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Heat-Induced Gelation of Globular Proteins: 4. Gelation Kinetics of Low pH β-Lactoglobulin Gels Gaynor M. Kavanagh,†,§ Allan H. Clark,‡ and Simon B Ross-Murphy*,† Division of Life Sciences, King’s College London, Franklin-Wilkins Building, 150 Stamford Street, London SE1 8WA, U.K., and Unilever Research, Colworth House, Sharnbrook, Bedford MK44 1lQ, U.K. Received March 28, 2000. In Final Form: July 11, 2000 Particulate gels, especially those formed from heating protein solutions, have been extensively investigated over the years. One focus of this work has been, for commercial reasons, on rather crude mixtures of the main milk protein components (whey isolates), β-lactoglobulin and R-lactalbumin. Moreover, most previous work has concentrated on examining structural and rheological properties of fully cured gels. In the present paper, a less pragmatic approach is adopted, and the gelation behavior of solutions of relatively pure β-Lg (under isothermal heating at 80 °C) are investigated over a range of concentrations and pH values (7, 3, 2.5, and 2). Both gel time and limiting (extrapolated) long-time modulus data were measured and were considered in light of currently available models for the gelation process. Whereas the gel time data was described best by a semiempirical model introduced by one of the authors, the modulus data could be quite adequately understood in terms of branching theory (cascade theory description) results, corresponding well with conclusions from structural studies of the gels using negative-staining electron microscopy. A fractal model was much less successful in this respect. Gel time data, it seems, reflect much more sensitively the details of network assembly in a particular case. Modulus-concentration data, on the other hand, are less determined by gel type and more universal.
1. Introduction In Parts 1 and 2 of this series, we have investigated the kinetics of gelation of globular protein solutions, in particular from bovine serum albumin, and related the behavior to a novel kinetic sol/gel “state” diagram.1,2 In Part 3,3 a number of molecular probes (TEM, FTIR, and wide-angle X-ray scattering) were used to contrast the properties of the highly fibrillar gels formed by heating low pH ( ∼2 (Figure 1). The curve fits are weighted toward the final experimental modulus values, although modification of the fit could be made to obtain estimates for gel times. The modulus results obtained using the extrapolation method are tabulated in Table 1 together with the final (i.e., longest-time) experimental values. Some specific observations are now outlined in advance of a more complete examination of the data. 4.3 Observations for Different Systems. 4.3.1 Gelation of β-Lactoglobulin at pH 7. Not unexpectedly, G′ and G′′ increase, and the corresponding gel times decrease, as the concentration increases. The difference between filtered and unfiltered (but concentration-corrected) (37) Kavanagh, G. M. Rheological and structural studies of denatured globular protein gels and sols. PhD Thesis, University of London, 1998.
samples was not highly significant, although results for the latter are more scattered, a fact attributable to the small amount of undissolved matter present in these gels. 4.3.2 Gelation of β-lactoglobulin at pH 3. Figure 3 shows the combined time sweep data obtained over the concentration range 7.5-17.0% w/w for β-Lg at pH 3. This shows all the G′ data increasing with respect to both concentration and time; a similar result was observed for the G′′ data, although these were typically about 10× smaller than the corresponding G′. The pH 3 results appeared to be more reproducible than the pH 7 data. 4.3.3 Gelation of β-lactoglobulin at pH 2.5 and pH 2. At the lower pH values 2.5 and 2, the stainless steel cover plate was fitted to the RFSII before commencing the mechanical measurements. Extra time was allowed to ensure the cover plate had reached the desired temperature. Fewer concentrations were examined at these pH values, as the main interest was in establishing C0. Figures 4 and 5 show the characteristic gelation curves at pHs of 2.5 and 2, respectively. 4.3.4 Gelation of β-lactoglobulin in D2O. Almost all previous gelation experiments on β-Lg have employed buffer solutions or deionized water with the pH then adjusted. Few workers have investigated the implications of performing gelation experiments in D2O, although many molecular techniques (IR, neutron scattering) are usually performed in this solvent, and it is assumed then that the gelling behavior will be the same. However, recent scattering experiments23 imply that for β-Lg, at least, the gelation kinetics could be 2-fold slower in D2O than in deionized water. In the present work, time sweep experiments were carried out on lactoglobulin samples prepared in D2O, the pD adjustment being performed using deuterium chloride instead of hydrochloric acid. Two pD values were examined: pD 7 (pH 6.6 on the pH meter) and pD 3. Figure 6 shows cure curves for two 15.0% lactoglobulin samples (pH, pD ) 7), one in D2O and the other in H2O. The gel times and final modulus values are quite different. This difference indicates that pD and pH cannot be considered equivalent in this particular pH/pD
Heat-Induced Gelation of Globular Proteins
Figure 7. Oscillatory shear (frequency) sweep for 15.8% β-Lg gel (80 °C, filtered, pH 7); G′ (squares), G′′ (circles), and η* (triangles).
