Team:Munich/Model

Modelling

Modelling in Biosciences is a powerful tool that allows us to get a deeper understanding of our system. It guided and assisted the design of our detection system which helped us saving a lot of time by avoiding dead-end designs. Rather simple models can already give fair amount of information about a system. From the very beginning we used simple models to simulate the kinetics of our enzyme cascade to develop intuition for the design of experiments. We further used our models to estimate detection limits using different cascades and reaction times.

Detection Limit

One major concern when dealing with the problem of diagnostics on patients is extracting 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 or non-invasive sample. First approximations from different papers already showed that virological samples show concentrations no higher than low pM and can even go as low as aM. Thus, we characterised the theoretical detection limit of the Cas13a RNase activity. In order to do this, we first estimated the rate constants for the different reactions to create a behavioural model. 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. The concentrations that can be used in Cas13a nucleotide reaction have an upper limit since it is known from literature that Cas proteins show activity independent of their activation mechanism at high concentrations. This is why the concentration of Cas13a in the detection system could not be increased to higher orders of magnitude. The chemical and differential equations for the model are shown below:

Figure 2: Kinetics of Cas13a using 1 nM and 10 nM at different target concentrations.

Figure 1: Kinetics of the Cas13a system using 1 nM concentrations of Cas13a and 10 nM crRNA at different target concentrations using the reaction cascade.

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- to two-digit nM concentrations of RNase alert, we employed a

Estimating the Detection Limit of the Reaction Cascade

In order to compare the detection limit of the Cas13a protein kinetics by itself with the detection limit implementing the reaction cascade of reverse transcription, recombinase polymerase amplification and in-vitro transcription, we created a model that takes exponential amplification of the target RNA into account. This was constructed in a way that amplification of the RNA signal takes in an exponential manner, illustrated in Figure. In order to assure that the amplification of the target RNA reaches an upper limit, we capped the amplification at 1000 nM. As visible in Figure XXX, the detection limit of the reaction cascade decreases the detection limit by approximately three orders of magnitude.

Figure 3: Schematic representation of the target RNA amplification during the estimation of the detection limit using the reaction cascade.

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 as commonly done in the literature:

(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 k_lysis being the rate constant of bacterial lysis, k_RNase 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.

Signal Amplification

We next developd a model for an RNA amplification circuit. Therefore, we chose a combination of reverse trancription to recombinase polymerase amplification (RPA) and in-vitro transcription. A scheme for the model is shown in Figure 4. Following Occami's razor, we tried to build the model with as few parameters as possible, breaking down the model into a reaction scheme that uses only 4 rate constants. In addition, we made the following assumptions to this model:
First, the RPA reaction is thought to be in the exponential region, independent of Primer concentration since we work in an environment of 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 lower nM region. 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.
The rate constants are the following:

rate constant	value	reference or rationale
k_lysis	0.000152 [1/s]	Liccardello & Nickerson, 1963; temperature dependency via Arrhenius
k_RPA	0.0008 [1/s]	experimental data fitted with exponential function
k_Tx	0.0000333 [1/s]	Berta Tinao, Master's Thesis, Technical University of Munich
K_RT	0.15 [1/s]	http://www.sygnis.com/hiv-reverse-transcriptase/ (11/2/2017)

The coupled ODEs for the signal amplification circuit can be described simply by:

(13)

Equations 12 + 13

(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.

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.

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 suggests that the reaction is limited by the T7 RNA Polymerase and might be increased in yield by adding higher concentrations of T7 RNA Polymerase

References

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Figure 2: Theoretical Detection Limit determined for the Cas13a system using concentrations of 1 nM Cas13a and 10 nM crRNA.