Analytical model of peptide mass cluster centres with applications
 Witold E Wolski^{1, 2}Email author,
 Malcolm Farrow^{1},
 AnneKatrin Emde^{2},
 Hans Lehrach^{4},
 Maciej Lalowski^{3} and
 Knut Reinert^{2}
DOI: 10.1186/14775956418
© Wolski et al; licensee BioMed Central Ltd. 2006
Received: 30 January 2006
Accepted: 23 September 2006
Published: 23 September 2006
Abstract
Background
The elemental composition of peptides results in formation of distinct, equidistantly spaced clusters across the mass range. The property of peptide mass clustering is used to calibrate peptide mass lists, to identify and remove nonpeptide peaks and for data reduction.
Results
We developed an analytical model of the peptide mass cluster centres. Inputs to the model included, the amino acid frequencies in the sequence database, the average length of the proteins in the database, the cleavage specificity of the proteolytic enzyme used and the cleavage probability. We examined the accuracy of our model by comparing it with the model based on an in silico sequence database digest. To identify the crucial parameters we analysed how the cluster centre location depends on the inputs. The distance to the nearest cluster was used to calibrate mass spectrometric peptide peaklists and to identify nonpeptide peaks.
Conclusion
The model introduced here enables us to predict the location of the peptide mass cluster centres. It explains how the location of the cluster centres depends on the input parameters. Fast and efficient calibration and filtering of nonpeptide peaks is achieved by a distance measure suggested by Wool and Smilansky.
Background
The mass spectrometric (MS) technique is widely used to identify proteins in biological samples [1–4]. The proteins are cleaved into peptides by a residue specific protease, e.g. trypsin. The resulting cleavage products can then be analysed by Peptide Mass Fingerprinting (PMF) [5] or subjected to MS/MS fragment ion analysis [6, 7], which both rely on the comparison of peptide or peptide fragment ion spectra with spectra simulated from protein sequence databases [8].
Masses of Atoms
Atom  monoisotopic  nominal  mass defect  

1  H  1.00782  1  0.00782 
2  C  12.00000  12  0.00000 
3  N  14.003074  14  0.003074 
4  0  15.99491  16  0.00032 
5  S  31.97207  32  0.00087 
Calibration
Mass spectrometric peptide peaklists of peptide mass finger print experiments [18] can be calibrated by comparing the location of measured peptide masses with the location of the peptide mass cluster centres. Gras et al. [19] suggested the use of maximum likelihood methods in order to determine the calibration coefficients a and b. They defined the likelihood function by:
$\sum _{i}P(a{m}_{i}+b,\Delta m)},\left(1\right)$
where m_{ i }is the ith mass in the peaklist, and Δm is a search window. P(m, Δm) is the probability to find a mass in [m, m + Δm] given the theoretical distribution of peptide masses. The parameters a, b for arg_{max} ∑_{ i }P(am_{ i }+ b, Δm) can then be used to calibrate the peaklists. The authors, however, do not provide information on whether P(m, Δm) was determined from the exact distribution of the peptide masses or if a model approximating the distribution was used. They also do not mention which algorithm was used to maximise the likelihood. They reported that a mass measurement accuracy of 0.2Da and better was obtained after calibration.
Wool and Smilansky [10] have used Discrete Fourier Transformation (DFT) to determine the frequency λ and phase ϕ of a peaklist or mass spectrum. By comparing the experimental λ and ϕ with the theoretical λ = 1.000495 and ϕ = 0, they determined the slope and intercept of the calibration function. The authors reported a 40 – 60% reduction of the mass measurement error. Furthermore, they presented a scoring scheme for sequence database searches. This scoring scheme approximates the probability P(m, Δm) to observe a peptide peak of mass m with given measurement error Δm.
Matrix noise filtration
The most widely used MALDI matrices for the analysis of peptides are 3,5Dimethoxy4hydroxycinnamic acid (synapic acid), alphaCyano4hydroxycinnamic acid (alpha cyano) [20] and 2,5dihydroxybenzoic acid (DHB) [21]. Unfortunately, clusters of matrix molecules can be ionised and cause peaks in the same mass range where peptide peaks are measured. Matrix aggregate formation can be minimised but not eliminated by adding ammonium acetate [21].
Some of the database search scoring schemes incorporate the number of signals (peaks) not assigned to a protein when computing the identification scores [22]. Therefore, the presence of matrix signals in MS spectra decreases the sensitivity of the MS spectra interpretation. Hence, the removal of peaks strongly deviating from the cluster centres is applied [21, 23]. The measure of deviation from cluster centres introduced here provides a simple tool to filter nonpeptide peaks.
Data reduction
A further application which employs the property of peptide mass clustering is the binning of the mass measurement range. By applying this technique the amount of data is reduced, thus increasing the speed with which the pairwise comparison of spectra can be made [24, 25].
All these applications require us to know the exact location of or the distance between the peptide mass cluster centres. The distance between the cluster centres, which we will henceforth call wavelength λ, is commonly computed by first generating an in silico digest of the database. Afterwards, the linear dependence between the decimal point and the integer part is determined by regression analysis, for a relatively small mass range of 500 to 1000Da [23]. Various authors report different values of the distance between clusters: Wool and Smilansky reported 1.000495 [10], Gay et al. 1.000455 [15], while Tabb et al. used a wavelength of 1.00057 [24].
In this work we present an analytical model allowing us to predict the mass of the peptide cluster centres. The parameters of the model include: the frequencies of the amino acids in the sequence database [26], the average protein length of the proteins in the database, the cleavage sites of the proteolytic enzyme and the cleavage probability. Based on this model we introduced a measure of deviation of peptide masses from the nearest cluster centre, which is a refinement of a measure proposed by Wool and Smilansky [10]. Using this distance measure, we developed a calibration procedure which employs least squares linear regression in order to determine the affine model of the mass measurement error and subsequently to calibrate the spectra. Using this method we reached higher calibration accuracy as reported by Wool and Smilansky [10], and Gras et al [19]. We used the same distance measure to identify and remove nonpeptide peaks prior to database searches performed by the Mascot search engine [22].
