formalism using binding polynomials

C H A P T E R F I V E

M

IS

*{

ethods

SN 0

DepaInstituBIFI)

Isothermal Titration Calorimetry:

General Formalism Using Binding

Polynomials

Ernesto Freire,* Arne Schon,* and Adrian Velazquez-Campoy†

Contents

1. I

in En

076-6

rtmente of, Uni

ntroduction

zymology, Volume 455 # 2009

879, DOI: 10.1016/S0076-6879(08)04205-5 All rig

t of Biology, Johns Hopkins University, Baltimore, Maryland, USABiocomputation and Physics of Complex Systems (BIFI), and Fundacion Aragon Iþversidad de Zaragoza, Zaragoza, Spain

Else

hts

D (

128

2. T
he Binding Polynomial 129
3. M
icroscopic Constants and Cooperativity 131
4. I
ndependent or Cooperative Binding? 132
5. A
nalysis of ITC Data Using Binding Polynomials 133
6. A
Typical Case: Macromolecule with Two Ligand-Binding Sites 135
7. D
ata Analysis 137
8. D
ata Interpretation 141
8
.1. In dependent ligand binding: Two identical binding sites 142
8
.2. In dependent ligand binding: Two nonidentical binding sites 143
8
.3. C ooperative ligand binding: Two identical binding sites 144
9. A
n Experimental Example 146
10. E
xperimental Situations from the Literature 147
11. M
acromolecule with Three Ligand-Binding Sites 150
12. C
onclusions 150
Appe
ndix 151
Ackn
owledgment 154
Refer
ences 154
Abstract

The theory of the binding polynomial constitutes a very powerful formalism by

which many experimental biological systems involving ligand binding can be

analyzed under a unified framework. The analysis of isothermal titration calo-

rimetry (ITC) data for systems possessing more than one binding site has been

cumbersome because it required the user to develop a binding model to fit the

data. Furthermore, in many instances, different binding models give rise to

vier Inc.

reserved.

ARAID-

127

128 Ernesto Freire et al.

identical binding isotherms, making it impossible to discriminate binding

mechanisms using binding data alone. One of the main advantages of the

binding polynomials is that experimental data can be analyzed by employing

a general model-free methodology that provides essential information about

the system behavior (e.g., whether there exists binding cooperativity, whether

the cooperativity is positive or negative, and the magnitude of the cooperative

energy). Data analysis utilizing binding polynomials yields a set of binding

association constants and enthalpy values that conserve their validity after

the correct model has been determined. In fact, once the correct model is

validated, the binding polynomial parameters can be immediately translated

into the model specific constants. In this chapter, we describe the general

binding polynomial formalism and provide specific theoretical and experimental

examples of its application to isothermal titration calorimetry.

1. Introduction

The introduction of the binding polynomial theory several decadesago by Jeffries Wyman provided a general statistical thermodynamic frame-work for studying ligand binding to macromolecules (Wyman, 1948, 1964.Wyman and Gill, 1990). Being equivalent to a partition function, thebinding polynomial contains all the information about the system and allowsderivation of all thermodynamic experimental observables (e.g., averagenumber of ligand molecules bound, average excess enthalpy). Contrary tomodel-dependent parameters, the parameters that define the binding poly-nomial have a general validity. Consequently, unless a binding model hasbeen validated, the binding polynomial should be the preferred startingpoint for the analysis of complex binding situations. There are experimentalsituations that cannot be assigned to a particular model. For two or morebinding sites, different binding models can give rise to mathematically equalbinding equations. In those cases, the discrimination between models can-not be made on the basis of binding data alone and requires extrathermo-dynamic arguments.

The binding polynomial represents the basis for a general, model-independent analysis of a binding experiment. The same methodology isapplicable to a system with one or any arbitrary number of binding siteswithout the user having to decide on any particular binding mechanism. It isthe preferred analysis protocol unless a specific binding model has beenvalidated for the system under consideration.

Among the experimental techniques employed for studying ligandbinding, isothermal titration calorimetry (ITC) exhibits several featuresthat render it a unique experimental tool: (1) the signal measured (heat ofreaction) is a universal probe, avoiding the use of nonnatural spectroscopiclabels; (2) the interacting molecules are in solution; and (3) it allowsdetermining simultaneously the association constant, the enthalpy, and the

Binding Polynomials in ITC 129

stoichiometry of binding in a single experiment. Accordingly, ITC data isideally suited to be analyzed using the binding polynomial formalism. Thischapter discusses the theoretical basis of the binding polynomials and theirapplication to the analysis of ITC data.

2. The Binding Polynomial

The equilibrium of a ligand with a macromolecule with n ligandbinding sites can be described in terms of two different sets of associationconstants, the overall association constants, bi, or, alternatively the stepwiseassociation constants, Ki:

Mþ iL ! MLi bi ¼½MLi�½M�½L�i

MLi�1 þ L ! MLi Ki ¼ ½MLi�½MLi�1�½L�

; ð5:1Þ

The two sets of descriptors are equivalent, and they are related throughthe following relationships:

bi ¼Yij¼1

Kj

Ki ¼ bibi�1

: ð5:2Þ

Because the stepwise binding constants and the overall association con-stants are related and can be transformed into each other by using Eq. (5.2),for convenience this chapter will use the overall association constants, bi.

In a binding experiment, the main parameter is the average number ofligand molecules bound per macromolecule, nLB, which can be calculated asa simple enumeration:

nLB ¼ ½L�B½M�T

¼Pni¼0

i½MLi�Pni¼0

½MLi�¼ ½ML� þ 2½ML2� þ 3½ML3� þ . . .

½M�þ ½ML�þ ½ML2�þ ½ML3� þ . . .;

ð5:3Þ


where n is the number of binding sites in the macromolecule, [M]T is thetotal concentration of macromolecule, and [L]B is the concentration ofligand bound to the macromolecule. According to its definition, nLB takesvalues between 0 and n. In terms of the overall association constants, thebinding parameter nLB can expressed as:

nLB ¼Pni¼0

ibi½L�i

Pni¼0

bi½L�i: ð5:4Þ

This equation is the so-called Adair’s equation, which was first used foranalyzing the oxygen binding to hemoglobin (Adair, 1925).

