  As a discrete approach to genetic regulatory networks, Boolean models provide
an essential qualitative description of the structure of interactions among
genes and proteins. Boolean models generally assume only two possible states
(expressed or not expressed) for each gene or protein in the network as well as
a high level of synchronization among the various regulatory processes. In this
paper, we discuss and compare two possible methods of adapting qualitative
models to incorporate the continuous-time character of regulatory networks. The
first method consists of introducing asynchronous updates in the Boolean model.
In the second method, we adopt the approach introduced by L. Glass to obtain a
set of piecewise linear differential equations which continuously describe the
states of each gene or protein in the network. We apply both methods to a
particular example: a Boolean model of the segment polarity gene network of
Drosophila melanogaster. We analyze the dynamics of the model, and provide a
theoretical characterization of the model's gene pattern prediction as a
function of the timescales of the various processes.
