Cyclic Negative Feedback Systems: What is the Chance of Oscillation?

Abstract

Many biological oscillators have a cyclic structure consisting of negative feedback loops. In this paper, we analyze the impact that the addition of a positive or a negative self-feedback loop has on the oscillatory behavior of the three negative feedback oscillators proposed by Tsai et al. (Science 231:126–129, 2008) where, in contrast with numerous oscillator models, the interactions between elements occur via the modulation of the degradation rates. Through analytical and computational studies we show that an additional self-feedback affects the oscillatory behavior. In the high-cooperativity limit, i.e., for large Hill coefficients, we derive exact analytical conditions for oscillations and show that the relative location between the dissociation constants of the Hill functions and the ratio of kinetic parameters determines the possibility of oscillatory activities. We compute analytically the probability of oscillations for the three models and show that the smallest domain of periodic behavior is obtained for the negative-plus-negative feedback system whereas the additional positive self-feedback loop does not modify significantly the chance to oscillate. We numerically investigate to what extent the properties obtained in the sharp situation applied in the smooth case. Results suggest that a switch-like coupling behavior, a time-scale separation, and a repressilator-type architecture with an even number of elements facilitate the emergence of sustained oscillations in biological systems. An additional positive self-feedback loop produces robustness and adaptability whereas an additional negative self-feedback loop reduces the chance to oscillate.

Appendices

Appendix 1: Jacobian Matrix of the Smooth System

For the negative feedback-only oscillator, the Jacobian matrix is given by

$$\begin{aligned} J_\mathrm{No} = \left( \begin{array}{c@{\quad }c@{\quad }c} -k_1 - k_2 S_1(C) &{} 0 &{} - \frac{k_2 n_1 K_1^{n_1} C^{n_1-1}}{( K_1^{n_1}+ C^{n_1})^2} A \\ - \frac{k_4 n_2 K_2^{n_2} A^{n_2-1}}{( K_2^{n_2}+ A^{n_2})^2} B &{}-k_3 - k_4 S_2(A) &{} 0 \\ 0 &{} - \frac{k_6 n_3 K_3^{n_3} B^{n_3-1}}{( K_3^{n_3}+ B^{n_3})^2} C &{} -k_5 - k_6 S_3(B) \end{array} \right) , \end{aligned}$$

where it is easy to see that each term is negative. For the NN and PN oscillators, only the term in the first row and first column differs and we have

$$\begin{aligned} (J_\mathrm{NN})_{11}&= -k_1 - k_2 S_1(C) - k_7 \frac{(n_4+1) K_4^{n_4} A^{n_4} + A^{2 n_4}}{( K_4^{n_4}+ A^{n_4})^2} \end{aligned}$$

that is always negative and

$$\begin{aligned} (J_\mathrm{PN})_{11}&= -k_1 - k_2 S_1(C) + k_7 \frac{n_4 K_4^{n_4} A^{n_4-1} - (n_4+1)K_4^{n_4} A^{n_4} - A^{2 n_4}}{( K_4^{n_4}+ A^{n_4})^2} \end{aligned}$$

which has a positive term. Other components are given by $(J_\mathrm{NN})_{ij} =(J_\mathrm{PN})_{ij} = (J_\mathrm{No})_{ij}$ for $(i,j) \ne (1,1)$

Appendix 2: Piecewise Linear Oscillators

Our analysis is based on the observation that each parameter $K_i$ defines a threshold plane dividing the phase space into rectangular boxes, the so-called regulatory domains. Inside each regulatory domain, the system is linear and the analysis is straightforward.

1.1 Fixed Points

The three different oscillators may admit fixed points depending on parameter values. Due to the discontinuity of the vector field, it is convenient to distinguish between two classes of fixed points: “regular” steady points and “singular” steady points. Regular steady points are defined following the well-established theory of smooth dynamical systems. Singular steady states are characterized by the fact that at least one of its components lies on a threshold and thus require a specific treatment.

