Differential System Approach to Gene Regulatory Network Modeling

Abstract

The system of ordinary differential equations that arises in models of genetic networks is considered. This system has both nonlinear and linear parts. As a nonlinear part, we use a piecewise linear function. This allows us to treat systems with multiple (up to ten) equations. Special attention is paid to the four-dimensional systems that have an attractor in the form of a periodic solution. In the final part, the 10th-dimensional system is considered, which describes a subnetwork of a larger network that arises in a practical biomedical problem.

1. Introduction

Ordinary differential equations play a significant role in the mathematical modeling of real-world events. There are multiple examples of the decisive impact of mathematical study on the process of elaborating an optimal solution in real deals [1,2,3]. On the other hand, the proper objects for modeling systems of ODE used are gene networks. More on this subject can be found in numerous publications [4,5,6,7]. In this article, we will focus on systems that model GRN, using ODE of the form

$$\left\{\begin{array}{c}{x}_1^{\prime }=f_1\left({\mu }_1(w_{11}{x}_1+\dots +w_{1n}{x}_n-{\theta }_1)\right)-v_1{x}_1,\hfill \\ x_2^{\prime }=f_2\left({\mu }_2(w_{21}{x}_1+\dots +w_{2n}{x}_n-{\theta }_2)\right)-v_2{x}_2,\hfill \\ \dots \hfill \\ x_n^{\prime }=f_n\left({\mu }_n(w_{n1}{x}_1+\dots +w_{nn}{x}_n-{\theta }_n)\right)-v_n{x}_n.\hfill \end{array}\right.$$

(1)

This is the n-dimensional version of the system, which originates from the Wilson–Cowan system

$$\left\{\begin{array}{c}{\displaystyle x_1^{\prime }=\frac{1}{1+e^{-{\mu }_1(w_{11}{x}_1+w_{12}{x}_2-{\theta }_1)}}-v_1{x}_1,}\hfill \\ {\displaystyle x_2^{\prime }=\frac{1}{1+e^{-{\mu }_2(w_{21}{x}_1+w_{22}{x}_2-{\theta }_2)}}-v_2{x}_2.}\hfill \end{array}\right.$$

(2)

The right side contains two terms, a nonlinear one and a linear term. Without the nonlinear term, the system possesses only exponentially decreasing solutions, and it is not interesting (but physically explainable: there is no evolution without interactions). This system was used to describe the dynamics of a neuronal system consisting of an activating population of neurons $x_i$ and an inhibitory population [8,9]. Later, it was extensively used to reveal the possible behaviors of complex networks, such as genetic regulatory networks and telecommunication ones [10,11]. The function $f\left(z\right)=1/(1+e^{-\mu z})$ in the system (2) is sigmoidal. Such functions are used in models of biological networks [12,13] due to the fact that realistic biological systems cannot achieve very large values. The linear part in systems (1) and (2) means that in the absence of a nonlinear part the functions $x_i\left(t\right)$ go to zero as the time t increases. The nonlinear part means the interrelation between elements of a network. Biological networks may be large, and consequently systems of the form (1) are then also large. Even the two-dimensional system (2) has ten parameters. They are difficult to study theoretically and difficult to study numerically. To simplify the system, piece-wise sigmoidal functions may be used. The main objective is to replace the logistic sigmoidal function with the piece-wise linear approximation. We have shown through examples that the structure of the phase space is preserved using PWL sigmoid. So PWL sigmoids preserve qualitative dynamics. Replacing logistics with PWL makes the system analyzable. The idea of using piece-wise linear functions when studying genetic networks was considered, for instance in [14]. Modern GRN theory still uses this idea. From a computational viewpoint, piece-wise linear sigmoids eliminate the stiffness and exponential nonlinearities associated with logistic functions. This leads to faster simulations, improved numerical stability, and better scalability to large networks. For GRNs with dozens or hundreds of genes, the reduction in computational cost is substantial.

