# On the Use of Correlation and MI as a Measure of Metabolite—Metabolite Association for Network Differential Connectivity Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Differential Connectivity Analysis on Experimental Data

#### 2.2. Type I Error

#### 2.3. Comparison of Correlation and MI on Simulated Data with Known Correlation Structure

#### 2.4. Comparison of Correlation and MI on Simulated Data from a Dynamic Model

#### 2.5. Relationship between Correlation and MI

## 3. Materials and Methods

#### 3.1. Association Measures

#### Correlation Indices

#### 3.2. MI

`infotheo`R package [49].

#### 3.2.1. Entropy of Empirical Probability Distribution

#### 3.2.2. Miller-Madow Asymptotic Bias Corrected Empirical Estimator

#### 3.2.3. Shrinkage Estimate of the Entropy of a Dirichlet Probability Distribution

#### 3.2.4. Schurmann-Grassberger Estimation

`infotheo`R package.

#### 3.3. Network Concepts

#### 3.3.1. Differential Network Analysis

#### 3.3.2. Permutation Tests to Assess Statistical Significance of Differential Connectivity

#### 3.4. Data Simulations

**Σ**equals the correlation matrix. Three different correlation structures were used as described in the following section.

#### 3.4.1. Toeplitz Correlation Structure

`simcorTop`provided in the supplementary material of Reference [56] and are available at pages.pomona.edu/ jsh04747/research/simcor.r. The parameters used were $k=1$, $\u03f5=0.01$, and edim = 2. Data matrices were generated using the R function

`mvrnorm`.

#### 3.4.2. Hub Correlation Structure

`simcor.H`provided by Hardin [56]. The parameters used were $k=2$, $\u03f5=0.01$, $\gamma =2$, size = (5,2) and edim = 2. Data matrices were generated using the R function

`mvrnorm`.

#### 3.4.3. Average

#### 3.5. Data Generation Using a Dynamic Metabolic Model

#### Simulation of Individual Metabolite Concentration Profiles

#### 3.6. Experimental Data

`missForest`[62].

#### 3.7. Software

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NF-κB | Nuclear Factor kappa-light-chain-enhancer of activated B cells |

