The Human Microbe–Disease Association Database (HMDAD) (Ma et al., 2017) is the first human microbe–disease association database established by Ma et al. through a lot of biological experiments. The database includes 483 experimentally tested and verified associations between 292 microbes and 39 diseases. We downloaded the data from HMDAD (http://www.cuilab.cn/hmdad), then removed redundant associations. Thus, 450 microbe–disease associations including 39 diseases and 292 microbes were obtained from 61 publications. As a result, a 39 × 292 dimensional adjacency matrix A is constructed. In addition, in the adjacency matrix A, the value of A [ i ] [ j ] is set to 1 if microbe m [ j ] is related to disease d [ i ] , otherwise, A [ i ] [ j ] is set to 0.

The abnormal subjective feeling or some objective pathological changes of patients caused by a series of abnormal changes in function, metabolism, and morphological structure in the process of disease are called symptoms. Some diseases, especially in the early stage of some diseases, may not be accompanied by symptoms and signs. The human symptoms–disease network (HSDN) has been constructed by Zhou et al. from PubMed (Wheeler et al., 2007; Zhou et al., 2014). Moreover, they used term frequency inverse document frequency (TF-IDF) (Salton et al., 1975) to measure the symptom-based disease similarity based on the co-occurrence frequency between a disease and a symptom. Based on these data, Chen et al. (2016a) extracted those symptom-based similarities of common diseases from HMDAD. Hence, symptom similarity SDM can be constructed.

As the microbe–disease association matrix is low rank, in other words, it is very sparse, microbe-disease association matrix can be split into two low-dimensional feature matrices, i.e., disease feature X and microbe Y. Then, Tikhonov regularization terms are used to avoid over-fitting. The elementary matrix factorization model is formulated as follows: min X , Y 1 2 ‖ P Ω ( X Y T − A ) ‖ F 2 + λ 1 2 ( ‖ X ‖ F 2 + ‖ Y ‖ F 2 ) (7) where   • F denotes the Frobenius norm, A F   = t r ( A T A ) = ∑ i = 1 m ∑ j = 1 n a i j 2 ,   tr ( A ) is the trace of matrix A, A F 2 = t r ( A T A ) = ∑ i = 1 m ∑ j = 1 n a i j 2 , λ 1 is the harmonic parameter to counterpoise the error term and the regularization terms, Ω is an index set of known association in matrix A, and Ρ Ω is defined as: ( P Ω ( I ) ) i j = { I i j , ( i , j ) ∈ Ω 0 , ( i , j ) ∉ Ω (8)

However, Eq. 7 does not involve prior information about diseases and microbes. Given a disease similarity matrix D and a microbe similarity matrix M, as X,Y can be considered as matrices containing disease and microbe potential characteristic vectors, respectively, X X T and Y Y T are expected to match D and M, respectively (Zheng et al., 2013; Cui et al., 2019). Therefore, Eq. 7 is extended to: min X , Y 1 2 ‖ P Ω ( X Y T − A ) ‖ F 2 + λ 1 2 ( ‖ X ‖ F 2 + ‖ Y ‖ F 2 ) + λ 2 2 ( ‖ D − X X T ‖ F 2 + ‖ M − Y Y T ‖ F 2 ) (9)

In order to incorporate multiple similarity measures, an MSBMF model can be proposed for predicting microbe–disease associations, which is formulated as follows: min X , Y , P , Q , Z 1 2 ‖ X Y T − Z ‖ F 2 + λ 1 2 ( ‖ X ‖ F 2 + ‖ Y ‖ F 2 )                     + λ 2 2 ( ‖ D m − X P T ‖ F 2 + ‖ M m − Y Q T ‖ F 2 )                     + λ 3 2 ( ‖ P ‖ F 2 + ‖ Q ‖ F 2 )                     s . t .       P Ω ( A ) = P Ω ( Z )                                 X ≥ 0 , Y ≥ 0 (10) D m and M m are multi-similarities matrices of diseases and microbes, respectively, and λ 1 , λ 2 , λ 3 are balancing parameters. Obviously, D m = [ K D , C D , S D M ] and M m = [ K M , C M ] , where P and Q are matrices including latent features representing disease similarity and microbe similarity, respectively. Z is an auxiliary matrix that helps to optimize. Furthermore, by introducing two splitting matrices S and T, Eq. 10 is transformed into: min X , Y , P , Q , S , T , Z 1 2 ‖ X Y T − Z ‖ F 2 + λ 1 2 ( ‖ X ‖ F 2 + ‖ Y ‖ F 2 )                     + λ 2 2 ( ‖ D m − X P T ‖ F 2 + ‖ M m − Y Q T ‖ F 2 )                     + λ 3 2 ( ‖ P ‖ F 2 + ‖ Q ‖ F 2 )                     s . t .       P Ω ( A ) = P Ω ( Z )                                   S = X , T = Y                                   S ≥ 0 , T ≥ 0 (11)

