Pattern formation and nuclear divisions are uncoupled in Drosophila segmentation: comparison of spatially discrete and continuous models

We study the relationship between pattern formation in Drosophila segment determination and nuclear structure by replacing the nuclei by a homogeneous continuum. Because this replacement cannot be performed experimentally, mathematical simulation is applied by transforming a previously published model of the segmentation system formulated in terms of explicit nuclear structure into partial differential equations. This transformation has changed the mathematical type of model equations and is therefore interpreted as a structural perturbation of the model. Parameter values are found for three continuum models by means of a new optimal steepest descent algorithm. Each of these models contains a different mathematical representation of nuclear divisions (mitoses). We obtained correct pattern dynamics from all of them, as well as from the model with explicit nuclear structure. This leads us to conclude that nuclear divisions are not coupled to pattern formation and serve only to populate the blastoderm with nuclei. We also investigate whether the calculated patterns in the developmental period modelled resemble their attractors, and find that they fail to do so. The implications of our results for models of biological pattern formation based on partial differential equations are discussed.