Article
Analysis of Liquid Chromatography-Mass Spectrometry Data with an Elastic Net Multivariate Curve Resolution Strategy for Sparse Spectral Recovery Daniel Wesley Cook, and Sarah C. Rutan Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.7b01832 • Publication Date (Web): 24 Jun 2017 Downloaded from http://pubs.acs.org on July 1, 2017
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Analytical Chemistry
Analysis of Liquid Chromatography-Mass Spectrometry Data with an Elastic Net Multivariate Curve Resolution Strategy for Sparse Spectral Recovery Daniel W. Cook and Sarah C. Rutan Department of Chemistry, Box 842006, Virginia Commonwealth University, Richmond, VA 23284-2006 ABSTRACT: Analysis of liquid chromatography-mass spectrometry (LC-MS) data requires the differentiation between a small number of relevant chemical signals and a larger amount of noise. This is often done based, at least partially, on a threshold which assumes that low intensity m/z signals arise from the noise. This eliminates low intensity fragments, isotopes, and adducts and may exclude relevant low-intensity compounds all together. This work describes the use of multivariate curve resolution-alternating least squares with an additional sparse regression step using elastic net (MCR-ENALS) to distinguish relevant m/z signals without the use of a harsh thresholding step, thus allowing for discovery of low intensity m/z signals corresponding to the analytes. This strategy is demonstrated first on a unit mass analysis of amphetamines in which relevant m/z signals are found at as low as a 0.1% intensity relative the molecular m/z peak. The incorporation of MCR-ENALS into our previously reported data reduction strategy for analysis of high resolution LC-MS is also demonstrated. Analysis based on only 0.3% of the original data set, while retaining low intensity isotope peaks, was accomplished without the use of thresholding, allowing for the application of MCR-ENALS to the high resolution LC-MS data.
INTRODUCTION The wealth of information generated by liquid chromatography-mass spectrometry (LC-MS) analyses allows for the resolution, identification, and quantitation of compounds in mixtures. The high selectivity of LC-MS also makes it ideal for –omics applications in which information is extracted from the raw data and submitted to further statistical analysis which can identify biomarkers. The extraction of information from LC-MS data is often a limiting step in the analysis due to the large amounts of data obtained during these analyses. It is crucial to be able to distinguish between relevant chemical signals and the noise, which often comprises the majority of the data. This peak detection step is often performed partially based on an intensity threshold in the m/z domain with the assumption that low intensity m/z signals are noise and large intensity m/z signals are masses of interest.1–3 This assumption is clearly incorrect for smaller intensity peaks arising from adducts fragments, isotopes, etc. and may even eliminate a signal all together if its intensity is less than the threshold. An alternative approach is to use multivariate curve resolution-alternating least squares (MCR-ALS), which aims to mathematically extract pure analyte chromatographic and spectral profiles for each analyte. Ideally, these spectral profiles would only contain a small number of masses because electrospray ionization is a soft ionization method producing a limited number of fragment ions. While MCR-ALS has been previously applied to LC-MS data analysis,4–15 the resolved spectral profiles can still be noisy. The least squares fitting of the spectral profiles performed during MCR-ALS assumes that each m/z value can potentially contribute to the spectral profile of each analyte. In reality a very small proportion of the m/z values correspond to a given
analyte and the intensity at all other m/z values should be zero for that analyte. This property is known as sparsity and more advanced regression tools are available for sparse data analysis. One such method is elastic net,16 which penalizes the ordinary least squares fitting to minimize the number of m/z values used to fit the analyte signal.17 This greatly simplifies the interpretation of the resolved spectra by eliminating the need for either a threshold on the resolved spectra or visual inspection of the spectral profiles to assign to assign masses. To our knowledge, our approach is one of the first implementations of elastic net in chemical data analysis18 and the first to use it in the context of mass spectrometric analysis. The work presented here describes our development of multivariate curve resolution-elastic net alternating least squares (MCR-ENALS) for LC-MS and applies it both to data at unit mass and incorporates it into our previously reported strategy for analyzing high resolution LC-MS (LC-HRMS) data.7
