Introduction

With the rapid advances in the human microbiome research, it becomes increasingly important to detect and quantify the changes occurring in the human microbiome, especially the changes in the microbiome associated with disease. Differentiating between the healthy microbiome (sampled from healthy individuals) and diseased microbiome (sampled from individuals with microbiome associated disease such as bacterial vaginosis) is essentially a problem of measuring the dissimilarity between two microbial communities. Naturally, diversity analysis with traditional biodiversity measures such as species richness, Shannon information entropy (Shannon index), Simpson’s index, have been playing a dominant role in the field of comparing the human microbiome across space and time, as well as between healthy and diseased samples (HMP Consortium 2012a, 2012b, Lozupone et al. 2012)1,2,3. While diversity measures such as Shannon index are certainly useful for measuring the dissimilarity between communities, and indeed they have been applied to characterize microbial community in nearly every 16s-rRNA sequencing based microbiome study (e.g., Abusleme et al. 2013, Fodor et al. 2012, Srinivasan et al. 2012, Kong et al. 2012, McHardy et al. 2013)4,5,6,7,8, the diversity analysis is not without shortcomings. One inherent issue associated with all diversity measures is that they ignore the interactions between species, and they are simply some incarnations of species abundance distributions (SAD) in the community. Consequently, diversity analysis cannot fully account for the contributions of species interactions in the microbiome.

Although still not widely applied to the study of human microbiome, network analysis has been widely applied to other fields of computational biology and bioinformatics, such as gene regulatory and signal transduction networks, protein interaction networks, metabolic networks, phylogenetic networks, and ecological networks (Junker & Schreiber 2008)9. Indeed, network analysis, which considers both species abundance and their interactions (links), has been anticipated to remedy or even solve the issues associated with traditional diversity measures. In the field of human microbiome research, Faust and his collaborators (Faust & Raes 2012, Faust et al. 2012, 2015)10,11,12 performed extensive pioneering works. Since their works (Faust & Raes 2012, Faust et al. 2012, 2015)10,11,12, quite a few applications of network analysis in the human microbiome have been reported (e.g., Barberán et al. 2014, Chow et al. 2014, de Menezes et al. 2015, Duran-pinedo et al. 2011, Fernandez et al. 2015, Imangaliyev et al. 2015, Ma et al. 2015, 2016, Hunt et al. 2015, Ramayo-Caldas et al. 2016)13,14,15,16,17,18,19,20,21,22. Nevertheless, primarily due to the limitation with the existing microbiome datasets—almost all available microbiome datasets that can be utilized to build microbiome network models are species (OTU) abundance data from 16s rRNA sequencing technology, most of the successful applications are ad hoc. Accordingly, the standard correlation network analysis technique could only demonstrate limited power of network analysis in the study of human microbiome. In such applications to human microbiome research, the main results from standard correlation network analysis are a suite of network properties, including a series of simple network motifs (Shannon et al. 2003, Csardi & Nepusz 2006, Junker & Schreiber 2008)9,23,24, which often fail to produce conclusive, and occasionally may even generate conflicting evidence for differentiating between healthy and diseased microbiomes. We argue that the failure is due to some implicit assumptions of standard correlation networks (e.g., Shannon et al. 2003, Csardi & Nepusz 2006, Junker & Schreiber 2008)9,23,24. Correlation networks assume that nodes are ‘homogenous’ other than the heterogeneity in their correlation levels with immediate neighbors. For instance, in the case of human microbiome correlation networks, nodes (OTUs) are homogenous except for their abundances, which determine their correlation levels with their immediate neighbors in the network. Due to this implicit assumption, nodes lost their ‘identities’ other than some special roles such as hubs, identified by network properties. In the case of human microbiome network, nodes of commensalists, facilitators, and opportunistic pathogens may have very different functional roles, but the computation of network properties in standard correlation networks and motif detection do not even consider the difference between positive and negative links. The former may include facilitators, and the latter may be suppressed by beneficial microbes in the human microbiome. It is obvious that positive (facilitative) and negative (inhibitive) interactions may have very different biomedical implications. On the other hand, those implications associated with the interaction modes may not be reflected by the existing definition of correlation network properties and or motifs. Hence, identifying new network properties and/or network motifs that can capture the heterogeneities of node roles such as MAO (most abundant OTU) or hubs, MDO (most dominant OTU)25, as well as interaction modes (positive vs. negative) should remedy the insufficiency of standard correlation networks in analyzing the microbiome network, and offer potentially powerful tool for differentiating between healthy and diseased microbiomes.

