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. We used simple models to simulate the kinetics of our enzyme cascade to develop intuition for the design of experiments. We then used our models to optimize the design of our reaction cascades for an improved detection limit and optimal lysis times. Our scripts can be found on our GitHub repository.
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:
||reference or rationale
||Mekler et al. (2016) Nucleic Acids Resarch
||Estimated to be slow in comparison to kcr
||Estimated to be fast in comparison to kcr
||Estimated from RPA-Tx amplification experiments
||Weitz et al. (2014) Nature Chemistry
As shown in Figure 1, our simulations are able to reproduce the behavior observed experimentally.
Figure 1: Kinetics of Cas13a using 1 nM Cas13a and 10 nM crRNA 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 1 nM Cas13a and 10 nM crRNA.
Improved Reaction Cascade
In collaboration with our wetlab team we developed a reaction cascade for sample pre-amplification by coupling reverse transcription to isothermal recombinase polymerase amplification and transcription (RT-RPA-TX), resulting in auto-catalysis of target RNA (Figure 3).
Figure 3: Scheme for the RT-RPA-TX amplification system.
In order to compare the detection limit of the Cas13a system alone with the detection limit of the amplified the reaction cascade, we expanded our model, assuming exponential amplification of the target RNA. As the amplification reaction saturates due to a depletion of resources, the amplification stops as soon as the target RNA level reaches an upper limit of 1000 nM (Figure 4).
Figure 4: Schematic representation of the target RNA amplification during the estimation of the detection limit using the reaction cascade.
The kinetics for the amplfication cascade coupled to Cas13a based detection are shown in Figure 5. Strikingly, the start of the reaction seems to be determined by the amplificaiton reaciton, while the consecutive phase is limited by the rate of Cas13a mediated cleavage.
As shown in Figure 2, the detection limit of the reaction cascade decreases by approximately three orders of magnitude. These simulations led us to implement our pre-amplification cascade into our CascAID system.
Figure 5: Kinetics of the Cas13a systemusing 1 nM Cas13a and 10 nM crRNA at different target concentrations 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-life 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-life using Arrhenius equation as commonly done in the literature:
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:
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:
Plugging equation 5 into equation 4 gives
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
with initial condition of
we get the final solution to the lysis equation:
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.
- Moody, C., et al. (2016). “A mathematical model of recombinase polymerase amplification under continuously stirred conditions.” Biochemical Engineering Journal 112: 193-201.
- Moody, C., et al. (2016). “A mathematical model of recombinase polymerase amplification under continuously stirred conditions.” Biochemical Engineering Journal 112(Supplement C): 193-201.
- Li, L., et al. (2011). “Kinetics of hydrothermal inactivation of endotoxins.” Appl Environ Microbiol 77(8): 2640-2647.
- Valente, W., et al. (2009). “A Kinetic Study of In Vitro Lysis of Mycobacterium smegmatis.” Chem Eng Sci 64(9): 1944-1952.
- Fabritz, H. (2007). “Autoclaves Qualification & Validation.” Experts Meeting in Baden
- Marras, S. A., et al. (2004). “Real-time measurement of in vitro transcription.” Nucleic Acids Res 32(9): e72.
- 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.
- 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.
- Rauhut, R. and G. Klug (1999). “mRNA degradation in bacteria.” FEMS Microbiol Rev 23(3): 353-370.
- Licciardello, J. J., & Nickerson, J. T. R. (1963). “Some Observations on Bacterial Thermal Death Time Curves.” Applied Microbiology, 11(6), 476–480.
- Deindoerfer, F. H. (1957). “Calculation of Heat Sterilization Times for Fermentation Media.” Applied Microbiology, 5(4), 221–228.
- M. Weitz, K. Jongmin, K. Kapsner, E. Winfree, E. Franco, and F.C. Simmel.
“Diversity in the Dynamical Behaviour of a Compartmentalized Programmable Biochemical Oscillator.”
(2014) Nature Chemistry 6(4): 295–302.
- V. Mekler1, L. Minakhin, E. Semenova, K. Kuznedelov and K. Severinov
"Kinetics of the CRISPR-Cas9 effector complex assembly and the role of 3′-terminal segment of guide RNA"
Nucleic Acids Research, Vol. 44(6): 2837–2845