Protein crystallization screening through scattering techniques

Advances in Colloidand Interface Science, 58 (1995) 57-86

57

Elsevier Science B.V. 00244 A

PROTEIN CRYSTALLIZATION SCREENING THROUGH SCATTERING TECHNIQUES YANNIS GEORGALIS a'*, J E N S SCHEILER a, JOACHIM FRANK b, MARIO DIKEOS SOUMPASIS c, WOLFRAM SAENGER a

aInstitut fi~r Kristallographie, Freie Universiti~t Berlin, Takustr. 6, Berlin, Germany bFritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany CBiocomputing Group, Max-Planck-Institut fi~r Biophysikalische Chemie, P.O. Box 2841, 37018 G6ttingen, Germany ABSTRACT We have employed dynamic light scattering and X-ray small-angle scattering for monitoring the cluster formation of hen-egg white lysozyme solutions under crystallization conditions. From time-resolved dynamic light scattering experiments we can obtain a set of observables that exhibit clear extrema, when simultaneously plotted as functions of protein and electrolyte molarity. Close to these extrema, lysozyme crystals appear rapidly and with high probability. Both dynamic light scattering and small-angle X-ray scattering retrieve, under optimal crystallization conditions, particles 12 to 15 times larger than monomeric lysozyme; they could correspond to stable critical nuclei. Scattering amplitude analysis provides also useful information that correlates well with the size observables. A combination of both techniques can thus be successfully employed for screening supersaturated protein solutions. INTRODUCTION C o n t r o l l e d g r o w t h of p r o t e i n c r y s t a l s is t h e m a j o r o b s t a c l e in o b t a i n i n g a t o m i c r e s o l u t i o n s t r u c t u r e s w i t h t r o u g h diffraction m e t h o d s . A t o m i c s t r u c t u r e s a r e in t u r n i n d i s p e n s a b l e for e s t a b l i s h i n g s t r u c t u r e - f u n c t i o n r e l a t i o n s h i p s in m o l e c u l a r biology a n d biotechnology. T h e c r y s t a l l i z a t i o n of b i o m o l e c u l e s is a p r o c e s s involving n u c l e a t i o n , c r y s t a l g r o w t h a n d c e s s a t i o n of g r o w t h . M a n y p a r a m e t e r s i n f l u e n c e n u c l e a t i o n a n d c r y s t a l g r o w t h , m a k i n g a priori p r e d i c t i o n s for s u i t a b l e * To whom correspondence should be addressed. Tel: +30-838-4588; Fax: +30-838-6702; E-mail: yannis@chemie, fu-berlin.de 0001-8686/95/$29.00

© 1995 -- Elsevier Science B.V. All rights reserved.

SSDI 0001-8686(95)00244-8

58 growth conditions difficult. Very few attempts were made until the early 90s for tackling this problem. Since then, diagnostics of protein crystallization turned out to be a challenging topic and several laboratories are involved in the problem*. Despite the plethora of articles that have appeared on this subject, there are only a few systematic screening attempts. The crystallization of a macromolecular solution is determined (i) by the effective interactions (or potentials of mean force) between the molecules and (ii) by kinetic factors that control both nucleation and growth. All these quantities depend strongly on the fundamental state variables of the system (e.g. species conformations, temperature and pressure) as well as on structural and physical parameters of its components (e.g. sizes, charges etc.). Derived thermodynamic quantities such as the macromolecular solubility may be useful indicators for finding good crystallization conditions, however low solubility is known as a necessary but often not as a sufficient condition for obtaining crystals. It has to be stressed that classical crystallization theories are applicable to one component or binary systems and for the case of homogeneous nucleation. The extrapolation of similar concepts to proteins, which crystallize anyway inhomogeneously, has been hardly checked experimentally, due to the enhanced complexity of molecular interactions. The protein concentration at equilibrium, i.e. the solubility, is a complex function of size, shape, charge distribution and initial protein concentration. Furthermore, temperature, pH, size, charge and concentration of the precipitating agent (inorganic salt, polymer or organic solvent) will influence to an unpredictable extent the initial stable cluster formation, nucleation and later growth stages [1-3]. If one wishes to put protein crystallization diagnostics in a sound frame it is essential to proceed with the simplest systems possible. The various empirical recipes chosen to enforce protein crystallization (e.g. salts or organic reagents, or high molecular weight polymers, and water) represent systems whose physicochemical properties are not understood in the first place. Even if the problem is constrained to a simple w a t e r electrolyte-protein system one deals, in the thermodynamic sense, with a very complex situation. Side effects, like non-specific binding or electrolyte induced structural changes, cannot at present be taken into account even by the most advanced simulation schemes. * For references on crystallization diagnostics see the proceedings that appeared in J. Crystal Growth 110 (1991), 122 (1992), as well as in Acta Cryst. D50(4) 1994.

59 The problem of protein crystallization can be understood as a subproblem of the wider and very complex protein-protein interactions problem. Our inability to either theoretically predict or experimentally tackle protein nucleation emerges, to a large extent, from our incomplete knowledge of interactions between proteins in the molecular level. In the present work we address the problem of screening crystallization conditions of lysozyme by dynamic light scattering (DLS) in a higher degree of detail than in previous publications [4-8]. We have monitored the cluster formation kinetics of lysozyme at different protein concentrations while simultaneously varying the ionic strength of the electrolyte NaC1. We have used as observables (i) the mean quasi-stationary hydrodynamic radius of the clusters (ii) the fractal dimension and (iii) the zero time-lag size of the clusters. All three observables can be obtained from time-resolved DLS experiments according to previously described procedures [4]. With some effort, tentative estimates of the numbers of monomers and fractals can also be extracted. These estimates provide additional evidence for the definition of the optimal crystallization regions. We show that all observables, when examined as simultaneous functions of protein and electrolyte molarity, show extrema that coincide with those solution conditions where ]ysozyme crystals are obtained within less than two days. Since the aggregation of lysozyme is very fast under supersaturation conditions, the kinetic observations made so far concern postnucleation. It is however mandatory to have estimates of the minimum particle size formed in the solution. In a series of small-angle X-ray scattering (SAXS) experiments we have independently redetermined the mean radius of gyration of the smallest particles present in crystallizing solutions. We find excellent agreement between those sizes and the zero time-lag extrapolated hydrodynamic radii. DLS and SAXS have been reviewed by several authors [9-13]. For ensembles of monodisperse particles, which are small in comparison with the employed wavelength, DLS provides direct means for determining the particle size in the infinite dilution limit. The formalism of SAXS is very similar to that of static light scattering. The major difference is the superior spatial resolution attained through the shorter wavelength of the X-rays. It has to be stressed that most physical-chemical approaches, including scattering techniques, have been conceived for working with stationary systems in the infinite dilution regime. Crystallization diagnostics of

