A general mechanism for 'inexact' phase differences in reaction-diffusion-advection systems

'Inexact' phase differences that may take any value in the range [0,π], between the chemical morphogens diffusing in an embryo, have been proposed [M.A. Russell, Dev. Biol. 108 (1985) 269] to improve the positional information theory [L. Wolpert, J. Theor. Biol. 25 (1969) 1] by encoding this information with higher resolution than that provided by other mechanisms. Reaction-diffusion systems, including Turing systems, show only 'exact' phase differences 0 and/or π. We demonstrate here that inexact phase differences arise naturally in reactive flows described by reaction-diffusion-advection equations and illustrate them by the stationary waves in open flows (flow- and diffusion-distributed structures FDS [R.A. Satnoianu, M. Menzinger, Phys. Rev. E 62 (2000) 113; R.A. Satnoianu, P.K. Maini, M. Menzinger, Physica D 160 (2001) 79] and travelling waves in differential-induced flow systems (DIFI) [A.B. Rovinsky, M. Menzinger, Phys. Rev. Lett. 70 (1993) 778; R.A. Satnoianu, J.H. Merkin, S.K. Scott, Physica D 124 (1998) 345]. The ability of cells in a developing organism to read phase differences in addition to morphogen concentrations would endow them with a robust mechanism for producing segmentation patterns that is richer, shows higher spatial resolution and is more stable than Turing's and Wolpert's positional information mechanisms