Modeling
General Assumption
1.Since the biochemical reactions in cells are too complex to be completely simulated, the interactions of intermediates throughout the whole process were ignored.
2.Considering that differentiated cells in vivo are under homeostatic condition, where the cell proliferation rate can be generally ignored, and the intercellular microenvironment remains stable.
3.We ignore the effect of the endogenous trim21 enzyme on the degradation of target protein in our study object.
The Protein Interaction Module
GFP PrePro: DocS-Coh2 interaction
This module concentrates on the plasmid expression and protein interaction. After the transfection of p18(pCMV-GFPnano-Coh2-P2A-Trim21-DocS) and pGFP(pCMV-GFP), mRNA1 and mRNA2 can be produced via DNA transcription, at the rate v1_1 and v1_2 respectively. Synthesized mRNAs are then translocated to the cytoplasm and subsequently translated into two protein recombinants (Trim21-DocS, Coh2-GFP nanobody) and GFP. The antigen-antibody reaction occurs thereafter, with GFP binding to GFPnanobody at the rate v3. DocS and Coh2 will form a dimer, for which we use parameter kf to characterize. Given that the interaction of DocS-Coh2 is reversible, kb is therefore introduced to quantify the dissociation process. With the formation of the ternary complex (substrate-carrying E3, SE3), the assembly of GFP PrePro degradation system is accomplished. To increase the accuracy of our model, we use v0_1, v0_2, v0_3, v0_4, v0_5 to represent the natural degradation of intermediate metabolites (mRNA1, mRNA2, Trim21-DocS, Coh2-GFPnanobody, GFP) in mammalian cells.
Figure 2. DocS-Coh2 interaction
The equations were formulated based on the law of mass action :
RiPrePro: FRB-FKBP interaction
This module simulates the formation of RiPrePro system with RiPrePro plasmid (pCMV-GFPnano-FKBP-P2A-Trim21-FRB) and pGFP (pCMV-GFP) as input. The biochemical reactions involved are similar to the DocS-Coh2 module we present above, while the interaction between FRB and FKBP serves as the major difference. Unlike the constitutive DocS-Coh2 dimerization, the protein FKBP and FRB does not form dimers automatically but can be dimerized in the presence of rapamycin. Such rapamycin-induced dimerization process indicates that the parameter kf and kb associated with the dimerization are no longer constant, but varying from the change of external factors such as the concentration of rapamycin. In the absence of rapamycin, the protein pairs dissociate, suggesting that the value of kf should be equal to zero and the switch is off. The rising level of rapamycin concentration switch on the RiPrePro, resulting in a higher value of kf. As such, a simplified mathematical representation of this module can be developed by adapting the previous DocS-Coh2 module.
Figure 3. FRB-FKBP interaction
This protein interaction module determines the quantitative relationship between the plasmid dosage (as the Input) and the ternary degradation complex (SE3, the Output), linking to the protein degradation module with SE3 serving as the input.
The Protein Degradation Module
On the basis of the model we demonstrated in iGEM 2018-19, the protein degradation module mainly simulates the degradation of GFP via ubiquitin-proteasome pathway. The ubiquitination process requires three types of enzyme: ubiquitin-activating enzymes (E1), ubiquitin-conjugating enzymes (E2), and ubiquitin ligases (E3). In the initial step, E1 activates ubiquitin (Ub), followed by the transfer of ubiquitin from E1 to E2. The forward and reverse reaction rate of two processes are denoted as v4f, v4b, v5f, v5b, respectively. The substrate-carrying E3 (SE3) transiently bind ubiquitin, with RING domain of E3 catalyzing the direct ubiquitin transfer from E2 to the substrate (S) at the rate v6_1 and forming the single-ubiquitin substrate (UbSE3), more ubiquitin molecules can be added to the substrate at the rate v6_n(n=2,3,…8), yielding a polyubiquitin chain (UbnSE3). With the dissociation of E3 at the rate v7_n(n=1,2,…8) and the removal of ubiquitin at the rate v8_n(n=1,2,…8), the substrate tagged by more than four ubiquitin molecules would be recognized and ultimately degraded by 26S proteasome, denoted by v9_n(n=4,5,…8) and v10_n(n=4,5,…8), respectively.
