Extraction and Integration of Genetic Networks from Short-Profile Omic Data Sets

Abstract

Mass spectrometry technologies are widely used in the fields of ionomics and metabolomics to simultaneously profile the intracellular concentrations of, e.g., amino acids or elements in genome-wide mutant libraries. These molecular or sub-molecular features are generally non-Gaussian and their covariance reveals patterns of correlations that reflect the system nature of the cell biochemistry and biology. Here, we introduce two similarity measures, the Mahalanobis cosine and the hybrid Mahalanobis cosine, that enforce information from the empirical covariance matrix of omics data from high-throughput screening and that can be used to quantify similarities between the profiled features of different mutants. We evaluate the performance of these similarity measures in the task of inferring and integrating genetic networks from short-profile ionomics/metabolomics data through an analysis of experimental data sets related to the ionome and the metabolome of the model organism S. cerevisiae. The study of the resulting ionome–metabolome Saccharomyces cerevisiae multilayer genetic network, which encodes multiple omic-specific levels of correlations between genes, shows that the proposed measures can provide an alternative description of relations between biological processes when compared to the commonly used Pearson’s correlation coefficient and have the potential to guide the construction of novel hypotheses on the function of uncharacterised genes.

1. Introduction

The development and reduction in cost of high-throughput technologies in the post-genomic era has made possible genome-wide screening experiments that measure the molecular phenotypes observed in response to single gene alterations, such as deletion, or as a result of an increase in expression of the protein coding sequence [1,2,3,4]. As a consequence, functional omics studies that go beyond the paradigm of functional genomics and that are aimed at investigating genotype–phenotype relations at different omic layers have been carried out. Metabolomic [5,6,7] and ionomic [8,9] profiling have been combined with high-throughput screening of yeast S. cerevisiae single-gene deletion libraries to quantify elements and amino acids (essential building blocks of the cell whose concentration is highly informative on the physiological state in response to genetic perturbations), and to subsequently assess correlations between different mutants on the basis of the measured profiles of biological features. Because these molecular or sub-molecular signatures can be mapped and associated to a consistent region of the genome, statistical inference techniques are often applied to extract genetic networks from correlations that will reflect the interplay between gene function, molecular signatures, and environmental factors. Among other methods such as Bayesian networks [10,11] and Gaussian graphical models [12,13,14] reconstruction, relevance network inference [15,16,17,18] is the most used technique, and it is based on the idea of enforcing a pairwise similarity/distance measure between genes in order to extract via similarity-based thresholding a network of genetic associations on which the modern tools and analysis techniques of network theory [19,20,21] can be applied to reveal functional patterns at a system level. To quantify similarities, the Pearson’s correlation coefficient [22,23] is commonly adopted, regardless of the assumptions that the data have to satisfy for it to be used (such as approximate normality and the absence of outliers). This is a practice that is justified by a consensus that its drawbacks are mainly theoretical and that it turns out to be useful in practice in many applications and analyses of real-world data sets [24,25,26,27]. However, comprehensive functional omic approaches that target and simultaneously measure a substantial fraction of the intracellular elements (ionomics), or a complete class of metabolites, (e.g., amino acids) [7], provide a number of features M profiled for each mutant that is much smaller than the total number N of mutants in the library (N order of the genome size), mathematically $M<<N$. In this paper, we discuss the potential pitfalls of using the Pearson correlation coefficient in this short-profile omics scenario. We propose two extensions of the cosine similarity, namely the Mahalanobis cosine similarity and the hybrid Mahalanobis cosine similarity that appear more suitable for quantifying phenotype similarities between deletion mutants in the applications described here. Starting from theoretical considerations on the characteristic structure of short-profile omic data, we develop and apply a methodology to quantify advantages these measures may have in the task of extracting biologically meaningful genetic networks. We do this by considering three experimental benchmark data sets of the ionome and the metabolome of the model organism S. cerevisiae and several large curated databases as ground truth for genetic annotations. Our testing procedure for the similarity measures evaluates two fundamental aspects of the process of information extraction that accompanies the inference of a genetic network, namely the performance in encoding relevant biological relationships in the resulting network topology, and the ability to capture potential new biological information when integrating networks from different omic layers.

2. Materials and Methods

2.1. Similarity Measures for Short Omic Profiles

Let Z be an $N\times M$ matrix describing N genome-scale observations of a set of M biological signatures (features)—for example, the intracellular concentration of M amino acids in N different mutant strains of S.cerevisiae in which a single open reading frame (ORF) has been deleted, so that $N>>M$. Let’s assume that these features are standardised, meaning that for each amino acid j, the distribution of intracellular concentrations observed across the genome has zero-mean and unit variance:

$$\begin{array}{cc}\hfill {\mu }_j& =\frac{{\sum }_{i=1}^N{z}_{ij}}{N}=0\forall j,\hfill \\ \hfill {\sigma }_j& =\sqrt{\frac{{\sum }_{i=1}^N{(z_{ij}-{\mu }_j)}^2}{N-1}}=1\forall j.\hfill \end{array}$$

For each molecular signature f, we can define a genome-wide feature vector $\mathbf{v}$ (columns of the matrix Z):

$${\mathbf{v}}^f=(z_{1f},\dots ,z_{Nf}),$$

(1)

and since each mutant strain is associated to a specific gene g in the genome, we can write for each gene its associated profile (rows of Z):

$${\mathbf{z}}^g=(z_{g1},\dots ,z_{gM}).$$

(2)