THEORY MCR-ALS MCR-ALS is a method by which a data array consisting of two (or more) modes of data (e.g., chromatographic and spectral) is decomposed into individual components corresponding to each contribution to the data. In terms of LC-MS data, each of these components contains the pure chromatographic and spectral profiles for a single compound. Additional components may also be found which correspond to background contributions in the data. Mathematically, MCR-ALS does this by modeling the data as bilinear contributions to the data, as shown in Eq. 1, where X is the J x K raw data matrix where J is the number of chromatographic time points and K is the number of m/z channels. C (J x N) and S (K x N) are matrices
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containing the pure chromatographic and spectral profiles of each of the N components, respectively. E (J x K) is a matrix containing the residual error and T indicates a transpose of the matrix. If multiple samples are being used in the analysis, the data first need to be rearranged to a matrix format by appending the samples along the chromatographic mode keeping the spectral mode intact.19,20 X = CST + E (1) C and S are solved for using least squares in an iterative fashion where an initial estimate of S is used to estimate C. This initial guess is most often obtained from the raw data by one of several existing methods. In this work the iterative orthogonal projection approach (IOPA) was used, which finds the most orthogonal spectra in the raw data. For more information, the reader is directed to previous reports.19,21,22 This estimate of C is used to update the estimate of S. This loop continues until E is sufficiently minimized or a maximum number of iterations is reached. Between each iteration constraints are implemented to ensure that the mathematical results are physically reasonable. The most popular of these constraints is non-negativity which incorporates the knowledge that pure chromatograms and spectral profiles cannot have negative intensities. Other constraints include unimodality in which chromatographic profiles are constrained to have a single maximum, local rank in which regions of individual chromatographic and/or spectral profiles are set to zero intensity based on the knowledge that no signal should be present at those specific times or m/z values, as well as many others. For a more in-depth description of the MCR-ALS method, the reader is directed to other sources.23,24 Elastic Net The mass spectral signature for a single chemical species results in a sparse spectrum, meaning that most of the m/z channels contain zero intensity and only a few m/z channels contain non-zero intensity. To encourage sparsity in the MCR-ALS results, an elastic net regression step was included in the analysis. The elastic net is a regularization method which derives from least absolute shrinkage and selection operator (LASSO) method.16,17,25 The LASSO method produces sparse results by minimizing the residual sum of squares of the regression – as is done in ordinary least squares (OLS) – but penalizes the regression by the sum of the absolute values of the regression coefficients (i.e., the L1-norm) to be less than a constant forcing most of the coefficients to zero.25,26 An extension of this idea is the elastic net introduced by Zou and Hastie in which the regression penalty consists of a compromise of the L1norm used in LASSO and the L2-norm as shown in Eq 2, where β is the vector of regression coefficients, and y and x are the responses and inputs, respectively.16 The variable α is key to the elastic net method as it is a tunable parameter allowing the user to control the degree of sparsity imposed on the results. When α = 1, the regression is subject solely to the L1norm, producing the LASSO. In the case that α = 0, the elastic net subject solely to the L2-norm, thus reducing to ridge regression, a penalized regression method which allows for inclusion of correlated regression coefficients. 1
βˆ = min β
p
N
p
∑ ( y − ∑ x β ) + ∑ (α β 2 2
i
i =1
ij
j =1
j
j =1
j
+ (1 − α ) β j2 ) (2)
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STRATEGY In the current work we investigate the implementation of an elastic net step within the MCR-ALS analysis of LC-MS datasets. This MCR-ENALS strategy can be described in four steps: 1) obtain initial estimates of component spectra via IOPA;19,21,22 2) MCR-ALS optimization of C and S; 3) elastic net fitting of each component corresponding to a chemical species; 4) second round of MCR-ALS with local rank constraints based on step 3. Steps 1 and 2 are carried out in the same way as described previously.7 The elastic net fitting in step 3 is performed component-wise. First, the spectral and chromatographic profiles of each component resulting from step 2 that are suspected to correspond to real chemical species are added back to the residuals, E, as shown in Eq 3.The resulting matrix, R, is subjected to elastic net to estimate the sparse spectral profile for this component, sEN. Eq 4 shows the optimization performed by the elastic net as it relates to this work.
