Superdiffusion dominates intracellular particle motion in the supercrowded cytoplasm of pathogenic Acanthamoeba castellanii

Acanthamoebae are free-living protists and human pathogens, whose cellular functions and pathogenicity strongly depend on the transport of intracellular vesicles and granules through the cytosol. Using high-speed live cell imaging in combination with single-particle tracking analysis, we show here that the motion of endogenous intracellular particles in the size range from a few hundred nanometers to several micrometers in Acanthamoeba castellanii is strongly superdiffusive and influenced by cell locomotion, cytoskeletal elements, and myosin II. We demonstrate that cell locomotion significantly contributes to intracellular particle motion, but is clearly not the only origin of superdiffusivity. By analyzing the contribution of microtubules, actin, and myosin II motors we show that myosin II is a major driving force of intracellular motion in A. castellanii. The cytoplasm of A. castellanii is supercrowded with intracellular vesicles and granules, such that significant intracellular motion can only be achieved by actively driven motion, while purely thermally driven diffusion is negligible.

presents the displacement distributions, P(x,t) and P(y,t), for the intracellular motion of normal acanthamoebae. Here, the intracellular motion r I (t) is a drift-free part in the tracked trajectory r(t), obtained from the r(t) by subtracting the centroid contribution from it. In the figure the displacement distributions at several times are plotted together with the Gaussian fit to the data. The results show that the intracellular motion reasonably follows Gaussian statistics, albeit the distributions are noisy at larger t. Given the fact that the intracellular motion r I (t) is superdiffusive (or subdiffusive) in the experiments, this supports the validity of our simple model describing the intracellular motion in the frame of fractional Brownian motion (i.e., the Gaussian anomalous diffusion process). 1

Velocity autocorrelation function (VACF)
In this section we explain in detail the velocity autocorrelation function considered in our study with additional supplementary plots. Assume that the particle motion r(t) tracked by video tracking is the superposition of the drift of acanthamoeba during crawling V d and the intrinsic intracellular motion r I such that r(t) = V d t + r I (t). (S1) The internal motion r I (t) explains the net motion of a particle inside the acanthamoeba that arises potentially from its passive diffusion and/or the active transport by motor proteins. Given an arbitrary time interval δt we define an average velocity for the tracked trajectory r(t) Since the shortest time interval in the experiment is the unit video frame interval t 0 ≈ 0.01 s and the particle motion occurs in the overdamped limit at this time scale, one should keep in mind that the above velocity defined from the overdamped motion is not the instantaneous velocity V (t) defined in the underdamped limit. As seen below in Supplementary Equation (S3), the square of the velocity in the overdamped limit V 2 δt (t) = r 2 (δt) /δt 2 is a quantity related to the mean square displacement while the instantaneous velocity in the underdamped limit gives the relation V 2 (t) ∼ k B T . In the former case the effect of temperature exists in the parameters such as diffusion constant and the diffusion exponent. The velocity autocorrelation function (VACF) is conventionally obtained from the ensemble-average of many trajectories. Using the velocity defined in Supplementary Equation (S2) the VACF C(∆) = V δt (∆ + t) · V δt (t) is given by where C rr (t 1 ,t 2 ) = r(t 1 ) · r(t 2 ) is the covariance of the position vector. Instead of this, alternatively, we obtain an individual VACF curve from a single trajectory by the definition For the ergodic diffusive processes considered in our study, we expect Thus, the time-averaged VACF obtained from Supplementary Equation (S4) should be the same as the conventional VACF, if all the tracked particles have the identical diffusion behavior. However, intracellular diffusion is typically heterogeneous and thus the time-averaged VACFs can have trajectory-to-trajectory variations.

