Mitochondria are essential organelles responsible for fulfilling vital functions that guarantee cell survival^{1}. These organelles are particularly dynamic and flexible, being able to redistribute themselves in the cytoplasm and to adopt different morphologies in response to chemical or physical signals^{2,3}. In recent years, there has been growing interest in understanding how mechanical forces impact on the morphological fluctuations and dynamics of mitochondria. In particular, the cytoskeleton has been identified as one of the mechano-regulatory agents of these organelles due to its close interaction with them^{2}.

In the cell, mitochondria transport, organization, and morphodynamics strongly depend on the cytoskeletal networks^{4,5,6,7,8}. Given their anchorage to cytoskeletal elements (i.e. microtubules, F-actin, and intermediates filaments), these organelles are consistently exposed to both compressive and tensile stresses within the cytoplasm^{9}. Thus, their dynamics and function are affected by the mechanical impulses transmitted to them^{3,10,11,12}. For instance, F-actin and microtubule-associated motor proteins (i.e. myosins and dynein/kinesin, respectively) produce piconewton forces^{13} that are transferred to mitochondria through adaptor complexes that tether mitochondria to cytoskeletal filaments^{14,15}. These mechanical impulses are able to deform mitochondria membranes and induce fission^{16,17,18}.

The metabolic activity of cells is closely linked to changes in mitochondrial morphology and their exposure to mechanical signals^{19}. Modifications in the biophysical properties of these organelles lead to the reorganization of the cellular chondriome. The relationship between mitochondrial mechanics and metabolism is bidirectional, yet our understanding of how mitochondria adapt mechanically in response to metabolic cues remains limited.

Understanding the mechanobiology of mitochondria in their physiological environment is challenging^{2}. In many cell types, they display elongated shapes^{20,21,22} and their mechanical behavior resembles that of semiflexible filaments^{7}. The theory of flexible and semiflexible filaments has provided useful analytical tools to describe polymer dynamics in viscous and viscoelastic environments, which include: center of mass motion, amplitudes of flexural modes, curvature, end to end length and spatial and temporal correlations^{23,24,25}. All these statistical quantities strongly depend on mechanical properties of the filament and the medium, as well as the presence of external active (*e.g.* molecular motors) or passive forces (*e.g.* thermal agitation or constraints). These models have been tested in many experimental systems, such as cilia and microtubules in living cells^{26,27,28,29,30}. In particular, within this framework mitochondria apparent persistence length was estimated in \(\sim 2 \upmu\)m in *Xenopus laevis* melanocytes^{22}, highlighting the bending plasticity of these organelles.

Inspired by these ideas, we present here a new methodology that allows determining the active and passive forces exerted on mitochondria within living cells. Additionally, we study the different transport regimes to explore whether these organelles are subjected to active motion, diffusive movement, and/or confinement. We used these tools to explore the mechanisms underlying the motion and deformation of mitochondria within *Xenopus laevis* melanocytes. Due to their large size (\(\sim\)50 \(\upmu\)m) and relatively small thickness away from the perinuclear region (\(\sim\) 3 \(\upmu\)m)^{5} these cells behave as a 2D system in the periphery and so they represent an ideal model for live-cell confocal microscopy. In addition, most mitochondria display simple, i.e. non-branching, filamentous shapes^{7}.

We assessed the influence of the cytoskeleton on mitochondrial motion by selectively disrupting cytoskeletal networks (i.e. microtubules, F-actin and vimentin filaments), as discussed in a previous work^{7}. The cytoskeleton regulates cytoplasmic rheology and transmits mechanical impulses to mitochondria through molecular motors and crosslinkers^{6,15,31}. Therefore, we hypothesized that these treatments would impair mitochondrial cytoskeleton-dependent confinement and mechanical jittering, allowing us to explore the mechanical crosstalk between them. Using the quantitative tools proposed in this work, our results shed new light on how the cytoskeleton-exerted forces modulate mitochondrial dynamics. We believe that this study presents a new approach to extend the knowledge of how the impact of mechanical cues can be quantified at the single organelle level.

### Quantitative description of the motion of a filamentous mitochondrion

The motion and deformation of an elongated mitochondrion can be described through its curvilinear shape (Fig. 1). In this approximation, the shape of the filament and its evolution in time is characterized by the position \(\vec {\textbf{r}} (s,t)\) of a material point *s* at time *t*, with \(0 \le s \le L\), where *L* is the natural length. Ideally, *s* and *t* are continuous, but in experiments (or numerical simulations) the shape is described by a set of \(N_b\) discrete material points \(s_i\) (beads), tracked at discrete times \(t = j\, \Delta \, t\), with \(\Delta \, t\) the sampling time, as represented in Fig. 1a.

