A field model of left-right asymmetries in the pattern regulation of a cell

Dynamical models can offer insights into important aspects of patterns that recur throughout the biological world, such as global symmetries, mirror images, and pattern reversals, etc. Building on phenomenological rules of pattern formation (e.g. E. M. Nelsen and J. Frankel, 1986, Dev. Biol. 114, 53), the author developed a mathematical model of a morphogenetic field, expressed as a vector field, which contains the information to specify a pattern. The dynamics of the field arises from the minimization of a nonlinear energy density functional, a quartic polynomial of the field together with field gradient effects. This model was successfully applied in detail to surface patterns observed in right-handed Tetrahymena, where reproducible regions of left- and right-handed domains appear during regulation. Counterpart experiments on left-handed patterned cells suggest an intrinsic asymmetry between right- and left-handed patterning in this biological system, causing left-handed cells to follow different regulation pathways from the right-handed cells. We show that the energy minimization approach can account for this globally different behaviour in a version of the model with small local asymmetry.