1 Introduction

Among the main challenges in mass spectrometric metabolomic analysis is the high-throughput analysis of metabolic features, their fast detection and annotation. By contrast to the screening of known, previously characterized, metabolic features in these data, the putative annotation of unknown features is often cumbersome and requires a lot of manual work, hindering the biological information retrieval of these data. High-resolution mass spectrometric data is often very rich in information content and metabolic conversions, and reactions can be derived from structural properties of features (Breitling et al. 2006). In addition to that, statistical associations between features (based on their intensity values) can be a valuable resource to find co-synthesized or co-regulated metabolites, which are synthesized in the same biosynthetic pathways. Given that an analysis tool within the R framework is still lacking that is integrating the two features of mass spectrometric information commonly acquired with mass spectrometers (m/z and intensity values), I developed MetNet to close this gap. The MetNet package comprises functionalities to infer network topologies from high-resolution mass spectrometry data. MetNet combines information from both structural data (differences in m/z values of features) and statistical associations (intensity values of features per sample) to propose putative metabolic networks that can be used for further exploration.

The idea of using high-resolution mass spectrometry data for network construction was first proposed in Breitling et al. (2006) and followed soon afterwards by a Cytoscape plugin, MetaNetter (Jourdan et al. 2007), that is based on the inference of metabolic networks on molecular weight differences and correlation (Pearson correlation and partial correlation).

Inspired by the paper of Marbach et al. (2012) different algorithms for network were implemented in MetNet to account for biases that are inherent in these statistical methods, followed by the calculation of a consensus adjacency matrix using the differently computed individual adjacency matrices.

The two main functionalities of the package include the creation of adjacency matrices from structural properties, based on losses/addition of functional groups defined by the user, and statistical associations. Currently, the following statistical models are implemented to infer a statistical adjacency matrix: Least absolute shrinkage and selection operator (LASSO, L1-norm regression, (Tibshirani 1994)), Random Forest (Breiman 2001), Pearson and Spearman correlation (including partial and semipartial correlation, see Steuer (2006) for a discussion on correlation-based metabolic networks), context likelihood of relatedness (CLR, (Faith et al. 2007)), the algorithm for the reconstruction of accurate cellular networks (ARACNE, (Margolin et al. 2006)) and constraint-based structure learning (Bayes, (Scutari 2010)). Since all of these methods have advantages and disadvantages, the user has the possibility to select several of these methods, compute adjacency matrices from these models and create a consensus matrix from the different statistical frameworks.

After creating the statistical and structural adjacency matrices these two matrices can be combined to form a consensus matrix that has information from both structural and statistical properties of the data. This can be followed by network analyses (e.g. calculation of topological parameters), integration with other data sources (e.g. genomic information or transcriptomic data) and/or visualization.

Central to MetNet is the AdjacencyMatrix class, derived from the SummarizedExperiment S4 class. The AdjacencyMatrix host the adjacency matrices creates during the different steps within the assays slot. They will furthermore store information on the type of the AdjacencyMatrix, i.e. if it was derived from structural or statistical properties or if it used the combined information from these layers (combine). It also stores information if the information was thresholded, e.g. by applying the rtCorrection or threshold function. Furthermore, the AdjacencyMatrix object stores information on if the graphs are directed or undirected (within the directed slot).

Questions and bugs

MetNet is currently under active development. If you discover any bugs, typos or develop ideas of improving MetNet feel free to raise an issue via Github or send a mail to the developer.

2 Prepare the environment and load the data

To install MetNet enter the following to the R console

if (!requireNamespace("BiocManager", quietly = TRUE))
    install.packages("BiocManager")

BiocManager::install("MetNet")

Before starting with the analysis, load the MetNet package. This will also load the required packages glmnet, stabs, GENIE3, mpmi, parmigene, Hmisc, ppcor and bnlearn that are needed for functions in the statistical adjacency matrix inference.

library(MetNet)

The data format that is compatible with the MetNet framework is a xcms/CAMERA output-like \(m~\times~n\) matrix, where columns denote the different samples \(n\) and where \(m\) features are present. In such a matrix, information about the masses of the features and quantitative information of the features (intensity or concentration values) are needed. The information about the m/z values has to be stored in a vector of length \(\vert m \vert\) in the column "mz".

MetNet does not impose any requirements for data normalization, filtering, etc. However, the user has to make sure that the data is properly preprocessed. These include division by internal standard, log2 or vsn transformation, noise filtering, removal of features that do not represent mass features/metabolites, removal of isotopes, etc.

