Mathematical modeling of three - dimensional genetic regulatory
networks using logistic and Gompertz functions
INNA SAMUILIK
1
, FELIX SADYRBAEV
1,2
, DIANA OGORELOVA
1
1
Department of Natural Sciences and Mathematics
Daugavpils University
Parades street1
LATVIA
2
Institute of Mathematics and Computer science
University of Latvia
Rainis boulevard 29
LATVIA
Abstract: Mathematical modeling is a method of cognition of the surrounding world in which the
description of the object is carried out in the language of mathematics, and the study of the model is
performed using certain mathematical methods. Mathematical models based on ordinary differential
equations (ODE) are used in the study of networks of different kinds, including the study of genetic
regulatory networks (GRN). The use of ODE makes it possible to predict the evolution of GRN in time.
Nonlinearity in these models is included in the form of a sigmoidal function. There are many of them,
and in the literature, there are models that use different sigmoidal functions. The article discusses the
models that use the logistic function and Gompertz function. The comparison of the results, related to
three-dimensional networks, has been made. The text is accompanied by examples and illustrations.
Key-Words: gene regulatory network, Gompertz function, logistic function, periodic solutions
Received: March 28, 2021. Revised: January 16, 2022. Accepted: February 5, 2022. Published: February 23, 2022.
1 Introduction
The main problem in mathematical modeling of a
dynamic system is to develop a model and then to
determine dependencies and coefficients in the
equations used in developing the model. For
complex dynamic systems, the determination of
the coefficients and dependencies in the model is
a nontrivial task.
In nonlinear models, we consider cycles, stable
and unstable regimes [11], a strange attractor and
chaos [1]. Cycles are regular impacts on the
economic mechanism with a period of one year
(autumn harvest, increased heating costs in the
winter season). Chaos, in socio-economic systems
and biological communities can be interpreted as
a natural form of competition. Artificial
elimination of chaos (in mechanics - due to a
large dissipation of energy, in the economy - due
to excessive regulation, high taxation, in society -
due to legislative restrictions) leads to the
elimination of complex dynamic regimes and the
transition to simple solutions, degradation of the
system. In mechanics, these are the usual simplest
periodic fluctuations, in economics - a situation of
stagnation. During the transition from chaos to
orderliness or after the loss of stability of the
previous regime, new stable non-trivial solutions
arise in mechanics as well as inprofitable
directions in the economy.
Consider the general form of writing the n-
dimensional dynamical system, that is expected to
model a genetic regulatory network,
𝑥=
⋯ −𝑣𝑥,
…
𝑥=
⋯ −𝑣𝑥,
or
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2022.17.12
Inna Samuilik, Felix Sadyrbaev, Diana Ogorelova
E-ISSN: 2224-2856
101
Volume 17, 2022
𝑥′= 𝑒( ⋯ )−𝑣𝑥,
…
𝑥′=𝑒( ⋯ )−𝑣𝑥,
(1)
where 𝜇>0,𝜃 and 𝑣>0 are parameters, and
𝑤 are elements of the 𝑛 × 𝑛 regulatory matrix
𝑊. The parameters of the GRN have the
following biological interpretations:
𝑣− the rate of degradation of the i-th gene
expression product;
𝑤− the connection weight or strength of control
of gene j on gene i. Positive values of 𝑤indicate
activating influences while negative values define
repressing influences;
𝜃−the influence of external input on gene i,
which modulates the gene’s sensitivity of
response to activating or repressing influences.
[13]
The sigmoidal functions 𝑓(𝑧)=
and
𝑓(𝑧)=𝑒()are used in (1).
Figure 1.Logistic function.
Figure 2. Gompertz function.
