A Browser‐Based Tool for Assessing Accuracy of Isothermal Titration Calorimetry‐Derived Parameters: K _d, ΔH°, and n

Abstract

Accurate determination of equilibrium dissociation constants (K _d) is essential for decision‐making in drug discovery and diagnostics development. Isothermal titration calorimetry (ITC), which requires no reactant labeling or immobilization, is commonly used to validate K _d values from high‐throughput screens. Yet, like other methods, ITC results can be skewed by systematic errors in reactant concentrations, a fact that is often overlooked, potentially leading to misinformed decisions. To address this, accuracy confidence intervals (ACI)‐ITC is developed, a browser‐based tool that calculates ACI for ITC‐derived parameters, offering probabilistic ranges for the true values of K _d, enthalpy change (ΔH°), and binding stoichiometry (n). Unlike traditional confidence intervals that consider only random errors, ACI‐ITC explicitly accounts for systematic errors, providing a more accurate framework to assess experimental reliability. Alongside a user‐friendly interface, it offers detailed guidance for determining uncertainties in concentrations and heat, which are critical inputs for assessing measurement accuracy. The tool's browser‐based accessibility (https://aci.sci.yorku.ca) eliminates the need for specialized installation, enabling cross‐platform compatibility and streamlining accuracy assessments. By highlighting the importance of systematic errors and providing a structured approach for their evaluation, ACI‐ITC supports more robust conclusions, fosters better‐informed decisions based on ITC measurements, and enhances the reliability of research findings.

Accurate K _d determination is crucial for drug discovery and diagnostics. Accuracy confidence intervals (ACI)‐ITC, a browser‐based tool, calculates ACI for isothermal titration calorimetry (ITC)‐derived parameters, considering systematic errors in concentration and heat measurements. By offering probabilistic ranges for true K _d, ΔH°, and stoichiometry value n, ACI‐ITC enhances experimental reliability, guiding better decisions, streamlining cross‐platform accuracy assessments, and improving trust in ITC‐based findings.

1. Introduction

Drugs are often molecules capable of noncovalently binding to therapeutic targets, typically proteins, to form stable molecular complexes.^{[ 1} , ² , ^{3 ]} The thermodynamic stability of these complexes is characterized by the equilibrium dissociation constant (K _d); lower K _d values correspond to more stable complexes between a drug candidate and its target. Accurate determination of K _d values is critical for drug discovery, as these values guide the selection of promising compounds for further development.^{[ 4} , ^{5 ]}

Drug discovery typically begins with screening large molecular libraries to identify potential binders to the target. High‐throughput techniques, such as surface plasmon resonance (SPR) and bio‐layer interferometry (BLI), are commonly used during this initial screening phase to obtain preliminary K _d values for extensive libraries of compounds.^{[ 6} , ^{7 ]} However, these techniques require immobilization of the target on a solid surface, which can alter the natural binding process and potentially compromise the accuracy of the resulting K _d values.^{[ 8} , ^{9 ]} Additional inaccuracies arise from the inability to optimize binding experiments for each compound when the library is screened using high‐throughput techniques.^{[ 10 ]} Consequently, accurate methods for confirming and refining K _d measurements are essential to ensure that only the most suitable drug candidates progress through the discovery pipeline and that they are not mistakenly overlooked.

When evaluating the quality of K _d data, both precision and accuracy are critical. Precision refers to the reproducibility of K _d measurements under identical conditions and is often expressed as the standard deviation (SD) of the determined values.^{[ 11} , ¹² , ^{13 ]} Accuracy, in contrast, reflects how close the experimentally determined K _d is to its true, fundamentally unknown value.^{[ 14} , ^{15 ]} Even highly precise measurements can be significantly inaccurate due to systematic errors in experimental variables, such as reactant concentrations or measured signals.^{[ 14} , ^{15 ]} These errors propagate through calculations, introducing unnoticed systematic deviations in K _d values and compromising decision‐making in drug development. Quantitative assessment of K _d accuracy is particularly important for techniques like isothermal titration calorimetry (ITC), which is widely used to validate K _d values obtained from high‐throughput methods but remains vulnerable to undetected inaccuracies if only precision is considered.

ITC is broadly regarded as the gold standard for validating K _d values determined through high‐throughput screening.^{[ 16} , ¹⁷ , ^{18 ]} Unlike many other methods, ITC quantifies binding thermodynamics directly in solution without requiring any reactant modifications such as labeling or immobilization.^{[ 17} , ¹⁹ , ^{20 ]} By measuring heat changes during binding events, ITC provides a comprehensive thermodynamic profile, including K _d, enthalpy change (ΔH°), and binding stoichiometry (n). Additionally, ITC instrumentation ensures highly reproducible titration and measurements, providing exceptional precision in the determined parameters.^{[ 17} , ^{21 ]} However, ITC has limitations, such as requiring relatively large quantities of reactants and having low throughput.^{[ 17 ]} Despite these drawbacks, ITC remains a critical tool for confirming K _d values derived from high‐throughput methods.