In the present study, we propose to define/detect simple triangle motifs with a three-level hierarchical scheme that consider node role, interaction type (+ or –) and the combination of the role and type. Following Occam’s razor principle, we hypothesized that those trios, arguably the simplest motif in the OTU correlation network of the human microbiome, should be sufficient to detect significant changes occurred in the OTU correlation network of human microbiomes such as those impacted by or associated with diseases. The arguments supporting our hypothesis include: (i) our trios inherit the advantage of network analysis and therefore are able to overcome the disadvantage of community diversity analysis, which ignores the interaction between OTUs, (ii) our trios are advantageous over the standard correlation network analysis by considering the node role and interaction type, and should be more sensitive to the changes than the standard properties and motifs of correlation networks. As to the reason why we do not search for more complex motifs is due to the reality that finding arbitrary size motifs is a computationally NP-hard problem (Betzler et al. 2011, Tran et al. 2014)26,27, which, in a layman’s interpretation, implies that the computational time for finding all motifs of arbitrary size may be astronomical numbers even with the fastest computers humans have built. Therefore, if simple trio motifs, which are computationally lightweight to detect, can fulfill the mission to differentiate the microbiomes impacted by significant disturbances such as disease, complex and expensive computations become unnecessary.

In existing literature, our work is similar to the triad, which is a sub-graph consisting of three nodes and possible lines between the nodes, in social network analysis (O’Malley & Marsden 2008, Kitts & Huang 2010)28,29. However, the triads in social network analysis are directional, and a total of 16 triads were defined to describe directed interactions between three individuals. In our trio definitions, although our interactions (links) are directionless due to the nature of correlation network, we take into account the role of nodes (e.g., MAO) and the type of interactions (positive vs. negative). In addition, we consider trios with MAO (most abundant OTU) handle, i.e., the MAO is connected to a trio with possibly one, two, or three links (i.e., the handle). We consider this type of ‘trios,’ which consists of 4 nodes actually and, strictly speaking, should be termed “quartos,” because we found that the MAO may have far reaching effect on the whole structure of a trio beyond the effect on its immediate (directly connected) neighbor nodes. With a three-level classification scheme of node role, interaction type, and their combinations (specifically, the existence/absence of MAO handle or MAO, the positive vs. negative interaction, as well as their permutations), a total of 19 trios, including 10 without MAO handle, and 9 with MAO handle. The former class was further classified into 10 types based on the existence of MAO in the trio, as well as permutation of positive and negative links, and the latter class was further classified into 9 types based on the number of links to MAO (1, 2, or 3) as well as the permutation of positive and negative links. We developed the trios-finder program (TFP) in Perl for detecting the trios and provided the source code in the Supplementary information. We demonstrate the concept and design of using the trios as potential in silico biomarkers, as well as their implementation with the TFP software, with six HMP datasets that cover microbiomes and associated diseases from five major body sites (gut, lung, oral, skin, vaginal).

The Method—Definitions and Computational Searching for Trios

Defining the trios

Formally, with the three-level classification scheme, we distinguish two classes, five categories, and 19 types of trios at the first, secondary, and tertiary level, respectively (Table 1). At the primary level of the classification, we distinguish two classes, the trios without MAO handle and trios with MAO handle. At the secondary level, the first class refers to the triangle motif that is not connected with an external MAO ‘handle’ and is further distinguished as two categories: trios with MAO (i.e., MAO is part of the trio) and trios without MAO (i.e., MAO is not part of the trio). The second class refers to the triangle motif that is connected with an external MAO and is further distinguished as three categories: single-link MAO handle, double-link MAO handle, and triple-link MAO handle, with one, two and three links to the node of MAO handle, respectively.

