The circadian signalling network in Arabidopsis thaliana generates oscillations with negative feedback loops of transcription factor binding, and regulates many developmental and physiological processes. Mathematical models have been proposed which describe the transcriptional and post-translational feedback loops generating oscillations, though do not incorporate the interactions with cytosolic messengers such as Ca2+ and cADPR, or with metabolites such as sucrose. The aim of this thesis is to use mathematical modelling to investigate how the circadian clock regulates [Ca2+]cyt, how a cADPR-based feedback loop modulates circadian oscillations, and also how the circadian clock perceives changes in the availability of sucrose. Delay linear systems of ordinary differential equations were used to demonstrate how [Ca2+]cyt is co-regulated by two pathways operating on different timescales, the first dependent on the transcription factor CIRCADIAN CLOCK ASSOCIATED 1 (CCA1), and a light/dark-dependent circadian clock-independent pathway. Simulated mutant analysis provided a potential role for the red-light sensing PHYTOCHROME A (PHYA) in the light-dependent pathway and offered predictions for the dynamics of [Ca2+]cyt when system components are removed. A proposed feedback loop between the second messenger cADPR and the circadian clock was investigated in silico using a pairwise parameter perturbation method on mathematical models of the Arabidopsis central oscillator. Experimental observations from transient and persistent manipulations to cADPR synthesis could be explained with time-varying parametric perturbations to mathematical oscillator models representing the effects of cADPR on clock gene expression, supporting the hypothesised cADPR-based feedback loop. Finally, the dependency of circadian oscillations in darkness on exogenous sucrose availability was investigated using the Three Loop model of the circadian clock. Mathematical analyses and experimental validation demonstrated the involvement of GIGANTEA (GI) in the sucrose-sensing by the circadian clock, either by a transcriptional or post-translational mechanism. Further experimental evidence is presented supporting the hypothesis that sucrose up-regulates GI transcription. This thesis identifies components of the circadian clock which serve as entry or exit points for cytosolic signalling and metabolic pathways involved in key physiological processes, and demonstrates the use of mathematics to generate non-intuitive hypotheses for subsequent experimental validation in complex biological systems.