Why do Mathematical Modelling?

Synthetic Biology is a discipline with an inherently large amount of unpredictability. There are many difficult to account for factors within the cell which can cause BioBrick parts to function in unexpected ways. In order to combat this uncertainty mathematical modelling techniques are often applied to make predictions about how BioBrick parts will operate to inform their design and characterisation before parts are obtained by DNA synthesis or other methods.

Modelling Methodology

A wide variety of modelling methodologies exist with adaptations to a myriad of different tasks. Here we would like to focus on the particular mechanistic modelling methodology that we applied in our DNA Degradation Switch, Semantic Containment Failure and Collaboration models. In each of these models we began with a collection of mass action equations which were used to derive a system of ordinary differential equations (ODE). By solving the resulting system using SciPy's ODE solver it is possible to keep track of the changing levels of different reactants involved over the model runtime.

Mass Action Equations

The law of Mass Action states that the rate of a reaction is proportional to the activities or concentrations of reactants and is used to describe situations of dynamic equilibrium which commonly arise in biology. Mass action equations describe the details of physical phenomena governed by this law. Mass action equations provide the starting point for our mechanistic models by describing the reactions in question and allowing the derivation of ODEs which are used to calculate the exact quantities of reactants over time. An example representation of mass action equations for protein expression are given in the example below.

Ordinary Differential Equations

By solving systems of ordinary differential equations (ODEs) it is possible to model chemical reactions by monitoring the quantities of reactants over time however it is not always easy for those unfamiliar with such techniques to understand the information conveyed by an ODE. The result of an ODE is a rate of change hence to formulate such an equation it is neccesary to know what actions may add or take away from a quantity at a particular time point. A simple example may involve keeping track of speed; the rate of change of which is acceleration. In this example suppose at any time point the pressure applied to the break applies some loss in speed (deceleration) and the pressure applied to the gas pedal applies some gain in speed (acceleration) then this would give rise to the following equation for acceleration:

Example

Now lets examine a more familiar biological scenario. Suppose we have some mass action equations describing protein expression within the cell:

In this scenario the transcription, translation and degradation rates correspond to activities of DNA and mRNA respectively. Now lets suppose we would like to derive an ordinary differential equation that will allow us to examine the rate of change of mRNA and by solving the mRNA levels over time. We can observe from the mass action equations that mRNA is added to by transcription based on the mass of DNA and transcription rate and that mRNA is removed by its degradation rate based on the mass of mRNA and hence we can derive the following ODE:

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