We will load here the object x_test that contains m/z values (in the column "mz"), together with the corresponding retention time (in the column "rt") and intensity values. We will use here the object x_test for guidance through the workflow of MetNet.

data("x_test", package = "MetNet")
x_test <- as.matrix(x_test)

3 Creating the structural adjacency

The function structural will create an AdjacencyMatrix object of type structural containing the adjacency matrices based on structural properties (m/z values) of the features.

The function expects a matrix with a column "mz" that contains the mass information of a feature (typically the m/z value). Furthermore, structural takes a data.frame object as argument transformation with the colnames "mass" and additional columns (e.g. "group", "formula" or "rt"). structural looks for transformations (in the sense of additions/losses of functional groups mediated by biochemical, enzymatic reactions) in the data using the mass information.

Following the work of Breitling et al. (2006) and Jourdan et al. (2007), molecular weight difference wX is defined by \(w_X = \vert w_A - w_B \vert\)

where wA is the molecular weight of substrate A, and wB is the molecular weight of product B (typically, m/z values will be used as a proxy for the molecular weight since the molecular weight is not directly derivable from mass spectrometric data). As exemplified in Jourdan et al. (2007), specific enzymatic reactions refer to specific changes in the molecular weight, e.g. carboxylation reactions will result in a mass difference of 43.98983 (molecular weight of CO2) between metabolic features.

The search space for these transformation is adjustable by the transformation argument in structural allowing to look for specific enzymatic transformations. Hereby, structural will take into account the ppm value, to adjust for inaccuracies in m/z values due to technical reasons according to the formula

\[ppm = \frac{m_{exp} - m_{calc}}{m_{exp}} \cdot 10^{-6}\]

with mexp the experimentally determined m/z value and mcalc the calculated accurate mass of a molecule. Within the function, a lower and upper range is calculated depending on the supplied ppm value, differences between the m/z feature values are calculated and matched against the "mass"es of the transformation argument. If any of the additions/losses defined in transformation is found in the data, it will be reported as an (unweighted) connection in the assay "binary" of the returned AdjacencyMatrix object.

Together with this assay, additional character adjacency matrices can be written to the assay slot of the AdjacencyMatri object. E.g. we can write the type of connection/transformation (derived e.g. from the column "group" in the transformation object) as a character matrix to the assay "group" by setting var = "group".

Before creating the structural AdjacencyMatrix, one must define the search space, i.e. the transformation that will be looked for in the mass spectrometric data, by creating here the transformations object.

## define the search space for biochemical transformation 
transformations <- rbind(
    c("Hydroxylation (-H)", "O", 15.9949146221, "-"),
    c("Malonyl group (-H2O)", "C3H2O3", 86.0003939305, "+"),
    c("D-ribose (-H2O) (ribosylation)", "C5H8O4", 132.0422587452, "-"),
    c("C6H10O6", "C6H10O6", 178.0477380536, "-"),
    c("Rhamnose (-H20)", "C6H10O4", 146.057910, "-"),
    c("Monosaccharide (-H2O)", "C6H10O5", 162.0528234315, "-"),
    c("Disaccharide (-H2O) #1", "C12H20O10", 324.105649, "-"),
    c("Disaccharide (-H2O) #2", "C12H20O11", 340.1005614851, "-"),
    c("Trisaccharide (-H2O)", "C18H30O15", 486.1584702945, "-"),
    c("Glucuronic acid (-H2O)", "C6H8O6", 176.0320879894, "?"),
    c("coumaroyl (-H2O)", "C9H6O2", 146.0367794368, "?"),
    c("feruloyl (-H2O)", "C9H6O2OCH2", 176.0473441231, "?"),
    c("sinapoyl (-H2O)", "C9H6O2OCH2OCH2", 206.0579088094, "?"),
    c("putrescine to spermidine (+C3H7N)", "C3H7N", 57.0578492299, "?"))

## convert to data frame
transformations <- data.frame(
    group = transformations[, 1],
    formula = transformations[, 2],
    mass = as.numeric(transformations[, 3]),
    rt = transformations[, 4])

The function structural will then check for those m/z differences that are stored in the column "mass" in the object transformations. To create the AdjacencyMatrix object derived from these structural information we enter

struct_adj <- structural(x = x_test, transformation = transformations, 
    var = c("group", "formula", "mass"), ppm = 10)

in the R console.

