Circadian clocks and phosphorylation: Insights from computational modeling

Evolution of casein kinase 1 and functional analysis of new doubletime mutants in Drosophila

Circadian clocks are timing devices that rhythmically adjust organism's behavior, physiology, and metabolism to the 24-h day-night cycle. Eukaryotic circadian clocks rely on several interlocked transcription-translation feedback loops, where protein stability is the key part of the delay between transcription and the appearance of the mature proteins within the feedback loops. In bilaterian animals, including mammals and insects, the circadian clock depends on a homologous set of proteins. Despite mostly conserved clock components among the fruit fly Drosophila and mammals, several lineage-specific differences exist. Here we have systematically explored the evolution and sequence variability of insect DBT proteins and their vertebrate homologs casein kinase 1 delta (CKIδ) and epsilon (CKIε), dated the origin and separation of CKIδ from CKIε, and identified at least three additional independent duplications of the CKIδ/ε gene in Petromyzon, Danio, and Xenopus. We determined conserved regions in DBT specific to Diptera, and functionally tested a subset of those in D. melanogaster. Replacement of Lysine K224 with acidic residues strongly impacts the free-running period even in heterozygous flies, whereas homozygous mutants are not viable. K224D mutants have a temperature compensation defect with longer free-running periods at higher temperatures, which is exactly the opposite trend of what was reported for corresponding mammalian mutants. All DBTs of dipteran insects contain the NKRQK motif at positions 220-224. The occurrence of this motif perfectly correlates with the presence of BRIDE OF DOUBLETIME, BDBT, in Diptera. BDBT is a non-canonical FK506-binding protein that physically interacts with Drosophila DBT. The phylogeny of FK506-binding proteins suggests that BDBT is either absent or highly modified in non-dipteran insects. In addition to in silico analysis of DBT/CKIδ/ε evolution and diversity, we have identified four novel casein kinase 1 genes specific to the Drosophila genus.

Analytical study of robustness of a negative feedback oscillator by multiparameter sensitivity

Background

One of the distinctive features of biological oscillators such as circadian clocks and cell cycles is robustness which is the ability to resume reliable operation in the face of different types of perturbations. In the previous study, we proposed multiparameter sensitivity (MPS) as an intelligible measure for robustness to fluctuations in kinetic parameters. Analytical solutions directly connect the mechanisms and kinetic parameters to dynamic properties such as period, amplitude and their associated MPSs. Although negative feedback loops are known as common structures to biological oscillators, the analytical solutions have not been presented for a general model of negative feedback oscillators.

Results

We present the analytical expressions for the period, amplitude and their associated MPSs for a general model of negative feedback oscillators. The analytical solutions are validated by comparing them with numerical solutions. The analytical solutions explicitly show how the dynamic properties depend on the kinetic parameters. The ratio of a threshold to the amplitude has a strong impact on the period MPS. As the ratio approaches to one, the MPS increases, indicating that the period becomes more sensitive to changes in kinetic parameters. We present the first mathematical proof that the distributed time-delay mechanism contributes to making the oscillation period robust to parameter fluctuations. The MPS decreases with an increase in the feedback loop length (i.e., the number of molecular species constituting the feedback loop).

Conclusions

Since a general model of negative feedback oscillators was employed, the results shown in this paper are expected to be true for many of biological oscillators. This study strongly supports that the hypothesis that phosphorylations of clock proteins contribute to the robustness of circadian rhythms. The analytical solutions give synthetic biologists some clues to design gene oscillators with robust and desired period.

Modeling and simulating the Arabidopsis thaliana circadian clock using XPP-AUTO

Circadian clocks are endogenous timekeepers that produce oscillations with a period of about one day. Their rhythmicity originates from complex gene regulatory networks at the cellular level. In the last decades, computational models have been proven to be a powerful tool in order to understand the dynamics and design principles of the complex regulatory circuitries underlying the circadian clocks of different organisms. We present the process of model development using a small and simplified two-gene regulatory network of the Arabidopsis circadian clock. Subsequently, we discuss important numerical techniques to analyze such a mathematical model using XPP-AUTO. We show how to solve deterministic and stochastic ordinary differential equations and how to compute bifurcation diagrams or simulate phase-shift experiments. We finally discuss the contributions of modeling to the understanding and dissection of the Arabidopsis circadian system.

Modelling the effect of phosphorylation on the circadian clock of Drosophila

It is by now well known that, at the molecular level, the core of the circadian clock of most living species is a negative feedback loop where some proteins inhibit their own transcription. However, it has recently been shown that post-translational processes, such as phosphorylations, are essential for a correct timing of the clock. Depending on which sites of a circadian protein are phosphorylated, different properties such as degradation, nuclear localization and repressing power can be altered. Furthermore, phosphorylation domains can be related in a positive way, giving rise to consecutive phosphorylations, or in a negative way, hindering phosphorylation at other domains. Here we present a simple mathematical model of a circadian protein having two mutually exclusive domains of phosphorylation. We show that the system has limit cycles that arise from a unique fixed point through a Hopf bifurcation. We find a set of parameters, with realistic values, for which the limit cycle has the same period as the wild type circadian oscillations of the fruit fly. The domains act as a switch, in the sense that alterations in their phosphorylation can alter the period of circadian oscillation in opposite ways, increasing or decreasing the period of the wild type oscillations. In particular, we show that our model is able to reproduce some of the experimental results found for switch-like phosphorylations of the PER protein of the circadian clock of the fly Drosophila melanogaster.

How molecular should your molecular model be? On the level of molecular detail required to simulate biological networks in systems and synthetic biology

The recent advance of genetic studies and the rapid accumulation of molecular data, together with the increasing performance of computers, led researchers to design more and more detailed mathematical models of biological systems. Many modeling approaches rely on ordinary differential equations (ODE) which are based on standard enzyme kinetics. Michaelis-Menten and Hill functions are indeed commonly used in dynamical models in systems and synthetic biology because they provide the necessary nonlinearity to make the dynamics nontrivial (i.e., limit-cycle oscillations or multistability). For most of the systems modeled, the actual molecular mechanism is unknown, and the enzyme equations should be regarded as phenomenological. In this chapter, we discuss the validity and accuracy of these approximations. In particular, we focus on the validity of the Michaelis-Menten function for open systems and on the use of Hill kinetics to describe transcription rates of regulated genes. Our discussion is illustrated by numerical simulations of prototype systems, including the Repressilator (a genetic oscillator) and the Toggle Switch model (a bistable system). We systematically compare the results obtained with the compact version (based on Michaelis-Menten and Hill functions) with its corresponding developed versions (based on "elementary" reaction steps and mass action laws). We also discuss the use of compact approaches to perform stochastic simulations (Gillespie algorithm). On the basis of these results, we argue that using compact models is suitable to model qualitatively biological systems.