Gelation of Globular Proteins BY M. P. TOMBS Unilever Research Laboratory, Colworth House, Sharnbrook, Bedfordshire Received 22nd October 1973 Globular proteins form gels as a resuIt of aggregation to form strands followed by interaction of the strands to form the gel mesh. An approximate (pore size, concentration) relationship can be predicted from selected models of the aggregation process, which is consistent with that determined from electron micrographs of gels. A limited random aggregation may plausibly lead to gelation, and the mode of aggregation is the main quantitative factor determining concentration requirements. Globular proteins form gels in conditions which would be expected to, and are often known to, disrupt the native structure of the protein. Thus, gels typically form when solutions are heated, or in urea solutions or comparable disrupting agents.Events may be complex; for example, glycinin will gel at 20°C in urea+alcohol mixtures ; these gels melt at 5O-6O0C, and then set again at 70-80°C to give irreversible thermostable gels. No doubt both covalent and non-covalent interchain links are involved. (“ Chain ” will always be the peptide chain. “ Strands ’’ are the rnesh- forming structures in the gel, which may not be single peptide chains). As a rule, to obtain gels from globular proteins requires concentrations an order of magnitude higher than, for example, from gelatin or the gel forming carbohydrates. It is sot obvious how more or less spherical particles could reasonably form a structure such as a gel mesh. Electron micrographs of gels (e.g., fig.1) show a mesh structure which may be imagined as built up from strands ; and the dimensions of the strands and the pore sizes show that they must arise by aggregation of the individual particles. This paper reports an investigation of some possible aggregation processes and attempts to predict expected pore sizes from models ranging from highly-oriented to random aggregation. STRAND LENGTHS: SIMPLE CASES IN A REGULAR CUBIC ARRAY Consider a peptide chain of weight 30 000 mol-’, containing about 280 amino-acid residues. The length of each residue is 3.67A, ignoring the terminals and any proline residues. Thus, the fully extended length of the chain is about 99OA and. taking the specific volume as 0.75 cm3 g-l, 1 cm3 of pure protein contains 2.66 x 1019 particles and a total length I of chain of 2.64 x 1014 cm (2.6 x lo9 krn!).We imagine chains like these linked end to end, at some volume concentration c, and folded into a regular cubic array contained in a centimetre cube. Ifp, is the pore size and d the thickness of the strand then, approximately, and, if p , is large compared with d, this further simplifies to from which for this casep; = 1.13 x 10-14/c cm. (pr+q2 = 311; (1) pr” = 311, (2) Here I is the total length of strands 158T available to form the mesh. At the other extreme, a compact spherical molecule of 30 000 mol-l would have a diameter of about 42 A, and if these aggregate like a " string of beads ", then I = 11.2 x 10l2 cm ~ m - ~ and p: = 2.68 x 10-13/ccm.In an intermediate case, e.g., after thermal disruption, the effective diameter might be about 78 A, and p f = 1.44 x 10-13/c cm. Thus, the concentration requirement to produce a gel of specified pore size is strikingly different depending on the postulated arrangement. Table 1 shows the concentrations required for a regular cubic array for p = 300 A in these three cases, and the pore sizes expected for c = 0.01. RANDOM AGGREGATIONS The simple cases considered above required a highly oriented interaction of a molecule (in three different configurations) to form strands, and then a regular arrangement of the strands. However, in addition to variable configurations of the particle, we must also consider both a possibly random aggregation of the initial particle to form the strands, and a possibly random arrangement of the strands so obtained to form the gel mesh.Protein molecules do not show completely random aggregation because the surface of protein molecules is not uniform with respect to the probability that contact will lead to adhesion, though we do not know how non-random it is. In general, random aggregation of spherical particles might be expected to produce larger, more or less spherical, particles. Of interest is the distribution of shapes of such an assembly of random aggregatec and the effect of a small degree of non-randomness, because