The amount of cellular proteins is a crucial parameter that is known to vary between cells as a function of the replicative passages, and can be important during physiological aging. The process of protein degradation is known to be performed by a series of enzymatic reactions, ranging from an initial step of protein ubiquitination to their final fragmentation by the proteasome. In this paper we propose a stochastic dynamical model of nuclear proteins concentration resulting from a balance between a constant production of proteins and their degradation by a cooperative enzymatic reaction. The predictions of this model are compared with experimental data obtained by fluorescence measurements of the amount of nuclear proteins in murine tail fibroblast (MTF) undergoing cellular senescence. Our model provides a three-parameter stationary distribution that is in good agreement with the experimental data even during the transition to the senescent state, where the nuclear protein concentration changes abruptly. The estimation of three parameters (cooperativity, saturation threshold, and maximal velocity of the reaction), and their evolution during replicative passages shows that only the maximal velocity varies significantly. Based on our modeling we speculate the reduction of functionality of the protein degradation mechanism as a possible competitive inhibition of the proteasome.