Figure 1 shows the logistic sigmoidal function $g\left(z\right)=1/(1+e^{-\mu z})$ ($\mu =4$) together with the piece-wise linear function

$$f\left(u\right)=\left\{\begin{array}{cc}0,\hfill & \hfill u\le -0.5;\\ u+0.5,\hfill & \hfill -0.5\le u\le 0.5;\\ 1,\hfill & \hfill u\ge 0.5.\end{array}\right.$$

(3)

It is assumed that large dimensional systems of the form (1) are easier to study numerically if the sigmoidal function is of the piece-wise linear form. Some comments on this matter are given in the conclusions section.

This idea needs to be confirmed. For confirmation and for further discussion, we compare the results obtained for both cases.

So, the structure of the paper is as follows. In Section 2, we recall the main facts about the systems of the form (1). We recall the meaning of multiple parameters in the system (1). In Section 3, we pay attention to some results mostly previously known about systems (1). In Section 4, we consider two-dimensional systems with piece-wise linear sigmoidal functions and compare the results to those obtained for GRN-systems. In Section 5, we consider the three-dimensional case. In Section 6, the four-dimensional system is considered. Section 7 is devoted to an example of a system that is not easy to treat numerically if the system uses a logistic sigmoidal function, but it is treatable for the piece-wise-linear case.

2. Main Facts About System (1)

1. The vector field at the border of the set $Q_n=\{x\in R^n:0\le x\le 1/v\}$ is directed inward. So, any trajectory is trapped by $Q_n$;

2. The nullclines of the system (1) are defined by the relations

$$\left\{\begin{array}{c}0=f_1\left({\mu }_1(w_{11}{x}_1+\dots +w_{1n}{x}_n-{\theta }_1)\right)-v_1{x}_1,\hfill \\ 0=f_2\left({\mu }_2(w_{21}{x}_1+\dots +w_{2n}{x}_n-{\theta }_2)\right)-v_2{x}_2,\hfill \\ \dots \hfill \\ 0=f_n\left({\mu }_n(w_{n1}{x}_1+\dots +w_{nn}{x}_n-{\theta }_n)\right)-v_n{x}_n.\hfill \end{array}\right.$$

(4)

The cross-points of the nullclines are the critical points (equilibria) of the system (1). They can be found by solving the system (4). This system has at least one solution. The existence of equilibria follows from general topological arguments (fixed-point theorem on a topological ball). Calculations of equilibria are to be made by standard tools making use of the fact that all equilibria are in the invariant set. All solutions of the system (1) are in the region $Q_n$. Since it continuously maps the topological ball $Q_n$ into itself,

3. The number of critical points is finite: it does not exceed nine when n = 2 or twenty-seven when $n=3$, and increases similarly for higher values of n;

4. The attracting sets for the system (1) always exist and are located in $Q_n.$ They can be in the form of stable critical points, stable periodic solutions (limit cycles) and chaotic formations in $Q_n$;

5. The behavior of solutions of the system (1) is heavily dependent on the complex of parameters, $w_{ij},$ that form the so-called regulatory matrix

$$W=\left(\begin{array}{cccc}{w}_{11}& w_{12}& \dots & w_{1n}\\ w_{21}& w_{22}& \dots & w_{2n}\\ \dots & & & \\ w_{n1}& w_{n2}& \dots & w_{nn}\end{array}\right).$$

(5)

The information on these properties of the system (1), as well as discussions and visualizations of particular cases, can be found in [15,16,17] and references therein.