60 biomolecules, even in their simplest form, depart from this ideal picture. At high protein and electrolyte concentrations rapid aggregation and cluster formation, render correct particle sizing difficult. MATERIALS AND METHODS The chemicals used in the present work were of analytical grade. Deionised water was obtained from a Milli-Q device (Millipore). Three times crystallized lysozyme was purchased from Sigma Chemical Co. (Deisenhofen, Germany) and was further exhaustively dialysed against water and lyophilized. All experiments were conducted in a 0.1 M Na-acetate, pH 4.2. NaC1 p.a. grade, was purchased from Merck (Darmstadt, Germany). Monodispersity of the preparations was controlled by DLS at several protein concentrations, in the absence of electrolyte (non-aggregation conditions) before each kinetic experiment. Protein and NaC1 were rapidly mixed in the appropriate ratio and filtered through Millex sterile filters, 0.22 ~m pore size, into standard cylindrical light scattering cells. Experiments were initiated within less than one minute after mixing protein with salt. D L S ~ D a t a Acquisition a n d E v a l u a t i o n

DLS measurements were conducted with an ALV/SP-86 spectrogoniometer (ALV, Langen, Germany) and the ALV-FAST/5000 digital autocorrelator boards at a scattering angle of 20 ° throughout. An Ar ÷ laser (operating wavelength 488 nm) was employed as light source. DLS provides direct means for determining the free-particle diffusion coefficient for ensembles of dilute, monodisperse particles. This is true only if the particles do not interact with each other and their hydrodynamic radius, R, is small compared with the employed wavelength (R < ~/20). The electric field autocorrelation function (ACF), G(1)(x), is a quantity that is proportional to the distribution of relaxation times x and scattering amplitudes of the examined components. The formulation of the field ACF involves an integration over the size distribution function to account for polydispersity and eventual polymodality Rmax

G(1)(z) ~

f N(R) M2(R) P(q) S(q) exp(-mR -1 q2 ~ret) d R Rmm

(1)

61 w h e r e m is a p r o p o r t i o n a l i t y c o n s t a n t , N(R) a n d M(R) d e n o t e n u m b e r a n d m a s s of particles w i t h r a d i u s R in t h e size r a n g e b e t w e e n Rmi n a n d Rmax. P(q) a n d S(q) d e n o t e t h e form a n d t h e static s t r u c t u r e factors [9] respectively, q is t h e s c a t t e r i n g vector equal w i t h (4~n/k) sin(0/2), n t h e refractive index a n d 0 t h e s c a t t e r i n g angle. Ideally, for n o n - i n t e r a c t i n g particle e n s e m b l e s , Eq. (1) delivers t h e z-average h y d r o d y n a m i c r a d i u s R t h a t is associated w i t h t h e t r a n s l a tional diffusion coefficient Dt, via t h e S t o k e s - E i n s t e i n e q u a t i o n

kBT D t - 6~T1R

(2)

w h e r e ~ is t h e viscosity of t h e solvent. For c o n c e n t r a t e d s u s p e n s i o n s t h e static s t r u c t u r e factor can be obt a i n e d from m e a s u r e m e n t s of t h e a n g u l a r d e p e n d e n c e of t h e total scatt e r e d light or it can be a p p r o x i m a t e d by k n o w n a p p r o a c h e s [14-16]. If all t h e t e r m s involved in Eq. (1) are available t h e A C F can be Laplace-inverted. T h u s , p o l y m o d a l i t y a n d polydispersity can be resolved w i t h a d e q u a t e precision for c o n c e n t r a t e d s u s p e n s i o n s as well. Some a d d i t i o n a l aspects t h a t s h o u l d not escape a t t e n t i o n in m i c r o s t r u c t u r e formation, involve a s y m m e t r y corrections a n d d e d u c t i o n of t h e correct m o m e n t s of t h e cluster size distribution. Brief descriptions of t h e s e issues are given in A p p e n d i x I. Kinetic e x p e r i m e n t s were conducted at a constant t e m p e r a t u r e (20 _+ 0.1)°C, a n d in a 0.1 M sodium-acetate buffer, pH 4.25. The protein concent r a t i o n s e x a m i n e d w e r e 0.72 mM, 1.55 m M a n d 2.4 m M w h e r e a s t h e NaC1 c o n c e n t r a t i o n w a s v a r i e d b e t w e e n 0.2 M a n d 0.8 M. E v e r y p r o t e i n concentration w a s m e a s u r e d at 11 different electrolyte molarities resulting in a total of 33 time-resolved experiments. Spectra were collected every 30 s for n e a r l y one hour. M e a n radii of t h e evolving clusters w e r e calculated by L a p l a c e - i n v e r s i o n w i t h a modified version of t h e p r o g r a m C O N T I N [16,17] i m p l e m e n t e d in a CONVEX-C220 supercomputer.

S A X S - - Data Acquisition and Evaluation D a t a collection w a s a c c o m p l i s h e d w i t h a K r a t k y s m a l l - a n g l e spect r o m e t e r (Anton Paar, Graz, Austria) u s i n g an X-ray g e n e r a t o r w i t h a sealed-off t u b e 40 kV/20 mA, CuK d radiation, w a v e l e n g t h k = 1.5418 A. A position-sensitive detector (4.7 cm l e n g t h - - 85 c h a n n e l s p e r cm), t h e

62 m u l t i c h a n n e l d a t a processor and the appropriate analysis software were obtained from B r a u n Co. (Munich, Germany). Absolute calibration of the intensities was m a d e with a s t a n d a r d Lupolen block. Solutions w e r e m e a s u r e d in Marck-capillaries (Mfiller Co., Berlin, G e r m a n y ) with a d i a m e t e r of 1 mm. Spectra were recorded at (20 + 0.1)°C for periods r a n g i n g b e t w e e n two to ten hours. The intensity pulses received in each c h a n n e l w e r e plotted as a function of the scattering vector. The p a r a m e t e r of i n t e r e s t in these experiments is the radius of gyration, Rg, t h a t can be r e g a r d e d as a direct m e a s u r e of the spatial extent of the particle. According to the Debye approximation [12,13] the scattered X-ray i n t e n s i t y can be expressed as

I(h) = ~ exp -

(3)

4~ w h e r e h = - : - 2 8 denotes the scattering vector. After subtracting the b a c k g r o u n d scattering intensity due to solvent, the scattering curves w e r e plotted in semilogarithmic, In I(h) vs. h 2, according to the Guinier procedure [12]. The radius of gyration can t h e n be obtained from the slope of t h e scattering curve at small angles and at the limit of zero angle. B e t t e r resolution, i.e. detection of larger species present in the solutions, can be accomplished by increasing the distance between the probe and t h e detector. For distances between 20 cm and 37 cm we could explore cluster sizes up to ca. 60 n m or 120 nm, respectively. The latter implies t h a t fractal s t r u c t u r e s growing in crystallizing solutions (they exceed 100 n m n e a r l y immediately) will not be "seen" in the SAXS experiment. Therefore the possibility of observing additional d o m i n a n t cluster populations, of low n u m b e r density and sizes between those of m o n o m e r s and fractals, is enhanced. .

.

