## Overview

###### In September and October, our team realized that we would encounter a problem when we couldn’t conduct the ELISA assay to measure for binding affinity between aptamer and biomarker. Therefore, we decided to mathematically model the interaction between our aptamers and biomarkers by using the Hill function, which is often used to measure the amount of ligand attached to a macromolecule.

## Equation Derivation

###### Our equation was derived from the rate equation commonly used in chemistry used to model the interaction between two elements:

###### For our experiment, we defined “A” as our aptamer, “B” as our biomarker and “AB” as our aptamer biomarker construct to model the interaction between aptamer and biomarker:

###### In our case, K_{on} would represent the process of aptamer and biomarker binding to form the aptamer biomarker construct, and Koff as the process by which the aptamer and biomarker construct unbinds and returns to its original state before reaching equilibrium.

###### Using the above in conjunction with the rate equation for reactions, the rate of which the aptamer and biomarker construct is formed can be thus represented by the following equation.

###### (Derivatives, or rates, are noted by a · (dot), and concentrations are represented by [] (brackets).)

###### Likewise, the rate of biomarker interaction at equilibrium (when the rate is 0) can be modelled in the following manner:

###### Thus, at equilibrium, the interaction can be written as the following by applying the above equation after adding the aptamer and biomarker concentrations to the other side:

###### Using algebra and simplification:

###### Now, we use the fact that the ratio of aptamer biomarker constructs is given by the amount of aptamer biomarkers construct divided by the total number of aptamers (including those currently binding). Collated with the expression K_{off} /K_{on} (which is equal to [A][B]/[AB] by definition) this gives:

###### K_{off}/ K_{on} represents the dissociation constant, or K_{d}. The dissociation constant represents the binding affinity between the aptamer and biomarker. Thus, a smaller dissociation constant signifies higher binding affinity.

###### Thus, the final equation to model the proportion of interaction of aptamer and biomarker forming the aptamer and biomarker construct can be written as:

### References:

######
- Libretexts. 2015. “Dissociation Constant.” Libretexts. August 9, 2015. https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/

Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Equilibria/Chemical_Equilibria/

Dissociation_Constant.
- Ortiz, A. 2013. “Derivation of Hill’s Equation from Scale Invariance.” Journal of Uncertain Systems. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.1068.2409&rep=rep1&type=pdf.
- “Engineering/Webinars - 2021.igem.org.” n.d. Accessed October 6, 2021. https://2021.igem.org/Engineering/Webinars.

Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Equilibria/Chemical_Equilibria/

Dissociation_Constant.