Modeling

Modeling is used as a tool to predict behaviors. These behaviors can range from simple one step processes to complex chains of reactions. The main goal of modeling is the same for all of these cases, however, and that is to provide an easy method for manipulation of variables to achieve an output. The model is especially crucial for our project because our project focuses on the behavior of PDGF and VEGF with respect to time.

The goal of modeling is to understand how the concentration of a particular molecule species varies once the inducing molecule is added with respect to time. The dependent relationship between each critical molecule of the genetic circuit is represented in the flow-chart on right.

Although setbacks in assembly prevented us from actually testing the complete circuit, we utilized ordinary differential equations to generate a model describing the behavior of each protein synthesis and employed the Michaelis-Menten Equation to describe the production of messenger RNAs.

where I represents the concentration of LuxI, R represents the concentration of LuxR (induced by the bidirectional promoter), A represents the concentration of N-acyl homoserine lactone (or AHL), M represents the concentration of the dimmer formed by AHL and LuxR, PDGF represents the concentration of platelet derived growth factor, and VEGF represents the concentration of vascular endothelial growth factor, kdeg is the degradation factor for each molecule. I0 is the initial concentration of LuxI, and G0 takes into account for the concentration of molecules that induce the bidirectional promoter.

Note that most of the unknown variables (such as I0, VA, KA, G0, VL, and KL) above could be measured using HPLC once the genetic circuit is constructed. One key assumption we have made is that the dimer is formed via the law of mass action at a constant rate, which means that the concentration of the dimer is a factor of the product of I and R.

Below is the output diagram:

 

Note: The diagram above was produced using hypothetical numbers, and is not an accurate prediction of the concentrations of VEGF and PDGF with respect to time. Instead, it qualitatively shows the general expected behavior of the genetic circuit.

What Modeling Can Tell Us

1. The model provides a basic shape of our expected results. Even if the steady state concentration is higher or lower than the predicted results, we expect a quick increase in the concentration of VEGF, which also means that the LuxR dimer is being formed to produce siRNA. The siRNA inhibits the translation of VGEF, and we see the graph of VGEF approach zero while PDGF is produced and sustained for all time. The graph for VEGF does not quite reach zero because even if siRNA is binding to VEGF mRNA, there is no guarantee that all of the mRNA is inhibited. Thus, there may be some residual VEGF mRNA being translated. This quantity, however, is not significant and may be ignored.
2. While the constants were being manipulated to produce the graphs, certain combinations of the reaction constants produced graphs that show negative production of VEGF and PDGF. Understanding that this is not physically possible allows us to discard these combinations as possible experimental results, and we can predict a range of possible constants from extensive model analysis.
3. As stated above, the model allows for easy manipulation of the input, which shows the robustness of the system and warns us of possible unbounded behavior by the system. Since angiogenesis is a delicate balance between organized growth and neovascularization, it is important that we retain control over the production of these growth factors, and a model can help us to determine these boundaries.
4. Since timing is an integral part of our project, a model allows us to predict reaction constants we want to achieve our desired output. For example, if we want VEGF production to be sustained for a few more hours, we can use the model to predict how much slower we want the LuxR dimer to be formed, and based on these predictions, we can modify our circuit to add a molecule that degrades the LuxR molecule at a slow rate or time the production of LuxI and LuxR so that they do not form the dimer immediately. The model also allows us to quantify the improvement in time compared to tissue engineering approaches without a lot of experimental data.