Model
Now we model the intracellular motion r I (t) = (x I (t), y I (t)) to follow fractional Brownian motion (FBM). 1 That is, each component motion in r I (t) is a correlated Gaussian processes characterized by the mean x I (t) = y I (t) = 0 and the covariance 1 From the mean squared displacement r 2 I (t) = 4K α t α , FBM is classified to subdiffusion for 0 < α < 1, diffusive for α = 1, and superdiffusive for 1 < α < 2. Using this statistical property we obtain the normalized velocity autocorrelation function /δt denotes the average velocity for the intracellular motion r I (t). We note that the relaxation profile of C(∆) is fully governed by the VACF of the intracellular part C I (∆) = V I δt (∆ + t) · V I δt (t) . Effect of α: The main advantage of studying C(∆) is that one obtains a basic information about the stochastic identity of the intracellular motion r I (t) from the tracked position r(t) without extracting r I from it. As we plot in Fig. 4 in the main text, C(∆) has three distinguished profiles depending on the range of α: (1) For subdiffusive intracellular motion with 0 < α < 1 the VACF has a dip at t = δt, owing to the anti-persistent intracellular motion, and then relaxes towards the positive saturation value.
(2) If the intracellular motion is a normal random walk with α = 1 the VACF has no relaxation for t ≥ δt. (3) If the intracellular motion is superdiffusive with 1 < α < 2 the VACF is always positive at all times and monotonically decreases from unity to the saturation value.
Supplementary Figure S2. Profiles of the theoretical VACF curves (S6) at varying the ratio Effect of the amoeba drift and time interval (V d , δt, and K α ): As the above VACF (S6) shows, the profile and the saturation ) of the VACF depend on the time interval δt and the exponent α as well as V d . Physically, such dependences are plausible; In Supplementary Equation (S6) the term V 2 d /4K α (δt) α−2 is the ratio between the two square displacements accomplished by the acanthamoeba's drift (V d δt) 2 and by the intracellular transport 4K α (δt) α over the time interval δt. In case this term is large, the effect of the acanthamoeba's drift dominates over the intracellular contribution. In the opposite situation where the acanthamoeba's drift is negligible (as we observed in the blebbistatin-treated amoeba), the VACF C(∆) is dominated by the VACF of the intracellular motion. In Supplementary Fig. S2 we plot the variation of VACF profiles against the change of these parameters for both superdiffusive and subdiffusive intracellular motions with the exponent α found in our experiment. Note that each curve converges to its own saturation value

Experiment
In the main text we present VACF curves for x(t) for the normal and drug-treated acanthamoeba using equation (S4) with δt = 10t 0 . Here we provide additional VACF curves. Time intervals: In Supplementary Fig. S3 we plot the same VACF curves (shown in the main text) for the normal, blebbistatin-, latrunculin A-, and nocodazole-treated acanthamoeba at various time intervals δt = t 0 , 5t 0 , 10t 0 , and 20t 0 . For comparison, the VACFs are plotted against the rescaled lag time ∆(= ∆/δt) because the C(∆) is always self-correlated up to ∆ = δt and then the regime for ∆ > δt tells about the correlation of a velocity between two time points. Note that at a given ∆ the number of data points plotted in the VACF of δt = nt 0 is n times that of the VACF of δt = t 0 . Because of this, for all the plotted VACFs always the one with δt = t 0 has the largest fluctuation and varies in a discontinuous manner. The results show that except for the case of δt = t 0 the rest VACFs exhibit consistent patterns. In fact in our study the VACF with δt = t 0 is not a proper choice in studying superdiffusive intracellular motion observed in the experiment. Different from the above theoretical model based on fractional Brownian motion, the experimentally tracked particle motion has a time-dependent diffusion exponent α = α(∆). As TA MSD curves (Figs. 3, 5, 7 and 8 in the main text) for the tracked particles show, the superdiffusive motion is observed only for ∆ ≥ O(10t 0 ) ≈ 0.1 (sec). At ∆ = t 0 the TA MSD curves exhibit a pattern of strong subdiffusive motion. Therefore at this shortest time interval (where the effect of the acanthamoeba drift motion is minimal) the VACFs have patterns clearly distinguished from the profile for superdiffusive motion. For our study the time interval of δt = 10t 0 seems to be a good choice for analyzing the superdiffusive intracellular motion while reducing the effect of the acanthamoeba drift in the VACF curve. Note that with our choice of δt = 10t 0 the ratio V 2 d /4K α (δt) α−2 safely lies in the range [0, 1] for all the experiments presented in this study. VACF curves for the intracellular motion r I (t): As supplement figures we plot in Supplementary  Fig. S4 the VACF curves C I (∆)/C I (0) for the intracellular motion r I (t), which is the trajectory of particles in the frame of the centriod of an acanthamoeba. Because the acanthamoeba is not a rigid body, all the relative trajectories from the centroid may not correspond to the true intracellular motion. Nevertheless our approach seems to be fairly reasonable, as seen in this plot; the obtained intracellular part r I (t) has a positive correlation in its VACF and its decay converges to zero.

Time-averaged displacement (drift)
To see the trajectory-to-trajectory variation of the particle drift due to the locomotion of the acanthamoeba we calculate the time-averaged displacement  for both x and y coordinate components. Supplementary Figure S5 presents the TA displacement traces for about 40 particles in a moving acanthamoeba. The results are compared with the TA displacement of the acanthamoeba itself via its centroid motion. Note that the averaged trace for the individual particles almost exactly follows the trace for the centroid at lag times ∆ < 3 (sec), (above which the disagreement between the two is because only a few trajectories were used for the averaging). This shows that on average the linear increase or decrease of TA displacements with ∆ is determined by the locomotion of the acanthamoeba. Then the scatter of individual TA displacements from the average is due to the intracellular particle motion independent from the drift. It is confirmed in Supplementary Fig. S6, in which the TA displacements of FBM with α = 0.5 (subdiffusion), 1 (normal), 1.8 (superdiffusion) are depicted. In all cases the TA displacements have symmetric dispersion from the average δ (∆) = 0.

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Further details on the experimental results