Differential Binding Models for Direct and Reverse ... - ACS Publications

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Differential Binding Models for Direct and Reverse Isothermal Titration Calorimetry Isaac Herrera and Mitchell A. Winnik* Chemistry Department, University of Toronto, 80 St. George Street, Toronto ON, Canada, M5S 3H6 S Supporting Information *

ABSTRACT: Isothermal titration calorimetry (ITC) is a technique to measure the stoichiometry and thermodynamics from binding experiments. Identifying an appropriate mathematical model to evaluate titration curves of receptors with multiple sites is challenging, particularly when the stoichiometry or binding mechanism is not available. In a recent theoretical study, we presented a differential binding model (DBM) to study calorimetry titrations independently of the interaction among the binding sites (Herrera, I.; Winnik, M. A. J. Phys. Chem. B 2013, 117, 8659−8672). Here, we build upon our DBM and show its practical application to evaluate calorimetry titrations of receptors with multiple sites independently of the titration direction. Specifically, we present a set of ordinary differential equations (ODEs) with the general form d[S]/dV that can be integrated numerically to calculate the equilibrium concentrations of free and bound species S at every injection step and, subsequently, to evaluate the volume-normalized heat signal (δQV = δq/dV) of direct and reverse calorimetry titrations. Additionally, we identify factors that influence the shape of the titration curve and can be used to optimize the initial concentrations of titrant and analyte. We demonstrate the flexibility of our updated DBM by applying these differentials and a global regression analysis to direct and reverse calorimetric titrations of gadolinium ions with multidentate ligands of increasing denticity, namely, diglycolic acid (DGA), citric acid (CIT), and nitrilotriacetic acid (NTA), and use statistical tests to validate the stoichiometries for the metal−ligand pairs studied.

1. INTRODUCTION Titrations or binding assays are widely used techniques to quantitatively characterize the stoichiometry and the affinity between receptor and ligand binding species. An additional step when studying a receptor−ligand binding system is to obtain the underlying thermodynamics. From the many analytical methods available to perform binding assays, isothermal titration calorimetry (ITC) can be used to simultaneously characterize the stoichiometry and the thermodynamics of binding systems. Modern calorimeters used in ITC offer a relatively high sensitivity to determine apparent association constants in the range from 103 M−1 to 109 M−1 for titrations that require less than 300 μL of sample at a concentration as low as 10−5 M.1 In addition, ITC allows a label-free and homogeneous approach to characterize receptor−ligand interactions, since the heat released or absorbed during the titration is the signal measured and the heat exchanged accompanies all physicochemical binding processes. As such, ITC has been extensively employed to obtain a complete stoichiometric and thermodynamic characterization of many physical, chemical, and biological binding systems.2 A significant challenge in ITC is to select and validate the models applied to the calorimetric data, particularly for titrations of receptors with multiple sites where the stoichiometry or binding mechanism is not available. Titrations © XXXX American Chemical Society

of simple binding systems can be analyzed using titration calorimetry software (e.g., Origin, NanoAnalyze, HypΔH,3 or AFFINImeter4) and free software such as SEDPHAT,5 IC ITC,6 or CHASM.7 These software packages include models constrained to represent receptors with a set number of binding sites and specific binding interactions. Additionally, a combination of direct and reverse titrations have been used to obtain more accurate information about the stoichiometry of a binding system.8,9 Global regression routines have also been used to help select and validate a binding model, particularly, when more than one of the available models can be applied to the same experimental data.10−13 For instance, SEDPHAT offers a unique platform to analyze the combined experimental data from calorimetric, spectrophotometric, sedimentation, and surface binding assays.14 While the models in the previous software packages cover many binding systems and global analysis increases one’s confidence in the binding parameters obtained, using a model constrained to a particular stoichiometry or binding interaction could potentially limit the evaluation and interpretation of the results. Received: September 21, 2015 Revised: January 5, 2016

A

DOI: 10.1021/acs.jpcb.5b09202 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B An alternative approach, sometimes referred to as “modelfree”, makes use of the binding polynomial formalism to analyze calorimetry titrations independently of the interaction between the binding sites in the receptor.15,16 In studies by our group and others, differential equations based on binding polynomials have been used to simulate and analyze calorimetric titrations of receptors with multiple binding sites.17−19 We refer to this class of differential equations to analyze titration experiments as differential binding models (DBMs). In a recent study, we derived a theoretical model for a direct titration where the receptor concentration remains constant and the ligand concentration increases with each injection of titrant. In practice, however, both ligand and receptor concentrations change due to the injection, dilution, and overflow of the binding species during the titration. As shown by Keeler et al., a coefficient can be included in the differential expression to adjust the rate of receptor−ligand binding for dilution effects.19 We apply an alternate approach to account for dilution effects by deriving and evaluating differential equations with respect the volume of titrant injected,20 which allows us to analyze both direct and reverse titrations with the same set of equations. Our primary goal in this work is to present a more robust framework to develop differential binding models based on the binding polynomial formalism. The updated framework provides general expressions that can be applied to titrations of receptors with multiple binding sites in any type of titration cell or titration direction. To validate our model, we examine the direct and reverse calorimetry titrations between gadolinium ions and complexing agents of increasing denticity such as diglycolic acid (DGA), citric acid (CIT), and nitrilotriacetic acid (NTA) in buffered solutions. In addition, we compare the results obtained by evaluating the same data with either differential or finite difference models to show that DBMs are a valid alternative to interpret ITC results. There are several advantages of using metal−ligand complexes to validate models for calorimetry titrations of receptors with multiple binding sites. First, the stoichiometry and affinity constant(s) depend on the coordination number of the metal ion and the denticity of the ligand. Lanthanide ions, in particular, have coordination numbers between 8 and 10,21 and the ligands used in this study have between 2 and 4 chelating groups per ligand molecule. Second, metal−ligand complexes have been characterized using similar titration techniques, which allows us to compare the enthalpy and affinity constants available to those obtained with the DBMs. Finally, the reagents used exhibit good solubility at neutral and acidic pH values, which allows us to easily reverse the direction of the titration. In comparison, proteins or nucleic acids form dimers at the concentrations required to perform reverse titrations,5 and the analysis of such titrations requires models based on multimeric binding polynomials.22 We use statistical tests, a global regression routine, and a homotropic binding polynomial to determine the stoichiometries of the metal−ligand complexes. With this approach, we probe the binding system with questions that include the following: How many binding sites are required to evaluate the metal−ligand titrations? What are the appropriate concentrations for the titrant and analyte species for the direct and reverse titrations?