To characterize the mechanics and motion of the filament, one of the most well-known quantities is the position of the center of mass (CM):

$$\begin{aligned} \vec {\textbf{r}}_{\text {CM}} (t)= \frac{1}{L} \int _{0}^{L} \vec {\textbf{r}} (s,t) \text {d} s \approx \frac{1}{N_b}\sum _{i=1}^{N_b} \vec {\textbf{r}} (s_i,t) \end{aligned}$$

(1)

The trajectory of the center of mass contains important information about the overall motion of the filament and its interaction with the environment, which can be rendered by its Mean Square Displacement (MSD):

$$\begin{aligned} \text {MSD} (\tau ) = \left\langle \left( \vec {\textbf{r}}_{\text {CM}} (t + \tau ) - \vec {\textbf{r}}_{\text {CM}} (t) \right) ^2 \right\rangle \end{aligned}$$

(2)

where \(\tau\) is the time-lag,e.g. \(\tau =j \Delta t\), and the brackets indicate an average on time *t*. Typically, the MSD as a function of time-lag is fitted by a power law^{32}:

$$\begin{aligned} \text {MSD} (\tau ) = D^* \tau ^{\alpha }. \end{aligned}$$

(3)

where \(D^*\) and \(\alpha\) are the generalized diffusion coefficient and exponent, respectively.

While \(D^*\) represents a rate of motility, the diffusion exponent \(\alpha\) constitutes a key global parameter that indicates the overall directionality of a trajectory. Moreover, it is related to the balance of active *vs.* passive forces acting on the organelle. In the absence of active processes, this exponent will take values smaller than 1 – a hallmark of sub-diffusion – like in ATP-depleted cells^{33} or in overcrowded environments^{34,35,36}, reflecting the viscoelastic nature of the microenvironment. In contrast, organelles driven by active forces lead to super-diffusion, and, in these conditions, \(\alpha\) values ranging 1.2–1.5 have been determined in a wide variety of systems^{37,38,39,40}. Yet, for a free filament in a viscous medium, subjected to thermal noise, a diffusive motion is expected \((\alpha = 1)\). In this case, the diffusion coefficient depends inversely on the medium’s viscosity and the filament’s length^{23}.

Nevertheless, a filament could deform in such a way that the center of mass remains still, but its shape is completely different after a certain time. Therefore, it is more informative to analyze the displacement of individual material points of a filament after a time lag \(\Delta \,t\) (Fig. 1a, right panel):

$$\begin{aligned} \Delta \vec {\textbf{r}} (s,t, \Delta \,t) = \vec {\textbf{r}} (s,t+ \Delta \,t)-\vec {\textbf{r}} (s,t) \end{aligned}$$

(4)

This is a very fluctuating quantity, so different averages can be performed to extract relevant information. Here, we consider the average of the square displacements along the filament as a function of time:

$$\begin{aligned} K(t) = \frac{1}{L} \int _{0}^{L} [ \Delta \vec {\textbf{r}} (s,t, \Delta \, t) ] ^2 \text {d} s = \frac{1}{N_b}\sum _{i=1}^{N_b} [ \Delta \vec {\textbf{r}}_i (t) ]^2 \end{aligned}$$

(5)

This is related to the average area swept by each material point of the filament in a time interval \(\Delta \,t\). The greater the fluctuation of the filament’s shape due to stochastic thermal (and non-thermal) forces, the greater the value of *K* will be, regardless of the movement of the center of mass, which can eventually be still. In Eq. 5, we keep implicit the dependence of *K* and \(\Delta \textbf{r}\) with \(\Delta \,t\), as it is typically the sampling time of the experiment.

For reasons that will become clear later, it is useful to also define the Cumulative Square Displacement (CSD) as:

$$\begin{aligned} \text {CSD} (t) = \sum _{j=1}^{t/\Delta t} K(j\Delta t) \end{aligned}$$

(6)

Since the CSD is the sum of positive quantities, it grows monotonically with time.

In order to explore how these two magnitudes, the MSD and the CSD, can unveil aspects of the dynamics underlying the motion of elongated mitochondria, we performed numerical simulations of a simple model system. We adapted a Worm-Like Chain Model^{30,41} to describe the temporal evolution of a semiflexible filament immersed in a viscous medium. Briefly, we consider a filament composed of \(N_b\) coupled beads immersed in a homogeneous viscous medium and subjected to thermal noise (see Methods and Supplementary Section 1 for details). Active forces are modeled as point-like impulses acting on individual beads following stochastic temporal dynamics. Figure 1b shows two representative filament configurations obtained from simulations in the absence or the presence of active forces.

