Evolutionary reconstruction algorithms produce models of the evolutionary history of proteins or species. Such algorithms are highly sensitive to their inputs: the sequences used and their alignments. Here, we asked whether the variance introduced by selecting different input sequences could be used to better identify accurate evolutionary models. We subsampled from available ortholog sequences and measured the distribution of observed relationships between paralogs produced across hundreds of models inferred from the subsamples. We observed two important phenomena. First, the reproducibility of an all-sequence, single-alignment reconstruction, measured by comparing topologies inferred from 90% subsamples, directly correlates with the accuracy of that single-alignment reconstruction, producing a measurable value for something that has been traditionally unknowable. Second, topologies that are most consistent with the observations made in the ensemble are more accurate and we present a meta algorithm that exploits this property to improve model accuracy. Orthology and paralogy are two forms of evolutionary homology between genetic sequences introduced by Walter Fitch  to distiWe propose to reduce the complexity of reconstruction, motivated by the fact that not all ancestral nodes in protein evolution are of equal interest. Since a typical collection of input sequences contains both orthologs and paralogs, the divergence topology for these sequences contains both duplication and speciation ancestral nodes. Depending on the protein family and the number of species represented in the collection, the fraction of speciation nodes may be quite high. Reconstructing speciation history from a single protein family is of little interest aside from identifying extremely rare evolutionary events, such as horizontal gene transfer and reciprocal paralog loss, through reconciliation of the protein phylogeny with the accepted species phylogeny . And yetGiven any high confidence ancestral node, we can imagine "factoring'' the search of topology space into two components: (a) topologies below the node in question, (b) everything else. This is not particularly helpful for phylogenetic reconstruction in general, since (a) and (b) still must be optimized jointly in order to determine the overall optimal phylogeny. However, if we were not concerned with inferring (a), we would be satisfied to integrate over the uncertainty of (a) to produce, in a sense, a marginal reconstruction of (b). Moreover, treating the root of (a) – the high confidence ancestral node – as a leaf in our reconstruction of (b), would allow us to simultaneously consider multiple sources of uncertainty in the "integrated out'' reconstruction (a): the selection of specific sequences descended from that ancestor to be included in the reconstruction, the alignment of those sequences, and the uncertainty in the inference of (a), given that alignment. Here, we describe an approach for such marginal reconstruction and demonstrate it by reconstructing histories of gene duplications, using common ancestors of ortholog sets as the roots of the integrated-out clades.We hypothesize that topology features that are robust across many phylogenetic models of divergence are more likely to be accurate and that we can use the variability of topology inference induced by sampling leaves under a set of ancestral nodes of interest to identify the topological features most consistent with robust phylogenetic signal. Testing this hypothesis requires a mechanism for capturing the uncertainty and incorporating it into phylogenetic reconstruction. Inspired by the findings of Salichos and Rokas, who showed that topologies of yeast divergence reconstructed from single genes, while rarely identical, shared much greater similarity than randomly generated topologies , we propIn this work, we present a framework for: 1) assessing the amount of uncertainty in the phylogenetic reconstruction of a protein family, 2) identifying topological features and determining the frequency with which they occur across reconstructions, and 3) quantifying the consistency of individual topologies with the identified features. Based on our findings, we propose an observable metric of reconstruction uncertainty, precision, which directly correlates with reconstruction accuracy. We then describe a meta-algorithm that identifies the topologies which are most consistent with topological features extracted from an ensemble of individual reconstructions. Finally, we demonstrate that for some reconstruction tasks, multiple models should be considered as equally likely, given the evidence. Our methodology, ASPEN (Accuracy through Subsampling of Protein EvolutioN), produces significantly more accurate topology models than those produced by a single-alignment reconstruction from all available sequences.We tested the effectiveness of our analysis and inference framework on simulated protein families and the LacI family of bacterial transcription factors. We turned to simulated sequence data, in addition to a real protein family, for two previously noted reasons . First, Evolution of real protein families is rarely that neat. Speciations can predate duplications. Loss of some paralogs in some lineages can result in different collections of paralogs appearing in different genomes, dramatically complicating orthology assignments. As a result, it can be difficult to determine with a high degree of confidence, whether the leaf sequences in a clade are strictly one-to-one orthologs, or contain some paralogs. In addition, duplication events intermingled with speciation events can produce co-orthologs: many-to-one orthology relationships between genes in post-duplication and pre-duplication species . FortunaThe framework for generating reduced topologies and quantifying reconstruction uncertainty is outlined in We quantified topology differences using the Robinson-Foulds symmetric distance metric , modifieWe refer to a topology inferred from an alignment of all available sequences as the "all-sequence'' topology. This is the topology most often used when considering a single reconstruction. Using multiple representative sequence sets obtained through subsampling allows us to characterize the uncertainty of ancestor divergence reconstruction arising from selection and, critically, alignment of descendant sequences. In this work, we used two different subsampling sizes to characterize uncertainty. First, we subsampled most, but not all, of the available sequences. The comparisons of these topologies to the all-sequence topology are effectively a measure of the reproducibility of the all-sequence reconstruction given highly similar representations of descendant divergence, but under uncertainty due to input sequence alignment. We refer to this quantity as precision, defined as:st genes , with thst genes . We settSince the specific alignment and phylogenetic inference algorithms used affect reconstruction , we test studies . L-INS-i studies . FastTre studies . BecauseWe made a striking observation when we compared the reproducibility of an all-sequence tree with its accuracy. Reproducibility, expressed as precision, was measured by comparing 50 trees created from ≈90% subsamples of available sequences . As showWe wished to understand why subsampling produces a metric that is correlated with accuracy, a hidden value, and to understand the potential generalizability of this observation. To explore this further, we compared the values of the topology search objective function of topologies (log-likelihoods), given an alignment:and under the different model used for simulating sequence evolution . We compared the topologies selected by FastTree2 for all-sequence alignments (the reference topology in our formulation) to true topologies using the log-likelihood difference metric over the all-sequence alignments. A negative value would indicate that the alternate (true) topology obtained a better log-likelihood score than the reference, but the search algorithm failed to identify it. We found no such cases among our synthetic families . While i-4, suggesting that the all-sequence topology is within the optimum well. As accuracy and precision decrease, and -4 to 10-3 worse than the optimum. Greater displacement of the optimum pushes the all-sequence topology out of the optimum well for more synthetic families, but does not increase their log-likelihood differences from the reference beyond the 10-4 – 10-3 range. We were surprised to discover that true topologies were always located on this plateau, outside of the optimum well on the all-sequence alignment likelihood surface  topology to the all-sequence  topology using the log-likelihood difference metric. In order to calculate the likelihood of the all-sequence topology given a subsampled alignment, the all-sequence topology was pruned down to the set of leaves contained in the subsampled alignment. Unlike the all-sequence vs. true topology comparisons , the ext surface . We suggGiven that subsampling produces new likelihood surfaces determined by the alignments of subsampled sequences, we wished to ask whether topology reconstruction accuracy is determined by the accuracy of the input sequences alignment , and whether precision of topology reconstruction is determined by the robustness with which the input sequences can be aligned. If this were the