2. THEORY Titrations are analyzed using mathematical expressions, or binding models, which relate the signal measured to the amount of titrant added and are based on a hypothetical mechanism for the interaction between receptor and ligand species. For instance, the calorimetric titration of a receptor (R) with Ni binding sites for a ligand (L) can be interpreted with a homotropic binding mechanism that includes the equilibrium reaction R + i L ⇄ RLi + heat, βi =

[RLi] [R][L]i

(1)

where the cumulative binding constants (βi) can be used in a partition function, or binding polynomial,23 to describe the distribution between free and bound species Ni

PR =

∑ βi [L]i

(2)

i=0

The binding polynomial PR is the basis for deriving and representing the binding and statistical properties of the system such as the average number of ligand species bound (N̅ RL ) and the binding capacity (BRL ), as well as the macroscopic properties of the titration such as the total concentrations of receptor (CR) and ligand (CL) species (see Supporting Information, eqs S2− S5). For example, the first and second derivatives of the binding potential, or RT ln PR, with respect to the chemical potential of free ligand correspond to N̅ RL and BRL, respectively, and are given by N

N̅LR =

∑i =i 0 iβi [L]i PR N

BLR

=

∑i =i 0 i 2βi [L]i PR

(3a)

⎛ ∑Ni iβ [L]i ⎞2 ⎟ − ⎜⎜ i = 0 i ⎟ PR ⎝ ⎠

(3b)

Similarly, the mole fraction αi can be expressed as the ratio of the concentration of a receptor bound to i ligand species ([RLi]) and the total receptor concentration CR, or in terms of PR as given by αi =

β [L]i [RLi] = i CR PR

(4)

In calorimetry titrations (eq 1), the total heat released or absorbed (q) is proportional to the cumulative enthalpy of binding (ΔHi), the molar concentration of bound species RLi, and the cell volume (V0), as given by Ni

q = V0 ∑ ΔHi[RLi] i=1

(5)

For ITC, in particular, the signal represents the change in the heat content of the titration cell (Δq) and the titration proceeds by injecting small aliquots of titrant (ΔV), where both terms can be expressed mathematically as differentials in the limit ΔV → 0. Hence, the binding models applied to ITC experiments can be expressed as differential equations and evaluated using either numerical or explicit approaches.17−19,24−28 Two numerical methods used to evaluate the differential heat are based on finite difference and differential expressions. With B


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The Journal of Physical Chemistry B finite difference expressions, the signal is evaluated as the change in total heat exchanged at injection point p (Δqp), which corresponds to

dC B C Syr − C B = B dV V0

Second, the equilibrium concentrations for the binding species and the dilution-adjusted concentrations for CR and CL are evaluated numerically by integrating the system of ordinary differential equations (ODEs) defined by eqs 9−11. This numerical evaluation can be applied to direct and reverse titrations after selecting the appropriate expressions for dCR/dV and dCL/dV and setting initial values for the ODEs (see Supporting Information, eqs S40−S43). In comparison, our theoretical model could only be integrated to represent direct titrations where CR remains constant. The heat signal in a calorimetry titration can be evaluated with the differential d[S]/dV after considering the mass and heat balances for the species in the calorimeter. In an overfill calorimeter, for instance, the material flowing out of the titration cell and represented by the terms −CA/V0 and −CB/V0 (eq 11) does not contribute to the heat measured.30,31 After combining eqs 8−11 and canceling out the heat contribution of the overflown titration mixture, we obtain a general expression to evaluate both direct and reverse calorimetry titrations in overfill titration cells

Ni

Δqp = V0 ∑ ΔHi·Δ[RLi]p

(6)

i=1

where the term Δ[RLi]p is calculated using the volume injected (ΔVp) and the difference in concentration between two consecutive injection points, as given by Δ[RLi]p = [RLi]p − (1 − ΔVp/V0)[RLi]p − 1

(7)

With differential expressions, the signal is evaluated as the ratio between the differential heat and a variable representing the progress of the titration. The resulting differential equation represents the change in the heat content of the titration cell with the addition of titrant. For example, the differential heat per volume of titrant injected (dQV) can be evaluated with the general expression N

dQ V =

i dq d[RLi] = V0 ∑ ΔHi dV dV i=1

(8)

Ni ⎛ ∂[RLi] ⎞ δQ VB → A = C BSyr ∑ ΔHi⎜ ⎟ ⎝ ∂C B ⎠C i=1

In this article, we derive differentials with the general form d[S]/dV, where [S] represents the concentration of the free or bound species in the titration cell, by implicitly differentiating mass-balance expressions as a function of the dilution of both receptor (dCR/dV) and ligand (dCL/dV) species. We express the differentials using terms derived from the binding polynomial (eqs 2−4). This representation allows us to evaluate titrations of receptors with any number of binding sites Ni or binding interaction.18 For instance, the change in the concentration of bound species RLi with the addition of titrant is represented as ⎛ ∂[RLi] ⎞ dC R d[RLi] ⎛ ∂[RLi] ⎞ dC L =⎜ +⎜ ⎟ · ⎟ · dV ⎝ ∂C L ⎠C dV ⎝ ∂C R ⎠C dV R

