Calibration of mass spectrometric peptide mass fingerprint data without specific external or internal calibrants
 Witold E Wolski^{1, 2, 5}Email author,
 Maciej Lalowski^{3},
 Peter Jungblut^{4} and
 Knut Reinert^{2}
DOI: 10.1186/147121056203
© Wolski et al; licensee BioMed Central Ltd. 2005
Received: 26 April 2005
Accepted: 15 August 2005
Published: 15 August 2005
Abstract
Background
Peptide Mass Fingerprinting (PMF) is a widely used mass spectrometry (MS) method of analysis of proteins and peptides. It relies on the comparison between experimentally determined and theoretical mass spectra. The PMF process requires calibration, usually performed with external or internal calibrants of known molecular masses.
Results
We have introduced two novel MS calibration methods. The first method utilises the local similarity of peptide maps generated after separation of complex protein samples by twodimensional gel electrophoresis. It computes a multiple peaklist alignment of the data set using a modified Minimum Spanning Tree (MST) algorithm. The second method exploits the idea that hundreds of MS samples are measured in parallel on one sample support. It improves the calibration coefficients by applying a twodimensional Thin Plate Splines (TPS) smoothing algorithm. We studied the novel calibration methods utilising data generated by three different MALDITOFMS instruments. We demonstrate that a PMF data set can be calibrated without resorting to external or relying on widely occurring internal calibrants. The methods developed here were implemented in R and are part of the BioConductor package mscalib available from http://www.bioconductor.org.
Conclusion
The MST calibration algorithm is well suited to calibrate MS spectra of protein samples resulting from twodimensional gel electrophoretic separation. The TPS based calibration algorithm might be used to correct systematic mass measurement errors observed for large MS sample supports. As compared to other methods, our combined MS spectra calibration strategy increases the peptide/protein identification rate by an additional 5 – 15%.
Background
Proteomics interalia focuses on the identification of peptides/proteins in complex biological samples [1]. Before the identification of the complex constituents, several separation steps are required to reduce the sample complexity. The classical separation method is the twodimensional gel electrophoresis [2–5], followed by excision of the detected spots from the gel, digestion with sequence specific proteases and extraction of the cleaved proteins [6, 7]. Mass Spectrometric (MS) analysis [8–13] of the resulting mixture of peptides yields a peptide mass fingerprint (PMF): a set of measured molecular masses of the proteolytic peptides derived from the analysed protein [14–16].
PMF commonly requires matrix assisted laser desorption/ionisation (MALDI) time of flight (TOF) instruments, capable of high throughput analysis of complex samples with minimal precleanup, high femtomolar range sensitivity and accuracy of peptide molecular mass determination up to 5 – 10 parts per million (ppm) [17–20]. Due to the high ion transmission of the TOF mass analyzer, this technique is more sensitive compared with other MS techniques. In relation to Electrospray ionisation (ESI) MS [21], MALDIMS is more tolerant to sample contamination resulting from salts and detergents often present in protein samples due to the separation method. MALDIMS and ESIMS have become the standard high throughput proteome analysis techniques in many research laboratories.
The experimental peptide mass lists are generated by the analysis of TOF spectra [22]. Ideally, the TOF is proportional to the square root of mass over charge . Thus, in order to transform the spectrum from TOF into m/z, two calibration constants A and B are necessary. These can be derived by measuring the flight times t of at least two different ions with known masses and fitting them such that . After the transformation from time into m/z, the monoisotopic peptide signals in the spectrum are identified and their intensity is determined by computational methods [23–26]. The lists of the first monoisotopic peptide peaks – further called peaklists – are used to identify the protein of interest. In order to assign the PMF to a protein in a sequence database, database search algorithms use the match (within a given measurement accuracy) of theoretical peptide masses computed from protein sequence databases [27] with observed MS masses [15, 16].
Usually the scoring schemes model the mass frequencies of the proteins and peptides in the sequence databases [24, 28–30]. Other properties to be considered include the different sensitivity of detection for individual peptides, known protein modifications, and/or possible mutations [23, 31–33], although generally, all popular search scores depend on the precise assignment of experimental to theoretical peptide masses.
