Difference between revisions of "Team:Fudan-TSI/Model"

When we were constructing the plasmid, we encountered a dilemma concerning how RT and Cre should be expressed. Firstly, we thought of putting them both under a same Lac operon so that their expression can be easily induced merely by one kind of inducer—IPTG. Meanwhile, we also considered using different inducers to achieve a more modular design which would be easier to control. As it would take a long time to test which induced expression scheme is better through experiments, we used modelling to test the two constructs. We modelled all the reactions involved and computed the yield of the desired product, i.e. recombined P_target. Through comparison of the yield acquired using these two induced expression schemes, we decided that the latter scheme should be employed for our system to perform better.

By common knowledge we can assume that, if the amount of RT and Cre needs to be different to achieve optimal yield, we should choose the second scheme and put them under different operons. On the contrary, if the yield reaches the maximum under the maximum amount of RT and Cre, the first scheme should be chosen.

In our initial attempt, we found that modelling all the reactions involved is rather difficult, as the reactions are in such a large number and all mixed together. This circumstance makes inspection of the reasonability of our models and parameters impossible. To overcome this issue, we decided to separate these reactions into three minor models and use the steady-state concentration of the substances derived from the previous model as the input of the next model. The three minor models are: induced expression model, reverse transcription model and Cre recombination model, corresponding to the 3 reaction steps in R-Evolution. The schematic diagram is shown in Fig. 1.

We first assumed that both genes encoding RT and Cre are placed together under a lac operon (Fig 2a). The repressor protein LacI is stably expressed in the cell, 2 molecules of LacI will form a dimer which binds to LacO DNA fragment and represses the expression of RT and Cre. When IPTG is added and transported into the cell, IPTG molecules will bind with LacI and inhibit its binding to LacO. In this way, RT and Cre can be rescued from suppression (Nikos et al.). Details of the substance names, parameter names and mathematical equations can be found in the appendix.

According to our modelling result, the amount of target protein (RT and Cre) will be extremely low when IPTG is not added (Fig. 2). The origin point represents the time when an E. coli comes into being through reproduction. As a result, the lac operon is not fully repressed by LacI dimer, causing a leakage expression of target protein (from 0 min to 1 min, Fig. 2b). After that, due to slow degradation rate of the target protein’s mRNA as well as the target protein itself, the amount of target protein will continue to accumulate to a certain amount (from 1 min to 5 min, Fig. 2b) after the lac operon is fully repressed. Finally, the degradation process removes target protein from the system (from 5 min to 50 min, Fig. 2b). When IPTG is added, we find that the concentration of protein product quickly rises (Fig. 2c). The steady-state concentration of target protein is 1.63 μM. This number will be used for further analysis.

From the first model, the concentration of both RT and Cre are acquired. The concentration of RT serves as input to the reverse transcription model. As the schematic diagram depicts (Fig. 3a), tRNA primer first binds with reverse transcriptase. When this complex binds with a certain fragment on the target sequence, which is called primer binding site (PBS), the reverse transcription will start and cDNA will be synthesized.

Although a more elaborate model of reverse transcription has been proposed by Kulpa et al, it includes many reactions whose kinetic properties are not well characterized. As a result, we simplified that model and came up with our own. Details of the substance names, parameter names and mathematical equations we used can be found in the appendix.

The modelling result is shown in Fig. 3b. It shows that the concentration of cDNA will accumulate at the presence of RT (whose initial concentration is 1.63 μM, computed by the induced expression model) and finally reach a steady-state of 66.5 nM. This number will be used for further analysis.

Our first assumption is that the genes encoding RT and Cre are both placed under lac operon and thus be expressed in the same amount. So now we are about to compute the yield of our desired product to identify whether this experimental setup is feasible. The model of the recombination process has been clearly described by Ehrlich et al. We made some changes to it according to our own experimental design. The schematic diagram is shown in Fig. 4a. Details of the substance names, parameter names and mathematical equations can be found in the appendix.

