood–based approach implemented in IQ-TREE. The original unrooted tree topology and branch lengths were used as input, and branch lengths were fixed using the -blfix option during reconstruction. Substitution models were specified as GTR + G4 for nucleotide sequences and JTT + G4 for amino acid sequences. After ancestral states were inferred on the unrooted trees, the trees were midpoint-rooted, and the information metric was calculated for all parent-child pairs except for the two branches directly descending from the root. This supplementary analysis was conducted to assess whether the choice of ancestral sequence inference method could affect the relative ranking of phylogenetic trees.

Phylogenetic Simulation for Testing the Information-Theoretic Metric

We conducted a phylogenetic simulation analysis to examine whether the information-theoretic metric can accurately identify the true phylogenetic tree. First, random tree topologies were generated under the Yule process (Harding, 1971). Based on each topology and the specified evolutionary model, nucleotide and amino acid sequences corresponding to the leaf nodes were simulated using AliSim (Ly-Trong et al., 2022), which is included in IQ-TREE 2 (Minh et al., 2020). Random trees were generated for two conditions, with the number of taxa N = 20 and N = 100. For each tree, all branch lengths were scaled so that the mean branch length became 0.02 or 0.05. For nucleotide sequences, sequence lengths of 300 and 1000 were used, and for amino acid sequences, lengths of 100 and 300 were used. The substitution models GTR + G4 and JTT + G4 were used for nucleotide and amino acid sequence simulations, respectively, using default parameter settings. Each simulated tree was regarded as the true phylogenetic tree. For each true tree, nine additional, similar but distinct topologies were generated by applying a single Subtree Pruning and Regrafting (SPR) operation. The same procedure was repeated using three and five successive SPR operations, generating nine additional perturbed trees for each case. The amount of information for all ten trees (one true tree and nine perturbed trees) was calculated using our metric and ranked in descending order. For each simulation replicate, we recorded whether the true tree achieved the highest information score (i.e., ranked first among the ten candidate trees). This entire procedure was repeated 100 times for each parameter combination. For each condition, the Top-1 identification rate was calculated as the proportion of replicates in which the true tree was ranked first. Two-sided 95% confidence intervals were computed using the Wilson score method.

Results

Comparison of Phylogenetic Trees by the Information-Theoretic Metric in Real Data

We introduced an information-theoretic metric that enables quantitative comparison of phylogenetic analysis methods by evaluating which methods construct phylogenetic trees with higher amounts of information. Phylogenetic trees inferred from the CYTB, 16S rRNA, and S gene nucleotide datasets are shown in Supplementary Figures S1-S3. Table 2 shows the amount of information in the phylogenetic trees by each method using the CYTB, 16S rRNA, and S gene datasets. For the CYTB gene, the amount of information in the BI tree with the GTR + G model was 0.83352, which was the highest compared to the trees obtained by the other methods. For the 16S rRNA gene, the ML tree with the GTR + G model had the highest amount of information, 0.84548. For the S gene, the BI tree with the HKY model showed the highest amount of information, 0.96945. In contrast, the ML tree inferred under the GTR + G model showed a substantially lower amount of information (0.87192) than the other five trees. According to the results of DIC for the BI phylogenetic inference of the S gene, the GTR + G model was the preferred model, not the HKY model (Table 3). Furthermore, the AIC and BIC results for the ML phylogenetic inference of the S gene recommended the GTR + G model over the K2P model (Table 3), although the amount of information in the GTR + G model ML tree was clearly lower than that in the other five trees, including the K2P model ML tree.

Table 2.

Comparison of the amount of information in the gene trees by each method

Method Evolutionary model Amount of information
CYTB gene 16S rRNA gene S gene
NJ K2P (pairwise deletion) 0.82663 0.83962 0.96936
K2P + Gap 0.82661 0.84050 0.96939
ML K2P 0.82994 0.84173 0.96931
GTR + G 0.83312 0.84548 0.87192
BI HKY 0.83025 0.84185 0.96945
GTR + G 0.83352 0.84545 0.96876
Random Max 0.80449 0.82320 0.72187
Random Ave 0.79735 0.79668 0.70486
Random Min 0.76435 0.77526 0.67689
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Values are rounded to the sixth decimal place and shown to the fifth decimal place; bold values indicate the highest amount of information, and Random Max, Ave, and Min represent the maximum, average, and minimum amount of information from 100 random phylogenetic trees

Table 3.

