Derive the Adair-type equation that expresses the fractional saturation ( Y ) of

Question

Derive the Adair-type equation that expresses the fractional saturation (Y) of a homotrimeric protein (one binding site per subunit) in terms of macroscopic association constants (e.g., K1', K2', and K3') and ligand concentration. Next, derive the equivalent expression but in terms of microscopic association constants (e.g., K1, K2, and K3) and ligand concentration. Show your work so that I can tell that you know the relationships between the macroscopic association constants and the corresponding microscopic association constants. Your work should include a complete binding scheme.

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Explanation / Answer

Protein–ligand interactions are fundamental to numerous biological processes, including hormone signaling, allosteric regulation, metabolism, and cell-to-cell communication. A key metric for such interactions is the dissociation constant (Kd), which is often determined in equilibrium binding titrations where the concentration of saturated receptor or bound ligand is measured as a function of the total ligand (LT) concentration. Methods used to obtain ligand binding information include various forms of spectroscopy [1], surface plasmon resonance [2], calorimetry [3], radioactivity [4], and assays that couple protein saturation to biological or chemical– enzymatic signals. Such measurements report on the concentration of bound and/or free ligand and are often interpreted as the fractional saturation (Y) of protein.

The simplest equations for fitting ligand binding data (cf. Eq. (1)) require knowledge of free ligand (L), which may be either measured directly or estimated by virtue of experimental designs in which L can be approximated by LT such as when protein concentration (ET) is very low relative to Kd or LT. Signal detection limits often preclude the use of such designs. When L is not known, the data are best modeled using an expression for Y in terms of LT, ET, and Kd (Eq. (2)). Eq. (2) (obtained from mass conservation laws and the quadratic formula) provides such an analytical expression for the elementary case of ligand binding to a single site (or multiple noninteracting sites). Using such an expression, a best-fit Kd value can be extracted from titration data by least-squares fitting.

For higher order processes, such as cooperative binding to a multimeric protein (where ligand occupancy at one protomer alters ligand affinity at the others), fitting methods must take into account each of the multiply liganded forms. Such behavior manifests experimentally as multiphasic saturation curves or nonlinear Hill and Scatchard plots, whose shapes indicate enhanced (positive cooperativity) or diminished (negative cooperativity) binding of successive ligands [5,6]. Among the many proteins and protein classes that exhibit homotropic cooperativity are the dimeric G-protein-coupled receptors [7], sulfotransferases [8], trimeric metabolic enzymes [9–12], monomeric cytochrome P450s [13], AcrB bacterial drug efflux pump [14], bacteriorhodopsin trimers [15], and DegS protease that regulates the Escherichia coli stress response [16].

The Hill [17,18] and Adair [19] equations (Eqs. (3) and (4)) address the cases where L is known and report cooperativity either indirectly as the Hill coefficient (n) or explicitly as Kd values (given by Ki for the ith ligand to bind) that reflect changes in affinity between successive binding steps. Derivation of parallel analytical equations in terms of LT is nontrivial given that the equilibrium concentration of any species of a dimeric or trimeric protein is the root of a cubic or quartic polynomial, respectively. Analytical solutions for higher order oligomers are not possible, and fitting the titrations of such systems requires the use of numerical methods [20]:

Y=LL+Kd

(1)

Y=(ET+LT+Kd)(ET+LT+Kd)24ETLT2ET

(2)

Y=LnLn+Kd

(3)

Y=LK1+2L2K1K22(1+LK1+L2K1K2)

(4)

Y=LK1+2L2K1K2+3L3K1K2K33(1+LK1+L2K1K2+L3K1K2K3)

(5)

Although analytical expressions for the roots of cubic and quartic functions have been known for 400 years [21–23], they require knowledge of advanced algebra. Recent articles have treated the cubic polynomial in ligand binding systems; Wang and Sigurskjold described competitive binding of two ligands to identical independent binding sites [24,25], and Whitesides and co-workers used exact analysis to describe binding of homodivalent ligands to monomeric proteins, producing expressions that apply equally to homodimeric proteins binding to monovalent ligands [26]. However, to our knowledge, application of the quartic polynomial to trimer–ligand binding has not yet been addressed, and no exact analysis has been applied to heteroligomers that exhibit homotropic cooperativity.

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