There are three genes under the control of the PlacO promoter including the msr, msd(rpobR), and RT, represented by variables xamount of msr , yamount of msd(rpobR) , and zamount of reverse transcriptase , respectively. These equations detail the production rate of mRNA, indicating their individual transcriptional rates . By subtracting the rate of degradation of mRNA times the amount of mRNA, of that respective gene, from the optimal transcription rate of that gene, the rate of transcription is obtained. We represented the degradation rate for each gene as the same variable due to the negligible difference between them.

=(-(·))
=(-(K_degradation·))
=( -(K_degradation·))

Below, the production rate of protein from mRNA, for each of the genes under the PlacO promoter is detailed. Multiplication of the rate of translation by the amount of mRNA of each gene indicates the optimal translation for each gene, under ideal conditions. However, to account for degradation of the protein over time, subtracting the rate of degradation times the amount of protein, helps to account for the discrepancy between ideal conditions and realistic, lab results.

=( ·X_m)-(K_{degradation_{x_p}}·)
=(K_translation·Y_m)-(K_{degradation_{y_p}}·)
=(K_translation·Z_m)-(K_{degradation_{z_p}}·)

Dynamics regulated by Van Promoter in front of Beta Recombinase

mRNA:

=( -(K_degradation·))

Protein:

=(K_translation·U_m)-(K_{degradation_{u_p}}·)

Cas9 Plasmid Dynamics

The second of the three plasmids in our system is PCas9CR4, which checks for the mutation from the SCRIBE system in the E. coli DH5 and kills the wild-type bacteria. In the PCas9CR4 plasmid, the tet operator is typically repressed by tetR. However, in the presence of ATC, the system is induced and Cas9 is expressed. We sought to capture this repressible system in terms of differential equations for a few reasons to better understand this how PCasCR4 works under different conditions.

Figure 1

α= maximal production rate of TetR

θ= the dissociation constant of TetR

The expression which represents the amount of Cas9 (s) produced is as follows:

=α^θⁿ⁄_{θⁿ+ramount of tetRⁿ}

Expanding upon this, other factors which contribute to the overall presence of Cas9 mRNA include the degradation rate of the Cas9 mRNA and amount of mRNA present.

=^θⁿ⁄_{θⁿ+r_pⁿ}-K_{degradation_{S_m}}·S_m

Then, the production rate of the Cas9 protein is dependent upon the rate of translation as well as the amount of Cas9 mRNA present. Subtracting by the degradation rate of the Cas9 mRNA times the amount of protein present will account for the less than optimal rate of production of tetR protein.

=K_translation· -K_{degradation_{N_m}}·N_p

The rate of production of the tetR mRNA is:

=·θⁿ⁄_{θⁿ+r_pⁿ}-K_{degradation_{R_m}}·

The rate of production of tetR protein under ideal conditions is equivalent to the rate of translation. However, to account for other factors which could impact the overall production of the protein, this equation takes into account the rate of degradation of tetR protein as well as the binding rate (β) of the inducer, ATc, to the tetR repressor protein and the amount of the ATc in the cell (a).

=K_{translation_{R_m}}-K_{degradation_{R_pamount of tetR protein}}·B_r- _{R_p}·

Optimally, the rate of ATc entering the cell would be determined by the rate of induction of ATc into the cell, which takes into account the concentration of ATc outside the cell times the rate of diffusion into the cell. But, to gain a more realistic sense of the rate of ATc entering the cell we account for the degradation rate of the ATc times the amount of ATc as well as the binding rate of the ATc to the tetR repressor protein.

ȧ=·-k_{degradation_a}·a-β_{R_p}·a

Steady State Analysis:

The system in equilibrium is modeled below:

1. β_{R_p}·a=c·d 2. β_{R_p}·a=(k_{translation_r} ⁄ k_{degradation_{r_m}}) · θⁿ⁄_{θⁿ+r_pⁿ}=c·d 3. =(1⁄K_{degradation_{S_p}})· (k_{translation_s} ⁄ k_{degradation_{s_m}}) ·α_s· θⁿ⁄_{θⁿ+r_pⁿ}=c·d ⁄ k_{degradation_{s_p}}

Graphs from MATLAB

In the absence of inducer, ATc, the tetR repressor typically binds to the tet operator, preventing transcription. However, when inoculating the cell with ATc, it binds to the tetR repressor, causing the repressor to unbind from the operator. This allows for the transcription and ultimate translation of Cas9. On the left, as the concentration of ATc increases outside of the cell, then it diffuses, there is a linear increase in the number of Cas9 molecules produced. This linear model was also seen in "Negative autoregulation linearizes the dose–response and suppresses the heterogeneity of gene expression" by Dmitry Nevozhay, Rhys M. Adams, Kevin F. Murphy, Krešimir Josić, and Gábor Balázsi.

The mRNA over time graph is used to show the changes in the Cas9 and tetR mRNA over time. Each of the graph lines are formulated using modeling procedures, meaning they consist of a production rate and degradation rate. The production rate for each is the product of their respective transcriptive rates times the hill coefficient for the mRNA present, and the degradation rate is the degradation rate of the mRNA times the amount of mRNA.

The protein over time graph is used to show the changes in the Cas9 and tetR proteins over time. Each of the graph lines are formulated using modeling procedures, meaning they consist of a production rate and degradation rate. The production rate for each is the product of their respective translation rates times the amount of mRNA present, and the degradation rate is the degradation rate of the protein times the amount of protein. The last part of each equation is the binding rate of the promoter times the promoter and the amount of protein present. This portion is included in order to take into account the degradation of the promotor that is promoting the protein.

The Cas9 complex graph shows the binding and degradation rate of the Cas9 to the Guide mRNA. The graph’s formula takes into account the production and degradation rates of the Cas9 protein and the guide RNA. It takes the changing values and plugs them into another equation that shows the difference between the production rate of the bound complex and the degradation of the bound complex.

This graph depicts the trend that results when Cas9 is introduced. Initially, the plate is assumed to start with half wild-type bacteria and half mutated bacteria. However, overtime, there is decay in the population of wild-type bacteria, which trends towards zero, as they are being cut and killed by the Cas9.

Conclusions for Modeling

Overall, we took the opportunity to use modeling to more closely characterize our plasmids by mathematically representing the kinetics of the subunits in the different plasmids. We used this data to get a foundational understanding of the way that our parts work and grew our knowledge from these mathematical representations. The characterization of each part of our plasmids helped mold our thought process throughout our experiment. So, we used a numerical and graphical mentality to view results from experiments in the lab which helped us to gauge the accuracy of our results. We were able to generally understand what our results should trend towards. If our results did not trend towards these values, we stopped and asked ourselves why they didn’t and what we could have changed in order to make them more numerically consistent next time. Our hope in creating these models was also that other iGEM teams who would like to use SCRIBE in conjunction with Cas9 could use our code to better characterize their system with any changes to their promoters, repressors, or genes.

Team:Florida/Model

Dynamics for PFF745 SCRIBE Plasmid

Dynamics regulated by PlacO promoter