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Revision as of 18:24, 20 October 2019
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- Overview
- Part I: Yield of recombined Ptarget
- Part II: Times of occurrence of recombined Ptarget
- Part III: Optimal induction time
- Complementary materials
- Reference
As we are designing a brand-new mutagenesis system inside E. coli, we want to demonstrate whether and under what condition it can work, so we turn to modelling to answer these questions. Our modelling work is comprised of 3 parts. 1) We used 3 deterministic models to describe the 3 reaction steps of our system—induced expression, reverse transcription and recombination. This allows us to compute and maximize the yield of the recombined Ptarget which in turn, contributes to the optimization of our experimental setup. 2) We simulated the recombination process stochastically and calculated the number of recombined products that occurred during one replication cycle of E. coli. 3) We combined the 3 reaction steps together using deterministic model and found that selecting the least efficient degradation tag for Cre is optimal.
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.
![](https://static.igem.org/mediawiki/2019/5/54/T--Fudan-TSI--Fig1.gif)
Three Grey boxes indicate three major reaction steps in R-Evolution. Arrows indicate the reaction that certain substance is involved. White arrows indicate the case in which substances that originally exist in E.coli act as inputs. Red arrows indicate the case in which intermediates, which are produced in the previous reaction, are generated or involved in next reaction process. The blue arrow indicates the final output that we would like to observe. Inducer – IPTG or aTc (anhydrotetracycline). RT – reverse transcriptase. Cre – Cre recombinase. cDNA – complementary DNA.
![](https://static.igem.org/mediawiki/2019/6/6f/T--Fudan-TSI--Formula1.gif)
![](https://static.igem.org/mediawiki/2019/archive/5/59/20191020044015%21T--Fudan-TSI--Fig2b%26c.gif)
a) Schematic diagram of the model. b) Dynamics of target protein. Horizontal axis shows the length of time. Vertical axis demonstrates the amount of protein (RT and Cre) within the system. RT and Cre are expressed under the same Lac operon. c) Dynamics of free lac operon. Horizontal axis shows the length of time. Vertical axis demonstrates the amount of free lac operon, i.e. the lac operon unbound by tetR dimer, within the system. The vertical magenta line indicates the moment when 50μM is added to the system.
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. The ODEs describing these processes are shown as follows. Details of the substance names, parameter names and chemical equations we used can be found in the appendix.
![](https://static.igem.org/mediawiki/2019/3/3f/T--Fudan-TSI--Formula2.gif)
![](https://static.igem.org/mediawiki/2019/3/31/T--Fudan-TSI--Fig3.gif)
a) Schematic diagram of the model. b) Dynamics of cDNA. Horizontal axis shows the length of time. Vertical axis demonstrates the amount of cDNA within the system.
![](https://static.igem.org/mediawiki/2019/e/e9/T--Fudan-TSI--Formula3.gif)
Unfortunately, we found that the amount of substances is too small. For example, the concentration of is only 10 nM, which means there are only about 5 molecules of 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.
Now the recombination step is modeled under the initial condition of 5 molecules of non-mutated , 3228 molecules of Cre and 5 molecules of cDNA (Fig 4b). The last two numbers are the outputs of previous models after going through some unit conversion steps.
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 varies greatly responding to different numbers of cDNA and Cre (Fig. 4d). When cDNA is confined to 5 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 20 molecules. Instead, the yield is maximized when the final Cre concentration is around 2 molecules (Fig 4e).
We then used stochastic modelling techniques to determine whether and how many recombined Ptargets will show up in one replication cycle of E. coli.
![](https://static.igem.org/mediawiki/2019/d/d8/T--Fudan-TSI--Fig4.gif)
a) Schematic diagram of the model. Ps, Un-recombined Ptarget. Pp, Recombined Ptarget. b-c) Recombination when Cre is expressed under Lac operon. Dynamics of the percentage of un-recombined/ recombined Ptarget among all Ptargets is shown in b. Horizontal axis shows the length of time (8 hours, corresponding to R-Evolution’s function period). The distribution of the percentage of substances at the steady-state is shown in c. d) Yield of recombined Ptarget at different initial number of cDNA and Cre. The yield of recombined Ptarget is calculated as the percentage of recombined Ptarget among all Ptargets. The horizontal white line corresponds to current situation where the initial number of cDNA is 5 molecules in one E.coli. e) Yield of recombined Ptarget at different initial number of Cre when initial number of cDNA is 5 molecules. f-g) Recombination when Cre is expressed under different operon. Dynamics of the percentage of un-recombined/ recombined Ptarget among all Ptargets is shown in f. The distribution of the percentage of substances at the steady-state is shown in g.
Step 1: Initialize the reaction system at \(t=0\) with rate constants \(c_1, c_2, ......, c_v\) as initial numbers of molecules \(x_1, x_2, ......, x_u\) corresponding to \(v\) reactions and \(u\) sustances (both reactants and products) involved in the reaction system.
Step 2: For each \(i=1,2,......,v\), calculate the hazard for the \(i^{th}\) type of reaction, denoted as \(h_i(x,c_i)\) based on current substance number \(x\).
Step 3: Calculate the combined reaction hazard \(h_0(x,c)=\Sigma_{i=1}^{v}h_i(x,c_i)\).
