Biomedical Engineering Reference
In-Depth Information
mutations
k
and not the position of these mutations in a string of length
L
is sig-
nificant, (3) the cell with
n
mutations is an absorbing state. With these assump-
tions one can write down an (
n
+ 1) x (
n
+ 1) transition matrix:
This is a flexible formulation, as it allows for either
genomic instability
, in
which B
0
> B
1
> ... > B
n
-1
, which describes how the incidence of mutations re-
duces the efficacy of the apoptotic response, or when B
0
< B
1
< ... < B
n
-1
, which
reflects an increasing probability of cells with more mutations undergoing effec-
tive surveillance. I will only discuss the case in which
q
=
q
1
=
q
2
= ... =
q
n
-1
and
B = B
1
= B
2
= ... = B
n
-1
.
The effects of apoptotic purging can be demonstrated by comparing the
waiting time for
k
=
n
of a non-apoptotic cell, assuming thereby that B
i
= 0 for
all
i
, and the alternative case with apoptosis as described above in which B
i
> 0
for all
i
.
The waiting without apoptosis for one cell in a tissue of
N
cells to obtain
n
mutations is given by
1
d
ยจ
T
=
(
(, )
n a
N
da
,
[17]
log(1 /
qn
)(
1)!
N
0
where ((.,.) is the incomplete Gamma function. The waiting time for a single
cell with apoptosis to obtain
n
mutations is given by
p
CB
(
+
p
)
1
T
=
o
.
[18]
BB
pqp
(
+
)(1
B B B
/(
p q
+
))
n
0
In the case without apoptosis, the waiting time depends inversely on the loga-
rithm of replication fidelity
q
. With apoptosis the waiting time grows exponen-
tially with
n
. Thus purging of damaged cells prolongs the waiting time to
tumorigenesis, and thereby increases the latency of cancer.
4.5. Spatial Compartmentalization of Predators and Prey:
Infectious Disease
Theoretical immunology is in large part based on reinterpretation of the
immune system as an interaction between predators and prey. Whereas in ecol-
