Crossover from a Linear to a Branched Growth Regime in the

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Cite This: Cryst. Growth Des. XXXX, XXX, XXX−XXX

Crossover from a Linear to a Branched Growth Regime in the Crystallization of Lysozyme R. J. Heigl,† M. Longo,† J. Stellbrink,‡ A. Radulescu,† R. Schweins,§ and T. E. Schrader*,† †

Forschungszentrum Jülich GmbH, Jülich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), Lichtenbergstrasse 1, 85748 Garching, Germany ‡ Forschungszentrum Jülich GmbH, Jülich Centre for Neutron Science JCNS-1 and Institute for Complex Systems ICS-1, 52425 Jülich, Germany § Institut Laue − Langevin (ILL), DS/LSS, 71 avenue des Martyrs, 38000 Grenoble, France S Supporting Information *

ABSTRACT: Using lysozyme as a crystallization model, existing intermediate clusters and aggregates have been previously identified as fractal systems using light scattering techniques. However, this has not been confirmed with neutron or X-ray scattering directly. In this work, we attempt to deepen our knowledge of the role of the fractal clusters during the crystallization process by following the evolution of the fractal dimension df from the early stage of the nucleation process. Indeed, three different scattering techniques have been used simultaneously on the same sample: dynamic light scattering, small-angle neutron scattering, and static light scattering. We focused on the optimal batch crystallization condition in order to obtain large crystals (30 mg/mL lysozyme concentration and 3 wt % sodium chloride at pD 4.75 at 298 K). The selected temperature reduces the nucleation speed allowing us to investigate in detail the very early stage of the crystallization process. A direct temporal change of the fractal dimension df during the initial growth phase of lysozyme was observed with df rising from 1.0 to 1.7 in the first 90 min after initiating the crystallization process. The early phase of crystallization shows remarkable similarities to simulations on colloid aggregation. Long-term dynamic light scattering measurements allowed us to gain some insight into how fractal clusters may contribute during the crystal growth process. These findings help to improve theoretical models of crystal growth and may lead to the growth of larger crystals through a better understanding of the initial nucleation phase.

I. INTRODUCTION Detailed information on the three-dimensional protein structure is an important requirement in biology, pharmacy, and biotechnology. Despite the efforts made at XFELs,1 the determination of high-resolution protein structures using X-ray diffraction still requires the crystallization of the protein samples. For many proteins, the question of whether there is a crystalline phase remains unanswered. Moreover, even if the protein is known to form crystals, the protein solution may form a liquid−liquid phase, gels, or amorphous debris under small variations of the typically small parameter window of the “right” crystallization conditions.2 In the case of neutron diffraction, protein crystals are inevitably needed, and their size poses an additional requirement and should exceed at least 0.1 mm3.3 Given the huge number of parameters governing the nucleation and crystal growth process, protein crystallization is regarded more as an art rather than a science. Often crystallization remains a trial and error process.4 The nucleation of proteins has turned out to be the thin edge of a wedge that determines the final structure and size of the growing crystals.5,6 In particular, this study chose crystallization conditions where large (>0.1 mm3) crystals form which are © XXXX American Chemical Society

even suitable for neutron protein crystallography; this has become an important tool in determining protonation states of amino acid side chains.7−11 The crystallization process varies widely in terms of size ranges, starting from monomers of a few nanometers in diameter up to crystals that can be seen by the naked eye on the order of millimeters. This poses a difficult challenge in terms of the methods employed for following the growth of the crystals. The most limiting factor in scattering experiments is the available momentum transfer range, which directly correlates with the particle size that can be observed. As probes, neutrons, visible light, and X-rays may be chosen. However, the radiation damage of X-rays makes it difficult to follow the nucleation process on the same sample continuously. Neutrons provide comparable information in the size range between 1 and 300 nm. For larger sizes, static light scattering supplements the q-range accessible for neutrons due to their limitations in flux. Static light scattering can be used to provide Received: October 12, 2017 Revised: January 15, 2018 Published: January 23, 2018 A