6. The standard analysis of critical points (the Jacobian analysis) can be done using the linearization. We show the main steps of the analysis on the example of a three-dimensional system [18]. Consider the system

$$\left\{\begin{array}{c}{\displaystyle x_1^{\prime }=\frac{1}{1+e^{-{\mu }_1(w_{11}{x}_1+w_{12}{x}_2+w_{13}{x}_3-{\theta }_1)}}-v_1{x}_1,}\hfill \\ \\ {\displaystyle x_2^{\prime }=\frac{1}{1+e^{-{\mu }_2(w_{21}{x}_1+w_{22}{x}_2+w_{23}{x}_3-{\theta }_2)}}-v_2{x}_2,}\hfill \\ \\ {\displaystyle x_3^{\prime }=\frac{1}{1+e^{-{\mu }_3(w_{31}{x}_1+w_{32}{x}_2+w_{33}{x}_3-{\theta }_3)}}-v_3{x}_3.}\hfill \end{array}\right.$$

(6)

The linearized system for any critical point $(x_1^{*},x_2^{*},x_3^{*})$

$$\left\{\begin{array}{c}{\displaystyle u_1^{\prime }=-v_1{u}_1+{\mu }_1{w}_{11}{g}_1{u}_1+{\mu }_1{w}_{12}{g}_1{u}_2+{\mu }_1{w}_{13}{g}_1{u}_3,}\hfill \\ {\displaystyle u_2^{\prime }=-v_2{u}_2+{\mu }_2{w}_{21}{g}_2{u}_1+{\mu }_2{w}_{22}{g}_2{u}_2+{\mu }_2{w}_{23}{g}_2{u}_3,}\hfill \\ {\displaystyle u_3^{\prime }=-v_3{u}_3+{\mu }_3{w}_{31}{g}_3{u}_1+{\mu }_3{w}_{32}{g}_3{u}_2+{\mu }_3{w}_{33}{g}_3{u}_3,}\hfill \end{array}\right.$$

where

$$g_1={\displaystyle \frac{e^{-{\mu }_1(w_{11}{x}_1^{*}+w_{12}{x}_2^{*}+w_{13}{x}_3^{*}-{\theta }_1)}}{{[1+e^{-{\mu }_1(w_{11}{x}_1^{*}+w_{12}{x}_2^{*}+w_{13}{x}_3^{*}-{\theta }_1)}]}^2}},$$

(7)

$$g_2={\displaystyle \frac{e^{-{\mu }_2(w_{21}{x}_1^{*}+w_{22}{x}_2^{*}+w_{23}{x}_3^{*}-{\theta }_2)}}{{[1+e^{-{\mu }_2(w_{21}{x}_1^{*}+w_{22}{x}_2^{*}+w_{23}{x}_3^{*}-{\theta }_2)}]}^2}},$$

(8)

$$g_3={\displaystyle \frac{e^{-{\mu }_3(w_{31}{x}_1^{*}+w_{32}{x}_2^{*}+w_{33}{x}_3^{*}-{\theta }_3)}}{{[1+e^{-{\mu }_3(w_{31}{x}_1^{*}+w_{32}{x}_2^{*}+w_{33}{x}_3^{*}-{\theta }_3)}]}^2}}.$$

(9)

One has

$$A-\lambda I=\left|\begin{array}{ccc}{\mu }_1{w}_{11}{g}_1-v_1-\lambda & {\mu }_1{w}_{12}{g}_1& {\mu }_1{w}_{13}{g}_1\\ {\mu }_2{w}_{21}{g}_2& {\mu }_2{w}_{22}{g}_2-v_2-\lambda & {\mu }_2{w}_{23}{g}_2\\ {\mu }_3{w}_{31}{g}_3& {\mu }_3{w}_{32}{g}_3& {\mu }_3{w}_{33}{g}_3-v_3-\lambda \end{array}\right|$$

and the characteristic equation is

$$\det |A-\lambda I|=-{\lambda }^3+{\lambda }^2(-v_1-v_2-v_3+{\mu }_1{w}_{11}{g}_1+{\mu }_2{w}_{22}{g}_2+{\mu }_3{w}_{33}{g}_3)+\lambda (g_1{v}_3{\mu }_1{w}_{11}+$$

$$+{\mu }_2{w}_{22}{g}_2{v}_3+g_1{g}_2{w}_{21}{\mu }_1{\mu }_2{w}_{12}-g_1{g}_2{w}_{11}{w}_{22}{\mu }_1{\mu }_2+g_1{g}_3{w}_{31}{w}_{13}{\mu }_1{\mu }_3-$$