region. (It is worth pointing out that no insoluble particles were seen when the sample was dissolved in deuterium oxide at this pD value.) In contrast, at pD 3, agreement was excellent, the results at pH 3 and pD 3 virtually overlapping (data not shown). 4.4 Frequency and Strain Dependence. Frequency spectra were measured over the ranges 10-2 to 102 rad/s for gels at pH 7 and 3 and 10-2 to 101 rad/s for gels at pH 2.5 and 2, using strains between 1 and 10%. A typical frequency sweep is shown in Figure 7. In general, gel mechanical spectra are insensitive to structural detail; for all pHs, G′ was 1-2 orders of magnitude greater than G′′, and both moduli showed a slight increase as the frequency increased. Some concentration dependence is expected, however, and as protein concentration increases, the difference in G′ and G′′ usually increases and the loss tangent (tan δ ) G′′/G′) decreases. Here, tan δ was found to be relatively independent of frequency, although it decreased with increasing concentration. At the lower concentration end, that is, 13.0% w/w for pH 7 and 7.9% w/w for pH 3, the storage moduli were 2 and, as indicated earlier, the basic assumption of Model 1 of random pair-wise cross-linking of bonding sites on denatured protein molecules is incapable of explaining such high exponents. Other workers facing much the same problem for polysaccharide gels and gelatin41 were forced to generalize the pairwise cross-linking description to involve the simultaneous coming together of several crosslinking sites. However, this was done at the expense of introducing high-order reaction kinetics, which seems improbable in the present β-Lg case. Interestingly, Model 2 fits the overall data of Figure 9 much better, although it does have arbitrary parameters. This model also allows estimates of C0 to be obtained, which can be compared with corresponding results estimated from modulus data (see below). For the pH 7 data, Model 1 appears to describe the gel time-concentration relationship rather better (Figure 10),
but the data are extremely scattered since many of the gel times fall between 10 and 100 s. Because of the time required for sample loading and thermal equilibration, we feel any value 1, and f (>2) is the functionality, the potential number of cross-linking sites per initial molecule.
β)
(f - 1)Rν (1 - R + Rν)
(4)
Here, ν is the so-called extinction probability, effectively the probability that a particular network chain is of finite length and always unity prior to gelation, obtained as the lowest positive root of the equation
ν ) (1 - R + Rν)f-1
(5)
Since Equation 5 is recursive, the overall solution has to be obtained numerically. The consequent modulus versus concentration model4-6,43 has three independent parameters, K, a, and f. Here, K is a ratio of rate constants in the irreversible gel model but is an equilibrium association constant if cross-linking equilibrium can be assumed; it can be calculated from R. The critical gel concentration Co, for example, is then given by
C0 )
M(f - 1) Kf(f - 2)2
(6)
The model is generally better at describing the long-time properties of growing networks than the initial stages of curing. At longer times, detailed kinetic history seems to be “forgotten” and a greater universality prevails. Two different computer programs were available to fit the cascade model (reversible or irreversible forms) to G′∞ (42) Gordon, M.; Ross-Murphy, S. B. Pure and Appl. Chem. 1975, 43, 1.
(40) Ross-Murphy, S. B. Carbohydr. Polym. 1990, 14, 281. (41) Oakenfull, D. J. Food. Sci. 1984, 49, 1103.
(43) Clark, A. H. In Food Structure and Behaviour; Blanshard, J. M. V., Lillford, P. J., Eds.; Academic Press: London, 1987; p 13.
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Figure 11. G′∞ versus concentration for pH 7, 3, 2.5, and 2, fitted using branching theory; both fitting methods produced the same results when f was constrained. For parameters, see Table 2: pH 7 (squares); pH 3 (circles); pH 2.5 (filled triangles); and pH 2.0 (open triangles).