While being highly precise, ITC is prone to inaccuracies in the determined K _d values due to systematic errors in experimental variables, such as analyte concentrations and measured heat.^{[ 15} , ²² , ^{23 ]} These inaccuracies can significantly affect the reliability of the resulting K _d values. Under typical operating conditions, ITC is recommended to be used at c‐values (the ratio between the true titrand concentration and the true K _d value) significantly exceeding unity to ensure sufficient curvature in the binding isotherm for reliable parameter fitting. However, multiple studies suggest that such elevated c‐values (>>1) can potentially amplify systematic errors in input variables, leading to significant deviations of the determined K _d values from their true values.^{[ 23} , ²⁴ , ^{25 ]} These findings indicate that even under standard experimental conditions, ITC‐derived K _d values may be more prone to inaccuracies than broadly appreciated. Without their quantitative assessment, such inaccuracies can result in misleading conclusions about binding affinities and compromise downstream decision‐making. By quantifying these inaccuracies, researchers can better leverage ITC's strengths while mitigating potential errors in critical decision‐making steps.

To address the challenges associated with assessing the accuracy of K _d values in the classic non‐titration approach (commonly used in techniques such as fluorescence anisotropy, microscale thermophoresis, BLI, and SPR), we recently developed the ACI‐Kd tool.^{[ 14 ]} The term accuracy confidence interval (ACI) refers to a confidence interval (CI) that specifically accounts for systematic errors, which is why the word “accuracy” is included in its name. ACI‐Kd effectively assesses systematic errors in non‐titration methods using analytical (non‐numerical) error propagation to assess K _d accuracy, avoiding the complexities of statistical numerical methods and offering a practical, accessible approach.^{[ 14 ]} Moreover, ACI‐Kd is implemented as a user‐friendly web application (https://aci.sci.yorku.ca), requiring only a web browser and making it broadly accessible.^{[ 14 ]} However, ACI‐Kd is not designed for ITC, which requires accounting for additional unknowns, such as ΔH° and n.

Monte Carlo simulations have long been proposed as a robust statistical method for assessing the accuracy of ITC‐derived K _d values.^{[ 24} , ²⁶ , ²⁷ , ^{28 ]} These simulations estimate CIs and error distributions by conducting numerous virtual experiments that incorporate random variations in experimental parameters, such as analyte concentrations and measured heat.^{[ 24} , ²⁶ , ²⁷ , ^{28 ]} By simulating a wide range of conditions, Monte Carlo methods can identify how systematic biases impact derived parameters.^{[ 24} , ²⁶ , ²⁷ , ^{28 ]}

While Monte Carlo simulations are relatively straightforward for propagating random errors (indicators of precision), they may be excessive for precision evaluation, as standard nonlinear regression already provides precision estimates in the form of SDs.^{[ 24} , ²⁶ , ²⁷ , ^{28 ]} In contrast, applying Monte Carlo simulations to assess K _d accuracy is fully justified but significantly more challenging, as it involves propagating systematic errors—less obvious and more complex parameters. Estimating these systematic errors requires a deep understanding of their nature, origins, and dependence on experimental design. This complexity, combined with the often‐overlooked potential for large inaccuracies in ITC‐derived K _d values and the technical expertise required to apply Monte Carlo techniques, likely explains their limited use. Simplified tools are essential to bridge these gaps and make accurate assessments accessible to a broader research community.

Here, we address the challenges of quantitatively assessing the accuracy of ITC‐derived K _d values. We present ACI‐ITC, a Monte Carlo‐based tool that calculates the ACI for K _d, ΔH°, and n. Unlike traditional Monte Carlo approaches, ACI‐ITC is optimized for usability and requires no programming or advanced statistical knowledge, making it accessible to a broad range of molecular scientists. To further enhance usability, ACI‐ITC is integrated into the ACI web application (https://aci.sci.yorku.ca), which requires no software installation and provides comprehensive reports that integrate accuracy with precision for K _d, ΔH°, and n, along with publication‐quality graphs. Additionally, we offer comprehensive instructions for estimating input parameters, such as systematic errors in concentrations and heat, to facilitate practical implementation.

By simplifying the complexities of propagating systematic errors, ACI‐ITC enables researchers to generate reliable data for K _d, ΔH°, and n from ITC experiments. This advancement enhances the reliability of ITC‐derived K _d values and their applications in drug discovery, allowing researchers to draw more accurate conclusions about molecular interactions.

