## 4 Discussion
Identity between a protein sequence containing post-translational modifications and a sequence of the same protein devoid of modifications computed based on Equation (5) (parameter I) has a similar physical sense as the identity computed using specialist software for matching protein sequences such as, for example, BLAST [65], Clustal Omega [66], or HMMER [67]. Equation (5) is very simplified compared to the aforementioned computer programs because it takes no account of insertion and deletion of protein fragments and considers only the equivalent of amino acid substitutions. Even though the analyzed proteins contain a relatively low number of modified amino acid residues, the changes noted in the predicted values of parameters describing proteins as precursors of bioactive peptides seem to be significant. The appearance of false-positive results for, e.g., four out of nine activities of fragments potentially released from \(\beta\)-casein, is likely (Table 2). The percentage of false-positive results obtained for particular types of activities varied within a broad range. FP = 0 means that the frequency of occurrence of bioactive fragments in protein sequences or the predicted frequency of release of bioactive fragments are identical for the proteins with post-translational modifications considered and for the same sequences analyzed without considering modifications (A = A\({}_{0}\) or A\({}_{\text{E}}\) = A\({}_{\text{E0}}\)). In turn, FP = 100% indicates that the value of the analyzed parameter is 0 for the proteins with post-translational modifications considered and differs from 0 for the same proteins analyzed without considering these modifications (A = 0 and A\({}_{0}\
eq\) 0 or A\({}_{\text{E}}\) = 0 and A\({}_{\text{E0}}\
eq\) 0). In the case of certain activities and certain genetic variants of individual casein fractions, the bioactive fragments do not occur either in modified or in unmodified sequences (A = 0 and A\({}_{0}\) = 0 or A\({}_{\text{E}}\) = 0 and A\({}_{\text{E0}}\) = 0). In such a case, computing the percentage of false-positive results is impossible. Predictions of frequent activity types (relatively high values of A and AE), such as inhibition of angiotensin-converting enzyme (EC 3.4.15.1) or inhibition of dipeptidylopeptidase IV (EC 3.4.14.5), and also a sum of all activities yield a relatively low percentage of false-positive results. In the case of the less known activities (relatively low values of A and A\({}_{\text{E}}\)), the percentage of false-negative results ranged from 0 to 100%. Among proteins summarized in Table 4, HLA-A is a protein of the immune system responsible for presenting antigens, e.g., those derived from viruses or neoplasms [68; 69; 70]. DPTIV (EC 4.4.14.5) is a commonly known proteolytic enzyme whose inhibitors are applied as anti-diabetic drugs [59; 71]. Recent studies have focused on DPTIV role in the course of also other diseases, such as Alzheimer's disease [72] or neoplasms [73]. The XIAP protein is ubiquitin ligase (EC 2.3.2.27), involved in apoptotic processes, modulation of inflammatory signals and immune response, and cell spreading, including tumor metastases [74; 75; 76; 77]. The LPAR1 and LPAR3 proteins are receptors of lysophosphatidic acid (PubChem CID: 5497152; ChEMBL ID: CHEMBL117021), which together with its receptors are involved in signal transfer in cells of, i.e., nervous, respiratory, and gastrointestinal systems [78]. In turn, the PLK1 (EC 2.7.11.21) and PRKCE (EC 2.7.11.13) proteins are kinases catalyzing the phosphorylation of amino acid residues (serine in particular) in proteins. Phosphorylation is the most common type of post-translational modifications of proteins in live organisms [79; 80]. BRCA1 (Breast cancer type 1 susceptibility protein) (EC 2.3.2.27) is a ligase catalyzing protein ubiquitination. Its malfunction due to, among other things, mutations has been implicated in the course of breast cancer [81; 82].