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Computational and Statistical Analysis of Metabolomics Data
Sheng Ren1,2, Anna A. Hinzman1,3, Emily L. Kang2, Rhonda D. Szczesniak6,7, L. Jason Lu1,3,4,5,7,*
2014.12.16 for Metabolomics
1. Division of Biomedical Informatics, Cincinnati Children’s Hospital Research Foundation,
3333 Burnet Avenue, Cincinnati, OH 45229-3026
2. Department of Mathematical Sciences, McMicken College of Arts & Sciences, University of
Cincinnati, 2815 Commons Way, Cincinnati, OH 45221-0025
3. Department of Biomedical Engineering, College of Medicine, University of Cincinnati, 231
Albert Sabin Way, Cincinnati, OH 45267-0524
4. Department of Environmental Health, College of Medicine, University of Cincinnati, 231
Albert Sabin Way, Cincinnati, OH 45267-0524
5. Department of Computer Science, College of Medicine, University of Cincinnati, 231 Albert
Sabin Way, Cincinnati, OH 45267-0524
6. Division of Pulmonary Medicine, Cincinnati Children’s Hospital Research Foundation, 3333
Burnet Avenue, Cincinnati, OH 45229-3026
7. Division of Biostatistics and Epidemiology, Cincinnati Children’s Hospital Research
Foundation, 3333 Burnet Avenue, Cincinnati, OH 45229-3026
* Corresponding authors:
Long J. Lu, Ph.D., Associate Professor
Division of Biomedical Informatics, MLC 7024
Cincinnati Children’s Hospital Research Foundation
3333 Burnet Avenue
Cincinnati, OH 45229
Phone: (513) 636-8720
Fax: (513) 636-2056
Email: long.lu@cchmc.org
Website: http://dragon.cchmc.org
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ABSTRACT
Metabolomics is the comprehensive study of small molecule metabolites in biological systems.
By assaying and analyzing thousands of metabolites in biological samples, it provides a whole
picture of metabolic status and biochemical events happening within an organism and has
become an increasingly powerful tool in the disease research. In metabolomics, it is common to
deal with large amounts of data generated by nuclear magnetic resonance (NMR) and/or mass
spectrometry (MS). Moreover, based on different goals and designs of studies, it may be
necessary to use a variety of data analysis methods or a combination of them in order to obtain
an accurate and comprehensive result. In this review, we intend to provide an overview of
computational and statistical methods that are commonly applied to analyze metabolomics data.
The review is divided into four sections. The first section will introduce the background and the
databases and resources available for metabolomics research. The second section will briefly
describe the principles of the two main experimental methods that produce metabolomics data:
MS and NMR, followed by the third section that describes the preprocessing of the data from
these two approaches. In the fourth and the most important section, we will review four main
types of analysis that can be performed on metabolomics data with examples in metabolomics.
These are unsupervised learning methods, supervised learning methods, pathway analysis
methods and analysis of time course metabolomics data. We conclude by providing a Table
summarizing the principles and tools that we discussed in this review.
Key words: computational, statistical, unsupervised learning, supervised learning, pathway
analysis, time course data
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INTRODUCTION
Omics is the study of the totality of biomolecules. Just as genomics is the analysis of a complete
genome, proteomics is the comprehensive analysis of proteins, and transcriptomics is the
comprehensive analysis of gene transcripts, metabolomics is the analysis of the complete set of
metabolites, or metabolome, in an organism (Griffin, Shockcor 2004; Oliver 2002). The
metabolome represents a large number of compounds including parts of amino acids, lipids,
organic acids, or nucleotides. Metabolites are used in or produced by chemical reactions, and
their levels can be regarded as the ultimate response of biological systems to genetic or
environmental changes. Therefore, it has been suggested that the metabolome is more sensitive
to systematic perturbations than the transcriptome and the proteome (Kell, Brown, Davey, Dunn,
Spasic, Oliver 2005).
Cellular processes involve specific metabolites for reactions. Studying and recording these
metabolites can lead to the discovery of biomarkers which are measureable biological
characteristics that can be used to diagnose, monitor, or predict the risk of diseases (Xia,
Broadhurst, Wilson, Wishart 2012). There are several approaches to studying the metabolome,
including target analysis, metabolic profiling and metabolic fingerprinting (Griffin, Shockcor
2004). Target analysis focuses on the quantification of a small number of known metabolites.
Metabolic profiling focuses on a larger set of unknown metabolites. Metabolic fingerprinting
focuses on the extracellular metabolites. Rather than studying individual metabolites,
metabolomics collects quantitative data over a large range of metabolites to obtain an overall
understanding of the metabolism associated with a specific condition (Kaddurah-Daouk,
Krishnan 2009).
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Discovering biomarkers through metabolomics will help diagnose, prevent, and produce
drugs for treatment of diseases, including cancer (Griffin, Shockcor 2004), cardiovascular
diseases (Griffin, Atherton, Shockcor, Atzori 2011), central nervous system diseases (Kaddurah-
Daouk, Krishnan 2009), diabetes (Wang-Sattler, Yu, Herder, Messias, Floegel, He, Heim et al.
2012) and cystic fibrosis (Wetmore, Joseloff, Pilewski, Lee, Lawton, Mitchell, Milburn et al.
2010). Metabolomics can be a minimally invasive procedure since data can be gathered from
plasma, urine, cerebrospinal fluid (CSF), or tissue extracts. It has also been used in studying
plants to understand cellular processes and to decode the function of genes, in studying animals
to discover biomarkers, in foods research, and in herbal medicines (Putri, Nakayama, Matsuda,
Uchikata, Kobayashi, Matsubara, Fukusaki 2013).
The idea behind metabolomics has been in existence since people have used the sweetness
of urine to detect high glucose in diabetes. In the 1960s, chromatographic separation techniques
made it possible to detect individual metabolites. Robinson and Pauling’s “Quantitative Analysis
of Urine Vapor and Breath by Gas-Liquid Partition Chromatography”, written in 1971, was the
first scientific article about metabolomics (Pauling, Robinson, Teranishi, Cary 1971). The word
“metabolome” was coined by Olivier et al. in 1998 and defined as the set of metabolites
synthesized by an organism (Oliver, Winson, Kell, Baganz 1998). Nicholson et al. first used the
word metabonomics in a publication in 1999 to mean “the quantitative measurement of the
dynamic multiparametric metabolic response of living systems to pathophysiological stimuli or
genetic modification” (Nicholson, Lindon, Holmes 1999). Griffin, in his paper (Griffin,
Shockcor 2004), suggested one of the best definitions of metabolomics given by Oliver is “the
complete set of metabolites/low-molecular-weight intermediates, which are context dependent,
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varying according to the physiology, developmental or pathological state of the cell, tissue, organ
or organism”.
DATABASES AND RESOURCES FOR METABOLOMICS
In 2004, the Metabolomics Society was established to promote the growth, use and
understanding of metabolomics in the life sciences. The Metabolomics Society later launched a
journal, Metabolomics, published by Springer. The Society now has a Twitter feed
(@MetabolomicsSoc), which provides news from the Metabolomics Society, its annual
international conference, and the Metabolomics journal. METLIN, the first metabolomics
database, was also established in 2004. In 2005, the Human Metabolome Project was launched to
find and catalogue all of the metabolites in human tissue and biofluids. This metabolite
information is kept in the Human Metabolome Database, which produced its first draft in 2007
(Wishart, Jewison, Guo, Wilson, Knox, Liu, Djoumbou et al. 2013). In recent years the number
of papers written about metabolomics has been increasing. More than 800 papers were written in
2009, compared to fewer than 50 in 2002 (Griffiths, Koal, Wang, Kohl, Enot, Deigner 2010). As
technologies for the quantification and analysis of metabolomics are adapted and improved, the
use of metabolomics is expected to continue to grow.
There are many databases containing metabolomics data, and each has different
information, ranging from NMR and MS spectra to metabolic pathways. The purpose of
metabolic databases is to organize the many metabolites in a way that helps researchers easily
identify and analyze metabolomics data. The information found in metabolite databases has
continuously been updated in recent years as metabolomics studies have become more widely
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conducted. Just as metabolomics is a new field and new approaches are still being discovered,
metabolomics databases are new and still improving. These databases contain various types of
information, including concentration, anatomical location, and related disorders. Among the
databases are the Human Metabolome Database (HMDB), MassBank, METLIN, lipid
metabolites and pathways strategy (LIPID MAPS), Madison metabolomics consortium database,
and Kyoto Encyclopedia of Genes and Genomes (KEGG).
HMDB contains detailed information for 40,444 metabolite entries with chemical, clinical,
and molecular biology/biochemistry data (Wishart, Jewison, Guo, Wilson, Knox, Liu, Djoumbou
et al. 2013). Each metabolite in the database includes a “Metabocard” with information including
molecular weights, spectra, associated diseases, and biochemical pathways. The purpose of
HMDB is to identify all of the metabolites in the human. The 39,293 spectra in MassBank are
useful for the chemical identification and structure interpretation of chemical compounds
detected by mass spectrometry (Horai, Arita, Kanaya, Nihei, Ikeda, Suwa, Ojima et al. 2010).
