Alkali Cold Gelation of Whey Proteins. Part II: Protein Concentration

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Alkali Cold Gelation of Whey Proteins. Part II: Protein Concentration Ruben Mercade-Prieto* and Sundaram Gunasekaran Biological Systems Engineering, University of Wisconsin;Madison, 460 Henry Mall, Madison, Wisconsin 53706. † Present address: School of Chemical Engineering, University of Birmingham, B15 2TT Birmingham, U.K. Received December 11, 2008. Revised Manuscript Received March 30, 2009 The effect of the whey protein isolate (WPI) concentration on the sol-gel-sol transition in alkali cold gelation was investigated at pH 11.6-13 using oscillatory rheometry. The elastic modulus increases quickly with time to reach a local maximum (G0 max), followed by a degelation step where the modulus decreases to a minimum value (G0 min). Depending on the pH, a second gelation step will occur. At the end of the first gelation step around G0 max, the system fulfilled the Winter-Chambon criterion of gelation. The analysis of the maximum moduli with the protein concentration shows that (i) there is a percolation concentration above which an elastic response is observed (∼6.8 wt %); (ii) there are two concentration regimes for G’0 max and G00 max above this concentration, where we have considered power-law and percolation equations; (iii) there is a crossover concentration between the two regimes (at ∼8 wt %) for both G0 max and G00 max when both moduli are equal, and this value is constant under all conditions tested (G0 max = G00 max ≈ 4 Pa). Therefore, alkali cold gelation is better represented using two concentrations regimes than one, as observed for other biopolymers.

Introduction

Materials and Methods

In the first part of the series, we have explored the novel rheology of alkali cold gelation of WPI at different gelation pH values.1 Although the gelation of many biopolymers follows a classic sol-gel transition, we observed sol-gel-sol(-gel) transitions with time at gelation pH >11.5. Here we study in more detail the first gelation step, the result of quick swelling and unfolding of protein aggregates that allows the formation of new noncovalent interactions. The fast gelation step and the subsequent degelation step are the major novelties of alkali cold gelation that have not been previously observed. In the present article, we report how both processes are affected by the whey protein concentration using oscillatory shear rheometry.2 Higher protein concentration critically effects gelation by reducing the gelation time and increasing the gel modulus. In previous studies on protein gelation, great emphasis has been placed on using theoretical models to understand such results.3-6 Some theoretical models are suited to the simulation of fibrillar aggregation and gelation,7 and others deal with particulate aggregation and gelation.8,9 The first gelation step in alkali cold gelation will be analyzed with established gelation models and compared with other gelation procedures (e.g., heat-set gelation). *To whom correspondence should be addressed. E-mail: rubenmp@ cantab.net. (1) Mercade-Prieto, R.; Gunasekaran, S. Langmuir 2009, (2) Kavanagh, G. M.; Ross-Murphy, S. B. Prog. Polym. Sci. 1998, 23, 533–562. (3) Kavanagh, G. M.; Clark, A. H.; Ross-Murphy, S. B. Langmuir 2000, 16, 9584–9594. (4) Gosal, W. S.; Clark, A. H.; Ross-Murphy, S. B. Biomacromolecules 2004, 5, 2420–2429. (5) Mehalebi, S.; Nicolai, T.; Durand, D. Soft Matter 2008, 4, 893–900. (6) Renkema, J. M. S.; van Vliet, T. Food Hydrocolloids 2004, 18, 483–487. (7) Clark, A. H.; Kavanagh, G. M.; Ross-Murphy, S. B. Food Hydrocolloids 2001, 15, 383–400. (8) Bromley, E. H. C.; Krebs, M. R. H.; Donald, A. M. Eur. Phys. J. E 2006, 21, 145–152. (9) Pouzot, M.; Nicolai, T.; Durand, D.; Benyahia, L. Macromolecules 2004, 37, 614–620.

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Sample Preparation. BioPure whey protein isolate (WPI), batch no. JE 139-6-420, was provided by Davisco Foods International, Inc. (Le Seur, MN). Well-homogenized 10 wt % WPI solutions in deionized water (pH ∼6.9) were held at 68.5 ( 0.1 C for 2 h; sodium azide was added (0.05 wt %) as a preservative after cooling to room temperature. Solutions were vacuum filtered to remove large impurities (>20 μm) and were stored at 4 C. The pH of the system was determined as in ref 1.

Gelation Rheology. Alkali cold-set gels were formed as follows: to 4 mL of preheated WPI solution in a test tube, equilibrated to room temperature, deionized water and 2 M NaOH were added to yield the desired WPI concentration and pH. Mixing was performed by quickly swirling with the tip of the pipet for ∼5 s, which greatly minimized the formation of bubbles. Immediately after, the solution was pipetted onto the plate of the rheometer. Cone (4) and plate geometry was used with a Bohlin C-VOR (Malvern) in controlled strain mode, at 0.01 strain and 1 rad/s, in the linear regime. The typical delay time between mixing and data collection was about 60 ( 10 s, and it is included in the Figures shown. Light mineral oil (Fisher, O121-1) was applied to avoid evaporation at the edges. Experiments were performed at 22 ( 1 C. Storage Time. To minimize the variability of the heat aggregation process, we considered making a few solutions that would be used for a long period of time (3-30 days). It was assumed that storage at low temperatures would not affect gelation. However, after analyzing all of the data collected, we observed some systematic behavior in samples stored at long gelation times: higher G0 max values, longer times to reach G0 max, lower phase angle δGP and frequency power-law index Δ at the gelation point, and lower critical concentrations (Table 1). These phenomena can be understood if aggregation reactions still occur during storage, resulting in larger aggregates. Future studies should consider this; here we just note the average storage time of the solutions when the measurements were made. The protein concentration sets at different gelation pH (Figure 5) are not distorted by this because each set was obtained from experiments performed in less than 2 days.

