Reaction kinetic pathway of reversible and irreversible ... · Reaction kinetic pathway of...

Lait 87 (2007) 301–315 Available online at:c© INRA, EDP Sciences, 2007 www.lelait-journal.orgDOI: 10.1051/lait:2007012

Original article

Reaction kinetic pathway of reversibleand irreversible thermal denaturation

of β-lactoglobulin

Alexander Tolkach*, Ulrich Kulozik

Institute for Food Process Engineering and Dairy Technology, Technical University Munich,Weihenstephaner Berg 1, 85354 Freising-Weihenstephan, Germany

Abstract – A reaction kinetic approach for the description of reversible (unfolding) and irreversible(aggregation) denaturation of β-lactoglobulin (β-Lg) was developed in this study. This novel ap-proach allows the calculation of the content of native, partially unfolded and irreversibly denaturedβ-Lg as a function of the heat treatment time. Moreover, it is able to give a mathematical inter-pretation of the sharp bend on the Arrhenius plot of the formal denaturation rate constant whichis characteristic of the denaturation behaviour of both major whey proteins, α-lactalbumin and β-lactoglobulin. The developed theoretical backgrounds were applied for the denaturation kinetics ofβ-Lg in a protein solution containing 50 g·L−1 of the isolated whey protein fraction and verified bymeans of differential scanning calorimetry.

β-lactoglobulin / thermal denaturation / unfolding / aggregation / reaction kinetic

摘摘摘要要要 – β-乳乳乳球球球蛋蛋蛋白白白可可可逆逆逆和和和不不不可可可逆逆逆热热热变变变性性性反反反应应应动动动力力力学学学。。。本文研究了β-乳球蛋白（β-Lg）可逆（去折叠）和不可逆（凝聚）热变性反应动力学方法。该种新方法可以根据热处理时间计算天然的、部分去折叠的和不可逆热变性的 β-乳球蛋白含量。此外,根据变性速率常数的阿累尼乌斯图中的拐点可以从数学上解释乳清蛋白中 α-乳白蛋白和 β-乳球蛋白的变性行为。此种新建立的理论方法已经用于解释浓度为 50 g·L−1 的乳清分离蛋白溶液中β-乳球蛋白的变性动力学,并采用差示扫描量热法进行了验证。

β-乳乳乳球球球蛋蛋蛋白白白 /热热热变变变性性性 /去去去折折折叠叠叠 /凝凝凝聚聚聚 /反反反应应应动动动力力力学学学

Résumé – Cinétique de la réaction de dénaturation réversible et irréversible de la β-lactoglobuline (β-Lg). Une approche de la cinétique de réaction pour décrire la dénaturation ré-versible (dépliement) et irréversible (agrégation) de la β-Lg a été développée dans cette étude. Cenouveau modèle cinétique permet de calculer la fraction de β-Lg native, partiellement dépliée etirréversiblement dénaturée en fonction de la durée de traitement thermique. De plus, l’approche ci-nétique développée est capable de donner une interprétation mathématique du brusque changementde pente de la constante de dénaturation en fonction de la température, caractéristique du comporte-ment de dénaturation des deux protéines sériques majeures, α-lactalbumine et β-lactoglobuline. Lemodèle théorique développé a été appliqué à l’étude de la cinétique de dénaturation de la β-Lg dansune solution protéique à 50 g·L−1 de la fraction de protéine sérique isolée, et vérifié par mesure encalorimétrie différentielle.

β-lactoglobuline / dénaturation thermique / dépliement / agrégation / cinétique de réaction

* Corresponding author (通讯作者): [email protected]

Article published by EDP Sciences and available at http://www.lelait-journal.org or http://dx.doi.org/10.1051/lait:2007012

http://www.edpsciences.org

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http://dx.doi.org/10.1051/lait:2007012

