Neuronal messenger ribonucleoprotein transport follows an aging Lévy walk

Abstract

Localization of messenger ribonucleoproteins (mRNPs) plays an essential role in the regulation of gene expression for long-term memory formation and neuronal development. Knowledge concerning the nature of neuronal mRNP transport is thus crucial for understanding how mRNPs are delivered to their target synapses. Here, we report experimental and theoretical evidence that the active transport dynamics of neuronal mRNPs, which is distinct from the previously reported motor-driven transport, follows an aging Lévy walk. Such nonergodic, transient superdiffusion occurs because of two competing dynamic phases: the motor-involved ballistic run and static localization of mRNPs. Our proposed Lévy walk model reproduces the experimentally extracted key dynamic characteristics of mRNPs with quantitative accuracy. Moreover, the aging status of mRNP particles in an experiment is inferred from the model. This study provides a predictive theoretical model for neuronal mRNP transport and offers insight into the active target search mechanism of mRNP particles in vivo.

Fig. 1

Schematic and experimental data showing motor-driven transport of mRNPs in a neuron. a An ensemble of mRNP complexes are transported by kinesin and dynein motor proteins along microtubules in the dendrite. In each dendrite, there are multiple microtubule tracks and thus several mRNP particles are efficiently transported at the same time to the target sites where they are localized. b A live-cell image showing fluorescently labeled β-actin mRNP complexes in a hippocampal neuron. Time-lapse images were taken with a time interval of t₀ = 0.1 s for the overall observation time of T = 60 s. The region of interest for analysis is marked in yellow. Scale bar, 10 μm. c A kymograph for an ensemble of mRNP particles obtained from an image similar to the yellow area in (b). Horizontal and vertical axes correspond to the elapsed time and the distance along the dendrite, respectively. For a few trajectories, their two distinct dynamic modes, rest and run, are denoted in green and red, respectively. The kymographs exhibit constant small-length scale fluctuations (see Supplementary Fig. 1 for more information). Scale bars, 10 s and 10 μm, respectively

Fig. 2

Stochastic transport dynamics of individual mRNP particles. a A simplified diagram describing an individual mRNP motion. Run and rest phases are repeated with randomly given sojourn times t_r and t_s for each. The flow diagram shows the distribution of the sojourn times. The direction of run is either anterograde or retrograde with equal probability and without memory of the previous run (Supplementary Table 1). The time t = 0 specifies the initial moment that an mRNP begins to be transported by motor proteins. In the experiment, the mRNP motion is observed from an arbitrary time t_a (aging time) and for the overall observation time T. b Experimental data of the run time pdf ψ_r(t). The dashed line represents the best fit to the data with Eq. 1. c Experimental data of the rest time pdf ψ_s(t). The rest times follow a power-law distribution (dashed line). d TA MSD curves $\overline{x^2\left(\tau \right)}=\frac{1}{T-\tau }{\int }_0^{T-\tau }{\left[x\left(\mathrm{\Delta }t+\tau \right)-x\left(\mathrm{\Delta }t\right)\right]}^2\mathrm{d}\mathrm{\Delta }t$ from individual trajectories. Thick line (blue) denotes the average curve over all TA MSDs. e EA MSD (red) $⟨x^2\left(\mathrm{\Delta }t\right)⟩={\sum }_{i=1}^N{\left[x_i\left(\mathrm{\Delta }t\right)-x_i\left(\mathrm{\Delta }t=0\right)\right]}^2\mathrm{∕}N$ plotted together with the average TA MSD curve (blue) shown in (d)

Fig. 3

Weak ergodicity breaking and aging in the simulation of truncated Lévy walk with rests. a Sample trajectories from the simulations with three different measurement initiation times t_a (see Fig. 2a). Top: Lévy walk trajectories when the measurement begins with the start of the process (t_a = 0). Middle: Trajectories when the measurement is initiated at t_a = 100 s. Bottom: Trajectories with the aging time t_a = 1800 s. In each panel, 436 trajectories are plotted, the majority of which are silent trajectories. b EA and TA MSD curves when the measurement initiation time is the same as the process start time (i.e., t_a = 0, top panel in (a)). Inequivalence between the EA and TA MSDs, called weak ergodicity breaking, is clearly visible in this case. For times longer than the run characteristic time, the scaling for EA MSD approaches $~{\left(\Delta t\right)}^{\alpha }$ while the scaling for TA MSD follows ~τ¹. c EA and TA MSD curves when the measurement is initiated at t_a = 1800 s after the start of the process (bottom panel in (a)). Owing to aging, the EA MSD also shows the same apparent Fickian scaling as in the TA MSD for τ_r < Δt < T. Note that aging obscures weak ergodicity breaking in this case

Fig. 4

Quantitative comparison of the experiment and simulation of the aging Lévy walk process. a Comparison of EA MSD curves for the aging Lévy walks at various aging times t_a = 0, 100, and 1800 s (green, red, and blue curves, respectively) with the EA MSD from the experiment (black circles, which is the same data shown as red circles in Fig. 2e). The simulation data at t_a = 100 s is in excellent agreement with the experimental data. b Comparison of TA MSD curves. The simulation data with aging time t_a = 100 s, unequivocally, match both the EA and TA MSDs from the experiment. c Individual TA MSD curves of the aging Lévy walk processes with the aging time t_a = 100 s. The thick blue curve is the average of the TA MSDs (gray curves). d Normalized amplitude scatter distributions ϕ(ξ) for the TA MSDs as a function of the rescaled variable $\xi =\overline{x^2}∕⟨\overline{x^2}⟩$. The experimental distribution from Fig. 2d (black circles) is compared with the theoretically expected distributions from the aging Lévy walk at three different aging times. e Aged probability density functions P(x, Δt) of mRNP particles from the experiment and the aging Lévy walk simulations. f A fraction q(t_a;T) of the trajectories showing no run at all in [t_a, t_a + T] as a function of t_a. Two theoretical curves for the aging Lévy walk were obtained from simulations with and without the uncertainty of ~1 μm (red and blue curves, respectively) for identifying the silent trajectories. Thus, the shaded region indicates the expected probabilities q(t_a;T) in theory. The horizontal dashed line represents the experimental values for mRNP particles. It can be seen in the plot that the expected aging time is about t_a = 100 s