Published on Web 4/20/2009

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Table 1. Characteristic Parameters of the Sol-Gel-Sol Transition in the Alkali Cold Gelation of WPI at Different Gelation pH Valuesa

A B C D E F G H I J a

pH

[WPI]c,high (wt %)

[WPI]c,low (wt %)

[WPI]c,high - [WPI]c,low (wt %)

11.60 11.84 11.84 12.07 12.07 12.07 12.55 12.55 12.81 13.00

8.3 7.8 8.9 7.4 7.9 8.5 7.2 8.1 7.8 8.2

6.8 7.0 7.6 5.9 6.6 7.6 5.9 7.0 6.8 6.7

1.6 0.8 1.4 1.5 0.9 1.3 1.3 1.1 1.1 1.6

average 8.0 ( 0.5 6.8 ( 0.6 1.2 ( 0.3 Superscripts a and b denote different preheated WPI solutions.

Results and Discussion Effect of Protein Concentration in a Sol-Gel-Sol Transition. Alkali cold gelation, like any gelation mechanism, is strongly affected by the protein concentration. Figure 1 shows the rheological profiles of alkali cold gelation at pH 11.84 and different WPI concentrations. At this pH, we have shown in the first part of the series that sol-gel-sol-gel transitions are observed with time. We have defined two characteristic parameters: G0 max, the local maximum elastic modulus value at the end of the first gelation step, and G0 min, the local minimum at the end of the degelation step. The initial gelation step is greatly diminished at lower concentrations, thus decreasing the value of G0 max. At low enough concentrations (7.6 wt % in Figure 1a), a G0 peak is not observed around the expected time (between the two dashed lines in Figure 1a). G0 remains unchanged with time and equals that of the liquid solution at alkaline pH, termed G0 L. An average value of ∼0.05 Pa was calculated from all experiments, although Figure 1a shows that it is highly scattered around 0.01-0.1 Pa. The G0 L value should be treated with caution because the low modulus values are at the low end of the rheometer specification. The degelation step after G0 max is also greatly affected by the concentration, resulting in lower G0 min at lower concentrations. G0 min collapses to G00 is observed only at higher concentrations. [WPI]c,high in Figure 1 (pH 11.84) is about 9 wt %, which is slightly different than protein concentration where sol-gel-sol transitions start to be observed (∼8.3 wt %), although at pH 12.07 it was found that they were quite similar at 8-8.5 wt %. 5794

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G0 max at [WPI]c,high (Pa)

time G0 max (s)

time G0 min ≈ [WPI]p p

storage (days)

5.0 ( 2 4.1 ( 2 5.4 ( 2 3.1 ( 1 3.2 ( 2 4.6 ( 2 4.0 ( 4 4.3 ( 1 4.6 ( 1 4.5 ( 2

460 ( 30 435 ( 75 290 ( 60 320 ( 60 310 ( 35 210 ( 30 190 ( 10 134 ( 5 130 ( 7 113 ( 7

NA 5.6 ( 0.9 6.8 ( 0.9 5.1 ( 1 5.0 ( 0.4 4.6 ( 1.2 3.0 ( 0.3 3.5 ( 0.4 3.0 ( 0.3 3.1 ( 0.3

24a 26a 7b 19a 13b 6b 18a 8b 9b 11b

4.2 ( 2

Figure 1. (a) Elastic modulus G0 and (b) phase angle δ during alkali cold gelation at pH 11.84 and different WPI concentrations. G0 L is the average modulus for solutions that do not gel at low concentrations (∼0.05 Pa); dashed lines in plot a represent the time to reach G0 max ( SD.

Frequency Dependence of the Sol-Gel-Sol Transition. What is and what is not a gel has provoked extensive debate in the past (e.g., Flory10). It is arguable that after ∼300 s, when G0 max is observed, a gel is formed. Of course, that depends on the definition of a gel. If the gelation criterion requires very solidlike behavior (e.g., frequency independence of G0 at ∼10G00 ),11 then we do not have a gel at G0 max. However, if a (10) Flory, P. J. Faraday Discuss. 1974, 57, 7–18. (11) Almdal, K.; Dyre, J.; Hvidt, S.; Kramer, O. Polym. Gels Networks 1993, 1, 5–17.

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gel is only required to have more elastic than viscous behavior (δ < 45) and to have moduli values higher than in the delay time (e.g., G0 > 1 Pa),4,7,12 then we do have a gel. A common gelation criterion used to determine the gelation point was developed by Winter and Chambon (WC)13,14 for chemically cross-linked gels, although it is also applicable to physical gels.15 The dynamic mechanical behavior at the gel point is characterized by a scaling relation between the dynamic moduli and the oscillation frequency ω G0 ðωÞ≈G00 ðωÞ≈ωΔ

ð1Þ

and a frequency-independent phase angle at the gel point (δGP) δGP ¼ Δπ=2 or tan δGP ¼ G0 =G00 ¼ tanðΔπ=2Þ

ð2Þ

and obtaining the same value of Δ with both equations. Percolation theory predicts 0.67 < Δ < 1, although lower values are commonly found.16 If gelation occurs when G0 = G00 , as in stoichiometrically balanced end-linking polymers, then Δ = 0.5.13 Muthukumar17 considered the screening of the excluded volume interaction in order to explain values of Δ smaller than those predicted by percolation theory, resulting in eq 3 to calculate the fractal dimension df. Δ ¼

dðd þ 2 -2df Þ 2ðd þ 2 -df Þ

ð3Þ

Alkali cold gelation experiments could not be performed at a low enough frequency to reach the plateau regime of the modulus. Such experiments require long sampling times, which makes the determination of the local maximum at the end of the first gelation step all but impossible. Here we are constrained by the fast kinetics of the process. It should be noted, therefore, that all of the moduli given are frequency-dependent. Figure 2 shows an alkali cold gelation experiment at pH 11.6 with a small degelation step. This experiment was stopped several times to perform frequency sweeps, as shown in the inset. It was observed that between 300 and 600 s the frequency dependency of G0 and G00 was the same and coincides with the value of Δ found from δGP at the same time. At longer times, G0 and G00 have different frequency dependencies. Ikeda18 also observed the WC criterion in the heat-induced gelation of WPI in the absence of salts. Another experiment, shown in Figure 3a, was performed at the same pH but using an older WPI solution (∼16 vs 4 days old, respectively). The WC criterion was also observed, but with a lower value of Δ (Figure S1 in the Supporting Information). Because the degelation step was a bit more pronounced here than in Figure 2, δ is observed to increase lightly after the gelation point at ∼300 s, and so did Δ calculated from G00 . In addition, full gelation experiments were performed at different frequencies, and the frequency dependence of G0 max and G00 max is shown in Figure 3b (filled points). Large scatter of the data is observed, but if a power-law relationship is assumed then the calculated values of Δ agree well with those performed with the (12) Horne, D. S. Int. Dairy J. 1999, 9, 261–268. (13) Chambon, F.; Petrovic, Z. S.; Macknight, W. J.; Winter, H. H. Macromolecules 1986, 19, 2146–2149. (14) Winter, H. H.; Chambon, F. J. Rheol. 1986, 30, 367–382. (15) Lin, Y. G.; Mallin, D. T.; Chien, J. C. W.; Winter, H. H. Macromolecules 1991, 24, 850–854. (16) Scanlan, J. C.; Winter, H. H. Macromolecules 1991, 24, 47–54. (17) Muthukumar, M. Macromolecules 1989, 22, 4656–4658. (18) Ikeda, S. Food Hydrocolloids 2003, 17, 399–406.