Let $X_\mathrm{ss} = (A_\mathrm{ss},B_\mathrm{ss},C_\mathrm{ss})$ be a fixed point. The regular fixed points of the No-oscillator and the corresponding conditions of existence are given by

$$\begin{aligned} \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} A_\mathrm{ss}=1 &{} \text {for} &{} C_\mathrm{ss}<K_1 &{} \text {and} &{} A_\mathrm{ss}=K_2^* &{} \text {otherwise}, \\ B_\mathrm{ss}=1 &{} \text {for} &{} A_\mathrm{ss}<K_2 &{} \text {and} &{} B_\mathrm{ss}=K_3^* &{} \text {otherwise}, \\ C_\mathrm{ss}=1 &{} \text {for} &{} B_\mathrm{ss}<K_3 &{} \text {and} &{} C_\mathrm{ss}=K_1^* &{} \text {otherwise}. \end{array}. \end{aligned}$$

It is easy to show that when a regular fixed point exists it is stable. Each species, $A,\,B,\hbox { and }C$, can formally admit two different values at its resting state and thus we can distinguish between eight analytically different regular fixed points. The different possible steady states are the so-called focal points (Glass and Pasternack 1978a, b; Mestl et al. 1995a, b) of the associated regulatory domain. If the focal point is inside its regulatory domain, it is a stable steady state of the system. Otherwise the system will leave the current regulatory domain and enter a new one that may have a different focal point.

The steady state $X_\mathrm{ss}$ is a singular fixed point if $0 \in \mathcal {F}(X_\mathrm{ss})$ where $\mathcal {F}$ is the multi-valued function obtained when the Heaviside function of the component value that lies on its threshold is allowed to vary in $(0,1)$. For the No-oscillator, the only singular fixed point is $(K_2,K_3,K_1)$ that exists when a solution $(\varTheta _A,\varTheta _B,\varTheta _C) \in [0,1]^3$ can be found to the following system:

$$\begin{aligned} k_1(1-K_3) - k_2 K_3 \varTheta _C = 0, \\ k_3(1-K_3) - k_2 K_3 \varTheta _A = 0,\\ k_5(1-K_3) - k_2 K_3 \varTheta _B = 0. \end{aligned}$$

We obtain the conditions

$$\begin{aligned} K_1^*< K_1 < 1, \quad K_2^*< K_2 < 1, \quad K_3^*< K_3 < 1, \end{aligned}$$

that (as we will show hereinafter) coincide with the conditions for oscillations. The state $(K_2,K_3,K_1)$ is the so-called loop characteristic state (Snoussi and Thomas 1993) of the No-oscillator. We show in “Stability of Origin” section of Appendix 2 that this singular fixed point is always unstable.

For the NN-oscillator, the values and the conditions for the existence of regular fixed points are identical to those obtained in the No-oscillator model for the two components $B_\mathrm{ss}\hbox { and }C_\mathrm{ss}$. For $A_\mathrm{ss}$, we get

$$\begin{aligned} \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} A_\mathrm{ss}=1 &{} \text {for} &{} C_\mathrm{ss}<K_1 &{} \text {and} &{} K_4 > 1 , \\ A_\mathrm{ss}=K_{4,b}^*&{} \text {for} &{} C_\mathrm{ss}<K_1 &{} \text {and} &{} K_4 < K_{4,b}^*, \\ A_\mathrm{ss}=K_{2}^*&{} \text {for} &{} C_\mathrm{ss}>K_1 &{} \text {and} &{} K_4 > K_{2}^* ,\\ A_\mathrm{ss}=K_{4,a}^*&{} \text {for} &{} C_\mathrm{ss}>K_1 &{} \text {and} &{} K_4 < K_{4,a}^* , \end{array} \end{aligned}$$

and, as for the No-oscillator, a regular fixed point, when it exists, is stable. Singular steady points with at least one component on a discontinuity plane may exist. As previously the loop characteristic state $(K_2,K_3,K_1)$ is a singular steady state, and for $K_4>K_2$, the conditions of existence remain the same than for the No-oscillator. If $K_4<K_2$ conditions of existence become

$$\begin{aligned} K_1^*< K_1 < 1, \quad K_{4,a}^*< K_2 < K_{4,b}^*, \quad K_3^*< K_3 < 1. \end{aligned}$$