302 A. Tolkach and U. Kulozik

1. INTRODUCTION

Bovine β-lactoglobulin (β-Lg) is a ma-jor whey protein with a molecular weightof 18.3 kg·mol−1 and average concentra-tion in milk or whey of 3.5 g·L−1 [37].Under physiological conditions the proteinexists as a dimer. It belongs to the proteinsof the lipocalin family that have an abil-ity to bind and transport small moleculeswithin a central hydrophobic core [28].The amino acid sequence of β-Lg com-prises 162 residues that marginally dif-fer depending on the genetic variant ofthe protein. Currently, 13 genetic variantsof β-Lg are known, whereas two variants,β-Lg B and β-Lg A (Gly64 → Asp64,Ala118 → Val118), are the most com-mon ones [28]. The molecular structure ofβ-Lg consists of 6–10% α-helix, 44–52%β-sheet, 8–10% turn and 32–35% of ran-dom coil [4,12,26]. The tertiary structure isstrongly dependent on temperature, pH andother environmental conditions and is sta-bilised by two disulphide bonds (Cys106-Cys 119 and Cys66-Cys160). β-Lg also hasa free thiol group (Cys121), which is nor-mally hidden by the α-helix in the nativestate (Fig. 1).

Due to its excellent functional proper-ties, this protein has found wide applica-tions in the food industry as a texturingagent and a structure stabiliser [23]. Thefunctional properties of β-Lg are stronglyaffected by its ability to denature and toreact with thermally stable proteins suchas caseins [5] as well as to form aggre-gates with diverse structures [32] or con-jugates with other milk, whey or egg pro-teins [22, 29, 39]. Thermal denaturation ofβ-Lg appears to be an essential techniqueto improve the functionality of this proteinfor its use in the different applications offood manufacture.

The mechanism of thermal denaturationof β-Lg is the subject of a considerablenumber of interesting studies [2,13,15,20,25, 26] that resulted in a number of hy-

pothetical models describing the thermalbehaviour of β-Lg in heated solutions as-sessed by different analytical techniques.By summarising the previous studies thethermal denaturation of β-Lg can be char-acterised as a multistage process, depictedin Figure 2. According to this schemethe native dimer of β-Lg (N2) dissoci-ates reversibly into native monomers (N)at temperatures above 40 ◦C. In the tem-perature range between 40 and 55 ◦Cthe monomer undergoes an intramolecu-lar transition into a so-called R-state (NR),which is especially characteristic for pHvalues above 7.5. The R-state of β-Lg dif-fers from the native state only by mi-nor conformational changes of some sidechains (especially the α-helix that masksthe free thiol group) and, consequently,by enhanced accessibility of the free thiolgroup. The major secondary structure ofthe molecule remains nearly unaltered andvery similar to the native one. The β-Lgmolecule in the R-state is able to formpolymers. However, the polymerisationrate is negligible.

When the temperature of the proteinsolution increases above 60 ◦C, the β-Lgmolecule undergoes a reversible partial un-folding and forms a thermally inducedmolten globule state (UMG). The criti-cal transition of β-Lg contains a com-plete unfolding of the α-helix structureand exposition of the free thiol group(Cys121) and some parts of the hydropho-bic core (see Fig. 1). However, at this pointpartially unfolded β-Lg molecules with ex-posed Cys119 can also be formed by in-tramolecular interchange reactions. Thisform is not able to refold into the na-tive state, which results in the formationof irreversible β-Lg monomers upon cool-ing [6, 30]. Via the hydrophobic bonds andthiol exchange, β-Lg molecules take partin the irreversible intermolecular interac-tions that result in the formation of poly-mers and aggregates (Un, Um). Most au-thors distinguish between polymerisation

Reaction kinetic pathway of the denaturation of β-Lg 303

C

N

IA

H

G

F

DC

B

E

Cys121

Cys -Cys66 160

Cys -Cys106 119

Figure 1. Molecular structure of β-Lg. Molecular graphics images were produced using the UCSFChimera package from the Resource for Biocomputing, Visualization, and Informatics at the Uni-versity of California, San Francisco (supported by NIH P41 RR-01081) [24]. Protein data bank file1B0O was used for the model construction [38].

2N

2N

R2N

MG2U

D2U

nU mU

m n

n 2,3,4,5= ~130−140 °C

~60−70 °C

~40−55 °C

~40 °C

Figure 2. Mechanism of the thermal denaturation of β-Lg.