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Figure 2. Alkali cold gelation at pH 11.60 and 9.55 wt % WPI and 1 rad/s where frequency sweeps were performed periodically (inset). The calculated Δ from frequency sweeps are shown at different times (filled points, diamonds for G0 , squares for G00 ). (Inset) Frequency sweep between 270 and 400 s. Best fit power-law frequency dependency Δ: 0.43 ( 0.02 for G0 and 0.42 ( 0.01 for G00 .

Figure 3. (a) Alkali cold gelation at pH 11.60 and 9.55 wt % WPI and 1 rad/s where frequency sweeps were performed periodically. The calculated Δ from frequency sweeps are shown at different times (filled points, diamonds for G0 and squares for G00 ). Open points are the power-law indices at Gmax (shown in plot b) and Gmin. (b) G0 max and G00 max in experiments performed at different oscillation frequencies ω at pH 11.60 (filled points) and pH 12.55 (open points). The time to reach G0 max was 440 ( 100 s at pH 11.6 and 140 ( 15 s at pH 12.55. Best-fit Δ: 0.32 ( 0.05 for G0 max and G00 max at pH 11.60; 0.34 ( 0.03 for G0 max and 0.20 ( 0.03 for G00 max at pH 12.55. Similar experiments at pH 11.84 are shown in Figure S1.

frequency sweeps (shown in Figure 3a as empty points). The Δ values for G0 min and G00 min are also in agreement with those found with frequency sweeps in a single experiment. The triple DOI: 10.1021/la804094n

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agreement of Δ at the gelation point (from δGP, from frequency sweeps in single experiments, and from experiments at different frequencies) was also observed at pH 11.84 (Figure S1a,b). The values of Δ at the gelation point were 0.3-0.4 from 10 data sets between pH 11.6 and 12, and similar Δ values have been reported in the literature.16,19 This results in a fractal dimension df of 2.11-2.22 for the structure of the incipient gel using eq 3. Δ was observed to decrease slightly in WPI solutions stored for longer times, which could be due to the presence of larger aggregates.16 Frequency-sweep experiments were not performed at low concentrations because there was little time to perform them. Thus, it is unclear whether the WC criterion is fulfilled over the whole concentration range studied, although studies with polymers show that it is but the Δ exponent is different.16 In chemically cross-linked poly(vinyl alcohol), the reduction of Δ at lower concentrations was followed by a crossover from G0 > G00 at the gelation point (Δ < 0.5) to G0 < G00 (Δ > 0.5).20 The former scenario is observed in Figures 2 and 3a, plus we also observe the crossover with the protein concentration (Figure 1b), which may imply, by analogy, that the WC criterion is still observed at low protein concentrations. It has been suggested that physical (weak) gels present G0 > G00 at the gel point,20 which is a reasonable description at the end of the first gelation step. We address one further question: is the WC criteria fulfilled at ∼300 s because of the gelation kinetics or because a G0 max value was achieved? We have shown previously1 that the time to reach G0 max is much lower at highly alkaline pH where sol-gelsol transitions are observed than at low alkaline pH. Figure 3b (open points) shows G0 max and G00 max at pH 12.55, which are reached after only 140 ( 15 s, in experiments performed at different frequencies. We observe here that the frequency dependences of G0 max and G00 max are different and a unique value of Δ is not achieved. This suggests that the gelation point after the first gelation step is related to the gelation kinetics, not to the existence of a maximum modulus value, as would be expected. There is an alternative explanation, other than WC, to what is observed at the end of the first gelation step: swollen microgels. Microgels are cross-linked polymer particles that swell in good solvent, and ionic microgels in particular swell at different pH values.21,22 These swollen particles can reach space-filling conditions, above which solidlike behavior is observed at low stresses.23 These swollen microgels present a similar frequency dependence of the modulus to that observed in Figure 2 (inset), yielding values of Δ between 0.1 and 0.4,22,24,25 similar to that in alkali cold gelation but do not present fractal structures. Therefore, eq 3 should not be used to calculate a nonexistent df. Considering that swelling is very important in the first gelation step, as discussed in part I,1 the behavior observed at ∼300 s may be due to a microgel-like sol-gel transition instead of WC-like gelation. whereas, we do not think that this is the case. In alkali cold gelation, eq 1 is fulfilled only at a specific time; the gel point;as in any other WC-like gelation, whereas eq 1 is (19) Aamer, K. A.; Sardinha, H.; Bhatia, S. R.; Tew, G. N. Biomaterials 2004, 25, 1087–1093. (20) Kjoniksen, A. L.; Nystrom, B. Macromolecules 1996, 29, 5215–5222. (21) Saunders, B. R.; Vincent, B. Adv. Polym. Sci. 1999, 80, 1–25. (22) Saunders, J. M.; Tong, T.; Le Maitre, C. L.; Freemont, T. J.; Saunders, B. R. Soft Matter 2007, 3, 486–494. (23) Ketz, R. J.; Prudhomme, R. K.; Graessley, W. W. Rheol. Acta 1988, 27, 531–539. (24) English, R. J.; Raghavan, S. R.; Jenkins, R. D.; Khan, S. A. J. Rheol. 1999, 43, 1175–1194. (25) Lally, S.; Mackenzie, P.; LeMaitre, C. L.; Freemont, T. J.; Saunders, B. R. J. Colloid Interface Sci. 2007, 316, 367–375.