Then, we use the alternating direction method of multipliers (ADMM) framework to solve Eq. 10. The augmented Lagrangian function is given by: ℓ ( X , Y , P , Q , S , T , Z ) = 1 2 ‖ X Y T − Z ‖ F 2 + λ 1 2 ( ‖ X ‖ F 2 + ‖ Y ‖ F 2 ) + λ 2 2 ( ‖ D m − X P T ‖ F 2 + ‖ M m − Y Q T ‖ F 2 ) + λ 3 2 ( ‖ P ‖ F 2 + ‖ Q ‖ F 2 )                 + 〈 Φ , X − S 〉 + 〈 Ψ , Y − T 〉 + μ 2 ( ‖ X − S ‖ F 2 + ‖ Y − T ‖ F 2 ) (12) where Φ and Ψ are the Lagrange multipliers, and μ is the penalty parameter. After k iteration, X k + 1 , Y k + 1 , P k + 1 , Q k + 1 , S k + 1 , T k + 1   a n d Z k + 1 will be computed. We adopt a scheme with gradually increasing learning rate to achieve fast convergence (Shang et al., 2018). After executing the MSBMF algorithm, a non-negative matrix M* is a predicted scores matrix. The scheme of MSBMF model is illustrated in Algorithm 1 .

Algorithm 1

MSBMF algorithm.

Input: the microbe–disease association matrix M, the multiply similarities of disease matrices D _m , the multiply similarities of microbe matrices M_m, subspace dimensionality r, parameters λ 1 , λ 2 and λ 3 .

The problem of microbe–disease associations prediction can be seen as a classification or regression problem, usually using cross-validation to evaluate the generalization capabilities of the new sample. In order to evaluate performance of our model, we carry out two kinds of computational experiments, including LOOCV and fivefold cross validation. In LOOCV, each confirmed microbe–disease association was chosen as a test sample in turn, and the rest of the associations were used to train. After executing MSBMFHMDA, the score of the test example would be ranked with the scores of candidate samples that were made up of all unconfirmed microbe–disease pairs. In fivefold cross validation, we first divided the known microbe–disease associations into five equal parts and later made each part as a test sample in turn and the remaining four parts of associations as training samples. Similarly, the score of each test sample would be ranked with the scores of candidate samples that were made up of all unconfirmed microbe–disease pairs. As the sample divisions may cause bias, we repeated the fivefold cross-validation 100 times to get an average value as the final result. As the predicted score that obtained a higher rank than the given threshold, our model is considered to make a successful prediction. Then according to diverse thresholds, we plotted the receiver operating characteristics (ROC) curve by computing the ratio of true positive rate (TPR, sensitivity) to false positive rate (FPR, 1-specificity). The AUC can be used to evaluate its predictive performance, where the AUC value of 1 represents perfect prediction ability, and the AUC value of 0.5 indicates random prediction performance (Chen et al., 2016a).

In our algorithm, the tunable parameters include the latent dimension r and the three coefficients λ 1 , λ 2 , and λ 3 . We set r = [ τ min (m, n)], where τ ∈ [ 0,1 ] and [ • ] denotes the rounding function. Because there are many parameters, they may lead to overfitting. So, we set λ 2 and λ 3 to the same value to prevent overfitting. Finally, three parameters need to be determined, including τ , λ 1 , and λ 2 .

We choose to adopt a “fixing one and determining the others” strategy. First, we set τ to 0.1 and then picked the values of λ 1 and λ 2 from {0.001, 0.01, 0.1, 1} by LOOCV in a standard dataset. Then, we fix the determined values of λ 1 and λ 2 , and selected τ from {0.1,0.3,0.5,0.7,0.9,1}. The computational results for determining the λ 1 and λ 2 are listed in Table 1. We can discover that the AUC value reach maximum when λ 1 = 0.1 and λ 2 = 0.01 . As shown in Table 2, our model furnishes approximately the same good performance when τ ≥ 0.7 . Therefore, we set τ = 0.7 .