METLIN is a repository of over 75,000 endogenous and exogenous metabolites from essentially
any living creature, including bacteria, plants and animals (Smith, O'Maille, Want, Qin, Trauger,
Brandon, Custodio et al. 2005). LIPID MAPS is not only the largest database of lipid molecular
structures, but the lipid maps resource contains information on the lipid proteome, quantitative
estimates of lipids in the human plasma, the first complete map of the macrophage lipidome, and
a host of tools for lipid biology, including mass spectrometry tools, structure tools, and pathway
tools (Fahy, Sud, Cotter, Subramaniam 2007). The Madison metabolomics consortium database
is a resource for metabolomics research based on nuclear magnetic resonance (NMR)
spectroscopy and mass spectrometry (MS) (Cui, Lewis, Hegeman, Anderson, Li, Schulte,
Westler et al. 2008). The current total number of compounds in the Madison metabolomics
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consortium database is 20,306. Finally, KEGG contains information about metabolic pathways
(Kanehisa 2002).
Metabolomics reporting and databases currently suffer from a lack of common language or
ontologies (Wishart 2007). The issue is further aggravated by the large number of different types
of instruments used in research, each of which has its own language. This makes working with
different instruments or other laboratories difficult. A possible solution (Wishart 2007) is
standardizing data by entering data into an electronic record-keeping system such as LIMS. The
establishment of common reporting standards and data formats would make it much easier to
compare and locate metabolomics data.
EXPERIMENTAL METHODS
While tools for transcriptomics and proteomics have made significant improvements in recent
years, tools for metabolomics are still emerging. No analytical tool can measure all of the
metabolites in an organism, but nuclear magnetic resonance (NMR) spectroscopy and mass
spectrometry (MS) combined come the closest. That is, using NMR and MS together may result
in more complete data than using them individually. NMR spectroscopy and MS are the most
common technologies used to collect data from biofluids or tissues. NMR can be used to identify
and quantify metabolites from complex mixtures. NMR spectroscopy relies on certain nuclei that
possess a magnetic spin and when placed inside a magnetic field can adopt different energy
levels that can be observed using radiofrequency waves (Griffin, Atherton, Shockcor, Atzori
2011). Proton NMR (1H NMR) is the most commonly used for metabolomics. NMR approaches
can typically detect 20-40 metabolites in tissue and 50 in urine samples (Griffin, Shockcor 2004).
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It is non-destructive because the sample does not come in contact with the detector, usually
occurs in a noninvasive manner, requires no chemical derivation, and can be easily reproduced
(Armitage, Barbas 2014). A major advantage of NMR is that the signal frequencies observed in
an NMR spectrum are directly proportional to the concentration of the nuclei in the sample
(Smolinska, Blanchet, Buydens, Wijmenga 2012). However, compared to MS, it has a lower
sensitivity, only medium to high abundance will be detected (Smolinska, Blanchet, Buydens,
Wijmenga 2012).
Mass spectrometry-based metabolomics is more commonly used than NMR, judging by the
number of publications annually that use each technique (Dettmer, Aronov, Hammock 2007). In
order to separate the makeup of a mixture, MS is always coupled to other separation techniques.
Among all hyphenated MS methods, gas chromatography MS (GC-MS) and liquid
chromatography MS (LC-MS) are most popular, as they can be used to detect low-concentration
metabolites. GC-MS can be applied to the analysis of low molecular weight metabolites, and it is
highly sensitive, quantitative and reproducible (Armitage, Barbas 2014). GC-MS is also
preferred in terms of cost and operational issues (Theodoridis, Gika, Want, Wilson 2012). It can
typically detect 1,000 metabolites (Griffin, Shockcor 2004). LC-MS, which can also be prefixed
with high (HPLC) or ultra-high (UPLC) performance, is suitable for the analysis of non-volatile
chemicals, therefore it is complementary to GC-MS (Armitage, Barbas 2014). It has a high
sensitivity and is less time consuming than GC-MS, but it can be more expensive (Griffin,
Shockcor 2004). One advantage of LC-MS is that it can separate and detect a wide range of
molecules and allows for the collection of both quantitative and structural information
(Theodoridis, Gika, Want, Wilson 2012).
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In addition to the three main stream methods mentioned above, there are also other
important spectroscopy and hyphenated methods. Among all spectroscopy methods, vibrational
spectroscopy is one of the oldest (Li, Wang, Nie, Zhang 2012). There are primarily two
vibrational methods utilized: Fourier-transform infrared spectrometry (FT-IR) and Raman
spectroscopy (RS). FT-IR is inexpensive and good for high-throughput screening but it is very
poor at distinguishing metabolites within a class of compounds (Griffin, Shockcor 2004) and
much less sensitive compared to MS (Patel, Patel, Patel, Rajput, Patel 2010). Moreover, even
combining HPLC with FTIR, the method HPLC-FTIR may also have the disadvantage of
yielding a low level of detailed molecular identifications (Nin, Izquierdo-García, Lorente 2012)
and the progress in this hyphenated technique is slow (Patel, Patel, Patel, Rajput, Patel 2010).
Raman spectroscopy is an extension of FT-IR and it has been used for the identification of
microorganisms of medical relevance (Dunn, Bailey, Johnson 2005). However, although there
are some advantage to RS over FT-IR, it has similar problems as FT-IR (Griffin, Shockcor
2004). Capillary electrophoresis mass spectrometry (CE-MS) is a powerful separation technique
for charged metabolites (Dettmer, Aronov, Hammock 2007) and has been predominantly used in
targeted metabolomics (Gika, Theodoridis, Plumb, Wilson 2014). However, since the analytical
system stability is not as high as in GC or LC–MS, it is not yet applied widely in global
metabolite profiling (Theodoridis, Gika, Want, Wilson 2012).
The experimental design is an important aspect to consider before conducting any
metabolomics experiments. It is a plan of data-gathering studies, which is constructed to control
process variation in the experiments and to ensure potential confounders are not present or are
well-characterized (Dunn, Wilson, Nicholls, Broadhurst 2012). The process variation in the
experiments can be introduced in the sample collection, storage and preparation steps. For
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example, sample collection time of day (Slupsky, Rankin, Wagner, Fu, Chang, Weljie, Saude et
al. 2007), storage and experiment temperature (Cao, Dong, Cai, Chen 2008; Lauridsen, Hansen,
Jaroszewski, Cornett 2007) all have an impact on the metabolite profile determined. These
conditions and procedures, if not standardized, may lead to spurious biomarkers being reported
and may account for a lack of reproducibility between laboratories (Emwas, Luchinat, Turano,
Tenori, Roy, Salek, Ryan et al. 2014). Therefore, a standard operating procedure is essential to
control variation introduced during the sample preparation process. There are some sample
procedures for NMR (Emwas, Luchinat, Turano, Tenori, Roy, Salek, Ryan et al. 2014) and MS
(Dunn, Broadhurst, Begley, Zelena, Francis-McIntyre, Anderson, Brown et al. 2011) studies.
Controlling for potential confounding factors is also critical and is better addressed in
experimental design (Broadhurst, Kell 2006). In metabolomics research, the confounding factors
are those variables that correlate with both response variables (e.g. disease status) and
metabolites concentrations. Such factors include but are not limited to age, gender (Emwas,
Luchinat, Turano, Tenori, Roy, Salek, Ryan et al. 2014), diet (Heinzmann, Brown, Chan,
Bictash, Dumas, Kochhar, Stamler et al. 2010), physical activity (Enea, Seguin, Petitpas-Mulliez,
Boildieu, Boisseau, Delpech, Diaz et al. 2010) and individual metabolic phenotypes (Assfalg,
Bertini, Colangiuli, Luchinat, Schäfer, Schütz, Spraul 2008). Those factors, if not properly
controlled, could lead to failure of discovering true significance or reporting spurious findings
(Dunn, Wilson, Nicholls, Broadhurst 2012). For human studies, we should control for
confounders first by defining more specific criteria in selecting subjects, since subjects are
heterogeneous with respect to demographic and lifestyle factors. This is especially important in
defining healthy control (Scalbert, Brennan, Fiehn, Hankemeier, Kristal, van Ommen, Pujos-
Guillot et al. 2009). Then, it is recommended to perform sample randomization in order to reduce
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the correlation between confounders and sample analysis order and instrument conditions (Dunn,
Wilson, Nicholls, Broadhurst 2012). More advanced statistical experimental design methods, for
example nested stratified proportional randomization and matched case-control design, can be
used when outcomes (e.g. disease status) are imbalance (Dunn, Wilson, Nicholls, Broadhurst
2012; Xia, Broadhurst, Wilson, Wishart 2012). For animal studies, where many confounding
factors can be well controlled, large number of samples are not needed compared to human
studies (Emwas, Luchinat, Turano, Tenori, Roy, Salek, Ryan et al. 2014). In fact, by using some
statistical experimental design methods, such as factorial design and randomized block design,
researchers can minimize the number of samples used and control most confounders at the same
time (Kilkenny, Parsons, Kadyszewski, Festing, Cuthill, Fry, Hutton et al. 2009). There are rich
literatures discussing experimental design in both statistical methodology (Box, Hunter, Hunter
1978; Montgomery 2008) and its applications in high throughput biological assays (Riter, Vitek,
Gooding, Hodge, Julian 2005; Rocke 2004).