Sigmoidal functions are monotonically increasing
from zero to unity and have a single inflection
point. They are many, but the above functions suit
well for the analysis and visualizations. A set of
coefficients 𝑤 form the so-called regulatory
matrix 𝑊 = 𝑤 … 𝑤
… … …
𝑤 … 𝑤.(2)
2 Three-element GRN
Consider the three-dimensional system for
Logistic function
⎩
⎪
⎨
⎪
⎧
𝑥′=1
1 + 𝑒( )−𝑣𝑥,
𝑥′=1
1 + 𝑒( )−𝑣𝑥,
𝑥′=1
1 + 𝑒( )−𝑣𝑥,(3)
and for Gompertz function
⎩
⎨
⎧
𝑥′= 𝑒 −𝑣𝑥,
𝑥′= 𝑒 −𝑣𝑥,
𝑥′= 𝑒 −𝑣𝑥. (4)
The nullclines for the system (3) are defined by
the relations
⎩
⎪
⎨
⎪
⎧
𝑥=
,
𝑥=
,
𝑥=
. (5)
For the system (4) the nullclines are defined by
the relations
⎩
⎪
⎨
⎪
⎧
𝑥=
𝑒 ,
𝑥=
𝑒 ,
𝑥=
𝑒 . (6)
2.1 Logistic function
Case 1. Stable limit cycles can exist in systems of
the form (3). Consider the system (3) with the
matrix 𝑊 = 0 1 0
−1 1 0
1 1 0.01(7)
and 𝜇=𝜇=5,𝜇=15; 𝑣=𝑣=𝑣=1;
𝜃=0.5,𝜃=0.04,𝜃=−0.5. Three nullclines
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2022.17.12
Inna Samuilik, Felix Sadyrbaev, Diana Ogorelova
E-ISSN: 2224-2856
102
Volume 17, 2022
Inna Samuilik, Felix Sadyrbaev
, Diana Ogorelova
are located as shown in Figure 3. Computations
and graphical results are performed using
Wolfram Mathematica.
Figure 3. Nullclines (𝑥−𝑟𝑒𝑑,𝑥− 𝑔𝑟𝑒𝑒𝑛, 𝑥−𝑏𝑙𝑢𝑒)
There is one critical point 𝑝:
(0.4018; 0.4204; 0.9987). Linearization around
this point provides us with the characteristic
numbers 𝜆 given in Table 1.
Table 1. The characteristic numbers λ
-
λ
1
λ
2
λ
3
p
1
-0.9999 0.8275-1.0261
𝑖
0.8275+1.0261
𝑖
Figure 4. Periodic solutions
Figure 5. The graphs of 𝑥(𝑡),𝑖=1,2,3
Case 2. Consider the system (3) with the matrix
𝑊 = 1.2 0 2
0.5 1 −0.5
−2 0.01 1 (8)
and 𝜇=𝜇=5,𝜇=15; 𝑣=𝑣=𝑣=1;
𝜃=1.0,𝜃=0.4,𝜃=−0.5. Three nullclines
are located as shown in Figure 6.
Figure 6. Nullclines (𝑥−𝑟𝑒𝑑,𝑥− 𝑔𝑟𝑒𝑒𝑛, 𝑥−𝑏𝑙𝑢𝑒)
There are three critical points 𝑝, 𝑝
and 𝑝: (0.4885; 0.0342; 0.2023),
(0.4890; 0.1297; 0.2022),
(0.4935; 0.9999; 0.2013). Linearization around
these points provides us with the characteristic
numbers 𝜆 given in Table 2.
5BCMF5IFDIBSBDUFSJTUJDOVNCFST
-
λ
1
λ
2
λ
3
p
1
-0.5026 0.1522-1.9782
𝑖
0.1522+1.9782
𝑖
p
2
0.6965 0.1511-1.9799
𝑖
0.1511+1.9799
𝑖
p
3
-0.9998 0.1518-1.9743
𝑖
0.1518+1.9743
𝑖
Figure 7. Periodic solutions
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2022.17.12
Inna Samuilik, Felix Sadyrbaev, Diana Ogorelova
E-ISSN: 2224-2856
103
Volume 17, 2022
Inna Samuilik, Felix Sadyrbaev
, Diana Ogorelova
Figure 8. The graphs of 𝑥(𝑡),𝑖=1,2,3
2.2 Gompertz function
Case 1. Stable limit cycles can exist in systems of
the form (4). Consider the system (4) with the
same matrix (7) and the same parameters 𝜇,𝑣 and
𝜃. Three nullclines are located as shown in Figure
9. Computations and graphical results are
performed using Wolfram Mathematica.
