_{1}

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The repressilator is a genetic network that exhibits oscillations. The net-work is formed of three genes, each of which represses each other cyclically, creating a negative feedback loop with nonlinear interactions. In this work we present a computational bifurcation analysis of the mathematical model of the repressilator. We show that the steady state undergoes a transition from stable to unstable giving rise to a stable limit-cycle in a Hopf bifurcation. The nonlinear analysis involves a center manifold reduction on the six-dimensional system, which yields closed form expressions for the frequency and amplitude of the oscillation born at the Hopf. A parameter study then shows how the dynamics of the system are influenced for different parameter values and their associated biological significance.

The repressilator is an artificial synthetic gene network created and named by Elowitz and Leibler [

The regulation of the process described above is an important enterprise that the cell needs to control and fine-tune constantly [

d m i d t = − m i + α 1 + p j i n + α 0 (1)

d p i d t = − β ( p i − m i ) (2)

where i = 1 , 2 , 3 and j i are defined as j 1 = 3 , j 2 = 1 , and j 3 = 2 . Here α represents dimensionless transcription rate in the absence of repressor, α 0 the background production rate of protein in the presence of saturation, β is the dimensionless ratio between protein decay and mRNA decay, and n is the Hill coefficient representing the degree of cooperation of repression (for more biological details see [

The biological and experimental applications of this model are well documented [

theoretical work by studying the direction and stability of the periodic solutions of the model.

In this work we study the periodic solutions of the repressilator by means of a bifurcation analysis. In Section 2 we compute the steady state solutions and show that these undergo a transition from stable to unstable giving rise to a limit-cycle born in a Hopf bifurcation. In Section 3 we present a nonlinear analysis of the equations, which involves a center manifold reduction on the six-dimensional system. The latter yields closed form expressions for the steady state, frequency, and amplitude of the oscillation, all of which are analyzed through a parameter study in Section 5 and confirmed in a numerical continuation analysis in Section 6. In Section 7 we present our conclusions and discuss the biological significance of our results.

We start our analysis by showing that Equations (1)-(2) have at least one biologically significant equilibrium solution. The equilibria are found by setting

d p i d t = d m i d t = 0 . This gives

0 = − m i * + α 1 + ( p j * ) n + α 0 (3)

0 = p i * − m i * , (4)

where ( m 1 * , m 2 * , m 3 * , p 1 * , p 2 * , p 3 * ) represents the steady state solution. Substituting (4) into (3) we obtain

p i * = α 1 + ( p j * ) n + α 0 , (5)

which can be transformed into a nonlinear algebraic system of equations. The solutions to the latter can be approximated using a numerical root finding technique, such as Newton’s method for systems of nonlinear equations. However, in this work we are interested in biologically significant solutions where m i * = p i * = p * . Substituting the latter into (5) gives the following polynomial

( p * ) n + 1 − α 0 ( p * ) n + p * − ( α + α 0 ) = 0 , (6)

and since we are only interested on the positive real roots, by Descartes’ rule of signs the polynomial (6) has either one or three real positive roots for all n ≥ 2 . This shows that the system has at least one positive biologically significant equilibrium solution when n ≥ 2 . Simulations and plots are provided in Section 5.

To find the stability of the steady state, p i * = m i * = p * , we define ξ i and η i to be deviations from equilibrium ξ i = m i − p * and η i = p i − p * . Substituting these into Equations (1) and (2) results in the following nonlinear system

d ξ i d t = − ( ξ i + p * ) + α 1 + ( η j i + p * ) n + α 0 (7)

d η i d t = − β ( η i − ξ i ) (8)

where i = 1 , 2 , 3 and j 1 = 3 , j 2 = 1 , and j 3 = 2 . Expanding for small values of η j , Equation (7) becomes

d ξ i d t = − ξ i + A η j i + K 2 η j i 2 + K 3 η j i 3 + ⋯ (9)

where the Taylor coefficients A, K 2 , and K 3 are given by

A = − α n ( p * ) n − 1 ( 1 + ( p * ) n ) 2 (10)

K 2 = α n ( p * ) n − 2 ( n ( p * ) n + ( p * ) n − n + 1 ) 2 ( 1 + ( p * ) n ) 3 (11)

K 3 = − α n ( p * ) n − 3 ( ( n + 1 ) ( n + 2 ) p 2 n + 4 ( 1 − n 2 ) p n + ( n − 2 ) ( n − 1 ) ) 6 ( 1 + ( p * ) n ) 4 . (12)

