A genetically encoded local learning rule enables physical learning in engineered bacteria

Training physical neural networks directly in matter remains difficult because most platforms do not store and update weights within the same physical substrate. Here we show that engineered Escherichia coli can implement a genetically encoded local learning rule acting on persistent biological memory. We introduce memregulons: bacterial memory elements in which coupled plasmids store an analogue weight as the relative abundance of P1 within the P1+P2 plasmid pool. Environmental inputs activate local promoters, and a shared sublethal kanamycin signal converts promoter activity into differential growth that lowers the stored P1 fraction. In single strains, flow-cytometry trajectories across distinct promoters support the predicted dependence of weight change on learning-channel activity and standing population variance. At the single-cell level, repeated negative learning reshapes the stored weight distribution as it approaches the lower boundary. In mixed populations and co-cultures, one shared negative learning signal selectively rewrites active memregulons, enabling externally routed supervised tic-tac-toe lessons. We then generalise the architecture across orthogonal chemical inputs and combinatorial promoters, and use experimentally measured updates in hybrid analyses of winner-take-all and nonlinear-classifier tasks. These results establish physical learning in living matter through local negative updates of genetically encoded analogue weights.