The binding polynomial is defined as the partition function, P, of thesystem, and therefore is the sum of the different species concentrationsrelative to that of the free macromolecule that is defined as the reference:

P ¼Xni¼0

½MLi�½M� ; ð5:5Þ

and it can be expressed in terms of the association constants:

P ¼Xni¼0

bi½L�i: ð5:6Þ

Therefore, the binding polynomial of a macromolecule with n ligandbinding sites is a finite power series (nth-order polynomial) in the freeligand concentration, each term representing the relative concentration ofa macromolecular species with a given number of bound ligands. Being Pthe partition function of the system, the thermodynamic parameters of thesystem are obtained from P as follows (Schellman, 1975; Wyman, 1964;Wyman and Gill, 1990):

nLB ¼ RT@ ln P

@mL

!T;p

¼ @ ln P

@ ln½L�

!T;p

hDGi ¼ �RT ln P

hDHi ¼ �R@ ln P

@ð1=TÞ

!p;½L�

¼ RT2 @ ln P

@T

!p;½L�

; ð5:7Þ


where mL is the chemical potential of the free ligand, <DG> and <DH>are the average excess molar Gibbs energy and enthalpy of binding atconstant pressure, p, taking the unliganded macromolecule as the referencestate. The last expression in Eq. (5.7) is equivalent to the Gibbs-Helmholtzequation. The temperature derivative of the association constants is eval-uated using the van’t Hoff equation, which links the temperature derivativeof a given association constant, bi, with the enthalpy change, DHi, associatedwith the equilibrium process.

The fraction or population of each species, Fi, can be obtained from theexpression of the binding polynomial:

Fi ¼ ½MLi�½M�T

¼ bi½L�iP

; ð5:8Þ

and these populations exhibit two properties: (1) their sum is equal to 1; and(2) as the system is saturated with an increasing concentration of ligand,Fi reaches a maximum when nLB equals i (Wyman and Gill, 1990). Theaverage number of ligand molecules bound per macromolecule and theexcess molar thermodynamic parameters may be expressed as:

hDGi ¼ �RT lnXni¼0

bi½L�i !

hDHi ¼

Xni¼0

bi½L�iDHi

P¼Xni¼0

FiDHi

nLB ¼

Xni¼0

bi½L�ii

P¼Xni¼0

Fii ¼ hii

; ð5:9Þ

where it can be clearly seen that <DH> and nLB are statistically weightedaverages of the enthalpy of binding, DHi, and the number of ligand mole-cules bound, i.

3. Microscopic Constants and Cooperativity

The overall and stepwise association constants are macroscopic associa-tion constants, and no mechanistic interpretation about the ligand bindingcan be inferred from them. Therefore, they are considered phenomenological


or model-free association constants. Besides macroscopic association con-stants, there is another type of association constants, the microscopic bindingconstants ki, related intrinsically with the ligand binding to the differentbinding sites, and therefore reflecting the intrinsic binding affinities to eachsite. In case of independent binding, they can be readily obtained from themacroscopic association constants, as the binding polynomial would factorizeinto n first-order polynomials with n negative real roots:

P ¼Xni¼0

bi½L�i ¼Yni¼1

ð1þ ki½L�Þ: ð5:10Þ

This conclusion represents a particular case of the more general statisticalthermodynamic result that the partition function of a system composed ofindependent subsystems is equal to the product of the partition functions forthe independent subsystems (P ¼Q

i

Pi).

In the case of nonindependent (cooperative) binding, the binding poly-nomial would not factorize, and in addition to the microscopic bindingconstants, interaction or cooperative constants must be included in thedescription. Factorability of the binding polynomial into n first-order poly-nomials with n negative real roots is not guaranteed. Accordingly, eventhough it is always possible to estimate values for the macroscopic associationconstants, it is not always possible to extract microscopic intrinsic associationconstants in a straightforward manner (Krell et al., 2007; Tochtrop et al.,2002). The cooperative system will be defined in terms of microscopic andinteraction constants mathematically related according to a specific model.

4. Independent or Cooperative Binding?

In the analysis of systems with two or more binding sites, one of themost important questions is to assess whether the sites are independent ofeach other or whether cooperative interactions affect the ligand affinityof different sites. A qualitative analysis can be performed in a straightforwardway once the overall association constants have been determined. Fora macromolecule with n binding sites, a set of n-1 parameters, ri (i ¼ 2,. . . , n), can be calculated from the macroscopic association constantsdetermined experimentally (Wyman and Gill, 1990; Wyman andPhillipson, 1974):


ri ¼

bin

i

� �

bi�1

n

i� 1

� �0BB@

1CCA

i

i� 1

; ð5:11Þ

with i¼ 2, . . . , n. A r value of 1 indicates that the binding sites are identicaland independent, because in such situation:

P ¼Xni¼0

bi½L�i ¼ ð1þ k½L�Þn ) bi ¼ n

i

� �ki; ð5:12Þ

and every parameter ri is equal to 1. Any deviation from 1 in the rparameters indicates that the binding sites are not identical or that theybehave cooperatively: (1) r values less than 1 indicate that not all thebinding sites are identical or that they may exhibit negative cooperativity;and (2) r values greater than 1 indicate that the binding exhibit positivecooperativity. Thus, the binding polynomials provide a way to characterizea binding reaction in a similar way to that employed in differential scanningcalorimetry in which the van’t Hoff-calorimetric enthalpy ratio is used foridentifying the two-state or non-two-state character of the reaction.

5. Analysis of ITC Data Using Binding

Polynomials

The binding polynomial (Eqs. (5.5) and (5.6)) provides the startingpoint in the analysis of ITC data. As in the standard analysis, the totalconcentration of ligand is written as the sum of the concentrations of freeand bound ligand, and expressed in terms of the binding polynomial:

½L�T ¼ ½L� þ ½L�B ¼ ½L� þ ½M�TnLB ¼ ½L� þ ½M�T@ ln P

@ ln½L� : ð5:13Þ

Eq. (5.13) is the basis for the analysis of the binding experiment; knowingthe total concentrations of ligand and macromolecule, the values of themacroscopic association constants (bi) will determine the free ligand


concentration and the concentration of each complex. The values of theassociation constants are obtained through nonlinear least squares regressionanalysis of the experimental binding data.

