CONTROL THEORETIC APPROACH TO ANALYSIS OF

RANDOM BRANCHING WALK MODELS ARISING IN

MOLECULAR BIOLOGY

Andrzej Swierniak

Silesian University of Technology, Department of Automatic Control, Akademicka 16, Gliwice, Poland

Keywords: Systems modelling, random branching walks, control applications in biology.

Abstract: We present two models of molecular processes described by infinite systems of first order differential

equations. These models result from branching random walk processes used to represent the evolution of

particles in these problems. Using asymptotic techniques based on Laplace transforms it is possible to

characterize the asymptotic behavior of telomeres shortening which is supposed to be the mechanism of

aging and evolution of cancer cells with increasing number of copies genes responsible for coding causing

drug removal or metabolisation. The analysis in both cases is possible because they could be represented by

systems with positive feedbacks.

1 PROBLEM STATEMENT

Shortening of telomeres is one of the supposed

mechanisms of cellular aging and death. The

hypothesis is that each time a cell divides it loses

pieces of its chromosome ends. These ends are

called telomeres and consist of repeated sequences

of nucleotides, telomere units. When a critical

number of telomere units is lost, the cell stops

dividing. Telomeres are assumed to consist of

telomere units repeated sequences of nucleotides.

When a chromosome replicates each newly

synthesized strand loses one telomere unit at one of

its ends. This means that the pair of daughter

chromosomes each has one old unchanged strand

and one new, one unit shorter. Once a critical

number of telomere units is lost a so called Hayflick

checkpoint is reached and the cell stops dividing.

Under this assumption, only the length of the

shortest telomere will matter and thus a chromosome

is said to be of type j if its shortest telomere has j

remaining units (Arino, Kimmel, Webb, 1995).

The amount of DNA per cell remains constant from

one generation to another because during each cell

cycle the entire content of DNA is duplicated and

then at each mitotic cell division the DNA is evenly

apportioned to two daughter cells. However, recent

experimental evidence shows that for a fraction of

DNA, its amount per cell and its structure undergo

continuous change. Gene amplification can be

enhanced by conditions that interfere with DNA

synthesis and is increased in some mutant and tumor

cells. Increased number of gene copies may produce

an increased amount of gene products and, in tumor

cells, confer resistance to chemotherapeutic drugs.

Amplification of oncogenes has been observed in

many human tumor cells and also may confer a

growth advantage on cells which overproduce the

oncogene products (for an overview see e.g. survey

in (Stark, 1993)).

We present models of this two phenomena using

branching random walk machinery. The asymptotic

properties of them could be found using methods of

Laplace transforms and spectral analysis.

Conclusions resulting from this analysis are general

because we demonstrate that the models could be

represented by the linear systems with positive

feedbacks and therefore we are able to use some

well known results from standard control theory of

infinite dimensional control systems.

217

Mihai D. (2008).

ON THE SAMPLING PERIOD IN STANDARD AND FUZZY CONTROL ALGORITHMS FOR SERVODRIVES - A Multicriterial Design and a Timing

Strategy for Constant Sampling.

In Proceedings of the Fifth International Conference on Informatics in Control, Automation and Robotics - SPSMC, pages 217-220

DOI: 10.5220/0001477502170220

Copyright

c

SciTePress

2 MODEL OF TELOMERE

SHORTENING

The simplest model of telomere shortening is due to

Levy et al.(1992). It is based on the following

assumptions:

1. Each chromosome consists of 2 strands: upper

and lower, each of them having 2 endings right

and left.

2. Number of telomere units on both endings may

be written as quadruple (a, b; c, d), where a and c

correspond to left and right ending of the upper

strand, while b and d correspond to left and right

ending of the lower one. The only possible

combinations are of the form (n–1, n; m, m) or

(n, n; m,m–1).

3. Cells having chromosomes described by a

quadruple (n–1, n; m, m) while dividing result in

progenies of types (n–1, n–1; m, m–1) and (n–1,

n; m, m). The similar rule takes place for the

second type leading to the situation in which one

of the progenies is always of the same type as the

parent cell while the other is missing two

sequences each on a different ending of a

different strand.

4. The process ends when telomere endings are

short enough; without loss of generality it may

be viewed as the case (n–1, n; 0, 0) or (0, 0; m,

m–1). In this case the cell does not divide and the

single progeny is identical with the parent.

