<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$R_{ij} (t) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mrow> <mi mathvariant="italic">ij</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> that measure how the system reacts to the injection of the substance <i>j</i> at time <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$t=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, by measuring the concentration of the substance <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$i \ne j$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$t >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In particular, we obtain a condition involving two reciprocal measurements (that is <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$R_{ij}(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mrow> <mi mathvariant="italic">ij</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$R_{ji}(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mrow> <mi mathvariant="italic">ji</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>) 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 by the graph associated to the network, as well as a stability property that guarantees that the chemical rates are not fine-tuned.</p>

Characterizing the Detailed Balance Property by Means of Measurements

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2026年4月3日

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

摘要

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) $ $R_{ij} (t)$ that measure how the system reacts to the injection of the substance j at time $t=0$ $t = 0$ , by measuring the concentration of the substance $i \ne j$ $i \neq j$ for $t >0$ $t > 0$ . In particular, we obtain a condition involving two reciprocal measurements (that is $R_{ij}(t)$ $R_{ij} (t)$ , $R_{ji}(t)$ $R_{ji} (t)$ ) 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 by the graph associated to the network, as well as a stability property that guarantees that the chemical rates are not fine-tuned.