Deciphering the ‘gut–brain axis’ through microbiome diversity

Alpha-diversity

Alpha-diversity is a subject-level metric summarising OTU abundance into a scalar value for each person. Consider a sample of n subjects and a vector    denoting the counts of the p OTUs for subject i . Alpha-diversity for subject i is:

where m is the order controlling the weight allocation among taxa. Varying m permits different alpha diversities. For example,    yields the observed species index    that counts the total number of species present, hence weighing more on the rare taxa and indexes the richness of microbe in total.9 On the other hand,    leads to the Simpson index    that assigns more weight towards abundant taxa, hence indicating species evenness.9 When    (1) is undefined but the resulting limit is the Shannon index,   .

Beta-diversity

Unlike the OTU abundance or alpha-diversity that describes features within each subject, beta-diversity is a between-subject attribute6 indexing the dissimilarity between any two individuals or a pair. Essentially, beta-diversity compares the feature differences between any pair    using a dissimilarity or distance metric, denoted as   , where   :    is a mapping from the high-dimensional OTUs to a scalar value for that pair. Different choices of    lead to distinct versions of beta-diversity, with the commonly used ones including Aitchison, Bray-Curtis, Jaccard, Unifrac, and so on. For example, the Aitchison beta-diversity10 is defined as:

By integrating information from high-dimensional features, beta-diversity represents a totality measure of dissimilarity between two subjects across all the OTUs, which constitutes a biologically relevant indicator of human health11 and merits interest. It is non-negative, with 0 (bigger values) indicating the same (very different) taxonomic abundances. Also, its dimension does not change with the number of taxonomic units    hence can be viewed as a dimension reduction.

Alpha-diversity

The alpha-diversity is composed of data from one subject; therefore, most standard statistical methods are readily applicable for the analysis of alpha-diversity. For example, popular tools for associating a phenotype with alpha-diversity include the Kruskal-Wallis H (or Mann-Whitney U) test12 for categorical variables and the Spearman’s correlation for continuous phenotype variables. When controlling for covariates (eg, demographics) is needed in more complex settings, the robust regression framework is often suggested to ensure valid inferences given the non-normality nature of alpha-diversity. For a study with size n , let    denote a response, and    an explanatory variable for the i -th subject. The semiparametric GLM characterising the within-subject relationship between    and    is:

where    is the inverse of some link functions. Compared with the classical parametric GLM, (3) is more flexible. It removes the distributional assumption on   , thus yielding valid inference even when the data deviate from such an assumption. Hence, it is especially suited for modelling the alpha-diversity by specifying    in (1). It is worth noting that the same method can be used if alpha-diversity is the predictor.

Beta-diversity

Given a main factor    with K levels for the group membership, let    and    denote the mean and variance of    for the k -th group. We can formalise the scientific question into a hypothesis to test the mean beta-diversity distance across the K groups:

However, unlike the case with alpha-diversity, the    involves the OTU abundances from a pair, so it introduces rather complex correlation structures that are difficult, if not impossible, to model using parametric form.

Perceived stress as a continuous response

Perceived stress was assessed by the standardised instruments of the Perceived Stress Scale (PSS)14; it can be viewed as continuous with larger values indicating higher levels of perceived stress. The distance-based model in (8) with predictors of beta-diversity   , alpha-diversity difference   , age difference    and the one-hot encoded gender    can be specified as   , where    denotes the difference score for the perceived stress of the i -th pair.

Here,    represents the mean difference between the perceived stress per unit difference in the beta-diversity   ,       indicates the directional mean difference between the perceived stress per unit difference in the alpha-diversity (age), and    is the mean difference between the perceived stress comparing male–female pairs with homogeneous gender pairs.

When the response and explanatory variable are both continuous, the between-subject regression will preserve their corresponding relationships among within-subject attributes. We demonstrate this in the continuous alpha-diversity (age) in figure 1. Shown at the top of figure 1 are the scatter plots with locally estimated scatterplot smoothing (LOESS) curves for the perceived stress (within-subject) with alpha-diversity and age, respectively. On average, the perceived stress did not correlate strongly with alpha-diversity, but it showed a negative relationship with age, which was confirmed by the univariate linear regression where the alpha-diversity effect was insignificant (t-statistic=−0.3145, p=0.448) but had a significant negative age effect (t-statistic=−1.1021, p<0.001). The bottom of figure 1 shows the scatter plots for their between-subject counterparts. As expected, the within-subject relationships were well-preserved in the between-subject regression.

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Shown at the top of table 1 are the results of perceived stress from the robust distance-based regression model. Per unit difference in a pair’s beta-diversity is associated with a 0.1457 (Wald=2.9752, p=0.085) unit difference in their perceived stress, suggesting that a larger difference between two subjects’ microbiome profiles implies more discrepancies in their stress levels. The mean (directional) stress score difference for any pair was −0.0270 per unit difference in their alpha-diversity but was not significant (Wald=0.1869, p=0.666). Per unit age difference was significantly associated with a 0.2520 unit decrease in the mean perceived stress difference (Wald=10.7857, p=0.001). This is expected since the scatter plots in figure 1 suggest that age is negatively related to perceived stress. The mean perceived stress difference comparing male–female pairs versus homogeneous gender pairs (male–male and female–female pairs) was 0.0218 (Wald=0.343, p=0.558); that is, we did not find strong evidence implicating perceived stress in distinct gender pairs being different from homogeneous gender pairs. Although signals between perceived stress and the microbiome composition were not strong, the negative association between age and stress shown here has been demonstrated in the literature.20

Positive states (traits) as a multivariate response

In mental health studies, some traits are evaluated as a composite outcome.9 Particularly, resilience, optimism, mental well-being and wisdom all belong to positive states (traits). We devised the distance-based regression notion to link this four-dimensional multivariate outcome    to the microbiome composition by constructing the composite pairwise outcomes with the Euclidean distance and adopting (8) with a log link to decipher their relationships.

At the bottom of table 1 are the results of the positive states. Significant associations were found between the mean distance (variability) in the positive states and beta-diversity (  , Wald=181.3488, p<0.001) but not with the variability in alpha-diversity (   Wald=0.5804, p=0.446). This non-significant alpha-diversity is similar to the case of perceived stress as the univariate outcome. Hence, the microbiome beta-diversity may be a more sensitive indicator in capturing the between-subject attributes of mental health than the alpha-diversity in this dataset. Age variability was only marginally significant (  , Wald=3.6775, p=0.055). The mean distance (variability) in the positive states for male–male pairs was    the distance for female–female pairs (Wald=1.1775, p=0.278); the mean distance (variability) for male–female pairs was    of that for female–female pairs (Wald=1.8043, p=0.179) but neither effect was significant. Hence, no significant gender effect was observed, similar to the case with perceived stress.

Taken together, those findings help support the existence of the microbiome–gut–brain axis17 19 and, more importantly, provide clinical implications for developing microbiota-related interventions to improve mental health and mitigate its related consequences. For example, given the positive association between beta-diversity and positive states, strategies that modulate patients’ microbiome composition may be beneficial to improve their mental health in general.