On the interaction of two different types of ligands binding to the same molecule part I: basics and the transfer of the decoupled sites representation to systems with n and one binding sites

Abstract

The decoupled sites representation (DSR) for one type of ligand allows to regard complex overall titration curves as sum of classical Henderson-Hasselbalch (HH) titration curves. In this work we transfer this theoretical approach to molecules with different types of interacting ligands (e.g. protons and electrons), prove the existence of decoupled systems for n 1 and one binding sites for two different ligands, and point out some difficulties and limits of this transfer. A major difference to the DSR for one type of ligand is the loss of uniqueness of the decoupled system. However, all decoupled systems share a unique set of microstate probabilities and each decoupled system corresponds to a certain permutation of these microstate probabilities. Moreover, we show that the titration curve of a certain binding site in the original system can be regarded as linear combination of the titration curves of the individual sites of the decoupled system if the weights of the linear combination are substituted by functions in the activity of the second ligand. In the underlying model with only pairwise interaction, an important observation of our theoretical investigation is the following: Even though the binding sites of ligand L 1 may not interact directly, they can show secondary interaction due to the interaction with the second type of ligand. This means, if the activity of the second ligand is fixed and we regard the 1-dimensional titration curve of an individual binding site for ligand L 1 depending on its activity, we may observe a strong deviation from the classical HH shape in spite of non-interacting sites for ligand L 1 .

On the interaction of different types of ligands binding to the same molecule Part II: systems with n to 2 and n to 3 binding sites

“…The binding energies g i can be calculated using the equations given by the coefficients with only one type of ligand. According to Vieta's formulas (for more details see Corollary 1, [7]):…”

“…with i ∈ {1, 2, 3, 4} and j ∈ {A, B}. A look at system (10) reveals that it consists of three systems of the type described in Corollary 1 of [7] and a system which is a sum of two systems of the same structure…”

“…We give a short summary to recall the basics. A more detailed description is given in the first part of this work [7].…”

“…In the first part of this work, we formulated the DSR for molecules with two different types of ligands as conjecture and highlighted the case of n 1 to one binding sites for the two different ligands [7]. In this part we will treat the cases (n, 2) and (n, 3) exemplarily to generate an intuition for the situation of (n 1 , n 2 ) ligand binding sites, for which certain statements are difficult to prove.…”

A derivation of the Grand Canonical Partition Function for systems with a finite number of binding sites using a Markov chain model for the dynamics of single molecules

“…We consider the following situation: A target molecule M has n binding sites for substance L. A certain amount of both substances is solvated in a liquidity at a much higher concentration of ligand L than of M, and the number of free ligand molecules can be measured. Thus, the difference between the number of free and the total number of ligand molecules allows us to determine the average number of ligand molecules L bound to a single target molecule M as a function of the concentration (or chemical activity) of L. Experiments of this kind are a classical procedure in chemistry and produce titration curves that characterize the overall binding of L to M. Titration curves for protons binding to aminoacids can be found in nearly every biochemistry textbook and have been studied for 100 years [2,4,7,8,13]. The mathematical model for titration curves is based on the binding polynomial (bp).…”