L

where the partial differential (∂[RLi]/∂CB)CA is given by eq 10a for a direct or ligand-to-receptor titration (LTR) and by eq 10b for a reverse or receptor-to-ligand titration (RTL). The differential heat per mole of titrant for an overfill titration cell δH, which is commonly used as the ordinate to plot most ITC experiments, is related to δQV as given by

(9)

(10a)

⎛ ⎛ ∂[RLi] ⎞ C N̅ R (i − N̅LR ) ⎞ ⎟ ⎜ ⎟ = αi⎜1 − R L [L] + C R ·BLR ⎠ ⎝ ∂C R ⎠C ⎝

(10b)

L

Incidentally, eq 10a is equivalent to our theoretical model18 since both terms were derived using a constant receptor concentration. The differential in eq 9 provides two main benefits over our theoretical model.18 First, the dilution expressions dCR/dV and dCL/dV can be independently adjusted to represent the titration setup used for the experiment. For example, the dilution differentials for titrations in overfill or perfusion cells (Figure S1a, Supporting Information),29 where the titration cell contains analyte A at an initial concentration C0A and the syringe contains titrant B at concentration CSyr B , are given by

dCA C =− A dV V0

δQ VB → A C BSyr

Ni

=

⎛ ∂[RLi] ⎞ ⎟ ⎝ ∂C B ⎠C

∑ ΔHi⎜ i=1

A

(13)

Equations 12 and 13 show that models derived at constant analyte concentration can be applied to calorimetry titrations in overfill cells when the partial differential term (∂[RLi]/∂CB)CA is evaluated using dilution-adjusted concentrations. We include the derivation steps for eqs 9−13 in the Supporting Information (eqs S1−S36). In a partial-fill titration cell (Figure S1b, Supporting Information), the moles of analyte in the titration cell remain constant and any change in concentration results from the volume of titrant injected and the dilution of analyte. Therefore, the total differential dQV in eq 8 can be applied to calorimetry titrations in partial-fill titrations cells. In this titration setup, both dCL/dV and dCR/dV dilution terms contribute to the heat signal. We include the complementary dilution expression for a partial-fill titration cell in the Supporting Information (eqs S24−S26).

where the partial differential terms in eq 9 are given, respectively, by

R

(12)

A

δHB → A =

⎛ ∂[RLi] ⎞ C α (i − N̅LR ) ⎜ ⎟ = R i [L] + C R ·BLR ⎝ ∂C L ⎠C

(11b)

3. EXPERIMENTAL SECTION 3.1. Materials. Chemicals of the highest purity available were used for the calorimetric titration. Nitrilotriacetic acid (NTA) (99%, Fulka), disodium hydrogen citrate sesquihydrate (CIT) (>99.9%), diglycolic acid (DGA) (98%, Aldrich), 4morpholineethanesulfonic acid monohydrate (MES) (>99.9%, Aldrich), sodium hydroxide (Sigma-Aldrich, TraceSELECT >99.99% trace metal basis), hydrochloric acid (TraceSELECT,

(11a) C


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where χ2 is the weighted sum of square residuals or chi-square value, ν represents the degrees of freedom, and f(Vp) represents the fitting curve for the differential heat per volume (eq 12) evaluated at injection point Vp and constrained for the appropriate titration direction, binding interaction, and number of binding sites. We followed three steps to evaluate f(Vp) with a numerical integration approach. First, we defined Ni pairs of initial guess values for the fitting parameters, which correspond to enthalpies of binding ΔHi and the cumulative binding constants βi for the binding sites. Second, we calculated the free ligand concentration [L] and the dilution adjusted concentrations for CR and CL by integrating a system of ordinary differential equations (ODEs) defined by the terms d[L]/dV, dCR/dV, and dCL/dV (eq S40a, Supporting Information) using the initial values for each term. This system of ODEs is used for direct and reverse titrations. In direct titrations, the initial concentrations correspond to

Aldrich), and gadolinium(III) chloride hexahydrate (99.99%, Aldrich) were purchased from Sigma-Aldrich. Doubly distilled water was purified through a Milli-Q water purification system (18 MΩ·cm). MES buffer (250 mM) was adjusted to pH 5.50 using dilute solutions of NaOH (1.0 M) and HCl (1.0 M). Stock solutions of Gd (10.0 mM), NTA (10.0 mM), CIT (10.0 mM), and DGA (10.0 mM) in MES buffer (100 mM) were prepared from the pure salts, readjusted to pH 5.50, and used as titrant solutions (i.e., placed in the syringe). The previous stock solutions were diluted in the range from 1.0 to 0.3 mM in MES buffer (100 mM, pH 5.50) and used as analyte solutions (i.e., placed in titration cell). Both titrant and analyte solutions were homogenized in a sonication bath (15 min) and degassed (15 min) prior to a titration experiment. 3.2. Isothermal Titration Calorimetry. Metal−ligand complexometric titrations were carried out in a MicroCal VPITC titration calorimeter (Piscataway, NJ, USA) as previously described.18 In short, the titration cell (V0 = 1.4037 mL) was rinsed with a 5% CONTRAD solution (2.0 mL), followed by D.I. water (20.0 mL), and finally a small volume of analyte (ca. 500 μL) to precondition the cell. Then, the sample cell was filled with a solution of analyte and the syringe (300 μL) was filled with a solution of titrant. In a direct titration experiment, the buffered Gd(III) solution (0.5 mM) was placed in the titration cell and NTA, CIT, or DGA (10.0 mM each) solutions were placed in the syringe. In an reverse titration experiment, a buffered solution of either NTA (0.5 mM), CIT (0.5 mM), or DGA (1.0 mM) was placed in the titration cell while the syringe was filled with Gd(III) (5.0 or 10.0 mM). The titration experiments were conducted at 25 ± 0.1 °C with an initial reference power of 41.84 μW (ca. 10 μcal·s−1) and a stirring rate of 400 rpm. Due to the passive diffusion of the titrant solution into the titration cell during equilibration time, the first injection point was 1−2 μL and this injection point was ignored during data analysis. The following injections points were 5 μL with a 360 s delay between injections and a 2 s data collection rate. A minimum of six titrations were performed for each metal−ligand pair, which included three direct titrations and three reverse titrations. 3.3. Data Analysis. The raw data thermograms, given by the differential heating power (dP) vs time (t), were processed using the software provided with the calorimeter (Origin 7.0). First, the automatic baseline generated in Origin 7.0 was smoothed using 2−4 cycles of a linear Savitzky-Golay noise filter. Then, the differential heat (δq) was calculated by integrating the injection peaks in the thermogram with a smoothed baseline. The differential heat and differential volume columns, labeled respectively as “DH” and “INVJ” in Origin 7.0, were transferred to IGOR Pro (WaveMetrics Inc., Lake Oswego, OR, USA) and used to calculate the differential heat per volume of titrant (δQV = δq/dV) and the total volume of titrant injected up to point p (Vp = ΣΔVp). From the titrations available for each metal−ligand pair in a titration direction, we calculated the averaged differential heat per volume (δQ̅ V,p) and the standard deviation (σp) of each injection point p. The experimental values for δQ̅ V,p were analyzed individually and globally. For the individual analysis, we used a weighted nonlinear regression algorithm to minimize the reduced chisquare value (χ2v ) as given by χν2

