A study of statistical error in isothermal titration calorimetry Joel Tellinghuisen * Department of Chemistry, Vanderbilt University, Nashville, TN 37235, USA Received 25 March 2003 Abstract In isothermal titration calorimetry (ITC), the two main sources of random (statistical) error are associated with the extraction of the heat q from the measured temperature changes and with the delivery of metered volumes of titrant. The former leads to uncertainty that is approximately constant and the latter to uncertainty that is proportional to q. The role of these errors in the analysis of ITC data by nonlinear least squares is examined for the case of 1:1 binding, M + X ¢ MX. The standard errors in the key parameters—the equilibrium constant K o and the enthalpy DH o —are assessed from the variance–covariance matrix computed for exactly fitting data. Monte Carlo calculations confirm that these ‘‘exact’’ estimates will normally suffice and show further that neglect of weights in the nonlinear fitting can result in significant loss of efficiency. The effects of the titrant volume error are strongly dependent on assumptions about the nature of this error: If it is random in the integral volume instead of the differential volume, correlated least-squares is required for proper analysis, and the parameter standard errors decrease with increasing number of titration steps rather than increase. Ó 2003 Elsevier Inc. All rights reserved. Keywords: ITC; Data analysis; Nonlinear least squares; Monte Carlo; Statistical errors The method of isothermal titration calorimetry (ITC) 1 is now widely used to obtain thermodynamics information about biochemical binding processes. In a typical application of this technique, one of the reac- tants (M), typically at a concentration of 1 mM, is contained in a reaction vessel of small volume (0.2– 2.0 mL), and the second reactant (the titrant X) is added stepwise to beyond the endpoint of the reaction. The temperature changes that occur after each injection of titrant are analyzed to obtain the heat q associated with the chemical changes from that injection, and the ex- periment thereby produces a titration curve of q vs ex- tent of reaction. Analysis of such titration curves yields the enthalpy change DH o and the equilibrium constant K o for the reaction [1–3]. Despite the burgeoning use of the ITC technique, 2 there has been surprisingly little effort directed toward understanding the role of statistical errors in ITC data and their effect on the estimates of DH o and K o , which are obtained by nonlinear least-squares (LS) analysis of the data [4]. Such information can be used to optimize the parameters chosen for a particular experiment with respect to the precision of determination of either DH o or K o or to find some judicious compromise between these two precisions. It is also of relevance to a better resolution of a simmering controversy over the extent to which the DH o values estimated from the temperature dependence of K o (the vanÕt Hoff DH o s) are consistent with those obtained directly from the analysis of the ITC titration curves [5–7]. The latter issue was in fact the impetus for the present work. In what follows, I investigate the role of statistical errors in the analysis of ITC data using the ‘‘exact’’ variance–covariance matrix V for the nonlinear fit model. This ‘‘exact’’ V is obtained for exactly fitting data of known statistical error structure and in fact is not exact in its prediction of the parameter errors for nonlinear fit models. However, Monte Carlo (MC) cal- culations on typically 10 5 equivalent simulated titration curves confirm that analysis of such data is in accord with a previously derived ‘‘10% rule of thumb’’ [8]: If the Analytical Biochemistry 321 (2003) 79–88 www.elsevier.com/locate/yabio ANALYTICAL BIOCHEMISTRY * Fax: 1-615-343-1234/322-4936. E-mail address: joel.tellinghuisen@vanderbilt.edu. 1 Abbreviations used: ITC, isothermal titration calorimetry; LS, least-squares; MC, Monte Carlo. 2 Ref. [1] has been cited over 800 times, with about 1/4 of these coming in the last 2 years alone. 0003-2697/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0003-2697(03)00406-8