Cytoskeleton proteins are filament structures that support a large number of important biological processes. These dynamic biopolymers exist in nonequilibrium conditions stimulated by hydrolysis chemical reactions in their monomers. Current theoretical methods provide a comprehensive picture of biochemical and biophysical processes in cytoskeleton proteins. However, the description is only qualitative under biologically relevant conditions because utilized theoretical mean-field models neglect correlations. We develop a new theoretical method to describe dynamic processes in cytoskeleton proteins that takes into account spatial correlations in the chemical composition of these biopolymers. Our approach is based on analysis of probabilities of different clusters of subunits. It allows us to obtain exact analytical expressions for a variety of dynamic properties of cytoskeleton filaments. By comparing theoretical predictions with Monte Carlo computer simulations, it is shown that our method provides a fully quantitative description of complex dynamic phenomena in cytoskeleton proteins under all conditions.