DOI: 10.1021/acs.cgd.7b01433 Cryst. Growth Des. XXXX, XXX, XXX−XXX

Crystal Growth & Design

Article

Figure 1. Schematic illustration of the dynamic and static light scattering setup at the sample stage (view from the top) of the small-angle neutron scattering instrument D11. Equal parts of the lysozyme solution and the sodium chloride solution are mixed and injected into the setup by a syringe pump. After passing through the mixer, the crystallization sample continues through the sample cell of the static light scattering device and flows into a rectangular flow through cell, where it is analyzed by neutron scattering and in situ dynamic light scattering simultaneously.

information in the size range between ca. 250 nm and 2 μm. Although neutron and static light scattering can be utilized for structure determination of the growing crystal nuclei, dynamic light scattering is used to report on the hydrodynamic radii in a range between 1 nm and 1 μm of the species in the crystallization process. It is also able to detect any unwanted aggregation. Here we also characterize the evolution of the sample system by measuring the diffusion constant of the involved particles with dynamic light scattering, thereby giving an overview of how many species are present. Another aspect is the speed of the formation process, which leads to the nuclei of critical size.12 If the speed of crystallization is too high, there is not enough time to acquire size and structural information from the intermediate particle sizes. However, if the speed is too slow, e.g., if it takes place over a period of weeks, an experiment cannot be reproduced within a reasonable time period. Since slight temperature or concentration variations change the speed of crystallization, it is important to record as much data as possible from each individual crystallization process simultaneously. This is realized in this study by measuring neutron diffraction and dynamic light scattering in the same cell (so-called in situ DLS on small angle neutron instruments). The same sample was also run through the static light scattering instrument to rule out concentration variations. In this work we focus on hen-egg white lysozyme, as it is one of the most studied models for protein nucleation and crystallization. For lysozyme, the phase diagram of solubility, including solutions with different pH and salt conditions, is well documented in the literature.13,14 Therefore, we were able to find the right compromise for the crystallization speed in deuterated mother liquor. Fortunately, this speed led to the formation of quite large crystals (>0.1 mm3). There is an open debate on the presence of intermediate clusters during the crystallization of hen-egg white lysozyme. There are studies dealing with the particle size distribution of aggregates and existing particle fractions during the nucleation process mostly based on dynamic light scattering. Georgalis et

al. provide detailed information about the growth of subnuclear clusters.15−17 By using a diffusion limited cluster aggregation (DLCA) model18 to fit the time-dependence of the hydrodynamic radii derived from the dynamic light scattering data, the fractal dimension df of these clusters is determined during the crystallization of hen-egg white lysozyme. These particles could be identified as mass fractals19 corresponding to the classification of Lin et al.20 On the other hand, there are authors who question the existence of such particles21,22 albeit as a consequence of the utilized measurement technique23 and due to the lack of available momentum transfer in most scattering experiments. Following the suggestion of Yau et al.,24 this work accounts for the structure of the nucleus. This study builds upon the findings of Georgalis et al.,19 who identified the fractal structure of lysozyme particles during nucleation, but it also determines the change of the particle structure during the nucleation of lysozyme. On the basis of the determination of a timedependent fractal dimension df(t), further insights into the development of the structure of nucleation agents could be achieved and may serve to improve theoretical models of crystal growth and nucleation.

II. MATERIALS AND METHODS Sample Preparation. The mother liquor analyzed in this work was produced by a mixture of equal parts of 60 mg/mL lysozyme sample and 6 wt % NaCl in D2O. Heavy water was used instead of normal water in order to diminish incoherent scattering of neutrons.25 This batch crystallization method yields a lysozyme concentration of 30 mg/mL (2.1 mM) and a final NaCl concentration of 3 wt % (0.5 M). The selected conditions belong to the optimal crystallization region for lysozyme (above 0.5 M NaCl and 1.55 mM Lysozyme).26 Above 0.5 M NaCl the screening of the surface negative charges is complete, and diffusing monomers are free to contact and form bonds.27 Both sample parts exhibit a pD of 4.75 (pD = pH + 0.4)28 in order to achieve a repulsive potential between the lysozyme molecules, preventing unwanted nucleation and to arrive at tetragonal lysozyme crystals.5,29 Lysozyme was directly dissolved in D2O, but not in a buffer, and the pD of 4.75 was reached by the addition of a B



