A proteolytic process where single or multiple peptides are generated by cutting peptides into smaller fragments can be modeled as a graph .

The nodes V correspond to the degraded and generated peptides and the edges E to the proteolytic reactions. Since proteolysis is an irreversible reaction under physiological conditions the edges in the graph are directed from the degraded to the generated peptides.

As mentioned above, one can distinguish two types of proteolytic reactions, exoproteolytic reactions, where a single amino acid is removed from one of the free termini of the peptide, and endoproteolytic reactions, where the targeted peptide is cleaved at a position between the N- and C-terminus. For exoproteolytic reactions we connect two nodes with a directed edge from node u to v if we can obtain the amino acid sequence of v by subtracting a single amino acid from the beginning or the end of the amino acid sequence of u. For endoproteolytic reactions this is not that easy. Since we need to connect three nodes (the peptide that is targeted u and the two resulting fragments v,w) we need to break the idea of one reaction equals one edge in the graph. To ensure that we still associate the reaction with single edge, we introduce pseudo-nodes , that represent the endoproteolytic process of cutting the peptide u at a specific position c. The pseudo-nodes can also be seen as representation of the endoprotease that cuts the peptide u at position c. We can now connect u to and associate all reaction specific information (e.g., reaction rate) with this single edge. We further connect to v and w with so called pseudo-edges.

Both reaction types are separately shown in Figure 3. An example with real peptide sequences and both reaction types is shown in Figure 2.

In the previous section we defined the degradation graph and its relation to proteolytic processes. Now we present an approach to construct this graph based on series of N mass spectra collected at different time points and a seed sequence S which we will also call base peptide from here on. Based on this input we try to identify signals in the mass spectra, that represent fragments of S produced by a proteolytic process. The seed sequence needs to be provided as input. It can for instance be the sequence of a known peptide probe that was incubated with an unknown mixture of proteases or a sequence taken from MS/MS identifications.

We shortly introduce some notation that eases the understanding of the following explanations. Given a node v in the degradation graph, denotes the amino acid sequence of the peptide associated with the node v. The length of the amino acid sequence is given by . with is the subsequence of the amino acid sequence from position a to position b. denotes the mass of the peptide associated with the node v. If we could identify a signal that corresponds to the peptide associated with v, we will denote it’s intensity with . The association between mass and intensity takes into account, that mass spectrometers measure only mass to charge ratios and therefore cannot distinguish peptides with equal mass. Therefore different peptides with equal mass can be associated to the same intensity value, without counting the signal twice in the later analysis. The set of all peptide masses in the graph is denoted by M. We further introduce a queue of nodes L, which is empty at the beginning of the construction.

The construction of the graph is divided into two parts, verification and extension, which are executed on each of the input spectra. Before we can execute these steps, we need to initialize the degradation graph. This is done by adding a node for the seed sequence to the degradation graph. Afterwards we start with the verification step for the first spectrum recorded at time point , followed by the extension step. This is repeated for each of the input spectra. The pseudocode for both parts is shown in the Supporting Information (Figure S1).

After we generated the model representing the proteolytic process, i.e., the degradation graph, the next task is to estimate the kinetic parameters of the underlying process. To achieve this we first generate a system of ordinary differential equations (ODE) based on a degradation graph as described in the following section. For this system we estimate the kinetic parameters based on the observed signal intensities.

The above presented approach to construct the degradation graph is greedy, i.e., it assumes that every signal in a spectrum that could match a subsequence of the base peptide is part of the proteolytic process and that every possible reaction occurred. This assumption is not always true. The signals could also originate from peptides with equal or at least similar masses as we have already seen in the previous section. But these peptides do not necessarily take part in the proteolytic reactions, that we want to model. We will call such peptides decoy peptides. Alternatively we may have multiple reactions to explain the formation of a peptide where only one is true. Hence, the degradation graph may contain peptides or reactions that did not occur in the actual underlying proteolytic process. To account for this we present a method to rank different subgraphs of an initial degradation graph with respect to their ability to explain observed data. Followed by a heuristic approach to construct a series of smaller models from the initially generated degradation graph without the need to compute every possible subgraph.

The above presented combination of degradation graph construction, parameter estimation and structure optimization requires a considerable amount of time, if the initial degradation graph is large. Therefore we now describe an approximation of the running time in the worst case. The run time of the initial degradation graph construction is determined by the number of verifications needed. Under the assumption that we would construct the complete degradation graph, i.e., all peptides are degraded in every possible way, one would create a degradation graph, which contains all possible substrings of the initial peptide sequence. Since we would need to verify each of this substrings once, the running time is in the worst case bounded by the maximal number of possible substrings of the initial peptide sequence. Given a seed sequence of length we can construct at most possible fragments, which could be checked in the spectrum. If we now analyze spectra we will have at most verifications.

