# Single-Particle Tracking Reveals Anti-Persistent Subdiffusion in Cell Extracts

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Microscopy and Single-Particle Tracking

#### 2.2. Xenopus Extract Preparation and Modification

## 3. Results and Discussion

#### 3.1. Calibration Experiments in Viscous Media

#### 3.2. Evaluation of Tracer Motion in Native Xenopus Extract

#### 3.3. From Native to Pharmaceutically Treated Xenopus Extracts

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Representative TA-MSDs for trajectories with length $N=70$ (randomly chosen from the ensemble) from experiments in glycerol–water mixtures (red thin lines) and in Xenopus extract (black thin lines), together with the respective ensemble-averaged TA-MSDs (colored thick lines). For better visibility, data for calibration experiments have been shifted upward tenfold. The scaling for normal diffusion (${\langle {r}^{2}\left(\tau \right)\rangle}_{t}\sim \tau $) is indicated by a red dashed line; vertical grey dashed lines indicate the fit region used to analyze individual TA-MSDs. (

**b**) The PDF of anomaly exponents, $p\left(\alpha \right)$, as obtained from fitting TA-MSDs in glycerol–water mixtures features a mean $\langle \alpha \rangle \approx 1$, irrespective of the trajectory length (black-grey histogram: $N=70$, blue histogram: $N=150$).

**Figure 2.**(

**a**) The PDF of anomaly exponents, $p\left(\alpha \right)$, as obtained from fitting TA-MSDs in the interval $\tau \in [0.05,0.3]$ s, features a mean $\langle \alpha \rangle \approx 0.9$, irrespective of the trajectory length (black-grey histogram: $N=70$, blue histogram: $N=150$). The considerable width of the PDF may not only reflect statistical fluctuations but is likely to also report on spatially varying material properties of the Xenopus extract. Performing a bootstrapping approach with geometric averaging (black-open histogram) confirms the slightly subdiffusive motion of particles, while an arithmetic averaging (red histogram) overestimates the mean scaling exponent; see also main text for discussion. Please note the logarithmic y-axis. (

**b**) The PDF of generalized diffusion coefficients, $p\left(K\right)$, shown here versus the average area covered in one second, $K\times 1{s}^{\alpha}$, features an almost lognormal shape (indicated by full lines) for trajectory lengths $N=70$ (grey/black) and $N=150$ (blue), with a slight tendency for lower mobilities in longer trajectories. Please see the main text for discussion. (

**c**) A scatter plot of trajectory-wise values of $\alpha $ and K (blue and grey symbols) highlights a correlation between these two quantities, in good agreement with results on simulated FBM trajectories with a Hurst coefficient $H=\alpha /2=0.45$ (red symbols). The black dashed line is an empiric guide for the eye. FBM simulation data have been shifted upward fivefold for better visibility.

**Figure 3.**The PDF of normalized increments taken within a period $\delta t$, shown here as $p\left(\right|\chi \left|\right)$, complies well with a standard Gaussian (black full line) for different choices of $\delta t$ (color-coded symbols). For $\delta t\ge 5\Delta t$ and $\left|\chi \right|>3$, consistently lower probabilities than the Gaussian benchmark are observed for unknown reasons.

**Figure 4.**The normalized VACF, $C\left(\xi \right)$, for different choices of $\delta t$ (color-coded symbols) shows excellent agreement with the FBM prediction [Equation (5)] when inserting the mean scaling exponent $\langle \alpha \rangle =0.9$ (full black line). In particular, a clearly negative value of $C(\xi =1)$ confirms an antipersistent random walk, most likely of the FBM type. No significant changes of the VACF minimum are seen for different $\delta t$, confirming that trajectories are not plagued by localization errors.

**Figure 5.**The PSD of individual trajectories (black and blue thin lines, representing trajectories with length $N=70$ and $N=150$, respectively) fluctuate around the ensemble-averaged PSD (thick colored lines). In both cases, the FBM prediction for a scaling $S\left(f\right)\sim 1/{f}^{1+\langle \alpha \rangle}$ (with $\langle \alpha \rangle =0.9$, dashed line) are nicely met. For better visibility, data for $N=150$ have been shifted upward 100-fold.

