Fibrillar β-Lactoglobulin Gels: Part 2. Dynamic Mechanical

Biomacromolecules 2004, 5, 2420-2429

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Fibrillar β-Lactoglobulin Gels: Part 2. Dynamic Mechanical Characterization of Heat-Set Systems Walraj S. Gosal,†,§ Allan H. Clark,*,‡,| and Simon B. Ross-Murphy† Department of Life Sciences, King’s College London, Franklin-Wilkins Building, 150 Stamford Street, London SE1 9NN, United Kingdom, and Unilever Research, Colworth House, Sharnbrook, Bedford MK44 1LQ, United Kingdom Received June 9, 2004; Revised Manuscript Received August 3, 2004

Oscillatory shear rheometry (mechanical spectroscopy) has been used to study the heat-set gelation of β-lactoglobulin at pH 2. Modulus-concentration relationships were obtained by extrapolating cure data to infinite time. In terms of theory, these fail to provide a clear distinction between the fractal description of biopolymer gels and the classical random f-functional polycondensation branching theory (cascade) approach, though the latter is preferred. Critical exponents for the sol-gel transition, derived from these data, are also discussed. Where gel time-concentration results are concerned the fractal model makes no predictions, and the cascade approach in its simplest form must be rejected in favor of a more sophisticated version involving delivery of fibrils by nucleation and growth into the random aggregation process. Over the limited concentration range accessed experimentally, cure data for the different β-lactoglobulin solutions, reduced to the universal form G′/G′inf versus t/tgel, superimposed well for samples heated both at 80 and 75 °C and for different batches of protein. Studies of the frequency responses of the fully cured gels confirm the validity of the gel description given to these materials, and a study of the temperature dependence of the frequency spectrum suggests a fall in the elastic component of the modulus as temperature decreases. This contrasts with what has been found for other heat-set globular protein gels such as those from serum albumin where the gel modulus increases at lower temperatures. The present results are in good agreement with more limited amounts of pH 2 β-lactoglobulin data published earlier, though some differences arise through a previous neglect of measurement “dead time”. Introduction A previous paper (Part 1) in this series1 has presented evidence of fibrillar aggregation by the milk protein β-lactoglobulin under conditions of heating (pH 2) and chemical denaturation. This evidence included the results of high resolution microscopy (TEM and AFM), X-ray diffraction, and spectroscopic investigations (IR/Raman). The results were compared with corresponding data for another globular protein well-known to form long linear aggregates on heating, namely insulin at acid pH. It was concluded that although the two forms of protein aggregation shared common features, such as the generation of increased intermolecular β-sheet during protein assembly, there remained significant differences: β-lactoglobulin generally formed “strings of beads” while insulin formed smoother, more ribbonlike polymers which were, themselves, capable of association to form higher-order structures. The insulin aggregates also contained a higher amount of intermolecular β-sheet, and * To whom correspondence should be addressed. Phone: +44 (0) 20 7848 4081. Thermal Fax: +44 (0) 20 7848 4082. Plain Paper Fax: +44 (0) 20 7848 4500. E-mail: [email protected]. † King’s College London. ‡ Unilever Research. § Present Address: Astbury Centre for Structural Molecular Biology, University of Leeds, Leeds LS2 9JT, United Kingdom. | Present Address: Department of Life Sciences, King’s College London, Franklin-Wilkins Building, 150 Stamford Street, London SE1 9NN, United Kingdom.

this was of a more organized, and extensive, form capable of generating a narrow (well-defined) X-ray diffraction peak. In all of these respects, the insulin fibrils resembled the amyloid protein aggregates2-5 associated with various disease states, whereas the linear β-lactoglobulin polymers were more like what have previously been described6-9 as “protofibrils”, i.e., aggregated forms of protein recognized as being in some way precursors to amyloid fibril formation. Compared to Part 1 of this series, the present paper examines β-lactoglobulin aggregation from a different perspective. The focus is on the thermally induced gelation of β-lactoglobulin at pH 2 which occurs when the protein is heated at higher concentrations than were considered previously. Although it is assumed that the same underlying phenomena of fibrillar self-assembly are involved, these aggregates are capable, at the higher concentrations, of forming a uniform three-dimensional network (transparent gel), via the usual transition from a viscous (or viscoelastic) sol to a viscoelastic soft solid (or gel). The emphasis is therefore on the linear viscoelastic properties of the β-lactoglobulin system at pH 2, and the technique employed to measure these properties is oscillatory shear rheometry, sometimes known as dynamic mechanical spectroscopy. The discussion centers on the measurement of gel cure data on heating the β-lactoglobulin at 75 and 80 °C and at several concentrations. Long-time limiting values of the gel modulus are obtained from the data by extrapola-