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continuously observed in microgels. Equation 1 is a transition state in WC-like gelation, while it is the final state in microgels. Once microgels are fully swollen, higher pH values do not change the frequency dependence,22 even at a high pH of 12.6.25 However, in alkali cold gelation at high pH, such as 12.55 in Figure 3b, G0 and G00 do not even have the same frequency dependency at their peak values; therefore, eq 1 is not fulfilled. In addition, the degelation phase cannot be explained considering microgel behavior because whey protein aggregates are stable and fully swollen with time.1 Finally, alkali cold-set gels have an effective volume fraction of aggregates that is always lower than the critical phase volume where there is close packing of particles. For monodisperse hard spheres, this critical value is 0.64; however, higher values are commonly found because of the particle size distribution and the deformability of the particles.26,27 However, the volume fraction calculated at 6.8 wt % after swelling is ∼0.37.28 Considering the high compressibility of highly denatured proteins, this value is far lower than that expected for microgel-like gelation. Our interpretation of the first sol-gel transition is that swelling is important because it allows the stranded whey protein aggregates to cross the overlapping concentration but it is the alkali denaturation that makes these swollen aggregates reactive to form noncovalent interactions, most probably hydrophobic in nature. This kinetic cross-linking process is not consistent with a microgel interpretation of the frequency data. Kinetics of the Characteristic Modulus. The effect of protein concentration on gelation kinetics is shown in Figure 4. The time to reach G0 max is very weakly dependent on protein concentration. Assuming a power-law relationship, power-law indices of around 1.1-1.6 are found in four data sets (one of them is shown in Figure 4 by open circles), which are barely statistically different than zero considering the scatter of the data and the small range of protein concentration. At high pH, in sol-gel-sol transitions, the time to reach G0 max is greatly reduced, and any effect of the protein concentration would be on the order of the sampling time (15 s). Therefore, with the accuracy and repeatability of the present data, we consider that the time to reach G0 max is independent of the WPI concentration, the average values of which are given in Table 1. However, the time to reach G0 min is strongly affected by the protein concentration. Figure 4 shows that power-law relationships can fit the data well, assuming that the time to reach G’min is ∼[WPI]p; the calculated power-law indices p for each data set are given in Table 1. The time to reach G0 min in sol-gel-sol transitions is different than in sol-gel-sol-gel transitions, as discussed in part I;11 the former is when G0 collapses to less than 0.1 Pa. The different methodology may be the reason behind the different p values calculated for the different regimes: 4.6-6.8 in sol-gel-sol-gel and 3-3.5 in sol-gel-sol transitions (Table 1). The small effect of the protein concentration with the time to reach G0 max suggests an uncooperative process, as would be the (26) Frith, W. J.; Lips, A. Adv. Colloid Interface Sci. 1995, 61, 161–189. (27) Adams, S.; Frith, W. J.; Stokes, J. R. J. Rheol. 2004, 48, 1195–1213. (28) This value is calculated as follows: We use the volume fraction of a 6.8 wt % solution that at the percolation concentration [WPI]c,low, considering a typical protein concentration in the whey powder of 90 wt % and a specific volume of the native protein of 0.75 mL/g (using β-lactoglobulin as reference), gives a volume fraction of 0.046 before swelling. The use of the specific volume for native conditions is justified because of the low heating temperatures used to make the aggregates. In part I, we showed that the effective diameter of the swollen aggregates is twice that at neutral pH. If we assume that the corresponding volume increase occurs fully in the protein aggregates, whichis highly unlikely because the aggregates are not spherical but strandlike, then the volume fraction after swelling has a maximum value of 0.046  23 = 0.37.

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over the whole concentration range examined. We assume the following relationships

Figure 4. Time to reach G0 max (open symbols) and G0 min (filled symbols) at pH 11.84 (triangles), 12.07 (circles), and 12.81 (squares). The time to reach G0 min at pH 12.81, a sol-gel-sol transition, corresponds to when G0 < 0.1 Pa. Lines are the best power-law fit to G0 min, yielding the following p exponents: 6.8, 5.0, and 3.0 at pH 11.84, 12.07, and 12.81, respectively (Table 1).

swelling of individual aggregates. However, the time to reach G0 min depends linearly on the destruction of the modulus (Figure 6 in ref 1) and therefore on the extent of cross-linking at G0 max. We have suggested that the modulus increase during the first gelation step is related to the formation of new noncovalent interactions, which are highly cooperative and would explain the high p values found here. For example, Bauer et al.29 reported a power index of 4.3 for the rate of aggregation of metastable βLg oligolmers at pH 8.7 through noncovalent interactions. The present values of p in the sol-gel-sol-gel transition are similar to those reported for the gelation time in the single-step gelation of βLg, obviously with the opposite sign (-5.5 to -6.4).4 Effect of Protein Concentration on the Gel Modulus. Here we characterize the protein concentration effect at the end of the first gelation step, using G0 max and G00 max. Figure 5a-e shows five data sets at different gelation pH values between 11.84 and 13. Following the description provided previously for Figure 1a, at low protein concentrations G0 is that for the liquid solution (G0 L); this limit is shown in Figure 5a-e as horizontal dashed lines. As the protein concentration is increased, a mechanical response starts to be observed in the rheometer (G0 > G0 L) showing the characteristic peak of the elastic modulus. The concentration when this starts to happen is termed [WPI]c,low. Above [WPI]c,low, G00 max is higher than G0 max, but the latter increases more steeply with the protein concentration until a point is reached where G0 max = G00 max. The concentration when this occurs is termed [WPI]c,high. Above this value, the expected G0 max > G00 max regime is observed. Therefore, there exists a critical concentration [WPI]c,low, and a crossover concentration [WPI]c,high, which is observed in 10 data sets at different pH values. The values determined are shown in Table 1, with a confidence interval of (0.2 wt %. In the present data, a clear trend in these critical concentrations with the gelation pH is not observed; the calculated average values are 6.8 and 8.0 wt % for [WPI]c,low and [WPI]c,high, respectively. Note that the difference between these two concentrations is also independent of the gelation pH (Table 1). The value of these concentrations is smaller for the WPI solutions stored for longer times, suggesting again that older solutions present larger aggregates. A log-log representation of the data, as in Figure 5, reveals that two linear regimes can be used to fit G0 max and G00 max (29) Bauer, R.; Carrotta, R.; Rischel, C.; Ogendal, L. Biophys. J. 2000, 79, 1030– 1038.

