Quantum Dynamics of Vibrationally-Assisted Electron Transfer beyond Condon approximation in the Ligand-Receptor Complex

TL;DR

The study develops a nonperturbative open-quantum-system framework to study vibrationally assisted electron transfer in the ACE2–Spike interface, using a donor-acceptor TLS coupled to a Spike vibrational mode and embedded in a protein-like environment. By employing the Non-Markovian Quantum State Diffusion approach, it captures memory effects and non-Condon modulations of the electronic coupling, revealing how diagonal and off-diagonal bath couplings reshape ET dynamics. In Markovian/diagonal regimes, ET follows Marcus–Jortner kinetics, while non-Markovian memory and structured baths preserve coherence and enable vibrational gating; off-diagonal (non-Condon) coupling yields pronounced coherent transfer and sharp frequency selectivity. The results imply that Spike-associated vibrations and environmental fluctuations could function as a molecular VA-ET switch, providing insight into quantum aspects of viral binding and suggesting strategies to modulate infectivity via spectral detuning or bath engineering.

Abstract

We investigate the quantum dynamics of ligand--receptor electron transfer and conformational response in a prototypical viral binding complex, using the SARS-CoV-2 Spike protein bound to the human ACE2 receptor as a model system. Treating the ACE2--Spike interface as an open quantum system embedded in a biological environment, we simulate how vibrational interactions and environmental memory reshape the coupled receptor--ligand dynamics and modulate vibrationally assisted electron transfer (VA-ET). Using a Non-Markovian Stochastic Schr"odinger Equation (NMSSE) approach, we simulate electron transfer between donor and acceptor states in ACE2 modulated by a specific vibrational mode of the Spike protein. The influence of environmental memory (non-Markovian dynamics) and non-Condon effects (vibrational modulation of electronic coupling) are analyzed in detail. In the Markovian limit with an Ohmic bath, population dynamics reduce to exponential kinetics, and extracted transfer rates agree with semiclassical Marcus--Jortner predictions in the appropriate regime. Beyond the Markovian, high-temperature limit, we observe clear deviations: non-exponential decay, coherent oscillatory features, and enhanced sensitivity to the vibrational frequency. Incorporating off-diagonal system--bath coupling alongside diagonal coupling shows that nuclear motion can dynamically gate electron tunneling, sharpening the frequency selectivity of the VA-ET mechanism. Finally, a structured (sub-Ohmic) environmental spectral density with long-lived correlations (``memory'') preserves electronic--vibrational coherence over longer times, amplifying vibrational selectivity under non-Condon coupling. Our results support the proposition that ACE2--Spike binding may exploit vibrational assistance and quantum coherence as a molecular recognition mechanism.

Figures (5)