case, then alignment accuracy and robustness would explain the observed precision-accuracy relationship for topology reconstruction. To test this, we measured the similarity between all-sequence and true alignments using the Guidance sum-of-pairs alignment metric , as implNext, we asked whether the reproducibility (robustness) of the all-sequence alignment could explain reproducibility of the inferred topology (precision). We used Guidance2  to deterNext, we sought to apply our sequence subsampling framework to characterize the region of topology space throughout which subsampled topologies are dispersed, and then to use this information to reconstruct topologies that are most consistent with observations across the subsampled ensemble. We arrived at a reconstruction approach that has three parts: (1) a framework to characterize features observed across an ensemble and quantify their occurrence, (2) a scoring function that reflects the consistency of a given topology with observations across the ensemble, and (3) an algorithm that exhaustively enumerates the highest-scoring topologies according to this metric. We call this approach ASPEN, for Accuracy through Subsampling of Protein EvolutioN.First, we required a way of representing topological features observed across multiple topologies and of quantifying the occurrence of those features. Since there is no obvious way to do this using the standard acyclic graph topology representation, we turned to an alternative representation  in termsWe can use the frequencies with which specific lengths of leaf-to-leaf paths present in a topology occur in an ensemble to reflect the consistency of that topology with observations across the ensemble, and to make comparisons between proposed topologies. ASPEN formalizes this into a scoring function expressed in terms of log-frequencies of leaf-to-leaf path lengths, This scoring function rewards incorporation of frequently observed path lengths and penalizes rarely observed ones.N-best topologies that are the most consistent topologies with observations across the ensemble. Given that objective, we created a branch-and-bound procedure to identify the top N topologies discussed here. A detailed description of the algorithm is available in Materials and methods. Briefly, branching occurs when a partially constructed topology can be extended by multiple internal nodes. An internal node is permitted as an extension only if every pairwise path completed by the proposed node . Next, we wished to explore whether ASPEN’s ranking system, based on likelihood according to the ensemble, identifies more accurate trees in general. We wanted to understand how log-frequency scoring facilitates identification of more accurate topologies than phylogenetic reconstructions from single alignments. Using the length of path between two leaves, we explored the connection between path length frequencies observed across an ensemble and differences between true, all-sequence, and ASPEN topologies.all paths with incorrect length in the all-sequence topology  remains surprisingly constant. The larger fraction of all paths for which the ensemble supports the true path length (height of blue bar segment) accounts for the much slower drop off in accuracy of all ASPEN topologies at lower precision, compared to the precipitous fall in the accuracy of all-sequence topologies. Unfortunately, the fraction of all paths for which the ensemble supports the incorrect all-sequence path length (height of white bar segment) also increases at lower precision, which explains why any drop off in ASPEN topology accuracy occurs at all. Nevertheless, the frequency of the most frequent path length and the breadth of the path length distribution provides a measure of confidence in the ensemble’s support of a particular path length. The aggregate of this confidence across all pairwise paths is reflected in the log-frequency score differences between ASPEN topologies.First, we compared the observation frequencies of path lengths between true and all-sequence topologies. Among paths on which the two topologies disagree, the length consistent with the true topology was observed more frequently than the length consistent with the all-sequence topology for half or more paths across all but the highest-precision bin . In the Next we examined the agreement, in terms of path lengths, between the top ASPEN topology, the true topology, and the all-sequence topology . As expeAs mentioned previously, LacI falls into an intermediate range of reconstruction difficulty. In this range, the 10 to 30 highest ranked ASPEN topologies are likely to be more accurate than any all-sequence reconstruction, based on our observations from synthetic protein families . Given tA comparison of all-sequence and top ASPEN topologies  illustraWe described a novel approach to analyzing and reconstructing divergence histories of protein families. Our approach is conceptually rooted in the decomposition of the topology search space at high confidence ancestral nodes, which are extremely likely to exist in the true topology, and takes advantage of the fact that complete divergence histories include ``nuisance'' segments, which provide little biological insight. Instead of reconstructing such segments, we propose integrating over the uncertainty of their reconstruction to produce more accurate "marginal'' reconstructions of the most interesting segments. Critically, our approach considers the uncertainty arising from input sequence selection and alignment, a historically thorny issue in phylogenetic analysis . The traSupplementary Material) to model among-site differences in substitution rate across the protein fold. To incorporate model violation into our phylogenetic reconstructions, we used the WAG substitution model with substitution rate heterogeneity modeled by a discrete gamma distribution and no mechanism to model heterotachy. However, the evolutionary histories of real protein families may be subject to a number of additional forms of model violation, such as among-site variation in amino acid frequencies, varying selection pressures leading to convergent evolution, a known cause of the long branch attraction artifact in phylogenetic reconstruction, and, crucially, non-independent evolution of sites.Given the potential usefulness of the correlation between accuracy and precision, it remains vitally important to ascertain the robustness of this relationship to varying degrees of mismatch between actual evolutionary processes which gave rise to extant sequences and the evolutionary model used in phylogenetic inference. In our sequence evolution simulations, we rescaled the grafted species tree, both as a whole and its individual segments, to model evolutionary rate differences between paralogs (heterotachy) and used the JTT amino acid substitution model with variable branch length scaling spanning a five-fold range  systematically higher accuracy than would typically be indicated by the corresponding degree of precision, or 2) systematically higher precision than the degree of accuracy. The former – In this study, we limited evolutionary simulations to a simplified scenario: divergence of 15 paralogs through duplication events, followed by divergence of orthologs through speciation without any further duplications or gene loss. This produced topologies with the divergence history of 66 species fully recapitulated under the common ancestor of each ortholog set. With only duplication nodes above ortholog common ancestors, and only speciation nodes below them, these 15 ancestors were the natural choice for roots of integrated-out clades in our analysis. Inference of ortholog common ancestors was nearly universally recapitulated across all subsamples for all 600 simulated families, so these nodes satisfied our high confidence criterion for decomposition of the topology search space.In real protein families, including the LacI family, duplications are typically intermingled with speciations. For distantly related species, where multiple duplications have occurred along both lineages since the species diverged, this can produce collections of paralogs without orthology relationships to each other. For more closely related species, where one or more duplications occurred along one of the two lineages, it can result in co-orthology, mapping multiple genes in a post-duplication genome to a single gene in a genome from the other lineage. In addition, loss of paralogs along some lineages can result in different numbers of paralogs in different species, with unclear orthology vs paralogy relationships. ASPEN analysis of any real protein family starts with selecting the ancestral nodes for search space decomposition. Any collection of internal nodes, regardless of the relationships among leaves within each clade or their relationships to leaves outside the clade, can be selected for this purpose. In our analysis of the LacI family, the leaf sets in integrated-out clades did not come from a uniform collection of species, suggesting that post-speciation duplications and/or gene loss occurred in the evolution of LacI paralogs. The only requirement is sufficient confidence in each ancestor’s existence, as evidenced by robust inference over any set of reasonable input alignments or alignment perturbations. In our opinion, ancestors with predominantly speciation nodes below them and predominantly