DATA PREPROCESSING
Data preprocessing plays an important role and can substantially affect subsequent statistical
analysis results. It takes place after the raw spectra are collected and serves as the link between
raw data and statistical analysis. NMR and MS spectra typically show differences in peak shape,
width, and position due to noise, sample differences or instrument factors (Blekherman,
Laubenbacher, Cortes, Mendes, Torti, Akman, Torti et al. 2011; Smolinska, Blanchet, Buydens,
Wijmenga 2012). The goal of preprocessing is to correct those differences for better
quantification of metabolites and improved comparability between different samples. Similar
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preprocessing considerations and methods can be applied to both MS and NMR (Vettukattil
2015).
Preprocessing for NMR typically includes baseline correction, alignment, binning,
normalization, and scaling (Smolinska, Blanchet, Buydens, Wijmenga 2012). Baseline correction
is a procedure to correct the distortion in the baseline caused by systematic artifacts. It is very
important since signal intensities are calculated with reference to the baseline (Vettukattil 2015).
Current automatic baseline correction methods are mostly based on polynomial fitting such as
local weighted scatter plot smoothing (Xi, Rocke 2008) and splines (Eilers, Marx 1996). After
baseline correction, some of unwanted spectral region are often removed, such as water and other
contaminations (Vettukattil 2015). Due to differences in instrumental factors, salt concentrations,
temperature and changes of pH, peak shifts can always been observed between samples.
Therefore, alignment must be performed in order to correct those shifts. Since most shifts in
NMR are local shifts, it is often insufficient to simply perform global alignment by spectral
referencing (Smolinska, Blanchet, Buydens, Wijmenga 2012). Several automatic methods like
icoshift (Savorani, Tomasi, Engelsen 2010) and correlation optimized warping (Tomasi, van den
Berg, Andersson 2004) can be used to perform local alignment. After automatic baseline
correction or alignment, it is recommended to visually inspect the processed spectra and one can
also choose to manually correct baseline and perform alignment (Vettukattil 2015). Binning (also
known as bucketing) is a dimension reduction technique, which divide the spectra into segments
and replace the data values within each bin by a representative value. It is a useful technique
when perfect alignment is hard to achieve (Smolinska, Blanchet, Buydens, Wijmenga 2012).
Traditional equal sized binning is not recommended since peaks can be split into two bins. Some
adaptive binning methods such as Gaussian binning (Anderson, Reo, DelRaso, Doom, Raymer
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2008) and adaptive binning using wavelet transform (Davis, Charlton, Godward, Jones, Harrison,
Wilson 2007) can overcome this difficulty to some extent. However, binning can reduce spectral
resolution, therefore, it may be better to avoid binning when spectral misalignment is not serious
or when identification of metabolites is more important (Vettukattil 2015). Normalization can
remove or correct for some systematic variations between samples, for example sample dilution
factors, which is a key factor in analysis of urinary metabolites (Smolinska, Blanchet, Buydens,
Wijmenga 2012), in order to make samples more comparable with each other. Typically,
normalization is a multiplication of every row (sample) by a sample specific constant (Craig,
Cloarec, Holmes, Nicholson, Lindon 2006). One popular normalization technique is total integral
normalization, where the total spectral intensity of each sample is the constant. When some of
the strong signals change considerably between samples, probabilistic quotient normalization can
offer more robust results than total integral normalization (Dieterle, Ross, Schlotterbeck, Senn
2006). Scaling, in metabolomics data analysis, often refers to the column operations that are
performed on each feature (spectral intensity or metabolite concentration) across all samples in
order to make the features more comparable. Scaling can affect the results of subsequent
statistical analysis and we briefly discussed this problem the Principal Component Analysis
below. Commonly used scaling methods include but not limited to autoscaling, Pareto scaling
and range scaling. More detailed discussion on these methods can be found in (van den Berg,
Hoefsloot, Westerhuis, Smilde, van der Werf 2006) and (Timmerman, Hoefsloot, Smilde,
Ceulemans 2015).
Data preprocessing for MS typically include noise filtering, baseline correction,
normalization, peak alignment, peak detection, peak quantification and spectral deconvolution.
One should note that not all methods use all of the processing steps listed above, nor do they
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necessarily perform them in the same order (Coombes, Tsavachidis, Morris, Baggerly, Hung,
Kuerer 2005). Although many preprocessing steps of MS are similar to NMR, there are still
some differences. First, a noise filtering step is often associated with MS data preprocessing to
improve peak detection (Blekherman, Laubenbacher, Cortes, Mendes, Torti, Akman, Torti et al.
2011). There are many different noise filters, such as Savitzky-Golay filter, Gaussian filters and
wavelet based filters, and wavelet based methods provides the best average performance (Yang,
He, Yu 2009), due to its adaptive, multi-scale nature (Coombes, Tsavachidis, Morris, Baggerly,
Hung, Kuerer 2005). Second, a de-isotoping step, which is specific to MS data, can be used to
cluster the isotopic peaks corresponding to the same compounds together to simplify the data
matrix (Vettukattil 2015). Third, deconvolution is an important step to separate overlapping
peaks in order to improve peak quantification. However, deconvolution also has the potential to
introduce errors and extra variability to the process (Coombes, Tsavachidis, Morris, Baggerly,
Hung, Kuerer 2005). There are many software tools available for NMR and MS data
preprocessing, a comprehensive summary of software tools can be found in (Vettukattil 2015).
OVERVIEW OF METABOLOMICS DATA ANALYSIS
There are two different approaches to processing metabolomics data: chemometrics and
quantitative metabolomics. For the former, we directly perform statistical analysis on spectral
patterns and signal intensity data and identify metabolites in the last step if needed. For the latter,
we identify all metabolites first and then analyze the metabolite’s data directly. Compared to
quantitative metabolomics, the key advantage of chemometrics profiling is its ability of
automated and nonbiased assessment of metabolites data. But it requires a large number of
spectra and strict sample uniformly, which are less concerned in quantitative metabolomics.
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Therefore, quantitative metabolomics is more amenable to human study or studies that require
less day to day monitoring (Matthiesen, SpringerLink (Online service) 2010). However, the data
analysis methods behind them are similar. In this section, we will discuss four different types of
data analysis methods. Note that these methods are not totally independent; they differ only by
serving different research purposes. Within each type of data analysis method, we select the most
basic, important and widely used models or methods based on published research and review
papers we found in metabolomics. The methods we selected cover most core methods currently
in use on metabolomics data analysis platforms, such as MetaboAnalyst (Xia, Mandal,
Sinelnikov, Broadhurst, Wishart 2012; Xia, Psychogios, Young, Wishart 2009). We also
included methods beyond the scope of such platforms. We gave brief introduction of the
background, models and algorithms, important facts and potential limitations for each method
that we discussed in detail, together with important references and illustrative examples. For the
methods not discussed in great detail, we listed a few key references. At the end, we briefly
summarized all methods discussed in Table 2 in order to offer readers a clear overview of these
methods.
Unsupervised Learning Methods
When we receive the data after pre-processing, we may wish to obtain a general idea of its
structure. Unsupervised learning methods allow us to discover the groups or trends in the data.
The word “unsupervised” here implies that the data we analyze is unlabeled with class
membership. The purpose of unsupervised learning is to summarize, explore and discover.
Therefore we may only need a few prior assumptions and a little prior knowledge of the data.
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Unsupervised learning is usually the first step in data analysis and can help visualize the data or
verify any unintended issues with the DOE. Among the many different unsupervised learning
methods, we will discuss four of the most commonly used methods in metabolomics data
analysis.
1. Principal Component Analysis (PCA)
If we have a high-dimensional dataset, e.g., dozens or hundreds of metabolites, peak locations, or
spectral bins for each subject, we may wish to find only a few combinations that best explain the
total variation in the original dataset. PCA is one of the most powerful methods to perform this
type of dimension reduction (Jolliffe 2005). The main objective of the PCA algorithm is to
replace all correlated variables by a much smaller number of uncorrelated variables, often
referred to as principal components (PCs), that still retains most of the information in the original
dataset (Jolliffe 2005). Although the number of PCs equals to the number of variables, only a
limited number of PCs are interpreted. Moreover, if the first few PCs can explain a large
proportion of variation in the data, we can visualize the data using a 2-dimensional or 3-
dimensional plot (sometimes called scores and loadings plots) (Fig. 1).
Before we perform PCA, it is recommended to standardize the variables (Jolliffe 2005).
This process consists of centering each of the vectors and standardizing each of them to have
variance equal to 1. By doing so, we actually perform PCA on sample correlation matrix instead
of sample covariance matrix. When the variables have widely differing variance or use different
units of measurement, the variables with the largest variances may dominate the first few PCs
(Jolliffe 2005). If those variances are biologically meaningful, we do not need to standardize all
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variables; otherwise it is highly recommended to perform standardization before performing
PCA (Johnson, Wichern 2007). We can calculate the sample PC scores matrix (denoted by)
of all subjects (denoted by, which is the original dataset with subjects and variables).
The process can be expressed by the following formulas is the
weight matrix (i.e., loading matrix), is the row of X, which represents the values of the
subject, is the column of , which represents the weights of all original variables on the
PC, is called the score of the subject on the PC. All of the PCs are uncorrelated
and their variances are eigenvalues of sample correlation matrix (or sample variance matrix if not
standardized). The largest eigenvalue corresponds to the first PC, the second largest eigenvalue
corresponds to the second PC, and so on. In order to measure the contribution of a PC to the total
sample variance, we use its corresponding eigenvalue divided by the summation of all
eigenvalues as a measure. This is the percentage of variance explained by the corresponding PC.
