Among the most fascinating time devices in biology are circadian clocks, which are found in organisms ranging from cyanobacteria and fungi, to plants, insects and animals. These clocks are biochemical oscillators that allow organisms to coordinate their metabolic and behavioural activities with the 24-hour cycle of day and night. Remarkably, these clocks can maintain stable rhythms for months or even years in the absence of any daily cue, such as light/dark or temperature cycles, from the environment. In multicellular organisms, the robustness might be explained by intercellular interactions, but it is now known that even unicellular organisms can have very stable rhythms. An excellent example is provided by the clock of the cyanobacterium Synechococcus elongatus, which is one of the best-characterized experimental model systems. This clock has a correlation time of several months, even though the clocks of the different cells in the population hardly interact with one another (Mihalcescu etal., Nature, 2004). This robustness is surprising, given the fact that experiments in recent years have vividly demonstrated that protein synthesis, which is required to sustain the clock, is highly stochastic. Clearly, the clock is designed in such a way that it has become resilient to the intrinsic stochasticity of the underlying biochemical reactions. We aim to elucidate the design principles that underlie the robustness of circadian clocks, with the cyanobacterium S. elongatus as a model system.
The Kai system
Cyanobacteria are the simplest organisms to use circadian clocks to anticipate the changes between day and night. In the cyanobacterium S. elongatus, the core components of the clock are the three proteins KaiA, KaiB and KaiC. In higher organisms, it had long been believed that circadian rhythms are primarily driven by transcriptional negative feedback. In 2005, however, the group of Kondo showed in a series of groundbreaking experiments that the three Kai proteins are sufficient to generate rhythmic phosphorylation of KaiC. They first showed that in vivo KaiC phosphorylation maintains a 24-hour rhythm even when KaiC synthesis is inhibited (Tomita et al, Science, 2005). They then showed that this rhythmic KaiC phosphorylation can be reconstituted in vitro in the presence of only KaiA, KaiB and ATP (Nakajima et al., Science, 2005). These experiments show that the phosphorylation cycle of KaiC is a key element of the clock, and we therefore set out to develop a mathematical model that can describe this cycle .
Model – the phenomenon of differential affinity
Our model is built on two key elements (Fig.1). First, we hypothesize that KaiC, which forms a hexamer, has a tendency to be cyclically phosphorylated and dephosphorylated as it flips between two conformational states. Second, the noisy oscillations of the individual KaiC hexamers are synchronized via the mechanism of differential affinity, whereby the laggards in the population outcompete the front runners for a limited supply of KaiA molecules, which stimulate KaiC phosphorylation. The slowest KaiC hexamers thus speedup, while the slowest are forced to slow down, causing the entire population to oscillate in phase.
Fig.1 Model of the phosphorylation cycle of KaiC. KaiC forms a hexamer, which can switch between an active conformational state (circles), in which the phosphorylation level tends to rise, and an inactive conformational state (squares), in which the phosphorylation level tends to fall. Nucleotide binding stabilizes the active state, while phosphorylation favors the inactive state. The interplay between the two creates a phosphorylation cycle of the individual KaiC hexamers. KaiA stimulates phosphorylation of KaiC that is in the active state. However, KaiA is sequestered by complexes containing KaiB and inactive KaiC. KaiC in the inactive state (the laggards) can thus delay the progress of fully dephosphorylated hexamers that have already switched back to the active state and are ready to be phosphorylated again (the front runners). This phenomenon of differential affinity leads to synchronization of the oscillations of the individual KaiC hexamers, creating macroscopic oscillations of the phosphorylation level.
The phase diagram of the model is shown in Fig.2. It is seen that, as long as a minimum amount of KaiB is present, neither the amplitude nor the period of the oscillations are much affected by the amount of KaiB. This can be understood by noticing that KaiB, when bound to inactive KaiC, mainly serves to sequester KaiA. The dependence of the oscillations on the level of KaiA is more interesting. For low KaiA concentrations, the system does not exhibit oscillations. When the KaiA concentration is raised to about half the KaiC concentration, the system starts to oscillate via a supercritical Hopf bifurcation. The period of the oscillations decreases monotonically with increasing KaiA concentration. In contrast, the amplitude first increases, but then decreases until the oscillations ultimately disappear. The latter is a direct consequence of the mechanism of differential affinity, which relies on KaiA sequestration – when the concentration of KaiA is high enough, KaiA can no longer be sequestered, causing the synchronization mechanism to break down. The prediction of the disappearance of oscillations when the KaiA concentration is raised has been confirmed experimentally (Rust et al., Science, 2007; Nakajima, FEBS Lett., 2010).
