Molecular structure of many polymers creates elastic rigidity that impacts polymers behavior. Understanding this impact is critical to address the physics describing variety of single-molecule experiments and biological processes. Utilizing analytical theories and numerical methods, we illustrate the effect of molecular elasticity on the behavior of single molecules employed in single-molecule experiments as well as the morphology of assemblages in protein self-assembly processes. In single-molecule experiments, we address the effect of thermal fluctuation, focusing on the equilibrium statistical behavior of specific class of underling single molecules (semiflexible polymers) to gain insight into the physics governing their behavior. To demonstrate the role that molecular elasticity plays in protein self-assembly processes, we focus on clathrin protein, a protein recruited by the cell wall for ingesting food particles during endocytosis. We demonstrate that molecular elasticity and binding affinity have a significant impact on the versatile equilibrium and nonequilibrium assemblages occurred in clathrin protein self-assembly process.