  During development of biological organisms, multiple complex structures are
formed. In many instances, these structures need to exhibit a high degree of
order to be functional, although many of their constituents are intrinsically
stochastic. Hence, it has been suggested that biological robustness ultimately
must rely on complex gene regulatory networks and clean-up mechanisms. Here we
explore developmental processes that have evolved inherent robustness against
stochasticity. In the context of the Drosophila eye disc, multiple optical
units, ommatidia, develop into crystal-like patterns. During the larva-to-pupa
stage of metamorphosis, the centers of the ommatidia are specified initially
through the diffusion of morphogens, followed by the specification of R8 cells.
Establishing the R8 cell is crucial in setting up the geometric, and
functional, relationships of cells within an ommatidium and among neighboring
ommatidia. Here we study a mathematical model of these spatio-temporal
processes in the presence of stochasticity, defining and applying measures that
quantify order within the resulting spatial patterns. We observe a universal
sigmoidal response to increasing transcriptional noise. Ordered patterns
persist up to a threshold noise level in the model parameters. As the noise is
further increased past a threshold point of no return, these ordered patterns
rapidly become disordered. Such robustness in development allows for the
accumulation of genetic variation without any observable changes in phenotype.
We argue that the observed sigmoidal dependence introduces robustness allowing
for sizable amounts of genetic variation and transcriptional noise to be
tolerated in natural populations without resulting in phenotype variation.