2. Results and Discussion

2.1. Browser‐Based ACI Calculator

To evaluate the accuracy of ITC‐derived binding parameters considering systematic errors in experimental variables, such as concentrations and measured heat changes, we first developed a Python program named ACI‐ITC. In ITC measurements, the heat change per injection is typically normalized to the heat change per mole of the injected target, denoted as the enthalpy change ΔH, with units of kcal/mol. To clarify, ΔH will be used throughout this text to refer to the “measured heat changes”, while ΔH° will specifically denote the enthalpy change per mole of the formed complex, which is one of the parameters determined in ITC experiments and reported.

The ACI‐ITC program employs Monte Carlo‐based bootstrapping to compute the ACI for three binding parameters—K _d, ΔH°, and n—with 95% confidence. A streamlined computational algorithm for ACI‐ITC is depicted in Figure  1 , while a more detailed version of the algorithm is presented in Figure S1. To ensure broad accessibility and ease of use for individuals with diverse technical backgrounds, we integrated the ACI‐ITC program into the browser‐based ACI web application (https://aci.sci.yorku.ca), providing a user‐friendly platform for the field and integrating other tools for assessing the accuracy of physicochemical parameters. This free web app is accessible from any browser on any device and requires no programming skills. It accepts ITC binding isotherms, represented as “ΔH versus Molar Ratio,” directly from commercial software integrated with the ITC instrument for calculations. The web app produces and downloads a report including the input and output data, a high‐quality binding curve “Cumulative ΔH versus Molar ratio” with bootstrapping fitting lines, and the frequency‐distribution histograms of the determined values of K _d, ΔH°, and n. An example of the graphs produced by ACI‐ITC is shown in Figure  2 . Notably, “Cumulative ΔH” refers to the total heat change measured up to the i‐th injection. It is the summation of the measured ΔH values for the first i injections at the molar ratio corresponding to the i‐th injection. From this point forward, we will denote the cumulative ΔH after the i‐th injection as ΔH _cum,i .

Figure 1.

Streamlined computational algorithm of the ACI‐ITC program.

Figure 2.

Example of graphs generated by the ACI‐ITC web application (http://aci.sci.yorku.ca): a) Bootstrap fitting results for an ITC binding curve “Cumulative ΔH versus Molar ratio”, where the red dots represent experimental data, and the gray‐shaded region contains bootstrap‐fitted curves. The ITC binding curve was derived from an ITC experiment using the DA‐3BP aptamer–dopamine binding pair. See the experimental demonstration section for more details. b) Frequency distributions of the determined K _d, ΔH°, and n, derived from 5000 bootstrap iterations. The y‐axis in each histogram represents the number of occurrences for each parameter value. The calculated ACI for K _d, ΔH°, and n with 95% confidence are indicated on the graphs.

2.2. Necessary Inputs for Computing ACI

To use the ACI‐ITC program, users must input the following information: the nominal concentration of the target in the syringe (T ₀), the nominal initial concentration of the ligand in the sample cell (L ₀), sample/reference cell volume (V ₀), injection volume per step (V _inj), total number of injections (N), the SDs corresponding to the (CIs) of the relative systematic errors in the target and ligand concentrations (δ T ₀/T ₀ and δ L ₀/L ₀, respectively), the SD for the CI of the relative systematic error in the measured heat changes (δ(ΔH)), and the desired number of bootstrap iterations. The definitions of δ T ₀/T ₀, δ L ₀/L ₀, and δ(ΔH), along with guidance on how they can be experimentally assessed, are provided in the next section.

The default minimum number of bootstrapping iterations is set at 2,000 to ensure data convergence. However, users can input a higher value to obtain more detailed frequency distributions for the determined parameters. After users enter or load the experimental binding isotherm “ΔH versus Molar ratio”, the web app completes the ACI calculations within ≈10 s for 2,000 bootstrapping iterations; the time increases with the increasing number of iterations.

The binding isotherm “ΔH versus Molar ratio” is a typical output from modern ITC instruments.^{[ 17 ]} In the ACI‐ITC program, this data is internally converted to the binding curve “ΔH _cum,i versus Molar ratio”, which is then fitted using Equation (1):

(1)

Here, Γ represents the molar ratio of target to ligand in the sample cell; ΔH _cum,max denotes the maximum cumulative ΔH value, which theoretically corresponds to Γ = ∞, at which all ligands in the sample cell bind to targets, forming complexes. K _d, ΔH _cum,max, and n are the three parameters to be determined. Notably, rather than representing binding stoichiometry, the parameter n is often interpreted as a correction factor for the ligand concentration in the 1:1 binding model.^{[ 29} , ^{30 ]} It is worth emphasizing that using the first derivative of Equation 1 (with respect to the amount of injected target) has been shown to offer advantages over directly fitting Equation 1.^{[ 31 ]} However, in this work, we use Equation 1 as the fitting equation because it is directly derived from the mass balance of analytes and the fundamental definitions of K _d and the measured bound/unbound fraction of the ligand.^{[ 26 ]} This formulation provides a clear and intuitive framework, making it the most easily interpretable fitting model for biochemists.

