a 1774 (2007) 985–994www.elsevier.com/locate/bbapap

Biochimica et Biophysica Act

Thermodynamic and kinetic characterization of the association oftriosephosphate isomerase: The role of diffusion

Hugo Nájera a,⁎, Leonardo Dagdug b, D. Alejandro Fernández-Velasco c,⁎

a Área Académica de Nutrición, Instituto de Ciencias de la Salud, Universidad Autónoma del Estado de Hidalgo, Pachuca, Hidalgo, Méxicob Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, México DF

c Laboratorio de Fisicoquímica e Ingeniería de Proteínas, Departamento de Bioquímica, Facultad de Medicina,Universidad Nacional Autónoma de México, México DF

Received 28 January 2007; received in revised form 26 May 2007; accepted 13 June 2007Available online 19 June 2007

Abstract

It is known that diffusion plays a central role in the folding of small monomeric proteins and in the rigid-body association of proteins,however, the role of diffusion in the association of the folding intermediates of oligomeric proteins has been scarcely explored. In this work,catalytic activity and fluorescence measurements were used to study the effect of viscosity in the unfolding and refolding of the homodimericenzyme triosephosphate isomerase from Saccharamyces cerevisiae. Two transitions were found by equilibrium and kinetic experiments,suggesting a three-state model with a monomeric intermediate. Glycerol barely affects ΔG0

fold whereas ΔG0assoc becomes more favourable in the

presence of the cosolvent. From 0 to 60% (v/v) glycerol, the association rate constant showed a near unitary dependence on solvent viscosity.However, at higher glycerol concentrations deviations from Kramers theory were observed. The dissociation rate constant showed a viscosityeffect much higher than one. This may be related to secondary effects such as short-range glycerol-induced repulsion between monomers.Nevertheless, after comparison under isostability conditions, a slope near one was also observed for the dissociation rate. These results stronglysuggest that the bimolecular association producing the native dimer is limited by diffusional events of the polypeptide chains through thesolvent.© 2007 Elsevier B.V. All rights reserved.

Keywords: Dimer; Diffusion; Glycerol; Protein stability; TIM barrel

1. Introduction

The acquisition of the tertiary and quaternary structure ofoligomeric proteins is characterized by interactions formedbetween residues that are far apart in the same chain and fromresidues involving different chains. During folding, thedevelopment of intramolecular and intermolecular interactionsmay be limited by diffusional motions of the polypeptide chain(s) through the solvent or by the internal friction of the chain(s).

Abbreviations: TIM, Triosephosphate isomerase; yTIM, TIM from Sac-charomyces cerevisiae; GdmCl, Guanidinium chloride; SCM, Spectral Centre ofMass⁎ Corresponding authors. H. Nájera is to be contacted at fax: +52 771

7172000x5114. D.A. Fernández-Velasco, fax: +52 55 56162419.E-mail addresses: hnajera@uaeh.edu.mx (H. Nájera),

fdaniel@servidor.unam.mx (D.A. Fernández-Velasco).

1570-9639/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.bbapap.2007.06.001

According to Kramer's theory [1] and models of protein foldingsuch as the diffusion-collision model [2], in reactions limited bydiffusion rates should be inversely proportional to solventviscosity (k α 1/η). The effect of viscosity on folding rateswas first studied in monomeric enzymes that show multistaterefolding kinetics, such as ribonuclease A [3,4], octopine dehy-drogenase [5], and aspartokinase-homoserine dehydrogenase[6]. All these proteins show an unfolding or refolding rate thatdecreases linearly with η [4–6] suggesting that diffusion plays acentral role in determining the rate of domain–domaininteractions. The analysis of viscosity effects in foldingreactions is sometimes not straightforward [7], because thepolyols used to modify the viscosity of the solution alsoincrease protein stability [8–10], this may lead to an increase inreaction rates upon viscogen addition. In order to separatestability from viscosity effects, it has been proposed that foldingrates should be analyzed at points of equivalent stability, i.e. at a

986 H. Nájera et al. / Biochimica et Biophysica Acta 1774 (2007) 985–994

concentration of denaturant and viscogen that fixes the stabilityof the protein to a given value [11,12]. This so-calledisostability analysis was first used in the analysis of theunfolding of the α subunit of tryptophan synthase (αTS) [11,12]and was further applied to the study of small two-state foldingproteins, such as the cold shock protein from Bacillus subtilis(CspB) [13], protein L [14] and monomeric and dimericversions of GCN4-p2′ [15]. In these studies reaction ratesanalyzed under isostability conditions were found to decreaselinearly with η. A conceptually-similar approach has beenrecently used to show that the folding of an SH3 domain is alsoviscosity dependent [16]. The isostability analysis has beenquestioned, since viscogens and denaturants may changeprotein stability by different mechanisms [17]. It has alsobeen suggested that the osmolytes used to increase the viscosityof the solvent may slow folding by prematurely collapsing thecoil [18]. A new experimental approach was followed bySchmid and colleagues, who used changes in pressure to showthat the folding and unfolding rates of CspB are viscosity-dependent near 50 °C, where the viscogen (ethylene glycol)does not affect the stability of the protein [19].

