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.)

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.

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:

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.

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.

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.

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.

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 :

Where...

m is mRNA concentration, p is Protein concentration, R is Repressor, k1 is Max Transcription Rate, k is the Repression Coefficient, n is number of repressors that need to cooperatively bind the promoter to trigger the inhibition of gene expression (Hill Coefficient), R is Repressor, d1 is mRNA degradation rate, d2 is Protein degradation rate

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.

At any given time, it is expected that the proteins would be expressed so the bacteria would fluoresce. This can be confirmed by looking at the bacteria after being constructed and observing that they are giving off light.

This means that an equation must be developed to find out what the intensity of fluorescence would be at that certain time. This consisted of of calculating the protein concentration at the time and using real life lab data of the fluorescence at that time period, the team could map that intensity to the protein concentration at that time.

When the fluorescence data received from the wet lab were graphed, a model was constructed, refined and optimised to demonstrate the trends shown from the real data gained from the labs. Originally, the data from the lab was the Fluorescence against Time but by using the Gene Transcription Regulation by Repressors model developed earlier ¹ , the team was able to estimate the protein concentration at a certain time periods.

Due to time constraints, rather than implementing the relationship directly from lab data, the data was fitted using a Polynomial Fit of Order 3 using Excel and an equation was calculated from these. These equations were directly plugged into the simulation. However, this is inaccurate as the R squared value was ... , suggesting that it doesn't fully capture the data trend.

These relationships were implemented into the simulation to give the expected spectra produced by each protein. This highlights another use: by adding or subtracting values from our fit, we can create a threshold for our Keys. This was essential when developing the Raw Data Simulator. ²

¹ See Gene Transcription Regulation by Repressors (CRISPRi) - Concentration over Time

² See Software

When constructing our proteins with our current method, there were 3 vectors we could order from.

However, in this proof of concept, order is irrelevant as the gene is either inhibited (1) or not (0). Using

Where n = 2 and r = 3, this gives us a total combination of 2 ³ {1,1,1} {1,1,0} {1,0,1} {1,0,0} {0,1,1} {0,1,0} {0,0,1} {0,0,0}

Randomness comes from the fact the system relies on Brownian Motion ¹ , a random process to create these combinations.

However, in order for a movement to fall under Brownian Motion, it must fulfill a condition where the process must have continuous paths. This is not true as once the structures begin to form, the paths stop (they do not collide off each other elastically, but rather, combine.) Furthermore, the bacterium might become biased towards options that put less metabolic stress on the bacterium, which results in selection. Alternatively, using metabolites to undergo transposition can improve randomness. ²

In order to aid, with the wet lab in what combinations they can expect, the team developed an Excel Spreadsheet where a user can simply input details of the construction and it would show what construction it would look like

Members of the public are encouraged to try it out and can use it to help with identifying how their spectra would look if they used the same proteins the project used

¹ Refer to https://statistics.stanford.edu/sites/default/files/EFS%20NSF%20149.pdf

² Refer https://link.springer.com/book/10.1007%2F978-1-4612-0459-6 for more information about Brownian Motion and Random Walk.