Fig. 1

Illustration of the conversion from the HomMTM spectral identification problem to the MSkSF problem. A deconvoluted HomMTM spectrum generated from two modified proteoforms of the protein GKGKLKAKE with one expected PTM: acetylation on K, is used as an example. a Each peak corresponds to a potential prefix residue mass of a proteoform of GKGKLKAKE satisfying that the prefix residue mass or its complementary suffix residue mass matches an experimental fragment mass. Potential masses for the prefix GKGKL matched to experimental masses are shown in the red dotted box. b A graph with 10 layers is constructed based on the masses in \(\mathcal {P}_{0}, \mathcal {P}_{1}, \ldots, \mathcal {P}_{10}\) and the peaks in (a). Each vertex in layer i, 0≤i≤10, corresponds to a mass in \(\mathcal {P}_{i}\) and those with dotted circles are removed because they are not on any path from the source to the sink. The capacity of a vertex is the ratio (shown in percentage) between the intensity of the mass and the sum of the intensities of all masses corresponding to vertices with solid circles in the same layer. The solution to the MSkSF problem is the two blue paths with flows 70 and 30 (in percentage), which correspond to two proteoforms GK[Acetylation]GK[Acetylation]LKAKE with relative abundance 70% and GKGK[Acetylation]LK[Acetylation]AKE with relative abundance 30%