The role of diffusion in protein–protein recognition is notclear [20,21], however, the expected dependence of associationrates on 1/η has been found in the association of cytochrome cand antibodies [22], thrombin and serpins [23], the S-peptideand the ribonuclease A [24], barnase and barstar [25], and in theassociation of the MYL arc repressor dimer [26]. On the otherhand, no viscosity-dependence was observed in the associationrate of the tetrameric protein p53 [27]. In spite of the prevalenceof obligate oligomeric proteins in nature [28], the role ofdiffusion in the association of folding intermediates has beenonly addressed in two oligomeric systems with opposite results[26,27]. In this work, we studied the effect of viscosity on thefolding pathway of the homodimeric enzyme Triosephosphateisomerase (TIM). TIM is a glycolytic enzyme, formed by twoidentical (β/α)8 barrels of around 27 kDa. The structure andcatalytic properties of Saccharomyces cerevisiae TIM (yTIM)have been explored in detail [29–34]. The reaction catalyzed byTIM is a diffusion-controlled process, limited by the rate atwhich glyceraldehydes-3-phosphate encounters, or departsfrom the active site [35]. Dimerization is required for TIMactivity, as deduced from kinetic [36–39] and equilibrium [40–42] folding studies or from monomers obtained by chemicalderivatization [43]. The catalytic activity of designed mono-meric mutants of TIM is at best 100 times lower than that ofwild-type TIM [44–46].

The urea-induced unfolding of human TIM [47] and theguanidinium chloride (GdmCl)-induced unfolding of rabbit TIM[48,49] have been described as two state processes atequilibrium. More complex reversible and irreversible schemes,including monomeric and dimeric intermediates, have also beenreported [40,42,50–52]. It is clear that quaternary interactionsare responsible for most of the stability of the native enzyme[41,42,47,48]. The unfolding/refolding of yTIM perturbed byGdmCl [41,53], urea [41] or temperature [54,55] involves amonomeric intermediate N ⇆ 2M ⇆ 2U. The intermediate ofyTIM observed in the folding/unfolding in GdmCl is an

expanded and inactive monomer, that possesses a considerableamount of secondary structure, whereas its fluorescence spectrashow that aromatic residues are partially buried from the solvent[41]. In order to investigate the role of diffusion in the foldingpathway of yTIM, in what follows, we determined the influenceof glycerol on the equilibrium and kinetic properties of yTIMunfolding/refolding.

2. Materials and methods

α-Glycerol phosphate dehydrogenase and GdmCl were from Roche,glycerol 99.5% was from Fluka, the others reagents were from Sigma ChemicalCo. Production, expression and purification of recombinant yTIM were carriedout as previously reported by Vázquez-Contreras et al. [56]. The concentrationof purified yTIM was measured using a molar absorption coefficient ε280 of26,664 M−1 cm−1 [57]. Protein solutions were concentrated by ultrafiltrationwith membranes of nitrocellulose, Amicon PM-10 (Aminco, Co).

Unless otherwise stated, renaturation and denaturation experiments werecarried out in 100 mM Triethanolamine, 10 mM EDTA, 1 mM DTT pH 7.4(Buffer A). Solutions with GdmCl and/or glycerol also contained 100 mMTriethanolamine, 10 mM EDTA, 1 mM DTT and were adjusted to pH 7.4.

Fluorescence measurements were acquired on an ISS PC1 spectrofluorom-eter (ISS Inc., Champaign, IL) the temperature of the cells (1 cm pathlength) wasmaintained at 25 °C±0.1. Unless otherwise stated, emission spectra wereobtained using an excitation wavelength of 280 nm (4 nm bandwidth); emissionwas monitored from 300 to 400 nm (8 nm bandwidth). For renaturation kinetics,the emission wavelength was 320 nm (λmax of native yTIM), excitation andemission bandwidths were 8 and 16 nm respectively. Reference samples withoutprotein were subtracted in all spectroscopic measurements. The fluorescencespectral centre of mass (SCM) was calculated from intensity data (Iλ) obtained atdifferent wavelengths using SCM=∑λIλ /∑ Iλ.

yTIM catalytic activity was measured as previously described, following theoxidation of NADH at 340 nm in a coupled-enzyme assay at 25 °C±0.1 in aBeckman DU 7500 spectrophotometer [41,58,59]. Catalytic activity measure-ments are complicated by the effect of high concentrations of GdmCl in theactivity of the coupling enzyme. In addition, since TIM is a very active enzyme,catalytic activitymeasurements must be carried out at 75 pM, a∼1000 fold lowerconcentration than those used for spectroscopic measurements. A two-stepdilution protocol was therefore devised to test that the ratio of active and inactivemolecules was not affected by the dilution steps: samples were first diluted to7.5 nM yTIM without changing the concentration of GdmCl or glycerol, aftermixing, an aliquot was immediately diluted in the activity media. Fromexperiments varying the time-span and yTIM concentration in the dilution steps,it was found that the ratio of active and inactive molecules is not affected by thedilution protocol [40,41].

