Circadian clocks in all organisms include feedback loops that generate rhythmic expression of key genes. We model the first such loop proposed for the clock of Arabidopsis thaliana, the experimental model species for circadian timing in higher plants. As for many biological systems, there are no experimental values for the parameters in our model, and the data available for parameter fitting is noisy and varied. To tackle this we constructed a cost function, which quantifies the agreement between our model and various key experimental features. We then undertook an efficient global search of parameter space, to test whether the proposed circuit can fit the experimental data. Using this approach we show that circadian clock models can function well with low cooperativity in transcriptional regulation, whereas high cooperativity has been a feature of previous (hand-fitted) clock models in other species. Our optimized solution for the Arabidopsis clock model fits several, but not all, of the key experimental features. We test the predicted effects of well-characterized mutations in the clock circuit and show the phases of the circadian cycle where additional components that are yet to be identified experimentally must be present to complete the circadian feedback loop.