Team:Lambert GA/Model
BACKGROUND
Overexpression is a prevalent issue in synthetic biology circuits and can create false positives. Last year, Lambert iGEM utilized the BBa_J23100, a strong promoter that created leakiness in a previous toehold switch. While constructing this year’s Toehold Biosensor for C. elegans, we decided to use a weaker promoter to prevent this “leakiness”. We changed our promoter from the strong constitutive BBa_J23100 to a medium promoter, BBa_J23106. We employed the Ribosomal Binding Site BBa_K2550200, the same as the one implemented in last year’s toehold switch. However, we needed to demonstrate that a difference in the strength of components of the transcriptional unit caused a change in protein production to validate our change of promoter. With the aim of establishing a relationship between the strength of promoter and RBS and the protein production from the gene enclosed, we set out to create a system of validating our assumptions.
PARAMETERS
We utilized the Carolina’s BioBuilder iTune Device Kit to obtain and test 10 unique cellular strains, each with a different combination of different strengths of promoter and RBS. The strain promoters and ribosomal binding sites encode the LacZ gene; naturally, the gene is repressed. The inhibition is relieved in the presence of the molecule Isopropyl-β-D-thiogalactoside (IPTG). For the purposes of our experiment, we insert IPTG into cells to begin transcription of the LacZ gene by a particular promoter, after which the mRNA is translated by a particular RBS. This produces the enzyme β-galactosidase (β-gal). When ortho-Nitrophenyl-β-galactoside (ONPG) is added, β-gal cleaves ONPG into ortho-Nitrophenol (ONP) and galactose. ONP creates a yellow color, the concentration of which is indicative of the total enzymatic activity of β-gal. We assume that evaluating the yellow coloration of ONP will give us a comprehensive measurement of the strength of each promoter/RBS combination. Below are the combinations of BioBricks and strength classifications.
| Strain # | Promoter Registry # | RBS Registry # | Relative strength Promoter/RBS |
|---|---|---|---|
| 2-R | BBa_J23115 | BBa_B0035 | Reference/Reference |
| 2-1 | BBa_J23113 | BBa_B0031 | Weak/Weak |
| 2-2 | BBa_J23113 | BBa_B0032 | Weak/Medium |
| 2-3 | BBa_J23113 | BBa_B0034 | Weak/Strong |
| 2-4 | BBa_J23106 | BBa_B0031 | Medium/Weak |
| 2-5 | BBa_J23106 | BBa_B0032 | Medium/Medium |
| 2-6 | BBa_J23106 | BBa_B0034 | Medium/Strong |
| 2-7 | BBa_J23119 | BBa_B0031 | Strong/Weak |
| 2-8 | BBa_J23119 | BBa_B0032 | Strong/Medium |
| 2-9 | BBa_J23119 | BBa_B0034 | Strong/Strong |
BioBrick parts were classified by Carolina Science iTune Device.
We varied both the promoter and the ribosomal binding site within our experiments to simulate differences in strength of both RBS and promoter. We utilized the available quantitative values for these biological components that could best capture their impact.
Promoter strength varies due to differences in nucleotide sequence between a particular promoter sequence and a consensus promoter. A promoter with more differences in nucleotide sequences than the consensus promoter is considered “weaker”. We quantified our promoter strengths with the measured values from the iGEM Parts Registry for Anderson Promoters.
Promoter Strengths
| BioBrick | Relative Strength | Promoter Value |
|---|---|---|
| BBa_J23119 | Strong | 1 |
| BBa_J23106 | Medium | 0.47 |
| BBa_J23113 | Weak | 0.01 |
| BBa_J23115 | Reference | 0.15 |
To quantify the strength of our ribosomal binding sites, we utilized the Salis Lab’s RBS Calculator. The RBS Calculator is a software that predicts translation rates of specific start codons within an mRNA sequence. We measure the translation rate of an mRNA sequence corresponding to the LacZ.GFP fusion, BBa_E0051. We analyzed four different RBSs, each attached to our tested promoters, to measure their translation rate of this protein.