2.1. Equilibrium experiments

In equilibrium experiments, unfolding samples were prepared and thenincubated for 48 h at 25 °C. Refolding samples were completely unfolded byincubation in 6 M GdmCl for 45 min. Thereafter, renaturation started by dilutionin buffer A containing lower concentrations of denaturant, with or withoutglycerol. Samples were then incubated for 48 h at 25 °C. Fluorescence spectraand catalytic activity were thereafter determined as described above.

Equilibrium unfolding experiments were analyzed according to a three-statemodel with a monomeric intermediate [41].

2U⇄Kfold

2M⇄Kassoc

N

The changes in Gibbs energy for monomer folding (ΔGfold) and monomerassociation (ΔGassoc) were assumed to vary linearly with denaturant concentra-tion x according to:

DGfold ¼ DG0fold þ mfoldx ð1Þ

DGassoc ¼ DG0assoc þ massocx ð2Þ

987H. Nájera et al. / Biochimica et Biophysica Acta 1774 (2007) 985–994

The molar fractions of subunits in the unfolded, monomeric and native states are

defined by fU ¼ UPt

; fM ¼ MPt

and fN ¼ 2NPt

, respectively, where the total

protein concentration (Pt) is expressed on a monomer basis (Pt=U+M+2N).

fM and fN can be expressed as a function of the equilibrium constants as:

fM ¼�ð1=Kfold þ 1Þ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1=Kfold þ 1Þ2 þ 8KassocPt

q4KassocPt

ð3Þ

fN ¼ 2ðfMÞ2KassocPt ð4Þ

with fU=1− fN− fM.In order to simultaneously fit catalytic activity and fluorescence data,

experimental data were normalized with a ¼ ½yðxÞ � yðxVÞ�½yðx ¼ 0Þ � yðxVÞ� where x′ is the

denaturant concentration for complete unfolding. The experimental observable(y) was assumed to be additive, y=yU fU+yM fM+yN fN. yU and yN were fixed to0 and 1 respectively. Since the intermediate is inactive yM for catalytic activitywas fixed to 0. yM for SCM was obtained as fitting parameter, considered as aconstant or a linear function of GdmCl. The parameters ΔG0

fold, mfold, ΔG0assoc,

massoc and yM were fitted to α(x) using Eqs. (3) and (4) [41].

2.2. Denaturation kinetics

Unfolding kinetics were initiated by adding native yTIM to a solutioncontaining the desired concentration of GdmCl and glycerol in Buffer A (37.5 or375 nM yTIM). Fluorescence and activity measurements were carried out asdescribed above.

For kinetic experiments, unfolding and dissociation rates constants wereobtained by fitting the data to the following exponential equation:

y ¼ y0 þ A1ekt ð5Þ

where t is time, k is the observed rate constant, y is the experimental observable(catalytic activity or fluorescence intensity), y0 is the value of the observable att=∞.

Extrapolation to absence of GdmCl was carried assuming a linear depen-dence of ln k on the concentration of GdmCl.

ln k ¼ ln k0 þ m‡½GdmC1� ð6Þk values obtained at low GdmCl concentrations were used to obtain k0dissocwhereas data for higher concentrations of denaturant were used to obtain k0unfold(see Results).

2.3. Renaturation kinetics

yTIM (18.75 μM ) was unfolded at 25 °C by incubation in buffer Acontaining 6MGdmCl for at least 45min. Renaturationwas started by dilution ofthe unfolded protein in a mixture that contained buffer A plus differentconcentrations of glycerol (0–75% v/v). Final yTIM concentration inrenaturation assays varied between 1.875 and 187.5 nM yTIM. The concentra-tion of GdmCl in renaturation conditions was fixed to 60 mM. Controlexperiments showed that catalytic activity was not affected by this concentrationof denaturant.

Catalytic activity measurements were carried out as follows: aliquots (0.4–40 μL) containing 75 pMol yTIM were withdrawn from the refolding media atthe indicated times, and diluted in the activity reaction mixture (1 mL finalvolume). Activity traces were linear with time, indicating that no reactivation ordeactivation took place during the activitymeasurements. Fluorescencemeasure-ments were carried out with 37.5 to 375 nM yTIM, using an excitationwavelength of 280 nm (8 nm bandwidth), and emission was monitored at 320 nm(16 nm bandwidth).

Under conditions that strongly favour folding and association, there are twopossible irreversible three-state models that describe the formation of dimericproduct (the native dimeric enzyme N) starting from monomeric reactants(unfolded monomers U) [60,61]. In the monomeric intermediate model, a uni-

bimolecular reaction is proposed where U folds in a first order reaction with rateconstant kfold to give the folded monomers (M) that then associate to give N in asecond order reaction described by kassoc i.e.

UYkfold

M 2MYkassoc

N

In refolding experiments the starting material is all in the unfolded state,therefore the initial conditions at t=0 are [U]=[U]0 [M]=0 and [N]=0.