The distribution of the intracellular concentration values of amino acids and other metabolism-related biomolecules measured across a collection of all possible specific mutants (single-gene knockout or overexpression) is usually centred around a typical value corresponding to the wild type phenotype (most of the mutations are neutral and produce a phenotype consistent with the one of the wild type), and exhibits long tails due to high/low concentration values associated to those minority of mutations that alter the cell function and produce a phenotype detected at the level of the molecular profile (examples of these distributions are shown in Figure A1, Figure A2 and Figure A3). Moreover the concentrations of these biomolecules are likely to vary in a correlated fashion because they are regulated by the network of metabolic reactions and functional processes within the cell. Therefore, we assume for our omic data set that the standardised feature vectors ${\mathbf{v}}^f$ are skewed ($H1$) and that some of them can be highly correlated ($H2$):

$$H1:|\frac{\frac{1}{N}{\sum }_{i=1}^N{(v_i^f-{\overline{\mathbf{v}}}^f)}^3}{{\left[\frac{1}{N-1}{\sum }_{i=1}^N{(v_i^f-{\overline{\mathbf{v}}}^f)}^2\right]}^{\frac{3}{2}}}|>>0\forall f,$$

(3)

and

$$H2:corr({\mathbf{v}}^f,{\mathbf{v}}^{f^{\prime }})=\left|\frac{({\mathbf{v}}^f-{\overline{\mathbf{v}}}^f)({\mathbf{v}}^{f^{\prime }}-{\overline{\mathbf{v}}}^{f^{\prime }})}{\sqrt{{({\mathbf{v}}^f-{\overline{\mathbf{v}}}^f)}^2}\sqrt{{({\mathbf{v}}^{f^{\prime }}-{\overline{\mathbf{v}}}^{f^{\prime }})}^2}}\right|>>0,\mathrm{for}\mathrm{some}f,f^{\prime }.$$

(4)

A way to put hypothesis $H2$ in practical terms is to say that, when measuring from the data the feature–feature covariance matrix C, we expect to find a substantial fraction of the matrix elements to be significantly larger than zero in absolute value; therefore, the features reveal a significant and extended pattern of pairwise correlations (examples of these patterns can be seen is Figure 1).

Taking into account these factors, each element of a generic profile ${\mathbf{z}}^g$, corresponding to the concentration value of a specific amino acid, is associated with a distinct, characteristic non-Gaussian, skewed empirical distribution, and the values in the profile array are correlated. It follows that the average value of the profile ${\overline{\mathbf{z}}}^g$ is likely to be substantially different from zero for a consistent fraction of mutants showing a phenotype (see Appendix C for a systematic analysis of the average profile value on simulated short-profile omic data). Therefore, if our goal is to find a reliable pairwise similarity score between mutant profiles to infer genetic associations from the data, the Pearson correlation coefficient ($PCC$) [22,23] might not be the best choice. For two genetic profiles $({\mathbf{z}}^g,{\mathbf{z}}^{g^{\prime }})$ the similarity in terms of $PCC$ is given by:

$$PCC({\mathbf{z}}^g,{\mathbf{z}}^{g^{\prime }})=\frac{{\sum }_{j=1}^M(z_{gj}-{\overline{\mathbf{z}}}^g)(z_{g^{\prime }j}-{\overline{\mathbf{z}}}^{g^{\prime }})}{\sqrt{{\sum }_{j=1}^M{(z_{gj}-{\overline{\mathbf{z}}}^g)}^2}\sqrt{{\sum }_{j=1}^M{(z_{g^{\prime }j}-{\overline{\mathbf{z}}}^{g^{\prime }})}^2}}.$$

(5)

$PCC$ is intrinsically multivariate (because of the arithmetic mean ${\overline{\mathbf{z}}}^g$), and it should be applied under the assumption that the vectors ${\mathbf{z}}^g$ contains uncorrelated values that are consistent with the same marginal Gaussian distribution. When we drop the multivariate terms, we have the cosine similarity, defined as:

$$cos({\mathbf{z}}^g,{\mathbf{z}}^{g^{\prime }})=\frac{{\sum }_{j=1}^M{z}_{gj}{z}_{g^{\prime }j}}{\sqrt{{\sum }_{j=1}^M{z}_{gj}^2}\sqrt{{\sum }_{j=1}^M{z}_{g^{\prime }j}^2}}.$$

(6)

If we see $\mathbf{z}$ as a generic vector in an M-dimensional space, then $cos({\mathbf{z}}^g,{\mathbf{z}}^{g^{\prime }})$ represents the angle between the two vectors ${\mathbf{z}}^g$ and ${\mathbf{z}}^{g^{\prime }}$. If our features f were Gaussian-like, standardised, and uncorrelated, then the values of the gene profiles would also be normally distributed, and in that scenario, for most of the genes, we would have ${\overline{\mathbf{z}}}^g\simeq 0$ and $PCC({\mathbf{z}}^g,{\mathbf{z}}^{g^{\prime }})\simeq cos({\mathbf{z}}^g,{\mathbf{z}}^{g^{\prime }})$; therefore, there would be no reason to prefer the cosine over the $PCC$ and vice versa (see Figure A4 and Figure A6 for a comparison between $cos$- and $PCC$-generated similarity scores from real short-profile omic data sets and from synthetic short-profile omic data).