As we set var = c("group", "formula", "mass"), the AdjacencyMatrix object will contain the assays "group", "formula", and "mass" that store the character adjacency matrices with the information defined in the columns of transformations.

3.1 Advanced topic: Creating a directed structural graph

By default, the structural AdjacencyMatrix object and the contained adjacency matrices are undirected (the argument in structural is set to directed = FALSE by default; i.e. the matrices are symmetric). MetNet, however, also allows to include the information on the directionality of the transformation (e.g. to distinguish between additions and losses). This behaviour can be specified by setting directed = TRUE:

struct_adj_dir <- structural(x = x_test, transformation = transformations, 
    var = c("group", "formula", "mass"), ppm = 10, directed = TRUE)

In the following we will visualize the results from the undirected and directed structural network.

We will set the mode of the igraph object to "directed" in both cases to make the distinction between the returned outputs of structural for setting directed = FALSE and directed = TRUE. Alternatively, we could also set the mode for the first igraph object (using the undirected output of structural) to "undirected" which results in an igraph object where the directionality of the edges is not retained.

g_undirected <- igraph::graph_from_adjacency_matrix(
    assay(struct_adj, "binary"), mode = "directed", weighted = NULL)
plot(g_undirected, edge.width = 1, edge.arrow.size = 0.5,
     vertex.label.cex = 0.5, edge.color = "grey")

g_directed <- igraph::graph_from_adjacency_matrix(
    assay(struct_adj_dir, "binary"), mode = "directed", weighted = NULL)
plot(g_directed, edge.width = 1, edge.arrow.size = 0.5,
     vertex.label.cex = 0.5, edge.color = "grey")

Advanced topic: Refining the structural adjacency (optional)

The retention time will differ depending on the chemical group added, e.g. an addition of a glycosyl group will usually result in a lower retentiom time in reverse-phase chromatography. This information can be used in refining the adjacency matrix derived from the structural matrix. The rtCorrection function does this check, if the predicted transformations correspond to the expected retention time shift, in an automated fashion. It requires information about the expected retention time shift in the data.frame passed to the transformation argument (in the "rt" column). Within this column, information about retention time shifts is encoded by "-", "+" and "?", which means the feature with higher m/z value has lower, higher or unknown retention time than the feature with the lower m/z value. The values for m/z and retention time will be taken from the object passed to the x argument. In case there is a discrepancy between the transformation and the retention time shift, the adjacency matrix at the specific position will be set to 0. rtCorrection will return the an AdjacencyMatrix object with updated adjacency matrices ("binary" and the additional character adjacecny matrices).

To account for retention time shifts we enter

struct_adj <- rtCorrection(am = struct_adj, x = x_test, 
    transformation = transformations, var = "group")

in the R console. The character "group" defined with var will serve here as the link between the assay "group" and the column in transformation to calculate the retention time discrepancies between feature pairs.

For data analysis a data.frame can be generated from AdjacencyMatrix objects by applying as.data.frame(). Further filtering displays only feature-pairs which were matched to a transformation.

struct_df <- as.data.frame(struct_adj)
struct_df <- struct_df[struct_df$binary == 1, ]

Some overview on the mass-difference distribution of the data can be observed using the mz_summary function. The number of determined mass differences can be displayed by using the mz_vis function.

mz_sum <- mz_summary(struct_adj, var = "group")
mz_vis(mz_sum, var = "group")

For larger data-sets, also a filter can be applied to visualize mass-difference above a defined threshold. A filter can be applied, by filter. Since the maximum count of any mass difference in struct_adj is 4, a filter of 5 results in 0 mass differences.

mz_summary(struct_adj, filter = 4)
##                               group    formula count
## 1                Hydroxylation (-H)          O    15
## 2              Malonyl group (-H2O)     C3H2O3    11
## 3             Monosaccharide (-H2O)    C6H10O5     6
## 4                   Rhamnose (-H20)    C6H10O4     9
## 6                   feruloyl (-H2O) C9H6O2OCH2     4
## 7 putrescine to spermidine (+C3H7N)      C3H7N     4
mz_summary(struct_adj, filter = 5)
##                   group formula count
## 1    Hydroxylation (-H)       O    15
## 2  Malonyl group (-H2O)  C3H2O3    11
## 3 Monosaccharide (-H2O) C6H10O5     6
## 4       Rhamnose (-H20) C6H10O4     9