10.1021/bm049660c CCC: $27.50 © 2004 American Chemical Society Published on Web 10/14/2004

Characterization of Heat-Set Systems

tion and critical gel times by a logarithmic discontinuity criterion. The behaviors of these quantities in relation to protein concentration and temperature are discussed in the light of current models10 of the protein gelation phenomenon and compared with limited previous data11 for the same protein again studied at pH 2. The mathematical form of the cure data is also examined in the light of these theories and whether, or not, these data can be reduced to master curve form is a particular concern. Finally, the frequency dependence of the linear viscoelastic shear modulus is described and its behavior on changing temperature. It is this frequency dependence which justifies the use of the term “gel” for the materials produced on heating β-lactoglobulin solutions under the conditions prescribed and allows these gels to be distinguished from viscoelastic fluids generated simply by the entanglement of filamentous aggregates. The study of the fibrillar aggregation and gelation of β-lactoglobulin is important for two reasons. First, the solgel transition, particularly in the presence of other proteins, is of practical industrial importance as a means of structuring fluids (e.g., in the food industry). Second, the filamentous aggregation of proteins is of general interest, as this phenomenon seems to be an important element in the generation of disease states (in particular, in amyloid disorders such as Alzheimer’s and v-CJD) through the malfunction of protein components. Indeed the literature in this area (see ref 1 for a limited review) is very much based around the so-called amyloid state which is recognized as a direct product of fibrillar protein aggregation. Materials and Methods In the present work, pseudo strain-controlled measurements were used to probe both the incipient and the long-time mechanical behavior of thermally induced β-lactoglobulin pH 2 fibrillar gels. A principal aim was to compare these data with results obtained previously11 with a strain controlledinstrument. In this previous study, cure-curves were followed over a range of pH, at a temperature of 80 °C, and parameters extracted from these were used in modeling exercises. The advantage of using the present methodology was 2-fold. First, a general increase in the pre-gel signal could (in theory) lead to a greater accuracy in the extraction of the sol-gel transition time tgel, especially at higher concentrations. This is essential when attempting to discriminate between different theoretical models. Second, a Peltier system was used to rapidly heat the solution above the unfolding temperature, thus increasing the reproducibility of the results. However, as is discussed below, this experimental approach is intrinsically more uncertain than the use of strain control and, although it is quite widely employed, not least because controlled-stress rheometers are usually ∼50% of the cost of an equivalent controlled-strain instrument, it is not usually to be recommended. Below, however, are described the precautions adopted in the present work to guarantee data quality. A preliminary account of these new experiments and their findings has already been published.12 Sample Preparation. Lyophilised β-lactoglobulin AB (Sigma Chemicals, Dorset, UK) from bovine milk (product code L-0130, lot number 20K7043) was left to dissolve in