duplication nodes above, as evidenced by leaf sequence species of origin, result in analysis with the most biological insight, but this is not a requirement. Although this step must be carried out individually for any protein family, based on the information researchers wish to obtain from the analysis, we provide some suggestions for how to apply our methodology to one’s protein family of interest.We recommend relying on existing genomic and/or functional annotations and average within-group vs. average out-of-group sequence similarity (separability) to identify candidate leaf subsets to integrate out. When dealing with large families, such as some protein domains, which can number in the dozens or hundreds of paralogs in vertebrate genomes, common ancestors of multiple closely related paralogs may be logical choices. ASPEN methodology can be used recursively to separately reconstruct the divergence of these paralogs – another advantage of decomposing the search space at high confidence nodes. Next, we recommend reconstructing an all-sequence topology and a small ensemble of densely subsampled topologies to assess confidence in the existence of selected ancestors, as well as to determine reconstruction precision and form an expectation of the ultimate accuracy of reconstruction. If selected leaf subsets are not reliably monophyletic across the ensemble, the selections need to be revisited until the common ancestor of each integrated out clade can be inferred with high confidence.Clearly, applicability of ASPEN analysis to any particular family hinges on the existence of such high confidence ancestors and the willingness to sacrifice reconstruction of clades below those ancestors. In part, this requirement can be satisfied by careful assembly of the input sequence collection: including as many ortholog or co-ortholog sequences as possible will maximize the chances of high confidence reconstruction of the common ancestors of those orthologs, provide large leaf subsets for subsampling, and mitigate the conditions under which the long branch attraction artifact occurs. Nevertheless, it is possible that individual leaf sequences or common ancestors of a few sequences (e.g. five or fewer) will not be robustly monophyletic with any other leaves. We recommend including these leaf sequences in every subsample and treating them  as leaves in ASPEN topology reconstruction. ASPEN analysis remains applicable given at least one integrated out clade, although we expect the usefulness of that analysis to diminish as more leaf sequences fall outside of such clades.N-best trees were produced – at the end of its run, but execution times and resource requirements can be substantial for large families. ASPEN’s reconstruction is parallelizable across multiple CPUs and we generally observed that the least difficult (most accurate) synthetic families were complete within an hour of total CPU time, whereas the most difficult problems could take up to 400 CPU hr. However, the vast majority of N-best trees were always identified early in the the run (within a few hours), with the remainder of the run spent almost exclusively rejecting worse topologies. Dispensing with the branch-and-bound guarantee, topology assembly can be truncated after a fraction, sometimes a very small fraction, of the full run time, retaining a nearly-complete collection of N-best trees. Additionally, there are possibilities for other heuristic approaches that could scale better with the inference complexity and the number of nodes than the current implementation of ASPEN that uses the log-likelihood with the ensemble as an objective function, now that it has been shown to be capable of identifying more accurate topologies.ASPEN’s branch-and-bound algorithm provides a powerful guarantee of completeness – that the Once ASPEN has produced a collection of likely topologies of ancestor divergence, researchers may want to obtain complete phylogenies for their input sequences by grafting topologies of integrated out clades at the corresponding leaves of ASPEN-reconstructed topologies and optimizing branch lengths for the resulting full topologies. For common ancestors of orthologs, we strongly recommend using the species divergence topology for the divergence of orthologs, unless truly compelling evidence to the contrary exists. In our analysis of simulated families, where ortholog evolution was simulated over the same species tree in each case, the correct speciation topology was never recovered for all 15 paralogs, underscoring the futility of reconstructing species divergence from single protein families. For common ancestors of multiple paralogs, ASPEN can be applied recursively to infer the divergence of those paralogs. After complete topologies have been assembled, branch lengths and other parameters can be optimized for any given sequence alignment, for example using RAxML .Traditionally, the effect of alignment uncertainty on phylogenetic reconstruction has been addressed through simultaneous inference of the alignment and the phylogeny, either through extensions of the approach used by progressive multiple sequence alignment algorithms: alternating iterative refinement of alignment and phylogeny , or throWe anticipate that, as a meta analysis approach to tree evaluation and reconstruction, ASPEN is likely to continue to boost the accuracy of individual alignment and tree reconstruction approaches, regardless of the specific underlying alignment and reconstruction algorithms or the application . Alternate statistical approaches are increasingly important with the advent of affordable genome sequencing and the resulting explosion in the number of sequenced and annotated species’ genomes . The curPhylogenetic inference is an extremely difficult problem, particularly for individual protein families, where short amino acid sequences contain limited phylogenetic signal. Here, we demonstrated that the sensitivity of inference and alignment algorithms to their inputs is a result of the failure of objective functions to accurately model the mutational events that underlie the divergence nodes in an evolutionary tree, which manifest in both alignment and inference procedures. At their core, phylogenetic trees are hypothetical models of a currently unknowable evolutionary history. George E.P. Box’s much quoted philosophy about models is relevant here: "Essentially, all models are wrong, but some are useful''. Extending this, we propose that integrating across a lot of models that are wrong in different ways, but right in the same way, yields more useful models of evolutionary divergence. This is based on the observation that topologies capturing more of the shared information across many incorrect models are significantly more accurate. However, we also find that, for especially difficult inference problems  it is important to consider multiple models as equally viable explanations of evolutionary history. In other words, for most real evolutionary inference problems, scientists should integrate their analyses over multiple plausible phylogenetic models. The methodology we presented facilitates both an overall assessment of accuracy of topology reconstruction for any particular problem, and, in light of that assessment, reconstruction of an appropriate number of maximally plausible topological models. Crucially, ASPEN is geared toward analysis of precisely the most difficult class ancestors: ancient ancestors of multiple paralogs.which inferred ancestors are reliable – both with respect to systematic biases and with respect to uncertainty due to various sources of random noise – and which are not. ASPEN topologies can be used for this purpose in a fashion similar to the traditional phylogenetic bootstrap, with the additional benefit of evaluating consistency of an ancestor’s inference across the most likely topologies.It should be noted that individual ancestral nodes can vary dramatically in their difficulty of inference, and many ancestors of great biological interest can be inferred with very high accuracy, despite low overall accuracy of topology inference for that family. Moreover, given high confidence in the existence of such an ancestor, reconstruction of that ancestor’s sequence is robust with respect to topological uncertainty for the clade below it . A critiDivergence reconstructions for well-studied protein domain families are relied upon extensively by the scientific community. For example, evolutionary trees of catalytic and recognition protein domains involved in signaling, including protein kinases  and phostimetree.org (10.6084/m9.figshare.5263885).We simulated 600 families, each containing 15 paralogs, with each paralog represented by 66 orthologs. First, we generated random 15-leaf phylogenies representing paralog divergence. Random phylogenies were generated with average branch lengths of 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 –100 phylogenies each. Next, the Ensembl Compara species tree topology  containitree.org , randomlWe started with a collection of 19 LacI paralogs represented by 28 to 192 orthologs . After iWe demonstrate equivalence of acyclic graph and path length matrix representations of individual topologies by presenting a procedure for interconverting between the two. Transformation of a topology into its path lengths matrix representation is trivially accomplished by counting internal