χ2 1 = = ν ν

⎛ δQ̅ − f (Vp) ⎞2 V ,p ⎟ ∑ ⎜⎜ ⎟ σp ⎠ p ⎝

C L = [L] = 0, and C R = C R0

(15)

whereas the initial concentrations for reverse titrations correspond to C L = [L] = C L0 , and C R = 0

(16)

Subsequently, we evaluated f(Vp) in eq 14 using the calculated concentrations for [L], CR, and CL and the fitting parameters. In Figure S2 of the Supporting Information, we show an example of the numerical integration approach for the direct titration of a receptor with one binding site. The numerical integration approach replaces the steps shown in our previous manuscript to calculate [L] using a root-finding algorithm and the dilution adjusted concentrations for CR and CL.18 We included two additional fitting parameters in the definition of f(Vp). First, the coefficient f R was used to adjust for small errors in the total receptor concentration or the active fraction of receptor,5 as given by

C R* = fR C R

(17)

Second, the enthalpy of dilution Δhd was used to compensate for the heat exchanged during the dilution of titrant species. The homotropic DBM for direct and reverse titrations was adapted for EXCEL using an open-source library for numerical integration of ODEs (ALGLIB)32 and is available upon request. For the global analysis, we utilized the MOTOFIT package33 for IGOR Pro to evaluate the direct and reverse titrations of each metal−ligand pair. In this case, the enthalpies of binding ΔHi and the cumulative binding constants βi were shared fitting parameters for both titration directions, while the receptor coefficients f R and the enthalpies of dilution Δhd were independently assigned to each titration direction. In MOTOFIT, we used two nonlinear regression methods sequentially to optimize the fitting parameters and to obtain a global minimum for χ2v . First, we obtained approximate values for the optimized binding parameters with a Genetic regression algorithm,34 which evaluates each fitting parameter within a range of testing values. The testing ranges were defined as 0.8− 1.2 for the receptor coefficients f R; − 10i − +10i kJ·mol−1 for ΔHi; 0 − 10i for log βi; and −1.0 − +1.0 kJ·mol−1 for the enthalpies of dilution ΔhdLTR and ΔhdRTL. After 100 iterations with the Genetic method, we continued the optimization with a Levenberg−Marquardt algorithm. The sequential use of these

(14) D


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The Journal of Physical Chemistry B two regression methods allowed us to reach an appropriate minimum in the global analysis that is independent of the initial value selected for the fitting parameters.33 For comparison, we analyzed the calorimetric titration curves with a finite difference model (eqs 6 and 7) constrained with the same number of binding sites, binding polynomial, and initial values for the fitting parameters as the differential model. Our intent here is to show that both models can provide similar fitted values when evaluating the same ITC data. In contrast with most studies by ITC, we evaluated δQV (eq 12) rather than δH (eq 13) to eliminate the inherent bias toward titrations with a higher concentration of titrant. This approach is important when using a weighted regression algorithm to analyze ITC experiments, where the statistical weight applied to each injection point (wp) is inversely proportional to the variance in the signal (i.e., wp = σp−2). The uncertainty in δQV, represented by the term σp(δQV), is a result of instrumental errors in the volume of titrant injected and the heat measured; hence, the statistical weights are given by wp(δQV) = σp(δQV)−2. In contrast, the statistical weights for δH include the error associated with the concentration of titrant in the syringe (cf. eq 13) and are given by wp(δH) = 2 (CSyr B /σp(δQV)) . Hence, titration setups with higher concentrations of titrant receive higher statistical weights during global analysis of δH. To prevent this bias when analyzing δH with a global regression algorithm, one could introduce a global −2 weight proportional to (CSyr for each titration setup.35 B ) 3.4. Selection and Validation of DBMs. As part of our analysis of ITC experiments with DBMs, we utilized homotropic DBMs constrained for Ni and Ni + 1 bindings sites, and F-tests to determine the maximum stoichiometry between receptor and ligand species. For the F-test,36 we calculated the chi-square ratio (Fχ) using the minimized chisquare values (χ2Ni and χ2Ni+1) and degrees of freedom (vNi and vNi+1) of the respective models with Ni and Ni + 1 stoichiometries, as given by Fχ =

χN2 − χN2 + 1 i

i

χN2 + 1 i

·

Figure 1. Individual and global analyses of direct and reverse calorimetric titrations between DGA and Gd with models constrained for two and three binding sites. (A) In the direct titrations, DGA (10 mM) was placed in the syringe and GdCl3 (0.5 mM) was in the cell. (B) In the reverse titrations, GdCl3 (10 mM) was placed in the syringe and DGA (1.0 mM) was in the cell. The top panels in each plot show the averaged δQV titration points from three titrations and the fitting curves indicate the individual analyses with models constrained for two (dotted line, blue) and three (solid line, red) binding sites. The residuals obtained with two-site and three-site models are shown in the middle and bottom panels, respectively; where the filled circles represent residuals from individual fits, and the open circles represent residuals from global fits. The baseline-corrected thermograms for one of the replicates in each titration direction are shown in Figure S3 of the Supporting Information.