R = E + c n s nT
(3)
p p 1 N sEN = min ∑ (cn,i − ∑ rij s j ) 2 + ∑ (α s j + (1 − α ) s 2j ) β j =1 j =1 2 i =1
(4)
This procedure is performed for each component for which sparsity is desired (i.e., components corresponding to true chemical species) and the final sparse spectral profile matrix, SEN, is obtained. Components which are not subjected to elastic net are equal in S and SEN. The elastic net method can introduce negative values in the spectra. If this occurs, the negative intensities are replaced with zeros. These spectral profiles, SEN, are then used as the initial estimate for the second round of MCR-ALS as well as the basis for a local rank constraint in this round of MCR-ALS. Any m/z channels with zero intensity in a spectral profile in SEN are constrained to maintain this zero intensity during the second round of MCR-ALS. The results of this second round of MCR-ALS are pure chromatographic profiles as well as spectral profiles containing only relevant masses. Sequential Binning In the present work the utility of elastic net for LC-MS data is demonstrated in a traditional MCR-ALS analysis at unit mass resolution as well as part of the sequential binning MCRALS strategy recently reported by our group.7 The sequential binning method aims to analyze data from LC coupled with high resolution MS (LC-HRMS) using a series of binning and MCR-ALS steps. Another noteworthy attempt at applying MCR-ALS to LC-HRMS data has been reported by Tauler et al.;11 however, their attempt requires the input of an intensity threshold which assumes low intensity masses correspond to noise. Our approach eliminates the need for such a threshold, allowing low intensity masses if they correspond to a chemical species.. First, the data are binned to unit mass resolution and MCR-ALS is performed. Masses are selected from the resolved spectral profiles corresponding to true chemical compounds. These masses are extracted from the raw data and binned at 0.1 u resolution. This process repeats until data are analyzed at 0.001 u resolution (or at the resolution provided by the instrument). The selection of masses to advance through each step was performed via an intensity threshold. In the present paper we eliminated this threshold by using the elastic net
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strategy described above to avoid excluding low-level spectral peaks while maintaining the reduction in data size.
EXPERIMENTAL Software All data analysis was performed in the MATLAB programming environment (R2016a; Mathworks, Inc., Natick, MA) on a Dell Precision T3600 desktop computer with an Intel Xeon E5−1620 CPU at 3.60 GHz and 32.0 GB of RAM. Data files were converted to mzXML files via msConvert, which is contained in the ProteoWizard suite.27 These mzXML files could then be imported directly into MATLAB using the Bioinformatics toolbox by Mathworks. Elastic net was performed using the ‘lasso’ command in the MATLAB statistics toolbox and MCR-ALS was performed using an in-house program described previously.7 Datasets Two datasets were used to demonstrate the proposed sparse MCR-ALS approach, both of which have been described in detail previously.7 As is common in MCR-ALS analyses,6,14,28– 31 both datasets were divided into chromatographic windows to reduce the complexity of each analysis. For the purposes of the current work, the results from a single chromatographic window from each dataset is reported. The first dataset consisted of a chromatographic window containing eight amphetamine compounds – shown in Fig 1 – split into three groups. Four calibration mixtures and two test mixtures for each group plus a blank were analyzed. One group had only one test mixture due to instrumental error. The composition and concentrations of each sample are listed in Supporting Information in Table S1. The second dataset consisted of bacterial lipid extracts from three separate strains of bacteria. Five replicates of each strain were run for a total of 15 samples. Both sample sets were analyzed with a Shimadzu LC system (Nexera series, Kyoto, Japan) coupled to an AB Sciex TripleTOF 5600 mass spectrometer (Concord, Ontario, Canada). Further experimental details are provided in our original report.7