There is no rule for how many PCs to keep; we usually make the decision by checking the
“variance explained” measure mentioned above or using a scree plot (Johnson, Wichern 2007) .
Here we use a simple example of microbial metabolomics (Hou, Braun, Michel, Klassen,
Adnani, Wyche, Bugni 2012) to illustrate the basic idea of how to use PCA. Other examples of
how PCA was used in metabolomics studies can be found in (Heather, Wang, West, Griffin
2013) and (Ramadan, Jacobs, Grigorov, Kochhar 2006). In this study, the authors used PCA to
perform strain selection and discover unique natural products. They assumed bacterial strains
producing the same secondary metabolites would group together. The PCA scores and loadings
plots of 47 strains are shown in Fig. 1.
In PCA, the scores plot is mainly used to discover groups while the loading plot is mainly
used to find variables that are responsible for separating the groups. In the loading plot, we
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mainly check the points that are further from the origin than most other points in the plot. For the
scores plot (Fig. 1a), we can see seven identifiable groups; these groups were identified by
human eye. In the loading plot (Fig. 1b), point 1 corresponds to a compound that is responsible
for separating group G7 from the other groups. We cannot judge which of the points in the
loading plot are responsible for separating subjects into groups using only these two plots;
instead, we should go back to the loading matrix to check the weights. Furthermore, the
groups shown in the PCA scores plots are not necessarily the biologically meaningful groups.
PCA often provides a clue for further investigation. Although the authors mentioned 74 PCs that
were generated, which explained 98% of the variation in the data set, they did not show how
much of the variance was explained by the first two PCs. Note that PCA may not be powerful if
the first few PCs cannot explain a large proportion of variability of the sample. For example, if
the first two PCs in the plot account for only 50% of the total variation, then the visualization
results may be misleading, so we cannot identify the groups of the strains only by graphs.
2. Clustering
Unlike PCA, clustering analysis explicitly aims at identifying groups in the original dataset. All
clustering algorithms group the subjects such that the subjects in the same group or cluster are
more similar to each other than to subjects in other groups. Different algorithms may use
different similarity measures, such as various distances and correlation coefficients. Among the
many different clustering methods, we will only introduce two most common methods in
metabolomics as well as in many other areas of data analysis.
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(1) K-means Clustering
K-means clustering is centroid-based clustering and is a type of partitioned clustering method
(Hartigan, Wong 1979). Here centroid-based indicates that each cluster can be represented by a
center vector, which may not be an observation in the original dataset, . Partitioning
requires each subject to appear in exactly one cluster. K-means clustering divides subjects into k
non-overlapping clusters such that each subject belongs to the nearest mean of the corresponding
cluster. If all of the variables are numerical, we generally choose Euclidean distance as the
metric that distinguishes between the subject and the center vector. When using Euclidean
distance fails to find meaningful clusters, we may consider using other distance metrics, for
example Mahalanobis distance, which has the form . It is clear that
Euclidean distance is a special case of Mahalanobis distance when A is identity matrix. In
general, A is a covariance matrix with unknown form, a general and efficient algorithm (Xing,
Jordan, Russell, Ng 2002) can be used to learn the parameters in A together with performing K-
means clustering. Variations of the K-means clustering algorithm include using median instead
of mean as the center vector and assigning weights to each variable. There are also some
drawbacks to K-means clustering methods. The major problem is that the number of clusters,
“K”, is an unknown parameter, and thus we must determine K before we employ the algorithm.
Visualization tools such as PCA, multidimensional scaling (MDS) and self-organizing map
(SOM) may help to determine K. There are also some statistical methods for estimating K, the
most widely used methods are gap statistic (Tibshirani, Walther, Hastie 2001) and weighted gap
statistic (Yan, Ye 2007). Another problem is that K-means assumes that subjects in each cluster
are distributed spherically around the center (Hamerly, Elkan 2003). This assumption may lead
to poor performance on data with outliers or with clusters of various sizes or non-globular shapes
20
(Ertöz, Steinbach, Kumar 2003). An adaptive Fuzzy c-means clustering (Gunderson 1982;
Gunderson 1983) can be used in these cases. Fuzzy c-means clustering (Bezdek, Coray,
Gunderson, Watson 1981; Dunn 1973) is an extension of K-means where each data point
belongs to multiple clusters to a certain degree, which is called membership value. The adaptive
Fuzzy c-varieties clustering algorithm (Gunderson 1983) which is based on (Gunderson 1982) is
a data dependent approach that can seek out cluster shapes and detect a mixture of clusters of
different shapes. Therefore, it removes the limitation of imposing non-representative structures
in K-means and Fuzzy c-means clustering. An alternative way to solve the arbitrary clusters
shape problem is using kernel K-means (Schölkopf, Smola, Müller 1998) and it was suggested in
(Jain 2010). Another limitation of K-means and other clustering methods is that some variables
may hardly reflect the underlying clustering structure (Timmerman, Ceulemans, Kiers, Vichi
2010). One possible way to solve that problem is performing K-means in reduced space (De
Soete, Carroll 1994). Many methods have been proposed to improve the original reduced K-
means (De Soete, Carroll 1994), including factorial K-means (Vichi, Kiers 2001) and subspace
K-means (Timmerman, Ceulemans, De Roover, Van Leeuwen 2013). An alternative solution is
using variable selection (Steinley, Brusco 2008) or variable weighting (Huang, Ng, Rong, Li
2005). An illustrative example of using K-means clustering on metabolites profiles to explore
dietary intake patterns can be found in (O'Sullivan, Gibney, Brennan 2011). More details of how
to use fuzzy c-means in metabolomics was explained in (Li, Lu, Tian, Gao, Kong, Xu 2009).
(2) Hierarchical Clustering
Hierarchical clustering (Johnson 1967) builds a hierarchy and uses a dendrogram to represent the
hierarchical structure. Unlike K-means clustering, hierarchical clustering does not provide a
21
single partition of the dataset. It only shows the nested clusters organized as a hierarchical tree
and lets the user decide the clusters. In order to form the hierarchical tree, we must choose the
similarity metric between pairs of subjects and pairs of clusters. The similarity metric between
two subjects is distance. Different clusters will form by using different distance functions.
Commonly used distance functions include Euclidean distance, Manhattan distance,
Mahalanobis distance and maximum distance. A general discussion of distance functions can be
found in (Jain, Murty, Flynn 1999). Based on the distance function we choose, we can construct
a distance matrix for all subjects before we perform hierarchical clustering. Then we need to
select a linkage function, which is the similarity metric for pairs of clusters. Different linkage
functions will lead to different clusters. Commonly-used linkage functions include single
linkage, complete linkage and average linkage. A general discussion of linkage functions can be
found in (Hastie, Tibshirani, Friedman, Hastie, Friedman, Tibshirani 2009). An advantage of
hierarchical clustering over K-means is that it does not stop at a special number of clusters
found, but will continue to split until every object in the dataset belongs to the same cluster.
Therefore, the hierarchical tree may provide some meaningful finding of the real structure of the
dataset. However, it also has some drawbacks, for example it may not be robust to outliers.
Hierarchical clustering is often used together with a heat map to visualize the data matrix.
Heat maps use different colors to represent different values in the data matrix. The values in the
data matrix can be either the value of some variable or some statistic, e.g., correlation coefficient
or p-value. We can add hierarchical clustering trees on the side or top of the heat map so that we
can clearly see the structure of the data. There is a good example of this kind of representation
from (Poroyko, Morowitz, Bell, Ulanov, Wang, Donovan, Bao et al. 2011).
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In this paper, the authors studied the effect of different diets on selecting different intestinal
microbial communities using a metabolomics approach. They used a heat map to show the
significant p-values associated with the relationship between metabolites and bacterial taxa in
piglet cecal content (see Supplementary Fig. 1). The dendrogram on the left side shows the
hierarchical structure of different genus of bacteria; the one on the top shows the hierarchical
structure of different metabolites. This graph helps us visualize the degree to which bacteria were
associated with the same or different metabolites. Another similar example of using hierarchical
clustering together with heat map representation can be found in (Draisma, Reijmers, Meulman,
van der Greef, Hankemeier, Boomsma 2013), which used hierarchical clustering to analyze
blood plasma lipid profiles of twins.
3. Self-Organizing Map (SOM)
SOM is a powerful tool to visualize high-dimensional data (Kohonen 1990); it can thus help us
visually discover the clusters in the data. It is an arrangement of nodes in a two-dimensional
(may also be 1D or 3D) grid. The nodes are vectors whose dimension is the same as input
vectors. Since SOM is a type of artificial neural network (ANN), the nodes are also called
neurons. Unlike other types of ANNs, SOM uses a neighborhood function to connect adjacent
neurons. The neighborhood function is a monotonically decreasing function of iterated times and
the distance between the neighborhood neurons and neuron that matches the input best. It defines
the region of influence that the input pattern has on the SOM and the most common choice of the
function is Gaussian function. More technical details can be found in (Kohonen 1998). In this
way, data points located closely in the original data space will be mapped to neurons nearby.
