Mechanical properties of globular proteins gels: 1. Incipient gelation behaviour

Mechanical properties of globular protein gels: 1. Incipient gelation behaviourt Robert K. Richardson and Simon B. Ross-Murphy* Unilever Research, Colworth Laboratory, Sharnbrook, Bedford MK44 1LQ, UK

(Received 30 July 1980)

It has long been known that globular protein molecules in concentrated aqueous solution can be converted into a different form by heating or use of denaturing agents. Under certain conditions of pH and ionic strength, elastic gels are formed. This work describes kinetic measurements of viscosity and elasticity close to the gel point (sol-gel transition) in bovine serum albumin solutions. These studies and optical rotation measurements near the gel point lend support to the conclusions of earlier structural studies on this system, that the gel is fibrillar in nature.

Introduction It has long been known ~-3 that globular protein molecules in concentrated aqueous solution can be converted into a different form by heating or by the action of denaturing agents such as urea. Boiling an egg provides an example of such a transformation, and in some cases a clear rubbery gel is formed, but very often (as in the egg example) the gel is opaque, or, in some cases, is best described as a coagulate. In all cases, however, the native protein has become substantially unfolded by heating, and has aggregated. This paper describes a series of experiments on the globular protein bovine serum albumin, a system which can form gels of different properties depending upon such factors as pH and ionic strength. In the present paper, however, we restrict the discussion to the clear elastic gels formed by heating the protein solution at 'natural' pH, 6.5, to temperatures just above the unfolding temperature, ~ 57°C. In particular, we are interested in measurements made close to the critical conditicns of temperature and concentration, when the gel just begins to form - - it is under these conditions that it should be most straightforward to relate the mechanism of gel formation to the mechanical properties of the gel. The fine structure of bovine serum albumin (BSA) gels has been extensively studied by other workers in this laboratory 4'5, in particular such physicochemical techniques as small angle X-ray scattering, high resolution infrared and laser Raman spectroscopy and transmission electron microscopy have been applied, and our work is complementary to this, although results will be drawn from these studies wherever appropriate. Most of the experimental techniques used in this work are simple, but we hope to illustrate the preliminary rheoiogical screening of this gelation system. In contrast to the present treatment of gels, an extensive study of the aggregation of subgelling concentrations of BSA by a light scattering technique has been carried out by Kamata 6. * To whom all correspondenceshould be addressed t Presentin part at the Meeting'CriticalPhenomenain Fluid Phases', Bristol, April 1979 0141-8130/81/050315-08502.00 © 1981 IPC BusinessPress

At the 1974 Faraday meeting, Flory 7 clearly delineated four classes of gels. (1) Well-ordered lamellar structures, including gel mesophases. (2) Covalent polymeric networks, completely disordered. (3) Polymer networks formed through physical aggregation; predominantly disordered, but with regions of local order. (4) Particulate disordered structures. In the second and third categories (and formally in the fourth s) gelation is shown to be associated with a critical degree of, for example, reaction of chemical bonds by condensation, refolding of helices or crosslinking of preformed linear chains. The Flory-Stockmayer theory of gelation and its extension and modifications 9-12 have been quite widely applied to the aggregation of biological polymers, including early studies on human serum albumin 13'14, and work on gelatin 15, (see Ref. 16), milk protein coagulates s't7 and on the fibrin-fibrinogen system I a,19. The aim of the present study was to establish to what extent this same treatment was applicable to the BSA gels. Further and more detailed studies on, for example, the long-time behaviour of BSA gels will be discussed in another paper 2°.

Experimental Bovine serum albumin (Sigma, cat. no. A4378; crystallized and lyophilized) was essentially salt-free and was used without further purification. Solutions of the BSA were made up in deionized water to ~ 10%, w/w; the pH of such a solution was 6.5 (_0.1). For the concentration dependence part of the study, the protein concentration was determined spectroscopically by measuring the intensity of u.v. absorption (El~m~sO~m= 6.6). Gel times and rates of gelation were determined using (a) a simple falling sphere viscometer and (b) a driven torsion pendulum. Kinetic studies of aggregation were carried out by monitoring optical rotation as a function of time.

