Microtubules and actin filaments are biopolymer molecules that are major components of cytoskeleton networks in biological cells. They play important roles in supporting fundamental cellular processes such as cell division, signaling, locomotion, and intracellular transport. In cells, cytoskeleton proteins function under nonequilibrium conditions that are powered by hydrolysis of adenosine triphosphate (ATP) or guanosine triphosphate (GTP) molecules attached to them. Although these biopolymers are critically important for all cellular processes, the mechanisms that govern their complex dynamics and force generation remain not well explained. One of the most difficult fundamental issues is to understand how different components of cytoskeleton proteins interact together. We develop an approximate theoretical approach for analyzing complex processes in cytoskeleton proteins that takes into account the multifilament structure, lateral interactions between parallel protofilaments, and the most relevant biochemical transitions during the biopolymer growth. It allows us to fully evaluate collective dynamic properties of cytoskeleton filaments as well as the effect of external forces on them. It is found that for the case of strong lateral interactions the stall force of the multifilament protein is a linear function of the number of protofilaments. However, for weak lateral interactions, deviations from the linearity are observed. We also show that stall forces, mean velocities, and dispersions are increasing functions of the lateral interactions. Physical-chemical explanations of these phenomena are presented. Our theoretical predictions are supported by extensive Monte Carlo computer simulations.