Figure 4. Ubiquitination and degradation of GFP
Following the law of mass action, the ODE equations can be defined as follow:
Result and Discussion
GFP PrePro
Figure 5. the side-by-side comparison of model prediction and experimental data
To validate the accuracy of our model, we challenged the model with different input plasmid levels and predict the degradation of GFP. As is shown in Figure 5A, 5B and 5C, simulation showed that multiple sets of model predictions match well with the corresponding wet-lab data, indicating that our model might have correctly reflected the whole processes of targeted protein degradation in our case.
Figure 6. the influence of plasmid dosage and ratio on GFP degradation rate
To find the optimal dosage of plasmids for in vitro experiments, further analysis of the plasmid ratio and dosage is performed. We first obtained the quantitative relationship between the plasmid ratio and the degradation efficiency under different GFP plasmid levels (Figure 6B). In order to further evaluate the stability of degradation under different plasmid ratios, we calculated the second derivative of the degradation efficiency with respect to the ratio. As is shown in Figure 6C, regardless of the GFP plasmid level, the second derivative values all tended to be zero when the plasmid ratio was greater than 1, which demonstrated that a relatively stationary phase could be reached in this circumstance. Based on the results above and to avoid dosage waste, we chose 1:1 as the optimal plasmid ratio. In order to further determine the optimal dosage of the plasmids, we revisited the wet-lab data. As is shown in Figure 6A, when introducing 0.25ug or 0.125ug GFP plasmid, an improved degradation performance could be observed in comparison to the case for 0.5 ug at the same p18 plasmid level. Given that the 0.25 ug curve manifested a stronger stability than that of 0.125ug, the 0.25ug dosage for GFP should be prioritized. Thus, we believed that 0.25ug dosage for both plasmids might be a good choice in the subsequent experiments.
Figure 7. the time-dependent characteristic of GFP PrePro system in silico and in vitro
In order to determine the experimental time of degradation of p18 and GFP, we simulated the function according to the experimental data, and drew the corresponding function curve. According to the above figure, we can see that the degradation rate of GFP is basically stable within 48 hours. In other words, the reaction between p18 and GFP was basically completed. Therefore, we think that the experimental group can take 48 hours as the node of experimental data collection, so as to obtain stable data and follow-up experimental observation results.
RiPrePro
In order to determine which parameter have the most direct and dramatic influence on GFP degradation process, we used the built-in sensitivity analysis program to calculate the time-dependent sensitivities (derivatives) of GFP with respect to each parameter. By checking the magnitude of the computed sensitivities integrated over time, we could observe that k3f, the rate parameter characterizing the dimerization of FRB and FKBP, manifested the highest sensitivity across the board, which suggested that GFP protein level was most sensitive to k3f (Figure 8A). In contrast, other sensitivities, especially those associated with the ubiquitination process, were significantly lower. Similar trends could be observed in Figure 8B, the curves for other parameters stayed flat compared to that of k3f. All these results demonstrated that k3f was most important in determining the GFP degradation.
In order to further verify the effects of variations in k3f (k) on GFP abundance, the value of k was scanned within a range. Simulation showed that when k was set to zero, the GFP abundance resulted in a significant high level, which corresponded to the situation that the rapamycin was absent. With the growth of k value, GFP level stabled at an obviously lower value, suggesting an ever-increasing degradation efficiency. Such results indicated that we could optimize the RiPrePro by increasing the value of k, or in other words, increasing the dosage of rapamycin or the number of FKBP domains. Considering the toxicity of rapamycin to cells, the future experiment focused on adding up the number of FKBP domains.
Figure 8. the sensitivity analysis of parameters in the model
Conclusion
In conclusion, we have developed the GFP PrePro model and the RiPrePro model to simulate the biochemical processes involved in PrePro-mediated target protein degradation. The simulation showed that our model matched well with wet-lab data and provide optimal dosage and ratio of plasmids, as well as the proper reaction time for in vitro experiments. Also, the simulation of the RiPrePro model suggested a great strategy by increasing the dosage of rapamycin or numbers of the FKBP domain to optimize the degradation capacity of RiPrePro, which directly defines the subsequent orientation of in vitro experiments and exerts positive implications on our future design.
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The Protein Interaction Module
The Protein Degradation Module
