Explicit time delay

Suppose that the rate of protein synthesis at present (at time t) depends on the concentration of protein at some time in the past (at time t-τ), where τ is the time delay required for transcription and translation. Then, the governing kinetic equation becomes

This model of protein synthesis and degradation was first studied in detail by Mackey and Glass in 1977²¹. For a proper choice of rate constants and time delay, this equation exhibits periodic oscillations, as illustrated in Fig. 1c. The time delay causes the negative feedback control repeatedly to overshoot and undershoot the steady state (Fig. 1d). For details on how to simulate Eq. (2) and all other models in this review, see supplementary information S1 (box).

By ‘proper choice of rate constants…’ we mean that, in order for Y(t) to oscillate, the kinetic parameters—S (signal strength), p (nonlinearity of feedback), K_m (nonlinearity of the removal step), and τ (duration of time delay)—must satisfy specific constraints, illustrated in Fig. 1e and f. For details on how these constraint curves are calculated, see supplementary information S2 (box). The constraints can be summarized in three requirements. (1) The time delay, τ, must be sufficiently long. (For fixed values of p and K_m, there is a minimum value of τ —call it τ_min—below which oscillations are impossible.) (2) The reaction rate laws must be sufficiently ‘nonlinear’. (Oscillations become easier—i.e., τ_min gets smaller—as either p or K_d/K_m increases.) (3) The rates of opposing processes must be appropriately balanced.

To understand the third requirement, we must look more closely at the axes in Figs. 1e and f. The value of ‘time delay’ plotted on the vertical axis is really a dimensionless combination of parameters:

The value of ‘Signal strength’ plotted on the horizontal axis is the dimensionless ratio

For fixed values of p and K_d/K_m, these ratios must lie above a specific curve plotted in the figures. For example, for p = 2 and K_d/K_m = 10 (‘modest’ nonlinearity of the rate laws for protein synthesis and degradation), these ratios must (roughly speaking) satisfy the inequalities: τ / T_degr > 2 and T_degr / T_syn > 1 (from the lowest curve in Fig. 1f). Estimating the time delay for transcription and translation to be about 20 min, we predict that, in order for the negative feedback loop to oscillate, the time scale for protein degradation must be < 10 min and the time scale for protein synthesis must be even shorter. If these conditions are satisfied, then the period of oscillation is (again, roughly speaking) between 2× and 4× the time delay, i.e., approximately 40—80 min.

In the remainder of this review, we intend to show that these four requirements (negative feedback, nonlinearity, time delay, and time-scale constraints) are quite generally true of all biochemical oscillators, provided the notion of ‘time delay’ is suitably generalized.

Time delay by a series of intermediates

Our simple model of negative feedback on gene expression, Eq. (2), exhibits sustained oscillations if there is a sufficiently long time delay between the action of the protein Y on the gene and the appearance of new protein molecules in the cytoplasm. We expressed this requirement as a discrete time delay, τ, in the kinetic equation. Maybe we can dispense with τ if we include the dynamics of mRNA in our model (Fig. 2a). To this end, we write a pair of kinetic equations for X = [mRNA], Y = [Protein]:

Open in a separate window

Figure 2

Multi-component, negative feedback oscillator

a| Negative feedback between mRNA and protein, as described by kinetic equations (3). b| Representative solutions (dashed curves) of the kinetic equations (3), for parameter values: p = 2, K_m/K_d =1, S/K_d =1, k₁ = k_dx = 0.1 min^-1, k_sy = k₂E_T/K_d = 1 min^-1. Notice that every trajectory spirals into the stable steady state located at the black circle. Curves a and b are ‘nullclines’ for differential equations (3), as explained in the text. The small arrows indicate the direction of motion of trajectories as they cross the nullclines. Notice that the nullclines in this figure are identical to the rate curves in Fig. 1b. c| The negative feedback loop taking into account transport of macromolecules between nucleus and cytoplasm. d| Sustained oscillations for the four-component loop in panel c. See supplementary information S1 (box) for details. e| Nonlinearity constraint. For the negative feedback loop to oscillate, p and K_d /K_m must be large enough. f| Time-scale balancing constraint. The half-lives of mRNA in the nucleus and of protein in the cytoplasm must lie in the shaded band in order for the negative feedback loop to oscillate.