RESULTS AND DISCUSSION

DLS In the DLS experiments we can clearly observe two particle populations, the first r e p r e s e n t i n g m o n o m e r s (or dimers) and the second mass fractal clusters [4,5], (Fig. 1). Within less t h a n two hours typical clusters r e a c h sizes above 1 tim. The m e a n cluster size kinetic records w e r e fitted

63 O

1.00

.......

I

........

I

........

I

........

I

........

0.75

0.50

c'-I

° "

0.25

30min 45 min 60 min

"N% ~o %~ k %

~

0.00 10 -5

10-6

10 -4

10 -3

10 .2

10-]

(a)

1.00

0.75

111 l

'

'

'''"'1

'

'

'''"'1

. . . .

'"'1

o °

f

'

'

'''"'1

5 min 30 min 45 min 60 min

0.50 Z 0.25

0.00 100 (b)

101

10 2

10 3

10 4

R [nm]

I?ig. 1. (a) Typical DLS spectra obtained at different times (inset) with 1.0 M NaC1, in 0.1 M acetate buffer pH 4.2 and 2.1 mM lysozyme. Data were collected at a scattering angle of 20 °. The solid line through the data is the CONTIN fit (best solution). (b) Particle size distributions obtained by Laplace inversion of the ACE. The smallest component corresponds, at low supersaturation, to monomeric lysozyme and its size remains stable throughout, R m = 2.09 nm. The larger components represent fractal clusters; their amplitude and relative peak position evolves with time.

64 2

i

~

I

7

I

~

1

~

i

~

~

,

~

[

,

~

""~'

104 0.75 M 103 []

D

0.70 M 10 2 o o0.6~

10 t A ~-, ,x:

0.60 M

10 o 10 l v

10 .2 10-3

10 .4

0.50 M

~A ~

--

5~

O~ 0.40

•

M

o

~

O

-'=

0

10-5

102

10 3 t[s]

Fig. 2. Time-resolved aggregation kinetics of lysozyme under crystallization conditions, Typical traces were obtained by incubating a 1.55 mM lysozyme solution with NaC1 and following the cluster size growth by DLS. The lines indicate non-linear least squares fits of Eq. (4) on the data, For the sake of clarity the ordinates are shifted by ascending powers of two between consecutive data sets, The number above each curve denotes electrolyte

molarity. according to a power-law expression [19,20] corresponding to diffusion limited cluster-cluster aggregation (DLCA) Rh(t) = Rh(O) [1 + ct]l/dr

(4)

65 w h e r e c is a constant involving the Smoluchowski collision rate, Rh(t) denotes the m e a n cluster radius at time t, R~,(0) can be identified as a zero time-lag extrapolated m e a n cluster radius and df the fractal dimension of the cluster. Typical selected fits of data sets deduced from Eq. (4) are depicted in Fig. 2. We have grouped d a t a together and recast the DLS results in illustrative t h r e e - d i m e n s i o n a l surface and spectral plots using s t a n d a r d inverse distance w e i g h t i n g presentation algorithms [21]. The obtained results are displayed in Figs. 3 to 5 as t h r e e dimensional plots followed by the respective spectral plots. All three observables, the quasi-stationary radius (Fig. 3), the fractal dimension (Fig. 4) and the zero time-lag extrapolated cluster radius (Fig. 5), display clear maxima. The highlighted regions in the spectral plots indicate t h a t protein and electrolyte molarities w h e r e crystals grow in the light scattering vials within less t h a n two days as j u d g e d by u n a i d e d eye. These m a x i m a define clearly the region above 0.5 M NaC1 and 1.55 mM lysozyme as the optimal crystallization region. We expect t h a t more detailed spacing of both electrolyte and protein concentration will result in improved figures. However~ the gross screening characteristics allow directly for a first order indexing of the observables. We will describe below in detail the features of these plots.

The "Quasi-stationary" Cluster Radius To have a unique m e a s u r e of every kinetic experiment, we have a v e r a g e d cluster radii collected between 50 min and 60 min after initiation of the experiments. This average, which involves some t w e n t y points, we h a v e arbitrarily t e r m e d as "quasi-stationary cluster radius", . Strictly speaking the system is always far from stationarity, but for short e n o u g h time-scales the approximation is justified. Fractals are expected to grow until they occupy the whole volume available, but in practice they grow only for some 12 h to 24 h before they start sedimenting, unless solutions undergo gelation. Since the DLS experiments cease to r e n d e r useful d a t a if the m e a n cluster size exceeds a couple of micrometers, such observations can be better accomplished with static light scattering preferably at low scattering angles [22,23]. Surface and spectral plots of are shown in Fig. 3. It is evident t h a t the behaviour of this surface is quite complex and the electrolyte concentration sections indicate a bell-shaped behaviour. In contrast, the

66

2.2 ~

f2.2

1.3 ~

~

;1.3

l~mhl0"9 t0.4

~

0.90.4 lgml

07

I[M]

0.4

~

1.15

CL[mM]

0.2 0.75 0.80

[um] 2.03+ 1.85 to 2.03 1.68 to 1.85 1.50 to 1.68 1.33 to 1.50

0.65 ~'0.50

1.15 to 1.33

Ima¢

0.97 to 1.15 0.80 to 0.97 0.62 to 0.80 0.45 to 0.62 0.27 to 0.45 0.09 to 0.27

0.35 0.20, 0.75

I. 5

1.55

1.95

2.35

CL [mMl Fig. 3. Surface and spectral plot of the "quasi-stationary" cluster r a d i u s of lysozyme fractals as a function oflysozyme, (CL) , and NaC1, (I), ionic strength. The e s t i m a t e s were o b t a i n e d by a v e r a g i n g the cluster radii recorded between 50 min and 60 rain. An o p t i m u m is confined b e t w e e n 1.25 mM to 1.85 mM protein and above 0.55 M electrolyte (see also [8]). This region correlates very well with microscopic observations on crystal growth w i t h i n one to two days.

67 protein concentration sections indicate a parabolic or slightly sigmoidal b e h a v i o u r in all other regions t h a n at the ends of the surface. In previous studies [4,5], w h e r e a more extended electrolyte r a n g e was employed, we have shown t h a t this behaviour is bell-shaped for both NaC1 and MgC12. If the full experiments are plotted, one can obtain a very detailed grid involving all 33 experiments consisting of 120 points each. The computations and data manipulations required for obtaining this very detailed grid are by far more tedious t h a n for the simple "quasi-stationary radius" since the latter involves only a fraction of points (33 vs. 3960 points). Both d a t a sets have been e x a m i n e d in the late stages of the work and the similarity b e t w e e n the t h r e e - d i m e n s i o n a l surfaces was found to be striking. The absolute values between the two grids did not differ more t h a n 11% at the very most. Therefore, the selection of the "quasi-stationary radius" as an observable is not as a r b i t r a r y as it m a y seem at the first place. These comparisons indicate also t h a t ambiguities due to (i) the choice of weights (ii) d a t a normalization and (iii) specific griding p a r a m e t e r s w h e n employing a limited n u m b e r of d a t a points can be objectively handled.

