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 system
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:
- pCopA sfGFP (PCG)
- pCopA CueR sfGFP (FCG)
- pCusC RFP (pCusC)
- pCusC CusR RFP (FCK)
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.
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 P1 is the half-saturating concentration of ligand, and so can be interpreted as an averaged dissociation constant. For Csp1 the fractional occupation is given by the following equation (see the table below for parameter values):
From this, we can estimate the amount of copper that is bound to the chelator for different copper concentration.
In order to develop the system of equations required to model each system, we had to work out each reaction which was occurring and determine its reaction rate constants
The following reactions involving mRNA occurred in all our systems:
The following reactions were involved in the pCopA system:
The following reactions were involved in the pCusC system:
Finally, these are reactions in which Csp1 is involved:
This table give the various parameter values we found in literature:
Here are the systems of equations we used to model each system
pCopA sfGFP (PCG):
pCopA CueR sfGFP (FCG):
pCusC RFP (pCusC):
pCusC CusR RFP (FCK):
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.
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 pCusC and FCK, 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:
The full graph from the plate reader experiment can be found here.
Although the parameter fitting should have yielded a general insight into the order of magnitude of the constants for K and n (alpha, beta and gamma are in arbitrary units), it turned out less helpful than it was hoped to be as, the fluorescence per cell data over time from the plate reader experiment decreased rather than increased, as can be seen from the graphs in PDF. The data fit only worked for pCusC, where it showed a satisfying increase in the exponential growth phase of E Coli.
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 ~10e11 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. Kbody 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:
Assuming the flow is laminar, the concentration will have a parabolic profile in the radial direction. Hence:
The solution to the equation (found by Laplace transforms) is:
The total number of moles of copper in the gut at any time is given by:
The second part is true as we are assuming the gut is axisymmetric. Hence:
The table below contains the values of all the various parameters:
Below is an animation showing the results of simulating the above equations. The simulation time was equal to the time a meal would spend in the gut (~11hours).
Anticlockwise, starting in at the top left: copper concentration profile as it moves along the gut; heat map showing copper distribution as it moves along the gut; bar chart showing the proportion of copper absorbed by the body and our bacteria; variation in gut copper content with time.
Model D. Beads
Beads are the mechanism that we will be using to supply the bacteria to the patient. The beads consist of a chitosan coating containing the bacteria to be administered to the patient. This chitosan coating is gradually degraded by the alkaline fluid it is submerged in. The time taken for the drug to be released once inside the patient is determined by how long it takes for this coating to be degraded.
This can be modelled as a diffusion process, the release of the bacteria into the gut occurs at an increasing rate with respect to time. The rate of diffusion depends on Km, the mass transfer coefficient.
We must assume that the thickness of the bead coating is such that none of the bacteria is released before reaching the intestine. The wet lab experimentation has demonstrated that the beads will survive the conditions of the stomach with no leakage from the bead so we can consider this assumption to be true. We also must assume that the beads are equally-sized and perfectly spherical and also that the intestine can be treated as a straight tube with an average flow speed of 1cm/s.(12)
The equation for diffusive mass transfer can be derived from Newton’s law of cooling and is:
Where Ab = total surface area of beads, Km = mass transfer coefficient, Cb = concentration of bacteria in bead, Cg = concentration of bacteria in gut
By conservation of mass we can equate the mass increase within the gut to the mass decrease within the beads:
Where Vg and Vb denote the volume of the intestine and the beads respectively and the dot notation on the concentration terms denotes 'rate of change'. Vg is approximately 3e-3 
We need to use the two relationships above to find a relationship between concentration of bacteria in the intestine, Cg and time, t.
Where Cb0 = initial concentration of bacteria in the bead.
Rearranging and integrating:
Where c is the constant of integration.
Using the initial conditions that at t=0, the initial concentration in the bead is zero, Cg=0:
The constant Km can be determined by the data obtained in the wet-lab. An experiment to measure the concentration at discrete time intervals was conducted to demonstrate the absorption of dye out of the beads. I fitted the results against an exponential curve in order to find the parameter Km (mass transfer coefficient). This gave an approximate value of the curve parameters in order to find the mass flow rate by rearranging eq.(1).
Now that we have obtained all necessary values we can see how long it will take for the bacteria in the beads to be released.
From the graphs we can see that it takes approximately 500 minutes (around 8.5 hours) for over 90% of the drug to fully diffuse into the gut. The average velocity of the fluid in the intestine is approximately 1cm/min, the distance that the bead will have travelled in this time is around 5 metres which is less than the distance of the intestine. The curves have a very similar nature, demonstrating that the time take for absorption is roughly the same regardless of the size of the dose. The conclusion of this model is that the beads designed in the lab will work.
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