For any given system, the general analysis procedure follows the samelines used for specific models and consists of the following steps: (1) definethe number of binding sites and corresponding binding polynomial; (2)calculate the total concentration of macromolecule and ligand for eachexperimental point; (3) solve (analytically or numerically) the ligand con-servation equation for each experimental point, assuming certain values ofthe macroscopic association constants; (4) calculate the concentration of thedifferent complexes for each experimental point, assuming certain valuesfor the association constants; (5) calculate the expected signal, assumingcertain values for the binding enthalpies, which are also floating parametersin the nonlinear least squares analysis; and (6) obtain the optimal set ofmacroscopic association constants that reproduce the experimental datausing an iterative method.

In ITC, the total concentrations of ligand and macromolecule in thecalorimetric cell after the injection k are given by:

½L�T;k ¼ ½L�0 1� 1� v

V

!k0@

1A

½M�T;k ¼ ½M�0 1� v

V

!k; ð5:14Þ

where [M]0 and [L]0 are the initial macromolecule concentration in the celland the concentration of ligand in the syringe, respectively, and V and v arethe cell volume and the injection volume, respectively. The average excessmolar enthalpy of the system can be calculated as previously mentioned, andthe total accumulated heat until injection k is given by:

Qk ¼ V½M�T;khDHik ¼ VXni¼1

DHi½MLi�k: ð5:15Þ

Then, the heat effect associated to the injection k, qk, is calculated from thedifference between the total heats after injection k and k � 1, that is, it isproportional to the change in the concentration of each macromolecule-ligand complex between injection k and k � 1:


qk ¼ Qk �Qk�1 1� v

V

� �¼ V ½M�T;khDHik � ½M�T;k�1hDHik�1 1� v

V

� �� ¼ V

Xni¼1

DHi ½MLi�k � ½MLi�k�1 1� v

V

� �� ; ð5:16Þ

where DHi is the enthalpy of formation of complex [MLi], and the concen-tration of each type of complex is calculated according to the fractioncorresponding to each species (Eq. (5.8)). The values for the macroscopicassociation constants (bi) and the binding enthalpies (DHi) are obtainedthrough nonlinear least squares regression analysis of the experimentalbinding data (qk).

6. A Typical Case: Macromolecule with Two

Ligand-Binding Sites

The binding polynomial for a macromolecule with two ligand bindingsites is equal to (see Fig. 5.1):

P ¼ 1þ b1½L� þ b2½L�2: ð5:17Þ

The average number of ligand molecules bound per macromolecule, nLB,and the average excess molar enthalpy, <DH>, are written in terms of themacroscopic association constants and binding enthalpies as follows:

kk2[L]2

2k[L]

1

Identicalcooperative

Nonidenticalindependent

Identicalindependent

Generalmodel

k1k2[L]2k2[L]2b2[L]2

k1[L]+k2[L]2k[L]b1[L]

111

Figure 5.1 Scheme for computing the binding polynomial for a macromolecule withtwo binding sites. The different liganded states are shown with their statistical factor orrelative concentration, taking the free macromolecule as the reference state. The sum ofthe different terms in each column provides the expression of the binding polynomialfor each model.


nLB ¼ b1½L� þ 2b2½L�21þ b1½L� þ b2½L�2

¼ F1 þ 2F2

hDHi ¼ b1DH1 þ b2½L�2DH2

1þ b1½L� þ b2½L�2¼ F1DH1 þ F2DH2

: ð5:18Þ

These expressions are completely general for any macromolecule with twoligand-binding sites. Even though the number of binding sites can be con-sidered a fitting variable, it is not recommended unless absolutely necessary.If it is done, it is necessary to perform a statistical F-test to determinewhether the improvement obtained by increasing the number of fittingparameters (two per binding site) actually reflects the nature of the system orthe trivial fact of increasing the number of adjustable parameters. Often, inthe analysis of ITC data the number of binding sites is considered as anadjustable parameter, yielding fractional values. In reality, the parameterbeing adjusted is the effective amount of active protein relative to thenominal value entered as protein concentration. In the analysis of ITCdata using binding polynomials, the effective protein concentration needsto be adjusted to correctly represent the number of binding sites, which is aninteger value. Once this is achieved, analysis of the ITC data providesaccurate bi’s and DHi’s values.

For a system with two binding sites there is only one cooperativeparameter r:

r2 ¼4b2b21

: ð5:19Þ

If r2 is equal to 1, the binding sites are identical and independent. If r2 is lessthan 1, the binding sites can be either independent but different or identicalbut with negative cooperativity; and if r2 is greater than 1, the binding sitesexhibit positive cooperativity. For cooperative binding the r2 parameter isequal to the cooperativity association constant k, as discussed subsequently.

If the two binding sites behave independently (i.e., if the binding to onesite does not influence the binding to the other site), the binding polynomialfactorizes into two first-order polynomials, each corresponding to a bindingsite with a microscopic binding constant ki (see Fig. 5.1):

P ¼ ð1þ k1½L�Þð1þ k2½L�Þ ¼ 1þ ðk1 þ k2Þ½L� þ k1k2½L�2: ð5:20Þ

If the two binding sites are identical (equal thermodynamic microscopicbinding parameters), then the binding polynomial simplifies to (see Fig. 5.1):


P ¼ ð1þ k½L�Þ2 ¼ 1þ 2k½L� þ k2½L�2; ð5:21Þ

where the factor 2 in the second term represents the degeneracy of the statewith one ligand bound.

If two identical binding sites show cooperativity, then the binding poly-nomial will not factorize due to the presence of the cooperativity constant,k (see Fig. 5.1):

P ¼ 1þ 2k½L� þ kk2½L�2: ð5:22Þ

The cooperativity constant reflects the energy penalty or gain due tosimultaneous ligand binding to both binding sites. A value of k greaterthan 1 means positive cooperativity (for equivalent degrees of saturation theconcentration of single liganded species is less than for independent bind-ing), whereas a value of k less than 1 means negative cooperativity (theconcentration of the single liganded species is higher than for independentbinding). Each situation would correspond to a different model; however,the binding polynomial representation written in terms of macroscopicassociation constants is the same in both cases. This example illustrates thevalue and generality of the binding polynomials.

7. Data Analysis

The equations described here have been implemented for an arbitrarynumber of binding sites in the analysis software distributed by manufacturersof isothermal titration calorimeters or can be employed as user-definedfitting functions using commercially available software. Sometimes, stepwiseassociation constants are estimated (e.g., MicroCal) rather than the overallassociation constants discussed here. In those cases, Eq. (5.2) should be usedto calculate them. Thermodynamic binding parameters (bi’s and DHi’s) areobtained through nonlinear least squares regression.

