Analytical Biochemistry 434 (2013) 233–241

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Analytical Biochemistry

journal homepage: www.elsevier .com/locate /yabio

Modeling complex equilibria in isothermal titration calorimetry experiments:Thermodynamic parameters estimation for a three-binding-site model

Vu H. Le a, Robert Buscaglia b, Jonathan B. Chaires b, Edwin A. Lewis a,⇑a Department of Chemistry, Mississippi State University, Mississippi State, MS 39762, USAb Department of Medicine, James G. Brown Cancer Center, University of Louisville, Louisville, KY 40202, USA

a r t i c l e i n f o

Article history:Received 9 August 2012Received in revised form 13 November 2012Accepted 28 November 2012Available online 19 December 2012

Keywords:ITCCalorimetryFittingAnalysisNonlinear regressionThree-binding-site modelMultiple binding sites

0003-2697/$ - see front matter � 2012 Elsevier Inc. Ahttp://dx.doi.org/10.1016/j.ab.2012.11.030

⇑ Corresponding author.E-mail address: Elewis@chemistry.msstate.edu (E.

a b s t r a c t

Isothermal titration calorimetry (ITC) is a powerful technique that can be used to estimate a complete setof thermodynamic parameters (e.g., Keq (or DG), DH, DS, and n) for a ligand-binding interaction describedby a thermodynamic model. Thermodynamic models are constructed by combining equilibrium constant,mass balance, and charge balance equations for the system under study. Commercial ITC instruments aresupplied with software that includes a number of simple interaction models, for example, one bindingsite, two binding sites, sequential sites, and n-independent binding sites. More complex models, forexample, three or more binding sites, one site with multiple binding mechanisms, linked equilibria, orequilibria involving macromolecular conformational selection through ligand binding, need to be devel-oped on a case-by-case basis by the ITC user. In this paper we provide an algorithm (and a link to ourMATLAB program) for the nonlinear regression analysis of a multiple-binding-site model with up to fouroverlapping binding equilibria. Error analysis demonstrates that fitting ITC data for multiple parameters(e.g., up to nine parameters in the three-binding-site model) yields thermodynamic parameters withacceptable accuracy.

� 2012 Elsevier Inc. All rights reserved.

Titration calorimetry has been used for the simultaneous deter-mination of K and DH for more than 40 years [1–4]. Isothermaltitration calorimetry is now routinely used to directly characterizethe thermodynamics of biopolymer binding interactions [5–13].Knowledge of the thermodynamic profiles for drug–receptor bind-ing interactions greatly enhances drug design and development[14–17]. Isothermal titration calorimetry (ITC) instruments (avail-able from GE Healthcare (Microcal) and TA Instruments (Calorim-etry Sciences)) have adequate sensitivity to measure heat effectsas small as 0.1 lcal, making it possible to directly determine bind-ing constants as large as 108 to 109 M�1. Even larger values for Kmay be estimated from competitive binding experiments [18,19].

To take full advantage of the powerful ITC technique, the usermust be able to design the optimum experiment, understand thenonlinear fitting process, and appreciate the uncertainties in thefitting parameters K, DH, and n. ITC experiment design and dataanalysis have been the subject of numerous publications [5,6,14–23]. Recent reviews of isothermal titration calorimetry describethe ease of use of modern microcalorimeters [18,19,24]. Severalpapers have described modern uses of isothermal titration calo-rimetry to study a broad range of chemical equilibria in numerousways [25–27]. For example, ITC studies are now being used to iden-

ll rights reserved.

A. Lewis).

tify possible binding mechanisms for ligand–DNA complexes basedon their thermodynamic signatures [28]. ITC experiments explor-ing iron binding to Escherichia coli ferritin were accompanied bya model describing the equations for three independent bindingsites [29], while ITC studies of histone nucleoplasm interactionswere best fit with a site-specific cooperative model including fourequilibrium constants and four enthalpy changes [30]. Examples ofthe use of ITC experiments to unravel the complicated bindingequilibria often occurring in biology are limited since the analysistools provided by the ITC industry cover only the simplest cases.

The improved sensitivity of the current ITC instruments has re-sulted in the ability to accurately estimate thermodynamic param-eters for multiple overlapping binding equilibria. Fig. 1 showsthree unique thermograms that might result from the titration ofa system exhibiting two overlapping binding processes. Fig. 1Band C were simulated from a model for a system having two bind-ing processes, wherein K1 is much greater than K2 and DH1 is notequal to DH2. These two thermograms show a clear distinction be-tween the first and the second binding process, which can be dif-ferentiated when fitting data based on the differences inenthalpy change. However, Fig. 1A also demonstrates a simulationfor a system exhibiting two binding sites in which the bindingaffinities have the same relative magnitude as the previous cases,K1� K2, but with DH1 equal to DH2. In this case, it is not possibleto distinguish the two overlapping equilibria and only the weaker

-10

-5

0

ΔH

(kc

al/m

ol)

-10

-5

0

ΔH

(kc

al/m

ol)

0 1 2 3-15

-10

-5

0

Mol Ratio

ΔH

(kc

al/m

ol)

A

B

C

Fig.1. Representative two-site ITC titration experiment thermograms. In all threecases, the stoichiometry demonstrates that two ligand molecules are binding to thereceptor molecule. (A) A system wherein the enthalpy changes for binding of bothligands are experimentally indistinguishable. (B) A system wherein the binding sitewith higher affinity is accompanied by a more exothermic enthalpy change. (C) Asystem wherein the higher binding affinity site demonstrates a smaller exothermicenthalpy change than the lower binding affinity site.