The Fractal Dimension Surface and spectral plots of the fractal dimension are shown in Fig. 4. The overall change of d / i s small but this is, on purely theoretical grounds [19,20], not unexpected. The fractal dimension will r e m a i n n e a r l y constant as far as one observes aggregation events in a distinct regime w i t h o u t crossing over to another. For pure DLCA aggregation one would expect a constant fractal dimension of 1.81 but lysozyme fractals exhibit deviations from this figure. The latter m a y involve some experimental u n c e r t a i n t y since the fractal dimension is less easy to determine. Multiple experiments u n d e r identical conditions suggested small s t a n d a r d errors. The fact t h a t the fractal dimension falls below the universal limit of 1.81, is probably due to counterion screening t h a t leads to cluster polarization effects [24,25].

The Zero Lag-time Extrapolated Cluster Radius It is also interesting to estimate the m i n i m u m size of stable clusters p r e s e n t in crystallizing solutions. A stable population of m e a n size equal to t h a t of the zero time-lag extrapolated cluster size, is a s s u m e d to be present.

68

1.80

1.70

df

d! .60

50 35

[mM] 0.20

0.75 df 1.80+ ==~ 1.78 to ~ i!~ :i~ i~ :I~ i~ i1.75 i to ii!~iiiiiil/1.73 to ~'~ ~ 1.71 to 1.68 to 1.66 to 1.63 to 1.61 to 1.59 to 1.56 to 1.54 to

0.75

1.15

1.55

1.95

1.80 1.78 1.75 1.73 1.71 1.68 1.66 1.63 1.61 1.59 1.56

2.35

CL [mM] Fig. 4. Surface and spectral plot of the fractal dimension, df, of lysozyme clusters obtained from Eq. (4). An optimal region is confined between 1.3 mM and 1.8 mM protein, and above 0.7 M electrolyte. Within these limits the fractal dimension of the clusters is 1.81 indicative of universal DLCA aggregation [19,20].

B9 In general, one faces a problem w h e n a t t e m p t i n g to c a p t u r e the f e a t u r e s of i n t e r m e d i a t e populations in DLS experiments. The s c a t t e r e d i n t e n s i t y is a function of both n u m b e r and size of the scatterers. In crystallizing ( s u p e r s a t u r a t e d ) solutions, is still p r e s e n t a very large n u m b e r of m o n o m e r s or dimers, 2.09 nm, to 2.65 nm. The s c a t t e r i n g process is however d o m i n a t e d by the presence of only a few t h o u s a n d s but c o m p a r a t i v e l y v e r y large, 200 n m to 1 gm, fractal clusters. The two populations can be, in t h e beginning of the aggregation, decoupled from each o t h e r by inverse Laplace t r a n s f o r m a t i o n , but the resolution of s c a t t e r e r s of i n t e r m e d i a t e sizes and n u m b e r densities is in principle, difficult or even impossible. Note also t h a t for s t a t i o n a r y s y s t e m s one can achieve n e a r l y a r b i t r a r y precision by sampling for long times. This is h o w e v e r not the case with n o n - s t a t i o n a r y systems, since only m e a n i n g less a v e r a g e s would be obtained. Therefore, no more t h a n two compon e n t s can be resolved with confidence, upon Laplace inversion of the ACF. F r o m the classical nucleation t h e o r y one expects t h a t the free e n e r g y of a n u c l e a t i n g cluster increases up to a critical size and is limited by a b a r r i e r t h a t has to be s u r m o u n t e d to t r a n s f o r m a liquid to a crystalline p h a s e [261. The size of these clusters is not expected to v a r y drastically until crystallization is completed. However, the extrapolated Rh(0) should not be identified as the critical size of a nucleus. The typical O s t w a l d Miers nucleation regime [27] observed by other investigators 128-30] using viruses and proteins larger t h a n lysozyme is not observed* in t h e s e experiments. This is due to the rapid diffusion of the lysozyme m o n o m e r s a n d the observed kinetics resembles postnucleation events. Rh(0) is expected to be, in general, larger t h a n the h y d r o d y n a m i c r a d i u s of a typical lysozyme nucleus. The m e a n size of the observed population varies drastically with lysozyme and electrolyte concentration. In Fig. 5, Rh(0) is plotted as a function of lysozyme and electrolyte molarity. A clear m a x i m u m is observed at 1.55 m M protein and a r o u n d 0.50 M electrolyte. Since at p r e s e n t direct theoretical comparisons of these observations are not available, we h a v e r e d e t e r m i n e d Rt,(0) with SAXS in an a t t e m p t to reproduce these results. * Occasionally, at low supersaturation the Ostwald-Miers regime may be observed as well. However, such experiments are subject to large uncertainties since the late nucleation events coincide with the times required for the thermostation of the sample. It is therefore difficult to conclude with certainty about the reliability of the first few records.

70

3529

Rh(0) [nm]

2.3

Rh(O) [nml

P

1

0.~

M1 0.20

0.75

0.80

Rh(O) [nm] 32.0+

29.3 to 32.0

0.65

ii !ii 26.7 24.0 21.3 18.7

~___~0.50

to to to to

29.3 26.7 24.0 21.3

16.0 to 18.7

13.4 to 16.0 10.7 to 13.4

0.35

8.0 to 10.7 0.20

0.75

1.15

1.55

1.95

2.35

5.4 to 8.0 2.7 to 5.4

CL ImM! Fig. 5. Surface and spectral plot of the zero-time lag e x t r a p o l a t e d m e a n r a d i u s R/,(0) of lysozyme clusters, obtained from Eq. (4). A very clear m a x i m u m is observed for 1.35 mM to 1.75 m M protein, a n d 0.4 M to 0.6 M electrolyte. This region correlates well with microscopic observations on crystal growth within less t h a n two days.

71

Size Determination of Stable Clusters with S A X S For identifying the m e a n size of the lysozyme clusters in solution we h a v e u n d e r t a k e n SAXS m e a s u r e m e n t s at two different ionic s t r e n g t h s and six different protein concentrations. The experiments w e r e conducted u n d e r conditions identical to those of the DLS e x p e r i m e n t using the s a m e lysozyme preparation. The Guinier plots were biphasic, indicating the existence of at least two distinct populations in solution. We have decoupled with linear-least squares a low molecular weight population and plotted these d a t a separately as a function of protein and electrolyte molarity. This species does not exhibit any time dependence. Their size, w h e n extrapolated to infinite dilution, is found equal to 1.96 nm, which compares favourably with the DLS result, 2.09 nm, for the h y d r a t e d lysozyme monomer. It has to be e m p h a s i z e d t h a t t h e r e is no w a y to decide from SAXS e x p e r i m e n t s w h e t h e r the sample is mono- or polydisperse. Therefore, each d a t a set can be e v a l u a t e d by a s s u m i n g either a monodisperse or a polydisperse composition. We have analysed the SAXS data by a s s u m i n g monodisperse ensembles. Because of the complexity of the problem, a more intensive analysis is at this stage, unjustified. The extraction of size estimates of the clusters by SAXS is by far more complex. For a t t a i n i n g an a d e q u a t e r e p r e s e n t a t i o n of the d a t a in the i n n e r portion of the scattering curves, the data were fitted with least squares after omitting the first one or two channels t h a t showed systematically u p w a r d curvature. Both experiments indicate m a x i m u m cluster sizes, R s ~ s , up to 16 times larger t h a n monomeric lysozyme. A comparison of this i n t e r m e d i a t e cluster size, present in crystallizing solutions by both DLS and SAXS, is depicted in Fig. 6. The observed e x t r e m a occur again a r o u n d 1.55 mM lysozyme, and resemble quite closely the zero-lag time extrapolated cluster m e a n sizes in the respective DLS experiments, Fig. 5. The Monomer to Dimer Transition Finally, the m e a n size of the species t h a t are usually assigned as m o n o m e r s in a DLS experiment was examined. Careful e x a m i n a t i o n of over 2500 spectra indicated t h a t the m e a n size of this species obeys a systematic t e n d e n c y t h a t m a y include useful information. Since dimerization is the r a t e limiting step for any kind of association, one expects