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de-ionized water (Milli-Q purification system, Millipore Limited, Watford, UK) for 1 h on a vibrating platform (Vibrax-VXR, IKA-Werke GmbH & Co., Staufen, Germany). The pH was monitored with the use of a combination microelectrode (Radiometer Limited, Crawley, UK) and the pH of native solutions was found to be 7.1 ( 0.05. The pH was adjusted from 7.1 to pH 2 ( 0.05, with the use of 1 M HCl (BDH., UK). Re-weighing of the samples after pH adjustment allowed calculation of the amount of HCl required to change the pH. Concentrations were thus corrected for the effects of acid addition (and indeed for the initial moisture content of the powdered protein sample). The resulting β-lactoglobulin solution was filtered with the use of a 0.2 µm syringe filter (Acrodisc PF 0.8/0.2 µm, Gelman Sciences, UK) to yield an optically clear liquid. UV absorption measurements of protein concentration confirmed that there was negligible loss of protein during filtration. Samples that were required for thermally induced gel experiments were degassed under vacuum (∼15 min) either with the use of a “T-tube” or by inserting a needle connected to a vacuum supply into the top of the vial enclosed with a SubaSeal cover seal (Aldrich, Dorset, UK). This step was necessary to prevent the formation of air bubbles during the heat-set experiments. All samples were used immediately after filtration and degassing. Dynamic Oscillatory Measurements. β-Lactoglobulin gel-formation induced by thermal methods was monitored in situ using pseudo-strain controlled measurements. For these experiments, 1.25 mL of sol was pipetted onto the plate of the rheometer (CSL 100, T.A Instruments, UK) at 25 °C, and oscillatory measurements (ω ) 1 rad/s) were made between 68 and 90 °C. The geometry used was a 40 mm 4° metallic cone, with a truncation length of 99 µm (T.A Instruments, U.K). The gap was set, and any excess solution was soaked away. A layer of silicone oil (Dow Corning, 200/ 1000 cs fluid 63011-4X, BDH, UK) was applied around the edge of the exposed sample to prevent evaporation. The expansion of the cone and plate was taken into account by either allowing the cone and plate to expand at the experimental temperature and presetting the gap at 25 °C to take this into account or by using expansion coefficients for the geometry and adjusting the gap at intervals ∼10 °C during the temperature ramp. No differences were found between these methods. To handle low pH samples at high temperatures, modifications were made to the CSL-100 instrument using accessories designed and constructed at the King’s College Instrument Development Unit. This included a ∼2 mm stainless steel false bottom plate, and a stainless steel cover-slip made in two parts. When used together with a solvent-trap, these modifications greatly limited evaporation and damage to the genuine bottom plate, allowing measurements to be made over several hours. A thin layer of petroleum jelly was applied over the cover-slip to further enhance its effectiveness. The thickness of the false bottom plate was critical as it determines the effectiveness of the Peltier temperature control system. To ensure correct temperature control, a silicon heat-sink paste (Dow Corning) was applied between the genuine and false plates, and the effect on the heating

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ramp was found to be negligible. The time-delay to reach the experimental temperature from an intermediate temperature of 70 °C was monitored for each experiment, the results being 60 ( 5 s at 75 °C, and 120 ( 8 s at 80 °C. Cure data were corrected by adding these delays to the time coordinate recorded by the mechanical spectrometer it being known that the sample actually started aggregating rapidly from about 70 °C onward. As was noted above, such pseudo strain-controlled measurements are generally to be avoided, since the sample stress-strain history may be more complex than simply sinusoidal, as the instrument control system “locks on” to the requested strain. Indeed, the same criticisms can be applied to the commonly used “multi-wave” controlled-strain experiment, although this is now quite widely utilized. However, there is no doubt that, at least potentially, controlled-stress instruments, even in the controlled-strain mode, can attain greater sensitivity than conventional controlled-strain instruments. This is because they are typically lighter and are able to monitor on the same measurement head, rather than having to separate the drive system from the transducer. (In a typical controlled-strain instrument, the drive head operates on one part of the sample geometry, say the plate of a cone-plate system, and the transducer head is attached to the other, say the cone.) In the present case, any real differences were assessed by comparison with the earlier, if more limited, data obtained with a controlled-strain instrument. Indeed, comparison of the two serves as a further check on the linearity (or otherwise) of the strain response. One final aspect is, we suggest, the apparently greater sensitivity of instruments in the pseudo controlled-strain mode to long-time instrument drift, as well as to the usual inertial artifacts. Results and Discussion Cure Curves. Cure curves (G′ versus time) were measured for gelling β-lactoglobulin solutions as described above. The data were obtained at two temperatures (75 and 80 °C), one batch of β-lactoglobulin being used in the 80 °C measurements and two distinct batches (but same Lot Number) at 75 °C. Typical results at 80 °C are presented in Figure 1 for concentrations in the range 8.3% to 13.8% w/w. Very similar data were obtained at 75 °C over similar concentration ranges. These latter cure curves will not be reproduced here but will be referred to later, as will comparable cure data from an earlier publication.11 The cure data of Figure 1 are typical13 of the sol-gel transitions of biopolymer solutions, the elastic component of the shear modulus starting small but undergoing a sudden increase at the critical gel time and then ultimately leveling off at longer times. The order of the curves in the Figure, i.e., of increasing gel times and decreasing plateau heights follows that of decreasing concentration. The behavior of G′′, the viscous component, which is not shown, was typical of a gel (G′′ , G′) at long times; but, interestingly, G′, though small initially, was also greater than G′′, i.e., even before the gel point. This “solidlike” behavior of the starting protein solutions has already been commented upon11 and has been observed for other gelling globular protein sys-

Gosal et al.

Figure 1. G′ versus time, cure data, for the β-lactoglobulin (batch 1) pH 2 system heated at 80 °C. For experimental conditions see text.

