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Stability and Hopf Bifurcation Analysis of a Gene Expression Model with Diffusion and Time Delay
We consider a model for gene expression with one or two time delays and diffusion. The local stability and delay-induced Hopf bifurcation are investigated. We also derive the formulas determining the direction and the stability of Hopf bifurcations by calculating the normal form on the center manifold.
The study on dynamics of a biological model is one of the dominant subjects in mathematical biology due to its universal existence and importance. In this paper, we consider a mathematical model of intracellular regulatory system which began with the work of Goodwin . Since then, many researchers developed this work [2–5]. However, it is Monk who developed the first mathematical model for the Hes1 system and validated it with biological data . In order to describe the intracellular process more precisely, he introduced time delays to account for the processes of transcription and translation. If we denote by and the concentrations of Hes1 mRNA and Hes1 protein, respectively, the basic reaction kinetics for this system can be expressed in the form of where . The parameter is the rate at which Hes1 protein is produced from Hes1 mRNA and and are the decay rates of Hes1 protein and Hes1 mRNA, respectively. is the rate of production of new mRNA molecules, with and as constants, to represent the rate of transcript initiation in the absence of Hes1 protein and the reference concentration of protein, respectively, and is the Hill coefficient. and represent the transcriptional and translational time delays. The units of the parameters are as follows. is measured in protein molecules per mRNA molecule per minute; and are measured in molecules per minute; is measured in mRNA molecules per diploid cell per minute; is measured in molecules and time delays and are measured in minute.
In order to reduce the number of parameters,  introduced transformations under which system (1) takes the following form: Recently, Zhang et al.  investigated the stability and Hopf bifurcation of the equilibrium of system (3). By using the method of multiple time scales, they also obtained the normal form on the center manifold of delay differential equations (3).
The diffusion process comes naturally in biology as pointed out by Murray in . Experimental data also suggests that many pathways exhibit oscillation in concentrations of substance involved, both temporally and spatially. For example, a type diffusion process appears in a process of assemblage of particles, which is due to the particles spread out as a result of an irregular individual particle's motion. In these cases, a partial differential equation or system of partial differential equations will be employed to describe the processes. Nowadays, models involving delays and also spatial diffusion are increasingly applied to the study due to more appropriate biological justification. Reference  gave plausible biological explanations for the delays appearing in the model (3). In what follows, spatial diffusion will be applied to the model (3) to gain new information about the precise spatiotemporal dynamics of mRNA and proteins. In fact, Sturrock et al.  derived a system of partial differential equations to capture the evolution in space and time of the variables in the Hes1 and p53-Mdm2 systems. In this paper, we extend previous mathematical model (3) into its spatial version by considering spatial interactions within the cell explicitly (of course, the clear biological explanation will be given behind the model (4) of promotion); namely, we consider the delayed reaction-diffusion system with the following initial and boundary conditions: where and are the diffusion coefficients for Hes1 mRNA and Hes1 protein, respectively, with unit such as cm/min. The initial function . The imposed Neumann boundary condition here implies that mRNA and protein are not exported across the nuclear membrane or the cell membrane.
Obviously, (4) is a system of reaction-diffusion equations modeling the spatiotemporal evolution of the Hes1 system. The same reaction kinetics from the ODE model (3) are retained but are now also coupled with diffusion to model explicitly protein and mRNA transport in a cell. That is to say, molecules move from the nucleus to the cytoplasm and from cytoplasm to nucleus across the nuclear membrane. Here, we use a system of PDEs (4) to reflect the reality that the mRNA is transcribed from DNA exclusively in the nucleus and that protein is translated from mRNA exclusively in the cytoplasm. The main advantage of using systems of PDEs (4) to model intracellular reactions is that the PDEs enable spatial effects to be examined explicitly. The main object of this paper is to investigate the effect of the delay and diffusion on the dynamics of system (4). In addition, in order to determine the direction and the stability of Hopf bifurcations, we use the normal form procedure for functional differential equations (FDEs) with diffusion due to Faria [12, 13].
This paper is organized as follows. In Section 2, stability of positive equilibrium and existence of Hopf bifurcation are studied using as a parameter. In Section 3, the effect of diffusion on the Hopf bifurcation will be investigated. Using the normal form technique for partial functional differential equations, the formulas for determining the direction and stability of Hopf bifurcation are presented in Section 4. Finally, in Section 5, we will illustrate the theoretical results by numerical simulations along with some discussion.
2. Stability of Positive Equilibrium and Existence of Hopf Bifurcation
It is easy to verify that system (4) has a unique positive equilibrium determined by
The object here is then to relate the dynamics of (3) and (4) in the neighborhood of , at the first critical point of the parameter . To this end, we let , . With the help of (5), after dropping the bars for simplicity of notation, (4) is transformed into the following system: with the origin as its equilibrium.
Let Define and as follows: where and are the nonnegative integers. Then system (6) can be rewritten as where For simplification of notation, we use for and for and is in a suitable Hilbert space By setting , we further write (9) as an abstract equation in : where and , and are given by for . Obviously, is a linear operator. The linearization of (12) is It has a characteristic equation given by where , , , and is the total time delay. When there is no diffusion effect, namely, , (15) can be written as which is equivalent to (12) of Zhang et al.’s work in , and the related stability and Hopf bifurcation have been investigated.
In what follows, we will analyze the effect of diffusion terms by the distribution of the roots of (15) with , . We first consider the case when the delay is zero. For (15), if , then we have Since , for any and , it is easy to verify that (17) has a pair of roots with negative real parts. And, for , we have the following lemma.
Lemma 1. Assume that holds. Then all the roots of the characteristic equation (15) have negative real part for .
