Chaos and hyperchaos in simple gene network with negative feedback and time delays.

Today there are examples that prove the existence of chaotic dynamics at all levels of organization of living systems, except intracellular, although such a possibility has been theoretically predicted. The lack of experimental evidence of chaos generation at the intracellular level in vivo may indicate that during evolution the cell got rid of chaos. This work allows the hypothesis that one of the possible mechanisms for avoiding chaos in gene networks can be a negative evolutionary selection, which prevents fixation or realization of regulatory circuits, creating too mild, from the biological point of view, conditions for the emergence of chaos. It has been shown that one of such circuits may be a combination of negative autoregulation of expression of transcription factors at the level of their synthesis and degradation. The presence of such a circuit results in formation of multiple branches of chaotic solutions as well as formation of hyperchaos with equal and sufficiently low values of the delayed argument that can be implemented not only in eukaryotic, but in prokaryotic cells.