234 Modeling Complex Equilibria in ITC Experiments / V.H. Le et al. / Anal. Biochem. 434 (2013) 233–241

binding process (K2) can be modeled. The equivalence of the bind-ing enthalpy changes for the two overlapping processes causes thissystem to be modeled as a single binding process with n = 2.

The three plots shown in Fig. 1 demonstrate the increased com-plexity that is often seen in ITC thermograms as the number ofbinding process is increased. ITC thermograms must be understoodprior to modeling the data. Understanding the binding profiles en-sures that the models being applied to the data give an accuraterepresentation of the chemical equilibria being observed. Further-more, error analysis of algorithms being applied to analyze ITCthermograms ensures best-fit solutions are accurate. Several pa-pers have been published on methods for evaluating the errorintroduced into ITC results from either experimental or data fittingconsiderations [32–39]. In general, the potential for using the ITCmethod to estimate the number of parameters needed to describemore complex thermodynamic binding models is underappreciated.

In this work, we describe the construction of algorithms thatcan be used to model ITC data obtained on systems exhibitingthree or more binding sites described with overlapping equilib-rium constants. We have also used simulated data and the MonteCarlo method [31,36,39] to evaluate the uncertainties and cross-correlation in the binding parameters (K, DH, and n). One- andtwo-site systems are used to compare our nonlinear fitting rou-tines to those in the program commercially available from GE

Healthcare (Microcal, Northampton, MA, USA; Origin 7.0). Simu-lated data are used to evaluate the three-binding-site algorithmimplemented in MATLAB (version R2012a, 2012; The MathWorks,Natick, MA, USA) for the accuracy of multiple parameters (K1–3,DH1–3, and n1–3) determined in an ITC experiment. The analysistechniques described in this paper can be extended to ITC data ob-tained on other complex systems if the user can construct theappropriate thermodynamic model for the binding interactionsand/or linked equilibria. One novel aspect of this work is that ourn-sites program for the analysis of systems having up to four bind-ing sites is described in detail in the Supplementary material andthe fully functional analysis program can be downloaded fromour web site (http://lewis.chemistry.msstate.edu/download.html).

Materials and methods

Multiple-binding-site model

Algorithms for modeling ITC thermograms demonstrating up tofour overlapping binding processes were developed using MATLABsoftware. The algorithms model the binding affinity, Kj; the molarenthalpy change, DHj; and the total stoichiometric ratio, nj; foreach of the j = 1 to 4 binding process using nonlinear regressiontechniques. The mass balance and equilibrium equations weremanipulated to produce an (n + 1)th degree polynomial for the n-binding sites, with free ligand as the indeterminate.

General-binding-site model

The thermodynamic model algorithms were developed from acombination of the appropriate mass balance and equilibrium con-stant expressions. Eqs. (1) and (2) are the generalized mass balanceand binding equilibrium equations, respectively. Each equation iswritten in a simplified form that can be expanded to include n-binding site:

Lt ¼ ½L� þMt

Xj¼n

1

ðnjHjÞ; ð1Þ

Kj ¼Hj

ð1�HjÞ½L�) Hj ¼

½L�Kj

1þ ½L�Kj: ð2Þ

Eq. (1) establishes that at any point in the titration, the total li-gand present in the reaction vessel (Lt) must be either bound to oneof the n-binding sites, Mt

Pj¼n1 ðnjHjÞ or free in the reaction vessel

([L]). The equilibrium constants in Eq. (2) have been rewritten toexpress the fraction of process j bound (Hj) as a function of thebinding affinity, Kj, and the presence of free ligand. Substitutionof Hj into Eq. (1) and expanding yields a (n + 1)th degree polyno-mial where [L] is the indeterminate:

½L�kþ1 þ ak½L�k þ ak�1½L�k�1 þ . . .þ a0 ¼ 0: ð3Þ

Eq. (3) represents a simplified form of this polynomial, where ai

represents the ith coefficient. The coefficients, ai, are derived fromligand concentration, macromolecule concentration, stoichiome-tric ratios, and binding affinities. Calculating the roots of the poly-nomial shown in Eq. (3) determines the concentration of freeligand present in the reaction vessel after the ith injection.