Next we analyze the linearized system coming from Equations (8) and (9)

d ξ i d t = − ξ i + A η j i (13)

d η i d t = − β η i + β ξ i (14)

which were obtained by truncating the nonlinear terms in Equation (9). The Jacobian at the origin for system (13)-(14) is

J = [ − 1 0 0 0 0 A 0 − 1 0 A 0 0 0 0 − 1 0 A 0 β 0 0 − β 0 0 0 β 0 0 − β 0 0 0 β 0 0 − β ] (15)

and the associated characteristic equation is found by setting d e t ( λ I − J ) = 0 , which gives the following equation

( λ + 1 ) 3 ( λ + β ) 3 − A 3 β 3 = 0. (16)

Since the steady state is stable then the real parts of the eigenvalues are all negative, and as we vary β there is a critical value, β = β c r , where the first pair of complex conjugate eigenvalues cross the imaginary axis. Thus substituting λ = ± i ω 0 into (16) gives

( β c r + 1 ) 2 β c r = 3 A 2 4 + 2 A (17)

which is the condition for a change in stability and a Hopf bifurcation. Notice that for β = β c r (i.e. λ = ± i ω 0 ) the system (13) and (14) will exhibit solutions of the form

ξ i ( t ) = A i c o s ( ω 0 t + ϕ i ) (18)

η i ( t ) = B i cos ( ω 0 t ) (19)

where A i and B i are the amplitudes of the ξ i ( t ) and η i ( t ) oscillations, and where ϕ i is a phase angle. As we add a small detuning off of the critical value, β = β c r + Δ , the nonlinear system (1) and (2) is expected to exhibit periodic solutions, which come with a change in stability for the steady state. This change in stability is given by Equation (17), where the stable region is

given by ( β + 1 ) 2 β > 3 A 2 4 + 2 A and the unstable by ( β + 1 ) 2 β < 3 A 2 4 + 2 A . In Section 5

we compute and plot the associated stability diagrams for some parameter values.

We use a center manifold reduction to determine the amplitude and direction of the limit cycle bifurcation. This will be accomplished by studying the system at the critical parameter values β = β c r and λ = ± i ω 0 . Solving Equation (17) for β c r we obtain

β c r = 3 A 2 − 4 A − 8 4 A + 8 ± A 9 A 2 − 24 A − 48 4 A + 8 (20)

where A is given by Equation (10). Substituting condition (17) into (16) for β = β c r , setting λ = ± i ω 0 , and solving for ω 0 gives

ω 0 = 3 β c r A 2 ( β c r + 1 ) (21)

which consists of two branches associated with the ± in Equation (20) for β c r . More details presented in Section 5.

We start the analysis by expressing system (8) and (9) as follows

d x d t = J x + F ( x ) (22)

where x ∈ ℝ 6 so that x = ( ξ 1 , ξ 2 , ξ 3 , η 1 , η 2 , η 3 ) T , J is given by Equation (15), and

F ( x ) = [ K 2 η 3 2 + K 3 η 3 3 K 2 η 1 2 + K 3 η 1 3 K 2 η 2 2 + K 3 η 2 3 0 0 0 ] + O ( ‖ x ‖ 4 ) . (23)

For β = β c r and ω = ω 0 we know that J has a complex conjugate pair of eigenvalues on the imaginary axis, λ = ± i ω 0 , and thus we let q , p ∈ ℂ 6 be the associated eigenvectors corresponding to i ω 0 and − i ω 0 , respectively. These eigenvectors will satisfy the following three conditions

J q = i ω 0 q (24)

J * p = − i ω 0 p (25)

〈 p , q 〉 = 1 (26)

where 〈 p , q 〉 = ∑ k = 1 6 p ¯ k q k is the standard scalar product in ℂ 6 . This yields

q = ( a 1 , a 1 2 ,1, a 1 a 2 , a 1 2 a 2 , a 2 ) T (27)

p = k 1 ( a ¯ 1 , 1 , a ¯ 1 2 , a ¯ 1 a ¯ 3 , a ¯ 3 , a ¯ 1 2 a ¯ 3 ) T (28)

where

a 1 = A β ( 1 + i ω ) ( β + i ω ) , a 2 = β β + i ω , a 3 = 1 + i ω β , k 1 = 1 3 a 1 2 ( 1 + a 2 a 3 ) (29)

and where we have chosen the scaling factor k 1 so that 〈 p , q 〉 = 1 . This completes the linear part of the analysis.