the total length of strand available to form the gel will depend on the shape-distribution curve of the aggregated particles. Highly oriented cases are relatively easy : the " string of beads " model described above is one such case, and the problem is to allow for the random arrangement of such oriented aggregates.Ogston has treated the problem of the distribution of spaces in a random array of fibres : he finds the mzan pore size p can be described approximately by (p)2 = 0.25/2. (3) The interpretation of pore size in this theory is the diameter of the largest sphere that could just be contained within the pore. Comparing this with eqn (2) for the regular array, and in a random array the mean pore size is smaller. The relative concentration requirements for different models found for the regular array remains unchanged in the random case and this must be generally true for the conversion of any regular array to the random case.Thus, eqn (3) (which is only approximate because it takes the gel strand thickness as negligible and this is not justified in many real gels) may be used to allow for the effects of random arrangement of strands, providing an estimate of I can be obtained. Table 1 includes calculations for the " string of beads '' strands in a random array. We have attempted to allow for random aggregation to form strands by finding the shape distribution curve, and the mean maximum dimension of the aggregates. Thus, the contributions made, on average, by each initial particle to the total possible strand length can be calculated, and I estimated for any initial number of particles. The model used was to start with a particle, and then add others randomly by using a matrix reference and random numbers, up to a specified number of particles n.To begin, we generated two-dimensional aggregates and found the maximum dimension 0 . 2 9 ~ ~ = p (4)160 GELATION OF GLOBULAR PROTEINS (particle centre to particle centre) of the aggregate. For a ten-particle aggregate, the maximum dimension distribution from 30 trials and a visualisation are shown in fig. 2. Half the particles would be more asymmetric than the one illustrated and one may imagine that gels such as those illustrated in fig. 1 could be built up from such aggre- gates (cf. ref. (3)). I 2 3 4 5 6 7 8 9 centre-to-centre distance FIG. 2.-Random two-dimensional aggregation. At the bottom, the distribution of maximum dimension of a ten-particle aggregate. The units are molecular diameters, measured centre to centre.The actual maximum dimension requires addition of one molecular diameter. At the top, a repre- sentation of an approximately average particle : half the distribution would be more asymmetric. A two-dimensional aggregate corresponds to a particular type of oriented inter- action : we were also able to obtain the maximum dimension for a three-dimensional case (see appendix for details). From the mean maximum dimension, the average contribution made by each initial particle to the available strand length was found, and then, by applying eqn (3), the (concentration, pore size) relationship for random aggregates in a random array. Some figures are given in table 1. They show, as expected, that the larger the aggregate the smaller the contribution from each particle.A weakness of these calculations is that we have made all aggregates the same size and the next step is to find the contribution when the particles are allowed to aggregate to a mean value of n, and also to introduce elements of non-randomness into the aggregation process. The model is not, therefore, particularly realistic ; its main interest is in revealing how a random aggregation might lead to gels of the type observed.A 0 C FIG. 1.-EIectron micrographs of embedded, sectioned gels. Positive stain with uranyl acetate. A, Arachin, c = 0.15 made by reducing the pH from 12 to 4. B, Arachin, c = 0.15 made by heating at 110°C at pH5 in 4 % (w/v) NaCI soIution. C. Bovine serum albumin, c = 0.02, made by [To facepage 160 dissolving protein in 8 M urea solution.TABLE 1 .-THE EFFECT OF CONFORMATION, MODE OF AGGREGATION AND RANDOM ARRANGEMENT OF STRANDS ON (PORE-SIZE, CONCENTRATION) RELATIONSHIPS mean maximum distance in aggregate (units of molecular diameters) - 3.713 5.289 6.695 7.274 8.065 averago contribution to strand length per particklA 990 78 42 24.94 46.3 19.8 13.2 10.76 8.68 7.6 36.7 24.5 20.0 16.1 14.1 c required for p = 3008( in a regular case : n = degres of aggregation dm F molecular diameter kmlcm3 array extended chain random coil compact sphere 2-d aggregate n = lodm = 42 A 2-d aggregate n = lOd, = 78 A 3-4 aggregate n = 1Odm = 42 A n = 20 n = 30 n = 4 Q n = 50 n = lOdm = 78A n = 20 It = 30 n = 4 0 n = 50 2 .6 4 ~ 1014 2.08 x 1013 11.2x 10l2 6 . 