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 58

system's regulation (e.g., dilution and degradation, translation and transcription, and
the effects of interaction between the system's components) take place in a single time
step. Although models constructed under such assumptions can play an important
role in the qualitative analysis of a system, such simple models lack the capability
of describing delayed interactions and systems for which not all degradation rates
are equal. Further, since those models assume only high and low levels of internal
or external lactose, they effectively bypass the bistability phenomenon occurring at
intermediate lactose concentrations.
We next consider Boolean models that can incorporate time delays, account for
different degradation rates, and can exhibit bistability. The approach we outline below
follows in most parts the general methodology proposed by Hinkelmann et al. in [ 16 ].
We begin with some background considerations.

1. Assume that a variable Y regulates the production of X . 1 As in Chapter 1, we use
X to denote the values of the Boolean variables X and Y at time t ,

where t is discrete, t

( t ) and Y ( t )

= 0 , 1 , 2 ,... Assume that Y ( t ) = 1 implies X ( t + 1 ) = 1;

that is, the presence of Y at time t implies that X will be present at time t 1.

Assume that the loss of X due to dilution and degradation occurs over the course
of several time steps. To account for this process, n additional Boolean variables
X old ( 1 ) ,

+

X old ( 2 ) ,…, X old ( n ) , are introduced (where n is a positive integer) with the
property that:
i.

If Y ( t ) = 0 and X ( t ) = 1, then X old ( 1 ) ( t + 1 ) = 1. A value of 1 for

X old ( 1 ) ( t + 1 ) indicates that the amount of X present at time t + 1 is already

reduced once by dilution and degradation.

ii. If Y ( t ) = 0 and X old ( i − 1 ) ( t ) = 1, for some i = 2 , 3 ,..., n , then X old ( i )

( t + 1 ) = 1. A value of 1 for X old ( i ) ( t + 1 ) indicates that the amount of X

1 is already reduced i times by dilution and degradation.
iii. The number of required “old” variables is determined by the number of
time steps needed to reduce the concentration of X below the discretization
threshold when there is no new production of X . For instance, assume that in
the absence of new production, the concentration of X needs to be diluted n
times before falling below the discretization threshold. Thus, X

present at time t +

( + ) =

t 1 1

( ) =

when either Y t 1 (that is , a new amount of X will be produced by time

t 1 (that is, previously available amounts

of X are still available at time t and have not yet been reduced n -fold). In
other words, X

+ 1) or when ( X ( t ) ∧ X old ( n ) ( t )) =

. We should note that
our approach here differs somewhat from that in [ 16 ]. We highlight the
differences in Section 2.7 .
2. To be able to model bistability we need to be able to distinguish between low,
medium, and high concentrations of lactose L . This can be achieved by intro-
ducing an additional variable L high such that L high

( t + 1 ) = Y ( t ) ∨ ( X ( t ) ∧ X old ( n ) ( t ))

= 1 implies L = 1. In this

1 In the models we consider later, Y will usually be a compound Boolean expression describing multiple
regulating factors for X .