Results and discussion
A simple way to predict the peptide mass cluster centres of a protein database
The first model of this dependence which we examined was m^{(M)} m^{(N)}= c_{1}·m^{(N)}. We fixed the intercept at 0, because a hypothetical peptide with a nominal mass of 0 must have a monoisotopic mass equal to 0. The slope coefficient c_{1}, determined by linear regression (cf. Methods) equalled 4.98·10^{4}(Figure 1, Panel A – red dashed line), which is a value similar to the values 4.95·10^{4} reported by Wool and Smilansky [10].
We were interested in determining the dependence between monoisotopic and nominal mass analytically.
For example, the monoisotopic mass (m^{(M)}) of hypothetical peptides built only of one amino acid i can be predicted, given their nominal mass (m^{(N)}) by ${m}_{i}^{\left(M\right)}$ = λ_{ i }${m}_{i}^{\left(N\right)}$ when λ_{ i }= ${m}_{i}^{\left(M\right)}$/${m}_{i}^{\left(N\right)}$. For peptides generated by random cleavage of protein sequences from a protein database this dependence is approximated by:
${\lambda}_{DB}=\frac{{\displaystyle {\sum}_{i\in AA}{f}_{i}{m}_{i}^{\left(M\right)}}}{{\displaystyle {\sum}_{i\in AA}{f}_{i}{m}_{i}^{\left(N\right)}}},\left(2\right)$
where f_{ i }is the frequency of the amino acid i in the database.
Now write ${m}_{i}^{\left(M\right)}$ = λ_{ DB }${m}_{i}^{\left(N\right)}$ + ε_{ i }. Substituting this is (2), it follows that ∑_{i∈AA}f_{ i }ε_{ i }= 0. Therefore, for an amino acid randomly selected from the database, with frequencies f_{ i }, the expectation of ε_{ i }is zero. Now consider a peptide made of a random selection of J amino acids, i(1),...,i(J). The ratio of monoisotopic to nominal mass for this peptide would be:
${\lambda}_{p}=\frac{{\displaystyle {\sum}_{j=1}^{J}{m}_{i\left(j\right)}^{M}}}{{\displaystyle {\sum}_{j=1}^{J}{m}_{i\left(j\right)}^{N}}}=\frac{{\lambda}_{DB}{\displaystyle {\sum}_{j=1}^{J}{m}_{i\left(j\right)}^{N}}+{\displaystyle {\sum}_{j=1}^{J}{\epsilon}_{i\left(j\right)}}}{{\displaystyle {\sum}_{j=1}^{J}{m}_{i\left(j\right)}^{N}}}.$
If ∑_{ i }ε_{i(j)}were uncorrelated with ${\left({\displaystyle {\sum}_{i}{m}_{i\left(j\right)}^{\left(N\right)}}\right)}^{1}$ for a random selection of amino acids, then λ_{ p }would have expectation λ_{ DB }. Of course, there may be a relationship between ε_{ i }and ${m}_{i}^{\left(N\right)}$ and we would wish to use any such relationship to improve prediction of ${m}_{i}^{\left(M\right)}$
When testing for the significance of the intercept coefficient in the regression model m_{ M }∝ λm_{ N }of a sequence specific (Tryptic) in silico database digest, we found that the intercept coefficient must be included into the model. Therefore, the extended model of the monoisotopic peptide mass cluster centres was:
m^{(M)}= c_{1}·m^{(N)}+ c_{0}. (3)
Subtracting m_{ N }from each side of Equation 3 we obtained Δ = m^{(M)} m^{(N)}= (c_{1}  1)·m^{(N)}+ c_{0}. The coefficients of the affine linear model of the cluster centres, determined using regression analysis of Δ = m^{(M)} m^{(N)}on m^{(N)}were c_{0} = 0.029 and (c_{1}  1) = 4.85·10^{4}.
The maximal difference between the prediction of m^{(M)}using m^{(M)}= 1.000499·m^{(N)}and m^{(M)}= 1.000485·m^{(N)}+ 0.029 is 0.022 Dalton for m^{(N)}∈ [600, 2500] Dalton.
The influence of the digestion enzyme on the wavelength of peptide mass clusters
In case of a complete sequence specific cleavage of proteins, the number of generated peptides is C_{ P }+ 1 peptides, given that C_{ P }is the number of cleavage sites per protein. The peptides generated from the terminus of the protein (further called terminal) will not bear a cleavage site residue R_{ C }at their end. All the other peptides, which we call internal, will have such a residue at their end. The fraction of the internal peptides f_{c,n}is given by
${f}_{c,n}=\frac{{C}_{P}n}{{C}_{P}+1n},\left(4\right)$
where n is the number of missed cleavages per protein. We approximate C_{ P }, for a sequence database, by:
${C}_{P}=\leftP\right\cdot \left({\displaystyle \sum {f}_{{R}_{C}}}\right),\left(5\right)$
where ${f}_{{R}_{C}}$ are the relative frequencies of the cleavage sites and P is the average protein length in the database. The fraction of the terminal peptides in case of n missed cleavages is given by 1  f_{c,n}. The fraction of cleavage site residues R_{ C }in a internal peptide of mass m_{pep}, with n missed cleavage sites is denoted f_{m,n}and approximated by:
${f}_{m,n}=\left(n+1\right)\frac{\overline{m}}{{m}_{\text{pep}}},\left(6\right)$
Frequencies of cleavage site residues, and all other residues, in peptides of mass m and of terminal, and internal, peptides.
