Using nonequilibrium molecular dynamics simulation, we study the thermal rectification of an asymmetric double-stranded molecular structure system. We investigate the influence of temperature, intra- and inter-chain interaction and the system size on the thermal rectification effect of the asymmetric double- chain system. It is reported that the intra- and the inter-chain interaction have an advantage on the rectification efficiency. The rectification efficiency goes to a stable value at a large system size. This phenomenon is also explained by the power spectra of the inter-particles, which is similar to the situation of the single chain system.