Figure 9. Nullclines (𝑥−𝑟𝑒𝑑,𝑥− 𝑔𝑟𝑒𝑒𝑛, 𝑥−𝑏𝑙𝑢𝑒)
There is one critical point 𝑝:
(0.4894; 0.5672; 0.9996). Linearization around
this point provides us with the characteristic
numbers 𝜆 given in Table 3.
5BCMF5IFDIBSBDUFSJTUJDOVNCFST
-
λ
1
λ
2
λ
3
p
1
-0.9999 9.346-4.9497
𝑖
9.346+4.9497
𝑖
Figure 10. Periodic solutions
Figure 11. The graphs of 𝑥(𝑡),𝑖=1,2,3
Case 2. Consider the system (4) with the same
matrix (8) and the same parameters 𝜇,𝑣 and 𝜃.
Figure 12. Nullclines (𝑥−𝑟𝑒𝑑,𝑥− 𝑔𝑟𝑒𝑒𝑛, 𝑥−𝑏𝑙𝑢𝑒)
There are three critical points 𝑝,𝑝 and 𝑝:
(0.4109; 0;0.2652),
(0.4124; 0.3169; 0.2647) and
(0.4156; 0.9999; 0.2637). Linearization around
these points provides us with the characteristic
numbers 𝜆 given in Table 4.
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2022.17.12
Inna Samuilik, Felix Sadyrbaev, Diana Ogorelova
E-ISSN: 2224-2856
104
Volume 17, 2022
Inna Samuilik, Felix Sadyrbaev
, Diana Ogorelova
Table 4. The characteristic numbers λ
- λ
1
λ
2
λ
3
p
1
−
1
3.2096-5.75024
𝑖
3.2096+5.75024
𝑖
p
2
12.3148 3.2286-5.73206
𝑖
3.2286+5.73206
𝑖
p
3
14.0087 3.2278-5.80828
𝑖
3.2278+5.80828
𝑖
Figure 13. Periodic solutions
Figure 14. The graphs of 𝑥(𝑡),𝑖=1,2,3
3 More on 3D systems
We will show now some differences when
applying both functions. Consider system (3) with
the following set of parameters: 𝑣=1, 𝜇=10,
𝜃=1.5,𝜃=−0.5,𝜃=0.32 and the
regulatory matrix is
𝑊 = 1 2 0
−2 1 0
0 0 1.(9)
This system is uncoupled. The 2-dimensional
system with the matrix
𝑊 =1 2
−2 1(10)
is known to have the periodic solution. The 1-
dimensional equation
𝑥′=
()−𝑥 has the nullcline which is
a union of two points. These points are seen in the
Figure 15.
Figure 15. Logistic function (blue) and Gompertz
function (red) for 𝜃=0.32.
The third nullcline for 3D-system with the logistic
function consist of two planes of the form 𝑥=𝑝
and 𝑥=𝑝, where 𝑝, are roots of the equation
𝑥=
(). They are two, one of them
corresponds to the tangent point of the blue graph
with the graph of the bisectrix. The 3D system has
two periodic solutions that locate in the planes
𝑥=𝑝 and 𝑥=𝑝. Only the second one is
attractive. Consider the 3D-system (4), where the
Gompertz function is used in a model, and all the
parameters are the same as above. Since the red
curve has three points of intersection with the
bisectrix, this system has three periodic solutions
in the planes 𝑥=𝑞,𝑖=1,2,3, where the points
𝑞, 𝑞, 𝑞 are roots of the equation𝑥=
𝑒(). The first and the third periodic
solutions are attractive. Do the same for both 3D-
systems where only the parameter 𝜃is set for
0.15. The result is seen in Figure 16.
Figure 16. Logistic function (blue) and Gompertz
function (red) for 𝜃=0.15.
