Proteomic waves in networks of transcriptional regulators

The protein production for a gene regulatory network model with activation-repression links (cascade) is analysed. In these networks this production depends on how proteins induce or repress the genes. Experiments show that networks of inducers or repressors exhibit bistability or oscillatory behaviour of protein production. Here we report a completely novel aspect, namely for different promoter activity functions, protein production (initially localised on a certain number of genes) can propagate to the others in a “solitonic” way. In particular, the chemical rate equation for the cascade can be solved exactly and in the case of big number of operator sites the proteomic signal along the gene network is given by a superposition of perturbed dark solitons of defocusing semidiscrete modified Korteweg de Vries equation.