Two novel calibration methods
In a high throughput setting [34, 35], where the samples are placed on a moving sample support, the calibration coefficients for transforming the TOF into m/z differ depending on sample position. This is due to deviations in plate flatness, sample topography changing the size of the acceleration region [34, 36], and alterations in the strength of the electric field on the sample support borders which influences the drift velocity of the ions [22]. Thus, when calibration constants determined from one position on the sample support are used to calibrate TOF spectra acquired on other positions (a procedure known as external calibration), the determined m/z values have errors of up to 500 ppm.
Calibration is usually performed using external [36–38] or internal calibrants [39, 40], which rely on known masses to calibrate the spectra to common coordinates. It must be stressed, that in some cases the signal of a reference compounds might be suppressed by the analyte molecules, thus precluding internal calibration. In other cases, the reference signal may partially overlap with an analyte signal, resulting in an erroneous assignment. A third category of calibration methods is based on the peptide mass rule [23, 24]. A major advantage of the latter method is that no internal calibrants are required to calibrate the peaklists. The limitation of this method is it's sensitivity to the presence of nonpeptide peaks in the spectra, and that it completely fails if the number of peptide peaks in peaklists are small [23, 24, 39]. Therefore, in practice this method usually is used only to precalibrate [24] or to support the results of internal calibration [26, 39].
We have developed two novel calibration methods for PMF data. Both calibration methods exploit similarities of peaklists due to closeness in the origin of the analysed samples. The first method combines the computation of dissimilarities [41] between peaklists with internal calibration. The second method employs spatial statistical methods [42] to model systematic changes of the calibrationmodel over the MALDI sample support. The major advantage of the presented methods originates from the fact that the MS calibration derives from samples without internal standards or external calibrants positioned on each sample support.
Evaluating the methods
To demonstrate the accuracy of our methods, we studied one sample set of 380 mass spectra, consisting of a part of the Arabidopsis thaliana proteome study [43]. For this purpose, a MALDI MS sample support in prestructured [35] (384well) microtitre plate format was used. The measurements were performed using the Autoflex MALDITOF MS [44] instrument.
To compare the performance of calibration methods described here with those already published [26, 39], we used two different data sets. The first set consisted of 1193 spectra deposited on four prestructured sample supports and measured on a Reflex MALDITOF MS [44] instrument (Reflex data set). Spectra were generated via mass spectrometric analysis of the Rhodopirellula baltica proteome (unpublished data). The second set was generated in connection with a proteome study of Mus musculus and consisted of 1882 spectra deposited on five prestructured sample supports and measured on an Ultraflex MALDITOF MS [44] instrument (Ultraflex data set).
During MS sample preparation of the Ultraflex data set, standard peptides of known masses (human Angiotensin I – 1, 296.6853Da, human ACTH (18–39) 2, 465.1989Da) were added before the measurement to the MS matrix. This was done because the data sets were optimised for the calibration methods, which required the internal calibrants. We examined if the standard peaks could be observed in more than 33% of spectra and if so, we removed the peaks matching these masses from the data set. This procedure was applied in order to simulate a data set not optimised for internal calibration.
The Rhodopirellula peptide peaklists were searched against a Pirelulla database [45] with 13, 331 predicted Open Reading Frames (ORFs). The Mus musculus samples underwent searches against the Mus musculus entries (69, 343 sequences) of the NCBI nonredundant protein database [46].
Results and discussion
Internal calibration using a precalibrated list of calibration masses
Internal calibration is a widely used method in mass spectrometry. This method fails however, either if no peaks matching known masses are present or if MS peak assignment is false. A detailed description of the application of internal calibration in a high throughputMS setting, addressing the two points is given by e.g. Chamrad et al. [39], Levander et al. [40] and Samuelson et al. [26]. In order to avoid the lack of MS peaks matching the known calibration masses the authors used a precompiled list, e.g. trypsin autolysis peaks and unidentified, frequently observed masses [47].
Chamrad et al. [39] initiated the calibration procedure with searches for matching masses using a relatively large search window and iterated it with an increased accuracy. In this scheme, a large search window allows false assignments for calibration masses to occur more frequently. If a false assignment occurs in the first iteration, then the determined calibration constants are false and the entire calibration would be wrong. In the next round of calibration, where a search for matching masses is performed with a higher mass accuracy, the calibration would also fail. To prevent this, the authors [26, 39] checked the obtained calibration coefficients against the peptide mass rule (PMrule) [24, 48] and stopped further calibration attempts where they disagreed substantially.
