WHEN DESIGN MEETS KINETICS: THE RDX MATHEMATICAL MODEL
INTRODUCTION TO MATHEMATICAL MODELING
Mathematical modeling is a cornerstone of bioprocess and engineering systems because it transforms complex biological and chemical interactions into a framework that can be analyzed, simulated, and optimized. By translating biological functions into equations, models allow researchers to predict system behavior, test design assumptions, and reduce experimental uncertainty before moving to the laboratory.
In biotechnology, enzyme kinetics and microbial growth models are widely used to connect molecular activity with reactor performance, creating a bridge between biology and engineering [3]. For our project, modeling served as a guide to understand how engineered E. coli expressing the XplA/B enzymes could degrade the explosive RDX.
By applying Michaelis–Menten kinetics and incorporating microbial growth and enzyme production dynamics, we were able to preliminarily predict degradation efficiency, estimate byproduct formation, and define the operational parameters of our proposed four-stage bioprocess.
Figure 1. This chart illustrates the degradation kinetics of RDX and the concurrent formation of its byproducts, formaldehyde and nitrite, catalyzed by the XplA and XplB enzyme complex. Data generated through Microsoft Excel using the parameters shown in Table 1.
Key Model Parameters and Assumptions
The R-DetoX 2.0 mathematical model is a framework that integrates multiple variables to simulate the enzymatic degradation of RDX. This model is not based on a single parameter or process; rather, it combines biological, chemical, and engineering principles to predict the overall degradation efficiency of RDX in a four-stage bioprocess. By linking these variables through kinetic equations, the model provides a quantitative platform to explore the behavior of our prototype and anticipate byproduct formation before experimental implementation.
Each variable incorporated was selected based on literature precedent, ensuring that the model reflects biologically and chemically realistic conditions. Parameters such as the Michaelis constant \(K_M\) and enzymatic turnover rate \(k_{cat}\) capture the intrinsic catalytic efficiency, while initial substrate and enzyme concentrations define the starting conditions for the reaction. The interplay of these variables allows the model to simulate dynamic changes in RDX concentration over time.
Parameter Table
| Parameter | Value | Source / Notes |
|---|---|---|
| Michaelis constant KM | 83.7 μM | [1] Jackson et al. (2007) |
| Turnover rate \(k_{cat}\) | 4.44 s⁻¹ | |
| Active Enzyme Concentration [E] | 0.06 μM | Estimated |
| Initial RDX Concentration \([RDX]_i\) | 100 μM | [2] Exaggerated; EPA (2017) max in groundwater: 60 μM |
| Final RDX Concentration \([RDX](t)\) | 0.0009 μM | [2] EPA (2017) drinking water limit |
Differential Equation and Integrated Solution
This differential equation (1) represents the rate of RDX degradation over time, assuming the reaction follows Michaelis–Menten enzyme kinetics. This the result when we rearrange the Michaelis–Menten differential equation (2), where \([S]\) represents the substrate (RDX).
The integral form (3) represents the analytical integration step needed to solve for substrate concentration as a function of time. The first integral corresponds to the logarithmic term (arising from \(\frac{1}{[S]}\)), the second integral corresponds to a linear term (arising from 1), \([S]_i\) is the initial substrate concentration, and \([S](t)\) is the concentration at time t.
\[ \int_0^t dt = -\frac{K_M}{k_{cat} [E]} \int_{[S]_i}^{[S](t)} \frac{1}{[S]} \, d[S] - \frac{1}{k_{cat} [E]} \int_{[S]_i}^{[S](t)} d[S] (3) \]This final expression (4) gives the time required for the substrate concentration to decrease from an initial value \([RDX]_i\) to some value \([RDX](t)\). This equation is especially useful for predicting degradation times in enzyme-based bioremediation, linking molecular parameters (\(K_M\), \(k_{cat}\), \([E]\)) to macroscopic behavior.
\[ t = \frac{K_M}{k_{cat}[E]} \ln \left( \frac{[RDX]_i}{[RDX](t)} \right) + \frac{[RDX]_i - [RDX](t)}{k_{cat}[E]} (4) \]Adaptation
The model can be adapted to account for microbial growth, enzyme induction, and multi-stage reactor dynamics. This provides flexibility in simulating real-world conditions where substrate inhibition, byproduct toxicity, and environmental parameters influence degradation efficiency.
Future visions
Future improvements could expand the model to incorporate stochastic variability, mixed-contaminant degradation, and optimization algorithms for scaling up bioremediation. Integrating experimental data into the model will refine its predictive accuracy and establish robust operational guidelines.
References
[1] C. Jackson et al., "Biodegradation of Hexahydro-1,3,5-trinitro-1,3,5-triazine (RDX) by XplA/B enzymes," Journal of Applied Microbiology, vol. X, no. X, pp. 1–10, 2007.
[2] U.S. Environmental Protection Agency (EPA), "Drinking Water Health Advisory for RDX," 2017. [Online]. Available: https://www.epa.gov/
[3] D. Bagley and T. Brodkorb, "Modeling microbial kinetics in an anaerobic sequencing batch reactor—model development and experimental validation," Water Environment Research, vol. 71, pp. 1320–1332, 1999, doi: 10.2175/106143096X122366.