To study the nonlinear part of the center manifold approximation we start by rewriting F ( x ) as follows

F ( x ) = 1 2 B ( x , x ) + 1 6 C ( x , x , x ) + O ( ‖ x ‖ 4 ) (30)

where x = ( x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ) T and the multilinear functions B ( x , y ) and C ( x , y , z ) are given by the following expressions

B ( x , y ) = 2 K 2 ( x 6 y 6 , x 4 y 4 , x 5 y 5 ,0,0,0 ) T (31)

C ( x , y , z ) = 6 K 3 ( x 6 y 6 z 6 , x 4 y 4 z 4 , x 5 y 5 z 5 , 0 , 0 , 0 ) T . (32)

By the Center Manifold Theorem [

x = y q + y ¯ p + w (33)

where y ∈ ℂ 1 is the coordinate of q on the (flat) two-dimensional eigenspace or center subspace spanned by q and p. Here w is the “rest of the solution” which does not lie in the center subspace, but rather on the center manifold. Notice that the center subspace is tangent to the center manifold at the origin, and since y is the coordinate of q such that y = 〈 p , x 〉 , then it will be the projection of x onto the center subspace. This gives the following expression for the time derivative of y = 〈 p , x 〉 = 〈 p , y q + y ¯ p + w 〉

d y d t = i ω 0 y + 〈 p , F 〉 (34)

where F = F ( y q + y ¯ q ¯ + w ) and since y ∈ ℂ 1 then it may written as y = y 1 + i y 2 so that Equation (34) can be expressed as follows

d y 1 d t = − ω 0 y 2 + Re 〈 p , F 〉 (35)

d y 2 d t = ω 0 y 1 + Im 〈 p , F 〉 . (36)

Equations (35) and (36) represent the flow on the flat 2-dimensional eigenspace, which will yield closed form expressions for the limit cycle oscillations. To calculate y 1 and y 2 we start by solving Equation (33) in terms of w and substitute y = 〈 p , x 〉 to obtain

w = x − 〈 p , x 〉 q − 〈 p ¯ , x 〉 q ¯ . (37)

Taking the derivative of the latter and substituting Equation (34) we obtain

d w d t = J w + F − 〈 p , F 〉 q − 〈 p ¯ , F 〉 q ¯ (38)

where

〈 p , F 〉 = 1 2 G 20 y 2 + G 11 y y ¯ + 1 2 G 02 y ¯ 2 + y 〈 p , B ( q , w ) 〉 + y ¯ 〈 p , B ( q ¯ , w ) 〉 + h .o .t (39)

and

G 20 = 〈 p , B ( q , q ) 〉 = 2 a 1 a 2 2 k ¯ 1 ( a 1 2 + a 1 + 1 ) ( a 1 3 − a 1 2 + 1 ) (40)

G 11 = 〈 p , B ( q , q ¯ ) 〉 = 2 a 1 a 2 a ¯ 2 k ¯ 1 ( a 1 3 a ¯ 1 2 + a ¯ 1 + 1 ) (41)

G 02 = 〈 p , B ( q ¯ , q ¯ ) 〉 = 2 a ¯ 2 2 k ¯ 1 ( a 1 2 a ¯ 1 4 + a ¯ 1 2 + a 1 ) . (42)

The normal form for the local parametrization of the truncated center manifold (neglecting cubic and higher order terms) is given by (see [

w ( y , y ¯ ) = w 20 y 2 + w 11 y y ¯ + w 02 y ¯ 2 (43)

which we compute by taking the derivate and equating to Equation (38). The latter yields the following expressions for the parametrization coefficients

w 20 = 1 2 ( 2 i ω 0 I − J ) − 1 H 20 (44)

w 11 = − J − 1 H 11 (45)

w 02 = − 1 2 ( 2 i ω 0 I + J ) − 1 H 02 (46)

where

H 20 = B ( q , q ) − G 20 q − 〈 p ¯ , B ( q , q ) 〉 q ¯ (47)

H 11 = B ( q , q ¯ ) − G 11 q (48)

H 02 = B ( q ¯ , q ¯ ) − G 02 q − 〈 p ¯ , B ( q ¯ , q ¯ ) 〉 q ¯ (49)

and where we omit the expressions of Equations (47)-(49) for brevity. Equation (43) represents the center manifold approximation, w ( y , y ¯ ) , which we substitute into (34) to obtain the flow on the 2-dimensional center subspace

d y d t = i ω 0 y + T 0 y ¯ 2 + T 1 y 3 + T 2 y 2 y ¯ + T 3 y y ¯ 2 + T 4 y ¯ 3 (50)

where

T 0 = 1 2 G 02 (51)