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G0max ≈½WPIn

ð4Þ

G00max ≈½WPIm

ð5Þ

where the power-law indices n and m are higher in the low-concentration regime than at higher concentrations. The two regimes for G0 max and G00 max cross over at the same concentration, which is, incidentally, the same concentration where G0 max = G00 max: [WPI]c,high. The calculated n and m parameters in the different data sets are shown in Table 2. With the present accuracy and repeatability, we do not observe that the gelation pH affects either n or m. The average values found are 14 and 26 for n and 9.8 and 14.4 for m in the low- and high-concentration regimes, respectively. G0 min also shows a very strong dependence on the protein concentration. Power-law indices around 20-40 are calculated at pH values where sol-gel-sol-gel transitions occur and are of the same order of magnitude as n in the low-concentration regime (Table 2), although G0 min is significantly more scattered than G0 max (results not shown). Considering that G0 min < 10 Pa, the same physics at [WPI]c,low < [WPI] < [WPI]c,high for G0 max may also describe G0 min. The two concentration regimes could be an artifact of the frequency dependence of the modulus or a specific case of alkali cold gelation. However, we are not the first to observe two power-law regimes with the protein concentration. Chen and Dickinson30 reported such behavior in the heat-induced gelation of WPI emulsions gels, where the n value at low concentrations was higher than at high concentrations, as observed here, except for one data set when the opposite was observed. Two other studies in the heat gelation of whey proteins have considered two power-law regimes.31,32 We further analyze the high-quality data of Kavanagh et al.3 for the heat-induced gelation of βLg at different pH values in the absence of salts, as shown in Figure 5f in a log-log representation. Note that these data correspond to the plateau elastic modulus at an (extrapolated) infinite time, G0 ¥, which is not frequency-dependent. We observe that two power-law regimes can also be considered in Figure 5f, despite the scatter at pH 7 and the few experimental points at pH 2 and 2.5. The best data set, that at pH 3, is very well represented by two regimes. The lower part of Table 2 shows the calculated n values from Figure 5f as well as others reported in the literature. Values around 4-7 are typical at high concentrations;4,33 much higher values are found at lower concentrations, as in alkali cold gelation. Puyol31 and Chen and Dickinson30 reported lower n values for the two regimes in gelation experiments at neutral pH, 3.2 and 6.5 for the former and 4.3 and 9.2 for the latter (for the high- and low-concentration regimes, respectively), which could be due to the presence of salts during gelation, 60 mM NaCl in the former and 50 mM TrisHCl in the latter. Assuming the existence of two regimes, the difficulty lies in the determination of [WPI]c,high which separates them. In the present experiments, it was quite easy because the power-law regime change is observed when G0 = G00 , but that does not occur in the other gelation procedures. Another remarkable feature of the (30) Chen, J. S.; Dickinson, E. J. Texture Stud. 1998, 29, 285–304. (31) Puyol, P.; Perez, M. D.; Horne, D. S. Food Hydrocolloids 2001, 15, 233–237. (32) Sagis, L. M. C.; Veerman, C.; Ganzevles, R.; Ramaekers, M.; Bolder, S. G.; van der Linden, E. Food Hydrocolloids 2002, 16, 207–213. (33) Ikeda, S.; Foegeding, E. A.; Hagiwara, T. Langmuir 1999, 15, 8584–8589.

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Figure 5. (a-e) WPI concentration dependence of G0 max and G00 max at pH 11.84-13. Red (G0 max, diamonds) and blue (G00 max, squares) straight lines are the best-fit power-law equations considering two regimes (dashed lines at low concentrations; continuous lines at high concentrations), with [WPI]c,high at their intersection. The black horizontal line is G0 L (Figure 1a). Continuous purple lines for G0 are fit considering the percolation model for the whole concentration range using [WPI]c,low. The best-fit power law indices n (for G0 max) and m (for G00 max), percolation exponents t, and the critical concentrations are given in Tables 1 and 2. (f) Data reported by Gosal et al.4 for the plateau modulus (G0 ¥) in the heat-induced gelation of βLg at different pH values. Notice that two power-law regimes, with a transition [βLg] in between, can also be considered in traditional heat-set gelation experiments.

crossover point [WPI]c,high is that it occurs at a constant G0 value in all of the data sets, ∼4.2 Pa (Table 1). This agrees with the observation of Chen and Dickinson30 that “the break points in the ln G0 versus ln Cp lines for the pure protein gels and emulsion gels occur at similar values of ln G0 ”. With the available data of heat-set gelation, a similar conclusion can be reached. We estimate from Figure 5f that G0 at the crossover concentration is ∼(0.8-2)  103 Pa. This range agrees well with experiments at pH 2 and 13 mM NaCl from Sagis et al.,32 ∼0.9  103 Pa, and at pH 6.7 and 60 mM NaCl from Puyol et al.,31 ∼103 Pa, and ∼(2-5)  103 Pa in emulsion gels.30 In particulate heat-set gelation (pH 7, 100 mM NaCl), a much lower modulus value is observed at the crossover concentration, ∼20 Pa.9 If the modulus 5798

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of the crossover concentration is verified to be (fairly) constant under similar gelation conditions, then it would be very useful to estimate the crossover concentration and the two regimes. Percolation Models. The two-regime approach that we discussed is quite unorthodox in protein gels, thus we subsequently discuss better-established models. We first consider the percolation model34,35 using G0max -G0L ≈ð½WPI=½WPIp -1Þt

ð6Þ

(34) de Gennes, P. G. J. Phys. Lett. 1976, 37, L1–L2. (35) Stauffer, D.; Coniglio, A.; Adam, M. Adv. Polym. Sci. 1982, 44, 103–158.