The rate equations for this reaction system are

d½U �dt

¼ �kfold U½ � ð7Þ

d½U �dt

¼ kfold U½ � � 2kassoc�M�2 ð8Þ

and

d½N �dt

¼ kassoc½M �2 ð9Þ

the full solution to this system is given by

½U � ¼ ½U �0s ð10Þ

M½ � ¼ ½U �0ffiffiffisj

rI1ð2

ffiffiffiffiffijs

p Þ � bK1ð2ffiffiffiffiffijs

p ÞI0ð2

ffiffiffiffiffijs

p þ bK0ð2ffiffiffiffiffijs

p Þ ð11Þ

and

N½ � ¼ 12

U �0 � U½ � � M½ �� �� ð12Þ

where τ=e−k fold t, κ≡2[U]0 kassoc/kfold, b ¼ I1ð2ffiffiffij

p ÞK1ð2

ffiffiffij

p Þ and Iv and Kv are the

modified Bessel functions of first and second kind of order v respectively.In the dimeric intermediate model, a bi-unimolecular reaction is proposed

where U associates in a second order reaction with rate constant k*assoc. to give anon native dimer (N*) that then folds to give N in a first order reaction describedby k*fold i.e.

UYk*assoc

N* N*Yk*fold

N

In refolding experiments the starting material is all in the unfolded state,therefore the initial conditions at t=0 are [U]= [U]0 [N*]=0 and [N]=0.

The rate equations for this reaction system are

d½U �dt

¼ �k*assoc½U �2 ð13Þ

d½N*�dt

¼ 2k*assoc½U �2 � k*fold N*½ � ð14Þ

d½N �dt

¼ k*fold N*½ � ð15Þ

the time courses for the three states are given by

U½ � ¼ ½U �01þ ½U �0k*assoct

ð16Þ

N*½ � ¼ 2k*assocsV

� kV1 sVexpð�k*foldtÞ � 1� �

� sVk*fold Eiðk3sVÞexpðk3sVÞð Þ Eiðk3Þexpðk3Þð Þh i

and

N½ � ¼ 12

½U �0 � ½U �� �� N*½ �

where τ′≡ (tk1′+1), k1′≡k*assoc[U]0, and k3≡−k*fold/k′1 and Ei is the exponentialintegral [60,61].

Fig. 2. Stability of yTIM as a function of glycerol concentration. yTIM refoldingexperiments in the presence of several glycerol concentrations were monitoredas in Fig. 1. From these experiments ΔG0

fold (○) and ΔG0assoc (•) were

calculated. The solid lines show the linear fit of ΔG values to glycerolconcentration given by: ΔG=ΔG0+m[glycerol] where ΔG0

fold=−17.3±3.5 kJmol−1 mglyc

fold=−0.26±1.1 kJ mol−1 M−1; ΔG0assoc=−69.9±2.0 kJ mol−1;

mglycassoc=−3.3±0.6 kJ mol−1 M−1.

988 H. Nájera et al. / Biochimica et Biophysica Acta 1774 (2007) 985–994

3. Results

3.1. Reversibility of yTIM refolding in the presence of glycerol

In order to study the effect of glycerol in the kinetics andenergetics of yTIM unfolding and refolding, it was necessary todetermine the effect of this cosolvent on the reversibility of thereaction. Therefore, yTIM was incubated in 6 M GdmCl for45 min., this lead to complete unfolding of yTIM as indicated bythe complete inactivation of the enzyme, a far-UV CD spectrumcharacteristic of unfolded proteins, a decrease in the quantumyield of the fluorescence spectrum, and a red shift in the SCM to351 nm [41]. Folding/association was then initiated by dilutionof the unfolded enzyme in buffer A containing up to 75% (v/v)glycerol and variable amounts of enzyme (1.875 nM–3.75 μMyTIM, residual GdmCl concentration=60 mM). 48 h after thestart of refolding, the catalytic activity and spectroscopicproperties of the enzyme were measured. The SCM, fluores-cence intensity and catalytic activity of native and renaturedsamples were very similar, indicating that yTIM renaturation inthe presence of high concentrations of glycerol is reversible (datanot shown).

3.2. yTIM stability in the presence of glycerol

The properties of yTIM in the presence of 40% (v/v) glyceroland different amounts of GdmCl were determined for unfolding