• Figure 1: Electron–transfer rates from MSSE in the diagonal Markov regime. Bath fluctuations are treated in the Markov limit (white–noise SSE) with diagonal system–bath coupling $L=\gamma_E\sigma_z$ (pure dephasing in the donor–acceptor basis) and an Ohmic spectral density with Drude–Lorentz cutoff. Parameters (all in eV unless noted): $\Delta=0.001$, $\omega_v=0.1487$, $T=300 K$; Ohmic $\alpha=0.05$, $\omega_c=0.5$ (bath reorganization $\lambda_{bath}==0.0125\ \mathrm{eV}$); Linblad coupling strength $\gamma_E=0.05$. The Spike Protein mode contributes a vibrational reorganization $\lambda_{mode}=2\gamma^2/\omega_v$, so the total reorganization is $\lambda_{\mathrm{tot}}(\gamma)=\lambda_{bath}+\lambda_{mode}$. (a) The rate $k_{\mathrm{ET}}(\epsilon)$ displays a bell–shaped dependence on the driving force with a maximum near the activationless condition $\epsilon\simeq\lambda_{\mathrm{tot}}$. The peak is relatively narrow and high because the SSE retains coherent donor–acceptor mixing and finite–$\Delta$ effects beyond the strict golden–rule limit. (b) The 2D map $k_{\mathrm{ET}}(\epsilon,\gamma)$ shows a ridge of enhanced transfer that tracks the activationless curve $\epsilon=\lambda_{baths}+2\gamma^2/\omega_v$ (white dashed). For small $\gamma$ the bath–induced reorganization is too weak and rates remain low; for very large $\gamma$ strong dressing (polaronic narrowing) and off–resonant bias suppress transfer. Between these limits, the brightest region aligns with $\epsilon\approx\lambda_{\mathrm{tot}}(\gamma)$.
• Figure 2: Electron transfer dynamics in weak system-bath coupling regime. (a) Diagonal coupling ($L=\sigma_z$) exhibits Marcus-type behavior with activationless ridge at $\epsilon = \lambda_{\text{tot}}$ (white dashed line), where maximum rates ($k_{ET}$) occur due to vanishing activation energy. (b) Off-diagonal coupling ($L=\sigma_x$) shows coherent tunneling with optimal rate band ($\epsilon \approx 0.12-0.16 \text{eV}$, $\gamma \approx 0.05-0.1 \text{eV}$) deviating from activationless condition. (c) Population dynamics reveal static behavior for $\sigma_z$ (blue) vs. coherent oscillations for $\sigma_x$ (orange). Parameters: $\Delta = 0.001 \text{eV}$, $\alpha = 0.0005$, $\omega_0 = 0.1 \text{eV} \approx \omega = 0.1487 \text{eV}$, $T = 290 \text{K}$, $\gamma_E = 0.05$. NMSSE terms govern distinct mechanisms: diagonal coupling utilizes stochastic and dissipative terms for incoherent hopping, while off-diagonal coupling leverages stochastic and coherence terms for quantum tunneling. Weak coupling preserves room-temperature coherence, demonstrating quantum-to-classical transition control through system-bath geometry.
• Figure 3: Population dynamics with a non-Markovian Ohmic bath: geometry ($L$) and electronic coupling ($\Delta$) control coherence. Dynamics are propagated with the non-Markovian SSE (NMSSE) using an Ohmic spectral density (Drude–Lorentz cutoff) with finite memory. Parameters (eV unless noted): $\epsilon=0.1487$, $\omega_v=0.1087$, $\Delta\in\{0.0001,0.001,0.01,0.1\}$, $\alpha=0.0005$, $\omega_c=0.5$, $\gamma_E=0.05$, $T=0.025\approx 290~\mathrm{K}$. The bath reorganization is small, $\lambda_{\mathrm{bath}}=\tfrac{1}{2}\alpha\omega_c=1.25\times10^{-4}~\mathrm{eV}$, so dynamics are dominated by system geometry. Curves show donor/acceptor populations for diagonal coupling $L=\gamma_E\sigma_z$ (solid/dashed blue) and off-diagonal coupling $L=\gamma_E\sigma_x$ (solid/dashed orange). (a) $\Delta=0.0001\,\mathrm{eV}$. With $L=\gamma_E\sigma_z$ (pure dephasing), populations remain essentially frozen (donor $\approx 1$, acceptor $\approx 0$). With $L=\gamma_E\sigma_x$, weak but clear coherent transfer occurs: a single underdamped swing moves population to the acceptor ($\sim 0.4$) before relaxing back as memory-assisted damping sets in. (b) $\Delta=0.001\,\mathrm{eV}$. The $\sigma_z$ case stays static; $\sigma_x$ shows a faster, larger excursion with smoother relaxation—coherence persists but damps within $\sim$ps due to finite bath memory at 290 K. (c) $\Delta=0.01\,\mathrm{eV}$. Small oscillatory ripples now appear even for $\sigma_z$ (Hamiltonian mixing by $\Delta$ leaks through despite dephasing). The $\sigma_x$ channel exhibits pronounced coherent exchange followed by damping to a donor-dominated steady state. (d) $\Delta=0.1\,\mathrm{eV}$. Strong early-time transients and visible oscillations occur for both $L=\gamma_E\sigma_z$ and $L=\gamma_E\sigma_x$, then relax toward a common quasi-steady population. The oscillation period shortens with $\Delta$, consistent with an effective Rabi frequency increasing with $\Delta$, while non-Markovian damping from the Ohmic kernel suppresses long-time coherence.
• Figure 4: Population dynamics with a structured non-Markovian bath: Δ-controlled crossover and geometry selectivity. Dynamics are propagated with the non-Markovian SSE (NMSSE) and an underdamped Brownian spectral density (structured bath). Parameters (eV unless noted): $\epsilon=0.1487$, $\omega_v=0.1487$, $\Delta\in\{0.0001,0.001,0.01,0.1\}$; structured bath $(\omega_0=0.1,\,\beta=0.005,\,\alpha=0.08)$; stochastic coupling $\gamma_E=0.01$; $T=0.025\approx 290~\mathrm{K}$. A narrow linewidth ($\beta=0.005$) yields long-lived oscillatory correlations at $\omega_0\simeq 0.1$ that imprint beats on the populations. Curves: donor/acceptor for diagonal coupling $L=\gamma_E\sigma_z$ (solid/dashed blue) and off-diagonal coupling $L=\gamma_E\sigma_x$ (solid/dashed orange). (a) $\Delta=0.0001\,\mathrm{eV}$. The structured memory produces clear underdamped oscillations in both geometries during the first picosecond. With $L=\sigma_z$, dephasing plus memory-driven localization leaves the donor near $\sim 0.7$ at late times. With $L=\sigma_x$, the bath modulates the tunneling path, creating a larger initial swing and relaxation toward a near-balanced state. (b) $\Delta=0.001\,\mathrm{eV}$. Increasing $\Delta$ strengthens Hamiltonian mixing. In $\sigma_z$, donor recovers steadily while acceptor drains; in $\sigma_x$, oscillations persist but damp to $\sim 0.5/0.5$ within a few ps. (c) $\Delta=0.01\,\mathrm{eV}$. Memory-induced beats remain visible up to $\sim 1.5~\mathrm{ps}$. The $\sigma_z$ channel relaxes to a donor-dominated state ($\gtrsim 0.85$), whereas $\sigma_x$ stays near equipartition ($\approx 0.5\pm0.05$). (d) $\Delta=0.1\,\mathrm{eV}$. Strong coherent transients with pronounced beating appear in both geometries.
• Figure 5: Population dynamics: (a) non-Markovian structured bath with $\gamma=0$ (no Spike coupling) vs (b) closed system with $\gamma=0.1$ (Spike coupled) but no bath. Common parameters (eV unless noted): $\epsilon=0.1487$, $\Delta=10^{-4}$, $\omega_v=0.1487$, $T= 290\,\mathrm{K}$. Time grid: $t\in[0,1000]$ with $\mathrm{d}t=0.10$ (window $4.427\,\mathrm{ps}$, step $0.443\,\mathrm{fs}$). (a) Structured non-Markovian bath (UBO):$\omega_0=0.1$, $\beta=0.005$, $\alpha=0.08$, $\gamma_E=0.01$, $M=512$. Bath correlation is oscillatory, with period $T_0\simeq 4.14/\omega_0\approx 41\,\mathrm{fs}$ and memory time $\tau_c\simeq \hbar/\beta\approx 830\,\mathrm{fs}$. $\gamma=0\Rightarrow \lambda_{\rm mode}=0$. (b) Closed spin–boson limit:$\gamma_E=0$, $\alpha=0$$\Rightarrow$ no stochastic drive or memory kernels (plain Schrödinger dynamics). Spike–mode coupling on: $\gamma=0.1\Rightarrow \lambda_{\rm mode}=2\gamma^2/\omega_v\approx 0.1345\,\mathrm{eV}$. Polaron narrowing of tunneling: $\Delta_{\rm eff}\approx \Delta\,e^{-2(\gamma/\omega_v)^2}\approx 4.0\times 10^{-5}\,\mathrm{eV}$. The bias strongly dominates: maximal unitary transfer $P_A^{\max}\approx (\Delta_{\rm eff}/\sqrt{\epsilon^2+\Delta_{\rm eff}^2})^2\sim 7\times 10^{-8}$ (visually flat).