chromatographic local rank constraints were utilized for this step. The local rank constraints set regions of individual chromatographic profiles to zero intensity based on visual inspection of where analyte signals should not be present. A ninecomponent model was found to best fit the data, corresponding to the eight amphetamines plus one background component. The eight amphetamine components were submitted to the elastic net to obtain sparse spectral profiles as described in the Theory section above. The sparse spectral matrix, SEN, was then used as the initial guess for the second round of MCRALS as well as the basis for the local rank constraint in the spectral dimension. The second round of MCR-ALS had identical non-negativity and chromatographic local rank constraints as the first MCR-ALS analysis as well as the same number of components. Sequential Binning with MCR-ENALS Both the amphetamine and bacterial lipid datasets were analyzed with the sequential binning MCR-ENALS method. The analysis started at unit mass resolution and continued until 0.001 u resolution was obtained, as dictated by the instrument. The amphetamine analysis used nine components at unit resolution and eight components in the subsequent binning levels. This reduction in the number of components was due to the fact that the major background ions were rejected after the first stage of the analysis. The constraints used were identical to those described in the section above. The bacterial analysis used seven components at unit mass and six components in the subsequent steps. Non-negativity and chromatographic local rank were applied.
RESULTS To demonstrate the ability of the elastic net MCR-ALS (MCR-ENALS) approach, a dataset consisting of mixtures of eight amphetamines was analyzed. Fig 2 shows the total ion current chromatogram for the chromatographic window analyzed. Multiple α values between 5 x 10-3 to 0.5 were tested in order to assess the performance of MCR-ENALS. As expected, with increasing values of α, the number of masses with non-zero intensity decreased as shown in Fig 3. It can be seen in Fig 4 that the resolved chromatographic profiles obtained with MCR-ALS and MCR-ENALS (α = 0.005) are virtually identical. Indeed, when calibration was performed using the area of the chromatographic profiles resolved with MCRENALS (at each individual α value), the quality of the calibrations were identical to that of MCR-ALS (data not shown).
1. Amphetamine (Amp) 5. 3,4-Methylenedioxymethamphetamine (MDMA) 2. 3,4-Methylenedioxyamphetamine (MDA) 6. Phentermine (Phent) 3. Methamphetamine (Mamp) 7. 4-Methyoxymethamphetamine (PMMA) 4. Methyoxyamphetamine (Moxy) 8. 3,4-Methylenedioxy-ethylamphetamine (MDE)
2
Figure 1. Names and structures of the eight amphetamines located in the chromatographic window analyzed.
MCR-ALS with Sparse Spectral Recovery The amphetamine dataset was first analyzed at unit mass resolution to demonstrate the MCR-ENALS strategy. In order to initiate MCR-ALS, spectral initial estimates were obtained using IOPA, which aims to extract the most dissimilar spectra from the raw data matrix.19,21,22 This initial estimate is then used to begin the first alternating least squares process as described in the Theory section above. Non-negativity and
7 3 1
4
8
5 6
Figure 2. Total ion current chromatogram for the window analyzed in the amphetamine dataset. The compound numbers correspond to the names and structures in Fig 1.
