Towards a robust criterion of anomalous diffusion

The exploration of dynamic properties in complex systems has significantly advanced due to single-particle tracking (SPT) techniques. SPT, used in passive and active microrheology, probes material properties and biological systems by tracking micron- and submicron-sized tracers or even single molecules. It's crucial in understanding structure-function relationships and intracellular transport. SPT data, unlike averaged data from FCS or FRAP, provides high-resolution trajectories, ideal for statistical analysis. Often, this analysis reveals anomalous diffusion, characterized by a power-law dependence of the mean squared displacement (MSD): (X²)~t^α, where α is the anomalous diffusion exponent. α = 1 indicates normal (Brownian) diffusion, α < 1 subdiffusion, and α > 1 superdiffusion. Precisely determining α and distinguishing between normal and anomalous diffusion is crucial for predicting system characteristics like reaction rates and relaxation dynamics. However, accurately fitting α from finite measured time series is challenging. Measurement noise can lead to erroneous conclusions about the diffusion type. Existing methods like mean-maximal excursion statistics, Bayesian maximum-likelihood methods, deep learning, and feature-based methods exist but have limitations. The variability of α values between trajectories due to finite statistics and environmental heterogeneity makes predictions unreliable, especially when α is close to 1. Experimental errors, primarily static (from photon counts) and dynamic (from exposure time) localization errors, further complicate the analysis. Static error adds a positive offset, while dynamic error adds a negative offset, perturbing short-time scale analysis. This paper proposes a new method to overcome these limitations.

The literature extensively covers anomalous diffusion, its various physical origins (trapping, viscoelasticity), and its observation in diverse systems. Several methods exist to analyze single-particle trajectories to quantify anomalous diffusion, including fitting the MSD to extract the anomalous exponent α. However, these methods are often affected by noise and finite-size effects. The impact of localization error on MSD analysis has also been explored. Several studies have looked at power spectral analysis of single trajectories for characterizing stochastic processes. The coefficient of variation of the single-trajectory power spectral density has been proposed as a potential criterion but hasn't been extensively studied in the presence of localization errors, which is the focus of this paper.

This study focuses on fractional Brownian motion (FBM), a Gaussian process suitable for modeling anomalous diffusion in various contexts. FBM is defined by its covariance function: ⟨X_tX_t⟩ = D(|t - t|^2H + |t|^2H - |t - t + t|^2H), where D is a proportionality factor and H (the Hurst exponent) determines the diffusion type (H < 1/2 subdiffusion, H = 1/2 normal diffusion, H > 1/2 superdiffusion, with α = 2H). The paper employs single-trajectory power spectral density (PSD) analysis, examining the random variable S(f, T) = (1/T²)∫^T₀ dt₁∫^T₀ dt₂ cos(f(t₁ - t₂))X_t1X_t2. The coefficient of variation γ(f, T) = σ(f, T)/μ(f, T) of the PDF of S(f, T) is the central focus. For pure FBM, γ approaches 1 for subdiffusion, √5/2 for normal diffusion, and √2 for superdiffusion at high frequencies. The study investigates how this criterion is affected by static and dynamic localization errors. Static error is modeled as an additive Ornstein-Uhlenbeck (OU) process, while dynamic error is incorporated by averaging over a finite exposure time. Analytical predictions for the coefficient of variation in the presence of static error are derived, and numerical simulations (1D and 2D) are performed to explore both static and dynamic errors and to compare to the theoretical predictions. The numerical simulations carefully model the experimental process for accurate comparison with experimental results. Experimental datasets were then analyzed, including those showing subdiffusion (telomeres, P-granules), normal diffusion (beads in water), and superdiffusion (cytoskeleton fluctuations, nanoparticles in hMSCs). The experimental setups for each dataset are carefully explained.

The study found that the coefficient of variation γ is a robust indicator of anomalous diffusion, particularly for subdiffusion. Analytical and numerical results show that γ converges to 1 for subdiffusion regardless of localization errors. This makes it a reliable method for detecting subdiffusion even with noisy data. For normal diffusion, both static and dynamic errors affect γ, leading to deviations from the expected value of √5/2. Specifically, static error causes a decrease, while dynamic error causes an increase. For superdiffusion, static error significantly affects γ, causing a decrease towards 1 at high frequencies. However, dynamic error has minimal effect on the limiting value of γ in the superdiffusive regime. The analytical expression derived for the correction term due to static error allows for controlling the effect of the measurement error in the superdiffusive regime. The experimental data analyses validate these findings across a wide range of systems and diffusion types. The subdiffusive experimental data of telomeres and P-granules consistently showed γ converging to 1, even with static localization errors. The normal diffusive data of beads in water exhibited an increase in γ at high frequencies due to dynamic error, and the addition of artificial static error caused the expected decrease in γ. The superdiffusive data from cytoskeleton fluctuations and nanoparticles showed a consistent convergence to √2, confirming the robustness of the criterion against localization errors in this regime. In all cases, analysis showed γ to be sensitive to transitions between diffusion regimes.

The results demonstrate that the coefficient of variation of the single-trajectory power spectral density provides a robust criterion for identifying anomalous diffusion, especially in the presence of experimental errors that often plague single-particle tracking experiments. The robustness of the method, particularly for subdiffusion, addresses a significant limitation of existing methods that are highly susceptible to noise and uncertainty. The distinct responses of γ to static and dynamic errors in different diffusion regimes provide valuable insights for interpreting experimental data and refining data analysis approaches. The findings highlight the importance of considering the types of measurement errors and their specific impacts on the analysis of anomalous diffusion.

This work establishes the coefficient of variation γ from single-trajectory power spectral density analysis as a robust criterion for anomalous diffusion. Its robustness is especially significant for subdiffusion and relatively high for superdiffusion, even in the presence of localization errors. The study's comprehensive analytical, numerical, and experimental approach strengthens the findings and their implications for various fields. Future research can focus on extending this method to other anomalous diffusion processes and incorporating it into machine learning frameworks for improved data analysis and interpretation.

While the study demonstrates the robustness of the proposed method, some limitations exist. The analytical treatment of dynamic error is less complete than that of static error due to mathematical complexity. The analysis primarily focuses on FBM, and further investigation is needed to explore the applicability of this criterion to other anomalous diffusion models. The experimental data cover a range of systems but not an exhaustive set of all possible scenarios, so caution is needed when extrapolating to entirely new systems. Finally, the choice of frequency range for assessing the limiting value of γ may need to be carefully considered for individual datasets.