Every node can thus be treated as an approximation of a local distribution of the original space,
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and the resulting map retains its topological structure. Before implementing SOM, one may
choose the number of nodes and the shape of grids, either hexagonal or rectangular. Using a
hexagonal grid implies that one node will have six bordering nodes. After numerous updating
cycles, each subject is finally assigned to a corresponding neuron, and neighboring neurons can
be treated as mini clusters. These mini clusters may give hints of metabolic patterns. In order to
see the clusters clearly, a unified distance matrix (U-matrix) representation can be constructed on
the top of SOM. The U-matrix nodes are located among all neighborhood neurons, and the color
codes of each node represent the average Euclidean distance among weight vectors of
neighboring neurons (Ultsch 2003). Therefore, the gaps between clusters can be shown by the
colors of U-matrix nodes.
There is an example of SOM with a U-matrix from in metabolomics research (Haddad,
Hiller, Frimmersdorf, Benkert, Schomburg, Jahn 2009). In their paper, the SOM (see
Supplementary Fig. 2) was trained with the metabolome data from three different fermentations
of C. glutamicum. The color codes show different Euclidean distances between each output node
and its four bordering nodes. White – purple represents short distance, which implies the subjects
(black points) have similar metabolic patterns, while green – yellow represents long distance,
which implies the subjects have different metabolic patterns. We can see that the clusters were
clearly separated by the green – yellow gaps. SOM have also been used to visualize metabolic
changes in breast cancer tissue (Beckonert, Monnerjahn, Bonk, Leibfritz 2003) and to improve
clustering of metabolic pathways (Milone, Stegmayer, López, Kamenetzky, Carrari 2014).
Supervised Learning Methods
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The purpose of supervised learning is different from that of unsupervised learning. Supervised
learning methods are widely used in discovering biomarkers, classification, and prediction, while
unsupervised learning methods cannot complete these tasks. However, these distinctions do not
imply that supervised methods are superior to unsupervised methods; rather, each was designed
to achieve different objectives of analysis. Supervised learning deals with problems or datasets
that have response variables. These variables can be either discrete or continuous. When the
variables are discrete, e.g., control group vs. diseased group, the problems are called
classification problems. When the variables are continuous, e.g., metabolite concentration or
gene expression level, the problems are called regression problems. The purpose of supervised
learning is to determine the association between the response variable and the predictors (often
referred to as covariates) and to make accurate predictions. It is called supervised learning
because one or more response variables are used to guide the training of the models. Usually
both a training step and a testing step are included. Supervised learning algorithms are applied on
the training dataset to fit a model, and then the testing dataset is used to evaluate the predictive
power. In these steps, we may encounter the following problems: How to extract or select better
predictors? How to evaluate the fitness and predictive power of the model? And what learning
methods and algorithms to choose?
For the first problem, the process of choosing relevant predictors is called feature selection
or variable selection. There are three main types of feature selection methods: Wrapper, Filter
and Embedded (Guyon, Elisseeff 2003). The Wrapper method scores subsets of variables by
running every trained model on the test dataset and selecting the model (subset of variables) with
the best performance. The Filter method scores subsets of variables by easy-to-compute
measures before training the models. The Embedded method, just as its name implies, completes
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feature selection and model construction at the same time. For the second problem, we first need
goodness of fit statistics to measure model fit and predictive power. Commonly used statistics
include but are not limited to: root mean square error (RMSE) for regression; sensitivity,
specificity and the area under the Receiver-Operating Characteristic (ROC) curve for binary
classification. In addition, we need test datasets to assess the predictive power and avoid over-
fitting issues. Ideally, model validation should be performed using independent test datasets;
however, gathering objective data can be expensive due to limited resources and other pragmatic
factors. Therefore, various resampling methods are often used in order to reuse the data
efficiently. These methods include cross validation, bootstrapping, jackknifing, randomization of
response variables and some others. Among all of them, bootstrapping and cross-validation are
used more often in validation supervised learning models (Hastie, Tibshirani, Friedman, Hastie,
Friedman, Tibshirani 2009). Commonly used cross-validation methods include k-fold validation
and random sub-sampling validation. Together with resampling methods, we can obtain a set of
goodness-of-fit statistics. By averaging them we can obtain a single statistics indicating the
fitness and predictive power of the model. For example, if the average of k RMSEs, which can
be the result of k-fold validation of model A, is lower than those average RMSEs of other
models, then we can conclude that model A is the better one under RMSE criteria. For the third
problem, there are many different supervised learning methods to choose from. Here we briefly
introduce two of the most widely used methods in metabolomics.
1. Partial Least Squares (PLS)
PLS (Wold 1966) is a method of solving linear models. A general linear model has the form
, where is the response variable, it can be a vector (one variable) or a matrix
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(several variables); is the design matrix whose columns represent variables and rows represent
observations; is the vector (matrix) of parameter coefficients and is the random error vector
(matrix) (Martens 1992). Generally, we use the ordinary least square solution of, which
is. However, in metabolomics analyses, we always have a large number of
variables, such as metabolites, peak locations, and spectral bins, but a relatively small number of
observations. Moreover, these variables may be linearly dependent, and thus it will be impossible
to use the conventional least squares method to solve for in the linear regression model, since
it is impossible to invert the singular matrix. At first, Principal Component Regression
(PCR) was introduced to solve this problem. Instead of using all original variables, PCR uses the
first few PCs from PCA to fit the linear regression model. But it is not clear whether those PCs
have high correlation with response variables or not. Therefore, PLS was introduced to tackle
this problem (Wold, Ruhe, Wold, Dunn 1984). PLS may also stand for projection to latent
structures, which implies how this method works. The underlying model of the PLS method has
the form (Wold, Sjöström, Eriksson 2001):
Similar to PCA, and are called and scores, which are matrices formed by latent
variables; and are called and loadings, which can be thought of as weight matrices;
and are residuals, which are the remaining amounts that cannot be explained by latent
variables. The latent variables, which can be thought of as factors, are linear combination of the
original and variables, i.e., for each latent variable and , and ; and
are called weight vectors. These latent variables may have chemical or biological meanings. The
PLS method finds a best set of variables that can explain most of the variation of . Namely,
we should find each latent variable t and u, such that, under some orthogonal conditions, their
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covariance reaches its maximum value (Abdi 2010). There are many variants of the PLS and
corresponding algorithms, which may have different orthogonal conditions and different
methods to estimate scores and loading matrices. It is important to note that PLS is different
from PCA and PCR. First, PCA is an unsupervised learning method while PCR and PLS are
supervised learning methods. Second, PCR uses the first few PCs in the PCA as predictors to fit
a latent variable regression. Thus, PCA only explains the variance in itself while a PLS model
tries to find the multi-dimensional direction in the space that explains the maximum variance
direction in the space. Therefore, the PLS method may often perform better than PCR.
The only parameter we need to specify in PLS is the number of components to keep. There
are two approaches. First, we can use plots to help us decide the components, e.g., the and
scores plot or R-square plot. Another approach is using resampling methods together with a
measure of goodness of fit or predictive power. We can select different numbers of components
and check their goodness of fit or predictive power. Since the PLS method is a dimension
reduction method itself, feature selection is not a required step in PLS. However, in order to
improve interpretation, robustness and precision, there are also some feature selection methods
that can be used with PLS. For example, we can use a two-sample t-test, a filter method, to select
variables before running PLS. Sparse PLS (SPLS), which is an embedded method, imposes
sparsity when constructing the direction vectors, thereby improves interpretation and achieves
good prediction performance simultaneously (Chun, Keleş 2010). Another method called
orthogonal projections to latent structures (OPLS) (Trygg, Wold 2002), can be embedded as an
integrated part of PLS modeling to remove systematic variation in that is orthogonal to , thus
also enhancing the interpretation of PLS.
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Although PLS was first designed to deal with regression problems, it can also be used in
classification problems. One popular method is called PLS-Discriminant Analysis (DA)
(Boulesteix 2004; Nguyen, Rocke 2002). In PLS-DA, is a vector whose value represents class
membership. When considering model validation in PLS or PLS-DA, Predicted Residual Sum of
Squares (PRESS), can be used in addition to the commonly used diagnostic methods
mentioned above. Note that is a measure of fitness of the model to the training data set while
and PRESS are used to evaluate the predictive power of the model. For the PLS-DA method,
it is recommended to use a double cross-validation procedure (Szymanska, Saccenti, Smilde,
Westerhuis 2012) along with the number of misclassifications and the area under the ROC curve
as diagnostic statistics. Using similar algorithms as PLS-DA, other variants of PLS, like SPLS
and OPLS mentioned above, can also be extended to classification problem, where they called
SPLS-DA (Chung, Keles 2010) and OPLS-DA (Bylesjö, Rantalainen, Cloarec, Nicholson,
Holmes, Trygg 2006).
Here we use an example to illustrate some application aspects of the PLS model (Kang,
Park, Shin, Lee, Oh, Ryu do, Hwang et al. 2011) (Fig. 2). The authors used OPLS-DA to classify
coronary heart failure (CHF) groups and control groups. Fig. 2a is a scores plot of the first two
components, which shows the similarities and dissimilarities of the subjects. In this plot, we can
see that the diseased and control groups can be clearly separated by the OPLS-DA model. Fig.
2b is the corresponding loadings plot. Different from a PCA loadings plot, the metabolites
identified are responsible for the classification. The upper section of Fig. 2b shows that
metabolites increased in the control group while the lower section shows that metabolites
increased in heart failure group. They ran the OPLS-DA model with NMR spectra data (which is
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the input data matrix ) and then identified the metabolites responsible for the separation using
results in Fig. 2b.
