5.14 Chemical Reaction Production Rate

Production rate constitutive equations provide a relation for \(\hat{\zeta}\) as a function of solid matrix strain, solute concentrations, and the concentrations of solid-bound molecular species.

Mass Action Forward

According to the law of mass action for forward reactions,

\[ \begin{equation} \hat{\zeta}=k\left(\theta,\mathbf{F},\rho_{r}^{\sigma}\right)\prod\limits_{\alpha}\left(c^{\alpha}\right)^{\nu_{R}^{\alpha}}\,.\label{eq525} \end{equation} \]

A constitutive relation for the specific reaction rate \(k\left(\theta,\mathbf{F},\rho_{r}^{\sigma}\right)\) must also be provided.

Mass Action Reversible

According to the law of mass action for reversible reactions,

\[ \begin{equation} \begin{aligned}\hat{\zeta}_{F} & =k_{F}\left(\theta,\mathbf{F},\rho_{r}^{\sigma}\right)\prod\limits_{\alpha}\left(c^{\alpha}\right)^{\nu_{R}^{\alpha}}\\ \hat{\zeta}_{R} & =k_{R}\left(\theta,\mathbf{F},\rho_{r}^{\sigma}\right)\prod\limits_{\alpha}\left(c^{\alpha}\right)^{\nu_{P}^{\alpha}}\\ \hat{\zeta} & =\hat{\zeta}_{F}-\hat{\zeta}_{R}=\hat{\zeta}_{F}\left[1-K_{c}\left(\theta,\mathbf{F},\rho_{r}^{\sigma}\right)\prod\limits_{\alpha}\left(c^{\alpha}\right)^{\nu^{\alpha}}\right]\,, \end{aligned} \label{eq526} \end{equation} \]

where \(K_{c}=k_{R}/k_{F}\) is a function that reduces to the equilibrium constant of the reversible reaction at chemical equilibrium (when \(\hat{\zeta}=0)\). Constitutive relations for the specific forward and reverse reaction rates, \(k_{F}\left(\theta,\mathbf{F},\rho_{r}^{\sigma}\right)\) and \(k_{R}\left(\theta,\mathbf{F},\rho_{r}^{\sigma}\right)\) respectively, must also be provided.

Michaelis-Menten

Michaelis-Menten is a model for enzyme kinetics as represented by the reactions

\[ \begin{equation} \mathcal{E}^{e}+\mathcal{E}^{s}\rightleftharpoons\mathcal{E}^{es}\to\mathcal{E}^{e}+\mathcal{E}^{p}\,,\label{eq527} \end{equation} \]

where \(\mathcal{E}^{e}\) is the enzyme, \(\mathcal{E}^{s}\) is the substrate, \(\mathcal{E}^{es}\) is the enzyme-substrate complex, and \(\mathcal{E}^{p}\) is the product. The molar mass supply \(\hat{c}^{p}\) producing \(\mathcal{E}^{p}\) is related to the concentration of the substrate \(\mathcal{E}^{s}\) via

\[ \begin{equation} \hat{c}^{p}=\frac{V_{max}c^{s}}{K_{m}+c^{s}}\,,\label{eq528} \end{equation} \]

where \(V_{max}\) is the maximum rate achieved by the system, at maximum (saturating) substrate concentrations. \(K_{m}\) is the substrate concentration at which the reaction rate is half of \(V_{max}\).

This relation may be derived by applying the law of mass action to the two reactions in \eqref{eq527}. under the simplifying assumption that the reversible reaction between the enzyme and substrate reaches steady state much faster than the subsequent forward reaction forming the product. If the first and second reactions are denoted by subscripts 1 and 2, respectively, the law of mass action for the first (reversible) and second (forwar) reaction produces

\[ \begin{equation} \begin{aligned}\hat{\zeta}_{1} & =k_{F1}c^{e}c^{s}-k_{R1}c^{es},\quad\hat{\zeta}_{2}=k_{F2}c^{es}\,,\\ \hat{c}^{s} & =-\hat{\zeta}_{1},\quad\hat{c}^{p}=\hat{\zeta}_{2},\quad\hat{c}^{es}=\hat{\zeta}_{1}-\hat{\zeta}_{2}\,. \end{aligned} \label{eq529} \end{equation} \]

The total enzyme concentration remains constant at \(c_{0}^{e}=c^{e}+c^{es}\), so that\(\hat{\zeta}_{1}=k_{F1}c_{0}^{e}c^{s}-\left(k_{F1}c^{s}+k_{R1}\right)c^{es}\). Assuming that the first reaction equilibrates much faster than the second is equivalent to letting \(\hat{\zeta}_{1}\approx0\), in which case

\[ \begin{equation} c^{es}\approx\frac{c_{0}^{e}c^{s}}{c^{s}+K_{m}}\,,\label{eq530} \end{equation} \]

where \(K_{m}=k_{R1}/k_{F1}\). Then,

\[ \hat{\zeta}_{2}=\frac{V_{\max}c^{s}}{c^{s}+K_{m}}\,, \]

where \(V_{\max}=k_{F2}c_{0}^{e}\) represents the maximum value of \(\hat{\zeta}_{2}\), when \(K_{m}\ll c^{s}\). In practice, choosing \(k_{F1}\gg k_{F2}\) can produce the desired effect.