(formation of polymers by covalent bonds)and aggregation (formation of aggregatesby non-covalent interactions) despite thefact that both processes occur simultane-ously, leading to the formation of mixedaggregates stabilised by both covalent andnon-covalent bonds. The aggregation andpolymerisation rate is strongly dependenton the environmental conditions and tem-perature [32, 33]. According to differen-tial scanning calorimetry (DSC) measure-ments, the partial unfolding and formationof the molten globule state is completed attemperatures around 90 ◦C [11]. A tem-perature increase up to 130–140 ◦C leadsto a complete destruction of the secondarystructure of the β-Lg molecule and irre-versible denaturation.

The ability to form a thermally inducedmolten globule state plays a very importantrole for technological properties of β-Lg.This partially unfolded β-Lg molecule isable to react with other partially unfoldedand aggregated or polymerised whey pro-teins forming particles and gels [32, 33];the reaction between partially unfoldedβ-Lg and casein micelles is exploited in yo-ghurt production [8,9] and must be avoidedduring hard cheese production due to theinhibition of rennet-induced gel formation.However, this complex formation might beused for the increase in cheese yield infresh cheese manufacture [37]. Interactionsbetween partially unfolded β-Lg and hotsurfaces during treatment of milk and milkproducts results in the formation of depositand fouling of heat transfer surfaces [7].

Due to the technological functions ofthe partially unfolded β-Lg describedabove, a kinetic modelling of thermallyinduced molten globule state formationand degradation, allowing the predictionof its concentration as a function of theheat treatment time, is of high signifi-cance. However, direct measurements ofthe content of the partially unfolded β-Lgin situ (e.g. during heat treatment) are dif-ficult or even impossible to perform. The

most disturbing factor is that the molec-ular structure of the thermally inducedmolten globule state of β-Lg is identi-cal to that of aggregated molecules andmost analytical techniques cannot distin-guish in situ between both protein statesexisting at the same time in the solu-tion. Analytical techniques such as DSCare able to monitor the unfolding processduring heat treatment; however, they arenot in a position to assess the degradationof partially unfolded β-Lg as a result ofthe aggregation reaction. Online monitor-ing of the denaturation reaction by parti-cle measurement seems not to be an effec-tive method since the existing techniquescan only assess the aggregation, but can-not distinguish between native and par-tially unfolded states of β-Lg molecules.Particle measurements, as well as tech-niques such as high-performance liquidchromatography (HPLC), gel permeationchromatography (GPC), ionic exchangechromatography (IEC), polyamidacryl gelelectrophoresis (PAGE), and isoelectric fo-cusing (IEF) for the assessment of the con-tents of native, monomeric and agglom-erated or polymerised proteins after heattreatment are also not able to provide anydirect information about the concentrationof the thermally induced molten globulestate due to its refolding after cooling ofthe protein solution before performing theanalysis.

Owing to the above-mentioned prob-lem of the analytical assessment of thethermally induced partially unfolded stateof β-Lg molecule, most studies with a fo-cus on the thermal denaturation of β-Lgdid apply analytical techniques to distin-guish between the native and the aggre-gated states of the molecule [16, 18, 34–36]. Therefore, the degradation of native ornative-like β-Lg was commonly modelledby formal kinetic means supposing the de-naturation to be a two-stage process (native→ denatured). According to this assump-tion the degradation rate of native protein


−dCHPLC

dtas a result of thermal denatura-

tion is directly proportional to the concen-tration of the protein CHPLC in power ofthe reaction order n with the denaturationrate constant (or denaturation velocity con-stant) k f as a proportional coefficient:

−dCHPLC

dt= k f (CHPLC)n . (1)

The formal reaction order n may not beseen as an order of a classical chemical re-action: the formal reaction order expressesin this case the complexity of the denat-uration process and usually takes valuesfor the denaturation reaction of β-Lg be-tween 1 and 2, commonly 1.5. After inte-gration of Equation (1), the following ex-pressions can be used for the prediction ofthe amount of native β-Lg after heat treat-ment (for n � 1):

CHPLC,t=(C(1−n)

HPLC,0+(n−1) · k f · t) 1/(1–n)

(2)(CHPLC,t

CHPLC,0

)(1−n)

= 1+ (n−1) ·C(n−1)HPLC,0 · k f · t.