Computer-simulated calorimetric titrations covering different represen-tative situations for a macromolecule with two ligand-binding sites areshown in Fig. 5.2. Nonlinear least squares analysis of these titrations interms of binding polynomials yields the following results:

(A) b1 ¼ 1.9�107 M�1, DH1 ¼10.0 kcal/mol, b2 ¼ 9.5�1013 M�2, andDH2 ¼ 20.0 kcal/mol.

(B) b1 ¼ 1.0�107 M�1, DH1 ¼10.0 kcal/mol, b2 ¼ 1.0�1012 M�2, andDH2 ¼ 15.0 kcal/mol.

Time (min)

dQ/d

t (m

cal/s

)Q

(kc

al/m

ol o

f in

ject

ant)

0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

2.0

4.0

6.0

8.0

10.0

0.0

2.0

4.0

6.0

8.0

10.0

0.0

0.5

1.0

1.5

0.0

0.5

1.0

1.5

Time (min)A

C D

B

0 30 60 90 120 150 0 30 60 90 120 150

dQ/d

t (m

cal/s

)Q

(kc

al/m

ol o

f in

ject

ant)

0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0.0

0.5

1.0

1.5

2.0

dQ/d

t (m

cal/s

)

[L]T/[M]T

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

[L]T/[M]T

Q (

kcal

/mol

of

inje

ctan

t)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0.0

0.5

1.0

1.5

2.0

dQ/d

t (m

cal/s

)Q

(kc

al/m

ol o

f in

ject

ant)

0 30 60 90 120 150 0 30 60 90 120 150

Figure 5.2 Simulated titrations for a macromolecule with two binding sites. Fourrepresentative cases are illustrated, covering the different possibilities.



(C) b1 ¼ 1.9�107 M�1, DH1 ¼10.0 kcal/mol, b2 ¼ 9.60�1014 M�2, andDH2 ¼ 25.0 kcal/mol.

(D) b1 ¼ 2.0�107 M�1, DH1 ¼10.0 kcal/mol, b2 ¼ 10.0�1012 M�2, andDH2 ¼ 20.0 kcal/mol.

Accordingly, ther2 values are 1.0 for (A), 0.04 for (B), 10.3 for (C), and 0.1for (D). Therefore, the first case corresponds to two identical and independentbinding sites, the cases (B) and (D) correspond to either two different inde-pendent binding sites, or two identical sites with negative cooperativity; and,case (C) corresponds to two binding sites with positive cooperativity.

In order to inspect the system behavior during the titration, the popula-tions of the different complexes are shown as a function of the bindingsaturation for each model (Fig. 5.3). Several characteristic can be highlighted:(1) the population of each liganded species, MLi, reaches a maximum whennLB ¼ i; (2) different independent binding sites behave similarly to identicalbinding sites with negative cooperativity; (3) different binding sites andidentical binding sites with negative cooperativity exhibit a greater concen-tration of single-ligand bound macromolecules than independent binding forequal overall binding saturation; (4) identical binding sites with positivecooperativity exhibit a lower concentration of single-ligand bound macro-molecules than independent binding for equal overall binding saturation.

0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

0.0

0.5

1.0

0.0

0.5

1.0

0.0

0.5

1.0

nLB

Fi =

[M

Li]/

[M] T

A

B

C

D

Figure 5.3 Fraction or population of each liganded species along the titrations shownin Fig. 5.2. The populations of free (solid), single-ligand bound (dashed), and double-ligand bound (dotted) macromolecules are calculated according to Eq. (5.8).


The binding polynomial P and the saturation function nLB for thetitrations in Fig. 5.2 are shown in Fig. 5.4 as a function of the free ligandconcentration. If the case for identical and independent sites is used asreference, a lower numerical value of the binding polynomial for differentbinding sites and negative cooperativity is observed. The opposite is

1E-8 1E-7 1E-6 1E-5 1E-4

100

101

102

103

104

105

106

P

[L] (M)

A

B

1E-8 1E-7 1E-6 1E-5 1E-4[L] (M)

0.0

0.5

1.0

1.5

2.0

n LB

Figure 5.4 (A) Binding polynomial as a function of the free ligand concentration foreach model along the titrations shown in Figs. 5.2–5.3: identical and independentbinding sites (solid), nonidentical and independent binding sites (dashed), identicalbinding sites with positive cooperativity (short dashed), identical binding sites withnegative cooperativity (dotted). (B) Number of ligand molecules bound per macro-molecule as a function of the free ligand concentration along the titrations shown inFigs. 5.2–5.3: identical and independent binding sites (solid), nonidentical and inde-pendent binding sites (dashed), identical binding sites with positive cooperativity (shortdashed), identical binding sites with negative cooperativity (dotted).


observed for the model with identical binding sites with positive coopera-tivity. This is an important conclusion because the binding polynomial canalways be calculated by numerical integration of nLB (see Eq. (5.7)) andcompared to the one expected for independent binding without the needfor fitting the data.

The binding saturation parameter nLB can be obtained by applyingEq. (5.7). In the case of different binding sites and negative cooperativity,two inflection points may be observed, indicating two binding events withsignificantly different binding affinity. If the two binding sites differ by lessthan a factor of ten in affinity, only one inflection point would be observedbut with a broader transition toward saturation.

The discussion can be extended by considering the binding capacity(@nLB/@ln[L]) and the slope of the Hill plot (log (nLB/(2 � nLB)) versus log[L]) at half saturation (nLB ¼ 1) (Di Cera et al., 1988; Hill, 1910; Schellman,1990; Wyman, 1964; Wyman and Gill, 1990; see also the appendix). Thesetwo parameters take a value of 0.5 and 1, respectively, if the ligand-bindingsites are identical and independent. The calculated values of these twoparameters for the titrations shown in Fig. 5.2 are (A) 0.5 and 1; (B) 0.17and 0.34; (C) 0.75 and 1.5; and (D) 0.24 and 0.48. The relative deviations ofthese two parameters from the values corresponding to the reference case(identical and independent binding sites) are (A) 0%, (B) �66%, (C) þ50%,and (D) �52%. If the relative change in the fractional population of theintermediate complexML, is calculated at half saturation (Fig. 5.3), the samevalues are obtained: (A) 0%, (B) �66%, (C) þ50%, and (D) �52%. Asdemonstrated in the appendix, the experimentally accessible Hill slope orbinding capacity can be used to estimate the population of intermediateliganded species and also the cooperative energy.