Int. J. Biol. Macromol, 1981, Vol. 3, October

315

Mechanical properties of protein gels (1): Robert K. Richardson and Simon B. Ross-Murphy ~stirrer

1 I

Inner q tube

tip of rnicrosphe~ ~pillary

'quickfit' ground gla: joint - Thermocouple leads 'out

A solution

TUBE less steel microsphere

just immersed in the solution and prevents solvent loss by evaporation and consequent concentration changes. The fall of the sphere is viewed through a travelling microscope and, using a stopwatch, the time taken, t, for the ball to fall a measured distance, s, is noted. Eventually, the rate of fall (distance/time = s/t) approaches zero and the gel time, defined as the time elapsed when s/t = 0, is noted. Rate of fall was converted to viscosity by applying Stokes Law with Fax6ns corrections 21. Viscosity in poise was given by 4.25 × t/s. Using this, satisfactory agreement was obtained with calibrations using standard silicone oils. The torsion pendulum (TP), Figure 2, consists of a rigid vertical mainframe from which is suspended, by a thin torsion wire, the bob. The latter is extended to form the inner cyclinder of the concentric cylinder sample container. The bob, outer diam. 4.5 mm, is aligned centrally in the cup assembly, int. diam. 6 mm. The effective length of the concentric cylinder assembly is 6 cm. Above this measuring region of the cell, the diameter of the inner cylinder is reduced, and that of the outer cylinder increased, so that measurement takes place in the 'depth' of the gel. The outer cylinder is housed in a water jacket, temperature controlled by means of an external water bath. By these means, any surface 'skin' effect is reduced and heat loss from the inner cylinder is minimized to reduce any tempera-

Water in

NOT TO SqAt.E

~ Torsiowinre

Figure 1 Falling Sphere Viscometer. Insert: ground glass capillary tube for centralizing microsphere: details in text The falling sphere viscometer (FSV) Figure 1, was a modified version of that used by Peniche-Covas et al.18. A is an inner glass tube (0.9 cm outer diam.) glass blown onto a Quickfit B19/24 cone (see Figure 1). B is an outer glass jacket connected to a Haake circulating water heater fitted with a contact thermometer. The temperature within the outer glass jacket was monitored by a copperconstantan thermocouple (near the water exit tube) connected to a high impe~lance digital millivoltmeter. The BSA solution was placed in the inner tube A at room temperature, above this solution is a capillary tube (0.6 cm outer diam.) with a ground glass tip (see insert) and down the centre of the tube is a length of stainless steel wire used as a stirrer. The water jacket is closed with a stopper and preheated, then tube A is placed inside B, a stopclock is started and the solution ( ~ 2 ml) approaches the temperature of the outer jacket. Within this outer water jacket there is a temperature differential of ,,,0.4°C (measured by another thermocouple) between the entrance and exit tubes of the circulating water. The temperature drop between A and B is ~0.1°C and the temperature gradient within the solution in A is -~0.2°C. The time for the solution in A to reach thermal equilibrium ( ~ 20°C to ~ 60°C) is ~ 90 s. A pair of magnets was used to raise the microsphere inside tube A, and by their manipulation, the stainless steel sphere, a precision 1/32 inch (=0.78 mm) ball bearing, was dropped from inside the cupped end of the capillary to encourage it to fall centrally. This capillary is

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Int. J. Biol. Macromol., 1981, Vol 3, October

\\\N~ \\\xq .u

Ring

Circular drive-

coil

o0._

jacket " ~ - ~

Thermocouple

I

I

Figure 2 Torsion pendulum, sectioned in vertical plane

Mechanical properties olprotein gels (1): Robert K. Richardson and Simon B. Ross-Murphy Results

