1 Introduction

A genetic regulatory network (GRN in

short) is a complex system that governs

the interactions among genes and their pro-

ducts, including proteins, RNA molecules,

and other regulatory elements, within a

living organism, [1]. It plays an important

role in controlling the development, growth,

and function of cells and tissues, as well as

coordinating various physiological processes

throughout an organism’s life, [2],[3]. Un-

derstanding genetic regulatory networks is

of great interest in various ﬁelds, including

developmental biology, systems biology,

and biotechnology, [4], [5].

Computational models, such as Boolean

networks and diﬀerential equations, are

employed to simulate and predict the be-

havior of genetic regulatory networks under

diﬀerent conditions, [6]. These models help

uncover the network’s underlying principles

and aid in the design of genetic engineering

strategies to modulate speciﬁc biological

processes for therapeutic or industrial

purposes,[7].

The dynamical system of the form











dt =

1 + e−µ1(w11x+w12y+w13z−θ1)−v1x,

dt =

1 + e−µ2(w21x+w22y+w23z−θ2)−v2y,

dt =

1 + e−µ3(w31x+w32y+w33z−θ3)−v3z,

(1)

is used to model genetic regulatory

networks, where µi,θiand viare the

parameters, wij are the coeﬃcients of the

so-called regulatory matrix

W=



w11 w12 w13

w21 w22 w23

w31 w32 w33



.(2)

Quasi-periodic solutions for a three-dimensional system in gene

regulatory network

OLGA KOZLOVSKA, INNA SAMUILIK

Department of Engineering Mathematics, Riga Technical University, LATVIA

Abstract: This work introduces a three-dimensional system with quasi-periodic solutions for special values of

parameters. The equations model the interactions between genes and their products. In gene regulatory

networks, quasi-periodic solutions refer to a specific type of temporal behavior observed in the system. We

show the dynamics of Lyapunov exponents. Visualizations are provided. It is important to note that the study

of gene regulatory networks is a complex interdisciplinary field that combines biology, mathematics, and

computer science.

Key-Words: gene regulatory network, Lyapunov exponents, quasi-periodic solution, nullclines, critical points

Received: November 9, 2022. Revised: August17, 2023. Accepted: September 18, 2023. Published: October 9, 2023.

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Such system was considered in the paper [8].

The parameters of the GRN have the follow-

ing biological interpretations:

•vi- degradation of the i-th gene expres-

sion product;

•wij - the connection weight or strength

of control of gene jon gene i. Positive

values of wij indicate activating inﬂu-

ences while negative values deﬁne re-

pressing inﬂuences;

•θi- inﬂuence of external input on gene

i, which modulates the gene’s sensitiv-

ity of response to activating or repress-

ing inﬂuences.

Deﬁnition 1.1. The j0th nullcline is the

geometric shape for which dxj

dt = 0. The

critical points of the system are located

where all of the nullclines intersect. In

a two-dimensional linear system, the

nullclines can be represented by two lines

on a two-dimensional plot; in a general

two-dimensional system they are arbitrary

curves.

The nullclines for the system are deﬁned by

the relations











v1x=1

1 + e−µ1(w11x+w12 y+w13 z−θ1),

v2y=1

1 + e−µ2(w21x+w22 y+w23 z−θ2),

v3z=1

1 + e−µ3(w31x+w32 y+w33 z−θ2).

(3)

Critical points are solutions of the system

(3).

Proposition 1.1. System (1) has at least

one equilibrium (critical point). All equi-

libria are located in the open box Q3:=

{(x, y, z) : 0 < x < 1

v1,0< y < 1

v2,0<

z < 1

v3}.

2 Critical points

The type of the critical point is determined

by the position of the roots of the char-

acteristic equation on the complex plane.

Two main possibilities exist: either the

three eigenvalues are real or two of them

are complex conjugates. A critical point

is stable if all eigenvalues have negative

real parts; it is unstable if at least one

eigenvalue has positive real part.

•Node. All eigenvalues are real and

have the same sign. The node is sta-

ble (unstable) when the eigenvalues are

negative (positive).

•Saddle. All eigenvalues are real and

at least one of them is positive and at

least one is negative. Saddles are al-

ways unstable.

•Focus −Node. It has one real eigen-

value and a pair of complex-conjugate

eigenvalues, and all eigenvalues have

real parts of the same sign. The critical

point is stable (unstable) when the sign

is negative (positive).

•Saddle −Focus. Negative real eigen-

value and complex eigenvalues with

positive real part, and positive real

eigenvalue and complex eigenvalues

with negative real part. This type of

critical point is unstable, [9].

3 Lyapunov exponents

Lyapunov exponents are a concept from

chaos theory and dynamical systems that

provide a quantitative measure of the

sensitivity to initial conditions in a chaotic

system, [10]. The concept is named after

the Russian mathematician Aleksandr

Lyapunov, who introduced it in the late

19th century.

Lyapunov exponents have applications

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in various ﬁelds, including weather fore-

casting, ﬂuid dynamics, biology, economics,

and cryptography. They are particularly

useful in studying chaotic systems, predict-

ing long-term behavior, and understanding

the stability of dynamical systems.

Calculating Lyapunov exponents can

be a complex task, especially for high-

dimensional systems, and often involves

numerical methods and simulations.

The generally accepted convention is to

write the Lyapunov exponents in descend-

ing order

λ1≥λ2≥. . . λn.

In 3D phase space, there exist four types

of attractors: stable points, limit cycles, 2D

tori and strange attractors, [11]. The fol-

lowing set of LEs characterizes possible dy-

namical situations to be met:

•(LE1, LE2, LE3)=(−,−,−) - stable

ﬁxed point;

•(LE1, LE2, LE3) = (0,−,−) - stable

limit cycle;

•(LE1, LE2, LE3) = (0,0,−) - stable 2D

tori;

•(LE1, LE2, LE3) = (+,0,−) - strange

attractor.