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Table 2. Calculated Exponents {n for G0 max (eq 4) and m for G0 max (eq 5)} Considering Two Power-Law Regimes and Using a Percolation Model (eq 6 for t)a,b pH A B C D E F G H I J

11.60 11.84 11.84 12.07 12.07 12.07 12.55 12.55 12.81 13.00 average

n high conc

n low conc

m high conc

15 ( 5 19 ( 4 12.8 ( 1 11.1 ( 1 13 ( 2 11 ( 1 14.4 ( 1 13.7 ( 1 16 ( 3

21 ( 1b 28 ( 5 28.4 ( 2 14 ( 2 24 ( 2 35 ( 4 22 ( 1 29.5 ( 2 28.8 ( 1 21.7 ( 1

12 ( 3 13.1 ( 2 8.9 ( 1 7.7 ( 1 7.9 ( 1 8.4 ( 1 9.8 ( 1 9.7 ( 1 11 ( 2

14 ( 3

26 ( 6

9.8 ( 2

m low conc

12.6 ( 1b

t using [WPI]c,low

15 ( 6 16.4 ( 1 10.3 ( 2 12.9 ( 1 15.4 ( 2 13.7 ( 1 15.5 ( 1 15.7 ( 1 14.4 ( 1

2.7 ( 0.3 NA 2.8 ( 0.15 2.4 ( 0.25 2.2 ( 0.2 2.4 ( 0.1 2.9 ( 0.1 2.4 ( 0.1 2.7 ( 0.06 3.0 ( 0.15

14.4 ( 2

2.6 ( 0.3

t using [WPI]p

[WPI]p (wt %)

8.5 ( 0.3 NA 4.6 ( 0.15 NA 3.9 ( 0.25 3.2 ( 0.1 3.7 ( 0.05 2.9 ( 0.05 2.7 ( 0.06 5.6 ( 0.15

5.1 NA 7.1 NA 7.2 6.3 5.6 6.9 6.8 5.8

7† 7.2 ( 1.6 42 ( 7 2.65 ( 0.25§ 3† 6.6 ( 0.2 21.8 ( 0.3 2.73 ( 0.08§ 2.7‡ 19.5 5.6 ( 0.1 12.5 ( 0.5 2.34 ( 0.16§ 2.5† † 2 6(1 11.0 ( 0.2 2.64 ( 0.12§ ‡ 2 6.4 11.5 a Values at neutral or acidic pH are reported or calculated from the literature: †ref 3, ‡ref 24, and §ref 4. b Set A was fit using a single power-law regime because few data points were obtained. The ( values are the standard error of the fitting.

where t is the critical exponent related to the geometry of the underlying lattice and [WPI]p is the percolation concentration. We have assumed, as usual, that the protein concentration can be used as the probability of bond formation. Several strategies have been presented to fit these two parameters with the experimental data, and overall the analysis of our data agrees well with the arguments presented by Gosal, Clark, and RossMurphy.4 If both parameters were set free to float, then t and [WPI]p were fit by (i) a least-squares method of the decimal log version of eq 6 and (ii) by maximizing the linearity (e.g., R2) of (G0 max)1/t ≈ [WPI].36 The best-fit values of method (i) are shown in Table 2, method (ii) gave very similar values (data reported as NA in Table 2 is when convergence was not achieved). We obtained t values of around 2.7 to 8.5, and as in ref 4, “the leastsquares surface was fairly ‘flat’ confirming the difficulties experienced during full parameter refinement and suggesting that the data could not support unique values for t”. To obtain reasonable t values, [WPI]p was fixed to the value of [WPI]c,low. Fairly constant values were now obtained between 2.2 and 3.0, which agree with those reported in ref 4 (symbol § in Table 2). The good agreement of eq 6 using [WPI]c,low is shown as the continuous loci in Figure 5. The unrealistically high t using methods (i) and (ii) correlate with the calculated [WPI]p far from [WPI]c,low. This correlation between the chosen value of [WPI]p and t is well known37 and reduces the applicability of freefloating methods. Only one data set yielded the same parameters with methods (i) and (ii) and by fixing [WPI]p = [WPI]c,low (pH 12.81 in Table 2); in all other sets, the t values are significantly higher. Many theoretical predictions have been performed on the value of t. Our average value of 2.6 ( 0.3 seems to agree with the predicted ∼2.7 by Martin et al.38 and disagrees with that predicted by de Gennes34 (∼1.9) and ∼2 for isotropic force percolation,39,40 which has recently been preferred in fibrillar protein systems.36,41 However, we agree again with Gosal et al.4 that it is difficult and statistically unsound to discriminate (36) van der Linden, E.; Sagis, L. M. C. Langmuir 2001, 17, 5821–5824. (37) Gordon, M.; Torkington, J. A. Pure Appl. Chem. 1981, 53, 1461–1478. (38) Martin, J. E.; Adolf, D.; Wilcoxon, J. P. Phys. Rev. Lett. 1988, 61, 2620. (39) Mitescu, C. D.; Musolf, M. J. J. Phys. Lett. 1983, 44, L679–L683. (40) Pandey, R. B.; Stauffer, D. Phys. Rev. Lett. 1983, 51, 527. (41) Veerman, C.; Ruis, H.; Sagis, L. M. C.; van der Linden, E. Biomacromolecules 2002, 3, 869–873.

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between models with the present data considering the difficulties involved in estimating [WPI]p. An additional critique of the universality of experimental t values is given elsewhere.42 Two-Regime Models. A pure percolation model is unlikely to explain the whole concentration dependence because it is strictly valid only very close to the percolation threshold. Many studies with globular proteins have reported good powerlaw behavior at concentrations far from the critical concentration.6,9,33,43-46 Thus, it is reasonable to combine both methods: the percolation model (Gpercolation) close to the percolation concentration and a power-law model (Gpower law) far from it. This approach has been successfully applied to two different systems: gelatin47 and dibenzylidene sorbitol gels in poly (propylene oxide).48 The experimental elastic modulus (Gexp) was well fit over the whole concentration range considering47-49 1=Gexp ¼ 1=Gpercolation þ 1=Gpower law

