Boolean Networks in Inference and Dynamic Modeling of Biological Systems at the Molecular and Physiological Level - Biological Networks - page 71

bell shaped distribution of the concentration of the even-stripped protein was
implemented by using three different thresholds levels.
Albert and Othmer (Albert et al . 2003) proposed a Boolean model of the
segment polarity gene network whose expression is initiated by the pair-rule
module. The synchronous model was able to reproduce the wild type gene
expression patterns, as well as the ectopic expression patterns observed in over-
expression experiments and various mutants, and determined that the system can
have six fixed points (steady states). The model gives important insights into the
network's ability to correct errors in the pre-pattern. An asynchronous model
(Chaves 2005) of the same network showed that very variable synthesis and
decay times can perturb the wild type pattern. Starting from the wild type initial
condition, all six steady states (Albert et al. 2003), including mutant patterns,
may occur with a certain frequency, with the incidence of the wild type steady
state being 57%. On the other hand, a separation of timescales between pre- and
posttranslational processes and a minimal pre-pattern ensure convergence to the
wild type expression pattern regardless of fluctuations. A consequent piece-wise
linear model that assumes individual activation thresholds and individual node
timescales found that 100% reproducibility of the wild-type developmental
process is ensured by two assumptions, (i) a separation of mRNA and protein
half-lives, and (ii) an activation threshold less than 0.5. These assumptions are
supported by evidence from sea urchin development (Davidson 2001).
The threshold Boolean model described in the introduction has been used to
study the cell-cycle regulatory network of the budding yeast, Saccharomyces
cerevisiae . The model can simulate the temporal evolution of protein states
corresponding to the sequence of phases in the cell cycle, namely G1, S, G2
and M, before reaching a large basin of attraction corresponding to the G1 phase
in which the cell grows and under appropriate conditions commits to division
(Li et al . 2004). The states corresponding to the biological cell cycle checkpoints
have large basins of attraction and the pathway through these checkpoints is an
attracting trajectory. Stability analysis of this model and its comparison with
random networks further support the conclusion that the cell cycle is robustly
designed and is stable against perturbations. (Li et al . 2004). Recently (Davidich
et al . 2008) transformed an existing continuous (ordinary differential equation
based) model of the fission yeast cell-cycle (Novak et al . 2001) into a Boolean
model in order to compare the two approaches. The Boolean model correctly
reproduced the sequence of events and the steady states of the system and its
robustness to perturbations of the initial conditions. The results confirm the idea
that some molecular control networks are so robustly designed that timing is not
a critical factor (Braunewell et al. 2007). The comparison between threshold