Simulations using mathematical models are helpful in quantitatively describing and predicting system behaviour through relating various elements of the system to each other. We aim to calculate the mRNA production based on light dosage. We expanded on the model from Jayaraman et al 2016 to include variables that account for transfection efficiency and variable plasmid copy number, which are related to measured signal strength and noise respectively.
The Base Model
The following ordinary differential equations are adapted for LACE from Jayaraman et al’s paper [1]. (A summary table of variables can be found at the end of this section.)
KSand δSdetermine the steepness of light activation and deactivation kinetics since they are the initial synthesis and degradation rates of mRNA production respectively. Assume steady state, so that the left side becomes 0. Let L=1 in order to induce the system. Then the equation simplifies to
The change in mRNA production of the gene of interest over time is defined by
Again, assume steady state. Replace KGOI mRNA with the equation found earlier.
[L] is the light activator coefficient, which can be defined by the Hill function model, taking the type of promoter into account.
The final equation that describes mRNA of a whole population at some point in time that responds to light dosage becomes
The values for each parameter are summarized in the tables below. The range of light dosage inputs were 0 to 4000, which reflects the possible grayscale light intensities of our LPA.
Parameters for the Hill Function
| Parameter | Value | Reference |
|---|---|---|
| k(a.u) expression possible/ greatest ΔΔct |
8.6 Based on Figure 2a, the maximum average mRNA expression was about 400. Interpreting that as fold difference, and the fact the fold difference is 2-ΔΔct , then that comes out to be 8.6. |
[2] |
| n | 2 Having one plasmid (dCas9) on the DNA increases the chance that CRY2-VP64/-VPR binds to activate the system. |
Estimate |
| K1 | 6.42228 W/m2 Experimental data: at 1500GS. Measured with a lux meter. | Experimental |
| α | 1 | Estimate |
| Parameter Abbr. | Parameter name | Parameter value | Units | Reference |
|---|---|---|---|---|
| KSInd | Increase in mRNA synthesis rate (inducible system) | 5 | nM h-1 | Estimate |
| δSInd | Decrease in mRNA synthesis rate (inducible system) | 1 | h-1 | Estimate |
| Kbasal mRNA GOI | Basal synthesis rate of mRNA GOI | 2.5, number chosen from the range 0.025-25 | nM h-1 | [3] |
| δmRNA GOI | Decay rate of mRNA GOI | 0.8, number chosen from the range 0.2 - 1.4 | h-1 | [3] |
If we assume that the relationship between mRNA production and the ctfound in the qPCR data is linear and maximumct induction at ~8.5 to reflect the value from Polstein et al [2], plotting the equation in Matlab gives the following graph.
Accounting for Transfection Efficiency and Plasmid Variation
Within a 24 well plate well, there is a variable amount of cells in each well due to pipetting errors and the inability to control for a truly homogenous solution when cells were resuspended in media before adding it to the wells. The same problem occurs when the DNA/Lipofectamine solution gets added. As a result, the cell population end up uptaking various amounts of plasmids. While our system requires the presence of two different plasmids to activate, it is possible that some cells end up taking in only the first plasmid, only the second plasmid, a mixture of both plasmids, or none at all. The signal generated from the cell population comes from this mixture, as modeled in the earlier section.
We were able to quantify how much signal we were getting from each plasmid in each well by dying one of the plasmids red and the other blue. With the red light emission filter set to FL3 and the blue light set to FL2 on the BD Accuri C6 software, the scatter plot can be gated as seen in the following figure. For this example, the lower left (LL) quadrant is a scatter plot of cells that do not have either plasmid, the upper left (UL) shows cells that only emit red light, the lower right (LR) are cells that only emit blue light, and the upper right (UR) are the cells that has both plasmids. The statistics summary also lists the mean and coefficient of variance for each type of light emission, which can be used to find the variance.
The 2015 Mary and William iGEM team derived an equation for the Fano Factor that describes transcriptional noise strength for any number of gene copies [4]. It is defined as σ2/μ- the variance over the mean of the whole population. They were able to show how slight fluctuations of plasmid numbers in cell populations with lower plasmid copy numbers have a larger impact compared to populations with higher copy numbers in the start. Combining with the transfection efficiency and the distributions together, we define a new scaling factor:
c1 and c2 are the fraction of cells that have one plasmid, whereas c3 is the fraction of cell that has both plasmids. Subscript P1 stands for one plasmid, P2 represents the second plasmid, and P12 is for both. k1 and k2 are constants to adjust the graph so that it remains within range of the expected ΔΔct. The calculations for the variances, means, and transfection efficiencies for the 2 plasmid dyed tests are linked here.
Transient Expression Over Time
Our LACE system is not integrated into the cell’s genome, which makes it transient in definition. During the initial phase, mRNA production of the endogenous genes increases exponentially as the cell machinery start to process the interaction between the CRY2 and dCas9. However, when the cell divides, the plasmids do not, so only one of them will retain the plasmid. Consequently, the concentration of plasmids gets diluted as the number of cells grow.
First, we need to calculate how the concentration of plasmids/cells change over time. If we assume exponential decay, the function for plasmid/cell can be modeled by C(t)=C0e-rt.
For the two plasmid system, we transfected a total of 2μg of DNA to 5*104 cells in one well. THis simplifies to 0.04ng/cell at t=0hrs. Assuming that none of the plasmids get degraded and the amount stays constant, only the total number of cells change over time. Taking CHO cells as an example, their doubling time is 20-24hrs. We can take this to mean there will be 2μg of DNA to 1*105 cells or 0.02ng/cell at 24 hours. Plugging in the initial condition and 24 hr time point, we can solve for the decay rate to be 0.029. Therefore, C(t)=0.04e-0.029t.
However, plasmid number doesn’t necessarily correlate to transcription rate. Assume that mRNA production relates to copy number that follows the following equation:
C is the plasmid/cell copy number from above, n is the Hill coefficient, and k1 and k2 are constants that are used to scale the initial point at 1.
Click to download the Matlab script and PDF file.
References
| [1] | Jayaraman, Premkumar et al. “Blue light-mediated transcriptional activation and repression of gene expression in bacteria.” Nucleic acids research vol. 44,14 (2016): 6994-7005. doi:10.1093/nar/gkw548 |
| [2] | Polstein, Lauren R, and Charles A Gersbach. “A light-inducible CRISPR-Cas9 system for control of endogenous gene activation.” Nature chemical biology vol. 11,3 (2015): 198-200. doi:10.1038/nchembio.1753 |
| [3] | BioNumbers, http://homepages.ulb.ac.be/~dgonze/BIONUMBERS/bionumbers.html. |
| [4] | Team:William and Mary, https://2015.igem.org/Team:William_and_Mary. |