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to ca. 20 s. The resulting flow rate was 2 mL/min during the injection process. Faster injection speeds were not possible due to a buildup of back pressure from the cuvette in the static light scattering (SLS) instrument.32 After the 20 s long injection phase the sample was not moving anymore. So, during the crystallization process the sample was standing still in the two cuvettes. Therefore, no great influence on the crystallization process is expected from the injection flow rate. The SANS measurements were performed at a neutron wavelength of 8 Å. Utilizing detector distances of 2, 8, and 20 m, the available momentum transfer q ranged from 2.3 × 10−3 to 0.24 Å−1. The flowthrough cell in the neutron beam had a path length of 1 mm and was 1 cm wide. The neutron beam was cut to a rectangular shape of 0.7 cm in width and 1 cm in height. The available momentum transfer could have been further optimized to lower q at a maximum detector distance of 40 m. However, we refrained to make use of it due to poor resulting data quality emerging from the low neutron flux and the small scattering intensity of the sample system. The SANS curve after 80 s, depicted in Figure 3, was created by averaging 70 scattering curves of seven successive measurement runs. In total, 7 times 4 min of neutron data was averaged. As we account for small differences between successive experiments, SANS data for later times were averaged over several time slices recorded in a single experiment instead of being averaged over several measurement runs. For a detector distance of 20 m, the SANS data were averaged over 20 min, for 8 m over 80 s and for 2 m over 30 s, respectively. Due to the high incoherent scattering cross section of the hydrogen, the SANS curves were put on an absolute scale using pure water as a flat reference scatterer. Dynamic light scattering was performed with a light scattering setup of our own design. The light of the helium neon laser, which exhibits an operating wavelength of 632.8 nm, is guided to the flow cell by optical fibers and imaged into the middle of the flow-through cell in the neutron beam by a collimator (Schäfter and Kirchhoff, Hamburg). Since we performed in situ dynamic light scattering measurements, this flow through cell is identical to the one in the neutron beam which was described above. The scattered light is collected by another collimator of the same type, which is adjusted under a fixed scattering angle of 91.4°, which was chosen because of the geometry constraints at the sample stage of D11. The scattered light is guided with optical fibers to a dual photomultiplier detector (ALV GmbH, Langen). The scattering angle was calculated using the size of the lysozyme dimers as calibration particles. This size could be determined by fitting the highq part of the SANS data. The duration of each DLS intensity autocorrelation measurement is 1 min. The signal is processed by an ALV 7004 correlator. Longer-term DLS measurements were obtained by mixing the sample without using the syringe pump system, because time resolution did not matter in this case. A Malvern Zetasizer Nano S particle sizer was employed in this case to measure the sample at a constant temperature of 298 K in a standard quartz glass cell. The Zetasizer uses a fixed scattering angle of 173°. Static light scattering was performed by a commercial device, the Wyatt Dawn Heleos II, which is capable of measuring the scattered light at 18 scattering angles simultaneously and enables the measurement of scattering curves in a momentum transfer range of 2.8 × 10−4 to 2.4 × 10−3 Å−1. The integration time was set to two seconds per data point. In the case of SLS, normalization was carried out using a flat scatterer to enable data treatment. The scattering curves of the SLS measurements were averaged over 1 min. Since one time slice is integrated for two seconds, 30 data sets were averaged for one scattering curve leading to a marginal statistical error, which is smaller than the symbols in Figure 4. All three measurement techniques were performed simultaneously on the same sample. The procedure also renders a collective starting time of the crystallization process, which simplifies data processing. The crystallization process was followed for 1 h in the in situ experiments at D11. The in situ experiments were repeated six times involving all three techniques and another 12 times involving only SANS and in situ DLS. The experiments performed at D11 demonstrated the reproducibility of the experimental procedure since the only attested differences between the