$$-g_1{g}_3{w}_{11}{w}_{33}{\mu }_1{\mu }_3+g_2{g}_3{w}_{32}{w}_{23}{\mu }_2{\mu }_3-g_2{g}_3{w}_{22}{w}_{33}{\mu }_2{\mu }_3-v_1(v_2+v_3-g_2{w}_{22}{\mu }_2-g_3{w}_{33}{\mu }_3)+$$

$$+v_2(-v_3+g_1{w}_{11}{\mu }_1+g_3{w}_{33}{\mu }_3))+v_1(v_2(-v_3+g_3{w}_{33}{\mu }_3)+g_2{\mu }_2(v_3{w}_{22}+g_3{w}_{32}{w}_{23}{\mu }_3-$$

$$g_3{w}_{22}{w}_{33}{\mu }_3\left)\right)+g_1{\mu }_3(v_2(v_3{w}_{11}+g_3(w_{31}{w}_{13}-w_{11}{w}_{33}){\mu }_3)+g_2{\mu }_2(v_3(w_{21}{w}_{12}-w_{11}{w}_{22})+$$

$$+g_3(-w_{31}{w}_{22}{w}_{13}+w_{21}{w}_{32}{w}_{13}+w_{31}{w}_{12}{w}_{23}-w_{11}{w}_{32}{w}_{23}-w_{21}{w}_{12}{w}_{33}+w_{11}{w}_{22}{w}_{33}){\mu }_3\left)\right)=0.$$

Biologically, ${\mu }_i$ controls how sharply gene i responds to its regulators; $w_{ij}$ is the strength and sign of regulation of gene i by gene j (for instance, $w_{ij}>0$ means activation, $w_{ij}<0$ means repression); $v_i$ is the degradation rate of gene product i; and ${\theta }_i$ is the activation threshold for gene i. Biologically, thresholds encode sensitivity. Mathematically, ${\mu }_i$ controls the slope of the sigmoid and $w_{ij}$ define the coupling matrix. They determine the fixed-point structure, stability (via Jacobian), existence of oscillations (e.g., negative feedback loops), and possibility of chaos (in 3D systems with strong nonlinear feedback); $v_i>0$ is the decay rate for $x_I\left(t\right)$. Without the nonlinearity $f_i$ the function $x_i\left(t\right)$ tends to zero; ${\theta }_i$ shifts the sigmoid horizontally. Changing ${\theta }_i$ moves the switching hyperplane, which affects the number of equilibria, their stability, location of attractors, and transitions between regimes. Thresholds are crucial for control over networks. Biologically ${\mu }_i>1$ encodes cooperative binding and sharp regulatory thresholds. This is for better cooperativity. Mathematically, ${\mu }_i>1$ creates nonlinearity strong enough for diverse behaviors, such as switching, bistability, and oscillations. $w_{ij}$ and ${\theta }_i$ have no restrictions; $v_i$ are positive, representing total decay without cooperation between elements of a network.

3. Two-Dimensional Systems: The Case of Logistic Sigmoidal Function

There are three main modes of behavior of solutions to the system

$$\left\{\begin{array}{c}{\displaystyle x_1^{\prime }=\frac{1}{1+e^{-{\mu }_1(w_{11}{x}_1+w_{12}{x}_2-{\theta }_1)}}-v_1{x}_1,}\hfill \\ {\displaystyle x_2^{\prime }=\frac{1}{1+e^{-{\mu }_2(w_{21}{x}_1+w_{22}{x}_2-{\theta }_2)}}-v_2{x}_2,}\hfill \end{array}\right.$$

(10)

namely activation, inhibition, and mixed case [19,20]. The respective vector fields are depicted in Figure 2, Figure 3 and Figure 4, where the parameters ${\mu }_i=4,{\theta }_1=0.5(w_{11}+w_{12}),{\theta }_2=0.5(w_{21}+w_{22})$, $v_1=v_2=1.$ They are characterized by regulatory matrices:

$$W=\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right),W=\left(\begin{array}{cc}1& -1\\ -1& 1\end{array}\right),W=\left(\begin{array}{cc}2& 1\\ -1& 2\end{array}\right).$$

