Guidelines for the Fitting of Anomalous Diffusion Mean Square Displacement Graphs from Single Particle Tracking Experiments
February
Guidelines for the Fitting of Anomalous Diffusion Mean Square Displacement Graphs from Single Particle Tracking Experiments
Eldad Kepten 0 1 2
Aleksander Weron 0 1 2
Grzegorz Sikora 0 1 2
Krzysztof Burnecki 0 1 2
Yuval Garini 0 1 2
0 1 Physics Department & Institute of Nanotechnology, Bar Ilan University , Ramat Gan , Israel , 2 Hugo Steinhaus Center, Institute of Mathematics and Computer Science, Wroclaw University of Technology , Wroclaw , Poland
1 Funding: E. Kepten and Y. Garini were supported in part by the Israel Centers of Research Excellence (ICORE) No. 1902/12, and Israel Science Foundation No. 51/12. K. Burnecki, G. Sikora and A. Weron were supported by the NCN Maestro Grant No. 2012/06/A/ ST1/00258. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript
2 Academic Editor: Yaakov Koby Levy, Weizmann Institute of Science , ISRAEL
Single particle tracking is an essential tool in the study of complex systems and biophysics and it is commonly analyzed by the time-averaged mean square displacement (MSD) of the diffusive trajectories. However, past work has shown that MSDs are susceptible to significant errors and biases, preventing the comparison and assessment of experimental studies. Here, we attempt to extract practical guidelines for the estimation of anomalous time averaged MSDs through the simulation of multiple scenarios with fractional Brownian motion as a representative of a large class of fractional ergodic processes. We extract the precision and accuracy of the fitted MSD for various anomalous exponents and measurement errors with respect to measurement length and maximum time lags. Based on the calculated precision maps, we present guidelines to improve accuracy in single particle studies. Importantly, we find that in some experimental conditions, the time averaged MSD should not be used as an estimator.
-
The analysis of single particle trajectories has become a standard procedure in the analysis of
experimental and theoretical systems [17]. In biological systems, that are intrinsically
stochastic in nature, single particles have been measured in all cellular environments and stages, both
in vivo and in vitro [817].
Since cellular environments are complex microscopic systems with a strong thermal
component [17], the motion of single particles, even if directed, incorporates a random diffusive
component, which must be characterized in order to build a physical picture of the system [1824].
A common tool by which the diffusion of a single particle is classified is the time averaged
mean square displacement (TAMSD) [1417, 2531]:
Competing Interests: The authors have declared
that no competing interests exist.
defined here for a trajectory x(t) of length L, taken at sampling time-intervals and the
averaging window is = n. For normal diffusion (not necessarily Brownian or Gaussian [32]) the
MSD is linear in time d2 t D1t, where D1 is the generalized diffusion coefficient which
includes all constant prefactors, depending on the diffusion mechanism.
The TAMSD may be of any functional form, but in many cases it is a power law function
over long times, d2 t Data [33, 34]. The anomalous exponent is related to fundamental
characteristics of the stochastic process, such as temporal correlations and the distribution of
particle steps and it is necessary for predicting the future particle motion, first passage times
and more [35].
There are various classes of anomalous diffusion and they all result from the breaking of the
assumptions behind normal Brownian diffusion, see [36] for a recent review. Continuous time
random walks (CTRW) which have long tailed jump distributions or waiting times between
jumps exhibit weak ergodicity breaking of a normal TAMSD. Variation in the surrounding
space may lead, among other models, to heterogeneous diffusion processes (HDP) and
obstructed diffusion, both with unique characteristics. If the stochastic process is not Markovian
and there is a temporal correlation between steps, another class of anomalous diffusion is
exhibited. Fractional Brownian motion (FBM) for example has self-similar Gaussian steps with
a correlation that decays as a power law. A general description of processes with temporal step
correlations can be obtained through the ARFIMA framework that generalizes fractional
dynamics through a discrete generating process [37].
The TAMSD is normally fitted through the logarithm of eqn. 1 as a function of up to a
maximal M, Fig. 1:
Fig 1. Fitting a time averaged MSD with various maximum time lags. A trajectory with = 0.7, L = 29, =
0.5 was simulated (black squares) and fitted for various M values. While the small M fitting (red M = 10 and
blue M = 50) underestimated , the large M (green M = 150) gives an overestimation. Clearly, selecting the
optimal M value is not trivial as both small and large values may lead to erroneous results. Graphically
assessing the qual (...truncated)