(3)Equation (3) is commonly used for graphi-cal interpretations since the formal denatu-ration rate constant can be easily obtainedfrom the slope of the line by a simple lin-ear regression. The denaturation rate con-stant k f is a function of the compositionof the solution and heat treatment temper-ature. The relationship between the formalvelocity constant and heating temperaturecan be mathematically expressed by theArrhenius equation:

ln k f = ln k f 0 − Ea

R · T · (4)

Contrary to a classical chemical reaction,which, according to the Arrhenius rela-tionship, can be described by means of astraight line when ln k f is plotted versus 1

T ,the velocity constant of the β-Lg denatura-tion shows a discontinuity in the plot, re-sulting in a sharp bend or edge [34]. The

bend temperature (or transition tempera-ture) divides the Arrhenius plot into twoparts: unfolding-limited (low temperaturerange) and aggregation-limited (high tem-perature range) reaction. The interpretationof such behaviour of the reaction rate con-stant is that at temperatures lower than thebend temperature the partial unfolding ofβ-Lg molecules is slower than their aggre-gation, so the whole denaturation reactionis limited by the unfolding subreactions.In contrast to that, at temperatures higherthan the bend temperature, the aggregationof the thermally induced molten globulestate is slower than its formation and, con-sequently, the resulting denaturation reac-tion of β-Lg is limited by the aggregationprocess. The position of the sharp bendstrongly depends on the pH, lactose con-tent and other compositional parameters.

The major advantage of the formal ki-netic description is that the formal reactionrate constant can be simply calculated fromexperimental data and used for the predic-tion of the degree of irreversible denatura-tion for every time-temperature combina-tion. However, the theoretical backgroundof the formal kinetic description does nottake into consideration the existence ofthe molten globule state and, consequently,is not able to separately characterise theunfolding process, polymer and aggrega-tion formation as well as giving a math-ematical interpretation of the sharp bendoccurrence. For this reason, a number ofother models for quantitative descriptionof the subreaction of β-Lg denaturationwere adapted or developed. At tempera-tures between 60 and 70 ◦C and mod-erate protein concentration the denatura-tion of β-Lg could be described in termsof classical polymerisation with initiation,propagation and termination steps [27],fractal theory [1], Fuchs-Smoluchovski ki-netics [13] and different numerical mod-els of colloidal aggregation [14, 15]. Thesemathematical models deliver a very goodnumerical interpolation of data obtained


by dynamic light scattering or photoncorrelation spectroscopy in the condi-tions mentioned above. Nevertheless, theyare mainly unable to give a quantitativedescription of thermally induced β-Lg de-naturation over a broad range of temper-atures as well as mathematical interpreta-tion of the sharp bend on the Arrheniusplot. Moreover, none of the models wasable to calculate the concentration of eachof the three β-Lg main states involved inthe denaturation process: native, partiallyunfolded and irreversibly denatured.

Based on this situation, the major objec-tive of this study was to develop a novelreaction kinetic approach that is able toseparately describe the reversible unfold-ing and irreversible denaturation of β-Lgin heat-treated solutions. Therefore, usingthis reaction kinetic approach, a derivationof mathematical expressions describing theconcentration of native, partially unfoldedand irreversible aggregated or polymerisedβ-Lg as a function of heat treatment timewas aimed at.

The model development was based onthe formal kinetic data generated by ex situmeasurements. In this study we decided touse HPLC for the assessment of β-Lg con-tent since this technique provides informa-tion about protein molecules remaining na-tive or refolded to the native state after heattreatment. Application of formal kineticdata for development of a novel reactionkinetic approach allows the considerationand interpretation of the phenomena suchas the sharp edge on the Arrhenius plot ob-served by prior kinetic studies.