(11)

For some specific choice of the matrix W, the system has a periodic solution. A stable limit cycle is demonstrated in Figure 4. If we reverse the time, this limit cycle becomes an unstable limit cycle for the symmetric system.

$$\left\{\begin{array}{c}{\displaystyle x_1^{\prime }=-\frac{1}{1+e^{-{\mu }_1(w_{11}{x}_1+w_{12}{x}_2-{\theta }_1)}}+v_1{x}_1,}\hfill \\ {\displaystyle x_2^{\prime }=-\frac{1}{1+e^{-{\mu }_2(w_{21}{x}_1+w_{22}{x}_2-{\theta }_2)}}+v_2{x}_2.}\hfill \end{array}\right.$$

(12)

The vector field for this system is shown in Figure 5.

4. Two-Dimensional Systems: The Case of Piece-Wise Linear Sigmoidal Function

First, we consider two-dimensional (2D) systems of ordinary differential equations (ODE) of the form

$$\left\{\begin{array}{c}{\displaystyle x_1^{\prime }=f(w_{11}{x}_1+w_{12}{x}_2-{\theta }_1)-x_1,}\hfill \\ {\displaystyle x_2^{\prime }=f(w_{21}{x}_1+w_{22}{x}_2-{\theta }_2)-x_2,}\hfill \end{array}\right.$$

(13)

where $f\left(u\right)$ is defined in (3). The nullclines are solutions of the system

$$\left\{\begin{array}{c}{\displaystyle 0=f(w_{11}{x}_1+w_{12}{x}_2-{\theta }_1)-x_1,}\hfill \\ {\displaystyle 0=f(w_{21}{x}_1+w_{22}{x}_2-{\theta }_2)-x_2.}\hfill \end{array}\right.$$

(14)

The nullclines are depicted in Figure 6.

Periodic Solutions in 2D Systems

For some specific choice of the matrix W, the system (13) has a periodic solution. This is depicted in Figure 7.

Let the matrix W be $\left\{\right(0,-2),(1,2\left)\right\}.$

The system

$$\left\{\begin{array}{c}{\displaystyle x_1^{\prime }=-f(w_{11}{x}_1+w_{12}{x}_2-{\theta }_1)+x_1,}\hfill \\ {\displaystyle x_2^{\prime }=-f(w_{21}{x}_1+w_{22}{x}_2-{\theta }_2)+x_2}\hfill \end{array}\right.$$

(15)

has the same set of parameters as the system (13). In Figure 8, we see the periodic solution of the system (15). The purpose of reversing time is to make the stable attractor in system (13) an unstable one in system (15). Then, both systems form the 4D system, where stable meets unstable. In the sequel, we are interested in the interaction of these systems under different conditions.

5. Three-Dimensional Systems

It is more difficult to find periodic solutions (represented by closed trajectories) considering three-dimensional systems of ODE in general and systems of form (1) in particular. Examples of periodic solutions for systems of the form

$$\left\{\begin{array}{c}{\displaystyle x_1^{\prime }=\frac{1}{1+e^{-{\mu }_1(k{x}_1-x_3-{\theta }_1)}}-x_1,}\hfill \\ \\ {\displaystyle x_2^{\prime }=\frac{1}{1+e^{-{\mu }_2(-x_1+k{x}_2-{\theta }_2)}}-x_2,}\hfill \\ \\ {\displaystyle x_3^{\prime }=\frac{1}{1+e^{-{\mu }_3(-x_2+k{x}_3-{\theta }_3)}}-x_3,}\hfill \end{array}\right.$$