We show in “Stability of Origin” section of Appendix 2 that this singular fixed point is always unstable. Moreover, an additional singular fixed point may occur on the discontinuity plane $A_\mathrm{ss}=K_4$. This singular fixed point is induced by the additional self-feedback loop and defines a second loop characteristic state of the oscillator. It is easy to show that this steady state exists and is stable for $K_{4,b}^*< K_4 < 1\hbox { and }C_\mathrm{ss}<K_1$ or for $K_{4,a}^*<K_4<1\hbox { and }C_\mathrm{ss}>K_1$. The A-component of the steady state, $A_\mathrm{ss}$, can formally take five different values, so that we distinguish between twenty analytically different stable fixed points for the NN-oscillator.

For the PN-oscillator, the steady state values $B_\mathrm{ss}\hbox { and }C_\mathrm{ss}$ and the corresponding conditions of existence remain the same. For $A_\mathrm{ss}$, we obtain the following values and conditions of existence:

$$\begin{aligned} \begin{array}{c@{\quad }c@{\quad }c} A_\mathrm{ss}=1 &{}\text {for} &{} C_\mathrm{ss}<K_1, \\ A_\mathrm{ss}=K_{2}^* &{}\text {for} &{} C_\mathrm{ss}>K_1 \quad \text {and} \quad K_4 > K_{2}^*, \\ A_\mathrm{ss}=K_{4}^* &{} \text {for}&{} C_\mathrm{ss}>K_1 \quad \text {and} \quad K_4< K_{4}^* . \end{array} \end{aligned}$$

The singular fixed point $(K_2,K_3,K_1)$ when it exists is unstable (see “Stability of Origin” section of Appendix 2). Moreover, as for the NN-oscillator, the additional self-feedback loop may induce the existence of a singular fixed point on the discontinuity plane $A_\mathrm{ss}=K_4$. It is easy to show that this singular fixed point is always unstable. To sum up, the PN-oscillator has twelve different forms of stable steady states. It is worth noting that for $C_\mathrm{ss}>K_1$ the two stable fixed points $A_\mathrm{ss}=K_{2}^*\hbox { and }A_\mathrm{ss}=K_{4}^*$ can coexist when $K_{2}^*<K_4< K_{4}^*$.

To summarize, when a regular fixed point exists, it is stable. In addition, singular fixed points may exist but are unstable except the singular fixed point of the NN-oscillator satisfying $A_\mathrm{ss}=K_4$. It is worth mentioning that, for a given set of parameters, a stable fixed point, when it exists, is unique except for the PN-oscillator where two stable fixed points may coexist.

1.2 Oscillations

Based on the fixed points analysis done in “Fixed Points” section of Appendix 2, it is possible to derive analytically the conditions on parameters of the system for the existence of stable fixed points. These conditions are monitored by the relative location between $K_i$, the unity, and the associated critical value $K_i^*$. For instance, for the No-oscillator, it is easy to check that when $K_1<K_1^*,\,K_2>K_2^*, \hbox { and }K_3<1$, the point $(K_2^*,1,K_1^*)$ is a stable fixed point. It is thus possible to derive exactly the sets of parameters for which there are no stable fixed points taking the complementary of the sets for which stable fixed points exist. These sets are given by (14), (15), and (16) for the three different oscillators, respectively. Since a trajectory cannot escape from the box $D=[0,1]^3$, an oscillatory pattern exists inside the box. It is suspected that this oscillatory activity corresponds to a limit cycle (see the discussion in the section 3.5.1).

It is worth noting that (i) the limit cycles do not occur through the destabilization of a fixed point undergoing an Andronov–Hopf bifurcation and therefore we are not limited here to small size limit cycles. (ii) Multistability between two stable fixed points exists only for the PN-oscillator: one fixed point satisfies $A_\mathrm{ss}=K_4^*$ and the other $A_\mathrm{ss}=K_2^*$. This is in agreement with the result stating that positive loops are responsible for multistability (Snoussi 1998). (iii) When a regular fixed point exists, it is stable and only singular fixed points can be unstable. In particular, the singular fixed point $(K_2,K_3,K_1)$ may exist for the three different oscillators and is always unstable. For the NN- and PN-oscillator an additional singular fixed point may occur on the switching surface $A=K_4$ and can be stable for the NN-oscillator whereas it is always unstable for the PN-oscillator.