nodes along each path between pairs of leaves . The revUsing the bottom-up procedure for reconstructing a single topology graph as a template, we developed an algorithm that uses a branch-and-bound strategy to construct the requested number of highest-scoring topologies according to the log-frequency scoring function . By analAs in the single-topology procedure, construction of internal nodes is triggered by path length entries encountered during list traversal, with one key difference. In single topology reconstruction, if a path length could be recapitulated by the introduction of an internal node, that node could be safely constructed because it was guaranteed to satisfy every other list entry. Since that guarantee no longer holds, multiple topologies are constructed simultaneously by allowing the construction sequence to branch . "AssembIn principle, branching and iteration alone yield every topology consistent with path lengths observed in the ensemble. In practice, this results in a combinatorial explosion of tracked assemblies, which must be carefully managed to allow construction to proceed to completion.First, branching to satisfy non-conflicting path lengths can lead to collisions between diverged construction sequences on later iterations . This ocSecond, even if each distinct topology is constructed once, in most cases reconstructing every topology consistent with observations from the ensemble, no matter how infrequent, is neither practical nor useful. Bounding, described in the next section, guarantees reconstruction of only the requested number of top-scoring topologies.Completed topologies are ranked according to their log-frequency score, with the ranking updated every time a new topology is finalized. The number of top scoring topologies to reconstruct, As more high-scoring topologies are constructed, the bounding criterion becomes more strict, allowing both more and earlier abandoning of unproductive construction sequences. The branch-and-bound strategy guarantees that the www.trex.uqam.ca (Random 15-leaf phylogenies were generated at .uqam.ca  using th.uqam.ca  under thhttps://doi.org/10.6084/m9.figshare.5263885.v1).All sequence simulation materials, including synthetic phylogenies, the template alignment, and indel-seq-gen control files, as well as simulated sequence alignments are available via Figshare  was assessed with Guidance2 software . BecauseThe Robinson-Foulds  (RF) metIn order to compare reconstructed ancestor divergence topologies, we had to modify the In https://github.com/NaegleLab/ASPEN (https://github.com/elifesciences-publications/ASPEN). ASPEN is implemented in python 2.7. The ASPEN development repository is publicly available at  In the interests of transparency, eLife includes the editorial decision letter and accompanying author responses. A lightly edited version of the letter sent to the authors after peer review is shown, indicating the most substantive concerns; minor comments are not usually included.eLife. Your article has been reviewed by peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Patricia Wittkopp as the Senior Editor. The following individuals involved in review of your submission have agreed to reveal their identity: Rama Ranganathan (Reviewer #1); Itamar Sela (Reviewer #2); Luke Wheeler (Reviewer #3).Thank you for submitting your article "ASPEN: A methodology for reconstructing protein evolution with improved accuracy using ensemble models" for consideration by The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.Summary:The study by Sloutsky and Neagle addresses the important problem of how to make good models of the evolutionary history of a protein family given extant sequences. This is a topical, important, and interesting problem, especially given the growing use of sequence databases for understanding the function and evolution of proteins. Protein sequences diverge through two processes that must be distinguished in making an accurate model of their history: speciation events which lead to orthologous sequences and duplication/divergence events that lead to paralogous sequences. The manuscript presents a meta-algorithm, ASPEN, that improves phylogenetic inference of protein families by subsampling sequences and constructing a topology which is supported by subsampled trees. Sampling alignment columns is a common practice which is used to assign bootstrap support values to phylogenetic trees splits, however, in this work the sampling is performed over rows (i.e. sequences). Notably, there is a correlation between "accuracy"  and "precision" , as defined in the manuscript. Subsampled topologies are incorporated into a subtle branch-to-bound procedure, to construct topology that is statistically supported by many subsamples, rather than a single all-sequence alignment. The resulting constructed topology is shown to be more accurate than all-sequence based topology using a simulated dataset. The developed meta-algorithm will be of considerable interest for scientists that study evolution of protein/gene families and represents an advance in the effort to improve phylogenetic reconstruction by accounting for uncertainties in inference algorithms.Essential revisions:1) A key general point here is the real difficulty of actually inferring the true specific sequence of mutational events that underlie branches in an evolutionary tree. A very important and insightful concept that is partially expressed in this paper is to question the relevance of wanting to know that information from a point of view of deducing any principles of molecular evolution. The idea that the most invariant aspects of the tree reconstruction are simultaneously the most accurate is interesting and sensible, but isn't what is being proposed here that they are also the most relevant? If so, then this should be discussed and defended in more detail in the conclusions. In this same theme, what do these results mean for the whole community using ancestral reconstruction to infer epistasis along branches of the inferred trees and using such studies to extract principles of protein evolution? And for the interesting claims to having "resurrected" true ancestral states from all-sequence tree reconstructions? More generally, some discussion of what can be claimed and what cannot in reconstruction-based studies given the advances presented here about accuracy and precision in the process of these reconstructions would be valuable to guide the scientific community.2) Analyses summarized in Figure 2 . Figure 2A suggests that low accuracy topologies result from poor alignments, and are not due to failure of the tree reconstruction algorithm to retrieve the optimal topology for the given alignment. This can be tested directly by analysis of alignments quality. Moreover, since the quality of the inferred tree depends to a large extent on the quality of the alignment, such an analysis can provide a simple explanation for the observed correlation between accuracy and precision. We suggest that you compare the agreement of all-sequence and true alignments in several accuracy bins, as well as perform comparisons of subsampled alignments between themselves and to all-sequence and true alignments.A plausible explanation for the accuracy-precision correlation is that different accuracy bins represent different levels of alignment problems complexities. Low complexity alignments are robust with respect to subsampling and are therefore associated with high precision. Complex alignment problems might produce very different subsampled alignments that result in different inferred topologies and low precision. Analysis of the alignments quality can directly support or disprove this hypothesis and better isolate the source of low accuracy .Questions about how different scenarios might affect the efficacy of the ASPEN approach.3) As noted in the paper, the simulated trees have a fixed number of paralogs. There are no new gene duplications that occur from the oldest ancestral state to the end-state used for reconstruction. How will variability in this feature in real biological trees effect the ASPEN reconstructions? What is the minimum number of "high confidence nodes" that one needs to make this work?4) We wonder about using JTT model (or another similar amino acid substitution matrix) to conduct the simulated evolution, because it assumes independent evolution of each site/column. Real proteins will have epistatic interactions due to physical and functional constraints. Does this added complexity change the relationship between precision and accuracy or have some other effect on the efficacy of ASPEN? For example, will there be a deviation in the precision vs. accuracy curve for real proteins (like LacI in this study) relative to simulated proteins, that wouldn't be detectable without knowing the true model?Question about software.5) We were pleased with the availability of data and materials used in the paper via online repositories. One thing that would be very beneficial, though not necessary for publication of this article, is a thorough documentation of the ASPEN software that the authors have implemented. It will be difficult for any would-be users to dissect how the software functions without documentation. What are the authors' plans regarding providing documentation for the software? Essential revisions:1) A key general point here is the real