The AdjacencyMatrix class allows storing further information on the features as putative annotations, database identifier, SMILES, etc. using rowData(). A data.frame containing the same rownames as the test data needs to be provided. The columns can store different information as annotations, identifier, etc. We will load the x_annotation file, which contains an example annotation and other identifier for feature x1856. All the other features contain NAs in corresponding columns.

data("x_annotation", package = "MetNet")
x_annotation <- as.data.frame(x_annotation)

## add annotations to the structural AdjacencyMatrix object
rowData(struct_adj) <- x_annotation

## display annotation for the feature "1856"
rowData(struct_adj)["x1856", ]
## DataFrame with 1 row and 11 columns
##       database_mz  database_identifier chemical_formula                 smiles
##       <character>          <character>      <character>            <character>
## x1856       308.2 N-caffeoylspermidine       C16H25N3O3 C=1(C=C(C(=CC1)O)O)/..
##                        inchi               inchikey metabolite_identification
##                  <character>            <character>               <character>
## x1856 InChI=1S/C16H25N3O3/.. AZSUJBAOTYNFDE-FNORW..                        NA
##       fragmentations modifications      charge    database
##          <character>   <character> <character> <character>
## x1856             NA            NA           1          NA

4 Creating the statistical adjacency

4.1 Creating weighted adjacency matrices using statistical

The function statistical will create an AdjacencyMatrix object of type statistical containing the adjacency matrices based on statistical associations. The function will create
weighted adjacency matrices using the statistical models defined by the model argument. Currently, the following models are available: LASSO (using stabs, (Hofner, Boccuto, and Göker 2015; Thomas et al. 2017)), Random Forest (using GENIE3, CLR, ARACNE (the two latter using the package mpmi to calculate Mutual Information using a nonparametric bias correction by Bias Corrected Mutual Information, and the functions clr and aracne.a from the parmigene package), Pearson and Spearman correlation (based on the psych package), partial and semipartial Pearson and Spearman correlation (using the ppcor package) and score-based structure learning returning the strength of the probabilistic relationships of the arcs of a Bayesian network, as learned from bootstrapped data (using the boot.strength with the Tabu greedy search as default from the bnlearn package (Scutari 2010)).

For further information on the different models take a look on the respective help pages of lasso, randomForest, clr, aracne, correlation and/or bayes. Arguments that are accepted by the respective underlying functions can be passed directly to the statistical function. In addition, arguments that are defined in the functions lasso, randomForest, clr, aracne, correlation and/or bayes can be passed to the functions.

4.2 Creating an unweighted adjacency matrix using threshold

From the statistical AdjacencyMatrix object the function threshold will create an AdjacencyMatrix object with the derived unweighted adjacency matrix from the weighted adjacency matrices unifying the information present from all statistical models. This unweighted adjacency matrix is stored in the assay "consensus".

In the following example, we will create a list of unweighted adjacency matrices using Pearson and Spearman correlation using the intensity values as input data.

x_int <- x_test[, 3:ncol(x_test)] |>
    as.matrix()
stat_adj <- statistical(x_int, model = c("pearson", "spearman"))
## [1] "pearson finished."
## [1] "spearman finished."

The reasoning behind this step is to circumvent disadvantages arising from each model and creating a statistically reliable topology that reflects the actual metabolic relations. threshold returns an unweighted adjacency matrix with connections inferred from the respective models (in the "consensus" assay).

There are four different types implemented how the unweighted adjacency matrix can be created: threshold, top1, top2, mean.

For type = "threshold", threshold values have to be defined in the args argument for the respective statistical model within the list entry filter. Values above or below these thresholds in each respective weighted adjacency matrix will be reported as present (1) or absent (0) in the returned unweighted adjacency matrix.

For the other three types (top1, top2, mean) the ranks per statistical model will be calculated and from each respective link the top1, top2 or mean rank across statistical models will be calculated (cf. (Hase et al. 2013)). The top n unique ranks (defined by the entry n in args) will be returned as links in the unweighted consensus adjacency matrix.

We will create here for all ways the thresholded AdjacencyMatrix objects of type statistical containing the consensus adjacency matrix.