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Table 1. Masses and corresponding relative intensities found for each amphetamine at α = 0.005 Amp
MDA
Mamp
Moxy
MDMA
Phent
PMMA
MDE
119
93%
105
2%
119
10%
121
12%
105
1%
105
3%
121
7%
135
1%
120
10%
133
2%
120
1%
133
0.03%
133
1%
133
43%
122
1%
163
9%
121
4%
135
5%
150
100%
136
0.03%
135
2%
134
5%
149
45%
164
1%
122
2%
162
1%
151
13%
137
0.02%
163
18%
135
2%
150
5%
178
4%
123
1%
163
100%
152
4%
149
100%
164
2%
136
1%
151
2%
207
1%
136
100%
164
13%
153
3%
164
0.02%
165
1%
137
1%
152
1%
208
100%
137
11%
165
4%
154
2%
165
0.05%
193
1%
138
0.4%
153
1%
209
13%
138
4%
166
2%
155
1%
166
19%
194
100%
139
0.2%
179
1%
210
3%
139
2%
167
2%
167
2%
195
12%
140
0.1%
180
100%
211
2%
140
1%
168
1%
168
1%
196
3%
150
100%
181
12%
212
2%
142
1%
180
52%
169
1%
197
2%
151
11%
182
3%
213
1%
181
6%
170
0.4%
198
2%
181
0.01%
183
2%
214
1%
182
2%
199
1%
182
0.1%
184
2%
215
1%
183
1%
200
1%
184
0.03%
185
1%
186
0.004%
186
1%
187
0.01%
208
0.02%
279
0.1%
+
*Bolded masses are the [M+H] mass; italicized masses represent interfering masses
mization of α is required, this parameter results in a more meaningful reduction of masses in comparison to a threshold which eliminates masses simply based on their intensity. The addition of the elastic net required an additional 13 iterations to obtain the final sparse component profiles. Table 1 lists all of the non-zero intensity masses in the resolved spectral profile for each amphetamine at α = 0.005. Most peaks can be
Figure 3. Reduction in the number of non-zero masses as a function of α for each compound. The first data point shows normal MCR-ALS with no sparsity imposed. Note the x-axis is logscaled.
While the chromatographic profiles are unaffected by the addition of elastic net analysis step, the spectra show drastic improvement. Figure 5 shows the resolved spectrum for one compound, phentermine. It can be seen in the top row of Fig 5 that MCR-ALS does not completely eliminate irrelevant masses. Using MCR-ENALS (rows two and three) most of the m/z values are set to zero, leaving only peaks corresponding to the molecular peak, a fragment peak, and their associated isotope peaks. The choice of α dictates the degree of sparsity imposed on the spectra. The second and third rows of Fig 5 represent α values of 0.005 and 0.5, respectively. While both eliminate irrelevant masses, at α = 0.5, isotope peaks are eliminated. An α value of 0.005 was chosen as the most appropriate for this analysis. The percent lack-of-fit at this α value was 3.70% versus 3.27% without the elastic net step. While opti-
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Time (min)
Time (min)
Figure 4. MCR-ALS resolved chromatographic profiles of the amphetamine window with normal MCR-ALS (left column) and elastic net with α = 0.005 (right column). Each panel is a single resolved component with all samples overlaid. One additional background component was included in the model, which is not shown here.
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Analytical Chemistry
Table 2. Amphetamine masses resolved at 0.001 m/z resolution using sequential binning with MCR-ENALS at α = 0.0025 (bold) and ordinary MCR-ALS (regular). Amp
MDA
Mamp 166.121
Moxy
MDMA
195.122
195.122
194.117
194.117 151.130 151.130 166.122 166.122 194.117 194.117
137.115 137.115 181.105 181.105 150.127 150.127
Phent
195.122 195.120 195.120 151.02
195.121
PMMA
MDE
195.121 209.136 209.136
194.118 181.141 181.141 208.133 208.133
164.078
180.098 180.138 180.138 178.086
136.112 136.112 180.101 180.101 119.085 119.085 150.099 150.099 163.075 163.075 178.086
150.098 150.098
166.122
120.089 120.089 164.078 164.078
149.096 149.069
149.096 149.096
164.078
119.086 119.086 163.075 163.075
136.111
133.064
166.122 121.064 121.064 163.075 163.075
133.065
121.064
151.130
135.043 133.065
121.065 121.065 119.086
150.125 163.073
133.064
119.085 150.132 150.127 150.127 149.095 137.114 136.112 134.104 133.101 133.101 121.064 119.087 105.069
*Italicized masses represent interfering masses; m/z taken from maximum of spectral peak
these peaks are less than 0.1% relative intensity and can easily be interpreted as being due to an insufficiently resolved compound. Subsequent local rank constraints could be used to aid in the resolution of these two compounds, as this is a targeted analysis. Several masses were found at low intensities which clearly correspond to the compounds of interest due to the use of elastic net rather than a hard noise threshold. For example, a noise threshold on the MCR-ALS resolved mass spectra would eliminate many of the isotope peaks found for Moxy and Phent.