RBS Classifications
| BioBrick | Relative Strength |
|---|---|
| BBa_B0031 | Weak |
| BBa_B0032 | Medium |
| BBa_B0034 | Strong |
| BBa_B0035 | Reference |
Strain Translation Rates
| BioBrick | Promoter Strength | RBS Strength | Translation Rate |
|---|---|---|---|
| BBa_J10050 | Weak | Weak | 571.38 |
| BBa_J10051 | Weak | Strong | 2346.02 |
| BBa_J10052 | Weak | Medium | 284.43 |
| BBa_J10053 | Medium | Weak | 307.05 |
| BBa_J10054 | Medium | Medium | 234.39 |
| BBa_J10055 | Medium | Strong | 1205.21 |
| BBa_J10056 | Strong | Weak | 683.05 |
| BBa_J10057 | Strong | Medium | 246.1 |
| BBa_J10058 | Strong | Strong | 1343.64 |
| - | Reference | Reference | 330.15 |
We theorized that the promoter constructs affected the translation rate of the RBSs. Below is graphed the relationship between the strength of the promoter and its effect on RBS.
This graph was made using GraphPad Prism. The values for promoter strength are from Registry of Standard Biological Parts, and translation rates were calculated with the RBS Calculator from the Salis Lab.
MODELS
Ordinary Differential Equation Model
An assumption we made in creating this model was that the enzymatic coloration of ONP was indicative of the transcriptional and translational strengths of our promoter and RBS, respectively. To predictively show that the change in promoter and RBS affinity will have a tangible effect on the production of ONP, we used an Ordinary Differential Equation Model with Law of Mass Action Kinetics to simulate biochemical reactions inside the cell and varied the transcription and translation coefficients to observe the effect on the production of ONP after a set amount of time (30 minutes). The goal of this model was not to accurately calculate the amount of ONP produced after a certain amount of time, but rather to observe the relative differences in production between the strengths. Since the Beta Galactosidase amount varied and would not always be in excess, we could not use Michaelis-Menten Kinetics and used a more complex Law Mass Action model for the enzyme reaction.
We assumed that since the culture had been growing for 24 hours prior to the test, the cell amount in the sample would have reached its logistic growth limit, so we kept the cell number constant and produced results for a single cell, which is proportional to the total output. We also assumed that since the concentrations of IPTG and ONPG were far in excess of what would be used in the reaction, the diffusion of these components into the cell would be high enough such that the concentrations of each are constant within the cell. Also, since IPTG prevents the binding of the inhibitor to the promoter, we consider the IPTG as binding to the promoter in the reactions.
This diagram depicts the biological production of ONP.
From the reaction diagram, we derive the following biochemical reactions:
These reactions model the construction of ONP.
| Parameter | Description | Value |
|---|---|---|
| Kf | IPTG - Promoter “Association” coefficient | 0.1 s-1 |
| Kd | IPTG - Promoter "Dissociation" coefficient | 0.08 s-1 |
| Kt | Transcription coefficient | Varied in Analysis |
| Kcf | β-Galactosidase.ONPG complex formation coefficient | 20.0 s-1 |
| Kcd | β-Galactosidase.ONPG complex dissociation coefficient | 5.0 s-1 |
| K2 | β-Galactosidase.ONPG cleavage and ONP and Galactose formation coefficient | 20.0 s-1 |
| Kmd | Coefficient of mRNA degradation | 0.041 s-1 |
| Mbd | Coefficient of β-Galactosidase degradation | 0.026 s-1 |
| Kl | Translatioin coefficient | Varied in Analysis |
From this, we get 7 differential Equations.
| Variable | Biochemical Species | Initital Value |
|---|---|---|
| n1 | Promoter | 12.0 molecules |
| n2 | Activated Promoter | 0.0 molecules |
| n3 | mRNA | 0.0 molecules |
| n4 | β-Galactosidase | 0.0 molecules |
| n5 | ONP | 0.0 molecules |
| n6 | β-Galactosidase.ONPG Complex | 0.0 molecules |
| n7 | Galactose | 0.0 molecules |
| ONPG | ONPG | 3.612E4 molecules (constant) |
| IPTG | IPTG | 3.612E7 molecules (constant) |
| RBS | RBS | 12.0 molecules (constant) |
These Differential Equations aim to model the reactions occurring in the production of ONP.
Defining this model in MATLAB and choosing mid-range values for the transcription and translation constants, we can simulate the amounts of each species over time.
mRNA Molecules vs. Time
β-gal Molecules vs. Time
β-gal.ONPG Complex vs. Time
Activated Promoter vs. Time
ONP Molecules vs. Time
Galactose Molecules vs. Time
Inactive Promoter vs. Time
These graphs pictured above illustrate Species vs. Time for Mid-range Transcription and Translation Constants.
To demonstrate the effects of these constants and therefore the effect of promoter and RBS strength on the ONP production, we wrote a script to simulate the results of the reaction after 30 minutes for a wide range of parameter values.
This is a 3D Graph of the ONP Expression as a function of Promoter and RBS strength, created using MATLAB.