ð7Þ

This harmonic mean approximation favors the smallest calculated modulus, which is assumed to be Gpercolation at low concentrations and Gpower law at high concentrations. For the harmonic mean (or half of it as considered in the previous relationship) to express each model in the different regimes fully and only an average of them around the transition concentration, Gpercolation and Gpower law have to be significantly different. However, that is not the case here, and this results in a very poor fit when applying eq 7 to the present data (results not shown). Although eq 7 may not be applicable in alkaline cold gelation, the idea to combine both models certainly has merit and should be investigated further. Finally, we note that the percolation range extends up to, in our notation, ([WPI]/[WPI]p - 1) ≈ 1 in the literature,47-49 whereas here the low-concentration regime (42) Ross-Murphy, S. B. Polym. Bull. 2007, 58, 119–126. (43) Verheul, M.; Roefs, S. Food Hydrocolloids 1998, 12, 17–24. (44) Alting, A. C.; Hamer, R. J.; De Kruif, C. G.; Visschers, R. W. J. Agric. Food Chem. 2003, 51, 3150–3156. (45) Maltais, A.; Remondetto, G. E.; Gonzalez, R.; Subirade, M. J. Food. Sci. 2005, 70, C67–C73. (46) Eleya, M. M. O.; Ko, S.; Gunasekaran, S. Food Hydrocolloids 2004, 18, 315–323. (47) Joly-Duhamel, C.; Hellio, D.; Ajdari, A.; Djabourov, M. Langmuir 2002, 18, 7158–7166. (48) Dumitras, M.; Friedrich, C. J. Rheol. 2004, 48, 1135–1146. (49) van der Linden, E.; Parker, A. Langmuir 2005, 21, 9792–9794.

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lasts up to ([WPI]/[WPI]p - 1) ≈ 0.1 - 0.25. In fact, our values fall inside the Ginzburg region where the critical regime applies.35 Theoretical support for two distinct regimes above the percolation concentration, with a crossover concentration in between, is given from simulations of semiflexible fibrillar polymers.50-52 The low-concentration regime, above the percolation concentration, is defined as a bending-dominated nonaffine (e.g., nonuniform strain) regime. In the high-concentration regime, the network behaves as a homogeneous elastic medium, which is dominated either by the compression and stretching of the filaments50,52 or by thermal/entropic effects depending on the filament concentration.50 In addition, the physics of the nonaffine/affine crossover is different than for the percolation transition.50 Of course, the exact predictions of these models, specifically focused on understand long linear actin fibrils, are not expected to be valid in alkali cold gelation. A clear example is the very different experimental and predicted n values in actin fibrils, around 2-2.5,53,54 and in the heat- or cold-set gelation of globular proteins, such as those presented in Table 2. Nevertheless, we suggest that a similar approach could be applicable in alkali cold gelation; following that, we observe the same phenomenological behavior with the protein concentration: a liquid regime, a percolation concentration, a sharp increase in G0 , a crossover concentration, and a second milder increase in G0 . The use of a percolation model at low concentrations has many theoretical advantages, following its success in describing many sol-gel transitions in biopolymers.36,55 Yet for the present data, a simple power-law regime is equally valid. Both models yield constant exponent values (n and t) with the gelation pH, although n is significantly easier and more robust to calculate than t because it does not depend on the percolation concentration. We have estimated [WPI]p (considered to be equal to [WPI]c,low) by extrapolating the concentration where G0 max = G0 L ≈ 0.05 Pa, and the rheometer used is far from appropriate at such low values. Hence, our estimates of [WPI]p should be considered to be rough approximations, and because of the high sensitivity of t with respect to [WPI]p, t may change significantly with more accurate measurements. If the average t value calculated here is valid (2.6 ( 0.3), then we wonder why the same value is found for the fibrillar and particulate heat-set gelation of βLg (Table 2), when the gelation mechanisms are very different in all three cases. Van der Linden and Sagis36 suggested that the different variety of gels formed could be used to test the predictions of the dependence of the scaling exponent on the mesostructure. So far, little dependence is observed. Fractal Models. The use of fractal models to describe the elastic modulus dependence on the protein concentration has been criticized because of (i) the failure to account for a critical gelation concentration and (ii) by the large variability of the calculated power-law indices n and therefore of the calculated fractal dimensions.4,36,42,56 The large variability of n is hardly surprising considering that most fractal models are applicable (50) Head, D. A.; Levine, A. J.; MacKintosh, F. C. Phys. Rev. E 2003, 68. (51) Head, D. A.; Levine, A. J.; MacKintosh, F. C. Phys. Rev. Lett. 2003, 91, 108102. (52) Wilhelm, J.; Frey, E. Phys. Rev. Lett. 2003, 91, 108103. :: (53) MacKintosh, F. C.; Kas, J.; Janmey, P. A. Phys. Rev. Lett. 1995, 75, 4425. (54) Gardel, M. L.; Shin, J. H.; MacKintosh, F. C.; Mahadevan, L.; Matsudaira, P.; Weitz, D. A. Science 2004, 304, 1301–1305. (55) Axelos, M. A. V.; Kolb, M. Phys. Rev. Lett. 1990, 64, 1457. (56) Mellema, M.; van Opheusden, J. H. J.; van Vliet, T. J. Rheol. 2002, 46, 11– 29. (57) Shih, W.-H.; Shih, W. Y.; Kim, S.-I.; Liu, J.; Aksay, I. A. Phys. Rev. A 1990, 42, 4772.

