University of Washington
We used ordinary differential equations expressing mass action kinetics to model temporal changes in expression levels of RNAs and proteins in each of many neighboring cells. We studied several different genetic modules, best characterized experimentally in fruit fly embryos, but operating as well in most complex animals. We asked whether the experimentally proven interactions among genes and their products in these modules could explain the spatio-temporal patterns of gene expression those modules are known to make during early fruit fly development -- patterns that prefigure major features of the future larval body. It takes many (50 to 100) parameters to quantify the strengths and functional forms of the various interactions among the genes and their products, but none has an experimentally measured value. We performed vast computations to sample huge "boxes" in high-dimensional parameter space to ascertain the measure of the set of points in parameter space that would cause the model network to mimic correctly the spatio-temporal pattern formation performance of the real network. The measure of this set of 'good' points turned out to be unexpectedly huge, corresponding to robustness of the network which would be astonishing were it not essential to make complex genetic modules functionally heritable.
The real networks we have studied are more complicated, and involve more genes,
than any sophomore applied mathematician would need
in a made-up model that creates the right spatial pattern. The simple made-up models we have constructed turn out not to be robust. We would like to understand, but don't yet understand, how natural selection crafted networks whose spatio-temporal pattern-formation repertoires seems to be encoded mysteriously in the topology of of the network's connections, rather than in the strengths/functional forms of those connections. "We" = George von Dassow, Eli Meir, Ed Munro, and me.