Team:Northwestern/Model
Model objectives
Biosensors, especially those involving multiple plasmids, are incredibly versatile tools for sensing and quantifying the concentration of a chemical or compound of interest. However, despite their versatility, biosensor output is dependent on many different parameters, variables and initial conditions. Due to this complexity, it can be difficult to narrow down the exact conditions needed to optimize a biosensing system based purely on the results of traditional laboratory experiments. By stimulating the behavior of the system under varying conditions using a mathematical model, it is possible to determine the most likely variables that are key to designing an effective biosensor.
Using knowledge gained from the 2015 Bielefeld Team’s modeling of heavy metal biosensors, we have designed a model for a two plasmid sensing system. Although our end goal is to create biosensors that can sense lead and hexavalent chromium, we have designed a modeling system based on the well-researched LuxR/pLux system for sensing AHL, since it is well characterized in literature and is our first step towards creating a more efficient biosensor. The goal of this mathematical model is to provide some insight into the following questions:
• How does the ratio between the concentrations of the activator and reporting plasmid effect the final GFP concentration?
• What is the optimal ratio of plasmids for GFP expression?
• What is the sensitivity of our sensor in regards to AHL
• How does the initial ratio of the activator and reporter plasmids affect the sensors AHL sensitivity?
Assumptions
Due to the complexity of our biosensing system, the following assumptions were made to simplify the model:
• The reaction occurs in large enough numbers that stochastic processes of transcription and translation can be modeling with continuous differential equations
• The rates of transcription for both plasmids are the same
• The rates of translation for both plasmids are the same
• The rates of RNA degradation for both plasmids are the same
• The rate of degradation of luxR and GFP are the same
• The equilibrium constants KD1,2 for AHL and LuxR are the same
• The availability or lack of reactants can be modeled by the supply of amino acids
Model breakdown
Step 1.) DNA Transcription of LuxR and mRNA degradation
The rate of transcription is described by Equation 1, where [LuxRDNA] is the concentration of the LuxR plasmid in the system in nM, [LuxRRNA] is the concentration of mRNA in the LuxR plasmid, Km is the rate of transcription and αm is the rate of mRNA degradation.
Step 2.) mRNA translation and protein degradation
The rate of translation, described by Equation 2 below, is similar to the rate of transcription above, where Kp is the rate of translation, αp is the rate of LuxR protein degradation, and [LuxRp] is the concentration of the LuxR protein. Additionally, the rate of translation is further constrained by the supply of amino acids, which is modeled by the ratio between the current amino acid concentration [A] over the amino acid saturation level AS.
Step 3.) Transcription of GFP
The transcription of GFP has a similar basic form to Equation 1, where [GFPDNA] is the concentration of the reporter plasmid in the system in nM, [GFPRNA] is the concentration of mRNA in the reporter plasmid, and Km and αm are the same rates of transcription and mRNA degradation. Additionally, the behavior of the transcription factor(LuxR) binding to the reporter plasmid is modeled by the quotient of K1D, the transcription factor equilibrium constant, over the concentration of LuxR, [LuxRp]. The concentration of AHL is also modeled by the quotient of its transcription factor equilibrium constant,K2D, so that when the concentration of AHL or LuxR is small, the rate of GFP transcription is also very small.
Step 4.) mRNA translation and production of GFP
The translation of mRNA and the production of GFP in Equation 4 is very similar to Equation 2, except that the concentration of LuxR mRNA and protein are replaced by concentration of GFP mRA [GFPRNA] and GFP protein [GFPp] .
Step 5.) Change of amino acids
The last part of our model describes the rate that amino acids are used up in the system, where β1 and β2 are the length in amino acids of LuxR and GFP respectively.
Model parameters
Parameters were selected from reported literature values, found on the online database Bionumbers [1].
Simulation results
First, the model was created using the differential equations and the parameters shown above. To check that the model makes sense, a simulation was run, as shown in Figure 1, with initial concentrations of 10μL of 10nM concentrations of the activator plasmid, represented by [LuxR], and the reporter plasmid, represented by [GFP]. With this simulation we can see that the GFP steady state concentration is around 5μM which is in the range of concentration that we are expecting. The time scale is a little fast, since we are expecting a time scale of a couple of hours based on Bielefeld’s Model and experimental results; however, this is most likely due to the fact that our model did not take into account the slow rate of diffusion. For our purposes, we can ignore this simplification for now. Overall, qualitatively our model appears valid, and we can move forward with more advanced simulations.