~,,~ 1500

•

--

I000

500

-

0

•

6

12 -2 I0 x Time (s}

•

18

24

Figure 3 Viscosity (poises) vs. time (seconds) for a run at 58,8C, 9°0, w/w, BSA (Point denoted A in Fi,qure 5)

ture gradient across the sample. A couple can be applied to the bob by means of a specially wound coil and magnetic assembly which, for a given constant current through the coil, gives a constant torque over a sufficiently large angular displacement. Strain detection is by means of a linear voltage differential transformer (LVDT); the whole assembly is contained in a draught-proof cabinet. An electronic feedback system from the strain detector pulses the pendulum at its resonant frequency. This frequency may be continuously displayed in digital form, or alternatively by monitoring this frequency using a transient recorder (Intertechnique) in the 'count mode'. In this way, a hard copy corresponding to the variation of G' with time during gelation was obtained using an equation of the f o r m n :

G' ~ KI(~ 2 - o9~)

In the first series of experiments, the temperature dependence of gelation was studied by making up aqueous BSA solutions of nominally identical 9~o, w/w, concentrations and measuring the time taken for the system to gel by monitoring the increasing viscosity as a function of time. Close to 60cC native BSA is known to partially unfold, the native protein is dumb-bell shaped with approximate dimensions 100 x 40 A. The 'unfolded' protein is not converted to a random coil but, rather, forms a more expanded dumb-bell 23, in which apparently hydrophobic 'sticky' patches are exposed. From early optical rotation, differential scanning calorimetry and circular dichroism studies 4, this unfolding is known to occur between 55 and 60°C, so initial experiments were carried out at a temperature close to this. Results for a typical run (that at 58.8 + 0.1 °C) are illustrated in Figure 3; this is the run designated A in Figure 5. At lower temperatures, the time taken to gel is so long that the viscous solution is very sensitive to shear and on stirring the viscosity is reduced. At the same time 'chanelling' may occur in the gel, as the sphere always falls through the same part of the tube. Figure 4 illustrates an experiment in which the viscosity was measured before and after stirring. The two curves illustrate how the gelation time is delayed, but gelation is not prevented. The actual gel time still falls within the illustrated errors limits of Figure 5. One advantage of the falling sphere method is the quite small shear rates involved, the maximum ~i' is given by TM 3F ~'maX=~r

where v is the terminal-fall velocity, and r the sphere radius. Thus at a viscosity of, say, 500 poise, V( = s/t) was 4.25/500 cm s- ~ (see Experimental) and 7,~,~was 0,16 s - 1 In Figure 5 the gel time measured at constant temperature is plotted logarithmically against the reciprocal of this absolute temperature (and temperature in °C). The gel time is extremely temperature-dependent, changing from 650 to 12 m over the temperature range 54 to 62°C. This

(1)

where K is an apparatus constant, I the moment of inertia of the bob, ~o the resonant frequency of oscillation in the gelling system with o9o the resonant frequency in air. For this apparatus, KI = 127.2 g c m - ~ and m o was 1.456 rad S

(2)

1500

-

I000

-

o

-1

Most measurements were performed close to the gel point (say G'<104 dynes cm-2). For the temperature dependence of final modulus, the gel was heated at 90cC for 3 h, when no further increase in modulus was found, and then cooled to room temperature. The elastic modulus increases in this system upon cooling (see Discussion). Optical rotation measurements were carried out on a Perkin-Elmer 241 polarimeter using the blue mercury line (435 nm). Solutions were thoroughly degassed before being placed in the temperature controlled cell, the temperature as before monitored by a dipping thermocouple. The optical rotation was measured to _ 1 × 1 0 - 3 degrees, and output to a coupled printer as a function of time.

•

o

g-

500

-

0

30

60

90

120

I(~2x Time (s)

Figure 4 As Figure 3; temperature 56.1'C; • before stirring: t3, after stirring to show the effect of shear on the 'gel time'

Int. J. Biol. Macromol., 1981, Vol. 3, October

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Mechanical properties of protein 9els (1)." Robert K. Richardson and Simon B. Ross-Murphy

BSA(59.4"C) 110,8% 1500

i0 4

G E t~

+.