The complexity of the parameter estimation procedure can be approximated by , where is the number of time points, i.e., the number of evaluated mass spectra, and is the number of unknown parameters, i.e., the number of edges in the graph minus the number of edges connecting pseudo- and real nodes. Given this the time required for the parameter estimation will decrease with the subgraphs getting smaller. Under the assumption that even the proposed heuristic could require the computation of each subgraph, we would need to trigger optimizations in the worst case.

To demonstrate that our approach is able to recover the correct sequence of proteolytic events, i.e., the degradation graph and the corresponding kinetic parameters, we simulated a data set based on the fibrinopeptide A (FPA) (Swiss-Prot:P02671[20–35]) degradation, as described in [16]. It consists of a series of exoproteolytic cuts at the N-terminus of FPA. The corresponding degradation graph is shown in Figure 4. For the proteolytic reactions we used a slightly modified version of the kinetic parameters as published in [16]. The modified parameters can be found in the ODE formulation of the Supporting Information (File S1).

The proteolytic system was simulated over a time of 5 h. We generated sampling points for the time series, five during the first hour of the incubation and the other five distributed equally over the remaining 4 hours. For these time points we generated five sets of mass spectra with increasing signal variability of , 10, 20, 30, and 40% of the original signal intensity. The impact of the signal variability on the time course of the peptide intensities is shown in Figure 5. Thereafter we applied our new method to estimate the model structure as well as the kinetic parameters for each of the five time series.

Our method succeeded to reconstruct the original degradation graph as it is shown in Figure 4. The scores computed for the reconstructed systems show a clear dependency on the noise added during the mass spectra simulation (see Figure 6). The relative error for the individual parameters of the system in relation to noise on the simulated data is shown in Figure 7. These experiments show that even in the presence of extensive noise a valid reconstruction of the original process is possible. Also the estimated parameter values have an acceptable agreement with the original parameters. With a signal variability of 30% the quality of the estimated parameters starts to decrease drastically. This could possibly be mitigated by increasing the number of sampling points.

To test our method in a complex setting where also endoproteolytic reactions occur, we simulated the degradation of several human plasma peptides (and peptide fragments) by multiple artificial endo- and exoproteases. The targeted peptides were fragments of endothelin 1 (Swiss-Prot:P05305[53–73]), angiotensin (Swiss-Prot:P01019[34–43]), and somatostatin-28 (Swiss-Prot:P61278[89–116]). The full set of reactions and the corresponding peptide sequences are shown in Figures 8, 9, and 10.

All three systems were again simulated over a time of 5 h. We generated sampling points from the time series. More sampling points were generated in the first hour of each time series, since during this time the systems change most. For all time points we generated mass spectra with a signal variability of . During the mass spectrometry simulation of the systems we added decoy peptides that have masses similar to possible fragments of the base peptides. Therefore we also applied our method to iteratively optimize the structure of the degradation graph.

Our method generally succeeded to reconstruct the originally simulated degradation graphs. In case of the angiotensin system the peptide (e) was misinterpreted as IHPFH. Since both terminal amino acids of its predecessor (Leucin and Isoleucin) have equal mass they cannot be distinguished by the mass spectrometer hence both solutions are equally good.

For all three systems the estimated parameters in comparison to the original parameters are shown in Tables 1, 2, and 3. In general, the recovered parameters are quite well estimated. The average relative deviation between the estimated and the real parameters is between and for the different experiments. It can be observed that the largest errors occur towards the end of the degradation process (e.g., for the somatostatin 28 system). This can be due to the late formation of the later products and with this the lack of enough data points to effectively estimate the reaction parameters. An extension of the sampling range beyond 5 h or an increased sampling rate could possibly solve this issue.

Figure 11 shows the extracted intensities of two characteristic somatostatin-28 fragments compared with predicted model intensities. As one can see the predicted model intensities and the simulated intensities show a good agreement in their dynamic behavior.

To demonstrate the applicability of our method to experimental data, we analyzed a data set where a fragment of beta-2-microglobulin (Swiss-Prot:P61769[77–97]) was incubated with different urine proteins. Manual inspection of the mass spectra combined with the analysis of MS/MS spectra lead to the assumption, that four endoproteolytic reactions at positions 7–10 occurred. We applied the presented method to the dataset to validate this assumption.

