We developed a mathematical model of the Arabidopsis circadian clock, including PRR7 and PRR9, which is able to predict several single, double and triple mutant phenotypes.Sensitivity Analysis was used to identify the properties and time sensing mechanisms of model structures.PRR7 and CCA1/LHY were identified as weak points of the mathematical model indicating where more experimental data is needed for further model development.Detailed dynamical studies showed that the timing of an evening light sensing element is essential for day length responsiveness
In recent years, molecular genetic techniques have revealed a complex network of components in the Arabidopsis circadian clock. Mathematical models allow for a detailed study of the dynamics and architecture of such complex gene networks leading to a better understanding of the genetic interactions. It is important to maintain a constant iteration with experimentation, to include novel components as they are discovered and use the updated model to design new experiments. This study develops a framework to introduce new components into the mathematical model of the Arabidopsis circadian clock accelerating the iterative model development process and gaining insight into the system's properties.
We used the interlocked feedback loop model published in Locke et al (2005) as the base model. In Arabidopsis, the first suggested regulatory loop involves the morning expressed transcription factors CIRCADIAN CLOCK-ASSOCIATED 1 (CCA1) and LATE ELONGATED HYPOCOTYL (LHY), and the evening expressed pseudo-response regulator TIMING OF CAB EXPRESSION (TOC1). The hypothetical component X had been introduced to realize a longer delay between gene expression of CCA1/LHY and TOC1. The introduction of Y was motivated by the need for a mechanism to reproduce the dampening short period rhythms of the cca1/lhy double mutant and to include an additional light input at the end of the day.
In this study, the new components pseudo-response regulators PRR7 and PRR9 were added in negative feedback loops based on the biological hypothesis that they are activated by LHY and in turn repress LHY transcription (Farré et al, 2005; Figure 1). We present three iterations steps of model development (Figure 1A–C).
A wide range of tools was used to establish and analyze new model structures. One of the challenges facing mathematical modeling of biological processes is parameter identification; they are notoriously difficult to determine experimentally. We established an optimization procedure based on an evolutionary strategy with a cost function mainly derived from wild-type characteristics. This ensured that the model was not restricted by a specific set of parameters and enabled us to use a large set of biological mutant information to assess the predictive capability of the model structure. Models were evaluated by means of an extended phenotype catalogue, allowing for an easy and fair comparison of the structures. We also carried out detailed simulation analysis of component interactions to identify weak points in the structure and suggest further modifications. Finally, we applied sensitivity analysis in a novel manner, using it to direct the model development. Sensitivity analysis provides quantitative measures of robustness; the two measures in this study were the traces of component concentrations over time (classical state sensitivities) and phase behavior (measured by the phase response curve). Three major results emerged from the model development process.
First, the iteration process helped us to learn about general characteristics of the system. We observed that the timing of Y expression is critical for evening light entrainment, which enables the system to respond to changes in day length. This is important for our understanding of the mechanism of light input to the clock and will add in the identification of biological candidates for this function. In addition, our results suggest that a detailed description of the mechanisms of genetic interactions is important for the systems behavior. We observed that the introduction of an experimentally based precise light regulation mechanism on PRR9 expression had a significant effect on the systems behavior.
Second, the final model structure (Figure 1C) was capable of predicting a wide range of mutant phenotypes, such as a reduction of TOC1 expression by RNAi (toc1RNAi), mutations in PRR7 and PRR9 and the novel mutant combinations prr9toc1RNAi and prr7prr9toc1RNAi. However, it was unable to predict the mutations in CCA1 and LHY.
Finally, sensitivity analysis identified the weak points of the system. The developed model structure was heavily based on the TOC1/Y feedback loop. This could explain the model's failure to predict the cca1lhy double mutant phenotype. More detailed information on the regulation of CCA1 and LHY expression will be important to achieve the right balance between the different regulatory loops in the mathematical model. This is in accordance with genetic studies that have identified several genes involved in the regulation of LHY and CCA1 expression. The identification of their mechanism of action will be necessary for the next model development.
In plants, as in animals, the core mechanism to retain rhythmic gene expression relies on the interaction of multiple feedback loops. In recent years, molecular genetic techniques have revealed a complex network of clock components in Arabidopsis. To gain insight into the dynamics of these interactions, new components need to be integrated into the mathematical model of the plant clock. Our approach accelerates the iterative process of model identification, to incorporate new components, and to systematically test different proposed structural hypotheses. Recent studies indicate that the pseudo-response regulators PRR7 and PRR9 play a key role in the core clock of Arabidopsis. We incorporate PRR7 and PRR9 into an existing model involving the transcription factors TIMING OF CAB (TOC1), LATE ELONGATED HYPOCOTYL (LHY) and CIRCADIAN CLOCK ASSOCIATED (CCA1). We propose candidate models based on experimental hypotheses and identify the computational models with the application of an optimization routine. Validation is accomplished through systematic analysis of various mutant phenotypes. We introduce and apply sensitivity analysis as a novel tool for analyzing and distinguishing the characteristics of proposed architectures, which also allows for further validation of the hypothesized structures.