Each enzyme in a reaction network model has a thermodynamic parameter called \(k_{eq}\), which represents how much energy the corresponding chemical reaction stores (\(k_{eq}\) less than 1) or releases (\(k_{eq}\) greater than 1) from its environment. This document explains how the laws of thermodynamics constrain these parameters and how Maud ensures that these constraints are satisfied.

Why the laws of thermodynamics impose constraints

Thermodynamic parameters are constrained in two main ways.

First, each thermodynamic parameter must agree with the kinetic parameters describing its enzyme, according to the Haldane relationships governing the enzyme’s mechanism. For example, the Haldane relationships for an enzyme with an ordered unibi mechanism are as follows:

\[k_{eq} = \frac{ k_{cat1}k_{ip}k_{q} }{k_{cat2}k_{ia}} = \frac{ k_{cat1}k_{p}k_{iq} }{k_{cat2}k_{a}}\]

Second, the thermodynamic parameters in a network must not jointly imply that it is possible to create or destroy energy simply by following a series of reactions round in a loop. This implies that, at equilibrium, the net change in Gibbs free energy due to the reactions in a loop should be exactly zero. In mathematical notation:

\[\Sigma_{i\in loop}\Delta G_i = 0\]

Since there is a one-to-one relationship between \(k_eq\) s and \(DeltaG\) s, this condition further constrains the feasible area of thermodynamic parameter space for networks with loops.

How Maud ensures thermodynamic consistency

In order to ensure that each enzyme’s kinetic and thermodynamic parameters agree, Maud ensures that one parameter per Haldane relationship is fixed based on the values of the other parameters. For example, in the ordered unibi case the \(k_{ip}\) and \(k_{iq}\) parameters are fixed as follows:

\[\begin{split}k_{ip} = \frac{k_{eq}k_{ia}k_{cat2}}{k_{q}k_{cat1}} \\ k_{iq} = \frac{k_{eq}k_{cat2}k_{a}}{k_{cat1}k_{p}}\end{split}\]

In order to avoid free energy loops, Maud generates \(k_{eq}\) parameters from \(\Delta G\) parameters according to the following equation:

\[\mathbf{k_{eq}} = \exp(\frac{\Delta G}{-RT})\]

where R is the universal gas constant and T is the temperature in kelvin (currently this is assumed to be 298). \(Delta G\) parameters, in turn, are generated as follows:

\[\Delta G = K\mathbf{b}\]

where \(\mathbf{b}\) of auxiliary basis parameters whose length is the same as the rank of the network’s stoichiometric matrix and \(K = Nullspace(Nullspace(S^{T})^{T})\) is a matrix generated from the network’s stoichiometric matrix \(S\) so as to ensure that \(\Delta G\) sums to zero for loops.

In the case where the network has no loops, the width of K will be a diagonal matrix and the basis parameters directly determine to the \(\Delta G\) s. If there are loops, there will be fewer basis parameters than \(\Delta G\) s.

Information about marginal values of \(Delta G\) - for example from in vitro measurements - is represented directly as prior distributions on the transformed \(Delta G\) parameters. Since the transformation from basis parameters to \(Delta G\) s is linear and the posterior only needs to be ascertained up to proportionality, there is no need for any adjustments to take into account the effect of this transformation.