The cosine similarity formula contains the Euclidean norm in the denominator; indeed, all Euclidean vector spaces are inner product spaces in which the notion of angle between two generic vectors is well defined (for more details, see Appendix A). Now we consider the following metric, the Mahalanobis distance [28], defined as:

$$d({\mathbf{z}}^g,{\mathbf{z}}^{g^{\prime }})=\sqrt{\left({\left({\mathbf{z}}^g\right)}^T{C}^{-1}{\mathbf{z}}^{g^{\prime }}\right)}$$

(7)

where C is the feature–feature covariance matrix estimated from the data. Since any covariance matrix is always semi-positive definite, it is easy to show (see Appendix A) that the Mahalanobis metric induces an inner product space and that it is perfectly legitimate to write the following Mahalanobis cosine similarity:

$$cosM({\mathbf{z}}^g,{\mathbf{z}}^{g^{\prime }})=\frac{{\sum }_{j=1}^M{z}_{gj}^{\prime }{z}_{g^{\prime }j}^{\prime }{\lambda }_j^{-1}}{\sqrt{{\sum }_{j=1}^M{z_{gj}^{\prime }}^2{\lambda }_j^{-1}}\sqrt{{\sum }_{j=1}^M{z_{g^{\prime }j}^{\prime }}^2{\lambda }_j^{-1}}}$$

(8)

where $z_{gj}^{\prime }$ are the elements of the profile vector ${\mathbf{z}}^g$ in the base where the covariance matrix is diagonal with eigenvalues ${\lambda }_j$. We found one application of the Mahalanobis cosine in computer vision (face recognition) [29] and, to our knowledge, this measure has never been applied in the field of bioinformatics or computational biology before. In this paper, we show that it is a suitable and powerful measure to extract genetic association networks from short-profile omic data sets. As an analogue to the pseudo-cosine [30], we also introduce the hybrid Mahalanobis cosine similarity, defined as:

$$cosH({\mathbf{z}}^g,{\mathbf{z}}^{g^{\prime }})=\frac{{\sum }_{j=1}^M{z}_{gj}^{\prime }{z}_{g^{\prime }j}^{\prime }}{\sqrt{{\sum }_{j=1}^M{z_{gj}^{\prime }}^2{\lambda }_j^{-1}}\sqrt{{\sum }_{j=1}^M{z_{g^{\prime }j}^{\prime }}^2{\lambda }_j^{-1}}}.$$

(9)

The $cosH$ is not a proper cosine function because the inner product in the formula corresponds to the Euclidean dot product while the norms are computed with the Mahalanobis norm. Therefore, in principle, $cosH$ is not bounded between −1 and 1. However, for practical applications, it is always possible to rebound the scores in the range [−1, 1] by dividing all the scores extracted from a given data set by the largest absolute score value. Equations (8) and (9) show how the two proposed similarity measures enforce information from the empirical feature–feature covariance matrix extracted from the entire set of data (in the $N>>M$ framework, the estimation of this matrix is accurate). When computing each pairwise similarity score between profiles, these measures take into account the geometry of the cloud of data points in the M-dimensional feature space (the covariance eigenvalues can be seen to describe a hyper-ellipse with axes of lengths $\left\{{\lambda }_i\right\}$), and, in analogy with the general relativity formalism [28], they dilate distances by the factors ${\lambda }_j^{-1}$ so as to penalise the directions in the feature space along which the covariance is low and the data points are less scattered.

2.2. Omic Benchmark Data Sets

Throughout the paper, we evaluate and compare the performance of $PCC$, $cos$, $cosM$, and $cosH$ in inferring genetic relevance networks from short-profile omic data sets. In particular, we focus on testing these measures in two fundamental tasks of omics data analysis, namely the retrieval of known biological information, and the detection of potential undiscovered functional associations between genes. To do that, we consider here three experimental benchmark data sets:

(1)

Yeast ionome ko (non-essential ORF knock-out mutants, iHUB [31,32,33]).

The yeast ionome knock-out (ko) data set contains population-average intracellular concentrations of 14 different elements (Ca, Cd, Co, Cu, Fe, K, Mg, Mn, Mo, Na, Ni, P, S, and Zn) quantified by means of inductively coupled plasma–mass spectrometry (ICP–MS) for a library of 4944 S. cerevisiae haploid mutant strains having a single non-essential open reading frame knocked out.This data set includes a total of 26,976 samples measured in 305 different plates. Most of the strains were measured in replicates of four (4207), 684 strains in replicates of eight, 48 strains in replicates of 12, and two strains in replicates of 16. Additionally, three control strains present in multiple trays were included in our analysis: YDL227C, 1620 replicates; YLR396C, 1224 replicates; YPR065W, 1224 replicates.

(2)

Yeast ionome oe (overexpression mutants, iHUB [31,32,33]).

The yeast ionome overexpression (oe) data set contains population-average intracellular concentrations of 17 different elements (As, Ca, Cd, Cl, Co, Cu, Fe, K, Mg, Mn, Mo, Na, Ni, P, S, Se, and Zn) quantified by means of inductively coupled plasma–mass spectrometry (ICP–MS) for a library of 5718 S. cerevisiae haploid mutant strains having a single essential or non-essential open reading frame overexpressed. This data set includes a total of 24,060 samples measured in 310 different plates. Most of the strains were measured in replicates of four (5426), 287 strains in replicates of eight, and five strains in replicates of 12.

(3)

Yeast metabolome aa (amino acid profile of non-essential ORF knock-out mutants [7]).