Table 1 The 19 types of trios generated from a three-level classification scheme.
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At the tertiary level classification, each of the five secondary level categories (trios without MAO handle, trios with MAO handle, single-link MAO handle, double-link MAO handle, and triple-link MAO handle) is further classified based on the signs (+ or −) of the interactions (correlations) within the trio or between the trio and handle. Detailed classification of the 19 trio types at the tertiary level, generated from the above-described three-level classification scheme is presented in the following Table 1. Among 19 trio types, four types in the category of trios without MAO are nothing particular and are detected in existing network analysis software packages such as Cytoscape (Shannon et al. 2003)23 and iGraph (Csardi & Nepusz 2006)24. To the best of our knowledge, the other 15 trios have not been investigated in the existing literature. Our focus will be centered on those 15 special trios.

As it is demonstrated below, even the 15 special trios are not created equal, and some of the theoretically possible triangle motifs are not detected in our datasets and may even be ‘prohibited.’ Some of the trios, especially the six types of trios in the category of trios with MAO were named as Type-1A, Type-1B, Type-2A, Type-2B, Type-3, and Type-4, respectively. These six types are identified at the tertiary level classification; the sign (+ or −) of interaction in trios in the microbiome network is considered. For example, the difference between Type-1A and Type-1B lies in the signs of two links connected with the MAO, i.e., (+ −) in Type-1A and (− −) in Type-1B.

Computational procedures

Our computational procedures for detecting the trios (triangle motifs) consist of the following four major steps: (i) computing OTU correlation coefficients (using either Spearman’s or Pearson’s definitions), (ii) filtering out false correlations with FDR (false discovery rate) adjustment, (iii) constructing the OTU (or species) correlation (interaction) networks with standard network analysis software packages such as Cytoscape (Shannon et al. 2003)23 or iGraph (Csardi & Nepusz 2006)24; and (iv) detecting the trios with home-made trio finder program (TFP) program, supplied in the online Supplementary document. The first two steps were actually implemented in a R-script CCFDR.r (Correlation Computing with False Discovery Rate) we provided in the online Supplementary materials. The R-script (CCFDR.r) calls the function “rcorr” from existing R-package Hmisc (https://cran.r-project.org/web/packages/Hmisc/) and the function “multiple.correction” from existing R-package EMA (https://cran.r-project.org/web/packages/EMA/index.html) to compute the correlation coefficients and filter out false correlations, respectively. The output from the CCFDR.r, i.e., correlation computing adjusted with FDR control, is feed into our Perl program TFP.pl, which completes the task of seeking and counting the various trio types. The following flowchart shows the computational procedures, and we further elaborate the possible issues involved in the procedures below.

A flowchart showing the four steps for implementing the trio-finding process:

Step (i): Compute the OTU (species) correlation coefficients with Spearman’s or Pearson’s definitions.

Step (ii): Filter out false correlation with FDR (false discovery rate) control with our CCFDR R-script.

Step (iii): Construct the OTU (or species) correlation (interaction) networks with standard network analysis software packages such as Cytoscape (Shannon et al. 2003)23 or iGraph (Csardi & Nepusz 2006)24. This step can be omitted if no network graphs are output.

Step (iv): Detecting the trios with our trio finder program (TFP.pl) (see the Supplement).

To construct SIN, we recommend using Spearman’s rank coefficient or occasionally Pearson’s correlation coefficient as demonstrated in Junker & Schreiber (2008)9, Ma et al. (2015, 2016)20,21.

