Known protein interaction networks have very particular properties. Old proteins tend to have more interactions than new ones. One of the best statistical representatives of this property is the node degree distribution (distribution of proteins having a given number of interactions). It has previously been shown that this distribution is very close to the sum of two distinct exponential components. In this paper, we asked: What are the possible mechanisms of evolution for such types of networks? To answer this question, we tested a kinetic model for simplified evolution of a protein interactome. Our proposed model considers the emergence of new genes and interactions and the loss of old ones. We assumed that there are generally two coexisting classes of proteins. Proteins constituting the first class are essential only for ecological adaptations and are easily lost when ecological conditions change. Proteins of the second class are essential for basic life processes and, hence, are always effectively protected against deletion. All proteins can transit between the above classes in both directions. We also assumed that the phenomenon of gene duplication is always related to ecological adaptation and that a new copy of a duplicated gene is not essential. According to this model, all proteins gain new interactions with a rate that preferentially increases with the number of interactions (the rich get richer). Proteins can also gain interactions because of duplication. Proteins lose their interactions both with and without the loss of partner genes.