Bistability in the Lactose Operon of Escherichia coli: A Comparison of Differential Equation and Boolean Network Models - Mathematical Concepts and Methods in Modern Biology - page 70

replicate this finding, indicating once again that bistability is a robust phenomenon
that arises from the network structure.
The Santillán model from Exercise 2.11 [ 3 ], considers external lactose as a param-
eter of the model and the function

(an increasing
function of L and a decreasing function of G e ) in the equation for M in Eqs. ( 2.60 )isa
possible way to model the dependence of the system on these parameters. In [ 24 ]the
authors provide a detailed comparison of a number of ODE models of the lac operon
that differ in the numbers of variables and type (stochastic or deterministic), including
the three ODE models considered in this chapter. Many of those models differ in the
ways the dependence of mRNA on external glucose and internal lactose is modeled.
It is noted in [ 24 ] that in some cases heuristic reasoning is used to propose Hill-type
equations for the function

P R ( L , G e ) = P D ( G e )P R ( A )

with a different level of detail. Establishing
that Boolean approximations of the minimal model from Eqs. ( 2.60 ) can describe
and explain bistability indicates that such differences are nonessential with regard to
bistability and do not impact this feature of the system.
It has been shown that bistability requires a direct positive feedback loop or an
indirect positive feedback loop, such as a double negative feedback loop [ 25 ]. How-
ever, a feedback loop alone is not enough for a system to exhibit bistable behavior. It
must also possess some type of nonlinearity within the feedback circuit. That is, some
of the proteins in the feedback circuit must respond to their upstream regulators in an
ultrasensitive manner [ 26 - 28 ]. The Hill coefficient is used to quantify the steepness
of this response. A Hill coefficient larger than one is considered to be the ultrasensi-
tive response [ 29 ]. In Boolean models this condition is automatically satisfied since
the On-Off switches in such models can be viewed as responses with very large Hill
coefficients.
The analysis of the Boolean models was done using the web-based software DVD
[ 18 ]. When the number of variables is small, DVD can be used to produce the entire
state space for the model. When the number of variables grows, obtaining the entire
state space is not feasible, but DVD computes and returns the model's fixed points
together with the number of connected components in state space. DVD's successor
ADAM [ 30 ], which handles a broader class of discrete models, including logical
models and Petri nets in addition to Boolean and polynomial models, can also be used.
We found that introducing multiple “old” variables into the Boolean models does
not appear to change the long-term behavior of the system and to impact bistability.
This is consistent with findings reported in [ 31 ] where the author presents amethod for
reducing Boolean networks and their wiring diagrams while preserving the set of fixed
points. The reduction is done by deleting vertices with no self-loops (that is, vertices
whose transition functions do not include them as inputs). If a vertex is to be removed
from the network, its transition function is substituted for the variable representing
this vertex in all of the other transition functions in the model. The idea is to reduce the
size of the network by providing a way to delete a vertex and “pass on” its functionality
to other variables in the network. A reduced model with the same fixed points as the
original model would be indicative of characteristics that emerge not as a result of
some specific system interactions but, instead, from the core network topology. For
instance, in [ 23 ] Veliz-Cuba and Stigler present examples of Boolean models of the

P R ( L , G e )