NMR | Nuclear magnetic resonance |

MI | MI |

MS | Mass spectrometry |

## References

- Tavassoly, I.; Goldfarb, J.; Iyengar, R. Systems biology primer: The basic methods and approaches. Essays Biochem.
**2018**. [Google Scholar] [CrossRef] - Vignoli, A.; Ghini, V.; Meoni, G.; Licari, C.; Takis, P.G.; Tenori, L.; Turano, P.; Luchinat, C. High-throughput metabolomics by 1D NMR. Angew. Chem. Int. Ed.
**2019**, 58, 968–994. [Google Scholar] [CrossRef] [PubMed] - Emwas, A.H.; Roy, R.; McKay, R.T.; Tenori, L.; Saccenti, E.; Gowda, G.; Raftery, D.; Alahmari, F.; Jaremko, L.; Jaremko, M.; et al. NMR spectroscopy for metabolomics research. Metabolites
**2019**, 9, 123. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ma’ayan, A. Introduction to network analysis in systems biology. Sci. Signal.
**2011**, 4, tr5. [Google Scholar] [CrossRef][Green Version] - Trudeau, R.J. Introduction to Graph Theory; Courier Corporation: Chelmsford, MA, USA, 2013. [Google Scholar]
- Rosato, A.; Tenori, L.; Cascante, M.; De Atauri Carulla, P.R.; Martins dos Santos, V.A.; Saccenti, E. From correlation to causation: Analysis of metabolomics data using systems biology approaches. Metabolomics
**2018**. [Google Scholar] [CrossRef][Green Version] - Saccenti, E.; Suarez-Diez, M.; Luchinat, C.; Santucci, C.; Tenori, L. Probabilistic networks of blood metabolites in healthy subjects as indicators of latent cardiovascular risk. J. Proteome Res.
**2015**, 14, 1101–1111. [Google Scholar] [CrossRef] - Jahagirdar, S.; Suarez-Diez, M.; Saccenti, E. Simulation and Reconstruction of metabolite-metabolite Association Networks Using a Metabolic Dynamic Model and Correlation Based Algorithms. J. Proteome Res.
**2019**, 18, 1099–1113. [Google Scholar] [CrossRef] - Vignoli, A.; Tenori, L.; Luchinat, C.; Saccenti, E. Age and sex effects on plasma metabolite association networks in healthy subjects. J. Proteome Res.
**2017**, 17, 97–107. [Google Scholar] [CrossRef][Green Version] - Vignoli, A.; Tenori, L.; Giusti, B.; Valente, S.; Carrabba, N.; Balzi, D.; Barchielli, A.; Marchionni, N.; Gensini, G.F.; Marcucci, R.; et al. Differential network analysis reveals metabolic determinants associated with mortality in acute myocardial infarction patients and suggest potential mechanisms underlying different clinical scores used to predict death. J. Proteome Res.
**2020**, 19, 949–961. [Google Scholar] [CrossRef] - Afzal, M.; Saccenti, E.; Madsen, M.; Hansen, M.B.; Hyldegaard, O.; Skrede, S.; Martins dos santos, V.; Norrby Teglund, A.; Svensson, M. Integrated univariate, multivariate and correlation-based network analyses reveal metabolite-specific effects on bacterial growth and biofilm formation in necrotizing soft tissue infections. J. Proteome Res.
**2019**. [Google Scholar] [CrossRef][Green Version] - Rist, M.J.; Roth, A.; Frommherz, L.; Weinert, C.H.; Krüger, R.; Merz, B.; Bunzel, D.; Mack, C.; Egert, B.; Bub, A.; et al. Metabolite patterns predicting sex and age in participants of the Karlsruhe Metabolomics and Nutrition (KarMeN) study. PLoS ONE
**2017**, 12, e0183228. [Google Scholar] [CrossRef] [PubMed] - Smith, R. A MI approach to calculating nonlinearity. Stat
**2015**, 4, 291–303. [Google Scholar] [CrossRef][Green Version] - Haug, K.; Cochrane, K.; Nainala, V.C.; Williams, M.; Chang, J.; Jayaseelan, K.V.; O’Donovan, C. MetaboLights: A resource evolving in response to the needs of its scientific community. Nucleic Acids Res.
**2019**. [Google Scholar] [CrossRef][Green Version] - Sud, M.; Fahy, E.; Cotter, D.; Azam, K.; Vadivelu, I.; Burant, C.; Edison, A.; Fiehn, O.; Higashi, R.; Nair, K.S.; et al. Metabolomics Workbench: An international repository for metabolomics data and metadata, metabolite standards, protocols, tutorials and training, and analysis tools. Nucleic Acids Res.
**2015**, 44, D463–D470. [Google Scholar] [CrossRef][Green Version] - Meyer, F.; Paarmann, D.; D’Souza, M.; Olson, R.; Glass, E.M.; Kubal, M.; Paczian, T.; Rodriguez, A.; Stevens, R.; Wilke, A.; et al. The metagenomics RAST server–a public resource for the automatic phylogenetic and functional analysis of metagenomes. BMC Bioinform.
**2008**, 9, 386. [Google Scholar] [CrossRef][Green Version] - Wehrens, R.; Franceschi, P. Meta-Statistics for Variable Selection: The R Package BioMark. J. Stat. Softw. Artic.
**2012**, 51, 1–18. [Google Scholar] [CrossRef][Green Version] - Cacciatore, S.; Tenori, L.; Luchinat, C.; Bennett, P.R.; MacIntyre, D.A. KODAMA: An R package for knowledge discovery and data mining. Bioinformatics
**2017**, 33, 621–623. [Google Scholar] [CrossRef][Green Version] - Rohart, F.; Gautier, B.; Singh, A.; Lê Cao, K.A. mixOmics: An R package for ‘omics feature selection and multiple data integration. PLoS Comput. Biol.
**2017**, 13, e1005752. [Google Scholar] [CrossRef][Green Version] - Mcnicholas, P.D.; Murphy, T.B. Parsimonious Gaussian mixture models. Stat. Comput.
**2008**, 18, 285–296. [Google Scholar] [CrossRef] - Ganna, A.; Salihovic, S.; Sundström, J.; Broeckling, C.D.; Hedman, Å.K.; Magnusson, P.K.; Pedersen, N.L.; Larsson, A.; Siegbahn, A.; Zilmer, M.; et al. Large-scale metabolomic profiling identifies novel biomarkers for incident coronary heart disease. PLoS Genet.
**2014**, 10, e1004801. [Google Scholar] [CrossRef] [PubMed] - Hilvo, M.; Gade, S.; Hyötyläinen, T.; Nekljudova, V.; Seppänen-Laakso, T.; Sysi-Aho, M.; Untch, M.; Huober, J.; von Minckwitz, G.; Denkert, C.; et al. Monounsaturated fatty acids in serum triacylglycerols are associated with response to neoadjuvant chemotherapy in breast cancer patients. Int. J. Cancer
**2014**, 134, 1725–1733. [Google Scholar] [CrossRef] [PubMed] - Stevens, V.L.; Wang, Y.; Carter, B.D.; Gaudet, M.M.; Gapstur, S.M. Serum metabolomic profiles associated with postmenopausal hormone use. Metabolomics
**2018**, 14, 97. [Google Scholar] [CrossRef] [PubMed] - Armstrong, C.W.; McGregor, N.R.; Lewis, D.P.; Butt, H.L.; Gooley, P.R. Metabolic profiling reveals anomalous energy metabolism and oxidative stress pathways in chronic fatigue syndrome patients. Metabolomics
**2015**, 11, 1626–1639. [Google Scholar] [CrossRef] - Thévenot, E.A.; Roux, A.; Xu, Y.; Ezan, E.; Junot, C. Analysis of the human adult urinary metabolome variations with age, body mass index, and gender by implementing a comprehensive workflow for univariate and OPLS statistical analyses. J. Proteome Res.
