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Our mutagenesis system uses the BL21 (DE3) <i>E. coli</i> strain transformed with two plasmids, a stringent plasmid named P<sub>target</sub> carrying the target sequence that we want to mutate, and a relaxed plasmid named P<sub>mutant</sub>, carrying the gene encoding the tools necessary for mutagenesis, i.e. reverse transcriptase (RT) and Cre. <br /><br /> | Our mutagenesis system uses the BL21 (DE3) <i>E. coli</i> strain transformed with two plasmids, a stringent plasmid named P<sub>target</sub> carrying the target sequence that we want to mutate, and a relaxed plasmid named P<sub>mutant</sub>, carrying the gene encoding the tools necessary for mutagenesis, i.e. reverse transcriptase (RT) and Cre. <br /><br /> | ||
− | As we are designing a brand-new mutagenesis system inside <i>E. coli</i>, 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. | + | As we are designing a brand-new mutagenesis system inside <i>E. coli</i>, 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 P<sub>target</sub> 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 <i>E. coli</i>. 3) We combined the 3 reaction steps together using deterministic model and found that selecting the least efficient degradation tag for Cre is optimal. |
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− | 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 | + | We first assumed that both genes encoding RT and Cre are placed together under a lac operon (<a href="#Fig2">Fig 2a</a>). 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 (<a href="#Ref1">Nikos et al.</a>). The ordinary differential equations (ODEs) describing these processes are shown as follows. Details of the substance names, parameter names and chemical equations can be found in the appendix.<br /><br /> |
− | According to our modelling result, the amount of target protein (RT and Cre) will be extremely low when IPTG is not added (Fig. | + | According to our modelling result, the amount of target protein (RT and Cre) will be extremely low when IPTG is not added (<a href="#Fig2">Fig. 2b</a>). 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, <a href="#Fig2">Fig. 2b&c</a>). 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 after the lac operon is fully repressed (from 1 min to 5 min, <a href="#Fig2">Fig. 2b&c</a>). Finally, the degradation process removes target protein from the system (from 5 min to 50 min, <a href="#Fig2">Fig. 2b</a>). When IPTG is added, we find that the concentration of protein product quickly rises as the repression of lac operon is quickly removed (<a href="#Fig2">Fig. 2b&c</a> from 50 min to 100 min ). The steady-state concentration is 6.70 μM. This number will be used for further analysis. |
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− | 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.<br /><br /> | + | 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 (<a href="#Fig3">Fig. 3a</a>), 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.<br /><br /> |
− | 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 | + | Although a more elaborate model of reverse transcription has been proposed by <a href="#Ref2">Kulpa et al.</a>, 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.<br /><br /> |
− | 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 | + | The modelling result is shown in <a href="#Fig3">Fig. 3b</a>. It shows that the concentration of cDNA will accumulate at the presence of RT (whose initial concentration is 6.70 μM, computed by the induced expression model) and finally reach a steady-state of 9.60 nM. This number will be used for further analysis. |
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− | 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 | + | 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 <a href="#Ref3">Ehrilich et al</a>. We made some changes to it according to our own experimental design. The schematic diagram is shown in <a href="#Fig4">Fig. 4a</a>. The ODEs describing these processes are shown as follows. Details of the substance names, parameter names and chemical equations can be found in the appendix.<br /><br /> |
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.<br /><br /> | 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.<br /><br /> | ||
− | Unfortunately, we found that the amount of substances is too small. For example, the concentration of | + | 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. The necessity of these conversions is clarified in the appendix.<br /><br /> |
− | Now the recombination step is modeled under the initial condition of 5 molecules of non-mutated | + | 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 (<a href="#Fig4">Fig 4b</a>). The last two numbers are the outputs of previous models after going through some unit conversion steps. |