The yeast metabolome (aa) data set contains population-average intracellular concentrations of 19 different amino acids (A, D, E, F, G, H, I, K, L, M, N, P, Q, R, S, T, V, W, Y) quantified by means of liquid chromatography–mass spectrometry (LC–MS) for a library of 4475 S. cerevisiae haploid mutant strains having a single non-essential open reading frame knocked out. This data set includes a total of 4653 samples measured in 12 different batches. Most of the strains are associated with a single sample (4324), 128 strains have two replicates, 19 strains have three replicates, and four strains have four replicates.

Information and details concerning these data sets, including quality control analyses, can be found in [7,33,34]. To allow for an unbiased comparison of the different measures performance across the different data sets, all data are processed using the same pipeline starting from the raw measured concentration values. The pipeline corrects the raw data for batch effects and extracts a characteristic concentration profile for each strain by sequentially applying the following operations: (i) log-transformation of the data, (ii) median plate normalisation, (iii) outlier detection and removal, (iv) extraction of a median profile from the replicates (where replicates are available) for each strain, and (v) standardisation of the concentration values (a detailed description of the pipeline can be found in Appendix B). For each data set, the final mutant-related feature profiles show how many standard deviations the concentrations deviate from the median concentration measured across all the strains in that data set. These data sets have been obtained using a similar experimental design, and they also present the general characteristic of short-profile omic data sets discussed in Section 2.1: in

Table 1. Characteristics of the benchmark data sets considered in the study (after processing raw data). AAFS: average absolute features’ skewness. $M/N$: number-of-features over the number-of-genes ratio.

	AAFS	$\mathit{M}/\mathit{N}$
Ionome (ko)	2	0.003
Ionome (oe)	1.45	0.003
Metabolome (aa)	1.15	0.004

, we report the average absolute feature skewness (AAFS) and the number-of-features over number-of-genes ratio $M/N$ (after data processing); in Figure 1, we show the patterns of feature–feature correlations extracted from each experimental data set using the Pearson correlation coefficient (note that in the the framework $N>>M$, the feature–feature covariance matrix can be reliably estimated from the empirical correlation matrix).

2.3. Genetic Networks Inference

The statistic of pairwise similarity score values depends in general on the specific similarity measure used to calculate the scores. In order to robustly compare the topology of the genetic relevance networks obtained with the different similarity measures of interest ($PCC$, $cos$, $cosM$, $cosH$) we adopt a rank-based thresholding of the scores that depends in turn on the more robust rank statistic of the scores. Our general inference methodology consists of the following steps:

• Score computation: we compute all pairwise similarity scores between the N genes of a given data set with each similarity measure of interest.
• Score ranking: we rank in descending value order each set of scores computed with a different similarity measure.
• Relevance network extraction: we retain the top-n ranked scores in each set to define for each similarity measure a genetic association network of N nodes and n links; the links correspond to the n highest values in the score rank statistic.

This procedure allows for a robust comparative analysis of the genetic networks inferred using the different measures at any fixed value of the relevance threshold n. In Figure 2, we show the topology of the genetic relevance networks inferred from the ionome (ko) data set for different values of the relevance threshold n when using $PCC$, $cosM$, and $cosH$. To improve visualisation, we plot only the connected subgraphs with a size larger than or equal to five nodes. An analysis of the false positive rate of associations for the inferred relevance networks is reported in Appendix D.

2.4. Multiplex Integration of Genetic Networks

Once we have defined the procedure to extract genetic networks from a single omic data set, we can integrate, within the multilayer network paradigm, the genetic associations from all three benchmark data sets. In particular, our integration methodology will focus on constructing multiplex networks [35,36,37,38]. For each of the similarity measures of interest, we will take the set of all genes profiled in at least one data set and we will construct a multiplex network of three layers, each containing the single network inferred from one of the data sets by retaining the top 100,000 most relevant associations.

A multiplex network $\mathbf{G}$ is mathematically defined by a set of nodes V and K single-network layers $G^{\alpha }=(V,E^{\alpha })$, with $\alpha =1,2,\dots ,K$ and $E_{\alpha }$ indicating the set of links in layer $\alpha$ (see Figure 3). The multiplex network framework allows us to efficiently encode associations between genes for which there is omic-specific evidence or evidence across multiple omic layers by means of the multilink concept [38,39,40]. A multilink that connects a pair of nodes $(i,j)$ is defined by a vector ${\overrightarrow{m}}^{ij}=(m_1^{ij},m_2^{ij},\dots ,m_K^{ij})$ which specifies all the layers $\alpha$ where those nodes are connected ($m_{\alpha }^{ij}=1$), and where they are not ($m_{\alpha }^{ij}=0$). The multilinks can characterise all the different ways in which a pair of nodes in the multiplex can be connected across the K layers. For example ${\overrightarrow{m}}^{ij}=(1,0,0)$ indicates that node i and j are connected in layer 1 only, while in the case ${\overrightarrow{m}}^{ij}=(1,0,1)$, they are connected in both layer 1 and 3. It is then possible to define the multidegree $k_i^{\overrightarrow{m}}$ of a generic node i as the number of neighbours that are connected via multilinks of type $\overrightarrow{m}$.