corresponding amount of 1 M sodium acetate. The sodium chloride solution was prepared in a deuterated 5 mM sodium acetate/acetic acid buffer. A major obstacle in performing crystallization experiments is that lysozyme tends to form aggregates depending on protein concentration, temperature, and pD. Those aggregates may serve as precrystalline seeds and could affect the nucleation rates in an unpredictable manner,19 leading to unreproducible experiments. In order to get rid of these precrystalline aggregates and other foreign matter such as dust, both the protein solution and the electrolyte solution were filtered with a pore size of 0.02 μm (Whatman, Anotop filter, ordering number 6809-4002) prior to the experiments. The monodispersity of the preparations was checked using dynamic light scattering without added electrolytes before each series of experiments. 30 The pure lysozyme solution was found to remain monodisperse and aggregate-free for at least 40 min as inferred from static and dynamic light scattering. Within these 40 min, the mother liquor has to be created by mixing equal parts of the protein and electrolyte solution or a fresh protein solution had to be prepared. Although the small (typically less than 2%) variation of protein concentration (as inferred from UV-spectroscopy) before and after the filtration indicates that the number of aggregates is small compared to the amount of lysozyme monomers, their influence on nucleation and growth rates is crucial. The purity of the lysozyme sample, obtained from Sigma-Aldrich (ordering number 62971), was checked by sizeexclusion chromatography and found to be absolutely flawless (see Supporting Information, Figure S1). This is why the lysozyme sample from this batch was used as received for all experiments discussed in this study. Commercial, instead of deuterated, lysozyme was selected in order to obtain greater scattering length density contrast between the lysozyme protein and the heavy water. Both sample parts were injected into the two quartz glass sample cells (the one in the static light scattering device and the one in the neutron beam, see Figure 1) by a syringe pump using fresh disposable plastic syringes for each crystallization run to ensure a clean and reproducible procedure. Furthermore, the batch crystallization method employed here was checked to see whether crystals actually were able to grow under the chosen conditions. Given the above-mentioned salt and lysozyme concentrations, the nucleation rate is far too high to enable the direct observation of the nucleating agents with light or neutron scattering at room temperature (294.5 K) or lower.31 In the case of lysozyme, the solubility increases with rising temperatures, and as a consequence the saturation decreases leading to slower nucleation rates. The time between the preparation of the mother liquor and the direct observation of the first microcrystals by naked eye lies between 1 h at a temperature of 294.5 K and more than 24 h at 298 K. Hence, the temperature of all the crystallization experiments was set close to the temperature of the neutron guide hall at the Institut Laue-Langevin Grenoble (ILL) at 298 K. This setting is close to the limiting conditions with a threshold temperature of about 300 K. Above this temperature, no crystallization occurs at all using a mother liquor with 30 mg/mL lysozyme and 3 wt % NaCl. Experimental Set-up. Figure 1 shows the schematic setup of the experiments performed at the ILL in Grenoble at the small angle neutron scattering (SANS) instrument D11. The syringe pump was loaded with two fresh disposable syringes filled with the lysozyme and the salt solution, respectively. As a mixer, a T-shaped valve was employed. The mixture was pumped first through the quartz glass cell of the static light scattering device (Wyatt Dawn Heleos II, Wayatt Technology Corporation, Santa Barbara, CA, USA) and then (connected by a 0.8 mm inner diameter Teflon tubing) through the quartz glass cell in the neutron beam, which the DLS device was also connected to. The excess sample ended up in a waste syringe on top of the flow through cell in the neutron beam. The tubing was always thoroughly rinsed with deionized water and then dried by flushing with ample amounts of pressurized dry air. Also, the waste syringe was always replaced by an empty fresh one for each crystallization run in order to avoid any back diffusion into the flow-through cell in the neutron beam. The time resolution was limited by the moving syringes C



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experiments are a consequence of small variations of the ambient temperature in the neutron guide hall of the ILL. Nevertheless, the apparent variations result in differences in the velocity of the nucleation process and appear to be easily scalable with the help of the in situ DLS data. Attempts were made to average the neutron scattering curves of different crystallization runs using a time scaling factor to account for the small variation in speed of the observed process due to the small variations in ambient temperature. However, it was found that the data did not improve significantly due to this averaging procedure. As a result, in the Results section, representative scattering curves from a single typical crystallization run are shown.

