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Characterizing the detailed balance property by means of measurements in chemical networks

Eugenia Franco, Bernhard Kepka, Juan J. L. Velázquez

Abstract

In this paper we study how to determine if a linear biochemical network satisfies the detailed balance condition, without knowing the details of all the reactions taking place in the network. To this end, we use the formalism of response functions $R_{ij} (t) $ that measure how the system reacts to the injection of the substance $j$ at time $t=0$, by measuring the concentration of the substance $i \neq j$ for $t >0$. In particular, we obtain a condition involving two reciprocal measurements (i.e.~$R_{ij}(t), \, R_{ji}(t)$) that is necessary, but not sufficient for the detailed balance condition to hold in the network. Moreover, we prove that this necessary condition is also sufficient if a topological condition is satisfied, as well as a stability property that guarantees that the chemical rates are not fine-tuned.

Characterizing the detailed balance property by means of measurements in chemical networks

Abstract

In this paper we study how to determine if a linear biochemical network satisfies the detailed balance condition, without knowing the details of all the reactions taking place in the network. To this end, we use the formalism of response functions that measure how the system reacts to the injection of the substance at time , by measuring the concentration of the substance for . In particular, we obtain a condition involving two reciprocal measurements (i.e.~) that is necessary, but not sufficient for the detailed balance condition to hold in the network. Moreover, we prove that this necessary condition is also sufficient if a topological condition is satisfied, as well as a stability property that guarantees that the chemical rates are not fine-tuned.
Paper Structure (22 sections, 22 theorems, 106 equations, 3 figures)

This paper contains 22 sections, 22 theorems, 106 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Plan of the paper
  3. Notations
  4. Linear biochemical systems and detailed balance
  5. Notions of graph theory
  6. Linear biochemical systems
  7. Detailed balance
  8. Lack of detailed balance in biochemical systems
  9. Pathwise detailed balance
  10. Pathwise detailed balance and pathwise symmetry
  11. Pathwise detailed balance and detailed balance
  12. Stability classes of chemical networks
  13. Stability classes
  14. Stability/instability of a property in a stability class
  15. Classes of graphs for which the pathwise detailed balance condition is stable
  16. ...and 7 more sections

Key Result

Theorem 1.1

Consider a biochemical network without cut vertices for which property eq:intro p(DB) holds and is stable under small, admissible perturbations of the chemical rates. Then, the detailed balance condition holds.

Figures (3)

  • Figure 1: Graph corresponding to the matrix $A$. In red and marked with the symbol $//$ we have the only reaction without detailed balance of the network.
  • Figure 2: Graph corresponding to the matrix $A$. In red and marked with the symbol $//$ we have the only reaction without detailed balance of the network.
  • Figure 3: Graph corresponding to the matrix $A$. In red and marked with the $//$ symbol we have the only reaction without detailed balance of the network.

Theorems & Definitions (58)

  • Theorem 1.1
  • Definition 2.1: Detailed balance
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4: Sufficient geometrical condition for detailed balance
  • Corollary 2.5
  • proof
  • Definition 3.1: Pathwise detailed balance
  • Definition 3.2: Pathwise symmetry
  • ...and 48 more