Bifurcation and Global Dynamics of Continuous and Discrete Competitive Models for Genetic Toggle Switches

Abstract

We investigate the asymptotic behavior of a proposed ordinary differential equation (ODE) model for Genetic Toggle switches from Gardner et. al. and I. Rajapakse and S. Smale: ${\displaystyle \frac{dx}{dt}}={\displaystyle \frac{a}{1+y^m}}-x$ and ${\displaystyle \frac{dy}{dt}}={\displaystyle \frac{b}{1+x^n}}-y$ where $a,b,m,n>0$ and $x\left(t\right),y\left(t\right)\ge 0$. We also investigate the asymptotic behavior of the Euler discretization of this system: $x_{n+1}=a_1{x}_n+{\displaystyle \frac{b_1}{1+y_n^m}}=f(x_n,y_n)$ and $y_{n+1}=a_2{y}_n+{\displaystyle \frac{b_2}{1+x_n^n}}=g(x_n,y_n),$ where $1-h=a_1$, $1-k=a_2$, $ah=b_1$ and $bk=b_2$, $a_1,a_2\in (0,1)$ and $h,k>0$ are steps of discretizations. Here, x and y represent protein concentrations at a particular time in both genes and $a,b,m,n>0$, respectively, above. We will apply the theory of competitive maps to find the basins of attractions of different equilibrium points and period-two solutions of systems of difference equations.

1. Introduction

The human genome contains 20,000 genes and there are at least 200 different cell types that make up the human body. Each cell type has the same genome and all share a common ancestor, namely the single zygote cell (fertilized egg) created at the time of conception. Currently, it is thought that only a subset of genes are active and it is the particular subset of active genes that determines the cell type. Thus, cell differentiation during development involves a process whereby certain genes are inactivated, while others are activated. One possible explanation for the process of differentiation proposed in [1,2,3] is based on bifurcation theory of differential equations [4]. In [1,3], the authors identify differentiation with bifurcations in the genetic regulatory network, which takes the form

$${\displaystyle \frac{d{x}_i}{dt}}=H_i\left(x_{i-1}\right)-{\alpha }_i{x}_i,i=1,2,\dots ,n{x}_0=x_n,$$

(1)

where $x_i$ is the expression level of the i-th gene, the $H_i$ are Hill-type functions, and the ${\alpha }_i$ are the degradation rate constants; ${\alpha }_i$ represents the natural degradation of protein in cells. Natural degradation is important for maintaining and regulating protein levels. When a gene is active, protein will be continuously produced. So, there must be a corresponding natural degradation to maintaining levels of expression for a particular gene. Note that even with naturally degrading protein and repression, it is still possible for a gene to produce its protein product but the binding of cellular machinery to produce the product is very unlikely, resulting in repression but not zero concentration. Also, in this model, we will be assuming for simplicity that the degradation of protein is constant. This equation describes the genetic regulation in a single cell type and hence n represents the number of genes in the genome; see [1]. In [1,2,3], the authors discussed in detail only the cases of two and three gene networks, as bifurcation and stability analysis becomes mathematically very complicated for more than three genes networks.

Knowing how a cell works is an important foundation to the fundamentals of biology. The DNA of a cell is considered to be like the blueprint of the cell and its regulation is, in part, what controls the cell’s functions [5,6,7,8]. Gene regulation and its dynamics are often conceptualized by considering them as being similar to electrical circuits. These are called biological circuits or specifically in this case a genetic circuit [5,7]. One of the important interactions of gene regulation is considering a genetic toggle switch circuit.

Gene toggle switches are a system where, for example, Gene A is expressed to create Protein A, which we can call Repressor A. Repressor A will inhibit the cell’s machinery to bind to Promoter B and thus inhibiting Gene B’s transcription and translation and therefore the eventual expression of Protein B. However, if Protein B is also a repressor, which we can call Repressor B, then the same interaction can occur vice versa, leading to the inhibition of the creation of Protein A. In other words, one gene will inhibit the other; see [9]. This creates a toggling effect similar to a light switch, depending on inducers to activate promoter regions of Promoter A or Promoter B.

For the purposes of this investigation we will consider a simple two-gene genetic toggle switch with respect to one gene producing a protein that can act as a repressor to another gene and a second gene producing a protein that can act as a repressor to the first. We will assume, for simplicity, the concentration of each repressor protein is proportional to the activity of the corresponding gene producing that protein, such that when there is a high concentration of the first repressor protein, then the first gene is active and if there is low concentration of the first repressor protein then the first gene has low activity or is fully repressed (depending on if the concentration is below the threshold of the proposed activation of said gene for interpretations). It is understood that mRNA levels and protein concentrations can differ when it comes to considering gene expression and the activity of a gene [10]. It is also acknowledged that gene expression and regulation can be controlled in many different ways, for example at the steps of transcription and or translation [6,9]. We will later consider the “environmental factors” that ultimately influence regulation as lumped parameters that describe the total net effects of gene expression as a whole, similar to the T.S. Gardner et al.’s genetic toggle switch system proposed in [11] and analyzed briefly in I. Rajapakse and S. Smale’s [1,3]. Our goal is to investigate the biological applications of mathematics results to understand the basis of select foundations of gene regulation dynamics.

In the next sections we will mathematically analyze equations that can describe gene expression through the repressor protein concentration of two competing genes. We will use a differential equation model from [1,3] and its Euler discretization, which will be the so-called competitive system; see [12,13]. We will apply the theory developed in [12,13,14,15,16,17], primarily in [13,15,16], to the derived discrete system to obtain global dynamics of all attractors and describe the basins of attraction of all attracting sets. It is worth mentioning that there are no results that describe the structure of the basins of attractions for competitive systems of differential equations. We offer Conjectures 1 and 2 with such a description. Then we will give biological interpretations of our results. The obtained discrete system is of interest in a theoretical sense as it is the first competitive system which possesses coexisting attracting equilibrium solutions and a large number of period-two solutions, when m and n are large. Our visualisations indicate that the symmetry of the coefficients, that is $a=b,m=n$ in the ODE case and $a_1=a_2,b_1=b_2,p=q$ in $\Delta E$ case, implies the symmetry of the basins of attractions.

2. Genetic Toggle Switch: Continuous ODE Model

The general form of the system of differential equations that can be used as a model for genetic toggle switch with two genes is given by

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dx}{dt}}& =& {\displaystyle F\left(y\right)-ax}\hfill \\ \hfill {\displaystyle \frac{dy}{dt}}& =& {\displaystyle G\left(x\right)-by,}\hfill \end{array}$$

(2)

where $F\left(y\right)$ and $G\left(x\right)$ are Hill-type functions, defined by

$$H_1\left(x\right)={\displaystyle \frac{\beta x^p}{K_1^p+x^p}},H_2\left(x\right)={\displaystyle \frac{\gamma }{1+{\left(\frac{x}{K_2}\right)}^q}},$$

(3)

where $K_1,K_2,p,q>0,$ and $\beta ,\gamma >0$; see [1]. Here, x represents the concentration of a transcription factor in a cell, H represents activation, and G represents repression [1]. The function $H_1\left(x\right)$ is also known as the Beverton–Holt function or Holling function of the first order and it is used in population dynamics and fisheries; see [18].

An equilibrium point $(\overline{x},\overline{y})$ of system (2) satisfies $F\left(y\right)=ax,G\left(x\right)=by$ or $ax=F\left(G\right(x)/b)$ and $by=G\left(F\right(y)/a)$. Such equations can be easily solved numerically if all parameters are constant.

