Deep Gene Networks and Response to Stress

Sergey Vakulenko,Dmitry Grigoriev

doi:10.3390/math9233028

Abstract

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.

Highlights

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 allows us to incorporate in the model a genetic regulation by a genetic code
It is clear that replicative stability is important for the evolution of cancer cells, viruses, for example, such as COVID 19, and bacteria such as E. coli
One can conclude that it is necessary to take into account cancer cell adaptivity and evolution, which allows us to describe cancer dormancy, an important effect in cancer treatment [41]

Summary

Introduction

We consider systems of differential equations with polynomial and rational nonlinearities and with a dependence on a discrete parameter (hybrid systems). Such systems arise in biological and ecological applications, where the discrete parameter allows us to incorporate in the model a genetic regulation by a genetic code. Our aim is to investigate the attractor stability under infinite sequences of shocks. It is inspired by the following idea of M. It is clear that replicative stability is important for the evolution of cancer cells, viruses, for example, such as COVID 19, and bacteria such as E. coli. There is a non-trivial question: how to estimate chances that such replicative evolution can continue eternally even under action of therapeutic agents? For COVID-19, it is an important question of will we have fourth or fifth pandemic waves, etc., or can the virus vanish? In this paper, we consider cancer cells as an example because our analyses are more applicable to this case

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