As is shown in the diagram, 2 Cre molecules bind with 1 loxP site successively, either on cDNA or P_target. Four Cre molecules will form a Holliday junction, and thus starting the recombination reaction. Two pairs of loxP will work together and complete the strand exchange between cDNA and P_target. After that, the recombined product is produced. What we are interested in is the percentage of recombined P_target among all P_targets in one E. coli. So, we turn to compute that percentage based on the model that we have established.

Unfortunately, we found that the amount of substances is too small. For example, the concentration of P_target is only 10 nM, which means there are only about 5 molecules of P_target in one cell. These small numbers caused some computational problems in Matlab when we were using its ODE solver (ode15s). To address this problem, we converted the units of the amount of the substances from mole per litter (M) to molecule. The units of the kinetic parameters are also converted accordingly. The necessity of these conversions is clarified in the appendix.

Now the recombination step is modeled under the initial condition of 5 molecules of non-mutated P_target, 785 molecules of Cre and 31 molecules of cDNA (Fig 4b). The last two numbers are the outputs of previous models after going through some unit conversion steps.

The result is disappointing. After a long period of reaction, no recombined P_target showed up. It is because there are too many Cre molecules so P_targets are all bounded by them and remain in the intermediate form. What’s more, P_target can't bind with T7 RNA polymerase and be transcribed as a consequence of Cre occupation. This leads to the system’s inability of undergoing further reverse transcription process, stopping cDNA’s production, resulting in a stop of the system, and rendering mutation accumulation impossible (Fig. 4c).

This result tells us that the number of Cre molecules needs to be much lower for the system to function. We then set out to determine how many Cre is optimal. After we fed the recombination model with cDNA and Cre at different concentrations, the problem seems to be clear as the yield of recombined P_target varies greatly responding to different numbers of cDNA and Cre (Fig. 4d). When cDNA is confined to 31 molecules, we will get no yield at all in the period of E. coli's replication cycle if the concentration of Cre is greater than 80 nanomoles. Instead, the yield is maximized when the final Cre concentration is around 27 molecules (Fig 4e).

Now we use the optimized number of Cre as the input to our third model. The result is shown in Fig. 4f, which is satisfactory. The recombined P_target finally occurs and P_target has a chance to bind with T7 RNA polymerase, which means mutated gene of interest could be transcribed and further mutated, thus making the accumulation of mutations possible (Fig 4g).

There is still something that is not well explained in our current model. The final percentage of recombined P_target is around 2.5%. The unit of the substance is molecules, so it means there is 0.125 recombined P_target in one cell, which is unrealistic. This problem reflects that converting the unit of substance into molecule when doing deterministic modelling cannot offer a precise description of the system’s status.

We then used stochastic modelling techniques to determine whether and how many recombined P_targets will show up in one replication cycle of E. coli.

We use Gillespie algorithm in stochastic modelling. Detailed description of this technique is described in the appendix. Although the algorithm is rather simple, basic mathematical skills is required to understand its theoretical basis. The result is shown in Fig. 5.

The result demonstrates that recombined P_targets do occur and two rounds of reverse transcription and recombination can take place in one replication cycle of E. coli (1200 s) (Fig 5a). On the contrary, no recombined P_target will come out within that period if the initial cDNA is 31 molecules and initial Cre is 785 molecules (Fig 5b). This again demonstrates the necessity of putting RT and Cre under different induction setups. The fluctuation of the number of recombined P_targets is due to the backward reaction that Cre can rebind with recombined P_target and reverting the action, making it not counted as recombined P_target by the algorithm.

In the first part, we demonstrated models which separate the 3 reaction steps and use the output from the preceding model as the input of the subsequent one. The previous setup successfully provided us with a clear insight into the reactions and dynamic changes of substances that underlie our mutagenesis system. However, this simplification doesn’t match real reaction situations. For example, when RT and Cre are expressed simultaneously upon induction, cDNA would bind with Cre and undergo recombination as soon as it is synthesized. This fact contradicts with our model assumption that recombination only takes place after cDNA has reached its steady-state concentration. To overcome this problem, we employed deterministic model to combine the separate steps together into one and better simulate real intracellular circumstances.