If we discard bistability, the sufficient conditions for oscillations are necessary. However, it is known that such a bistability-exclusion can be broken by additional interactions that modify the cyclic nature or the monotonicity of the model and allow for multistability between a fixed point and a limit cycle, chaotic solutions or dynamics not allowed in $R^2$. For instance bistability may occur in two-side interaction systems, i.e., there exists $j$ such that $\dot{x}_j = f_j(x_{j},x_{j-1},x_{j+1})$ (Li et al. 2012), but not chaotic solutions (Elkhader 1992). However, if the monotonicity property fails then chaotic solutions may appear (Di Cera et al. 1989). Moreover, a subcritical Hopf bifurcation generating bistability can be obtained in monotone negative feedback systems with a variable self-feedback loop (Hasty et al. 2002) indicating that bistability-exclusion probably requires monotonicity conditions on the self-interaction term. However, bistability often occurs in a narrow region of parameter space (see (Hasty et al. 2002) for instance) and one can expect that the probability of oscillations derived from steady state analysis constitutes a good approximation.

1.3 Calculation of $P(K_2<K^*_{4,a})$

We can easely check that

$$\begin{aligned} P(K_2<K^*_{4,a}) = \frac{1}{\bar{K}_2 \bar{k}_1 \bar{k}_2 \bar{k}_7} \left( G( \bar{k}_2 + \bar{k}_7) - G(\bar{k}_7) - G(\bar{k}_2) + G(0) \right) \end{aligned}$$

(33)

where

$$\begin{aligned} G(u) = \int \limits _{0}^{\bar{k}_1} x(x+u) \ln (x+u) \mathrm{d}x. \end{aligned}$$

Integrating by parts we obtain

$$\begin{aligned} G(u) = \frac{(\bar{k}_1+u)^2}{3} \left( \bar{k}_1 - \frac{u}{2}\right) \ln (\bar{k}_1+u) + \frac{u^3}{6} \ln (u) - \frac{\bar{k}_1}{36} \left( 3 \bar{k}_1 u - 6 u^2 + 4 \bar{k}_1^2\right) \end{aligned}$$

that completes the analytical expression of $P(K_2<K^*_{4,a})$.

Let $r_\mathrm{c}=k_1/k_2\hbox { and }r_\mathrm{s}=k_1/k_7$, analytical calculations give

$$\begin{aligned} P(K_2<K^*_{4,a})&= \frac{1}{K} \bigg ( \frac{1}{3} + \frac{r_\mathrm{s}}{6 r_\mathrm{c}^2} \ln r_\mathrm{c} + \frac{r_\mathrm{c}}{6 r_\mathrm{s}^2} \ln r_\mathrm{s} + \frac{r_\mathrm{c}r_\mathrm{s}}{6} \left( \frac{1}{r_\mathrm{c}}+ \frac{1}{r_\mathrm{s}}\right) ^3 \ln \left( \frac{1}{r_\mathrm{c}}+ \frac{1}{r_\mathrm{s}}\right) \\&\quad +\, \frac{r_\mathrm{s}}{3} \left( \frac{1}{r_\mathrm{c}} +1 \right) ^2 \left( \frac{1}{2}-r_\mathrm{c}\right) \ln \left( 1 + \frac{1}{r_\mathrm{c}}\right) \\&\quad +\, \frac{r_\mathrm{c}}{3} \left( \frac{1}{r_\mathrm{s}} + 1 \right) ^2 \left( \frac{1}{2}-r_\mathrm{s}\right) \ln \left( 1 + \frac{1}{r_\mathrm{s}}\right) \\&\quad -\, \frac{1}{3} \left( \frac{1}{r_\mathrm{c}}+\frac{1}{r_\mathrm{s}}+1 \right) ^2 \left( \frac{r_\mathrm{s}}{2}+\frac{r_\mathrm{c}}{2}-r_\mathrm{c} r_\mathrm{s} \right) \ln \left( 1 + \frac{1}{r_\mathrm{c}}+ \frac{1}{r_\mathrm{s}} \right) \bigg ). \end{aligned}$$

