Mechanisms of Gene Regulation: Boolean Network Models of the Lactose Operon in Escherichia coli - Mathematical Concepts and Methods in Modern Biology - page 10

Table 1.1 Tables of values for the basic Boolean operations: Logical AND ( ∧ ) ,

logical OR ( and NOT (an overbar identifies the negation of the Boolean

variable underneath).
Logical AND: z

∨ )

= x ∧ y Logical OR: z = x ∨ y Logical NOT: z = y

Input Output Input Output Input Output

x y z x y z y z

0 0 0 0 0 0 0 1

0 1 0 0 1 1 1 0

1 0 0 1 0 1

1 1 1 1 1 1

of the protein it makes would be thousands of times higher than the trace amounts
that are present when the gene is silent, with a clear distinction between present and
absent (1 or 0). Throughout this chapter, we will assume without further mention that
a Boolean value of 0 indicates a concentration near the basal level while a Boolean
value of 1 signifies a markedly higher concentration.
In Boolean models, the dynamical evolution of the system is described using
Boolean functions defined in terms of the model variables and the logical operators
AND (denoted by the symbol

, and NOT (denoted
by an overbar on the variable it negates). The tables of values for these Boolean
operators are presented in Table 1.1 .
In the context of modeling network dynamics, it is often useful to consider the
following intuitive interpretation for the operations AND and OR: if the components
x and y of the system influence (control) a third component z , then z

∧ ), OR (denoted by the symbol ∨ )

= x ∧ y means

that x and y need to be simultaneously present (have values 1) to affect z y

means that x and y influence z independently and z is affected when x OR y (or both)
are present. In the absence of any parentheses, the order of precedence is this: logical
NOT has highest precedence, followed by the logical AND, followed by the logical
OR.
Sometimes, if we know the value for one of the operands in the AND / OR opera-
tions, we may not need to evaluate the other operand to be able to determine the value
for the operation. For instance, if A

; z = x ∨

0 regardless of the value
of B. In general, if at least one of the operands of the AND operation is zero, the
resulting value for the operation is zero no matter what the value of the other operand
is. In a similar way, if one of the operands of the OR operation has value 1, the value
for the operation is 1 regardless of the value of the other operand. This is known as
short-circuit evaluation .
Example 1.1. Assume the Boolean variables A , B , C , and D have valu es A

= 0 , C = A ∧ B =

= 0 ,

B = 1 , C = 1 , D = 1. Determine the value of the Boolean expression ( D ∨ B ) ∧

B .
The expression in the parentheses will be evaluated first and in order to do this, D
must be co mputed because NOT has higher pre cedence than OR. Since D

A ∧ C ∨

= 1, the

value of D = 0. Since B = 1, the value of ( D ∨ B ) = 1. Next, following the rules