Compute-Intensive Simulations for Cellular Models - Parallel Computing for Bioinformatics and Computational Biology - page 95

are large enough to send some cells down the lysogenic pathway (with high concen-
tration of CI but low concentration of cro), whereas other cells proceed down the lytic
pathway (with high concentration of cro but low concentration of CI). The predictions
of the model agree quite well with experimental results.
This is pioneering work on the role of fluctuations in gene regulation. Apparently
the lysogenic or lytic path in individual cells results from the inevitable fluctuations
in the temporal pattern of protein concentration growth. The central conclusion is that
fluctuations cannot always be viewed as simply small perturbations as they can, in
fact, induce different developmental pathways.
By using a model with full binding configurations and experimental data, Tian and
Burrage realized bistability by introducing threshold values into amathematical model
[19]. This approach represents an attempt to describe the regulatory mechanisms in a
genetic regulatory network under the influence of intrinsic noise in the framework of
continuous models. The threshold values are used for indicating the effect of positive
and negative feedback regulatory mechanisms. For example, the synthesis rate S x is
now a function of the concentration of CI, given by

x 1 ,

S x , x ≥

S x =

S x x
x 1

x < x 1 . ,

This functional reaction rate is used to represent different developmental stages due
to different concentrations of proteins in the system. Using these mathematical rep-
resentations, Tian and Burrage constructed a quantitative model for describing the
evolutional pathways of λ phage that agrees well with experimental results.
In addition, a stochastic model has been introduced for describing switching in
induction from the lysogenic pathway to the lysis pathway. A stochastic degradation
rate has been used to represent intrinsic noise so that the first equation in (4.11) is
(see [19])

k x x d W(t) .

d x = S x P x (x , y) d t − k x x d t − (4.12)

Figure 4.3 gives two simulations of this model with k x = 1.0 at 40-70min. A suc-

cessful switching is given in the left figure while the right figure is a simulation of
unsuccessful switching. Numerical simulations have also been used to predict the pro-
portionally induced λ phage through a large number of simulations and this percentage
is consistent with the experimental data.
The power of the stochastic kinetics approach lies in its completeness and attention
to detail. This might lead, for example, to more rapid hypothesis testing, by indicating
which experiments would be expected to distinguish most sharply among the com-
peting hypotheses. The drawback of the simulations is the large computational load
compared with other methods. But that is the motivation for us to discuss parallel
computation in the following sections.