Amphiphilic monomers in polar solvents can self-assemble into lyotropic liquid crystal (LLC) bicontinuous cubic structures under the right composition and temperature conditions. After cross-linking, the resulting polymer membranes with 3D continuous uniform channels are excellent candidates for filtration applications. Designing such membranes with desired physical and chemical properties requires molecular level understanding of the structure, which can be obtained through molecular modeling. However, building molecular models of bicontinuous cubic structures is challenging due to their narrow regime of stability and the difficulty of self-assembly of large unit cells in molecular simulations. We developed a protocol for building stable bicontinuous cubic unit cells involving both parameterization and assembly of the components. We validated theoretical structure against experimental results for one such LLC monomer and provide insight into the structure missing in experimental data, as well as demonstrate the qualitative nature of water and solute transport through these membranes.