Any dissipative dynamical system will have

at least one negative exponent, the sum of

all of exponents is negative, [12],[14],[15].

In general, a system has a set of Lyapunov

exponents, each characterizing the average

stretching or shrinking of phase space in a

particular direction, [13].

3.0.1 Properties of Lyapunov expo-

nents

1. The number of Lyapunov exponents is

equal to the number of phase space di-

mensions, or the order of the system

of diﬀerential equations. They are ar-

ranged in descending order [15].

2. The largest Lyapunov exponent of a

stable system does not exceed zero [16].

3. A chaotic system has at least one

positive Lyapunov exponent, and the

more positive the largest Lyapunov

exponent, the more unpredictable the

system is [16].

4. To have a dissipative dynamical sys-

tem, the values of all Lyapunov expo-

nents should sum to a negative number

[15].

5. A hyperchaotic system is deﬁned as a

chaotic system with at least two pos-

itive Lyapunov exponents. Combined

with one null exponent and one nega-

tive exponent, the minimal dimension

for a hyperchaotic system is four [15].

Proposition 3.1. The largest Lyapunov ex-

ponent of a stable system does not exceed

zero, [16].

Proposition 3.2. Only dissipative dynami-

cal systems have attractors, [9].

4 Quasi-periodic solu-

tions

Quasi-periodicity refers to a pattern or

behavior that exhibits some level of perio-

dicity but is not perfectly periodic. In other

words, it shows repetitive characteristics,

but the repetition is not strictly regular.

This concept is encountered in various

ﬁelds, including physics, mathematics,

signal processing, and even in everyday life,

[17]. They can describe the behavior of

certain celestial objects in astronomy, vibra-

tions in mechanical systems, or oscillations

in chemical reactions. Quasi-periodicity is a

valuable concept that helps us understand

and model complex systems that exhibit

both regular and irregular behaviors.

Quasi-periodicity describes in dynamical

systems solutions, which neither exhibit a

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ﬁnite period length nor are chaotic, [18].

Quasi-periodic behavior can be challenging

to visualize and analyze. Advanced mathe-

matical tools and numerical simulations are

often used to study and understand these

solutions.

Deﬁnition 4.1. Quasi-periodic solutions

are characterized by a discrete frequency

spectrum, which does not consists of integer

multiples of one single base frequency.

The spectrum consists of linear combi-

nations of at least two independent base

frequencies,[18].

Two zero Lapunov values in 3D systems

indicates quasi-periodic dynamic, [19].

4.1 Example

Consider the system (1) and the regulatory

matrix

W=



0.025 0.5 1.14

−0.9 0.2 5.5881

0.08 −1 2



(4)

v1= 0.2505, v2= 0.1407, v3= 0.4305; µ1=

4.6, µ2= 4.2, µ3= 7; θ1= 0.565, θ2=

0.8355, θ3= 0.49.Let initial conditions

be (0; 0.5; 0.1).There is the critical

point P= (3.94; 1.30; 0.67). Charac-

teristic values for critical point Pare

λ1=−0.23, λ2,3= 1.21 ±1.87i. The

type of critical point is a saddle-focus.

The nullclines of the system (1) with the

regulatory matrix (4) are considered in

Figure 1.

The trajectory of the system (1) with

the regulatory matrix (4) is considered in

Figure 2.

Solutions (x(t), y(t), z(t)) of the system (1)

with the regulatory matrix (4) are shown

in Figure 3.

The dynamics of Lyapunov exponents are

shown in Figure 4. Lyapunov exponents

are λ1= 0.00; λ2= 0.00; λ3=−0.05;

The presence of two zero values indicates

the quasi-periodicality of the dynamics.

Figure 1: The nullclines of the system (1) with

the regulatory matrix (4).

0.5

1.0

1.5

2.0

0.0

0.5

1.0

0.0

0.2

0.4

Figure 2: The trajectory of the system (1) with

the regulatory matrix (4).

100

150

200

250

300

350

400

0.5

1.0

1.5

2.0

2.5

8x, y, z<

Figure 3: Solutions (x(t), y(t), z(t)) of the

system (1) with the regulatory matrix (4)

Figure 3 demonstrate a discrete frequency

spectrum, which does not consists of inte-

ger multiples of one single base frequency.

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0 2000 4000 6000 8000 10000

-0.4

-0.3

-0.2

-0.1

0.0

Steps

LCEs

Figure 4: The dynamics of Lyapunov expo-

nents.

Figure 2 demonstrate complicated trajec-

tory motion. Despite trajectories complex

motion, the quasi-periodic motion is pre-

dictable. Trajectories starting close to each

other stay close to each other, and the long-

term prognosis is guaranteed.

5 Conclusions

The three-dimensional GRN system is con-

sidered. The dynamics of Lyapunov expo-

nents are shown. The nullclines of the sys-

tem (1) with the regulatory matrix (4) are

shown. Also the trajectory of the system

(1) with the regulatory matrix (4) is shown.

Getting quasy-periodicity in 3D GRN sys-

tem is very important.

The next step of devolopment of GRN sys-

tem after quasy-periodicity is chaotic be-

havior. Chaotic behavior in 3D cases in

GRN occur extremely rare, [20].

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The authors equally contributed in the

present research, at all stages from the

formulation of the problem to the ﬁnal

ﬁndings and solution.

Sources of Funding for Research

Presented in a Scientiﬁc Article or

Scientiﬁc Article Itself

No funding was received for conducting

this study.

Conﬂicts of Interest

The authors have no conﬂicts of interest to

declare that are relevant to the content of

this article.

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