8. Data Interpretation

As discussed earlier, the precise binding mechanism for systems with twoor more binding sites is difficult to derive and usually requires extrathermody-namic information. Statistical fitting of the data to amodel does not validate theappropriateness of the model. Even for hemoglobin, the most widely studiedbinding system in history, there are still lingering questions about the exactoxygen-binding mechanism (Holt and Ackers, 2005; Ackers and Holt, 2006).

Even a macromolecule with two binding sites presents at least sixdifferent possible binding mechanisms: (1) two identical and independentbinding sites, (2) two identical and negatively cooperative binding sites,(3) two identical and positively cooperative binding sites, (4) two noniden-tical and independent binding sites, (5) two different and negatively coop-erative binding sites, and (6) two different and positively cooperative


binding sites. Not all these cases are distinguishable experimentally (i.e.,some give rise to exactly the same binding curve and extrathermodynamicinformation is required to elucidate the binding mechanism). For example,a macromolecule with two different binding sites exhibiting positivecooperative might resemble a macromolecule with two identical and inde-pendent binding sites, because both features will have compensating effects.Also, as mentioned earlier, a macromolecule with two different and inde-pendent binding sites is mathematically equivalent to a macromolecule withtwo identical and cooperative binding sites with negative cooperativity.This result can be demonstrated by considering Eqs. (20)–(22), andobtaining the mathematical relationship between k1, k2 , k, and k:

k1 ¼ kð1þ ffiffiffiffiffiffiffiffiffiffiffi1� k

p Þk2 ¼ kð1� ffiffiffiffiffiffiffiffiffiffiffi

1� kp Þ : ð5:23Þ

If k ¼ 1, the system is noncooperative and corresponds to a macromoleculewith two identical binding sites. If k < 1, the cooperative system isequivalent to a macromolecule with two different and independent bindingsites. If k> 1, the cooperative system cannot be equated with a system withdifferent sites. Consequently, the distinction between a macromoleculewith different and independent binding sites and a macromolecule withidentical and negatively cooperative binding sites cannot be made frombinding data. Different binding models for a macromolecule with twobinding sites are briefly reviewed in the following subsections.

8.1. Independent ligand binding: Two identical binding sites

This case represents the reference model, and the other models may beconsidered deviations from independency. The binding polynomial for amacromolecule with two identical and independent binding sites is given byEq. (5.21), from which the average number of ligand molecules bound permacromolecule is:

nLB ¼ 2k½L� þ 2k2½L�21þ 2k½L� þ k2½L�2 ¼ F1 þ 2F2; ð5:24Þ

and the average excess molar enthalpy of the system is:

hDHi ¼ 2k½L�Dhþ 2k2½L�2Dh1þ 2k½L� þ k2½L�2 ¼ F1Dhþ F22Dh; ð5:25Þ


where Dh is the binding enthalpy to any of the two binding sites. Theconcentration of any macromolecular state is calculated as follows:

½ML� ¼ ½M�T2k½L�

1þ 2k½L� þ k2½L�2 ¼ ½M�TF1

½ML2� ¼ ½M�Tk2½L�2

1þ 2k½L� þ k2½L�2 ¼ ½M�TF2: ð5:26Þ

8.2. Independent ligand binding: Two nonidenticalbinding sites

The binding polynomial for a macromolecule with two different andindependent binding sites is given by Eq. (5.20), from which the averagenumber of ligand molecules bound per macromolecule is:

nLB ¼ ðk1 þ k2Þ½L� þ 2k1k2½L�21þ ðk1 þ k2Þ½L� þ k1k2½L�2

¼ F1 þ F2 þ 2F12; ð5:27Þ

where F1 and F2 are the populations of macromolecule with only one ligandbound in either binding site (ML and LM), and F12 is the population ofmacromolecule with two ligands bound (LML). The average excess molarenthalpy of the system is:

hDHi ¼ k1½L�Dh1 þ k2½L�Dh2 þ k1k2½L�2ðDh1 þ Dh2Þ1þ ðk1 þ k2Þ½L� þ k1k2½L�2

¼ F1Dh1 þ F2Dh2 þ F12ðDh1 þ Dh2Þ; ð5:28Þ

where Dh1 and Dh2 are the binding enthalpies for the two binding sites. Theconcentration of any complex is calculated as follows:

½ML� ¼ ½M�Tk1½L�

1þ ðk1 þ k2Þ½L� þ k1k2½L�2¼ ½M�TF1

½LM� ¼ ½M�Tk2½L�


½LML� ¼ ½M�Tk1k2½L�2


: ð5:29Þ


8.3. Cooperative ligand binding: Two identical binding sites

The binding polynomial for a macromolecule with two identical andindependent binding sites is given by Eq. (5.22), from which the averagenumber of ligand molecules bound per macromolecule is:

nLB ¼ 2k½L� þ 2kk2½L�21þ 2k½L� þ kk2½L�2 ¼ F1 þ 2F2; ð5:30Þ

and the average excess molar enthalpy of the system is:

hDHi ¼ 2k½L�Dhþ kk2½L�2ð2Dhþ D�Þ1þ 2k½L� þ kk½L�2

¼ F1Dhþ F2ð2Dhþ D�Þ;ð5:31Þ

where Dh and D� are the binding enthalpy for any of the two binding sitesand the cooperativity enthalpy, respectively. The concentration of anycomplex is calculated as follows:

½ML� ¼ ½M�T2k½L�

1þ 2k½L� þ kk2½L�2 ¼ ½M�TF1

½LML� ¼ ½M�Tkk2½L�2

1þ 2k½L� þ kk2½L�2 ¼ ½M�TF2: ð5:32Þ

The comparison of the binding polynomial and the average excess molarenthalpy written in terms of the overall and the microscopic parametersprovides links between the overall and the microscopic parameters. In thecase of a macromolecule with two identical and independent binding sites:

b1 ¼ 2k

b2 ¼ k2

DH1 ¼ DhDH2 ¼ 2Dh

; ð5:33Þ

and the microscopic parameters can be calculated from the overallparameters:


k ¼ b12¼

ffiffiffiffiffib2

pDh ¼ DH1 ¼ DH2

2

: ð5:34Þ

In the case of a macromolecule with two nonidentical and independentbinding sites:

b1 ¼ k1 þ k2b2 ¼ k1k2

DH1 ¼ k1Dh1 þ k2Dh2k1 þ k2

DH2 ¼ Dh1 þ Dh2

; ð5:35Þ

and the microscopic parameters can be calculated from the overall parameters:

k1 ¼ b12

1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4b2

b21

s0@

1A

k2 ¼ b12

1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4b2

b21

s0@

1A

Dh1 ¼ ðk1 þ k2ÞDH1 � k2DH2

k1 � k2

Dh2 ¼ k1DH2 � ðk1 þ k2ÞDH1

k1 � k2

: ð5:36Þ

Finally, in the case of a macromolecule with two identical and dependentbinding sites:

b1 ¼ 2k

b2 ¼ kk2

DH1 ¼ DhDH2 ¼ 2Dhþ D�

; ð5:37Þ

and the microscopic parameters can be calculated from the overall parameters:


k ¼ b12

k ¼ 4b2b21

Dh ¼ DH1

D� ¼ DH2 � 2DH1

: ð5:38Þ

The preceding equations allow for estimation of the microscopic thermo-dynamic parameters from the macroscopic thermodynamic parameters oncea specific model has been validated. Because model validation is sometimesdifficult, it is advisable to publish macroscopic association constants (b’s) thathave a universal validity.

9. An Experimental Example

The binding of ferric ions to ovotransferrin is an example of a reactionthat involves two sites. Fig. 5.5 shows a microcalorimetric titration ofovotransferrin with ferric ions, chelated with nitrilotriacetate, at 25 �C,100 mM HEPES, pH 7.5. Nonlinear least squares analysis of the experi-mental data using the binding polynomials formalism yields: b1 ¼1.7�106 M�1, DH1 ¼ �7.4 kcal/mol, b2 ¼ 1.1�1012 M�2 and DH2 ¼�12.2 kcal/mol. The calculated value for r2 is 1.5, suggesting positivecooperativity between the two sites, in agreement with earlier observations(Taniguchi et al., 1990). The enthalpy changes are also in good agreementwith a previous calorimetric study (Lin et al., 1991). A cooperative indexr2 of 1.5 is not very high and consistent with a cooperative energy of only�0.2 kcal/mol, and a degree of cooperativity of 10% (that is, the maximalpopulation of the single liganded state ML is 45%, and the Hill slope at halfsaturation is 1.1). In fact, the data can also be fitted to a model of twodifferent and independent sites, albeit with a slightly larger sum of squaredresiduals. Lin and coworkers (Lin et al., 1991) also noticed that the model oftwo different independent sites could not produce a good fit unless thenumber of sites was allowed to vary independently of each other. On theother hand, because the two binding sites are different, compensatory effectsare possible (i.e., the positive cooperativity might be higher than theobserved as a result of the masking effect of two sites with different intrinsicaffinities).

The results with ovotransferrin help to illustrate a frequently encounteredsituation. It is common for researchers to use a model with two different andindependent binding sites as a starting point in the analysis of a system with

0 1 2 3 4−8.0

−6.0

−4.0

−2.0

0.0

−1.5

−1.0

−0.5

0.0

0.5

0 30 60 90 120 150

Time (min)

dQ/d

t (m

cal/s

)

[Fe+3-NTA]T/[OT]T

Q (kc

al/m

ol o

f in

ject

ant)

Figure 5.5 Binding of ferric ions, chelated with nitrilotriacetate, to ovotransferrinmeasured by isothermal titration calorimetry at 25 �C using a high-precision VP-ITCtitration calorimetric system from MicroCal, LLC (Northampton, MA). The upperpanel shows the raw data for the titration of �1.4 mL of 25 mM ovotransferrin with1.03 mM ferric ions in steps of 5 mL (100 mM HEPES, pH 7.5). The lower panel showsthe integrated heats per mole of Fe3þ after subtraction of the heats of dilution (filledcircles). The values for the affinities and enthalpy changes according to a bindingpolynomial for a molecule with two sites are obtained from a non-linear regression ofthe data (solid line).


two binding sites. This practice is dangerous because a model of different andindependent binding sites cannot account for positive cooperativity. Werecommend a fit with a binding polynomial as a starting point.

10. Experimental Situations from the

Literature

Other examples of ITC studies of macromolecules with two ligand-binding sites can be found in the literature. Several illustrative casescorresponding to different scenarios have been selected and are shown inFig. 5.6. The published experimental titration data were digitally extractedand analyzed following the procedure outlined in this work.

0 1 2 3 4

0 1 2 3 4 5

−10

−5

0

A

C D

B

Q (

kcal

/mol

of

inje

ctan

t)

0 1 2 3 4

−10

−5

0

Q (

kcal

/mol

of

inje

ctan

t)

[a-methyl fucoside]T/[RSL]T [Fe(III)-NTA]T/[HT]T

0

2

4

6

Q (

kcal

/mol

of

inje

ctan

t)

[cAMP]T/[cAMP RP]T

−4

0

4Q

(kc

al/m

ol o

f in

ject

ant)

[TBPan]T/[d(T4G4T4G4)]T

0 1 2 3 4

Figure 5.6 Experimental calorimetric titrations with macromolecules with twoligand-binding sites: (A) a–methyl fucoside binding to fucose-binding lectin, RSL,from Ralstonia solanacearum (data extracted from Kostlanova et al., 2005); (B)nitrilotriacetate-chelated ferric ion binding to human transferrin, HT (data extractedfrom Lin et al., 1993); (C) cAMP binding to cAMP receptor protein, cAMP RP, fromEscherichia coli (data extracted from Gorshkova et al., 1995); (D) telomere binding alphaprotein n-terminal domain, TBPan, from Oxytricha nova binding to a single-strandtelomere fragment (data extracted from Buczek and Horvath, 2006). The values forthe affinities and enthalpy changes according to a binding polynomial for a moleculewith two sites are obtained from a nonlinear regression of the data (solid line), and r2 iscalculated from bi values.