Levander et al. [40] introduced an adaptive method to eliminate lowsensitivity autoproteolysis trypsin peaks from the calibration mass list if no highsensitivity trypsin peaks e.g. (842.5099Da, 1045.5642Da, 2211.1046Da) were found to decrease the chance of false matches. Unfortunately, this method could only be applied for "tryptic" calibration peaks.
Figures 1A &1B demonstrate the limitations of a calibration list compiled from ubiquitous masses of the whole data set. One can recognise that out of three abundant masses (in red, Figure 1A), only two can be practically used for calibration. Specifically, the first and the third abundant mass in the list of ubiquitous masses (Figure 1A) match simultaneously two peaks in peaklist 3, 4 and 5 (Figure 1B). Thus, out of five peaklists only three could be calibrated. The second calibration mass is also of no use, since it is the only calibration mass in the peaklists 1 and 2 (although these peaklists do contain other shared masses). This illustrates that the usage of a global calibration list may fail to calibrate a set of peaklists.
Finding the optimal multiple peaklist alignment using a modified Minimum Spanning Tree (MST) algorithm
In order to bypass the limitations imposed by global calibration we used an observation made by Schmidt et al. [49]. They noticed that protein samples excised from highresolution 2Dgels are usually not ideally separated and therefore exhibit local similarities. Compiling a calibration list of abundant masses from a whole data set obtained from a 2Dgel does not differentiate local spectra similarities. For example peaklists 1, 2 and 3 (Figure 1B) share peaks, which were not recognised as ubiquitous masses and hence not used further for calibration using a global calibration list. The peaklist pairs (2,3) and (1,3) shared more than one peak, thus allowing an easy calibration.
In the MST algorithm, the vertices are joined by edges of smallest dissimilarity. Consequently, the MST algorithm connects all peaklists in the data set in the way that the length of the path from the peaklist of origin (root of the tree: peaklist 3 in Figure 1D) to any peaklist in the data set is minimal. The algorithm for computing the agglomerative clustering using the single linkage method [51, 52] works similarly like the MST algorithm and therefore the dendrogram (Figure 1C) provides (as read from bottom to top) the order, by which the peaklist pairs were chosen. The horizontal lines joining two dendrogram tree branches were drawn at the height of the value of the minimal dissimilarity of two peaklists in either branch.
Finally, the algorithm returns a list of coefficients and a measure of confidence for all peaklists equalling the smallest similarity in the path from s to v.
The stripcharts of mass ranges including peaks of the trypsin autolysis products 842.508 and 2, 211.100 are presented in Figure 2C_{1} and C_{2}. One can observe that the MSTmethod works robustly on raw data with a mass measurement error of up to ± 0.7Da (black crosses), even if the search for matching peaks when computing the similarities and calibration coefficients was performed within a much smaller window of ± 0.45Da. Notably, if the maximal error among two peaklists is much larger than the search window, it is still possible to find a path, thus allowing alignment of two extreme peaklists.
Due to the fact that all peaklists were aligned to the peaklist of origin s, which did not necessarily match to the theoretical trypsin autolysis masses, a final correction was required to calibrate the whole tree to the theoretical coordinate system before database searches (not shown).
Determining the calibration model of the sample support using ThinPlate Spline interpolation (TPS)
Because a large part of the MME is of systematic origin and depends on the sample support position, the mapping of the calibration coefficients across the entire MALDI plate was introduced by Gobom et al. [36] and Moskovets et al. [38]. The calibration coefficients were determined using a standard mixture of peptides with known masses. Subsequently, the calibration coefficients were used during MS analysis in order to correct for the masses measured afterwards on the same plate.
However, it is unbiased to assume a smooth transition between adjacent positions of the sample support. For example, Figure 2B demonstrates that the slope coefficient of the sample calibrationmodel obtained by the MST calibration methods increases for samples close to the support border. This change is due to alterations in the electric field E (Equation 1) influencing the flight velocity given by
where s_{ a }is the size of the acceleration region, z is the ion charge and m is the mass of the ion. We determined the systematic change of the slope using the ThinPlate Spline (TPS) interpolation method [42, 53]. At first, we computed the TPS with a degree of smoothing λ = 5·10^{2} (see Equation 15). Calibration models with slope coefficient c_{1} that varies more than ± 1·10^{4} or with intercept coefficient c_{0} varying more than 0.2Da from the one predicted by the TPS were discarded. Using the remaining calibration models, the TPS was recomputed with smaller degree of smoothing λ = 1·10^{3}. Figure 3B, demonstrates the Colour scheme coded slope coefficient c_{1}, as estimated by the refined TPS. This model resembles the one generated by the MST method (Figure 2B). We corrected the peaklists masses (black cross hairs, Figure 3C), using the TPS values as estimates of the slope coefficients, and as intercept estimate we used the average intercept of all coefficients of the refined calibration models to obtain the calibrated masses (red circles).
