Stronger findings from mass spectral data through multipeak modeling
 Tommi Suvitaival^{1},
 Simon Rogers^{2} and
 Samuel Kaski^{1, 3}Email author
DOI: 10.1186/1471210515208
© Suvitaival et al.; licensee BioMed Central Ltd. 2014
Received: 19 March 2014
Accepted: 12 June 2014
Published: 19 June 2014
Abstract
Background
Mass spectrometrybased metabolomic analysis depends upon the identification of spectral peaks by their mass and retention time. Statistical analysis that follows the identification currently relies on one main peak of each compound. However, a compound present in the sample typically produces several spectral peaks due to its isotopic properties and the ionization process of the mass spectrometer device. In this work, we investigate the extent to which these additional peaks can be used to increase the statistical strength of differential analysis.
Results
We present a Bayesian approach for integrating data of multiple detected peaks that come from one compound. We demonstrate the approach through a simulated experiment and validate it on ultra performance liquid chromatographymass spectrometry (UPLCMS) experiments for metabolomics and lipidomics. Peaks that are likely to be associated with one compound can be clustered by the similarity of their chromatographic shape. Changes of concentration between sample groups can be inferred more accurately when multiple peaks are available.
Conclusions
When the samplesize is limited, the proposed multipeak approach improves the accuracy at inferring covariate effects. An R implementation and data are available at http://research.ics.aalto.fi/mi/software/peakANOVA/.
Keywords
ANOVAtype modeling Bayesian modeling Clustering Mass spectrometry Metabolomics Lipidomics Nonparametric BayesBackground
The study of changes in the levels of metabolites and lipids has become essential for the comprehensive understanding of human health [1]. Chromatographycoupled mass spectrometry (MS) techniques have become the standard method for characterizing the human metabolome [2] and lipidome [3]. The technique generates a spectrum of peaks describing the sample in the plane defined by the retention time from the chromatograph and the masstocharge ratio from the mass spectrometer. Each peak in this plane is either generated by an ion arising from one of the compounds present in the sample, or is an artifact of the measurement without association to any of the compounds. The association between the peaks and compounds is unknown a priori. The produced peak data are noisy: First, sample preparation introduces sources of uncertainty that propagate to the analysis [4]. Second, the accuracy of the device is limited [5] and it produces biases. Third, peak identification, annotation and preprocessing steps produce additional layers of uncertainty [6]. The effect of errors at all these levels is exacerbated by the “small n, large p” problem: experiments cover a very limited number of samples, n, while the set of compounds measured, p, is potentially large.
To solve the problem we introduce the following assumptions about the generative process of the data within a Bayesian model: First, samples carry betweengroup differences in their compound concentrations and the differences arise from responses to controlled covariates. Second, multiple observed spectral peaks follow an identical generative process and their heights are a noisy reflection of the true concentration level of the compound. Third, shapes of the peaks from one compound are generated through an identical process following the properties of the measurement device, and thus these shapes are highly similar.
The approach presented in this paper consists of two stages of computational inference: (1) peaks that share a compound as their generative source are clustered together, and (2) the responses to controlled covariates of the experiment are inferred on these clusters of peaks.
The clustering part of the approach is based on a nonparametric Bayesian Dirichlet process model [14]. To improve the performance of this model, we have redefined the prior distributions from a normal distribution to a beta distribution to improve the match to the peak shape similarity observations.
The model for inferring the responses to covariates operates on clusters inferred in the first part. A Bayesian multiway model [13] is suitable for this task. This model itself could be used for clustering summarized mass spectral intensity data, but in this work, we demonstrate that the clustering can be done more accurately based upon the similarity of chromatographic peak shapes.
Methods
respectively, where N, P and K are their respective total numbers. We use bold capital, bold noncapital and nonbold noncapital symbols to refer to matrices, vectors and scalars, respectively (e.g., V, v and v).
Clusters of peaks based on the similarity
Following earlier work [14], we measure the similarity between the shapes of two peaks by their Pearson correlation computed over a window of retention time after a standard peak alignment [15] across the samples. Truncating negative values to zero, this leads to a distinct similarity matrix Q_{i··}∈[0,1]^{P×P} for each sample i. In the notation, the operator “ ·” indicates that the entire dimension of the array is included, not only the single item j. Since a peak is not necessarily observed in every sample, there may be missing values in the matrices. Therefore, we construct an additional mask R∈{0,1}^{N×P×P} with binary values ${r}_{\mathit{\text{ij}}{j}^{\prime}}$ indicating whether the peak pair (j,j^{′}) in sample i appears together within the window where the similarity is measured and whether both of the peaks are observed. An unidentified peak may still be present in the sample below the limit of detection of the mass spectrometer. However, then it is not useful for the inference of covariate effects and, thus, is treated as missing.