72

35

'

F

'

'

'

'

'

I

. . . . .

I

. . . . .

I

SAXS 30

'

DLS

[]

0.55 M

"

0.55 M

o

0.75 M

•

0.65 M

•

0.75 M

25 ,

r

t

I

20

15 'A

10

5

,

I

0.6

I

,

,

,

,

I

1.2

I

,

,

,

,

I

1.8

~

~

J

,

I~

T

,

2.4

CL [mM] Fig. 6. Comparison of the m i n i m u m cluster size, R/~(0), as a function of lysozyme concentration o b t a i n e d from DLS and SAXS experiments. Note t h a t the e x t r e m a shown by SAXS r e s e m b l e closely those d e t e r m i n e d by the more detailed DLS experiment.

t h a t these species will behave in a way that reflects bulk properties of the examined solution. Therefore, at some critical protein and electrolyte molarity one would expect the formation of stable oligomeric species having a size larger than that of the hydrated monomer. The data plotted as surface and spectral plots (Fig. 7) indicate an increment of the mean monomer size, Ro, with increasing protein and electrolyte molarity. These data reproduce accurately those obtained in [8] although the latter experiments covered a wider range of conditions

73

2.70

2.70

2.53

2.53

R0

!.35

2.3~

[nml

inml

.18

2.1

00 35

2.( 0.1

[raM] 0.20

0.75

ROlnmi

~

0.75

1.15

1.55

1.95

2.65+ 2.59 to 2.54 to 2.48 to 2.43 to 2.37 to 2.32 to 2.27 to 2.21 to 2.16 to 2.10 to 2.05 to

2.65 2.59 2.54 2.48 2.43 2.37 2.32 2.27 2.21 2.16 2.10

2.35

CL ImMl Fig-. 7. Surface and spectral plot of the lysozyme monomer to stable lysozyme d i m e r transition. S t a b l e dimers are formed with higher probability above 1.95 mM lysozyme and 0.65 M NaC1. I n t e r m e d i a t e sizes m a y r e p r e s e n t rapidly a s s o c i a t i n g - d i s s o c i a t i n g species due to incomplete electrostatic screening.

74 and w e r e purposely designed for studying molecular interactions of lysozyme d u r i n g the crystallization process. For free diffusing spheres, the m o n o m e r radius is 77.8% of t h a t of the stable dimer. The largest sizes detected in the experiments do not exceed ca. 2.68 nm, a size t h a t could correspond to stable dimers. L a r g e r oligomers, m a y exhibit sizes t h a t are only slightly higher t h a n the later figure, d e p e n d i n g on their m u t u a l orientation. Precise m e a s u r e m e n t s of larger oligomers are however very difficult due to the non-stationary n a t u r e of the e x a m i n e d samples. Particles with such sizes of about 2.65 a p p e a r above 0.65 M NaC1 and 1.95 mM lysozyme. It is however questionable if the i n t e r m e d i a t e s b e t w e e n pure m o n o m e r s and dimers r e p r e s e n t stable species. The s a m e question has been raised for the dimerization of bovine pancreatic trypsin inhibitor [31, 32]. There, such a behaviour was attributed to an either rapid m o n o m e r to d i m e r association-dissociation or, to the prevalence of van der Waals interactions. In general these concepts are more complex t h a n one m a y t h i n k and their detailed discussion is out of the scope of the p r e s e n t work. We can conclude t h a t the transition from monomeric species to stable dimers occurs as a r a t h e r smooth function of the solution s u p e r s a t u r a tion. The stable d i m e r region is again observed within the r e a l m of crystal growth; above 1.95 mM lysozyme and 0.55 M electrolyte.

Amplitude Analysis One can a s s u m e t h a t the total scattered intensity results from two major contributions, n a m e l y scattering from the smallest species and scattering from fractal clusters. One m a y t h e n be t e m p t e d to compute t e n t a t i v e estimates of the n u m b e r s of scattering species from the above e x p e r i m e n t s a s s u m i n g t h a t only "monomers" and fractals are present in the e x a m i n e d solutions. This assumption is r a t h e r crude since it neglects the free nucleating species and p r e s u m e s irreversibility. Neglecting the n u c l e a t i n g species is not expected to influence the relative tendencies since their n u m b e r is small compared to the n u m b e r of monomers and fractals. F u r t h e r , the irreversibility assumption is justified within the f r a m e of fractal cluster formation theory. The tentative estimates obtained from the scattering amplitude analysis can again be used for defining the optimal crystallization region for lysozyme based on criteria similar to those described above.

75 As can be seen from Eq. (1), the n u m b e r of s c a t t e r e r s is explicitly included in t h e k e r n e l of the expression for the ACF. The particle size distribution function of each species can be obtained after Laplace-transformation of the ACF spectra. For small h a r d spheres, due to the i n t e n s i t y p r e s e r v e d weighting of the ACF, one usually considers the m a s s - s q u a r e d w e i g h t e d average, P(q) M2(R) N(R), as a function of the radius R. For the "monomeric" species one can simply recast this t e r m as R GN(R) w h e r e a s for fractal s t r u c t u r e s as P(q) S(q)N(R)R'2C/i due to the mass-size scaling power-law, M(R) o
N() ~ N,,,(t) + n(t) N / ( t ) + .

