Combinatorial Probabilities of Multiple Fragmentation Events Explain Polypeptide MS/MS Intensity Distribution, Overrepresentation of Smaller Fragments, and Missing Middle of Top-Down MS.

RATIONALE: Lyon and coworkers demonstrated that multiple random fragmentation events can bias intensity distributions toward smaller terminal fragment ions. With any high-yield method of dissociation, such as high-energy collisional activation, this phenomenon can greatly influence the intensity distribution of product ions in MS/MS spectra. Previously, multiple fragmentation events were simulated with computationally intensive stochastic studies. The goal of this work is to provide a mathematical model that explains the results obtained from computational stochastic studies. METHODS: We present a probabilistic model that predicts terminal and internal product ion intensities based on polypeptide size and the number of fragmentation events. This model is validated by demonstrating convergence with the previous stochastic model and described that the intensity trend from smaller fragments to larger fragments follows the probability of multiple fragmentation events on peptide backbone. RESULTS: Under the stated assumptions, the analytical expressions formally demonstrate that the "missing middle" is a necessary mathematical consequence of multiple fragmentation events, and they reproduce the stochastic simulation results with exact agreement while reducing computation time from hours to less than 1 s. Our method consistently offers greater accuracy and mathematically proves the mechanistic necessity of "missing middle" phenomenon in top-down MS. The abundances of fragments from stochastic simulation distribute around the average value calculated by the probability formula within twofold coefficient variance (%CV). CONCLUSION: The closed-form probability expressions accurately describe the combinatorial consequences of multiple fragmentation events under the stated assumptions. Because they are computationally inexpensive and differentiable, these expressions could be integrated with sequence-dependent fragmentation propensities in future machine learning models for MS/MS spectral prediction.