Team:Oxford/InterLab
Interlab Study
The field of synthetic biology relies heavily on mathematical models that help simulate and predict the behaviour of biological systems. Our team developed a mathematical framework to simulate the effectiveness of designs and analyse the performance of our bacteria.
We developed our models for the following four main purposes:
1.Simulating reactions in the bacterial cell to predict its copper chelation efficiency.
2.Characterisation of promoters by parameter fittings to the standard Hill function.
3.Simulating the advection of copper along the gut
4.Simulating the spread and settlement of bacteria in the gut
In the first two models, we applied mathematical methods of modelling chemical reaction, transcription of repressed promoters and chelation, using MATLAB to solve the ordinary differential equations (ODEs) and fitting parameters to the standard Hill function. In the latter two models, we applied fluid dynamics to model advection and diffusion.
Overview of our systems
Our goal is to create a chelator generating system that complements copper homeostasis. In order to make the system copper responsive, we developed four variations of copper binding systems, which shares the back bone circuit described below.
We developed four variations of promoters that can be incorporated into our system. Modelling was used in order to simulate and later to characterise the behaviour of these promoters to demonstrate which of the promoters are suitable for the use of the project.
The four promoters are described below:
1. pCopA
2. pCopA with Feedback
3. pCusC
4. pCusC with Feedback
For each of the promoters, we developed a kinetic model to simulate / analyse its behaviour.
For more information about the parts and sequences, please visit our Parts page.
Model A. Reaction Kinetics
To predict copper chelation efficiency of our bacteria, we developed kinetic models to simulate reactions in the bacterial cell for each of our four promoters.
Method
In order to simulate the transition of different quantities, eventually reaching equilibrium, we used ordinary differential equations (ODEs) that can be solved by MATLAB.
Chemical reactions such as
can be modelled in a set differential equations
However, this is true only under an assumption that the chemical bindings are uncooperative - independent to each other.
Our chelators, Csp1 and MymT has 52 and 8 copper binding sites respectively. These bindings are cooperative, and a close approximation is formalised in the Hill function shown where Y denotes the fractional saturation of total copper binding sites.
The constant K is the half-saturating concentration of ligand, and so can be interpreted as an averaged dissociation constant. For Csp1, ¬ K¬_Cu = 1.3 * 10^-17 M, n = 2.4. [1]
From this, we can estimate the amount of copper that is bound to the chelator for different copper concentration.
[1] Nicolas Vita et al., 2015. A four-helix bundle stores copper for methane.
For modelling gene expression regulated by transcription-factor binding, we calculated the fraction of promoters in each possible state. To simplify the simulation when we don’t know the exact curve of copper concentration - transcription rate, we made the following assumption. (Later, we improved our model by developing model B)
For instance, for system 1 and 2, repressor CueR, copper and promoter binds according to the following:
Here, the promoter can either be in state P, P.CueR or P.CueR.Cu, where state P.CueR.Cu is the active state. Therefore, once we calculate the fraction of promoters that are in state P.CueR.Cu, we can figure out the rate of transcription.
Fraction in state P.CueR.Cu:
Therefore, we can model mRNA transcription as:
where the second term is for degradation and dilution.
