A stochastic, cantilever approach to the evaluation of solution phase thermodynamic quantities

doi:10.1073/pnas.0606604104

A stochastic, cantilever approach to the evaluation of solution phase thermodynamic quantities

*Department of Chemistry, Duke University, Durham, NC 27708-0346; and
^†Department of Mechanical Engineering and Material Science, Duke University, Durham, NC 27708-0271

Edited by George M. Whitesides, Harvard University, Cambridge, MA, and approved December 6, 2006 (received for review August 1, 2006)

Abstract

A cantilever device based on competitive binding of an immobilized receptor to immobilized and soluble ligand and capable of measuring solution-phase thermodynamic quantities is described. Through multiple binary queries, the device stochastically measures the probability of the formation of a bound complex between immobilized protein and immobilized ligand as a function of soluble ligand concentration. The resulting binding isotherm is described by a binding polynomial consisting of the activities of soluble and immobilized ligand and binding constants for the association of immobilized protein with free and immobilized ligand. Evaluation of the polynomial reveals an association constant for the formation of a complex between immobilized ligand and immobilized protein close to that for the formation of complex between soluble protein and soluble ligand. The methodology lays the foundation for construction of practical portable sensing devices.

Footnotes

^‡To whom correspondence should be addressed. E-mail: eric.toone{at}duke.edu
Author contributions: P.E.M., R.L.C., and E.J.T. designed research; P.W.S. and G.L. performed research; and P.W.S., R.L.C., and E.J.T. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS direct submission.
This article contains supporting information online at www.pnas.org/cgi/content/full/0606604104/DC1.
↵ § Presumably, the interaction between thiol and nickel(II) is precluded by the kinetics of thiolate desorption from the gold surface: Over the time course of an experiment, the change in concentration of thiolate in solution is approximately zero.
↵ ¶ Although the protein is bound to a surface, so too is the protein–ligand complex. The activity coefficient of the immobilized protein is presumably similar to that of the immobilized protein-free ligand complex, leading to a cancellation of this term in the equilibrium quotient, K _if. This cancellation should lead to a constant that is very similar to the solution-phase value K _ff. Calorimetric data are presented in SI Fig. 6.
↵ ‖ The disordered glycol brush layer allows for Brownian motion of the tethered ligand within a volume determined by the length of the linker that extends above the alkanethiol monolayer. A linker with a length of 3 nm scribes a hemispheric volume of 5.7 × 10⁻²³ liters, a volume that contains exactly (N_A)⁻¹ moles of ligand. Division provides a bound ligand concentration of 3 × 10⁻² M.

↵ ‖ Even though the oligoethylene glycol linker traps one tethered ligand within a calculable volume, the ligand is not homogenously distributed throughout this volume. In the absence of excluded volume effects from adjacent chains, the motion of the polymer head-group is dictated by a worm-like chain model, producing a nonuniform probability of ligand localization across the volume. This distribution presents a problem for the unambiguous determination of the concentration of immobilized ligand because concentrations are, by definition, homogenous. However, the nonidealities introduced by tethering ligand to a surface are absorbed into the activity coefficient.
Abbreviation:

AFM,

atomic force microscopy.