A l E o I000

A

i0 3

%

500

102

I

[

1-

I

I

1

I

61

60

59

58

57

56

55

T(°C)

,"

y 12

Figure 5 Time for gelation (log scale) vs. temperature (scale is linear in 1/absolute temperature), at constant protein concentration (9~o, w/w)

I __......--r-24 36

I 48

[ 60

'l 72

IO:~x Time is}

Figure

6

S t o r a g e m o d u l u s G' vs. t i m e f r o m t o r s i o n p e n d u l u m :

concentrations as given plot is effectively an Arrhenius plot (at constant con-

centration), and if a single kinetic process is involved, the points would lie on single straight line. By analysing the data by various models, e.g. a single straight line, second and third order polynomials, and the model of two intersecting straight lines, the latter model 24 gives the most significant fit. This invites the hypothesis that the two processes, unfolding and aggregation, are distinguishable. Above ~57°C the unfolding is fast, effectively spontaneous, and the aggregation is a slower process, while below this temperature the unfolding is rate determining (and here the gels formed are slightly opalescent, whereas at higher temperatures they are transparent). In the concentration-dependent part of the study, the temperature was maintained at 59.4°C, above the spontaneous unfolding temperature indicated by the above discussion. The gel time was measured both by the FSV, as described above, and as the time when a finite storage modulus was just detected, at an oscillatory frequency of ,--1.5 rad s - 1 The effect of concentration on gelation behaviour is very marked. Figure 6 shows traces of G' as a function of time for protein concentrations from 8 to 10.8%. As expected, as the protein concentration is increased the gel time is reduced, but the slope dG/dt is increased. Figure 7 compares the results from the torsion pendulum and from the falling sphere viscometer. The torsion pendulum, which monitors the formation of the gel at a finite frequency, but very low strain (typically < l~,0), indicates a gel time systematically less than that measured by the falling sphere viscometer, which mea-

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Int. J. Biol. Macromol., 1981, Vol 3, October

iO4

103

p-

102

-

I0 ~ 6

I

I

1

I

1

1

8

I0

12

14

16

18

20

Protein cencentrotion (%)

Figure 7 Time for gelation (log scale) vs. concentration: o. falling sphere viscometer: o. torsion pendulum (1 Hz)

Mechanical properties of protein gels (1): Robert K. Richardson and Simon B. Ross-Murphy i0 -i

otr = - (¢t,- ~to)/C

i0 -2

(the rotation becomes more negative) and (see Discussion) assuming a simple nth order kinetic scheme

_.-..

,

iO -3

iO -4

d ~ ] = k , ( 1 - [ct])"

f

i

1o- 5

0

5.0

c~!

(4)

with k, the nth order rate constant. Where [ct] is the reduced rotation:

la I

(3)

(5)

[a] = ~,/~

f

r

J

I0.0

15.0

20.0

with ~ the asymptotic value of a, as t--. ~ . Integrating (4) with (5):

Conc. (% protein) '

a . = a = [ 1 - { ( n - 1 ) k . t + l } l/~-"]

Figure 8 As Figure 7, but reciprocal gel time plotted, the solid line is taken from Ref. 26 without adjustable parameters sures at a low rate of strain, 7, but at a higher absolute strain (deformation). The exact determination of gel times is sometimes difficult in systems, such as the present one, where noncovalent 'bonding' is involved, because all measurements are performed at finite 7 and 7. Frequency scans of G' as a function of time, for example, indicate that even in covalently bonded systems quite low frequencies (~0.7 rad s- 1) are required to eliminate anelastic effects, so that G'(og) can be equated with GOthe equilibrium modulus 25. Nevertheless, the agreement between the results obtained with the FSV and the TP is encouraging, and the extrapolated 'critical concentration', co from Figure 8 is seen to be 6.8 + 0.1% under these experimental conditions. This agrees well with the results of recent measurements on the protein soybean globulin, 6:6~/o26, by Soviet workers, cq= 6 ~ extrapolated from recent work on ovalbumin gels ~, anci~the shghtly higher c o of -.~8% obtained in this laboratory for ct-chymotrypsin. The concentration dependence of the initial rate of gelation is extremely pronounced, as illustrated in Figure 6. For example, over a limited range of low concentrations dG'/dt varies at ~'~c27, and even at higher concentrations (~20%) the rate varies with concentration to approximately the sixth power. Such a variation is illustrative of the 'critical' regime just as the gel is forming, for example Bisschops 27 found d G ' / d t - c 22 in measurements of poly(acrylonitrile) in dimethylformamide. Optical rotation experiments were carried out by monitoring the change in o.r. as a function of time for different gelling concentrations of BSA (8-12%) at the same temperature as was used in the above experiments, 59.4°C. The degassed solution was transferred to a preheated 2 cm quartz cell, and the rotation recorded on a printer unit at preset time intervals. The change in o.r. with time was monitored just after the preheated cell is filled with protein; initially the curve is sigmoidal as the change in rotation occurs some 90 s before the solution reaches thermal equilibrium. The effective zero time of the process was then determined graphically. Smooth kinetic curves were usually obtained, and the results analysed as below. If the observed rotation at time t is ct,, and the initial rotation (proportional to the protein concentration, c) is ao, then:

n#l

This function was fitted to observed values of at, dependent variable, for various t, by non-linear least squares, and the best values of z%, n and k, were obtained using the Fletcher-Marquardt algorithm (NAG E04 GAF) 2s. Results liar n and k, for different concentrations are given in Table 1.

Discussion The kinetics of protein unfolding are complex, the simplest models regard the process at first order but with a rdte 'constant' varying during the course of the reaction as, for example, the natural pH of the protein solution changes 29. The collision of two partially unfolded protein molecules, the first stage in intermolecular aggregation, should give rise to second order kinetics, as should the intermolecular aggregate-aggregate reaction steps, while intramolecular reaction within aggregate is first order. A simple scheme to illustrate this is: p

k, ,-

native protein

A+U+

~U + U

k~l ,.___~

unfolded

ka2

-~A + A~, A

aggregate

(6)

and if the aggregation is rate (r) determining, we have r, • ra~ "" ra2 "~r,, where k, is the rate constant for unfolding, k,1 and k,2 are the aggregation rate constants, and k, the intramolecular rate constant. For this very simple model the overall reaction order should lie between 1 and 2 and, at higher concentrations, when intermolecular steps are

Table 1 Run

Concentration (%, w/w)

n

103 x k,

K01 K02 Kll K12 KI3 K14 K15 K 16 K17

9.0 9.0 9.14 11.32 8.28 8.60 12.10 10.06 12.12

1.99 1.36 1.40 1.40 2.06 1.62 1.65 1.47 1.13

2.49 3.35 3.17 3.31 3.26 2.95 3.94 3.25 2.74

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Mechanical properties of protein gels (1): Robert K. Richardson and Simon B. Ross-Murphy

0

80

01 O1

0

0

l 0

lO

0

0

lo

0

0

lo

0 lO

O1

lO

1 o

70

olO 0

0

% eo lo

60

lo • o 0

"~

50

0 o

:= E

lO

4O •

O

0

3O •

•

20

0

0

0

I0

I

I

I

I

I

I

I

1

I

l

I

I

200

400

600

800

I000

1200

1400

1600

1800

2000

2200

2400

2600

Time (s)

Figure

Superposition of two kinetic runs of optical rotation against time; data is for K01 and K02 (see Table 1)

9

,of

and it is difficult to determine them very accurately.

Figure 9 illustrates this: the pomts from two different runs

0'9 1-

_

•

~

•

•

o t0.7 0.6

~ 0.5 ~L 0.4 0.5 0.2 0.1 I 0

I 4

t 8

I 12

1 16

I 20

24

I 32

I 36

I 40

I 44

48

162 x Time(s)

Figure 10 Rescaled kinetic runs using calculated values for [cc]~. All except the filled squares (lowest Concentration) fall close to a single master curve favoured compared to the intramolecular, it should approach 2. Table 1 illustrates that, overall, the reaction order does lie between 1 and 2, but the range of concentrations is too small to deduce anything of v a l u e about change in reaction order from inter and intramolecular competition. In any case, the parameters k, and n are not independent