Based on the determined ΔH _cum,max, the enthalpy change per mole of the formed complex (ΔH°) is calculated with Equation (2):

$$\hspace{0.17em}\Delta H°=\frac{\Delta H_{\text{cum},\text{max}}{V}_{\text{inj}}{T}_0}{n{V}_0{L}_0}$$ (2)

where all parameters retain their definitions as described above.

Therefore, except for the SDs for the CIs of the relative systematic errors in the experimental variables, i.e., δ T ₀/T ₀, δ L ₀/L ₀, and δ(ΔH), all other inputs are either known based on the experimental design or obtained from standard ITC measurements. Next, we will explore how to experimentally determine δ T ₀/T ₀, δ L ₀/L ₀, and δ(ΔH).

2.3. Assessing CIs of Systematic Errors in Variables

The systematic errors in ITC‐derived parameters K _d, ΔH°, and n arise from inaccuracies in experimental variables, including analyte concentrations, detected heat changes, injection volumes, and sample cell volume. However, because the systematic errors associated with injection and sample cell volumes are tied to inherent instrumental accuracies that cannot be mitigated and measured by experimentalists, this work focuses solely on assessing the systematic error ranges for analyte concentrations and detected heat changes, and on propagating them into the ACI of the determined parameters. Disregarding systematic errors in the injection and sample cell volumes leads to the resulting ACI being narrower than the real ACI.

The actual systematic errors in the experimental variables are inherently unknown. If these errors were known, assessing the accuracy of the parameters would be unnecessary, as the input values could be mathematically corrected to yield accurate parameters. Therefore, only the CIs of the systematic errors in the experimental variables can be determined.

As has been explained in detail elsewhere,^{[ 12} , ^{14 ]} for the classic non‐titration approaches, the CIs of relative systematic errors in analyte concentrations can be estimated by determining the relative random errors in the concentrations of stock solutions. The conversion between relative random errors and CIs for systematic errors of the variables used in K _d determination requires a valid assumption under specific experimental conditions.^{[ 12} , ^{14 ]} First, all established approaches, such as assessing reagent purity, minimizing adsorption, and instrument calibrations, should be employed to minimize systematic errors in variables, including analyte concentrations and measured signal (e.g., heat changes in ITC experiments).^{[ 12 ]} Given the experimental similarities between K _d determination using ITC and the classic non‐titration methods, the error‐minimization approaches detailed in our previous work,^{[ 12 ]} are applicable for minimizing systematic errors of variables in ITC experiments.

After minimizing systematic errors in analyte concentrations and measured signals, it is reasonable to assume that the CIs for the remaining relative systematic errors in analyte concentrations (i.e., ΔT ₀/T ₀ and ΔL ₀/L ₀) are primarily governed by the relative random errors observed across independently prepared sample solutions (i.e., δ T ₀/T ₀ and δ L ₀/L ₀). In non‐titration equilibrium approaches, these sample solutions refer to independently prepared stock solutions of the target and ligand used to prepare each equilibrium mixture.^{[ 12} , ^{14 ]} In contrast, for ITC experiments, the relevant solutions are those loaded into the syringe and sample cell, namely, the concentrated target solution and the ligand solution, used consistently throughout the entire titration.^{[ 17 ]} As a result, the random error associated with preparing the target and ligand solutions, typically involving multiple steps, effectively manifests as the systematic error in analyte concentrations within a single ITC experiment. The CI for the systematic error in the measured heat change (ΔH) is primarily determined by the random errors in baseline adjustments, as a single baseline adjustment is applied to the entire dataset from an ITC titration experiment.^{[ 32} , ^{33 ]}

Experimentally, after minimizing the systematic errors in analyte concentrations, δ T ₀/T ₀ and δ L ₀/L ₀ can be determined by measuring spectroscopic signals (e.g., absorbance) of the target and ligand across multiple independent sample preparations.^{[ 12} , ^{14 ]} In contrast, assessing CIs for heat changes remains considerably more challenging. Ideally, the buffer compositions of the target and ligand should be carefully matched to minimize heat changes arising from buffer mixing.^{[ 34 ]} Moreover, control titration experiments should be meticulously designed and executed to account for heat changes unrelated to binding events, such as those caused by dilution or stirring.^{[ 18 ]} In addition to these experimental optimizations and data corrections, the selection of the baseline for ITC thermograms during data analysis plays a critical role in the accuracy of the determined heat change for each injection.^{[ 32} , ^{33 ]} Multiple approaches have been developed to improve the accuracy of baseline determination;^{[ 32} , ³³ , ^{35 ]} however, the true baseline is fundamentally unknown, and its selection is often subject to methodological or subjective biases. Unlike some other experimental parameters, the baseline cannot be determined independently through control experiments,^{[ 32 ]} and no robust method currently exists to rigorously assess the CI of baseline offsets. Nevertheless, reasonable estimates can be made based on the maximum heat pulse detected in the titration experiment,^{[ 15 ]} which will be discussed in the next section.