In Fig. 2b we plot the ‘nullclines’ of this pair of nonlinear differential equations. The X-nullcline (curve a in Fig. 2b) is the locus of points in the (X,Y) plane where the rate of mRNA synthesis is exactly balanced by the rate of mRNA degradation, that is, where $X=\frac{k_{1}S}{k_{dx}}\frac{K_{\mathit{\text{d}}}^{\mathit{\text{p}}}}{K_{\mathit{\text{d}}}^{\mathit{\text{p}}}+Y^p}$. Along the X-nullcline, dX/dt = 0 and trajectories move horizontally in the (X,Y) plane because there is no change in the X direction but there may be change in the Y direction. The Y-nullcline (curve b) is the locus of points where the rate of protein synthesis is balanced by its rate of degradation, that is, where $X=\frac{k_2{E}_T}{k_{sy}}\frac{Y}{K_m+Y}$. Along the Y-nullcline, dY/dt = 0 and trajectories move vertically in the (X,Y) plane. Where the nullclines intersect (the solid circle in Fig. 2b), the trajectory comes to rest at a steady state (both dX/dt = 0 and dY/dt = 0). The sample trajectories in Fig. 2b (the dashed lines) spiral into the stable steady state. Although the system of reactions may exhibit damped oscillations on the way to the steady state, sustained oscillations in this simple gene regulatory circuit are impossible²². Hence, adding mRNA to the model does not do away with the requirement for an explicit time delay.

Before giving up the idea of replacing the time delay by a series of intermediates in the negative feedback loop, let's recognize that (in eukaryotes) the mRNA and protein molecules need to be transported between nucleus and cytoplasm (Fig. 2c). Equation (3) is readily expanded to four variables: mRNA and protein in the nucleus (X_n, Y_n) and in the cytoplasm (X_c, Y_c). The four-variable negative feedback loop oscillates as naturally as a pendulum (Fig. 2d)!

The constraint diagrams in Fig. 2 underscore the conclusions of the previous section. Figure 2e shows that the kinetic rate laws must be sufficiently nonlinear (p and/or K_d/K_m large enough). Figure 2f shows that the turnover rates for mRNA in the nucleus (k_dxn = 0.693/half-life of X_n) and for protein in the cytoplasm (k_dyc = 0.693/half-life of Y_c) must be properly balanced. Furthermore, if either k_dxn → ∞ or k_dyc → ∞, the negative feedback loop reduces to three components, (X_c, Y_c, Y_n) or (X_n, X_c, Y_n) respectively, and yet oscillations are still possible. Hence, we conclude that oscillations are impossible in a two-component negative feedback loop (Fig. 2b) but possible in a three-component negative feedback loop (Fig. 2f)²².

This mechanism (negative feedback on gene expression with three or more components in the feedback loop), which we have used to illustrate the four basic principles of biochemical oscillations, was first put forward by Brian Goodwin^{23, 24} in mid-1960s as a model for periodic enzyme synthesis in bacteria. Our calculations in this section show that, with an effective time delay (transcription + translation + reaction intermediates) of about 20 min, this model gives a period of about 1 h, which is close to the observed periods of such rhythms²⁵. This mechanism has also been a favorite model for circadian rhythms in flies and mammals, governed by the PER protein, which moves into the nucleus and blocks expression of the PER gene^{26, 27}. Of course, to get a period of 24 h, the time delay for the feedback signal must be considerably longer than the delay expected for transcription, translation and nuclear transport. It is believed that PER undergoes slow post-translational modifications (phosphorylations) in the cytoplasm before it returns to the nucleus²⁰.

The possibility of sustained oscillations in a three-component negative feedback loop was used by Elowitz and Leibler¹¹ to design the ‘Repressilator’, a synthetic gene regulatory network in E. coli consisting of three operons, each one expressing a protein that represses the next operon in the loop. The successful engineering of the Repressilator was a foundational triumph of the nascent field of ‘synthetic biology’ and a vindication of the theoretical ideas of Goodwin^{23, 24}, Griffith²², Goldbeter²⁶ and others.