However, to utilize the correlation coefficients for constructing the species or OTU correlation networks, there are two issues that should be addressed first: one is the choice of either the relative abundance or actual OTU reads and another is to filter out the false correlations in the raw correlation coefficient values in consideration of the rising risk of false correlations from multiple testing (i.e., simultaneously testing multiple null hypotheses or the significance of multiple correlation coefficients) with sequence data30,31,32. Both steps are usually necessary to ensure proper construction of the underlying OTU (species) correlation networks (SCN), also known as species interaction networks (SIN) as often termed in macro-ecology.

Regarding the utilization of OTU reads for computing the correlation coefficients, our recommendation is that, when the numbers of sequence reads from different samples are approximately equal, the OTU reads can be utilized directly to compute the correlation coefficients; when the numbers of sequences reads from different samples are significantly different, the relative abundances should be utilized instead. The usage of OTU reads directly has an advantage over the relative abundance since the former can avoid the potential error from decimal conversion in calculating the relative abundance (i.e., the OTU reads for a particular OTU or species divided by the total number of reads of all OTUs in the sample). Our pre-experiment tests found that, although both relative abundance and absolute abundance (raw OTU reads) may produce different results when the numbers of sequencing reads across samples are different, the trend of trios is rather robust. In this study, we use the relative abundance (i.e., the reads of a particular OTU in a sample divided by the total reads in the sample) to be fail-safe. Alternatively, if one does not wish to use relative abundance, sub-sampling (i.e., randomly choose the same number of reads from each sample, e.g., 5000 reads from each sample) may be utilized to deal with the issues associated with unbalanced sample sizes.

To deal with the rising chances that some tests will tend to pass falsely when simultaneously testing multiple null hypotheses (i.e., the significance of many correlation relationships) in 16s-rRNA sequence data, we suggest correcting the p-values associated with the correlation coefficients (from either Spearman’s or Pearson’s methods) with the FDR-BH algorithm (Benjamini-Hochberg standard false discovery rate correction) (Benjamini and Hochberg’s 1995)30. The procedures with the FDR control have been implemented in several R packages, and we choose to use the R-package EMA, which implemented FDR-BH algorithm by Servant et al. (https://cran.r-project.org/web/packages/EMA/index.html)31 . Specifically, we called the “multiple.correction” function from the EMA package in our own R-script “CCFDR.r”, which also called another function rcorr from the R-package Hmisc (https://cran.r-project.org/web/packages/Hmisc/) to compute the Spearman’s or Pearson’s correlation coefficients. Our R-script “CCFDR.r” essentially implemented the first two steps in previous flowchart and its output is feed into our Perl program TFP.pl (Trios-Finder Program), which completes the trios-finding function outlined in step 4 in the previous flowchart. Both CCFDR.r and TFP.pl are supplied in the online Supplementary materials.

In summary, after dealing with the above-described two potential issues with our “CCFDR.r” R-script (i.e., eliminating the side effect of unbalanced of sample sizes and filtering out false correlations) we use the remaining correlation relationships (i.e., Spearman’s rank correlation coefficient (ρ) in this study) with a threshold of p ≤ 0.05 to build OTU correlation networks for the healthy and diseased microbiome samples, respectively. From the OTU correlation networks, the trios defined in Table 1 are sought out and counted with our homemade Perl-program TFP.pl.

In this article, we present our methodology and hypothesis based on the trios that are associated with the MAO (most abundant OTU) in the microbial species interaction network to simplify the presentation. It is noted that our methodology presented here regarding the special node can be readily extended to other nodes in SIN with some special biomedical or computational implications. We have also applied the MDO (most dominant OTU) and hub associated trios elsewhere (Ma & Ellison 2017a)25 with the same computational procedures presented here, but the detailed approaches are only reported in this article.

To compare the distribution of the above-defined trios in the healthy and diseased microbiome samples, we define RDHT (the ratio of disease to healthy trios), the number of a particular trio type in the disease treatment divided by the number of the same trio type in the healthy counterpart. Nevertheless, caution should be taken to use RDHT for diagnostic purpose since the magnitude of different trio types may be different. In addition, the identity of trio members may also be of critical biomedical significance.