The mass measurement error
Mass Measurement Error. Standard deviation (S_{ N }) observed for the trytpic autolysis peaks 842.508 and 2211.1. Raw data; TPS – ThinPlate Spline (TPS) calibrated data; TPSMST – The data, which undergone ThinPlate Spline (TPS)(preprocessing), followed by Maximum Spanning Tree (MST) calibration
Calibration  Mass  S_{ N }[Da]  S_{ N }[ppm] 

Raw data  842.508  0.1  118 
Raw data  2211.1  0.3  135 
TPS  842.508  0.03  37 
TPS  2211.1  0.057  26 
TPSMST  842.508  0.012  14.5 
TPSMST  2211.1  0.01  4.6 
The optimal size of the search window
Figure 5 and Table 1 demonstrate that it is possible to reduce the mass measurement error to approximately ± 10 ppm for most of the peaklists in a dataset consisting of 380 spectra, by applying the TPSMST calibration sequence. Nevertheless, in this dataset one can observe peaklists that do not exhibit such high mass measurement accuracy. Consequently, if the database searches were performed with a search window of 10 ppm, these PLs would not be identified.
Prior to the database searches we removed all masses that occur in more than 8% of spectra, as it significantly increased the identification rate [39, 40] (cf. Methods – Filtering of ubiquitous masses prior to database search). The sequence data base search was performed using the Mascot [55] search software version 1.8.1. We interfaced the search server from within R using the inhouse developed R package msmascot [56].
Combining different calibration methods and their comparison
All parameters were fitted to a data set optimised for internal calibration, measured on an Autoflex MALDITOF MS [44] instrument. We applied the calibration methods introduced (MST and TPS based calibration) without changing the parameters to two sample sets obtained using two different instruments, namely a Reflex MALDITOF MS and a Ultraflex MALDITOF MS instrument. This was executed to illustrate that our methods are robust with respect to different instruments even if the parameters were not optimised for the respective machines.
Calibration sequences. LR/PR – linear regression on peptide rule, IC – Internal calibration with two iterations. (Bruker Reflex – mass measurement error (MME) window of 450 and 250 ppm, Bruker Ultraflex 250 and 125 ppm); MST – MST calibration method computed with an search window of ± 0.4Da; TPSIC – Preprocessing (TPS calibration) and subsequent internal calibration with a MME window of 250 ppm; TPSMST – preprocessing and an MST with a search window of ± 0.25Da;
Abbreviation  Description  

1  LR/PR  peptide rule calibration. 
2  IC  internal calibration 450 ppm and 250 ppm. 
3  MST  minimum spanning tree calibration. 
4  TPS  LR/PR and subsequent thinplate spline (TPS) calibration. 
5  TPSIC  TPS calibration and subsequent internal calibration. 
6  TPSMST  TPS calibration and subsequent MST calibration. 
The only calibration method for which parameters were optimised with respect to the instrument was the standard internal calibration (IC) method, which employs a precompiled calibration list of theoretical trypsin autolysis peaks and a calibrated set of ubiquitous masses (cf. Methods – Standard internal calibration). In case of the peptide rule based calibration (LR/PR) method we applied an additional filtering of the calibrationmodels. Only models with an intercept coefficient c_{0} satisfying 0.4Da <c_{0} < 0.4Da and slope coefficients c_{1} with 5·10^{3} <c_{1} < 5·10^{3} were kept. In order to avoid falsely calibrated peaklists we performed the filtering.