The Jacobian matrix J at an equilibrium point $E(\overline{x},\overline{y})$ is

$$J\left(E\right)=\left[\begin{array}{cc}-a& F^{\prime }\left(\overline{y}\right)\\ G^{\prime }\left(\overline{x}\right)& -b\end{array}\right],$$

and the characteristic equation of $J\left(E\right)$ is

$$(\lambda +a)(\lambda +b)-F^{\prime }\left(\overline{y}\right)G^{\prime }\left(\overline{x}\right)=0,$$

with the eigenvalues

$${\lambda }_{1,2}=-{\displaystyle \frac{1}{2}}\left(a+b\pm \sqrt{{(a-b)}^2+4F^{\prime }\left(\overline{y}\right)G^{\prime }\left(\overline{x}\right)}\right).$$

Thus, both eigenvalues are real when $F^{\prime }\left(\overline{y}\right)G^{\prime }\left(\overline{x}\right)>0$, which will happen when F and G are repressors and also when ${(a-b)}^2+4F^{\prime }\left(\overline{y}\right)G^{\prime }\left(\overline{x}\right)\ge 0$. Indeed, when F and G are repressors they have negative derivatives. In these cases, system (2) can not exhibit the Hopf bifurcation [4,19]. In view of Bendixson’s theorem (see [20]), system (2) can not have any periodic solutions as its divergence is $div\left(F\right(y)-ax,G(x)-by)=-a-b<0$ in the whole plane. As shown in [1,3], system (2) can exhibit the pitchfork bifurcation for some specific choice of Hill’s functions F and G. The pitchfork bifurcation happens for the value of parameters when det $J\left(E\right)=ab-F^{\prime }\left(\overline{y}\right)G^{\prime }\left(\overline{x}\right)=0$. The bifurcation parameters in [1,3] are taken to be $m,n$ and then the equation $ab-F^{\prime }\left(\overline{y}\right)G^{\prime }\left(\overline{x}\right)=0$ was solved for a specific equation (5). In addition, an equilibrium $E(\overline{x},\overline{y})$ is locally stable if $ab>F^{\prime }\left(\overline{y}\right)G^{\prime }\left(\overline{x}\right)$; see [20].

System (2) is a cooperative (respectively competitive) system of differential equations if both functions $F\left(y\right)$ and $G\left(x\right)$ are non-decreasing (respectively non-increasing), with simple behavior if the functions F and G are bounded. In this case, we have the following result:

Theorem 1.

Consider system (2) subject to the conditions $a,b>0$ and assume that the functions $F\left(y\right),G\left(x\right)$ are differentiable and bounded such that $0\le F\left(y\right)\le M_1,0\le G\left(x\right)\le M_2$. If both functions $F\left(y\right)$ and $G\left(x\right)$ are non-decreasing or non-increasing, then every solution of system (2) is bounded and converges to an equilibrium.

Proof of Theorem 1.

The proof will follow from Theorem 3 in [21] or Theorem 3.4.1 in [22] if we show that every solution of the system (2) is bounded. Rewriting the first equation of (2) as $dx/dt+ax=F\left(y\right)$, multiplying with $e^{at}$, integrating from 0 to t, and solving for $x\left(t\right)$, we obtain

$$x\left(t\right)=x\left(0\right)e^{-at}+e^{-at}{\int }_0^t{e}^{as}F\left(y\left(s\right)\right)ds.$$

Similarly we can rewrite the second equation of system (2) as an integral equation and thus get the following equivalent system of integral equations:

$$\begin{array}{c}\hfill {\displaystyle x\left(t\right)=x\left(0\right)e^{-at}+e^{-at}{\int }_0^t{e}^{as}F\left(y\left(s\right)\right)ds}\\ \hfill {\displaystyle y\left(t\right)=y\left(0\right)e^{-bt}+e^{-bt}{\int }_0^t{e}^{bs}G\left(x\left(s\right)\right)ds.}\end{array}$$

(4)

The equations of system (4) imply

$$\begin{array}{c}\hfill {\displaystyle 0\le x\left(t\right)\le x\left(0\right)e^{-at}+e^{-at}{\int }_0^t{e}^{as}{M}_{1}ds\le x\left(0\right)+\frac{M_1}{a}}\\ \hfill {\displaystyle 0\le y\left(t\right)\le y\left(0\right)e^{-bt}+e^{-bt}{\int }_0^t{e}^{bs}{M}_{2}ds\le y\left(0\right)+\frac{M_2}{b}.}\end{array}$$

Now, an application of Theorem 3 in [21] will complete the proof. □

Remark 1.

Both Hill’s functions are positive and bounded as $0\le H_1\left(x\right)\le \beta$ and $0\le H_2\left(x\right)\le \gamma$. So, Theorem 1 applies when both functions F and G are either $H_1\left(x\right)$ or $H_2\left(y\right)$ and every solution of system (2) converges to an equilibrium. Then the main problem now is to find the basins of attractions of different equilibrium points in the case where the system has several equilibrium points.

Remark 2.

Part of Theorem 1 about the boundedness of solutions can be extended to the more general system (1). In this case we have the following results, for which proof is analogous to the proof of Theorem 1.

Corollary 1.

Consider system (1) subject to the conditions ${\alpha }_i>0$ and $0\le H_i\left(u\right)\le M_i,i=1,2,\dots ,n$. Then every solution of system (1) is bounded.

Remark 3.

Theorem 1 is a very specific result which holds for two-dimensional cooperative or competitive systems of differential equations. In the case of higher dimensional systems, such a result is not true and dynamical behavior in such cases could be very different, as was mentioned in [1]. Even for a three-dimensional system of differential equations (1), some locally stable periodic solutions are possible as so called Hopf bifurcation might happen; see [1,4].

Now, we consider the setting from [11] which was later analyzed in [1], consisting of two genes described as a system of differential equations

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dx}{dt}}& =& {\displaystyle \frac{a}{1+y^m}-x}\hfill \\ \hfill {\displaystyle \frac{dy}{dt}}& =& {\displaystyle \frac{b}{1+x^n}-y,}\hfill \end{array}$$

(5)

where $a,b,m,n>0$ and $x\left(t\right),y\left(t\right)\ge 0$.

First we will consider the example in [1], where $a=b=2$, resulting in the following system:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dx}{dt}}& =& {\displaystyle \frac{2}{1+y^m}-x}\hfill \\ \hfill {\displaystyle \frac{dy}{dt}}& =& {\displaystyle \frac{2}{1+x^n}-y,}\hfill \end{array}$$

(6)

which has a simple equilibrium solution $E_1(1,1)$.

The Jacobian, $J_2$ for system (6) at any equilibrium solution $E(x,y)$ is:

$$J_2\left(E\right)=\left[\begin{array}{cc}-1& -{\displaystyle \frac{2m{y}^{m-1}}{{\left(y^m+1\right)}^2}}\\ -{\displaystyle \frac{2n{x}^{n-1}}{{\left(x^n+1\right)}^2}}& -1\end{array}\right].$$

The characteristic polynomial of $J_2\left(E\right)$ at the equilibrium E is given by:

$$char\left(J_2\right)={\lambda }^2+2\lambda +1-{\displaystyle \frac{4mn{y}^{m-1}{x}^{n-1}}{{(1+y^m)}^2{(1+x^n)}^2}}$$

and the eigenvalues are

$${\lambda }_{1,2}=-1\pm {\displaystyle \frac{2\sqrt{mn}{x}^{\frac{n-1}{2}}{y}^{\frac{m-1}{2}}}{(1+y^m)(1+x^n)}}.$$

In particular, at the equilibrium point $E_1(1,1)$ we have ${\lambda }_{1,2}=-1\pm {\displaystyle \frac{\sqrt{mn}}{2}}$, showing that this equilibrium can be either locally asymptotically stable, saddle point, or non-hyperbolic depending on m and n.

