Intramolecular electron transport in biological systems is typically described as a diffusive hopping process, according to the semi-classical rate theories of Marcus and Hopfield combined with classical Pauli-type master equations. However, the possibility that non-trivial quantum mechanical effects could play a functional role in the transport dynamics in certain biomolecular processes has attracted increasing attention. Here, we extend the quantum mechanical model of open system dynamics by the Lindblad equation to a key biological component, the long chains of redox centers based on iron-sulfur clusters or heme groups that are widespread in many biological organisms, where they realize the cellular respiration. This approach allows to explore a wide range of physical parameters, showing key features of electron transport in these multi-domain protein structures. We pay particular attention to heat and entropy transfer between the electrons and the protein bath, which constitutes a benchmark of physical realism for the models. Electron currents, average transfer times and relative efficiency of the transport process are also explicitly characterized.
Intramolecular electron transport in biological systems is typically described as a diffusive hopping process, according to the semi-classical rate theories of Marcus and Hopfield combined with classical Pauli-type master equations. However, the possibility that non-trivial quantum mechanical effects could play a functional role in the transport dynamics in certain biomolecular processes has attracted increasing attention. Here, we extend the quantum mechanical model of open system dynamics by the Lindblad equation to a key biological component, the long chains of redox centers based on iron-sulfur clusters or heme groups that are widespread in many biological organisms, where they realize the cellular respiration. This approach allows to explore a wide range of physical parameters, showing key features of electron transport in these multi-domain protein structures. We pay particular attention to heat and entropy transfer between the electrons and the protein bath, which constitutes a benchmark of physical realism for the models. Electron currents, average transfer times and relative efficiency of the transport process are also explicitly characterized.
Quantum biology, as introduced in the previous chapter, mainly studies the dynamical influence of quantum effects in biological systems. In processes such as exciton transport in photosynthetic complexes, radical pair spin dynamics in magnetoreception, and photo-induced retinal isomerization in the rhodopsin protein, a quantum description is a necessity rather than an option. The quantum modelling of biological processes is not limited to solving the Schrödinger equation for an isolated molecular structure. Natural systems are open to the exchange of particles, energy or information with their surrounding environments that often have complex structures. Therefore the theory of open quantum systems plays a key role in dynamical modelling of quantum-biological systems. Research in quantum biology and open quantum system theory have found a bilateral relationship. Quantum biology employs open quantum system methods to a great extent while serving as a new paradigm for development of advanced formalisms for non-equilibrium biological processes. In this chapter, we overview the basic concepts of quantum mechanics and approaches to open quantum system (or decoherence ) dynamics. Here, we do not intend to discuss all aspects of about a century-old theory of open quantum systems that dates back to the original work of Paul Dirac on atomic radiative emission and absorption (Dirac, 1927). Instead, we mainly focus on the integro-differential equations that are commonly used for modelling quantum-biological systems. Interested readers can learn more about open quantum systems in various books and review articles in both physics and chemistry literature, including the references (Kraus, 1983; Breuer and Petruccione, 2002; Kubo et al. , 2003; Weiss, 2008; May and Kühn, 2011).
Based on the references [P1–P6], we investigate in this cumulative PhD thesis the dynamics and transport in closed and open interacting spin chain models. We focus on (i) the connection between the open and the closed system dynamics [P1, P2], (ii) the influence of various system parameters on the transport behavior in closed and open systems [P3, P5], and in addtion (iii) the dynamics and transport in closed models [P4, P6]. As for (i), we explore the connection of the dynamics in open systems, modeled by the Lindblad equation, and the correlation functions in the closed system [P1]. Building on this connection for magnetization transport, we study whether the time evolution of the open quantum system can be described in terms of classical correlation functions generated by the Hamiltonian equations of motion for real vectors [P2]. Second, we study in (ii) the influence of different parameters on the diffusion constants in closed and open systems [P3, P5]. In Ref. [P3], we study the impact of integrability-breaking perturbations in both closed and open spin chain systems. While we find a continuous change of the diffusion constant with perturbation strength in the closed system, in the open system, in the weak-perturbation limit, we indicate the possibility of a diverging diffusion constant. Also, we investigate the influence of the system-bath coupling on the transport behavior in boundary-driven open systems [P5]. We observe that the diffusion constants depend on the system-bath coupling and that in the nonequilibrium steady state, they inevitably retain finite-size effects, limiting the computation of long-time transport coefficients. Additional for (iii), in Ref. [P4], we first examine the impact of perturbations on the dynamics in closed spin chain models. We employ projector-operator techniques to understand, on perturbative grounds, the nontrivial damping of the standard correlation function. Second, we also discuss a dynamical quantum typicality approach for studying the energy transport in bipartite systems at low temperatures [P6].
Quantum coherence in biological systems is often assumed to be thermodynamically expensive to maintain, but its actual energy cost has never been quantified. We compute the thermodynamic cost of quantum-coherent energy transfer in the FMO photosynthetic complex using Lindblad master equation simulations. At optimal ENAQT dephasing (γ = 100 cm⁻¹), the transport efficiency reaches 29.4%, the Holevo channel capacity peaks at 2.17 bits, and the Landauer ratio—the energy cost per successfully transmitted bit relative to the fundamental thermodynamic limit kT ln 2—is ~ 390×. This is approximately 104 times more efficient than classical CMOS computing (~ 107× Landauer). The entropy production rate peaks at 1.77 kB/ps at γ = 300 cm⁻¹, and the von Neumann entropy saturates at Smax = 2.45 bits at steady state. These results provide the first quantitative thermodynamic characterization of quantum-coherent energy transfer in a pigment-protein complex.
We investigate nonequilibrium two-electron transfer in a model redox system represented by a two-site extended Hubbard model and embedded in a dissipative environment. The influence of the electron-electron interactions and the coupling to a dissipative bosonic bath on the electron transfer is studied in different temperature regimes. At high temperatures, Marcus transfer rates are evaluated, and at low temperatures, we calculate equilibrium and nonequilibrium population probabilities of the donor and acceptor with the nonperturbative numerical renormalization group approach. We obtain the nonequilibrium dynamics of the system prepared in an initial state of two electrons at the donor site and identify conditions under which the electron transfer involves one concerted two-electron step or two sequential single-electron steps. The rates of the sequential transfer depend nonmonotonically on the difference between the intersite and on-site Coulomb interaction, which become renormalized in the presence of the bosonic bath. If this difference is much larger than the hopping matrix element, the temperature as well as the reorganization energy, simultaneous transfer of both electrons between donor and acceptor can be observed.
Findings related to quantum mechanics over the last century have had a pivotal role in our understanding of key physical, chemical, and biological features. Using manipulations of Schrödinger’s wavefunction, we describe electron tunneling through orbits. We extend our findings to a biological setting where we theorize an imbalance in exothermic biochemical reactions may contribute to protein damage. Additionally, we posit the cellular environment encourages electron binding among higher principal quantum orbits than traditionally assumed. This model can be one of many explanations for the sporadic nature of neurodegenerative disease where protein damage, misfolding, or aggregation is a key characteristic.
Findings related to quantum mechanics over the last century have had a pivotal role in our understanding of key physical, chemical, and biological features. Using manipulations of Schrödinger’s wavefunction, we describe electron tunneling through orbits. We extend our findings to a biological setting where we theorize an imbalance in exothermic biochemical reactions may contribute to protein damage. Additionally, we posit the cellular environment encourages electron binding among higher principal quantum orbits than traditionally assumed. This model can be one of many explanations for the sporadic nature of neurodegenerative disease where protein damage, misfolding, or aggregation is a key characteristic.