A brief description of the breast microbiome datasets

This study included two datasets, one for breast milk microbiome and the other for breast tissue microbiome (Table 1). The breast milk microbiome dataset consisted of 126 samples, including 44 samples from healthy controls, 44 samples from mothers during mastitis symptoms, and 38 samples from mothers after mastitis symptoms cessation. The DNA was extracted from breast milk, and V3-V4 regions of 16 S rRNA were sequenced using Illumina MiSeq platform [17]. The breast tissue microbiome dataset consists of two groups: normal tissues (47 samples) and tumor tissues (47 samples). The normal and tumor tissue samples were simultaneously obtained from the same female patients with primary invasive breast cancer during breast cancer surgery [29]. V1-V3 regions of 16 S rRNA were sequenced using Illumina MiSeq platform. More detailed information on these datasets is provided in Boix-Amorós et al. (2020) and Kim et al. (2021) [17, 29]. Sequencing data and assemblies are publicly available at NCBI’s GenBank and SRA databases under BioProject PRJEB34421 and PRJEB37724. 16 S rRNA sequencing data were classified to the species-level and genus-level using Kraken 2 v2.1.2 with default parameters (--confidence 0) against the reference sequence databases [30,31,32]. After taxonomic classification, the species abundance was estimated using Bracken v2.6 [33, 34].

Table 1 Brief information on the microbiome datasets of breast milk and breast tissue samplesBeta-diversity in hill numbers

In present study, microbiome diversity was quantified using Hill numbers, which provides a more general framework for measuring biodiversity [35, 36]. Within this framework, gamma diversity (qDγ) can be decomposed multiplicatively into independent alpha (qDα) and beta (qDβ) diversity.

Assume that each group or treatment is a meta-community consisting of N sample microbial communities (microbiomes) and S taxa. Let yij is the abundance of the ith taxa in jth community, i = 1, 2, …, S, j = 1, 2, …, N. The alpha diversity (qDα) measure the within-community diversity and is defined for q ≠ 1 as:

$$^q} = ^S ^q} } \right)^}$$

(1)

where pi is the relative abundance of the ith taxa, and q is the order number of diversity. The formula when q = 1is:

$$^1} = \mathop }}\limits_ ^q} = exp\left( ^S }\left( } \right)} } \right)$$

(2)

The parameter q determines the sensitivity of the measure to the taxa abundances. Hill numbers at different q orders correspond to special ecological diversity indices. For example, 0D is equal to taxa richness, 1D represents the exponential of the Shannon index, 2D represents the reciprocal of the Simpson index. The diversity in Hill numbers is adjusted for sensitivity to taxa abundance through the diversity order q. The larger the diversity order q, the more sensitive qD is to taxa with high abundance. When q = 0, species abundance is not considered, making the measure equivalent to taxa richness. At q = 1, all taxa are weighted equally by their frequency, reflecting the diversity of common taxa. At q = 2–3, greater weight is given to abundant taxa, thus reflecting the diversity of dominant taxa.

The gamma diversity (qDγ) measures the total diversity of the meta-community and is defined as:

$$^q} = ^S _i}} \right)}^q}} } \right)^}$$

(3)

in which \(\: = \left( ^N }} } \right)/\left( ^S ^N }} } } \right)\).

The beta diversity measures the between-community diversity, which is the ratio of gamma to alpha:

Similarity indices from beta-diversity

We also estimated four similarity indices between communities within each treatment (meta-community), including Cq, Uq, Sq and Vq. These similarity indices can be obtained by the transformations of beta-diversity in Hill numbers (Eq. 4), and illuminate different aspects of similarity.

The similarity Cq quantifies the effective average proportion of shared taxa (overlap) per community, which takes the following form:

$$ = \frac^q})}^} - ^}}}^}}}$$

(5)

The similarity Uq quantifies the effective average proportion of shared taxa (overlap) in the meta-community, which is defined as:

$$\: = \frac\,})}^} - ^}}}^}}}$$

(6)

The similarity Sq is homogeneity measure, which quantifies the proportion of meta-community diversity contained in the average community. That is :

$$\:\:\: = \frac\,} - 1/N\:}}}$$

(7)

The similarity Vq measures the relative taxa turnover rate per community, i.e.,

$$\: = \frac\,}\:}}}$$

(8)

Detecting AKP effects with beta-diversity and similarity indices in hill numbers

Beta diversity reflects the heterogeneity in community composition, while the similarity indices are the opposite. AKP effect of the animal microbiome is manifested by greater differences in the microbiome composition of dysbiotic individuals compared to healthy individuals, that is, the microbiome of the dysbiotic individuals is more heterogeneous than that of the healthy individuals, or less similar than that of the healthy individuals [1, 8]. In this study, the beta-diversity and four similarity indices in Hill numbers were used as metrics to determine AKP effect. We used Wilcoxon rank sum test to examine differences in beta-diversity or similarity indices between two treatments. A p-value of < 0.05 indicated significant difference in measures between two treatments.

Shared species analysis (SSA)

The decrease in the number of shared taxa between samples can also reveal rising heterogeneity in microbiome composition. Let SS be the set whose members are the number of taxa shared between each pair-wise samples in the treatment. The non-parametric Wilcoxon test was used to examine whether there was a significant difference in the values of the SS set between the two treatments. If the SS values of the dysbiotic treatment is significantly lower (higher) than that of the healthy treatment (p < 0.05 from Wilcoxon test), then the test reveals AKP (anti-AKP) effects; otherwise, dysbiosis has no effect on the microbiome.