Bivariate change point detection in direction and speed of cell organelle movement
Authors:   , Plomer Solveig 
Published: January 2025

In this dissertation a new model for the description of cell organelle movement is defined and a test and change point detection algorithm for changes in the model parameters is proposed.
The description of movement patterns can be important for the understanding of various biological processes on multiple scales. One of the primary goals is to understand the causes of change between movement patterns. On the micro scale, movement patterns of cell organelles and swimming micro-organisms such as cells are investigated. To learn more about the different movement types of cell organelles, as well as the changes between these types, we analyse the movement of two specific types of cell organelles in the root of the plant Arabidopsis thaliana, Plastids and Peroxisomes. Interestingly, while all tracks were recorded in three dimensions, over 90% of the tracks show more than 95% of their movement variability in only two dimensions. We therefore focus on the two-dimensional projection of the movements, allowing comparability to approaches for animal movement pattern analysis. In this data set of organelle movement we observe visually prominent, linear movement structures with seemingly piecewise constant movement direction and speed. Since the different sections of the movement could be associated with different movement mechanisms such as movement through cytoplasmic streaming or transport along intracellular filaments, it is of interest to detect the change points (CPs) in the direction and speed of the movement.
The most widely used time discrete models are variants of the random walk. Most often used, particularly in animal studies, are so-called correlated random walks, which are parametrised by a random turning angle relative to the previous direction. Correlated random walks are useful to model movement where the difference between highly directed and undirected movement is of interest. By fitting HMM models to the observed organelle movement we show that a biased random walk (BRW), parametrised via absolute directions, may be more appropriate to model sections of different but constant movement direction. The BRW is therefore used as a reference model. However, our findings indicate that a BRW shows a higher variability in the movement direction and may thus move less strictly along a linear structure than can be observed in many organelle tracks.
Therefore, we define a new model termed linear walk (LW) with less variability around an expected position. Both models, the BRW and LW, have the same process of expected positions, which is parametrised by two parameters, the movement direction and the step length, where the direction is the angle relative to the x-axis. The models differ in regard to the variability around the expected position. In the BRW, independent random increments are summed up, while in the LW an independent, random error is added to each expected position. Due to this definition the changes in movement direction and step length can be described independently. The expectation of the increments in both models can be parameterised by the movement direction and step length or alternatively by a two dimensional expectation. The maximum likelihood estimators of the model parameters in both models and proof of their strong consistency is provided. Note that for the expectation of the increments in the BRW the estimator is the classical mean, while in the LW the estimator is a weighted average.
In the context of the BRW model a known MOSUM approach for the bivariate detection of change points is easily adapted to our setting. In this approach a double window is shifted over the movement track. The difference of the bivariate expectation of increments in the left and right window half is estimated, leading to a process of differences that fluctuates around the origin, but shows systematic deviation in the neighborhood around a CP. Therefore the maximum deviation of the process of differences from the origin is used as a test statistic. The rejection threshold is obtained via simulation of a limit process. CPs are estimated successively by identifying local deviations of the process of differences from the origin.
In the LW model the MOSUM has inconvenient properties due to the dependency structure of the increments. Therefore a moving kernel approach is proposed where the maximum likelihood estimators for the expectation in the LW model are used instead of the classical mean. For this approach the proof of the weak convergence of the process of differences assuming the true variance is provided. For both approaches simulation studies concerning the test power and precision are provided.
Since the CPs in both approaches are detected within the expectation, we propose a graphical technique to classify the detected change points into CPs in movement direction and step length and apply this technique to the observed organelle movement.