Class 1: Delayed Negative Feedback Loops

By delayed negative feedback we refer to 3 or more components connected in a single loop by positive and negative links, with an odd number of inhibitory links. Delayed negative feedback is often used to model oscillatory responses in molecular cell biology. Besides circadian oscillations of PER protein in fruit flies (mentioned earlier), other examples include oscillations of p53 in response to ionizing radiation (p53 → MDM2 mRNA → MDM2 protein p53)^37-40 and oscillations of NF-κB in response to stimulation by tumor necrosis factor (NF-κB → IκB mRNA → IκB protein NF-κB)^40-43.

Because this regulatory motif has 3 or more components, it cannot be adequately represented by a two-variable state space, as in Figs. 2b and ​and3b.3b. Nonetheless, it is instructive to plot trajectories of the basic motif (X → Y → Z X) in the XY plane (Fig. 5a, left) and in the XZ plane (Fig. 5a, right), and to compare these plots to the case of a two-component negative feedback loop (X → YX) in Fig. 2b. In that figure, curves a and b (called ‘nullclines’) indicate places where the flow of the reaction system is horizontal (in the Y direction only) and where it is vertical (in the X direction only). (Please notice that we always plot the activator X on the vertical axis.) These ‘flow indicators’ force the trajectories (the dashed curves in Fig. 2b) to spiral into the steady state. For the case of a delayed feedback loop, the trajectories do not obey the flow indicators (Fig. 5a). If we are plotting X vs. Y (Fig. 5a, left panel), the limit cycle trajectory does not cross curve a in a horizontal direction, because the rate of synthesis of X depends on Y concentration some time in the past. If we are plotting X vs. Z (Fig. 5a, right panel), the limit cycle trajectory does not cross curve b in a vertical direction, because the rate of synthesis of Z depends on X concentration some time in the past. In either case, the delay allows the trajectory to form a closed orbit around the steady state instead of spiraling into it.

Class 2: Amplified Negative Feedback Loops

In Fig. 3 we considered a special case where the inhibitor (the protein) is amplified by a positive feedback loop. It should be obvious that activator amplification may be just as effective. For the case of activator amplification (Fig. 5b, left panel), the X-nullcline (curve a) is ‘kinked’ by the positive feedback loop. For inhibitor amplification (Fig. 5b, right panel), the Y-nullcline (curve b) is ‘kinked’. In either case, trajectories are now forced to wheel around the steady state onto a closed orbit (curve c, a sustained oscillation).

Where do the ‘kinks’ come from? A two-component positive feedback loop (W → X → W or W X W) can respond in a bistable manner to inhibition by Y. (In this case—Fig. 5b, left—we are thinking of Y as signal strength and X as the response variable.) If Y is large, then X will be small, for sure, and if Y is small, then X may be large. But for intermediate values of Y, the steady state concentration of X can be either large or small, depending upon how the system got to the intermediate value of Y. This bistability is reflected in the Z-shaped nullcline (curve a) on the left of Fig. 5b. For the case of inhibitor amplification (Fig. 5b, right), we think of X as signal strength and Y as response variable. In this case, Y can be a multi-valued function of X, and the Y-nullcline (curve b) becomes N-shaped. In either case, the negative feedback between X and Y forces trajectories to rotate clockwise on our standard XY plane, and the positive feedback loop puts kinks in one of the nullclines to prevent trajectories from spiraling into the steady state.

This class of oscillators appears commonly in the literature, from the earliest models of chemical oscillations ^44-46 to recent models of mitosis-promoting factor (MPF) in frog egg extracts^{47, 48}. The latter case is an activator-amplified negative feedback loop, with W=Cdc25, X=MPF and Y=Cdc20. An inhibitor-amplified negative feedback loop (X=Cdc10, Y=Cig2, Z=Rum1) has been used by Novak & Tyson⁴⁹ to model endoreplication (periodic DNA synthesis in the absence of cell division) in mutant fission yeast cells. Recently, a synthetic oscillator based on an activator-amplified negative feedback loop has been built in bacteria by Jeff Hasty's group⁵⁰.