Test Results and Discussion

Test dataset description

Largely following the sampling scheme of NIH human microbiome project, we selected six datasets that represent the microbiomes sampled from five major body sites (gut, lung, oral, skin, and vagina). Except for gut that is represented by two datasets (HIV and IBD), each microbiome site is represented by one dataset, with six datasets in total. A brief description on the six datasets is summarized in Table 2, and detailed information on individual dataset is referred to the original publication noted in Table 2.

Table 2 Datasets utilized to develop and test the TFP (trio finder program)
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For each of the six case studies representing the five major microbiome sites, we constructed separate species interaction networks (SIN) with the 16s-rRNA sequence dataset from each treatment in the six case studies. For example, with BV dataset (Srinivasan et al.)6, we built two SINs: one with the 16s-rRNA microbiome samples from BV group and another with the samples from healthy group. We followed the 4-step computational procedure described in the previous section and built 15 networks in total for the 15 treatments of the healthy and diseased microbiome groups, covering five major microbiome sites (gut, lung, oral, skin, and vaginal) and representing several diseases including HIV-infection, inflammatory bowel disease (IBD), periodontitis, cystic fibrosis (CF), Atopic Dermatitis (AD), and bacterial vaginosis (BV), as detailed in previous Table 2.

After getting respective SIN for each of the 15 treatments of the six case studies, we utilized our homemade trio finder program (TFP.pl) program, supplied in the Supplementary document, to compute the 19 triangle motifs or trio types defined in previous Table 1. The results from TFP computing are listed in the following Table 3 for the class of “trios without MAO handle” and Table 4 for the class of “trios with MAO handle,” respectively.

Table 3 The number of various trios in the class of “Trios without MAO handle” in the SIN.
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Table 4 The number of various trios in the class of “Trios with MAO handle” in the SIN.
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The class of “Trios without MAO handle”

In consideration of the existence or absence of MAO handle connected with trios, the 19 triangle motifs are classified into two classes (explained in Table 1) trios without MAO handle (upper section in Table 1 and discussed in this sub-section further) and trios with MAO handle (bottom section in Table 1 and discussed in the next sub-section).

As displayed in Table 3, the class of trios without MAO handle is further distinguished as two categories: trios with MAO and trios without MAO. The category of trios without MAO, including four types that differ from each other by the signs of trio links [(– – –), (+ – –), (+ + –), (+ + +)], although listed for comparative purpose, may be less important than the category of trios with MAO for the following reason: because MAO is not involved apparently, the number of trios in this category may be too many to focus on for further etiological studies in practice.

Given the particular significance of the category of trios with MAO, we further distinguish four types (Type 1–4) at the tertiary level classification, depending on the signs (+ or –) of the trio links. Due to the special role of MAO, we further distinguish two sub-types for Type-1 (Type-1A & Type-1B) and Type-2 (Type-2A & Type 2B), respectively, at the tertiary level by noting the ‘position’ of MAO in the trio (see Table 1). This classification results in the six types in the category of trios with MAO, i.e., Type-1A, Type-1B, Type-2A, Type-2B, Type-3, and Type-4 (also see previous Table 1).

Table 3 shows that theoretically possible Type-2 and Type-3 were not detected in our case studies. The counterpart types in the category of trios without MAO were not detected either. Actually, the apparent prohibition of both Type-2 and Type-3 is not difficult to explain by their internal interactions. Type-2, which has three links with (+ + –) interaction relationships, may be hard to sustain because a third link of negative (–) interaction would be ‘coerced’ to follow the ‘mainstream norm’ of two other collaborative relationships. Similarly, a trio consisting of three totally opposing nodes is unlikely to sustain because they would most likely ‘destroy’ each other. Obviously, regardless whether or not MAO is involved, the arguments regarding (+ + –) and (– – –) hold; hence in both trios with and without MAO, these two patterns may not be sustainable.