In Fig. 5.6A, RSL is a fucose-binding lectin from Ralstonia solanacearum,a gram-negative b-protobacterium causing lethal wilt in plants. RSL is atrimer, each monomer containing two carbohydrate-binding sites(Kostlanova et al., 2005). Nonlinear least squares analysis of the experimen-tal data using the binding polynomials formalism yields: b1 ¼ 3.0�106 M�1,DH1 ¼ �10.1 kcal/mol, b2 ¼ 2.4�1012 M�2 and DH2 ¼ �19.1 kcal/mol.Although the binding sites in each subunit are slightly different, they canbe considered identical and independent for a–methyl fucoside binding(r2 ¼ 1.1). Data analysis with any model different from the identicaland independent binding sites results in overparameterization and highparameter dependency.


In Fig. 5.6B, human transferrin, HT, is structurally very similar toovotransferrin: two different structural domains, each one containing aniron-binding site. Nitrilotriacetate-chelated ferric ion binding to humantransferrin exhibits the same features as ovotransferrin (Lin et al., 1993).Nonlinear least squares analysis of the experimental data using the bindingpolynomials formalism yields: b1 ¼ 1.0�107 M�1, DH1 ¼ �9.5 kcal/mol,b2 ¼ 5.4�1013 M�2 and DH2 ¼ �15.0 kcal/mol. As in the case of ovo-transferrin, the two binding sites are slightly different and iron bindingshows low positive cooperativity (r2 ¼ 2.2). This cooperative effect corre-sponds to a cooperative Gibbs energy of �0.5 kcal/mol, and a degree ofcooperativity of 17% (i.e., the maximal population of the single ligandedstate ML is 41.5%, and the Hill slope at half saturation is 1.17). Analysisusing a model of two different and independent sites is also possible but witha poorer fit as a result.

Fig. 5.6C shows cAMP receptor protein, cAMP RP, a homodimericprotein that binds DNA after undergoing a conformational change inducedby cAMP binding. Each identical subunit presents a cAMP binding domainand a DNAbinding domain (Gorshkova et al., 1995). Nonlinear least squaresanalysis of the experimental data using the binding polynomials formalismyields: b1 ¼ 5.5�104 M�1, DH1 ¼ �1.9 kcal/mol, b2 ¼ 4.2�109 M�2 andDH2 ¼ 9.9 kcal/mol. cAMP binding to cAMP receptor protein showspositive cooperativity (r2 ¼ 5.4). This cooperative effect corresponds to acooperative Gibbs energy of �1 kcal/mol, and a degree of cooperativity of40% (i.e., the maximal population of the single liganded stateML is 30%, andthe Hill slope at half saturation is 1.4). If the analysis is performed using themodel with two nonidentical and independent binding sites, the result is apoorer fit with fractional stoichiometries. In this case, however, there is noreason for using such a model as the protein is known to be a homodimerwith two binding sites that are identical in the absence of ligand.

Fig. 5.6D shows how telomere-binding alpha protein n-terminaldomain, TBPan, from the ciliate Oxytricha nova binds to single-strandDNA repeats at telomeres (Buczek and Horvath, 2006). Nonlinear leastsquares analysis of the experimental data using the binding polynomialsformalism yields: b1 ¼ 2.5�107 M�1, DH1 ¼ 3.4 kcal/mol, b2 ¼3.3�1012 M�2 and DH2 ¼ �2.5 kcal/mol. TBPan binding to a single-strand telomere fragment d(T4G4T4G4) is consistent with either two bind-ing sites in the fragment with negative cooperativity or two binding sitesnonidentical and independent (r2 ¼ 0.022). If the cooperative model isassumed, the cooperative effect corresponds to a cooperative Gibbs energyof þ2.1 kcal/mol, and a degree of cooperativity of�74% (i.e., the maximalpopulation of the single liganded state ML is 87%, and the Hill slope at halfsaturation is 0.26). If the model with two nonidentical and independentbinding sites is assumed, the affinities of the two binding sites differ by afactor of 200, approximately. Given that the DNA fragment is not


completely symmetric (end effects), the model with nonidentical and inde-pendent binding sites would be preferred. However, given the small size ofthe DNA fragment, steric or other unfavorable interactions responsible forthe negative cooperativity may arise when two proteins are bound to thesame oligonucleotide.

11. Macromolecule with Three Ligand-Binding

Sites

As an additional example, three different possible models for a macro-molecule with three ligand-binding sites are illustrated in Fig. 5.7. In thiscase, a set of two parameters, r2 and r3 (see Eq. (5.11)), provides informa-tion about the behavior of the binding sites in the macromolecule.

The data analysis based on the binding polynomial formalism usingoverall association constants is fairly simple and straightforward. It requiressolving a (nþ 1)th-order polynomial equation on the free ligand concentra-tion, which can be easily done numerically (e.g., Newton-Raphson, secant,or bisection methods) for n � 2 using commercially available software.

12. Conclusions

The binding polynomial provides a general framework for describingligand-binding equilibria to biological macromolecules using tools fromstatistical thermodynamics. The methodology can be easily applied to nearly

3k1k2[L]2k1k2[L]2+k1k3[L]2

+ k2k3[L]23k2[L]2b2[L]2

k13k2k3[L]3

3k[L]

1

Identicalcooperative

Nonidenticalindependent

Identicalindependent

Generalmodel

k1k2k3[L]3k3[L]3b3[L]3

k1[L]+k2[L]+k3[L]3k[L]b1[L]

111

Figure 5.7 Scheme for computing the binding polynomial for a macromolecule withthree binding sites. The different liganded states are shown with their statistical factor orrelative concentration taking the free macromolecule as the reference state. The sum ofthe different terms in each column provides the expression of the binding polynomialfor each model.


any kind of system. Binding experiments can be analyzed using a model-freemethodology, which allows determination of phenomenological overallmacroscopic association constants, bi, and binding enthalpies, DHi. Theparticular values of a set of parameters ri, provide information on the ligandbinding process: identical or nonidentical, independent or cooperativebinding sites. Once a binding model is developed, the microscopic bindingparameters, ki and Dhi, and their relationships with the macroscopic bindingparameters can be employed to describe in detail the ligand binding to themacromolecule. These relationships can be derived for more complicatedcases using the procedures outlined in this chapter.