6 4 ~ loi2 12.3 x 10l2 5.28 x 10l2 3 .5 2 ~ 10'' 2.88 x 10l2 2 . 3 2 ~ 10l2 2.02x loi2 9.79x 10l2 6 . 5 4 ~ 10l2 5.34x 10l2 4 . 2 9 ~ 10l2 3 . 7 6 ~ 10l2 1.26 x 10-3 1.6 x 0.297 x lo-' 0.501 x 10-1 0 . 2 7 ~ 10-1 0.63 x lo-' 0.95 x lo-' 1.15 x 10-1 1.43 x 10-1 1.64x 10-1 0.340~ 10-1 0.509~ 10-l 0.623 x 10-l 0.776~ 10-1 0.885 x lo-' c required for p = 300 8( in a random array 1.04~ 10-4 1.33 x 10-3 0.247~ 0.417~ 0.225~ 0.524~ lo-' 0.786~ 0.961 x lo-' 1 . 1 9 ~ 1.37 x 0.282~ lo-' 0.423 x 0.518 x 0.645 x lo-' 0 . 7 4 ~ p obtained from c = 0.1 or 0.01, a random array A A 9.7 31 34.6 109 47.2 149 61 193 45 142 68.8 217 84 266 93 294 104 328 105 351 51 159 62 195 68 216 76 241 81 257 p for c = 0.01, for a regular array A 107 375 513 ' 665 489 2 750 5 917 v1 101 3 1131 1210 548 672 744 83 1 886I62 GELATION OF GLOBULAR PROTEINS The rcL.ults arc 5uniniariscd in table 1.The concentrations required to produce a p value of 300& and the pore sizes expected from 1 (v/v) and 10 % (v/v) solu- tions, were calculated for a random array by using eqn (3), or for a regiilar array by eqn (2). Surprisingly, the requirements for a "string of beads '' model and an M = 10 random aggregate are not very different. However, the absolute concentra- tion requirements for the random array are uniformly lower than general experience suggests : for the regular array, requiring an order of magnitude larger concentrations, these requirements are of the right order. Any case where the strands are not ran- doiii1y arranged should fall between these limits, subject to the approximations already noted.ELECTRON MICROGRAPHS OF GELS Over the past few years we have collected electron micrographs of a number of gcls, made by methacrylate embedding and sectioning. Quantitative data from such photographs are obviously suspect. The most readily available parameters are a mean pore size and a mean strand diameter but as fig. 1 shows, these are not casy to obtain. In particular, it is easy to miss small pores, while diiiiensional changes during embedding lead to errors in concentration. Both these effects make i t likely that thc estimated p values are liable to substantial errors. Some results are given in t3ble 2, where I has been calculated froin a suitable aggregation iiiodel a i d from an cstiniatc of dmade directly from the photograph.In view of the approximation used, the agrccnient hetween predicted and estimated value is as good as can be expected, though it is interesting that predicted values are almost always lower than those found. TABLE 2.-cOMPARISON OF MEASURED AND PREDICTED PORE SIZES ex~ullplc c glycinin hcatcd 100" 0.038 i n 4 (x NaCl 0.075 0.1 13 0.15 albumin 0.01 5 arachin 0.15 arachin f 0.15 niensured measured r? d I" P b I C ( P ) 545 100 4.Sx 10'' 229 7 . 6 7 ~ 10'' 180 333 188 2 . 7 ~ 10" 304 1 5 . 1 5 ~ 10" 128 211 134 8 . 0 3 ~ 10'' 176 2 2 . 8 3 ~ 10'' 105 320 100 1 3 . 9 ~ 1 0 ' ~ 115 3 . 0 3 ~ 1 0 ~ ~ 91 330 100 1.89x 10" 363 3 . 0 3 ~ 10'' 287 200 100 1 8 . 9 ~ 10'' 115 3 . 0 3 ~ 10" 91 1000 500 0 . 7 6 ~ 10" 570 - - calculated from c = hr(0.5c/)2 ; 0 from eyn (4) ; C from 3 - 4 ti = 50, CIM = 42 A case in table I in all examplcs ; d:f as shown in fig.1, C , Bt A. The main conc1usion, that the postulated mode of zggregation lias a mzjor quantitative effect on the (pore-size, concentration) relationship, follows from the nature of eqn (3), which appears to be consistent with the limited experimental data, at least to an order of magnitude. Also, it seems reasonable to explain the relatively high concentration required to obtain gels from globular protejns as dependent on such effects. It is perhaps surprising that random aggregations can lead to gel formation, but this follows from the shape distribution curves of limited random aggregates. It also scenis reasonable to suppose that as the aggregates increase in size, the eventual result will be IL population of less favourable shapes and fewer particles which would then form coagulates. Seen in this way, gelation is a particular kind of limited aggregation process.Albumin heated at different temper;itures shows m;irkcdly different modes