We name Type-1 (+ – –) triangle motif as dominantly inhibitive trio given that negative interactions form majority in the system. In Type-1A sub-type, MAO takes a mixed strategy, collaborating with one and competing with another node in the trio. We term Type-1A as dominantly inhibitive trio with mixed strategy. In the Type-1B sub-type, MAO competes with both nodes in the trio simultaneously, and we term this type dominantly inhibitive trio with pure strategy. Among the six tested cases displayed in Table 3, in the cases of skin and oral, no Type-1A trio was detected; in the cases of IBD and lung, no Type-1B was detected. In the other cases, the RDHT of Type-1 ranged from 0 to infinity. That is, disease may raise or lower the number of Type-1 trios depending on the type of microbiome and its associated disease, possibly on other factors, and the difference can be exploited to detect the impact of diseases.

We name Type-4 (+ + +) triangle motif as fully facilitative trio given that positive interactions are the sole interaction in this type of trio system. It is also the most abundant triangle motif among the four types in the category of trios with MAO.

In summary, the results in Table 3 suggest that dominantly inhibitive trio (i.e., Type-1, including both Type-1A with mixed strategy and Type-1B with pure strategy) and fully facilitative trio (Type-4) possess the potential to act as in silico biomarker for differentiating the healthy and diseased microbiomes. As to the criteria for differentiating disease from healthy microbiome, we previously defined the ratio of disease to healthy trio (RDHT) as indicator of the changes, but actual application of the indicator is individual case specific, depending on the types of microbiome, disease, and possibly other factors. In fact, the taxonomic identities and biological characteristics (such as anaerobes or opportunistic pathogens) of trio nodes should play a rather important role in deciphering the mechanism of specific trio formation as we demonstrate elsewhere.

The class of “Trios with MAO handle”

In the class of trios with MAO handle, MAO is connected with the trio as a ‘handle’ rather than as a constituent node of the trio. In contrast with the previously discussed class of trios without MAO handle, there is no ‘prohibited’ trio in the trios with MAO handle. Therefore, all three categories (SLM, DLM, and TLM) including nine types (classified at the tertiary level by considering the link signs) are practically possible. Note that the trios in this class actually do not contain MAO because usually MAO is unique in microbiome network.

The results in Tables 5 and 6 suggest that the range of RDHT spans from zero to infinity. In a half of the cases (48%, or 13 out of 27 cases in the last three columns of Table 5), RDHT exceeded one, that is, diseases tend to raise the number of trios in the class of trios with MAO handle. Furthermore in 44% of the cases (12 out of 27), the RDHT exceeded 10, i.e., diseases caused more than 10 times increase in the number of trios; in 1/3 of the cases (9 out of 27), the RDHT reaches infinity, i.e., the trios occurred only in diseased microbiome networks. Given the striking differences in RDHT among different microbiome-disease treatments, we consider this class of trios also possesses the potential to act as in silico biomarker for assessing the effects of diseases on the human microbiome. Since it seems that the numbers of trios in this class are far greater than those in the class of trios without MAO handle, we believe that the previously identified fully facilitative trios and dominantly inhibitive trios may have an advantage over this class in exploring the mechanisms of disease effects. Another argument in support of our opinion is that the fully facilitative trios and dominantly inhibitive trios are simpler with three nodes only.