Model
We assume that the peaks are generated through a Dirichlet process [16]: there is an unknown number of clusters and an unknown and variable number of peaks that arise from each of the clusters. Peaks are assumed to have a oneoutofmany association: each peak is associated with only one of the unknown clusters. With these basic assumptions, we can infer the PbyK clustering matrix V from the data Q. Value v_{ j k }=1 in the clustering matrix V assigns the peak j to the cluster k. To make the following equations more compact, we use an additional variable, ${\epsilon}_{j{j}^{\prime}}={\mathbf{v}}_{j\xb7}{\mathbf{v}}_{{j}^{\prime}\xb7}^{\text{T}}\in \{0,1\}$, which is an inner product of the cluster indicator vectors of the peaks j and j^{′}, to denote whether the two peaks come from the same or different clusters (1 or 0, respectively).
becomes a product over all peak pairs and samples following the distributional assumption of Equation 2.
the process may create a new cluster with the index K+1 and only the peak j inside. Then, the likelihood term is weighted by the Dirichlet process concentration parameter α_{DP}, which can be seen as a pseudocount for the number of peaks outside the current K clusters.
Inference
We infer the posterior distribution of the clustering via Gibbs sampling, which results in a set of S samples of the clustering V^{(s)}, s=1,…,S, from the true posterior distribution p(VQ,R). The computational complexity of a Gibbs iteration is $\mathcal{O}\left(K{P}^{2}\right)$. Further analysis can operate on the entire posterior distribution of the clustering through integration, or on a point estimate of the distribution. We follow earlier work [18] and acquire a point estimate of the posterior distribution of the clustering through finding the leastsquares clustering (Section 1 in Additional file 1).
Covariate effects based on peak heights
Having inferred the grouping of similar peaks into clusters that each correspond to a compound, we infer the differences in concentrations between sample groups for each cluster given the peak height data $\mathbf{X}\in {\mathbb{R}}^{P\times N}$ and the clustering V. Again, some values in the data may be missing.
Model
except for the first level, l=1, which is defined as the baseline level and thus is fixed to zero. The model is not limited to one covariate: the clusterspecific mean ${\mathbf{x}}_{i\xb7}^{\text{lat}}$ can be expressed as a sum of effects of multiple covariates and their interaction effects (Section 1 in Additional file 1). Further, the model is readily extensible for dependent covariate effects [19].
follows a scaled inverse χ^{2} distribution with n_{0} prior samples and a scale ${\sigma}_{0}^{2}$.
Inference and analysis
We infer the covariate effects via Gibbs sampling. Now the clustering matrix V has been learned earlier, and is thus taken as known in the model. Computational complexity of a Gibbs iteration is $\mathcal{O}\left(\mathit{\text{NP}}{K}^{2}\right)$. The clustering and the covariate effects can be inferred overnight on a standard desktop computer for a typicalsized data set. The posterior distributions of the covariate effects α are descriptive of the differences between the sample groups and, thus, interesting from the analysis point of view. To assess the significance of the difference between a sample group, c=l>1, and the control group, c=1, for a cluster k, we can study the posterior probability of the effect α_{ k l } being greater or less than zero.
Comparison methods
 1.
the multipeak approach using both peak shape and height information, as proposed in this work (nonparametric clustering of peaks by their shape similarity, inference of covariate effects on the clusters based on the height of the peaks),
 2.
the multipeak approach using peak height information only [13] (clustering of peaks and inference of covariate effects based on the height of the peaks only),
 3.
the typical singlepeak approach (inference of covariate effects by the height of the strongest annotated peak only).
For the studied real data sets, we discovered that peak height information alone is not enough for clustering the peaks into individual compounds due to the high level of noise and strong correlations between compounds. Thus, for real data we compared Model 1 to Model 3 and highlight the benefit gained by using peak shape information.
Model 2 assumes the generative Gaussian latent variable model of the Equations 7–10 for the intensity observations X and a uniform multinomial prior for the clustering of the peaks. The clustering is inferred by Gibbs sampling together with the covariate effects.
The Kronecker delta function ${\delta}_{{a}_{i},l}$ selects the samples that have the covariate level l by receiving the value 1, when a_{ i }=l, and 0, otherwise. When the data are logtransformed, the mean difference corresponds to the fold change computed in many analysis platforms such as MZmine [15] and XCMS [6].
Experiments
We demonstrate the performance of the proposed method through three experiments: a simulated data experiment, a spikein benchmark experiment with known changes in concentrations, and a gene silencing experiment with measurements of the lipidome of cancer cells.