3. Results

The process of inference of a genetic network from a given data set while gradually increasing the relevance threshold (Figure 2) can be regarded as a reverse percolation process on the network, in which edges are sequentially added one at a time between the same set of N vertices [41,42]. This fact allows us to obtain insights about the inference power and performance of the similarity measures based on network theory considerations. Our first analysis focuses on three network descriptors, namely (i) the number of connected components—that is, the number of connected subgraphs in the network; (ii) the size of the largest component (LC)—that is, the number of nodes present in the largest connected component of the graph; and (iii) the average local clustering coefficient [43,44] of the largest component—that is, the average fraction of triangles over triplets closed by a node in the LC. The clustering coefficient measures how well the nodes in a graph tend to cluster together, and it works as an indicator for the network modularity [45,46,47]. In Figure 4, we plot the number of connected components, the size of the LC, and the average clustering coefficient of the LC measured from the inferred networks in the top n genetic associations ranked according to the different measures considered. Interestingly, the curves reveal that:

• By increasing the relevance threshold n and considering more and more links in the inferred networks, $cosM$ tends to generate faster than $PCC$ and than $cos$ a giant component whose node size is a finite fraction of the genome size N (yellow curves in top and middle panels, the number of connected components goes to one, and the size of the LC increases asymptotically to N), with a relatively (comparable to $PCC$ and $cos$) low level of clustering of the nodes (bottom panels).
• On the other hand, $cosH$ tends to aggregate genes in few connected components or modules (purple curves, top panels) which grow in parallel with relatively comparable sizes (middle panels, the size of the LC is far below N), and are highly clustered (purple marks in bottom panels, high clustering coefficient).

This analysis suggests that the measures that enforce information from the feature covariance matrix, namely $cosM$ and $cosH$, represent two alternative ways of operating at the genome scale: given a certain number n of reliable associations, $cosM$ is able to infer relations that involve a larger set of genes, thus retrieving faster biological information at the global genetic network scale, while $cosH$ can best retrieve information at the local genetic network scale by selecting associations within several isolated clusters of genes.

3.1. Retrieval of Known Biological Information

The above considerations are based on the assumption that the links inferred contain relevant biological information. We therefore designed a testing regime that quantifies the performance of the proposed measures in the task of retrieving previously known biological information. By assuming that a certain amount of known biological information is contained in each of the benchmark data sets, we want to quantify which similarity measure can better recall that information. We consider separately five curated databases as ground-truth for genetic associations:

(1)

Protein–Protein Interactions (PPI) from STRING [48] (v11, database scores only).

(2)

BIOGRID [49] (v3.4.158, genetic interactions only).

(3)

Protein Complex Consensus [50] (co-occurrence in protein complexes).

(4)

KEGG Biological Pathways [51] (Release 90.1, co-occurrence in metabolic pathways).

(5)

Yeast Net Gold Standard [52] (v3, Gene Ontology based gold-standard associations).

For each of the benchmark data sets, we compute the percentage of corresponding ground-truth links that are found in the associated networks (true positive rate, or recall) extracted at different values of the relevance threshold n, and compare those numbers across the similarity measures of interest used in the inference process. Of note, this approach is conceptually different from a receiver operating characteristic curve analysis (a study of the true positive associations rate against false negative associations rate in function of the threshold n), that would evaluate the diagnostic ability of each omic data set in recovering each specific type of biological associations alone. For any set of ground-truth associations (single database), such analysis would disregard any amount of potential new information extracted from a given data set (which is highly dependent on the specific type of omic data considered and on the associated experimental conditions—for example, the type of media used to grow the yeast mutants) as well as information that is relative to other databases.

Results are shown in Figure 5. For the top n scores values considered, $n\in [1000,10,000,100,000]$, the true positive rate (TPR) of retrieved associations is reported. The red stars mark the measures that performed best with at least a 10% gain in performance with respect to the second best (that is, the difference in TPR value between the best measure and second best measure divided by the second best TPR). In

Table 2. Best performing similarity measures and associated gain in performance in the task of recalling associations from protein–protein interactions (PPI), co-occurrence in protein complexes, co-occurrence in metabolic pathways (KEGG), genetic interactions (BIOGRID), and associations based on GO Ontology Terms (YeastNet GS).The gain in performance is defined as the difference in percentage between the best and the second best true positive rate obtained with the measures under study on a data set.

			Ionome (ko)
	Top 1000 Scores		Top 10,000 Scores		Top 100,000 Scores
	Best Performance	Gain	Best Performance	Gain	Best Performance	Gain
PPI	cosH	18.1%	PCC	7.7%	PCC	2.9%
Protein Complexes	cosM	100%	cosM	12.5%	cosH	2.1%
KEGG Pathways	cosH	140.8%	cosH	39.2%	cosH	1%
BIOGRID	cosM	50%	cosM	3%	cosM	5.9%
YeastNet GS	cos	100%	cosH	21.2%	PCC	3.7%
			Ionome (oe)
	Top 1000 Scores		Top 10,000 Scores		Top 100,000 Scores
	Best Performance	Gain	Best Performance	Gain	Best Performance	Gain
PPI	cosM	60%	PCC	16%	PCC	2.9%
Protein Complexes	cosH	inf	cosM	50%	cosH	2.1%
KEGG Pathways	cosM	7.7%	cosM	3.5%	cosH	1%
BIOGRID	cosH	7.1%	cosM	2.5%	cosM	5.9%
YeastNet GS	cos	14.3%	PCC	2%	PCC	3.7%
			Metabolome (aa)
	Top 1000 Scores		Top 10,000 Scores		Top 100,000 Scores
	Best Performance	Gain	Best Performance	Gain	Best Performance	Gain
PPI	cosM	146.7%	cosM	91.1%	cosM	41.8%
Protein Complexes	cosM	200%	cosM	80%	cosM	35.1%
KEGG Pathways	cosM	44%	cosM	8.6%	cosM	4.8%
BIOGRID	PCC	0%	cos	0.7%	cos	1.7%
YeastNet GS	cosH	16.7%	cos	8%	cosH	1.8%