Because the baseline is selected only once for the entire titration, any random error in its selection propagates as an absolute systematic error in the measured heat changes. This systematic error impacts all heat changes used for downstream data analysis, introducing bias and compromising the overall accuracy of the analysis. The lack of a reliable method to assess the CI of baseline offsets remains a critical challenge in the ITC field, directly impacting the accuracy of binding thermodynamic measurements. Researchers are encouraged to develop innovative approaches to address this gap, advancing the accuracy and reliability of ITC data analysis.

It is important to emphasize that in conventional ITC data analysis using least squares regression (e.g., Levenberg–Marquardt algorithm), all of the systematic errors discussed above are typically ignored (assumed to be zero).^{[ 17 ]} Furthermore, the frequency (probability) distributions of the determined parameters are conventionally assumed to follow a Gaussian shape.^{[ 36 ]} However, when accounting for the CIs associated with systematic errors in the variables, these assumptions no longer hold true. As a result, the frequency distributions of the determined parameters can change significantly.

2.4. Minimum Values of Systematic Errors in Input Variables

Modern ITC instruments deliver highly precise injection volumes, resulting in precise values of T ₀ and L ₀, as well as their molar ratios, with relative random errors typically within 1%–2%.^{[ 37 ]} However, the accuracies of T ₀ and L ₀ are fundamentally constrained by the limitations of sample‐preparation methods. These limitations stem from the inherent precision of preparation instruments (e.g., analytical balances and pipettes) and unavoidable reagent impurities. Consequently, the relative systematic errors in these concentrations (|ΔL ₀/L ₀| and |ΔT ₀/T ₀|) are unlikely to fall below 5%.^{[ 38 ]}

With a well‐calibrated ITC instrument and careful baseline adjustments, systematic errors in measured heat changes (|Δ(ΔH)|) can typically be maintained within 2% of the maximum absolute value of observed ΔH.^{[ 15 ]} This maximum value is often observed in the first reliable measurement (commonly corresponding to the second injection) and is denoted here as |ΔH ₁|. Based on this, it is reasonable to assume that the probability distribution of Δ(ΔH) is normal and has a SD (random error) of δ(ΔH) = 0.02|ΔH ₁|. Users can therefore input δ(ΔH) = 0.02|ΔH ₁| as the minimum SD for measured heat changes until a robust method for assessing CIs of baseline offsets becomes available.

For inputting information into the ACI‐ITC program, users should provide δ T ₀/T ₀ and δ L ₀/L ₀ values based on the relative random errors determined from multiple independent target and ligand solution preparations, with minimum values of 0.05 (5%). In cases of retrospective data where CIs for systematic errors in concentrations are unavailable, users may reasonably assume δ T ₀/T ₀ = δ L ₀/L ₀ = 0.05 as the minimum value. At present, due to the lack of reliable methods for determining CIs for detected heat changes, users are advised to input δ(ΔH) = 0.02|ΔH ₁| as a practical estimate to compute the narrower limits for the ACI of K _d, ΔH°, and n. The ACI‐ITC utilizes the mentioned values as preset defaults that can be changed by users.

2.5. Verification of ACI‐ITC

We rigorously verified the ACI‐ITC program and its web application using synthetic ITC data generated with a Python‐based simulator. These simulations utilized predefined (true) values for K _d, ΔH°, and n (n = 1), initial concentrations L ₀ and T ₀, and their associated systematic errors, as well as the systematic errors in the heat change per injection. The complete verification process and results are detailed in Note S1. The outcomes from the ACI‐ITC web app for the synthetic data were in full agreement with theoretical predictions, confirming the reliability and accuracy of both the ACI‐ITC program and its web app (http://aci.sci.yorku.ca).

2.6. Application of ACI‐ITC to Experimental Data

Following the verification of the ACI‐ITC web app, we applied it to evaluate the ACI for the parameters determined in real ITC experiments. Our goal was to compare the ACI of K _d, ΔH°, and n computed by ACI‐ITC and the CIs of these parameters computed in conventional data analysis utilizing the Levenberg–Marquardt algorithm. In this study, we conducted ITC experiments using a MicroCal Auto‐ITC200 instrument to investigate the binding interactions of two molecular pairs: Theo2201 aptamer–theophylline and DA‐3BP aptamer–dopamine. An additional application of the ACI‐ITC tool to retrospective ITC data is provided in the Supporting Information (Note S2).