νNi + 1 νNi − νNi + 1

(18)

Initially, the term Ni was set to the number of inflection points on the titration curve. However, the model with Ni + 1 binding sites was adopted in its place when Fχ > FCrit,0.05 for both titration directions. We repeated the F-tests with the updated stoichiometry values until Fχ < FCrit,0.05 for one of the titration directions, which signaled that the Ni + 1 model was overfitting the titration curve while the Ni model was statistically valid to a 95% confidence level. For some F-tests, it was necessary to fix f R to 1.0 during the analyses with the Ni and Ni + 1 models.

represent the equivalence points for the binary (DGA−Gd) and ternary (DGA2−Gd) complexes, respectively. Hence, the DBM applied to the Gd−DGA titrations was constrained initially to two sites (Ni = 2) and later extended to include three and four sites (Ni = 3, 4). In Figure 1, the fitting curves obtained with a two-site model (dotted lines) show good agreement to the experimental points and match closely the fitting curves obtained with a three-site model (solid lines). Upon closer examination of the residuals, we observe strong oscillating patterns for the residuals of a twosite model (central panels, filled circles). Comparatively, a three-site model (bottom panels, filled circles) gives smaller residuals and weaker oscillating patterns, which suggests that Gd ions also form DGA3−Gd complexes. The global-regression residuals (open circles) show good agreement with the

4. RESULTS AND DISCUSSION 4.1. Evaluating and Validating the Number of Binding Sites. To evaluate calorimetric titrations of receptors with unknown stoichiometry, we first define a hypothetical number of binding sites Ni from the number of inflection points on the titration curves. In the direct and reverse titrations between diglycolic acid (DGA) and Gd (Figure 1), one can see two major inflection points, corresponding to a ligand-to-receptor mole ratio (ΦLTR) of 1.0 and 2.0 for the direct titration (Figure 1A), and a receptor-to-ligand mole ratio (ΦRTL) near 0.5 and 1.0 for the reverse titration (Figure 1B). These mole ratios E


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Table 1. Binding and Statistical Parameters of the Individual and Global Fits for Direct and Reverse Titrations between DGA and Gd two sitesa,b binding parameters (units) fR log β1 (−log M) log β2 (−2·log M) log β3 (−3·log M) ΔH1 (kJ·mol−1) ΔH2 (kJ·mol−1) ΔH3 (kJ·mol−1) χν 2

direct

three sitesa,c

reverse

1.058 ± 0.001

1.084 ± 0.001

5.76 ± 0.01 9.94 ± 0.01

5.44 ± 0.004 9.31 ± 0.01

2.63 ± 0.02 −8.53 ± 0.04

2.80 ± 0.01 −10.49 ± 0.04

20.9

33.1

global 1.064 1.073 5.56 9.59

± ± ± ±

0.001d 0.001e 0.004 0.01

2.87 ± 0.01 −9.25 ± 0.03 50.1

direct 1.023 ± 0.002 6.48 ± 0.03 11.39 ± 0.05 14.45 ± 0.07 2.55 ± 0.02 −6.17 ± 0.05 −14.41 ± 0.10 0.23

reverse 0.989 ± 0.002 6.51 ± 0.04 11.40 ± 0.07 14.39 ± 0.09 2.81 ± 0.05 −5.96 ± 0.05 −15.04 ± 0.39 0.78

global 1.022 ± 0.001d 0.992 ± 0.001e 6.46 ± 0.02 11.30 ± 0.03 14.33 ± 0.04 2.54 ± 0.01 −6.30 ± 0.03 −14.68 ± 0.09 0.91

a

Confidence intervals correspond to one standard deviation. bCorrelation matrices for DBMs with Ni = 2 are shown in Table S1 of the Supporting Information. cCorrelation matrices for DBMs with Ni = 3 are shown in Table S2 of the Supporting Information. df RLTR from direct or ligand-toreceptor (LTR) titration. ef RRTL from reverse or receptor-to-ligand (RTL) titration.

titrations and the corresponding correlation matrices for the models where f R = 1.0 and Ni = 2−4 are shown in the Supporting Information (Tables S6−S8). First, the Fχ ratios obtained when comparing the individual analyses with two-site and three-site models were 1.05 × 103 and 5.11 × 103 for the direct and reverse titrations, respectively (Table S9). Then, the Fχ ratios obtained when comparing three-site and four-site models were 19.9 and 2.23 for the direct and reverse titrations, respectively. The latter value was lower than a critical value of 3.19; thus, the DBM with Ni = 3 and f R = 1.0 was statistically valid to a 95% confidence level for a DGA−Gd binding system in comparison to models with Ni = 2 and Ni = 4. We also evaluated the stoichiometries of the NTA−Gd and CIT−Gd binding systems by applying DBMs with two and three binding sites to both titration directions and subsequently performing F-tests. The titration curves for the NTA−Gd and CIT−Gd binding systems have two inflection points near ΦLTR of 1.0 and 2.0 for the direct titrations and ΦRTL of 0.5 and 1.0 for the reverse titrations (Supporting Information, Figures S4 and S6 respectively). These inflection points suggest that the stoichiometries in the DMB should be initially set to Ni = 2. The fitting results and correlation matrices for the NTA−Gd titrations using DBMs with f R = 1.0 and Ni = 2 or 3 are shown in Tables S10 and S11, respectively, whereas those for the CIT−Gd titrations using similar constraints are shown in Tables S13 and S14 of the Supporting Information. From the F-tests, we determined that models with Ni = 2 were statistically valid to a 95% confidence level in relation to models with Ni = 3, when f R was fixed to 1.0 (see Supporting Information, Tables S12 and S15). The literature data available indicate that Gd forms ternary complexes with both ligands (i.e., Gd−L2).38 From these results, we established that the adequate protocol to validate the stoichiometry of receptors was to evaluate DBMs where the receptor coefficient f R was held constant. Consequently, we require accurate concentrations for the analyte and titrant species when validating receptor−ligand stoichiometries with F-tests. After validation of the stoichiometries, we refined the binding parameters by allowing the coefficients f R to vary during the global analysis of direct and reverse titrations (Tables S16 and S17). 4.2. Optimizing the Analyte and Titrant Concentrations. In ITC, the shape of the titration curve and the accuracy of the fitted binding parameters are influenced by the initial concentration of titrant and analyte species. For example, direct titrations of receptors with one binding site have a

