The macromolecules that encode and translate information in living systems, DNA and RNA, exhibit distinctive structural asymmetries, including homochirality or mirror image asymmetry and $3 - 5$ directionality, that are invariant across all life forms. The evolutionary advantages of these broken symmetries remain unknown. Here we utilize a very simple model of hypothetical self-replicating polymers to show that asymmetric autocatalytic polymers are more successful in self-replication compared to their symmetric counterparts in the Darwinian competition for space and common substrates. This broken-symmetry property, called asymmetric cooperativity, arises with the maximization of a replication potential, where the catalytic influence of inter-strand bonds on their left and right neighbors is unequal. Asymmetric cooperativity also leads to tentative, qualitative and simple evolution-based explanations for a number of other properties of DNA that include four nucleotide alphabet, three nucleotide codons, circular genomes, helicity, anti-parallel double-strand orientation, heteromolecular base-pairing, asymmetric base compositions, and palindromic instability, apart from the structural asymmetries mentioned above. Our model results and tentative explanations are consistent with multiple lines of experimental evidence, which include evidence for the presence of asymmetric cooperativity in DNA.