On the gain of entrainment in the $n$-dimensional ribosome flow model

TL;DR

This paper investigates the gain of entrainment (GOE) in the $n$-dimensional ribosome flow model (RFM) under jointly $T$-periodic translation-rate inputs. It introduces a novel approach based on integrating along the $T$-periodic solution and using centered state variables with higher-order moments, enabling rigorous analysis beyond previously solved cases ($n=1,2$). The authors derive several non-GOE results, including six explicit rate-structure conditions (e.g., odd $n$ with paired internal rates) under which the average production with periodic inputs cannot exceed the constant-rate case, suggesting that constant rates often optimize long-run protein production. The work provides a rigorous framework for understanding when periodic regulation can fail to improve performance, with implications for translation control strategies and for related contractive, periodic dynamical systems. The discussion also outlines connections to TASEP, potential extensions to stochastic settings, and directions for future research on GOE in higher-dimensional RFMs.

Abstract

The ribosome flow model (RFM) is a phenomenological model for the flow of particles along a 1D chain of $n$ sites. It has been extensively used to study ribosome flow along the mRNA molecule during translation. When the transition rates along the chain are time-varying and jointly $T$-periodic the RFM entrains, i.e., every trajectory of the RFM converges to a unique $T$-periodic solution that depends on the transition rates, but not on the initial condition. In general, entrainment to periodic excitations like the 24h solar day or the 50Hz frequency of the electric grid is important in numerous natural and artificial systems. An interesting question, called the gain of entrainment (GOE) in the RFM, is whether proper coordination of the periodic translation rates along the mRNA can lead to a larger average protein production rate. Analyzing the GOE in the RFM is non-trivial and partial results exist only for the RFM with dimensions $n=1,2$. We use a new approach to derive several results on the GOE in the general $n$-dimensional RFM. Perhaps surprisingly, we rigorously characterize several cases where there is no GOE, so to maximize the average production rate in these cases, the best choice is to use constant transition rates all along the chain.

Figures (4)

• Figure 1: Periodic solution $\gamma(t)$ as a function of $t$ for the system in Example \ref{['exa:tanh']}.
• Figure 2: Trajectories $x_i(t)$ as a function of $t$ in an RFM with $n=3$ and $2\pi$-periodic rates. Every state-variable converges to a periodic pattern, with period $2\pi$.
• Figure 3: Moving average of the production rate $u_4 x_4$ in an RFM with $n=4$ for different choices of $1$-periodic transition rates.
• Figure 4: Plot of $\gamma_4(t)$ over a single period for several simulations using different transition rates with the same average values.

Theorems & Definitions (13)

• Theorem 1
• Proposition 2
• Remark 1
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Corollary 5
• proof