Modeling diurnal rhythms with an array of phase dynamic oscillators

Behavior of living organisms is strongly modulated by light especially by the day and night cycle giving rise to a cyclic pattern of activities. Such a pattern helps the organism to coordinate their activities and maintain a balance between what could be performed during the 'day' and what could be relegated to 'night'. This cyclic pattern, called the 'Circadian Rhythm', is a biological phenomenon observed in a large number of organisms ranging from unicellular bacteria to human beings and is present in data collected at various levels viz. transcriptome, proteome etc. In this paper, our goal is to analyze transcriptome data from Cyanothece, a photosynthetic cyanobacteria, for the purpose of discovering genes whose expressions are rhythmic, especially those for which these rhythms have a 24 hours cycle. Subsequently we propose a model with a network of three phase oscillators for each one of the twenty four hours cycle. Each of the three phase oscillators is chosen to maintain a phase difference of 120 degrees between each other. All the oscillators are connected to an internal clock that is designed to maintain a phase activity close to a master clock derived using KaiC proteins. In Cyanobacteria it is believed that the KaiC proteins provide the internal rhythm. The model parameters, viz. connection strengths between the master clock and peripheral oscillators and the parameters computing the linear combinations of the oscillator phase variables, are optimized to provide a close match to the observed gene expressions even when the frequency of the internal clock and the natural frequencies of the oscillators vary within a certain range. As a final step, the oscillator network model has been used to isolate genes, and hence the associated subprocesses, whose expression cycles are robust with respect to variations in the oscillator frequencies.