, we report details of the best recall performance with its associated performance gain. Overall, the true positive rates are very low, which is expected because it is hard to reconstruct the exact ground truth networks, especially using 1000 or 10,000 top scores; however, the estimated false positive rates of associations are at least an order of magnitude smaller (see Appendix D). In the case of the metabolome (aa), which revealed the most extended and prominent correlation pattern across the set of features (Figure 1), $cosM$ appeared to consistently outperform the other metrics in recalling protein–protein interactions, protein complexes associations, and biological pathways co-occurrences (links between genes annotated to the same metabolic pathway). In the case of the ionome data sets, in which the level of correlation between features was found to be lower, we observed that for the top 100,000 associations considered, the recalling of the similarity measures is comparable; however, when considering lower relevance thresholds (1000 and 10,000 associations), $cosM$ and $cosH$ outperformed the other measures in several cases. For the ionome (ko), $cosM$ and $cosH$ consistently performed best in recalling information about protein complexes and biological pathways, respectively. In particular, when only a few highly reliable associations were considered (top 1000 scores), the gain in performance of $cosH$ and $cosM$ was often extremely high (see Table 2). The Pearson correlation coefficient appeared to perform effectively better only in one single case (PPI, top 10,000 scores).

3.2. The Ionome–Metabolome Multiplex Genetic Network of the Yeast S. cerevisiae

When we constructed the multiplex network from the three single-network layers inferred from each data set using the top 100,000 significant associations, we observed that the multilink statistics did not differ in terms of orders of magnitude across the similarity measures considered (Figure 6A), with the exception of the multilinks of the type ionome.oe-metabolome.aa for which $cosM$ provided the largest number of links (2537) and $cosH$ the lowest (685). Each measure, however, had a substantial fraction of characteristic multilinks that did not overlap with the other measures (Figure 6B). The lowest specificity was observed for $PCC$ with only 26.8% of characteristic ionome.ko-metabolome.aa multilinks, while the highest specificity was observed with $cosM$ (respectively 92.9% of ionome.ko-ionome.oe-metabolome.aa multilinks and 77.8% of ionome.oe-metabolome.aa multilinks that are not in common with other measures). Furthermore, we checked the content of relevant biological information of non-layer-specific (NLS) multilinks by considering a subset of GOSlim Biological Process Ontology terms from the SGD database [53] that are related to metabolic processes and cellular ion homeostasis maintenance [34,54]. When we looked at the four subnetworks defined by each specific class of NLS multilinks, we found that those constructed with $cosM$ included the largest number of genes annotated to the ontology categories considered (Figure 6C). $cosM$ also yielded the largest number of ionome.ko-metabolome.aa multilinks between the annotated genes considered, while for the two multilinks classes ionome.oe-metabolome.aa and ionome.ko-ionome.oe, $cosH$ revealed the largest number of connections among the annotated genes (Figure 6A).

These results obtained in the multiplex framework were consistent with the previous results and supported the observation that $cosM$ best captures global information at the genome scale by spanning in its NLS subnetworks the largest number of annotated genes, while $cosH$ could reveal larger, well-clustered neighbourhoods of fewer annotated genes, thus better characterising them at a local network scale. The proposed measures also revealed alternative relations between different biological processes compared to the Pearson’s correlation coefficient. In Figure 7, the flow of multilinks between the different ontology classes selected for the ionome.oe-metabolome.aa $cosH$ subnetwork and for the ionome.ko-metabolome.aa$cosM$ subnetwork is depicted, together with the analogous $PCC$ flows. The large number of genetic associations belonging to these two NLS multilink types (see Figure 6A) also reflected a richer scenario in terms of the flow of multilinks between biological processes. $cosM$ revealed, for example, associations between ion transport genes and genes annotated to endosomal transport, nucleobase-containing compound transport, and vacuole organization, while $cosH$ revealed associations between Golgi vesicle transport genes and several ontologies including Golgi vesicle transport, endosomal transport, ion transport, mitochondrion organization, nucleobase-containing compound transport, transmembrane transport, the cellular amino acid metabolic process, generation of precursor metabolites and energy, and the lipid metabolic process that were not captured by the $PCC$.

Figure 8A provides a clear example of a gene, LST7, whose functional role falls at the interface between metabolic processes and ion homeostasis, that is highlighted by the multilink flow analysis. The ionome.ko-metabolome.aa neighbourhood of LST7 spanned by $cosM$ (Figure 8A, left) contains GTR1 and MHE1 (EGO1). This triad is known to be involved in the regulation of TOR signalling; indeed, Gtr1 and its membrane-tethering subunits Ego1 are part of the EGO complex that binds the vacuolar membrane and subunits of TORC1. The Lst4-Lst7 complex has been recently shown [55] to attach to the vacuolar membrane next to the GTPase complex Gtr1-Gtr2 under amino acid starvation, and to transiently interact with Gtr1-Gtr2, thereby entailing TORC1 activation and Lst4-Lst7 release from the membrane, under subsequent amino acids stimulation (for example, glutamine). This machinery is reflected by the significantly altered amino acid profiles of the genes (Figure 8A, bottom-right) with an overabundance of several species, including glutamine, and none of the ions overrepresented (Figure 8A, top-right), suggesting that the loss of function of these genes induces in the cell a state of stationary amino acids stimulation. The ion profiles also shows a significant decrease in sodium concentration in the cell that reflects an interplay between the amino acid induced signalling of TORC1 and pH homeostasis: physical interactions from high-throughput data of the protein-fragment complementation assay [56] indeed suggest association of Ego1 and Gtr1 with Vma8, the D subunit of the V1 peripheral membrane domain of the vacuolar ATPase (V-ATPase) that is essential for maintaining ion homeostasis. In humans, Gtr1 signalling has also been shown to depend on interactions with the vacuolar V-ATPase [57,58]).