hydrodynamic radius of this particle species was determined to be approximately 50 nm. The chosen duration of 1 min to record one intensity autocorrelation curve was too long to be able to detect smaller particles because they were growing too fast. The presence of just two aggregate species compared to the higher number which Georgalis et al. observed in similar crystallization conditions15,16 could be due to the higher temperature selected in this experimental set up in order to better follow the nucleation process. Indeed, we also observed three different monodisperse aggregate species in the same crystallization conditions at 294 K (data not shown). For a lysozyme concentration of 30 mg/mL, the initial particle number density of the lysozyme dimers is about NDimers = 6 × 1017 cm−3. Together with the amplitude of the autocorrelation function and an expected increase of the scattered light intensity by 6 orders of magnitude with each order of magnitude in particle radius, the particle number of the lysozyme oligomers is about NOligomers = 1 × 106 cm−3 10 min after initiation of the crystallization. The growth pattern shown in the top right inset of Figure 2 based on the increase of the time constant of the second slope has already been observed by Georgalis et al. under comparable crystallization conditions in H2O instead of D2O.19 The mean cluster size as a function of time scales as a power law according to the diffusion limited cluster aggregation (DLCA)18,34,35

III. RESULTS Short-Term Kinetic Analysis. First the sample system was characterized by dynamic light scattering in order to account for any participating particles during crystallization. Figure 2

R h(t ) = R h,0(1 + ct )1/ d f

(3)

where c is the Smoluchowski rate constant c = 4kBTN0/3η, kBT is the thermal energy, η is the viscosity, and N0 is the number density of constituent particles (dimers in our case) at time zero. The hydrodynamic radius Rh,0 = 2.7 nm of the seed particles was fixed, which is derived from the radius of gyration r0 of the stable lysozyme dimers evaluated by the spherical form factor fit of the scattering curves measured by SANS (R h,0/r0 = 1.29 for hard spheres). By fitting the estimated hydrodynamic radius using dynamic light scattering as a function of time with eq 3, we obtained a fractal dimension of df = 1.85 ± 0.07, which is a dimensionless quantity. As we will discuss later, this result is characteristic of the DLCA regime, where more open and less compact clusters compared to that of a typical reaction cluster limited aggregation (RCLA) regime have been experimentally observed by means of transmission electron micrographs.36,37 In the inset of Figure 2, a good agreement between the diffusion limited cluster aggregation model and the experimental data is shown. This accounts for only an averaged fractal dimension in time, which is why we are using SANS in combination with SLS to account for structural changes during the formation of nuclei leading to a time dependent df(t). Short-Term SANS and SLS Results. Figure 3 shows the scattering curves of SLS and the SANS measurements as the function of the whole explored q-window. Both the curves were put on the same scale by a single scaling factor, constant for all times. The scaling factor accounts for the different contrasts of light and neutron scattering. Although the scaling factor could in principle be computed based on the measurement of the change of the refractive index with concentration, it has been determined for the time slice at 80 s where the scattering curves of SANS and SLS overlap each other in a certain intermediate q-range. It was kept constant for all further time slices. For q larger than 0.04 Å−1, the scattering curve resembles the form factor of the lysozyme dimers. However, the structure

Figure 2. In-situ time-resolved dynamic light scattering spectra of the crystallization of hen egg white lysozyme at three different times. The data points describe the first-order autocorrelation function g(1)(τ) as a function of the delay time. The red lines are fits to the experimental data. Inset: accretion process of the lysozyme oligomers, determined by conversion of the fitted time constant of the slower exponential decay into a hydrodynamic radius. The error bars shown in the inset reflect the uncertainty in the determination of the hydrodynamic radius and refer to the error of the respective exponential decay fit. The linearity of the hydrodynamic radius trend as a function of the time corroborates the DLCA regime. The full line is a fit for the experimental data using eq 3.

shows the temporal evolution of the lysozyme aggregates observed by in situ dynamic light scattering under the given conditions. There are three different time slices depicted in this graph. A double exponential decay function is shown to be a good fit of all DLS curves, displaying two separate lysozyme species during the early crystallization process. The good agreement with the double exponential decay function suggests the presence of a monodisperse distribution of the oligomeric and dimeric lysozyme aggregates. The hydrodynamic radius may then be evaluated with the help of the Stokes−Einstein relation.33 The initial decay of the intensity autocorrelation function results in a time constant, which remains approximately constant throughout the whole crystallization process. It is interpreted as stemming from the diffusion of lysozyme dimers, which appear just after the addition of NaCl. As a consequence, the hydrodynamic radius Rh = 2.7 nm of this mixture, which is represented by the first exponential decay in Figure 2, remains constant. As opposed to this, the second decay exhibits an increasing time constant, which represents the growth of an additional particle species. The value of the initial D



