We discuss theoretical and computational methodologies for quantitatively describing how cell-membrane topologies are actively mediated and manipulated by intracellular protein assemblies. Such scenarios are ubiquitous in intracellular trafficking mechanisms, i.e., active transport mechanisms characterized by vesicle nucleation and budding of the cell membrane orchestrated by protein-interaction networks. We will describe the development and application of the kinetic Monte Carlo time-dependent Ginzburg Landau (KMC-TDGL) algorithm for unified dynamics of curvature-inducing proteins and membranes. We will also describe the surface-evolution method which predicts minimum energy conformations of highly curved axis-symmetric membrane structures and an extension of the TDGL method which relaxes the assumption of axis-symmetry. Together, these methods enable us to quantify thermal effects in membrane mediated events. We will briefly focus on an application to clathrin-dependent endocytosis.