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Directed Evolution

Objectives

1.Quantify the copy efficiency of positive and negative clones. The expression of TrpR was simulated by a mathematical model with tryptophan binding, the inhibition of TrpR was described by the ODE equation, and then the expression of Taq polymerase was characterized using GFP. Finally, using R, the difference in replication efficiency between positive and negative clones was simulated to obtain the final quantitative function.

2.After obtaining the difference in replication efficiency, the number of cycles used to select the final target is estimated by simulating cycles of directed evolution.


3.Based on the model results, mathematically simulate the entire pathway and mathematically validating the feasibility of directed evolution. Integration with experimental data, evaluation models.

Design

The general tryptophan repressor (TrpR) is not specific, and we wanted to obtain mutant that could stably suppress tryptophan (Trp) expression. Therefore, we apply the compartmentalized partnered replication (CPR) to perform positive and negative selection. Each one positive or negative selection is called a cycle (Figure 1).


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FIGURE 1
In vivo circuit diagrams used for positive and negative CPR selections (W. Ellefson et al., 2018).

The tryptophan repressor (TrpR) undergoes transcription, translation and dimerization to eventually form a dimer. The dimeric form of the tryptophan repressor (TrpR) has one Trp or Br-Trp binding site per monomer, each binding to a Trp or Br-Trp (Figure 2). In the positive selection environment, only tryptophan but not bromotryptophan is present. If TrpR is the repressor that we want to bind strongly with the tryptophan (Trp), then it is possible to inhibit the expression of the Lamda repressor after binding with tryptophan. The role of the Lamda repressor is to inhibit the expression of Taq polymerase. Hence, as Lambda repressor is inhibited, Taq expression will increase. To measure the amount of Taq polmerase, we construct pNEG- sfGFP and pPOS-sfGFP which replace Taq polymerase gene with sfGFP to characterize the expression of Taq polymerase.


In the negative screening environment, symmetrical to the positive screening, only bromotryptophan (Br-Trp) is present without tryptophan (Trp). If TrpR is the repressor we want that binds weakly to the bromotryptophan (Br-Trp), it does not inhibit Taq expression, allowing the increase in the weakly chromotryptophan-binding repressor that we want, characterized by GFP.


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FIGURE 2
The TrpR regulatory program controls the negative-feedback inhibition during L-tryptophan biosynthesis (W. Ellefson et al., 2018).

Model-Positive Selection

Reaction I \(TrpR2\) Synthesis

The synthesis of tryptophan repressor is divided into three steps: transcription, translation and dimerization. In the following ODE equation describing TrpR transcription, \(mRNA_{TrpR}\) is the concentration of TrpR mRNA; \(k_{TC1}\) is the transcription rate of DNA and \(k_{DedM}\) is the degradation rate of mRNA.

$$ \frac{\text{d}\left[ mRNA_{TrpR} \right]}{\text{dt}}=k_{TC1}DNA_{TrpR}-k_{DegM}\left[ mRNA_{TrpR} \right] $$ In the translation equation (2), \(k_{TL1}\) is the translation rate. \(k_{DegP}\) repersents the degration rate of TrpR. \(K_{Di1}\) is TrpR dimerizing rate and thus \(2k_{Di1}{[TrpR]}^2\) indicates that the resulting molecule will undergo dimerization. $$ \frac{\text{d}\left[ TrpR \right]}{\text{dt}}=k_{TL1}\left[ mRNA_{TrpR} \right] -k_{DegP}\left[ TrpR \right] -2k_{Di1}\left[ TrpR \right] ^2 $$ The dimerization process is the formation of a dimer from two TrpR . What’s more, as is to be described in Reaction2 $$ \frac{\text{d}\left[ TrpR2 \right]}{\text{dt}}=k_{Di1}\left[ TrpR \right] ^2-k_{DegP}\left[ TrpR2 \right] +k_{Dis-Asso}\left[ TrpR2-Trp \right] $$
Reaction II \(TrpR2\) combine with Trp
Dimerisation leads to a conformational change of active site of each TrpR, enabling each monomer to bind to a molecule of either Trp . Therefore, TrpR2 can bind to two Trp molecules, forming TrpR2-Trp or TrpR2-BT (Figure 3) $$ \left[ TrpR2 \right] +2\left[ Trp \right] \left[ TrpR2-T \right] $$ Blog One
FIGURE 3
TrpR2 bind with two molecules of Trp or Br-Trp