Fig. 1. Unfolding and refolding of yTIM in the presence of 40% (v/v) glyceroland different concentrations of GdmCl. Normalized changes (α) for unfolding(closed symbols) and refolding (open symbols) of yTIM samples (0.75 μM).Catalytic activity (•,○) and SCM (▪,□) measurements were carried out afterincubation for 48 h at 25 °C. The simultaneous fit of catalytic activity and SCMdata to a three state model (see Materials and methods) gives for unfolding data:ΔG0

assoc=−87.8±4.9 kJ mol−1, massoc=48.7±4.4 kJ mol−1 M−1; ΔG0fold=

−26.3±6.7 kJ mol−1, mfold=10.5±2.7 kJ mol−1 M−1. A similar fit carried outon refolding data gives: ΔG0

assoc=−86.7±0.3 kJ mol−1, massoc=51.2±0.3 kJmol−1 M−1. ΔG0

fold=−22.6±7.0 kJ mol−1, mfold=8.4±2.9 kJ mol−1 M−1. Theglobal fit for unfolding and refolding data (solid lines) gives: ΔG0

assoc=−89.1±0.3 kJ mol−1, massoc=51.8±0.3 kJ mol−1 M−1. ΔG0

fold=−21.2±6.8 kJ mol−1,mfold=8.2±2.8 kJ mol−1 M−1.

and refolding experiments (Fig. 1). The SCMof the fluorescencespectra showed two transitions, the first one was observed at lowconcentrations of denaturant (0.9 to 1.2 M GdmCl), followedwith a shoulder at 1.2–1.9 MGdmCl, finally, a second transitiontook place between 1.9 and 3.25 M GdmCl. On the other hand,activity changed in a single transition coincident with the firsttransition observed by SCM (Fig. 1). The pattern shown in Fig. 1is similar to that obtained in the absence of glycerol, where amonomeric intermediate was characterized [41]. Fig. 1 data werecorrectly described by a three-state model with a monomericintermediate (solid lines in Fig. 1) suggesting that the presence ofglycerol does not modify the general features of the foldingpathway of yTIM. It is noted, that the activity data for unfoldingand refolding present a slight hysteresis at low concentrations ofdenaturant (Fig. 1). This effect was previously observed in theabsence of glycerol [41], and is due to the very slow dissociationof the native dimer. In spite of the observed hysteresis, ΔG0

values obtained from refolding and unfolding experiments werevery similar (see Fig. 1 legend). As previously observed in theabsence of glycerol, most of the stability of the enzyme comesfrom ΔG0

assoc.Refolding experiments similar to that shown in Fig. 1 were

carried out at different concentrations of glycerol (data notshown) these SCM and catalytic activity data were also fitted tothe three-state model. It was found that glycerol barely affectsΔG0

fold, whereas ΔG0assoc is more favourable in the presence of

the viscogenic agent (Fig. 2, Table 1).

3.3. Kinetics of folding and association (U–N)

Catalytic activity measurements were used to follow thereassociation kinetics of yTIM after complete unfolding of the

Table 1Thermodynamic parameters of yTIM refolding in the presence of glycerol a

Glycerol (%) ΔGkinfold

b (kJ mol−1) ΔG0fold (kJ mol−1) mfold (kJ mol−1 M−1) ΔGkin

assocc (kJ mol−1) ΔG0

assoc (kJ mol−1) massoc (kJ mol−1 M−1) yM

0d −20.6 −16.6±0.7 12.4±0.6 −75.8 −70.3±1.1 69.9±2.4 0.59±0.0220 −21.8 −16.7±4.4 8.7±4.5 −80.2 −82.1±1.3 57.9±0.8 0.63±0.0740 −20.5 −22.7±7.0 8.4±5.9 −82.3 −86.8±0.3 51.2±0.3 0.56±0.03a Thermodynamic parameters were calculated from catalytic activity and fluorescence SCM data obtained from equilibrium experiments carried out in the refolding

direction (open symbols in Fig. 1). Similar experiments were used to determineΔG0fold, mfold,ΔG0

assoc andmassoc in the presence of 20% glycerol. yM is the normalizedSCM of the monomeric intermediate.b ΔGkin

fold=−RT ln (kfold/kunfold).c ΔGkin

assoc=−RT ln (kassoc/kdissoc) kunfold and kdissoc were obtained from Fig. 6 data whereas kfold and kassoc were obtained from Figs. 3 and 4.d ΔG values in the absence of glycerol from ref. [41].

989H. Nájera et al. / Biochimica et Biophysica Acta 1774 (2007) 985–994

enzyme (see Materials and methods). After dilution of the un-folded enzyme into buffer (residual GdmCl concentration=60 mM), yTIM samples progressively recovered activity (Fig.3A). It has been shown that during refolding, the native dimer isthe only catalytically active state [36,37,39]. Because yTIMreactivation from unfolded monomers must include a bimole-cular process, protein concentration was used as a variable tostudy the kinetics of reactivation. The regain of catalytic activitywas faster with increasing protein concentration (Fig. 3A),indicating that the rate-limiting step is the association reaction.Fluorescence measurements were then used to follow therecovery of tertiary structure upon renaturation. It was found(Fig. 3B) that fluorescence intensity was close to the native valuewithin the time of manual mixing, the regain of catalytic activitywas much slower than the regain of fluorescence, indicating thatafter some seconds of refolding, a kinetic intermediate waspresent. This inactive intermediate can be monomeric (uni-bimolecular pathway) or dimeric (bi-unimolecular pathway).Since the native dimer is the only active state, both modelsaccurately fitted the fraction of active molecules (Fig. 3A),however, the bi-unimolecular pathway predicts a smalleraccumulation of the folding intermediate and a slower decayof the unfolded species than the uni-bimolecular model (Fig.3B), since the quantum yield of the unfolded enzyme is muchsmaller than that of the native enzyme [41], the highfluorescence intensity observed at short times is more compa-tible with the uni-bimolecular model (Fig. 3B). Fig. 3A data,obtained at several yTIM concentrations, were thereforesimultaneously fitted to the monomeric intermediate model.The folding and association rates obtained were kfold =1.43×10−2 ±7.8×10−3 s−1 and kassoc=6.70×10

5±2.81×105

M−1 s−1 (Table 2). It should be noted that there is no lag in theappearance of catalytic activity, therefore the value reported forkfold is only an upper limit.