Step 4: Simulate the time to the next reaction, \(t^{'}\) , which is a random quantity that obeys exponential distribution with parameter \(\lambda\).
Step 5: Put \(t:=t+t^{'}\).
Step 6: Simulate the reaction index, \(j\). The probability that the \(j^{th}\) reaction can occur is \(\frac{h_i(x,c_i)}{h_0(x,c)}, i=1,2,......,v.\).
Step 7: Update \(x\) according to reaction \(j\), which means putting \(x:=x+S^{(j)}\), where \(S^{(j)}\) denotes the \(j^{th}\) colomn of the stoichiometry matrix \(S\). The \(j^{th}\) column of denotes the change in substance number caused by the \(j^{th}\) reaction.
Step 8: Record time \(t\) and current substance number \(x\).
Step 9: If \(t\ \text{<}\ T_{max}\), return to step 2. \(T_{max}\) corresponds to the maximum duration of the reaction set by the user.
Step 10: Plot the result to see the dynamic of the quantity of the substance that you are interested in.
Although the algorithm is rather simple, basic mathematical skills is required to understand its theoretical basis. You may consult the book written by Wilkinson and Darren J. for a thorough understanding. The result is shown in Fig. 5.
The result demonstrates that recombined Ptargets 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 will come out within that period if the initial cDNA is 5 molecules and initial Cre is 3228 molecules (Fig 5b). This again demonstrates the necessity of putting RT and Cre under different induction setups. The fluctuation of the number of recombined Ptarget is due to the backward reaction that Cre can rebind with recombined and reverting the action, making it not counted as recombined Ptarget by the algorithm.
![](https://static.igem.org/mediawiki/2019/5/5f/T--Fudan-TSI--Fig5bc.gif)
Horizontal axis shows the length of time (20min, corresponding to the duration of 1 E.coli replication cycle). Vertical axis demonstrates the number of recombined Ptarget. The initial number of Cre is 2 molecules in a, 3228 molecules in b.
We intend to use the models described in Part I, combined with aTc induction model proposed by Steel et al. to compute the yield of recombined Ptarget under different degradation rate of Cre (the reason why Tet operon is used has been elaborated in Part I; the schematic diagram of this process is shown in Fig 6a. The ODEs describing these processes are shown as follows. Details of the substance names, parameter names and chemical equations we used can be found in the appendix.
![](https://static.igem.org/mediawiki/2019/9/9c/T--Fudan-TSI--Formula4.gif)
To overcome this issue, we decided to combine all three minor models together and calculate the expected output.
As a result of the impreciseness of the basic assumption of the models in part I, we only gave a qualitative conclusion that the amount of RT and Cre should be different. Here we need to quantify how Cre degradation rate and steady-state concentration affects the yield of recombined Ptarget. That’s why we employed deterministic model here to combine the separate steps together into one and better simulate real intracellular circumstances.
By combining the models that have been talked above, we revealed the reason why the degradation tag with a moderate degradation rate, which can’t be too high or too low, should be selected (Fig 6b): under appropriate inducer concentration (20~22uM), when the degradation rate is relatively low (below 0.1 min-1), the yield of recombined Ptarget will increase according to the increase of Cre degradation rate, but when that rate is sufficiently high (above 0.1 min-1), the increase of Cre degradation rate will do harm to the yield of recombined Ptarget.
The average degradation rate acquired from literature is 0.2 min-1(Nikos et al.) and the degradation rate of Cre when tagged with the most efficient degradation tag is 0.69 min-1. Within this range of degradation rate, the maximum yield of recombined Ptarget will decrease according to the increase of Cre degradation efficiency (Fig 6c). So we decided to use the least efficient degradation tag.
We also revealed the dynamic change of the recombined Ptarget. It will continuously accumulate within Cre function period (Fig 6d). However, the concentration remains to be low within that period, due to Cre degradation (Fig 6e).
Finally, there is another interesting phenomenon that is worth mentioning. From Fig 6b and Fig 6c, we can find that for each degradation tag rate greater than 0.2 min-1, there exits a range of aTc dosage that can make the yield of recombined relatively big. Also, decreased degradation efficiency enlarges that range. This discovery provides us with another reason for using less efficient degradation tag in that it can increase the robustness of our mutagenesis system by decreasing its sensitivity to the change of inducer dosage.
![](https://static.igem.org/mediawiki/2019/archive/8/84/20191020042934%21T--Fudan-TSI--Fig6.gif)
a) Chemical reactions involved in Cre induced expression. b) Yield of recombined Ptarget at different Cre degradation rate and aTc dosage. The white line on the left corresponding to the case where the degradation rate of Cre is 0.2 min-1. The white line on the left corresponding to the case where the degradation rate of Cre is 0.69 min-1. c) Yield of recombined Ptarget at different Cre degradation rate and aTc dosage (3D plot). The range of Cre degradation rate is 0.2~0.69 min-1. d) Dynamics of yield of recombined Ptarget at the Cre degradation rate of 0.2 min-1 and the initial 22uM aTc dosage. e) Dynamics of yield of recombined Ptarget at the Cre degradation rate of 0.2 min-1 and the initial 22uM aTc dosage.
- [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]. 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.
- [4]. Wilkinson, Darren J. Stochastic Modelling for Systems Biology, Second Edition[M]. Crc Press, 2011.
- [5]. 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.