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A turnpike property in an eigenvalue optimization problem

Adam Kaminer, Thomas Kriecherbauer, Lars Grüne, Michael Margaliot

TL;DR

The paper studies a constrained eigenvalue optimization problem tied to a nonlinear ODE model of mRNA translation, proving that the optimal parameter vector exhibits a discrete turnpike structure with a long central region of near-constant values. The authors develop a spectral-analysis-based approach, reducing the problem to a two-dimensional hyperbolic-map dynamics that yields an explicit exponential convergence rate $\alpha=2-\sqrt{3}$ toward the bulk and establish a concentration turnpike for large systems. They show that the minimal Perron root $\bar{\sigma}$ converges to 2 as the system size grows, implying the optimal steady-state production rate tends to $1/4$, and provide numerical evidence validating the theoretical predictions for large $n$. The work offers a novel connection between turnpike phenomena and eigenvalue optimization, with potential implications for translation efficiency modeling and spectral-design problems in physics and engineering, and suggests pursuing optimal-control reformulations and extensions to cyclic budgets.

Abstract

We consider a constrained eigenvalue optimization problem that arises in an important nonlinear dynamical model for mRNA translation in the cell. We prove that the ordered list of optimal parameters admits a turnpike property, namely, it includes three parts with the first and third part relatively short, and the values in the middle part are all approximately equal. Turnpike properties have attracted considerable attention in econometrics and optimal control theory, but to the best of our knowledge this is the first rigorous proof of such a structure in an eigenvalue optimization problem.

A turnpike property in an eigenvalue optimization problem

TL;DR

The paper studies a constrained eigenvalue optimization problem tied to a nonlinear ODE model of mRNA translation, proving that the optimal parameter vector exhibits a discrete turnpike structure with a long central region of near-constant values. The authors develop a spectral-analysis-based approach, reducing the problem to a two-dimensional hyperbolic-map dynamics that yields an explicit exponential convergence rate toward the bulk and establish a concentration turnpike for large systems. They show that the minimal Perron root converges to 2 as the system size grows, implying the optimal steady-state production rate tends to , and provide numerical evidence validating the theoretical predictions for large . The work offers a novel connection between turnpike phenomena and eigenvalue optimization, with potential implications for translation efficiency modeling and spectral-design problems in physics and engineering, and suggests pursuing optimal-control reformulations and extensions to cyclic budgets.

Abstract

We consider a constrained eigenvalue optimization problem that arises in an important nonlinear dynamical model for mRNA translation in the cell. We prove that the ordered list of optimal parameters admits a turnpike property, namely, it includes three parts with the first and third part relatively short, and the values in the middle part are all approximately equal. Turnpike properties have attracted considerable attention in econometrics and optimal control theory, but to the best of our knowledge this is the first rigorous proof of such a structure in an eigenvalue optimization problem.
Paper Structure (11 sections, 10 theorems, 125 equations, 4 figures)

This paper contains 11 sections, 10 theorems, 125 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Eigenvalue optimization problem
  3. Maximizing protein production rate in the ribosome flow model
  4. Minimizing the maximal characteristic frequency in a linear chain
  5. Main results
  6. Proofs
  7. Proof of Thm. \ref{['thm:mainj']}
  8. Simplifying Eq. \ref{['eq:sps']}
  9. Hyperbolic map formulation
  10. Numerical examples
  11. Discussion

Key Result

Theorem 12

Fix $n\in\mathbb N$ with $n>1$. Then the optimal parameters $\bar{\lambda}=\bar{\lambda}(n)$ and the minimal Perron root $\bar{\sigma}=\bar{\sigma}(n)$ satisfy: and if $n$ is even then and if $n$ is odd then Furthermore, if $n\geq 36$ then

Figures (4)

  • Figure 1: Optimal $\bar{\lambda}_i$s as a function of $i$ for $n=100$.
  • Figure 2: Graphical representation of Eq. \ref{['eq:eq_psati']}. The volume of the four boxes is equal.
  • Figure 3: Optimal steady state densities $\bar{e}_i$ as a function of $i$ for $n=100$.
  • Figure 4: The value $M(n)$ in \ref{['eq:maxM']} as a function of $n\in\{2,\dots,35\}$.

Theorems & Definitions (34)

  • Remark 2
  • Example 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Example 8
  • Example 9
  • Remark 10
  • Remark 11
  • ...and 24 more