Proof. If the conclusion is not true, namely, (15) admits at least one root with , then we obtain Separating the real and imaginary parts yields It implies that namely, Since , we can easily verify That is, . Notice that , for , , gives It is a contradiction. Thus the conclusion follows.
Theorem 2. Assume that holds. Then the equilibrium point of system (4) is asymptotically stable for .
3. Effects of Diffusion on the Hopf Bifurcation
Assume that is a purely imaginary root of (15); then we have which implies that where Equation (25) implies that should be negative for some . It is equivalent to the fact that should be less than 0 as it is easy to see that has the same sign as that of . Rewrite into the following form: It is obviously a quadratic polynomial in terms of . Equation (28) implies that there exists a such that if and only if . Furthermore, we have From (29), we obtain that, for each , (25) has only one positive real root , which is given by
We now can make the following conclusion.
Lemma 3. Assume that is not true. Then (15) has a pair of purely imaginary roots for each and has no purely imaginary roots for , where and are defined as above.
In the rest of this section, we will discuss the case of .
From (24), we have Then, for , define In the following, we will order the sequence of depending on the diffusion coefficients and for .
Lemma 4. If and holds, then
Proof. If , from (30), we have Let ; it is easy to verify Thus, according to (32), we obtain In addition, From (35), we know that is decreasing with respect to . Then we obtain if . Clearly, , so we have
According to Lemma 4 and the continuous dependence of on and , we summarize the following lemma.
Lemma 5. For any , there exists an such that, for any satisfying .
Lemma 6. If and satisfy the condition in Lemma 5, then, for and , .
Combining the above analysis and the qualitative theory of partial functional differential equations in , we have the following results on the stability of equilibrium of system (4) and existence of Hopf bifurcation near .
Theorem 7. Assume that , satisfy the condition in Lemma 5. Then, for system (4),(i)the positive equilibrium is asymptotically stable for and unstable for ;(ii)it undergoes Hopf bifurcations near the positive equilibrium at for and .
Remark 8. If , Theorem 2 and Theorem 7 are the conclusion of Theorem 1 in . That is, assuming either condition or , Theorem 2 and Theorem 7 show that the local stability of for is the same for system (3) and system (4). Here we can know the effect of the diffusion coefficients and , or, in order to more clearly understand the effect of diffusion, we take , , , and then we get , by calculation. In the absence of diffusion, , it is known that the equilibrium of system (3) is stable when according to Theorem 1 in . In the presence of diffusion, for example, taking , , which implies , we know that the equilibrium of system (4) is stable for by Theorem 2.
4. The Direction and Stability of Hopf Bifurcation
In this section, we assume the hypotheses of Theorem 7 hold and . For the case of , which is not our concern in this paper, the calculation of the normal form should follow the method developed in . By using the normal form method in  for partial differential equations with time delay, we will investigate the stability of these Hopf bifurcations. For standard notations and classical results on partial functional differential equations, please refer to [12, 13, 17]. More details on techniques for computing the normal form can also be found in recent work .
In the following, we denote any one of these critical values by at which the characteristic equation (15) has a pair of simply purely imaginary roots . Let , and consider only the case is the set of eigenvalues on the imaginary axis of the infinitesimal generator associated with the flow of Equation (42) is now written as where , for . The eigenvalues of on are , , , with corresponding normalized eigenfunctions , where Let . Assume that and Then linear PFDE (44) restricted to is equivalent to the FDE on with the characteristic equation given by (15).
Suppose that there exists a such that when , (15) for fixed has a pair of purely imaginary roots and all other roots of (15) have negative real parts. Define such that and the adjoint bilinear form on , , For (48) with fixed , choose a basis for the adjoint space and a basis for its the eigenspace as follows: such that , where is a identity matrix. Then we can easily have with Following the standard procedure in , especially , using the decomposition , , , we decompose (45) as where here and throughout this section we refer to [12, 18] for results and explanations of several notations involved.
Consider the formal Taylor expansion where is the th Fréchet derivative of . Then (55) can be written as where and are given by in . Then (45) has a normal form on the center manifold of the origin at , written as where and , , are given by in . The normal form procedure will show that these terms have the form where is the conjugate of . Consider the following: where with where, for , , while and , are determined by the following equations: where So the normal form (45) on the center manifold has the form Next we will derive the normal form in the real coordinates. To this end, let , , and then the polar coordinates , . We finally reach here , . Then, from , we know that the number tells the bifurcation direction and the stability of bifurcating periodic solution. (i)When , it is a supercritical bifurcation and the bifurcating periodic solution is stable.(ii)When , it is a subcritical bifurcation and the bifurcating periodic solution is unstable.
5. Numerical Simulation and Discussion
In this section, we present some numerical simulations to system (4). These simulations are used to support our theoretical results. Similar to [6, 7, 20], the values of parameters are taken from the published experimental and theoretical results in our simulations where, , , and . and are taken as in  for original model (1).
Taking , , , , and , then the positive equilibrium . From (30) and (32), we obtain the critical value for time delay, . In this case, the parameters satisfy . By Theorem 7, the positive equilibrium is asymptotically stable for . Figure 1 is the numerical simulation of system (4) for .
When the delay increasingly crosses through the critical value , the positive equilibrium loses its stability and the Hopf bifurcation occurs. Taking , Figure 2 is the numerical simulation results of system (4). It is consistent with the theoretical results.
In Figure 2, we fix and the other parameter values are the same as Figure 2. Then we get Figure 3 which shows that the oscillation will sustain when the time delay is much greater than the critical value .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The research is supported by the National Natural Science Foundation of China (nos. 11101076 and 11371087), the Shanghai Committee of Science and Technology (nos. 11ZR1400200 and 12ZR1400100), the Fundamental Research Funds for the Central Universities (2232011D-35), and China Scholarship Council.
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Copyright © 2014 Yahong Peng and Tonghua Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.