Class 3: Incoherently Amplified Negative Feedback Loops

By rewiring an activator-amplified negative feedback loop (D), we create new regulatory motifs (C and E)

that may also have the potential to oscillate. Each of the motifs (C) and (E) has a two-component positive feedback loop (− −, in both C and E) embedded in a three-component negative feedback loop (+ + − in C and − − − in E). Motif (C) also has the characteristic that X inhibits W directly and activates W indirectly (through Y). This characteristic is called an incoherent feedforward loop (C′). Motif (E) can also be redrawn as an incoherent feedforward loop (E′).

Hence, we describe these motifs as incoherently amplified negative feedback loops (NFLs). In both motifs C′ and E′, the embedded positive feedback loop is (− −). There are two other incoherently amplified NFLs based on an embedded (+ +) feedback loop. The four cases are shown in Fig. 5c. In each case, the NFL may, of course, oscillate in its own right, but the additional positive feedback loop adds bistability and robustness to the mechanism⁵¹.

The earliest models of glycolytic oscillations ^52-54 belong to this class of oscillators (Fig. 5c, fourth motif). In this motif, Z → X refers to the biochemical reaction that converts fructose-6-phosphate + ATP (Z) into fructose-1,6-bisphosphate + ADP (X). The enzyme that catalyzes this reaction, phosphofructokinase (species Y in the motif), is activated by ADP (X → Y), and the active form of Y promotes both the removal of Z (Y Z) and the production of X (Y → X).

The Martiel-Goldbeter⁵⁵ model of cyclic AMP oscillations in slime mold cells is another example of the fourth motif in Fig. 5c. In this case, intracellular cAMP is the activator (X) and extracellular cAMP is the incoherent signaler (Y). Extracellular cAMP binds to a membrane receptor that quickly activates the enzyme that synthesizes cAMP from ATP inside the cell. This is the fast activatory signal from Y to X. Furthermore, extracellular cAMP pushes the membrane receptor (Z) into an inactive state, which only slowly recovers activity after cAMP is destroyed by extracellular phosphodiesterase. This is the slow inhibitory signal from Y to X. Tyson & Novak^{56, 57} use incoherently amplified(- -)NFLs (the first and second motifs in Fig. 5c) to model cell cycle transitions (M/G1 and G1/S, respectively). To model oscillations in p53 (a transcription factor that coordinates intracellular responses to DNA damage), Ciliberto et al.⁵⁸ used an incoherently amplified(− −)NFL and Zhang et al.⁵⁹ suggested an alternative mechanism based on an incoherently amplified(+ +)NFL (the first model in their Fig. 1).

Other possibilities?

There are three other regulatory motifs with 3 components, 4 links and a topology similar to motifs (C′) and (E′):

In these motifs, we use circles to indicate interactions of either sign (+ or -); two white circles must have the same sign, whereas black and white circles have opposite signs.

Motif (F) is a ‘coherently repressed’ NFL, consisting of two negative feedback loops, one with three components and the other with two components. Clearly, if the link X → Y is weak enough, the three-component NFL might oscillate on its own. Addition of the two-component NFL dampens the propensity of the three-component NFL to oscillate.

Motif (G) is a different sort of incoherently amplified NFL; it differs from the Class 3 motifs in Fig. 5c in that the positive loop has three components (rather than two) and the negative loop has two components (rather than three). Everything we have said so far might lead us to expect oscillations in this motif under the right choice of nonlinearities and rate constants. However, it is possible to show that motif (G) cannot generate oscillations for any choice of nonlinear rate equations or any parameter settings (see supplementary information S3 (box)).

Motif (H) has two positive feedback loops, one with two components and one with three. We might expect this regulatory motif to exhibit bistability but not oscillations, and indeed this is the case (see supplementary information S4 (box)).

JJT acknowledges financial support from the National Institutes of Health (R01-GM07898 and R01-GM079207) and the hospitality of Merton College, Oxford, during the writing of this review. B.N. acknowledges support from the BBSRC (UK) and from the EC FP7 (201142). Our understanding of biochemical oscillations has developed over many years of delightful conversations with Albert Goldbeter, Lee Segel, Art Winfree, Michael Mackey and Leon Glass.