difficulty of actually inferring the true specific sequence of mutational events that underlie branches in an evolutionary tree. A very important and insightful concept that is partially expressed in this paper is to question the relevance of wanting to know that information from a point of view of deducing any principles of molecular evolution. The idea that the most invariant aspects of the tree reconstruction are simultaneously the most accurate is interesting and sensible, but isn't what is being proposed here that they are also the most relevant? If so, then this should be discussed and defended in more detail in the conclusions. In this same theme, what do these results mean for the whole community using ancestral reconstruction to infer epistasis along branches of the inferred trees and using such studies to extract principles of protein evolution? And for the interesting claims to having "resurrected" true ancestral states from all-sequence tree reconstructions? More generally, some discussion of what can be claimed and what cannot in reconstruction-based studies given the advances presented here about accuracy and precision in the process of these reconstructions would be valuable to guide the scientific community.This is a wonderful summary of the problems phylogenetic inference presents. In fact, we appreciate this summary so much, we almost directly quoted the first line. We have added a new Conclusions section that summarizes this work in context of the field at large.We have moved one of the original Discussion paragraphs to the new Conclusions section . Additionally, we have added the following two paragraphs to address the essence of the request here.“Phylogenetic inference is an extremely difficult problem, particularly for individual protein families, where short amino acid sequences contain limited phylogenetic signal. Here we demonstrated that the sensitivity of inference and alignment algorithms to their inputs is a result of the failure of objective functions to accurately model the mutational events that underlie the divergence nodes in an evolutionary tree, which manifest in both alignment and inference procedures. At their core, phylogenetic trees are hypothetical models of a currently unknowable evolutionary history. George E.P. Box's much quoted philosophy about models is relevant here:“Essentially, all models are wrong, but some are useful''. Extending this, we propose that integrating across a lot of models that are wrong in different ways, but right in the same way, yields more useful models of evolutionary divergence. This is based on the observation that topologies capturing more of the shared information across many incorrect models are significantly more accurate. However, we also find that, for especially difficult inference problems  it is important to consider multiple models as equally viable explanations of evolutionary history. In other words, for most real evolutionary inference problems, scientists should integrate their analyses over multiple plausible phylogenetic models. The methodology we presented facilitates both an overall assessment of accuracy of topology reconstruction for any particular problem, and, in light of that assessment, reconstruction of an appropriate number of maximally plausible topological models. Crucially, ASPEN is geared toward analysis of precisely the most difficult class ancestors: ancient ancestors of multiple paralogs.which inferred ancestors are reliable – both with respect to systematic biases and with respect to uncertainty due to various sources of random noise – and which are not. ASPEN topologies can be used for this purpose in a fashion similar to the traditional phylogenetic bootstrap, with the additional benefit of evaluating consistency of an ancestor's inference across the most likely topologies.”It should be noted that individual ancestral nodes can vary dramatically in their difficulty of inference, and many ancestors of great biological interest can be inferred with very high accuracy, despite low overall accuracy of topology inference for that family. Moreover, given high confidence in the existence of such an ancestor, reconstruction of that ancestor's sequence is robust with respect to topological uncertainty for the clade below it . A critical step in evolutionary analysis, then, is to determine 2) Analyses summarized in Figure 2 . Figure 2A suggests that low accuracy topologies result from poor alignments, and are not due to failure of the tree reconstruction algorithm to retrieve the optimal topology for the given alignment. This can be tested directly by analysis of alignments quality. Moreover, since the quality of the inferred tree depends to a large extent on the quality of the alignment, such an analysis can provide a simple explanation for the observed correlation between accuracy and precision. We suggest that you compare the agreement of all-sequence and true alignments in several accuracy bins, as well as perform comparisons of subsampled alignments between themselves and to all-sequence and true alignments.A plausible explanation for the accuracy-precision correlation is that different accuracy bins represent different levels of alignment problems complexities. Low complexity alignments are robust with respect to subsampling and are therefore associated with high precision. Complex alignment problems might produce very different subsampled alignments that result in different inferred topologies and low precision. Analysis of the alignments quality can directly support or disprove this hypothesis and better isolate the source of low accuracy .Thanks for the great suggestions. At the level of perturbation we are performing, both alignment and topology search are affected and interconnected in their ability to identify models that best describe the data . In the first draft of the manuscript we knew that alignment error alone cannot explain the phenomenon identified here, since, for families with higher precision, topology inference cannot identify the true topology, given the true alignment: the all-sequence topology  has higher likelihood under the JTT model and the true alignment than does the true topology. Under these conditions the scoring procedure in the topology inference algorithm scores inaccurate topologies as better than true topologies, even when given the true alignment and substitution model. We did not sufficiently describe those results in the original manuscript and have expanded on that point.We prefer the term “alignment robustness” to “alignment quality”, because the latter seems too subjective. After all, different alignments can serve better or worse for different purposes. In order to address the root question here – can alignment robustness explain the observed relationship between topology precision and topology accuracy? – we performed the suggested experiment and compared the all-seq alignments to the true alignments  using the Guidance/Guidance2 sum-of-pairs similarity metric. Indeed, there is a correlation between alignment accuracy and topology accuracy (Pearson Correlation of 0.80), but this correlation is weaker than that of topology precision and topology accuracy (Pearson Correlation of 0.91). The lower correlation is driven by the large deviations in alignment accuracy and a compression in this scale – no all-sequence alignment is completely accurate, yet these can ultimately yield completely accurate trees, and most of the mid-accuracy/mid-precision trees come from all-sequence alignments that are almost wholly different than the true alignment.Comparing alignments of subsampled sequences to each other or to the all-sequence or true alignment is problematic, as each subsample has a different collection of sequences. We were concerned that resulting similarity might conflate the intended understanding with the additional variability of randomness of the characteristics in that particular subsample. Instead, we asked whether the reproducibility of the all-sequence alignment  could explain reproducibility of inferred topologies (precision). We used Guidance2 to measure the all-sequence alignment robustness and found that this is almost perfectly correlated with the all-sequence alignment accuracy (Pearson’s Correlation of 0.97), but does not capture any additional variance of topology accuracy . Hence, one could use the reproducibility of the all-sequence alignment to predict the accuracy of the all-sequence alignment (a hidden variable) to a great extent, but that value does not translate as precisely to the resulting topology accuracy as does topology precision. We believe this is likely due to the nuances of how and where alignment errors are located and how they ultimately influence the resulting topology.In short, the answer to the question asked by the reviewers is yes – alignment accuracy correlates with tree accuracy and, since alignment reproducibility correlates almost perfectly with alignment accuracy, alignment reproducibility also correlates with topology accuracy. However, an emergent relationship exists between topology precision and topology accuracy that gives additional predictive power to topology precision that is not fully explained by either alignment accuracy or alignment reproducibility.We have added a new figure and edited the section “Subsampling produces an observable measure of accuracy” to clarify the roles and