residuals for individual analyses (filled circles) on models that have the same stoichiometry. The binding parameters obtained through global analyses have lower or equal confidence intervals than those obtained from individual analyses (Table 1). Consequently, we cannot validate the maximum stoichiometry between Gd and DGA based on the agreement between individual and global residuals or based on the increased confidence of the global fitting parameters. To validate the stoichiometry between Gd and DGA, we compared the optimized binding parameters of models with 2− 4 binding sites and performed F-tests using the chi-squared values of each model. For instance, the relative errors for the fitted parameters f R were 5.8−8.4% with a two-site model and 0.8−2.3% with a three-site model when compared to an ideal value of f R = 1.0 (Table 1 and Tables S1−S2 in the Supporting Information). Additionally, the χ2v values, which are a measure of the error to the fit, are considerably lower with a three-site model compared to a two-site model. Although χ2v values lower than 1.0 could represent an overparameterized model,36 the low χ2v values in the Ni = 3 model suggest that the standard deviation of the replicate measurements was overestimated.37 In combination, the values for f R and χ2v demonstrate that a three-site model is a better representation of the binding system, and this result agrees with literature data available for the DGA−Gd complexes.38 We include the correlation matrices for the individual and global analyses for the titrations between Gd and DGA in the Supporting Tables S1−S4. For the F-tests, we calculated the Fχ ratios (eq 18) using chivalues from the individual fits with the two-site (ν = 50) and three-site (ν = 48) models. We obtained Fχ ratios of 4.54 × 103 and 1.03 × 103 for the direct and reverse titrations, respectively, which were considerably higher than a critical value FCrit,0.05 = 3.20. Next, the F-tests from the three-site (ν = 48) and four-site (ν = 46) models gave Fχ ratios of 15.4 and 6.97 for the direct and reverse titrations, respectively (Table S5). Although these Fχ ratios were also higher than a critical value of 3.20, the binding enthalpy for the hypothetical fourth binding site had nearly the same magnitude as its standard deviation (i.e., ΔH4 = −30.0 ± 22.8 kJ·mol−1 in Table S4) and the optimized values changed depending on the iterations used to minimize χ2v . Hence, the DBM with Ni = 4 was likely overfitting the titration curves between Gd and DGA. We performed additional F-tests on the DGA−Gd titrations using the fitting results for DBMs where the receptor coefficient f R was constrained to 1.0. The fitting results for the DGA−Gd F


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Figure 2. Differential binding curves and mole fraction distributions for the binding species in the direct and reverse titrations between DGA (L) and Gd (R). The dash-n-dotted lines represent the receptor species bound to n ligands and the dotted lines represent free ligand species. (A,B) The fitting curves in Figure 1 for a model with Ni = 3 are a weighted sum of the differential binding curves (∂[RLi]/ ∂CB)CA and the cumulative enthalpies of binding ΔHi in Table 1 for the direct and reverse titrations, respectively. (C) In a direct titration, the receptor saturation increases with the addition of ligand where free receptor (α0) is transformed sequentially into binary (α1), ternary (α2), and quaternary (α3) complexes, which also suggests that a sequential-site model can be applied to the DGA−Gd binding system (Table S3). (D) In a reverse titration, the degree of receptor saturation decreases with the addition of receptor and the major species at the beginning are ternary and quaternary complexes. This effect of increasing saturation for direct titrations and decreasing saturation for reverse titrations has been reported for spectroscopic titrations of fluorescent binding species.40

and is similar to expressions for the limiting value δH on receptors with multiple sites.39 The product β1C0R in eq 19 is equivalent to c from the Wiseman isotherm. Consequently, the differential (∂[RL]/∂CR)CL has an initial value higher than 0.9 when β1C0R ≥ 10, as shown for the direct titration of Gd with DGA ligand, where β1C0Gd = 1.5 × 103 (Figure 2A). Typically, a calorimetry titration curve for a receptor with one binding site and a c value higher than 1 × 103 is expected to have a sharp inflection point at the equilibrium point.1 However, this was not observed for the direct titration between Gd and DGA (Figure 1A). For receptors with multiple binding sites, we noted that the sigmoidicity of the i-th inflection point in the titration curve with the exception of the final inflection point (i.e., i < Ni) can be approximated with the ratio

recommended range of c values between 10 and 1000, where c is equivalent to the product of the binding constant and the initial receptor concentration (i.e., c = KC0R).1 In this range of c values, the receptor requires approximately 2 mol equiv of ligand to reach saturation, and analysis of the sigmoidal titration curve gives accurate values for the stoichiometry, enthalpy, and binding constant. In contrast, titrations curves with c much higher than 1000 have a rectangular shape and provide inaccurate values for the binding constant, whereas titrations curves with c much lower than 10 exhibit asymptotic behavior and provide inaccurate values for the stoichiometry. An important feature of the DBM approach is that the partial differentials (∂[RLi]/∂CL)CR and (∂[RLi]/∂CR)CL (eqs 10a,10b) can be used to find optimal analyte and titrant concentrations. As shown with eqs 12 and 13, the normalized heat signals are proportional to the weighted sum of the partial differentials (∂[RLi]/∂CB)CA with respect to the total concentration of titrant. Therefore, we can estimate the initial heat signal in a calorimetry titration by evaluating the limiting value of the partial differentials in eq 10. In a direct titration, the initial heat is proportional to the limiting value of the partial differential for the binary complex (∂[RL]/∂CL)CR, which combined with eq 12 gives lim

