q2-gglasso

This is a QIIME 2 plugin which contains algorithms for solving General Graphical Lasso (GGLasso) problems, including single, multiple, as well as latent

Graphical Lasso problems.

Docs | Examples

For details on QIIME 2, see https://qiime2.org.

Installation

Example of Atacama soil microbiome analysis

Welcome to this tutorial on using QIIME 2 for analyzing soil samples from the Atacama Desert in northern Chile. This tutorial assumes that you have already installed QIIME 2, but if you haven't, you can follow the instructions from the docs.

The Atacama Desert is known to be one of the most arid locations on Earth, with some areas receiving less than a millimeter of rain per decade. Despite such extreme conditions, the soil in the Atacama Desert is known to harbor a variety of microbial life. In this tutorial, we will explore how to use graphical models for analyzing microbial compositions in soil samples from the Atacama Desert.

Specifically, we will demonstrate the application of Single graphical lasso (SGL), adaptive SGL, and SGL + low-rank, to illustrate how covariates are related to microbial compositions.

Let's get started!

Compositional data

In the following tutorial we will work with 130 ASVs written into count table of the following format:

	ASV 1	ASV 2	ASV 3	...	ASV 130
sample 1	-0.167770	-0.167770	-0.167770	...	-0.167770
sample 2	-0.436407	-0.436407	-0.436407	...	-0.436407
sample 3	-0.229753	1.967471	-0.229753	...	-0.229753
sample 4	-0.424645	4.097143	3.264234	...	-0.424645
...	...	...	...	...	-0.353991
sample 53	-0.384811	4.069537	3.443831	-0.384811	-0.384811

Please note that preprocessing steps, such as the center log-ratio transformation of the count table and scaling metadata, have been omitted from this tutorial, but you can find these steps in the documentation linked here.

Figure 1. Correlation between ASVs in Atacama soil microbiome.

Metadata

This section presents a description and basic statistical analysis of the covariates under investigation. For more comprehensive information about the research, readers are referred to the original paper.

feature	mean	std	min	max	description (units)
elevation	2825	1014.23	895	4697	height above sea level (m)
extract-concen	2.92	5.96	0.01	33.49	concentrarion of DNA before PCR (µg/ml)
amplicon-concentration	9.54	6.81	0.12	19.2	concentration of DNA after PCR (µg/ml)
depth	2	0.46	1	3	level of sampling A/B/C (0–60/60–220/220–340 cm)
ph	7.05	2.53	0	9.36	potential of hydrogen (log)
toc	693.8	1958.49	0	16449	total organic carbon (μg/g)
ec	0.72	1.26	0	6.08	electric conductivity (S/m in SI)
percentcover	1.82	3.05	0	8.8	vegetation coverage (% per $m^2$)
average-soil-relative-humidity	63.27	33.54	0	100	average soil humidity (%)
relative-humidity-soil-high	78.51	32.09	0	100	soil humidity at the top of a sample (%)
relative-humidity-soil-low	43.62	32.58	0	100	soil humidity at the bottom of a sample (%)
percent-relative-humidity-soil-100	37.86	39.45	0	100	relative soil humidity across all depth levels (%)
average-soil-temperature	15.72	5.8	0	23.61	average soil temperature (t°)
temperature-soil-high	23.61	6.82	0	35.21	soil temperature at the top of a sample (t°)
temperature-soil-low	7.24	5.96	-2.57	18.33	soil temperature at the bottom of a sample (t°)

Figure 2 illustrates the correlation between the covariates, it is clear that some of them are highly correlated and thus can be disregarded. For instance, in the subsequent analysis of humidity and temperature, their average values shall suffice for our purposes.

Figure 2. Correlation between Atacama covariates.

SGL

Figure 3. Single graphical lasso (SGL) solution.

