Stability of Gene Regulatory Networks Modeled by Generalized Proportional Caputo Fractional Differential Equations.

A model of gene regulatory networks with generalized proportional Caputo fractional derivatives is set up, and stability properties are studied. Initially, some properties of absolute value Lyapunov functions and quadratic Lyapunov functions are discussed, and also, their application to fractional order systems and the advantage of quadratic functions are pointed out. The equilibrium of the generalized proportional Caputo fractional model and its generalized exponential stability are defined, and sufficient conditions for the generalized exponential stability and asymptotic stability of the equilibrium are obtained. As a special case, the stability of the equilibrium of the Caputo fractional model is discussed. Several examples are provided to illustrate our theoretical results and the influence of the type of fractional derivative on the stability behavior of the equilibrium.

1. Introduction

Gene expression is the process where the hereditary code of a gene is used for synthesizing proteins and producing the structures of the cell. Genes that code for amino acid sequences are named ‘structural genes’. Gene expression processes include two main stages known as ‘Transcription and translations’. Transcription is the creating of messenger RNA (mRNA) by the enzyme RNA polymerase and the processing of the resulting mRNA molecule. A gene regulatory network consists of a number of genes interacting by proteins. Mathematical models of gene regulatory networks are described and studied in several papers (see, for example, [1,2], for fractional order [3,4,5,6], and with delays [7,8]).

Recently, fractional calculus, fractional derivatives, and fractional integrals of various types have been extensively studied and applied in mathematical modeling. The memory property of fractional derivatives makes them well suited in modeling and describing the complex nature of real-world problems, in comparison to local derivatives (see, for example [9,10,11]).

In this paper, a gene regulated model with the generalized proportional Caputo fractional derivative is set up, and the equilibrium is defined. The generalized exponential stability is introduced and studied via the application of Lyapunov functions and their generalized Caputo proportional fractional derivatives. Generalized proportional Caputo fractional derivatives were recently introduced (see [12,13]); this type of derivative is a generalization of the Caputo fractional derivative, and their application provides wider possibilities for modeling adequately the complexity of real-world problems. The stability of fractional order systems with a proportional Caputo fractional derivatives is quite recent (see, for example, [14,15]). In this paper, some properties of absolute values of Lyapunov functions and their fractional derivatives are discussed, and several examples are provided to illustrate the properties. The advantages of the application of the quadratic Lyapunov functions are considered, and sufficient conditions for generalized exponential stability and asymptotic stability are obtained. Several examples are provided to illustrate the theoretical results and the dependence of the fractional derivative on the behavior of the solutions.

2. Notes on Fractional Calculus

We recall the definitions needed in this paper, namely fractional integrals and derivatives (cf. [13]):

The generalized proportional fractional integral of a function is defined by (as long as all integrals are well defined)

The generalized Caputo proportional fractional derivative of a function is defined by (as long as all integrals are well defined) where .

Note that the generalized proportional Caputo fractional derivative is defined for

If

We say

Let

([ holds.

We will use the following property of the Mittag–Leffler function with one parameter, defined by with the gamma function.

([

[

3. Some Comments on Properties of the Fractional Derivatives of Lyapunov Functions

One of the most applicable Lyapunov functions is the absolute values Lyapunov function. In connection with this, we will give and discuss some results about their fractional derivatives.

In [17], the following result is proved:

([ is a continuously differentiable function, and the following relation holds almost everywhere:

This result is applied by many authors to study the stability of various types of Caputo fractional differential equations and models. For example, this equality is applied in the proof of global Mittag–Leffler stability for fractional-order gene regulatory networks in [3], and to study global uniform asymptotical stability for fractional-order gene regulatory networks with delays in [4,18]. Unfortunately, this equality is not satisfied for all continuously differentiable functions, and we will demonstrate this with an example.

Let

For any

Let

Therefore, for the Caputo fractional derivative, the equality is not true for all

Note that in the proof of ([5], Theorem 1), the inequality of the type is applied to prove the equality . Unfortunately, this inequality is not true for all functions. We will illustrate this with an example:

Consider the Caputo fractional differential equation with

Use and obtain the bounds for the solution:

For

Note a similar situation occurs when the generalized proportional Caputo fractional derivative is applied. We will illustrate this with an example.

