Markovian Kinetics Research Articles

Multisite reversible association in membranes and solutions: From non-Markovian to Markovian kinetics.

The role of diffusion on the kinetics of reversible association to a macromolecule with two inequivalent sites is studied. Previously, we found that, in the simplest possible description, it is not sufficient to just renormalize the rate constants of chemical kinetics, but one must introduce direct transitions between the bound states in the kinetic scheme. The physical reason for this is that a molecule that just dissociated from one site can directly rebind to the other rather than diffuse away into the bulk. Such a simple description is not valid in two dimensions because reactants can never diffuse away into the bulk. In this work, we consider a variety of more sophisticated implementations of our recent general theory that are valid in both two and three dimensions. We compare the predicted time dependence of the concentrations for a wide range of parameters and establish the range of validity of various levels of the general theory.

Non-Markovian intracellular transport with sub-diffusion and run-length dependent detachment rate.

Intracellular transport of organelles is fundamental to cell function and health. The mounting evidence suggests that this transport is in fact anomalous. However, the reasons for the anomaly is still under debate. We examined experimental trajectories of organelles inside a living cell and propose a mathematical model that describes the previously reported transition from sub-diffusive to super-diffusive motion. In order to explain super-diffusive behaviour at long times, we introduce non-Markovian detachment kinetics of the cargo: the rate of detachment is inversely proportional to the time since the last attachment. Recently, we observed the non-Markovian detachment rate experimentally in eukaryotic cells. Here we further discuss different scenarios of how this effective non-Markovian detachment rate could arise. The non-Markovian model is successful in simultaneously describing the time averaged variance (the time averaged mean squared displacement corrected for directed motion), the mean first passage time of trajectories and the multiple peaks observed in the distributions of cargo velocities. We argue that non-Markovian kinetics could be biologically beneficial compared to the Markovian kinetics commonly used for modelling, by increasing the average distance the cargoes travel when a microtubule is blocked by other filaments. In turn, sub-diffusion allows cargoes to reach neighbouring filaments with higher probability, which promotes active motion along the microtubules.

Generalized quantum kinetic expansion: Time scale separation between intra-cluster and inter-cluster kinetics.

For a general two-cluster network, a new methodology of the cluster-based generalized quantum kinetic expansion (GQKE) is developed in the matrix formalism under two initial conditions: the local cluster equilibrium and system-bath factorized states. For each initial condition, the site population evolution follows exactly a distinct closed equation, where all the four terms involved are systematically expanded over inter-cluster couplings. For the system-bath factorized initial state, the numerical investigation of the two models, a biased (2, 1)-site system and an unbiased (2, 2)-site system, verifies the reliability of the GQKE and the relevance of higher-order corrections. The time-integrated site-to-site rates and the time evolution of site population reveal the time scale separation between intra-cluster and inter-cluster kinetics. The population evolution of aggregated clusters can be quantitatively described by the approximate cluster Markovian kinetics.

Minimal Model of Quantum Kinetic Clusters for the Energy-Transfer Network of a Light-Harvesting Protein Complex.

The energy absorbed in a light-harvesting protein complex is often transferred collectively through aggregated chromophore clusters. For population evolution of chromophores, the time-integrated effective rate matrix allows us to construct quantum kinetic clusters quantitatively and determine the reduced cluster-cluster transfer rates systematically, thus defining a minimal model of energy-transfer kinetics. For Fenna-Matthews-Olson (FMO) and light-havrvesting complex II (LCHII) monomers, quantum Markovian kinetics of clusters can accurately reproduce the overall energy-transfer process in the long-time scale. The dominant energy-transfer pathways are identified in the picture of aggregated clusters. The chromophores distributed extensively in various clusters can assist a fast and long-range energy transfer.

Integral, unified and Markovian theories of biexcitonic photoionization

The non-Markovian alternative to a conventional rate description of the biexcitonic photoinduced processes in solutions was deduced from integral encounter theory at sufficiently low concentration of excitations. This is the unified theory (UT) of photoionization followed by charge separation and recombination.The UT formalism allowed us to study the initial distribution of charged products of non-linear (biexcitonic) ionization as well as ion accumulation and recombination after δ-pulse excitation. The Markovian equations were obtained from UT as a limiting case and their phenomenological rate constants were given the microscopic definitions, which related them to the charge transfer rates and encounter diffusion of reactants. In the case of short lived (singlet) excitations the non-Markovian kinetics was shown to be more accurate and detailed than oversimplified Markovian kinetics. Only for long lived (triplet) excitations the non-Markovian and Markovian results are qualitatively similar, provided that the properly defined rate constants are used in the conventional rate equations.

Markovian Statistics of Nonhomogeneous Copolymer Systems

Abstract Nonhomogeneous copolymer systems that follow Markovian kinetics on a microscale will give non-Markovian sequence distributions in the polymer produced. This is because each microdomain is statistically independent of the others and the assumption of statistical stationarity does not apply. Equations that consider non-homogeneity, however, do apply and can reconcile Markovian kinetics and non-Markovian sequence distributions. Failures of systems to follow lower order Markovian mechanisms based on distribution data should be reviewed for nonhomogeneity. Complex mathematical models that can never be validated experimentally should only be considered if nonhomogeneity fails to explain the data adequately.

Ergodic state with least mean recurrence time in Markovian kinetics

We present a proof of the theorem that the ergodic state with the least mean recurrence time in an irreducible Markov process having countable states is identical, with probability unity, to the state with the least irreversible decay rate if neither microscopic reversibility nor the doubly stochastic property holds.

Nonsteady stochastic properties of Markovian kinetics and macroscopic fluctuations

A stochastic evolution following a nonlinear Langevin equation with one degree of freedom was examined by a machine computation. The autocorrelation function of macroscopic fluctuations fixed by the time average instead of by the ensemble average is observed to have a long-time tail exhibiting a divergence of their power spectrum in the low-frequency limit. Such a divergence prevents the establishment of strongly steady stochastic process of fluctuations. The present results confirm that nonlinear Markovian kinetics keeps everlasting macroscopic fluctuations.

Ergodicity of observable and ergodic hypothesis in Markovian kinetics

The time average of any observable, which evolves following an irreducible Markov process with countable states each of which is a persistent state, equals the ensemble average of the same quantity if and only if the corresponding master equation does not exhibit the accumulation of its eigenvalues around the infinitesimal neighborhood of the point with the value zero and if the steady distribution as the eigenfunction with the eigenvalue zero is uniform. The nonuniform steady distribution invalidates the ergodicity of observable. The principle of a priori equal weight is identical to the ergodicity of observable.