2. MATERIALS AND METHODS

β-Lg AB was obtained from pasteurisedskim milk using the procedure describedby Maubois et al. [21]. After freeze-drying,the β-Lg powder was dissolved in ultrafil-tration permeate (molecular weight cut-offof the UF membrane 10 kg·mol−1, pH 6.8,

lactose content 45 g·L−1) to obtain a so-lution containing 50 g·L−1β-Lg and leftovernight at 4 ◦C. The concentration of50 g·L−1 was chosen due to the intendedDSC measurements in order to obtain amaximal signal/noise ratio.

For the thermal treatment and the eval-uation of kinetic data two different sys-tems were used. For temperatures below100 ◦C a heat treatment in stainless steeltubes (inner diameter of 3 mm and lengthof 258 mm) in a water bath was used. Thegeometry of the tubes allowed heating upthe sample within 10 s by application ofanother water bath for pre-heating (as arule the temperature of the pre-heating bathwas 10 ◦C higher than the target tempera-ture). Following heat treatment the sampleswere cooled down immediately in a waterbath with melting ice.

For heating temperatures above 100 ◦Cand holding times longer than 60 s a steam-heated pressure vessel was used. The ther-mal treatment was carried out in the samestainless steel tubes as used for the wa-ter bath experiments. In order to minimisethe heating-up time the steam pressure dur-ing the heating-up was adjusted such that atemperature of 10 ◦C above the intendedheat treatment temperature was realised.

The β-Lg content was determined byHPLC as described in detail by Beyerand Kessler [3]. According to this method,only native β-Lg which showed an identi-cal retention time to the standard fractionsobtained from Sigma (β-Lg A: L7889,β-Lg B: L8005) could be detected. Anychanges in the tertiary structure of the pro-tein resulted in a shift of the retentiontime and, consequently, these monomericproteins with altered structures could notbe detected as native by this method. Thedenatured β-Lg was separated from thesolution by shifting the pH value to 4.6 fol-lowed by a centrifugation step (10 000× g,10 min). The supernatant containing na-tive β-Lg was filtered through a mem-brane filter (pore ø 0.45 μm, Chromafil�


nN nU Dn

Cooling

HPLC

kagg α

Unfolding Aggregation

Native by HPLC Denatured by HPLC

Figure 3. Denaturation of β-Lg as a three-state process.

RC-45/25, Macherey-Nagel, Düren, Ger-many).

The DSC measurement was carried outusing the DSC Q1000 (TA Instruments,Alzenau, Germany). The pans used forthe DSC measurements were high-volumepans made of stainless steel with a samplevolume of 75 mg. The reference samplewas UF permeate. The heating rate usedwas 5 K·min−1.

3. RESULTS AND DISCUSSION

3.1. Development of themathematical model

According to the scheme in Figure 2the heat-induced denaturation of β-Lg attemperatures above 60 ◦C can be sim-ply characterised by a three-state modelthat involves native molecules of β-Lg,a thermally induced molten globule state(reversible unfolded molecules) and irre-versibly denatured (aggregated or poly-merised) molecules. Since the aggregationof the R-state is negligible, all reversiblereactions in Figure 2 can be substitutedby one irreversible unfolding step. Thus,for the kinetic approach this process canbe simply described by the reaction equa-tion depicted in Figure 3 which shows the

heat denaturation as a two-step process: af-ter reversible unfolding of the native β-Lgmolecule (N) follows an irreversible aggre-gation or polymerisation of the thermallyinduced molten globule state (U) with for-mation of covalent and non-covalent boundprotein polymers or aggregates (Dn). Thestoichiometric coefficient n corresponds tothe formal reaction order n in Equation (1)and expresses the complexity of the poly-merisation and aggregation processes.