PLS method has been successfully applied in numerous metabolomics studies for disease
classification and biomarker identification (Marzetti, Landi, Marini, Cesari, Buford, Manini,
Onder et al. 2014; Velagapudi, Hezaveh, Reigstad, Gopalacharyulu, Yetukuri, Islam, Felin et al.
2010; Zhang, Gowda, Asiago, Shanaiah, Barbas, Raftery 2008). Note that the PLS method
sometimes can also be used as a dimension reduction (feature selection) tool rather than as a
classification method (Bu, Li, Zeng, Yang, Yang 2007).
2. Support Vector Machine (SVM)
Since metabolomics data is represented in matrix form, every subject is a row vector; thus, each
subject can be viewed as a point in a p-dimensional space where p is the number of variables. If
we can separate the data into two groups, intuitively we can find a “gap” between these two
groups in the p-dimensional space. SVM tries to find such a gap that is as wide as possible
(Cortes, Vapnik 1995). The margins for the gap are defined by support vectors, i.e., the points
located on the margin. SVM is trained to determine the support vectors. The boundary in the
middle of the gap that separates the data is called the separating hyper plane. The prediction is
done by deciding to which side of the hyper plane new subjects (observations) belong.
The original SVM algorithm is a linear classifier, which means it can only produce a hyper
plane (p-1 dimension plane in p-dimensional space) to classify the data. We aim to find the
largest margin, i.e., the largest distance between two groups, which can be solved by quadratic
programming. The related mathematical expression of this problem has been documented by
Bishop (Bishop 2006). However, it is quite common that the data cannot be linearly separated,
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i.e., a separating hyper plane does not exist. In this case, we can use kernel trick to map the
original data to a higher dimensional space so that it can be linearly separated in that space.
Kernel trick or kernel substitution is very useful in extending algorithms. It substitutes the inner
product (linear kernel) with other kernels. Commonly used kernels include the polynomial kernel
and the Gaussian kernel (Bishop 2006).
Another problem is the stability of the algorithm. If there are outliers or mislabeled data,
the original SVM may give an unsatisfactory classification result. In this case, we can use SVM
with a soft margin to solve this problem. The soft margin SVM allows some misclassification in
the training step by adding slack variables to the original objective function. This modification
changes our objective from maximizing the margin between two groups to maximizing the
margin as cleanly as possible. Here “clean” means only a few misclassified subjects.
Since SVM is a well regularized method, it does not always require a feature selection step.
However, there are some feature selection methods that can be used to enhance the performance
of SVM and lower its computational cost. Examples include the recursive feature elimination
(RFE) method and L1 norm SVM (Guan, Zhou, Hampton, Benigno, Walker, Gray, McDonald et
al. 2009). As discussed in the PLS section, similar validation methods and diagnostic measures
can be applied to the SVM algorithm. Moreover, these validation methods and diagnostic
measures can help us select optimum parameters such as which kernel to choose and its
parameters.
Compared to the PLS-DA method, one minor disadvantage of SVM is that it is often
difficult to visualize and interpret the classification result using plots; especially when the
number of variables is large. However, for classification purposes, we still recommended SVM
over most other methods (Mahadevan, Shah, Marrie, Slupsky 2008), and SVM has been widely
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used for classification and prediction in metabolomics research, especially in cancer research
(Guan, Zhou, Hampton, Benigno, Walker, Gray, McDonald et al. 2009; Henneges, Bullinger,
Fux, Friese, Seeger, Neubauer, Laufer et al. 2009; Stretch, Eastman, Mandal, Eisner, Wishart,
Mourtzakis, Prado et al. 2012). Moreover, SVM can also be used in regression problems, where
it is called support vector regression (SVR) (Brereton, Lloyd 2010). Detailed discussions on
SVR and its applications in metabolomics and chemometrics can be found in (Li, Liang, Xu
2009).
Pathway analysis methods
Pathway analysis allows us to detect the biological mechanisms in which identified metabolites
are involved. Some metabolic pathway analysis methods are directly borrowed from gene
pathway analysis, e.g., over-representation analysis (ORA) and enrichment score. Here we
provide a brief introduction to ORA, Functional Class Scoring (FCS) and some pathway
simulation methods.
1. Over-Representation Analysis (ORA)
During the research there may be cases where we have a list of metabolites identified and we
only want to know which pathways are involved in the samples being studied. There are many
metabolic pathway databases available on the internet. In this case the pathways are already
specified and we only need to test which pathway is significantly involved based on the available
samples. This kind of pathway analysis is called knowledgebase-driven pathway analysis
(Khatri, Sirota, Butte 2012). Among all of the knowledgebase-driven pathway methods, ORA is
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well-known and the simplest. ORA is used to test whether pathways are significantly different
between two study groups. Before performing ORA, we should have a list of metabolites
showing significant differences between two groups. This can be done by using two sample tests,
e.g., t-tests or nonparametric tests, for all metabolites. Then we select metabolites whose
significance reaches a predetermined threshold for false discovery rates (FDRs) or p-values.
Next, we can perform ORA, which is equivalent to a 2×2 contingency table (Table 1) test in
statistics. After we obtain all related pathways from knowledgebase, we can count the number of
metabolites in or not in both a known pathway and the list, then perform a statistical test for
whether this pathway is significantly involved for each known pathway. The most frequently
used tests include the chi-square test, which requires larger sample sizes, and Fisher’s exact test,
which is more appropriate for smaller cell counts in the table and uses a hypergeometric
distribution (Agresti 2014).
2. Functional Class Scoring (FCS)
ORA is simple to perform, but it has several drawbacks. First, much information is lost since
only the most significant metabolites are used and the rest are ignored, and only the number of
identified metabolites is considered. Second, the optimal threshold is unclear. Third, it assumes
improper independence. For example, it assumes that each metabolite and pathway is
independent of others, but in reality, this assumption may not be valid. Therefore, another class
of methods called Functional Class Scoring (FCS) was proposed to address some of the
limitations in ORA. A general framework of univariate FCS methods works as follows: First,
obtain single-metabolite statistics (e.g., t-statistic and z-statistic) by computing differential
expression of individual metabolites. Second, aggregate those single-metabolite statistics to
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compute a pathway level statistic. This pathway level statistic can be univariate or multivariate.
Commonly used univariate pathway level statistics include mean, median and enrichment score
(Holmans 2010). For multivariate statistics, a widely used statistic is Hotelling’s statistic,
which has an F distribution under the null hypothesis (Johnson, Wichern 2007). The final step is
hypothesis testing. There are two kinds of null hypothesis: competitive and self-contained. A
competitive test considers metabolites both within and outside of the pathway, while a self-
contained test ignores metabolites that are not in the pathway. In other words, for a competitive
test, the null hypothesis being tested is that the association between the specific pathway and
disease is average; while for a self-contained test, the null hypothesis is that there is no
association between the specific pathway and disease (Holmans 2010). For the multivariate
statistics, the null hypothesis is self-contained since the null hypothesis is that there is no
association between metabolites in the pathway and the phenotype (Holmans 2010). Although
multivariate statistics take the correlation of different metabolites into account, they may not be
necessarily more powerful than univariate statistics (Khatri, Sirota, Butte 2012). There are also
some drawbacks of the FCS method. If two pathways have the same metabolites, FCS will give
the same result. That is, FCS does not take the reactions among metabolites (topology structure)
into account. One way to address this problem is to use a correlation measure, such as the
Pearson correlation coefficient, to help us choose the most suitable pathway; another method is
to use pathway reconstruction.
3. Metabolic pathway reconstruction and simulation
Metabolic pathway/network reconstruction and simulation are a batch of methods used to refine
or construct metabolic networks. A reconstruction collects all of the relevant metabolic
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information of an organism and compiles it in a mathematical model. The relevant metabolic
information includes all related known chemical reactions, previously constructed networks,
experimental data, and related research results. After compiling the data into a model, we can
obtain the output of the system and then use it to refine our model and perform the simulation
iteratively. If we have the knowledge of all involved metabolites, then we can enhance the
predictive capacity of the reconstructed models by connecting the metabolites within the
pathways. The pathway models can be roughly classified into one of two categories: static
(stoichiometric network models) and kinetic models. We will first discuss static models and then
give a brief introduction on kinetic modeling.
The mathematical model behind static models is a linear system. If we treat the metabolic
network as a system, then, based on mass conservation of internal metabolites within a system,
we can express the reaction network by a stoichiometric matrix (). Each element of ()
represents the coefficient of metabolite involved in reaction . At a steady state, we have
, since there is no accumulation of internal metabolites in the system (Schilling, Schuster,
Palsson, Heinrich 1999). This linear system is called the flux-balance equation. Here
represents fluxes through the associated reactions in . These linear equations define the entire
reaction network; all of the solutions to this linear system are valid steady-state flux
distributions. In general, is an matrix where the number of columns is larger than the
number of rows ( ), which means there are more reactions than metabolites (Schilling,
Schuster, Palsson, Heinrich 1999). Moreover, is a full rank matrix, which means
. Therefore, there are multiple solutions to this linear system. Different solutions define
different pathways. Based on different research purposes, we may impose different constraints
on the linear system and obtain different types of solutions.
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Here we introduce three kinds of solutions that are the most widely used in metabolic
pathway analysis. The first type of solution is called elementary modes. In addition to the flux-
balance equation, we add the constraints that all fluxes are greater than 0. Then, by applying the
convex analysis method, we can find a solution set called elementary modes (EM) if the
following properties are satisfied:
(i) Uniqueness: The solution set is unique for a given network.
