Examples of periodic biological oscillators: transition to a six-dimensional system

INNA SAMUILIK

1,2

, FELIX SADYRBAEV

2,3

,VALENTIN SENGILEYEV

Institute of Applied Mathematics

Riga Technical University

Zunda krastmala 10

LATVIA

Department of Natural Sciences and Mathematics

Daugavpils University

Parades street1

LATVIA

Institute of Mathematics and Computer science

University of Latvia

Rainis boulevard 29

LATVIA

Abstract: We study a genetic model (including gene regulatory networks) consisting of a system of

several ordinary differential equations. This system contains a number of parameters and depends on the

regulatory matrix that describes the interactions in this multicomponent network. The question of the

attracting sets of this system, which depending on the parameters and elements of the regulatory matrix,

isconsidered. The consideration is mainly geometric, which makes it possible to identify and classify

possible network interactions. The system of differential equations contains a sigmoidal function, which

allows taking into account the peculiarities of the network response to external influences. As a

sigmoidal function, a logistic function is chosen, which is convenient for computer analysis. The

question of constructing attractors in a system of arbitrary dimension is considered by constructing a

block regulatory matrix, the blocks of which correspond to systems of lower dimension and have been

studied earlier. The method is demonstrated with an example of a three-dimensional system, which is

used to construct a system of dimensions twice as large. The presentation is provided with illustrations

obtained as a result of computer calculations, and allowing, without going into details, to understand the

formulation of the issue and ways to solve the problems that arise in this case.

Key-Words: gene regulatory network, attractors, logistic function, numerical analysis

Received: March 19, 2021. Revised: March 13, 2022. Accepted: April 10, 2022. Published: May 5, 2022.

1 Introduction

The dynamics of gene regulatory networks can be

described by a system of ordinary differential

equations in the form

⎩

⎪

⎨

⎪

⎧𝑥󰆒=1

1 + 𝑒(   ⋯ )−𝑥,

…

𝑥󰆒=1

1 + 𝑒(   ⋯ )−𝑥,

(1)

where𝑥corresponds to the expression of the

protein by the 𝑖-th element of the system,𝜇 and

𝜃are parametershaving biological meaning.

Interaction between network elements (classified

as activation, inhibition or no influence) is

described by the regulatory matrix

𝑊 = 𝑤 … 𝑤

… … …

𝑤 … 𝑤,(2)

where the positivity (negativity) of the element

𝑤 means the activation (inhibition) of the i-th

gene by the j-th gene.Zero value means no

influence. The combined result of these

WSEAS TRANSACTIONS on COMPUTER RESEARCH

DOI: 10.37394/232018.2022.10.7

Inna Samuilik, Felix Sadyrbaev, Valentin Sengileyev

E-ISSN: 2415-1521

Volume 10, 2022

interactions forms the network response to

changes in the external environment.

Among these reactions, periodic reactions are

distinguished, which are often caused by periodic

processes in the environment. These periodic

reactions are formed by bio-oscillators. Our goal

is to give examples of bio-oscillators in

mathematical models of gene networks. These

examples are reflections of specific types of

network interactions. These specific types

correspond to certain regulatory matrices. The

system of differential equations describing the

gene network has an infinite set of solutions,

which correspond to trajectories in the space of

variables 𝑥. We are especially interested in the

case of periodic attractors that arise from stable

periodic solutions of the system.

It should be noted that the system (1) is also used

in modeling processes of various natures. One of

its earliest appearances in the literature is an

article by Wilson and Cowan, which describes the

behavior of groups of neurons.[15] In the future,

the system repeatedly appeared as an element of

dynamic models of gene networks.[1], [2], [3],

[10], [11], [14]. This system was also used to

build the optimal topology of telecommunication

networks.[9]In this case, the behavior of gene

networks was taken as a model. The literature on

issues related to the modeling of complex

networks such as gene networks and/or

telecommunications has hundreds of articles and

continues to grow. Fairly complete lists of articles

can be found in the reviews [8], [12], [13], [4].

The works [1], [7], [16] are devoted to qualitative

and/or numerical analysis of systems of type (1).

Periodic solutions and related issues were

considered in [6], [7]. Applications in medicine of

system (1) are the subject of works [5], [14].

2 Three-dimensional system

Consider the three-dimensional system (1) with

regulatory matrix

𝑊 = 𝑘 0 −1

−1 𝑘 0

0 −1 𝑘.(3)

This matrix contains the inhibitory cycle

𝑊 = 0 0 −1

−1 0 0

0 −1 0(4)

and auto-activation

𝑊 = 𝑘 0 0

0 𝑘 0

0 0 𝑘.(5)

For values 𝑘 from the interval(0.36;2) the three-

dimensional system is

⎩

⎪

⎨

⎪

⎧

𝑥=

   ,

𝑥=

     ,

𝑥=

   . (6)

It has a periodic solution that attracts the other

solutions. Figure 1 and Figure 2 show the periodic

solutions of the system (6) for 𝑘=1 and 𝑘=

0.5.The values of other parameters are as

follows𝜇=5; 𝜃=

.

Figure 1. Periodic solutions, 𝑘=1.

Figure 2. Periodic solutions, 𝑘=0.5.

The specified three-dimensional periodic solution

arises as a result of a three-dimensional

Andronov-Hopf bifurcation. For small values of

WSEAS TRANSACTIONS on COMPUTER RESEARCH

DOI: 10.37394/232018.2022.10.7

Inna Samuilik, Felix Sadyrbaev, Valentin Sengileyev

E-ISSN: 2415-1521

Volume 10, 2022

𝑘, the attractor of the system (6) exists in the form

of a single critical point of the type of stable focus

and attraction in the remaining dimension. As 𝑘

increases, the point type changes to an unstable

focus. The result is a three-dimensional limit

cycle.