Evaluation measures
Evaluation of the performance on real data sets is not a trivial task, as there is no ground truth available: neither the identity of the peaks nor the true effect sizes are known. Thus, we also used spikein data, where the true covariate effects are known, although only a small number of the peaks are annotated.
For the simulated and benchmark experiments, we computed the mean squared error (MSE) between inferred and true covariate effects as an evaluation metric. As a result of the logtransformation of the intensity data, we were quantifying relative changes between sample groups, independent of the average height of each peak. In the model, we thus assumed that the change is preserved across the peaks of one compound, in relative terms. The significance of the difference in the MSE of the proposed approach and the comparison method was tested by the paired onesided ttest. The false discovery rate was controlled by the BenjaminiHochberg stepup procedure [20]. Additionally for the simulated experiment, we studied the inference of the statistical significance of effects, since the true distribution of the data was known.
To assess the sensitivity of the approaches to noise in natural lipidomic data lacking a ground truth, we used two types of indirect evaluation: First, we studied the consistency of the inferred covariate effects given a prior assumption about their similarity. Second, we examined the robustness of the inferred covariate effects to noise. Finally, we demonstrated differences between the multipeak and singlepeak approaches through examples of qualitative analysis of annotated peak clusters.
Simulated data
We started by investigating the performance of the proposed approach on synthetic data, where the true covariate effects are known. We focused on a usual task in exploratory analysis of biological data: the detection of significant nonzero covariate effects. We measured the performance by the accuracy at this task—the ratio of true positive and true negative significant differences among all effects. We used the 95% posterior quantiles to determine the significance. Additionally, we compared the approaches by the MSE to the true effects and studied the performance of the two clustering models by computing the normalized information distance (NID) [21] to the true clustering.
The approaches were tested across a set of potential experimental settings to study how the observation of additional peaks and samples affects the performance. Simulated data were generated by assuming the latent structure of Model 1. The following data parameters were varied on a grid: samplesize N=2×{3,7,15} and peakspecific noise σ^{2}={1,5}. Additionally, the number of peaks per cluster was varied between 3, 7 and 15. Covariate effects α_{·2}=[2,1,0.5,0,0,0,0] were generated for each data set. The experiment was repeated 100 times with independent data sets. The results are reported in the Results and discussion section.
Benchmark data with known changes in concentrations
The benchmark data set of apple samples [22] includes a set of annotated spikein compounds with increases of 20, 40 or 100% in concentrations. We started with the raw spectral data [23] in order to acquire the shapes of the peaks in addition to their heights. The mass spectra were preprocessed using MZmine 2 [15] (Section 4 in Additional file 1). We used standard preprocessing methodology also used in the original publications of the data sets, thus maintaining the focus of the work on the potential benefit gained from the multiple peaks. The compared approaches were on the same line in terms of the data.
We evaluated the approaches by the MSE between inferred and true covariate effects. If the cluster contained multiple annotated peaks, the effect of each annotated peak was evaluated separately for the singlepeak approach. Clusters with no annotated peaks were considered to have a 0% true effect and the effect of the singlepeak approach was inferred based on the strongest peak of the cluster.
Lipidomic data from a gene silencing study
The data come from a recent experiment [24] to study the effects of gene silencing treatments on lipidomic profiles and growth of breast cancer tissue. Driven by the lack of ground truth about the covariate effects, we evaluated the inferred effects indirectly in two ways: (1) by quantifying the consistency of the effects within a lipid family and (2) by quantifying the robustness of the magnitudes of the inferred effects across the lipidome in presence of additional noise. Additionally, we investigated the stability of the inferred clustering on the data and qualitatively analyzed differences between the covariate effects of single peaks and the effects inferred on clusters of peaks by Model 1.
The data included 32 lipidomic profiles of breast cancer cells from the ZR751 cell line. We inferred the effects of seven distinct silencing interventions on metabolism regulating genes (Section 5 in Additional file 1) at two time points. The raw spectra were preprocessed with MZmine 2 as described in the original publication [24], in addition to which the shape similarities of the peaks were computed.
Consistency of effect signs
In the first task, we quantified the consistency as the accuracy at predicting the covariate effect of a test lipid given the model on the covariate effects of other lipids of the same family. For instance, we predicted the effect of a gene silencing treatment on the sphingomyelin SM(d18:1/22:0) based on the sphingomyelin compounds in the training set. We examined the sign of the effect instead of the absolute effect, since even within a family of lipids the changes have a high variance and thus cannot be reliably predicted without imposing additional information about the biological system.