V →0

δQ VLTR

=

C LSyr ΔH1

ci =

1+

βi − 1·βi + 1

=

Ki Ki−1

(20)

where Ki is the stepwise binding constant for a receptor with i ligands. For example, the calculated ci values for the first and second inflection points in the direct titration between Gd and DGA are c1 = 37.2 and c2 = 70.8, respectively. The sigmoidal shape of the curve at the final inflection point for a receptor with Ni binding sites is given by

β1C R0 β1C R0

(βi )2

c Ni =

(19)

βN C R0 i

βN − 1 i

G

= K NiC R0

(21) DOI: 10.1021/acs.jpcb.5b09202 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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Table 2. Binding and Statistical Parameters from the Analysis of Gd−NTA and Gd−CIT Binding Systems with Differential (DBM) and Finite-Difference Models (FDM) NTA−Gda DBM binding parameters

direct

log β1 (−log M) log β2 (−2·log M) ΔH1 (kJ·mol−1) ΔH2 (kJ·mol−1) χν2

7.43 ± 12.14 ± 6.52 ± −16.54 ± 2.31

FDM

reverse 0.02 0.02 0.02 0.08

7.52 ± 12.19 ± 6.73 ± −16.94 ± 3.31

0.02 0.03 0.05 0.09

global 7.54 ± 12.25 ± 6.48 ± −16.70 ± 3.72

direct 0.01 0.02 0.02 0.06

7.30 ± 11.98 ± 6.56 ± −17.45 ± 7.44

reverse 0.02 0.02 0.02 0.08

7.45 ± 12.12 ± 7.16 ± −16.10 ± 9.10

0.03 0.04 0.06 0.11

global 7.41 ± 0.01 12.10 ± 0.02 6.59 ± 0.02 −16.79 ± 0.06 11.41

CIT−Gda DBM directb

binding parameters

7.07 ± 11.21 ± −1.85 ± −18.44 ± 1.92

log β1 (−log M) log β2 (−2·log M) ΔH1 (kJ·mol−1) ΔH2 (kJ·mol−1) χν2 a

0.03 0.03 0.01 0.06

FDM

reverse 6.88 ± 10.93 ± −1.59 ± −19.62 ± 2.26

0.02 0.03 0.03 0.13

directc

global 6.93 ± 11.04 ± −1.79 ± −19.05 ± 4.31

0.01 0.01 0.01 0.04

6.98 ± 11.09 ± −2.02 ± −19.27 ± 3.52

reverse 0.03 0.03 0.01 0.08

6.83 ± 10.86 ± −1.32 ± −19.16 ± 2.57

0.02 0.03 0.03 0.13

global 6.97 ± 11.10 ± −1.88 ± −18.79 ± 7.73

0.01 0.01 0.01 0.05

Confidence intervals correspond to one standard deviation. bData reprocessed from ref 18 using eq 12. cData reprocessed from ref 18 using eq 6.

initial contribution calculated for the Gd−DGA3 complex was lower than 20% (data not shown). Using the binding constants obtained in the direct titration (see Table 1) where K3 = β3/β2 ≈ 1000 M−1, we calculated an improved initial concentration of 1.0 mM for DGA. As shown in Figure 2D, the ternary and quaternary mole fractions are nearly equivalent at the beginning of the reverse titration, and the initial values of the partial differentials (∂[RLi]/∂CR)CL in Figure 2B are proportional to the respective mole fractions. Using a similar strategy, concentrations C0L ≥ 10/KNi guarantee that at least 90% of the initial heat signal corresponds to the saturated complex as shown for the reverse titrations of NTA and CIT with Gd (see Supporting Information, Figures S5 and S7). 4.3. Comparing Differential and Finite Difference Models. Finite difference models (FDMs) have been widely applied in many studies and software packages to evaluate the differential heat from ITC experiments, in part, due to the ease in implementing and evaluating these expressions (cf. eqs 6 and 7).29 On the other hand, differential binding models (DBMs) have been considered cumbersome and impractical.16,41 Concurrently with other groups, we have simplified the derivation of DBMs by implicitly differentiating expressions based on binding polynomials.17−19 This approach has broadened the scope of DBMs by obtaining expressions that can be applied to receptors with multiple sites and any type of binding interaction,18 to receptors with multiple sites in the presence of competing ligand species,19 and, as shown in this article, to direct and reverse titrations of receptors with multiple sites. One advantage of using DBMs over FDMs to analyze calorimetry titrations is that the averaged heat δQ̅ V and its corresponding experimental error σp are evaluated independently for each injection point (eq 14). In comparison, FDMs require two experimental values and their associated errors, which could potentially increase the uncertainty of the fitting parameters. To investigate how either differential or finitedifference models influence the fitting parameters and the residuals obtained through nonlinear regression, we analyzed the direct and reverse titrations between NTA and CIT with Gd using both numerical approaches. The finite difference

where KNi is the stepwise binding constant of the saturated receptor. Hence, only the sigmoidicity of the final inflection point in a direct titration can be adjusted by changing the initial receptor concentration C0R. For example, the calculated cNi value for the direct titration of Gd with DGA at an initial concentration C0Gd = 0.5 mM is c3 = 0.6 which explains the asymptotic behavior toward the end of the titration curve. In comparison, the calculated cNi values for the direct titrations of Gd with either NTA or CIT are c2 = 25.6 and 6.9, respectively, which explains the sigmoidal shape of the curve at the second inflection point in each direct titration (Figures S4C and S6C). In a reverse calorimetry titration, the initial heat signal corresponds to the weighted sum of the enthalpies and the mole fractions, as given by Ni

