The mathematical modeling theory outlined on this page was integrated in the creation of our application, to ensure reliable results.

# Relating Relative Promoter Units to Transcription Rate

Assumption:

The first major assumption, is declaring "Promoter Activity" , to be a direct model of transcription rate. "Promoter Activity" is the number of RNAP molecules that clear the final base pair of a promoter (with units of [PoPS] = Polymerase per second).

Thus,

Where,

Thus, through simple substitution:

The modelling can further be simplified through a series of assumptions:
1) GFP expressed from test and standard promoters, have equivalent maturation rates, as they mature under the same conditions:

2) Since both the test and standard promoters are carried on the same plasmid backbone, assume they have the same average copy number:

3) Since promoters have been standardized to have identical transcription initiation sites (predicted) and identical sequences downstream of the site, we expect them to produce the same mRNA sequences.
Therefore,
Expect transcribed mRNA to be identical, implying mRNA degradation rates are equivalent:

Thus, we assume translational rates of immature GFP from some mRNA are equal:

4) Assume that immature GFP is stable. Therefore, protein degradation is negligable compared to dilution due to cell growth.
Thus:

Thus,

and if

Then,

Therefore, we assume the difference between growth rates of cells containing the test promoter construct and cells containing the standard promoter construct, is negligible compared to the maturation rate of GFP.

Thus, for the purposes of QGEM's modelling;

# Relating Transcription Rate to Proteins Produced

Assumption:
CsgA alone and CsgA-fusions (such as CsgA-AFP8 and CsgA-SpyTag) behave similarly in terms of transcription in the cells.

The transcription rate of the CsgA gene is modeled as follows:

where,

Thus, by the stated assumption, the transcription rate of CsgA-fusions can be modeled with the same equation.

# RBS Strength to Translation Rate

Altering the RBS strength of the ribosome binding site in a plasmid, can be completed by altering the genetic makeup of the RBS. This was modeled through the utilization of the RBS Calculator created by Dr. Howard Salis, at Penn State University. Using the forward implementation of the calculator, one input's their protein coding sequence, organism, and target translation initiation rate. The calculator then outputs the RBS sequence required to achieve such a translation initiation rate under the conditions outlined.

# Translation Rate to Proteins Produced

The effect translation rate has on the protein production, was accounted for in the Salis Lab RBS Calculator Code. Thus, by simply declaring the desired translation initiation rate, the code properly took into account the impacts on protein production.

# Is There a Maximum Threshold of Secretion?

Is diffusion the rate limiting step? Do CsgA pile up in the cell?

After 1 second, a spherical particle of radius 10 nanometers, has a displacement of 6.6 micrometers due to Brownian motion in water at room temperature. Determined via;

Assuming this model represents movement of CsgA, diffusion is not the rate limiting step.

The degradation rate of mRNA is on the order of 2 to 5 minutes.
The average protein half-life is approximately 23 hours.

Therefore, degradation rates do not impact curli protein production, as this would imply most of the curli proteins being degenerated during biofilm formation (which can take days). Given this is not the case, and curli protein are fully present in our biofilm, the degradation rate of curli proteins is negligible.

# Anderson Promoter Library

The Anderson Promoter Library was utilized by QGEM, given the specific measured promoter strengths, as seen in the table below. By incorporating these known strengths into our ratiometric program (as a factor of transcription rate), QGEM was able to precisely model the outcome ratios of proteins in our biofilm.

The promoters/parts, BBa_J23101 through BBa_J23119, are a family of constitutive promoters - BBa_J23119 being the consensus promoter sequences. Their sequences simply differ in the -35 and -10 consensus regions.

The promoter strengths listed are relative promoter strengths - hence, having units [RPU]. The relative promoter strength of external promoters, those not found in the Anderson Library, can be measured via a procedure found with the RPU Measurement Kit, provided by iGEM. QGEM tried to utilize this measurement kit, creating our own family of T7 promoters. Unfortunately, the T7 promoter failed the test, and no conclusive measurements were obtained pertaining to RPU strength.