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only far from the minimum gelation concentration.9,57,58 Combination with a percolation model, as described previously, can satisfactorily eliminate the variability of n and include the critical gelation concentration. Points (i) and (ii) can also be solved by considering a second power-law regime at low concentrations, as performed here (n values at high concentrations are very similar; see Table 2). Ross-Murphy42 suggested that the limiting power law is seen only for ([WPI]/[WPI]p - 1) J10. Under the two-regimes approach, we have shown in the previous section that the high-concentration regime, which starts after [WPI]c,high, is observed at much lower values (∼0.1 - 0.25), and even in the data of Figure 5f, using the [WPI]p from ref 3, we find {([WPI]/[WPI]p) - 1} ≈ 0.15-0.5. Nevertheless, we acknowledge that the concentration range studied here is too small for us to make any conclusions at much higher concentrations. We analyze alkali cold gelation above [WPI]c,high with the fractal gel models of Shih et al.57 and Wu and Morbidelli.58 We consider that the weak-link regime at G0 max is the most appropriate to use (e.g., covalently cross-linked aggregates are linked with weak interactions), but either the strong- or weak-link regime yields very high fractal dimensions of 2.6-2.9, much higher than expected for a reaction or diffusion-limited aggregation (1.7-2.2).33 These large values are consistent with previous estimates of weak-link gels,58,59 although they have been found in turbid particulate gels,60 not in transparent fine-stranded gels such as those formed in alkali cold gelation. However, the strongest criticism of fractal models is that homogeneous transparent gels do not have a fractal structure over any significant length scale61,62 and therefore such models should not be used.5 Discussion. The large n values that we found may reflect the intrinsic differences between alkali cold gelation and heator cold-set gelation at neutral or acidic pH. Conditions where fine-stranded gels are formed (pH far from the pI, low salt concentrations) systematically show n values higher than when particulate gels are formed (close to pI, high salt concentrations).6,31,59,63 For example, n values for egg white are 3.3, 2.9, 2.8, 4.2, and 6.9 at pH 3, 5, 7, 9, and 11 (pI ∼5), respectively.46 For WPI gels at pH 7, n is 5.4 and 2.7 for 25 and 100 mM NaCl.33 High n values may arise, for example, if instead of assuming a binary reaction of functional groups3 we consider that six whey peptides interact, through hydrophobic interactions, with one whey protein.64 In addition, a comparison of alkali cold gelation exponents with those of other gelation procedures may not be appropriate because of the extensive protein denaturation and aggregate swelling occurring in the former. However, large n values could be an artifact caused by the frequency dependency of G0 max and G00 max. Ross-Murphy42 has argued strongly against the existence of two power-law regimes in the gelation of methylcellulose.65 He asserted that a log G versus log C gives a continuous curve with limit exponents of ¥ very close to the gelation point and ∼1.5-2 at very large values. However, we believe that the claim that this (58) Wu, H.; Morbidelli, M. Langmuir 2001, 17, 1030–1036. (59) Hagiwara, T.; Kumagai, H.; Nakamura, K. Food Hydrocolloids 1998, 12, 29–36. (60) Hagiwara, T.; Kumagai, H.; Matsunaga, T. J. Agric. Food Chem. 1997, 45, 3807–3812. (61) Renard, D.; Axelos, M. A. V.; Boue, F.; Lefebvre, J. Biopolymers 1996, 39, 149–159. (62) Nicolai, T.; Pouzot, M.; Durand, D.; Weijers, M.; Visschers, R. W. Europhys. Lett. 2006, 73, 299–305. (63) Weijers, M.; Sagis, L. M. C.; Veerman, C.; Sperber, B.; van der Linden, E. Food Hydrocolloids 2002, 16, 269–276. (64) Creusot, N.; Gruppen, H. J. Agric. Food Chem. 2007, 55, 2474–2481. (65) Wang, Q.; Li, L. Carbohydr. Polym. 2005, 62, 232–238.

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relationship is continuous still lacks experimental evidence. We certainly do not see it in our experiments. However, we claim that there is a crossover concentration, [WPI]c,high: (i) the transition of the two power-law regimes for both G0 max and G00 max occurs at [WPI]c,high; (ii) G0 max equals G00 max at [WPI]c,high; (iii) the value of G0 max = G00 max at [WPI]c,high is constant under all conditions tested, ∼4.2 Pa; and (iv) at protein concentrations lower than [WPI]c,high, a gel does not eventually form after the degelation step. The existence of a crossover concentration goes against the idea of a continuous curve of log G versus log C. Our assumption of two power-law regimes is based on empirical simplicity, following the scaling equations in nongelled polymeric systems.66 We argue for the existence of two regimes; we do not emphasize their nature (e.g., percolation vs power law) because there is not enough information to make a decision. In fact, everything still remains to be explained with respect to the value of the exponents; existing fractal models are clearly of no use here. Nevertheless, we are not the first to empirically observe two power-law regimes in whey protein gels.30-32 In fact, two other studies with biopolymers;agarose67 and lyzozyme68; show good agreement considering two power-law regimes, although their calculated n values are much lower because of their fibrillar morphology. The existence of a high-n regime at low concentrations was assigned by Ramzi et al.67 to the existence of loose chains (totally disconnected chains or connected only by one end to the network), which do not contribute to the network elasticity, a nonaffine regime. This observation is consistent with fibrillar theoretical studies that predict the existence of a crossover concentration.50-52 Yan et al.68 referred (66) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (67) Ramzi, M.; Rochas, C.; Guenet, J. M. Macromolecules 1998, 31, 6106– 6111. (68) Yan, H.; Frielinghaus, B.; Nykanen, A.; Ruokolainen, J.; Saiani, A.; Miller, A. F. Soft Matter 2008, 4, 1313–1325. (69) te Nijenhuis, K. Colloid Polym. Sci. 1981, 259, 522–535.

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to [WPI]c,high as the macroscopic critical gelation concentration. The fact that in alkali cold gelation [WPI]c,high corresponds to the point where G0 max = G00 max, a traditional way to establish the gelation point,69 suggests that such a description is also quite appropriate here.

Conclusions We have characterized the rheological properties at the end of the first gelation step in sol-gel-sol(-gel) alkali cold gelation, focusing on the behavior of the maximum elastic and viscous modulus values, G0 max and G00 max. Around the time that this maximum is observed, the system is at the gelation point as defined by the Winter-Chambon criterion, the fulfilment of eqs 1 and 2.The WPI concentration has little effect on the time to reach G0 max, but it strongly affects the time to reach G0 min. We relate the first phenomenon to the uncooperative process of aggregate swelling and the second to the formation/destruction of noncovalent interactions. The WPI concentration has a strong effect on the modulus, higher than previously observed under other whey gelation conditions. Two concentration regimes and two critical concentrations were observed, which agrees well with theoretical and experimental studies in fibrillar gelation. This approach is also reasonable for traditional heat-induced gelation. Good agreement was also found under a percolation approach, but why similar t exponents were found compared to other types of protein gelation remains to be explained. Acknowledgment. R.M.-P. gratefully thanks Dr. Alex Routh for helpful discussions on microgels. Supporting Information Available: Frequency dependency of the alkali cold gelation at pH 11.84. This material is available free of charge via the Internet at http://pubs. acs.org.

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