The identification rates were defined as the number of identified samples by at least one of the calibration sequences divided by the number of samples submitted for searches
where CS_{ i }indicates the set of identified samples by one of the calibration sequences (Table 2), and #{A} denotes the number of elements in a set A. The identification rates were 74%, 87%, 79%, 85% for the Pirellula (Reflex) data set, with an overall identification rate of 82%, whereas for the Mus musculus (Ultraflex) data set they were 51%, 72%, 35%, 51%, 27%, with an overall identification rate of 58%. The lower identification rate of the Mus musculus data set can possibly be explained by the fact that it was matched with a larger database. Therefore, more matching peaks are required to make significant assignments to a data base entry.
In order to directly compare the identification rates for both data sets and each calibration sequence, we computed the relative identification rate. It was defined as the ratio of the number of identified samples calibrated by a sequence (numerator) and of the number of identified samples, which could be identified by at least one method (denominator):
Only in one case of one data set was a single calibration sequence TPSMST (see Table 2) able to identify all peaklists (100% identification rate) and therefore it completely dominated over the other methods (black line, Figure 7A). In the case of the Ultraflex data set (Figure 7B) we observed that the TPSMST method had the highest identification rate, while in Reflex data set (Figure 7A) it achieved the highest performance for approximately half of the data sets.
Figure 7C illustrates the averaged relative identification rate of the calibration methods for the Ultraflex and Autoflex data sets. In addition, it demonstrates that the ordering of the calibration methods according to the relative identification rate does not depend on the value of the Probability Based Mowse Score [55] (PBMS) used as identification threshold. The dashed lines (Figure 5) indicate the identification rates obtained for a PBMS 5 units higher than the one used to identify the samples with a 0.5% significance level (continuous lines).
Interestingly, the TPS smoothing method resulted in an overall higher identification rate than the other methods tested on raw data (peptide rule based calibration, internal calibration, MSTcalibration), except for one case of the Ultraflex data set. Furthermore, a combination of the internal calibration with TPS calibration (TPSIC) did not increase either the sequence coverage (dashed lines) or the identification rate of the TPS method applied alone.
In two out of the four Reflex data sets, the MST method applied on TPSprocessed data (PTPS Figure 7A, dashed lines) slightly decreased the sequence coverage indicating a reduction of calibration accuracy. For the Ultraflex data sets, the sequence coverage correlated well with the identification rate and the TPSMSTmethod accomplished the highest performance.
Moreover, if similar identification rates of the peptide rule based calibration and the internal calibration were observed, the peptide rule based calibration method provided higher sequence coverage (Figure 7B). This could be explained by the fact that the peptide rule based method calibrated well the peaklists possessing many peptide peaks. Such peaklists potentially contain the higher sequence coverage.
The BioConductor package mscalib
All of the calibration methods are part of the mscalib programme, which is available as a BioConductor [59] package. The Bioconductor project is an initiative for the collaborative creation of extensible software for computational biology and bioinformatics [58]. The scripts carrying out the calibration sequences tested, can be found in the subdirectory/samples of the package. Furthermore, in the same directory and in the directory/doc there are two vignettes [60] with detailed descriptions of two selected calibration sequences.
Conclusion
While the methods described in this study significantly improve the calibration of raw data, they do not perform better than other published calibration routines which reduce the MME to 10 ppm or below. The real advantage of the methods described here is that they are not dependent on the presence of internal or external calibrants, required to correct for the affine component of the MME. Furthermore, the calibration methods described in this study allow a larger fraction of peaklists in the datasets to be calibrated than the reference internal calibration method would do.
The TPS method deals with systematic detrimental calibration effects that are due to imperfections in the geometry of the electric field over the MALDI sample plates. Usage of TPS calibration results in up to 10% higher identification rates, at least for the Bruker mass spectrometers, than the internal calibration. The TPS calibration procedure enables, for most of the samples deposited on the sample support, to obtain mass accuracy in the range of ± 0.1Da. Moreover, the TPS method does not require the presence of internal calibrants since it relies on calibration coefficients acquired from a calibration method based on the peptide mass rule.
The MST method is able to increase the identification rates obtained by the TPSmethod for protein samples separated by a 2DGel electrophoretic procedure. Furthermore, the parameters optimised for one instrument (Autoflex) can be directly utilised for other instruments (Reflex, Ultraflex).
In this work, we have only examined a version of the MST algorithm that builds a single tree for all peaklists. This is adequate if the data are a set of peaklists with smooth transitions in the similarity values. If this is not the case, it might be more appropriate to compute a forest of several MSTs. We have examined, however, only a single peaklist similarity measure (Equation 10) for peaklists calibration. It is possible that better similarity measures can still be generated and subsequently applied for peaklists calibration.
