Analytical Model of Resonant Quantum Excitation Transport in Molecular Chains at finite Temperatures: Application of Integral Transforms

TL;DR

This work analyzes resonance-driven excitation transport in a finite biomolecular chain subjected to thermal fluctuations. Using a Holstein/Lang-Firsov framework with mean-field treatment, it derives coupled correlation-function dynamics for right- and left-side segments and solves them analytically via Laplace transforms, yielding expressions for $V_n(t)$ in terms of modified Chebyshev polynomials $D_n(x)$ and the probability $p_n(t)=|V_n(t)|^2$. The results reveal that the excitation can coherently appear at distant sites with residence times that grow with temperature $\theta$, the polaron coupling $S$, and chain geometry, while the arrival timing also depends on the adiabatic parameter $B$ and chain length. This provides a quantitative basis for predicting how intramolecular excitations might perturb local biochemical functions and offers a route to estimate vibron–phonon interaction constants from observed migration patterns. The framework thus bridges quantum-coherent transport theory and biomolecular function under physiologically relevant conditions, with potential implications for understanding and controlling excitation-driven processes in proteins, DNA, and related systems.

Abstract

This study investigates the potential impact of intramolecular excitations on the active regions of biomolecular chains, which may play a role in physiological processes within living cells. We assumed that an excitation localized in a specific chain segment can modify its physical properties (e.g., local charge distribution or electric dipole moments), thereby altering its role in biochemical processes. As a consequence, the biochemical functionality of the molecular chain may be altered, or even disrupted. Moreover, quantum resonance effects may cause an excitation induced at one structural element to delocalize and appear at a distant site, potentially affecting the functionality of regions located far from the site where the excitation was initially induced. To investigate this phenomenon, we developed and analyzed a theoretical model in which a single excitation is induced in a particular structural element of a finite molecular chain in thermal equilibrium with its environment. The interaction between the excitation and the thermal oscillations of the chain was taken into account. Differential equations for the correlation functions were derived and solved analytically using integral transformations, providing information on the probability of finding the excitation at each site of the chain. The results show that both the probability of finding the excitation at distant sites and its residence time depend on the chain's physical characteristics, temperature, and initial excitation location.

Figures (8)

• Figure 1: Understanding the excitation migration through BmC is particularly important in the bioenergetics of living cells. The figure shows a schematic representation of energy quantum transport along the polypeptide MC.
• Figure 2: A schematic representation of the considered MC structure. The SE where the excitation is initially induced is labeled as $0$. On the left side of the $0$--th SE, there are $M$ SEs, numbered from 1 to $M$, while on the right side, there are $N$ SEs, numbered from 1 to $N$.
• Figure 3: Time probability distribution $p_9(\tau)=|V_9(\tau)|^2$ of excitation appearance on last ($9$th) node, for different values of system temperatures. Here, $S=0.3$ and $B=0.1$.
• Figure 4: Time probability distribution $p_9(\tau)=|V_9(\tau)|^2$ of excitation appearance on last ($9$th) node, for different values $S$. Here, $B=0.1$ and $\theta=4$.
• Figure 5: Time distribution of the excitation probability on the last node of the MC. Parameters: $S=0.3$, $B=0.1$, $\theta=4$. Excitation was induced at the second SE from the left.