Deep Gene Networks and Response to Stress

We consider systems of differential equations with polynomial and rational nonlinearities and with a dependence on a discrete parameter. Such systems arise in biological and ecological applications, where the discrete parameter can be interpreted as a genetic code. The genetic code defines system responses to external perturbations. We suppose that these responses are defined by deep networks. We investigate the stability of attractors of our systems under sequences of perturbations (for example, stresses induced by environmental changes), and we introduce a new concept of biosystem stability via gene regulation. We show that if the gene regulation is absent, then biosystems sooner or later collapse under fluctuations. By a genetic regulation, one can provide attractor stability for large times. Therefore, in the framework of our model, we prove the Gromov–Carbone hypothesis that evolution by replication makes biosystems robust against random fluctuations. We apply these results to a model of cancer immune therapy.