Refolding experiments were also carried out in the presenceof glycerol (Fig. 4), the recovery of activity is severely sloweddown, indicating that the association rate is decreased by thiscosolvent. In addition, a lag period in the appearance of catalyticactivity was also observed at high concentrations of glycerol(Fig. 4 inset), indicating that the folding of the monomericintermediate is also delayed by this cosolvent. The folding andassociation rates obtained after fitting experiments carried out atvarious yTIM and glycerol concentrations to the monomericintermediate model (Table 2), indicate that in spite of the

stabilizing effect of glycerol, both kfold and kassoc decrease withincreasing concentrations of the viscogen.

3.4. Kinetics of dissociation and unfolding (N–U)

The kinetics of yTIM dissociation/unfolding were studied bydilution of the native enzyme in solutions containing 0.5 to3.5 M GdmCl and 0–40% (v/v) glycerol. In all cases, changes incatalytic activity (Fig. 5A) or fluorescence intensity (Fig. 5B)were well fitted to single exponential decays. The plot of ln kversus GdmCl concentration showed two slopes (Fig. 6).Parallel folding pathways or other kinetic models can be used toexplain the change in slope observed in Fig. 6. For parsimony,we assume that the two observed slopes report sequentialkinetic events that correspond to the steps observed in bothequilibrium and refolding kinetic experiments. In what follows,we discuss our arguments for this assumption. Equilibriumexperiments indicate that the more stable state at lowconcentrations of GdmCl (0.5–1.2 M ) is the monomeric inter-mediate (Fig. 1), therefore, activity changes following dilutionof the native enzyme into this range of GdmCl concentrations(Fig. 5A) should reflect dimer dissociation. The dissociationrate constant in the absence of denaturant (k0dissoc) was thereforecalculated from the extrapolation of these data to zerodenaturant concentration (Table 3), this assumes that in thisrange of GdmCl concentrations kdissoc is the main component ofkobs and that ln kdissoc varies linearly with denaturantconcentration. On the other hand, a different slope is observedat higher concentrations of GdmCl (Fig. 6), indicating thatmonomer unfolding becomes the rate-limiting step in this range(2.0–3.5 M) of GdmCl concentrations (Fig. 5B). The unfoldingrate constant in the absence of GdmCl (k0unfold) was obtainedfrom the linear extrapolation to zero denaturant of the ratesobtained in this range of denaturant concentrations (Table 3). Itwas found that both kdissoc and kunfold decrease with glycerolconcentration, as expected from the favourable effect ofglycerol on yTIM stability.

4. Discussion

4.1. Comparison of equilibrium and kinetic data

Table 1 shows a comparison between Gibbs energy changesobtained from kinetic experiments (ΔGkin

assoc and ΔGkinfold) and

Table 2Kinetic parameters of yTIM folding/association in the presence of glycerol a

Glycerol % (v/v) kfold (s−1) kassoc (M

−1·s−1)

0 (1.43±0.78)×10−2 (6.70±2.81)×105

20 (9.83±3.83)×10−3 (3.72±2.05)×105

40 (2.88±2.02)×10−3 (1.83±0.86)×105

60 (1.08±0.99)×10−3 (5.67±3.31)×104

75 (2.24±1.66)×10−4 (3.15±0.92)×104

a Rate constants were obtained from the simultaneous fit of refoldingexperiments (Figs. 3 and 4) carried out at different yTIM concentrations (1.875–187.5 nM to the unibimolecular model (Eqs. (10)–(12)).

Fig. 4. Kinetics of yTIM reactivation in the presence of glycerol. UnfoldedyTIM was refolded by dilution in buffer A at a final yTIM concentration187.5 nM, containing 0 (•), 20 (○), 40 (▪) 60 (□) and 75% (v/v) glycerol (▴),GdmCl concentration was adjusted to 60 mM. For each concentration ofglycerol, experiments carried out at yTIM concentrations varying from1.875 nM to 187.5 nM, were fitted to the uni-bimolecular model (solid lines,Table 2). The inset shows an amplification of the first experimental points.