Once the concentration of free ligand is determined, substitu-tion of [L] into Eq. (2) yields the fraction of site n bound after theith injection. The total heat produced from the start of the titrationthrough the ith injection, Qi, can be calculated using Eq. (4),

Qi ¼ MtVo

Xi¼n

1

nihiDHi; ð4Þ

Modeling Complex Equilibria in ITC Experiments / V.H. Le et al. / Anal. Biochem. 434 (2013) 233–241 235

where V0 is the active calorimeter cell volume and whereMtVo

Pi¼n1 nihiDHi is the total reaction heat, calculated as the sum

of the heat produced in all of the overlapping binding reactionsfrom the start of the titration through the ith injection. Eq. (5) isthen used to calculate the differential heat produced during theith injection, DQi,

DQ i ¼ Q i � Q i�1; ð5Þ

where Qi and Qi�1 represent the total heats produced from the startof the titration through injection points i and i � 1; and Mt is themacromolecule concentration in the reaction vessel (calorimetercell) as corrected for dilution and for displacement from the calo-rimeter cell as titrant is added. Eq. (5) assumes that all reaction heatis sensed by the calorimeter and that no heat is lost up the fill tubeor owing to reactions occurring outside of the calorimeter cell andthus incompletely sensed by the calorimeter. DQ can be changedto include heat produced from reactions occurring outside of thereaction vessel. One such assumption is that the heat produced out-side of the calorimeter cell (i.e., in the fill tube) is measured withonly half the efficiency as that inside of the reaction vessel. A cor-rection term for the heat being produced outside of the reactionvessel can be easily introduced into the fitting algorithms.

The algorithms used here introduced a correction term for theheat being produced outside of the reaction vessel by restrictingthe concentration of macromolecule (Mt) and the concentrationof the ligand (Lt) after each injection. Eqs. (6) and (7) are the iter-ative functions used to determine the concentrations of macromol-ecule and ligand after the ith injection. Mt,i and Lt,i are theconcentrations of macromolecule and ligand, respectively, afterthe ith injection; Li is the concentration of ligand being injectedinto the cell; Vi is the injection volume; V0 is the active cell volume.The initial conditions for Mt and Xt are Mt,0 = Mi and Xt,0 = 0, whereMi is the concentration of macromolecule loaded into the cellinitially:

Mt;i ¼ Mt;i�1 � ðViMt;i�1=V0Þ; ð6Þ

Table 1Parameters used for the generation of two-competitive-processes simulated ITC titrationcompetitive-site model: best fit parameters. (C) Case 2, two-competitive-site model: aver

Parameter Case 1

K1 2.00 � 106

DH1 (kcal/mol) �9.000n1 1.000K2 1.00 � 105

DH2 (kcal/mol) �3.000n2 1.000

Parameter Case 1 Case

Origin MATLAB Origin

K1 2.00 � 106 1.98 � 106 2.02 �DH1 (kcal/mol) �8.99 �8.94 �12.0n1 0.99 1.00 0.99K2 0.99 � 105 0.98 � 105 0.94 �DH2 (kcal/mol) �3.12 �3.09 �5.01n2 0.98 0.99 0.99Parameter Origin

Average SD

K1 1.991 � 108 0.068 �DH1 (kcal/mol) �12.008 ±0.001n1 1.005 ±0.001K2 0.995 � 106 ±0.003 �DH2 (kcal/mol) �5.004 ±0.002n2 1.005 ±0.001

Best-fit parameters (K1, DH1, n1, K2, DH2, and n2 values) are listed for the Origin 7.0 and Mprocesses test cases. The comparison shows that both programs result in the same beanalyses of the Case 2 two-site test data using both algorithms. The standard deviations

Lt;i ¼ ½LiVi þ Lt;ið1� ViÞ�=V0: ð7Þ

Eqs. (6) and (7) were constructed to accurately mimic the con-centrations of ligand and macromolecule present in the active cellvolume of the calorimeter. After each injection, the macromoleculeconcentration is lowered by the amount of macromolecule leavingthe active cell volume. Similarly, the concentration of ligand is in-creased after each injection, but after the initial injection, a correc-tion is also made for the amount of ligand that leaves the active cellvolume. These equations assume that the stirring rate is efficientenough that the ligand is accurately mixed into the active cell vol-ume directly after the injection.

The difference between the heats calculated for DQi (Eq. (5))and those measured during an ITC experiment was used to deter-mine the goodness of fit. A Levenberg–Marquardt nonlinear regres-sion model common to MATLAB was used to determine best-fitparameters. (No other minimization routines were tried.) By allow-ing the model to iterate over the parameters (K1, . . ., Kn; n1, . . ., nn;DH1, . . ., DHn), a solution is found such that the v2 merit function isminimized by steepest descent and quadratic minimization.

Algorithm comparison

To compare the MATLAB and Origin 7.0 nonlinear regressionanalysis programs, the same simulated data sets were fit with bothprograms. Each set of simulated data was generated using 25 injec-tions of 5 ll with a ligand concentration of 1 mM and a macromol-ecule concentration of 40 lM at 298 K. Randomly generatednormally distributed noise, 0.1 lcal, was added to the simulatedthermograms prior to fitting with either Origin 7.0 or MATLAB rou-tines. Solutions resulting from the MATLAB algorithms-developedtwo-binding-site model were compared to solutions obtainedusing Microcal Origin 7.0 to ensure that the resulting best-fitparameters were in agreement between the two nonlinear regres-sion analysis programs. Simulated ITC thermograms were createdfor two different cases: two binding sites and three binding sites.

data. (A) Two-competitive-site model: ITC test data simulation parameters. (B) Two-age Monte Carlo parameters and standard deviations.