Complete utilisation of microtitre plates and sample supports is not only rational with respect to increased accuracy of the TPS method, but also with respect to the idea of high throughput experiments – maximal utilisation of energy and resources. Dense excision of spots from 2Dgels not only increases the performance of the MST method, but also identifies novel proteins. Hence, the main contribution of this manuscript is to present two calibration methods, compatible with the principle of high throughput sample processing and aims to identify a maximum of the proteins resolved on 2Dgels.
However, no single "bestcalibration" method exists. Each of the methods utilises different properties of the peaklists. Consequently, applying these methods in parallel and determining the total (union) of the identified samples provides the highest identification rate.
Methods
Data sets
All PMF MS spectra derive from tryptic protein digests of individually excised protein spots. For this purpose, the whole tissue/cell protein extracts of the former mentioned organisms were separated by twodimensional (2D) gel electrophoresis [4] and visualised with MS compatible Coomassie brilliant blue G250 [43]. The MALDITOF MS analysis was performed using delayed ion extraction and by employing the MALDI AnchorChip ™targets (Bruker Daltonics, Bremen, Germany). Positively charged ions in the range of 700 – 4, 500 m/z were recorded. Subsequently, the SNAP algorithm of the XTOF spectrum analysis software (Bruker Daltonics, Bremen, Germany) detected the monoisotopic masses of the measured peptides. The sum of the detected monoisotopic masses constitutes the raw peaklist. Before affine mass calibration, mass measurement errors which can be described by higher order polynomials and determined using external calibration (cf. Methods: External Calibration), were removed. Processed peaklists were then used for the protein database searches with the Mascot search software (Version 1.8.1) [55], employing a mass accuracy of ± 0.1Da. Methionine oxidation was set as a variable and carbamidomethylation of cysteine residues as fixed modification. We allowed only one missed proteolytic cleavage site in the analysis.
Describing the Mass Measurement Error (MME) and predicting the correct mass
A mass difference can be described either in absolute Δ_{ A }= m_{ y } m_{ x }[m/z] or in relative Δ_{ R }= (m_{ y } m_{ x })·10^{6}/m_{ y }[ppm] units. The masses in two peaklists X, Y were compared to each other and we considered two peaks to match, in the case of the absolute error if Δ_{ A }<a[m/z] and in the case of the relative errors if Δ_{ R }<a[ppm]. If we plotted Δ_{ A }or Δ_{ R }as a function of m_{theo}, we observed, besides a white noise component ε ∝ N(0, σ^{2}), a systematic dependence. This dependence was modelled using a function . Given we corrected the experimental masses using the equations:
depending on whether the relative or absolute error was used, to obtain corrected masses m_{corr}.
Affine MME model
If only one matching peak was found or the mass range enclosed by the matching masses was small (e.g. less than 200Da), as a remedy one can fix:

the intercept at 0, if absolute difference Δ_{ A }[Da],

the slope coefficient at 0, if relative difference Δ_{ R }[ppm]
and determine the slope or intercept respectively from the data.
To correct the experimental masses m_{exp} we used Equation 5 for the absolute differences Δ_{ A }of matching peaks and Equation 4 in case of relative differences Δ_{ R }.
The difference between theoretical and measured masses is called a mass measurement error MME, while the alignment of m_{exp} on m_{theo} an internal calibration [23, 54, 61].
Determining ubiquitous masses and their filtering
B_{ j }= [x_{0} + (j  1)h, x_{0} + jh], with j ∈ 1,..., l, (6)
where l = (max(M)  x_{0}) mod h. Formally the histogram of counts f is given by [62]
where n represented the number of masses in M. If a bin had more counts than a given threshold, the average mass of all peaks in the bin was computed. In the case of two adjacent or overlapping bins B_{1}, B_{2} with a significant number of counts c, we first computed a weighted average of the bin midpoints using the number of counts in each bin as weight
where m_{1} and m_{2} are the bin midpoints. Afterwards, the average mass of all peaks in the range m ± h/2 was computed. All peaks with mass m ∈ [ ± h/2] were subsequently removed from the data set. Using two overlapping histograms allows the detection of clusters that are scattered over two adjacent bins in one of the histograms. Different ways to determine ubiquitous masses were used and reported by Levender et al. [40] and Kreitler [63].