(5)

E q u a t i o n (5) should obey the conditions: N,,,(O) = No, n(0) = 0 and Nf(O) = 0. After infinite time it is theoretically expected t h a t N o = N,,~(~) = 0 a n d N/.(~) = 1 and n(t) = N o (if one imagines a huge fractal t h a t fills up all the available space). The l a t t e r is in general not true, unless gelation occurs and monomeric species can still be observed in DLS e x p e r i m e n t s for several days even after the s e d i m e n t a t i o n of fractals and the appearance of the crystalline phase. In Fig. 8 the relative n u m b e r o f " m o n o m e r s " is plotted as a function of protein and electrolyte molarity. The ordinate has been normalized to a c o n v e n i e n t scale b e t w e e n 0 and 1. It can be seen t h a t the n u m b e r of free "monomers" in solution r e a c h e s a distinct m i n i m u m a r o u n d 1.55 mM of protein and 0.65 M of electrolyte. If crystallization conditions are close to optimal the n u m b e r of m o n o m e r s is indeed expected to be m i n i m i z e d since t h e y h a v e a h i g h e r probability to be incorporated onto nuclei t h a n

76

1.0

0.8 In Nm

In Nm 0.~ O.

0 0.~

al 0.80 2.35 °'8

In Nm

0.35

0.20 0.75

1.15

1.55

1.95

2.35

0.99+ 0.91 to ii. IIII;IZi i!ii=iiii0.84 to 0.76 to 0.68 to 0.61 to 0.53 to 0.45 to 0.38 to 0.30 to 0.22 to 0.15 to

0.99 0.91 0.84 0.76 0.68 0.61 0.53 0.45 0.38 0.30 0.22

CL [mM] Fig, 8. Surface a n d spectral semilogarithmic plots of the relative n u m b e r of monomers, Arm, in lysozyme crystallizing solutions. A clear m i n i m u m is observed at 1.55 mM protein a n d 0.65 M electrolyte. The relative n u m b e r of free monomers in solution is m i n i m i z e d t h r o u g h incorporation on small nuclei. Please note the changes u n d e r t a k e n at the origin of the x - y axes for the sake of clarity.

77 move freely in the solution. These figures agree well with the o p t i m u m r a n g e of observables defined above and in Ref. [8]~ on the s a m e subject. The relative n u m b e r of fractals Nf is plotted in Fig. 9. W h e r e a s the n u m b e r of observed fractals develops smoothly almost over the entire protein-electrolyte surface, a distinct drop is observed above 1.25 mM protein and 0.30 M electrolyte. This behaviour can be easily understood if one considers fractal and nuclei formation as two competing processes (see discussion below). At optimal regions the n u m b e r of fractals is expected to be minimized t h u s favouring nucleation.

Association Between Nucleation a n d Fractal F o r m a t i o n

Fractal cluster formation is typically observed in systems developing far from equilibrium. Kinetic models are employed for the i n t e r p r e t a t i o n of the obtained d a t a t h a t do not allow for s t r u c t u r a l r e a r r a n g e m e n t s , and therefore m i n i m u m energy configurations cannot be deduced. On the contrary, t h e r m o d y n a m i c models describe growth p h e n o m e n a close to equilibrium via compact s t r u c t u r e formation. Classical phase transition models (nucleation and growth or spinodal decomposition) are based on t h e r m o d y n a m i c s . The formation of possibly fractal structures in kinetic growth processes has only been recently considered [33]. For a consistent picture associating nucleation with fractal cluster formation, one can imagine the whole scenario starting w i t h the nucleation b u r s t w h e n protein and electrolyte come into contact. N u c l e u s - n u cleus collisions t h e n trigger catastrophic r a n d o m aggregation t h a t dominates the light scattering process in the early stages. However a minority of nuclei m a n a g e s to escape the "Smoluchowski sink" and put crystal formation through. The question w h e t h e r the nuclei formed at the early stages are compact or ramified is not easy to answer. The equilibrium shape of a cluster is usually approximated as a smooth sphere. This is certainly not correct since it is k n o w n t h a t cubes or polyhedra can r e p r e s e n t b e t t e r lattice forming shapes. In such a case geometrical form factors can account for the deviations from spherical shape [34]. To mention an example, if one assumes cubes instead of spheres for the equilibrium shape, the n u m b e r of molecules present in the critical nucleus is expected to be twice as large. Nuclei having a m e a n size of 25 n m will involve up to 1700 monomers, a s s u m i n g a spherical shape.

78

In Nf

1.0

1.0

0.8

El.8

0.I

P.6 in Nf .4

.

2

0

) 75

0.

imMl 0.80

2.35

In Nf 1.00+

0.75

1.15

1.55

1.95

2.35

0.91 iiiiii iiiiijiii0.82 0.73 0.64 0.55 0.45 0.36 0.27 0.18 0.09 0.00

to to to to to to to to to to to

1.00 0.91 0.82 0.73 0.64 0.55 0.45 0.36 0.27 0.18 0.09

C L [raM! Fig. 9. Surface and spectral semilogarithmic plots of the relative number of fractals, N/; in lysozyme crystallizing solutions. A broad minimum is observed above 1.55 mM lysozyme and 0.30 M electrolyte. The relative number of fractals is minimized if smaller compact clusters are consumed in nuclei formation. Please note the changes undertaken at the origin of the x - y axes for the sake of clarity.

79 F u r t h e r , there is no direct experimental evidence t h a t these nuclei are indeed compact. Theoretical studies [35-37] have shown t h a t deviations from the classical picture are not improbable. W h e n small compact clusters evolve in time, both the n u m b e r of contact points and the probability of collision will increase. Such clusters will t h e n fail to i n t e r p e n e t r a t e each other as if t h e y were small. Therefore, the growth of more t e n u o u s s t r u c t u r e s with lower fractal dimensions, as time proceeds, is not unexpected. Very recently, some interesting observations were m a d e by low- and wide-angle X-ray scattering on zeolites I38]. These e x p e r i m e n t s indicate t h a t nucleation, in contrast to classical theory, occurs via formation of small fractal structures. These clusters undergo r e s t r u c t u r i n g and after r e a r r a n g e m e n t s a dense microcrystalline phase appears. W h e r e a s it would be r a t h e r p r e m a t u r e to extrapolate such properties to proteins, these observations r e n d e r indirect support to our findings. Although it is difficult to define the exact borders between nucleation and postnucleation, we believe t h a t our light scattering e x p e r i m e n t s capture well the postnucleation events. Nucleation, in the classical sense, is probably so fast for lysozyme t h a t it escapes observation. Unless special precautions are taken, the information is lost. However, we have recently shown t h a t pertinent, albeit not always easy to interpret, information concerning the fate of protein crystallizing solutions can be extracted w i t h DLS techniques [7,391.