Case 2 Case 3

2.00 � 108 2.00 � 108

�12.000 �3.0001.000 1.0001.00 � 106 1.00 � 106

�5.000 �12.0001.000 1.000

2 Case 3

MATLAB Origin MATLAB

108 1.99 � 108 2.04 � 108 2.02 � 108

1 �11.93 �2.99 �2.981.00 0.99 1.00

106 0.98 � 106 1.02 � 106 1.00 � 106

�4.97 �12.03 �11.931.00 0.99 1.00

MATLAB

Average SD

108 1.944 � 108 0.066 � 108

�11.950 ±0.0011.010 ±0.001

106 0.990 � 106 ±0.003 � 106

�4.950 ±0.0021.013 ±0.001

ATLAB solutions for the nonlinear regression analysis of the three two-competitive-st-fit parameters. Average best-fit parameters are also listed for 100 Monte Carlo

(SD) are also listed for the six average parameters for both methods.

Table 2Parameters used for the generation of simulated ITC titration data for the three-competitive-processes model. (A) Three-competitive-sites model: simulation param-eters. (B) Three-binding-sites algorithm: Monte Carlo analysis ‘‘best-fit’’ parameters.

Model parameter Site 1 Site 2 Site 3

Ki (M�1) 1.0 � 108 1.0 � 106 1.0 � 104

DHi (kcal/mol) �12.0 �8.0 �4.0ni 1.0 1.0 1.0

Model parameter Monte Carlo mean 95% confidence interval

Minimum Maximum

K1 (M�1) 1.06 � 108 1.02 � 108 1.09 � 108

K2 (M�1) 1.06 � 106 1.02 � 106 1.10 � 106

K3 (M�1) 1.11 � 104 1.03 � 104 1.19 � 104

DH1 (kcal/mol) �12.00 �11.999 �12.001DH2 (kcal/mol) �7.99 �7.993 �8.002DH3 (kcal/mol) �4.24 �4.147 �4.339n1 0.99 0.999 1.000n2 0.99 0.999 1.000n3 1.08 1.052 1.101

The simulated data were used in the Monte Carlo analysis of the three-competitive-processes fitting algorithm. Resulting Monte Carlo mean parameter values and 95%confidence intervals were determined for each of the nine parameters (K1–3, DH1–3,and n1–3) obtained as a solution for the three-competitive-processes model with theMATLAB code given in the Supplementary material. The Monte Carlo analysisdemonstrates that all of the three-competitive-processes model parameters arewell defined and have acceptable error. A correlation between n and DH wasobserved. The 95% confidence intervals show that the equilibrium constants, molarenthalpy changes, and stoichiometric ratios are well determined for all three pro-cesses. The binding affinities for each process are slightly overestimated and thethird-process molar enthalpy change, DH3, and stoichiometry, n3, exhibit the largestuncertainties.

236 Modeling Complex Equilibria in ITC Experiments / V.H. Le et al. / Anal. Biochem. 434 (2013) 233–241

The parameters used to simulate ITC thermograms and the result-ing best-fit parameters for the two-binding-site model are given inTable 1. The parameters used to simulate ITC test data for thethree-binding-site case are given in Table 2, in which the best-fitparameters are listed only for the MATLAB program. (Origin 7.0is limited to solving equilibria for two or fewer reactions althoughthe sequential model in Origin could be transformed and used tosimulate thermograms for three independent sites.)

Monte Carlo analysis

Monte Carlo analysis [31,36,39] was used to estimate the uncer-tainties for each of the nine parameters (K1–3, DH1–3, n1–3) requiredto fit the three-binding-site model. The Monte Carlo analysis in-volved the creation of a perfect data set with the MATLAB algo-rithm. This perfect data set was then used to create 1000 virtualdata sets by adding random normally distributed noise to eachdata point in the perfect data set. The ITC instrument noise was ta-ken to be ±0.1 lcal per injection [18].

The MATLAB algorithm was then used to fit each of the 1000virtual ITC experiments and to obtain a set of best-fit parametersfor each of the virtual experiments. The mean value for each ofthe nine fitting parameters was then calculated from the 1000 setsof best-fit values obtained in the Monte Carlo procedure.

Error analysis

Two-dimensional error plots for each of the nine parameters in-volved in modeling the three binding sites were generated. The er-ror plot for parameter X was established by fixing the value of Xand allowing all other parameters to be iterated until a best-fitsolution was found. Fixed values for parameter X were chosen suchthat Dlog(X) = log(X) � log(perfect value) had a range of �3 to 3.The error plot was produced by plotting Dlog(parameter) versuslog(error). Transformation of the data using logs condenses the

data and exaggerates the local and global minima being observed.These error plots were used to evaluate the interdependence of theparameters and to establish the range of acceptable initial param-eter guess values required in modeling three binding sites.