The side view, created Using MATLAB.
As we can see, the graph has a rough tilted hyperbolic paraboloid shape and shows the trend that increasing Promoter and RBS strengths both have appreciable effects on the ONP production from the cell. Specifically, the expression becomes exponentially higher with both constants increasing proportionally. Unaccounted for variables show some degree to which the promotor has a greater effect on the outcome as we will see in the following regression model.
Multivariate Regression Model
An ANOVA test in SAS was performed on the data from the tuning lab. ANOVA assesses the significance of one or more factors, which in this project are the nine different combinations of promoter strength and RBS translation rate, by comparing the responses at different factor levels. This test is appropriate for the data collected because the factor levels do not have overlapping membership and can be considered independent. The null hypothesis is that there is no difference between factor level means, while the alternative hypothesis is that at least one is different.
Four assumptions were taken into account:
- The expression is continuous and normally distributed for each factor.
- Factors levels are independent of each other and do not overlap.
- There are no major outliers.
- Testing for unequal variances determines which version of the one-way ANOVA is appropriate.
| Source | DF | Sum of Squares | Mean Square | F Value | Pr>F |
|---|---|---|---|---|---|
| Model | 6 | 22.22921752 | 3.70486959 | 4.58 | 0.0006 |
| Error | 65 | 52.54432998 | 0.80837431 | - | - |
| Corrected Total | 71 | 74.77354750 | - | - | - |
The p-value associated with the F statistic is 0.006. Given alpha=0.05, we reject the null hypothesis, concluding that there is a statistically significant difference in levels of ONP expression across varied promoter strengths and RBS translation rates.
| Source | DF | Anova SS | Mean Square | F Value | Pr>F |
|---|---|---|---|---|---|
| Promoter | 3 | 17.30860619 | 5.76953540 | 7.14 | 0.0003 |
| RBS | 3 | 4.92061132 | 1.64020377 | 2.03 | 0.1185 |
This graph was made using GraphPad. This graph shows the B-gal expression levels for all 10 iTune strains.
The test result shows that the translation rate is not an important factor in the determination of expression of ONP; however, the promoter is a significant factor. In other words, expression values are statistically significantly different with different promoters, but changing the RBS did not affect the expression significantly.
This graph was made using GraphPad. The graph shows the expected expression versus the actual expression in terms of miller units.
To represent how the strength of the promoter impacts expression ONP, we built a simple multivariate linear regression on the data without considering many theoretical assumptions. The algorithm selected the promoter strength as the only significant variable in the model. The expression of ONP can be modeled by the expression:
From the positive coefficient of the promoter, we concluded that higher promoter strength yields higher expression.
Biological systems are extremely complicated, and a model developed from a small sample with very limited additional information may yield values inconsistent with what was observed; however, the impact of the strength of the promoter on the expression of ONP should be valid.
REFERENCES
[1] One-way Analysis of Variance (ANOVA) in SAS. (2019, June 10). Retrieved October 15, 2019, from https://stat-methods.com/home/one-way-anova-sas/.
[2] Anderson, J. C. (2019). Promoters/Catalog/Anderson. Retrieved October 19, 2019, from http://parts.igem.org/Promoters/Catalog/Anderson.
[3] Salis, H. M., Mirsky, E. A., & Voigt, C. A. (2009, October 4). Automated design of synthetic ribosome binding sites to control protein expression. Retrieved October 17, 2019, from https://www.nature.com/articles/nbt.1568.
[4] Borujeni, E., Amin, Salis, & M., H. (2013, November 14). Translation rate is controlled by coupled trade-offs between site accessibility, selective RNA unfolding and sliding at upstream standby sites. Retrieved October 17, 2019, from https://academic.oup.com/nar/article/42/4/2646/2435298.
[5] Espah Borujeni, A., & Salis, H. M. (2016, June 8). Translation Initiation is Controlled by RNA Folding Kinetics via a Ribosome Drafting Mechanism. Retrieved October 10, 2019, from https://www.ncbi.nlm.nih.gov/pubmed/27199273.
[6] Espah Borujeni, Amin, Daniel, Farasat, Iman, Smith, … M., H. (2017, February 1). Precise quantification of translation inhibition by mRNA structures that overlap with the ribosomal footprint in N-terminal coding sequences. Retrieved October 10, 2019, from https://academic.oup.com/nar/article/45/9/5437/2965383.
[7] Sana, S. (2019). Kinetics of B-Gal and inhibition of ACHE. Retrieved from https://www.slideshare.net/SurayyaSana/enzyme-kinetics-of-gal-45084595