Catalytic competence bounds the decay rates of a thermally formed radical pair

Technical Summary: Catalytic Competence Bounds the Decay Rates of a Thermally Formed Radical Pair

Problem Statement
The radical-pair mechanism (RPM) is the established model for how weak magnetic fields, such as the geomagnetic field (~50 µT), can influence chemical reaction yields, notably in avian magnetoreception via photoexcited cryptochromes. A recurring question is whether this mechanism can operate in dark-active flavoenzymes, such as monoamine oxidases (MAOs), which degrade neurotransmitters. For a flavoenzyme to respond to a geomagnetic field, its catalytic cycle must pass through a radical pair that lives long enough for the field to induce spin dynamics. However, if the radical pair is formed thermally from a closed-shell enzyme-substrate complex and must carry the catalytic flux back to the product, the rates of its decay are constrained by thermodynamics. Previous work has established limits based on spin relaxation and exchange couplings, but it has not derived the fundamental constraints imposed by the requirement of catalytic turnover itself.

Methodology
The authors derive theoretical bounds on the decay rates of a radical pair formed thermally and then perform a numerical search to determine the maximum possible magnetic field effect (MFE) under these constraints.

1. Theoretical Derivation:

   • The study considers a single-electron-transfer (SET) scheme where a closed-shell complex ($E \cdot S$) forms a singlet radical pair ($[Fl^{\bullet-} RNH_2^{\bullet+}]$).
   • The pair can return to the ground state via singlet back-transfer (rate $k_S$) or proceed to product via a spin-independent onward reaction (rate $k_P$).
   • Using the master equation with Haberkorn reaction terms and assuming trace-preserving, unital, and positivity-preserving relaxation, the authors apply the principle of detailed balance.
   • They derive an inequality for the onward yield ($\Phi_P$): $\Phi_P \le 4k_P/k_S$. This bound holds for any spin Hamiltonian and admissible relaxation.
   • By requiring the turnover number ($k_{cat}$) to be maintained, they establish lower bounds for the decay rates: $k_S \ge 4k_{min}(\Delta)$ and $k_P \ge k_{min}(\Delta)$, where $k_{min}(\Delta) = k_{cat} e^{\Delta/k_B T}$ and $\Delta$ is the free energy of the radical pair relative to the closed-shell state.

2. Numerical Search:

   • A spin model for a flavin–aminium radical pair was constructed, including hyperfine couplings (flavin N5, N10, and aminium protons) and dipolar couplings.
   • The authors performed a grid search over exchange coupling ($J$), magnetic field orientation, and dipolar scaling factors to find the maximum MFE ($F^*$) for specific pairs of $(k_S, k_P)$.
   • This search was repeated with anisotropic hyperfine tensors and electron spin relaxation (dephasing) to test robustness.
   • Finally, the theoretical bounds were combined with the numerical results to define $G^*(\Delta)$, the upper bound on the field effect for a "competent" pair (one capable of carrying the catalytic flux).

3. Thermodynamic Estimation:

   • The authors estimated the free energy ($\Delta$) of the radical pair in monoamine oxidases using measured flavin redox potentials and a proxy potential for dialkylamine radical cations in water.
   • Continuum electrostatic models were used to estimate the shift in $\Delta$ when moving from bulk water to the enzyme active site.

Key Contributions and Results

• Catalytic Competence Bounds: The paper establishes that for a thermally formed radical pair carrying catalytic flux, detailed balance imposes a strict lower bound on decay rates. Specifically, the singlet back-transfer rate must exceed $4k_{cat}e^{\Delta/k_B T}$ and the onward rate must exceed $k_{cat}e^{\Delta/k_B T}$. These bounds depend only on the driving force ($\Delta$) and turnover number, independent of electronic couplings or reorganization energies.
• Maximum Field Effect: In the numerical search for a flavin–aminium model with $k_{cat} = 1 \text{ s}^{-1}$, the largest magnetic field effect found among pairs satisfying the catalytic bounds drops below 0.1% at $\Delta = 0.47 \text{ eV}$ (0.50 eV with anisotropic tensors). The effect remains below this threshold up to the grid limit of 0.61 eV. Electron spin relaxation does not raise these thresholds.
• Application to Monoamine Oxidases: Using measured potentials, the authors estimate that an amine-to-flavin pair in MAO lies 0.86–1.44 eV above the closed-shell state in water. Even with favorable continuum corrections for the active site, the pair is estimated to be around 0.49 eV. This places the pair near or above the threshold where the field effect becomes negligible (<0.1%).
• Robustness: The derived bounds and the resulting thresholds are shown to be robust against variations in the spin Hamiltonian, including anisotropic hyperfine tensors and electron spin relaxation.

Significance and Claims
The paper claims to define a fundamental constraint on the radical-pair mechanism in dark-active enzymes. It demonstrates that the requirement for catalytic turnover inherently limits the lifetime of the radical pair, thereby bounding the time available for a weak geomagnetic field to act.

The authors conclude that for monoamine oxidases, the thermodynamic driving force ($\Delta$) likely places the radical pair outside the range where a measurable geomagnetic field effect (>0.1%) is possible, unless the active site significantly stabilizes the pair (lowering $\Delta$ to ~0.49 eV) more than current estimates suggest. The paper does not rule out the mechanism entirely but suggests that for these specific enzymes, the thermodynamic cost of forming the radical pair is likely too high to allow for a significant magnetic field response under the derived constraints. The authors note that determining the exact transfer free energies or the one-electron flavin potential with substrate bound is necessary to settle the question definitively.