lim δQ VRTL = C RSyr ∑ ΔHiαi0

V →0

i=0

(22)

where the initial mole fractions α0i are evaluated using the initial ligand concentration C0L (see eq S44 in the Supporting Information). As a result, the experimental parameter C0L can be adjusted to select the receptor−ligand complex that is evaluated at the beginning of the titration. For example, a concentration C0L ≤ 1/β1 results in a titration where the RL complex is the major contributor to the initial heat signal. Conversely, an initial ligand concentration given by C L0 ≥ βN − 1/βN , or ≥ 1/K Ni i

i

(23)

results in at least a 50% contribution from the fully saturated complex. Reverse calorimetry titrations have been typically used to independently measure enthalpies and binding constants of receptors with multiple binding sites and to validate receptor− ligand stoichiometries.8 Equation 22 reveals that reverse calorimetry titrations can also be used to selectively measure the binding parameters for one of the complexes based on the initial concentration of ligand C0L. We tested this rationale to optimize the signal of the fully saturated complex on the reverse titration between DGA and Gd (Figure 1B). During our preliminary titrations, we evaluated a titration of DGA (0.3 mM) with Gd (10.0 mM), but the H


Article


parameter can be adjusted to control the shape of the final inflection point. In a reverse titration, the initial ligand concentration can be adjusted to increase the heat contribution of the saturated complex at the beginning of the titration. Finally, by comparing the fitting results from the differential and finite difference models, we show that both approaches provide good fits to the experimental data. However, differential models provide a better fit to the titration curves when studying receptors with strong binding interactions such as those observed for NTA and CIT. We hope the DBMs presented here allow others to study complex binding systems where stoichiometry or the interactions among the binding sites in the receptor are not known beforehand. With the increased flexibility of the DBMs also comes the responsibility to validate each model selected. Thus, we suggest that one should evaluate the goodness-of-fit for the models using statistical tests.

routine was based on eqs 6 and 7 and utilized the same number of binding sites and binding polynomial as the DBM in eq 12. We show a summary of the fitting parameters obtained with differential models and finite difference models (eq 6) constrained for two binding sites (Ni = 2) (Table 2). For clarity, we show the stoichiometry coefficients, the heats of dilution, and the correlation matrices for the global analysis in the Supporting Information (Tables S16 and S17). Both differential and finite-difference models produce similar results, particularly, when comparing the binding parameters obtained through global analysis of direct and reverse titrations. However, the reduced chi-square values χ2v are lower with DBMs than with FDMs. Experimentally, χ2v values between 2 and 5 signify a good fit to the data.42 Both DBMs and FDMs provide good fits for CIT−Gd complexes. However, DBMs provide better statistical results when analyzing receptors with stronger binding interactions, such as the NTA−Gd binding system. A possible explanation for this effect may be found by rewriting the general expression to evaluate Δ[RL]i (eq 7) in the following form Δ[RLi]p = [RLi]p − [RLi]p − 1 + (ΔVp/V0)[RLi]p − 1

6. ACKNOWLEDGMENTS The authors thank NSERC Canada, DVS Sciences (now Fluidigm Canada, Markham, ON) and the Province of Ontario for their support of this research. We also would like to thank Dr. Gerald Guerin and Professor Heiko Heerklotz for their valuable comments and discussions about this manuscript. Finally, we would like to thank Dr. John D. Simon (University of Virginia, USA), Dr. Lian Hong (Catalent Pharma Solutions, USA), and Dr. Peter Westh (Roskilde University, Denmark) for generously providing examples of analysis routines based on the finite-difference approach.

(24)

where the difference [RLi]p − [RLi]p−1 implies a linear change in the concentrations between injections p and p − 1, and the term (ΔVp/V0)[RLi]p−1 accounts for the material flowing out of the titration cell. The curves that describe the concentrations of binding species RLi follow similar profiles as the curves for their respective mole fractions αi. As shown in Figures S5A and S7A, the mole fraction α1 increases linearly before and after the inflection point located at ΦLTR = 1.0. However, the mole fraction α1 and its respective concentration [RL] does not follow a linear trend as the titration proceeds through ΦLTR = 1.0. Thus, the values calculated with the terms [RLi]p − [RLi]p−1 for i = 1 are slightly less accurate near a sharp inflection point and thus results in higher χ2v values.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.5b09202. General derivation procedure and numerical integration approach to evaluate the DBM. Supporting Tables with correlation matrices. (PDF)

5. CONCLUSIONS In this Article, we develop the framework for the application of DBMs to analyze titration calorimetry curves. These DBMs take into account the dilution of both receptor and ligand species in the calorimetric titration of homotropic binding systems and can be readily applied to direct and reverse ITC experiments in an overfill calorimeter by selecting the expression for the partial differential (∂[RLi]/∂CB)CA where titrant B is injected into analyte A. The solutions to the DBMs are based on the binding polynomial formalism, which allowed us to seamlessly change the number of binding sites considered in the model. We validated the number of sites for each DBM against gadolinium−ligand binding systems with known stoichiometry values. Here, we noted that the F-tests provide a result that agrees with the literature data available for each complex when the fitting parameter f R, used to adjust for small errors in the total receptor concentration, has a constant value of 1.0. Hence, accurate concentrations for the titrant and analyte species are required when testing binding systems where the receptor−ligand stoichiometry is not accurately known. Furthermore, we show that the differentials (∂[RLi]/∂CB)CA can be used to optimize the initial concentrations of receptor and analyte in the titration cell. In a direct titration for a receptor with multiple sites, the ci parameters can be used to predict the shape of the titration curve, whereas the cNi

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

REFERENCES

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