Difference between revisions of "Team:UNOTT/Modelling"

Biological insight had told us we need a model with constant gene expression. Investigating models from literature ¹ so see which model would satisfy these conditions, and it was found the constitutive gene expression model was suitable to guide the model.

The first step was to take the general model from literature and apply it in our scenario using the proteins (GFP, ECHP, RFP.)

^{Figure 1} $$ \color{white}{ sfGFP \underset{Transcriptin}{\rightarrow} mRNA \underset{Translation}{\rightarrow} sfGFP } $$

The equation above describes the process of which the gene undergoes transcription to produce mRNA. The mRNA carries the genetic information copied from the DNA which codes for protein. The expression of protein, can therefore, be measured by the fluorescence which is the desired output of the system.

^{Figure 2} $$ \color{white}{ mRNA \underset{Degradation}{\rightarrow} \oslash } $$ $$ \color{white}{ sfGFP \underset{Degradation}{\rightarrow} \oslash } $$

The two equations above state the same time, the concentration of protein and mRNA would undergo degradation which means the concentration would drop. However, since there is always protein and mRNA being created, over time, the creation and degradation keep the concentration constant. ²

We can apply Law of Mass Action combine both equations for the concentration of protein and mRNA over time. This model can be described as:

^{Figure 3} $$ \color{white}{ mRNA = k_{1} -d _{1 } mRNA } $$ $$ \color{white}{ Protein = k_{2} \cdot mRNA - d_{2} \cdot Protein } $$

Where...

• mRNA is the concentration of mRNA
• Protein is the concentration of Protein
• k ₁ is the constitutive transcription rate. This represents the number of mRNA molecules produced per gene, per unit of time.
• d ₁ is the mRNA degradation rate
• k ₂ is the translation rate. This represents the number of protein molecules produced per mRNA molecule, per unit of time.
• d ₂ is the protein degradation rate.

This is important because we can use this model to calculate the concentration of proteins we can expect over time. This is useful as we can use this information to calculate the total emitted light spectra during the time period which is what we are looking for in our system. However, the constants and variables are individual for each protein and which means parameters for each protein would need to be found. These constants were found using literature ³ (for GFP) and lab results (the rest.)

¹ GB Stan, 20137. Modeling in Biology. London, the United Kingdom: Imperial College London. p, pp.59-65.

² See Non-Inhibited conditions from Figure 5 Gene Transcription Regulation by Repressors (CRISPRi) - Concentration over Time

³ See Relationship between Max Fluorescence and Protein Concentration for more details

The next step in developing our simulation was to calculate our protein concentration at any given time when using CRISPRi. Discussion with wet-lab revealed our method would be using CRISPRi as a repressor, which works by inhibiting the expression of one or more genes by binding to the promoter region ¹ . The expanded mRNA and Protein concentration models from the Constitutive Gene Expression Model ² were modified to include the element of repression from the CRISPRi inhibition.

This system can be described as above. Where gRNA(i), Cas9, and mRNA are produced constitutively with their associated rates of production kc, kg, and k0 respectively. The Cas9 and gRNA(i) will undergo an inrreversible association to form Cas9:gRNA(i) at rate kf, which in turn inhibits the production of mRNA and reduce the production of Fluorescent protein (k1). All molecules spontaneously degrade and diffuse away at their own ssociated rate. (i) will account for us having multiple gRNAs and just as many fluorescent proteins i.e. i=3 with three fluorescent proteins and subsequent set of three gRNAs.

The system can be described by the following 5 ordinary differential equations, defining how the concentration of each variable will change at any given change in time. Equations 1, 2 and 3 are derived from Farasat et al.(2016), which comprehensively investigated the rates at which CRISPR-Cas9 can cleave DNA targets.

$$ \color{white}{(1) \frac{dgRNA,i}{dt} = k_{g,i} – δ_{dg} \cdot gRNA,i – k_{f} \cdot Cas9 \cdot gRNA,i} $$

The above equation details the change in gRNA concentration per unit time, also extending along index i. At any given time, the concentration of gRNA(i) will be increased by its production (kgi), and decreased by its association with cas9 at rate kf, relative to it's concentration, and it will also degrade and diffuse away at rate δdg.

$$ \color{white}{(2) \frac{dCas9}{dt} = k_{c} – δ_{dc} \cdot Cas9 – k_{f} \cdot Cas9 \cdot \underset{i}{∑}gRNA,i} $$

This equation details the change in Cas9 protein per unit time. It will be increased by its production (kc) and reduced by its degradation (δdc), and again it's association to gRNA(s). This will be proportioal the sum of all the gRNA's along i, accounting for the competition for Cas9.

$$ \color{white}{(3) \frac{dCas9:gRNA,i}{dt} = k_{f} \cdot Cas9:gRNA,i – δ_{dcg} } $$

This equation details the change in concentration of the Cas9 associated with gRNA(i). This is simply the rate of formation from before, minus its degredation.

$$ \color{white}{(4) \frac{dmRNA,i}{dt} = k_{0} \cdot \frac{1}{1+k{m} \cdot Cas9:gRNA,i} −δ_{dm} \cdot mRNA,i} $$

This equation details the change in mRNA(i), which is very similar to the equation seen earlier when describing transciption. This is produced at a rate k0, but it is also inhibited by Cas9:grna(i), so there is a standard inhibition function which will reduce rate k0 as Cas9:gRNA(i) increases. It is also simply reduced by it's degradation and diffusion rate δdm.

$$ \color{white}{(5) \frac{dFP,i}{dt} = k_{1} \cdot mRNA,i – δ_{dp} \cdot FP,i} $$

This details the rate of translation and is the same as before; only changes to protein translation are increased proportionally to mRNA(i) and reduced by it's degradation and diffusion δdp. ³ :

The value for these constants and variables were taken from literature and calculating them ⁴ but later, adjusted to the lab results.

Figure 6

Figure 6 shows the structure which underwent CRISPRi inhibition are expected to produce lower concentration of the protein whose expression were are inhibiting. This is important as it means the team can calculate concentration of proteins which are inhibited and compare them to the control conditions as well as giving the correct concentration for the simulation.

Furthermore, by having a model which can calculate the protein concentration at any given time, we can deduce how much fluorescence is being emitted at that time period by the bacteria

⁴ See Relationship between Max Fluorescence and Protein Concentration

Farasat, I. and Salis, H.M., 2016. A biophysical model of CRISPR/Cas9 activity for rational design of genome editing and gene regulation. PLoS computational biology, 12(1), p.e1004724.