## type = "threshold" 

## the assayNames in stat_adj are used to define the filter criteria
assayNames(stat_adj)
## [1] "pearson_coef"    "pearson_pvalue"  "spearman_coef"   "spearman_pvalue"
## return edges with positive Pearson correlation coefficients > 0.95
args_thr <- list(filter = "pearson_coef > 0.95")

## return edges with positive Spearman correlation coefficients > 0.95
args_thr <- list(filter = "spearman_coef > 0.95")

## return edges with absolute Pearson correlation coefficients > 0.95 and 
## associated p-values <= 0.05
args_thr <- list(filter = "abs(pearson_coef) > 0.95 & pearson_pvalue <= 0.05")

## return edges with absolute Pearson OR Spearman correlation coefficients > 0.95
args_thr <- list(filter = "abs(pearson_coef) > 0.95 | abs(spearman_coef) > 0.95")

## return edges with absolute Pearson AND Spearman correlation coefficients > 0.95
args_thr <- list(filter = "abs(pearson_coef) > 0.95 & abs(spearman_coef) > 0.95")

## pass the filtering criteria to the args argument and set type to "threshold"
stat_adj_thr <- threshold(am = stat_adj, type = "threshold", 
    args = args_thr)

## alternatively, use the types "top1", "top2", "mean"
## retrieve the feature pairs which have the 100 highest coefficients
args_top <- list(n = 100)
## type = "top1" 
stat_adj_top1 <- threshold(am = stat_adj, type = "top1", 
    args = args_top)

## type = "top2"
stat_adj_top2 <- threshold(am = stat_adj, type = "top2", 
    args = args_top)
 
## type = "mean"
stat_adj_mean <- threshold(am = stat_adj, type = "mean", 
    args = args_top)

5 Combining the structural and statistical matrix

After creating the structural and statistical AdjacencyMatrix objects, the two objects are combined. The function combine will combine these two objects and create an AdjacencyMatrix object of type combine. The function accepts the arguments am_structural and am_statistical for the respective AdjacencyMatrix objects. Please note that for am_structural the AdjacencyMatrix obtained via structural or rtCorrection can be used, while for am_statistical the AdjacencyMatrix from threshold has to be used. The edges that are present both in the binary assay and the consensus assay will be reported within the combine_binary assay in a combine AdjacencyMatrix object. If there are additional assays in the AdjacencyMatrix of type structural these matrices will be stored in the AdjacencyMatrix of type combine and will contain the edges that are supported by the combine_binary adjacency matrix (the other edges are set to "").

We will use here the thresholded statistical AdjacencyMatrix from type = "mean" to combine it with the structural AdjacencyMatrix, stat_adj:

comb_adj <- combine(am_structural = struct_adj, am_statistical = stat_adj_mean)

6 Visualization and further analyses

To display the created consensus adjacency matrix, existing visualization tools available in the R framework can be employed or any other visualization tool after exporting the consensus matrix as a text file. In this example, we will use the igraph (Csardi and Nepusz 2006) package to visualize the adjacency matrix.

We use here the assay "combine_binary" from the AdjacencyMatrix of type combine and pass it to the graph_from_adjacency_matrix function:

adj <- assay(comb_adj, "combine_binary")
g <- igraph::graph_from_adjacency_matrix(adj, mode = "undirected")
plot(g, edge.width = 2, vertex.label.cex = 0.5, edge.color = "grey")
_Ab initio_ network inferred from structural and quantitative mass spectrometry data. Vertices are connected that are separated by given metabolic transformation and statistical association.

Figure 1: Ab initio network inferred from structural and quantitative mass spectrometry data
Vertices are connected that are separated by given metabolic transformation and statistical association.

Furthermore, the network can be analysed by network analysis techniques (topological parameters such as centrality, degree, clustering indices) that are implemented in different packages in R (e.g. igraph or sna) or other software tools outside of the R environment.

Appendix

Session information

All software and respective versions to build this vignette are listed here:

## R version 4.1.3 (2022-03-10)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 20.04.4 LTS
## 
## Matrix products: default
## BLAS:   /home/biocbuild/bbs-3.14-bioc/R/lib/libRblas.so
## LAPACK: /home/biocbuild/bbs-3.14-bioc/R/lib/libRlapack.so
## 
## locale:
##  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
##  [3] LC_TIME=en_GB              LC_COLLATE=C              
##  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
##  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
##  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       
## 
## attached base packages:
## [1] stats4    stats     graphics  grDevices utils     datasets  methods  
## [8] base     
## 
## other attached packages:
##  [1] MetNet_1.12.2               SummarizedExperiment_1.24.0
##  [3] Biobase_2.54.0              GenomicRanges_1.46.1       
##  [5] GenomeInfoDb_1.30.1         IRanges_2.28.0             
##  [7] MatrixGenerics_1.6.0        matrixStats_0.61.0         
##  [9] S4Vectors_0.32.4            BiocGenerics_0.40.0        
## [11] knitr_1.38                  BiocStyle_2.22.0           
## 
## loaded via a namespace (and not attached):
##  [1] sass_0.4.1             tidyr_1.2.0            jsonlite_1.8.0        
##  [4] tmvnsim_1.0-2          bslib_0.3.1            assertthat_0.2.1      
##  [7] BiocManager_1.30.16    highr_0.9              GenomeInfoDbData_1.2.7
## [10] yaml_2.3.5             stabs_0.6-4            pillar_1.7.0          
## [13] lattice_0.20-45        glue_1.6.2             digest_0.6.29         
## [16] XVector_0.34.0         colorspace_2.0-3       htmltools_0.5.2       
## [19] Matrix_1.4-1           psych_2.2.3            pkgconfig_2.0.3       
## [22] magick_2.7.3           bookdown_0.25          zlibbioc_1.40.0       
## [25] purrr_0.3.4            scales_1.1.1           GENIE3_1.16.0         
## [28] BiocParallel_1.28.3    tibble_3.1.6           farver_2.1.0          
## [31] generics_0.1.2         ggplot2_3.3.5          ellipsis_0.3.2        
## [34] ppcor_1.1              cli_3.2.0              mnormt_2.0.2          
## [37] magrittr_2.0.3         crayon_1.5.1           evaluate_0.15         
## [40] fansi_1.0.3            nlme_3.1-157           MASS_7.3-56           
## [43] tools_4.1.3            lifecycle_1.0.1        bnlearn_4.7.1         
## [46] stringr_1.4.0          munsell_0.5.0          mpmi_0.43.2           
## [49] DelayedArray_0.20.0    compiler_4.1.3         jquerylib_0.1.4       
## [52] rlang_1.0.2            grid_4.1.3             RCurl_1.98-1.6        
## [55] igraph_1.3.0           labeling_0.4.2         bitops_1.0-7          
## [58] rmarkdown_2.13         gtable_0.3.0           DBI_1.1.2             
## [61] R6_2.5.1               dplyr_1.0.8            fastmap_1.1.0         
## [64] utf8_1.2.2             parmigene_1.1.0        KernSmooth_2.23-20    
## [67] stringi_1.7.6          Rcpp_1.0.8.3           parallel_4.1.3        
## [70] vctrs_0.4.0            tidyselect_1.1.2       xfun_0.30

Transformations

The list of transformations is taken from Breitling et al. (2006). The numerical m/z values were calculated by using the structural formula and the Biological Magnetic Resonance Data Bank web tool.