**2015**, 14, 3322–3335. [Google Scholar] [CrossRef] - Zheng, X.; Huang, F.; Zhao, A.; Lei, S.; Zhang, Y.; Xie, G.; Chen, T.; Qu, C.; Rajani, C.; Dong, B.; et al. Bile acid is a significant host factor shaping the gut microbiome of diet-induced obese mice. BMC Biol.
**2017**, 15, 120. [Google Scholar] [CrossRef] [PubMed][Green Version] - Fahrmann, J.F.; Kim, K.; DeFelice, B.C.; Taylor, S.L.; Gandara, D.R.; Yoneda, K.Y.; Cooke, D.T.; Fiehn, O.; Kelly, K.; Miyamoto, S. Investigation of metabolomic blood biomarkers for detection of adenocarcinoma lung cancer. Cancer Epidemiol. Prev. Biomark.
**2015**, 24, 1716–1723. [Google Scholar] [CrossRef] [PubMed][Green Version] - Sakanaka, A.; Kuboniwa, M.; Hashino, E.; Bamba, T.; Fukusaki, E.; Amano, A. Distinct signatures of dental plaque metabolic byproducts dictated by periodontal inflammatory status. Sci. Rep.
**2017**, 7, 42818. [Google Scholar] [CrossRef] - Franzosa, E.A.; Sirota-Madi, A.; Avila-Pacheco, J.; Fornelos, N.; Haiser, H.J.; Reinker, S.; Vatanen, T.; Hall, A.B.; Mallick, H.; McIver, L.J.; et al. Gut microbiome structure and metabolic activity in inflammatory bowel disease. Nat. Microbiol.
**2019**, 4, 293. [Google Scholar] [CrossRef] - Chan, A.W.; Mercier, P.; Schiller, D.; Bailey, R.; Robbins, S.; Eurich, D.T.; Sawyer, M.B.; Broadhurst, D. 1 H-NMR urinary metabolomic profiling for diagnosis of gastric cancer. Br. J. Cancer
**2016**, 114, 59. [Google Scholar] [CrossRef] - Eisner, R.; Stretch, C.; Eastman, T.; Xia, J.; Hau, D.; Damaraju, S.; Greiner, R.; Wishart, D.S.; Baracos, V.E. Learning to predict cancer-associated skeletal muscle wasting from 1 H-NMR profiles of urinary metabolites. Metabolomics
**2011**, 7, 25–34. [Google Scholar] [CrossRef] - Lusczek, E.R.; Lexcen, D.R.; Witowski, N.E.; Mulier, K.E.; Beilman, G. Urinary metabolic network analysis in trauma, hemorrhagic shock, and resuscitation. Metabolomics
**2013**, 9, 223–235. [Google Scholar] [CrossRef] - Powers, R.K.; Sullivan, K.D.; Culp-Hill, R.; Ludwig, M.P.; Smith, K.P.; Waugh, K.A.; Minter, R.; Tuttle, K.D.; Lewis, H.C.; Rachubinski, A.L.; et al. Trisomy 21 activates the kynurenine pathway via increased dosage of interferon receptors. Nature Commun.
**2019**, 10, 4766. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bernini, P.; Bertini, I.; Luchinat, C.; Nepi, S.; Saccenti, E.; Schäfer, H.; Schütz, B.; Spraul, M.; Tenori, L. Individual human phenotypes in metabolic space and time. J. Proteome Res.
**2009**, 8, 4264–4271. [Google Scholar] [CrossRef] [PubMed] - Caldana, C.; Degenkolbe, T.; Cuadros-Inostroza, A.; Klie, S.; Sulpice, R.; Leisse, A.; Steinhauser, D.; Fernie, A.R.; Willmitzer, L.; Hannah, M.A. High-density kinetic analysis of the metabolomic and transcriptomic response of Arabidopsis to eight environmental conditions. Plant J.
**2011**, 67, 869–884. [Google Scholar] [CrossRef] [PubMed] - Khan, J.; Wei, J.S.; Ringner, M.; Saal, L.H.; Ladanyi, M.; Westermann, F.; Berthold, F.; Schwab, M.; Antonescu, C.R.; Peterson, C.; et al. Classification and diagnostic prediction of cancers using gene expression profiling and artificial neural networks. Nat. Med.
**2001**, 7, 673. [Google Scholar] [CrossRef] - Bushel, P.R.; Wolfinger, R.D.; Gibson, G. Simultaneous clustering of gene expression data with clinical chemistry and pathological evaluations reveals phenotypic prototypes. BMC Syst. Biol.
**2007**, 1, 15. [Google Scholar] [CrossRef][Green Version] - Stanley, D.; Geier, M.S.; Hughes, R.J.; Denman, S.E.; Moore, R.J. Highly variable microbiota development in the chicken gastrointestinal tract. PLoS ONE
**2013**, 8, e84290. [Google Scholar] [CrossRef][Green Version] - Forina, M.; Armanino, C.; Lanteri, S.; Tiscornia, E. Classification of olive oils from their fatty acid composition. Food research and data analysis. In Proceedings of the IUFoST Symposium, Oslo, Norway, 20–23 September 1982; Martens, H., Russwurm, H., Jr., Eds.; Applied Science Publishers: London, UK, 1983. [Google Scholar]
- Streuli, H. Der heutige stand der kaffeechemie. In Proceedings of the ASSIC, 6e, Colloque, Bogota, Colombia, 4–5 October 1973; Volume 61. [Google Scholar]
- Forina, M.; Armanino, C.; Castino, M.; Ubigli, M. Multivariate data analysis as a discriminating method of the origin of wines. Vitis
**1986**, 25, 189–201. [Google Scholar] - Nemenman, I.; Bialek, W.; Van Steveninck, R.D.R. Entropy and information in neural spike trains: Progress on the sampling problem. Phys. Rev. E
**2004**, 69, 056111. [Google Scholar] [CrossRef][Green Version] - Gelfand, I.M.; Yaglom, A.M. Calculation of amount of information about a random function contained in another such function. Am. Math. Soc. Transl.
**1957**, 2, 199–246. [Google Scholar] - Kendall, M.G. Rank Correlation Methods; Griffin: London, UK, 1948. [Google Scholar]
- Zimmerman, D.W.; Zumbo, B.D.; Williams, R.H. Bias in estimation and hypothesis testing of correlation. Psicológica
**2003**, 24, 133–158. [Google Scholar] - Pearson, K. VII. Note on regression and inheritance in the case of two parents. Proc. R. Soc. Lond.
**1895**, 58, 240–242. [Google Scholar] - Spearman, C. Measurement of association, Part II. Correction of ‘systematic deviations’. Am. J. Psychol.
**1904**, 15, 88–101. [Google Scholar] - Kullback, S.; Leibler, R.A. On information and sufficiency. Ann. Math. Stat.
**1951**, 22, 79–86. [Google Scholar] [CrossRef] - Meyer, P.E. Information-Theoretic Variable Selection and Network Inference from Microarray Data; Universite Libre de Bruxelles: Brussels, Belgium, 2008. [Google Scholar]
- Paninski, L. Estimation of entropy and MI. Neural Comput.