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Int. J. Biol. Macromol., 1981, Vol 3, October

at nominally identical (9~) concentrations are plotted. In this case, k, and n are 1.99 and 2.49 x 10 -3 for the run designated by squares and k, and n are 1.36 and 3.35 x 10-3 for the run designated as circles in Figure 9. Figure 10 shows the result of rescaling. The seven runs K l l to K17 are plotted in their reduced form [~] (the experimental values, but using the optimized ct~ values) vs. time. Clearly, second order concentration effects are small except at the lowest concentration, where the time taken to gel and the rather turbid nature of the gel, make the result,~ less reliable. One of the earliest approaches to the effect of the kinetics of thermal denaturation upon the aggregation of serum albumin was that given in a series of papers by Kratochvil, Munk and Sedlh~ek 13'14. They showed that only part of the unfolded protein surface is involved in aggregation and that the kinetics of aggregation obey the same laws as the kinetics of chemical polycondensation propounded by Flory 7. They established that the predominant mode of aggregation is via essentially two aggregating sites, but with a small proportion of monovalent and polyvalent sites on the protein (see Ref. 6). Their studies were restricted to molecular weight measurements on solutions where the protein concentration was ~ 0.5%. Consequently, they did not produce gels, but they managed to show that the aggregation was not of the classical Smoluchowski type, i.e. collision of monomers 3°, but is effectively accelerated by the 'aggregation of aggregates', see, for example, RetL 8. It is this process which is required if a gel, a species of nominally infinite

Mechanical properties of protein gels (1): Robert K. Richardson and Simon B. Ross-Murphy molecular weight, is to be formed, and a critical 'gel point' conversion follows as a consequence of this latter (Flory) theory. This critical conversion, ~c, is related to the 'functionality', .[; the number of possible reactive (aggregation) sites on the protein, by ~,.(J- 1)= 1

17)

Thus, for a linear random polycondensation (f=2) a gel (infinite molecular weight) is formed only at ~,.= 1, i.e. when all possible aggregation sites have been used (an unreasonable proposition on kinetic grounds). Our strategy will be to calculate f or, more exactly, its effective weight average J~., by determining :t,,. The fact that gelation occurs in a finite time precludes an entirely linear aggregation mode; our problem is to relate. ~,. to an observable property of the aggregation process. In our case, we make the postulate that the observed change in optical rotation is a quantitative measure of the amount of aggregation or, more precisely, to the proportion of prospective sites which at any time have become involved in aggregate formation. Thus, by relating the time taken to form a gel (see Figure 7) to the proportion ~ , / ~ ( = [~]) and by equating [~],. to :~,. in the equation we can calculate the apparent weightaverage functionalityfw (a similar technique was applied by Peniche-Covas et at.15). For example, at 10~o the gel time of the protein solution is > 13 min from the torsion pendulum and <22 min from the falling sphere viscometer. The discrepancy appears large but the calculation is not very sensitive to this, leading to 2.4 >fw > 2.3. In the range of concentrations studied in the current work 2.5 >]~. > 2.05 (decreasing with concentration), which agrees quite well with the value 2.05 calculated by Kratochvil, Munk and Sedlli~ek 13'14 from the same model, but applied to the molecular weight dependence of the aggregation at the non-gelling concentrations of HSA used in their work, and also with the value of ~2.1 calculated by M/iller and Burchard 18"19 for the conversion of the rod-like fibrinogen to fibrin from measurements of the angular dependence of scattered light. Conversely, in the coagulative aggregation of milk proteins where a higher functionality would be expected, Parker and Dalgleish successful fitted both light scattering and gel fraction data to a trifunctional model 8'17. Thus, whilst the incipient gelation can be treated by the model of random polycondensation of protein molecules in an essentially linear fashion, but with occasional branching or crosslinking, the necessary extension of this treatment to calculate the elastic modulus of the gelling system is not so straightforward. The equilibrium elastic modulus of many systems measured close to the gel point has been shown to agree well with values calculated from the statistical theory of rubber elasticity as extended by Case and Scanlan 3~'32. This may be written: (see, for example, Ref. 33) Go -

gN,,RT Vmol

(8)

where R is the gas constant, Tis absolute temperature, Vmo~ is the volume per mole of 'primary chain' (a reciprocal concentration) and N,, is the number of effective junctions or, more rigorously, the number of elastically active

