Difference between revisions of "Team:Munich/Model"

Modelling in Biosciences is a powerful tool that allows one to get a deeper understanding of one's system. We mainly used Modelling to help with the design of our device. By this, we could avoid spending time on dead-end-designs that otherwise might have cost us a significant amount of time. Rather simple models can already give fair amount of information about one's system. That is why we decided at an early stage to incorporate Modelling in our device design.

Detection Limit

One major concern when dealing with the problem of diagnostics on patients is obtaining the sample with which detection can actually be performed. Since we wanted our method to be non-invasive, one concern that we needed to deal with is the concentration of pathogens and thus detectable RNA in the patients mucus. First approximations from different papers already showed that virological samples show concentrations no higher than low pM and can even go as low as fM. Thus, we characterised the theoretical detection limit of the Cas13a RNAse activity. In order to do this, we first fitted parameters using experimental data to the model shown below and used these in target RNA concentration dependent simulations. The results are shown in Figure 1. It shows that the detection limit in the time range of an hour is approximately one- to two-digit nM region. Due to this result, our initial design of applying the lysed and purified RNA sample directly on the detection paper strip had to be discarded. Instead, we had to explore amplification methods we could perform upstream in the detection process.
As a side note, the detection limit could most probably have been pushed a bit to lower concentrations by using higher concentrations in Cas13a and crRNA, but by doing this production cost per paperstrip would have increased a lot. Also, it is known from literature that Cas proteins at high concentrations show activity independent of their activation mechanism which is why the concentration of Cas13a in the detection system could not be increased by higher orders of magnitude.

Figure 1: Theoretical Detection Limit determined for the Cas13a system using 20 nM concentrations of Cas13a and crRNA.

Lysis on Chip

We modelled the lysis process on chip to get an idea of how long lysis would need to take place in order to release enough RNA for downstream amplification. For this, we constructed a very simplistic model for bacterial cell lysis. In this, we estimated the rate constants for cell lysis by common colony PCR protocols which use a 10 minute lysis step at 95 °C for thermolysis. Thus, we considered a half-time of Bacteria of 2 minutes at 95 °C. This would result in a lysis efficiency of 96.875%. Starting from this estimation, we considered the rate constant of lysis and thus the half-time using Arrhenius equation.

Equations 1+2

where R is the gas constant and k describe the rate constant at Temperature T_1 and Temperature T_2. The Arrhenius energy E_A was fitted to a barrier that follows the common rule of thumb that lysis should increase twice in efficiency every temperature increase of 10 °C. The model for lysis is shown below:

Effect of lysis temperature on the lysis efficiency of bacterial cells and Determination of the released concentration of target RNA from lysis assuming a ratio of 30 RNA molecules per cell.

The full model can then be described by the coupled ordinary differential equations:
Equations 3+4

The full model at 95 °C looks as follows:

Signal Amplification

For the simulation of an amplification system, we circuit amplifying an RNA system. Therefore, we couple a Reverse Trancription to an isothermal PCR-like amplification called Recombinase Polymerase Amplification (RPA) and do In-Vitro Transcription from the build template. A scheme for the model is shown in Figure 2. For simplicity, we made assumptions to this model:
First, the RPA reaction is thought to be in the linear region, independent of Primer concentration since we work in an environment of very high primer and dNTP concentrations (up to 1000 nM) and only want to reach RNA concentration within the range of the detection limit of our Cas13a protein, which is in the nM region. Therefore, since we are amplifying the RNA by Transcription from the cDNA, this assumption is reasonable. The same argument goes for the In-Vitro Transcription; since we are in an environment of excessive rNTP concentrations, thus first order approximation is valid.
Rate constants were approximated by experiments or taken from literature. The only rate constant that was not available was the rate of Reverse Transcription. We, thus, took producer's information about commercial RT kits and estimated from these very conservatively (two orders of magnitude less in reaction speed) to not be biased in the simulation by overfitting parameters.
The rate constants are the following: COUNT ALL 4 RATE CONSTANTS

Figure 2: Scheme for the RT-RPA-Tx Amplification system

Figure 3: Target RNA concentration dependent on initial concentrations to determine the cycle time in RT-RPA-Tx needed for reaching the Cas13a detection limit of 10 nM (red line).

The overall dynamics of the RT-RPA-Tx system are shown below for several starting concentrations of RNA.

Theoretical Detection Limit using the Amplification Circuit and Cas13a Detection

Since the reasoning behind using an amplification method was to bring down the detection limit, a new theoretical detection limit of the device may be determined combining model of lysis and isothermal amplification. For this, a reasonable cycle time for point-of-care application of one hour was chosen.

Determining Cycle times to reach 10 nM Detection Limit using Amplification Circuit. Red dashed line marks the end of the thermolysis

When comparing this to cycle times needed for reaching the detection limit at 65 °C, one sees that lysis temperatures is not very important to the amplification and only results in a slight shift to longer time scales. This is reasonable, since RPA, and PCR in general, are enormously sensitive methods, and thus only need few templates to show a signal. Also, when comparing the concentrations in the temperature screen above, one can observe that the concentrations of RNA within the sample only change insignificantly, all showing concentrations that range within three-digit attomolar region or higher.