For the Theo2201 aptamer–theophylline system, the nominal initial aptamer concentration in the sample cell ([Theo2201]₀, nominal L ₀) was 60 μM, while the nominal initial theophylline concentration in the syringe ([theophylline]₀, nominal T ₀) was 936 μM. The high purity of both the aptamer and theophylline, with no detectable impurities, was confirmed by liquid chromatography–mass spectrometry (LC–MS) analysis (Notes S3.1 and S3.2). Using a spectrophotometer (NanoDrop 1000, Thermo Scientific), the relative random error associated with independently prepared theophylline solutions at 936 μM was determined to be 6% (Note S4.1). Since experimentally determining the batch‐to‐batch variability of the aptamer concentration was impractical, we assigned a minimum practical value of 5%, as discussed in a previous section, as the SD of the CI for the relative systematic error in aptamer concentration when calculating the ACI of the fitted parameters, despite the relative random error from preparations made from the same stock being only 4% (Note S4.2). The maximum observed absolute enthalpy change (|ΔH ₁|) during the titration was 30.3 kcal mol⁻¹. To set a lower bound for the uncertainty in ΔH, we used δ(ΔH) = 0.02 × 30.3 ≈ 0.61 kcal mol⁻¹.

The ITC experiment involved 19 injections, each of 2 μL, into a 200 μL sample cell. Using the experimental binding isotherm (“ΔH vs Molar ratio”) and inputting all the values listed above into the ACI‐ITC web application, we calculated the ACI for K _d, ΔH°, and n, along with their frequency distributions (Figure  3a). In all ACI‐ITC applications presented in this study, results were generated using 5,000 bootstrap iterations. For comparison, the 95% CIs of these parameters were also determined using conventional least‐squares (L‐S) nonlinear regression with the Levenberg–Marquardt algorithm, assuming Gaussian probability distributions.

Figure 3.

Comparison of the probability density distributions for the ACI calculated using ACI‐ITC versus the CIs obtained from conventional least‐squares (L‐S) regression for a) the Theo2201 aptamer–theophylline binding pair and b) the DA‐3BP aptamer–dopamine binding pair. ACI histograms were generated from 5,000 bootstrap iterations. Gaussian probability distributions were constructed using the mean and SD values obtained from L‐S nonlinear regression of the ITC binding isotherms using the Levenberg–Marquardt algorithm. The mean values and the boundaries of the 95% CIs determined by both data analysis approaches are indicated on the graphs. Skewness values are reported in each panel to indicate distribution asymmetry.

The results, summarized in Figure 3a, demonstrate that ignoring the systematic error contributions to variable uncertainties in conventional L‐S analysis can significantly underestimate the uncertainties (i.e., the CI widths) of ITC‐derived parameters. For instance, the 95% ACI for K _d calculated by ACI‐ITC was 73–320 nM, with a mean of 180 nM and a non‐Gaussian distribution with a skewness of 0.65. In contrast, the CI derived from L‐S analysis was much narrower, at 120–200 nM with a mean of 160 nM, incorrectly suggesting a Gaussian distribution. Understanding the correct probability distribution of the equilibrium dissociation constant K _d is essential for accurately ranking drug candidates, as binding affinity comparisons rely on this distribution to conclude which ligand binds the target tightly with a higher probability.^{[ 39 ]} This result emphasizes the importance of incorporating systematic error considerations to ensure reliable parameter estimation in ITC analysis. Similar implications apply to the determination of the other two parameters: ΔH° and n.

To further investigate the differences between ACI and conventionally determined CIs, we applied ACI‐ITC to an ITC experiment examining the dopamine (DA)‐3BP aptamer–dopamine binding pair.^{[ 40 ]} The nominal initial concentrations were L ₀ = [DA‐3BP]₀ = 450 μM and T ₀ = [dopamine]₀ = 7,020 μM. The purity of the DA‐3BP aptamer was determined to be 96% by LC–MS (Note S3.3), resulting in a corrected DA‐3BP concentration in the ITC cell of 432 μM. Immediately after preparation, the high purity of the dopamine solution was confirmed by LC–MS (Note S3.4). Given the susceptibility of dopamine to oxidation, its stability over the duration of the ITC experiment was assessed by monitoring the time‐dependent decrease in absorbance at 280 nm (OD_{2 8 0}) over a 90‐minute period (Note S3.5). The results showed no significant change in OD_{2 8 0}, indicating that dopamine oxidation was negligible under the experimental conditions and unlikely to impact the effective dopamine concentration during the measurement. This evaluation strategy can be generalized to other unstable reagents (e.g., glutathione). In such cases, freshly prepared solutions should be used immediately before measurements, and degradation kinetics should be experimentally characterized, such as by monitoring absorbance at characteristic wavelengths or assessing purity over time using analytical methods like in situ LC–MS, especially when anti‐degradation measures (e.g., pH adjustment or stabilizing agents) are impractical. For retrospective analyses lacking experimental stability data, concentration corrections may be estimated based on published degradation kinetics.

