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Active-Matter models commonly consider particles with overdamped dynamics subject to a force (speed) with constant modulus and random direction. Some models also include random noise in particle displacement (a Wiener process), resulting in diffusive motion at short time scales. On the other hand, Ornstein–Uhlenbeck processes apply Langevin dynamics to the particles’ velocity and predict motion that is not diffusive at short time scales. Experiments show that migrating cells have gradually varying speeds at intermediate and long time scales, with short-time diffusive behavior. While Ornstein–Uhlenbeck processes can describe the moderate-and long-time speed variation, Active-Matter models for over-damped particles can explain the short-time diffusive behavior. Isotropic models cannot explain both regimes, because short-time diffusion renders instantaneous velocity ill-defined, and prevents the use of dynamical equations that require velocity time-derivatives. On the other hand, both models correctly describe some of the different temporal regimes seen in migrating biological cells and must, in the appropriate limit, yield the same observable predictions. Here we propose and solve analytically an Anisotropic Ornstein–Uhlenbeck process for polarized particles, with Langevin dynamics governing the particle’s movement in the polarization direction and a Wiener process governing displacement in the orthogonal direction. Our characterization provides a theoretically robust way to compare movement in dimensionless simulations to movement in experiments in which measurements have meaningful space and time units. We also propose an approach to deal with inevitable finite-precision effects in experiments and simulations.
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Aging power spectrum of membrane protein transport and other subordinated random walks
Abstract Single-particle tracking offers detailed information about the motion of molecules in complex environments such as those encountered in live cells, but the interpretation of experimental data is challenging. One of the most powerful tools in the characterization of random processes is the power spectral density. However, because anomalous diffusion processes in complex systems are usually not stationary, the traditional Wiener-Khinchin theorem for the analysis of power spectral densities is invalid. Here, we employ a recently developed tool named aging Wiener-Khinchin theorem to derive the power spectral density of fractional Brownian motion coexisting with a scale-free continuous time random walk, the two most typical anomalous diffusion processes. Using this analysis, we characterize the motion of voltage-gated sodium channels on the surface of hippocampal neurons. Our results show aging where the power spectral density can either increase or decrease with observation time depending on the specific parameters of both underlying processes.
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- Award ID(s):
- 2102832
- PAR ID:
- 10307941
- Publisher / Repository:
- Nature Publishing Group
- Date Published:
- Journal Name:
- Nature Communications
- Volume:
- 12
- Issue:
- 1
- ISSN:
- 2041-1723
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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{"Abstract":["Datasets generated in the report "Aging power spectrum of membrane protein transport and other subordinated random walks". Included data are:<\/p>\n\nNumerical simulations <\/strong>\nRWdata1.mat: 10,000 realizations, subordinated random walk with Hurst exponent, H<\/em>=0.3 and \\(\\alpha\\)=0.4.\nRWdata3.mat: 10,000 realizations, subordinated random walk with Hurst exponent, H<\/em>=0.7 and \\(\\alpha\\)=0.4.\nRWdata8.mat: 5,000 realizations, subordinated random walk with Hurst exponent, H<\/em>=0.75 and \\(\\alpha\\)=0.8.\nRWdataCTRW.mat: 10,000 realizations, continuous time random walk (CTRW), \\(\\alpha\\)=0.7.<\/p>\n\nSpectra of simulations<\/strong>\nPSDdata1.mat: Power spectral density (PSD) of a subordinated random walk with Hurst exponent, H<\/em>=0.3 and \\(\\alpha\\)=0.4. Five different realization times are used to compute the PDS: 2^8, 2^10, 2^12, 2^14, and 2^16.\nPSDdata3.mat: PSD of a subordinated random walk with Hurst exponent, H<\/em>=0.7 and \\(\\alpha\\)=0.4. Five different realization times are used to compute the PDS: 2^8, 2^10, 2^12, 2^14, and 2^16.\nPSDdata8.mat: PSD of a subordinated random walk with Hurst exponent, H<\/em>=0.75 and \\(\\alpha\\)=0.8. Four different realization times are used to compute the PDS: 2^15, 2^16, 2^17, and 2^18.\nPSDs_CTRW.mat: PSD of a continuous-time random walk (CTRW), \\(\\alpha\\)=0.7. Five different realization times are used to compute the PDS: 2^8, 2^10, 2^12, 2^14, and 2^16.<\/p>\n\nExperimental data of Nav1.6 channels in the soma of hippocampal neurons<\/strong>\nNavMSDtimes.csv: ensemble-averaged (EA) MSD and time-averaged (TA) MSD. The TA-MSD is measured for three observation times, 64, 128, and 256 frames (3.2, 6.4, and 12.8 s).\nNavPSD.csv: Power spectral density (PSD) measured for three observation times, 64, 128, and 256 frames.<\/p>"],"Other":["We acknowledge the support of the National Science Foundation grant 2102832 (to DK) and Israel Science Foundation grant 1898/17 (to EB).","{"references": ["Fox, Z.R., Barkai, E. & Krapf, D. Aging power spectrum of membrane protein transport and other subordinated random walks. Nat Commun 12, 6162 (2021). https://doi.org/10.1038/s41467-021-26465-8"]}"]}more » « less
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