Model selection for the gene data by AIC/BIC (ML) and DIC (BI)

Method Evolutionary model CYTB gene 16S rRNA gene S gene
AIC BIC DIC AIC BIC DIC AIC BIC DIC
ML K2P 30,048 30,056 35,868 36,229 48,081 49,626
GTR + G 25,892 26,266 31,944 32,349 46,427 48,023
BI HKY 29,546 35,353 47,380
GTR + G 25,915 31,954 46,878
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Values in bold show the lowest of the two candidate models

Phylogenetic trees inferred from the CYTB protein and S glycoprotein datasets are shown in Supplementary Figures S4 and S5. Table 4 shows the amount of information in the phylogenetic trees by each method using the CYTB protein and S glycoprotein datasets. For the CYTB protein, both the NJ tree with the JC model (pairwise deletion of gap sites) and the NJ tree with the JC + Gap model had the highest amount of information of 0.95228 compared to the trees obtained by other methods. The amino acid alignment length of the CYTB protein was 382 with almost no gaps. Therefore, the NJ trees based on the JC model and the JC + Gap model had the same topology and almost the same branch lengths. For the S glycoprotein, the NJ tree with the JC + Gap model had the highest amount of information, 0.98078. In contrast, the ML tree inferred under the JTT + G model showed a lower amount of information (0.90807) than the other five trees. The AIC and BIC results for the ML phylogenetic inference of the S glycoprotein recommended the JTT + G model over the Dayhoff model (Table 5), although the amount of information in the JTT + G model ML tree was lower than that in the other five trees, including the Dayhoff model ML tree. Also, the DIC results for the BI phylogenetic inference of the S glycoprotein indicated that the JTT + G model was the recommended model instead of the Poisson model (Table 5), although the amount of information in the Poisson model BI tree was higher than that in the JTT + G model BI tree.

Table 4.

Comparison of the amount of information in the protein trees by each method

Method Evolutionary model Amount of information
CYTB protein S glycoprotein
NJ JC (pairwise deletion) 0.95228 0.98031
JC + Gap 0.95228 0.98078
ML Dayhoff 0.95095 0.97692
JTT + G 0.95153 0.90807
BI Poisson 0.95119 0.98038
JTT + G 0.95223 0.97690
Random Max 0.93010 0.80192
Random Ave 0.92658 0.77919
Random Min 0.91016 0.76011
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Values are rounded to the sixth decimal place and shown to the fifth decimal place; bold values indicate the highest amount of information, and Random Max, Ave, and Min represent the maximum, average, and minimum amount of information from 100 random phylogenetic trees

Table 5.

Model selection for the protein data by AIC/BIC (ML) and by DIC (BI)

Method Evolutionary model CYTB protein S glycoprotein
AIC BIC DIC AIC BIC DIC
ML Dayhoff 8611 8868 27,858 29,149
JTT + G 7912 8173 27,077 28,373
BI Poisson 9520 29,164
JTT + G 7942 27,480
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Values in bold show the lowest of the two candidate models

In all five datasets, the amount of information in the phylogenetic trees generated by each of the phylogenetic analysis methods used in this study was greater than the maximum amount of information in 100 randomly generated phylogenetic trees.

In addition to the primary analysis using our weighted Hamming distance method, we performed a supplementary ancestral sequence reconstruction (ML-ASR) for each phylogenetic tree constructed by NJ, ML, and BI (Supplementary Table S1). Across the five datasets, CYTB gene, 16S rRNA gene, S gene, CYTB protein, and S glycoprotein, the ML-ASR results showed amounts of information comparable to those obtained with the weighted Hamming distance method. Specifically, for the nucleotide datasets, average amounts of information were 0.878 (SD = 0.00026) for CYTB, 0.855 (SD = 0.00006) for 16S rRNA, and 0.910 (SD = 0.00018) for S gene. For the amino acid datasets, the averages were 0.974 (SD = 0.00014) for CYTB protein and 0.891 (SD = 0.00004) for S glycoprotein. By comparison, the weighted Hamming distance method produced averages ranging from 0.830 to 0.967, but with larger standard deviations (0.00051–0.03358) than ML-ASR, reflecting greater variability and sensitivity to sequence differences among trees. Notably, in the S gene and S glycoprotein datasets, ML trees inferred under complex evolutionary models exhibited very long internal branches. Using the weighted Hamming distance approach for internal node inference, this resulted in a clear reduction in the amount of information of these trees. In contrast, the ML-ASR-based metric, computed on the fixed-branch-length unrooted trees, did not capture these effects, and the amounts of information were similar across trees.

