Circadian (~24 h) rhythms are generated by transcription-translation feedback loops whose components are largely known; a 2014 modeling study by DeWoskin, Geng, Stinchcombe and Forger argued that what determines timekeeping behavior is not the parts but their interactions, at three scales: transcription factors regulating multiple promoter sites, clock proteins assembling into multi-protein complexes, and neurons coupling through the neuropeptide VIP in the mammalian suprachiasmatic nucleus (SCN). The paper closed with a concrete validation agenda: (i) build a synthetic clock in which a single protein acts both as activator and repressor at different sites, to test the long-period slowing and chaos predicted specifically for that design; (ii) determine biochemically the in-vivo assembly topology of the Drosophila NMO-PER-TIM-SGG timekeeping complex, in particular whether it uses the chain-like pairwise adjacencies (NMO-PER, PER-TIM, TIM-SGG) the models found most favorable to oscillations; (iii) measure the phase response of individual isolated SCN neurons to VIP and test the prediction that such neurons are bistable, transient VIP pulses switching them into or out of rhythmicity; (iv) incorporate bidirectional coupling between electrophysiological membrane dynamics and molecular clocks into multicellular SCN models; and (v) replace the numerical observations by mathematical analysis.

Background

Circadian (~24 h) rhythms in organisms from fungi to mammals are generated by transcription-translation feedback loops (TTFLs). In the mammalian clock, the activator complex BMAL1::CLOCK binds short DNA regulatory motifs called E-boxes in the promoters of the Per and Cry genes, driving their transcription; the PER and CRY protein products assemble into complexes that return to the nucleus and switch off their own activator, closing a negative feedback loop that takes about a day to traverse. Two features of this picture set the stage for the present problem. First, promoter regulation is richer than a single on/off switch: many clock-associated transcription factors act in both roles. In the fungus Neurospora, the core repressor FRQ also stabilizes its own activator and thereby acts as an activator (Schafmeier et al. 2006); the Neurospora photoreceptor VIVID likewise acts as both activator and repressor; and the bacterial factor AraC represses or activates depending on which of its DNA binding sites it occupies. AraC was used as the activating arm of a synthetic bacterial oscillator (Stricker et al. 2008), but no synthetic clock has used one protein in both roles. Second, the repressing arm works through multi-protein complexes whose assembly order in vivo is poorly known. In Drosophila, the PERIOD (PER) and TIMELESS (TIM) proteins assemble with the kinases NEMO (NMO) and SHAGGY (SGG, the fly glycogen synthase kinase-3) into a complex that represses the positive arm; PER phosphorylation by NMO initiates a time-delay circuit that sets clock speed (Chiu et al. 2011). Because combinatorial complex formation generates many possible intermediate species, most clock models lump repression into a phenomenological Hill function (a sigmoidal decreasing function of repressor concentration) rather than modeling assembly explicitly; the alternative, protein sequestration, in which repressor and activator stoichiometrically neutralize each other, produces qualitatively different dynamics (Kim 2016). In mammals the clock resides in roughly 20,000 neurons of the suprachiasmatic nucleus (SCN) of the hypothalamus. Individual SCN neurons are self-sustained cellular oscillators that synchronize through intercellular signals, most importantly vasoactive intestinal polypeptide (VIP), released by about 20% of SCN neurons and received through the VPAC2 receptor on nearly all of them. A standard experimental observable is the phase response curve (PRC): the steady phase shift of a rhythm produced by a brief perturbation applied at each phase of the cycle. DeWoskin, Geng, Stinchcombe and Forger (2014) built models at all three scales. (i) Promoter level: in a TTFL where one clock protein binds two regulatory sites, two independent repressing sites (as in the E-boxes of the Per1 promoter) give a monotonic transcription regulation function; this design can be bistable in the sense of hard excitation, where a stable steady state and a stable oscillatory state coexist (two alternative stable behaviors for identical parameters), but it cannot produce chaos. If one site activates while the other represses, the loop admits an additional equilibrium point that the monotonic repressing-only regulation lacks: the system can spend long intervals near this equilibrium and thus develop very long periods, and can exhibit deterministic chaos (aperiodic, irregular dynamics with sensitive dependence on initial conditions), something known since the 1980s only in more elaborately regulated biochemical models (Decroly and Goldbeter 1982). (ii) Complex level: in models of the Drosophila NMO-PER-TIM-SGG complex, oscillations could be obtained under each of the assembly schemes considered, but which pairwise bindings are allowed during assembly strongly affects how readily the TTFL oscillates, and the binding scheme most likely to show oscillations coincides with the pairwise interactions known at the time: NMO binds PER, PER binds TIM, TIM binds SGG. (iii) Network level: extending the detailed Kim-Forger stoichiometric-balance clock model (Kim and Forger 2012) to a multicellular SCN model with VIP signaling, and refitting parameters so that cell-cell coupling matches experimental data, changed the model's dynamics: isolated (uncoupled) cells became bistable, each capable of settling into a rhythmic or an arrhythmic state, while network coupling removed this bistability and made the network robust to genetic and external perturbations. This matched single-cell data on nondeterministic rhythm generation in identified SCN neurons (Webb et al. 2009) and on paracrine resuscitation of cycling (Maywood et al. 2011). The paper's Discussion states that these numerical claims need experimental and mathematical validation, and enumerates the specific open directions that constitute the present problem.