R _{noncleavage}  R _{cleavage}  Peptide type  

(1  f_{m,n})  f _{m,n}  f _{c,n}  internal 
(1  f_{m,n  1})  f _{m,n  1}  1  f_{c,n}  terminal 
In the case of internal peptides, the average contribution of the amino acid residues to the peptide mass is the weighted sum:
$\begin{array}{cc}{m}_{{R}_{C},n}^{(\ast )}=\left(1{f}_{m,n}\right)\cdot {m}_{none}+{f}_{m,n}\cdot {m}_{{R}_{c}}& \left(7\right)\\ ={m}_{none}+{f}_{m,n}\cdot \left({m}_{{R}_{C}}{m}_{none}\right),& \left(8\right)\end{array}$
where
${m}_{none}={\displaystyle \sum _{i\in R\backslash {R}_{C}}{f}_{i}\cdot {m}_{i},}\left(9\right)$
is the average mass of non cleavage residues, and:
${m}_{{R}_{C}}={\displaystyle \sum _{i\in {R}_{C}}{f}_{i}\cdot {m}_{i\cdot}}\left(10\right)$
is the average mass of the cleavage site residues R_{ C }. Finally, the wavelength of internal peptides is presented as:
${\lambda}_{{R}_{C},n}^{m}=\frac{{m}_{{R}_{C},n}^{\left(M\right)}}{{m}_{{R}_{C},n}^{\left(N\right)}}\left(11\right)$
The wavelength of terminal peptides was determined by: ${\lambda}_{{R}_{C},m}^{\left(n1\right)}=\frac{{m}_{{R}_{C},n1}^{\left(M\right)}}{{m}_{{R}_{C},n1}^{\left(N\right)}}$.
The wavelength λ of all peptides at a mass m with exactly n missed cleavages is given by:
${\lambda}_{{R}_{C},n}^{m,\ast}=\frac{{m}_{{R}_{C},n}^{\left(M\right),\ast}}{{m}_{{R}_{C},n}^{\left(N\right),\ast}}\left(12\right)$
where
$\begin{array}{llll}{m}_{{R}_{C},n}^{[MN],\ast}\hfill & =\hfill & {f}_{c,n}\cdot {m}_{{R}_{C},n}^{[MN]}+(1{f}_{c,n})\cdot {m}_{{R}_{C},n1}^{[MN]}\hfill & \left(13\right)\hfill \\ =\hfill & {m}_{none}+\left({m}_{{R}_{C}}{m}_{none}\right)\cdot \left({f}_{c,n}{f}_{m,n}+{f}_{m,(n1)}{f}_{c,n}{f}_{m,(n1)}\right)\hfill & \left(14\right)\hfill \\ \underset{\text{withEquation6}}{\underset{\u23df}{=}}\hfill & {m}_{none}+\frac{\overline{m}}{m}\left({f}_{c,n}+n\right)\left({m}_{{R}_{C}}{m}_{none}\right)\hfill & \left(15\right)\hfill \\ \underset{\text{withEquation4}}{\underset{\u23df}{=}}\hfill & {m}_{none}+\left(\frac{{C}_{p}n}{{C}_{p}+1n}+n\right)\cdot \frac{\overline{m}}{m}\cdot \left({m}_{{R}_{C}}{m}_{none}\right)\hfill & \left(16\right)\hfill \end{array}$
is the weighted sum of the mass of the terminal peptides (with frequency 1  f_{c,n}) and the internal peptides (with frequency f_{c,n}).
Cleavage probability p_{ c }In practice, the cleavage probability will depend on various factors, for example on the incubation time and the efficiency of the protease used. The probability to generate a peptide with n ∈ 0...∞ missed cleavage sites, given the cleavage probability p_{ c }can be modelled using the geometric distribution:
P(n, p_{ c }) = (1  p_{ c })^{ n }·p_{ c } (17)
Furthermore,
$\sum _{n=0}^{\infty}{\left(1{p}_{c}\right)}^{n}}\cdot {p}_{c}=1\left(18\right)$
holds. Hence, given the cleavage probability is p_{ c }and cleavage residues R_{ C }, we express the peptide mass by:
${m}_{{R}_{C},{p}_{c}}^{\ast}={m}_{none}+{\displaystyle \sum _{n=0}^{\infty}{\left(1{p}_{c}\right)}^{n}}\cdot {p}_{c}\cdot \left({m}_{{R}_{C}}{m}_{none}\right)\cdot {S}_{n},\left(19\right)$
where
S_{ n }= (f_{c,n}f_{m,n}+ f_{m,(n1)} f_{c,n}f_{m,(n1)}). (20)
Therefore, the wavelength λ of peptides if the cleavage probability is p_{ c }is given by:
${\lambda}_{{R}_{C},{p}_{c}}^{m,\ast}=\frac{{m}_{{R}_{C},{p}_{c}}^{\left(M\right),\ast}}{{m}_{{R}_{C},{p}_{c}}^{\left(N\right),\ast}}\left(21\right)$
The monoisotopic mass as a function of the nominal mass can be expressed by:
$\begin{array}{llll}{m}^{(M)}\hfill & =\hfill & {\lambda}_{{R}_{C},{p}_{c}}^{(m),\ast}\cdot {m}^{(N)}\hfill & \left(22\right)\hfill \\ =\hfill & \frac{{m}_{{R}_{C},{p}_{c}}^{(M),\ast}\cdot {m}^{(N)}}{{m}_{{R}_{C},{p}_{c}}^{(N),\ast}}\hfill & \left(23\right)\hfill \\ \underset{\text{withEq}\text{.20and4}}{\underset{\u23df}{=}}\hfill & \frac{{m}_{none}^{\left(M\right)}\cdot {m}^{\left(N\right)}+{\displaystyle {\sum}_{n=0}^{\infty}{\left(1{p}_{c}\right)}^{n}\cdot {p}_{c}\cdot \left({m}_{{R}_{C}}^{\left(M\right)}{m}_{none}^{\left(M\right)}\right)\cdot \overline{m}\left({f}_{c,n}+n\right)}}{{m}_{none}^{\left(N\right)}+{\displaystyle {\sum}_{n=0}^{\infty}{\left(1{p}_{c}\right)}^{n}\cdot {p}_{c}\cdot \left({m}_{{R}_{C}}^{\left(N\right)}{m}_{none}^{\left(N\right)}\right)\cdot \frac{\overline{m}}{{m}^{\left(N\right)}}\left({f}_{c,n}+n\right)}}\hfill & \left(24\right)\hfill \\ \underset{\text{for}{m}^{\left(N\right)}\gg \overline{m}}{\underset{\u23df}{\approx}}\hfill & \frac{{m}_{none}^{(M)}\cdot {m}^{(N)}}{{m}_{none}^{(N)}}+\frac{{\displaystyle {\sum}_{n=0}^{\infty}{\left(1{p}_{c}\right)}^{n}\cdot {p}_{c}\cdot ({m}_{{R}_{C}}^{(M)}{m}_{none}^{(M)}})\cdot \overline{m}\left({f}_{c,n}+n\right)}{{m}_{none}^{(M)}}\hfill & \left(25\right)\hfill \end{array}$
${c}_{1}=\frac{3000\cdot {\lambda}_{{R}_{C},{p}_{c}}^{\left(3000\right),\ast}500\cdot {\lambda}_{{R}_{C},{p}_{c}}^{\left(500\right),\ast}}{3000500}=1.000482,\left(26\right)$
and intercept coefficient
${c}_{0}=500\cdot \left({\lambda}_{{R}_{C},{p}_{c}}^{\left(500\right),\ast}1\right){c}_{1}\cdot 500=\mathrm{0.029.}\left(27\right)$
These coefficients are in good agreement with the slope and intercept determined by linear regression for the in silico sequence database digest (Figure 1).
