# Optimal Translation Along a Circular mRNA

## TL;DR

This paper introduces a spectral method for optimizing translation efficiency in circular mRNA models, enabling stable computation and convex optimization of ribosomal flow for genetic engineering.

## Contribution

It derives a new spectral representation for the RFMR model, facilitating efficient and stable optimization of translation rates in circular mRNA.

## Key findings

- Spectral representation simplifies optimal rate computation.
- Optimal production rate is a strictly concave function.
- Optimization becomes a convex problem with a unique solution.

## Abstract

The ribosome flow model on a ring (RFMR) is a deterministic model for translation of a circularized mRNA. We derive a new spectral representation for the optimal steady-state production rate and the corresponding optimal steady-state ribosomal density in the RFMR. This representation has several important advantages. First, it provides a simple and numerically stable algorithm for determining the optimal values even in very long rings. Second, it enables efficient computation of the sensitivity of the optimal production rate to small changes in the transition rates along the mRNA. Third, it implies that the optimal steady-state production rate is a strictly concave function of the transition rates. Thus maximizing the optimal steady-state production rate with respect to the rates, under an affine constraint on the rates becomes a convex optimization problem that admits a unique solution, which can be determined numerically using highly efficient algorithms. This optimization problem is important, for example, when re-engineering heterologous genes in a host organism. We describe the implications of our results to this and other aspects of translation.