Abstract
Periodic oscillations play an important role in many biomedical systems. Proper functioning of biological systems that respond to periodic signals requires the ability to synchronize with the periodic excitation. For example, the sleep/wake cycle is a manifestation of an internal timing system that synchronizes to the solar day. In the terminology of systems theory, the biological system must entrain or phase-lock to the periodic excitation. Entrainment is also important in synthetic biology. For example, connecting several artificial biological systems that entrain to a common clock may lead to a well-functioning modular system. The cell-cycle is a periodic program that regulates DNA synthesis and cell division. Recent biological studies suggest that cell-cycle related genes entrain to this periodic program at the gene translation level, leading to periodically-varying protein levels of these genes. The ribosome flow model (RFM) is a deterministic model obtained via a mean-field approximation of a stochastic model from statistical physics that has been used to model numerous processes including ribosome flow along the mRNA. Here we analyze the RFM under the assumption that the initiation and/or transition rates vary periodically with a common period . We show that the ribosome distribution profile in the RFM entrains to this periodic excitation. In particular, the protein synthesis pattern converges to a unique periodic solution with period . To the best of our knowledge, this is the first proof of entrainment in a mathematical model for translation that encapsulates aspects such as initiation and termination rates, ribosomal movement and interactions, and non-homogeneous elongation speeds along the mRNA. Our results support the conjecture that periodic oscillations in tRNA levels and other factors related to the translation process can induce periodic oscillations in protein levels, and may suggest a new approach for re-engineering genetic systems to obtain a desired, periodic, protein synthesis rate.
Highlights
External and internal periodic oscillations play an important role in intracellular and extracellular biomedical systems and have attracted enormous research interest
There exists a minimal Tw0 such that all these functions are T -periodic. We refer to this case as the periodic ribosome flow model (PRFM)
Remark 1 Note that the PRFM includes in particular the case where some of these rates are constant, as a constant function is Tperiodic for every T
Summary
External and internal periodic oscillations play an important role in intracellular and extracellular biomedical systems and have attracted enormous research interest (see e.g. [1] and the references therein). External and internal periodic oscillations play an important role in intracellular and extracellular biomedical systems and have attracted enormous research interest Proper functioning of cells that are exposed to such periodic signals requires internal biological mechanisms that are able to synchronize with the periodic excitation. In the terminology of systems theory, the biological system must entrain or phase-lock to the periodic excitation. In response to a periodic excitation with period T the system’s internal state converges to a periodic signal with period T. Entrainment in biological systems (sometimes called phase locking [2]) and, more generally, biological oscillators and rhythms have recently attracted enormous attention The sleep/wake cycle is a manifestation of an internal timing system that entrains to the 24 hours period of the solar day using a visual pathway connecting the retina to the suprachiasmatic nucleus (SCN) [6]
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