(16)

were presented in reference [21]. Consider the system with piece-wise sigmoidal functions

$$\left\{\begin{array}{c}{\displaystyle x_1^{\prime }=f_1(w_{11}{x}_1+w_{12}{x}_2+w_{13}{x}_3-{\theta }_1)-x_1,}\hfill \\ {\displaystyle x_2^{\prime }=f_2(w_{21}{x}_1+w_{22}{x}_2+w_{23}{x}_3-{\theta }_2)-x_2,}\hfill \\ {\displaystyle x_3^{\prime }=f_3(w_{31}{x}_1+w_{32}{x}_2+w_{33}{x}_3-{\theta }_3)-x_3.}\hfill \end{array}\right.$$

(17)

Let the matrix W be

$$W=\left(\begin{array}{ccc}-2& 1& -2\\ -1& 2& 1\\ 0& -2& 2\end{array}\right).$$

(18)

The respective system (17) has a periodic solution. The trajectory is depicted in Figure 9. Consider the system (6) with marix (18), where ${\mu }_1=4, {\mu }_2=4, {\mu }_3=4$, $v_1=v_2=v_3=1$ and

$${\theta }_1={\displaystyle \frac{w_{11}+w_{12}+w_{13}}{2}},$$

$${\theta }_2={\displaystyle \frac{w_{21}+w_{22}+w_{23}}{2}},$$

$${\theta }_3={\displaystyle \frac{w_{31}+w_{32}+w_{33}}{2}}.$$

Linearization around critical point $(0.5,0.5,0.5)$ provides us with the characteristic numbers ${\lambda }_1=3; {\lambda }_2=1-1.73205i; {\lambda }_3=1+1.73205i$. The type of critical point $(0.5,0.5,0.5)$ is a saddle-focus. The trajectories of system (6) and (17) are depicted in Figure 9. Wolfram Mathematica was used for computations and visualizations.

We provide an example of a system that has an equilibrium, transforming to the limit cycle. We model the same system, changing the sigmoidal function with the piece-wise linear approximation. The fixed points, and the transforming limit cycle, are preserving the limit cycle and are preserving in a piece-wise sigmoid approach.

Another example concerns the three-dimensional system

$$\left\{\begin{array}{c}{\displaystyle x_1^{\prime }=-f_1(w_{11}{x}_1+w_{12}{x}_2+w_{13}{x}_3-{\theta }_1)+x_1,}\hfill \\ {\displaystyle x_2^{\prime }=-f_2(w_{21}{x}_1+w_{22}{x}_2+w_{23}{x}_3-{\theta }_2)+x_2,}\hfill \\ {\displaystyle x_3^{\prime }=-f_3(w_{31}{x}_1+w_{32}{x}_2+w_{33}{x}_3-{\theta }_3)+x_3,}\hfill \end{array}\right.$$

(19)

where $f\left(z\right)$ is a piece-wise linear sigmoidal function defined in (3). This system defines the vector field that is directed to the exterior of the set $Q_3.$ So the existence of an attracting set in $Q_3$ is not supposed.

The matrix W is

$$W=\left(\begin{array}{ccc}2& 2& -2\\ -2& 2& 1\\ -2& 2& 2\end{array}\right).$$

(20)

This system has a periodic solution. The respective trajectory is depicted in Figure 10.

6. Four-Dimensional Systems

Consider the four-dimensional system of the form

$$\left\{\begin{array}{c}{\displaystyle x_1^{\prime }=f_1(w_{11}{x}_1+w_{12}{x}_2+w_{13}{x}_3+w_{14}{x}_4-{\theta }_1)-x_1,}\hfill \\ {\displaystyle x_2^{\prime }=f_2(w_{21}{x}_1+w_{22}{x}_2+w_{23}{x}_3+w_{24}{x}_4-{\theta }_2)-x_2,}\hfill \\ {\displaystyle x_3^{\prime }=f_3(w_{31}{x}_1+w_{32}{x}_2+w_{33}{x}_3+w_{34}{x}_4-{\theta }_3)-x_3}\hfill \\ {\displaystyle x_4^{\prime }=f_4(w_{41}{x}_1+w_{42}{x}_2+w_{43}{x}_3+w_{44}{x}_4-{\theta }_4)-x_4,}\hfill \end{array}\right.$$

(21)

with the regulatory matrix

$$W=\left(\begin{array}{cccc}2& 1& 0& 2\\ -1& 2& -0.4& 0\\ 0& 0.5& 2& 1\\ -2.8& 0& -1& 2\end{array}\right).$$

(22)

Set the initial conditions to $(0.4,0.6,0.4,0.6).$ The respective solution tends towards the limit cycle. The projection onto the $(x_1,x_3,x_4)$-subspace is depicted in Figure 11.