Figure 2. Data of Figure 1 re-plotted as log G′ versus 1/time. Solid lines indicate extrapolations used to estimate long-time limiting modulus values G′inf.

tems.14 It makes gel time estimation a more empirical operation than it might otherwise have been (i.e., the WinterChambon criterion15 could not be applied). Another feature of the cure data in Figure 1 is the “dipdown” in the modulus which is visible at long times for most of the concentrations studied. This appears to be an artifact caused by slippage of the gel samples after long times of heating at an elevated temperature. This may indicate eventual gel shrinkage and syneresis but creep of the silicone oil between the gel surface and the rheometer plates provides the most likely explanation. Concentration Dependence of the Extrapolated Modulus (G′inf). Figure 2 shows the data of Figure 1 re-plotted in the form log G′ versus 1/time. Extrapolation of data in this form to 1/time ) 0, where this can be done convincingly, provides an estimate for G′ at very long times, i.e., G′inf. In an earlier paper,11 it was argued, we believe persuasively, that such an extrapolation is essential if subsequent data are to be tested against physical models. Despite this, few workers make such an extrapolation. In this previous paper, a slightly different form of extrapolation procedure was used to that adopted here but one essentially equivalent to the current method10 which is, we believe, rather simpler to perform. In the present case, as the figure shows, the extrapolation is, nonetheless, not totally straightforward, because of the reduction in the modulus at long times mentioned above.

Characterization of Heat-Set Systems

Figure 3. log-log displays of G′inf versus concentration data (80 °C) for the current β-lactoglobulin sample (batch 1) and for a similar sample studied in ref 11. Best fitting power law (solid lines) and cascade (open circles) models are also shown together with corresponding power law indices and predicted (cascade) critical concentrations.

Figure 4. log-log displays of G′inf versus concentration data (75 °C) for current β-lactoglobulin sample (batches 1 and 2). Best fitting power law (solid lines) and cascade (open circles) models are also shown together with corresponding power law indices and predicted (cascade) critical concentrations.

Since this is almost certainly an artifact, linear extrapolation was carried out as shown, using linear portions of the plots occurring prior to the beginning of the dip-down effect. It has to be admitted, however, that this procedure could lead to overestimation of the limiting modulus values, particularly for the lowest concentration data where the cure curves may simply be leveling off in a genuine way. However, overall, the extrapolation method used seemed to be the fairest, and most objective, procedure possible under the circumstances, and it was applied in an analogous way to the 75 °C data (Batches 1 and 2) and to the previously published 80 °C data.11 The limiting modulus results obtained at 80 °C (including those from the cure data of ref 11) are plotted against concentration (log-log display) in Figure 3 and for both batches at 75 °C, in Figure 4. Indications are also given of the best fitting versions of theoretical models for this relationship based (a) on the constant power law fractal approach16,17 and (b) the classical mean field cascade random branching description.10,18-21 It is clear from the figures that, although all of the new G′inf data could be described equally well by either of the models, the 80 °C data from ref 11

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convincingly favors the cascade theory and, by implication, the existence of a finite critical concentration (C0) for gelation (7.1% w/w). The branching theory approach is also favored on physical grounds, for the fractal interpretation implies the absence of a concentration limit for gelation (critical concentration) which is counter to experience, and it derives arbitrarily varying fractal dimensions (from the power law indices shown in the Figures) for the gel network structures. In this last respect, the fractal description seems at odds with the uniform fibrillar networks expected10,22-24 for such finestranded protein gels, these networks being unlikely to be self-similar over a significant length scale. To arrive at a conclusive discrimination between the two models, however, the measurement of G′inf would have to be extended to cover a wider concentration range than considered here, i.e., to include measurements at both higher and lower concentrations. This is not normally feasible for globular protein systems as, in terms of the former objective, the terminal slopes in modulus-concentration plots are not reached until C exceeds 10C0 (or even a greater multiple) when there is no longer any significant sol fraction. For the present systems where C0 is ∼5-7% w/w, such a region is almost impossible to access practically on account of the very high protein concentration implied. Accessing accurate data at low concentrations is just as difficult owing to the long heating periods involved (risk of drying effects). Indeed, in the present work, lower concentration samples were studied and, although estimates of gel times could be obtained for these, the establishment of reliable modulus data at long times proved impossible. A final comment on the data in Figures 3 and 4 relates to the variation of results between samples. The results for the two different batches of β-lactoglobulin at 75 °C are reasonably consistent but are certainly not identical. This is not surprising as batches of the protein supplied commercially could vary somewhat, particularly in relation to residual amounts of salt. The difference between the present 80 °C results and those based on the 80 °C data of ref 11 are larger, however. Batch variation will again play a part in this, particularly as a different Lot Number was involved, but this may also reflect the different experimental procedures applied. The published data of ref 11 were obtained using a different temperature history (sample introduced between preheated plates). Protein gels are not usually equilibrium systems and differences in thermal history, in particular, are known to influence their final structural and mechanical properties.23 As Figure 3 shows, however, despite these differences, the estimates obtained for the critical concentrations (5.2 and 7.1% w/w) using branching theory are not greatly different. Critical Exponents and Modulus Behavior. A further issue regarding the concentration dependence of the modulus is the relationship between the latter, again represented by its extrapolated value G′inf, and the difference between the ratio C/C0 and unity. According to percolation theory,25 there should be a dependence of the form G′inf proportional to (p - pc)γ or, more appropriately for the present analysis, (p/pc - 1)γ. Here, p and pc are, respectively, the probability of bond formation and the critical value for this probability at