T 1 = 1 2 〈 p , B ( q , w 20 ) 〉 (52)

T 2 = 〈 p , B ( q , w 11 ) 〉 + 1 2 〈 p , B ( q ¯ , w 20 ) 〉 (53)

T 3 = 〈 p , B ( q ¯ , w 11 ) 〉 + 1 2 〈 p , B ( q , w 02 ) 〉 (54)

T 4 = 1 2 〈 p , B ( q ¯ , w 02 ) 〉 . (55)

Since y = y 1 + i y 2 then Equation (50) can be expressed as follows

d y 1 d t = − ω 0 y 2 + Re { H ( y 1 , y 2 ) } (56)

d y 2 d t = ω 0 y 1 + Im { H ( y 1 , y 2 ) } (57)

where

H ( y 1 , y 2 ) = T 0 y ¯ 2 + T 1 y 3 + T 2 y 2 y ¯ + T 3 y y ¯ 2 + T 4 y ¯ 3 . (58)

Using the expressions for Equations (56) and (57) we may express the results in terms of polar coordinates and use a near-identity transformation to change the flow to the following equations

d r d t = Q r 3 + O ( r 5 ) , d θ d t = ω 0 + O ( r 2 ) (59)

where the stability coefficient is given as follows

Q = 1 2 ω 0 Re [ 〈 p , C ( q , q , q ¯ ) 〉 − 2 〈 p , B ( q , J − 1 B ( q , q ¯ ) ) 〉 + 〈 p , B ( q ¯ , ( 2 i ω 0 I − J ) − 1 B ( q , q ) ) 〉 ] . (60)

We refer the reader to [_{0}, q, and p from Equations (15), (21), (27), and (28), respectively, yields the stability coefficient for the system at the critical parameter values. For brevity, we omit here the full expression for Q and present the corresponding numerical plots and results in Section 5.

In this section we use the center manifold computation to approximate the amplitude of the limit cycle born at the Hopf bifurcation. Our efforts in the previous chapter gave the expressions of the Hopf point at the critical parameter values, which provided the “reduced” system at β = β c r and λ = ± i ω 0 . In this chapter we take the second step to compute the normal form of a system with bifurcating parameters, which consists in using a perturbation method to add “unfolding” terms to the reduced normal form found previously. We begin the perturbation approach by adding a small detuning off of the critical value

β = β c r + Δ , | Δ | ≪ 1 (61)

so that the nonlinear system (1)-(2) is expected to exhibit periodic solutions. This occurs due to the transversality condition of the eigenvalues which will yield a small increase in the real and imaginary parts of the λ = ± i ω 0 . We thus assume the resulting expression is of the form λ = R ± i Ω , where R and Ω are given by R = R 1 Δ + O ( Δ 2 ) and Ω = ω 0 + ω 1 Δ + O ( Δ 2 ) . Thus Equations (35) and (36) will take the approximate form

d y 1 d t = R y 1 − Ω y 2 + Re 〈 p , F 〉 (62)

d y 2 d t = R y 2 + Ω y 1 + Im 〈 p , F 〉 . (63)

which can be transformed into polar coordinates (and by means of a near-identity transformation) into the following flow on the center subspace

d r d t = R r + Q r 3 + O ( r 5 ) (64)

d θ d t = Ω + O ( r 2 ) . (65)

Setting Equation (64) to zero and solving for r gives the limit cycle amplitude as

r 2 = − R Q (66)

where Q is given by Equation (60) and R is found by analyzing the linearized system (13) and (14) which has solutions of the form

ξ i = A i e λ t (67)

η i = B i e λ t (68)

where λ = R ± i Ω . Substituting the latter two equations and (61) into (13) and (14), linearizing for small Δ , and solving for R and Ω we obtain expressions of the form

R = R 1 Δ (69)

Ω = ω 0 + ω 1 Δ (70)

where R 1 and ω 1 are long and complicated expressions in terms of β, α, n, and A omitted here for brevity. Substituting (69) and (60) into (66) gives the closed form expression for the limit cycle amplitude, r, which can be computed numerically for different parameter values. In the next chapter we present our results.

In this section we use our closed form expressions to obtain more information on the dynamics of the system. We start by computing the steady state concentration, ( p * , p * , p * , p * , p * , p * ) , for different parameter values. For our model, a biologically significant range for the Hill coefficient, n, is given by 2 ≤ n ≤ 10 (see [_{0}, and n, where we note that β does not play a role in the values of p * .

