Project

Modelling

In our mathematical model we developed a system to characterize each of the two (las and rhl) plasmids. Simple detection is fairly straightforward. The engineered E. coli cells will express R-proteins (LasR and RhlR) constitutively. In the presence of PAI-1 and PAI-2, the R-proteins and the autoinducers will dimerize which results in the induction of the induced promoter. Upon induction the induced promoters will express the reporter genes. The purpose of modelling our system is to understand on a fundamental level which factors, bio-chemical species, rate constants and parameters affect the system. By understanding the mechanics of the system and mathematically modelling it, we can better understand and characterize our findings.

Our modeling approach describes the time evolution of concentrations of the relevant molecules as a system of first-order, nonlinear, ordinary differential equations. The associated variable and constants relevant to the generic model are detailed in the table below. Additionally, the [] indicate concentrations.

We have developed two distinct biosensor systems which can function independently of one another. They are the Las and Rhl sensor systems. However, each system can be modeled using a similar approach, implementing a series differential equations. The general model accounts for the production of the R-protein from the plasmid, diffusion of the autoinducer into the cell, and finally, the transcriptional activation and production of fluorescent reporter protein. A graphical representation of the biochemical system can be found below in Figure 1.

Figure 1: The general model scheme that represents both Las and Rhl sensor systems that we created.

The R-protein/autoinducer dimer (D) can act as a transcription factor and bind to the induced promoter (IP), which induces the expression of the reporter at the rate r6 and degrades back to D and IP at the rate r13. Transcription and translation are described as a single step that follows a hill function, yielding:

The R-protein is produced by the translation of the R-protein mRNA (RmRNA) at a rate r5 and degrades at the rate r10. Moreover, the R-protein can forward dimerize at the rate r1 and reverse at rate r2. RmRNA is transcribed at the rate r3 by the constitutive promoter (CP) and degrades at the rate r4,

Upon the binding of D to IP at the rate r6, GFP mRNA (GmRNA) is transcribed. GmRNA degrades at the rate r9 and is translated to GFP at the rate r7. GFP degrades at the rate r8,

The autoinducer PAI-1 diffuses passively into the cell as a result of the concentration gradient, cell volume, surface area and membrane thickness which establish the equation mass transfer1. The intracellular (A1i) and extracellular (A1e) PAI-1 degrade at the rate r11 and r12,

The general model will now be applied to the Las and Rhl systems in the figure below. The increasing rate numbers are just indicative of independent reactions and rate constants for each reaction.

Figure 2: Application of the general model to the Las and the Rhl system.

The main purpose of conducting the sensitivity analysis is to understand how the variation and output of our model can be attributed to different inputs (to the model). This analysis will demonstrate to us which biochemical species and/or parameters are critical to the model and how it functions.

The first set of sensitivity analysis was conducted with the autoinducer, R-protein, Dimer, GFP mRNA, and GFP designated as outputs. The inputs were the autoinducer, R-protein, Dimer, GFP mRNA, GFP, constitutive promoter, and the induced promoter as illustrated in figure 3. The GFP concentration is sensitive to the R-protein/autoinducer dimer, and the GFP mRNA. Additionally, GFP mRNA is even more sensitive to changes in the dimer complex than GFP, autoinducers, the R-protein and the free induced promoter.

Figure 3: Application of the general model to the Las and the Rhl system.

The next analysis compares the same output factors mentioned before with with new input parameters in figure 4. The new input parameters are k1, k2, k3, k4, k5, k10, k15 whose function can be observed above in figure 1. The data in figure 4 suggests that the among all the output factors, the R-protein is the most sensitive to changes in the rate constants. The constants with the greatest influence on the R-protein are k3, k4, k5 (function found in figure 1). However, it is important to note that the scale in this figure goes to 10⁴, which suggests that k3 may be the most influential parameter.

Figure 4: Application of the general model to the Las and the Rhl system.

The last sensitivity analysis matrix found in figure 5 uses a different set of input parameters (relative to Figures 3 and 4). The output factors however, stay the same. In this case, the input parameters k6, k7, k8, k9, k11, k12, k13, and the induced promoter whose function can be found in figure 1. The only output species affected by the input parameters is the autoinducer. The autoinducer is most affected by rate constants k11 and k12 which are the degradation rate constants of the intracellular and extracellular autoinducers.

Figure 5: Application of the general model to the Las and the Rhl system.

By conducting sensitivity analysis, we have identified the key parameters and biochemical species which can affect the system much more influentially than the others.