(ii) Non-decomposability: Each solution in the solution set consists of the minimum number of
reactions that it needs to exist as a functional unit. If any reaction in a solution set were removed,
the whole solution set could not operate as a functional unit.
(iii) The solution set is the set of all routes through a metabolic network consistent with the
second property (Papin, Stelling, Price, Klamt, Schuster, Palsson 2004).
The second type of solution is called extreme pathways (EP). By convex analysis, the
solution set is called the extreme pathways if it is under the same constraints of EM and follows
properties below:
(i) Uniqueness: The solution set is unique for a given network.
(ii) Non-decomposability: Each solution in the solution set consists of the minimum number of
reactions that it needs to exist as a functional unit.
(iii) This solution set is the systemically independent subset of the elementary modes; that is, no
solution in the set can be represented as a nonnegative linear combination of any other solutions
in the solution set, namely, they are convex basis vectors (Papin, Stelling, Price, Klamt, Schuster,
Palsson 2004).
By using these two kinds of solutions, we can analyze or construct metabolic pathways and
networks. Note that we may have a finite number of solutions for EM and EP, which means that
36
we may obtain several different pathways. A numerical example of the calculation and use of the
EM and EP can be found in (Förster, Gombert, Nielsen 2002). The key difference between EM
and EP is that they treat internal reversible and irreversible reactions differently. EP analysis
decouples all internal reversible reactions into forward and reverse directions while EM analysis
accounts for reaction directionality through a series of rules in the corresponding calculations of
the modes. Moreover, EPs are subsets of EMs, i.e., the numbers of extreme pathways are smaller
(potentially much smaller) than or equal to the number of elementary modes.
Another important solution corresponds to an independent analysis method called Flux
Balance Analysis (FBA). FBA differs from EM or EP by imposing more constraints and an
objective function. Depending on the purpose of the research being performed, we have different
problems of interest pertaining to the pathway. For example, we may want to maximize or
minimize the flux of certain reactions; or want to limit some flux to a certain interval and see
how the pathway changes. Therefore we need to impose an objective function on the linear
system. The problem is thus transformed into an optimization problem. We can use linear
programming to solve this problem; it does not matter how many linear constraints we have to
impose on the flux vector. Note that FBA generally gives only one solution. This is in contrast
to EM and EP, which give several solutions. Fig. 3 shows how to perform an FBA (Raman,
Chandra 2009). First, we specify the reaction network that contains all metabolites and detailed
information on all possible reactions (Fig. 3, 1st step). Internal fluxes are denoted by
and exchange fluxes are denoted by . Then, after building the linear system
based on the network structure (Fig. 3, 2nd and 3rd steps), we can add a biologically relevant
objective function and relevant constraints (Fig. 3, 4th and 5th steps). The remainder of the task is
37
linear programming (Fig. 3, last step), which can be accomplished using software packages, such
as MATLAB COBRA toolbox (Becker, Feist, Mo, Hannum, Palsson, Herrgard 2007).
The pathway constructed using the methods above cannot show the dynamic state such as
regulatory effects, how the enzymes work, or whether the pathway is in stable steady state.
Therefore, we may use a kinetic model to simulate the metabolomics network (Tomar, De 2013).
A kinetic reaction network model can be described by ordinary or partial differential equations
(ODE or PDE, respectively). For example, we can simulate the network based on the following
simple ODE (Steuer 2007).
;
is the time-dependent concentration vector of m internal metabolites, represents the
Michaelis-Menten kinetics parameters, and S is the stoichiometric matrix. is a vector of
enzyme-kinetic rate equations that consists of nonlinear functions of and . We can see that if
we let the left-hand-side term equal zero (indicating that it is at a steady state) and let be a flux
vector, then the equation is exactly the flux balance equation. Given an initial condition of ,
the value of the kinetic parameters and the rate equations , we can simulate the data through the
ODE given above. A common choice for the rate equations (for every in) is
,
where is the maximal reaction velocity. However, sometimes we do not know the explicit
form of the rate equations or it is difficult to estimate the kinetic parameters k. In these cases, we
can use the structural kinetic modeling method (SKM) (Steuer 2007). The SKM method uses the
Jacobian matrix as a local linear approximation of the rate equations. The Jacobian matrix
consists of all first order partial derivatives of the rate equations and it can be rewritten and
estimated using the SKM method (Wiechert 2002) and (Steuer 2007).
38
Analysis methods for time course data
Variables, for example the concentration of metabolites, may change with time, thereby creating
a time dimension in the dataset. Unsupervised learning and data visualization tools are still
initially useful for giving us a general idea of the data structure. We can also use visualization
tools such as PCA, SOM, and heat maps with a hierarchical clustering structure to detect patterns
and groups/clusters of the data. The only difference is that we should include a time dimension.
In addition, by drawing profile graphs we can check the profiles of metabolites or subjects for
different clusters. However, if we want to compare the temporal profiles (similar or different
patterns of change) of metabolites between different subjects or groups of subjects, we need to
introduce statistical methods different from those described above. Among different statistical
methods that can be used to analyze time course data (Smilde, Westerhuis, Hoefsloot, Bijlsma,
Rubingh, Vis, Jellema et al. 2010), we only introduce analysis of variance (ANOVA) based
methods.
If we are analyzing one variable over time, e.g., the expression level of a protein or
concentration of a metabolite, and we want to test whether the temporal profiles of this variable
are significantly different under different experimental conditions, a natural choice is to use two-
way ANOVA, which is often used when studying the effects of different treatments in chemical
or biological experiments. Here we show the basics of this ANOVA model. In metabolomics
research, the experimental condition (α) and time effect (β) can be treated as two fixed effects.
The general linear model for a two-way ANOVA is:
39
where refers to the measurement obtained from the subject at the time point under
the condition ( ); α and are fixed effects
corresponding to condition and time; a and b are the total number of levels for each effect; n is
the number of replicates (often corresponding to subjects) for each combination of condition and
time effects; µ is the overall (grand) mean. In many applications involving the traditional two-
way ANOVA, are assumed to be independent random errors following a normal distribution,
denoted by (Kutner 2005).
Although it seems reasonable to use a two-way ANOVA to analyze time course data, the
data may be better described by a repeated measures (RM) model. The main difference between
the two-way ANOVA and the RM model is that the RM model has a subject error term (),
which takes variation within the group or subject into consideration. The following is the model
of RM:
Where . are subject effects (within group error)
which follow a normal distribution
and are independent of random error . Other
notations are the same as the two-way ANOVA model mentioned above. In metabolomics
research, there are often some differences between subjects even within the same groups. If the
subjects vary a great deal within each group, then the RM model will be more powerful than the
simple two-way ANOVA analysis (Milliken, Johnson 2009). Since each subject is repeatedly
assessed in most time course studies, we strongly recommend to use the RM model instead of
two-way ANOVA. With the RM model, the user can specify a variety of correlation structures
for the measurement error. Compound-symmetric correlation is often assumed in repeated
measures analyses. This covariance structure for the allows for within-subject correlation
40
that is common over time. If the correlation is expected to decay over time, it is advantageous to
consider autoregressive covariance structures (Brockwell, Davis 2002) or exponential covariance
functions (Szczesniak, McPhail, Duan, Macaluso, Amin, Clancy 2013).
Inference procedure for RM model is quite similar to ANOVA. We can calculate the
statistics and p-values by decomposing the total sum of squares (SST) into different parts as
shown in the following formula:
SST= SS (α) +SS () +SS (α) +SS () +SSE
After validating all assumptions (normality, independence and homogeneity of the variances),
we turn to look at the ANOVA table (Kutner 2005; Milliken, Johnson 2009). The F statistics
each follow an F distribution under the null hypothesis (corresponding effects are all zero) with
corresponding numerator degrees of freedom. Therefore, we have three effects to test: two main
effects (α and ) and an interaction effect. Note that we must always test the interaction between
α and first, since a significant interaction may mask the significance of main effects and
influence the explanation of data. The null hypothesis for the interaction effect is: for
all i and j. If the interaction effect is significant, it implies that the temporal profiles for different
groups (experimental conditions) are different. Usually, the estimated treatment means plots (and
many other plots) will give us a straightforward explanation of main and interaction effects
(Milliken, Johnson 2009).
If the response variable Y is a matrix, then we will need to analyze multiple metabolites
simultaneously, while taking their correlation structure into consideration. In this case, we need
another method called ANOVA-simultaneous component analysis (ASCA). ASCA is a
generalization of ANOVA from the univariate case to the multivariate case. Statistically,
traditional MANOVA is a generalization of ANOVA. However, MANOVA will not work if the
41
covariance matrices are singular or the assumption of multivariate normality is violated. The idea
behind ASCA comes from principal component analysis, which decomposes the original data
matrix into a component score and a loading matrix plus an error term. The following is the full
model (Smilde, Jansen, Hoefsloot, Lamers, van der Greef, Timmerman 2005):
With the following constraints:
; . H denotes the number of groups and denotes the number of
replicates in group h. is a matrix where denotes the number of available time
points and J denotes the number of variables. is a K dimensional vector of ones and is a J
dimensional vector of overall means of variables, so each row in the matrix represents the
overall mean of the variables. is the matrix of “time” effect; represents the “time” and
treatment interaction effect; is the interaction of treatment, time, and subject; is the
matrix of residuals. The corresponding matrices are loading matrices. Unlike in ANOVA, we
will use the scores plots of the first few components for each effect ( matrices, a.k.a., sub-
model) to detect the time main effect or interaction effect; and we can use the corresponding
loadings plots to detect which variables are responsible for the variation. There are some
examples of ASCA score plots from (Nueda, Conesa, Westerhuis, Hoefsloot, Smilde, Talon,
Ferrer 2007).
