9.4: Kinetic rate laws

We can organize our thoughts on the kinetics of microbial reactions by considering rate laws for microbial growth. One of the best known rate laws is the Monod equation defined by Monod (1949). The equation relates the rate of growth of a microbial population to the concentration of a limiting resource. If the limiting nutrient is an electron donor, we can write: \[r = k_{+} [X] F_{D}\]where \(k_{+}\) is the rate constant for the forward reaction \(\left(\text{mol (g biomass)}^{-1} \text{s}^{-1}\right)\), \([X]\) represents biomass concentration \(\left( \text{(g biomass) (kg} \cdot \text{H}_{2} \text{O)}^{-1}\right)\), and \(F_{D}\) is a kinetic factor for electron donation. Biomass is included because the overall rate of a microbial reaction varies directly with the size of the microbial population that is actively catalyzing it. As the abundance of active cells increases, more enzymes can be produced, and more catalysis can occur.

\(F_{D}\) can be expanded as: \[F_{D} = \frac{m_{D}}{m_{D} + K_{D}}\]where \(m_{D}\) is the electron donor concentration (molal) and \(K_{D}\) is the half-saturation constant (molal) corresponding to the electron donor. Thus, the kinetic factor considers substrate availability and the kinetic properties of the microorganism’s enzymes.

The dual-Monod equation is a slightly expanded version that accounts for not one but two limiting resources. For example, the following rate equation includes factors for the electron donor \(\left(F_{D}\right)\) and electron acceptor \(\left(F_{A}\right)\): \[r = k_{+} [X] F_{D} F_{A}\]

Here \(F_{A}\) is defined in an equivalent manner as \(F_{D}\) (equation \((\PageIndex{2})\)), except the substrate considered is the electron acceptor rather than the electron donor: \[F_{A} = \frac{m_{A}}{m_{A} + K_{A}}\]where \(m_{A}\) is electron acceptor concentration (molal) and \(K_{A}\) is the half saturation constant for the electron acceptor (molal).

The Monod equation works well for resource-rich environments where only a small number of factors vary. However, in geochemical systems, resources are often limited and conditions can be spatially and temporally variable. In such systems, therefore it may be necessary to use rate expressions that also account for thermodynamic controls (Jin and Bethke, 2005). To that end, the Jin and Bethke model (2007, 2005, 2003, 2002) provides an example of a rate model that expands the Monod equation further by adding a thermodynamic potential factor \(\left(F_{T}\right)\): \[r = k_{+} [X] F_{D} F_{A} F_{T}\]

The thermodynamic potential factor \(\left(F_{T}\right)\) is unitless and defined as follows: \[F_{T} = 1 - \exp \left(- \frac{\Delta G_{A} - m \Delta G_{P}}{\chi RT}\right)\]

where \(\Delta G_{A}\) is energy available in the environment for the microbial reaction as described in Section 8.4 and repeated here for convenience: \[\Delta G_{A} = - \Delta G_{r}\]

\(m\) is the number of ATPs produced (unitless), \(\Delta G_{P}\) is the free energy change of ATP synthesis \(\left(\text{kJ mol}^{-1}\right)\), \(\chi\) is the average number of times the rate determining step occurs (unitless), \(R\) is the gas constant \(\left(\text{kJ mol}^{-1} \text{K}^{-1}\right)\), and \(T\) is temperature \((\text{K})\). As also discussed in Section 8.4, \(m \Delta G_{P}\) quantifies energy captured by the cell \(\left(\Delta G_{C}\right)\) and thus the numerator in equation \(\PageIndex{6}\) is usable energy \((\Delta G_{U})\). Thus, thermodynamic potential factor equation can also be written as: \[F_{T} = 1 - \exp \left(- \frac{\Delta G_{U}}{\chi RT}\right)\]

For each of these rate laws, reaction rates can then be related to change in biomass concentration according to: \[\frac{d[X]}{dt} = Yr - D[X]\]where \(Y\) is the amount of biomass growth relative to the amount of substrate consumed \(\left(\text{(g biomass) (mol substrate)}^{-1}\right)\), which is known as growth yield, and \(D\) is the biomass decay constant \(\left(\text{s}^{-1}\right)\), which accounts for the rate at which cells die.

By these rate laws, \(F_{D}\) and \(F_{A}\) approach values of one as substrate concentrations increase. Similarly, if the amount of energy available \((\Delta G_{A})\) is much greater than the amount captured by the cell \((m \Delta G_{P})\), the exponential term in equation \(\PageIndex{6}\) approaches zero and the overall value of \(F_{T}\) approaches one. If the value of any of these factors, \(F_{D}\), \(F_{A}\), or \(F_{T}\), is one, then that factor does not affect the rate of the reaction. Thus, if the values of all three factors is 1, all three factors have no influence on the reaction rate. In this case, the rate is simply equal to the product of the rate constant and biomass concentration \(\left(k_{+} [X]\right)\), which is the maximum rate of the reaction \(\left(V_{\text{max}}\right)\) (Jin and Bethke, 2002).

---