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the gel point. Thus, if we make the usual assumption26 that p/pc ≈ C/C0 , this gives G′inf proportional to (C/C0 - 1)γ as C f C0. In the percolation model, the so-called critical exponent γ is “universal” to the geometry of the underlying lattice graph of the percolation process. This should be carefully distinguished from the power law exponents referred to in the last section which relate to dependence on C or C/C0 alone. In the literature a number of γ values exist, but the critical exponent for a 3D gelation process is typically given within percolation theory as ∼1.7.25 This contrasts with the classical Bethe lattice value of 3.18 Over the last 20 years, there has been much discussion of the “universality” of this approach, and perhaps it is fair to say that there is still no real agreement because, as Stauffer suggests,27 there is “...a failure of cooperation between physics and chemistry...”. Nevertheless, there have been many attempts to apply such a formalization to data from gelling systems, particularly to examples much more closely following ideal behavior than the globular protein aggregation systems studied here.28,29 The problem with this calculation, however, is that, as pointed out by Gordon and Torkington in 1981,30 the value of exponent γ is very sensitive to the precise value of p/pc adopted. In percolation theory, pc usually depends on the lattice, but in fitting to data, it is normally estimated from experimental results. In situations of the type addressed in the present study, it appears that most workers follow an analogous procedure; that is, they first estimate C0 and then evaluate C/C0 values from this. However, as the paper cited above demonstrates,30 the value of γ extracted is likely to change with, and hence be correlated to, the value of C0 adopted. Alternatively, values of γ can be assumed, and the critical concentration calculated back, but whatever the approach, without great care, it is clearly susceptible to error. Recently, it has been suggested that γ for protein gelformation belongs to a certain “universality class” which gives γ ∼ 2.7 when C0 is assumed for a variety of protein systems.31 This is reasonable, provided that the strategy for determining C0 is made explicit and can be accepted as satisfactory. It appears that, whatever approach is adopted, there is a clear correlation between C0 and γ, as predicted,30 with higher values of C0 giving lower values of γ and vice versa. For the present exercise, the strategy has been adopted of fitting the function G′inf ) κ(C/C0 - 1)γ by least squares, minimizing the difference {log(G′inf,exp) - log(G′inf,calc)}, where the subscripts “exp” and “calc” refer respectively to the experimentally extrapolated modulus values described previously and the values calculated from the above formula. The variable parameters include the proportionality constant κ, C0, and γ. Adopting log(G′inf) values for the fit is consistent with the approach used in the fits to branching theory described earlier. It does, of course, implicitly weigh (inversely) against the higher modulus values, but since the theory is meant to apply in the limit C f C0, this seems an inherent advantage. It also fits the usually plotted function log(G′), rather than the linear version G′, and it is well accepted that rheological parameters of interest are always plotted logarithmically.

Gosal et al.

Figure 5. (a) Critical exponent plot for heat-set (80 °C) β-lactoglobulin gels. Open squares: pH 7, filled circles: pH 3, filled triangles: pH 2.5, open triangles: pH 2 - no added salt except during pH adjustment, data from ref 11. Open circles: pH 7, 0.1 M NaCl, data from ref 33. Dotted lines, where present, represent 95% confidence intervals. (b) As in (a), but for all pH 2 data. Open triangles: ref 11, open circles: current 80 °C, filled triangles: current 75 °C Batch 1, open squares: current 75 °C Batch 2.

