Precise protein sequencing and folding are believed to generate the structure and chemical diversity of natural channels(1,2), both of which are essential to synthetically achieve proton transport performance comparable to that seen in natural systems. Geometrically defined channels have been fabricated using peptides, DNAs, carbon nanotubes, sequence-defined polymers and organic frameworks(3-13). However, none of these channels rivals the performance observed in their natural counterparts. Here we show that without forming an atomically structured channel, four-monomer-based random heteropolymers (RHPs)(14) can mimic membrane proteins and exhibit selective proton transport across lipid bilayers at a rate similar to those of natural proton channels. Statistical control over the monomer distribution in an RHP leads to segmental heterogeneity in hydrophobicity, which facilitates the insertion of single RHPs into the lipid bilayers. It also results in bilayer-spanning segments containing polar monomers that promote the formation of hydrogen-bonded chains(15,16) for proton transport. Our study demonstrates the importance of the adaptability that is enabled by statistical similarity among RHP chains and of the modularity provided by the chemical diversity of monomers, to achieve uniform behaviour in heterogeneous systems. Our results also validate statistical randomness as an unexplored approach to realize protein-like behaviour at the single-polymer-chain level in a predictable manner.