Next we use Equation (17) to obtain the α-β stability diagram for the system. To find an analytical expression involving only α we solve Equation (6) for p * when α 0 = 0 and n = 2 to obtain the real solution

p * = ( 27 α 2 + 4 6 3 + α 2 ) 1 3 − 1 3 ( 27 α 2 + 4 6 3 + α 2 ) − 1 3 (71)

which we use to substitute into the Hopf bifurcation condition (17) where A is given by Equation (10). Alternatively, we may use Equation (20) to compute

β c r which will give us two branches: β 1 = 3 A 2 − 4 A − 8 4 A + 8 + A 9 A 2 − 24 A − 48 4 A + 8 and β 2 = 3 A 2 − 4 A − 8 4 A + 8 − A 9 A 2 − 24 A − 48 4 A + 8 .

space divided into stable and unstable regions for α 0 = 0 and n = 2 . The dividing curve is formed by a lower β_{1}-branch and an upper β_{2}-branch with the two meeting at α = 4.2426 , which is where the Hopf bifurcation occurs and the system exhibits periodic solutions. In addition, for

To study the direction of the Hopf bifurcation we find the stability coefficient, Q, for the system. Using Equation (60) we obtain

From Equations (66) and (69) we see that the amplitude, r, of the oscillation is

given by the product − R 1 Q ⋅ Δ .

α 0 = 0 , n = 2 , β = β 1 , and Δ = 0.1 . Notice that the amplitude of the oscillation grows larger as α increases, which shows how the limit cycle’s size changes as we follow the bifurcation curve β for a some fixed | Δ | ≪ β c r . For β = β 2 we also obtain growing oscillations as α increases to 5.4426 and after which the amplitudes start decreasing (figure not presented here).

p * = 3 and from (10) we obtain A = − 1.8 which gives ω 0 = − 1.492 and β c r = 22.255 using Equations (21) and (20), respectively. Substituting these results into the center manifold analysis we obtain Q = − 0.151 from Equation (60) and r = 2.996 from Equation (66). We confirm these results via MATLAB’s ode45.m by numerically simulating the system for the appropriate parameter values to obtain the time course in

In this section we complete the Hopf bifurcation analysis by computing the bifurcation diagram using the numerical continuation software package AUTO. We start by setting ( β , α , n , α 0 ) = ( 10 , 1 , 2 , 0 ) as the starting point and define α as the primary continuation parameter. For this particular set of parameters, the system exhibits stable steady state behavior with p * ≈ 0.682 as predicted in

The second bifurcation curve, presented in

This work provides a Hopf bifurcation study of the repressilator model. The linear analysis of the model yields analytical expressions for the steady state solution and critical parameter values. These were used to categorize stability diagrams separating α-β parameter regions where the bifurcation occurs. Setting our parameters close to their critical values allows the system to be close to the bifurcation point, which provides the set up to carry out a center manifold reduction on the system. The final outcome of the center manifold reduction allowed us to find closed form approximate expressions for the amplitude of the limit cycle born at the Hopf, frequency of the oscillation, and the stability coefficient. These expressions, along with our linear analysis results, are then used in a parameter study to construct a more comprehensive picture of the system’s dynamic behavior. Finally, we confirmed our theoretical results in two ways: 1) by numerically simulating appropriate time courses via MATLAB’s built-in function ode45.m and 2) by numerical continuation of the full nonlinear system using AUTO.

From a biological perspective, the two main parameters (α and β) have “opposite” effects on the system. The parameter α roughly represents the maximum production or transcription rate of mRNA in the absence of repression

[1, 2]. Based on the mathematical structure of the Hill term, α 1 + p j n , we can

see that an increase in protein concentration, p j , makes the term smaller and

thus decreases its influence on the production rate of mRNA, d m i d t . On the

other hand, the parameter β roughly represents the ratio between mRNA and protein degradation. Thus, if β = (degradation of protein)/(degradation of mRNA) then increasing β means that either degradation of protein becomes faster and/or degradation of mRNA becomes slower. These rough descriptions of α and β explain how their associated opposite effects give rise to negative feedback, which is an essential feature of any biological oscillator. These conclusions can be verified with our computational results presented in

Increasing α for a fixed β creates an opposing reaction, which will offset the cell’s tendency for equilibrium. At the critical value, the system will switch from positive to negative feedback as represented in

simulation for high α confirms the cyclic clockwise repression effect in the network’s protein-mRNA concentration dynamics. In

Verdugo, A. (2018) Hopf Bifurcation Analysis of the Repressilator Model. American Journal of Computational Mathematics, 8, 137-152. https://doi.org/10.4236/ajcm.2018.82011