Standard internal calibration – Alignment to a precompiled list of calibration masses
Instead of using a predefined list of calibration masses, we chose the calibration masses adaptively. The calibration list consisted of ubiquitous masses determined for the data set (cf. Determining ubiquitous masses). Some of the peaks in the list of ubiquitous masses could be assigned to tryptic autolysis products.
These matches were used to calibrate the abundant masses. The peaklists in the data set were then aligned to the calibrated list of ubiquitous masses.
Filtering of ubiquitous masses prior to database search
We removed ubiquitous masses that occurred in more than 7.7% of peaklists [39, 40]. Filtering of ubiquitous masses was performed on a calibrated set of peaklists. As a result, we could use a small bandwidth of h = 0.2Da (Equation 6) to determine ubiquitous masses. Next, we checked which of them can be assigned with a significant Probability Based Mascot Score (PBMS) to a sequence database entry and subsequently removed these masses from the filtering list. Abundant masses assigned to a database entry usually result from proteins multiply detected on a 2Dgel. The multiple identification is due to different localisation of the protein on the 2Dgel caused by: protein modifications (phosphorylation, glycosylation), different splice variants or by partial protein degradation. Finally, we removed all peaks within the range ± 0.1Da around the ubiquitous masses.
Linear regression and peptide mass rule algorithm
Wolski et al. (publication in preparation) defined the distance measure
which computes given λ_{ DB }(the average peptide cluster distance for a sequence database DB against which the search is performed, e.g. λ_{ DB }= 1.000495) the deviation of a peptide mass difference m_{ i } m_{ j } from the closest monoisotopic mass predicted by the PMrule [48]. If there was a linear dependence between m_{ i } m_{ j } and d_{ λ }(m_{ i }, m_{ j }), then it was caused by the slope of the MME. If we computed all differences m_{ j } m_{ i } and d_{ λ }(m_{ i }, m_{ j }) for peak pairs m_{ i }, m_{ j }with m_{ i }, m_{ j } < 1400, we could determine the slope coefficient c_{1} using linear regression, while fixing the intercept to zero [64]. In order to make the prediction robust against e.g. nonpeptide peaks, we used a robust linear regression [65]. We removed the slope by multiplying each mass m_{ i }in the peaklist by (1  c_{1}). Next, we identified the intercept, which was the average of the distance d_{ λ }(m_{ i }, 0), and corrected for it.
External calibration
In order to model higher order systematic changes of mass dependent differences Δ of experimental m_{exp} and reference masses m_{theo}, the measurements must be evenly distributed over the whole measurement range [37, 66]. To model the dependence Δ ∝ m we used a cubic smoothing spline function [67, 68], given by Δ = f(m) + ε_{ i }, where f is a smooth function, and ε_{ i }~ N(0, σ^{2}).
In our study, we used an implementation of the smoothing spline function, provided by B.D. Ripley and Martin Mächler (based on Fortran code of T. Hastie and R. Tibshirani) as part of the Rstats package. Other nonparametric regression methods like local polynomial regression [69] generated similar results for all types of instruments used in this study.
To obtain equidistantly spaced measurements of known masses, External calibration was employed. Some sample spots on the sample support are dedicated to calibration only. Calibration samples, of polymer mixtures [36], which yield equidistant peaks were used to precisely estimate the massdependent difference function.
Similarity/quality measures for internal calibration
Peaklists can be easily aligned if they contain many matching peaks and the masses of these peaks span a wide mass range. The alignment of a peaklist pair (X, Y) fails if no matching peaks are found. We described these properties mathematically by the following similarity measure:
where n represented the number of matches, while m_{ i }and m_{ j }were the masses of matching peaks. This measure computed the sum of all mass differences of the matching peaks. The power p could be used to weight the large differences stronger.
Alignment of a set of peaklist using a Minimum Spanning Tree
To align a whole dataset to a single peaklist and to align the peaklists with the highest similarity given by Equation 10, we computed for all peaklists pairs a distance matrix D by casting the similarities into dissimilarities. This distance matrix can be represented by a complete, weighted graph G, where the vertices V correspond to peaklists and the edges are weighted with the pairwise dissimilarity. To connect all vertices in the graph G with edges e of maximal similarity, the DijkstraPrim algorithm for finding the Minimum Spanning Tree(MST) [50] was implemented. We present here a modified version of this algorithm (see Figure 8). The algorithm was modified with respect to the starting conditions. As a startingvertex s we chose a vertex incident to an edge of smallest distance. In addition to the MST tree T, the algorithm returns also a list of calibration coefficients C, which align all peaklists V in the data set to the starting vertex (peaklist) s, and a list with connection weights W.