Performance of the Information-Theoretic Metric in Simulations

In the phylogenetic simulation analysis, we evaluated the ability of the proposed information-theoretic metric to distinguish the true topology from alternative candidates. For each simulated true tree, its score was compared against nine alternative topologies generated by successive SPR operations (m = 1, 3, or 5), totaling ten candidate trees for each comparison set. As illustrated in Fig. 1, the metric achieved a high Top-1 identification rate, frequently reaching 1.0 (100% accuracy) as the number of SPR operations increased. This trend was robust regardless of the sequence type (nucleotide or amino acid), the number of taxa (N = 20 or N = 100), the mean branch length (0.02 or 0.05), or the sequence length (300 or 1000 nt for nucleotide sequences; 100 or 300 aa for amino acid sequences). Furthermore, the narrow 95% confidence intervals shown in Fig. 1 confirm the statistical reliability of the metric, demonstrating that the information-theoretic differences effectively distinguish the true topology from increasingly divergent alternatives.

Fig. 1.

Fig. 1

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Top-1 identification accuracy of the true topology. The accuracy of the metric was evaluated by its ability to identify the true topology among candidates perturbed by m SPR operations (m = 1, 3, 5). Simulations were performed with varying numbers of taxa (N = 20, 100) and mean branch lengths (MBL = 0.02, 0.05). Each panel displays the Top-1 identification rate for nucleotide sequences (L = 300, 1000) or amino acid sequences (L = 100, 300). Each data point represents the proportion of 100 simulation replicates in which the true tree was correctly ranked as the top topology, with shaded regions indicating the 95% confidence intervals

Discussion

We proposed a method to infer ancestral sequences and introduced an information-theoretic metric to evaluate in a unified way which method of phylogenetic analysis can construct a phylogenetic tree that maximizes information transmission.

Branch length in a phylogenetic tree represents time or genetic difference in evolution. Branch length estimation is also an important factor in phylogenetic tree construction. In a phylogenetic tree, the shorter the length of the branches between a parent and its children, the higher the degree of similarity between their sequences. Thus, focusing on an internal node and considering the sum of branch lengths from that internal node to each leaf node, we can infer that leaf node sequences with shorter sums resemble that internal node sequence. In the inference of ancestral sequences (internal node sequences), we calculated the sum of branch lengths leading to each leaf node for each internal node, and gave greater weight to shorter sums so that closely related leaf sequences were more strongly reflected in the internal node sequence. Ancestral sequences are inferred by minimizing the sum of the weighted Hamming distances with the aligned sequences assigned to the leaves of the tree. In this method, the execution time is only proportional to the number of sequences (species) \:n and the sequence length \:k, respectively, so it can be completed in linear time of Inline graphic.

Our metric can evaluate which phylogenetic tree is optimal from an information-theoretic point of view by determining and comparing the amount of information in rooted phylogenetic trees generated by different methods in terms of the transmission of information between molecular sequences along the evolutionary process. This study used the midpoint of the longest leaf-to-leaf length as the root to convert the unrooted phylogenetic trees to rooted phylogenetic trees. However, even when the root is determined by the outgroup, it is possible to calculate the amount of information in the trees.

To examine whether the choice of ancestral reconstruction framework affects the amount of information, we conducted supplementary analyses using maximum likelihood-based ancestral sequence reconstruction (ML-ASR). Across all five datasets, ML-ASR produced amounts of information that were generally comparable across trees and exhibited very small variance, and in several cases identical values were obtained for different trees. In contrast, the weighted Hamming distance-based approach showed larger variability among trees, and in some cases yielded higher amounts of information than ML-ASR. When trees were strictly ranked, differences in ordering were observed between ML-ASR and the weighted Hamming distance-based approach. Therefore, the information metric should be interpreted in conjunction with the ancestral inference framework used. The tree with the highest amount of information provides an objective evaluation of the information transmission from ancestral to descendant nodes, reflecting the underlying evolutionary relationships. Nevertheless, marginal differences in the amount of information between highly competitive empirical trees should be interpreted with caution, as these minor variations may reflect stochastic sampling noise inherent to finite alignment lengths rather than definitive biological or evolutionary superiority.

