Aging power spectrum of membrane protein transport and other subordinated random walks

Many biological, physical and social systems exhibit correlations in their dynamics across multiple time scales. Such correlations manifest themselves as a broad spectrum of relaxation times and a 1/f decay in the power spectrum, which reflects temporal self-similarity in the dynamics. Such spectra are observed in nanoscale devices, network traffic, earthquakes, heartbeat dynamics, DNA base sequences, climate and ecology, among many settings. Most processes of this kind are non-stationary, and the basic spectrum is best described at low frequencies with a formula S(ω) ∼ ω tmz where ω = 2πf is the frequency, tm is the measurement time, z is a non-zero exponent, and 0 < β ≤ 2.
This insight has motivated a new theorem – the so-called aging Wiener-Khinchin theorem – as a replacement for the celebrated Wiener-Khinchin theorem valid for stationary processes and widely applicable to systems that do not exhibit 1/f noise. But many questions remain. If the aging Wiener-Khinchin theorem relates the aging power spectrum with z ≠ 0 to a non-stationary correlation function, what is this correlation function? In a recent paper, LML Fellow Eli Barkai and colleagues explore this matter in biological data, demonstrating the applicability of the aging Wiener-Khinchin theorem and a corresponding calculation of the correlation function with experimental recordings of the power spectra of the dynamics of ion channels in the plasma membrane of mammalian cells.
As they point out, the emergence of 1/f noise has triggered significant interest in biological environments. In particular, the internal dynamics in globular proteins are self-similar and the autocorrelation function is aging over an astonishing thirteen decades in time. These fluctuations play essential roles in cellular functions involving molecular interactions such as gene regulation. This behaviour is widespread, ranging from the dynamics of proteins within cell membranes to the scaling behaviour of heartbeat time series. Nevertheless, it remains a challenge to measure how aging affects the spectrum of recorded 1/f noise in real systems.
In the recent work, Barkai and colleagues address the spectral content of processes with scale free relaxation times, using both theoretical modelling and experimental validation. They first demonstrate the usefulness of the aging Wiener-Khinchin theorem and then, more importantly, demonstrate how the aging exponent z and the spectral exponent β relate to the underlying processes. Depending on whether the process is negatively or positively correlated, they find vastly different frequency decays of the power spectrum. Thus, the work demonstrates how the aging Wiener-Khinchin theorem can be used to classify widely different classes of dynamics. By analyzing the dynamics of voltage-gated sodium channels on the somatic membrane of hippocampal neurons, the researchers also demonstrate how this basic approach works in the laboratory. This work not only validates the aging Wiener-Khinchin theorem as an emerging tool in spectral analysis, but also – in one biological context – unravels the meaning of the exponents describing the aging and the frequency decay.
The paper is available as a pre-print at

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