MATH MODELING

After experimenting with our aptamer solutions (please see the Results page for more information), we were to create a standard curve using standardized solutions of buffer spiked with our biomarkers. The Hill equation (more details below) was to be used to derive a graph comparing [PLn] (the total concentration of aptamer-biomarker complex) and the relative fluorescence reading by plugging in [L] (predetermined concentration of aptamer solution used) and [P0] (predetermined concentration of biomarker in standard solution), in addition to the Kd values from literature and our experiments in order to derive [PLn].

With linear regression, it is possible derive an equation for the curves comparing fluorescence to [PLn]. Furthermore, these curves allow us to determine the concentrations of unknown solutions, using a process elucidated in the next few sections.

ELONA Assay Quantification and Verification of Dissociation Constant

We used a modified ELISA assay (also known as an ELONA assay) pioneered by iGEM Madrid 2018 and a process pioneered by Al-Hamaoy et al to experimentally calculate the dissociation constants of all Mucin and CA 15.3 aptamers. We first incubated the biotinylated aptamers and the biomarkers, then conducted an ELONA assay with the resulting supernatant to figure out the amount of unbound aptamer, and then use that to figure out the concentration of biomarker-aptamer complex formed in the original solution. We were able to then determine the Kd values using the Hill equation and Scatchard plot, as well as non-linear regression. The results are shown in the table below:
For more information, please refer to our lab notebook.

Conversion of Fluorescence to Biomarker Concentration Using Hill Equation

The Hill’s Equation is a useful equation to determine the amount of protein (biomarker) bound to the ligand (aptamer). Since it is theoretically unlikely that all of the biomarkers will bind to the aptamers, we will need a method to calculate the concentration of the biomarker that is bound to the aptamer. Only using the standard curve will result in an underestimation of the concentration of the biomarker. Hill’s Equation helps us determine the original concentration of biomarker by relating the biomarker concentration with the aptamer concentration and proportions of the pattern-biomarker complex (the dissociation constant). Although the dissociation constant was to be obtained from the Scatchard Plot, test results were inconclusive. Therefore, the Kd value was derived from 3D modeling through UNAFold


The Hill’s Equation is shown above. [PLn] is the concentration of protein (biomarker) bound to ligand (aptamer), [P0] is the total concentration of biomarkers, [L] is the total ligand (aptamer) concentration, and Kd is the apparent dissociation constant. The dissociation constant (Kd) describes the ability of a compound to dissociate or break down to its constituent components. In our case, it describes the ability of the biomarker to break away from the aptamer.

To create a standard curve, we prepared 100 microliters of the 1000 micromolar aptamer solution in 12 wells of the 96-well plate. Then, we put in 100 microliters of 1000 micromolar of biomarker in the first well and then conducted a serial dilution. After placing it into the 96-well plate reader, we were able to obtain data for our standard curve (fluorescence vs. concentration).


We were able to get a total of 12 total data points with calibration on 475nm excitation peak and 580-640 emission peak. We then were able to plot the line of best fit and find the equation of the standard curve.


Above is the standard curve obtained from the serial dilution. The equation for the line of best fit is shown above the graph and the vertical error bars are depicted. To determine the bound concentraiton of the biomarker, the equation of the standard curve was initially rearranged in terms of the fluorescence value and then plugged into the Hill’s Equation. The Hill’s Equation was then used with the Kd found through 3D modeling.


Where F is the fluorescence value (RFU) from the standard curve, [L] is the aptamer concentration, and [PLn] is the concentration of the aptamer-biomarker complex. Using this equation, we are able to calculate the concentration of the biomarker bound to the aptamer.

Scatchard Plot

Theoretically, in order to obtain the Kd value, we would have needed to use the scatchard plot in conjunction with ELISA Assay. The Scatchard Plot (1949) utilizes a linearization procedure to calculate the constant which helps quantify the interactions between molecules. Below is the equation to the scatchard plot.


References

  1. Al Hamaoy, Russul R., et al. “A New Method for the Determination of Dissociation Constant (kd) on the Binding of CA19-9 to Its Antibody in Type 2diabetic Patients by Enzyme Linked Immunosorbent Assay (ELISA) with Some Modifications.” International Journal of Advanced Research, vol. 4, no. 4, Apr. 2016, pp. 710–19.
  2. “How Do Experimentalists Use Mathematical Models?” Practically Science, 28 Apr. 2013, https://www.practicallyscience.com/how-do-experimentalists-use-mathematical-models/.
  3. “Understanding Ligand-Receptor Dose Response Curves.” Practically Science, 28 Feb. 2013, https://www.practicallyscience.com/understanding-ligand-receptor-dose-response-curves/.
  4. Wikipedia contributors. “Hill Equation (biochemistry).” Wikipedia, The Free Encyclopedia, 2 Oct. 2022, https://en.wikipedia.org/w/index.php?title=Hill_equation_(biochemistry)&oldid=1113583285.
  5. Zhu, Chao, et al. “Dual-Competitive Lateral Flow Aptasensor for Detection of Aflatoxin B1 in Food and Feedstuffs.” Journal of Hazardous Materials, vol. 344, Feb. 2018, pp. 249–57.