Finally, in order to illustrate how enforcing these measures can help construct novel hypotheses on the function of uncharacterised genes, we selected examples of genes from the characteristic flows discussed above that contained in their network neighbourhood genes of unknown function (Figure 8B–D). The $cosM$ ionome.ko-metabolome.aa neighbourhood of the plasma membrane transport gene QDR2 (Figure 8B, left) suggests an association with the plasma membrane gene SYG1 of unknown function which is also connected with the uncharacterised membrane protein YIL054W. Interestingly, the knockout ionome profiles of QDR2 (which is known to contribute to potassium homeostasis and whose expression is regulated by copper), SYG1, and YIL054W display evident signatures including a characteristic decrease in cadmium and an increase in sulphur, and variable levels of accumulation of manganese and sodium and decumulation of molybdenum. The metabolic profiles show a general overabundance of all amino acid species including leucine and lysine. When we performed transcription factor enrichment analysis (YeastEnricher online tool [59]), we consistently found significant regulation by CAD1, a cadmium resistance gene, involved in stress responses and iron metabolism (adjusted p-value 0.04956) and LEU3, involved in amino acids biosynthesis, (adjusted p-value 0.04956). The QDR2 neighbourhood also contains KGD1, which is associated with LEU3 and annotated to the lysine degradation pathway (KEGG 2018), and FKH1, which is annotated to DNA-binding transcription activator activity, RNA polymerase II-specific together with CAD1 and LEU3, suggesting that SYG1 and YIL054W might take part in membrane transport processes involving QDR2, that are orchestrated by LEU3 and CAD1 according to cellular amino acids and cations availability.

Another interesting example is gene TPI1 (Figure 8C), a glycolytic enzyme which contains in its $cosM$ multiplex neighbourhood the uncharacterised proteins YDR514C and YMR102C and the transport gene ATP19, subunit of the the mitochondrial F1F0 ATP synthase. The mRNA half-life of TPI1 is regulated by iron availability which is decreased in the ion knock-out profiles of the strains while molybdenum is variably decumulated. This behaviour is characteristic of mRNA sequestrated by P-bodies (processing bodies), which are sites where, under stress conditions, nontranslating mRNA is degraded or stored to return back to translated when the cell enters stress recovery [60]. TPI1 has been experimentally observed to localise in P-bodies under glucose depletion in S. cerevisiae via chemical cross-linking coupled to affinity purification (cCLAP) [61], together with other subunits of the F1F0 ATP synthase complex (ATP11, ATP14, ATP20). YDR514C and YMR102C have been observed to physically interact in S. cerevisiae with Ccr4, the core subunit of the Ccr4-Not complex, which is involved in the regulation of translation and decay of specific mRNAs and is the main cytoplasmic deadenylase in S cerevisiae, and with its associated protein Dhh1 [62]. Ccr4 and Dhh1 associate with mRNAs whose abundance increases during nutrient starvation, and those that fluctuate during metabolic and oxygen consumption cycles. Moreover, there is experimental evidence that YMR102C mRNA is sequestrated by P-bodies under glucose stress, Ca ${}^{2+}$ stress, and Na ${}^{+}$ stress [61]. Given that YDR514C, ATP19, and TPI localise to mitochondria, and that inhibition of TPI1 is known to stimulate the pentose phosphate pathway and to increase antioxidative metabolism [63], the scenario described suggests the idea that YDR514C and YMR102C, together with TPI1, might be activated in the cell to reshape and switch the metabolism in response to ions/nutrients related stress that might cause mitochondrial disruption and aerobic inefficiency. This hypothesis is supported by the ion overexpression profiles of these genes, in which the concentration level of potential toxic species like Cd and As is very low, as well as by experimental evidence that YMR102C is involved in galactose metabolism in the haploid S cerevisiae Cd-resistant strain (EC9-8, that tolerates high levels of cadmium) [64], and significantly overexpressed in the ethanol-tolerant strain Y-50316 [65].

As a last example of hypothesis formulation, we considered the $cosH$ multiplex of the Golgi transport gene GCS1 (Figure 8D, left). We found that HRT3, YLR352W, and ROY1 physically interact with Skp1 to form SCF-ubiquitin ligase complexes similar to F-box proteins, despite the fact that they lack an identifiable F-box domain [66]. Gcs1 contains a ArfGAP1 lipid packing sensor (ALPS) motif that binds to lipid membranes to recruit coat complexes whose role is to generate carrier vesicles that mediate transport of proteins and lipids between intracellular compartments. This ALPS motif couples the activity of GCS1 with the curvature of lipid membranes and allows GCS1 to control the assembly and the dynamics of the COPI coat complex, analogously to its human homologue Arf1GAP1 [67]. Interestingly, NUP133, which is found in the neighbourhood of GCS1, is also a membrane curvature sensors and it encodes the same ALPS motif in the Nup84p subcomplex of the nuclear pore complex (NCP) [68]. Moreover, it has been shown that the F-box protein Rcy1 is required for recycling of Snc1 to the Golgi, although its precise mechanism is unknown [69]. An intriguing hypothesis suggested by the ionome–metabolome multilayer is that HRT3, ROY1, and YLR352W might take part in machinery controlling recycling of plasma membrane proteins similarly to RCY1. The ability of the Rcy1-Skp1 complex to recycle independently of the cullin subunit makes this hypothesis plausible even in the absence of recognised E3 activity by Hrt3, Ylr352w, and Roy1. These pseudo-F-box-Skp1 complexes could be specifically activated through different stress factors. Indeed, other genes in the subnetwork include general regulators of ion homeostasis (boron efflux transporter BOR1 and the potassium transport system TRK1). NVJ1, which promotes the formation of ER–vacuole junctions that can expand in response to starvation and regulate the production of lipid droplets [70]. TGL3, whose expression is known to be reduced in the absence of lipid droplets. PDR3, a gene of the conserved pleiotropic drug resistance (PDR) pathway, that, together with PDR1, regulates more than half of the known pumps transporting potentially harmful chemicals outside of the cell membrane, is known to be overexpressed in response to the loss of CCW12 (also present in the subnetwork) which is crucial for wall integrity [71]. GLO1 is also of interest because is known to be regulated by osmotic stress and to process glutathione, a sulfur compound that is synthesised in yeast as a cadmium detoxifying agent and whose synthesis has been recently shown to be mediated by an SCF ligase complex (SCF-Met30) [72].