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2022.17.12
Inna Samuilik, Felix Sadyrbaev, Diana Ogorelova
E-ISSN: 2224-2856
105
Volume 17, 2022
Inna Samuilik, Felix Sadyrbaev
, Diana Ogorelova
Now the first 3D system with the logistic function
has one periodic solution, which is attractive. The
second 3D system with Gompertz function has
two periodic solutions corresponding to two roots
of the equation 𝑥=𝑒(), 𝜃=0.15.
We conclude that for the set of parameters listed
above, where only 𝜃 varies, the following is true.
For 0.15 <𝜃< 0.32 the first 3D system (3) has
one periodic solution which is attractive. The
second 3D-system (4) has three periodic
solutions. Of them two periodic solutions are
stable. The structure of phase spaces for the two
systems is completely different despite of the fact
that they use the same set of parameters.
Figure 17. Logistic function, the attractive
solution, 𝜃=0.24
Figure 18. Gompertz function,
three periodic solutions, 𝜃=0.24
4 Conclusions
The Gompertz function resembles a logistic
function, both sigmoidal functions have a lot in
common, but also a lot of differences. In the
Gompertz function, growth deceleration does not
occur as fast as its acceleration. Both functions are
activation functions. In this paper we approach
mathematical models of genetic networks. The
same set of ordinary differential equations appear
in models of telecommunication networks and
neuronal networks. The systems of ODE are
quasi-linear and nonlinearities are represented by
sigmoidal functions. To which extent the results
obtained can coincide and/or differ if different
sigmoidal functions are used? We got partial
answer to this question. We have studied two
systems with the same sets of parameters (and
they are many). The only difference was that in
the first system the logistic function was used,
while in the second one it was substituted by
Gompertz function. Both functions are quite
similar, but the second one uses double exponent
and this makes it rapidly going to limits.
Nevertheless, if both systems had a single critical
point of the non-attractive nature, as expected,
both had a stable periodic solution. For a different
regulatory matrix (8) both systems had three
critical points of non-attractive nature. Both
systems had periodic solutions and behavior of
solutions tending to this periodic one, is quite
similar (Fig. 5 and Fig. 11). The section 3 is
devoted to differences that can occur when
applying both functions to the same model. The
uncoupled systems were considered with the
specific third nullcline, which could be in the
form of one or three 𝑥-planes. Only one
parameter,𝜃, was allowed to vary. On a relatively
long interval, 𝜃 ∊ (0.15,0.32), significant
differencies between two systems were observed.
The reason is that the third nullcline is defined by
the equation of the form 𝑥=𝑓(𝑥), where 𝑓 can
be logistic or Gompertz function. For 𝜃 in the
above interval these equations have different
number of roots. This leads to substantial
differencies in the strucrture of attractors in both
systems. Stable periodic solutions serve as
attractors in both systems, but their number (and
location) is different. This may lead to
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2022.17.12
Inna Samuilik, Felix Sadyrbaev, Diana Ogorelova
E-ISSN: 2224-2856
106
Volume 17, 2022
Inna Samuilik, Felix Sadyrbaev, Diana Ogorelova
misunderstanding in interpretation of behavior of
modeled networks.
Acknowledgements:
ESF Project No. 8.2.2.0/20/I/003 ”Strengthening
of Professional Competence of Daugavpils
University Academic Personnel of Strategic
Specialization Branches 3rd Call”
References
[1]. A.Das, A.B.Roy, Pritha Das. Chaos in a three
dimensional neural network. Applied Mathematical
Modelling, 24(2000), 511-522.
[2]. A.Das, A.B.Roy, Pritha Das. (2002) Chaos in a Three
Dimensional General Model of NeuralNetwork.