By traversing the edges in T, we reached each vertex in G, starting at s via edges with the highest possible calibration similarity (smallest distance). This is because we picked D(uv) with the smallest possible distance (Figure 8, line 5).
To align peaklist v to the starting peaklist s we needed to determine the coefficients C(v, s) of the difference function (Equation 5). We could obtain them from the coefficients C(v, u) and C(u, s) of the pairwise difference function and by:
where e.g. denotes the slope coefficient, and the intercept of the function .
Proof
The masses of the peaklist pairs (v, u) as well as (u, s) can be aligned given the C(v, u) and C(u, s) using the equations
Hence,
C(v, s) was computed online using Equation 11 while growing the tree (Figure 8, line 8). Subsequently, the algorithm returned a list C of calibration constants, where C(v, s) described the calibration coefficients allowing to transform peaklist v into the coordinate system of the peaklist of origin s.
In order to gain more confidence in the calibration constants in C, the MST algorithm was iterated n times. For computing the consecutive. T_{ i }, C_{ i }, W_{ i }, D_{ i }with i = 2,..., n we applied the dissimilarity matrix D_{i1}and set as a starting vertex s_{ i }= s_{1} – the vertex incident to the edge of highest similarity in D_{1}. The returned T_{ i }, C_{ i }, W_{ i }, D_{ i }differed since we removed in iteration i  1 each visited edge (Figure 8, line 6).
The calibration constants C_{ i }(v, s) with i = 1,.., n should ideally be the same. It is known that C_{ i }(v, s) differ due to alignment errors. Therefore, we computed a weighted average of the coefficients of the difference model. As weight of each model C_{ i }(v, s) we utilised the smallest pairwise calibration similarity W_{ i }(v) (Figure 8, line 9), on the path from s to v:
We applied the calibration constants in C_{ w }to align all peaklists to the peaklist s.
Appendix
Thinplate spline
The thinplate spline is the twodimensional analogue to the cubic spline in one dimension [42, 71]. Let v_{ i }denote one of the error model coefficients, e.g. intercept, at a target location (x_{ i }, y_{ i }). A thinplate spline f(x, y) is a smooth function which interpolates a surface that is fixed at the landmark points P_{ i }= (x_{ i }, y_{ i }) at a specific height h_{ i }A thinplate spline interpolation function can be written as
where U(r) = r^{2} ln(r) is the radial basis function with . This equation is used to predict an unknown v for location (x, y), and is the unique solution [42, 71] which minimises the equation:
This quantity was called the bending energy of the thinplate spline function. If noise in the determined coefficients v_{ i }is detected, one may wish to relax the exact interpolation requirement (Equation 14). This can be accomplished by multiplying equation 14 with a regularization parameter λ, a positive scalar, and by adding the residual sum of squares, which gives:
Again, as in case of the cubic smoothing spline with the parameter λ, the degree of smoothing can be determined. In our study, we utilised an implementation of the TPS [72], according to Doug Nychka [53].
Abbreviations
 MME:

mass measurement error
 MST:

minimum spanning tree.
 MS:

Mass Spectrometry.
 TOF:

Time of Flight.
 MALDI:

Matrix Assisted Laser Desorption Ionization.
 mod:

modulo operator.
 TPS:

Thin plate spline.
Declarations
Acknowledgements
We would like to thank the members of Algorithmic Bioinformatics group at FUBerlin for valuable discussion, especially Dr. Clemens Gröpl. We would like to thank Dr. Johan Gobom, Dr. Patrick Giavalisco and Thomas Kreitler for providing the PMFMS data and for valuable discussion. We thank Carole Procter, Stale Nygard, Richard Boys and Daniel Henderson for proofreading the manuscript. We thank Prof. Dr. Hans Lehrach, at whose department part of the work was performed. This project was funded by the National Genome Research Network (NGFN) of the German Ministry for Education and Research (BMBF), and the Max Planck Society.
Authors’ Affiliations
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