Now, let us consider a general case from system (5). Let $E_{+}$ be the equilibrium point of system (5) where $a,b>0$ and $J_{ab}$ are the Jacobian of a linearization of system (5) given as:

$$J_{ab}=\left[\begin{array}{cc}-1& -{\displaystyle \frac{am{y}^{m-1}}{{\left(y^m+1\right)}^2}}\\ -{\displaystyle \frac{bn{x}^{n-1}}{{\left(x^n+1\right)}^2}}& -1\end{array}\right].$$

The characteristic polynomial at the equilibrium $E_{+}$ is given by:

$${\lambda }^2+2\lambda +1-{\displaystyle \frac{abmn{y}^{m-1}{x}^{n-1}}{{(1+y^m)}^2{(1+x^n)}^2}}$$

and the eigenvalues are

$${\lambda }_{1,2}=-1\pm {\displaystyle \frac{\sqrt{abmn}{x}^{\frac{n-1}{2}}{y}^{\frac{m-1}{2}}}{(1+y^m)(1+x^n)}}.$$

Thus, Hopf bifurcation cannot happen.

We can continue to do additional analysis on $J_{ab}\left(E_{+}\right)$ using a Poincaré stability diagram [20]. We have $Tr\left(J_{ab}{|}_{E_{+}}\right)=-2$ and $det\left(J_{ab}{|}_{E_{+}}\right)=1-{\displaystyle \frac{abmn{y}^{m-1}{x}^{n-1}}{{(1+y^m)}^2{(1+x^n)}^2}}$.

The following are possible equilibrium classifications with respect to their local stability character:

• Spiral Sink is the region determined by equations:

  $$1-{\displaystyle \frac{abmn{y}^{m-1}{x}^{n-1}}{{(1+y^m)}^2{(1+x^n)}^2}})>1\mathrm{and}abmn{y}^{m-1}{x}^{n-1}<0,$$

  which implies that a or b must be negative. This case is not feasible.
• Degenerative Sink

  $$1-{\displaystyle \frac{abmn{y}^{m-1}{x}^{n-1}}{{(1+y^m)}^2{(1+x^n)}^2}})=1\mathrm{and}abmn{y}^{m-1}{x}^{n-1}=0.$$
  Then x or y, or both, are 0 which is not feasible.
• Sink

  $$1>{\displaystyle \frac{abmn{y}^{m-1}{x}^{n-1}}{{(1+y^m)}^2{(1+x^n)}^2}})>0\mathrm{or}{(1+y^m)}^2{(1+x^n)}^2>abmn{y}^{m-1}{x}^{n-1}.$$
  This region is given as

  $$(1+y^m)(1+x^n)y^{{\displaystyle \frac{1-m}{2}}}{x}^{{\displaystyle \frac{1-n}{2}}}>\sqrt{abmn}.$$
• Line of Stable Fixed points

  $$1={\displaystyle \frac{abmn{y}^{m-1}{x}^{n-1}}{{(1+y^m)}^2{(1+x^n)}^2}}\mathrm{or}{(1+y^m)}^2{(1+x^n)}^2=abmn{y}^{m-1}{x}^{n-1}.$$
• Saddle point

  $${\displaystyle \frac{abmn{y}^{m-1}{x}^{n-1}}{{(1+y^m)}^2{(1+x^n)}^2}}>1\mathrm{or}{(1+y^m)}^2{(1+x^n)}^2<abmn{y}^{m-1}{x}^{n-1}.$$
  This region is given as

  $$(1+y^m)(1+x^n)y^{{\displaystyle \frac{1-m}{2}}}{x}^{{\displaystyle \frac{1-n}{2}}}<\sqrt{abmn}.$$

We conclude this section with two conjectures on the form and structure of the basins of attractions of different equilibrium solutions. As we commented in Remark 1 the main problem for two-dimensional cooperative or competitive systems is determining the basins of attractions of all equilibrium solutions and, in particular, of those which are locally stable. We will state our conjectures for the general two-dimensional competitive systems.

Our simulations will clearly illustrate the results in these conjectures.

Conjecture 1.

Consider the competitive systems of differential equations

$${\displaystyle \frac{dx}{dt}}=f(x,y),{\displaystyle \frac{dy}{dt}}=g(x,y),$$

(7)

where ${\displaystyle \frac{\partial f}{\partial y}}\le 0,{\displaystyle \frac{\partial g}{\partial x}}\le 0$. Assume that all solutions are bounded and so converge to the equilibrium solutions. Assume that system (7) has three equilibrium points in south-east ordering such that $E_1⪯E_2⪯E_3$ and that $E_1$ and $E_3$ are locally asymptotically stable and $E_2$ is a saddle point. Then there exists the global stable manifold $W^s\left(E_2\right)$, which is a graph of a continuous and non-decreasing function passing through $E_2$, which separates the first quadrant into two regions $W^{+}\left(E_2\right)=\{(x,y):(x,y)⪯(p,q)\}$ and $W^{-}\left(E_2\right)=\{(x,y):(p,q)⪯(x,y)\}$ for some point $(p,q)\in W^s\left(E_2\right)$.

Conjecture 2.

Consider the competitive systems of differential equations (7) subject to the conditions of Conjecture 1. Then, the set $W^{+}\left(E_2\right)$ is an immediate basin of attraction of $E_1$ while the set $W^{-}\left(E_2\right)$ is an immediate basin of attraction of $E_3$. In addition, there exists the global unstable manifold $W^u\left(E_2\right)$, which is a graph of a continuous and non-increasing function passing through all three points $E_1,E_2,E_3$. All solutions in both basins of attractions $W^{+}\left(E_1\right)$ and $W^{-}\left(E_3\right)$ are approaching the global unstable manifold $W^u\left(E_2\right)$ when they are approaching their respective attractors. Thus, $W^u\left(E_2\right)$ is a carrying simplex of this system. The complete basins of attractions are the union of preimages of immediate basins of attraction. Here the immediate basin of attraction is a part of the full basin of attraction that contains the equilibrium; see [14].

Conjectures 1 and 2 are formulated for the cooperative system of differential equations in [23].

3. Genetic Toggle Switch: Discrete $\Delta \mathit{E}$ Model

Euler discretization of system (2) gives the following system of difference equations:

$$\begin{array}{ccc}\hfill x_{n+1}& =& {\displaystyle a_1{x}_n+b_{1}F\left(y_n\right)}\hfill \\ \hfill y_{n+1}& =& {\displaystyle a_2{y}_n+b_{2}G\left(x_n\right),n=0,1,\dots ,}\hfill \end{array}$$

(8)

where $a_1=1-ah,b_1=h,a_2=1-bk,b_2=k$, and where $h,k>0$ are the steps of discretization which satisfies $ah<1,bk<1$. The conditions on the steps sizes h and k seem to be natural and give the limitations of the size of steps. Here they are also essential in order to preserve the competitive character of the discretized map. If at least one of these conditions is not satisfied the corresponding map is not competitive and so its dynamics might be quite different.

If both $F\left(y\right)$ and $G\left(x\right)$ are non-decreasing functions, such as Hill’s function of type $H_1$, system (8) is cooperative, while if both functions $F\left(y\right)$ and $G\left(x\right)$ are non-increasing functions, such as Hill’s function of type $H_2$, system (8) is competitive. In this paper we will focus on the competitive case and leave the cooperative case for future research. There is an extensive theory developed for these systems (see [12,13,15]), where it was shown that the global dynamics in the hyperbolic case and some non-hyperbolic cases =can be derived from the local dynamics of the equilibrium points and the period-two points. In addition, we have obtained precise information about the basins of attraction of the equilibrium points and the period-two solutions.

The theory for the case when the functions $F\left(y\right)$ and $G\left(x\right)$ are monotonic functions of different monotonic types, such as Hill’s function of different types, is not well developed at this time and it may be a topic of future research.