(assumed ≈ G) versus concentration, C, data. One of these used the Simplex method to optimize parameters globally, the other the Newton-Raphson algorithm to carry out an iterative nonlinear least-squares minimization.44 The first program was used to obtain initial parameter estimates and the second to examine the nature of minima and to obtain final estimates for parameters and their standard deviations/correlation coefficients. M was assumed ∼18 000, the MW of the basic β-Lg monomer unit, though in some cases the dimer value of 36 000 might have been more appropriate. In fact, the parameters a and K increase in proportion to M, so the influence of changing M is easily accounted for. Fitting was performed using 15 data points at pH 7, 14 data points at pH 3, and 6 data points at pHs 2.5 and 2. Reasonable estimates (based on previous globular protein work)5,10 were introduced for the functionality, the front factor, and K, and these were then optimized using the Simplex program. Final optimum values for these parameters (and for the related critical concentration Co) were obtained for each pH data set, together with ∑∆2, an indicator of the quality of fit defined as ∑(log GiOBS - log GiCALC)2. Additional constrained fits were carried out using the Newton-Raphson algorithm, with the functionality held constant at a series of values in the range 2.1-1000 and the remaining parameters, K, a (and hence the dependent Co) allowed to vary. For each f, optimum values for K and a were determined together with standard deviations and correlation coefficients and the qualityof-fit indicator. Although for a matching value of f, this second approach led to essentially the same parameter results as program one, it allowed a clearer judgment to be made of how uniquely determined the functionality obtained by the global search (Simplex method) actually was. This point will be returned to later. The best fits to experimental data for each pH are plotted on single graphs of log G′∞ versus log C in Figure 11. As was established from the gelation time versus concentration fits and from earlier work on other protein systems, heat-set globular protein gels can differ substantially in Co depending on gelling conditions. For β-Lg, the critical (44) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes; Cambridge University Press: Cambridge, U.K., 1986.
Kavanagh et al.
Figure 12. As in Figure 11 but plotted versus C/C0.
concentration at pH 7 is clearly larger than at acid pH. The fits to modulus-concentration data at acid pH values are particularly good at the lower concentrations measured but less so as concentration increases. The apparent exponent varies from a very high value near C0, to a limiting, and much lower, value at high concentrations. Critical concentration estimates allowed direct comparison of data for β-Lg gels formed at different pH values by plotting this in reduced form, that is, versus C/C0. As Figure 12 shows, consistent with the observed relative values for the critical concentrations at the different pHs, the steepest curve was obtained for pH 7, followed by pH 3, pH 2.5, and finally pH 2. From Table 2, it is clear that the parameters resulting from f-constrained fitting make physical sense for β-Lg gels. Low final f values were obtained, particularly for the acid pH systems (2.5 and 3.0), the maximum value being 6 for the pH 7 data. Consistent with this, denatured protein molecules are not expected to have large numbers of binding sites (i.e., large f). The a values obtained for all pH data sets (∼5 < a < ∼50) are also sensible. Deviations from ideal entropic network behavior (a ) 1) should become apparent through an increase in a. For example, the presence of short stiff network chains is expected to cause a to increase;4,5 in the present case, the value of a was indeed highest (ca. 50) for the pH 7 gels whose networks are known to be based on rather short, thick, and presumably inflexible network strands.1 Interestingly, the a value dropped from ∼12.8 to ∼6.7 on moving from pH 3 to 2, consistent with the fact that TEM negative staining experiments show increasingly extensive linear aggregation over this pH range.3 Subsequent data analysis using the constrained functionality approach6 is less encouraging, a reflection of the very shallow minima in parameter space. For all the data sets, increasing the functionality to higher values than those obtained initially led to very nearly as good fits (slightly better for very high fixed functionality values). This reflects a strong correlation between K, a, and f, the correlation between K and f being particularly reflected in the near constant value of Co obtained for all f, rather than any more physical explanation such as a change in protein conformation. The critical concentrations are thus the only quantities unambiguously determined in the analyses. Despite this, the low functionality “minima” do seem to accord best with other information about the networks, which implies that some resolution of the problem can be achieved by introducing extra evidence as a constraint. Another line of attack is to cover a wider
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Table 2. Tabulated Results for Each pH Data Set Obtained from Branching Theory (Equation 3) Fits with Constrained Functionality Together with Estimated Errorsa calculated values (ESD) Co (% w/w)
∑∆2
(5.0) (2.3) (1.5) (1.06) (1.03)
12.17 12.17 12.16 12.15 12.15
0.858 0.859 0.864 0.873 0.874
10.1 4.6 2.92 1.97 1.89
(0.4) (0.2) (0.1) (0.08) (0.08)
6.86 6.86 6.85 6.82 6.82
0.173 0.175 0.170 0.156 0.154
(1.3) (0.2) (0.03) (