contributions of alignment vs. topology scoring in the precision/accuracy.Questions about how different scenarios might affect the efficacy of the ASPEN approach.3) As noted in the paper, the simulated trees have a fixed number of paralogs. There are no new gene duplications that occur from the oldest ancestral state to the end-state used for reconstruction. How will variability in this feature in real biological trees effect the ASPEN reconstructions? What is the minimum number of "high confidence nodes" that one needs to make this work?It is certainly true that real protein families may have additional complexity of speciations, duplications, and gene loss that are not reflected in the specific set of evolutionary circumstances under which we simulated evolution. However, these nuances of duplication/speciation/loss will not affect ASPEN reconstructions, unless one has consistently mis-assigned or cannot confidently assign such a paralog to a node – i.e. it only matters whether subsampling under that node produces consistently monophyletic topology reconstructions. The nature of the node does affect the analysis or the reconstruction.The Results subsection “Defining protein families for testing” is expanded, and now states:“Evolution of real protein families is rarely that neat. Speciations can predate duplications. Loss of some paralogs in some lineages can result in different collections of paralogs appearing in different genomes, dramatically complicating orthology assignments. […] Therefore, the analysis based on this simulation experiment is applicable without loss of generality.”Additionally, we have expanded theDiscussion subsection called “Selection of ancestral nodes for topology search space decomposition”,detailing of the impacts and implications of selection of nodes. We also provide some practical recommendations for identifying high-confidence nodes and handling poorly-behaved leaf sequences.4) We wonder about using JTT model (or another similar amino acid substitution matrix) to conduct the simulated evolution, because it assumes independent evolution of each site/column. Real proteins will have epistatic interactions due to physical and functional constraints. Does this added complexity change the relationship between precision and accuracy or have some other effect on the efficacy of ASPEN? For example, will there be a deviation in the precision vs. accuracy curve for real proteins (like LacI in this study) relative to simulated proteins, that wouldn't be detectable without knowing the true model?This is a really interesting and complex question. We believe that a detailed answer to this question requires a careful and extensive study beyond the scope of this manuscript. Answering this question requires using a generative model that incorporates constraints amongst sequence sites reflective of constraints experienced by real protein families. To date, all attempts at this have relied on physical models of protein structure with rather simplistic assumptions (e.g. that the only constraints arise from protein folding requirements) . Despite their reductionism, are far too computationally intensive for the kind of large-scale sequence evolution simulation we carried out and for phylogenetic reconstruction of the simulated families. In addition, at the moment these models are not available as implementations that can be incorporated into the ASPEN analysis workflow.The study of protein evolution as a whole would tremendously benefit from a more realistic treatment of evolutionary constraints experienced by proteins than what is offered by the commonly used substitution models. Algorithmic improvements to make physics based-models more tractable and incorporation of such models into popular sequence simulation and phylogenetic inference software will be necessary to make them more penetrant in the field.At this point, we have included our thoughts on this process as part of the new Discussion subsection “Generality of the accuracy-precision relationship”, where we propose that non-independence would ultimately manifest as less overall information on which to build the model, leading to a more complex/difficult inference problem, but not substantively affecting the accuracy-precision relationship.New Discussion subsection titled “Generality of the accuracy-precision relationship” covers this topic and others.Question about software.5) We were pleased with the availability of data and materials used in the paper via online repositories. One thing that would be very beneficial, though not necessary for publication of this article, is a thorough documentation of the ASPEN software that the authors have implemented. It will be difficult for any would-be users to dissect how the software functions without documentation. What are the authors' plans regarding providing documentation for the software?Thank you for the suggestion. We have created a clean repository to accompany this publication (moving the development code to another repository in GitHub) and are in the process of documenting these tools. We did not wish to delay the evaluation of this revision and so this new repository is not yet fully populated. However, it will be complete by the end of September. In this clean repository, we will focus on the most complex aspect of this project: converting many inferred trees into a pairwise distance distribution and performing ASPEN reconstruction of best topologies according that distribution. We will also provide some scripts as examples of how to subsample for the creation of the family of trees. However, we highly recommend users to select their own methods for alignment and inference, and subset size as they see fit. In addition, we hope that upon publication of this manuscript, our ability to secure future funding for development of ASPEN will be improved, allowing us to improve usability and flexibility of ASPEN as a software tool.


A larger amount of sequence data in private and public databases produced by next-generation sequencing put new challenges due to limitation associated with the alignment-based method for sequence comparison. So, there is a high need for faster sequence analysis algorithms. In this study, we developed an alignment-free algorithm for faster sequence analysis. The novelty of our approach is the inclusion of fuzzy integral with Markov chain for sequence analysis in the alignment-free model. The method estimate the parameters of a Markov chain by considering the frequencies of occurrence of all possible nucleotide pairs from each DNA sequence. These estimated Markov chain parameters were used to calculate similarity among all pairwise combinations of DNA sequences based on a fuzzy integral algorithm. This matrix is used as an input for the neighbor program in the PHYLIP package for phylogenetic tree construction. Our method was tested on eight benchmark datasets and on in-house generated datasets  and 16 s rDNA sequences of 40 bacterial isolates from plant interior). The results indicate that the fuzzy integral algorithm is an efficient and feasible alignment-free method for sequence analysis on the genomic scale. The alignment-free methods are not only used in phylogenetic studies5, but also for metagenomics11, analysis of regulatory elements14, protein classification16, sequence assembly17, isoform quantification from transcriptome data18, and to identify biomarkers in diagnostic tests19. The alignment-free methods fall into two broad categories: methods based on k-mer or word frequency, and methods based on match length20. Methods based on k-mer or word frequency are quite popular and studied extensively. The k-mer based methods were developed to compare DNA sequences, in which it counts the frequencies of substrings with k letters occurring in respective sequences21. In recent past, a lot of k-mer based methods have been proposed and implemented in sequence analysis and phylogeny, such as, feature frequency profile (FFP)22, return time distribution (RTD)23, frequency chaos game representation (FCGR)24, an improved complete composition vector method (ICCV)25, composition vector (CV)26 and complete composition vector (CCV)27. For sequence comparison, ICCV method is more efficient and robust compared to CV and CCV methods. The other category of the alignment-free method is based on match lengths, where it employs the similarity of substrings between two sequences31. Examples of match length methods are, k-mismatch average common substring32, average common substring28, Kr – method28, etc. These methods are commonly used for string processing in computer science. In this study, we propose to use fuzzy integral33 to analyze DNA sequences based on a Markov chain34, which can be categorised as k-mer or word frequency method. The fuzzy integral similarity method36 assigns a similarity score between two DNA sequences based on the estimated parameters of a Markov chain. A DNA sequence consists of four characters . By taking the state space as S = {A, T, G, C}, we used the k-th step transition probability matrix, a fuzzy measure37 and fuzzy integral to describe the DNA sequences. We used the fuzzy integral similarity to obtain a distance matrix, which was used in the neighbor program in the PHYLIP package38 to construct a phylogenetic tree. The similar fuzzy integral similarity approach was taken by36. However in36, the method of feature