Problem Statement

The problem is the validation and formalization agenda posed in the Discussion of DeWoskin et al. (2014): to establish whether the interaction-level mechanisms identified by their models are real properties of circadian timekeeping systems rather than modeling artifacts, at the three scales the paper studied. The Discussion poses the following open directions, each separately motivated in the paper: (1) Promoter scale - construct a synthetic clock in which a single protein under negative feedback acts as an activator at one binding site and a repressor at another, and determine whether the behaviors predicted specifically for this design occur: oscillations with very long periods, arising from the additional equilibrium point that the monotonic repressing-only two-site regulation lacks, and deterministic chaos. (2) Protein-complex scale - determine biochemically which assembly mechanism the Drosophila NMO-PER-TIM-SGG timekeeping complex uses in vivo, in particular whether assembly proceeds through the chain-like pairwise adjacencies (NMO-PER, PER-TIM, TIM-SGG) that the models found most favorable to sustained oscillations, or through some alternative binding scheme. (3) Cellular and network scale - measure the phase response of individual, isolated SCN neurons to VIP at single-cell resolution, and test the model prediction that such neurons are bistable, so that transient VIP pulses can switch them into or out of rhythmicity while network coupling removes the bistability. (4) Extend the multicellular SCN model with bidirectional coupling between electrophysiological membrane dynamics and the intracellular molecular clocks, an aspect of rhythm generation that the source model did not consider even though in the actual SCN there is a strong connection between the molecular clock and the electrical activity of the cells, and that its authors planned to explore in future work; separately, the paper identifies as an important area for future work the study of the differences in phase among neurons in the SCN network, since heterogeneity of period and phase within the SCN has been shown to be important for the coding of seasonal changes in light and dark cycles and such period differences have been suggested to drive the spatio-temporal 'phase wave' seen in SCN slices. (5) The results of the paper are numerical, and further mathematical analysis could expand them: replace the numerical observations with mathematical analysis for the model families of the paper - the promoter-level TTFL with a protein binding two regulatory sites (in both the repressing-only and the one-activator/one-repressor forms), the feedback loops with explicit protein-complex assembly, and the refitted multicellular SCN model - covering the bifurcation structure, the existence or absence of chaos, and parameter dependences such as the way transcription rates control how long the oscillator spends near an equilibrium point and hence the period. The directions are linked: (1), (2) and (5) validate and formalize the two intracellular interaction mechanisms; (3) and (4) validate and complete the intercellular coupling mechanism; together they address the paper's central claim that timekeeping interactions are tuned towards proper function. All five directions belong to the problem and each must be addressed for it to be resolved, and the two parts of direction (4) - extending the model with bidirectional membrane-molecular coupling, and studying neuronal period/phase heterogeneity and the phase wave - are jointly required within that direction. A result that settles only some of the directions is partial progress, not a resolution of the problem.

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