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And how can it be used?

Mathematical modelling provides a powerful tool for scientists of all disciplines, allowing inspection and manipulation of a system in ways which are unachievable in the lab. In the context of biology, we can use mathematical models to study the behaviour of a single cell or an entire ecosystem. In fact, inspecting a mathematical model is very much like a laboratory experiment – the main difference being that in modelling, the environment is artificial.

Computational Methods

When coupled with computational power, we can use mathematical modelling to produce intricate simulations of a very fine scale. This is exactly what the iGEM team has done this year, in producing a comprehensive model of the plaques and microglia in the brain. With many millions of cells present, mathematical tricks and techniques have been employed in order to allow computation of this highly complex system. For the end-user, this scaffolding is hidden from view: the final result will be a user-friendly windows application.

An early iteration of the application is shown in the image below. This early iteration indicates what the final result will be – a large number of numerical inputs are processed in a way representing the biological mechanics of the system, generating a set of numerical outputs. For example, these outputs could be the coordinates of a single microglia cell with corresponding time values, providing insight into how these cells move. Since almost every numerical factor can be altered by the user, the model can also be used to predict how a particular change would affect the system at large.

Additionally, the final model will include graphical capabilities, allowing visualisation of the system at an increased time rate. Aside from being very interesting, this visualisation will provide qualitative insight in a way which pure numerical data cannot.

Method Overview

Ideally, the brain would be modelled taking every relevant cell and process into consideration. However, this would require an unrealistically large level of computing power, and so approximations must be made.

The brain is modelled as a 3D homogenous static medium, throughout which small ‘plaques’ are fixed. As far as the model is concerned, these plaques cause Alzheimer’s, and their removal would result in the reversal of the disease’s effects. At the start of the model (t=0), a number of microglia cells will be introduced to the system at a number of distinct points. These will then move throughout the brain, will the overall goal of binding to and removing the plaques.

Initially, we planned to model the entire system numerically, with the position of each plaque and microglia cell modelled individually. However, it quickly became apparent that due to the vast number of cells present, this approach would not be viable. We wanted to retain a microscopic element in the model, while also taking the entire brain into consideration. With this in mind, a two-level system was proposed:

1. A microscopic simulation of a single section of the brain. Every plaque and microglia cell will be modelled individually, with the state of the system regularly sampled (roughly every minute). The system will be sufficiently small to allow this to take place, whilst being large enough for its results to have significance.

2. A macroscopic simulation of the entire brain. The brain will be divided into a large number of ‘compartments’, which will each be considered separately. Each plaque and microglia cell will no longer be modelled individually; instead, in each time interval one compartment will pass some of its properties (number of microglia, concentration of protease, etc.) to the adjacent compartments, and vice versa. This property exchange will be quantified by results from the level 1 model.

Development Process

Producing a complex piece of software is a difficult task which requires precise organisation and a clear development process. Here is the step-by-step process generally followed while developing the model:

1. Establish theory. Before any modelling can take place, it is essential that the background material has been properly researched and understood. Here, relevant equations are found and most of the pen-to-paper mathematics takes place.

2. Model using spreadsheets. MS Excel is a powerful tool during the early stages of developing a mathematical model. Every variable’s value can be monitored using a large table of data, VBA provides programming integration, and data can easily be plotted.

3. Translate into C# (‘see-sharp’). This programming language is extremely powerful, and offers many benefits over a spreadsheet. However, inspecting variables becomes more difficult, and errors are more easily hidden within a mass of computer code. This is why it is important to establish the modelling techniques first.

4. Add a user interface. This will be constructed in MS Visual Studio, using the C# code written in step three.