Table 5 The RDHT (ratio of disease to healthy trios) of category-specific total trios in the human microbiome computed from Tables 3 & 4.
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Table 6 The RDHT (ratio of disease to healthy trios) of Type-1, 2, 3, 4 trios (i.e., Trios with MAO but no MAO handle) in the human microbiome computed from Table 3.
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General patterns of the trio differences between the healthy and diseased microbiomes

In the following, we further look into general patterns by cataloging the 19 trios into five categories and summing up the total trios of each category in Table 5. In Table 5, besides listing the microbiome sites and healthy/disease treatments in the first two columns, the remaining five columns display the respective ratios of the trios in diseased network to those in healthy network for each of the five categories, i.e., RDHT for: trios with MAO, trios without MAO, single-link with MAO (SLM), double-link with MAO (DLM) and triple-link with MAO (TLM), respectively. That is, for each category, we define and compute the ratio of diseased to healthy trios (RDHT). In the ideal scenario when disease has no impact on the microbiome, the RDHT should be 1. If the ratio is larger than 1, then it indicates that the disease may raise the number of trios in the specific category; vice versa, it indicates that the disease may lower the number of trios if the RDHT is smaller than 1.

Table 5 shows that in approximately 47% cases (21 out of 45) disease caused a decrease in the number of trios. Specifically, the decline of trios occurred mostly in two diseases: HIV and periodontitis. The RDHT values range from zero to infinity; the occurrence of zero or near zero (≤0.1) counts to 9, and that of infinity reaches 29% (13 out of 45). The number of RDHT exceeding 10 (i.e., disease caused more than 10 times of increase in the trios) approaches to 38% (17 out of 45). In these cases, disease led to a significant increase in the number of trios.

More interesting insights can be found by looking into the third level classification—considering the sign (+ , –) of interactions (links) in the trios, as well as the ‘position’ of signs (see Table 1). Table 6 summarized the RDHT of Type-1, 2, 3, 4 trios from the information presented in Tables 3 & 4 to further reveal patterns and trends embedded in the six trios that are associated with MAO but without a MAO handle (see Table 1). Of course, the undetected Type-2 and Type-3 appear to be “prohibited” in our case studies as explained previously.

Among the six cases we analyzed, except for the CF-lung case, which we cannot draw a definite conclusion due to data insufficiency, there were three cases (IBD-gut, AD-skin, and BV-vaginal) that displayed disease-up-regulated trios trend, and two cases (HIV-gut and periodontitis-oral) displayed disease-down-regulated trend. Although further accumulating test cases is certainly meaningful, this splitting trend of the up or down of trios does not affect the testing of our primary hypothesis—whether or not the trios we defined can differentiate between healthy and diseased microbiomes. This is because the validity of our hypothesis hinges on the level of difference or gap in the trio numbers (i.e., RDHT) rather than on the sign or direction of the difference (rise or decline). Indeed, we believe that the variable sign of the difference among microbiomes may simply be a biomedical reality.

Finally, we suggest that, among 19 types of trios we initially propose to test, 12 are indeed promising as in silico biomarkers. The six trio types we excluded are the four types in the category of “trios without MAO”, and three types (Type-2A, Type-2B, & Type-3) in the category of “trios with MAO”. The reason they are excluded is either because they are either too abundant (to be indicative) or too rare (not detected) in both healthy and diseased samples, to be indicative. Indeed, the entire category of “trios without MAO” is excluded, not only because they are too abundant to be indicative, but also because they lack special node (in this study, the MAO). We demonstrated that the following 12 types or categories are the most promising: Type 1A (dominantly inhibitive trio with mixed strategy), Type 1B (dominantly inhibitive trio with pure strategy), Type-4 (fully facilitative), SLM (including 2 types), DLM (3 types), and TLM (4 types). We particularly favor the first three types, and give them the special names in particular in consideration that they are simpler and less abundant (in general) than the four-nodes SLM, DLM, and TLM. This may give them an advantage in further studying their etiological implications experimentally. As mentioned previously, two further improvements can be made to reveal potentially more meaningful biomedical insights. One is to look up the taxonomic identities or biomedical characteristics such as the trios of anaerobes, and another is to replace the MAO with other special network nodes such as MDO (most dominant OTU) or hub nodes. We will demonstrate these additional improvements elsewhere.

Data accessibility

The datasets utilized in this study are available in the original studies cited in Table 2. The study does not involve any experiments involving humans and/or animals.