**2003**, 15, 1191–1253. [Google Scholar] [CrossRef][Green Version] - Schäfer, J.; Strimmer, K. An empirical Bayes approach to inferring large-scale gene association networks. Bioinformatics
**2005**, 21, 754–764. [Google Scholar] [CrossRef] - Schäfer, J.; Strimmer, K. A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics. Stat. Appl. Genet. Mol. Biol.
**2005**, 4. [Google Scholar] [CrossRef][Green Version] - Schürmann, T.; Grassberger, P. Entropy estimation of symbol sequences. Chaos: Interdiscip. J. Nonlinear Sci.
**1996**, 6, 414–427. [Google Scholar] [CrossRef] - Wu, L.; Neskovic, P.; Reyes, E.; Festa, E.; William, H. Classifying n-back EEG data using entropy and MI features. In Proceedings of the ESANN, Bruges, Belgium, 26–28 April 2017; pp. 61–66. [Google Scholar]
- Guo, Y.; Hastie, T.; Tibshirani, R. Regularized linear discriminant analysis and its application in microarrays. Biostatistics
**2006**, 8, 86–100. [Google Scholar] [CrossRef][Green Version] - Hardin, J.; Garcia, S.R.; Golan, D. A method for generating realistic correlation matrices. Ann. Appl. Stat.
**2013**, 7, 1733–1762. [Google Scholar] [CrossRef] - Ghosh, S.; Henderson, S.G. Behavior of the NORTA method for correlated random vector generation as the dimension increases. ACM Trans. Model. Comput. Simul. (TOMACS)
**2003**, 13, 276–294. [Google Scholar] [CrossRef] - Lewandowski, D.; Kurowicka, D.; Joe, H. Generating random correlation matrices based on vines and extended onion method. J. Multivar. Anal.
**2009**, 100, 1989–2001. [Google Scholar] [CrossRef][Green Version] - Malik-Sheriff, R.S.; Glont, M.; Nguyen, T.V.N.; Tiwari, K.; Roberts, M.G.; Xavier, A.; Vu, M.T.; Men, J.; Maire, M.; Kananathan, S.; et al. BioModels—15 years of sharing computational models in life science. Nucleic Acids Res.
**2019**. [Google Scholar] [CrossRef] [PubMed][Green Version] - Sharp, G.C.; Ma, H.; Saunders, P.T.; Norman, J.E. A computational model of lipopolysaccharide-induced nuclear factor kappa B activation: A key signalling pathway in infection-induced preterm labour. PLoS ONE
**2013**, 8, e70180. [Google Scholar] [CrossRef] [PubMed] - Mendez, K.M.; Reinke, S.N.; Broadhurst, D.I. A comparative evaluation of the generalised predictive ability of eight machine learning algorithms across ten clinical metabolomics data sets for binary classification. Metabolomics
**2019**, 15, 150. [Google Scholar] [CrossRef][Green Version] - Stekhoven, D.J.; Bühlmann, P. MissForest—Non-parametric missing value imputation for mixed-type data. Bioinformatics
**2011**, 28, 112–118. [Google Scholar] [CrossRef][Green Version] - R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2013. [Google Scholar]
- MATLAB. Version 9.5.0 (R2018b); The MathWorks Inc.: Natick, MA, USA, 2018. [Google Scholar]
- RPython Core Team. Python: A Dynamic, Open Source Programming Language; Python Software Foundation: Wilmington, DE, USA, 2015. [Google Scholar]
- Zhao, J.; Zhou, Y.; Zhang, X.; Chen, L. Part MI for quantifying direct associations in networks. Proc. Natl. Acad. Sci. USA
**2016**, 113, 5130–5135. [Google Scholar] [CrossRef][Green Version] - Cover, T.M.; Thomas, J.A. Elements of Information Theory; John Wiley & Sons: Hoboken, NJ, USA, 2012. [Google Scholar]
- Steuer, R.; Kurths, J.; Daub, C.O.; Weise, J.; Selbig, J. The MI: Detecting and evaluating dependencies between variables. Bioinformatics
**2002**, 18, S231–S240. [Google Scholar] [CrossRef][Green Version] - Lindlöf, A.; Lubovac, Z. Simulations of simple artificial genetic networks reveal features in the use of Relevance Networks. Silico Biol.
**2005**, 5, 239–249. [Google Scholar] - Song, L.; Langfelder, P.; Horvath, S. Comparison of co-expression measures: MI, correlation, and model based indices. BMC Bioinform.
**2012**, 13, 328. [Google Scholar] [CrossRef][Green Version] - Numata, J.; Ebenhöh, O.; Knapp, E.W. Measuring correlations in metabolomic networks with mutual information. In Genome Informatics 2008: Genome Informatics Series Vol. 20; World Scientific: Singapore, 2008; pp. 112–122. [Google Scholar]
- You, Y.; Liang, D.; Wei, R.; Li, M.; Li, Y.; Wang, J.; Wang, X.; Zheng, X.; Jia, W.; Chen, T. Evaluation of metabolite-microbe correlation detection methods. Anal. Biochem.
**2019**, 567, 106–111. [Google Scholar] [CrossRef] [PubMed] - Kraskov, A.; Stögbauer, H.; Grassberger, P. Estimating MI. Phys. Rev. E
**2004**, 69, 066138. [Google Scholar] [CrossRef] [PubMed][Green Version] - Matsuda, H. Physical nature of higher-order MI: Intrinsic correlations and frustration. Phys. Rev. E
**2000**, 62, 3096. [Google Scholar] [CrossRef] [PubMed] - Camacho, D.; De La Fuente, A.; Mendes, P. The origin of correlations in metabolomics data. Metabolomics
**2005**, 1, 53–63. [Google Scholar] [CrossRef] - Saccenti, E.; Hendriks, M.H.; Smilde, A.K. Corruption of the Pearson correlation coefficient by measurement error and its estimation, bias, and correction under different error models. Sci. Rep. (Nat. Publ. Group)
**2020**, 10, 1–9. [Google Scholar] [CrossRef][Green Version] - Saccenti, E. Correlation patterns in experimental data are affected by normalization procedures: Consequences for data analysis and network inference. J. Proteome Res.
**2017**, 16, 619–634. [Google Scholar] [CrossRef] - Mason, M.J.; Fan, G.; Plath, K.; Zhou, Q.; Horvath, S. Signed weighted gene co-expression network analysis of transcriptional regulation in murine embryonic stem cells. BMC Genom.
**2009**, 10, 327. [Google Scholar] [CrossRef][Green Version] - Doquire, G.; Verleysen, M. A Comparison of Multivariate MI Estimators for Feature Selection. In Proceedings of the 1st International Conference on Pattern Recognition Applications and Methods, Vilamoura, Portugal, 6–8 February 2012; pp. 176–185. [Google Scholar]

