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.

Snap shot of interactive beta-galactosidase kinetics model

Figure 1.Snap shot of interactive beta-galactosidase kinetics model.

Snap shot of interactive horseradish peroxidase kinetics model

Figure 2.Snap shot of interactive horseradish peroxidase kinetics model.

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 its 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.

Figure 3. Representation of our agent based model, with the simulation before hitting run on the left, and after the simulation on the right. The bound TALES, shown in blue, would be where the blue line on our prototype would appear if this was a test strip.

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.


Figure 4. Model results of sensitivity tuning for TALE test strip.

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.

Full System Kinetic Model

With a preliminary understanding of how our system works we proceeded to quantitatively model our entire system. To do this we used the kinetic constants found in our experiments from the TALE and prussian blue ferritin characterization to build a deterministic model in Scilab. We modelled the binding of an immobilized TALE to target DNA in solution and then the subsequent binding of one of our FerriTALEs. We then used the Michaelis-Menton kinetics of our prussian blue ferritin to calculate how quickly it converts TMB into the coloured product that we can see. A list of our constants and variables can be seen in the table below.

Variable Description   Constant Value
IT Immobilized TALE on nitrocellulose   IT dissociation constant 293 nM
D Target DNA Seqeunce   FT dissociation constant 66 nM
ID Immobilized DNA   IR catalytic constant 5118 min-1
FT FerriTALE      
IR Immobilized FerriTALE on captured DNA      
X Coloured output chemical      

Table 1.Variables and constants used in our deterministic model.

When we assembled our reactions into differential equations we had 6 equations to cover the change in our 6 variables over time. These can be seen below in the screenshot of the Scilab model.


Figure 5. Screenshot of the code for the Scilab model showing the differential equations governing the change of our 6 variables over time.

When this equation was run with the amount of DNA present in a super shedding cow the output chemical, X, changes over time as shown below.


Figure 6. Change in X, the output chemical of our prussian blue ferritin reaction, over time in a system with the amount of target DNA present in a super shedding cow. The red line indicates the concentration at which the reaction becomes visible to the naked eye.

The red line on the figure above shows when the reaction reaches 22.176 µM, which is the concentration at which a blue dot or line becomes visible. This value was calculated based on the kinetic parameters we determined in our characterization of prussian blue ferritin. This takes approximately 4.9 minutes, meaning that we will be able to see a visible response from a super shedding cow in less than 5 minutes!