Results

Algorithm comparisons

ITC experiments were simulated for multiple two-site models.These simulated data (including random noise) were fit using bothOrigin 7.0 and the MATLAB algorithms. This comparison was doneto ensure that our MATLAB algorithms yielded a solution that wasin agreement with the solution obtained with Origin 7.0. The best-fit parameters obtained with both programs, for all of the simu-lated ITC experiments, are in excellent agreement with one anotherand returned parameters close to the parameters used to create thesimulated ITC data sets.

Monte Carlo analysis

A Monte Carlo analysis was used to estimate the uncertainty ineach of the thermodynamic parameters determined in modelingsimulated ITC data for a system exhibiting three binding sites. Be-cause modeling a three-process system is not currently imple-mented in any of the commercially available ITC data analysisprograms, a rigorous determination of the uncertainties in thebest-fit parameters was conducted in lieu of a direct comparisonwith other approaches or solutions.

Table 2 gives the mean value and 95% confidence interval foreach parameter estimated by the Monte Carlo method. Comparisonof the mean Monte Carlo parameters with the test data simulationparameters (shown in Table 2A) demonstrates that the MATLABalgorithm finds a solution minimum in which all nine of the bind-ing parameters are well determined. As seen from Table 2, each ofthe binding constants (K1, K2, K3) has a 95% confidence interval thatis slightly skewed to higher values than the values used to simulatethe ITC data. Overestimating the binding constants is typical forbinding algorithms and can be attributed to the number of datapoints in the slope of the thermogram and the interdependenceof the binding affinity on the other parameters. The molar enthalpychanges and stoichiometric ratios for processes 1 and 2 have 95%confidence intervals that are equally distributed around the noise-less values. This indicates that DH1, n1, DH2, and n2 are determinedwith the greatest accuracy. Because the stoichiometric ratios arewell determined for the two processes with the higher affinitiesand larger heats, the algorithms correctly identify the solutionsfor accurate estimates of DH1, n1, DH2, and n2. However, the 95%confidence interval for DH3 (and/or n3) demonstrates that thesebest-fit parameters contain the largest error of the nine modelparameters. DH3 is skewed toward more exothermic values. Thiserror results from the inability of the binding algorithm to deter-mine an accurate endpoint for the third process where the heat sig-nal and integrated heats between successive injections are small.

Error surfaces

A two-dimensional error plot was created for each of the nineparameters required to fit the three-binding-sites model. MATLABwas used to fix the value of the parameter of interest while allow-ing all other parameters to be optimized by nonlinear regression.The error plots were constructed by plotting Dlog(X) versus log(er-ror), where Dlog(X) = log(X) � log(noiseless value). The error plotsdemonstrate the interdependence of each parameter, shown bythe presence of multiple local minima in several of the parameters.

-3 -2 -1 0 1 2

-30

-20

-10

0

ΔLog (K2)

Lo

g (

Err

or)

Fig.2. Error plot produced by holding the value of K2 constant while all other three-site parameters were optimized using nonlinear regression. The value of K2 rangedfrom 101 to 108 to cover a large enough range of values (i.e., �3 < Dlog(K2) < 2).Error plots were produced for all nine parameters, each showing several localminima and a global minimum at Dlog(X) = 0. The error plot shown illustrates theinterdependence of parameters by the local minimum found at �1.5. Error plots fornk and DHk show similar local minima due to their interdependence. The error plotdemonstrates that appropriate starting positions are necessary to determine anoptimal best-fit solution.

Modeling Complex Equilibria in ITC Experiments / V.H. Le et al. / Anal. Biochem. 434 (2013) 233–241 237

However, the parameters are all found to have global minima atDlog(X) equal to 0. This implies that the algorithm finds the correctminima when the starting parameters are adequately set.

A typical parameter 2-D error plot is shown in Fig. 2. This figureshows the plot of Dlog(K2) versus log(error), where Dlog(K2) =log(K2) � log(1 � 106). As can be seen from the K2 error plot, atleast one local minimum exists that would result in a solutionnot corresponding to the global minimum. Local minima may bethe result of the interdependence of one or more of the fittingparameters. For K2, the shallow local minimum occurring at bind-ing affinities 2 orders of magnitude larger than the correct valueresults from the algorithm finding a two-site solution wherein K2

and K3 are indistinguishable. This indicates the codependence ofthe binding affinity values. Hence, it is up to the individual user

Fig.3. Error plot produced by holding the values of n3 and DH3 constant while all otherranged from 0 to �10 kcal/mol and the value of n3 ranged from 0 to 2, to include the bevalues for the two parameters. (B) The magnified error surface near the best-fit values f

applying these algorithms to use an appropriate number of bindingprocesses to arrive at an optimal thermodynamic solution that isconsistent with the ITC thermograms (i.e., fitting the experimentaldata within the expected or observed experimental error). Further-more, as seen from Fig. 3, which shows a 3-D error plot for the twoparameters DH3 and n3, it may be necessary for the individual userto enter appropriate starting values for each parameter being eval-uated and to verify that the minimum solution found correspondsto a global minimum and represents a plausible solution for the ac-tual chemical equilibria occurring in the system.