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min attached to the SANS data implies a particle concentration 5 −3 of the lysozyme oligomers of NSLS Oligomers = 9 × 10 cm , which is 6 −3 very close to NOligomers = 1 × 10 cm evaluated from DLS amplitudes. Due to the small scattering of the lysozyme solution in the SANS experiment, the data must be averaged in order to improve on the statistical error for large detector distances. The averaged SANS data indeed exhibits an increase in cross section after 40 min, which directly overlaps with the scattering curves recorded using SLS. This clearly demonstrates the conformity of the two methods. In particular, it shows that the time evolution of crystallization is the same in the two cuvettes, the SANS cuvette and the SLS cuvette. For smaller detector distances the statistical error does not affect the data quality. The covered momentum transfer range only shows scattering from the form factor of the lysozyme dimers which do not vary in size or shape.39 This agrees with the constant hydrodynamic radius of the lysozyme dimers observed with the in situ DLS method. At 298 K the rate of crystallization is too slow to detect some Bragg peaks from small crystals in the first hour after the onset of the nucleation. In the next section, a fitting model to describe the recorded static light scattering and neutron scattering data will be established: The measured coherent differential scattering cross section is proportional to the static structure factor S(q), which is the Fourier transform of the pair correlation function for the centers of the constituent particles, g(r) and may be expressed as

Figure 3. Temporal evolution of scattering curves during nucleation on an extended momentum transfer range enabled by static light scattering in unison with small-angle neutron scattering. The SLS and SANS spectra are put on the same scale by using a single scaling factor, constant for all times. The error bars attached to the SANS data points represent the uncertainty of the measurement in terms of the statistical error in the coherent differential scattering cross section per unit volume. The errors in momentum transfer were calculated based on geometrical considerations and the wavelength spread of the incoming neutron beam at the Instrument D11. They are too small to be visible on the scale of the above figure. Open symbols show SANS data; half open symbols show SLS data.

factor peak in the intermediate q region measured by Stradner et al.38 is absent due to the missing repelling force of the lysozyme monomers at this NaCl concentration. As another consequence of adding the salt solution, a structure factor of the additional particle species found for q-values smaller than 5 × 10−3 Å−1 emerges, supporting the existence of the lysozyme fractal cluster observed by means of the DLS experiments. The variation of the scattering curves in this region indicates a change in particle structure. After 80 s, the SLS data indicate the presence of large particles, which grow even larger, leaving the observable momentum transfer range within the first 10 min. We attribute this to a filling artifact of the static light scattering cell: because a linear interpolation of the growth rate of these large particles would lead to the observation of crystals within the first 2 h. Since we cannot observe crystals with the naked eye even after 24 h, we ascribe this either to an artifact arising from the mixing procedure or a considerable decrease in crystallization speed. Nevertheless, we cannot fully rule out that these particles could be considered as crystal seeds in the latter case. Because of its momentum transfer range, the SANS experiment is ideal to account for particles in a size range of 5−200 nm. But the concentration of the lysozyme oligomers in the early phase of nucleation is too small to generate a detectable neutron scattering signal for a measurement duration of a few minutes. As an example, in order to reach a signal-tonoise ratio of 1:1 for a lysozyme cluster of 50 nm radius, a minimal number density of clusters of Nmin. = 5 × 108 cm−3 would be necessary, which is 2 orders of magnitude higher than the number density of the lysozyme oligomers observed, i.e., NOligomers = 1 × 106 cm−3. (Nmin. has been evaluated applying the scattering contrast of lysozyme reported by Kohlbrecher et al.39) Static light scattering is more sensitive and enables a timedependent structure analysis of the lysozyme oligomers from the start. The coherent differential scattering cross section per unit volume of the scattering curve measured with SLS after 10

dΣ = NOligomersv02(ρs − ρ0 )2 f (q)S(q) dΩ

(4)

where ρs is the scattering-length density averaged over a lysozyme dimer particle of volume v0, ρ0 is the scattering-length density of the embedding medium, NOligomers is the number density of the fractal objects, and f(q) = A(q)2 is the singleparticle form factor. By considering hard sphere scattering, the amplitude may be written as ⎛ sin(qr ) − qr cos(qr ) ⎞ 0 0 0 ⎟ A(q) = 3⎜ (qr0)3 ⎝ ⎠