Let us examine the discovered connections among ASVs at Figure 4. By definition of inverse covariance, the relationship between ASV 18, ASV 51, ASV 46, ASV 13, ASV 7, and ASV 5 is conditionally independent, implying that these relationships do not affect one another. Nevertheless, in reality, we are aware that the microbial compositions are frequently influenced by the environment. Therefore, we should contemplate the possibility of the existence of additional covariates and their potential impact on these associations. Furthermore, it is worth investigating whether these associations will be altered or remain unchanged due to the introduction of new covariates.

Figure 4. Bacterial associations in SGL solution.

Adaptive SGL

Figure 5. Adaptive SGL solution.

Let us now incorporate information regarding the covariates in our model and evaluate the correlation between ASVs and various selected covariates such as temperature, humidity, pH, and others. As some of these covariates have a profound effect on microbial compositions, we will utilize an adaptive SGL model which only penalizes the associations between ASVs and does not penalize associations between ASVs and covariates. By examining the changes in the relationship between a hub node of ASV 18 identified through the previous SGL model, we can observe that the relationship between ASV 18 and ASV 53 was influenced by covariates such as average humidity, average temperature, elevation, and the concentration of DNA in a sample prior to PCR. However, with the adaptive SGL model, the spurious relationship between ASV 18 and ASV 53 was eliminated.

Figure 6. Association between ASVs and covariates in adaptive SGL solution.

SGL + low-rank

Figure 7. SGL with latent variables solution.

Frequently, due to privacy concerns, covariate information may not be available, and only microbial data may be at hand. In such cases, it may be of interest to determine whether the effects of the covariates on microbial associations can be detected without having access to covariate information. It is indeed possible to do so by employing an SGL model with latent variables. As shown in Figure 8, the use of latent variables, which are accounted for by the low-rank of the solution, resulted in the disappearance of spurious correlations between ASV 18 and ASVs 51, 46, and 5, and ASV 18 is no longer a hub node.

Figure 8. Association between ASVs and covariates in adaptive SGL solution.

Model comparison

To demonstrate the correlation between ASVs (amplicon sequence variants) and covariates, we can compute the l1-norm of the covariates and arrange our solution matrix in descending order based on this norm. Figure 9 reveals certain ASVs located at the top, which are solely associated with the covariates. Among these ASVs, we can observe the presence of ASV_18, which we previously identified.

Figure 9. Inverse covariance sorted by l1-norm of the covariates.

The Graphical Lasso solution is of the form Θ−𝐿 where Θ is sparse and 𝐿 has low rank. We use the low rank component of the Graphical Lasso solution in order to do a PCA by the following eigendecomposition

$L = V \Sigma V^T$

where the columns of 𝑉 are the orthonormal eigenvecors and Σ is diagonal containing the eigenvalues. Denote the columns of 𝑉 corresponding only to positive eigenvalues with $\tilde{V} \in \mathbb{R}^{p\times r}$ and $\tilde{\Sigma} \in \mathbb{R}^{r\times r}$ accordingly, where $r=\mathrm{rank}(L)$. Then we have

$L = \tilde{V} \tilde{\Sigma} \tilde{V}^T$

Now we project the data matrix $X\in \mathbb{R}^{p\times N}$ onto the eigenspaces of $L^{-1}$ which are the same as of $𝐿$ - by computing

$U := X^T \tilde{V}\tilde{\Sigma}$

Figure 10. PCA of Atacama soil microbiome

In Figure 11, a comparison is presented between the outcomes of principal component analysis (PCA) of the adaptive graphical lasso solution and the low-rank graphical lasso solution. The results indicate a strong correlation between average temperature and the principal components in both cases, suggesting that a single graphical lasso with low-rank may be appropriate in scenarios where only compositional data is available and additional covariates are unknown, i.e., SGL + low-rank is capable of accounting for the impact of hidden confounders.

Figure 11. PCA comparison

Figure 12. Correlation between temperature, ASV 18 and ASV 51.

Figure 13. Inverse covariance sorted by l1-norm of low-rank principal components (PCs).