When the environment matters most: polarizable-embedding QM/MM for multi-photon absorption in red fluorescent proteins.

Multi-photon absorption in fluorescent proteins (FPs) is highly sensitive to the surrounding protein environment, yet most computational studies treat the chromophore in isolation, which limits the understanding of protein-specific non-linear response. This work presents a polarizable-embedding (PE), quantum mechanical/molecular mechanical (QM/MM), investigation of two- and three-photon absorption (2PA and 3PA) across seven red fluorescent proteins, including DsRed, mCherry, mPlum, mStrawberry, TagRFP, mRFP1, and mKate. The calculated 2PA cross sections (σ2PA) show strong agreement with experiment, reproducing both the overall magnitude and the relative ordering across the protein series. The proteins predicted to exhibit the largest σ2PA values, DsRed, TagRFP, and mKate, match the experimental ranking, with maximum absolute deviations of 23 GM (DsRed) and 31 GM (TagRFP). For the remaining proteins, the deviations do not exceed 6 GM. Furthermore, the computed trends correlate with changes in permanent dipole moments (|Δμ|), supporting the physical consistency of the PE description in capturing environment-driven charge redistribution. Absolute 3PA cross sections (σ3PA) are reported for all investigated FPs. Although direct quantitative comparison with experiment is limited to action cross sections, the predicted ordering for the available systems is consistent with reported measurements. Moreover, the computed 3PA trends closely follow those obtained for 2PA, preserving the same general ranking of non-linear response across the FPs. Overall, this work demonstrates that PE is essential for reproducing multi-photon absorption magnitudes in FPs and provides the first description of both 2PA and 3PA across multiple red fluorescent proteins.