The stopping criteria of the MSBMF algorithm are f k ≤ t o l 1 and | f k + 1 − f k | max { 1 , | f k | } ≤ t o l 2 , where f k = ‖ S k + 1 T k + 1 − S k T k ‖ F ‖ S k T k ‖ F and t o l 1 , t o l 2 are the given tolerances. Here, according to the related studies (Yang et al., 2021), we set t o l 1 = 2 × 10 − 3 and t o l 2 = 10 − 4 .

In this section, we consider several state-of-the-art microbe–disease association prediction methods and make comparisons to demonstrate superior performance of our proposed method MSBMFHMDA. We compare it with KATZHMDA, BiRWMP, and NBLPIHMDA based on the dataset of known microbe–disease associations. As illustrated in the following Figure 2 and Table 3, MSBMFHMDA yields best performance in LOOCV, achieving an AUC score of 0.9186, while KATZHMDA, BiRWMP, and NBLPIHMDA produce AUC scores of 0.8382, 0.8637, and 0.8777, respectively. As demonstrated in the following Figure 3, in the framework of fivefold cross validation, MSBMFHMDA can achieve a reliable AUC of 0.9043 ± 0.0048, which is better than the AUC achieved by KATZHMDA (0.8301 ± 0.0033), BiRWMP (0.8522 ± 0.0054), and NBLPIHMDA (0.8958 ± 0.0027).

In this section, we concentrate on the sensitivity analysis for λ 1 , λ 2 , and τ in LOOCV. As we all know, when λ 1 = 0.1 , λ 2 = 0.01 , and τ = 0.7 , our model can realize excellent performance. We vary one parameter and keep the rest of the two parameters fixed to observe how the parameter benefits the AUC value.

As shown in Figure 4, the AUC can achieve the best values when λ 1 = 0.1. In the same way, Figure 5 indicates the best AUC on λ 2 = 0.01. Finally, the effect of parameter τ on the prediction accuracy is discussed. Figure 6 shows the AUC values of MSBMF with different τ . When τ > 0.7, the trend of AUC is becoming steady. If τ continue to increase to 0.9 or 1, our model will not only generate overfitting but also increases the computational complexity.

Microbes are closely related to human health, and it is meaningful to explore whether microbes are associated with disease. In order to investigate into disease-causing microbes and further measure the prediction performance of our model, we selected three kinds of common microbe-induced diseases as cases for the analysis, namely, asthma, inflammatory bowel disease, and type 1 diabetes. The scores of the top 10 disease-related microbes are published in Supplementary Tables S1–S3, respectively.

Asthma is short for bronchial asthma, a heterogeneous disease characterized by chronic airway inflammation and airway hyper-responsiveness (Lemanske and Busse, 2010). The key features of asthma include chronic inflammation of the airway, high responsiveness of the airway to a variety of stimulators, limited variable reversible flow, and a series of changes with the course of the disease, namely, airway reconstruction (Çalışkan et al., 2013). Asthma is one of the most common chronic diseases in the world, with about 300 million people worldwide and about 45 million asthma patients in China, and there is a trend year by year. Epidemiological studies have shown that early exposure to microbes may determine the composition of the microbiome, which can help prevent allergies or cause the development of asthma. Asthma had been demonstrated to be closely associated with microbes by a number of research (Gilstrap and Kraft, 2013). In this section, though the there is implementation of our model to infer the novel asthma-related microbes, we published evidence for the top 10 potential asthma-related microbes predicted by MSBMFHMDA in Table 4.

Inflammatory bowel disease (IBD) is a group of chronic non-specific intestinal inflammatory diseases that have no etiology, including ulcerative colitis and Crohn’s disease (D’Aoust et al., 2017). In this paper, we selected IBD as one of our case studies to evaluate the performance of our model. As illustrated in the following Table 5, there are 10 out of these top 10 microbes predicted by MSBMFHMDA that have been substantiated to be associated with IBD.

Type 2 diabetes mellitus (T2D), also known as adult-onset diabetes, is characterized by a rise in blood sugar and a relative lack of insulin production because of a decline in the ability of insulin to help glucose enter cells for metabolism, a metabolic disorder resulting from a disorder of glucose metabolism (Furet et al., 2010). We took T2D as a case study for potential T2DM-related microbe prediction, and as illustrated in the following Table 6, 8 out of the top 10 predicted microbes were confirmed by experimental reports.