If the temperature of the protein solutiondrops under 60 ◦C, the protein molecules inthe partially unfolded state U refold to thenative state N, whereas the molecules inthe irreversible denatured state Dn are un-able to refold. Therefore, analytical tech-niques such as HPLC, which detect onlythe amount of the native proteins in thesolution after heat treatment, assess allprotein molecules that were in the na-tive and reversibly unfolded states at themoment when the temperature dropped be-low 60 ◦C:

CHPLC = CN +CU . (5)

The formation of the molten globule stateduring thermal treatment of protein solu-tion can be characterised by equilibriumbetween native and partially unfolded β-Lgthat quantitatively can be described by an


unfolding degree α:

α =CU

CN + CU=

CU

CHPLC(6)

orCU = α ·CHPLC (7)

and, consequently,

CN = (1 − α) · CHPLC . (8)

The irreversible aggregation and polymeri-sation of the molten globule state of β-Lgcan be described by an aggregation veloc-ity constant kagg. Thus, the degradation rateof partially unfolded β-Lg as a result ofthe aggregation process can be expressedas follows:

−dCU,agg

dt= kagg (CU)n . (9)

From Figure 3 it can be seen that the de-crease in β-Lg content assessed as nativeby HPLC is equal to that caused by ag-gregation or polymerisation of the moltenglobule state:

dCHPLC

dt=

dCU,agg

dt· (10)

Therefore, Equation (9) can be rewritten asfollows:

−dCHPLC

dt= kagg (CU)n . (11)

By taking into consideration the rela-tion between the concentration of partiallyunfolded β-Lg and β-Lg being assessedby HPLC as native (Eq. (7)), differentialEquation (11) can be rewritten using theHPLC-assessable content of β-Lg only:

−dCHPLC

dt= kagg (α ·CHPLC)n

= kagg αn (CHPLC)n . (12)

By integration of Equation (12) the contentof β-Lg assessed as native by HPLC can be

calculated as a function of the heat treat-ment time:

CHPLC,t =(C(1−n)

HPLC,0

+ (n − 1) · αn · kagg · t) 1/(1–n)

. (13)

By taking into consideration Equations (7)and (8) a calculation of the concentrationof unfolded CU and native CNβ-Lg duringheat treatment can be obtained from Equa-tion (13):

CU,t = α(C(1−n)

HPLC,0

+ (n − 1) · αn · kagg · t)1/(1−n)

(14)

CN,t = (1 − α)(C(1−n)

HPLC,0

+ (n − 1) · αn · kagg · t)1/(1−n)

. (15)

Equations (14) and (15) show the relation-ships between the heat treatment time ofthe protein solution and the concentrationof native N and partially unfolded form Uof β-Lg. The content of the irreversibly ag-gregated or polymerised β-Lg can be cal-culated as the difference between the β-Lgconcentration assessable by HPLC and thestarting β-Lg concentration. It can clearlybe seen that the requirement for the useof the above-mentioned equations is theknowledge of the degree of unfolding αand aggregation rate constant kagg.

The calculation of both α and kagg canbe carried out by drawing an analogy be-tween the formal kinetic equation in differ-ential form (Eq. (1)) and the reaction ki-netic expression (Eq. (12)), which show arelationship between the rate of decrease ofβ-Lg that can be assessed by HPLC as na-tive and its concentration. From the com-parison of both equations it follows:

k f = αn · kagg. (16)

Or, in logarithmic form:

ln(k f

)= n lnα + ln

(kagg

). (17)


0

n ln α

ln (T)α

ln k (T)agg

1/T

ln X

Aggregation limitedtemperature area

Unfolding limited temperature area

α=1, ln α=0 0<α<1, ln α<0

Figure 4. Interpretation of the sharp bend on the Arrhenius plot according to Equation (17).

Equations (16) and (17) show that the for-mal kinetic rate constant of thermal denat-uration of β-Lg can be seen as a directcombination of unfolding and aggregationterms. This allows a mathematical explana-tion of the sharp edge on the Arrhenius plotof the formal velocity constant (Fig. 4): thevalue of the unfolding degree α varies be-tween 1 (all protein molecules are com-pletely unfolded, high temperature area)and 0 (all protein molecules are native,low temperature area). Consequently, thevalue of the term lnα varies between 0 and−∞. In the aggregation-limited area the un-folding of β-Lg is completed so that onlythe unfolded and the irreversibly denaturedβ-Lg molecules exist in the solution. Theunfolding degree α takes the value of 1and, consequently, the term n lnα = 0.Therefore, the formal velocity constant ofβ-Lg denaturation takes a value of the ag-gregation velocity constant:

ln(k f

)= ln

(kagg

). (18)

