Difference between revisions of "Team:Munich/Model"

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 other 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.

Our wetlab experiments indicated that the detection limit of the Cas13a RNase activity is in the range of 10 nM. Using our kinetic data, we estimated the rate constants for the different reactions to create a simple ODE model.
The chemical and differential equations for the model are shown below:

As shown in Figure 1, our simulations are able to reproduce the behavior observed experimentally.

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

Next, we analysed the amount of readout RNA that was cleaved after 30 minutes for varying target concentrations. As shown in Figure 2, the curve follows a sigmoidal behavior and suggests a detection limit in the range of 10 nM. 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. Since it is known from literature that Cas proteins show activity independent of their activation mechanism at high concentrations, we could not increase the concentration of Cas13a to improve the sensitivity. Instead, we explored amplification methods upstream in the detection process.

Figure 2: Estimated detection limit determined for the Cas13a system using 20 nM concentrations of Cas13a and crRNA.

Figure 3: Kinetics of the Cas13a system using 1 nM concentrations of Cas13a and 10 nM crRNA at different target concentrations using the reaction cascade.

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.

Figure 4: Scheme for the RT-RPA-Tx Amplification system.

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.

References

1. Moody, C., et al. (2016). “A mathematical model of recombinase polymerase amplification under continuously stirred conditions.” Biochemical Engineering Journal 112: 193-201.
2. Moody, C., et al. (2016). “A mathematical model of recombinase polymerase amplification under continuously stirred conditions.” Biochemical Engineering Journal 112(Supplement C): 193-201.
3. Li, L., et al. (2011). “Kinetics of hydrothermal inactivation of endotoxins.” Appl Environ Microbiol 77(8): 2640-2647.
4. Valente, W., et al. (2009). “A Kinetic Study of In Vitro Lysis of Mycobacterium smegmatis.” Chem Eng Sci 64(9): 1944-1952.
5. Fabritz, H. (2007). “Autoclaves Qualification & Validation.” Experts Meeting in Baden
6. Marras, S. A., et al. (2004). “Real-time measurement of in vitro transcription.” Nucleic Acids Res 32(9): e72.
7. Mafart, P., et al. (2002). “On calculating sterility in thermal preservation methods: application of the Weibull frequency distribution model.” Int J Food Microbiol 72(1-2): 107-113.
8. Chiruta, J. (2000). “Thermat Sterilisation Kinetics of Bacteria as Influenced by Combined Temperature and pH in Continuous Processing of Liquid.” Thesis, The Universit of Adelaide Department of Chemical Engineering Faculty of Engineering.
9. Rauhut, R. and G. Klug (1999). “mRNA degradation in bacteria.” FEMS Microbiol Rev 23(3): 353-370.
10. Licciardello, J. J., & Nickerson, J. T. R. (1963). “Some Observations on Bacterial Thermal Death Time Curves.” Applied Microbiology, 11(6), 476–480.
11. Deindoerfer, F. H. (1957). “Calculation of Heat Sterilization Times for Fermentation Media.” Applied Microbiology, 5(4), 221–228.