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In living cell or its nucleus, the motions of molecules are complicated due to the large crowding and expected heterogeneity of the intracellular environment. Randomness in cellular systems can be either spatial (anomalous) or temporal (heterogeneous). In order to separate both processes, we introduce anomalous random walks on fractals that represented crowded environments. We report the use of numerical simulation and experimental data of single-molecule detection by fluorescence fluctuation microscopy for detecting resolution limits of different mobile fractions in crowded environment of living cells. We simulate the time scale behavior of diffusion times τD(τ) for one component, e.g. the fast mobile fraction, and a second component, e.g. the slow mobile fraction. The less the anomalous exponent α the higher the geometric crowding of the underlying structure of motion that is quantified by the ratio of the Hausdorff dimension and the walk exponent d f /dw and specific for the type of crowding generator used. The simulated diffusion time decreases for smaller values of α ≠ 1 but increases for a larger time scale τ at a given value of α ≠ 1. The effect of translational anomalous motion is substantially greater if α differs much from 1. An α value close to 1 contributes little to the time dependence of subdiffusive motions. Thus, quantitative determination of molecular weights from measured diffusion times and apparent diffusion coefficients, respectively, in temporal auto- and crosscorrelation analyses and from time-dependent fluorescence imaging data are difficult to interpret and biased in crowded environments of living cells and their cellular compartments; anomalous dynamics on different time scales τ must be coupled with the quantitative analysis of how experimental parameters change with predictions from simulated subdiffusive dynamics of molecular motions and mechanistic models. We first demonstrate that the crowding exponent α also determines the resolution of differences in diffusion times between two components in addition to photophyscial parameters well-known for normal motion in dilute solution. The resolution limit between two different kinds of single molecule species is also analyzed under translational anomalous motion with broken ergodicity. We apply our theoretical predictions of diffusion times and lower limits for the time resolution of two components to fluorescence images in human prostate cancer cells transfected with GFP-Ago2 and GFP-Ago1. In order to mimic heterogeneous behavior in crowded environments of living cells, we need to introduce so-called continuous time random walks (CTRW). CTRWs were originally performed on regular lattice. This purely stochastic molecule behavior leads to subdiffusive motion with broken ergodicity in our simulations. For the first time, we are able to quantitatively differentiate between anomalous motion without broken ergodicity and anomalous motion with broken ergodicity in time-dependent fluorescence microscopy data sets of living cells. Since the experi...

Library of Congress Cataloging-in-Publication Data Baumann, Gerd. Symmetry anatysis of differential equations with Mathematica 1Gerd Baumann.p. cm. Includes bibliographical references and indexes.

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We present a new approach to distinguish between non-ergodic and ergodic behavior. Performing ensemble averaging in a subpopulation of individual molecules leads to a mean value that can be similar to the mean value obtained in an ergodic system. The averaging is carried out by minimizing the variation between the sum of the temporal averaged mean square deviation of the simulated data with respect to the logarithmic scaling behavior of the subpopulation. For this reason, we first introduce a kind of Continuous Time Random Walks (CTRW), which we call Limited Continuous Time Random Walks (LCTRW) on fractal support. The random waiting time distributions are sampled at points which fulfill the condition N < 1, where N is the Poisson probability of finding a single molecule in the femtoliter-sized observation volume ΔV at the single-molecule level. Given a subpopulation of different single molecules of the same kind, the ratio T/ Tm between the measurement time T and the meaningful time Tm, which is the time for observing just one and the same single molecule, is the experimentally accessible quantity that allows to compare different molecule numbers in the subpopulation. In addition, the mean square displacement traveled by the molecule during the time t is determined by an upper limit of the geometric dimension of the living cell or its nucleus.

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Based on classical mean-field approximation using the diffusion equation for ergodic normal motion of single 24-nm and 100-nm nanospheres, we simulated and measured molecule number counting in fluorescence fluctuation microscopy. The 3D-measurement set included a single molecule, or an ensemble average of single molecules, an observation volume DeltaV and a local environment, e.g. aqueous solution. For the molecule number N << 1 per DeltaV, there was only one molecule at a time inside DeltaV or no molecule. The mean rate k of re-entries defined by k = N/tau(dif) was independent of the geometry of DeltaV but depended on the size of DeltaV and the diffusive properties tau(dif). The length distribution l of single-molecule trajectories inside DeltaV and the measured photon count rates I obeyed power laws with anomalous exponent kappa =-1.32 approximately -4/3.