With decreasing temperature the unfold-ing degree α declines and the term n lnαin Equation (17) takes a negative value.Consequently, the relationship betweenthe formal velocity constant and recip-rocal absolute temperature deviates fromthe linear, which results in a sharp edgeon the Arrhenius plot. Therefore, in theunfolding-limited temperature area the de-gree of unfolding α can be calculated fromthe difference between the formal velocityconstant k f and aggregation rate constantkagg extrapolated into the unfolded limitedarea:

α = exp

⎛⎜⎜⎜⎜⎜⎜⎝ln

(k f

)− ln

(kagg

)n

⎞⎟⎟⎟⎟⎟⎟⎠ . (19)

Knowledge of the unfolding degree αand the aggregation rate constant kagg

allows the calculation of the contentof native, reversibly unfolded and ir-reversibly denatured β-Lg according tothe mathematical expressions mentionedabove (Eqs. (14)–(15)).


1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

030

0H

eat t

reat

men

t tim

e [s

]

(C/C) [s] HPLC,tHPLC,0−0.5

600

900

1200

1500

1800

2100

80 °

C63

°C

68 °

C75

°C

70 °

C

1.0

1.5

2.0

2.5

3.0

3.5

050

100

150

200

250

300

Hea

t tre

atm

ent t

ime

[s]

(C/C) [s] HPLC,tHPLC,0−0.5

105

°C

120

°C11

0 °C

85 °

C88

°C

95 °

C90

°C

Fig

ure

5.Im

pact

ofth

ehe

attr

eatm

entt

ime

and

tem

pera

ture

onth

elo

ssofβ

-Lg

asse

ssed

asna

tive

byH

PL

C.


3.2. Experimental results

Figure 5 shows the impact of the heattreatment time and temperature on thedegradation of native β-Lg content as-sessed by HPLC after cooling down thesamples. The relative decrease in β-Lgcontent is depicted according to Equa-tion (3), whereby the formal denaturationrate constant can be calculated from theslope of the straight line. The formal or-der n of the denaturation reaction wasfound to be 1.5 since the results couldbe linearised by plotting

(CHPLC,t

CHPLC,0

)−0.5ver-

sus the heat treatment time t. This for-mal reaction order of the β-Lg denatu-ration reaction can be well explained bymeans of the rapid aggregation theory ofcolloidal solutions [31] or according to theBodenstein principle of polymer formationreaction [27].

It is evident that with increasedtemperature the slope of the line and,consequently, the denaturation rate con-stant increases. The influence of the heattreatment temperature on the formal de-naturation velocity constant according tothe Arrhenius equation (Eq. (4)) is shownin Figure 6. It can be clearly seen thatthe relationship between the reciprocalabsolute temperature T and the naturallogarithm of the formal velocity constantk f shows a discontinuity in the form of asharp bend with a transition temperature of90 ◦C, dividing the Arrhenius plot into theunfolding-limited and aggregation-limitedareas. According to the interpretationgiven in Figure 4 the unfolding reaction ofβ-Lg and the formation of the molten glob-ule state must be finished at a temperaturebelow 90 ◦C. This hypothesis could beproven by a DSC measurement depictedin Figure 6 that shows an endothermicheat flow generated by the unfolding ofthe β-Lg molecules. It can be seen that theheat flow stops at a temperature of 87 ◦C,indicating that the unfolding is completed.

Table I shows a calculation of the un-folding degree α using the formal kineticdenaturation rate constant according toEquation (19). As can be seen, the β-Lgmolecules start to unfold significantlyat a temperature at or above 65 ◦C,which is in agreement with a numberof studies on thermal denaturation of β-Lg [10, 11, 17, 19]. At temperatures below60 ◦C the unfolding degree is insufficientfor the irreversible aggregation reaction tobe started: the probability of a collision be-tween two unfolded β-Lg molecules is ex-tremely low under these conditions.

Knowing the degree of unfolding α andthe aggregation rate constant kagg it is pos-sible to predict the content of the native andunfolded β-Lg according to Equations (14)and (15). Figure 7 depicts the influenceof the heat treatment time on the con-centration of β-Lg assessed as native byHPLC after cooling (measured values) aswell as the concentration of the unfoldedand native β-Lg (calculated values) at twoheating temperatures: 80 ◦C (unfolding-limited) and 110 ◦C (aggregation-limited).