In their paper, they mainly discussed the application of ASCA on time course microarray
data. As an example, in the score plots of their simulation study (see Supplementary Fig. 3), sub-
42
model a represents the time main effect and sub-model represents the treatment effect
(treatment main effect and its interaction with time effect). The percentages on the left show how
much the components in the sub-model (principal components kept in this matrix) explain the
variation in the corresponding effect (sub-model). The first plot shows the positive time main
effect exists; the following two show differences between subjects on the same treatment. Note
that in this example, the time-treatment effect was not modeled independently, which is different
from that in the original paper of ASCA (Smilde, Jansen, Hoefsloot, Lamers, van der Greef,
Timmerman 2005).
There are many other methods for analyzing time course data that we did not discuss in this
paper, such as the time-series data analysis (ARMA model) (Smilde, Westerhuis, Hoefsloot,
Bijlsma, Rubingh, Vis, Jellema et al. 2010). The ARMA model makes sense only when we have
many more than just two or three time points. If our dataset has many time points and we wish to
find and compare the profile curves, then a functional based method (Berk, Ebbels, Montana
2011) may be a good choice. The paper proposed a smoothing splines mixed effects model that
treats each longitudinal measurement as a smooth function of time and uses a functional t-type
test statistic to quantify the difference between two sets of curves. See (VanDyke, Ren,
Sucharew, Miodovnik, Rosenn, Khoury 2012) for a biomedical application related to this
approach. Furthermore, since metabolomics is well suited to longitudinal studies, if we have
many time points and experimental conditions, we can use the Hierarchical Linear Model to fit
the data. It treats the time profile as a function and the parameters of the time profile function as
random variables (Jansen, Hoefsloot, Boelens, van der Greef, Smilde 2004).
43
CONCLUSIONS
Metabolomics is a rapidly growing field that has greatly improved our understanding of the
metabolic mechanisms behind biological processes as well as human diseases. Its broad goal is
to understand how the overall metabolism of an organism has been changed under different
conditions. Metabolomics has been used to study diseases such as cystic fibrosis, central nervous
system diseases, cancer, diabetes, and cardiac disease. Using metabolomics could lead to the
discovery of more accurate biomarkers that will help diagnose, prevent, and monitor the risk of
disease. This review briefly introduced the background of metabolomics, NMR and MS
strategies, and data pre-processing. We then placed our main focus on the data analysis of
metabolomics and described mainstream data analysis methods in current metabolomics
research. These include unsupervised learning methods, supervised learning methods, pathway
analysis methods and time course data analysis. Finally, in Table 2, we summarized the key
points of the methods discussed, as well as some basic methods such as fold change and two
sample t-test that were not included in this review. We hope our review will be a useful reference
for researchers without this type of background in data analysis.
44
Acknowledgement
We would like to express our great appreciation to Dr. Lilliam Ambroggio and Dr. Lindsey
Romick-Rosendale for their valuable and constructive suggestions to our review. Their
willingness to give their time so generously has been very much appreciated. This study is
funded by the NIH grant R01 HL116226 to RDS and LJL.
45
Permissions
For the figures cited in the manuscript and supplementary materials, we have obtained
permissions from copy right owners for both the print and online format. We can send those
permissions to Metabolomics when needed.
46
Conflict of interest statement
Sheng Ren, Anna A. Hinzman, Emily L. Kang, Rhonda D. Szczesniak and L. Jason Lu declare
that we have no conflict of interest and we have included separately signed conflict of interest
forms in this manuscript.
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63
Table 1. ORA analysis table
Metabolites on list
Metabolites not on list
Subtotal
Metabolites in the pathway
a
b
a+b
Metabolites not in the pathway
c
d
c+d
Subtotal
a+c
b+d
64
Table 2. Summary of commonly used metabolomics data analysis methods
Type of
analysis
Basic Methods
Goal
Application
Input
Output
Softwarea
Basic
statistical
testing
Fold Change
Biomarker
discovery;
feature selection
for supervised
learning
Fold change of metabolite
concentrations between two
groups
Data tables with class
membership: each row
represents one subject and
each column represents
concentration of a
metabolite/ MS and NMR
peak lists or spectral bins
Lists of selected
metabolites with p-
values, volcano plot
(for two groups)
MetaboAnalystb,
R and MATLAB
Statistical
Testing
Two sample
tests (e.g. t-
test)
Identify significantly
different expressed
metabolites between two
groups
ANOVA
with post-
hoc analysis
Identify significantly
different expressed
metabolites for multiple
groups
Unsupervised
learning
Principal Component
Analysis (PCA)
Data grouping
and
visualization
Reduce dimensionality and
check clusters visually.
Same as above, but no
need for class
memberships
Scores and loading
plots for visualization
MetaboAnalyst,
R and MATLAB
Clustering
K-means,
fuzzy c-
means, K-
means in
reduced
space
Group the subjects into k
different clusters.
Cluster labels for
each subject
Hierarchical
Show how subjects form
different clusters.
Heatmap,
dendrogram
Self-organizing Map
(SOM)
Reduce dimensionality and
check clusters visually.
SOM
65
Type of
analysis
Basic Methods
Goal
Application
Input
Output
Softwarea
Supervised
learning -
Classification
Support Vector Machine
(SVM)
Predict class
membership
(disease
diagnostic),
biomarker
discovery
All of them are classification
methods suitable for
metabolomics research; need
to compare their performance
in real data analysis
Same as above, but class
memberships are needed
A prediction model
with selected
metabolites as
predictors
MetaboAnalyst,
R,and MATLAB
Partial Least Square –
Discriminant Analysis
(PLS - DA), OPLS-DA
and SPLS-DA
Supervised
learning -
Regression
Support Vector
Regression (SVR)
Predict
continuous
variables,
calibration,
biomarker
discovery
All of them are regression
methods suitable for
metabolomics research; need
to compare their performance
in real data analysis
The response variables
should be continuous
vector or matrix; for
calibration problem, the
response variables are
concentration and
covariates are spectral
intensity
A prediction model
with selected
variables as
predictors
Partial Least Square
(PLS), OPLS and SPLS
Pathway
analysis
Over-representation
Analysis (ORA)
Find
biologically
meaningful
metabolite sets
or pathways that
are associated
with certain
diseases or
characters
Find related pathways or
metabolite sets using
statistical testing
Significant metabolites
list and reference
pathways
A list of selected
pathways with their
p-values or FDRs
MetaboAnalyst,
Bioconductorc
Functional class scoring
(FCS) /
Enrichment Analysis
Metabolites list with
concentration and
reference pathways or
metabolite sets
Elementary modes/
Extreme pathways
Simulation and
pathway
reconstruction
Analyze or reconstruct
metabolic pathways
Pathway, stoichiometric
matrix
Several different
possible flux
distributions
MATLAB
66
Type of
analysis
Basic Methods
Goal
Application
Input
Output
Softwarea
Flux Balance Analysis
Find the flux distribution by
maximizing/ minimizing a
reaction based on some
constraints
Pathway, stoichiometric
matrix, constraints, and an
objective function
One unique flux
distribution
MATLAB
Kinetic reaction network
model
Simulate the dynamic
behavior of metabolites
reaction networks
Pathway, stoichiometric
matrix, Kinetic parameter
and rate equation
Simulation results of
network kinetics
Time course
data analysis
Analysis
of
variance
(ANOVA)
Fixed effects
ANOVA
Test time
dependent
effects; compare
time profiles for
metabolites of
different
subjects
Two or more given factors
Data tables with labels
(class membership
indicators): each row
represents the observation
value of one subject at
one time point and each
column represents
metabolites concentration/
MS and NMR peak lists
or spectral bins
Testing results of
time dependent
effects for each
metabolite, time
profile plots
MetaboAnalyst,
R
Repeated
measures
ANOVA method when we
have multiple within subjects
measurements (measured at
different time points)
ANOVA-simultaneous
component analysis
(ASCA)
ANOVA for multivariate
response
Scores and loading
plots of sub models,
time profiles plots
Functional based method
(e.g. smoothing splines
mixed effects model)
Estimate sets of curves
(metabolic profiles) and
quantify their difference
Testing results of the
differences in time
profiles between
groups for each
metabolite
Time series models (e.g.
ARMA model)
Model the
dynamic
property (e.g.
biorhythm) of
metabolites data
Long time series data
modeling
A model that best
describe the dynamic
property of
underlying biological
process
67
a. We only picked some commonly used software packages or web tools in metabolomics. b. MetaboAnalyst is an easy-to-use web based tool that
covers most basic computational and statistical methods for metabolomics (Xia, Mandal, Sinelnikov, Broadhurst, Wishart 2012; Xia, Psychogios,
Young, Wishart 2009). c. Bioconductor was built on R and provided many useful packages for bioinformatics (Gentleman, Carey, Bates, Bolstad,
Dettling, Dudoit, Ellis et al. 2004), including packages for metabolic pathway analysis (Luo, Brouwer 2013; Zhang, Wiemann 2009).
