Biomedical Engineering Reference
In-Depth Information
signal transduction system, and the concentration of active enzyme (
A
) interfac-
ing with the propulsive flagellum is the system output. A
n
adaptive system has
the characteristic that the steady-state concentration of
A a
is independent of
l
.
The key to the robust adaptive property is to make the modification and un-
modification transformation of
E
dependent only on the concentration of
A
. Yi
et al. (65) point out that this adaptive property of the network is a consequence
of
integral feedback control
. In mathematical terms:
()
x
=
a
,
[15]
aa akl
=
=
(
x a
)
.
[16]
1
Here the time integral of the system error (
x
), t
he
difference between the actual
output (
a
1
) and the desired equilibrium output
( )
a
, is fed back into the system.
The parameter
k
is the gain of the system. In this way one obtains robust asymp-
totic tracking of variations in input
l
.
4.4. Antiredundancy through Apoptosis in Neoplastic Lineages
Tumorigenesis marks the onset of unregulated cell proliferation. In most
long-lived mammals, progress towards tumorigenesis involves the cumulative
loss of important regulatory genes monitoring the genetic state of defective cells.
An important class of regulatory genes are the
tumor suppressor genes
(35,36),
which respond to mutations by inducing programmed cell death (apoptosis) or
repairing damaged DNA. Apoptosis represents a strategy of antiredundancy or
purging, in which defective cells are removed and subsequently replaced by the
descendants of healthy cells in the surrounding tissue. Purging as a mechanism
of robustness thus depends crucially on population sizes large enough to allow
for the replacement of eliminated cells.
Plotkin and Nowak (46) have modeled the waiting time for dividing cells
undergoing mutation and mutation-induced apoptosis to reach a tumorigenic
state. Assume that
L
genes in the genome of dividing cells regulate healthy cell
cycle function. For each cell, count the number of mutations in
L
and call it
k
.
When the value of
k
=
n
, the cell is tumorigenic. During each cell division a cell
with
k
mutations can divide and remain in the same state with a probability
q
k
or
mutate with a probability
p
k
= 1 -
q
k
. Any cell with
k
1 mutations is under the
surveillance of tumor suppressor genes and can be induced into apoptosis with a
probability B
k
. Apoptosis will fail with a probability C
k
= 1 - B
k
.
These probabilities can be used to construct a Markovian model of cancer
progression, with three important assumptions: (1) there are no population dy-
namics—cell populations are of a large fixed size with no fixation of mutant
lineages, (2) there are symmetric mutations such that only the total number of