The map that corresponds to system (8) is

$$T(x,y)=\left[\begin{array}{c}{a}_{1}x+b_{1}F\left(y\right)\\ b_{2}G\left(x\right)+a_{2}y\end{array}\right].$$

(9)

The Jacobian matrix of this map at the equilibrium point $E(\overline{x},\overline{y})$ is

$$J\left(E\right)=\left[\begin{array}{cc}{a}_1& b_1{F}^{\prime }\left(\overline{y}\right)\\ b_2{G}^{\prime }\left(\overline{x}\right)& a_2\end{array}\right],$$

and the characteristic polynomial is

$${\lambda }^2-(a_1+a_2)\lambda +a_1{a}_2-b_1{b}_2{G}^{\prime }\left(\overline{x}\right)F^{\prime }\left(\overline{y}\right).$$

The eigenvalues of $J\left(E\right)$ are

$${\lambda }_{1,2}={\displaystyle \frac{1}{2}}\left(a_1+a_2\pm \sqrt{{(a_1-a_2)}^2-4b_1{b}_2{G}^{\prime }\left(\overline{x}\right)F^{\prime }\left(\overline{y}\right)}\right).$$

Remark 4.

We can assume that $a_1<1,a_2<1$ in system (8). On the one hand, this follows from our discretization. On the other hand, if $a_1\ge 1$, then $x_n$ is a non-decreasing sequence and so it has a finite limit $\overline{x}$ or ∞. In either case the second equation in system (8) becomes asymptotically first-order linear equation $y_{n+1}=a_2{y}_n+b_{2}G\left({lim}_{n\to \infty }{x}_n\right)$ which is solvable and has $y_n$ which either goes to the finite limit $\overline{y}$ or to ∞. The similar analysis leads to an analogous result for the case if $a_2\ge 1$. See [18].

The local stability analysis of any equilibrium solution E of system (8) in the realistic case $a_1<1,a_2<1$, gives:

Lemma 1.

Let $E(\overline{x},\overline{y})$ be any equilibrium of a competitive system ($c>0$) (8) and assume that $a_1<1,a_2<1$.

The equilibrium solution E is

(1). 

Locally asymptotically stable if

$$c<(1-a_1)(1-a_2)andc<1-a_2;$$

(10)

(2). 

Locally a saddle point if $1-a_2<c\le 1+a_2$ or

$$(1+a_1)(1+a_2)>c,1+a_2<c,2(1+a_2)or(1-a_1)(1-a_2)<c\le 1-a_2;$$

(11)

(3). 

A non-hyperbolic point if

$$(1-a_1)(1-a_2)=c,a_2+c<1or1+a_2<c<2+2a_2,(1+a_1)(1+a_2)=c.$$

(12)

Proof. 

We will check Jury’s conditions from Theorem 2.12 in [24]. See also [25]. We have that $Tr\left(J\left(E\right)\right)=a_1+a_2,Det\left(J\left(E\right)\right)=a_1{a}_2-c$.

(1).

E is locally asymptotically stable if $\left|Tr\right(J\left(E\right)\left)\right|<1+Det\left(J\right(E\left)\right)<2$, which, after some straightforward and tedious calculations, leads to the conditions (10).

(2).

E is locally a saddle point if $\left|Tr\right(J\left(E\right)\left)\right|>|1+Det(J\left(E\right)\left)\right|$, which, after some straightforward calculations, leads to the condition (11).

(3).

E is locally a non-hyperbolic point if $\left|Tr\right(J\left(E\right)\left)\right|=|1+Det(J\left(E\right)\left)\right|$ or $Det\left(J\right(E\left)\right)$$=1,\left|Tr\right(J\left(E\right)\left)\right|\le 2$, which leads to the condition (12) in a straightforward matter.

□

Remark 5.

When system (8) is cooperative or competitive, the bifurcation parameter becomes $(1-a_1)(1-a_2)-c$, where $c=b_1{b}_2{F}^{\prime }\left(\overline{x}\right)G^{\prime }\left(\overline{y}\right)$. In this case, Naimark–Sacker bifurcation is not possible but pitchfork and period doubling bifurcations are feasible. If we calculate the bifurcation parameter $(1-a_1)(1-a_2)-c$, for the system (8) we obtain that $(1-a_1)(1-a_2)-c=hk(ab-F^{\prime }\left(\overline{x}\right)G^{\prime }\left(\overline{y}\right))$, which is exactly $hkdetJ\left(E\right)$ for the system of differential equations (2). So, two bifurcation parameters are closely related, showing that local stability happens for close values of parameters. For instance, both systems have a saddle point if $detJ\left(E\right)<0$ and locally stable equilibrium point if $detJ\left(E\right)>0$, where an additional condition $c<1-a_2$ should be satisfied in discrete cases. Since $Tr\left(J\right(E)=-a-b<0$ for the system of differential equations (2), the equilibrium point for such a system is stable, when $detJ\left(E\right)=ab-F^{\prime }\left(\overline{x}\right)G^{\prime }\left(\overline{y}\right)>0$. Certainly, there are some extra conditions for local dynamics for system (8), which might be the artefacts of discretizations.

In the sequel we consider the competitive variant of system (8) where we assume that both functions $G\left(x\right),F\left(y\right)$ are of $H_2$ types; that is, we consider the system

$$\begin{array}{ccc}\hfill x_{n+1}& =& {\displaystyle a_1{x}_n+\frac{b_1}{1+y_n^p}}\hfill \\ \hfill y_{n+1}& =& {\displaystyle \frac{b_2}{1+x_n^q}+a_2{y}_n,n=0,1,\dots ,}\hfill \end{array}$$

(13)

where $a_1,a_2\in (0,1)$ and $p,q>0$

So let $T_1(x,y)$ be a corresponding map given as the following:

$$T_1(x,y)=\left[\begin{array}{c}{a}_{1}x+{\displaystyle \frac{b_1}{1+y^p}}\\ {\displaystyle \frac{b_2}{1+x^q}}+a_{2}y\end{array}\right].$$

(14)

So the Jacobian is given as the following:

$$J_{T_1}(x,y)=\left[\begin{array}{cc}{a}_1& {\displaystyle \frac{-b_{1}p{y}^{p-1}}{{(1+y^p)}^2}}\\ {\displaystyle \frac{-b_{2}q{x}^{q-1}}{{(1+x^q)}^2}}& a_2\end{array}\right].$$

(15)

Finding the equilibrium solution leads to the following. Let $T(\overline{x},\overline{y})=(\overline{x},\overline{y})$ so that we have that

$$\begin{array}{ccccc}\hfill a_1\overline{x}+{\displaystyle \frac{b_1}{1+{\overline{y}}^p}}=\overline{x}& ⟺\hfill & \hfill (1-a_1)x(1+{\overline{y}}^p)=b_1& ⟺\hfill & \hfill 1+{\overline{y}}^p={\displaystyle \frac{b_1}{(1-a_1)x}}\\ \hfill a_2\overline{y}+{\displaystyle \frac{b_2}{1+{\overline{x}}^q}}=\overline{y}& ⟺\hfill & \hfill (1-a_2)\overline{y}(1+{\overline{x}}^q)=b_2& ⟺\hfill & \hfill 1+{\overline{x}}^q={\displaystyle \frac{b_2}{(1-a_2)\overline{y}}}.\end{array}$$

We have that $Tr\left[J_{T_1}(\overline{x},\overline{y})\right]=a_1+a_2$ and $Det\left[J_T(\overline{x},\overline{y})\right]=a_1{a}_2-c$ where $c={\displaystyle \frac{b_1{b}_{2}pq{\overline{x}}^{q-1}{\overline{y}}^{p-1}}{{(1+{\overline{x}}^q)}^2{(1+{\overline{y}}^p)}^2}}$.