The mass spectra for two time points (t = 7 h and t = 24 h) are shown in Figure 1. In the mass spectrum for t = 24 h the peaks for the fragments generated by the four endoproteolytic reactions as well as the base peptide are annotated. A figure showing all mass spectra is included in the Supporting Information (Figure S2).

The preprocessed spectra were analyzed by our method to identify the optimal degradation graph for the given mass spectra. The initially constructed degradation graph contains all four manually confirmed endoproteolytic cuts as well as four additionally not manually annotated exoproteolytic and one additional endoproteolytic cut. The complete degradation graph is shown in Figure 12. The unvalidated proteolytic reactions and fragments are represented as dashed nodes and lines.

It can further be seen from Figure 12 that the fragments generated by the validated endoproteolytic cuts are interconnected by exoproteolytic reactions. Although these reactions are possible, they are very unlikely and hence should be removed during the optimization. To reflect this the previously described selection of initial values was applied. Due to the lack of sampling points for the actual reactions, which took place between the last two sampling points, we have chosen the low quality weighting factors for this analysis ( and ).

Optimizing the degradation graph structure results in a list of subgraphs ranked by their scores. The scores varied widely with the different generated structures. A figure showing the development of the score is included in the Supporting Information (Figure S3). Since the correct solution is unknown, we need to inspect the list and the different proposed solutions. As expected, based on the manual validation (see above), the degradation graph with the highest score contains the four endoproteolytic cuts at positions . The unvalidated side reactions (see the dashed nodes and edges in Figure 12) were mostly removed, just two exoproteolytic reactions (fragment i to g and g to d) are still included, but have an estimated reaction rate of . Although these reactions are still included in the degradation graph they have effectively no influence on the system and thereby can be neglected.

Figure 13 shows the observed and the predicted intensities for a subset of the peptide fragments of the degradation graph with the best score. The time courses of all peptides are shown in the Supporting Information (Figure S4). It can be seen that the measured intensities not always agree with the predicted intensity course, but they seem to show a comparable behavior. More time points especially in the time from to hours and an improved quantification (e.g., via a spiked in control sample) could further improve the results.

In this paper we presented a new method to model any proteolytic process as a degradation graph including an algorithm to construct the degradation graph based on mass spectrometry time series data. The degradation graph can easily be translated into a system of ordinary differential equations, which can be used to estimate the kinetic parameters of the proteolytic process. We further proposed an approach to optimize the initially constructed graph in the presence of decoy and overlapping signals. It is based on a score, that is used to rank the optimized and the original degradation graphs in their ability to explain the actually observed data. Using simulated data we have shown that our approach is able to compute good estimates for the kinetic parameters of the ODE systems even in the presence of noise and decoy signals. With a careful preparation of the samples using accepted standard operating procedures [29] the variability of the mass spectrometry data is below the observed boundary of the presented method. Applied to real data our approach reconstructed manually validated endoproteolytic reactions and removed unvalidated reactions and peptides from the graph.

We are aware of other biochemical approaches that give a much more robust and exact estimate for the reaction rates, but most of these methods rely on a much more time consuming measurement of the reactants and their concentration and, more importantly, often require prior knowledge of all reactants, which is not necessary for our method.

Applications for this method can be to identify and characterize unknown proteases and the estimated reaction kinetics can possibly be used to classify between different sample categories as it was done in [17]. With the ability to handle also false identifications the method can even be used in complex samples.

Future directions are an extensive validation of the proposed approach on real data. Another by now unsolved issue is the handling of unobserved peptides, i.e., peptides that participate in the reactions, but are not observable in the mass spectra. This can be due to different reasons e.g., the peptide cannot be ionized by the mass spectrometer or the degradation process is so fast, that the generated peptide is degraded before it can be measured. This problem can be handled by a modification of the construction algorithm for the degradation graph, as long as a downstream peptide is again observable. Also the handling of more than one seed sequence would be favorable. Finally a robust integration of MS/MS identifications into the method could further improve its performance. This could be done in two ways: One could use MS/MS identifications during the initial construction of the degradation graph in combination with the already used PMF approach, as an additional and more reliable way to identify possible fragments of the peptide probe. Furthermore one could integrate the MS/MS identification and its score into the scoring function by penalizing the removal of highly scored identifications.

The proposed score could also be improved in future development. The approach would benefit from a score that does not require specific scaling parameters for the different components. It would remove the step of optimizing the scaling parameters. First experiments using a scaled least-squares residual were carried out on simulated data sets. Those have shown promising but not yet comparable results.

The presented method is available on request. The whole approach is integrated into the proteomics.net platform [30]. The estimation procedure requires POEM which is available, for academic use, on request from the Computational Systems Biology Group, Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) (http://www.zib.de/en/numerik/csb.html).