Quantitative analysis of peroxisome tracks using a Hidden Markov Model

Abstract

Diffusion and mobility are essential for cellular functions, as molecules are usually distributed throughout the cell and have to meet to interact and perform their function. This also involves the cytosolic migration of cellular organelles. However, observing such diffusion and interaction dynamics is challenging due to the high spatial and temporal resolution required and the accurate analysis of the diffusional tracks. The latter is especially important when identifying anomalous diffusion events, such as directed motions, which are often rare. Here, we investigate the migration modes of peroxisome organelles in the cytosol of living cells. Peroxisomes predominantly migrate randomly, but occasionally they bind to the cell's microtubular network and perform directed migration, which is difficult to quantify, and so far, accurate analysis of switching between these migration modes is missing. We set out to solve this limitation by experiments and analysis with high statistical accuracy. Specifically, we collect temporal diffusion tracks of thousands of individual peroxisomes in the HEK 293 cell line using two-dimensional spinning disc fluorescence microscopy at a high acquisition rate of 10 frames/s. We use a Hidden Markov Model with two hidden states to (1) automatically identify directed migration segments of the tracks and (2) quantify the migration properties for comparison between states and between different experimental conditions. Comparing different cellular conditions, we show that the knockout of the peroxisomal membrane protein PEX14 leads to a decrease in the directed movement due to a lowered binding probability to the microtubule. However, it does not eradicate binding, highlighting further microtubule-binding mechanisms of peroxisomes than via PEX14. In contrast, structural changes of the microtubular network explain perceived eradication of directed movement by disassembly of microtubules by Nocodazole-treatment.

Figure 1

Overview of the HMM formulation. (a) Piece of a generic track that visualizes how a series of coordinates convert to the observables instantaneous speed ${\left|\left|\mathit{v}\right|\right|}_t$ and turning angle ${\alpha }_t$. (b) Graphical representation of the HMM with states $s_1$ (random migration mode) and $s_2$ (directed migration), transition probabilities $T_{1,2}$ and $T_{2,1}$ between the two states, and conversion into the parameters ${\iota }_t$ and ${\alpha }_t$. (c) Representative probability density functions (pdfs) $p\left({\iota }_t,|,s_i\right)$ for the dimensionless instantaneous speed ${\iota }_t$ distributions of peroxisomes in the respective states $s_1$ and $s_2$ (Eq. 2) calculated for parameters ${\mu }_{\iota ,1}=-0.93$, ${\mu }_{\iota ,2}=0.06$, ${\sigma }_{\iota ,1}=0.72$ and ${\sigma }_{\iota ,1}=0.50$. (d) The pdfs $p\left(\alpha |,s_i\right)$ for the turning angle and ${\alpha }_t$ distributions in the respective states $s_1$ and $s_2$ calculated from Eqs. 3 and 4 with ${\sigma }_{\alpha ,2}={26}^{\circ }$.

Figure 2

MSD analysis of peroxisome tracks across conditions. (a) MSD curves for the three conditions display similar dynamics across conditions. Shaded regions indicate standard deviation across cells in each condition. (b) Fitted values ${\alpha }_{\mathit{MSD}}$ of the exponent in $MSD\sim t^{{\alpha }_{\mathit{MSD}}}$ as quantification of potential anomalous diffusion dynamics, revealing similar values across conditions.

Figure 3

HMM validation on synthetic data. (a) Comparison of ground truth states and predicted states after fitting the parameters with Baum–Welch and determining the state of each time step using the Viterbi algorithm. (b) Performance measure of accurate state assignment using the Viterbi algorithm for each time step. (c) The effect on the F1 measure when disturbing the fitted parameter ${\mathit{\mu }}_{\iota }$ by adding normal distributed noise with zero mean and standard deviation $\sigma$ before predicting the states using the Viterbi algorithm.

Figure 4

Fitting of HMM parameters. Values of ${\mu }_{\iota }$ (a) and ${\sigma }_{\iota }$ (b) after fitting the parameters to 52 manually selected norm tracks with mixed migration modes. (c) The ratio of steps in states $s_1$ and $s_2$ expressed by the values of ${\pi }_i$ from the HMM after applying it to all tracks. (d) The values of the transition matrix after fitting it to all tracks. (e, f) The ratio of tracks showing consecutive directed steps as a function of the minimum number of consecutive steps required. In (f) we have a zoomed-in version of the curves in (e) to highlight the abolishment of tracks with more than six consecutive directed steps in the noc condition. Shaded regions indicate 95% confidence intervals obtained by bootstrapping. Stars represent the significance level after Bonferroni corrected p value: n.s.: q > 0.05, ***: q < 0.001. For panels (c) and (d), significance is only indicated in one of the states. The same significance pattern naturally holds for the other state since all ${\pi }_i$ and the rows in $\mathit{T}$ add up to one.