Next, consider the system

$$\left\{\begin{array}{c}{\displaystyle x_1^{\prime }=f_1(w_{11}{x}_1+w_{12}{x}_2+w_{13}{x}_3+w_{14}{x}_4-{\theta }_1)-v_1{x}_1,}\hfill \\ {\displaystyle x_2^{\prime }=f_2(w_{21}{x}_1+w_{22}{x}_2+w_{23}{x}_3+w_{24}{x}_4-{\theta }_2)-v_2{x}_2,}\hfill \\ {\displaystyle x_3^{\prime }=-f_3(w_{31}{x}_1+w_{32}{x}_2+w_{33}{x}_3+w_{34}{x}_4-{\theta }_3)+v_3{x}_3,}\hfill \\ {\displaystyle x_4^{\prime }=-f_4(w_{41}{x}_1+w_{42}{x}_2+w_{43}{x}_3+w_{44}{x}_4-{\theta }_4)+v_4{x}_4.}\hfill \end{array}\right.$$

(23)

This models the coexistence of a stable 2D limit cycle defined by the first two equations in (23) with the unstable one defined by the remaining two equations.

When combining both systems into a single 4-dimensional one, we cannot be certain of the existence of a periodic solution. Nevertheless, it (the periodic solution) can be found numerically. For the matrix

$$W=\left(\begin{array}{cccc}2& 1& 0& -2\\ -1& 2& 3& 0\\ 0& 0.7& 2& 1\\ 0.5& 0& -1& 2\end{array}\right)$$

(24)

it can be found numerically. The $(x_1,x_3,x_4)$-projections of two solutions starting at $(0.4,0.6,0.4,0.6)$ and $(0.3,0,0.3,0.2)$, respectively, and tending to the periodic solution are shown in Figure 12 below.

These matrices were chosen on the basis of the experience gained from conducting numerous computational and analytical experiments. For example, 2-dimensional matrices with a certain structure $\{+,-,+,+\}$ provide rotation of a vector field around the critical point, and a symmetrical structure $\{+,+,-,+\}$ changes the direction of rotation to the opposite. Four-dimensional matrices were constructed using two-dimensional blocks with a specified direction of rotation. These blocks are arranged along the main diagonal, while two remaining blocks are zero-filled. The resulting system is uncoupled. To make it coupled, suitable non-zero elements were added. The short answer is that a combination of numerical experiments with the results of analytical analysis, together with some intuitional suggestions, is behind the choice of the regulatory matrices. It can be predicted that the role of the third component will increase along with the dimension of a network and the corresponding matrix. The matrices (18), (20) were selected at random, scanning several examples. The matrices (22) and (24) were chosen as block-diagonal matrices, providing the desired behavior of the corresponding 2D systems, and then supplemented with nonzero off-diagonal elements, making the system coupled.

7. Example

For example, we take the model obtained by [16,22], which treats leukemia as dictating the trajectory to a wrong attractor. The model contains 60 genes, and the respective system, where the sigmoidal function is Hill’s function, consists of 60 equations. We put $+1$ for activation and $-1$ for inhibition. As an exception, we use $-2$ in the last row to test the inhibition in the 10th element. We consider the subsystem shown in Figure 13, which contains 10 elements; accordingly, the system consists of 10 equations

$$\left\{\begin{array}{c}{\displaystyle x_1^{\prime }=f_1(w_{11}{x}_1+\dots +w_{1,10}{x}_{10}-{\theta }_1)-v_1{x}_1,}\hfill \\ \dots \hfill \\ {\displaystyle x_{10}^{\prime }=f_{10}(w_{10,1}{x}_1+\dots +w_{10,10}{x}_{10}-{\theta }_{10})-v_{10}{x}_{10},}\hfill \end{array}\right.$$

