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} $$ sfGFP \underset{Transcriptin}{\rightarrow} mRNA \underset{Translation}{\rightarrow} sfGFP $$

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} $$ mRNA \underset{Degradation}{\rightarrow} \oslash $$ $$ sfGFP \underset{Degradation}{\rightarrow} \oslash $$

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} $$ mRNA = k_{1} -d _{1 } mRNA $$ $$ Protein = k_{2} \cdot mRNA - d_{2} \cdot Protein $$

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