A non-linear field model of pattern formation: Intercalation in morphalactic regulation

Abstract

Useful insights into pattern formation problems in regulating biological systems have been gained from the concept of positional information . In particular, the polar co-ordinate model of positional information ( French et al. , 1976 , Science 193 , 969–981) and its relatives (e.g. Bryant et al. , 1981 , Science 212 , 993–1002; Lewis, 1981 , J. theor. Biol. 88 , 371–392; Mitthenthal, 1981 , Dev. Biol. 88 , 15–26; Winfree, 1980 , The Geometry of Biological Time . New York: Springer) have helped to provide a qualitative framework for understanding epimorphic regulation, which involves localized growth and pattern formation at a discontinuity (e.g. at a cut surface). On the other hand, these models lack the formal structure to deal quantitatively with regulation; in particular, they are inadequate to treat morphallactic regulation, in which reorganization of the biological system occurs as a consequence of, e.g. changes in its size, rather than a distinct “discontinuity in the positional values”. To overcome this limitation, we propose a morphogenetic field model of pattern formation. We define a simple vector field (morphogenetic field) with generative dynamics arising from the minimization of a non-linear energy functional based on the positional information idea of an “optimal spacing of positional values” and an additional “smoothness” condition. As the system size is changed, transitions to solutions with pattern reversal regions take place, suggesting how reverse intercalation phenomena can arise in morphallactic regulation even without the presence of a discontinuity, as is observed in Tetrahymena doublets regulating to singlets ( Nelsen & Frankel, 1986 , Dev. Biol. 114 , 53–71; Brandts & Trainor, 1990 , J. theor. Biol. 146 , 57–86). We view the success of our model as support for the unification of the formalism, phenomenology and concepts of physical theory with the foundations of theory in biology.