Hopf bifurcation in the full repressilator equations

In this paper, we prove that the full repressilator equations in dimension six undergo a supercritical Hopf bifurcation. Copyright © 2014 John Wiley & Sons, Ltd.


Introduction
Oscillatory networks are a particular kind of regulatory molecular networks, that is, collections of interacting molecules in a cell. The regulatory oscillators can be used to study abnormalities of a process in the cell, from sleep disorders to cancer. So, they attract significant attention among biologists and biophysicists. There are many implementations of artificial oscillatory networks (see, e.g., [1][2][3][4][5][6][7]). One of them is the repressilator [8]. Its genetic implementation uses three proteins that cyclically repress the synthesis of one another. The following system of DEs describes the behavior of the repressilator: Here, u, v, and w are proportional to the protein concentration, while m i are proportional to the concentration of mRNA corresponding to those proteins. The nonlinear function f .x/ D1 Cx n reflects synthesis of the mRNAs from the DNA controlled by regulatory elements. The parameter˛0 represents uncontrolled part of the mRNA synthesis, and it is usually small. The explicit inclusion of the mRNA concentration variables into the model is given byˇ. Given that in generalˇ 1 and˛0 is very small, we consider˛0 D "a andˇD "b, where a and b are positive constants and " > 0 is sufficiently small. So, system (1) becomes In the papers [9,10], the authors consider a reduced system of dimension three. The reduction assumes that the three excluded variables, that is, m i , evolve an order of magnitude faster than the other three. In [9], the authors prove that in the reduced system exhibits a supercritical Hopf bifurcation. The existence of a Hopf bifurcation in the reduced system does not imply that the full system, in dimension six, also has a supercritical Hopf bifurcation. It just gives an indication about its existence. Here, in this work, we consider the full system in dimension six, and we extend the results of [9,10] on the supercritical Hopf bifurcation to the six-dimensional differential system (2). Our main result is the following one.

Theorem 1
Let n > 2 be an integer. The point .r 0 , r 0 , r 0 , r 0 , r 0 , r 0 / with is an equilibrium of the differential system (2) with The eigenvalues of the linear part of system (2) at this equilibrium are Moreover, there is a single supercritical Hopf bifurcation at˛D˛b if , and there exists a small " 0 > 0, such that for˛b if <˛<˛b if C " 0 , the system (2) has a stable limit cycle. The repressilator differential system depends on five parameters a, b,˛, ", and n and on six variables m 1 , m 2 , m 3 , u, v, and w. From Theorem 1, it follows that the supercritical Hopf bifurcation that such system exhibits takes place when n > 2 on the hypersurfacę inside the five-dimensional parameter space, recall that " is a small parameter. We must remark that the reduced repressilator differential system studied in [3,4] depends only on two parameters˛and n and on three variables m 1 , m 2 , and m 3 . From Proposition 1 of [3], we obtain that the supercritical Hopf bifurcation that such reduced system exhibits takes place when n > 2 on the curvę in the plane of parameters.

Proof of Theorem 1
It is clear that system (2) has the equilibrium p 0 D .r 0 , r 0 , r 0 , r 0 , r 0 , r 0 / where r 0 is solution of the equation From (3), we have that˛D .r 0 "a/ 1 C r n 0 . So, substituting˛in the linear part of system (2) at the equilibrium p 0 , we obtain We impose that the real part of the eigenvalues " . 2C.n 2˙i C O " 2 is zero, and we obtain Substituting (4) in (3), we obtain˛b Substituting (4) in M and computing the eigenvalues, we obtain˙" The linearization of (2) at p 0 has a pair of conjugate purely imaginary eigenvalues, and the other four eigenvalues have negative real part. This is the setting for a Hopf bifurcation. We can expect to see a small-amplitude limit cycle branching from the fixed point p 0 . It remains to compute the first Lyapunov coefficient`1.p 0 / of (2) near p 0 . When`1.p 0 / < 0, the point p 0 is a weak focus of system (2) restricted to the central manifold of p 0 , and the limit cycle that emerges from p 0 is stable. In this case, we say that the Hopf bifurcation is supercritical. When`1.p 0 / > 0, the point p 0 is also a weak focus of system (2) restricted to the central manifold of p 0 , but the limit cycle that emerges from p 0 is unstable. In this second case, we say that the Hopf bifurcation is subcritical.
Here, we use the following result presented on page 180 of the book by Kuznetsov [11] for computing`1.p 0 /.

Lemma 2
Let P x D F.x/ be a differential system having p 0 as an equilibrium point. Consider the third-order Taylor approximation of F around p 0 given by F.
Assume that A has a pair of purely imaginary eigenvalues˙ i. Let q be the eigenvector of A corresponding to the eigenvalue i, normalized so that q q D 1, where q is the conjugate vector of q. Let p be the adjoint eigenvector such that A T p D ip and p q D 1. If I denotes the 6 6 identity matrix, theǹ In our case, the linear part of system (2) at the equilibrium p 0 is We have that A has an eigenvalue " p 3bi C O " 2 . Now, we compute the bilinear and trilinear functions B and C. Considering the vector field .f 1 , f 2 , f 3 , f 4 , f 5 , f 6 / associated to the differential system (2), we observe that all second and third derivatives vanish except @ 2 f1 @v 2 , @ 2 f2 @w 2 , @ 2 f3 @u 2 , @ 3 f1 @v 3 , @ 3 f2 @w 3 , and @ 3 f3 @u 3 . Computing these derivatives, taking into account (4) and (5), we obtain that So, the bilinear function B is given by B..x 1 , y 1 , z 1 , u 1 , v 1 , w 1 /, .x 2 , y 2 , z 2 , u 2 , v 2 , w 2 // D . v 1 v 2 , w 1 w 2 , u 1 u 2 , 0, 0, 0/ and the trilinear function C is given by the expression Computing the normalized eigenvector q of A, associated to the eigenvalue " p 3bi C O " 2 , we obtain The normalized adjoint eigenvector of the transpose matrix A with the eigenvalue " p 3bi is According to Lemma 2, in order to compute`1.p 0 /, we need to compute first A 1 and 2 p 3b"iI A Á 1 . We have that  .n C 7/.n 2/ C O."/ As we said before,`1.p 0 / < 0 implies that we have a supercritical Hopf bifurcation at˛D˛b if , so there exists " 0 > 0, such that for bif <˛<˛b if C " 0 , the system (2) has a stable limit cycle.

Conclusions
The repressilator model is an implementation of an artificial oscillatory network used for studying the collections of interacting molecules in a cell. The model is given by a six-dimensional differential system. Buse et al. published two nice papers [9, 10] analyzing a reduced system of dimension three. In this reduced system, they show the existence of a supercritical Hopf bifurcation. Because the reduction is reasonable, we may expect that such supercritical Hopf bifurcation must also occur in the actual six-dimensional differential system. We prove that this is the case.

Appendix
The matrix 2 p 3b"iI A Á 1 is given by B @ C 11 C 12 C 13 C 14 C 15 C 16 C 21 C 22 C 23 C 24 C 25 C 26 C 31 C 32 C 33 C 34 C 35 C 36 C 41 C 42 C 43 C 44 C 45 C 46 C 51 C 52 C 53 C 54 C 55 C 56 C 61 C 62 C 63 C 64 C 65 C 66 where C ij D A ij C B ij , for i, j D 1, 2, : : : , 6.