Furthermore, we observed that the intercept c_{0} will be positive if ${m}_{{R}_{C}}$ > m_{ none }, zero or negative otherwise. The slope c_{1} equals λ_{ none }= $\frac{{m}_{none}^{\left(M\right)}}{{m}_{none}^{\left(N\right)}}$, for large m^{(N)}, because the frequency of the cleavage site residues R_{ C }decreases with increasing peptide length:
$\underset{\leftPep\right\to \infty}{\mathrm{lim}}{f}_{m,n}\propto \underset{{m}_{pep}\to \infty}{\mathrm{lim}}\frac{\left(n+1\right)\overline{m}}{{m}^{\left(N\right)}}=0.$
Figure 4, panel B, displays the difference between the line (c_{1} + 1)·m^{(M)}+ c_{0} and the prediction made using Equation 3. For the mass range m ∈ (500, 4000) where peptide masses for peptide mass fingerprinting are acquired this difference is minimal.
The coefficients c_{0} and c_{1} do not depend on the mass of the peptides. Due to this feature, we are going to use the affine model c_{1}m^{(N)}+ c_{0} to predict the peptide mass cluster centres in the applications discussed later. This simplified model is also in agreement with the affine model (Equation 3), which has been fitted by linear regression to the in silico database digest in order to explain the dependency of the peptide mass cluster centres on the nominal mass.
Error of the model
For a moving window of 100Da we computed the maximum and minimum (orange), third and first quartile (red), median (blue) and mean(gree) of $\overline{\Delta}$(cluster). The combinatorial restriction decreases with increasing mass and for peptide masses greater than 1000Da it is negligible. However, $\overline{\Delta}$(cluster) increases again for masses greater than 2500Da because peptide masses may deviate more strongly from the cluster centres and furthermore much fewer long peptides are generated.
The type of distribution around the cluster centres
In order to remove nonpeptide peaks prior to database search, filtering thresholds have to be chosen. In Figure 3 the orange line visualises the standard deviation while the green lines show the 1% and 99% quantiles of Δ^{ ppm }(m) = Δ(m)/m·10^{6} computed for a mass window of 15Da. In addition the dotted, dashed, and dot dashed line show the deviation Δ^{ ppm }(m), at which clusters of mass spectrometric matrices are expected.
Sensitivity analysis
The input parameters to the model of the peptide mass cluster centres included:

f_{ i }– frequencies of the amino acids.

cleavage specificity of the protease R_{ C }

P – Protein length

p_{ c }– cleavage probability
Cleavage sites of proteolytic enzymes [36]
Enzyme  R _{ C }  

1  Trypsin/P  K,R/P 
2  Arg.C  R/P 
3  CNBR + Trypsin  F, Y, M 
4  LysC  K/P 
5  PepsinA  F, L 
6  CNBr  M 
A measure of distance to cluster centres
Given an experimentally determined m_{ M }we were interested to estimate the deviation Δ from the closest predicted cluster centre. The model of the monoisotopic mass is:
c_{0} + c_{1}·m_{ N }+ Δ = m_{ M }, (28)
where c_{0}, c_{1} can be obtained using the Equations 27 and 26, m_{ N }is the nominal mass (an integer).
Therefore, for a given m_{ M }, c_{0} and c_{1} we can determine the deviation Δ from the closest cluster centre of smaller mass by using the modulo operator as suggested by Wool and Smilansky [10]:
(m_{ M } c_{0})(modc_{1}) = (c_{1}·m + Δ)(modc_{1}) = Δ. (29)
However, in order to determine the distance to the closest cluster centre we considered two cases:
${\Delta}_{\lambda}\left({m}_{i},0\right)=\{\begin{array}{lll}\left({m}_{i}{c}_{0}\right)\left(\mathrm{mod}{c}_{1}\right)\hfill & \text{if}\hfill & \left({m}_{i}{c}_{0}\right)\left(\mathrm{mod}{\lambda}_{none}\right)<0.5\hfill \\ 1+\left({m}_{i}{c}_{0}\right)\left(\mathrm{mod}{c}_{1}\right)\hfill & \text{otherwise}.\hfill \end{array}\left(30\right)$
The units of Δ_{ λ }(m_{ i }, 0) are in [m/z]. The magenta dot dashed curves in Figure 3 indicate the maximum detectable distance from cluster centres in ppm (±0.5Da/m·10^{6}[ppm]). Deviations from the cluster centres outside the range enclosed by these two curves are assigned to the wrong cluster. In case of theoretical peptide masses and experimental masses calibrated to high precision, such distances are observed only for masses greater than 2500Da. Fortunately, the majority of tryptic peptide masses detected in a mass spectrometric peptide fingerprint experiment are below this mass.
Applications
Linear regression on peptide mass rule LR/PR
The limitations of calibration methods based on the property of peptide mass clustering are a mass accuracy of only 0.2Da, its sensitivity to nonpeptide peaks in the spectra, and that it completely fails if the number of peptide peaks in the peak list is small [10, 14, 19]. Hence, in practice, the method is used to confirm the results of internal calibration only [14, 29]. However, the advantage of the calibration methods based on the property of peptide mass clustering, over other calibration methods [12], is that no internal or external calibrants are required in order to calibrate the peptide mass lists.