Applying this method to our earlier published data11 for heat-set (80 °C) β-lactoglobulin gels covering the pH range 2-7 produces a good fit, with the parameter γ lying in the range 2.2-2.8. Within this region, there does not seem to be any obvious discontinuity in γ values, although, from EM measurements on sub-gelling systems, this region corresponds to the pH driven transition from essentially particulate gels (pH 7) to increasingly fibrillar systems as the pH falls below 3.32 For example, at pH 7, the best value of this exponent is ∼2.65 ( 0.25 (standard error, SE); for pH 3, ∼2.73 ( 0.08; pH 2.5, ∼2.34 ( 0.16; and pH 2, ∼2.64 ( 0.12. These results are illustrated in Figure 5a. Also included in this figure are data from a very recent study of β-lactoglobulin gelation at pH 7 by Pouzot and co-workers,33 which has the advantage for data fitting purposes of having 19 data points extending to much lower modulus. This series gives a γ slope of 3.13 ( 0.07, remarkably close to the classical value. The disadvantage of this data set, however, is that no extrapolation to infinite time has been carried out, so the ordinate here is G′ not G′inf . Thus, kinetic factors will tend to influence the low modulus data preferentially and tend to shift the data from congruency. Some of the differences, however, may also reflect the effect of added salt on gelation profiles. Unfortunately, on making a similar analysis for the new pH 2 data described in the present work, there is the problem

Characterization of Heat-Set Systems

that there are no G′inf values < ∼103 Pa. This had the consequence that attempts to fit all three of the relevant parameters κ, γ, and C0, simultaneously, proved impossible, essentially because there were no measurements sufficiently close to C0. In consequence, to proceed further, two types of calculation were examined. First, we assumed that C0 was always close to the value suggested from the branching theory fit. When this was done, the results for γ were not far from the values cited earlier for the data from ref 11. Consistency with these results was thus achieved. In fact, the values obtained for γ were slightly smaller, lying in the range 2.262.41. As Figure 5b shows, plots of the final results, based on this assumption, lie close to, but not completely congruent with, those from the earlier pH 2 experiments. In the figure, the straight line drawn with a slope of 2.39 ( (SE) 0.18 is a fit to all pH 2 data, past and present. One problem with constraining C0 to the results estimated using the branching (cascade) theory model is that these γ values would inevitably be biased toward the classical Bethe lattice result as this lattice underlies the branching theory. Consequently, an alternative approach was adopted in which C0 was allowed to “float”, that is to take on a range of values, and the global best fitting model with respect to κ and γ then sought. In this way, the original unsuccessful threeparameter fit was re-examined from a different perspective. However, when this method was implemented, the results were essentially unphysical, for example for the 75 °C, Batch 2 data, the best fitting model, in terms of the minimum sum of squared differences, corresponded to C0 ∼ 1.3% w/w and an exponent γ ∼ 5.1. In reality, the least-squares surface was fairly “flat” confirming the difficulties experienced during full parameter refinement and suggesting that the data could not support unique values for γ. In other words, for the new pH 2 data some extra assumption about C0 has to be made to fix the critical exponent. The overall conclusion for heat-set β-lactoglobulin gels is, therefore, that where the exponents can be estimated directly from data (here the data from refs 11 and 33) they appear to lie between 2 and 3 and where they cannot (present pH 2 data), a similar range is credible, if not conclusively proved. Such exponent values are appreciably higher than earlier percolation estimates and could be consistent with the Bethe lattice value of 3 though values for γ, smaller than 3, could also be explained by the classical approach if network wastage is taken into account. Interestingly, also, is the fact that these results are not far from some of the estimates suggested by Van der Linden and Sagis.31 However, in our view, the uncertainties in making such estimates, already outlined, prevent us from supporting any particular model, nor do these estimates more than hint that a change in network structure from particulate to fibrillar would alter the exponent significantly. This is probably somewhat surprising to those who treat the network-forming units as fractal objects, but perhaps it reflects the percolation sensitivity to branch-point structure rather than to overall particulate “shape”. Finally, a further uncertainty, to add to those above, is that critical exponents are defined to be critical only when data are accessed within the critical region. This, in turn, is

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Figure 6. log-log display of a typical G′ versus time cure curve (80 °C: 9.7% w/w β-lactoglobulin pH 2) indicating logarithmic discontinuity method used in calculating gel times.

defined by various criteria such as that of Ginzburg. Evaluation of the extent of the Ginzburg region remains the realm of the renormalization group theorist,34 but a useful practical guide is given by Stauffer, Coniglio, and Adam,25 who suggest 10-2