Binding cooperativity in a macromolecule with several binding sitesemerges as a result of interactions between ligand binding sites and/or aconformational equilibrium modulated by ligand binding between confor-mations with different ligand-binding affinities (Wyman and Gill, 1990).Cooperativity is reflected in a dependency of the thermodynamic bindingparameters for each binding site on the occupancy of the other binding sites.Homotropic interactions occur if the binding sites bind the same type ofligand, whereas heterotropic interactions occur if the binding sites binddifferent types of ligand (Velazquez-Campoy et al., 2006). This chapter hasaddressed the description and analysis of homotropic interactions only. Thedescription and analysis of heterotropic interactions is rather more complex,even for simple systems (Velazquez-Campoy et al., 2006).

Appendix

For a protein with two binding sites the number of ligand moleculesbound per macromolecule is given by:

nLB ¼ b1½L� þ 2b2½L�21þ b1½L� þ b2½L�2

; ð5:39Þ

and the concentration of ligand required for achieving half saturation isgiven by:

½L�nLB¼1 ¼1ffiffiffiffiffib2

p : ð5:40Þ

The fractional population of the complex ML is given by:

½ML�½M�T

¼ b1½L�1þ b1½L� þ b2½L�2

; ð5:41Þ


which at half saturation takes the value:

½ML�½M�T

jnLB¼1 ¼b1ffiffiffiffib2

p

2þ b1ffiffiffiffib2

p : ð5:42Þ

Traditionally, two indexes have been used for expressing numerically thedegree of cooperativity: the binding capacity (@nLB/@ln[L])) (Di Cera et al.,1988; Schellman, 1990; Wyman, 1964; Wyman and Gill, 1990) and the Hillslope, nH, which is the slope of the Hill plot (log (nLB/(2 � nLB)) vs. log [L])(Hill, 1910; Schellman, 1990; Wyman, 1964; Wyman and Gill, 1990). Thebinding capacity is a measure of the ability of the macromolecule for acceptingor delivering large quantities of ligand for relatively small changes in ligandconcentration, and it is equal to the fluctuation (<i2>�<i>2) or variance inthe number of ligand molecules bound to the macromolecule (Di Cera et al.,1988; Schellman, 1990; Wyman, 1964; Wyman and Gill, 1990). The param-eter nH is equal to the ratio between the observed binding capacity and the onecorresponding to identical and independent binding sites (Schellman, 1990;Wyman, 1964;Wyman and Gill, 1990). Both parameters represent a measureof the efficiency of the biochemical signal transduction (response of the systemto changes in ligand concentration). Because the binding capacity and the Hillslope are functions of the ligand concentration, they are usually reported asvalues at half saturation.

For a macromolecule with two ligand-binding sites, the binding capacityis given by:

@nLB@ ln½L� ¼

b1½L� þ 4b2½L�2 þ b1b2½L�3ð1þ b1½L� þ b2½L�2Þ2

; ð5:43Þ

which takes values between 0 (maximal negative cooperativity) and 1(maximal positive cooperativity), and at half saturation takes the value(Wyman, 1967):

@nLB@ ln½L�

��nLB¼1

¼ 2


p : ð5:44Þ

The binding capacity is related to the Hill slope (Schellman, 1990; Wyman,1964; Wyman and Gill, 1990):

@nLB@ ln½L� ¼ nHnLB 1� nLB

2

� �: ð5:45Þ


The Hill slope takes values between 0 (maximal negative cooperativity) and2 (maximal positive cooperativity), and at half saturation it will be twice asbig as the binding capacity:

@nLB@ ln½L�

��nLB¼1

¼ 1

2nH

��nLB¼1

: ð5:46Þ

The binding capacity and the slope of the Hill plot at half saturation givequalitative and quantitative information on the ligand-binding process: (1) ifthe binding sites are identical and independent, these two parameters havevalues of 0.5 and 1, respectively; (2) if the binding sites are nonidentical andindependent, or show negative cooperativity, they have values less than 0.5and 1, respectively; and (3) if the binding sites show positive cooperativity,they have values greater than 0.5 and 1, respectively. In addition, therelative deviation of these two parameters from the values correspondingto identical and independent sites coincides with the relative deviation ofthe population of intermediate liganded states, as shown subsequently.

The relative change in the fractional population of complex ML at halfsaturation taking as a reference the case with identical and independentbinding sites is given by:

½ML�½M�T

��nLB¼1;identþindep

� ½ML�½M�T

��nLB¼1

½ML�½M�T


¼2� b1ffiffiffiffi

b2p


p ¼ffiffiffik

p � 1ffiffiffik

p þ 1; ð5:47Þ

where we have used the identity k ¼ 4b2/b12. On the other hand, therelative change in binding capacity (or the Hill slope, as they are propor-tional) of complex ML at half saturation taking as a reference the case withidentical and independent binding sites is also:

@nLB@ ln½L�

��nLB¼1

� @nLB@ ln½L�


@nLB@ ln½L�


¼nH

��nLB¼1

� nH


nH


¼2� b1ffiffiffiffiffi

b2p

2þ b1ffiffiffiffiffib2

p ¼ffiffiffik

p � 1ffiffiffik

p þ 1:

ð5:48Þ

Therefore, the relative change of the binding capacity or the Hill slope at halfsaturation, taking the values of 0.5 and 1.0 as reference values (for the case of


identical and independent binding sites), is equal to the relative change in thefractional population of the intermediate liganded state ML at half saturation,taking a fractional population of 50% as a reference value (for the case ofidentical and independent binding sites). Thus, the connection betweencooperativity and changes in the population of intermediate states is quanti-tative (Eqs. (5.44)–(5.48)): positive cooperativity (k> 1) causes a reduction inthe fractional population of ML, whereas negative cooperativity (k < 1)causes an increase in the fractional population of ML, from a 50% populationat half saturation for identical and independent binding sites. For example, if amacromolecule with two ligand-binding sites exhibits a Hill slope of 1.35 athalf saturation, then, the binding sites present 35% cooperativity, and themaximal population of the intermediate liganded state ML is 32.5% (0.65 �50%) at half saturation. Furthermore, the cooperative association constant is k¼ 4.3, and the corresponding cooperative Gibbs energy is �0.9 kcal/mol.

ACKNOWLEDGMENT

We acknowledge financial support from grant SAF2004-07722 (Ministry of Education andScience) to A.V.-C., and grants from the National Institutes of Health (GM56550 andGM57144) and the National Science Foundation (MCB0641252) to E.F. A.V.-C. wassupported by a Ramon y Cajal Research Contract from the Spanish Ministry of Scienceand Technology, and Fundacion Aragon IþD (Diputacion General de Aragon).

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