1.4 Stability of the Origin

For the No-oscillator, the stability of singular steady state $(K_2,K_3,K_1)$ is determined by the stability of the origin of

$$\begin{aligned} \dot{x}&= k_1 (1 - K_2 - x) - k_2 (x+K_2) \varTheta (z), \\ \dot{y}&= k_3 (1 - K_3 - y) - k_4 (y+K_3) \varTheta (x), \\ \dot{z}&= k_5 (1 - K_1 - z) - k_6 (z+K_1) \varTheta (y). \end{aligned}$$

which is given by the study of the trajectories of the system

$$\begin{aligned} \dot{x}&= k_1 (1 - K_2) - k_2 K_2 \varTheta (z), \\ \dot{y}&= k_3 (1 - K_3) - k_4 K_3 \varTheta (x), \\ \dot{z}&= k_5 (1 - K_1) - k_6 K_1 \varTheta (y). \end{aligned}$$

We define

$$\begin{aligned} \alpha _i&= k_{2i-1} (1-K_{i+1}), \\ \beta _i&= (k_{2i-1} + k_{2i}) K_{i+1} - k_{2i-1}, \end{aligned}$$

for $i=1,2,3$ where we set here $K_4=K_1$ for convenience. Conditions for the existence of a singular fixed point at the origin lead to $\alpha _i>0\hbox { and }\beta _i>0$. The dynamics can be rewritten as

$$\begin{aligned} \dot{x}&= \alpha _1 \ \ \text {if} \ z<0 \ \text {and} \ - \beta _1 \ \text {otherwise}, \\ \dot{y}&= \alpha _2 \ \ \text {if} \ x<0 \ \text {and} \ - \beta _2 \ \text {otherwise}, \\ \dot{z}&= \alpha _3 \ \ \text {if} \ y<0 \ \text {and} \ - \beta _3 \ \text {otherwise}. \end{aligned}$$

and we have sgn$(\dot{x})=-\text {sgn}(z)$, sgn$(\dot{y})=-\text {sgn}(x)$, and sgn$(\dot{z})=-\text {sgn}(y)$. Note that a similar system is studied in Farcot and Gouzé (2009) but with different assumptions on the parameters. Here we will show using basic calculus that the “local” system is unstable.

The trajectories make revolutions around the origin and pass many times into the plane $x=0$, intersecting it for $y>0$ (and $z<0$) and for $y<0$ (and $z>0$) for one revolution. After possibly a transient, the sign of the components defining the trajectory will follow the cycle $(+,+,-) \rightarrow (+,-,-) \rightarrow (+,-,+) \rightarrow (-,-,+) \rightarrow (-,+,+) \rightarrow (-,+,-)$ (see Fig. 6 where $1$ corresponds to $+\hbox { and }0$ to $-$).

The trajectory of the system defines a mapping of the half plane $P_0\,(x_0=0,\,y_0>0,\,z_0<0)$ into itself. The solution with initial value $x_0=0,\,y_0>0,\,z_0<0$ lies first in the region $(110)$, i.e., $(x>0,\,y>0,\,z<0)$, where the solution has the form

$$\begin{aligned} x(t) = \alpha _1 t, \quad y(t) = y_0 - \beta _2 t, \quad z(t) = z_0 - \beta _3 t. \end{aligned}$$