The first part of our model presents to us the optimal amount of Cre that should exist in the system, but leaves us with a problem concerning when Cre should be induced to achieve the greatest recombination efficiency. We first asked ourselves: can Cre function after cDNA accumulates to its steady-state, just as our previous model assumes? After inspecting the time required for the cDNA accumulation step, we found that this isn’t the case. The time needed for cDNA accumulation is close to the time length of a single E. coli replication cycle (1200 s). So if recombination happens only after cDNA reached its steady-state concentration, it does not happen at all. This can be explained by the substance division process when 1 parent E. coli reproduces into 2 child E. coli cells. As a result, when cDNA nearly reaches steady-state concentration in the parent E. coli, its concentration will consecutively be reduced by half in child E. coli, which breaks the steady-state again.

After realizing the fact that recombination cannot take place at steady-state cDNA concentration, we are faced with the second question: when should Cre be induced in one E. coli replication cycle, to enable the maximized percentage of recombined P_target? One possible answer is to induce the expression of Cre at the same time when RT is induced through a different inducer aTc (anhydrotetracycline). Under this method, recombination can occur throughout E. coli replication cycle, and thus has the longest duration. Adding the two inducers simultaneously in real experimental setup will further decreas the labor work of applying R-Evolution as well. However, at initial stages when cDNA concentration is minimized due to low concentration of RT and resulting in a low rate of cDNA synthesis process (reverse transcription), recombination efficiency will be at its minimal. To resolve this problem, we would like to find out whether there exists a certain time point that maximizes the recombination efficiency in one E. coli replication cycle by facilitating sufficient time for recombination as well as moderate initial reverse transcription.

By combining previous models (Part I. induced expression model, reverse transcription model, recombination model) and using the aTc induction model proposed by Steel et al. to simulate the Cre induced expression process (the schematic diagram of this process is shown in Fig 6a. Details of the substance names, parameter names and mathematical equations can be found in the appendix), we confirm that the optimal recombination efficiency will be achieved when expression of RT and Cre is induced at the same time point (the origin point represents the moment when IPTG is added to initiate RT expression, with 50 μM IPTG dosage and 1.75 μM aTc dosage), characterized by the maximized percentage of recombined product at the 20th minute (Fig 6 b-d, modeling different moment of aTc induction—the 5th min, 10th min, 15th min in b & c).

In the deterministic model, we combined the three minor models proposed previously and assessed the mutagenesis system in whole. Through this addition, we achieved a better simulation of the real intracellular reactions and answered the question of when Cre should be induced for the highest level of recombination efficiency to be obtaine

• [1]: Stamatakis M, Mantzaris N V. Comparison of Deterministic and Stochastic Models of the lac Operon Genetic Network[J]. Biophysical Journal, 2009, 96(3):887-906.
• [2]: Kulpa, D. Determination of the site of first strand transfer during Moloney murine leukemia virus reverse transcription and identification of strand transfer-associated reverse transcriptase errors[J]. EMBO (European Molecular Biology Organization) Journal, 1997, 16(4):856-865.
• [3]: Lanchy J M, Ehresmann C, Le Grice S F, et al. Binding and kinetic properties of HIV-1 reverse transcriptase markedly differ during initiation and elongation of reverse transcription.[J]. The EMBO Journal, 1996, 15(24):7178-7187.
• [4]: Kati W M, Johnson K A, Jerva L F, et al. Mechanism and fidelity of HIV reverse transcriptase[J]. Journal of Biological Chemistry, 1993, 267(36):25988-25997.
• [5]: Ringrose L, Lounnas V, Ehrlich L, et al. Comparative kinetic analysis of FLP and cre recombinases: mathematical models for DNA binding and recombination[J]. Journal of Molecular Biology, 1998, 284(2):0-384.
• [6]: Harris A W K, Kelly C L, Steel H, et al. The autorepressor: A case study of the importance of model selection[C]. Decision & Control. IEEE, 2018.