Simple stochastic model for the evolution of protein lengths.

We analyze a simple discrete-time stochastic process for the theoretical modeling of the evolution of protein lengths. At every step of the process, a new protein is produced as a modification of one of the proteins already existing, and its length is assumed to be a random variable that depends only on the length of the originating protein. Thus a random recursive tree is produced over the natural numbers. If (quasi) scale invariance is assumed, the length distribution in a single history tends to a log-normal form with a specific signature of the deviations from exact Gaussianity. Comparison with the very large Similarity Matrix of Proteins database shows good agreement.