Let

Case 1.1. Let and

Case 1.2. Let and

Therefore, for the generalized proportional Caputo fractional derivative, the equality is not true for all

We will now prove the correct result. To be general, we will consider the generalized proportional Caputo fractional derivative:

Let holds.

For any we get and

□

In the case of the Caputo fractional derivative, we obtain the following result:

Let holds.

The proof of Corollary 2 follows from Lemma 2 with .

If the function

When Lyapunov functions are applied to differential equations, including the absolute values Lyapunov function, the type of derivatives of Lyapunov functions is very important and it depends on the type of derivatives in the differential equations.

Let be a solution of the scalar fractional differential equation . In the literature, several types of fractional derivatives of Lyapunov function are applied. Consider the following derivative, which is called by some authors the upper right-hand derivative in Caputo’s sense or Caputo-type fractional derivative:

We will illustrate some of properties of the absolute values Lyapunov function and its derivative given by the above definition with an example.

Let where

However, the derivative

From Example 1, Example 2, and Example 4, it could be seen that in the case of Caputo fractional differential equations, for the Lyapunov function we have:

If its Caputo fractional derivative is applied, then the inequality holds in the general case (see Example 1–Case 2). According to Lemma 2, the equality is true only in a particular case;

If its Caputo-type fractional derivative

Before the application of the equality (

The situation mentioned in Remark 4 is true also for generalized proportional Caputo fractional differential equations and the absolute values Lyapunov function.

According to the above discussions, in this section, the Lyapunov function of the type is appropriate to apply to fractional differential equations using the following result:

[ holds.

Note that in the special case

Note that quadratic Lyapunov functions for Caputo fractional order time-delayed gene regulatory networks are applied in [

4. Statement of the Problem

In this paper, we will consider a class of fractional order gene regulatory networks modeled by a generalized proportional Caputo fractional derivative for , : where , denote the concentrations of messenger ribonucleic acid (mRNA) and protein of the j-th node at time t, respectively, and are degradation velocities of mRNA and protein, respectively, is the translation rate, the functions represent the activator initiates of protein of mRNA, and the coupling matrix of the network is described by and , where is the set of all repressors of gene j.

Commonly, the activator functions

Note that the model (

Introduce the following assumptions:

(A1) The activator functions are increasing and there exist constants such that for any with the inequalities hold.

(A2) There exist positive constants such that the coefficients in (9) satisfy the inequalities

From Assumption (A1), it follows that Lemma 2 is applicable to the solutions of (

From Lemma 1, it follows that the generalized proportional Caputo fractional derivative of a nonzero constant is not zero, and applying Corollary 1, we introduce the following definition.

The couple of functions with

Note that in the case of Caputo fractional derivative (

The equilibrium where

Use the transformations

Then, (9) can be written in the form where

The system (11) has a zero equilibrium.

The goal of our paper is to study the exponential and asymptotic stability of the equilibrium of (9); equivalently, we also study the stability properties of the zero solution of the IVP for FrDE (11).

We will apply quadratic Lyapunov functions, and in connection with this, we will use the following result:

([ holds. Then, where

([ holds, where where

Let the assumptions (A1) and (A2) be satisfied, and assume that there exists an equilibrium

The generalized exponentially stability of equilibrium of the model (9) is equivalent to the generalized exponential stability of the zero solution of (11).

Consider the Lyapunov function where

Let be a solution of (11). According to Lemma 3, we obtain

From assumption (A2), it follows that and thus where

According to Lemma 5, the inequality holds, where and or

□

Let the conditions of Theorem 1 be satisfied. Then, the equilibrium of the model (

5. Applications

Application 1

We will consider the model of three repressor-protein concentrations, , and their corresponding mRNA concentrations, , which are defined and studied in [21] when the kinetics of the system are determined by ordinary differential equations. To have a more appropriate model, we will adopt this model and use generalized proportional Caputo fractional derivatives; i.e., we will consider the model where (see [21]):System (19) is similar to (9) with , , and for , .

