- Simulating the cell internal structure using Delaunay triangulation (2010)
- The goal of this project is to develop a framework for a cell that takes in consideration its internal structure, using an agent-based approach. In this framework, a cell was simulated as many sub-particles interacting to each other. This sub-particles can, in principle, represent any internal structure from the cell (organelles, etc). In the model discussed here, two types of sub-particles were used: membrane sub-particles and cytosolic elements. A kinetic and dynamic Delaunay triangulation was used in order to define the neighborhood relations between the sub-particles. However, it was soon noted that the relations defined by the Delaunay triangulation were not suitable to define the interactions between membrane sub-particles. The cell membrane is a lipid bilayer, and does not present any long range interactions between their sub-particles. This means that the membrane particles should not be able to interact in a long range. Instead, their interactions should be confined to the two-dimensional surface supposedly formed by the membrane. A method to select, from the original three-dimensional triangulations, connections restricted to the two-dimensional surface formed by the cell membrane was then developed. The algorithm uses as starting point the three-dimensional Delaunay triangulation involving both internal and membrane sub-particles. From this triangulation, only the subset of connections between membrane sub-particles was considered. Since the cell is full of internal particles, the collection of the membrane particles' connections will resemble the surface to be obtained, even though it will still have many connections that do not belong to the restricted triangulation on the surface. This "thick surface" was called a quasi-surface. The following step was to refine the quasi-surface, cutting out some of the connections so that the ones left made a proper surface triangulation with the membrane points. For that, the quasi-surface was separated in clusters. Clusters are defined as areas on the quasi-surface that are not yet properly triangulated on a two-dimensional surface. Each of the clusters was then re-triangulated independently, using re-triangulation methods also developed during this work. The interactions between cytosolic elements was given by a Lennard-Jones potential, as well as the interactions between cytosolic elements and membrane particles. Between only membrane particles, the interactions were given by an elastic interaction. For each particle, the equation of motion was written. The algorithm chosen to solve the equations of motion was the Verlet algorithm. Since the cytosol can be approximated as a gel, it is reasonable to suppose that the sub-cellular particles are moving in an overdamped environment. Therefore, an overdamped approximation was used for all interactions. Additionally, an adaptive algorithm was used in order to define the size of the time step used in each interaction. After the method to re-triangulate the membrane points was implemented, the time needed to re-triangulate a single cluster was studied, followed by an analysis on how the time needed to re-triangulate each point in a cluster varied with the cluster size. The frequency of appearance for each cluster size was also compared, as this information is necessary to guarantee that the total time needed by to re-triangulate a cell is convergent. At last, the total time spent re-triangulating a surface was plotted, as well as a scaling for the total re-triangulation time with the variation. Even though there is still a lot to be done, the work presented here is an important step on the way to the main goal of this project: to create an agent-based framework that not only allows the simulation of any sub-cellular structure of interest but also provides meaningful interaction relations to particles belonging to the cell membrane.