Theoretical study of ultrafast electron transfer in Arabidopsis thaliana cryptochrome: A comparative study of mode specific contributions and perturbative methods

Cryptochromes are blue light photoreceptors that mediate key biological processes through photoinduced electron transfer (ET). We present a theoretical study of the second ET step along the conserved tryptophan triad in Arabidopsis thaliana cryptochrome, focusing on how vibrational environments affect the ultrafast electron transfer. Using the numerically exact low temperature quantum Fokker-Planck equation (LT-QFPE) within the hierarchical equations of motion (HEOM) framework, we resolve the contributions of individual Brownian oscillator modes: low frequency modes with large reorganization energies markedly slow the ET rate, whereas weakly damped high frequency modes generate oscillations linked to electronic-vibrational coherence. The LT-QFPE results are also used to benchmark three second order perturbative methods: the non-equilibrium Fermi's golden rule (NE-FGR), non-interacting blip approximation (NIBA), and the time local generalized quantum master equation (GQME). Although all approximate methods reproduce correct long time ET rates, they fail to capture short time non-equilibrium dynamics. These findings clarify how spectral width and bath correlation time, in addition to total reorganization energy, govern the validity of perturbative treatments and highlight the need for numerically exact methods such as HEOM in modeling non-equilibrium cryptochrome ET dynamics.