network chains; (4 is the front factor which, from recent experimental evidence, appears to be close to unity 3.. Theoretically, the maximum value of N,, is 1.5, and thus a maximum value of Go can be calculated. However, this is typically an order of magnitude higher than was found in the course of the present experiments, even when the protein gels were heated for longer times at much higher temperatures, until there was no further increase in G(t), i.e. 'fully cured'. It is important to separate the different terms in equation (8) if we are to analyse why it fails so drastically in this case. The expression for the gel elastic modulus can be split into two parts: (1) a statistical count of the number of elastically active chains: (2) the factor which gives the elastic modulus per active chain. Term (1) is equal to N,,, which can be directly related to [c~] and to time (for details see e.g. Ref. 33). Simply, for a second order kinetic scheme for aggregation we have:

des] dt -kz{1-~)2

19)

so that: t-

c( k2(l --~)

(10)

and for the simplest model of a trifunctional monomer aggregating we have:

1)3

N, =-3-(2 - ~ -] '

(11)

2\

so that, as expected, N e - 0 for [~] < [~]~ - 1/2. From equations (10) and (11) we can plot N,, as a function of time, and show that it has the form indicated by, for example, the experiments illustrated in Figure 6. On the other hand, the concentration dependence of elastic modulus is much more pronounced than that indicated by equation (8) (see Figure 7) - - see also Experimental (and the temperature dependence of the fully cured gels is the opposite to that predicted, i.e. approximately, G oc T - 5, when Tis in K - - see Figure 12). Equation (8) was originally derived for the model of an entropic rubber, that is to say, a material in which the application of strain reduces the number of macromolecular configurations and the chain is assumed to deform affinely. This model is inconsistent with the structure of the protein gel deduced from the microstructural techniques *'5 and the present work, which tends to support a more fibrillar type of gel network, and certainly little conformational freedom is to be expected in such a network, built up from protein molecules. Other workers 3 have applied the clagsical model of rubber physics to cured disulphide crosslinked ovalbumin gels, and calculated the 'number of crosslinks' and the molecular weight between crosslinks both from dynamic measurements and from equilibrium swelling measurements. This sytem differs from ours because of the presence of chemical crosslinks, but evel~ in this case the swelling measurements seem to be more consistent. The whole treatment of rubber elasticity in terms of 'molecular weight between crosslinks' has been criticized recently, and especially for

Int. J. Biol. Macromol., 1981, Vol. 3, October

321

Mechanical properties of protein gels (1): Robert K. Richardson and Simon B. Ross-Murphy gels of low modulus, the Case-Scanlan treatment is more reliable. It would appear, however, that a new model is required to describe elasticity in gels of the present type, where the predominant contribution to the higher free energy state of the deformed gel is enthalpic rather than entropic. The ideal entropic model for a rubber is a similar physical model to the ideal gas, so that for this case [see equation (8)3, the modulus is predicted to be directly proportional to absolute temperature. Whilst even for ideal networks there is invariably an energy contribution to elasticity because of the conformational energetics of real chains, experimentally the modulus either increases with, or is almost independent of, absolute temperature. In contrast, for the present BSA gels, both G' and G" the loss modulus, fall with increasing temperature, once the initial heat setting has been carried to 'completion'. Further details will be given in a later publication 2°. Such a pronounced negative temperature coefficient, dG'/dT, is surprising, although results somewhat similar to this a r e observed in swollen gels of poly(vinyl chloride) ( P V C ) 35. It would appear however that a new theoretical model (see e.g., Ref. 36) is required to fully describe elasticity in the case of both physical and chemical crosslinking of stiff chains and fibrillar networks, which occur quite widely in biophysical systems.

References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Conclusions The use of measurements close to the gel point has been discussed in BSA gels; further measurements should cover the effect of such factors as pH and ionic strength on the final gel modulus after extended heating, and also the long-time behaviour, using creep measurements. Until recently, the theological properties of globular protein gels have not been studied nearly as extensively as have the gels formed from proteins with more extended chain conformations such as gelatin, whose gelation behaviour is much more ideal, and in which the elastic properties are demonstrably entropic. Acknowledgements We thank our colleagues Dr D. A. Rees and Dr A. H. Clark for encouragement, advice and discussions, and Mrs Lesley Linger for assistance with the diagrams.

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