Discussion

Multiple-binding-site models

The theory used for developing the three-processes bindingmodel and fitting algorithm can be extended to model multiple-binding-site models of almost any order. The challenge in model-ing such complex systems can be attributed to two factors. Thefirst is the ability to determine the roots of the polynomial (Eq.(3)) used for calculating the concentration of free ligand withinthe reaction vessel. For modeling simpler binding models, includ-ing one or two binding sites, the polynomial expands only to a qua-dratic or cubic equation, respectively. Although closed-formsolutions can be obtained for both quadratic and cubic equations,the solution for the cubic equation can be quite cumbersome. Formore complex cases, such as three or more binding sites, aclosed-form solution for the polynomial does not exist. Softwaresuch as MATLAB can provide numerical methods for efficient andaccurate means of determining the correct root of Eq. (3). It is pos-sible to simplify the determination of roots from Eq. (3) based onchemical restrictions. The restriction that the concentration of freeligand in the cell cannot fall below 0 or exceed the injected ligandconcentration allows for the root-finding processes to efficientlydetermine the correct root of the polynomial.

Second, using the smallest correct number of reactions andintroducing restrictions to the binding algorithm can greatly in-crease the efficiency of complex algorithms. Knowing that the ther-modynamic model accurately describes the system under study(i.e., includes only chemical equilibria actually observed in the sys-

three-site parameters were optimized using nonlinear regression. The value of DH3

st-fit values for each of these parameters. (A) The error plot for the whole range ofor the parameters DH3 and n3 at the global minimum.

0 1 2 3 4 5

-12.5

-10.0

-7.5

-5.0

-2.5

0.0

Mol Ratio

ΔH (

kcal

/mol

)

Fig.4. ITC data for the reaction of a cationic porphyrin, TMPyP4, with the WT c-MYC27-mer polypurine P1 promoter sequence G-quadruplex at low ionic strength(20 mM K+ in BPES buffer at pH 7.0). The lower ionic strength data must be fit to thethree-binding-site model. The best-fit parameters are K1 = 3.9 � 108 M�1,K2 = 4.2 � 107 M�1, K3 = 2.9 � 106 M�1, DH1 = �0.2 kcal/mol, DH2 = �12.6 kcal/mol,DH3 = �5.9 kcal/mol, and n1 = 1, n2 = 2, n3 = 1.

238 Modeling Complex Equilibria in ITC Experiments / V.H. Le et al. / Anal. Biochem. 434 (2013) 233–241

tem) allows for the minimum number of fitting parameters to bedetermined by the nonlinear regression analysis. Similarly, if anyrestrictions can be placed on the binding algorithm to reduce thenumber of modeled parameters (e.g., fixing any of the parametersusing experimentally determined values), the uncertainty in thebest-fit parameters can be dramatically reduced. In particular,restrictions placed on the binding stoichiometry, n, are generallyhelpful for improving fitting efficiency.

The stoichiometric ratio, n, can also greatly influence the en-thalpy change, DH, of the reaction because these two variablesare indirectly proportional. That is, if the DH of the reaction is in-creased by an order of magnitude the stoichiometric ratio will de-crease proportionally. Caution must be taken when using thesecomplex algorithms to ensure that the proper minimum is beingapproached. Because of the compensation of stoichiometries andenthalpy changes, it is possible for the algorithm to enter a localminimum at which the enthalpy change becomes increasinglylarge to compensate for underestimated binding affinities and/orstoichiometric ratios. The relationship between DH and n can beseen in Eq. (4), which is the only equation in which DH appears. Be-cause DH enters linearly into the binding models, it is possible tofurther improve the binding algorithms through linearization ofthe DH parameter.

Monte Carlo evaluation

Monte Carlo evaluation is a valuable technique for determiningthe statistical uncertainties in best-fit binding parameters. Theuncertainties estimated from a Monte Carlo analysis for one possi-ble three-binding-site model in which DH1 < DH2 < DH3 is dis-cussed below. The Monte Carlo analysis demonstrated a strongcorrelation between DH and n when determining best-fit parame-ters. Other combinations of molar enthalpy changes not simulatedin this work may result in different parameter uncertainties. How-ever, the largest factor when evaluating uncertainties in the ITCmethod is the size of the raw heats produced from the reactionsoccurring in the calorimeter. Because the standard deviation ineach injection interval heat, DQi, is ±0.1 lcal, larger heats are lessaffected by experimental error in the integrated injection heats.The results given here from the Monte Carlo analysis of the fitparameter uncertainties are expected to be typical for modeling asystem with three binding sites. The parameters with the mostuncertainty are typically n3 and DH3 because these parametersare sensitive to the endpoint of the titration. Thermodynamicparameters obtained from titrations exhibiting a well-defined end-point (i.e., having a large value for Kn) are well determined with thealgorithms developed here. Furthermore, the above algorithms, ifextended beyond three binding sites, can be used to evaluate evenmore complex systems. The Monte Carlo method provides a rigor-ous test of the reliability of the best-fit parameters obtained in thenonlinear regression modeling of the three-binding-site system.The Monte Carlo method can be used to estimate the uncertaintiesin parameters obtained in the nonlinear regression modeling ofexperimental data in the same manner as used here for testingthe nonlinear fitting of simulated data, but should incorporateexperimentally determined error.

Error analysis

The error plots produced for all parameters involved in model-ing three binding sites clearly demonstrated the interdependenceof binding parameters (Fig. 2). Because several local minima areobserved for each parameter, stress is placed on the user to accu-rately enter reasonable starting parameters to avoid gettingtrapped in a local minimum. It is clear that several parametershave strong interdependence (i.e., K2 vs K3, DH3 vs n3) and must

be given starting values that will ensure that the true global min-imum is found. The user can check the validity of the solution ob-tained from nonlinear regression analysis by entering severalstarting locations for parameters that demonstrate large variabil-ity. If the same minimum is reached regardless of starting posi-tions, it is more likely that the solution represents a globalminimum. The user must take care to ensure that the solution bothis descriptive of the system and represents a plausible thermody-namic and chemical result. However, it must be remembered thatthe binding model chosen strongly influences the best-fit parame-ters obtained in the nonlinear regression analysis of ITC data, andbinding parameters will accurately reflect the system only whenthe correct model is chosen.

Example systems exhibiting three binding sites

Biological systems exhibiting multiple overlaps may be morecommon than previously thought. Several ITC studies of thesemore complex systems have been recently reported[29,30,40,42]. We present three examples of quadruplex DNAbinding either a small ligand or a protein that further illustratethe utility of our MATLAB n-binding sites ITC data analysisprogram.

ITC data for the titration of a 27-mer c-MYC promoter sequenceG-quadruplex construct (having the sequence 50-TGGGGAGGGTGGGGAGGGTGGGGAAGG) at low ionic strength,20 mM K+, with the cationic porphyrin ligand TMPyP4 (5, 10, 15,20-meso-tetra-(N-methyl-4-pyridyl) porphine) are shown inFig. 4. This wild-type (WT) c-MYC promoter sequence quadruplexbinds four ligands at saturation. At physiologic ionic strength(150 mM KCl), the four binding sites in the model are representedby two sets of two sites with each set having a characteristic affin-ity. In effect, two ligands bind to the two high-affinity sites and twoligands bind to the low-affinity sites [40]. We have assigned thehigh-affinity sites to end stacking on the two ends of the G-tetradstack, while the two lower-affinity sites were assigned to the twointercalation sites located between the first and second and thesecond and third G-tetrads. At low ionic strength (20 mM KCl), amore complicated model is required, having three sites exhibitingthree different affinities. We have suggested that at the lower ionicstrength the two ends of the G-quadruplex are no longer equiva-lent and the overhanging single strand is stiffened because of elec-trostatic repulsion. Based on the three-binding-site fit of the ITCdata, the first TMPyP4 binds to one end, the next two ligands bind

0 1 2 3 4 5

-14

-12

-10

-8

-6

-4

-2

0

Mol Ratio

ΔH (

kcal

/mol

)

Fig.5. ITC data for the reaction of a cationic porphyrin, TMPyP4, with the 1:6:1 c-MYC 32-mer polypurine P1 promoter sequence G-quadruplex capped with apartially complementary 17-mer 5-T capping oligonucleotide in100 mM K+/BPESbuffer at pH 7.0. These ITC data have been corrected for ligand-induced folding anddilution heat effects and fit to the three-binding-site model. The best-fit parametersare K1 = 7.4 � 109 M�1, K2 = 5.2 � 107 M�1, K3 = 7.0 � 105 M�1, DH1 = �7.0 kcal/mol,DH2 = �13.7 kcal/mol, DH3 = �9.9 kcal/mol, and n1 = 1.5, n2 = 1, n3 = 1.5.

Modeling Complex Equilibria in ITC Experiments / V.H. Le et al. / Anal. Biochem. 434 (2013) 233–241 239

in the two intercalation sites, and the fourth TMPyP4 binds to thesterically hindered low-affinity end of the G-quadruplex. Whetherthis structural model is correct or not, the ITC thermogram ob-served under low-salt conditions is well fit with the three-bind-ing-site model and cannot be fit with any simpler model.