International Journal of Bifurcation and Chaos, 12, 2271-
2281. http://dx.doi.org/10.1142/S0218127402005820
[3] F. Sadyrbaev, I.Samuilik, V.Sengileyev, "On Modelling
of Genetic Regulatory Networks," WSEAS Transactions on
Electronics, vol. 12, pp. 73-80, 2021
[4] Le-Zhi Wang, Ri-Qi Su, Zi-Gang Huang,XiaoWang,
WenXuWang,CelsoGrebogiandYingChengLai,Ageometrica
lapproachto controland control abilityofnonlinear
dynamicalnetworks.Nature Communications, Volume7,
Article number:11323(2016),
DOI:10.1038/ncomms11323
[5] H.D.Jong. Modeling and Simulation of
GeneticRegulatorySystems: A Literature Review
, J.
Comput Biol. 2002;9(1):67-103, DOI:
10.1089/10665270252833208
[6] Y. Koizumi et al. Adaptive Virtual NetworkTopology
Control Based on Attractor Selection.Journal of
Lightwave Technology(ISSN
: 0733-8724), Vol.28
(06/2010), Issue 11, pp.1720-
1731DOI:10.1109/JLT.2010.2048412
[7] S. Atslega, F. Sadyrbaev, I. Samuilik. On Modelling Of
Complex Networks. Engineering for Rural
Development(ISSN 1691-5976), 2021, pp. 10091014.
[8] F. Sadyrbaev, I. Samuilik. Mathematical Modelling of
Genetic Regulatory Networks. In: 2nd International Baku
Conference on Scientific Research: The Book of Full Texts.
Vol.1, Azerbaijan, Baku, 28-30 April, 2021. Baku: IKSAD
GLOBAL Publications, 2021, pp.463-469. ISBN 978-605-
70554-6-0.
[9] F. Sadyrbaev, I. Samuilik. On the hierarchy of attractors
in dynamical models of complex networks.19
Intern.Confer.Numer.Analys.andAppl.Mathematics,
Rhodes, Greece, 20-26 September 2021, To appear in AIP
Conference Proceedings. https:
//aip.scitation.org/journal/apc
[10]. N.Vijesh, S. Kumar Chakrabarti, J. Sreekumar.
Modeling of gene regulatory networks: A review. J.
Biomedical Science and Engineering, 2013, 6, 223-231.
http://dx.doi.org/10.4236/jbise.2013.62A027
[11] I. Samuilik, F. Sadyrbaev, “Mathematical Modelling
of Leukemia Treatment”, WSEAS Transactions on
Computers, 20:274-281. DOI: 10.37394/23205.2021.20.30
[12] E. Ott, (2002). Chaos in Dynamical Systems (2nd ed.).
Cambridge: Cambridge University Press.
doi:10.1017/CBO9780511803260
[13] I. Samuilik, F. Sadyrbaev, “Modelling Three
Dimensional Gene Regulatory Networks”, WSEAS
Transactions on Systems and Control 16:755-763. DOI:
10.37394/23203.2021.16.67
[14] J. C. Sprott. Elegant Chaos Algebraically Simple
Chaotic Flows. World Scientific Publishing Company,
2010, 302 pages. https://doi.org/10.1142/7183
[15] S. Mukherjee, S. K. Palit, D. K. Bhattacharya. Is one
dimensional Poincar map sufficient to describe the chaotic
dynamics of a three dimensional system?. Applied
Mathematics and Computation, Volume 219, Issue 23,
2013, Pages 11056-11064, ISSN 0096-3003.
https://doi.org/10.1016/j.amc.2013.04.043
Contribution of individual authors to
the creation of a scientific article (ghost
writing policy)
Allauthorshavecontributedequallyto creation
nofthisarticle.
Sources of funding for research
presented in a scientific article or
scientific article itself
Creative Commons Attribution License
4.0
(Attribution 4.0 International, CC BY
4.0)
This article is published under the terms of the
CreativeCommons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.
en_US
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2022.17.12
Inna Samuilik, Felix Sadyrbaev, Diana Ogorelova
E-ISSN: 2224-2856
107
Volume 17, 2022
ESF Project No. 8.2.2.0/20/I/003 ”Strengthening of
Professional Competence of Daugavpils University
Academic Personnel of Strategic Specialization
Branches 3rd Call”