We predicted the signs of the covariate effects for test lipids in a threefold crossvalidation setting with 100 randomizations. The examined lipids included the annotated members from the three most abundant families of lipids that had two or more peaks identified with the clustering model (Section 5 in Additional file 1).
Further, we studied the influence of noise to the consistency by adding independent normally distributed noise (from σ=0 to σ=10) on the peak intensity observations. Added noise variance σ=1 was equal to the existing original variance in the data, and the upper bound for the signaltonoise ratio then was 0.5 (Additional file 1: Table S4).
Robustness of effect magnitudes
To evaluate the inferred effects at the scale of the entire observed lipidome, we examined the consistency of inferred covariate effects between the original and noiseadded data sets. This experiment simulated the situation where the true effects are known (effects from the original data set), but the data based on which the effects are inferred are noisy (the addednoise data set). To measure the consistency, we computed the Spearman correlation between the covariate effects inferred from the original and the addednoise data sets. We studied all clusters with two or more peaks, constituting 20% of the clusters.
Results and discussion
Simulated data
On a high level of noise (σ^{2}=5), only Model 1 worked (Figure 4b). The reason for the failure of Model 2 was the imperfectly inferred clustering (Figure 4d). The good performance of Model 1 resulted from the clustering step, which is robust to noise in the peak heights. The peak shape similarity gave strong evidence for inferring the clusters already from a single sample.
Benchmark data with known changes in concentrations
In the first demonstration on real UPLCMS data [22], we show that Model 1 can infer the artificial perturbations in a spikein experiment more accurately than the singlepeak approach.
In the positive ion mode, the model inferred 794 clusters, among which 135 clusters included more than one peak. Seven clusters included annotated peaks from the spikein compounds, four of which included more than one annotated peak (Additional file 1: Table S2). Peaks from two compounds were distributed to two and four clusters, respectively. In the negative ion mode, the model inferred 367 clusters, among which 113 clusters were nonsingletons. Three clusters included annotated peaks from the spikein compounds, all of these clusters included more than one annotated peak and all peaks from one compound were clustered together. In both the ion modes, all clusters with annotated peaks were specific to one compound.
Lipidomic data from a gene silencing study
In the second demonstration on real UPLCMS data [24], we show that Model 1 can infer more consistent covariate effects in two ways even though the true effects are unknown.
Consistency and robustness of effects
Stability
Since the proposed approach is sensitive to the inferred clustering of the data, we analyzed the stability of the inferred clustering on biological data, using the lipidomic gene silencing data as a case study. We tested the influence of the concentration parameter α_{DP} in the Dirichlet process clustering model. The clustering result for the lipidomic gene silencing data was robust to changes in the magnitude of the concentration parameter (Additional file 1: Figure S2). As expected, the number of clusters increased, when the preset value of the concentration parameter increased, but the relative change was small.
Qualitative analysis
Conclusions
We have empirically demonstrated that a modelbased integration of multiple peaks can lead to an improved accuracy in the inference of covariate effects, and we introduced a novel method for this task. While the samplesize is always restricted by external constraints such as the experiment budget or the availability of suitable patients, the inference based on multiple peaks gives a shortcut to extracting more information from the limited set of samples, thereby directly addressing the “small n, large p” problem. However, some types of systematic measurement error, such as some batch effects, escape this treatment and can only be reduced by introducing independent replicates. Based on the results presented in this work, we argue that additional peaks are especially useful when the signaltonoise ratio is low and the differences between sample groups are small.
We suggest that all the detected peaks that can be associated with a compound should be taken into account in the comparative analysis. This is possible through the twostep generative modeling approach presented in this work: (1) by identifying the peaks that can be associated with one compound through clustering the peaks based on their shape similarity and (2) by the inference of covariate effects on the clusters, each representing one compound.
Abbreviations
 ANOVA:

Analysis of variance
 Cer:

Ceramide
 SM:

Sphingomyelin
 UPLCMS:

Ultra performance liquid chromatographymass spectrometry.
Declarations
Acknowledgements
The authors would like to thank Sandra Castillo, Peddinti V. Gopalacharyulu, Mika Hilvo and Matej Orešič for providing data and for useful discussions. The authors would also like to thank Rónán Daly and Joe Wandy for useful discussions. This work was supported by the Academy of Finland (Finnish Centre of Excellence in Computational Inference Research COIN, 251170; Computational Modeling of the Biological Effects of Chemicals, 140057), the Finnish Foundation for Technology Promotion (to TS) and the Finnish Doctoral Programme in Computational Sciences FICS (to TS). The calculations presented in the work were performed using computer resources within the Aalto University School of Science "ScienceIT" project.
Authors’ Affiliations
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