Fig. 3. Kinetics of yTIM reactivation. (A) Time-course reactivation of yTIM as afunction of protein concentration. Unfolded yTIM (incubated for 45 min in 6 MGdmCl) was refolded by dilution in buffer A. Final GdmCl concentration wasadjusted to 60 mM and yTIM concentration to 1.875 nM (•), 7.5 nM (▪),37.5 nM (▴) or 187.5 nM (⁎). Catalytic activity was measured at the indicatedtimes. Activity values are expressed as fraction of active yTIM molecules. Thesimultaneous fit for the data obtained at all yTIM concentrations to the uni-bimolecular model (solid lines) gives kfold =1.43×10

−2 s−1 and kassoc=6.70×105 M− 1 s− 1 (see Materials and methods). (B) yTIM refolding(187.5 nM) was followed by catalytic activity (▪) or fluorescence intensity at320 nm (□). The solid and broken lines show the fit of catalytic activity data tothe uni-bimolecular and bi-unimolecular models, respectively. Predictions fromthe uni-bimolecular model for the monomeric intermediate (•••) andunfolded monomers (+++) and those from the bi-unimolecular model for thedimeric intermediate (○○○) and unfolded monomers (×××) are also shown.

990 H. Nájera et al. / Biochimica et Biophysica Acta 1774 (2007) 985–994

those obtained from equilibrium experiments (ΔG0fold and

ΔG0assoc). The reasonable agreement between kinetic and

equilibrium parameters indicates that the three state modelwith a monomeric intermediate is a suitable framework for theanalysis of viscosity effects. However, analysis of equilibriumdata (Fig. 1, Table 1 and ref. [41]) indicates that the conversionbetween the monomeric intermediate (yM=0.59±0.02) and the

native dimer (yN=1.0) should be observable by changes influorescence; this is not the case, the kinetic intermediateobserved within the first minutes of refolding, shows native-likefluorescence intensity (Fig. 3B). This may be due to the presenceof a fast folding kinetic intermediate, with native-like fluores-cence properties, not stable in equilibrium conditions. Althoughwe have no direct experimental evidence for the presence of thisintermediate, results obtained using a variety of experimentalmethodologies indicate that the assembly of the triosephosphateisomerase monomer and other β/α barrels is not a two-stateprocess [49,62–74]. The free energy change assigned tomonomer folding (ΔG0

fold) may therefore contain contributionsfrom more than a single event, consequently, it is not possible toassign a precise molecular event to kunfold and kfold. Hence,viscosity effects will be discussed only for the bimolecular step.

4.2. Glycerol effects on viscosity and stability

Both kassoc and kdissoc showed an inverse correlation with η(Fig. 7). Indeed, association data up to η/η0=13.8 (0 to 60%

Fig. 5. Kinetics of yTIM dissociation/unfolding. (A) Native yTIM was diluted in1.0 M GdmCl (○), 20% (v/v) glycerol and 1.1 M GdmCl (▪) or 40% (v/v)glycerol and 1.1 M GdmCl (△), at a final yTIM concentration of 75 nM.Thereafter, catalytic activity was determined at the indicated times. (B) NativeyTIM was diluted in 2 (○), 2.5 (▪) or 3.0 MGdmCl (△) at a final concentrationof 37.5 nM yTIM, thereafter fluorescence intensity was determined. Solid linesare fits to single exponential decays.

Fig. 6. Dissociation and unfolding rate constants as a function of GdmCl.Kinetic constants were obtained from catalytic activity measurements (closedsymbols) or from fluorescence experiments (open symbols). Experimental datawere obtained in the absence (circles) of cosolvent, or in the presence of 20%(squares) or 40% (v/v) glycerol (triangles). Data obtained at low or highconcentrations of GdmCl were fitted assuming a linear dependence of ln k on theconcentration of GdmCl (Eq. (6), see Table 3 for results).

Table 3Kinetic parameters of yTIM dissociation/unfolding in the presence of glycerol a

Glycerol% (v/v)

k0dissoc(s−1)

m‡dissoc

(M−1)k0unfold(s−1)

m‡unfold

(M−1)

0 (3.54±1.60)×10−8 5.53±0.48 (3.56±1.27)×10−6 2.91±0.1320 (3.29±3.03)×10−9 6.53±0.61 (1.47±1.72)×10−7 2.96±0.0440 (7.06±4.11)×10−10 7.13±0.43 (7.47±6.11)×10−8 3.29±0.31a k values obtained at low GdmCl concentrations (Fig. 6 closed symbols) were

fitted to Eq. (6) to obtain k0dissoc and m‡dissoc whereas data for higher

concentrations of denaturant (Fig. 6 open symbols) were fitted to Eq. (6) toobtain k0unfold and m‡

unfold.