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We use Gillespie algorithm in stochastic modelling. The procedure of this algorithm[7] is shown as follows in the form of pseudocode:<br /><br /> | We use Gillespie algorithm in stochastic modelling. The procedure of this algorithm[7] is shown as follows in the form of pseudocode:<br /><br /> | ||
Step 1: Initialize the reaction system at \(t=0\) with rate constants \(c1, c2, ......, cv\) as initial numbers of molecules \(x1, x2, ......, xu\) corresponding to \(v\) reactions and \(u\) sustances (both reactants and products) involved in the reaction system.<br /><br /> | Step 1: Initialize the reaction system at \(t=0\) with rate constants \(c1, c2, ......, cv\) as initial numbers of molecules \(x1, x2, ......, xu\) corresponding to \(v\) reactions and \(u\) sustances (both reactants and products) involved in the reaction system.<br /><br /> | ||
− | Step 2: For each \(i=1,2,......,v\), calculate the hazard for the i<sup>th</sup | + | Step 2: For each \(i=1,2,......,v\), calculate the hazard for the i<sup>th</sup> type of reaction, denoted as \(h_i(x,c_i)\) based on current substance number x.<br /><br /> |
Step 3: Calculate the combined reaction hazard \(h_0(x,c)=\sigma_{i=1}^{v}h_i(x,c_i)\).<br /><br /> | Step 3: Calculate the combined reaction hazard \(h_0(x,c)=\sigma_{i=1}^{v}h_i(x,c_i)\).<br /><br /> | ||
Step 4: Simulate the time to the next reaction, \(t^'\) , which is a random quantity that obeys exponential distribution with parameter \(\lambda\).<br /><br /> | Step 4: Simulate the time to the next reaction, \(t^'\) , which is a random quantity that obeys exponential distribution with parameter \(\lambda\).<br /><br /> | ||
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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 \(P_{target}\). That’s why we employed deterministic model here to combine the separate steps together into one and better simulate real intracellular circumstances.<br /><br /> | 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 \(P_{target}\). That’s why we employed deterministic model here to combine the separate steps together into one and better simulate real intracellular circumstances.<br /><br /> | ||
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 (<a href="#Fig6">Fig 6a</a>).: under appropriate inducer concentration (20~22uM), when the degradation rate is relatively low (below 0.1 min^{-1}), the yield of recombined \(P_{target}\) 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 \(P_{target}\).<br /><br /> | 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 (<a href="#Fig6">Fig 6a</a>).: under appropriate inducer concentration (20~22uM), when the degradation rate is relatively low (below 0.1 min^{-1}), the yield of recombined \(P_{target}\) 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 \(P_{target}\).<br /><br /> | ||
− | The average degradation rate acquired from literature is 0.2 min^{-1}<sup><a href="#Ref1">[1]</a></sup | + | The average degradation rate acquired from literature is 0.2 min^{-1}<sup><a href="#Ref1">[1]</a></sup> 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 \(P_{target}\) will decrease according to the increase of Cre degradation efficiency (<a href="#Fig6">Fig 6b</a>). So we decided to use the least efficient degradation tag.<br /><br /> |
We also revealed the dynamic change of the recombined \(P_{target}\). It will continuously accumulate within Cre function period (<a href="#Fig6">Fig. 6c</a>). However, the concentration remains to be low within that period, due to Cre degradation (<a href="#Fig6">Fig. 6d</a>).<br /><br /> | We also revealed the dynamic change of the recombined \(P_{target}\). It will continuously accumulate within Cre function period (<a href="#Fig6">Fig. 6c</a>). However, the concentration remains to be low within that period, due to Cre degradation (<a href="#Fig6">Fig. 6d</a>).<br /><br /> | ||
Finally, there is another interesting phenomenon that is worth mentioning. From <a href="#Fig6">Fig. 6a</a> and <a href="#Fig6">Fig. 6b</a>, 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. | Finally, there is another interesting phenomenon that is worth mentioning. From <a href="#Fig6">Fig. 6a</a> and <a href="#Fig6">Fig. 6b</a>, 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. | ||
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− | <b>Figure 6. | + | <b>Figure 6. Whole process simulation considering Cre degradation tag.</b> <br /> a) 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}. b) 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}. c) Dynamics of yield of recombined \(P_{target}\) at the Cre degradation rate of 0.2 min^{-1} and the initial 22uM aTc dosage. d) Dynamics of yield of recombined \(P_{target}\) at the Cre degradation rate of 0.2 min^{-1} and the initial 22uM aTc dosage. |
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Revision as of 19:27, 19 October 2019