The relative random error in dopamine concentration, based on independently prepared solutions measured by UV spectrophotometry, was determined to be 11% (Note S4.3). Since experimental determination of batch‐to‐batch variability in aptamer concentration was impractical, a conservative SD of 5% was assigned for the relative systematic error in aptamer concentration when calculating the ACI, as discussed earlier. This estimate exceeds the observed 3% random variability in concentration measurements from the same stock (Note S4.4). Based on the maximum observed absolute heat change (ΔH ₁) in the experiment, the uncertainty in heat measurement, δ(ΔH), was set to 0.80 kcal/mol. The corrected binding isotherm was then analyzed using both ACI‐ITC and conventional least‐squares (L‐S) regression methods, with comparative results presented in Figure 3b.

The analysis of data in Figure 3b reveals that while the ACI distribution can, in some cases, appear nearly symmetrical (|skewness| < 0.5), significant differences often persist between the ACI and the conventionally determined CI. For instance, in analyzing the ITC binding isotherm for the DA‐3BP aptamer–dopamine pair, the ACI distribution exhibited a relatively small positive skewness of 0.49, indicating near symmetry. The 95% confidence ACI for K _d spanned a range of 9.0–20 μM, with a width of 11 μM, 38% wider than the 95% CI determined by L‐S regression, which spanned 15–23 μM with a width of 8 μM. More importantly, the mean of the ACI (14 μM) was substantially lower than that of the conventional CI (19 μM), representing a 26% difference. These results underscore a considerable divergence between the two CIs, emphasizing that accounting for systematic errors in experimental variables during ITC data analysis can significantly alter the probability distribution of K _d.

Notably, while the mean values of ACI and L‐S determined CI for ΔH° and n were similar for both binding pairs, the ACI were often considerably wider. In this study, we place greater emphasis on K _d, as it is a critical parameter in early‐stage drug discovery, particularly for ranking drug candidates. The potential impact of inaccuracies in ΔH° and n is not extensively discussed here.

In Figure 3, it is observed that, for both examples, the uncertainties in concentrations and measured heats have a much greater impact on the determined mean K _d values than on the mean values of ΔH° and n. This phenomenon can be qualitatively explained by examining Equation (1) and (2). The parameter n acts as a correction factor associated with the ligand concentration (L ₀), while ΔH° is derived from ΔH _cum,max, which is constrained by the extrapolation of the binding curve. Therefore, whether using conventional L‐S regression or ACI‐ITC, the mean values of ΔH° and n (i.e., the values close to the ones with the highest probabilities) are primarily governed by the nominal concentrations and measured heats, which have limited freedom to vary. In contrast, K _d is a parameter that is not directly tied to the concentrations or measured heats and thus retains greater flexibility to adjust during fitting. As a result, the distribution of K _d is much more sensitive to uncertainties in these variables, and its mean value can shift significantly when uncertainties in concentrations and heats are taken into account.

By addressing both random errors (reflected in data point fluctuations) and systematic errors (such as inaccuracies in concentrations and heat measurements), ACI‐ITC provides a more reliable determination of ITC‐derived parameters. Figure 3 highlights the significant consequences of disregarding systematic errors in analyte concentrations and heat changes, which can lead to misjudged CIs for K _d. Such oversights may result in either overestimation or underestimation of K _d, or misinterpretation of its frequency (probability) distribution. In the context of drug‐lead selection, these inaccuracies can disrupt the affinity ranking of drug candidates, potentially leading to the preference of weaker binders while overlooking stronger binders—decisions that could result in costly mistakes.

To compare two widened K _d probability distributions determined by ACI‐ITC, it is useful to first transform the K _d values to a logarithmic scale. This transformation makes the distributions more symmetrical and facilitates direct comparison.^{[ 13 ]} After transformation, a probabilistic comparison can be performed by quantifying the likelihood that one binder exhibits a lower/higher K _d than the other, based on their respective distributions. This approach allows for a more meaningful assessment than relying solely on overlapping CIs, particularly when the distributions are broad and partially overlapping. A detailed development of statistical comparison tools for such non‐Gaussian distributions is beyond the scope of the current study and will be addressed in future work.

Overall, the findings in this work highlight the importance of assessing ACI for ITC‐derived parameters. Relying solely on conventional least‐squares nonlinear regression, which neglects systematic errors, may lead to misleading conclusions. To ensure accurate and reliable interpretations, ITC‐derived parameters should be evaluated within the framework of ACI.

3. Conclusions

To enhance the reliability of ITC‐derived parameters, we developed a Monte Carlo‐based program, ACI‐ITC, which calculates ACI for key parameters commonly determined by ITC: K _d, ΔH°, and n. To ensure accessibility and ease of use for all ITC practitioners, we integrated ACI‐ITC into the ACI‐Kd web app (http://aci.sci.yorku.ca), creating a browser‐based tool.