G-quadruplex ligand studies have typically been limited to sin-gle-stranded constructs. To better model the G-quadruplex motif

Fig.6. Screen capture of the MATLAB user interface showing the three-binding-sites fit oTel-22. Experimental parameters, [L], [M], calorimeter cell volume, injection volume, nuscreen. Starting parameters ni, Ki, and DHi are entered in the table shown midleft on thbottom left of the screen. The right side of the screen shows the ITC data (corrected for hdownloaded at http://lewis.chemistry.msstate.edu/download.html.

in the presence of its complementary strand we have constructedcapped G-quadruplex constructs in which the C-rich strand hasbeen replaced with a shorter partially complementary strand hav-ing the ability to form short flanking duplex regions on either sideof the quadruplex and with a short noninteraction region of four,five, or six T’s bridging the quadruplex in the capped construct.ITC data for the TMPyP4 titration of one such construct are shownin Fig. 5. The interaction of TMPyP4 with the ‘‘capped’’ quadruplexis more complicated than the interaction with the singled-strandedquadruplex having the same sequence. In effect, the cap results in asmall amount of unfolding of the G-quadruplex and in the case ofthe WT sequence results in a shift in the equilibrium between the1:6:1 and the 1:2:1 foldamers. The TMPyP4 titration of the five-Tcapped 1:6:1 mutant c-MYC 32-mer promoter sequence is fit to athree-binding-site model after the data have been corrected forsome ligand-induced quadruplex folding and dilution effects. Thethree binding events must be the result of mixed end bindingand intercalation interactions.

We do not currently understand the stoichiometry 1.5:1.0:1.5;however, we are continuing to search for structural models thatare consistent with the titration results and the best-fit parametersshown in Fig. 5. (The ITC data shown here for the titration of thecapped quadruplex with TMPyP4 are unpublished and providedhere as an example of the need for the analysis of ITC data for sys-tems exhibiting multiple overlapping equilibria.)

Human telomeric DNA contains tandem repeats of the DNA se-quence (TTAGGG) and is capable of forming G-quadruplexes.Unfolding of the telomeric G-quadruplex DNA leads to elongationof the telomere by TERT in vitro. UP1 is the proteolytic fragmentof the hnRNP A1 protein that is responsible for several gene-regu-latory processes at the posttranscriptional level, including mRNA

f the ITC data obtained for the addition of UP1 to the human telomere quadruplex,mber of injections, and temperature, are shown in the upper left quadrant of the

e screen capture. Best-fit parameters and the standard deviations are shown at theeat of dilution) and the best-fit line through the data. The MATLAB program can be

Fig.7. Thermochemical cycle (Hess’ Law diagram) for the binding of 2 mol of UP1 to the Tel-22 G-quadruplex. The values of DH1, DH2, and DH3 are obtained directly from thethree-binding-sites model fit to the forward titration data for the addition of UP1 to Tel-22. The value for DHreverse is obtained from the ‘‘model-free’’ reverse titration. Thevalue of DH⁄DSC is obtained as the difference between DH1 and DH2 and compared to the DHDSC value obtained for the thermal denaturation of the naked Tel-22 G-quadruplex. The value of DH3 is obtained directly from the three-binding-sites model and compared to the difference between DH2 and DHreverse.

240 Modeling Complex Equilibria in ITC Experiments / V.H. Le et al. / Anal. Biochem. 434 (2013) 233–241

alternative splicing, transportation, and pre-miRNA processing. Acrystal structure of UP1 complexed with a partial human telomericsequence d(TTAGGG)2 revealed that the interaction between UP1and the telomeric DNA is established through three nucleobases,TAG. The TAG binding motif located at several corners of G-quar-tets was found to be critical for UP1 binding and initiating unfold-ing of the G-quadruplex structure. ITC experiments, including bothforward and reverse titrations, and DSC experiments were used toprobe how UP1 interacts with human telomeric G-quadruplex DNAin Na+ buffer conditions. ITC data and the three binding sites fitfrom a forward titration of Tel-22 with UP1 are shown in Fig. 6.These studies reveal that UP1 binds and unfolds the Tel-22 G-quadruplex DNA at a 2:1 molar ratio. The binding involves looprecognition plus unfolding plus single-strand binding for a secondUP1 molecule. This binding, unfolding, binding process is describedby the thermochemical cycle shown in Fig. 7. (The UP1/Tel-22 dataare unpublished and are provided by Dr. David Graves at the Uni-versity of Alabama at Birmingham, Department of Chemistry.)

Further models

The models developed here assume that the only chemical equi-libria being observed are reactions involving the binding of a ligandmolecule to one or more specific binding sites in a macromolecule(Eq. (2)). In these systems, each type of binding site competes withany other binding sites present in the macromolecule and thebinding at any site occurs with unitary stoichiometry.

We have shown that a large number of thermodynamic param-eters may be accurately estimated from an ITC experiment, e.g., thenine parameters in the three-binding-site model were all welldetermined. The larger the number of parameters that need to bedetermined, the more uncertainty will be introduced in parameterdetermination. There will always be some benefit in designing theexperiments or in combining data from complementary studies tolimit the number of parameters that are fit. For example the en-thalpy change for the strongest binding reaction may be best deter-mined in so-called ‘‘model-free’’ experiments [41], while thethermodynamic parameters for the weakest binding process maybe best determined in additional reverse titration experiments inwhich the low-affinity complexes are more highly populated

[30]. In conclusion, we would like to state that ITC data can be fitwith acceptable accuracy to complex thermodynamic models hav-ing as many as three or more overlapping binding equilibria.

Acknowledgments

The work was supported in part by Grant CA35635 from the Na-tional Cancer Institute (to J.B.C.) and by the James Graham BrownFoundation.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.ab.2012.11.030.

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