(25)

where the parameters are

$${\theta }_i={\displaystyle \frac{{\sum }_{j=1}^{10}{w}_{ij}}{2}}.$$

The sigmoidal function is the piece-wise-linear one. Now we have to choose the regulatory matrix for this system. It is a matrix $10\times 10$ that corresponds to the chosen subsystem. Blue squares indicate inhibition, and red squares represent activation.

The resulting matrix $10\times 10$ is

$$W=\left(\begin{array}{cccccccccc}-1& 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& -1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& -1& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& -2& 1& -2\end{array}\right).$$

(26)

The use of piece-wise sigmoidal functions makes it possible to calculate the solutions. Ten solutions of system (25) with matrix (26) were obtained, which evidently tends towards a constant limit. Full analysis of the 60th dimensional system remains challenging. The graphs of these solutions are shown in Figure 14.

Several experiments were carried out, and the corresponding results are presented for different initial conditions.

Experiment 1 was performed with the initial data $(0.10,0.14,0.18,0.22,0.26,0.30,0.34,0.38,0.42,0.46)$ and the limits were defined as $(0.46,0.5,0.5,0.5,0.5,0.5,0.34,0.5,0.42,0.47)$.

In the following experiment, the initial point is $(0.0,0.9,0.0,0.9,0.26,0.30,0.34,0.38,0.42,0.46)$, while the limits remained unchanged, i.e., $(0.46,0.5,0.5,0.5,0.5,0.5,0.34,0.5,0.42,0.47)$.

In the next experiment, the initial data set is $(0.8,0.7,0.6,0.4,0.6,0.8,0.8,0.8,0.8,0.8)$, with the limits defined as $(0.65,0.5,0.5,0.5,0.5,0.5,0.8,0.5,0.8,0.6)$.

Finally, in the last experiment, the initial data set is $(0.6,0.5,0.6,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.6,0.5)$, with the limits defined as $(0.55,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.34,0.5,0.42,0.47)$. In the experiments presented, different limiting values are observed, which suggests the existence of multiple stable states.

8. Conclusions

Periodic solutions other than the constant ones are important attractors in 2D genetic systems. They can be used to construct attractors in 3D systems and to build 4D attractors in 4D systems. By perturbation of the 4D regulatory matrices, multiple 4D systems can be obtained, exhibiting interesting properties. The multi-fragmented attractors can be constructed for 4D systems with the property that, in the system with perturbed regulatory matrix solutions, tends towards a fragment of the 4D attractor. The choice of the fragment depends on the initial values of a solution [23].

In dynamic models of GRN, the logistic sigmoidal functions are often used, which can also be represented as hyperbolic tangent functions. For systems of high dimensions, which are actual [16], the numerical study becomes difficult due to computational problems. To make numerical research easier, exponential functions may be excluded from the models. The piece-wise linear sigmoidal function provides a suitable approximation to the logistic one. Models using piece-wise linear sigmoidal functions show the existence of periodic solutions in 2D, 3D and 4D systems for suitably selected matrices $W.$ Moreover, the existence of periodic solutions can be obtained for systems that involve equations in the standard form and part of an equation with reversed sign on the right side.

In turn, the systems obtained in this paper can be used to construct systems of higher dimensions. However, the main problem, as usual, is to interpret the behavior obtained in real biological networks. Future research may further explore systems that combine equations in the standard form with equations whose right sides have reversed signs, in order to better understand the emergence of periodic solutions in such mixed dynamical systems. An important direction for future research is the extension of the proposed methodology to the analysis of systems with more than ten equations, which may provide a pathway toward understanding the global dynamical organization of biologically realistic gene regulatory networks.