Motivation

Migrating cells mainly move via pseudopoda, which often take the form of two elongated protrusions which curve in on themselves to encapsulate the medium surrounding the cell, and then proceed to expel the encapsulated medium via osmosis. Another means of cellular movement is through a process called blebbing, where the membrane and cortex detach, creating an outward facing round protrusion at the edge of which a new membrane forms. Understanding how such cellular propulsion occurs has clear applications in medicine, as it enables us to obtain a better understanding of how cells behave. As yet, blebbing is not fully understood and models with ever increasing complexity are being sought to better explain how and why it occurs. In a recent paper, Collier et al [1] proposed a deterministic framework within which blebbing was modelled succesfully .

Approach

In this project, weaimed to develop a stochastic framework with which to model cellular blebbing. In particular, we use pixel intensity to form a probabilistic framework (making use of machine learning) modelling where blebs are likely to form. One succesful and popular machine learning model which gives a probabilistic prediction for unobserved events given a data set of observed events is the Hidden Markov Model (henceforth HMM). It was first introduced in the 1960s with the main aim of catering for such situations. Further details about how this model works can be found in the HMM Section. Using this model, we aim to approximate the probability of a bleb forming at a specific location based on the observed pixel intensities. Before implementing the HMM on the actual data, a prototype data set was developed on which we could test and ensure that the HMM was capturing the desired information. The prototype data allowed for a deeper understanding of what is required for the HMM to operate, as well as illustrating its limitations. A detailed account of the prototype and the conclusions drawn from it can be found in Prototype Model Section. In order to use the actual data in a meaningful way, pre-processing of the raw data (images) was required to ensure that it could be used in the HMM. This was achieved via a process of data cleaning and smoothing, to ensure that the relationship between pixel intensities and bleb site could be explored at each time-step via the HMM. More information regarding the image pre-processing procedure can be found in Image Analysis Section.

Results

Having pre-processed the data as discussed in the previous section, we ran the HMM on the post-processed actual data to obtain some resulting matrices. The first of the two matrices obtained is T, whose entries provide us with the probability of the state of the hidden event changing from i to j. The second matrix E provides us with the probability of the observed event being in state i given that the hidden state is j. In order to reduce the number of parameters needed to be estimated and to obtain more reliable results, we discretised these intensities as follows:

where we take to denote the actual pixel intensity, taking values in . The hidden state describes how likely it is for the bleb to occur, with values ranging from 0 to 3, where higher values denote a higher probability of blebbing.

Below are the resulting matrices obtained after running the HMM on a particular pixel.

The results above pertain to one particular pixel chosen from dataset A, whose hidden state never assumed the value 3 and thus the HMM was run solely with hidden states 0, 1 and 2. The matrices above provide some insight into the underlying behaviour for some states, whereas for others the results are of varying reliability. The second and third rows in matrix T are unlikely to be accurate representations of the real probabilities solely due to the infrequency of events. A similar trend appears in E where the third row reflects the above-mentioned shortcoming. The second row also suffers from the same set back as the third. However, there seems to be a trend which would make sense, namely, the probability seems to concentrate on the lower two-pixel intensities (excluding the 0 intensity state), which is in line with what one might expect. Indeed, if the pixel intensity is lower, then the membrane is somewhat ''weaker" at this point and thus can be broken down more easily, therefore making it more prone to undergo blebbing. Furthermore, one would expect that given no blebbing event occurred, the pixel intensity would most likely either be 0 as the pixel does not reside within the cell, or else have a higher pixel intensity indicating that the membrane is ''stronger", and thus less susceptible to blebbing. Naturally, such conclusions would be somewhat more clear-cut if the HMM were to be ran on more data.

Acknowledgments

We would like to thank our project supervisors Dr. Bjorn Stinner and Dr. Stefan Adams for their help and guidance throughout the project. We are also grateful to Prof. Till Bretschneider and Dr. Sharon Collier for their time and helpful insights. The project was funded by EPSRC and the University of Warwick.

Links:

Homepage

HMM

Prototype Model

Image Analysis

Code

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H. Bharth
J. Kiedrowski
J. Sant
J. Thomas

Supervised by:
Dr Björn Stinner
Dr Stefan Adams