Investigating Vesicle Dynamics and the Effect of Cytoskeletal Fiber Attachments Using Computer-based Simulations

Abstract

Vesicle transport is essential for eukaryotic cells, for example in plant cells, growth and cell division are controlled by the transport and fusion of vesicles containing cell wall materials. Hence, understanding vesicle dynamics is vital to advance our knowledge of how plant cells grow. The goal of this project is to test the hypothesis that attaching an actin filament will alter the diffusive properties of a vesicle. We will also test the effect of increasing the length or number of attached actin filaments. To test this hypothesis, we decided on using the software Cytosim because it is designed to simulate systems of fibers with associated vesicles. For this project, we have developed a custom R script. This script supervises Cytosim simulations, parses the output of the simulations, runs calculations, and plots the results. We are validating the correctness of the diffusion engine by using Mean Square Displacement analysis of a single vesicle. To do this, we analyze the diffusion of the vesicle through different time steps averaged over many simulations. We then compare our obtained diffusion coefficient against a theoretically derived diffusion coefficient of a vesicle in identical conditions. Afterward, we will attach an actin filament, and evaluate how the diffusion coefficient is affected by filament length; this will allow us to determine the thresholds for when the length of the filament starts affecting diffusive properties and when it begins to have diminishing effects.

Introduction

Vesicle transport is essential for eukaryotic cells. For example in plant cells, growth and cell division are controlled by the transport and fusion of vesicles containing cell wall materials. Hence, understanding vesicle dynamics is vital to advance our knowledge of how plant cells grow. The goal of the project is to further our understanding of vesicle dynamics by simulating vesicles and filaments in different configurations. We can evaluate these configurations through the use of the software Cytosim, which is designed to simulate systems of fibers with associated vesicles. One of the first things we must do to be sure that we can rely on the findings of our simulations is to verify the diffusion engine present in Cytosim. Afterwards we can work on simulating the effect that a single filament has on a vesicle's diffusion coefficient.

Methods

To verify that the diffusion engine present in Cytosim is working properly, we have to see that its behavior matches what should be expected. This means that we have to compare the simulation-derived Diffusion Coefficient to the theoretically-derived Diffusion Coefficient.


We can use the below form of the Einstein-Stokes equation for estimating the Diffusion Coefficient of a spherical particle in a fluid. For all simulations and calculations in this project we chose a kT of 0.0042 and a η of 0.013.


To find the simulated diffusion coefficient for a vesicle, we ran fifty (50) simulations for each configuration and each simulation for one-hundred (100) seconds. We then used a recursive method to find the displacement of the vesicle at all possible time-steps (Tau). This means that for a simulation of one-hundred (100) seconds, we have 5050 datapoints, and for each vesicle size (a specific configuration), we have a total of 252,500 datapoints.


Due to the nature of our data collection, the number of datapoints for a given Tau decreases as our Tau increases. Therefore for the purposes of calculating our Mean Squared Displacement (MSD), we restrict our dataset to displacements with a Tau of 20 or under. We then calculate and plot the mean squared displacement, and find a fit for our dataset.


The diffusion coefficient for our vesicle in a two-dimensional space is a quarter (¼) of the value of the slope of our fit. We developed an R script to perform a parameter scan and generate the Cytosim configuration files for each configuration, run the simulations, parse the outputs of the simulation report, calculate the associated values, and create a plot (Fig. 1). All simulations were performed on consumer-level personal computers running Linux.

Einstein-Stokes Equatiopn

D = Diffusion Coefficient

Boltzmann Constant:

T = Temperature

η = Viscosity

R = Radius of Sphere


Fig 1. The resulting boxplot of mean square displacement values over 50 simulations.
Fig 1. The resulting boxplot of mean square displacement values over 50 simulations.

Results

Fig 2. Grid of Squared Displacement at Tau 1-20, with Varying Vesicle Radii and Cell Sizes.​

To determine which cell size to use, we tested vesicles of varying sizes with cells of varying sizes. This is a grid of plots, showing the mean square displacement at Tau values of 1 to 20. The red line is the linear regression model for the dataset from the configuration (Cell Size and Vesicle).​

    We found that we require a relatively large cell relative to the size of the vesicle in order to retrieve reliable data from a simulation. This is because if the vesicle is too small, it will have a higher diffusion coefficient, which can result in the vesicle reaching the cell boundaries, which will restrict its movement.
Fig 2. Grid of Squared Displacement at Tau 1-20, with Varying Vesicle Radii and Cell Sizes.

To determine which cell size to use, we tested vesicles of varying sizes with cells of varying sizes. This is a grid of plots, showing the mean square displacement at Tau values of 1 to 20. The red line is the linear regression model for the dataset from the configuration (Cell Size and Vesicle).


We found that we require a relatively large cell relative to the size of the vesicle in order to retrieve reliable data from a simulation. This is because if the vesicle is too small, it will have a higher diffusion coefficient, which can result in the vesicle reaching the cell boundaries, which will restrict its movement.

Fig 3. Diffusion Coefficient of Vesicles with Varying Radii. We then simulated vesicles of varying sizes in Cytosim with a cell size of 40μm. For each size, we plotted each simulation-derived diffusion coefficient alongside its theoretically-derived diffusion coefficient. The results instills confidence in the diffusion engine present in Cytosim.
Fig 3. Diffusion Coefficient of Vesicles with Varying Radii.
We then simulated vesicles of varying sizes in Cytosim with a cell size of 40μm. For each size, we plotted each simulation-derived diffusion coefficient alongside its theoretically-derived diffusion coefficient. The results instills confidence in the diffusion engine present in Cytosim.
Fig 4. Diffusion Coefficient of Filaments at Varying Lengths on a Logarithmic Scale. After our previous test the next step was to find the diffusive properties of a filament. We used the same methods as for the vesicle but we looked at the displacement of the center of the filament. This resulting plot has both axis on a logarithmic scale. The theory for the diffusion coefficient of a filament is complex and is beyond the scope of this work.
Fig 4. Diffusion Coefficient of Filaments at Varying Lengths on a Logarithmic Scale.
After our previous test the next step was to find the diffusive properties of a filament. We used the same methods as for the vesicle but we looked at the displacement of the center of the filament. This resulting plot has both axis on a logarithmic scale. The theory for the diffusion coefficient of a filament is complex and is beyond the scope of this work.
Fig 5. Logarithmic Diffusion Coefficient vs. Logarithmic Filament Length.​

The final step was to add a filament of varying lengths to a 50nm in radius vesicle, and plot the resulting diffusion coefficients. In addition we plotted the diffusion coefficient of the vesicle without any filaments, and we also plotted the diffusion coefficient of just a filament.
Fig 5. Logarithmic Diffusion Coefficient vs. Logarithmic Filament Length.​
The final step was to add a filament of varying lengths to a 50nm in radius vesicle, and plot the resulting diffusion coefficients. In addition we plotted the diffusion coefficient of the vesicle without any filaments, and we also plotted the diffusion coefficient of just a filament.

Conclusions

Future work

Future work will involve not only multiple filaments on a vesicle, but also testing clusters of vesicles with multiple fibers. In addition we will also begin running more simulations per configuration to add more confidence to our results. This will greatly increase the time complexity for the simulations, which will require us to develop a multi-threaded multi-computer infrastructure for running and supervising multiple simulations at a time. In addition, we may explore the ongoing work of modeling the diffusive properties of a filament and compare it to simulations in Cytosim.