We propose here a novel method for the calibration of PMF data, based on robust linear regression and the distance measure introduced in the Equation 30. To determine the slope of the mass measurement error we computed the deviation from the peptide mass rule for every pair of peak masses (m_{ i }, m_{ j }) within a peaklist, employing the following equation:
${\Delta}_{\lambda}\left({m}_{i},{m}_{j}\right)=\{\begin{array}{lll}{m}_{i}{m}_{j}\left(\mathrm{mod}{\lambda}_{none}\right)\hfill & \text{if}\hfill & {m}_{i}{m}_{j}\left(\mathrm{mod}{\lambda}_{none}\right)<0.5\hfill \\ 1+\left({m}_{i}{m}_{j}\left(\mathrm{mod}{\lambda}_{none}\right)\right)\hfill & \text{otherwise}\text{.}\hfill \end{array}\left(31\right)$
m_{ corrected }= m_{ experimental }·(1  $\widehat{c}$_{1})
To determine the intercept coefficient of the mass measurement error we subsequently computed Δ_{ λ }(m_{ corrected }, 0) (using Equation 30), for all peaklist masses. Figure 8, Panel B shows the distribution of Δ_{ λ }(m_{ i }, 0) before correcting for the slope error (gray histogram) and afterwards (black histogram). The red vertical line indicates the mean $\overline{\Delta}$_{ λ }(m_{ i }, 0), computed for the corrected data, which we used to approximate the intercept $\widehat{c}$_{0} of the mass measurement error.
The strip charts (Figure 8, Panel C and D) visualises the experimental masses of two trypsin peptides 842.508Da and 2211.100Da observed in most of the samples of the dataset with 380 peaklists. The result of LR/PR calibration (red circles) is compared with raw masses (gray triangles) and the output of the Wool and Smilansky calibration method (blue crosses). The LR/PRmethod is able to calibrate mass spectrometric peaklists to an accuracy of 0.1Da. This measurement accuracy surpasses the other published calibration methods [10, 19] at least twofold.
Filtering of nonpeptide peaks using the peptide mass rule
 413[ppm] = (λ_{ C } λ_{ DB })·10^{6} < Δ_{ Δ }(m, 0)·10^{6}/m = ${\Delta}_{\lambda}^{ppm}$ (m, 0) < (λ_{ L } λ_{ DB }) = 241[ppm], (32)
where λ_{ DB }= 1.000511 (Equation 2). We used the relative deviation of Δ^{ ppm }from the cluster centre in parts per million instead of using absolute values.
Figure 3 shows that only very short peptides approach the lower bound of 413ppm. This is due to the low frequency of Cysteine (C). The high frequencies of K, L, I (whose λ ≈ 1.00074) mean that the theoretical upper bound of 241ppm can indeed be reached by some peptides with a mass of ≈ l000Da. Peptides of higher mass never approach the upper and lower theoretical bound due to the rapidly decreasing probability to consist of K, L or I, or of C only. The lines for the standard deviation of S_{ N }(orange lines) and of the 1% and 99% quantile (green lines) in Figure 3 indicate that it is an exceedingly rare event to encounter a peptide mass for which ${\Delta}_{\lambda}^{ppm}$(m, 0) will deviate more than 200ppm from the peptide cluster centre predicted by our model. Therefore, we use 200ppm as a filtering threshold. An essential requirement, to apply this filtering method successfully is that peaklist must be calibrated to high precision [12].
Results for filtering of nonpeptide masses.
Arabidopsis t.  Rhodopirelulla b.  Mus musculus  

1  Identification no PR filtering  423  1009  872 
2  Identification with PR filtering  432  1017  894 
3  Change in identification (Percent)  2.13  0.79  2.52 
4  Total nr. of samples*  818  1169  1709 
5  Nr. samples with PBMS increase  240  622  724 
6  Nr. samples with no change of PBMS  571  542  982 
7  Nr. samples with PBMS decrease  7  5  3 
8  Percent increase of PBMS score  29.34  53.21  42.36 
9  Percent decrease of PBMS score  0.86  0.43  0.18 
We concluded that nonpeptide peak filtering increases the sensitivity of protein identification if using the PBMS scoring schema. However, to which extend these results can be reproduced is dependent on the database search algorithm used.
Conclusion
We introduced here a simple model to predict the cluster centres of peptide masses. The input parameters of the model can be easily determined for the sequence databases. We studied how these parameters influence the location of cluster centres, concluding that the cleavage specificity of the enzyme used for peptide digestion and the cleavage probability are the main factors. The change of the cluster centre location due to changes in average protein length or due to variability of amino acid frequencies among the databases is relatively small. However, our analysis also illustrates that, due to combinatorial constraints, the location of the cluster centres for masses smaller than l000Da can differ from the average location. Based on the model of the peptide mass cluster centres we derived a measure to determine the deviation of an experimental peptide mass from the nearest cluster centre. We used this distance measure to calibrate the peptide peaklists and to recognise nonpeptide peaks. The calibration method, linear regression on peptide rule, is a robust and accurate method to calibrate single peak lists without resorting to internal calibrants. With this method higher calibration precision was obtained in comparison to other calibration methods, which also employ the property of peptide mass clustering.
The same distance measure was used to recognise nonpeptide peaks and to remove them from the peaklists. Due to their removal, an increase of the identification rate of up to 2.5% for the PBMS scoring schema was observed.
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 aforementioned organisms were separated by twodimensional (2D) gel electrophoresis [33] and visualised with MS compatible Coomassie brilliant blue G250 [32]. The MALDITOF MS analysis was performed using a delayed ion extraction and by employing the MALDI AnchorChip ™targets (Bruker Daltonics, Bremen, Germany). Positively charged ions in the m/z range of 700 – 4, 500m/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.
Calibration
In order to perform filtering of nonpeptide peaks the dataset must be calibrated to high mass measurement accuracy. To align the dataset we used a calibration sequence [12] consisting of several calibration procedures.
First calibration using external calibration samples was performed in order to remove higher order terms of the mass measurement error [11]. Next, the affine mass measurement error of all samples on the sample support was determined by linear regression on the peptide mass rule introduced here. Subsequently, the thin plate splines were used to model the mass measurement error in dependence of the sample support positions to calibrate the spectra. Finally, the spectra were aligned using a modified spanning tree algorithm [12].
