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Modelling
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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.
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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.
Since our detector has shown to be sensitive enough to detect one-digit nM concentrations of RNase alert, the needed concentration of
protein and crRNA could be downscaled, as only few nM of cleaved RNase alert were needed to get a read-out. The concentrations
were then pushed from the initial model which included 20 nM Cas13a and 20 nM crRNA to 1 nM Cas13a and 10 nM crRNA.
The plot below shows that the Theoretical Detection Limit still stays equal, bearing in mind that detection occurs once ~6 nM RNase alert
have been cleaved.
Figure 2: Theoretical Detection Limit determined for the Cas13a system using concentrations of 1 nM Cas13a and 10 nM crRNA.
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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:
 (1) (2)
with rate constants k1 and k2 at temperature T1 and T2
and Boltzmann constant R.
where R is the gas constant and k1 and k2 are the rate constant at temperature T1 and T2
The activation energy difference E_A was fitted to a barrier that follows the common rule of thumb that lysis should increase twice in
efficiency for every temperature increase of 10 °C. The model for lysis is shown derived in the following:
The full model can then be described by the coupled ordinary differential equations:
 (3) (4)
with klysis being the rate constant of bacterial lysis, kRNase the rate constant
of RNA degradation, count of target RNA [targetRNA] and count of bacteria Baks(t).
The solution to equation 3 is of course simply:
 (5)
Plugging equation 5 into equation 4 gives
 (6)
where ratio determines the copy number of a target RNA in a single cell. This differential equation has the form and thus the analytical solution:
Equation 7 + 8
 (7) (8)
with initial condition of
 (10)
we get the final solution to the lysis equation:
 (11)
The full model at different temperatures looks as follows:
Figure 3: 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.
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Signal Amplification
For the simulation of an amplification system, we developd a model for a circuit amplifying an RNA. 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 4.
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
The coupled ODEs for the signal amplification circuit can be described simply by:
Equations 12 + 13
Figure 4: Scheme for the RT-RPA-Tx Amplification system.
Figure 5: 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.
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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 95 °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. Also, this model works with the statement in the literature that as little as 10 templates are enough to trigger amplification through RPA.
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Signal Amplification Measurement in RPATx
When we performed time-dependent measurements of crRNA in a RPATx Ansatz, we measured saturation of T7 RNA Polymerase already at 0.2 nM template DNA. The reaction
kinetics and thus the formation of RNA showed pseudo-first order dynamics with a rate constant of 97 ng/min transcribed RNA. Compared to the literature (https://www.biosciencetechnology.com/article/2003/09/maximizing-yield-full-length-rna-vitro-transcription-reaction) this is not even the bottleneck since In-Vitro Transcription reactions can yield up to 400 μg in 4 hours. This led us to try out a concentration series of different template concentration and try whether we could detect the extracted RNA with Cas13a.
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References
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