Mass spectrometry (MS) is a fundamental technology of analytical chemistry for measuring the structure of molecules, with many application fields such as clinical biomarker analysis or pharmacokinetics. In the context of proteomic analysis with MS, the superposition of the isotopic patterns of different proteins, in various charge-states produces MS spectra difficult to decipher. The complexity of the pattern models and the large size of the data again increase the difficulty of the analysis step. In this paper, we propose to formulate the problem of proteins characterization as the estimation of a positive-valued sparse signal thanks to a dictionary-based approach relying on the protein averagine concept. A proximal primal-dual splitting convex optimization method is considered to solve the resulting variational problem. Moreover, the large size of the dictionary matrix is circumvented by proposing a suitable block circulant approximation of it, allowing to limit the computational burden of the method. Numerical experiments on synthetic and real MS datasets illustrate the good performance of our approach.