Team:Kent/Modeling

Modeling

Modeling, i.e. the mathematical description of a physical system, is an important aspect of the scientific method as it allows us to quantitatively understand the system and to make predictions. Specifically, a model helps understanding how changes in any parameter affect the system; this is especially important when the relevance of some parameters is unknown. We chose to develop a simulation model of the system studied in our project, which consists of a self-assembling biological structure that we thought would be exciting and informative to quantify and visualize in an interactive way. We made the computer code of the model publicly available, so that other teams can play around and build upon it.

We used the Monte Carlo method to simulate the diffusion ^[3] of monomers inside an E. coli cell, their transit through the cell membrane and the production of amyloid nanowires outside the cell. A typical E coli cell has a length, \(l=2 \mu m \) and a diameter, \(d=1 \mu m \). We take a small observation cube, which is a portion of both the inside of the cell and the bulk outside the cell; we can extrapolate this to describe the whole system. Simulating just a portion has the advantage of requiring less computational power.

The monomers initially start at the bottom of the observation volume and are allowed to stochastically diffuse inside the cell and go through the cell membrane. The binding site seeds are located on top of the cells membrane and when a monomer gets close enough to the binding site it may bind and form a link in the chain. Over time the chains can grow to lengths in the range of 60nm to \(100 \mu m \) ^[1][5][10], these chains are not necessarily straight and persistence length parameter dictates the angles at which particles can bind. The monomer can only interact with the monomer attached to the chain, i.e. there is no branching.

Periodic and reflective boundary conditions are applied in the model. When a particle leaves through the side of the observation volume, we can assume that another particle enters through the other side. When a particle reaches the bottom of the observation volume we can assume that the particle is reflected. This allows us to reproduce behaviour similar to the bulk.

The number of monomers in the system is not necessarily constant; monomers can be created and degraded. However, Hall [8] (2003) argued with the two extreme cases that either; the monomer are generated and degraded on a time scale much slower than amyloid growth so the number of particles are constant; or that the amyloid growth occurs on a time scale much slower than monomer degradation so we can refer to the free concentration of the monomer as constant.

The Monte Carlo simulation was implemented in Matlab and visualized using Visual Molecular Dynamics (VMD).