CONCLUSIONS The combined use of DLS and SAXS allows a first order analysis of lysozyme aggregation during nucleation and growth. The obtained results show t h a t the observed clusters are fractals formed by smaller aggregates. The growth of these clusters can be described by power-law kinetics [4,5,7,8] and the appearance of crystals within short times is reinforced w h e n aggregation takes place in the DLCA regime, at least u n d e r the e x a m i n e d conditions. Cases w h e r e aggregation takes place in different regimes (reaction limited or cross-over aggregation regimes i.e. while using (NH4)2SO 4 as a precipitant should be t r e a t e d carefully keeping always in mind t h a t several notorious effects m a y modify the value of the fractal dimension. The growth kinetics can be obtained w i t h fair precision; kinetics can be rapidly deduced during the early stages of

80 the reaction (i. e. the first one to two hours) and can be recast into surface and spectral plots for visualizing the apparent tendencies. We have persuasively shown that the obtained observables show pronounced extrema that coincide with those solution conditions where tetragonal, well-diffracting, lysozyme crystals are obtained within less than two days. Therefore critical crystallization regions indeed exist, and they can be captured even when observations are made in the post-nucleation phase. The empirical correlation between the occurrence of extrema in cluster formation and crystal growth indicates that the observables derived from the scattering experiment may be identified as useful tools for rapidly screening multiple crystallization set-ups by using DLS instrumentation. Such experiments may help in general to understand metastability that is an intrinsically dynamic process. Due to the different spatial resolution of the DLS and SAXS experiments, it is easier to determine by SAXS small stable aggregates in nucleating solutions. Similar size estimates are obtained by DLS after extrapolating the growth kinetics to zero time-lag. This is an important aspect since it ensures that one observes cluster-cluster rather than monomer-cluster aggregation. Therefore, the seeding particles that build the fractals are not lysozyme monomers but, with very high probability, nuclei. It is understandable that, due to the small size of lysozyme, capturing the nucleation burst with methods like time-resolved DLS or SAXS may be extremely difficult. The times required for completion of the nucleation burst, may be orders of magnitudes shorter than the dead time of the experiment. A comparison with experimentally measured solubilities of lysozyme with two different cations Na ÷ and Mg 2÷ [2] was also undertaken. The results are displayed in Figs. 10a and 10b as surface and spectral plots. At conditions comparable with ours these plots indicate that the solubility minima, i.e. the regions where the probability for obtaining crystals is increased, are in either case quite close (see also Fig. 3 in Ref. [5]) but they do not coincide with the regions defined by the scattering experiment and the appearance of lysozyme crystals. Solubility maxima and minima alternate in the plots. Whereas such effects can be clearly seen in nearly all data sets measured by the authors, in the plots displayed herein such tendencies are clearer for Mg 2÷ whereas Na ÷ show a broader range of low solubilities. The conclusion drawn from such three-dimensional plots is that solubilities are complex functions of both protein and electrolyte concentration (packing and electrostatic screening).

81

1.2

1.2

0.9

0.9

k.6 S

S 0.(

1.3

.

,0 .33

0 8

q

CL 0.2

1.53

S NaCI 1.1+ 1.0 to 1.1 0.9 to 1.0 0.8 to 0.9 0.7 to 0.8 0.6 to 0.7 0.5 to 0.6 0.4 to 0.5 0.3 to 0.4 0.2 to 0.3 0.1 to 0.2 0.0 to 0.1

1.53

1.13

0.73

0.33 0.2 (a)

212

4.2

6.2

8.2

CL [mM]

Fig. 10. Solubilities computed from Table I of Ret\ [2]. (a) Surface and spectral plots of spectrophotometrically determined solubility S, of lysozyme solutions incubated with NaC1. Minima are identified at 2.01 raM, 4.2 mM and 7.1 mM lysozyme and correspond to 1.07 M, 0.73 M and 0.47 M NaC1, respectively.

82

0.32

0.32

0.27

D.27

S0.2]

~.21 S .16

0.1

10 20

0.]

1

[mM] 3.5

3.40

3 4a

S MgCi2 0.30+ 0.28 to 0.30 0.27 to 0.28 (!i~ 0.25 to 0.27 0.23 to 0.25 0.22 to 0.23 0.20 to 0.22 0.19 to 0.20 0.17 to 0.19 0.15 to 0.17 0.14 to 0.15 0.12 to 0.14 l.O

(b)

1.5

2.0

2.5

3.0

3.5

CL lmMl

Fig. 10 continued. (b) Surface and spectral plots of the relative solubility S, of lysozyme solutions i n c u b a t e d with MgC12. M i n i m a are identified at 2.19 mM, 2.27 mM a n d 3.36 m M lysozyme and correspond to 1.88 M, 3.14 M and 1.39 M MgC12, respectively. The l a s t v a l u e compares favourably with the e x p e r i m e n t s depicted in Fig. 3 of Ref. [5]. Please note t h e changes u n d e r t a k e n at the origin of the x - y axes for the sake of clarity.

83 T h e e x t e n s i v e e x p e r i m e n t a l d a t a d i s c u s s e d in this w o r k p r o v i d e a first p l a t f o r m for i m p r o v i n g o u r u n d e r s t a n d i n g on p r o t e i n c r y s t a l l i z a t i o n a n d p r e d i c t i v e t h e o r e t i c a l f r a m e w o r k s . T h e o r e t i c a l d e s c r i p t i o n s of p r o t e i n c r y s t a l l i z a t i o n a r e h i g h l y d e s i r a b l e . T h e r e one h a s to cope w i t h a s i m u l t a n e o u s d e s c r i p t i o n of n u c l e a t i o n a n d f r a c t a l c l u s t e r g r o w t h , in a m a n n e r s i m i l a r to t h a t p r o p o s e d b y B i n d e r a n d S t a u f e r I40], n e a r l y t w e n t y y e a r s ago. T h e first o r d e r t h e o r e t i c a l m o d e l l i n g of t h e l y s o z y m e NaC1 s y s t e m , in t h e region of t h e r m o d y n a m i c s t a t e s d e s c r i b e d h e r e will be p u b l i s h e d in a s u b s e q u e n t c o m m u n i c a t i o n ( S o u m p a s i s a n d Georgalis, in p r e p a r a t i o n ) .

ACKNOWLEDGEMENTS T h e f i n a n c i a l s u p p o r t to Y.G. r e c e i v e d from t h e E u r o p e a n S p a c e Agency (ESA/ESTEC), the Freie Universitfit Berlin and the Deutsche F o r s c h u n g s g e m e i n s c h a f t (Sa. 196/26-1) a n d to D.M.S. from t h e B u n d e s m i n i s t e r i u m ffir F o r s c h u n g u n d Technologic, is d e e p l y a c k n o w l e d g e d . T h a n k s b e l o n g to Dr. W. Vogel for his s u p e r v i s i o n a n d a s s i s t a n c e w i t h t h e X - r a y s a n d to Drs. P.R. Wills, a n d P. Z i e l e n k i e w i c z for critical s u g g e s t i o n s on t h e m a n u s c r i p t .

