Overview
The mathematical model of the proximity dependent ligation involves the following molecules/initial reactants:
The mathematical model of the proximity dependent ligation involves the following complexes, as defined in the methodology section:
At chemical equilibrium, the total gibbs free energy of the system will be minimized. The equation for the Gibbs Free Energy of the aptamer-protein interaction is given by:
G =(Go+RTln(Q)), where G denotes the gibbs free energy of the system, Go denotes the free energy change of the reaction under standard conditions (standard gibbs free energy of formation), R denotes the ideal gas constant (8.314 Jk-1 mol-1), T denotes the temperature in Kelvin, and Q denotes the reaction quotient, which is given by Q=[product(s)][reactant(s)], at the given state. Because aptamer interactions are often defined in terms of dissociation constants (kd), rather than by the standard gibbs free energy of formation (Go), we will define Go in terms of kd to calculate the gibbs free energy change. Because Go = -RTln(1/kd) at equilibrium, we can use kd to find Go.
Restrictions will be placed on this equation before the minimum gibbs free energy of the system is calculated. Our main restrictions are the mass balance equations. In other words, the concentration of aptamer one fed into the reaction (as reactants) should equal the amount that comes out of the reaction in the form of complexes. The same logic can be applied to the other three reactants (aptamer 2, connector, and protein), to provide a total of four restrictions (mass balance equations). Our final restrictions involve the fact that all concentrations have to be greater than or equal to zero (check “Code for Mathematica” for details).
Once our restrictions are in place, we will employ a built-in function on mathematica to find the minimum gibbs free energy of the system. The syntax for the minimization function is given by: Minimize[{f,cons},{x,y,…}]. We will find the minimum gibbs free energy of the system for several different concentrations of the reactants, in order to find the concentration that optimizes the aptamer-protein interaction.