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We demonstrate that these rather strange asymptotic
characteristics and the generality of their form is
related to the positive feedback which could be
discovered in all the three models of telomere
shortenings
.
3 MODEL OF GENE
AMPLIFICATION
We consider a population of neoplastic cells
stratified into subpopulations of cells of different
types, labeled by numbers i = 0, 1, 2 , ... . If the
biological process considered is gene amplification,
then cells of different types are identified with
different numbers of copies of the drug resistance
gene and differing levels of resistance. Cells of type
0, with no copies of the gene, are sensitive to the
cytostatic agent. Due to the mutational event the
sensitive cell of type 0 can acquire a copy of gene
that makes it resistant to the agent. Likewise, the
division of resistant cells can result in the change of
the number of gene copies. The resistant
subpopulation consists of cells of types i = 1, 2 , ... .
The probability of mutational event in a sensitive
cell is of several orders smaller than the probability
of the change in number of gene copies in a resistant
cell. Since we do not limit the number of gene
copies per cell, the number of different cell types is
denumerably infinite.
The hypotheses are as follows:
1. The lifespans of all cells are independent
exponentially distributed random variables with
means 1/
λ
i
for cells of type i.
2. A cell of type i ≥ 1 may mutate in a short time
interval (t, t+dt) into a type i+1 cell with
probability b
i
dt +o(dt) and into type i−1 cell
with probability d
i
dt + o(dt).
3. A cell of type i = 0 may mutate in a short time
interval (t, t+dt) into a type 1 cell with
probability
α
dt + o(dt), where
α
is several
orders of magnitude smaller than any of b
i
s or
d
i
s, i.e.
α
<< min(d
i
, b
i
), i ≥ 1. (15)
4. The chemotherapeutic agent affects cells of
different types differently. It is assumed that its
action results in fraction u
i
of ineffective
divisions in cells of type i.
5. The process is initiated at time t = 0 by a
population of cells of different types.
The mathematical model has the following form:
]
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
≥
+++−=
+++−=
+−−=
−+
K
&
K
&
&
2
),()()()()()(
)()()()()()(
)()()()(21)(
11
02111
1000
i
tbNtdNtNdbtNtN
tNtdNtNdbtNtN
tdNtNtNtutN
iiiii
λ
αλ
αλ
(16)
where N
i
(t) denotes the expected number of cells of
type i at time t, and we assume the simplest case, in
which the resistant cells are insensitive to drug's
action, and there are no differences between
parameters of cells of different type (b
i
= b > 0,
d
i
= d > 0,
λ
i
=
λ
> 0, u
i
= 0, i ≥ 1,
λ
0
=
λ
, u
0
= u).
The first step in the analysis is to evaluate the fate of
the drug resistant subpopulation without a constant
inflow from the drug sensitive subpopulation. In
other words we assume that it is possible to destroy
completely the sensitive subpopulation and we are
interested only how the drug resistant cancer cells
will develop. The analysis can be limited in this case
to equations with i ≥ 1. The asymptotic behavior of
the DNA repeats may be analyzed using inverse
Laplace transforms and asymptotic formulae for
integration of special functions for the case where
the initial condition contained only one nonzero
element N
1
(0) = 1, while N
i
(0) = 0, i > 1. It is
possible to extend that approach to the case of two
or more non-zero elements. The solution decays
exponentially to zero in this case, as t → ∞ for:
d > 0, b > 0,
λ
> 0, d > b, (17)
λ
>− bd (18)
To analyze the conditions under which it is possible
to eradicate the tumor or in other words to ensure
that the entire tumor population converges to zero
we may represent the model (16) in the form of the
closed-loop system with two components. One part
of this system is infinite dimensional and linear and
represents the drug resistant subpopulation. The
second part of the system is given by the first
bilinear equation of the model and describes
behavior of the drug sensitive subpopulation. The
model may be viewed as a system with positive
feedback stability of which may be analyzed using
generalized Nyquist type criterion (Swierniak, et al.
, 1999) in the case when we assume a constant
therapy protocol. The analysis for other protocols
could be also performed using more sophisticated
tools of stability analysis.
In the similar way we may consider more general
models of anticancer therapy under evolving drug
CONTROL THEORETIC APPROACH TO ANALYSIS OF RANDOM BRANCHING WALK MODELS ARISING IN
MOLECULAR BIOLOGY
219