Comparison of cellular oscillations driven by noise or deterministic mechanisms under cell-size scaling

Ultradian cycles are frequently observed in biological systems. They serve important roles in regulating, for example, cell fate and the development of the organism. Many mathematical models have been developed to analyze their behavior. Generally, these models can be classified into two classes: Deterministic models that generate oscillatory behavior by incorporating time delays or Hopf bifurcations, and stochastic models that generate oscillatory behavior by noise driven resonance. However, it is still unclear which of these two mechanisms applies to cellular oscillations. In this paper, we show through theoretical analysis and numerical simulation that we can distinguish which of these two mechanisms govern cellular oscillations, by measuring statistics of oscillation amplitudes for cells of different sizes. We found that, for oscillations driven deterministically, the normalized average amplitude is constant with respect to cell size, while the coefficient of variation of the amplitude scales with cell size with an exponent of -0.5. On the other hand, for oscillations driven stochastically, the coefficient of variation of the amplitude is constant with respect to cell size, while the normalized average amplitude scales with cell size with an exponent of -0.5. Our results provide a theoretical basis to discern whether a particular oscillatory behavior is governed by a deterministic or stochastic mechanism.