Medical University of Graz, Riesstrasse 58a/5, A-8047 Graz, Austria.
Curr Pharm Biotechnol. 2011 May;12(5):824-33. doi: 10.2174/138920111795470949.
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/ T(m) between the measurement time T and the meaningful time T(m), 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.
我们提出了一种新方法来区分非遍历和遍历行为。在个体分子的子群体中进行集合平均会导致平均值与在遍历系统中获得的平均值相似。平均化是通过最小化模拟数据的时间平均均方偏差的总和与子群体的对数标度行为之间的变化来实现的。出于这个原因，我们首先引入了一种连续时间随机游走（CTRW），我们称之为分形支撑上的有限连续时间随机游走（LCTRW）。随机等待时间分布在满足条件 N<1 的点上进行采样，其中 N 是在单分子水平上在 femtoliter 大小的观察体积 ΔV 中找到单个分子的泊松概率。给定相同类型的不同单个分子的子群体，测量时间 T 与有意义时间 T(m)之间的比率 T/T(m)是实验可访问的量，允许比较子群体中不同分子数量。此外，分子在时间 t 内移动的均方位移由活细胞或其核的几何尺寸的上限确定。
