Formation of a Morphogen Gradient - Acceleration by Quantum Walks

Abstract

The establishment of graded concentration distributions of certain signalling molecules, called morphogens, is crucial for the successful embryonic development of multicellular organisms. The morphogen system that has been studied in the greatest detail is the Bicoid (Bcd) gradient established during the early embryonic development of fruit flies and provides the embryo cells with the necessary positional information for the proper implementation of the body plan of the organism. Both continuous as well as discrete-state classical random-walk models fail to provide a completely satisfactory explanation of the dynamics of Bcd gradient formation within the developmentally relevant timescales. By presuming the existence of an arbitrary chirality degree of freedom that characterizes each Bcd molecule, we here therefore develop a discrete-time quantum walk model of the process. Our approach is based on the recent suggestion that high temperatures and random interactions need not necessarily destroy many-body quantum coherence. In this way we are led to a system of non-classical master equations bearing an interference term that can be numerically solved on a computer. Our analysis predicts a large value of diffusion rate for the Bcd system in the very early nuclear division cycles that needs to be verified experimentally. Additionally, with the chirality and position degrees of freedom at our disposal entanglement shows up as a natural consequence in the system. Taking inspiration from a very recent demonstration of the existence of entanglement phenomenon in a hot, strongly-interacting atomic system, the discussion takes the point of view that the emergence of geometry in the embryo system is consequence of quantum entanglement that needs further understanding and analysis.