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The adaptation property in non-equilibrium chemical systems

E. Franco, J. J. L. Velázquez

TL;DR

The paper investigates when biological-like adaptation can arise in chemical networks, asking whether energy expenditure is required or if detailed-balance (passive) systems can exhibit robust adaptation. It develops a formal framework for signalling systems built on mass-action kinetics with external boundary forcing and analyzes adaptation through network topology and conserved quantities. The main contributions show that robust adaptation in closed, detailed-balance networks is generically impossible unless a stringent factorization of conservation laws holds, while open, energy-exchanging networks can achieve robust adaptation under the same balance condition. It also connects these theoretical results to classical and canonical models of adaptation (Segel–Goldbeter–Devreotes–Knox, Barkai–Leibler) and to gene-expression scenarios, clarifying when and how robust adaptation can emerge or fail in passive versus active regimes.

Abstract

The goal of this paper is to understand if the property of adaptation, which is a typical property of many biochemical systems, can be achieved only by biological systems that actively consume energy or if it can be achieved also by passive systems. We prove that, unless the conserved quantities of a signalling system satisfy a very specific factorization assumption, adaptation cannot be achieved in a robust manner (i.e. in a stable manner under perturbations of the chemical reaction rates) by systems that satisfy the detailed balance property, hence that are passive, and that do not exchange substances or energy with the environment. We also prove that robust adaptation can be achieved by systems that satisfy the detailed balance property, but exchange substances with the environment.

The adaptation property in non-equilibrium chemical systems

TL;DR

The paper investigates when biological-like adaptation can arise in chemical networks, asking whether energy expenditure is required or if detailed-balance (passive) systems can exhibit robust adaptation. It develops a formal framework for signalling systems built on mass-action kinetics with external boundary forcing and analyzes adaptation through network topology and conserved quantities. The main contributions show that robust adaptation in closed, detailed-balance networks is generically impossible unless a stringent factorization of conservation laws holds, while open, energy-exchanging networks can achieve robust adaptation under the same balance condition. It also connects these theoretical results to classical and canonical models of adaptation (Segel–Goldbeter–Devreotes–Knox, Barkai–Leibler) and to gene-expression scenarios, clarifying when and how robust adaptation can emerge or fail in passive versus active regimes.

Abstract

The goal of this paper is to understand if the property of adaptation, which is a typical property of many biochemical systems, can be achieved only by biological systems that actively consume energy or if it can be achieved also by passive systems. We prove that, unless the conserved quantities of a signalling system satisfy a very specific factorization assumption, adaptation cannot be achieved in a robust manner (i.e. in a stable manner under perturbations of the chemical reaction rates) by systems that satisfy the detailed balance property, hence that are passive, and that do not exchange substances or energy with the environment. We also prove that robust adaptation can be achieved by systems that satisfy the detailed balance property, but exchange substances with the environment.

Paper Structure

This paper contains 15 sections, 9 theorems, 121 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Signalling systems
  4. Kinetic systems and their conservation laws
  5. Detailed balance property of kinetic systems
  6. Signalling systems
  7. Adaptation
  8. Definition of adaptation
  9. Response of the network to changes in the signal concentration
  10. Stable adaptation in non-conservative systems that satisfy the detailed balance property
  11. Stable adaptation is impossible in closed signalling systems
  12. Some models exhibiting the adaptation property
  13. Some classical models of adaptation with fine tuned parameters
  14. The Barkai-Leibler model of robust adaptation for bacterial chemotaxis
  15. An adaptation model of gene expression

Key Result

Theorem 1.1

Assume that a signalling system is closed. Then the signalling system does not satisfy the adaptation property unless the reaction rates are fine tuned or the conserved quantities satisfy a factorization assumption.

Figures (2)

  • Figure 1: On the left we plot the function $f$, describing the change in time of the concentration of signal. The function $f$ tends to a constant value as $t\rightarrow \infty$. On the right we plot $n_p(t)$, which is the concentration of product, for a signalling system that satisfies the adaptation property. Notice that the concentration of product changes as the signal changes. However it returns to the pre-signal values as time tends to infinity.
  • Figure 2: An example of signalling system that is conservative, i.e. it exchanges only the signal (yellow star) with the environment. Notice that the signals binds to the receptors of the signalling system.

Theorems & Definitions (22)

  • Theorem 1.1: Stable adaptation is impossible in closed systems
  • Theorem 1.2: Response in a connected system
  • Theorem 1.3: Adaptation in kinetic systems with the detailed balance property
  • Definition 2.1: Kinetic system
  • Definition 2.2: Set of conservation laws
  • Lemma 2.3
  • Definition 2.4: Detailed balance property
  • Lemma 2.5
  • Definition 2.6
  • Proposition 2.8
  • ...and 12 more