**Figure 6.**The coefficient of variation of individual PSDs with respect to the ensemble mean, $\gamma \left(f\right)$, for normally diffusive trajectories from calibration experiments (red line) clearly assumes higher values than those for trajectories from the Xenopus extract (blue and black lines), irrespective of the trajectory length, N. As predicted for FBM, these subdiffusive SPT data converge toward $\gamma =1$, whereas normally diffusive data from calibration experiments converge to the predicted value $\gamma =\sqrt{5}/2$. Both are clearly distinct from the prediction for superdiffusive FBM motion, $\gamma =\sqrt{2}$. For convenience, frequencies f were made dimensionless by multiplication with the total time $T=N\Delta t$ covered in each trajectory.

**Figure 7.**Representative fluorescence images of beads (upper panel) and microtubules (lower panel) in native and pharmaceutically treated Xenopus extracts (see Materials and Methods for details); scale bars indicate 10 $\mathsf{\mu}$m. While native extracts feature a significant amount of microtubule filaments (left column), the addition of nocodazole completely eradicates these higher-order structures (right column). In contrast, stabilizing microtubules by taxol further enhances the ‘filament jungle’ (middle column).

**Table 1.**Summary of glycerol concentrations (weight percent) in glycerol–water mixtures, along with the respective viscosities $\eta $, and predicted diffusion constants D. Average scaling exponents $\langle \alpha \rangle $ (found via fitting all TA-MSDs for trajectories of 20 nm radius particles as described in the main text, followed by averaging the individual values of $\alpha $) are near to unity and mean diffusion coefficients $\langle K\rangle $ (also obtained by averaging the results for individual TA-MSDs) compare favorably to the predicted values of D. Result for trajectories with length $N=70$ and $N=150$ are given in upper and lower lines, respectively. The ensemble size of evaluated trajectories for the respective condition is given by M.

glyc. | $\mathit{\eta}$ [Pas] | D [$\mathsf{\mu}{\mathbf{m}}^{2}/\mathbf{s}$] | $\langle \mathit{\alpha}\rangle $ | $\langle \mathit{K}\rangle $ [$\mathsf{\mu}{\mathbf{m}}^{2}/{\mathbf{s}}^{\mathit{\alpha}}$] | M |
---|---|---|---|---|---|

70% | 0.016 | 0.67 | 1.00 | 0.56 | 929 |

0.99 | 0.45 | 67 | |||

75% | 0.023 | 0.46 | 1.01 | 0.42 | 1513 |

1.03 | 0.35 | 185 | |||

80% | 0.035 | 0.30 | 1.01 | 0.28 | 2158 |

1.03 | 0.24 | 250 | |||

85% | 0.055 | 0.19 | 1.01 | 0.20 | 4426 |

1.02 | 0.18 | 749 | |||

90% | 0.096 | 0.11 | 1.00 | 0.13 | 4668 |

1.02 | 0.12 | 980 |

**Table 2.**Summary of results found for different conditions of Xenopus extracts. Data for trajectories with length $N=70$ and $N=150$ are given in upper and lower lines, respectively. The ensemble size for the respective condition is given by M.

cond. | $\langle \mathit{\alpha}\rangle $ | $\langle \mathit{K}\rangle \phantom{\rule{3.33333pt}{0ex}}[\mathsf{\mu}{\mathbf{m}}^{2}/{\mathbf{s}}^{\mathit{\alpha}}]$ | M |
---|---|---|---|

untreated | 0.89 | 0.39 | 3085 |

0.88 | 0.27 | 582 | |

+taxol | 0.83 | 0.33 | 2548 |

0.81 | 0.25 | 606 | |

+nocodazol | 0.89 | 0.37 | 1126 |

0.81 | 0.17 | 130 | |

+ATP$\gamma $S +GTP$\gamma $S | 0.85 | 0.31 | 915 |

0.81 | 0.10 | 189 | |

+ATP$\gamma $S +GTP$\gamma $S +noc. | 0.88 | 0.35 | 2646 |

0.83 | 0.23 | 471 |

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Speckner, K.; Weiss, M.
Single-Particle Tracking Reveals Anti-Persistent Subdiffusion in Cell Extracts. *Entropy* **2021**, *23*, 892.
https://doi.org/10.3390/e23070892

**AMA Style**

Speckner K, Weiss M.
Single-Particle Tracking Reveals Anti-Persistent Subdiffusion in Cell Extracts. *Entropy*. 2021; 23(7):892.
https://doi.org/10.3390/e23070892

**Chicago/Turabian Style**

Speckner, Konstantin, and Matthias Weiss.
2021. "Single-Particle Tracking Reveals Anti-Persistent Subdiffusion in Cell Extracts" *Entropy* 23, no. 7: 892.
https://doi.org/10.3390/e23070892