As a general remark, besides the proposed hypotheses, the clusters of genes discussed, which are not present in the corresponding $PCC$ multilayer network (not shown), confirmed that these measures have the potential to reveal aspects of the complex interplay between metabolism, ion homeostasis, and molecular transport, and can be used as novel analytical tools to quantify genetic similarity on the base of the altered levels of nutrients, such as amino acids and ions, that can be used as footprints of the global impact of altered genetic functions (loss or overexpression) in the cell.

4. Discussion

Network science has played a fundamental role in the development of the field of systems biology by providing analytical tools to reveal and characterise protein and genetic interaction maps and the relations between metabolic pathways and functional landscapes of many biological systems [73,74,75]. However, with the advent of the multi-omics era there is recognition that single isolated biological networks are insufficient to describe functional genetic patterns that arise from the multiple levels of complexity of the cell (genome, epigenome, transcriptome, metabolome, proteome, lipidome, ionome) [76]. Network analysis can provide advanced and powerful mathematical frameworks, such as multi-layer networks, to integrate multiple omics data efficiently and in the most intuitive way. In this article, we have focused on the problem of inferring and integrating association networks between genes from omic data sets containing a relatively small number (order $O\left(10\right)$) of biological signatures, profiled for almost all single non-essential gene mutants. These signatures contain comprehensive information on the intracellular concentration of elements or of classes of metabolites, and they present patterns of correlations that reflect those biological and biochemical processes inside the cell in which these concentrations play a fundamental role. The importance of these omic data lies in the fact that the associated studies and methodologies have been proposed as functional omic approaches alternative to the classic functional genomics that can reveal undiscovered relations between genes encoded in the specific omic-related signatures. Extracting informative genetic association networks from these types of short-profile omic data sets is the starting point to apply the tools and algorithms of modern graph theory for revealing new potential functional relations between genes, and from a mathematical perspective this translates into the fundamental issue of assessing reliable similarity scores between the signature vectors. Here, we proposed two pairwise-similarity measures, namely the Mahalanobis cosine, that, to the extent of our knowledge, has never been used before in computational biology, and what we defined as the hybrid Mahalanobis cosine. These two measures can be seen as extensions of the cosine similarity that enforce in different ways additional information from the empirical covariance matrix estimated from the entire set of data under study and can therefore be regarded as providing omic intrinsic or omic adjusted correlations when used to analyse data sets that are at the omic scale. We tested these measures in two fundamental tasks: (1) the inference of genetic relevance networks that can encode in their topology already known biological relationships, and (2) network-based multi-omic integration of short-profile omic data sets, for which multiple evidence of connection across the layers of a multiplex network can indicate potential undiscovered genetic associations. To do that, we developed a methodology that combines extraction and integration of relevance genetic networks based on the robust rank statistic of the similarity scores with cross-referencing from large curated databases of metabolic pathways, genetic interactions, protein–protein interactions, protein complexes co-occurrence, and GO Ontology annotations. We evaluated and compared the performance of the proposed measures against the widely used Pearson correlation coefficient and the standard cosine similarity using three experimental data sets of the ionome and metabolome of the model organism S. cerevisiae. Our construction and analysis of the first ionome–metabolome multiplex genetic network of the yeast S. cerevisiae indicates that the proposed covariance-based similarity measures, when utilised in the tasks of genetic network inference and network-based omics integration, have the potential to capture alternative and/or additional levels of relations between biological pathways and processes, and they can help to elucidate the function of uncharacterised genes through the inferred multilayer network topology.

The pipeline developed to pre-process the data deliberately removed all samples with any outlier or missing value in their concentration profile. Future work will be directed at investigating the robustness of the two measures proposed (Mahalanobis cosine and hybrid Mahalanobis cosine) to the presence of outliers and to missing value imputation. These questions would require, within the short-profile omic data framework, a combined theoretical and computational analysis of the perturbation spectrum of the eigenvalues of the covariance matrix of multivariate distributions with long tails and correlated dimensions.

As a final remark, the similarity measures proposed define corresponding distance measures through the simple transformation $d=(1+s)/2$, therefore they can be straightforwardly used in machine learning applications—for example, implemented as alternative metrics into state-of-the-art dimensionality reduction algorithms, such as t-SNE [77] and UMAP [78], and clustering algorithms, including density-based tools [79,80].