Limitations in linearized analyses of binding equilibria: binding of TNP-ATP to the H₄-H₅ loop of Na/K-ATPase

Abstract

Binding of TNP-ATP [2′,3′-O-(2,4,6-trinitrophenyl)adenosine 5′-triphosphate, a fluorescent analogue of ATP] to the K605 protein was studied. This is an isolated N-domain in the cytoplasmic loop of the Na/K-ATPase α-subunit, lying between membrane-spanning segments 4 and 5 (sequence L³⁵⁴-I⁶⁰⁴). A titration equation is derived that explicitly makes it possible to relate the fluorescence of TNP-ATP and K605 solutions to total probe concentration in the sample. Using this, it is possible to obtain the value of the dissociation constant from the titration experiment without resorting to the Scatchard plot, which is far from optimal from the statistical point of view. Using the new formula with non-linear regression analysis, a value of the dissociation constant K _D for TNP-ATP binding to the K605 protein of 3.03±0.28 μM at 22 °C was obtained. A series of fits to simulated data with added noise demonstrated clearly the advantage of non-linear regression using the new formula over the commonly employed linear regression using the Scatchard plot. The procedure presented is generally applicable to binding assays using changes in the fluorescence of ligands on binding.

Appendix

A: normalized fluorescence intensity

On multiplication of both sides by (γ−1)(F*−[L]_T), Eq. (9) becomes:

$$ (\gamma - 1)K_{\rm{D}} (F^* - \;K_{\rm{D}} {\rm{[L]}}_{\rm{T}} ) = (\gamma [{\rm{L]}}_{\rm{T}} - F^* )\left\{ {(\gamma - 1)[{\rm{P]}}_{\rm{T}} - F^* + \;[{\rm{L]}}_{\rm{T}} } \right\} $$

(A1)

hence the final quadratic equation:

$$ (F^* - \;[{\rm{L]}}_{\rm{T}} )^2 - (\gamma - 1)([{\rm{L]}}_{\rm{T}} + [{\rm{P]}}_{\rm{T}} + K_{\rm{D}} )(F^* - \;[{\rm{L]}}_{\rm{T}} ) + (\gamma - 1)^2 [{\rm{L]}}_{\rm{T}} {\rm{[P]}}_{\rm{T}} = 0 $$

(A2)

is usefully obtained, leading to:

$$ F_{1,2}^* = [{\rm L]}_{\rm T} + {{\gamma - 1} \over 2}\left( {[{\rm L]}_{\rm T} + [{\rm P]}_{\rm T} + K_{\rm D} \pm \sqrt {\left( {[{\rm L]}_{\rm T} + [{\rm P]}_{\rm T} + K_{\rm D} } \right)^2 - 4[{\rm L]}_{\rm T} {\rm [P]}_{\rm T} } } \right) $$

(A3)

One can easily verify that it is the "minus square-root" alternative that satisfies the initial condition F*=0 for [L]_T=0, as well as the result of titration experiments performed in the absence of protein, i.e., F*=[L]_T for [P]_T=0.

B: assessment of the γ factor

On substituting [P]_T=0 into Eq. (A3), i.e., for protein-free samples, the normalized fluorescence intensity F* equates with the ligand concentration:

$$ F^* = [{\rm{L]}}_{\rm{T}} $$

(B1)

The other limiting situation can be reached with samples of very high protein concentration, such that [P]_T>>K _D. In the initial phase of a titration experiment performed with such a sample, when both [L]_T and [PL] are very low compared to [P]_T, the ratio of occupied binding sites is negligible, i.e. [P]≈[P]_T. The equilibrium of Eq. (1) then leads to the following conclusion:

$$ {{{\rm [L]}} \over {{\rm [PL]}}} = {{K_{\rm D} } \over {[{\rm P]}_{\rm T} }} < < 1 $$

(B2)

which is equivalent to the statement that practically all ligands are present in the bound form. The intensity of fluorescence from samples containing protein at high concentration is thus given by:

$$ F^* = \gamma [{\rm{L]}}_{\rm{T}} $$

(B3)

The value of the γ factor is thus obtained as the ratio of initial slopes in F* versus [P]_T plots that have been obtained for a sample containing concentrated protein and no protein, respectively.