This study provides practical guidance on determining the critical inputs required for calculating the ACI of ITC‐derived parameters. These inputs include the CIs of systematic errors in analyte concentrations and measured heat changes. The CIs for analyte concentrations can be estimated by analyzing spectroscopic signals of the target and ligand across multiple independently prepared samples, following minimization of systematic errors, e.g., correcting concentrations based on measured purities. This approach is appropriate because ITC experiments typically employ a single concentrated target solution and a single ligand solution throughout the titration. In cases where independent sample preparation is not feasible, for example, when only a single stock solution or batch (e.g., a DNA aptamer batch from a manufacturer) is available, the CI for the reagent concentration can be estimated by propagating the known systematic uncertainties of the instruments used in sample preparation, combined with a reasonable assumption for batch‐to‐batch variability.

The CI for systematic error in measured heat changes is primarily influenced by the accuracy of baseline adjustment, as a single baseline adjustment is applied to the entire dataset in an ITC titration. Although a rigorously established method for assessing the CI of baseline offsets is currently lacking, reasonable estimates can be derived based on the maximum heat pulse observed during the titration experiment. These estimates serve as a practical approach to account for baseline‐related systematic errors in ACI calculations.

By evaluating the sources of systematic errors in analyte concentrations and detected heat changes, we estimated that the minimum SD input for the CIs of relative systematic errors in analyte concentrations should be set at 5%, while the minimum SD for the CIs of relative systematic errors in heat changes should be set at 2% of the maximum absolute value of the measured heat pulse. These values provide essential inputs for assessing narrower limits of the ACI of ITC‐derived parameters, particularly when analyzing retrospective data. They are set as default values in the ACI‐ITC web tool.

The ACI‐ITC web tool was rigorously validated using synthetic ITC data and subsequently applied to real experimental datasets to demonstrate its practical utility. The experimental demonstration also illustrates the procedures for minimizing systematic errors and for estimating the CIs of the remaining systematic errors in reagent concentrations. The results reveal that the conventional least‐squares fitting of ITC binding isotherms, which neglects systematic errors in variables, can underestimate the uncertainties of ITC‐derived parameters and misrepresent their probability distributions. While the current work suggests initial strategies for comparing the widened ACI, further studies are needed to develop comprehensive ranking methods.

Overall, the findings in this work highlight the crucial importance of assessing ACI to ensure accurate and reliable interpretation of ITC‐derived parameters. By enabling the incorporation of systematic error considerations, the browser‐based tool of ACI‐ITC substantially improves the reliability of ITC data analysis and strengthens the conclusions drawn from these analyses. Its application is expected to greatly enhance the accuracy and confidence in ITC‐derived parameter estimations across a wide range of experimental conditions.

4. Experimental Section

4.1.

4.1.1.

Materials

All DNA aptamer samples were obtained from Integrated DNA Technologies (IDT) (Coralville, IA, USA) as a lyophilized powder and used without further purification. Salts and buffer reagents were obtained from BioShop (Burlington, ON, Canada). All small‐molecule targets were obtained from Sigma Aldrich (Oakville, ON, Canada). DNA samples were dissolved in distilled deionized H₂O (ddH₂O), exchanged three times using a 3 kDa molecular weight cut‐off concentrator with 1.0 M NaCl, and washed three times with ddH₂O. The concentrations of the aptamers were determined by measuring the OD₂₆₀ in a UV–vis spectrometer using the extinction coefficients provided by IDT.

ITC Experimental Procedure

ITC was performed using a MicroCal‐ITC 200 (Malvern, Worcestershire, UK) instrument. Samples were degassed and centrifuged to remove any bubbles and aggregates before setting up the experiment. Titrations were performed with the aptamer samples (ligand) loaded in the cell, and the target as the titrant was in the syringe. Binding experiments were performed at 20 °C with both the DA‐3BP aptamer (5’‐ACG TCA GTT TGA AGG TTC GTT CGC AGG TGT GGA GTG ACG T‐3’),^{[ 41 ]} and the Theo2201 aptamer (5’‐GAC GAC GAT TGT GGT CTA TTC ATA GGC GTC CGC TGA GTC GTC‐3’).^{[ 42 ]} ITC was performed with the DA‐3BP concentration set at 450 μM and the dopamine (target) concentration at 7,020 μM (7.02 mM). The Theo2201 aptamer concentration was set at 60 μM and the theophylline (target) concentration at 936 μM.

A typical binding experiment consisted of 19 successive injections spaced every 180 s, apart where the injection volume was 2 μL with 10 μcal s⁻¹ reference power and a 750 rpm stir speed. The data was preliminarily analyzed using Microcal PEAQ software to produce the binding isotherm “ΔH versus Molar ratio”. The software automatically sets the baseline and integrates every peak from the baseline, but manual integration is often needed, as software integration is not always performing optimally.

Conflict of Interest

The authors declare no conflict of interest.

Supporting information

Supplementary Material

Data Availability Statement

The data that support the findings of this study are available in an on‐line depository (https://doi.org/10.6084/m9.figshare.28503245).