Considering both association and seperation of TrpR2-Trp or TrpR2-BT, we have the following equation: $$ \frac{\text{d}\left[ TrpR2-Trp \right]}{\text{dt}}=k_{AR\_T}\left[ TrpR2 \right] \left[ Trp \right] ^2-k_{sepe2}\left[ TrpR2-Trp \right] $$ Where the association and the separation rate of TrpR2-Trp (or TrpR2-BT) is \(k_{AR\_T}\) (or \(k_{AR\_BT}\) ) and \(k_{sepe2}\) respectively.
While during the formation of TrpR2-Trp or TrpR2-BT, the saaociation between TrpR2 and Trp or Br-Trp only takes place when the Trp concentration reaches a certain range. Therefore, a constant term [\(Trp_0\)] or [\(Br-Trp_0\)] is added. Hence we get the corrected equations: $$ \frac{\text{d}\left[ TrpR2-T \right]}{\text{dt}}=k_{AR\_T}\left[ TrpR2 \right] \frac{\left[ Trp \right] ^2}{\left[ Trp \right] ^2+\left[ Trp_0 \right] ^2}-k_{sepe2}\left[ TrpR2-Trp \right] $$
Reaction III \(TrpR2\)-Trp suppress the transcription of CI
The repression of cI transcription by Trp2-Trp could be represented by modifying transcription rate with a Hill equation: ： $$ \frac{\text{d}\left[ mRNA_{CI} \right]}{\text{dt}}=\frac{k_{TC2}}{1+\left( \frac{\left[ TrpR2-T \right]}{K_{Hill1}} \right) ^n}DNA_{CI}-k_{DegM}\left[ mRNA_{CI} \right] $$ The following two equations describe the process of CI protein translation (8) and dimerization (9)： $$ \frac{\text{d}\left[ CI \right]}{\text{dt}}=k_{TL2}\left[ mRNA_{CI} \right] -k_{DegP}\left[ CI \right] $$ $$ \frac{\text{d}\left[ CI2 \right]}{\text{dt}}=k_{Di2}\left[ CI \right] ^2-k_{DegP}\left[ CI2 \right] $$
Reaction IV CI protein inhibit the expression of Taq
Similarly, the inhibition of Taq expression by cI also follows Hill equation shown in equation (10). The expression of Taq mRNA is represented by equation (11) : $$ \frac{\text{d}\left[ mRNA_{Taq} \right]}{\text{dt}}=\frac{k_{TC3}}{1+\left( \frac{\left[ CI2 \right]}{K_{Hill2}} \right) ^n}DNA_{Taq}-k_{DegM}\left[ mRNA_{Taq} \right] $$ $$ \frac{\text{d}\left[ Taq \right]}{\text{dt}}=k_{TL3}\left[ mRNA_{Taq} \right] -k_{DegP}\left[ Taq \right] $$
Corrected Reaction IV CI protein inhibit the expression of
Since GFP is ultimately used to characterize the amount of Taq, we corrected Reaction 4 by replacing Taq with GFP. Below is the corrected Reaction4. $$ \frac{\text{d}\left[ mRNA_{GFP} \right]}{\text{dt}}=\frac{k_{TC4}}{1+\left( \frac{\left[ CI \right]}{K_{Hill2}} \right) ^n}DNA_{GFP}-k_{DegM}\left[ mRNA_{GFP} \right] $$ $$ \frac{\text{d}\left[ GFP \right]}{\text{dt}}=k_{TL2}\left[ mRNA_{GFP} \right] -k_{DegP}\left[ GFP \right] $$

Model-Negative Selection

The negative cycle is essentially the same as the positive cycle, the difference being that the substrate in the reaction environment is changed from tryptophan to bromotryptophan. Therefore, the only difference from the equation of the positive cycle is the replacement of Trp with Br-Trp. All reaction equations are as follows：

Reaction I \(TrpR^2\) Synthesis

$$ \frac{\text{d}\left[ mRNA_{TrpR} \right]}{\text{dt}}=k_{TC1}DNA_{TrpR}-k_{DegM}\left[ mRNA_{TrpR} \right] $$
$$ \frac{\text{d}\left[ TrpR \right]}{\text{dt}}=k_{TL1}\left[ mRNA_{TrpR} \right] -k_{DegP}\left[ TrpR \right] $$
$$ \frac{\text{d}\left[ TrpR2 \right]}{\text{dt}}=k_{Di}\left[ TrpR \right] ^2-k_{Sepe1}\left[ TrpR2 \right] $$
Reaction2：\(TrpR^2\) combine with Br-Trp

$$ \left[ TrpR2 \right] +2\left[ Br-Trp \right] \left[ TrpR2-TB \right] $$
$$ \frac{\text{d}\left[ TrpR2-TB \right]}{\text{dt}}=k_{AR\_BT}\left[ TrpR2 \right] \frac{\left[ Br-Trp \right] ^2}{\left[ Br-Trp \right] ^2+\left[ Br-Trp_0 \right] ^2}-k_{sepe2}\left[ TrpR2-TB \right] $$
Reaction3: \(TrpR^2\)-TB suppress the representation of CI

