Quantitative Modelling

Enzyme kinetics for potential reporter proteins

In the beginning of the project we were looking for a reporter system. The wet lab team has made a list of possible reporters that can be used in vitro. We wanted to ensure that the reporter system gave us a rapid output, was stable and did not require cellular machinery to work. We wanted to test this in silico before we tested it in the wetlab. We wanted to ensure there was concrete evidence to support our choice of one reporter enzyme over another. Given that it is much easier to do simulations using models than to do experiments in the lab, we simulated horseradish peroxidase (HRP) and beta-galactosidase which are one of the most rapid reporters known in the literature. We programmed differential equations modelling enzyme kinetics in Mathematica to produce a interactive plot of output over time, with parameters such as initial enzyme concentration and initial substrate concentration being the parameters that can be adjusted.

Two screen shots of the interactive models are attached here (see Figures 1 and 2). By changing the sliders, the corresponding parameters will change, whose response is plotted in blue. By comparing to the red curve, which is plotted with those adjustable parameters held at fixed values, we can immediately see the effect of those parameters, such that a range of combination of the parameters can be considered for the evaluation of the reporters. To see the functional interactive models, click here.

After careful examination of the models under various parameters, we decided to use HRP because it is fast and give accurate response, evident from the kinetic models. However it turns out that we couldn't produce HRP in E. coli. After some more literature search we found a substitue of horseradish peroxidase, ferritin, which is known to has fast kinetics and accurate response just like horseradish peroxidase.

Agent Based Modelling

For any detection system, the sensitivity and specificity are of great importance. Ideally, one should determine these parameters from repeated experiments, but to shorten the development time and to provide some insight on directions of further improvements, we developed an agent based model to describe the behaviour of the system. We have been able to challenge this model with varying the amount of target DNA that the TALEs in our system would bind to, as well as the amount of TALEs that would be on our strip. Through this we have been able to predict the amount of protein that needs to be loaded onto our strips before we started our prototyping.

How was our model created?

A simple 2D Monte-Carlo method was used for the simulation of the DNA movement along the strip, as well as for the action of the TALEs binding the DNA. Once a TALE binds to the DNA it is rendered unable to bind to any further DNA, effectively simulating saturation. For the quantitative output generated by the simulation, the number of target DNA binding to the TALEs is tallied with the number of non-target DNA that binds to the TALEs. To determine the sensitivity of our system we set the probability of target DNA binding to TALEs equal to 1, with non-target DNA having a probability of binding equal to 0. In future iterations we plan on altering these parameters such that we can determine the trade-off between sensitivity and specificity in our system. These calculated parameters will also be tested in the lab, allowing us to feed our experimental data back into our model in an iterative fashion.

A 2D region of the test strip, viewed from the top, was partitioned into a grid of squares such that a single TALE (0.1 nm by 0.1 nm, represented by the colour black) will be occupying a single square. The spaces on the left side of the grid are designated as the DNA loading zone, where the DNA species (red for target DNA, green for non-target DNA) will initially be located. The remaining spaces on the right will be initially allocated for the TALEs, representing the test line on our prototype. DNA then moves from its current square to its neighbouring squares, one square at a time, where the probability of the movement is set in such a way so that moving from left to right is favoured. This was designed to simulate the flow of fluid that will be used in our prototype and is currently used in home pregnancy tests. When DNA enters a square occupied by a TALE, the probability of binding to that TALE is determined by whether the DNA is target or non-target. If binding occurs, the TALE becomes saturated and additional DNA cannot bind, represented visually by the TALE changing from black to blue. After the simulation has run it's course, the number of TALEs bound by a positive DNA are tallied against the number of TALEs bound by a negative DNA, from which sensitivity and specificity can be calculated.

What did our model show?

We initially tested our model by varying the number of target DNA sequences vs non-target DNA and the number of TALEs bound to our test strip. After performing 3 runs from each of our 25 conditions (5 for both the TALEs and the target DNA levels), for a total processing time of approximately 6 hours, we obtained the number of saturated TALEs in a 50 x 50 grid square representing our prototype for 75 runs. This data would have taken us approximately 4 weeks (based on the construction time of our recent prototypes) to do in the lab. The data from this was analyzed and is represented as a 3D surface below in Figure 4. Based on the graph, it seems that the main determinant in the sensitivity of our system will be the amount of TALEs on the test strip, which makes intuitive sense.

We have deployed our agent based model into an application playable by the Mathematica CDF Player, and it can be found here, along with our enzyme kinetic models. The mathematica code for our model can also be found at this location.

Future Directions

The model that we have developed is very preliminary, currently unable to fully predict either the sensitivity or the specificity. It has given us a good starting point for determining the design of our prototype though, as it showed that the amount of TALEs present is extremely important to our system. In the future we aim to increase the scope of our model by adding more realistic probabilities for the binding reactions of our DNA sequences, as well as by including additional DNA sequences instead of an almost perfect two species system.