**Figure 1.**Graphical illustration of the concept of metabolite connectivity and differential connectivity. An ideal unweighted metabolite-metabolite association network involving 10 metabolites is shown under two different conditions. Metabolite i is connected with a different number of metabolites (represented by the existence of an edge ) in the two conditions. The connectivity ${\chi}_{i}$ of metabolite i is given by the number of connecting edges (for a generalisation for weighted association networks, see Equation (28) in Section 3.3): 6 under condition 1 and 2 under condition 2. The differential connectivity of metabolite i is given by $\Delta {\chi}_{i}={\chi}_{i}^{G1}-{\chi}_{i}^{G2}=6-2=4$, as described in Equation (29).

**Figure 2.**Graphical illustration of differential connectivity analysis. Given two data sets ${\mathbf{X}}_{1}$ and ${\mathbf{X}}_{2}$ of size ${n}_{1}\times p$ and ${n}_{2}\times p$ with ${n}_{1}$ possibly different from ${n}_{2}$, weighted association matrices are built using either correlation (${\mathbf{C}}_{1}$ and ${\mathbf{C}}_{2}$, for ${\mathbf{X}}_{1}$ and ${\mathbf{X}}_{2}$, respectively) or Mutual Information (MI) ($\mathbf{M}{\mathbf{I}}_{1}$ and $\mathbf{M}{\mathbf{I}}_{2}$). Weighted (metabolite) connectivity is then calculated as described in Equation (28) for group 1 and group 2 as ${\chi}_{i}^{G1}$ and ${\chi}_{i}^{G2}$. The differential connectivity is given by $\Delta {\chi}_{i}={\chi}_{i}^{G1}-{\chi}_{i}^{G2}$, and it is calculated using both correlation and MI. Significance is then assessed using a permutation test.