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(v/v) glycerol) gave a slope near to one (0.85±0.01), as expectedfor a diffusion-controlled reaction. However, at higher viscos-ities a clear deviation from linear behavior was observed.Regarding dissociation data, the slope obtained (15.4±4.3) ismuch higher than one. This slope may reflect (a) secondaryeffects related to preferential hydration, such as short-rangeglycerol-induced repulsion between monomers [75] and refer-ences within] and/or (b) the effect of glycerol in yTIM stability.Regarding the latter possibility, the stabilizing effect ofviscogens may result from an increase in folding rates, adecrease in unfolding rates or both. Therefore, it has beenproposed that kinetic rates obtained at different viscositiesshould be compared at a fixed stability. This is the basis of the so-

called isostability analysis [12–15]. The glycerol-induced in-crease inΔGassoc (Fig. 2) may result from a decrease in kdissoc, anincrease in kassoc or both. However, up to η/η0=13.83 kassocshowed the behaviour expected for a diffusion-limited reaction,therefore, we propose that the stabilizing effect of glycerol ismainly due to a decrease in kdissoc. Consequently, kdissoc at afixed stability value were calculated as follows: The isostabilitypoint was set at −70.3 kJ mol−1, (the value of ΔGassoc in theabsence of glycerol see Table 1). Equilibrium massoc values(Table 1) were then used to determine the concentration ofGdmCl that decreases ΔGassoc in the presence of glycerol to−70.3 kJ mol−1 (0.189 M GdmCl for 20% (v/v) glycerol and0.321 M for 40% (v/v) glycerol). Thereafter, k0dissoc and m‡

dissoc

(Table 3) were used to calculate the value of kdissoc at theaforementioned concentrations of denaturant. The slopeobtained for the isostability-corrected dissociation rate constantsagainst viscosity was 1.1±0.3 (Fig. 7), very close to thatexpected for a diffusion-controlled reaction. A similar findingwas observed for the association and dissociation of ribonu-clease A and the S peptide [24], where the association rate

Fig. 7. Viscosity dependence of yTIM association and dissociation. k0/kη andη/η0 are the ratios of solvent viscosities and rate constants in the presence andabsence of glycerol, respectively. Symbols used are (▪) for k0 assoc/kassoc, (△)for k0 dissoc/kη dissoc and (□) for isostability-corrected k0 dissoc/kη dissoc data. Forkassoc data up to a relative viscosity of 13.83, the slope is 0.85±0.01. For kdissocdata the slope is 15.4±4.3. After the isostability correction, the slope for kdissocdata is 1.1±0.3.

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showed the expected dependence on viscosity, and thedissociation rate showed a viscosity dependence higher thanthat predicted for a diffusion-controlled reaction. These authorsapplied the isostability analysis assuming that the stabilizingeffect of the viscogen (ethylenglycol) was due only todeceleration of refolding [24].

4.3. Diffusion and protein–protein association

The second order rate constants for the association of TIMmonomers (kassoc) was first determined for rabbit TIM. From thevalue obtained (3×105 M−1 s−1), it was suggested that TIMassociation may be diffusion−limited [37]. Similar rates werereported for the kassoc of TIMs from Trypanosoma brucei(0.2×105 M−1 s−1), Trypanosoma cruzi (2.6×105 M−1 s−1)[52], and for yTIM in the present work (6.7×105 M−1·s−1).These values are similar to the basal association rate of barnaseand barstar [25]. Higher values for protein/protein associationrates, have been reported for the dimeric version of the coiledcoil GCN4-p2 [15] and for the association of MYL Arcmonomers [26]. Most of these rates are smaller than thosecalculated using the Smoluchowski equation i.e., for thecollision of two uniformly reactive spheres, that yields ratesfor protein–protein association on the order of 109–1010

M−1 s−1 [76]. The interface region is normally restricted to ahighly specific region of the protein; therefore, severalapproaches have been followed to account for the constraintsimposed by the proper alignment of the interacting proteins.These include the addition of geometric constraints [77],Brownian dynamics simulations [20] and the diffusionalassociation of free energy landscapes [78]. The experimentalvalue obtained in this work for the association rate of yTIMmonomers (7×105 M−1 s−1) is in the range of the rates

computed (104–106 M−1 s−1) using the latter approach, thatneglects long-range interactions, free energy barriers, and land-scape ruggedness [78].

The viscosity dependence of the bimolecular step isindicative of significant hydration in the transition state [16],the expanded nature of the monomeric intermediate of yTIM[41] may therefore be required for the efficient association of thesubunits, increasing the probability of successful collisions.

Diffusion control provides an upper limit for reaction rates,however, because of orientational constraints, a reaction neednot necessarily be fast in this limit [79]. In fact, the rate ofviscosity-dependent conformational transitions observed inproteins varies several orders of magnitude, spanning from fastand localized conformational changes [80] to elementary foldingsteps [13,14,19], the relatively slow docking of prefoldeddomains in large proteins [5,6,11,12] or the rigid-body asso-ciation of proteins [22,23]. The results presented in this workindicate that diffusion plays a central role in the association offolding intermediates, and suggest that protein oligomerizationmay be modeled using diffusion-collision models.

Acknowledgments

We thank Prof. J. Knowles for the gift of the yTIM gene andProf. A. Gómez-Puyou for valuable suggestions and revision ofthe manuscript. This work was partially supported by grantsfrom CONACYT (46298-Q) and PROMEP/103.5/03/2568 HNand CONACyT (43592-Q, 41328-Q) DAFV.

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