transformations <- rbind(
    c("Alanine", "C3H5NO", "71.0371137878"),
    c("Arginine", "C6H12N4O", "156.1011110281"),
    c("Asparagine", "C4H6N2O2", "114.0429274472"),
    c("Guanosine 5-diphosphate (-H2O)", "C10H13N5O10P2", "425.0137646843"),
    c("Guanosine 5-monophosphate (-H2O)", "C10H12N5O7P", "345.0474342759"),
    c("Guanine (-H)", "C5H4N5O", "150.0415847765"),
    c("Aspartic acid", "C4H5NO3", "115.0269430320"),
    c("Guanosine (-H2O)", "C10H11N5O4", "265.0811038675"),
    c("Cysteine", "C3H5NOS", "103.0091844778"),
    c("Deoxythymidine 5'-diphosphate (-H2O)", "C10H14N2O10P2", "384.01236770"),
    c("Cystine", "C6H10N2O3S2", "222.0132835777"),
    c("Thymidine (-H2O)", "C10H12N2O4", "224.0797068840"),
    c("Glutamic acid", "C5H7NO3", "129.0425930962"),
    c("Thymine (-H)", "C5H5N2O2", "125.0351024151"),
    c("Glutamine", "C5H8N2O2", "128.0585775114"),
    c("Thymidine 5'-monophosphate (-H2O)", "C10H13N2O7P", "304.0460372924"),
    c("Glycine", "C2H3NO", "57.0214637236"),
    c("Uridine 5'-diphosphate (-H2O)", "C9H12N2O11P2", "385.9916322587"),
    c("Histidine", "C6H7N3O", "137.0589118624"),
    c("Uridine 5'-monophosphate (-H2O)", "C9H11N2O8P", "306.0253018503"),
    c("Isoleucine", "C6H11NO", "113.0840639804"),
    c("Uracil (-H)", "C4H3N2O2", "111.0194523509"),
    c("Leucine", "C6H11NO", "113.0840639804"),
    c("Uridine (-H2O)", "C9H10N2O5", "226.0589714419"),
    c("Lysine", "C6H12N2O", "128.0949630177"),
    c("Acetylation (-H)", "C2H3O2", "59.0133043405"),
    c("Methionine", "C5H9NOS", "131.0404846062"),
    c("Acetylation (-H2O)", "C2H2O",  "42.0105646863"),
    c("Phenylalanine", "C9H9NO",  "147.0684139162"),
    c("C2H2", "C2H2", "26.0156500642"),
    c("Proline", "C5H7NO", "97.0527638520"),
    c("Carboxylation", "CO2", "43.9898292442"),
    c("Serine", "C3H5NO2", "87.0320284099"),
    c("CHO2", "CHO2", "44.9976542763"),
    c("Threonine",  "C4H7NO2",  "101.0476784741"),
    c("Condensation/dehydration", "H2O", "18.0105646863"),
    c("Tryptophan", "C11H10N2O",  "186.0793129535"),
    c("Diphosphate", "H3O6P2", "160.9404858489"),
    c("Tyrosine", "C9H9NO2", "163.0633285383"),
    c("Ethyl addition (-H2O)", "C2H4", "28.0313001284"),
    c("Valine", "C5H9NO",  "99.0684139162"),
    c("Formic Acid (-H2O)", "CO", "27.9949146221"),
    c("Acetotacetate (-H2O)",  "C4H4O2", "84.0211293726"),
    c("Glyoxylate (-H2O)", "C2O2",  "55.9898292442"),
    c("Acetone (-H)", "C3H5O", "57.0340397826"),
    c("Hydrogenation/dehydrogenation", "H2", "2.0156500642"),
    c("Adenylate (-H2O)", "C10H12N5O6P", "329.0525196538"),
    c("Hydroxylation (-H)", "O", "15.9949146221"),
    c("Biotinyl (-H)", "C10H15N2O3S", "243.0803380482"),
    c("Inorganic phosphate", "P", "30.9737615100"),
    c("Biotinyl (-H2O)", "C10H14N2O2S", "226.0775983940"),
    c("Ketol group (-H2O)", "C2H2O", "42.0105646863"),
    c("Carbamoyl P transfer (-H2PO4)", "CH2ON", "44.0136386915"),
    c("Methanol (-H2O)", "CH2", "14.0156500642"),
    c("Co-enzyme A (-H)", "C21H34N7O16P3S", "765.0995583014"),
    c("Phosphate", "HPO3", "79.9663304084"),
    c("Co-enzyme A (-H2O)", "C21H33N7O15P3S", "748.0968186472"),
    c("Primary amine", "NH2", "16.0187240694"),
    c("Glutathione (-H2O)", "C10H15N3O5S", "289.0732412976"),
    c("Pyrophosphate", "PP", "61.9475230200"),
    c("Isoprene addition (-H)", "C5H7", "67.0547752247"),
    c("Secondary amine", "NH", "15.0108990373"),
    c("Malonyl group (-H2O)", "C3H2O3", "86.0003939305"),
    c("Sulfate (-H2O)", "SO3", "79.9568145563"),
    c("Palmitoylation (-H2O)", "C16H30O", "238.2296655851"),
    c("Tertiary amine", "N", "14.0030740052"),
    c("Pyridoxal phosphate (-H2O)", "C8H8NO5P", "229.0140088825"),
    c("C6H10O5", "C6H10O5", "162.0528234315"),
    c("Urea addition (-H)", "CH3N2O", "59.0245377288"),
    c("C6H10O6", "C6H10O6", "178.0477380536"),
    c("Adenine (-H)", "C5H4N5", "134.0466701544"),
    c("D-ribose (-H2O) (ribosylation)", "C5H8O4", "132.0422587452"),
    c("Adenosine (-H2O)", "C10H11N5O3", "249.0861892454"),
    c("Disaccharide (-H2O) #1", "C12H20O10", "324.105649"),
    c("Disaccharide (-H2O) #2", "C12H20O11", "340.1005614851"),
    c("Adenosine 5'-diphosphate (-H2O)", "C10H13N5O9P2", "409.0188500622"),
    c("Glucose-N-phosphate (-H2O)", "C6H11O8P", "242.0191538399"),
    c("Adenosine 5'-monophosphate (-H2O)", "C10H12N5O6P", "329.0525196538"),
    c("Glucuronic acid (-H2O)", "C6H8O6", "176.0320879894"),
    c("Cytidine 5'-diphosphate (-H2O)", "C9H13N3O10P2", "385.0076166739"),
    c("Monosaccharide (-H2O)", "C6H10O5", "162.0528234315"),
    c("Cytidine 5'-monophsophate (-H2O)", "C9H12N3O7P", "305.0412862655"),
    c("Trisaccharide (-H2O)", "C18H30O15", "486.1584702945"),
    c("Cytosine (-H)", "C4H4N3O",  "110.0354367661"))