Figure 5. MCR-ALS resolved spectral profiles for phentermine. The top panel shows the resulting spectral profile using normal MCR-ALS and the second and third rows display elastic net spectral profile at α= 0.005 and 0.5, respectively. The inset plots show the spectral baseline to highlight the reduction in zero-intensity masses.
assigned as the molecular peak, a fragment peak, or an isotope of either of these. For example, the peaks found for phentermine at 150, 133, and 105 m/z correspond to the molecular ion and two fragment ions, respectively. A few peaks were found that cannot be assigned such as the phentermine peaks at 181 m/z and above which are likely are due to imperfect resolution with MCR-ALS and correspond to the chromatographically overlapped peak (compound 8, MDMA). The intensities of
Sequential Binning MCR-ENALS Because MCR-ENALS allows for the detection of relevant masses without the use of a threshold, it can be used as an adjunct to our previously reported sequential binning method.7 In the original report of the sequential binning method a threshold was required to select masses which would be included in each step. In the current work, elastic net was used to select masses from each round of MCR-ALS to submit to the next round at higher resolution. First, the amphetamine dataset was analyzed using sequential binning MCR-ENALS. An α value of 0.0025 was selected for each step of the analysis. The final spectral peaks resolved at 0.001 m/z are listed in bold font in Table 2. Unlike the lower resolution steps in which each m/z is represented by a single spike, at 0.001 u resolution each m/z is represented by a peak consisting of several m/z points. The masses listed in Table 2 are taken from the maximum of these peaks. Compared to the unit mass analysis described above, fewer isotope peaks are resolved; however, the molecular peak and fragment peaks are easily detected with high mass precision. Compared to the masses found using ordinary MCR-ALS as in the previous report
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(Table 2, regular font), the present approach finds slightly fewer masses and in general the masses found are more likely to be associated with true ion peaks (i.e., isotopes, fragments, etc.) rather than background noise. The found masses correlate nicely with the proposed fragmentation patterns shown in Supporting Information Figure S1. One of the major advantages to the MCR-ENALS approach is the ease at which these masses can be identified. When MCR-ENALS is utilized detection of spectral peaks is performed simply by selecting m/z values with non-zero intensities. With the ordinary MCRALS approach, almost every m/z included in the final step has non-zero intensity, thus detection of relevant m/z values must be performed by an intensity threshold and/or analysis of spectral peak shape taking into account that each true spectral peak should consist of a minimum number of points with a roughly Gaussian shape. In addition to the amphetamine dataset, data from a bacterial lipid analysis were used to demonstrate sequential binning with MCR-ENALS. Figure 6 shows the total ion current chromatogram and the resolved profiles for this data. A total of seven components were needed to fit the data at the final level of binning, with two of these components corresponding to background contributions. While five chemical components were resolved, closer analysis revealed that two components shared almost identical chromatographic profiles and very similar spectral profiles indicating that these components are actually the same component that has been split between two components. This was due to instrumental limitations on the precision of the mass axis causing the location of mass spectral peaks for this compound shift slightly at the 0.001 m/z level of resolution. MCR-ALS requires consistent spectral profiles between each scan causing this compound to resolve into two separate components. This is simple to detect, however, by visual inspection showing identical retention times and only very slight (