Figure 1. Proof of concept graph of mRNA and GFP concentrations over time
Next, we wanted to use our model to explore how the ratio between the concentrations of the activator and reporting plasmid effect the final GFP concentration and what ratio optimizes GFP expression for the system. This question was simulated by varying the concentration of the LuxR plasmid in the system for a chosen initial concentration of reporter plasmid of 10nM. Figure 2 below on the left demonstrates the GFP concentration over time for the various concentrations. From Figure 2(left), we can see that as the initial concentration of LuxR gets smaller, the steady state GFP concentration increases. To further explore this issue, we graphed the steady state GFP concentration for various concentrations of LuxR. From Figue 2(right), we can observe that when there is very little LuxR, little to no GFP is produced, which matches our intuition of our understanding of the system. Additionally, matching our results from Figure 2(left), as the concentration of LuxR decreases from equimolar with initial reporter plasmid concentration, the steady state GFP levels increase and plateau between LuxR concentrations of 0.1nM and 0.01nM at a GFP concentration of 8.34μM.
Figure 2. Graph of GFP concentration for varying levels of [LuxR] when [GFP]=10nM. Steady state GFP concentrations(right) and GFP concentrations vs time(left) for varying levels of [LuxR] are observed
It it interesting to note that in Figure 2(right) the GFP steady state concentration stagnates over about seven orders of magnitude, where the LuxR plasmid is not appreciatively competing for amino acids, but its concentration is sufficient enough to activate protein synthesis. This most likely a consequence of our initial assumption that KD1 and KD2 were equal. Additionally, the concentration of [LuxR] is so small that we cannot realistically model the system in a real continuum regime. For these small values of [LuxR] our ODE becomes stiff and incalculable.
Finally, we used our model to explore the sensitivity of our sensor in regards to AHL and how that sensitivity changes with different initial ratios of the two plasmids. In Figure 3 below, we plotted the GFP steady state equilibrium concentrations for various levels of AHL and ratios of activator and reporter plasmids. Similar to the results of Figure 2, this simulations shows that higher ratios of activator plasmid to reporter plasmid have a lower sensitivity to AHL and a lower maximum level of GFP when AHL is saturating the system. We can see that between initial concentration of LuxR of 0.1nM and 0.001nM, the sensitivity of the system reaches a maximum around 10^-5nM. However, when [LuxR] is 0.001nM or smaller, the slope of GFP concentration as AHL is increased is very steep. Whereas, around [LuxR]=0.1nM, the rate of increase of GFP concentration as [AHL] increase is less steep, which is ideal for a sensor which can quantify the amount of AHL in a system.
Figure 3. Sensitivity of the sensor to AHL when [GFP]=10nM and [LuxR] is varied
Conclusions
In conclusion, we have developed a theoretical model mapping the GFP production of a two plasmid activator system for sensing AHL. From the results of the Figure 2 we can observe that GFP concentration reaches its peak when the ratio between the Reporter plasmid and the activator plasmid is 1000:1. This is vastly different from our initial guess of equimolar plasmids concentrations, so moving forward we can try to reduce the initial concentration of [LuxR] to get a stronger signal from our sensor.
From our simulations on sensitivity, we can be see that a larger ratio between the reporter and activator plasmid leads to a higher sensitivity to AHL. However, when the concentration of [LuxR] becomes very small relative to the reporter plasmid, the concentration of GFP increases very steeply as AHl concentration increases. Therefore, we can see that if we want a sense that is more quantitative, we should use a ratio of 1000:1 instead of 10,000 :1, since the change GFP level is less abrupt as AHL levels increase. Overall, this model proviedd our team with a better understanding of a two plasmid biosensing system and the challenging that we will have to tackle in the future to tune our system to our desired quantitative values of AHL.
References
1. Key Numbers for Cell Biologists. (n.d.). Retrieved October 3, 2018, from http://bionumbers.hms.harvard.edu/keynumbers.aspx