First, we recovered the trajectories of the center of mass from the simulated shapes and computed the MSD, displayed in Fig. 1c. Normal diffusion behavior is recovered in the absence of active forces, while a super-diffusive regime is obtained in their presence, as predicted by previous studies^{23}.

Then, we also calculated *K* and CSD from the filament coordinates, shown in Fig. 1d. Interestingly, the behavior of *K* for both simulations is very similar (*i.e.* their medians are not significantly different), except for the presence of a few outlier data points in the simulations with active forces. The effect of these extreme values is more evident in the CSD, where jumps are observed on top of a linear behavior obtained in the absence of active forces. Even more relevant, the occurrence of these events correlates with the periods when these forces are turned on, thus providing a potentially useful tool to uncover the temporal pattern of the active forces acting on the filament.

These results illustrate that both magnitudes, MSD and CSD, describe different but complementary aspects of the motion of a semiflexible filament in the presence of active forces. In the next sections, we will evaluate their performance with experimental data obtained from the tracking of mitochondria in living cells.

### The Mean Square Displacement of the mitochondria center of mass reveals the coexistence of different motion regimes

The shapes of filamentous mitochondria were recovered from confocal microscopy images of *X. laevis* melanocytes, as explained in Methods. Briefly, the organelles were considered as curvilinear filaments and a set of discretized coordinates were obtained for each frame of the time-lapse using a tracking algorithm (Fig. 2a and Supplementary Video S1).

Cells were treated with nocodazole (NOC) or latrunculin-B (LAT) to depolymerize microtubules or F-actin, respectively, or transiently transfected with a dominant negative vimentin mutant (mCherry-vim(1-138), \(\hbox {VIM}^-\)) that resulted in the depletion of this intermediate filaments network.

For each experimental condition, the trajectory of the center of mass of individual organelles was obtained, and the corresponding MSD was computed. An anomalous diffusion model (Eq. 3) was fitted to the MSD time courses, giving the parameters \(\alpha\) and \(D^*\). In all the evaluated experimental conditions, we identified mitochondria exhibiting super-diffusive (\(\alpha >1\)), diffusive (\(\alpha =1\)), and sub-diffusive (\(\alpha <1\)) behaviors. Super- and sub-diffusive regimes could be explained by the presence of active forces^{23} and confinement of the organelles in the cytoplasm^{25}, respectively. Figure 2b displays three representative examples of these cases (Supplementary Videos S2-S4), while Fig. 2c shows all the experimental MSD trajectories obtained for control condition (CTRL) to illustrate the wide dispersion of the observed different motion regimes.

Figure 2d shows that the distribution of \(\alpha\) obtained for control condition displays a more pronounced symmetry, in contrast to the results obtained for cells treated with latrunculin-B, which displayed a bimodal distribution (Supplementary Section 2). We found that this bi-modality is compatible with an enhancement of the super-diffusive population that could be attributed to an increase in the frequency and/or intensity of active forces that reach mitochondria in the absence of F-actin. On the contrary, disruption of the vimentin network reduced the directionality of mitochondria trajectories, i.e. \(\alpha\) values were lower than those observed in control cells, with a sub-diffusive behavior on average. This result could be interpreted considering that vimentin filaments may act as a physical network/mesh that confines organelles close to microtubules or F-actin. This proximity would increase the time mitochondria remain near those filaments, thus promoting motor binding and enhancing active transport^{38,42,43}.

On the other hand, the generalized diffusion coefficients (\(D^*\)) displayed asymmetric distributions and heavy tails towards high values of \(D^*\) (Fig. 2e). Mitochondria from cells with a partially depolymerized microtubule network presented less variability than those recorded in cells without F-actin, which exhibit greater dispersion. Interestingly, although non-significant differences were observed for the distributions of \(\alpha\) between NOC-treated and control cells (Fig. 2d), mitochondria \(D^*\) values were significantly lower in that condition.