REFERENCES 1 Z. Kam, H.B. Shore and G. Feher, J. Mol. Biol., 123 (1978) 539. 2 M.M. Ri~s-Kautt and A.F. Ducruix, J. Biol. Chem., 263 (1989) 745. 3 A.F. Ducruix and R. Gi~ge (Eds.), Crystallization of Nucleic Acids and Proteins. IRL Press, Oxford, 1992. 4 Y. Georgalis, A. Zouni, W. Eberstein and W. Saenger, J. Crystal Growth 126 (1993) 245. 5 Y. Georgalis and W. Saenger, Adv. Colloid Interface Sci., 46 (1993) 165. 6 Y. Georgalis, J. Schiller, W. Eberstein and W. Saenger, in: M.M. Novak (Ed.), Fractals in the Natural and Applied Sciences. IFIP (A-41), pp. 139, Elsevier Science, Amsterdam, 1994. 7 W. Eberstein, Y. Georgalis and W. Saenger, Eur. Biopbys. J., 22 (1993) 359. 8 W. Eberstein, Y. Georgalis and W. Saenger, J. Crystal Growth 143, (1994) 71. 9 S.K. Schmitz, An Introduction to Dynamic Light Scattering by Macromolecules. Academic Press, New York, 1990. 10 B. Chu, Dynamic Light Scattering. Academic Press, 1991. 11 W. Brown (Ed.), Dynamic Light Scattering, The Method and Some Applications. Oxford Science Publications, 1993.

84 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

O. Glatter and O. Kratky (Eds.), Small Angle X-Ray Scattering Academic Press, London, 1982. O. Glatter, in: P. Lindner and Th. Zemb (Eds.), Neutron, X-Ray and Light Scattering. North-Holland, Amsterdam, 1991. M.E. Fisher and R.J. Burford, Phys. Rev. A, 156 (2) (1967) 583. H.M. Lindsay, R. Klein, D.A. Weitz, M.Y. Lin and P. Meakin, Phys. Rev. A, 38 (5) (1988) 2614. H.M. Lindsay, R. Klein, D.A. Weitz, M.Y. Lin and P. Meakin, Phys. Rev. A, 39 (6) (1989) 3112. S.W. Provencher, Comp. Phys. Comm., 27 (1982) 213. S.W. Provencher, Comp. Phys. Comm., 27 (1982) 229. M.Y. Lin, H.M. Lindsay, D.A. Weitz, R.C. Ball, R. Klein and P. Meakin, Nature (London), 339 (1989) 360. M.Y. Lin, H.M. Lindsay, D.A. Weitz, R.C. Ball, R. Klein and P. Meakin, Proc. R. Soc. Lond. A, 423 (1989) 71. Stanford Graphics Presentation and Analysis Package, 3-D Visions. F. Ferri, M. Giglio, E. Paganini und U. Perini, Europhys. Lett., 7 (1988) 599. M. Carpineti, F. Ferri, M. Giglio, E. Paganini and U. Perini, Phys. Rev. A, 42 (12) (1990) 7347. R. Jullien, Phys. Rev. Lett., 55 (1985) 1697. R. Jullien, J. Phys. A, 19 (1986) 2129. D.W. Oxtoby, Adv. Chem. Phys., 70 (1988) 263. I.M. Lifschitz and V.V. Slyojov, J. Phys. Chem. Solids, 19 (1965) 35. A.J. Malkin, J. Cheung and A. McPherson, J. Crystal Growth, 126 (1993) 545. A.J. Malkin and A. McPherson, J. Crystal Growth, 126 (1993) 555. A.J. Malkin and A. McPherson, J. Crystal Growth, 1232 (1993) 128. P.R. Wills and Y. Georgalis, J. Phys. Chem., 85 (26) (1981) 3978. W.H. Gallagher and C.K. Woodward, Biopolymers, 28 (1989) 2001. M. Carpineti and M. Giglio, Phys. Rev. Lett., 68 (22) (1992) 3327. K.H. Lieser, Angew. Chem. 81 (6) (1969) 206. D.W. Heermann and W. Klein, Phys. Rev. Lett., 50 (14) (1982) 1062. W. Klein and F. Leyvraz, Phys. Rev. Lett., 57 (22) (1986) 2845. J. Yang, H. Gould and R.D. Mountain, J. Chem. Phys., 93 (1) (1990) 711. W.H. Dokter, H.F. van Garderen, T.P.M. Beelen, R.A. van Santen and W. Bras, Angew. Chem., 107 (1) (1995) 122. W. Heinrichs, Y. Georgalis, H.-J. Sch6nert and W. Saenger, J. Crystal Growth, 133 (1993) 196. K. Binder and D. Staufer, Adv. Phys., 25 (1976) 343. P. Meakin, in: C. Domb and J.L. Lebowitz (Eds.), Phase Transitions and Critical Phenomena, Vol. 12. Academic Press, New York, 1988, 351 pp.

85 APPENDIX I

Decoupling of Rotational Motions F r a c t a l s are a s y m m e t r i c structures, and the contributions of both t r a n s l a t i o n a l and rotational motions to the ACF m u s t be t a k e n into account. The field ACF of anisotropic, inhomogeneous scatterers, has been calculated w i t h the a s s u m p t i o n t h a t rotational a n d t r a n s l a t i o n a l motions are uncoupled. C o m p u t e r g e n e r a t e d and e x p e r i m e n t a l l y produced DLCA aggregates show typical a s y m m e t r y ratios of about 1.7, a value t h a t indicates t h a t this a s s u m p t i o n is valid. If the size of the clusters is large*, rotational motions are expected to c o n t r i b u t e t h r o u g h the cluster polydispersity to the ACF because t h e y are q i n d e p e n d e n t .

g(ll(-c) =

~, N M 2 exp(- q2 Dt ) ~_~Sl(q ) exp [- l(l + 1) D r ~]

(6)

~_, N M 2 S/(q) w h e r e D t a n d D r denote the t r a n s l a t i o n a l and rotational diffusion coefficient of the fractal with radius R respectively, and the s u m m a t i o n e x t e n d s over all clusters c h a r a c t e r i z e d by this radius

kBT D r = 8rcT1R3

(7)

E a c h t e r m in the second sum, in Eq. (6), which corresponds to the rotational contribution, is w e i g h t e d with the factor Sl(q) t h a t can be obtained after a multipole expansion of the s t r u c t u r e factor [15,16]

S(q) = ~. Sl(q)

(8)

The l a t t e r s u m has been computed up to the s e v e n t h t e r m from c o m p u t e r g e n e r a t e d DLCA aggregates. E q u a t i o n (8) allows t h e n the r e c o n s t r u c t i o n of the s c a t t e r e d field ACF as follows: * For simplification we have assumed that the hydrodynamic radius is equal with the radius of gyration. For DLCA clusters the two radii differ by some 10%.

86

g(1)(z) -

~_~N M 2 exp [-

q2 Deff X S(q)]

(9)

~_, N M 2 S(q) where Deft.= D t f (qR) denotes an effective diffusion coefficient including both translational and rotational diffusional motions. We have previously shown [4] how an approximate correction can be applied to account for coupling of the rotational motions. One can then obtain estimates of the translational diffusion coefficient, Dt, and therefore Rh, with adequate precision. While currently, the employed structure factors seem to adequately describe both diffusion and reaction limiting aggregation regimes, there may be a disadvantage in their use. Namely, the large-scale computer simulations where they originate from, are valid for ideal clusters composed of a limited number of particles and they do not consider the effects of internal rotations within the cluster (restructuring), which may modify the fractal dimension. Large-scale computer simulations [41] have indeed shown that changes of the fractal dimension are to be expected when clusters undergo internal rotations.