ABSTRACT
Transitions mediated by a bilinear intermolecular coupling to the nearby environment play the fundamental role in proper operation of biological molecular structures. This is supported by a range of examples from photo-synthesizing clusters to ligand-sensitive receptors to voltage-dependent ionic channels to ATP-regulated molecular motors, etc. As a rule, these transitions are accompanied by the generation or annihilation of the environmental vibration quanta (phonons). But in many cases, one has also to take into account the multitude of stochastic additions to an averaged position of energy levels due to thermodynamic fluctuations. Nowadays, a phenomenological approach for modeling elementary
acts of energy transformation processes at the macroscopic level is well recognized. At the same time, a more satisfying kinetic treatment of relaxation and fluctuation processes accompanying the environment-induced transitions in biomolecules on the micro-to nanoscales is now in active progress. In this overview, some recent results on the microscopic approach to time evolution of a few-state quantum molecular system are outlined. We point mainly to the power of a kinetics framework gaining popularity in microscopic community to provide a physically rigorous description of relaxation transitions between the energy levels on different timescales including those of nanoscale fluctuations. Within this framework, the respective expression for a transition rate constant generally valid on the wide time scale takes account of the most significant dynamic, stochastic, kinetic, and thermodynamic (quasi-static) parameters. Here, especial attention should be given to a correct reduction, in the frame of kinetic equation approach, of that expression to the effectively slower times by making it an average over both the steady-state fluctuations of system’s energy levels and the equilibrium vibrations of the environment. Performance of the framework in some simple important cases, particularly in the case of the two-level biomolecular system, would be also desirable to elicit. For this case, we show how the steady-state level population and the rate of inter-level transitions will explicitly depend on the ambient temperature and particle concentration. Finally, we present an implementation of the results provided for two nanoscale effects that is, the temperature “independent” decay for transitions between the quasi-isoenergetic levels by virtue of their fluctuations, and the “negative” cooperativity for sigmoid distribution of meta-stable state population because of the system irreversibility. 9.1 IntroductionNanobiophysics can be thought of as the physics-based framework for microscopically guiding a functioning of chemically stable biomolecular structures (proteins and nucleic acids and/or their components and assemblies) operating in their changeful noisy environments on the nanometer length scale [39, 47]. Generally, biomolecular structures are supramolecular nanoobjects assembled of the different molecular components or groups in
three dimensions, of the size from one to tens of nanometers [47, 73, 74]. However, the size alone is not a key determinant for the existing bio-nanoscopic effects. What is more important is that the biomolecular structures possess those specific energy spectra which are intermediate between the discrete energy-level spectrum of the separate atom or the molecule and the broad band energy continuum of the bulk matter [24]. Moreover, the biomolecule is not static in time and the state as it would be for the thermodynamically isolated system [19]. Rather, being open to the environment and exchanging of energy with it and particles in the sub-to supra-terahertz frequency range, biomolecular structure does constantly evolve within from far shorter to much longer time scales, according to the Liouville-Von Neumann equation for the density matrix of the whole system [63].In biophysics, using the concept of the whole system is diverse and depends on the context of the problem it appeals to solve. It comprises as basic factors intrinsic for the dynamic and kinetic properties of the biomolecular structure as well as relevant features of the surroundings, including apparatus impacts and influences of extrinsic controls. However, formally, one can compositionally refer the whole biophysical system to the closed one (C), and then partition it into its microscopic (molecular) nonequilibrium part and macroscopic (environmental) equilibrium part such as C = A + B, with A being a few-level open system of interest, while B is the heat bath modeled by an infinite set of non-interacting harmonic oscillators at the temperature T. Also, the thermodynamic fluctuations of structure groups are assumed to randomly perturb the positions of biomolecular energy levels of part A. This provides complementary conditions for averaging an evolution over the stochastic fluctuations in the sub-terahertz frequency range [18, 48]. Decomposition of the whole system is important not only for physics [40, 51, 72, 75, 76] but also in chemical and biological applications (cf. e.g., [21-23, 32, 45, 52, 54-61, 68, 69]). However, its implementation in biomolecular structures is not straightforward and has some basic aspects [18, 19, 25, 44, 48, 70].One aspect is that, to function in noisy environments, biomolecular structures should be physically stable. This means that to be accurate in a few-level representation of the state energy spectrum of molecular part A, one should maintain the randomly
