Oscillatory dynamics is ubiquitous in biology, yet the complexity of the underlying signaling pathways often makes quantitative understanding of such systems difficult. An ideal model system for understanding biochemical oscillators is the Kai system, a post-translational circadian oscillator found in the cyanobacterium Synechococcus elongatus. The Kai oscillator consists of three soluble proteins named KaiA, KaiB, and KaiC, and their periodic interactions, which are driven by ATP hydrolysis in KaiC, generate a near-24-h period oscillation of KaiC phosphorylation states. The Kai oscillator interacts with cellular metabolic conditions, in the form of ATP/ADP concentrations, that can act as external timing cues, such that the period remains robust while the amplitude and phase of the clock adjust. How the clock achieves such robust yet tunable dynamics is not fully understood. Understanding the interaction between KaiA and KaiC is key to understanding how the clock senses metabolic conditions. KaiA is a nucleotide-exchange factor that regulates the nucleotide-bound state of KaiC and thus the direction of its reversible phosphotransferase activity. In this work, I dissect the KaiA--KaiC interaction at two levels. First, I fit a detailed kinetic model of KaiA--KaiC interactions to experimental kinetic data using Bayesian Markov chain Monte Carlo. This model revealed ultrasensitivity in KaiC phosphorylation as a result of nucleotide-dependent KaiA binding affinity, which I argue is important for period stability. Second, I use molecular dynamics simulations to probe the molecular mechanism of ADP release from KaiC, which reveals coupling between the nucleotide-binding pockets and the A-loop, an unstructured region in KaiC that binds to KaiA. This provides a hypothesis for how KaiA acts as a nucleotide-exchange factor.