According to the reaction equation de-picted in Figure 3 the unfolding of β-Lgoccurs rapidly until the equilibrium be-tween the native and the molten globulestates of the β-Lg molecules is established.The ratio between the concentration of thepartially unfolded and native β-Lg underthese conditions is given by the unfold-ing degree α. The decreasing amount ofthe molten globule state of β-Lg due tothe aggregation or polymerisation reactionis balanced by the partial unfolding of thenative molecules, with the result that theachieved equilibrium between both statesremains unchanged as long as the heatingtemperature stays constant (80 ◦C, Fig. 7).Heat treatment of the β-Lg solution at atemperature in the aggregation-limited area(110 ◦C, Fig. 7) leads to a partial unfoldingof all available β-Lg molecules so that onlyirreversible aggregation takes place. Theamount of the partially unfolded protein in


−16

−14

−12

−10

−8

−6

−4

−2

2.5 2.6 2.7 2.8 2.9 3.0

60708090Temperature [°C]

1/T 10 [K ] ·3 −1

100

lnk

(kin

gL

s)

ff

−0.

5−

0.5

−1

120 110

End

o

Ea = 313.9 kJ.molk = 98.9

Ea = 80.8 k = 21.7

unf

unf

agg

agg

−1

0

0

kJ.mol−1

Figure 6. Impact of the heating temperature on the formal denaturation rate constant combined withthe DSC thermogram of the β-Lg solution.

Table I. Calculation of the unfolding degree of β-Lg on the basis of the formal kinetic parameters.

Temperature kf ln(kf ) ln(kagg) ln(kf ) − ln(kagg) lnα α

120 4.5 × 10−2 –3.09 –3.09 0 0 1110 2.4 × 10−2 –3.732 –3.732 0 0 1105 1.7 × 10−2 –4.066 –4.066 0 0 195 8.6 × 10−3 –4.761 –4.761 0 0 190 5.9 × 10−3 –5.122 –5.122 0 0 188 3.7 × 10−3 –5.602 –5.27 –0.332 –0.22 0.8085 1.5 × 10−3 –6.479 –5.50 –0.983 –0.65 0.5280 3.4 × 10−4 –7.972 –5.88 –2.092 –1.39 0.2575 7.4 × 10−5 –9.509 –6.28 –3.233 –2.15 0.1270 1.5 × 10−5 –11.090 –6.68 –4.407 –2.94 0.0568 8.0 × 10−6 –11.735 –6.85 –4.886 –3.26 0.0463 1.5 × 10−6 –13.382 –7.27 –6.109 –4.076 0.0260∗ 5.6 × 10−7 –14.394 –7.53 –6.861 –4.576 0.01

* Interpolated value.


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 300 600 900 1200 1500 1800 2100

Heating time [s]

C/C

[-]

tH

PLC

,0

HPLC: - 80 °C - 110 °C- 80 °CNative:

Unfolded: - 80 °C- 110 °C- 110 °C

Figure 7. Influence of the heat treatment time on the loss of unfolded, native and HPLC-assessable β-Lg.

this case is equal to the amount of β-Lg as-sessable by HPLC.

4. CONCLUSIONSAND OUTLOOK

It was shown that the novel reaction ki-netic approach for quantitative descriptionof reversible and irreversible denaturationof β-Lg developed above allows the cal-culation of the content of the native, par-tially unfolded and irreversibly denaturedβ-Lg as a function of heating time. Thepresented kinetic model can be used for in-dependent characterisation of the unfold-ing and aggregation reactions of β-Lg, aswell as other proteins showing a sharp edgeon the Arrhenius plot under different envi-ronmental conditions. The ability of the de-naturation model to quantitatively describethe formation of the molten globule state

of globular proteins can be used to model,and for a better interpretation of DSC mea-surements or the results of other analyticaltechniques. However, deeper insights intothe thermodynamics of the unfolding andaggregation reactions are necessary for thatpurpose.

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