Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rate-cost tradeoffs in continuous-time control with a biomolecular application

Yorie Nakahira, Fangzhou Xiao, Victoria Kostina, John C. Doyle

TL;DR

We analyze continuous-time control under communication constraints for a generalized Ornstein-Uhlenbeck process with additive and multiplicative control, aiming to bound the stationary variance via a data-rate constraint. A continuous-time rate-cost function $R(D)$ and channel capacity $C$ are defined, and a fundamental bound $R_e(D) \ge \frac{E[\sigma^2]}{2D} - E[\mu]$ is derived, with equality achieved by linear Gaussian policies. The work then maps the biomolecular setting to birth-death dynamics and the chemical Langevin approximation, deriving a bound on the stationary Fano factor $F_X$ in terms of $C$ and degradation efficiency $\gamma_X$, with equality under an AWGN channel. These results provide a principled converse linking information-rate limits to achievable control performance and offer guidance for information-efficient biomolecular feedback, while leaving open questions for more general noisy channels. $R(D)$ and $C$ serve as core design criteria for achieving reliable control under communication constraints in biochemical contexts.

Abstract

This paper focuses on rate-limited control of the generalized Ornstein-Uhlenbeck process where the control action can be either multiplicative or additive, and the noise variance can depend on the control action. We derive a lower bound on the data rate necessary to achieve the desired control cost. The lower bound is attained with equality if the control is performed via an additive white Gaussian channel. The system model approximates the dynamics of a discrete-state molecular birth-death process, and the result has direct implications on the control of a biomolecular system via chemical reactions, where the multiplicative control corresponds to the degradation rate, the additive control corresponds to the production rate, and the control objective is to decrease the fluctuations of the controlled molecular species around their desired concentration levels.

Rate-cost tradeoffs in continuous-time control with a biomolecular application

TL;DR

We analyze continuous-time control under communication constraints for a generalized Ornstein-Uhlenbeck process with additive and multiplicative control, aiming to bound the stationary variance via a data-rate constraint. A continuous-time rate-cost function and channel capacity are defined, and a fundamental bound is derived, with equality achieved by linear Gaussian policies. The work then maps the biomolecular setting to birth-death dynamics and the chemical Langevin approximation, deriving a bound on the stationary Fano factor in terms of and degradation efficiency , with equality under an AWGN channel. These results provide a principled converse linking information-rate limits to achievable control performance and offer guidance for information-efficient biomolecular feedback, while leaving open questions for more general noisy channels. and serve as core design criteria for achieving reliable control under communication constraints in biochemical contexts.

Abstract

This paper focuses on rate-limited control of the generalized Ornstein-Uhlenbeck process where the control action can be either multiplicative or additive, and the noise variance can depend on the control action. We derive a lower bound on the data rate necessary to achieve the desired control cost. The lower bound is attained with equality if the control is performed via an additive white Gaussian channel. The system model approximates the dynamics of a discrete-state molecular birth-death process, and the result has direct implications on the control of a biomolecular system via chemical reactions, where the multiplicative control corresponds to the degradation rate, the additive control corresponds to the production rate, and the control objective is to decrease the fluctuations of the controlled molecular species around their desired concentration levels.
Paper Structure (14 sections, 6 theorems, 56 equations, 1 figure)

This paper contains 14 sections, 6 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Continuous-time control
  3. Operational problem setup
  4. Directed mutual information
  5. Rate-cost function
  6. Channel capacity
  7. Main result
  8. Biomolecular control
  9. Conclusion
  10. Proof of Proposition \ref{['prop:continuous-time-separation']} and Proposition \ref{['prop:total']}
  11. Proof of Theorem \ref{['thm:Rd']}
  12. Rate-distortion function for the discrete-time system
  13. Rate-distortion function for the continuous-time system: proof of Theorem \ref{['thm:Rd']}
  14. Biomolecular control: proofs

Key Result

Proposition 1

The rate-cost function cannot be smaller than the rate-distortion function:

Figures (1)

  • Figure 1: The continuous-time control system.

Theorems & Definitions (18)

  • Definition 1: Directed information in continuous time weissman2013directed
  • Definition 2: Causally conditioned directed information in continuous time
  • Definition 3: Continuous-time rate-cost function
  • Definition 4: Continuous-time rate-distortion function
  • Proposition 1
  • Definition 5: Continuous-time channel capacity weissman2013directed
  • Proposition 2
  • Theorem 1
  • proof
  • Proposition 3
  • ...and 8 more