**Figure 3.**Four different data patterns obtained by plotting the simulated concentration of two metabolites,

**A**and

**B**, on which Gaussian experimental noise has been added. (

**A**) Positive linear relationship, $\rho =0.96$ (Pearson’s correlation); (

**B**) Negative linear relationship, $\rho =-0.96$; (

**C**) Sine-wave relationship, $\rho =0$ ; (

**D**) Bell-shaped relationship, $\rho =0.28$. In all cases, the MI is 1.32 nats (or 1.90 bits). One nat is the information content of the uniform distribution on the interval $[0,e]$ where e is the basis of the natural logarithm. This figure is an adaptation from Table 1 from Reference [13].

**Figure 4.**Type I error for the permutation test used to assess the statistical significance of metabolite connectivity. Two data sets ${\mathbf{X}}_{1}$ and ${\mathbf{X}}_{2}$ are generate of size $n\times 20$ under an uncorrelated multivariate model (${\mathbf{X}}_{1}\approx N(0,\mathbf{I})$. Differential connectivity is calculated as described in Equations (28) and (29) and assessed with a permutation test at the $\alpha =0.05$ significance level. The overall procedure is repeated 100 times.

**Figure 5.**Median of the significant differentially connected variables on all simulated data sets per known correlation ρ per sample size n.

**Figure 6.**Behaviour of the NF-κB dynamic model. (

**A**) Time concentration profiles for model perturbation with $\u03f5>1$. (

**B**) Original model. (

**C**) Time concentration profiles for model perturbation with $\u03f5<1$. Different colours correspond to different metabolite time profiles. The vertical lines indicate the time sampling point.

**Figure 7.**Median of the significant differentially connected variables on data simulated using the NF-κB dynamic model as a function of the model perturbation ϵ and the sample size n.

**Figure 8.**(

**A**) MI $MI({X}_{1},{X}_{2})$ of two bivariate variables ${X}_{1}$, ${X}_{2}$ linearly correlated with correlation ρ as a function of ρ. The two curves intersect at approximately $\rho =0.916$. (

**B**) MI versus Pearson’s correlation from data simulated with an average correlation of 0.6 (beta simulation). (

**C**) MI versus Pearson’s correlation from experimental data (data set 3 from Table 1).

**Figure 9.**MI versus Pearson’s correlation from data simulated with the NK-kB dynamic model when with (

**A**) $\u03f5=0.1$ and (

**B**) $\u03f5=10$.

**Table 1.**Correlation and MI indicate the number of features found to be statistically significantly differentially connected (at the $\alpha =0.05$ level using correlation and MI as measure of association). Only in correlation and Only in MI denote differentially connected features found only using correlation and MI, respectively. Overlap indicates those found by both methods. The number of observation (No. observations) is $n={n}_{1}+{n}_{2}$, where ${n}_{1}$ and ${n}_{2}$ is the sample size of group 1 and 2, respectively. Study IDs starting with MTBL indicate data available in Metabolights database [14] (www.ebi.ac.uk/metabolights), while those starting with ST indicate data available in the Metabolomics Workbench database [15] (www.metabolomicsworkbench.org). Data set No. 27 was obtained from the RAST database [16] (www.mg-rast.org). Data sets without study ID were derived either from the original publications or from R packages within which they were distributed: BioMark [17], kodama [18], MixOmics [19], and pgmm [20]. Abbreviations: CD, Crohn’s disease; CFS, Cronic fatigue syndrome; E Estrogen; E+P, Estrogen + Progesterone; ES, Ewing sarcoma; IBD, Inflammatory bowel disease; MA, microarray; RMS, Rhabdomyosarcoma; UC, Ulcertive colitis. For data set 24 and 25, the superscripts ‘+’ and ‘−’ indicate the 250 most (the least, respectively) expressed genes, and the superscript r indicates a random selecion of 500 genes.

No. Differentially Connected Features | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

No. | Study ID | Ref. | Platform | Type | No. Observations | No. Features | Design | Correlatio | MI | Only in Corr | Only in MI | Overlap |

1 | MTBLS90 | [21] | LC–MS | Plasma | 968 (485/483) | 189 | Sex (M/F) | 132 | 101 | 68 | 37 | 64 |

2 | MTBLS92 | [22] | LC–MS | Plasma | 253 (142/111) | 138 | Chemotherapy (before/after) | 138 | 12 | 126 | 0 | 12 |

3 | MTBLS136 | [23] | LC–MS | Serum | 668 (337/331) | 371 | Homone (E/E+P) | 255 | 125 | 167 | 37 | 88 |

4 | MTBLS161 | [24] | NMR | Serum | 59 (34/25) | 30 | CFS (case/control) | 14 | 12 | 6 | 4 | 8 |

5 | MTBLS404 | [25] | LC–MS | Urine | 184 (101/83) | 120 | Sex (M/F) | 105 | 58 | 51 | 4 | 54 |

6 | MTBLS547 | [26] | LC–MS | Caecal | 97 (46/51) | 35 | High fat diet (case/control) | 35 | 4 | 31 | 0 | 4 |

7 | ST000369 | [27] | GC–MS | Serum | 80 (49/31) | 181 | Adenocarcinoma/Healthy | 181 | 69 | 112 | 0 | 69 |

8 | ST000496 | [28] | GC–MS | Saliva | 100 (50/50) | 69 | Debridement (pre/post) | 59 | 31 | 32 | 4 | 27 |

9 | ST001000 | [29] | LC–MS | Stool | 121 (68/53) | 124 | IBD (CD/UC) | 96 | 79 | 33 | 16 | 63 |

10 | ST001047 | [30] | NMR | Urine | 83 (43/40) | 149 | Gastric cancer/healthy | 109 | 85 | 42 | 18 | 67 |

11 | ST000061 | GC-MS | Tissue | 118 (59/59) | 157 | subcutaeus/visceral fat | 156 | 83 | 73 | 0 | 83 | |

12 | [31] | NMR | Urine | 50 (25/25) | 200 | cachexia (case/control) | 163 | 57 | 115 | 9 | 48 | |

13 | [31] | NMR | Urine | 77 (47/30) | 63 | cachexia (case/control) | 63 | 33 | 30 | 0 | 33 | |

14 | [31] | NMR | Urine | 60 (30/30) | 63 | cachexia (case/control) | 55 | 43 | 15 | 3 | 40 | |

15 | [12] | GC-MS | Plasma | 291(172/119) | 128 | Sex (M/F) | 128 | 23 | 105 | 0 | 23 | |

16 | [12] | GC-MS | Plasma | 200 (100/100) | 128 | Sex (M/F) | 103 | 51 | 56 | 4 | 47 | |

17 | [12] | GC-MS | Urine | 301 (129/172) | 324 | Sex (M/F) | 256 | 143 | 136 | 23 | 120 | |

18 | MTBLS123 | [32] | NMR | Urine | 151 (79/72) | 63 | Shock (pre/post) | 63 | 9 | 54 | 0 | 9 |

19 | ST001243 | [33] | GC-MS | Plasma | 98 (48/50) | 69 | Trisomy 21 (yes/no) | 69 | 28 | 41 | 0 | 28 |