transformations <- data.frame(group = transformations[, 1], 
            formula = transformations[, 2],
            mass = as.numeric(transformations[, 3]))

References

Breiman, L. 2001. “Random Forests.” Machine Learning 45: 5–32. https://doi.org/10.1023/A:1010933404324.

Breitling, R., S. Ritchie, D. Goodenowe, M.L. Stewart, and M.P. Barrett. 2006. “Ab Initio Prediction of Metabolic Networks Using Fourier Transform Mass Spectrometry Data.” Metabolomics 2: 155–64. https://doi.org/10.1007/s11306-006-0029-z.

Csardi, G., and T. Nepusz. 2006. “The Igraph Software Package for Complex Network Research.” InterJournal Complex Systems: 1695. http://igraph.org.

Faith, J.J., B. Hayete, J.T. Thaden, I. Mogno, J. Wierzbowski, G. Cottarel, S. Kasif, J.J. Collins, and T.S. Gardner. 2007. “Large-Scale Mapping and Validation of Escherichia Coli Transcriptional Regulation from a Compendium of Expression Profiles.” Plos Biology 5: e8. https://doi.org/10.1371/journal.pbio.0050008.

Hase, T., S. Ghosh, R. Yamanaka, and H. Kitano. 2013. “Harnessing Diversity Towards the Reconstructing of Large Scale Gene Regulatory Networks.” PLoS Computational Biology 9: e1003361. https://doi.org/10.1371/journal.pcbi.1003361.

Hofner, B., L. Boccuto, and M. Göker. 2015. “Controlling False Discoveries in High-Dimensional Situations: Boosting with Stability Selection.” BMC Bioinformatics 16: 144. https://doi.org/10.1186/s12859-015-0575-3.

Jourdan, F., R. Breitling, M.P. Barrett, and D. Gilbert. 2007. “MetaNetter: Inference and Visualization of High-Resolution Metabolomic Networks.” Bioinformatics 24: 143–45. https://doi.org/10.1093/bioinformatics/btm536.

Marbach, D., J.C. Costello, R. Küffner, N.M. Vega, R.J. Prill, D.M. Camacho, K.R. Allison, et al. 2012. “Wisdom of Crowds for Robust Gene Network Inference.” Nature Methods 9: 796–804. https://doi.org/10.1038/nmeth.2016.

Margolin, A.A., I. Nemenman, K. Basso, C. Wiggins, G. Stolovitzky, R. Dalla Favera, and A. Califano. 2006. “ARACNE: An Algorithm for the Reconstruction of Gene Regulatory Networks in a Mammalian Cellular Context.” BMC Bioinformatics 7: S7. https://doi.org/10.1186/1471-2105-7-S1-S7.

Scutari, M. 2010. “Learning Bayesian Networks with the bnlearn R Package.” Journal of Statistical Software 35: 1–22. https://doi.org/10.18637/jss.v035.i03.

Steuer, R. 2006. “On the Analysis and Interpretation of Correlations in Metabolomic Data.” Briefings in Bioinformatics 7: 151–58. https://doi.org/10.1093/bib/bbl009.

Thomas, J., A. Mayr, B. Bischl, M. Schmid, A. Smith, and B. Hofner. 2017. “Gradient Boosting for Distributional Regression - Faster Tuning and Improved Variable Selection via Noncyclical Updates.” Statistics and Computing 28: 673–87. https://doi.org/10.1007/s11222-017-9754-6.

Tibshirani, R. 1994. “Regression Shrinkage and Selection via the Lasso.” Journal of the Royal Statistical Society. Series B (Methodological) 58: 267–88.