Anomalous diffusion of single particles in cytoplasm

Abstract

The crowded intracellular environment poses a formidable challenge to experimental and theoretical analyses of intracellular transport mechanisms. Our measurements of single-particle trajectories in cytoplasm and their random-walk interpretations elucidate two of these mechanisms: molecular diffusion in crowded environments and cytoskeletal transport along microtubules. We employed acousto-optic deflector microscopy to map out the three-dimensional trajectories of microspheres migrating in the cytosolic fraction of a cellular extract. Classical Brownian motion (BM), continuous time random walk, and fractional BM were alternatively used to represent these trajectories. The comparison of the experimental and numerical data demonstrates that cytoskeletal transport along microtubules and diffusion in the cytosolic fraction exhibit anomalous (nonFickian) behavior and posses statistically distinct signatures. Among the three random-walk models used, continuous time random walk provides the best representation of diffusion, whereas microtubular transport is accurately modeled with fractional BM.

Figure 1

Summary of data collection and methods. (A) Block diagram of the microscope used in this work. Custom PC software controls all microscope functions and finalizes data acquisition. After scan parameters are entered, the scan program is sent to a field-programmable gate array (FPGA) as a metalanguage string of hexadecimal characters and saved into onboard memory. A Start command is sent with the number of repeats to begin a scan by driving a direct digital synthesis (DDS) board, producing a series of frequencies and chirp rates directing the acquisition of the volume, while a concurrent trigger signal is sent to the data acquisition oscilloscope. A laser diode (LD) is directed by the acousto-optic deflectors (AOD), through a telescope tube (TT), and reflected by a dichroic mirror (DM) onto the back-aperture of an objective (OBJ). The light emitted by the sample is collected on a photomultiplier tube (PMT) and converted into an image on the PC. (B) Example image from the microscope. Note that high and low concentrations, as seen here, are common, and that all trajectories are taken from single molecules that never overlap. (Dashed line) The z cut shown to the right. (C–E) Example trajectories of a bead in a (C) buffer solution; (D) extract; and (E) extract treated with nocodazole.

Figure 2

Time courses of mean-square displacement. (A and B) Comparison of averaged trajectories for diffusion in cellular extract, buffer solution, and cellular extract treated with nocodazole. (Symbols) Experimental data. (Solid lines) Fit using Eq. 6. (A) Short lag-time analysis. (B) Long lag-time analysis. (C) Comparison of random walk models to experimental results. Note that a time-averaged MSD of CTRW trajectories is inappropriate, therefore there is no comparison to the experimental condition of extract with nocodazole.

Figure 3

Time-averaged mean-square displacement distributions for all experiments. Snapshots of the distribution ${\varphi }_{{\xi }_i}\left(\text{Ξ}\right)$ for the three experimental conditions at four different lag times Δ. In the case of buffer, 28 trajectories are analyzed, in the untreated extract case, 40 are analyzed, and in the case of extract treated with nocodazole, 31 are analyzed. All figure axes mirror those in the bottom left, but are removed for clarity. The trend, in the cases of Buffer and Extract + noc (nocodazole), is independent of lag time whereas the case of Extract shows a shifting peak with increasing lag time. To see this trend more clearly, compare this figure with Fig. 4, which plots the distribution for many values of Δ.

Figure 4

Temporal evolution of time-averaged MSD distributions. Comparison of distribution ${\varphi }_{{\xi }_i}\left(\text{Ξ}\right)$ for the three experimental and the three modeling conditions. BM denotes Brownian motion, CTRW denotes continuous time random walk, and fBM denotes fractional Brownian motion. In the case of the experimental data, although there is noise due to the limited number of trajectories, the trends in each case are distinctive. Furthermore, the three modeling conditions shown are quite similar to the paired experimental conditions, suggesting these processes are good representations of the biological process.