Fig.2 The period (left panel) and the amplitude (right panel) of the oscillations in the phosphorylation level of KaiC as a function of the concentration of KaiA and KaiB. It is seen that both the period and the amplitude are essentially independent of the concentration of KaiB, provided enough KaiB is present to sequester KaiA. In contrast, the oscillations show a non-monotic dependence on the KaiA concentration, but also at high KaiA concentrations.
The coupling between the protein modification and the protein synthesis cycle
While the ground-breaking experiments of the Kondo group in 2005 opened the possibility that the circadian clock might be driven by a protein modification cycle only, in 2008 the same group showed that when in vivo the protein modification cycle was impeded, the concentration of KaiC still oscillated with a period of 24 hours. This unambiguously demonstrated that in vivo the clock is driven by a combination of a protein synthesis cycle and a protein modification cycle. The natural question that arose was thus: why is the clock based on two oscillators?
Fig.3 (A) PPC-TTC model: A TTC of kaiBC expression (orange background) interacts with a KaiC PPC (yellow background). KaiC is a hexamer that, in our model, switches between an active conformational state (circles) in which its phosphorylation level tends to rise and an inactive state (squares) in which it tends to fall (see also Fig.1). Active KaiC activates RpaA, whereas inactive KaiC inactivates RpaA; active RpaA (red) activates kaiBC expression, leading (after a delay) to the injection of fully phosphorylated KaiC into the PPC. (B) PPC-in vitro model: Only the PPC is present, and the concentration of each Kai protein is constant. (C) PPC-in vivo model: Only the PPC is present, but all Kai proteins are now continually synthesized and degraded, with rates that are constant in time. (D) TTC-only model: Only the TTC is present, and KaiC is always in a highly phosphorylated state.
To address this question, we have studied the robustness of four models (Fig.3) : 1) the PPC-in-vitro model (Fig.3B), described above; 2) the PPC-in-vivo model (Fig.3C); 3) the PPC-TTC-only model (Fig.3A); 4) the TTC-only model (Fig.3D). In the PPC-in-vitro model (Fig.3B), the total number of each Kai protein is constant—they are neither produced nor destroyed—and only the PPC is operative. In this case the PPC is highly robust against noise arising from the intrinsic stochasticity of chemical reactions. Even for reaction volumes smaller than the typical volume of a cyanobacterium, the correlation time is longer than that observed experimentally. Living cells, however, constantly grow and divide, and proteins must thus be synthesized to balance dilution. In fact, dilution can be thought of as introducing an effective protein degradation rate set by the cell doubling time. We therefore studied next the PPC-in vivo model (Fig.3C), which describes a PPC in which the Kai proteins are produced and degraded with rates that are constant in time. The simulations revealed that protein synthesis and decay dramatically reduce the viability of the PPC (Fig.4). This is because the constant synthesis of proteins, which are all created in the same phosphorylation state, necessarily injects KaiC with the “wrong” phosphorylation level at certain phases of the cycle. One role of the TTC is thus to introduce proteins only when the phosphorylation state of the freshly made KaiC matches that of the PPC. Our simulations of the PPC-TTC model (Fig.3A), which combines a PPC and a TTC, revealed that a TTC can indeed greatly enhance the robustness of the PPC, yielding correlation times consistent with those measured experimentally (Fig.4). Finally, we considered whether the PPC is needed at all, or whether one could build an equally good circadian clock using only a TTC; to this end, we studied the TTC-only model (Fig. 3D). We found that it is possible to construct a TTC with a period of 24 h and the observed correlation time of a few months (Fig.4). However, this comes at the expense of very high protein synthesis and decay rates, which impose an extra energetic burden on the cell. Our results thus suggest that a PPC allows for a more robust oscillator at a lower cost. Although our models are simplified, we argue that our qualitative results are unavoidable consequences of the interaction between a circadian clock and cell growth and so should hold far more generally.
Fig.4 Correlation number of cycles n1∕2 as a function of degradation rate μ for the PPC-in vivo (Fig. 3C), TTC-only (Fig. 3D), and PPC-TTC (Fig. 3A) models. Clearly, a PPC in combination with a TTC generates robust rhythms over a wide range of degradation rates.
- Jeroen S. van Zon , David K. Lubensky , Pim R. H. Altena and Pieter Rein ten Wolde , An allosteric model of circadian KaiC phosphorylation PNAS 104, 7420–7425 (2007)
- David Zwicker , David K. Lubensky and Pieter Rein ten Wolde , Robust circadian clocks from coupled protein-modification and transcription–translation cycles PNAS 107, 52: 22540-22545 (2010)