Nonlinear Dynamical Modeling of Ionizing Radiation-Induced Redox Activity in L-DOPA-Derived Eumelanin from Radiotrophic Fungi

This paper presents a rigorous mathematical framework for modeling ionizing radiation-induced redox activity in L-DOPA-derived eumelanin from radiotrophic fungi. A lattice-manifold formalism is developed to characterize electronic transitions in eumelanin's $\pi$-electron system under $\gamma$-ray exposure (100 keV--2 MeV). Through differential geometry, relationships between manifold curvature and transition probabilities are established, demonstrating how the delocalized nature of $\pi$-electrons enhances radiation absorption. Operator algebra and sheaf cohomology are employed to characterize the electron transfer network, while critical scattering angles for Compton interactions are derived, revealing that a single $\gamma$-ray can induce multiple excitation and ionization events. The redox cycling between hydroquinone (H$_2$Q), semiquinone (SQ$^{\bullet-}$), and quinone (Q) states is analyzed as a nonlinear dynamical system, wherein radiation perturbation destabilizes equilibrium points and generates limit cycles via Poincaré-Bendixson dynamics. Precise criteria for sustainable radiotrophy are formulated, redox buffer capacity is quantified, and radiation dose-response relationships are established. The mathematical model provides a theoretical foundation for understanding how melanized fungi harness radiation energy through enhanced redox cycling in extreme environments.