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

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.

. Deviations between the exact results and the asymptotic approximations for the PSD. The deviations are presented for the four cases discussed in the text as the ratio between the exact hypergeometric function and the 1/ω β approximation for the lowest 30 natural frequencies.
SUPPLEMENTARY NOTE 1. POWER SPECTRAL DENSITY (PSD) The PSD of a time-dependent signal x(t) is defined as where the angle brackets denote averaging over an infinitely large ensemble, i.e., the expected value. The Wiener Khinchin theorem provides a connection between the PSD and the autocorrelation function in stationary processes. Namely, this theorem states that the PSD is the Fourier transform of the autocorrelation function, where C(τ ) = x(t)x(t + τ ) is the (ensemble-averaged) autocorrelation function.
In order to solve the correlation function, we derive the fractional moments n ν (t) and ∆n ν (τ ; t) , where the relevant case for us is ν = 2H. The former is directly obtained from χ n (t), the distribution of the number of steps up to time t, in the continuous approximation The distribution χ n (t) is given by [1] where L α (t) is the one sided Lévy function of order α. Then, Eq. 3 yields The Laplace transform of the one-sided Lévy function is L{L α (t)} = exp[−s α ], thus the fractional moment of the number of steps in Laplace domain is Finally, we can use Tauberian theorem [1] to obtain the inverse Laplace transform, Along the same lines, the second term in Eq. 20 is the fractional moment In order to solve for the third term in the ACF, the fractional moment ∆n ν (τ ; t a ) is obtained by considering Eq. 7 and the forward waiting time t F for the first step of CTRW after the aging time t a . The distribution of forward waiting times is [1] so that The fractional moments n ν (t) and ∆n ν (τ ; t) have been computed previously [2]. The solution for ∆n ν (τ ; t) given in Eq. 25 of Ref. [2] has a different form but it is equivalent to Eq. 10 above.

SUPPLEMENTARY NOTE 3. EXACT SOLUTIONS FOR THE AUTOCORRELATION FUNCTIONS AND PSD OF SUBORDINATED PROCESSES
A. Continuous time random walk (2H = 1) We start with the solutions for a traditional CTRW [3,4], i.e., H = 0.5. The ensemble-averaged autocorrelation function is which, for the case H = 0.5, becomes and, using Eq. 7, or the well know expression for the mean number of jumps in the interval (0, t), n(t) at long times [1], The ensemble averaged autocorrelation function in Eq. 15 has the form C EA = t α φ EA and, thus, we obtain the time-averaged autocorrelation function with the scaling function Using the time-averaged autocorrelation function in Eq. 17 and the aging Wiener-Khinchin theorem, we obtain the power spectral density of the CTRW, where 1 F 2 (a; b 1 , b 2 ; z) refers to the generalized hypergeometric function. When α > 1 the mean waiting time exists and the CTRW statistics revert in the long time to those of Brownian motion. In particular, replacing α = 1 and ωt m = 2πk we find 1 F 2 1; 2, 5/2; −(ωt m ) 2 /4 = 6/(ωt m ) 2 and, thus, the PSD in Eq. 18 is that of standard Brownian motion, which is independent of t m .

Autocorrelation function
When H = 0.5, the process has positively correlated increments for H > 0.5 and negatively correlated increments when H < 0.5. The autocorrelation function C EA in Eq. 13 is where ∆n(τ ; t) is the number of steps between the aged time t and t + τ . From Eqs. 7, 8, and 10, the terms in Eq. 20 are found to be where 2 F 1 (a 1 , a 2 ; b; z) is the Gaussian hypergeometric function. We have defined Note that in the specific case that H = 0.5, these constants revert to b = γ = α. Using a different formalism, ∆n ν (τ ; t) has been previously derived [2,5]. These previous results were expressed in terms of incomplete beta functions but they are equivalent to ours. The ensemble-averaged autocorrelation function, Eq. 20, is thus given by with which gives the EA-MSD when τ = 0; x 2 (t) = 2∆x 2 n 2H (t) = 2Dt γ . The ensemble-averaged autocorrelation function in Eq. 25 has the form C EA (t, τ ) = t γ φ EA (τ /t), which implies the time-averaged autocorrelation function is of the form C TA (t m , τ ) = t γ m φ TA (τ /t m ) [6]. Defining y = τ /t m , we can find the scaling function (see Methods section in main text) Numerical simulations are observed to agree with analytical results for both H < 1/2 and H > 1/2 in Supplementary Figs. 1a and 1b, respectively.

Power spectral density
We see that the subordinated process with correlated increments shows that C TA = t γ m φ TA (τ /t m ).The aging Wiener-Khinchin theorem gives the average power spectral density for the natural frequencies ωt m = 2πk with k a non-negative integer, For the process subordinated to fractional Brownian motion, the time average autocorrelation function is found to be given by Therefore, to obtain S , we compute the following three integrals with the notationω = ωt m and notingω = 2πk, 1 0 (1 − y) 1+γ cos(ωy)dy = 1 2 + γ 1 F 2 1; where 2 F 3 (a 1 , a 2 ; b 1 , b 2 , b 3 ; z) is also a generalized hypergeometric function, leading to S(ω, t m ) = 2Dt 1+γ m 1 (1 + γ)(2 + γ) 1