Mascot database search
Processed peaklists were then used for the protein database searches with the Mascot search software (Version 1.8.1) [22], 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.
Sequence databases
Protein lengths and amino acid frequencies (one letter code) for nine in the nine databases, length – average protein length in database, reference – database reference; f_{ i }– amino acid frequencies
Organ izm  length  f _{ F }  f _{ S }  f _{ T }  f _{ N }  f _{ K }  f _{ Y }  f _{ E }  f _{ V }  f _{ Q }  f _{ M } 

Arabidopsis t.  422.40  4.27  9.01  5.11  4.41  6.36  2.86  6.74  6.69  3.52  2.44 
Drosophila m.  506.20  3.48  8.33  5.68  4.80  5.70  2.91  6.41  5.88  5.21  2.33 
Escherichia coli  300.30  3.86  6.25  5.67  4.26  4.59  2.96  5.65  6.91  4.40  2.67 
Homo sapiens  360.40  3.61  8.61  5.55  3.55  5.54  2.86  6.81  6.02  4.80  2.12 
Mus musculus  378.30  3.74  8.58  5.55  3.59  5.71  2.88  6.75  6.11  4.74  2.22 
Rattus norvegicus  484.40  3.81  8.33  5.52  3.59  5.62  2.74  6.77  6.32  4.64  2.28 
Saccharomyces c.  447.00  4.47  9.02  5.93  6.18  7.26  3.41  6.43  5.58  3.94  2.10 
Rhodopirellula b.  314.70  3.70  7.37  5.85  3.37  3.44  2.09  6.02  7.05  4.04  2.43 
SwissProt DB  367.90  4.03  6.89  5.47  4.22  5.93  3.09  6.59  6.70  3.93  2.38 
Mean  397.96  3.89  8.04  5.59  4.22  5.57  2.87  6.46  6.36  4.36  2.33 
SD  71.90  0.32  0.98  0.24  0.88  1.07  0.35  0.39  0.50  0.54  0.18 
Min  300.30  3.48  6.25  5.11  3.37  3.44  2.09  5.65  5.58  3.52  2.10 
Max  506.20  4.47  9.02  5.93  6.18  7.26  3.41  6.81  7.05  5.21  2.67 
reference  f _{ C }  f _{ L }  f _{ A }  f _{ W }  f _{ P }  f _{ H }  f _{ D }  f _{ R }  f _{ I }  f _{ G }  
Arabidopsis t.  [34]  1.80  9.52  6.36  1.26  4.80  2.28  5.43  5.39  5.34  6.41 
Drosophila m.  [34]  1.95  9.02  7.36  1.00  5.46  2.64  5.18  5.53  4.96  6.17 
Escherichia coli  [34]  1.17  10.23  9.27  1.50  4.32  2.22  5.21  5.54  5.94  7.38 
Homo sapiens  [34]  2.24  9.78  6.98  1.35  6.22  2.51  4.73  5.64  4.28  6.80 
Mus musculus  [34]  2.29  9.92  6.86  1.29  6.03  2.57  4.76  5.51  4.38  6.54 
Rattus norvegicus  [34]  2.29  10.07  6.88  1.25  5.97  2.58  4.77  5.59  4.51  6.49 
Saccharomyces c.  [34]  1.30  9.52  5.51  1.04  4.39  2.18  5.76  4.41  6.58  5.00 
Rhodopirellula b.  [37]  1.27  9.31  9.25  1.54  5.33  2.31  6.23  6.96  4.95  7.48 
SwissProt  [27]  1.57  9.63  7.80  1.17  4.86  2.27  5.30  5.29  5.92  6.94 
Mean  1.76  9.67  7.36  1.27  5.26  2.40  5.26  5.54  5.21  6.58  
SD  0.45  0.38  1.25  0.18  0.71  0.18  0.50  0.65  0.80  0.74  
Min  1.17  9.02  5.51  1.00  4.32  2.18  4.73  4.41  4.28  5.00  
Max  2.29  10.23  9.27  1.54  6.22  2.64  6.23  6.96  6.58  7.48 
In silico protein digestion
The theoretical digestion of the protein databases was done with ProtDigest [35], a command line program taking a protein sequence database file in fasta format and cleavage specificities as input. Other optional input parameters included fixed as well as variable modifications and number of missed cleavages. The output file contains all theoretically resulting peptides with their corresponding masses.
Regression analysis
The complete tryptic insilico digest of the SwissProt [27] database generated more than 7 million peptides. In order to compute the slope coefficient we were sampling 500 times 10000 monoisotopic and corresponding nominal masses. For each sample we fitted the affine linear model with and without fixed intercept using linear regression. The slope and intercept coefficients in Figure 1 are the medians of these 500 samples.