$$ \frac{\text{d}\left[ mRNA_{taq} \right]}{\text{dt}}=\frac{k_{TC3}}{1+\left( \frac{\left[ TrpR2-TB \right]}{K_{Hill1}} \right) ^n}DNA_{CI}-k_{DegM}\left[ mRNA_{taq} \right] $$
$$ \frac{\text{d}\left[ taq \right]}{\text{dt}}=k_{TL3}\left[ mRNA_{taq} \right] -k_{DegP}\left[ taq \right] $$
Reaction4: CI protein inhibit the expression of

$$ \frac{\text{d}\left[ mRNA_{GFP} \right]}{\text{dt}}=\frac{k_{TC4}}{1+\left( \frac{\left[ TrpR2-TB \right]}{K_{Hill1}} \right) ^n}DNA_{GFP}-k_{DegM}\left[ mRNA_{GFP} \right] $$
$$ \frac{\text{d}\left[ GFP \right]}{\text{dt}}=k_{TL2}\left[ mRNA_{GFP} \right] -k_{DegP}\left[ GFP \right] $$

Outcome

We employed ordinary differential equation to model the transcription and translation dynamics of CPR selection process, containing both positive and negative selection. We used the ODE solver“deSolve”in R to simulate our selection system.


As shown by the graph (Figure 2.4A), within 2 hours, TrpR reaches its plateau at about 150 and at the same time TrpR gets rapid polymerization formation into TrpR2, which peaks at Time = 20 h, 75 . Compared with TrpR and TrpR2, TrpR2-T's formation process is much lower, reaching it saturated at Time = 40h, 25 uM. In the CI, CI2 graph (Figure 2.4B), CI quickly reaches its peak at Time = 8 h, 3.8 and then it go through a steady but slow decreasing process to about 1.7 uM. While for the CI2 protein, it has the similar trend with CI and its peak and steady state concentration is half of that of CI.


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FIGURE 4
A) TrpR, TrpR2 and TrpR2-T expression changed with time B) CI and CI2 expression changed with time C) GFP expression changed with time D) CI2 and GFP expression changed with increased Trp concentration.

While for the GFP graph (Figure 2.4 C), we could see in the first day (within 24h), GFP quickly peaks at about 2 μm, while after that it experiences a significant drop to no more than 0.5 μm. And soon, GFP quickly get translation. Apart from that, we model the influence of Trp's concentration in directed evolution system. The x-axis means Trp increase from to 1, y-axis means relatively expression level (Figure 2.4D). In the figure, we can see, when Trp's concentration is under 20×10^-3 μm. CI2 protein is at its saturate state. When Trp increases to 30×10^-1μm , CI2 experiences a significant drop, from 2.0 μm to 0.5 μm. In the figure, GFP's relative expression process is contrary to CI2 trend. Below Trp concentration 20×10^-3 μm, GFP's expression is close to 0. But as trp's concentration increases.
Our model may help us know better about this directional evolution system. As is seen in the results, our model shows how the positive and negative screening system works. TrpR and TrpR2 quickly reach its saturate state, with proper Trp concentration, TrpR2 and Trp's polymer can generate, which can inhibit the expression of CI2. And as a result, Taq enzyme can express. What's more, with the increase of Trp concentration, the relative expression of GFP has a similar trend with experiment data in the lab. According to this, we could estimate the combined rate between TrpR2 and Trp, which has a huge benefit to our gene design.

Comparing to the experimental data

In our experiments, we assumed a constant tryptophan concentration. Since GFP expression is affected by tryptophan-controlled gene expression circuit, GFP concentrations could be predicted by tryptophan concentrations supplied in the beginning (Figure 5). As the concentration of tryptophan supplied increases, the final concentration of GFP decreases following a logistic decay. Meanwhile, the decline slows down when the concentration of tryptophan is over 0.5 mM.


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FIGURE 5
Predicted model based on modeling result

We calculated the regression curve of the experimental data using the function “geom_smooth” in package “tidyverse” to conduct a LOESS regression curve and found the correlation between tryptophan concentration and RPU/OD600, a direct reflection of GFP concentration, also follows a logistic decay pattern (Figure 6).


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FIGURE 6
LOESS regression curve of experimental data

Comparing theoretical model with experimental data, we found that our model could make a fairly accurate prediction on GFP expression level based on tryptophan supplied. Our model is not sensitive when the concentration of Trp is above 10^ (-2) and under 10^ (-0.5) without showing the trend of decreasing slowly. However, our model shows accuracy when the concentration of Trp is above 10^ (-0.5) and under 10^ (-2) .