Using the equilibrium equations we can simplify the expression for determinants as

$$Det\left[J_T(\overline{x},\overline{y})\right]=a_1{a}_2-{\displaystyle \frac{pq{\overline{x}}^{q-1}{\overline{y}}^{p-1}{\overline{x}}^2{\overline{y}}^2{(1-a_1)}^2{(1-a_2)}^2}{b_1{b}_2}}=a_1{a}_2-K{\overline{x}}^{q+1}{\overline{y}}^{p+1}$$

where $K={\displaystyle \frac{pq{(1-a_1)}^2{(1-a_2)}^2}{b_1{b}_2}}$.

Using Lemma 1, the equilibrium $E(\overline{x},\overline{y})$ is locally asymptotically stable if and only if $c<(1-a_1)(1-a_2)$, which, after some straightforward calculations, gives:

$$(b_1-(1-a_1)\overline{x})(b_2-(1-a_2)\overline{y})<{\displaystyle \frac{b_1{b}_2}{pq}}.$$

Lemma 2.

If $a_1,a_2\in (0,1)$ and the functions $G\left(x\right)$ and $F\left(y\right)$ are bounded then every solution of system (8) is bounded.

Proof of Lemma 2.

Indeed the first equation of (8) implies

$$x_{n+1}=a_1{x}_n+F\left(y\right)\le a_1{x}_n+B_1,$$

where $B_1$ is an upper bound for $F\left(y\right)$. By using a difference inequalities method (see [26]) we obtain that $x_n\le u_n$ where $u_n$ satisfies the difference equation

$$u_{n+1}=a_1{u}_n+B_1$$

of which the solution is

$$u_n=a_1^n{C}_1+{\displaystyle \frac{B_1}{1-a_1}}\le {\displaystyle \frac{B_1}{1-a_1}}+ϵ$$

since $a_1\in (0,1)$. Thus

$$x_n\le u_n\le {\displaystyle \frac{B_1}{1-a_1}}+ϵ=U_x.$$

(16)

In an analogous way, we can prove that $y_n\le v_n$, where $v_n$ satisfies the difference equation

$$v_{n+1}=a_2{v}_n+B_2,$$

where $B_2$ is an upper bound for the function $G\left(x\right)$. This implies

$$y_n\le v_n\le {\displaystyle \frac{B_2}{1-a_2}}+ϵ=U_y.$$

(17)

Now, using (17) in system (8) yields

$$x_{n+1}=a_1{x}_n+F\left(y_n\right)\ge a_1{x}_n+L_1$$

where $L_1$ is a lower bound for $F\left(y\right)$. By using difference inequalities, we obtain that

$$x_n\ge l_n$$

where $l_n$ satisfies the difference equation

$$l_{n+1}=a_1{l}_n+L_1$$

(18)

that is, $l_n=C_1{a}_1^n+{\displaystyle \frac{L_1}{1-a_1}}$. Thus, $x_n\ge l_n\ge {\displaystyle \frac{L_1}{1-a_1}}=L_x$.

In an analogous way, we can prove that $y_n\ge {\displaystyle \frac{L_2}{1-a_2}}=L_y$, where $L_2$ is a lower bound for function $G\left(x\right)$. □

Remark 6.

Lemma 2 about the boundedness of solutions can be extended to a more general system

$$x_{n+1}^{\left(i\right)}=a_i{x}_n^{\left(i\right)}+F_i\left(x_n^{(i-1)}\right),i=1,2,\dots ,n{x}_n^{\left(0\right)}=x_n^{\left(n\right)}.$$

(19)

In this case we have the following results, for which proof is analogous to the proof of Lemma 2.

Corollary 2.

Consider system (19) and $x_n^{\left(i\right)}$ represents the $i^{th}$ gene in the gene network, subject to the conditions $1>a_i>0$, $L_i\le F_i\left(x_n^{(i-1)}\right)\le B_i,i=1,2,\dots ,n$, and $x_n^{\left(0\right)}=x_n^{\left(n\right)}$. Then every solution of system is bounded.

Symmetric Case

Special symmetric case $a_1=a_2$, $b_1=b_2$, $p=q$ is interesting. Then the line $y=x$ is an invariant set and so, based on the theory developed in [13,15], is the stable manifold of the fixed point $(\overline{x},\overline{y})$. In this case, the fixed point $(\overline{x},\overline{y})$ of the map satisfies:

$$(1-a_1)x(1+y^p)=b_1,(1-a_1)y(1+x^p)=b_1,$$

that is

$$x(1+y^p)={\displaystyle \frac{b_1}{1-a_1}}=y(1+x^p)\Rightarrow {\displaystyle \frac{x}{1+x^p}}={\displaystyle \frac{y}{1+y^p}}.$$

One solution of this system is $x=y$ where x is a unique positive solution of

$$x^{p+1}+x-{\displaystyle \frac{b_1}{1-a_1}}=0.$$

Indeed, $f\left(x\right)=x^{p+1}+x$ is an increasing function for $x>0$ with $f\left(0\right)=0$ and ${lim}_{x\to \infty }f\left(x\right)=\infty$. So, $f\left(x\right)={\displaystyle \frac{b_1}{1-a_1}}$ in exactly one positive point $\overline{x}>0$.

Now the dynamics on the line $y=x$ are one-dimensional and governed by the difference equation:

$$t_{n+1}=a_1{t}_n+{\displaystyle \frac{b_1}{1+t_n^p}},n=0,1,\dots .$$

(20)

4. Competitive Two-Gene Network Biological Relevance

Our analysis of the genetic toggle switch model via the ODE proposed in [1,3] suggests that locally an equilibrium, i.e., homeostasis, can be achieved with constant conditions or parameters where gene expression is stable. More specifically, all solutions, as well as progression of gene expression over time, will converge to an equilibrium, implying that no matter the environmental factors (represented by the parameters) of the biologically relevant competitive gene toggle switch system, it will always achieve some sort of homeostasis. We can also describe the forward trajectory of the gene activity over time, typically attracting a carrying simplex of a saddle point equilibrium to move towards another attracting homeostatic state. Through manipulation of parameters, and changing the “environmental factors” of the net result of gene expression, the system can model different equilibrium states for gene expression via repression protein concentrations, using our stated assumptions above. Each of these homeostatic equilibria, with the exception of the degenerate attractor(s), can be achieved where both genes are expressed, even if one gene is expressed at very low levels. This suggests that, even if a promoter region is being inhibited, there still can be low, or in some cases very low, levels of gene expression below a particular threshold, which could be considered active in affecting the system which is effectively inhibited. This system can be used to represent symmetric toggle switches, or the parameters can be adjusted to model asymmetric toggle systems.

Now we discretize the system and consider the genetic toggle switch in view of the $\Delta E$ model. We found that there are different conditions to test where a homeostatic state occurs based on lumped environmental parameters and on our concentrations of repressor proteins. Similarly to the continuous case, we have shown that no matter the environmental factors, the competitive gene toggle switch will have a bounded homeostasis. This implies that for both the continuous and discrete models the genetic expression controlled by the toggle switch will not “explode” in expression or activity. This is biologically supported by, for example, restraints on energy consumption or material availability [9]. So, it is possible to have minimal and maximal concentrations, suggesting constraints to the regulated gene expression. Analysis of this model and its equilibrium points also suggests that the repressor protein concentrations at homeostasis will not be zero as previously stated; there will be low levels of protein expressed even if a gene is being inhibited. With this perspective of the model, the emergence of periodic or cyclic steady states can occur. These states can be attracting, repelling or provide a carrying simplex for the initial values of the system to an attracting equilibrium or equilibria for the system. It is feasible to assume it is possible that the genetic toggle switch, under certain constant parameters or conditions, may oscillate on its own or, if an initial condition is given to the system, the forward orbit is oscillatory but evolves to homeostasis which is not periodic. This also includes the prospect of the toggle switch being able to pass through or be repelled from cyclic phases to reach either a periodic or singular equilibrium. The experimental evidence of the existence of period-two solutions is still missing, so they could possibly be the artefacts of discretization. The biological implications of the period-two solutions suggest that a toggle switch has the ability to oscillate given particular environmental conditions. It is a possibility that the continuous system model is not all-encompassing on the phenomena that may occur within a genetic toggle switch that can naturally toggle being induced by the environment, given as lumped parameters as discussed. Many interactions within the genome are still being studied and understood within the view of biology. This perspective will be a new avenue to investigate to describe genetics, where the continuous models may not be as good for modeling natural oscillations as the discretized version will.