Appendix
Wool and Smilanskys algorithm
Wool and Smilansky [10] use a Discrete Fourier Transform (DFT) to determine the calibration coefficients. The wavelength λ of a peptide peaklist can be determined by convolution. The "time domain" is the peaklist X with masses x_{ i }. We computed the amplitude A (Equation 36) for a small range of frequencies (ω ~ f = 1/λ around λ_{ theo }. We scanned the range λ ∈ λ_{ theo }± 0.0005 in steps of 5·10^{7} computing, for each λ, the real part (Equation 35), the imaginary part (Equation 34) and the amplitude A(ω) (Equation 36):
f = 1/λ ω = 2πf, (33)
$\Im \left(\omega \right)={\displaystyle \sum _{i}sin\left(\omega {x}_{i}\right),}\left(34\right)$
$\Re \left(\omega \right)={\displaystyle \sum _{i}cos\left(\omega {x}_{i}\right),}\left(35\right)$
$A\left(\omega \right)=\sqrt{\Im {\left(\omega \right)}^{2}+\Re {\left(\omega \right)}^{2}}.\left(36\right)$
The wavelength of the masses in the peaklist is the λ at the maximum of A(ω). The phase for this ω_{0} = ω_{max A(ω)}can be determined by:
${\varphi}_{0}=\varphi \left({\omega}_{\mathrm{max}A\left(\omega \right)}\right)=\mathrm{arctan}\left(\frac{\Im {\left({\omega}_{0}\right)}^{2}}{\Re {\left({\omega}_{0}\right)}^{2}}\right)\cdot \left(37\right)$
The peak centres are at the line:
$\stackrel{\prime}{M}=\frac{2\cdot \pi}{{\omega}_{0}}\cdot N+\frac{{\varphi}_{0}}{{\omega}_{0}}\text{where}N=1,2,\mathrm{...},n.\left(38\right)$
But they should be on the line:
M = λ_{ theo }* N. (39)
Solving Equation 38 for N and substituting N in the Equation 39 yields the Equation:
$M=\frac{{\lambda}_{theo}\cdot {\omega}_{0}}{2\cdot \pi}(\stackrel{\prime}{M}\frac{{\varphi}_{0}}{{\omega}_{0}}),\left(40\right)$
$\begin{array}{cc}\alpha =\frac{{\lambda}_{theo}\cdot {\omega}_{0}}{2\cdot \pi}\text{and}\beta =\frac{{\varphi}_{0}}{{\omega}_{0}}\text{and}& \left(41\right)\\ {m}_{corr}=\alpha ({m}_{exp}\beta )=\alpha {m}_{exp}\alpha \beta ,& \left(42\right)\end{array}$
m_{ corr }= α(m_{ exp } β) = αm_{ exp } αβ, (42)
which can be used to correct the masses. This is an affine linear model with two coefficients α and αβ.
Abbreviation
 PBMS:

Probability based Mascot score
 DFT:

Discrete Fourier Transformation
 m/z:

mass over charge
Declarations
Acknowledgements
I would like to thank the members of Algorithmic Bioinformatics group of Prof. Knut Reinert at the FUBerlin for valuable discussion, especially Andreas Döring and Dr. Clemens Gröpl. Many thanks also to Dr. Johan Gobom, Dr. Patrick Giavalisco for providing the PMFMS data and for valuable discussion. This project was partially funded by the National Genome Research Network (NGFN) of the German Ministry for Education and Research (BMBF).
Authors’ Affiliations
References
 Fenyo D: Identifying the proteome: software tools. Current Opinion in Biotechnology 2000, 11: 391–395. 10.1016/S09581669(00)001154View ArticlePubMedGoogle Scholar
 Griffin TJ, Aebersold R: Advances in proteome analysis by mass spectrometry. J Biol Chem 2001, 276: 45497–500. 10.1074/jbc.R100014200View ArticlePubMedGoogle Scholar
 Patterson SD: Data analysis – the Achilles heel of proteomics. Nat Biotechnol 2003, 21(3):221–2. 10.1038/nbt0303221View ArticlePubMedGoogle Scholar
 Aebersold R, Mann M: Mass spectrometrybased proteomics. Nature 2003, 422(6928):198–207. 10.1038/nature01511View ArticlePubMedGoogle Scholar
 Mann M, Hojrup P, Roepstorff P: Use of mass spectrometric molecular weight information to identify proteins in sequence databases. Biol Mass Spectrom 1993, 22(6):338–345. 10.1002/bms.1200220605View ArticlePubMedGoogle Scholar
 Johnson R, Martin S, Biemann K, Stults J, Watson J: Novel Fragmentation Process of Peptides by CollisionInduced Decomposition in a Tandem Mass Spectrometer: Differentiation of Leucine and Isoleucine. Analytical Chemistry 1987, 59(21):2621–2625. 10.1021/ac00148a019View ArticlePubMedGoogle Scholar
 Smith RD, Loo JA, Edmonds CG, Barinaga CJ, Udseth HR: New developments in biochemical mass spectrometry: electrospray ionization. Anal Chem 1990, 62(9):882–99. 10.1021/ac00208a002View ArticlePubMedGoogle Scholar
 Apweiler R, Bairoch A, Wu CH: Protein sequence databases. Curr Opin Chem Biol 2004, 8: 76–80. 10.1016/j.cbpa.2003.12.004View ArticlePubMedGoogle Scholar
 Gentzel M, Kocher T, Ponnusamy S, Wilm M: Preprocessing of tandem mass spectrometric data to support automatic protein identification. Proteomics 2003, 3(8):1597–610. 10.1002/pmic.200300486View ArticlePubMedGoogle Scholar
 Wool A, Smilansky Z: Precalibration of matrixassisted laser desorption/ionizationtime of flight spectra for peptide mass fingerprinting. Proteomics 2002, 2(10):1365–1373. 10.1002/16159861(200210)2:10<1365::AIDPROT1365>3.0.CO;29View ArticlePubMedGoogle Scholar
 Gobom J, Mueller M, Egelhofer V, Theiss D, Lehrach H, Nordhoff E: A calibration method that simplifies and improves accurate determination of peptide molecular masses by MALDITOF MS. Anal Chem 2002, 74(15):3915–3923. [(eng)] 10.1021/ac011203oView ArticlePubMedGoogle Scholar
 Wolski WE, Lalowski M, Jungblut P, Reinert K: Calibration of mass spectrometric peptide mass fingerprint data without specific external or internal calibrants. BMC Bioinformatics 2005, 6: 203. [http://www.biomedcentral.com/1471–2105/6/203] 10.1186/147121056203PubMed CentralView ArticlePubMedGoogle Scholar
 Levander F, Rognvaldsson T, Samuelsson J, James P: Automated methods for improved protein identification by peptide mass fingerprinting. Proteomics 2004, 4(9):2594–601. 10.1002/pmic.200300804View ArticlePubMedGoogle Scholar
