Entrainment to periodic initiation and transition rates in a computational model for gene translation

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 T. 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 T. 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.

Figure 1

Upper part: the elongation rates of codons and the initiation rate are -periodic, for example, due to signals related to the cell-cycle. Lower part: in the RFM, this is modeled by -periodic rates and yielding the PRFM. Our main result shows that consequently the translation rate and ribosomal densities (the s) converge to a unique -periodic solution.

Figure 2. State-variables (t) [solid line]; (t) [dashed]; and (t) [dotted] (y-axis) as a function of time (x-axis) in Example 1.

All state-variable converge to a periodic signal with period .

Figure 3. State-variables (t) [solid line]; (t) [dashed]; and (t) [dotted] (y-axis) as a function of time (x-axis) in Example 2.

The initiation rate is periodic with period , while the transition rates are constant and relatively small. All state-variable converge to a periodic signal with period , but the amplitude of the oscillations is considerably attenuated as it passes through the mRNA chain.

Figure 4. State-variables (t) [solid line]; (t) [dashed]; and (t) [dotted] (y-axis) as a function of time (x-axis) in Example 3.

The initiation and transition rates are periodic with a common period , but with added random noise. It may be seen that each state-variable converges to a periodic signal with period , but with added noise.

Figure 5. Simulation of TASEP with periodically time-varying rates.

The plot shows the averaged occupancy over time segments as a function of the time segment.