Focusing on the S gene and S glycoprotein datasets, the weighted Hamming distance-based analysis revealed that the amount of information was higher for ML trees obtained with simple evolutionary models than for those obtained with complex models, even though AIC and BIC supported the complex models. In contrast, when ancestral sequences were inferred using ML-ASR, this difference between simple and complex ML models was largely diminished, with only very minor differences remaining. In both datasets, the ML trees inferred under the complex evolutionary models had extremely long internal branch lengths, which were not observed in the other trees. Although the overall topologies of the trees appeared largely similar across methods, the scale of branch lengths in ML trees under complex models was markedly different, which in our weighted Hamming distance-based framework resulted in a significant reduction in the contribution of the descendant leaf sequences to the inference of internal nodes. Human coronaviruses show a tendency for recombination in structural genes such as the S gene (Pollett et al. 2021; Müller et al. 2022). ML phylogenetic inference can produce artificially long branches when recombinant lineages are present, as shown previously (Turakhia et al. 2022). Furthermore, under complex substitution models, branch lengths, particularly internal branches, tend to increase (Ekman and Blaalid 2011). Because complex models have many parameters, ML inference can overfit random substitution patterns in the data, leading to spurious branch lengths or topologies as an overfitting artifact (Lartillot and Philippe 2004).

The weighted Hamming distance-based approach is a heuristic method based on discrete character states and does not explicitly model continuous-time substitution processes or site-specific rate variation. Consequently, it may be less suited for datasets with extreme sequence divergence or strong saturation effects in either nucleotide or amino acid sequences, reflecting a trade-off between biological realism and computational efficiency. Nevertheless, our weighted Hamming distance-based approach provides a practical and computationally efficient metric for comparing phylogenetic trees, and it tends to capture relative differences among trees more clearly than ML-ASR, highlighting differences in tree rankings more distinctly. Although the present study applied the proposed information-theoretic metric to single-gene and single-protein datasets, its primary objective was to establish a conceptual and mathematical framework for evaluating phylogenetic trees. The metric is independent of the tree inference procedure and has linear computational complexity with respect to both the number of sequences (\:n) and the sequence length (\:k), i.e., Inline graphic. Therefore, as long as phylogenetic trees can be inferred from genome-wide datasets using existing methods, the metric can be directly applied, and its applicability naturally extends to genome-scale data.

Existing substitution-model selection tools are generally designed for specific phylogenetic inference approaches, such as maximum likelihood, Bayesian, or genetic difference-based methods. While they provide useful guidance for choosing models within each approach, their results are often difficult to compare directly across different methods. In contrast, our information-theoretic metric offers a unified way to evaluate phylogenetic trees, allowing consistent comparison of trees inferred using different approaches by quantifying their amount of information. In addition to existing criteria for evaluating evolutionary model selection, our information-theoretic metric is likely to enable more quantitative and in-depth phylogenetic inference. In the current situation, where it is challenging to determine which phylogenetic method is most suitable for large and diverse sequence datasets, and which resulting tree most appropriately represents the underlying evolutionary relationships, this metric may contribute not only to understanding organismal evolution and classification but also to gaining insight into life phenomena closely linked to information transmission.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary Material 1 (83.6KB, pdf)
Supplementary Material 2 (5.9MB, pdf)

Acknowledgements

Keiko Sato is very grateful to Masanori Ohya for his guidance and discussions on the information-theoretic approach needed to quantify life information as a science.

Author Contributions

All authors contributed to the conception and design of the study, the theoretical framework, the data analysis, and the interpretation and discussion of the findings. Takuma Nishimaki wrote the script to calculate the amount of information in phylogenetic trees and contributed to the revision of the manuscript. Keiko Sato contributed to the drafting and revision of the manuscript. All authors approved the final manuscript.

Funding

Open Access funding provided by Tokyo University of Science. This work was partially supported by JSPS KAKENHI Grant Number JP23K05880.

Data Availability

Datasets, accession numbers, and script to calculate the amount of information in phylogenetic trees associated with this paper are available on GitHub at https://github.com/TUS-Satolab/PhyloInfo/.

Declarations

Conflict of interest

The authors declare no conflict of interest.

Footnotes

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

Supplementary Material 1 (83.6KB, pdf)
Supplementary Material 2 (5.9MB, pdf)

Data Availability Statement

Datasets, accession numbers, and script to calculate the amount of information in phylogenetic trees associated with this paper are available on GitHub at https://github.com/TUS-Satolab/PhyloInfo/.

Articles from Journal of Molecular Evolution are provided here courtesy of Springer

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