fluctuating energy levels, in average, stationary and keep the relaxation transitions between them, in detail, balanced. To form the time-dependent position for energy levels of the microscopic system and to transit from one level to the other, two processes are well-separated in time-from tens of femtoseconds to picoseconds, for the first, and from milliseconds to seconds (or even hours), for the second [62]. Therefore, stretching on the nanoscale description to the lager meso-length scale in effect provides biomolecular structures with the hierarchy of largely extended kinetic processes. This hierarchy will follow the time evolution not only at short times, at which the level position is considered fixed, but also at intermediate times, during which the stochastization of the level position is completed, as well as at far longer times, subsequent to which only relaxation transitions between randomly fluctuating energy levels do occur [54].Another aspect is the choice of a correct mathematical apparatus. The most rigorous method of the microscopic description of the evolution of a finite-level system in contact with the macroscopic environment is based on the nonequilibrium density matrix theory [1, 4, 45, 54]. Adaptation of this theory to systems revealing randomization of the position of energy levels uses various treatments, e.g., [21-23, 55-60, 68] and references therein. Quite general treatment is the projection-operator method of Nakajima [51] and Zwanzig [75, 76]. It allows one to derive the generalized master equation (GME) for the density matrix r0(t) = trB r(t) (the race is over the states of system B) of an open quantum system A subjected to time-dependent (random or regular) external fields and a heat bath B [2, 23, 76]. The density matrix r(t) refers to the whole system C and satisfies the Liouville-Von Neumann equation( )= – ( ) ( ),t iL t tr r (9.1)where L(t) = (1/ћ)[H(t), ...] is the Liouville superoperator (ћ is the Planck constant) related to the Hamiltonian of the C system,0 B( )= ( )+ + .H t H t H V (9.2)The idea of the method is that the interaction V between the small system A and large (macroscopic) system B with the respective
Hamiltonians H0(t) and HB, is assumed weak enough to hardly effect the distribution of non-interacting states in B. Hence, independent of the distribution of energy levels in system A, system B remains in the thermal equilibrium, which is characterized by the equilibrium density matrix B B B B B B= exp(– / ) / exp(– / )H k T tr H k Tr (kB is the Boltzmann constant). Just this fact allows one to refer a large system B to the heat bath (or reservoir). Moreover, in the high accuracy, r(t) can be factorized by the nonequilibrium density matrix r0(t) of A and the equilibrium density matrix rB of the B in the form r(t) = r0(t) rB.For analytical reasons, a calculus of r0(t) describing the temporal behavior of even a few-level open system whose Hamiltonian can occasionally depend on time via some time-dependent energy coupling (owing, e.g., to intrinsic random fluctuations or extrinsic regular forces), is generally intractable [61]. Furthermore, in chaotic systems the form of H0(t) is almost unpredictable. Only some statistical properties of the random temporal behavior of microscopic energies can be described sufficiently accurately [50]. Therefore, to find the solution of GME for r0(t) implies assuming a hierarchy of time scales for the dynamics of transition processes in system A [69]. Physically, a time evolution of the open quantum system A is well described by the ensemble-averaged probability (population) for the system to be in the m-th quantum state as( )= ( ) ,m mP t p t (9.3)Here ... denotes the stochastic averaging of stochastically non-averaged populations ( )0( )= | ( )| ,dmp t m t m r (9.4)while m...m is the statistical ensemble averaging of the diagonal elements ( )0 0ˆ( )= ( )d dt T tr r (( )0 0ˆ( )= ( )d dt T tr is the projection operator that separates any operator into diagonal components) [61]. Thus, the main challenge is to derive the master equation for Pm(t) (9.3) by using GME for ( )0 0ˆ( )= ( )d dt T tr r.