The trajectory intersects the plane $y=0$ at time $t_a=y_0/\beta _2$ at the point

$$\begin{aligned} x_a = \alpha _1/\beta _2 y_0, \ \ y_a = 0, \ \ z_a = z_0 - \beta _3/\beta _2 y_0, \end{aligned}$$

and enters into the domain $100$. Using similar arguments, we compute the different reaching times, $t_a, \ldots , t_f$, of the different regions ($101,\,001,\,011,\,010,\hbox { and }110$, respectively) together with the corresponding intersection points. We find that from the point $x_0=0,\,y_0>0,\,z_0<0$ in $P_0$ the trajectory first goes back into $P_0$ at the point $x_f=0,\,y_f,\,z_f$ where

$$\begin{aligned} \left( \begin{array}{l} y_f \\ z_f \end{array} \right) = \left( \begin{array}{c@{\quad }c} a_{11} &{} a_{12} \\ a_{21} &{} a_{22} \end{array} \right) \left( \begin{array}{l} y_0 \\ z_0 \end{array} \right) \end{aligned}$$

(34)

with

$$\begin{aligned} a_{11}&= 3 + u_1 + u_2 + u_3 + u_1 u_2 + u_1 u_3 + u_2 u_3 + u_1 u_2 u_3 + \frac{1}{u_1} + \frac{1}{u_3} + \frac{1}{u_1 u_3} ,\\ a_{12}&= - \frac{\beta _2}{\beta _3} \left( 2 + u_1 + u_2 + u_1 u_2 + \frac{1}{u_1} + \frac{1}{u_3} + \frac{1}{u_1 u_3} \right) ,\\ a_{21}&= - \frac{\beta _3}{\beta _2} \left( 1 + u_3 + \frac{u_3}{u_2} + \frac{2}{u_2} + \frac{1}{u_1 u_2} + \frac{1}{u_2 u_3} + \frac{1}{u_1 u_2 u_3}\right) ,\\ a_{22}&= 1 + \frac{1}{u_2} + \frac{1}{u_1 u_2} + \frac{1}{u_2 u_3} + \frac{1}{u_1 u_2 u_3}, \end{aligned}$$

where we set $u_i=\alpha _i/ \beta _i$.

Equation (34) defines a 2D-linear mapping. Let $\lambda _1\hbox { and }\lambda _2$ be the two associated eigenvalues. Since we have $\vert \lambda _1 \vert + \vert \lambda _2 \vert \ge \vert \lambda _1+\lambda _2 \vert = \vert a_{11} + a_{22} \vert > 4$ then at least one eigenvalue is greater than 1 in absolute value. Therefore, the origin is unstable. Numerical investigations suggest than one eigenvalue is large whereas the other is less than one , in absolute value, indicating a saddle configuration and the existence of a stable manifold associated with the origin.

For the two others oscillators, the NN-oscillator and the PN-oscillator, the situation remains the same for $K_4 > K_2$. When $K_4 < K_2$ we define $\alpha _{1,\mathrm{NN}}=\alpha _1 - k_7 K_2\hbox { and }\beta _{1,\mathrm{NN}} = \beta _1 + k_7 K_2$ for the NN-oscillator and $\alpha _{1,\mathrm{PN}}=\alpha _1 + k_7 (1-K_2)\hbox { and }\beta _{1,\mathrm{PN}} = \beta _1 - k_7 (1-K_2)$ for the PN-oscillator. Conditions for the existence of the singular fixed point $(K_2,K_3,K_1)$ give $\alpha _{1,\mathrm{NN}}>0,\,\beta _{1,\mathrm{NN}}>0\hbox { and }\alpha _{1,\mathrm{PN}}>0,\,\beta _{1,\mathrm{PN}}>0$. Study of stability proceeds along the same lines than for the No-oscillator and we show that the origin is unstable.

Appendix 3: The Random Parameter Sets

For the numerical simulation of the smooth oscillators, we used the same parameter distributions as in Tsai et al. (2008). All parameters are dimensionless and (H) holds (see 22) with:

• $K=4$, i.e., we used $K_i\sim U(0,4),\,i=1,2,3,4$,

• $k=10$, i.e., we used $k_1,k_3 \sim U(0,10)$,

• $k_c=1000$; i.e., we used $k_2,k_4,k_6 \sim U(0,1000)$

except for $k_5$ that has been fixed to $1$. For the NN-oscillator and PN-oscillator, we used $k_7\sim U(0,100)$. Each Hill coefficient follows an uniform distribution over $(1 , 4)$.