Biology Reference
In-Depth Information
2. Xie and his colleagues (Lu et al. 1998; Min et al. 2005) demonstrated that a
single molecule of cholesterol oxidase exhibits not one but a set of over 30 rate
constants under an identical experimental condition (see Fig.
11.24
), which
phenomenon being known as
dynamic heterogeneity
(Zwanzig 1990; Bagshaw
and Cherny 2006).
Unlike a crisp machine which exists in well-defined, unambiguous internal states
that can be characterized in terms of a string of N digits or, geometrically speaking,
in terms of the 2
N
vertices of an N-dimensional hypercube (as discussed in Sect.
machine has many more internal states than a crisp machine of an equal dimension,
because each of the N variables of a fuzzy machine can assume numerical values
between 0 and 1. Thus, the internal states of a fuzzy machine can be represented by
the points located in the interior of the N-dimensional hypercube (e.g., see points A
and F in Fig.
5.8
), which are clearly more numerous than the number of the vertices
of the hypercube.
In Fig.
7.8
, a simplified description of an enzymic catalysis is given utilizing a
two-dimensional fuzzy cube, where the states of the enzyme and the chemical
subsystem (consisting of the substrate and the product bound to enzyme active site)
are represented in the
x
- and
y
-axis, respectively. The conformation of the enzyme
can assume any of the states between 0 (or A) and 1 (or B), while the chemical
subsystem can assume only one of the three states denoted as 0 (or a), ½ (or a & b),
and 1 (or b). The combined system of the enzyme and the chemical subsystem can
assume any of the states along the zigzag line that connects vertices (0, 0) and (1, 1)
and hence lies in the interior of the two-dimensional hypercube, a characteristic
property of a fuzzy machine.
The theoretical rationale as to why enzymes are fuzzy rather than crisp machines
is not yet known. One possibility is that, due to their small physical size, enzymes
cannot avoid the randomizing influence of thermal fluctuations, making it impossi-
ble for them to occupy any crisp (i.e., 0 or 1) internal states. We may refer to this
concept as the
Principle of Fuzzy Molecular Machines
(PFMM) which may be
stated as:
Molecular machines cannot reside on the vertices of the N-dimensional hypercube, where
N is the number of the binary questions required to characterize the internal states of the
machine. (7.27)
Alternatively, PFMM may be stated as follows utilizing the concept of the
Kosko entropy defined in Eq.
5.24
:
The Kosko entropy of molecular machines cannot be zero.
(7.28)
Or,
It is impossible to measure the single-molecule catalytic constant of an enzyme with an
arbitrary accuracy.
(7.29)
Statements 7.27-7.29 may be viewed as the different expressions of a new
principle in enzymology that has emerged from single-molecule enzyme