3Six-dimensional system

Consider the 𝟔×𝟔 regulatory matrix

𝑊=

⎝

⎜

⎛

𝑘 0 −1 0 0 0

−1 𝑘 0 0 0 0

0 −1 𝑘 0 0 0

0 0 0 𝑘 0 −1

0 0 0 −1 𝑘 0

0 0 0 0 1 𝑘

⎠

⎟

⎞

, (7)

where the 3D diagonal blocks correspond to the

matrix (3).

Let 𝑘=1.The system (7) has an attractor as a

periodic solution generated by the three-

dimensional periodic solution shown in the Figure

1.The projections of this periodic attractor onto

three-dimensional subspaces are shown in Figure

3, Figure 4 and Figure 5.

Figure 3. The projection of the attractor on

(𝑥,𝑥,𝑥)

Figure 4. The projection of the attractor on

(𝑥,𝑥,𝑥)

Figure 5. The projection of the attractor on

(𝑥,𝑥,𝑥)

Consider the 𝟔×𝟔 regulatory matrix

𝑊=

⎝

⎜

⎛

𝑘0 −1 0 0 0

−1 𝑘0 0 0 0

0 −1 𝑘0 0 0

0 0 0 𝑘0 −1

0 0 0 −1 𝑘0

0 0 0 0 1 𝑘

⎠

⎟

⎞

,(8)

where 𝑘=1 and 𝑘=0.5. The system (8) has

an attractor generated by two periodic solutions

shown in the Figure 1 and Figure 2. The

projections of this periodic attractor onto three-

dimensional subspaces are shown in Figure 6,

Figure 7 and Figure 8.

WSEAS TRANSACTIONS on COMPUTER RESEARCH

DOI: 10.37394/232018.2022.10.7

Inna Samuilik, Felix Sadyrbaev, Valentin Sengileyev

E-ISSN: 2415-1521

Volume 10, 2022

Figure 6. The projection of the attractor on

(𝑥,𝑥,𝑥)

Figure 7. The projection of the attractor on

(𝑥,𝑥,𝑥)

Figure 8. The projection of the attractor on

(𝑥,𝑥,𝑥)

The projections of this attractor (black) are

repeatedly shown in figures 9, 10, 11 together

with the solution (red), which starts close to the

attractor and has the initial conditions

𝑥(0)=0; 𝑥(0)=0.4; 𝑥(0)=0.1;

𝑥(0)=0.2; 𝑥(0)=0.1; 𝑥(0)=0.1.

Figure 9. The projection of the attractor on

(𝑥,𝑥,𝑥)

Figure 10. The projection of the attractor on

(𝑥,𝑥,𝑥)

Figure 11. The projection of the attractor on

(𝑥,𝑥,𝑥)

Further study of attractors and solutions can be

carried out using the perturbation of the regulatory

matrix 𝑊.

Consider the matrix

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Inna Samuilik, Felix Sadyrbaev, Valentin Sengileyev

E-ISSN: 2415-1521

Volume 10, 2022

𝑊=

⎝

⎜

⎛

𝑘0 −1 0 0 𝑚

−1 𝑘0 0 0 0

0 −1 𝑘0 0 0

0 0 0 𝑘0 −1

0 0 0 −1 𝑘0

−𝑚 0 0 0 1 𝑘

⎠

⎟

⎞

,(9)

which differs from the previous matrix (8) only by the

presence of non-zero elements (𝑚 and – 𝑚) at the end

points of the secondary diagonal. Numerical

experiments gave the following results. At small values

of 𝑚, a picture similar to that shown in Figure 6, Figure

7, and Figure 8 is preserved, which corresponds to the

ideas about the structural stability of systems

encountered in the theory of gene networks. With a

further increase in 𝑚(𝑡𝑜 𝑚 < 0.5), the form of

threedimensional projections, while remaining regular,

changes significantly. A further increase in 𝑚 leads to

an increase in the irregularity and chaotic behavior of

the solutions.

4 Conclusions

For systems (1) used in the mathematical modeling of

gene networks, it is true: 1) There is an invariant set in

the phase space: the vector field defined by the system

is directed inside this set. This follows from the

properties of sigmoidal functions used in systems (1); 2)

There is always an equilibrium (critical point). There

may be several critical points, but apart from degenerate

cases, a finite number only; 3) An attractor in the system

(1) can have the form of several stable equilibria

(critical points); 4) An attractor can exist in the form of

an attracting closed trajectory (limit cycle); 5) The

system of type (1) of any dimension can be constructed

with attractors of different types. In particular, for any

dimension, a system of the form (1) can be constructed

with a periodic attractor; 6) Сonstructing such a system,

block-type regulatory matrices are used. In this case, the

blocks correspond to systems of lower dimension with

attractors installed in them; 7) By perturbation of such

matrices, it is possible obtain and study systems with

chaotic behavior of solutions.

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WSEAS TRANSACTIONS on COMPUTER RESEARCH

DOI: 10.37394/232018.2022.10.7

WSEAS TRANSACTIONS on COMPUTER RESEARCH

DOI: 10.37394/232018.2022.10.7

Inna Samuilik, Felix Sadyrbaev, Valentin Sengileyev

E-ISSN: 2415-1521

Volume 10, 2022