20 | MTBLS147 | [9] | NMR | Plasma | 370 (185/185) | 417 | Sex (M/F) | 417 | 414 | 3 | 0 | 414 |

21 | KODAMA | [34] | NMR | Urine | 80(40/40) | 490 | Subject (A/B) | 459 | 293 | 187 | 21 | 272 |

22 | [35] | GC-MS | Plant | 70 (35/35) | 67 | Light/Dark | 37 | 19 | 22 | 4 | 15 | |

23 | BioMark | [17] | LC–MS | Apple | 20 (10/10) | 198 | Treated/Untreated | 124 | 58 | 83 | 17 | 41 |

24 | MixOmics | [36] | MA | Cell | 43 (23/20)^{−} | 250 | Sarcoma (RMS/ES) | 250 | 18 | 232 | 0 | 18 |

25 | MixOmics | [36] | MA | Cell | 43 (23/20)^{+} | 250 | Sarcoma (RMS/ES) | 250 | 8 | 242 | 0 | 8 |

26 | MixOmics | [37] | MA | Cell | 32 (16/16)^{r} | 500 | High/Low dose | 405 | 279 | 170 | 44 | 235 |

27 | 4537568.3-776.3 | [38] | 16S seq | Faeces | 145 (71/74) | 243 | Flock (A/B) | 241 | 150 | 91 | 0 | 150 |

28 | pgmm | [39] | Chemical assay | Oil | 50 (25/25) | 7 | Region (A/B) | 4 | 0 | 4 | 0 | 0 |

29 | pgmm | [40] | Chemical assay | Coffee | 43 (36/7) | 12 | Variety (Arabica/Robusta) | 4 | 11 | 0 | 7 | 4 |

30 | pgmm | [41] | Chemical assay | Wine | 130 (59/71) | 27 | Type (Barolo/Grignolino) | 8 | 10 | 5 | 7 | 3 |

**Table 2.**Results of pathway enrichment for data set 12 and 25 from Table 1 based on the sets of metabolite found to be differentially connected using correlation or MI as measure of metabolite-metabolite association. FDR: False discovery rate. Empty cells indicate that no metabolite was found to be associated with the given pathway.

Pathway Enrichment Based On | ||||

Data Set 12 | Correlation | MI | ||

Pathway | Raw P | FDR | Raw p | FDR |

Aminoacyl-tRNA biosynthesis | 3 × 10^{−12} | 3 × 10^{−12} | 0.0006 | 0.05 |

Valine, leucine and isoleucine biosynthesis | 3 × 10^{−5} | 0.001 | ||

Alanine, aspartate and glutamate metabolism | 3 × 10^{−5} | 0.002 | ||

Arginine biosynthesis | 0.0004 | 0.008 | 0.006 | 0.18 |

Glyoxylate and dicarboxylate metabolism | 0.001 | 0.020 | 0.25 | 1.00 |

Glycine, serine and threonine metabolism | 0.002 | 0.020 | 0.03 | 0.72 |

Citrate cycle (TCA cycle) | 0.002 | 0.020 | ||

Phenylalanine metabolism | 0.002 | 0.020 | 0.09 | 0.91 |

Phenylalanine, tyrosine and tryptophan biosynthesis | 0.004 | 0.040 | ||

Pathway Enrichment Based On | ||||

Data Set 25 | Correlation | MI | ||

Pathway | Raw P | FDR | Raw p | FDR |

Citrate cycle (TCA cycle) | 3 × 10^{−5} | 0.004 | ||

Alanine, aspartate and glutamate metabolism | 0.0004 | 0.016 | 0.15 | 1 |

Glyoxylate and dicarboxylate metabolism | 0.001 | 0.020 | 0.17 | 1 |

Glycine, serine and threonine metabolism | 0.001 | 0.020 | 0.18 | 1 |

Histidine metabolism | 0.002 | 0.036 | 0.09 | 1 |

Tyrosine metabolism | 0.004 | 0.050 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jahagirdar, S.; Saccenti, E. On the Use of Correlation and MI as a Measure of Metabolite—Metabolite Association for Network Differential Connectivity Analysis. *Metabolites* **2020**, *10*, 171.
https://doi.org/10.3390/metabo10040171

**AMA Style**

Jahagirdar S, Saccenti E. On the Use of Correlation and MI as a Measure of Metabolite—Metabolite Association for Network Differential Connectivity Analysis. *Metabolites*. 2020; 10(4):171.
https://doi.org/10.3390/metabo10040171

**Chicago/Turabian Style**

Jahagirdar, Sanjeevan, and Edoardo Saccenti. 2020. "On the Use of Correlation and MI as a Measure of Metabolite—Metabolite Association for Network Differential Connectivity Analysis" *Metabolites* 10, no. 4: 171.
https://doi.org/10.3390/metabo10040171