Analysis of parameter resolution from derivatives of binding isotherms

Optimizing experimental parameters in isothermal titration calorimetry: variable volume procedures

In the study of 1:1 binding, M + X right <==> MX, isothermal titration calorimetry is generally thought to be limited to reactions in which the key parameter, c = K[M]0, can be set in the range 1-1000. In fact, the range of applicability can be extended by a factor of 10-100 at the upper end and as much as 10(5) at the lower, with certain provisos. The present work emphasizes the low-c regime, with the key heat parameter, h identical with DeltaH degrees [M]0, low, as well. Successful determination of K and DeltaH degrees in this region requires that the titration be extended to large excesses of titrant X over titrate M, and then the reaction heat is distributed strongly in favor of the early injections. With decreasing c, DeltaH degrees and the stoichiometry parameter n (often called site number) also become highly correlated and individually indeterminate. However, the product DeltaH degrees x n ( identical with Hn) is well-determined, so if n is known from other information, both K and DeltaH degrees can be determined to quite low c. By varying the titrant volume from injection to injection, one can significantly reduce the uncertainties in the estimated K and Hn values, permitting determination of K to better than 10% and Hn within 3% down to c = 10(-4), even for the low h value of 0.1 cal/L. The titrant volume optimization algorithm yields best results for the minimal number of injections - three when n is fitted, two when it is fixed. At low c, the resulting volume distributions depend nearly exponentially on injection number. This observation facilitates the derivation of similar, near-optimal volume distributions for five- and four-injection procedures that offer two statistical degrees of freedom for analysis. The volume optimization results are tested on the Ba2+/18-crown-6 ether complexation reaction at c = 0.1 and h = 0.16 cal/L, illustrating some practical complications but confirming the utility of the variable-volume protocol.

Optimizing Experimental Parameters in Isothermal Titration Calorimetry

In isothermal titration calorimetry, the statistical precisions with which the equilibrium constant (K) and reaction enthalpy (DeltaH degrees ) can be estimated from data for 1:1 binding depend on a number of quantities, key among them being the products c identical with K[M](0) and h identical with DeltaH degrees [M](0), the stoichiometry range (R(m)(), ratio of total titrant X to total titrate M after the last injection), and the number of injections of titrant. A study of the statistical errors as functions of these quantities leads to the following prescription for optimizing throughput and precision: (1) Make 10 injections of titrant. (2) Set the concentrations in accord with the empirical equation R(m)() = 6.4/c(0.2) + 13/c (but no smaller than 1.1). (3) Make the starting concentration [M](0) as large as possible within the large-signal limits of the instrumentation but limited to c < 10(3) for estimating K. With this procedure, both K and [M](0) are predicted to have relative standard errors <1% over large ranges of K. Systematic errors in the concentrations, [X](0) and [M](0), are fully compensated by the "site number" or stoichiometry parameter (n). On the other hand, altering and freezing any of the fit parameters leads to a deterioration of the fit quality and to predictable changes in the other parameters. Fit divergence at very small c is avoidable through a simple redefinition of the fit parameters; however, unless n can be fixed from other information, DeltaH degrees may be statistically ill-defined in this region.

Statistical error in isothermal titration calorimetry: Variance function estimation from generalized least squares

The method of generalized least squares (GLS) is used to assess the variance function for isothermal titration calorimetry (ITC) data collected for the 1:1 complexation of Ba(2+) with 18-crown-6 ether. In the GLS method, the least squares (LS) residuals from the data fit are themselves fitted to a variance function, with iterative adjustment of the weighting function in the data analysis to produce consistency. The data are treated in a pooled fashion, providing 321 fitted residuals from 35 data sets in the final analysis. Heteroscedasticity (nonconstant variance) is clearly indicated. Data error terms proportional to q(i) and q(i)/v are well defined statistically, where q(i) is the heat from the ith injection of titrant and v is the injected volume. The statistical significance of the variance function parameters is confirmed through Monte Carlo calculations that mimic the actual data set. For the data in question, which fall mostly in the range of q(i)=100-2000 microcal, the contributions to the data variance from the terms in q(i)(2) typically exceed the background constant term for q(i)>300 microcal and v<10 microl. Conversely, this means that in reactions with q(i) much less than this, heteroscedasticity is not a significant problem. Accordingly, in such cases the standard unweighted fitting procedures provide reliable results for the key parameters, K and DeltaH(degrees) and their statistical errors. These results also support an important earlier finding: in most ITC work on 1:1 binding processes, the optimal number of injections is 7-10, which is a factor of 3 smaller than the current norm. For high-q reactions, where weighting is needed for optimal LS analysis, tips are given for using the weighting option in the commercial software commonly employed to process ITC data.

Calorimetric vs. van't Hoff binding enthalpies from isothermal titration calorimetry: Ba2+-crown ether complexation

The 1:1 complexation reaction between Ba(2+) and 18-crown-6 ether is re-examined using isothermal titration calorimetry (ITC), with the goal of clarifying previously reported discrepancies between reaction enthalpies estimated directly (calorimetric) and indirectly, from the temperature dependence of the reaction equilibrium constant K (van't Hoff). The ITC thermograms are analyzed using three different non-linear fit models based on different assumptions about the data error: constant, proportional to the heat and proportional but correlated. The statistics of the fitting indicate a preference for the proportional error model, in agreement with expectations for the conditions of the experiment, where uncertainties in the delivered titrant volume should dominate. With attention to proper procedures for propagating statistical error in the van't Hoff analysis, the differences between Delta H(cal) and Delta H(vH) are deemed statistically significant. In addition, statistically significant differences are observed for the Delta H(cal) estimates obtained for two different sources of Ba(2+), BaCl(2) and Ba(NO(3))(2). The effects are tentatively attributed to deficiencies in the standard procedure in ITC of subtracting a blank obtained for pure titrant from the thermogram obtained for the sample.

A study of statistical error in isothermal titration calorimetry

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 right arrow over left arrow MX. The standard errors in the key parameters-the equilibrium constant Ko and the enthalpy DeltaHo-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.