Nonetheless, we wanted to better explore the mechanisms underlying the differences in mitochondria motion, so we further analyzed the data displaying apparent normal diffusion regimes, i.e. \(\alpha \simeq 1\). We compared the apparent diffusion coefficient values attained in the different experimental conditions (Supplementary Fig. S5 and Table S4) and found that microtubules and F-actin have contrary effects on mitochondria mobility, as previously reported^{7}. The lower mobility observed when microtubules are partially depolymerized suggests a decrease in the active forces jittering mitochondria, supporting previous results showing that mechanical impulses are mainly transmitted to these organelles by the microtubule cytoskeleton^{4,6,11}. In contrast, LAT-treated cells displayed larger diffusion coefficient values, which can be associated with an increase of the active forces and a reduction of the effective viscosity, suggesting that F-actin would confine mitochondria and shield them from mechanical stimuli.

### The analysis of the Cumulative Square Displacement allows inferring the active forces temporal pattern acting on mitochondria.

Although the analysis of the MSD gives valuable information on the mitochondria global motion regimes, these results fall short of a complete characterization of the forces acting on the organelles. We propose the study of the Cumulative Square Displacement (CSD) (Eq. 6) as a complementary tool to the MSD.

Figure 3a shows the time course of the square displacements *K* and the CSD of representative mitochondria corresponding to the four explored experimental conditions (see Supplementary Videos S5-S8). Similarly to what was observed in the numerical simulations, the *K* data points fluctuate around an average value, except for a few outliers. These extreme values are spotted as jumps in the CSD curves. Consistently with the results found in the numerical examples, we propose that these shifts are originated by large mechanical impulses, *i.e.* kicks, acting on the organelles.

To explore this possibility, and taking advantage that the used *X. laevis* cell line expresses fluorescent microtubules – EGFP-XTP^{44,45}– we examined the movies of mitochondria and tried to identify moments when an organelle deforms considerably or is transported along these filaments. In most cases, these pushes correlate with jumps in the CSD curves. An example is shown in Fig. 3b and Supplementary Video S4. It is observed that for times *t* < 100 s, the mitochondrion slightly fluctuates, but around times 100 s and 150 s, rapid successive displacements of the organelle’s position occur, which is reflected in a vertical shift in the CSD curve. After that, the mitochondrion returns to a slow-motion regime. From the video, the origin of the driving force is not clear, but it might be caused by molecular motors driving the organelle. However, the push from the previous stationary motion is indisputable, highlighting the feasibility of using these events as force detectors.

To perform a systematic study of the data, an automated procedure was developed to detect outliers from the *K* distribution and group them into single events. The algorithm is based on a threshold criterion and was tested with numerical simulations to tune the threshold value that better reproduced the simulated force pattern (Methods, Supplementary Section 3, and Supplementary Figs. S7-S10). Although one could argue that the choice of this threshold is arbitrary, we will show that the procedure allowed us to compare the results in the treated cells with respect to the control condition., *i.e.* relative measurement.

We applied this routine to compute the number of events in the trajectories of mitochondria in *X. laevis* melanocytes, for each experimental condition. It is important to recall that *K* depends implicitly on \(\Delta \,t\), so both the stationary and outlier values of *K*, change if different time lags are considered. Therefore, to compare the results between different trajectories, it is crucial that they have very similar sampling times.

The average frequency of events was obtained as the quotient between the total number of events and the total tracking duration (Table 1). A bootstrapping procedure was applied to compute the confidence interval (see Methods). Since the number of detected events is sensitive to the data sampling time, as discussed above, Fig. 3c displays the mean frequency of each experimental condition relative to the control one obtained using the same sampling time.

Interestingly, the three cytoskeletal networks have an impact on the number of active impulses suffered by mitochondria. For cells with partially depolymerized microtubules (NOC) or in the absence of vimentin filaments (\(\hbox {VIM}^-\)) the frequency of active events is reduced with respect to control cells. On the other hand, the absence of F-actin (LAT) resulted in an increased number of events per trajectory.

Also, we explored the stationary fluctuations of *K*. We computed the median of *K*(*t*) for each trajectory and analyzed their distribution under different experimental conditions (Supplementary Fig. S11). Since outliers (\(K^*\)) are rare events, the median represents a robust estimate of the stationary value of *K*(*t*). We also calculated the mean of \(K_s=K-K^*\), and the results were consistent with the ones obtained for the median. Notably, we found a shift towards larger values of *K* for latrunculin-B-treated cells and for the vimentin mutant.

Altogether, these results go in the same direction as those obtained with the MSD analysis and support the hypothesis that microtubules are the main source of mechanical stimulation for mitochondria, while F-actin acts as a force absorber/dampener. Additionally, the reduction of active events in the vimentin mutant condition reinforces the notion that intermediate filaments confine mitochondria to remain in close proximity to microtubules, thereby favoring their mechanical contact.