These analyses can provide insight into developing, designing and studying cell regulation within the context of gene regulation.

5. Simulations

Simulations for both ODE Genetic Toggle Switch (5) and the difference equation Genetic Toggle Switch (13) are shown below. For the ODE system in Figure 1, Figure 2, Figure 3 and Figure 4, we use NDSolve of Mathematica 13.2 and for various $x_{initial},y_{initial},a,b>0$ and $m,n$ are positive integers with the exception of Figure 4. Figure 5 details the time series of each system for Figure 1, Figure 2, Figure 3 and Figure 4. Then, for the difference equations system in Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15, we observed basins of attraction using Mathematica 13.2 package Dynamica from [24].

5.1. ODE Model

In this part we provide several examples of system (5) with more than one equilibrium solution and the corresponding basins of attractions, which will clearly illustrate Theorem 1 and Conjectures 1 and 2. Here we can consider symmetric $a=b,m=n$ vs. non-symmetric cases: $a\ne b$ and/or $m\ne n$. In many simulations we performed the solutions in cases symmetric with respect to the first quadrant bisector, while in the non-symmetric case they are not symmetric.

Figure 1. This figure illustrates Theorem 1 and Conjecture 1. The left plot shows the phase space and the right plot is a particular solution given the initial conditions. There are two locally stable equilibria and one saddle point and two basins of attractions of locally stable equilibrium points are separated by stable manifold of the middle saddle point equilibrium.

Figure 1. This figure illustrates Theorem 1 and Conjecture 1. The left plot shows the phase space and the right plot is a particular solution given the initial conditions. There are two locally stable equilibria and one saddle point and two basins of attractions of locally stable equilibrium points are separated by stable manifold of the middle saddle point equilibrium.

Figure 2. Similar situation as in Figure 1 with an extra equilibrium solution outside of feasibility region. The extra equilibrium solution is a saddle point. The picture is symmetric, as we expect, as parameters are symmetric. The other figures of the solutions of the differential equations are not symmetric, as the controlling parameters $a,b,m,n$ are not symmetric. Many other choices for the symmetric case $a=b,m=n$ result in symmetric solutions with respect to the first quadrant bisector.

Figure 2. Similar situation as in Figure 1 with an extra equilibrium solution outside of feasibility region. The extra equilibrium solution is a saddle point. The picture is symmetric, as we expect, as parameters are symmetric. The other figures of the solutions of the differential equations are not symmetric, as the controlling parameters $a,b,m,n$ are not symmetric. Many other choices for the symmetric case $a=b,m=n$ result in symmetric solutions with respect to the first quadrant bisector.

Figure 3. Similar situation as in Figure 1.

Figure 3. Similar situation as in Figure 1.

Figure 4. An example of system (5) with one equilibrium solution, which is globally asymptotically stable.

Figure 4. An example of system (5) with one equilibrium solution, which is globally asymptotically stable.

Figure 5. This illustrates the figures above where $x\left[t\right]$ and $y\left[t\right]$ are plotted along the time axis from the previous Figure 1, Figure 2, Figure 3 and Figure 4. The corresponding parameters are given within the figure.

Figure 5. This illustrates the figures above where $x\left[t\right]$ and $y\left[t\right]$ are plotted along the time axis from the previous Figure 1, Figure 2, Figure 3 and Figure 4. The corresponding parameters are given within the figure.

5.2. $\Delta E$ Model

In this part we provide the numerical and visual evidence for Theorems 2 and 3. We show different equilibrium solutions and period-two solutions together with their basins of attractions. More precisely, in every case we will see the immediate basin of attraction as well as parts of the preimages of the immediate basin of attraction that form a complete basin of attraction. The biologically relevant basins of attractions are those in the first quadrant and the pre-images in other qudrants are of mathematical interest. The structure of the complete basin of attraction for competitive map given in [13] could be very complex, yet the immediate basins of attraction are, in most cases, simple and belong to the first quadrant of the initial conditions $(x_0,y_0)$. Let us recall that two points in $\mathbf{x}=(x_1,x_2),\mathbf{y}=(y_1,y_2)\in {\mathbb{R}}^2$ are in south-east ordering $\mathbf{x}{⪯}_{SE}\mathbf{y}$, if $x_1\le x_2,y_1\ge y_2$. The structure of a basin of attraction for a non-competitive map is even more complicated and is given in several papers by J. A. Yorke and his collaborators; see [27,28,29,30,31]. In many cases the boundaries of the basins of attractions of non-competitive maps are fractals, which certainly is not the case for competitive maps, where the boundaries of the basins of attraction are continuous non-decreasing curves.

Remark 7.

When the middle equilibrium in the south-east ordering (SE ordering) is a repeller then it is observed that there exists period-two solutions and when it is a saddle point then there is no emergence of period-two solutions; see [15]. This shows that there is a relationship between the appearance of period-two solutions and the local stability analysis for the middle equilibrium point in SE ordering.

Theorem 2.

(a) 

Assume that system (13) has a single equilibrium point E. Then E is globally asymptotically stable.

(b) 

Assume that system (13) has $2k+1,k\ge 1$ equilibrium points in south-east ordering such that $E_1{⪯}_{SE}{E}_2{⪯}_{SE}{E}_3\dots {⪯}_{SE}{E}_{2k+1}$ and that $E_1,E_3,\dots ,E_{2k+1}$ are locally asymptotically stable and $E_2,E_4,\dots ,E_{2k}$ are either saddle points or repellers that belong to an invariant set C, which is the graph of a continuous non-decreasing function. Assume that system (13) has no period-two solutions. When $E_{2m}$ is a saddle point for any $2\le m\le k$, then there exists the global stable manifold $W^s\left(E_{2m}\right)$, which is a graph of a continuous and non-decreasing function passing through $E_{2m}$, which separates the first quadrant into two regions $W^{+}\left(E_{2m}\right)=\{(x,y):(x,y){⪯}_{SE}(p,q)\}$ and $W^{-}\left(E_{2m}\right)=\{(x,y):(p,q){⪯}_{SE}(x,y)\}$ for some point $(p,q)\in W^s\left(E_{2m}\right)$. The set $W^{+}\left(E_{2m}\right)$ is an immediate basin of attraction of $E_{2m-1}$ while the set $W^{-}\left(E_{2m}\right)$ is an immediate basin of attraction of $E_{2m+1}$. When $E_{2m}$ is a repeller, then the set C plays a role of $W^s\left(E_{2m}\right)$.

Proof. 

The statements about the global stable and unstable manifolds follows from Theorems 3.1 and 3.2 in [15]. The results describing the basins of attraction follow from the results in [13]. If the system (13) is symmetric then the global stable manifold mentioned in the result is a bisector of the first quadrant; see Remark 8. See Figure 6 and Figure 7. The special case $k=1$ is interesting and easy to comprehend. □

Remark 8.

In view of the symmetric nature of system (13), all Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 are perfectly symmetric for the symmetric choice of parameters. Figure 6 and Figure 13, Figure 14 and Figure 15 are asymmetric due to the asymmetric choice of parameters. Similar results for cooperative systems have been proved in [23]; however, their biological implications are different since, in the cooperative case, 0 is a locally asymptotically stable equilibrium which implies the presence of Allee’s effect.

Theorem 3.