vector extraction from the DNA sequences is different from our method. In both our method and36, the extracted features are used as an input for the fuzzy integral similarity analysis. The proposed method is tested on 18S rDNA sequences from 11 Arbuscular mycorrhizal fungi isolates and 16S rDNA sequences from 40 bacterial isolates, and also tested on the following benchmark datasets, 41 mammalian mitochondrial genomes, 59 ebolavirus complete genomes, 30 coronavirus whole genomes, 30 bacterial whole genomes, 48 Hepatitis E virus (HEV) whole genomes, 24 Eutherian mammals sequences, 58 genome datasets from different species and 29 Escherichia/Shigella complete genomes. The method was also tested on large mammalian dataset. In addition, we used receiver operating characteristic (ROC)41 curve for measuring the performance of our method to compare the other alignment-free methods from Alfree repository2. The consistency can also be seen from the statistical analysis such as AUC (area under the ROC) values, calculated from ROC curves provided in Supplementary Material.Phylogenetic tree analysis and comparative studies of taxa are essential parts of modern molecular biology. Phylogenetic reconstruction and comparative sequence analysis traditionally depend on multiple or pairwise sequence alignments. However, various limitations are encountered when analyzing large datasets using an alignment based approach. Whole genome alignment of higher eukaryotes can exceed computational resources. Moreover, factors such as the combinatorics of genomic rearrangements and duplications make the alignment of entire genomes impossible. Therefore, the alignable homologous segments of the genomes under study have to be identified in the initial steps. Recently, large amounts of sequence data produced by next-generation sequencing techniques have become available in private and public databases, which has created new challenges due to the limitations associated with alignment based approaches. This plethora of sequence information increases the computation and time requirements for genome comparisons in computational biology. Therefore, there is a high need for faster sequence analysis algorithms. For this, various methods have been proposed to overcome the limitations of alignment based approachP = [pij] denote the transition probability matrix of a discrete-time Markov chain34. Each state transition probability pij is defined as follows:Xn indicates the actual state at time n and si is the ith state of S distinct states. In the context of a DNA sequence, the number of states is S = 4 which corresponds to the four nucleotide symbol set Let 42. If the total number of each adjacent nucleotide pair  in the sequence is denoted by st-step transition probability from state si to state sj is estimated assi and ending with nucleotide sj.Since the transition probabilities are unknown initially, they must be estimated based on the observed sequence. Here, we estimate the parameters of the Markov chain by taking the frequencies of occurrence of all possible nucleotide pairs for each sequencej after k steps from state i. Each state transition probability Presented above is the 1-step Markov chain. The k-step Markov chain can be calculated through the 1-step Markov chain, which is known as the Chapman-Kolmogorov process. Let The state transition probabilities are subject toA, B and C, the following identity is known: A as Xn+k = sj, B as Xn+t = sr and C as Xn = si, we haveFor any three events, Hence, the matrix with the elements k plays an important role in capturing rich evolutionary information from the DNA sequence. In the context of a DNA sequence, the kth-step transition probability can be written as:The selection of step Which is subject to entclass1pt{minima43 is particularly suitable for modelling imprecise data, whereas fuzzy integral is highly appropriate for representing the interaction among different information sources. The concept of fuzzy integral with respect to a fuzzy measure has been proposed by Sugeno in 197444. In this section, we propose the use of the fuzzy integral incorporating with the transition probability matrix, where the elements of transition probability matrix are taken as fuzzy membership degree.Fuzzy set theorykth-step nucleotides starting from nucleotide si and ending with nucleotide sj estimated from the observed sequence.Let X, Y ⊆ F and R(F) be the power set of F. A fuzzy measure μ is a real valued function:Let μ: R(F) →  satisfies the given condition,μ(ϕ) = 0 and μ(F) = 1.μ(X) ≤ μ(Y) if X ⊆ Y.μ let yij in determining the set F. By definition of the fuzzy measure μ, the measure of the union of two disjointed subsets cannot be directly computed from their disjointed component measures. In other words, the fuzzy measure value of a given subset is not simply the summation of the measures of its elements. Therefore, to define a fuzzy measure, we need to know the fuzzy densities of each element of the measured set and the measure of each combination. This measure can be provided by an expert or extracted from the problem definition. However, when dealing with a set of numerous elements, this job may become noisy, tedious or even unfeasible. A possible solution to this problem is to use a λ – fuzzy measure. A λ – fuzzy measure45 fulfills the criteria of a fuzzy measure, and has an additional property: for all For a fuzzy measure λi can be calculated by solving:Furthermore, For solving Eqs  and 7, wh: F →  represent a function that maps each element of F to its observed evidence. Suppose that F should be reordered so that the decreasing order relationship holds, and further investigation will be based on the modified relationship. Let μ: R(F) →  be a fuzzy measure. Then, the fuzzy integral of h with respect to the fuzzy measure μ isLet 46.The fuzzy integral considers the significance provided by every element of a given set, and the importance of each subset of elements  plays an important role in its decision-making process. The combination of the extracted information and the importance of the provided source convert the fuzzy integral to an appropriate form for information fusion. This theory has the potential to address uncertainties associated with issues related to data extraction and their processing procedures. Therefore, the theory has been widely implemented in feature extraction and classificationkth-step nucleotide pair frequency with respect to the conservation level of the position between two sequences. In our case, the kth-step nucleotide pair frequency at all sixteen positions in the transition probability matrix is taken as the fuzzy membership degree.The fuzzy integral similarity is based on the distance of the kth-step transition probability matrices. The fuzzy integral function find the similarity level of the nucleotide pairs between kth-step transition probability matrices. We constructed a fuzzy integral function h, which is given as:yij ∈ {(AA)k, (AT)k, (AG)k, (AC)k, (TA)k, (TT)k, (TG)k, (TC)k, (GA)k, (GT)k, (GG)k, (GC)k, (CA)k, (CT)k, (CG)k, (CC)k}.Let kth-step transition probability matrices Additionally, the fuzzy measure function find the maximum level of conservation of the nucleotide pairs between λ – fuzzy measure μ using the fuzzy density of each element μij.Taking advantage of the properties explained above, we can construct a yij ∈ {(AA)k, (AT)k, (AG)k, (AC)k, (TA)k, (TT)k, (TG)k, (TC)k, (GA)k, (GT)k, (GG)k, (GC)k, (CA)k, (CT)k, (CG)k, (CC)k}. At this stage, we should apply Eq.  and (b) of the fuzzy measure. After generating h and μ, we obtained the fuzzy integral similarity by applying Eq.  and stage(2) for phylogenetic construction:O(m3nk + nl), where l is the average length of the DNA sequences, n is the total number of DNA sequences, m is the number of bases and k is the kth-step transition probability matrix. In the second stage, we calculate the fuzzy integral similarity between the two m × m transition probability matrices generated in stage (1). The time complexity of stage (2) is O(m2m). In the third stage, we integrate stage (1) and stage (2) to generate a distance matrix. Let k = h be an optimal step that satisfies condition (8) in algorithm (3). Therefore, the total time complexity to generate the final distance matrix at the hth optimal step is:To determine the time complexity of a given algorithm, we assume that all operations took the same unit of time. The whole computational process consists of three stages. In the first stage, we calculate the transition probability matrix from the raw DNA sequences. The time complexity of stage (1) is m = 4. Hence, the time complexity of our proposed algorithm are O(nh + nl + hn2).Since we are calculating the computational complexity for DNA sequences, the number of bases  is k using the method described above. We increased step k until we obtained the same distance matrix for two consecutive distinct k, . Therefore, we considered k = h an optimal step and generate the phylogenetic tree at step k = h using the PHYLIP package. Here, we use the UPGMA (Unweighted Pair Group Method with Arithmetic Mean) approach in the PHYLIP package38 to generate the phylogenetic tree. To test the effectiveness of our proposed approach, we selected ten sets of test data: (i) 18S rDNA sequences from 11 Arbuscular mycorrhizal fungi isolates, (ii) 16S rDNA sequences from 40 bacterial isolates, (iii) 41 mammalian