Team:YiYe-China/Model
Model and Dry Lab
The idea of this project is to construct riboregulatory that would regulate fluorescence protein expression based on trigger concentration as a mean of visually presenting differences in methylation degree. The motifs of this model include extraction of human DNA sample from fecal samples, capturing degree of methylation into the DNA sequence through bisulfite conversion, amplification of T7-trigger sequence using strand-specific PCR, and cell-free system expression of both trigger and toehold construct.
Figure 1. Overview of reactions
Model construction and dry lab are crucial to understanding the behavior of the constructs. Dry lab produces results that might be costly in wet labs and enables prediction of outcomes under different sets of conditions.
Thermodynamics Model
As an inseparable part of modeling switch, thermodynamical simulation and analysis were performed on the toehold and trigger designed. The secondary structure prediction of both sequences was performed on NuPack (NUPACK: Nucleic Acid Package). The RNA-RNA interaction prediction was performed on ViennaWebServer (ViennaRNA Web Services (univie.ac.at))[3][4] using RNAup and the Andronescu model, 2007, with the probability of being unpaired in hairpin loops included. The two RNA binds favorably, ranging from positions 1 to 30 in the longer sequence and from positions 4 to 33 in the short sequence.
Figure 2. Left: secondary structure of trigger RNA. Right: secondary structure of toehold-mcherry construct.
Table 1. Results of interaction prediction. 
The equilibrium constant Ka can be deduced from the total free energy of binding.
∆G= -RTln(K) (1) K= exp((-∆G)/RT) (2)
The equilibrium constant will be crucial for the mathematical model of the cell-free system.
Mathematical Model
In order to explore the difference mCherry expression level in response to a gradient of methylation level, mathematical modeling was performed and computed using Python. The input, also the independent variable, is identified to be methylation degree, which is designated as m. The final output and dependent variable would be mcherry concentration. Ideally fluorescence level should be set as the final output. However, there is not an adequate and generalized scale for concentration-dependent fluorescence scale, while scaling of fluorescence level could potentially be engineered in future projects.
Figure 3. Illustration of the goal of modelling.
The model was constructed based on the major reactions, which are distinguished by their differences in reaction environment. It is assumed that residue from previous reactions should be absent and these four reactions do not affect each other in terms of the environment of reaction. Hence, the four reactions are modelled separately. This model in general assumes no loss due to experimental processes. The model aims to explore the responses to input in linear gradients, so the modeling of reactions before the cell-free system steps are conceptualized in a linear fashion for simplicity.
Extraction of Human DNA
The human DNA content is denoted by [S] in this model. Based on literature, there is not an agreed value on how much in average will human DNA content be extracted from fecal samples. In contrast, the average value varies for control group and CRC patients (controls: 0.46 ng/µl, CRC: 15.05 ng/µl) [1]. The human DNA content may not be a constant in the first place, but for the sake of generalized modeling, 1 ng/ul is taken as the value of the parameter.
Bisulfite conversion
Bisulfite conversion is itself a process equivalent to translation in terms of converting one form of information to another. Since the process is basically a 1-to-1 conversion, there is no loss or gain with quantitative significance. However, for the sake of PCR amplification modeling, the template sequence availability must be calculated.
Figure 4. Written notes on the fraction.
Only a part of the human DNA content is TFPI2 DNA, while not every part of TFPI2 can be methylated (or unlikely to be methylated). The design of trigger sequence and the toehold also considers that there are regions methylated on a more concentrated and frequent basis across CRC cases. Both dilution of the actual template concentration is considered by introducing two constants, prop_D and prop_M, where prop_D is the proportion of human DNA that is TFPI2 and prop_M is the proportion of TFPI2 sequence can be highly methylated. Both constants can be adjusted for dry lab. The model assumes that degree of methylation is directly proportional to the number of available templates.
[Tem]=[S]*prop_D*prop_M*m (3)
PCR
For the sake of simplification, several assumptions are made. 1) Asynchronous DNA replication from original DNA strands due to potential overlapping of primer-matching regions are neglected at this point. i.e. Replications of all available templates are considered to be synchronous and only initiated during cycle 1. Cycle 2 through 40 will not produce new strands from the original sequences. 2) Limitations and concentration-dependent dilution due to limited resources are not considered. 3) The prediction of copy number were based on the exponential model of PCR in which one template corresponds to one lineage of replication by the power of 2. 4) Any replication initiated from .
Then the expression for the amplified copy number after PCR is:
[t] = [Tem]* (2*2n-2-2) (4) Where [t] is trigger concentration N = 40 according to experimental protocol.
The formula is developed from the summation of geometric sequences with ratio of 2, while the exponents are determined as n-2 since valid copies of the desired sequence only appear after 2 cycles.
Figure 5. Illustration of PCR. Only the two dsDNA in orange boxes are valid copies.
Cell-free system:
The cell-free system involves four reactions during each time step: the transcription of trigger RNA. The transcription of toehold-mcherry construct, the mass-action driven toehold-trigger binding, and the translation of mcherry upon toehold activation. At the beginning of mathematical modeling, decision on the type of model was to be determined. In general, biochemical reactions can be conceptualized in two aspects: deterministic or stochastic; synchronous or asynchronous. For the sake of simplicity, noises were neglected, and synchronous model was constructed, while one publication [2] indicates that deterministic models fit cell-free system better based on the concentrated environment of reactants, which also suggests relatively low level of noises. Borrowing ideas from the same paper that models T7 cell-free GFP expression, the modeling of transcription and translation reactions using ODE will use Michaelis-Menten expressions with prefactors for translation and transcription saturation. There is no term for mRNA degradation due to absence of nuclease activity in the cell-free system.