of aggregation, leading to either gelation or coagulation.' In this case, the interactionM. P. TOMBS 163 alters from a highly oriented one to a less oriented one. Both larger random aggre- gates, and an increase in the randomness of aggregation can lead to a smaller effective strand length, and coagulation rather than gelation. More detailed models of aggregation processes could be constructed, and best pursued in conjunction with gel measurements on a well-characterised protein where some information on surface structure is available. In the relatively rare cases where highly oriented interactions, such as interchain disulphides, predominate it should be possible to test eqn (3) on the random arrangement of strands without the additional complications introduced by a random aggregation to form the strands.I am much indebted to Mr. J. M. Stubbs for electron micrographs and particularly to Mr. L. J. Aspinall for calculations on three-dimensional aggregates, and the appen- dix. M. P. Tombs in Proteins us Human Foods, ed. R. A. Laurie (Butterworths, London, 1970). A. G. Ogston, Trans. Faraday Soc., 1958,54, 1754. J. Fessler, Nature, 1956, 177, 439. APPENDIX COMPUTER SIMULATION OF THE RANDOM CLUMPING OF SPHERES The model used is as follows. An initial specified clump, of touching unit-diameter spheres, is considered to be stationary throughout the simulation.Other unit diameter spheres travelling in straight-line paths collide with the clump, and possibly stick to it. All such collisions are assumed equally likely to result in sticking. The path directions are assumed random, in the following sense. For an arbitrary spherical region, stationary with respect to the initial clump, each successive intersection of this region by the projected path of the centre of a travelling sphere is equally likely to be from any angle in 3-d space ; and for a particular angle the probability distribution of the point of intersection on any circle, which is a projection of the spherical region from this direction, is uniform. The probability of a collision resulting in sticking does not need to be known since this only affects the time-course of clump growth and not the distribution of clump shapes which is of interest.The probability that a new sphere sticks at a particular point on the current clump is proportional to Ez=, (probability that the nth collision is the next collision to result in sticking) x (probability that the nth collision is at this specified point), and hence is propor- tional to the probability that the nth collision is at the specified point which is independent of n. The algorithm adds spheres successively to the initially specified clump until the required size is reached. At each addition, the following steps are carried out. (These exploit the fact that in order for a sphere in the clump to be hit by a travelling sphere, the path of the latter’s centre must intersect in an “extended sphere”, with a diameter of two units concentric with the sphere in the clump.) (1) The equation of a “ containing sphere ” which contains all the ‘‘ extended spheres ” in the current clump is calculated.(2) A random direction in 3-d space is selected. (3) The equations of the circular projections from this direction of the ‘‘ containing sphere ” and the “ extended spheres ” are calculated. (4) A random point is selected in the circle which is the projection of the “ containing sphere.” (5) If this point does not fall in any of the circles which are the projections of the ‘‘ extended spheres ”, a return to step (2) is made. Otherwise, the point of first contact is calculated, and the new sphere is added to the clump. All probability-density functions are generated via transformation of pseudo-random uniform variates Ui in the range (0, 1). Successive values are produced by scaling down integers calculated from The first value is calculated by reference to the clock. Z(n+ 1) = (n)x 16 807 (Mod 2 31)164 GELATION OF GLOBULAR PROTEINS A direction in 3-d space can be defined by the two spherical polar co-ordinates 4,8. In order that the selected directions should have the required distribution values of 4 and 8 are chosen such that (b = 2nu1; 8 = cos-' (1-2u2). A point in a circle can be defined by the polar co-ordinates Y, 8. In order to produce a uniform distribution over the circle, values of r and 8 are chosen such that where R is the radius of the containing sphere. 8 = 2 m 3 ; r = R(u~)*,