Peptide bonds (N-C=O) play a key role in metabolic processes since they link amino acids into peptide chains or proteins. Recently, several molecules containing peptide-like bonds have been detected across multiple environments in the interstellar medium (ISM), growing the need to fully understand their chemistry and their role in forming larger pre-biotic molecules. We present a comprehensive study of the chemistry of three molecules containing peptide-like bonds: HNCO, NH$_2$CHO, and CH$_3$NCO. We also included other CHNO isomers (HCNO, HOCN), and C$_2$H$_3$NO isomers (CH$_3$OCN, CH$_3$CNO) to the study. We have used the \uclchem gas-grain chemical code and included in our chemical network all possible formation/destruction pathways of these peptide-like molecules recently investigated either by theoretical calculations or in laboratory experiments. Our predictions are compared to observations obtained toward the proto-star IRAS16293$-$2422 and the L1544 pre-stellar core. Our results show that some key reactions involving the CHNO and C$_2$H$_3$NO isomers need to be modified to match the observations. Consistently with recent laboratory findings, hydrogenation is unlikely to produce NH$_2$CHO on grain surfaces, while a combination of radical-radical surface reactions and gas-phase reactions is a better alternative. In addition, better results are obtained for NH$_2$CHO when a slightly higher activation energy of 25$\,$K is considered for the gas-phase reaction $\rm NH_2 + H_2CO \rightarrow NH_2CHO + H$. Finally, our modelling shows that the observed correlation between NH$_2$CHO and HNCO in star-forming regions may come from the fact that HNCO and NH$_2$CHO react to temperature in the same manner rather than from a direct chemical link between the two species.Read less <