The transformation takes the form:

⎩

⎨

⎧

−−−→

−→

−

)1,;1,1(

),;,1(

),;,1(

mmnn

mmnn

mmnn

(1)

⎩

⎨

⎧

−−−→

−→

−

)1,1;,1(

)1,;,(

)1,;,(

mmnn

mmnn

mmnn

(2)

)0,0;,1()0,0;,1( nnnn −→− (3)

)1,;0,0()1,;0,0( −→− mmmm (4)

We can observe that such "two-dimensional" process

may be simplified by introducing indices k and l

denoting total number of units on both upper and

lower strand for left and right endings respectively.

Denoting:

⎩

⎨

⎧

−−

−

=

appears),;,1(if12

appears)1,;,(if2

mmnnn

mmnnn

k

(5)

⎩

⎨

⎧

−−

−

=

appears)1,;,(if12

appears),;,1(if2

mmnnm

mmnnm

l

(6)

the feasible transformations are as follows:

⎩

⎨

⎧

−−→

→

)1,1(

),(

),(

lk

lk

lk

(7)

)0,()0,( kk → (8)

),0(),0( ll → (9)

Defining i = min(k, l) leads to the simplest form of

the admissible transitions:

⎩

⎨

⎧

−→

→

1i

i

i

(10)

and

0 → 0 (11)

Index i describing the state of the cell in the sense of

the telomere's length may be called the type of the

cell. Dynamics of this model could be represented

by a system of state different equations the

asymptotic behavior of which has a polynomial form

as a function of the number of generation.

Deterministic model treats all cells as homogeneous,

not taking into account its variability dealing mainly

with different life time. The simplest approaching to

real world is to treat cell doubling times as random

variables with exponential distribution characterized

by the same parameter

α

. The evolution process

becomes a branching random walk with an expected

number of cells of type j originated of the ancestor

of type i denoted by N

ij

(t) given by the following

state equation:

0),()(

1

≥≥=

+

jitNtN

ijij

α

&

(12)

For finite number of nonzero initial conditions:

N

i

(0) > 0, i ≤ M (13)

we have:

)0(

)!(

)(

i

M

ji

ji

j

N

ji

t

tN

∑

=

−

−

=

α

(14)

where N

j

(t) is an average number of cells in the state

j.

Once more the solution (exact solution and not only

asymptotic expansion as it has been the case in the

previously discussed discrete model) has a form of

polynomial function of time. Moreover if we assume

that the random variables representing doubling time

has an arbitrary distribution the same in each

generation the asymptotic formula for the average

number of cells in all states could be also given by

(14) with the parameter of exponential distribution

substituted by the inverse of the average doubling

time resulting from the assumed distribution.

ICINCO 2008 - International Conference on Informatics in Control, Automation and Robotics

218

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

resistance such as a multi-drug chemotherapy,

models including phase specificity in the sensitive

compartment or models which take into account

partial sensitivity of some neoplastic subpopulations

(Swierniak, Smieja, 2005).

4 CONCLUSION REMARKS

In this paper we have studied asymptotic properties

of two models of molecular processes each of them

modeled by the random branching walk models. The

properties of these models are strictly related with

their structure which when considered from system

theoretic point of view includes always the positive

feedback. Moreover although the models have the

form of infinite dimensional state equations linear or

bilinear the asymptotic analysis may be performed

rigorously using control theoretic tools resulting

from the closed loop structure of these models. Yet

another molecular process which could be analyzed

using similar techniques is the evolution of tandem

repeats in microsatellite DNA Once more random

branching walk could be used as a basis for the

model construction. Nevertheless in this case there is

no positive feedback which has been used by us to

simplify the asymptotic analysis of the two

processes considered in this paper.

REFERENCES

Arino O., Kimmel M., Webb G.F. 1995. Mathematical

modeling of telemore sequences, J. Theoretical

Biology, v.177, 45-57.

Kimmel M., Swierniak A., Polanski A., 1998. Infinite-

dimensional model of evolution of drug resistance of

cancer cells, J. Mathematical Systems, Estimation, and

Control, v.8, 1-16.

Levy

M.Z., Allstrop R.C., Futchert A.B., Grieder C.W.,

Harley C.B. , 1992. Telomere end-replication problem

and cell aging, J. Molec. Biol., v.225, 951-960.

Stark

G.R. , 1993. Regulation and mechanisms of

mammalian gene amplification, Adv. Cancer Res., v.

61, 87-113.

Swierniak A., Polanski A., Kimmel M., Bobrowski A.,

Smieja J. , 1999. Qualitative analysis of controlled

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Swierniak A., Smieja J. , 2005. Analysis and optimization

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