mitochondrial genomes, (iv) 59 ebolavirus complete genomes, (v) 30 coronavirus whole genomes, (vi) 30 bacterial whole genomes, (vii) 48 Hepatitis E virus (HEV) whole genomes, (viii) 24 Eutherian mammals sequences, (ix) 58 genome datasets from different species and (x) 29 Escherichia/Shigella complete genomes. We compared our tree with the tree generated by the previously published method using same datasets.To check the performance of the proposed method, it was tested on different datasets. Some datasets are small sized and others are medium sized. The length of sequences ranges from seven thousands to several millions base pairs. In order to compare and analyze various genomic data, we generated a distance matrix using Eq.  for eachglomeromycota. Arbuscular mycorrhizas fungi help plants to capture nutrients, such as phosphorus, sulfur, nitrogen and micronutrients, from soil. The development of arbuscular mycorrhizal symbiosis is believed to have played a crucial role in the initial colonization of plants on land and in the evolution of vascular plants48. We built a phylogenetic tree , except one genus of rhizophagus ). All glomus genera belonging to family glomeraceae were clustered in cluster (b). All gigaspora genera belonging to family gigasporaceae were clustered in cluster (c). While comparing the tree prepared by our method  or endomycorrhiza) is a type of mycorrhiza in which the fungus infects vascular plants by penetrating the cortical cells of the root. Arbuscular mycorrhizas are characterized by the formation of unique structures (arbuscules) and vesicles, these fungi belong to phylum ree Fig.  using thhod Fig.  using th50. The 40 bacterial sequences were obtained from pure cultures of endophytic bacteria isolated from the surface sterilized mature endosperms of six rice varieties. The rice seeds were collected from two different locations in north-east India: North-Lakhimpur, Assam, Aizawl and Mizoram. Genomic DNA was extracted from the pure cultures, and the full length 16S rDNA sequences were amplified using the primer pair 27f (5′-AGAGTTTGATYMTGGCTCAG) and 1492r (5′-TACCTTGTTAYGACTT). The amplicons were sequenced in an Applied Biosystems sequencer using the BigDye terminator method. To minimize sequencing error, we also used two internal primers 533f/805r along with 27 f/1492r. We assembled the contigs based on their phred scores (15) using the Codon Code aligner v7.0.1, BioEdit and SeqTrace v0.9 software51. The contigs were checked for the presence of any chimeras in mothur v1.35.1 and were aligned for identification against the NCBI reference rRNA database using the blastn algorithm. Information, including the accession numbers, phyla, classes, orders and families for the 40 bacterial isolates are listed in Table k = 6. The tree  and clade B (green)  were placed together in one clade. However, C. plantarum and M. proteolyticum were placed as sister groups, and the other C. plantarum sequence was positioned as an outlier. When we compared our method  consisted of 14 sequences. hod Fig.  with Cluhod Fig. , we founk = 6, the 41 species were correctly divided into eight groups: Primates (red), Cetacea (green), Artiodactyla (pink), Perissodactyla (light green), Rodentia (black), Lagomorpha (dark red), Carnivore (blue), and Erinaceomorpha (grey). The cat species in our approach was clustered with the Artiodactyla group. We compared the phylogenetic tree  with nearly 16500 nucleotides Table . The treree Fig.  generateree Fig.  and S4 cree Fig.  generateEbolavirus genus includes five species: Bundibugyo virus (BDBV), Reston virus (RESTV), Ebola virus , Sudan virus (SUDV), and Tai Forest virus (TAFV)53. Ebola viruses are single-strand negative sense RNA viruses. Each ebolavirus genome encodes seven proteins in which glycoprotein is the only viral protein on the surface of ebolavirus. The first case of human, infected by EBOV, was reported in 1976 in Zaire (currently the Democratic Republic of the Congo (DRC))54. We applied our proposed method to generate the phylogenetic tree  pandemic in 1976–1977 were clustered together as a clade. The EBOV strains in DRC pandemic in 2007 were clustered together with the exception EBOV_2007_KC242788, which was clustered with Zaire (DRC) in 1976–1977. The EBOV strains in Guinea epidemic in 2014 were clustered together as a clade. The three EBOV strains from the 1995 outbreak in Zaire (DRC) formed a clade. SUDV and RESTV formed separate clades. BDBV and TAFV viruses were positioned together. Our result was in consensus with the result generated using multiple encoding vector method52 , II(grey), III(blue) and IV(green)) except few sequences. Phylogenetic tree . This virus is characterized as non-enveloped, single-stranded RNA virus with nearly 7200 nucleotides Table . The acu63  and lactoferrin (green) sequences from 24 vertebrates as a benchmark dataset63 Table . Vertebr63 Table  using thk = 8. As shown in Fig. To verify the clustering efficiency of our method on extremely divergent sequences from different organisms, we randomly collected genomes of varying length from different datasets from Tables k = 7. As shown in Fig. 64 as a reference which is based on concatenated alignments of the 2034 core genes and used the maximum likelihood method to infer the phylogenetic relationships. We calculated Robinson-Foulds distance (RF-distance)65 of the tree produced by our method against the benchmark tree64. The RF-distance is often used to compare two trees of closely related species. Since, the species in this dataset (29 complete genomes from the genera Escherichia/Shigella) are closely related organisms, therefore, we employed the RF-distance, which evaluates the topological congruence between an inferred tree and a benchmark tree. We also collected the generated RF-distances from the previous study66. RF = 0 indicates that, the test-tree topology is completely similar to that of the benchmark tree, while similarity level decreases as the RF increases. As shown in Fig. The other benchmark dataset used in this study was 29 complete genomes from the genera Escherichia/Shigella with more than 1 million nucleotides Table . We applThis study focused on fuzzy integral similarity technique based on Markov chain and applied this algorithm to phylogenetic tree analysis. Sequence comparison is one of the most useful and widely practiced methods in bioinformatics and computational biology. Alignment based methods perform well if the genetic sequences are homologous. High mutation rates and genetic recombination brings in a limitation of the alignment based method. Also at the genomic scale, alignment based methods become impractical due to their computational complexity. Alignment-free methods are of great value, because they reduce the technical constraints of alignments. We constructed a transition probability matrix using a Markov chain for each DNA sequences without performing prior alignment at the genomic scale. The fuzzy integral similarity technique is a method that can calculate similarity score between two transition probability matrices of DNA sequences. The main advantage of our approach is that, it does not require any additional parameters, which makes it fully automated and robust. We implemented and tested our method on suitable datasets.All programs are implemented on a linux server with 384 GB RAM with 24 dual core processor. Our proposed approach is fast in computational speed Table  compare 41 , monkey (Macaca mulatta), chimpanzee (Pan troglodytes), gorilla (Gorilla gorilla), horse , mouse (Mus musculus), dog (Canis familiaris), opossum (Monodelphis domesticus), and platypus (Ornithorhynchus anatinus). The length of the chromosomes X in these organisms ranges approximately from 6 to 147 Mb. Applying our method, we generated the phylogenetic tree (Fig. k = 8. Our method took only 18 seconds for generating transition probability matrix. Phylogenetic tree (Fig. 52, mouse clustered with primates, while dog and horse came together in a clade. Figure 68. This similarity may have placed horse and mouse in a sister group. Dog and oppossum formed a distinct clade, while platypus separated as an outgroup.Our proposed method is faster and has the potential to build phylogenetic tree for large sized genomes, such as, mammalian genome. Mammalian genomes are divided into several chromosomes. In this study, we selected chromosome X to do the phylogenetic analysis, details are given in Table ree Fig.  of nine ree Fig.  generateBased on the results generated by our developed method, we found that our method performed well on divergent sequences, rather than closely related sequences. Therefore, this approach would be beneficial for the users to generate hypothesis that can be investigated in further detail with subsequent analysis. Before continuing research for further development of our method, we must keep in mind that, this approach is a probabilistic measure in nature, and can be modified by incorporating more information, such as, nucleotides substitution, insertion, deletion, genetic recombination, Physicochemical Properties of nucleotides etc., in sequences. Overall, our goal in this study was to introduce a new methodology and a new tool to the comparative genomics research community. This proposed work can be used to guide the development of more powerful measures for sequence comparison.Dataset 1Dataset 2Dataset 3Dataset 4Dataset 5