 Chamrad DC, Koerting G, Gobom J, Thiele H, Klose J, Meyer HE, Blueggel M: Interpretation of mass spectrometry data for highthroughput proteomics. Anal Bioanal Chem 2003, 376(7):1014–22. 10.1007/s002160031995xView ArticlePubMedGoogle Scholar
 Gay S, Binz PA, Hochstrasser DF, Appel RD: Modeling peptide mass fingerprinting data using the atomic composition of peptides. Electrophoresis 1999, 20(18):3527–3534. [(eng)] 10.1002/(SICI)15222683(19991201)20:18<3527::AIDELPS3527>3.0.CO;29View ArticlePubMedGoogle Scholar
 Wikipedia is a Webbased, freecontent encyclopedia2004. [http://www.wikipedia.org]
 Giles J: Internet encyclopaedias go head to head. Nature 2005, 438(7070):900–1. 10.1038/438900aView ArticlePubMedGoogle Scholar
 Pappin DJC, Hojrup P, Bleasby AJ: Rapid identification of proteins by peptidemass fingerprinting. Curr Biol 1993, 3: 327–332. 10.1016/09609822(93)90195TView ArticlePubMedGoogle Scholar
 Gras R, Muller M, Gasteiger E, Gay S, Binz PA, Bienvenut W, Hoogland C, Sanchez JC, Bairoch A, Hochstrasser DF, Appel RD: Improving protein identification from peptide mass fingerprinting through a parameterized multilevel scoring algorithm and an optimized peak detection. Electrophoresis 1999, 20(18):3535–3550. [(eng)] 10.1002/(SICI)15222683(19991201)20:18<3535::AIDELPS3535>3.0.CO;2JView ArticlePubMedGoogle Scholar
 Gobom J, Schürenberg M, Mueller M, Theiss D, Lehrach H, Nordhoff E: Alphacyano4hydroxycinnamic acid affinity sample preparation. A protocol for MALDIMS peptide analysis in proteomics. Analytical Chemistry 2001, 73(3):434–438. 10.1021/ac001241sView ArticlePubMedGoogle Scholar
 Zhen Y, Xu N, Richardson B, Becklin R, Savage JR, Blake K, Peltier JM: Development of an LCMALDI method for the analysis of protein complexes. J Am Soc Mass Spectrom 2004, 15(6):803–22. 10.1016/j.jasms.2004.02.004View ArticlePubMedGoogle Scholar
 Perkins DN, Pappin DJ, Creasy DM, Cottrell JS: Probabilitybased protein identification by searching sequence databases using mass spectrometry data. Electrophoresis 1999, 20(18):3551–3567. 10.1002/(SICI)15222683(19991201)20:18<3551::AIDELPS3551>3.0.CO;22View ArticlePubMedGoogle Scholar
 Schmidt F, Schmid M, Jungblut PR, Mattow J, Facius A, Pleissner KP: Iterative data analysis is the key for exhaustive analysis of peptide mass fingerprints from proteins separated by twodimensional electrophoresis. J Am Soc Mass Spectrom 2003, 14(9):943–56. 10.1016/S10440305(03)003453View ArticlePubMedGoogle Scholar
 Tabb DL, MacCoss MJ, Wu CC, Anderson SD, Yates JRr: Similarity among tandem mass spectra from proteomic experiments: detection, significance, and utility. Anal Chem 2003, 75(10):2470–7. 10.1021/ac026424oView ArticlePubMedGoogle Scholar
 Wolski WE, Lalowski M, Martus P, Herwig R, Giavalisco P, Gobom J, Sickmann A, Lehrach H, Reinert K: Transformation and other factors of the peptide mass spectrometry pairwise peaklist comparison process. BMC Bioinformatics 2005, 6: 285. 10.1186/147121056285PubMed CentralView ArticlePubMedGoogle Scholar
 Cagney G, Amiri S, Premawaradena T, Lindo M, Emili A: In silico proteome analysis to facilitate proteomics experiments using mass spectrometry. Proteome Science 2003, 1: 5. [http://www.Proteomesci.com/content/1/1/5] 10.1186/1477595615PubMed CentralView ArticlePubMedGoogle Scholar
 Bairoch A, Apweiler R, Wu CH, Barker WC, Boeckmann B, Ferro S, Gasteiger E, Huang H, Lopez R, Magrane M, Martin MJ, Natale DA, O'Donovan C, Redaschi N, Yeh LSL: The Universal Protein Resource (UniProt). Nucleic Acids Res 2005, (33 Database):D154–9.
 Becker RA, Chambers JM, Wilks AR: The New S Language. London: Chapman & Hall; 1988.Google Scholar
 Samuelsson J, Dalevi D, Levander F, Rognvaldsson T: Modular, Scriptable, and Automated Analysis Tools for HighThroughput Peptide Mass Fingerprinting. Bioinformatics 2004.Google Scholar
 Venables WN, Ripley BD:Modern Applied Statistics with S. Fourth edition. Springer; 2002. [ISBN 0–387–95457–0] [http://www.stats.ox.ac.uk/pub/MASS4/]View ArticleGoogle Scholar
 Bruker Daltonics – enabling life science tools based on mass spectrometry2004. [http://www.bdal.com]
 Giavalisco P, Nordhoff E, Kreitler T, Kloeppel KD, Lehrach H, Klose J, Gobom J: Proteome Analysis of Arabidopsis Thaliana by 2D Electrophoresis and Matrix Assisted Laser Desorption/Ionization Time of Flight Mass Spectrometry. [To appear in Proteomics]
 Klose J, Kobalz U: Twodimensional electrophoresis of proteins: an updated protocol and implications for a functional analysis of the genome. Electrophoresis 1995, 16(6):1034–59. 10.1002/elps.11501601175View ArticlePubMedGoogle Scholar
 Pruitt KD, Tatusova T, Maglott DR: NCBI Reference Sequence project: update and current status. Nucleic Acids Res 2003, 31: 34–7. 10.1093/nar/gkg111PubMed CentralView ArticlePubMedGoogle Scholar
 Emde AK: Protein Sequence Digester.2004. [http://www.inf.fuberlin.de/~emde/]Google Scholar
 Mascot2005. [http://www.matrixscience.com]
 Glockner FO, Kube M, Bauer M, Teeling H, Lombardot T, Ludwig W, Gade D, Beck A, Borzym K, Heitmann K, Rabus R, Schlesner H, Amann R, Reinhardt R: Complete genome sequence of the marine planctomycete Pirellula sp. strain 1. Proc Natl Acad Sci USA 2003, 100(14):8298–303. 10.1073/pnas.1431443100PubMed CentralView ArticlePubMedGoogle Scholar
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