iGEM Oxford 2016 - Cure for Copper

MODELLING

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.

Copper chelation within E Coli

We first modelled how copper within the bacterial cell will be chelated, assuming that there wouldn’t be any further intake. The following is the simulated results for each of the four promoters.

Total copper chelation

Then, we simulated how external copper concentration will change with copper chelation within E Coli. Initial external copper concentration (simulated in the gut) was assumed to be around 10uM – we didn't assume any further intake of copper.

Discussion

We found that the maximum transcription rate of mRNA is the limiting factor to the production of chelators, and although the chelators are effective in themselves, as chelators are not reusable in our system, reduction of copper is still proportional to the number of chelators produced.

The feedback system wasn’t as effective as we have expected, mainly because the feedback system comes into effect only when copper concentration is low.

As there are three layers of reactions involved in system 3 and 4, the response was not as swift as the other two. However, this is mainly due to the fact that we didn’t know the exact steady state amount of CueR and CueR*.

Model B. Promoter Characterisation

Although these models where useful for qualitatively describing the variations in species concentration, they were unable to provide accurate predictions for expression levels since many of the model parameters were unknown (or so imprecise, they were of little use).

Hence the next stage of our modelling attempted to take the data from plate reader experiments and fit our promoter models to it. Fitting each individual parameter would have been extremely difficult, since our experiments only provided data on the system as a whole rather than individual reactions like transcription and translation. Thus we reduced each system to a single equation which describes the rate of protein production under each promoter:

The first term represents leakage expression due to random fluctuations in the binding and unbinding of repressor molecules to the promoter. The second term accounts for the promoter’s sensitivity to copper. K, the half saturating constant, gives the copper concentration at which the promoter allows expression at half the maximum rate. The Hill coefficient, n, provides a measure of how quickly the expression rate varies with increasing copper concentration. The final term represents the combined effects of degradation and dilution of the protein concentration.

For a given copper concentration (Cui) this simplifies to:

This has an exact solution:

Hence we can determine a set of K1 and K2 for each copper concentration by using a least squares optimisation to fit this curve to the data. Since K2=delta regardless of the copper concentration, the values of K2 should be relatively consistent. To determine the other parameters we can use another least squares optimisation to fit the following equation to the values of K1:

Below is a table of our fitted parameters for each promoter:

Model C. Copper Advection

We wanted to gain an understanding of how effective our bacteria would be in the gut. To do so we first considered how many copper ions each bacterium could absorb.

Assuming each bacterium can produce 1000 Csp1 molecules throughout its lifetime, this means that each bacterium will can absorb 52,000 copper ions. We can determine how many moles of copper this is:

If one litre of intestinal fluid contains ~108 bacteria, this means that, on average, every twenty minutes, the concentration of copper will drop by:

And hence the rate of change in concentration throughout the gut is:

In order to use this figure to make predictions about the efficiency of our treatment, we next developed a partial differential model (PDE) of the gut. This involved treating the gut as an axisymmetric cylindrical pipe along which intestinal fluid flows at a constant average speed (this ignores the effects of peristalsis).

We arrived at the following PDE which describes how the concentration of copper would evolve with time and with distance along a streamline through the gut.

The time derivative describes how the concentration at each point in the gut varies with time. On the right hand side, the first term describes the change in copper concentration due to flow down the gut. The final term represents the combined absorbing effects of our bacteria and the body. K_body is the rate of absorption of copper by the gut naturally such that 40% of the copper ingested is absorbed.

The stomach empties into the gut exponentially with a half-time of 15 minutes. Hence the boundary condition for the above equation is: