Hopf Bifurcation of a Gene-Protein Network Module with Reaction Diffusion and Delay Effects

The infinite dimensional partial delay differential equation is set forth and delay difference state feedback control is considered to describe the cell cycle growth in eukaryotic cell cycles. Hopf bifurcation occurs as varying free pa-rameters and time delay continuously and the multi-layer oscillation phenomena of the homogeneous steady state of a simple gene-protein network module is investigated. Normal form is derived based on normal formal analysis technique combined with center manifold theory, which is further to compute the bifurcating direction and the stability of bifurcation periodical solutions underlying Hopf bifurcation. Finally, the numerical simulation oscillation phenomena is in coincidence with the theoretical analysis results.


Introduction
With the background knowledge of mathematical description of nonlinear dy- As is well known, Michaelis-Menten rate law describes the activation feedback regulation function of gene-protein with the action of protein factors [10]. In addition, to drive the downstream events to generate gene protein production, enzymins reaction formed by binding itself to Cdk2 monomer. Time delay is incorporated into the phosphate groups while binding to target proteins to active protein phosphorylation process [11]. Therefore, the activation regulation with Cdk2 dimer dynamics is dominated by the hill function The simulation work is finished by DDE-Biftool software which is applied to do dynamic analysis of delay differential equations with high technique. As shown in Figure 1  Based on the fundamental theory of functional differential equation [12] [13] [14], people developed Lyapunov-Schmidt dimensional reduction scheme to compute the bifurcating direction of periodical solutions underlying Hopf bifurcation. Therefore, people applied center manifold theory in functional differential equation to compute the stability of bifurcating periodical solution [15] [16] [17] [18]. As for the partial functional differential equations, the well known center manifold theory has also been developed to further apply in normal form analysis near equilibrium solution and herein we adopt Teresa Faria's method [19]. In Faria's paper, based on theory of the autonomous functional differential equations, the analysis technique is addressed for calculating coefficients of normal form on center manifold.
The whole paper is organized as listed. In section 2, with homogenous Neumann conditions, the mathematical model of cell cycle growth model is described with reaction diffusion effects. In section 3, Hopf bifurcation is tracked as varying time delay and free parameter continuously. In section 4, based on the fundamental theory of partial functional differential equations, the normal form is computed with center manifold analytical technique, and finally the numerical simulation verifies the correctness of theoretical results.

The Mathematical Description with Diffusion Effects
With homogenous Neumann conditions, Equation (1.1) with free diffusion effect is modeled by with the definition we assume X is the Hilbert space with inner product The boundary and initial condition of Hopf bifurcation occurs as varying free parameter and time delay, and the bi- Specially or not, we discuss Hopf bifurcation of system (2.1).

Hopf Bifurcation Analysis
The Taylor expansion of its truncation form of Equation (3.2) is written as It is verified that Equation (3.3) satisfies the following general condition: (H4) L can be extended to a bounded linear operator from BC to X wherein with sup norm form.

Hopf Bifurcation
Based on the fundamental theory of partial functional differential equations as stated by [19], the linear differential operator (3.3) exists the unique solution which satisfies initial condition. And the generated strong continuous semigroup composed of solution operators has infinitesimal generator A: The operator A has only its point spectrum, with It is well known that the eigenvalue problem has eigenvalues for 0,1, 2, k =  . With the aids of the above analysis, stability property for the positive equilibrium solution is plotted as shown in Figure 2

Normal Form Computation
The periodical solution arise near Hopf point. Based on the known center mani- The linear version of Equation (4.1) is rewritten as And the generated strong continuous semigroup composed of solution operators has infinitesimal generator A: The corresponding adjoint operator * A defined on the conjugate space The adjoint bilinear form Suppose k P is the eigensubspace corresponding to Λ , then the phase space C can be decomposed into Define the projection operator : herein, k Q is the complement subspace of K P . For any BC φ ∈ , we can write and ( ) ( ) Alike FDE reduction method, we want to enlarge the phase space in such a way that Equation (4.1) can be written as an anstract form of ODE on Banach space BC. For any With the infinitesimal generator Aφ given in Equation (4.7), the extension of : , then the linear part is transformed into the following form, Further, considering the nonlinear part ( ) F ⋅ , the dimensional reduction system of Equation (4.1) is written as

Modern Nonlinear Theory and Application
Note that herein, we suppose that the multiplication of the vector   =     means the multiplication between row elements, that is Based on the above analysis, we have the following theorem, Respectively, the stable bifurcating periodical solution arise at Hopf point 1 2 ,   coexistence of equilibrium solutions are observed and Hopf bifurcation occurs at 1 2 , E E . In Figure 4, the observed oscillating periodical solutions are induced due to the instability transition of equilibrium solution. As shown in Figure 4(a) and  Figure   4(c) and Figure 4(d) respectively. Near 2 E , the periodical solution is also observed as shown in Figure 5(b) which is bifurcated from the corresponding equilibrium solution. The equilibrium solution is asymptotically stable as observed in Figure 5  oscillation with different initial phase is observed in Figure 5

Conclusion
The partial delay differential equation of gene reaction protein equation was set forth. The stability dynamics and Hopf bifurcation was analyzed underlying the feedback control of state difference between present state and its past time state.
Without diffusion effects, using DDE-Biftool software, the bifurcating homoclinc  solution was derived as the time period tends to infinity with continuation of periodic solution as varying free parameter. The continuation of homoclinc orbit becomes a possible job with application of DDE-Biftool software, as shown in Figure 6(a) and Figure 6(b). With diffusion effects, Hopf bifurcation phenomena were further analyzed and the multi-layer periodical oscillation phenomena were discovered. Combined with center manifold technique, the bifurcating direction of periodical solution was analyzed with norm form analysis method.