(a) 

Assume that system (13) has three equilibrium points in south-east ordering such that $E_1{⪯}_{SE}{E}_2{⪯}_{SE}{E}_3$ and that $E_1$ and $E_3$ are locally asymptotically stable and $E_2$ is a repeller. Further, assume that there is a period-two solution $(p,T(p\left)\right)$, which is a repeller. Assume that system (13) has no period-four solutions. Then, there exists a continuous, non-decreasing curve C passing through $E_2,p,T\left(p\right)$, which separates the first quadrant into two sets, which are the immediate basins of attraction of $E_1$ and $E_3$. If the system is a symmetric set, C is a bisector of the first quadrant. See Figure 8.

(b) 

Assume that system (13) has three equilibrium points in south-east ordering such that $E_1{⪯}_{SE}{E}_2{⪯}_{SE}{E}_3$ and that $E_1$ and $E_3$ are locally asymptotically stable and $E_2$ is a repeller. Further, assume that there are three period-two solutions $(p_i,T\left(p_i\right))$ such that $p_1{⪯}_{SE}{p}_3{⪯}_{SE}{p}_2$ and that $(p_i,T\left(p_i\right)),i=1,3$ are saddle points and $(p_2,T\left(p_2\right))$ is locally stable. Assume that system (13) has no period-four solutions.

Then there exist the global stable manifolds $W^s\left(p_i\right)$, $W^s\left(T\left(p_i\right)\right),i=1,3$, which are the graphs of continuous and non-decreasing functions passing through $E_2$, which separates the first quadrant into three regions: $W^{+}\left(p_1\right)=\{(x,y):(x,y){⪯}_{SE}(p,q)\}$ and for some point $(p,q)\in W^s\left(p_1\right)\cup W^s\left(T\left(p_1\right)\right)$, $W^{-}\left(p_2\right)=\{(x,y):(p,q){⪯}_{SE}(x,y)\}$ for some point $(p,q)\in W^s\left(p_2\right)\cup W^s\left(T\left(p_2\right)\right)$ and the region between two stable manifolds, precisely the set between $W^s\left(p_1\right)\cup W^s\left(T\left(p_1\right)\right)$ and $W^s\left(p_2\right)\cup W^s\left(T\left(p_2\right)\right)$. The last set is the immediate basin of attraction of the period-two solution $(p_2,T\left(p_2\right))$. The sets $W^{+}\left(p_1\right)$ and $W^{-}\left(p_2\right)$ are the immediate basins of attraction of the equilibrium points $E_1$ and $E_3$.

Proof. 

The statements about the global stable and unstable manifolds follows from Theorems 3.2 and 3.3 in [15]. The statements about the immediate basins of attraction follow from the results in [13]. The structure of the complete basin of attraction is described in [13]. See Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14. □

Remark 9.

Theorems 2 and 3 continue to hold for most general competitive systems (8). In fact, the more general global dynamics with many equilibrium solutions with local stability alternate between locally stable and saddle points, according to the results from [13,15]. In fact, the global dynamics described in Theorems 2 and 3 replicate in the case of a large number of equilibrium solutions with described local dynamics. Similarly, Theorem 3 hold for situations when there are more than three equilibrium solutions and three period-two solutions, when the local stability of neighboring solutions and period-two solutions alternates between local stability and saddle point. For illustration, see Figure 11. Similar results for cooperative systems have been proved in [23].

In all the figures that are following the locally stable equilibrium points or period-two points are colored in green, the saddle point equilibium and period-two points are colored in yellow and the repelling equilibrium points or period-two points are colored in red.

Figure 6. This figure illustrates Theorem 2. There are two locally stable fixed points on the axis and one saddle point on the bisector. The dark blue region is the immediate basin of attraction of $E_1$ and the tan region is the immediate basin of attraction of $E_3$.

Figure 6. This figure illustrates Theorem 2. There are two locally stable fixed points on the axis and one saddle point on the bisector. The dark blue region is the immediate basin of attraction of $E_1$ and the tan region is the immediate basin of attraction of $E_3$.

Figure 7. This figure illustrates Theorem 2. There are two locally stable fixed points on the axis and one saddle point on the bisector. For P1 and P2 Solutions: The brown region is the basin of attraction of $E_1$ and the tan region is the basin of attraction of $E_3$.

Figure 7. This figure illustrates Theorem 2. There are two locally stable fixed points on the axis and one saddle point on the bisector. For P1 and P2 Solutions: The brown region is the basin of attraction of $E_1$ and the tan region is the basin of attraction of $E_3$.

Figure 8. This figure illustrates Theorem 3. There are three equilibrium points on the left figure and a single period-two solution on the bisector. The full basin of attractions of two equilibrium points are visualized as preimages of the immediate basins of attractions. For P1 Solutions: The brown region is the basin of attraction of $E_1$ and the tan region is the basin of attraction of $E_3$. For P2 Solutions: The tan region is the basin of attraction of $E_1$ and the orange region is the basin of attraction of $E_3$.

Figure 8. This figure illustrates Theorem 3. There are three equilibrium points on the left figure and a single period-two solution on the bisector. The full basin of attractions of two equilibrium points are visualized as preimages of the immediate basins of attractions. For P1 Solutions: The brown region is the basin of attraction of $E_1$ and the tan region is the basin of attraction of $E_3$. For P2 Solutions: The tan region is the basin of attraction of $E_1$ and the orange region is the basin of attraction of $E_3$.

Figure 9. This figure illustrates Theorem 3. There are three equilibrium points in the left figure and two period-two solutions in the right figure. The full basins of attractions of the equilibrium points are visualized as preimages of the immediate basins of attractions. For P1 Solutions: The brown region is the basin of attraction of $E_1$ and the tan region is the basin of attraction of $E_3$. For P2 Solutions: The orange region is the basin of attraction of $E_1$, the brown region is the basin of attraction of $E_3$, and the white region is the basin of attraction for $\{p_2,T\left(p_2\right)\}$.

Figure 9. This figure illustrates Theorem 3. There are three equilibrium points in the left figure and two period-two solutions in the right figure. The full basins of attractions of the equilibrium points are visualized as preimages of the immediate basins of attractions. For P1 Solutions: The brown region is the basin of attraction of $E_1$ and the tan region is the basin of attraction of $E_3$. For P2 Solutions: The orange region is the basin of attraction of $E_1$, the brown region is the basin of attraction of $E_3$, and the white region is the basin of attraction for $\{p_2,T\left(p_2\right)\}$.

Figure 10. This figure illustrates Theorem 3. There are three feasible fixed points $E_2,E_3,E_4$ and one feasible period-two solution, which is a saddle point. In addition, there are four nonfeasible period-two solutions. For P1 Solutions: The orange region is the basin of attraction of $E_2$ and the tan region is the basin of attraction of $E_4$. For P2 Solutions: The orange region is the basin of attraction of $E_2$ and the brown region is the basin of attraction of $E_4$.

Figure 10. This figure illustrates Theorem 3. There are three feasible fixed points $E_2,E_3,E_4$ and one feasible period-two solution, which is a saddle point. In addition, there are four nonfeasible period-two solutions. For P1 Solutions: The orange region is the basin of attraction of $E_2$ and the tan region is the basin of attraction of $E_4$. For P2 Solutions: The orange region is the basin of attraction of $E_2$ and the brown region is the basin of attraction of $E_4$.

Figure 11. This figure illustrates both Theorems 2 and 3. There are two locally stable fixed points on the axis and one saddle point on the bisector in the first quadrant with corresponding basins of attractions, which represent the biologically feasible solutions. There is also one non-feasible equilibrium solution in the third quadrant and five non-feasible period-two solutions. All solutions are given with the corresponding basins of attractions. For P1 Solutions: The oragne region is the basin of attraction of $E_2$ and the tan region is the basin of attraction of $E_4$. For P2 Solutions: The light blue region is the basin of attraction of $E_2$, the dark blue region is the basin of attraction of $E_4$, the brown region is the basin of attraction of $p_6$, and the tan region is the basin of attraction for $T\left(p_6\right)$.