Where Toehold is equivalent to the toehold-mcherry construct, and toehold_on stands for activated toehold upon triggerRNA binding. ODEs were adapted from Stögbauer et al, 2012. Parameters and values are listed in table2.
During early stage of model construction, attempts of removing the two prefactors were performed. However, without the saturation prefactors, the level of toehold RNA will spike unreasonable.
Since both the paper and the project uses T7 cell-free system, parameters related to transcription can be directly borrowed. The translation-related parameters for mCherry were challenges to this model. Ideally, the parameters for mcherry should be determined aside from those of GFP provided in the paper. However, wet lab data on mcherry expression over a reasonable gradient was not found, while data tends to be on other engineered variants of mcherry. Kmat for mcherry was successfully found by looking up mcherry maturation time. Based on the similarity in amino acid sequence length relatively similar secondary structure, it is assumed that translation related parameters were similar for GFP and mcherry. This assumption could have led to discrepancy between the actual mcherry concentration and the modeled concentration. Yet, this may not be a significant problem if the goal of this modeling is to explore how concentration gradient in trigger (as well as methylation degree) would affect the mcherry concentration.
Table 2. Parameters and corresponding values.
Corresponding to the ODE which tells us about the change in chemical concentration per timestep, we also need to way to calculate the amount of toehold activated per time-step. The assumption is that the entire cell-free reaction only goes forward and accumulates.
Based on the equilibrium constant and the expression, we know that the ratio between activated toehold, trigger, and toehold construct is kept constant during the entire cell-free reaction process. Hence, based on mass-action, if toehold and trigger RNA are constantly added into the system through transcription, the amount of activated toehold would either remain the same or increase. Given the ratio of reactant and products, the following expression can be written. After reorganizing the terms, we get a quadratic equation.
Figure 6. Illustration for deriving the quadratic equation.
The quadratic formula and the ODE are solved in python using np.roots() and step-wise integration over a specified timestep as illustrated by expression (12).
X_(t+1)=dX/dt+X_t (12)
Results
Figure 7. Flowchart for overviewing the required calculations.
Based on the model, the inputs maintain a linearly gradient before entrance into the cell-free system. As a potential direction of improvement, the model can be adjusted to consider saturation problem in PCR process such as density-dependent changes in amplification effect by considering length of the primer and overlapping of the actual primer recognition sites as well as the effect of limited reagents added.
Figure 8. Illustration of the linear gradient.
The dry lab attempted comparison between direct mcherry expression and that using toehold switch. The result is shown in figure 9, where certain extent of suppression of expression can be observed upon addition of toehold.
Figure 9. Expression level comparison.
Additionally, mcherry concentrations over a specific timespan under different methylation degree are also plotted as shown in figure 10 and 11. The predicted mcherry concentration gradient shows a non-linear and non-crossing pattern, indicating successful differentiation between different methylation degrees. Yet, differences in mcherry concentrations tend to diminish as methylation degree increases, which may suggest the need for adding another regulatory unit, fluorescence level scaling, or another fluorescence protein with a different color and a opposite expression pattern (i.e. expression level decreases with increased methylation degree).
Reference: [1] Lima, J., Teixeira, Y., Pimenta, C., Felipe, A. V., Silva, T. D., Junior, E., Saad, S. S., Deak, E., Murray, H., & Manoukian Forones, N. (2019). Fecal Genetic Mutations and Human DNA in Colorectal Cancer and Polyps Patients. Asian Pacific journal of cancer prevention : APJCP, 20(10), 2929–2934. https://doi.org/10.31557/APJCP.2019.20.10.2929 [2] Stögbauer, T., Windhager, L., Zimmer, R., & Rädler, J. O. (2012). Experiment and mathematical modeling of gene expression dynamics in a cell-free system. Integrative Biology, 4(5), 494-501. [3] Gruber AR, Lorenz R, Bernhart SH, Neuböck R, Hofacker IL. The Vienna RNA Websuite. Nucleic Acids Res. 2008 [4] U.Mueckstein, H. Tafer, J. Hackermueller, S.H. Bernhart, P.F. Stadler, and I.L. Hofacker "Thermodynamics of RNA-RNA Binding." Bioinformatics, 22(10), pp 1177-1182, 2006 [5] Macdonald, P. J., Chen, Y., & Mueller, J. D. (2012). Chromophore maturation and fluorescence fluctuation spectroscopy of fluorescent proteins in a cell-free expression system. Analytical biochemistry, 421(1), 291-298. [6] https://www.magnoliasci.com
Model and Dry Lab
Model construction and dry lab are crucial to understanding the behavior of the constructs. Dry lab produces results that might be costly in wet labs and enables prediction of outcomes under different sets of conditions.
- Thermodynamics Model
-
Mathematical Model
Extraction of Human DNA
Bisulfite conversion
PCR
Cell-free system
- Results