Model

CV-induction System

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We use crystal violet operator to improve our protein expressions. CV-induction system is both low-cost and highly efficient. The operator system is mainly composed of EilR gene (repressor protein) and Pjexd inducible promotor. When extracellular crystal violet is applied, the molecule will bind to EilR protein and release it from the promotor, allowing the downstream gene, which is our target protein, to express.

Based on chemical equilibrium, with our characterization data, we construct a model to illustrate the properties of this induction system as well as predicting the final protein yield.

(1) For the first part, we model the expression of EilR protein.

where Me refers to quantity of mRNA of the repressor protein EilR, c refers to the plasmid copy within a cell, aR refer to the promotor strength, and d1 refers to the degradation rate of the mRNA.

E refers to the quantity of protein EilR, bR refers to the translation strength, and d2 refers to the degradation rate of the protein.

(2) Then, we focus on the interactions between CV molecules, EilR proteins, and inducible promotor.

We use chemical equilibrium to predict of these parts, and C refers to Crystal Violet, E refers to EilR, and P refers to promotor.

(3) Finally, we predict the concentration of target protein.

where Mg refers to the quantity of the mRNA of GFP, cR refers to the transcription strength, and d3 refers to the degradation rate of the mRNA.

where G refers to the quantity of protein GFP, dR refers to the translation strength, and d4 refers to the degradation rate of GFP.

After supposing "t" levels off to infinity, where the the concentration of modeled mRNAs and proteins levels off to a constant value, and combining and solving for the constants through differential equations based on fitting characterization curve, we reset the variable to "c", the concentration of crystal violet, and plot the expression of protein vs CV concentration graph.

Figure 1. The GFP/OD vs CV curve

The value of the combination of the constants is shown below:

Figure 2. The value of combination of constants

The whole solving process is shown below:

CV model construction process

We'd like to thank Qirui Da for debugging our model, solving the differential equations, and plotting the curve.

The protein length and concentration modeling

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ProQC (Protein Quality Control) system is able to reduce the amount of useless protein expressions by truncated mRNA, thus increasing the efficiency of translation by preventing from wasting amino acids and recycling ribosomes. The system is composed of a “switch”, which is placed in front of the ORF and pairing up by itself to form stem apex structure on RBS, preventing ribosome binding and translation from happening, and a “trigger”, which is placed behind ORF, untying the “switch” stem apex, making the mRNA to form a loop, freeing the RBS, and thus initiating translation. Because only when the mRNA is at its full length, it can successfully start translation, ProQC system prevent from truncated mRNA from being translated and ensure the quality of yielded proteins.

Inspired by the mechanism of ProQC system, we designed a model, in which we predict the correlation between the length of the protein and the concentration of it in response to the aspect that ProQC system can affect. Because of time and apparatus limitations, we cannot contruct the plasmids and conduct the characterization. The following writings are all guidelines that can function as an instruction for further experiments.

The theoretical modeling process:

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Plasmid layout
lacI promoter-lacI-terminator T7-lacO-Cas12b-T7 terminator

Bacteria growth kinetics
OD600 = 0.6, add IPTG to induced Cas12b expression At OD600 = 0.6, cells are in exponential growth phase! Exponential growth phase is equivalent to steady state In biological system, we assume that all reactions are first order For example, mRNA --- degradation, degradation rate = kdegradation [mRNA]

Modeling details

• T7-lacO-Cas12b-T7 terminator
• Gene Cas12b is transcribed from DNA to mRNA; mRNA is translated to protein
• From DNA to mRNA, there’s a constant transcription rate
• From RNA to protein, there’s a constant translation rate

First, from DNA to RNA

Where c is the copy number of the plasmid; pfull is the fraction of full-length mRNA; aR is the mRNA transcription rate; km is the degradation rate of full-length mRNA; [mRNA]full is the concentration of full-length mRNA

At steady state,

Moreover, pfull is a function of the length of mRNA transcript

Therefore, we re-write pfull as f (L), a function that we don’t know

Second, from RNA to protein at steady state

In conclusion, the concentration of Cas12b is a function of L solely.

Therefore, the only unknown is the details of this function f (L), which is just the problem we want to solve in our modelling section.

Approach to solve f (L)

We fuse this construct: T7-sfGFP-GOI-mCherry-T7 terminator

We design GOI with different length (vioA, vioAB, vioABC, vioABCD, vioABCDE). Then we measure the kinetics of sfGFP and mCherry. The expected tendency is that sfGFP holds constant for these five constructs, whereas the fluorescence of mCherry decreases as L increases. Because fluorescence is directly proportional to [mCherry], we could use the mCherry fluorescence to represent [mCherry].

Finally, we would generate a scatter plot showing the relationship between fluorescence of mCherry and the length of transcript to solve for f (L).

The impact of this model is that we could predict the fraction of full-length mRNA given a specific transcript length.