Figure 11. This figure illustrates both Theorems 2 and 3. There are two locally stable fixed points on the axis and one saddle point on the bisector in the first quadrant with corresponding basins of attractions, which represent the biologically feasible solutions. There is also one non-feasible equilibrium solution in the third quadrant and five non-feasible period-two solutions. All solutions are given with the corresponding basins of attractions. For P1 Solutions: The oragne region is the basin of attraction of $E_2$ and the tan region is the basin of attraction of $E_4$. For P2 Solutions: The light blue region is the basin of attraction of $E_2$, the dark blue region is the basin of attraction of $E_4$, the brown region is the basin of attraction of $p_6$, and the tan region is the basin of attraction for $T\left(p_6\right)$.

Figure 12. This figure illustrates Theorem 3. There are two locally stable fixed points on the axis and one saddle point on the bisector in the first quadrant with corresponding basins of attractions, which represent the biologically feasible solutions. There are also three feasible period-two solutions. All solutions are given with the corresponding basins of attractions. For P1 Solutions: The brown region is the basin of attraction of $E_1$ and the tan region is the basin of attraction of $E_3$. For P2 Solutions: The dark blue region is the basin of attraction of $E_1$, the light blue region is the basin of attraction of $E_3$, the orange region is the basin of attraction to $p3$, and the tan region is the basin of attraction to $T\left(p_3\right)$.

Figure 12. This figure illustrates Theorem 3. There are two locally stable fixed points on the axis and one saddle point on the bisector in the first quadrant with corresponding basins of attractions, which represent the biologically feasible solutions. There are also three feasible period-two solutions. All solutions are given with the corresponding basins of attractions. For P1 Solutions: The brown region is the basin of attraction of $E_1$ and the tan region is the basin of attraction of $E_3$. For P2 Solutions: The dark blue region is the basin of attraction of $E_1$, the light blue region is the basin of attraction of $E_3$, the orange region is the basin of attraction to $p3$, and the tan region is the basin of attraction to $T\left(p_3\right)$.

Figure 13. This figure illustrates Theorem 3. There are two locally stable fixed points on the axis and one saddle point on the bisector in the first quadrant with corresponding basins of attractions, which represent the biologically feasible solutions. There are also three feasible period-two solutions. All solutions are given with the corresponding basins of attractions. For P1 Solutions: The brown region is the basin of attraction of $E_1$ and the tan region is the basin of attraction of $E_3$. For P2 Solutions: The light brown region is the basin of attraction of $E_1$, the dark blue region is the basin of attraction of $E_3$, the brown region is the basin of attraction for $p_2$, and the tan region is the bain of attraction for $T\left(p_2\right)$.

Figure 13. This figure illustrates Theorem 3. There are two locally stable fixed points on the axis and one saddle point on the bisector in the first quadrant with corresponding basins of attractions, which represent the biologically feasible solutions. There are also three feasible period-two solutions. All solutions are given with the corresponding basins of attractions. For P1 Solutions: The brown region is the basin of attraction of $E_1$ and the tan region is the basin of attraction of $E_3$. For P2 Solutions: The light brown region is the basin of attraction of $E_1$, the dark blue region is the basin of attraction of $E_3$, the brown region is the basin of attraction for $p_2$, and the tan region is the bain of attraction for $T\left(p_2\right)$.

Figure 14. This figure illustrates Theorem 3. The case of asymmetric parameters. There are two locally stable fixed points and one repeller between them in the first quadrant with corresponding basins of attractions, which represent the biologically feasible solutions. There are also three feasible period-two solutions, which must exist to generate the boundaries of the basins of attractions [13]. The blue region of the picture on the left is an immediate basin of attraction of locally stable period-two solution. For P1 Solutions: The brown region is the basin of attraction of $E_1$ and the tan region is the basin of attraction of $E_3$. For P2 Solutions: The light brown region is the basin of attraction of $E_1$, the dark blue region is the basin of attraction of $E_3$, the brown region is the basin of attraction for $p_2$, and the tan region is the bain of attraction for $T\left(p_2\right)$.

Figure 14. This figure illustrates Theorem 3. The case of asymmetric parameters. There are two locally stable fixed points and one repeller between them in the first quadrant with corresponding basins of attractions, which represent the biologically feasible solutions. There are also three feasible period-two solutions, which must exist to generate the boundaries of the basins of attractions [13]. The blue region of the picture on the left is an immediate basin of attraction of locally stable period-two solution. For P1 Solutions: The brown region is the basin of attraction of $E_1$ and the tan region is the basin of attraction of $E_3$. For P2 Solutions: The light brown region is the basin of attraction of $E_1$, the dark blue region is the basin of attraction of $E_3$, the brown region is the basin of attraction for $p_2$, and the tan region is the bain of attraction for $T\left(p_2\right)$.

Figure 15. This figure illustrates Theorem 3. The case of asymmetric parameters. There are two locally stable fixed points and one repeller between them in the first quadrant with corresponding basins of attractions. There are also five feasible period-two solutions, which must exist to generate the boundaries of the basins of attractions [13]. For P1 Solutions: The orange region is the basin of attraction of $E_1$ and the tan region is the basin of attraction of $E_3$. For P2 Solutions: The tan region is the basin of attraction of $E_1$ and the dark blue region is the basin of attraction of $E_3$.

Figure 15. This figure illustrates Theorem 3. The case of asymmetric parameters. There are two locally stable fixed points and one repeller between them in the first quadrant with corresponding basins of attractions. There are also five feasible period-two solutions, which must exist to generate the boundaries of the basins of attractions [13]. For P1 Solutions: The orange region is the basin of attraction of $E_1$ and the tan region is the basin of attraction of $E_3$. For P2 Solutions: The tan region is the basin of attraction of $E_1$ and the dark blue region is the basin of attraction of $E_3$.

6. Conclusions

In this paper we discuss differentiation with bifurcations in the genetic regulatory network, given with either a system of ODEs (1) or with the corresponding Euler discretization (19). We prove some facts regarding the boundedness of all solutions for the two systems. Then we discuss in great detail the model of two genes. In the case of the ODE two-gene model we obtained the ultimate result given in Theorem 1, which was missing in [1,3]. This result, together with Conjectures 1 and 2, provides the global dynamics in this case, with the precise description of the basins of attractions of every hyperbolic equilibrium. The corresponding two-dimensional discrete model (8) has similar behavior in many cases where all solutions converge to the equilibrium solutions. However, discretization introduces the existence of period-two solutions, some of which could be the attractors with the corresponding basins of attractions. See Figure 8, Figure 9, Figure 10 and Figure 11. In particular, Figure 11 contains five different feasible period-two solutions. The color coding in all figures is as follows: attracting equilibrium solutions or attracting period-two solutions are green, saddle point equilibrium solutions or saddle point period-two solutions are yellow and repelling equilibrium solutions or repelling period-two solutions are red. The feasible region for the gene toggle switch model is the first quadrant, which contains the immediate basins of attractions. The other quadrants contain the pre-images of the immediate basins of attractions.

We believe that future research will be directed towards multigene models, such as the three-gene toggle switch model. Some basic results were obtained in [1,3] on the existence of Hopf bifurcation for the model (1), where $n=3$ and $H_i\left(u\right)={\gamma }_i/(1+u^q),{\gamma }_i,q>0,i=1,2,3$. The appearance of Hopf bifurcation implies the existence of a locally stable periodic solution. Under some specific conditions, all solutions of (1) may converge to equilibrium solutions and our research will be directed towards finding such conditions and finding a structure of basins of attractions of locally stable equilibrium solutions. In the case of a discrete version (19) of this system of differential equations, the global dynamics are more complicated in the case of three or more gene models and such results are missing. This is an area where more results are needed.