The number of protein copies per cell produced from a given promoter type during continuous growth is in the presence of saturating amounts of repressor and in its absence;

is the ratio of the protein decay rate to the mRNA decay rate;

n is a Hill coefficient.

Take and . Thus, , , and and i.e., assumptions (A1) and (A2) are satisfied. According to Theorem 1, if there exists an equilibrium of (19), then it is generalized exponential stable.

Case 1. Caputo fractional derivative, i.e., . The equilibrium is a solution of the system

The system (20) has a solution for every value of and .

Consider a particular case of and . Then, the equilibrium is .

The solutions are given in Figure 2 (left), and the solutions , are given in Figure 2 (right). It could be seen that all components of the solution approach the equilibrium .

Figure 2

Convergence of the solutions of (19) with to the equilibrium .

Note that problem (19) is considered in [3] with . However, in this case, the equilibrium is , which does not correspond to the provided graphs.

Let . Then, the equilibrium is .

The solutions are given in Figure 3 (left) and the solutions , are given in Figure 3 (right). It could be seen that all components of the solution approach .

Figure 3

Convergence of the solution of (19) with to the equilibrium .

Case 2. Generalized proportional Caputo fractional derivative, i.e., .

Since and , the equilibrium is a solution of the system

Case 2.1. Let . Then, the system (21) has zero solution w.r.t. and the system (19) have a zero equilibrium. The solutions are given in Figure 4 (left) and the solutions are given in Figure 4 (right). It could be seen that all components approach the equilibrium 0.

Figure 4

Solution of system (19) with .

Case 2.2. Let . Then, the system (21) has no solution w.r.t. , and the system (19) has no equilibrium, and we could not apply Theorem 1.

Application 2

Consider the model of three repressor-protein concentrations, , and their corresponding mRNA concentrations, , (19) with the activator functions ; i.e., consider with . The system (8) has a zero equilibrium. Take and . Thus, , and assumptions (A1) and (A2) are satisfied. According to Theorem 1, the zero equilibrium is generalized exponential stable. The graphs of the solutions and , of system (22) are given in Figure 5 (left) and Figure 5 (right), respectively, with , , , , , with initial values .

Figure 5

Solution of system (22) with .

Application 3

Consider the general model describing the dynamics of the interacting defects in the genome and in the proteome with the generalized proportional fractional derivative: where is the coupling rate constant characterizing the regulation of gene expression by the proteins, K is the average number of genes regulated by any single protein and represents a simple measure of the overall connectivity of the genetic network, c reflects the combined efficiency of proteolysis and heat shock response systems, mediating the degradation and refolding of misfolded proteins, respectively, whereas characterizes the DNA repair rate, the “force” terms, and characterize the proteome and genome damage rates, respectively, and G is the total genome size.

Let and . Then, the model (23) has equilibrium

Model (23) is in the form of (9) with , , , . Thus, , and and i.e., assumptions (A1) and (A2) are satisfied if and . In other words, the DNA repair rate and the expressome (proteome, metabolome) turnover rate, c, have to be large enough. In Figure 6, the graphs of the solution are given with , , and the initial values , . Then, the equilibrium is

Figure 6

Convergence of the solution of (23) to the equilibrium .

Note that the model (23) in the case of ordinary derivatives is studied in [2] with the more restrictive assumption .

6. Conclusions

A new gene-regulated model is set up. The dynamics is decribed by the generalized proportional Caputo fractional derivative. The equilibrium is defined in an appropriate way. In the general case, the classical definition of the equilibrium differs. The generalized exponential stability is introduced and studied via the application of Lyapunov functions and their generalized Caputo proportional fractional derivatives. In connection with the application of Lyapunov functions to fractional type models, some properties of absolute values Lyapunov functions and their fractional derivatives are discussed. Several examples are provided to illustrate the properties. The advantages of the application of the quadratic Lyapunov functions are considered, and sufficient conditions for generalized exponential stability are obtained. Some examples illustrate the theoretical results and the dependence of the fractional derivative on the behavior of the solutions.