Non-Gaussianity in single-particle tracking: Use of kurtosis to learn the characteristics of a cage-type potential

Nonlinear interaction of membrane proteins with cytoskeleton and membrane leads to non-Gaussian structure of their displacement probability distribution. We propose a statistical analysis technique for learning the characteristics of the nonlinear potential from the time dependence of the cumulants of the displacement distribution. The efficiency of the approach is demonstrated on the analysis of the kurtosis of the displacement distribution of the particle traveling on a membrane in a cage-type potential. Results of numerical simulations are supported by analytical predictions. We show that the approach allows robust identification of some characteristics of the potential for the much lower temporal resolution compared with the mean-square displacement analysis and we demonstrate robustness to experimental errors in determining the particle positions.

Figure 1

Simulation of the trajectory of a diffusing protein in triangular compartments of size 600 nm during 10 s with a time resolution 10 ms. The dashed lines indicate the borders between compartments.

Figure 2

Kurtosis of the displacements along $x$ for three different compartment sizes calculated with the resolution $0.1$ ms. The inset shows the kurtosis scaled as $1/L$ for the resolution $1\mu$s and a function $f\left(t\right)\approx t^{1/2}$ (thin black line) which shows that the kurtosis scales as $t^{1/2}$ for small $t$.

Figure 3

Kurtosis for three different compartment sizes $L$ as a function of rescaled time $t/L^2$ showing $L^2$ scaling of the position of the minima. The time resolution of each simulation was $0.1$ ms and total length was ${10}^4$ s.

Figure 4

Kurtosis along $x$ for three different resolutions with $L=300$ nm. The kurtosis for pure diffusion (no barriers) is also shown with the resolution $0.1$ ms. Error bars correspond to $0.1$-ms resolution. The inset shows MSD for diffusion with no compartments (short dashed line) and for simulations with the resolutions $0.1$ ms (long dashed line) and 10 ms (solid line).

Figure 5

Kurtosis along $x$ for different fractions of randomly removed barriers for simulation of 100-s duration for triangular compartments with $L=300$ nm and 0.1 ms resolution.

Figure 6

Kurtosis along $x$ for different levels of the simulated experimental noise (simulated as the error in the measurement of particle position) in position of particles for simulation of 100-s duration and the same parameters as in Fig. 5. The simulated noise is taken to be Gaussian with the variance ${\gamma }^2.$

Figure 7

Dependence of the position of the second kurtosis peak for the square compartments of size $L$ from simulations (crosses connected by dashed lines line) vs the dependence (4) with $O(L^2/D)=1.216L^2/D$ (solid line). The default parameter values $\sigma =5$ nm and $H=7$ are used.