(5)

If the q−1 is much smaller than the cluster dimension, ζ, the fractal scaling is described by the power-law regime:20

S(q) ≈ q−d f if qζ ≫ 1

(6)

For realistic systems, this ideal scattering law will be modified by the upper and lower bounds on the length scale, where fractal correlations persist.40 The lower cutoff corresponds to the radius of the small particles r0, represented by the stable lysozyme dimers in our case. Furthermore, in a real threedimensional system of aggregates, the range of the scale invariance is limited at large length scales by the finite size of the clusters ζ or by the length scale at which the cluster density approaches the average density.41 This upper limit is described by a cutoff function h(r /ζ )and the g(r) can be written as

g (r ) ∝ r d f − 3h(r /ζ ) If an exponential is used to describe the cutoff function, the analytic form of the structure factor, which can be used to fit the data, evaluated with these limitations yields42 E


Crystal Growth & Design S(q) = 1 +

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fitting of the SLS scattering curves by the reduced structure factor S(q)red. Contrary to the DLS method, static light scattering can directly probe the structure of the clusters and determine their fractal dimension.20 Nevertheless, the value of df = 1.85 ± 0.07 as obtained from the DLS data by fitting the time-dependence of the hydrodynamic radius of the clusters is in reasonable agreement with the final value for the fractal dimension of 1.73 as seen by the static light scattering curves. The resulting time-dependent fractal dimension df(t) obtained by fitting the SLS data with the power-law regime I(q) ∝ q−df and the reduced structure factor fit are depicted in Figure 5. The error bar describes solely the error resulting from

C(df − 1)Γ(df − 1)ζ d f (1 + q2ζ 2)1/2 qζ (1 + q2ζ 2)d f /2

sin[(df − 1) arctan(qζ )] df − 1

(7)

where C is a constant. Whereas eq 4 provides excellent conformance with the merged data sets of SANS and SLS, it happens to be quite sensitive to the form factor. The scattering curves recorded with SANS overlap each other for q greater than 0.04 Å−1, indicating that the form factor of the small lysozyme dimers does not change at all during the SANS measurements of typically 40 min duration. However, the structure factor changes significantly, albeit this is mostly visible in the range of the momentum transfer covered by SLS. Since the change of fractal dimension is predominantly dependent on the variations in SLS data, we use a reduced form of eq 7. This reduced form of eq 7 is used to determine the fractal dimension solely based on SLS data. Here the form factor of the stable dimers does not lie within the momentum transfer range of static light scattering. We may therefore omit it and the prefactor NOligomersv02(ρs − ρ0) 2 may be replaced by a constant C*, which also contains the constant scaling factor used for merging the SANS and SLS scattering curves. Thus, the Raleigh ratio of the SLS measurements may be fitted with the reduced structure factor20 (1 − d f )/2 ⎛ q 2ζ 2 ⎞ S(q)red = C*⎜1 + 2 ⎟ κ ⎠ ⎝

Figure 5. Evolution of the fractal dimension df determined by the fitting of the static light scattering curves with the structure factor S(q)red and the power law regime I(q) ∝ q−df . The development of the fractal structure indicates rod or chain-like particles at the beginning.43 As time elapses the fractal dimension rises, indicating a more space-filling behavior.71

sin[(df − 1) arctan(qζκ −1)] (df − 1)qζκ −1

where κ =

3 (d 2 f

(8)

− 1) . ζ is representing the finite size of the

clusters and serves as the upper cutoff. This cutoff could also be determined from the measured hydrodynamic radius Rh derived by the DLS method. It was found to agree with the upper cutoff stemming from the fit parameter ζ. As mentioned previously, the change of fractal dimension is predominantly seen in the SLS data. An increase of the fractal dimension from 1.0 to 1.73 is observed in the first 90 min. Figure 4 shows a selection of the scattering curves depicting the

the fitting carried out by the given approaches. After the initial 20 min, a good agreement between the two fitting models is observed. As we are considering small q values, the qζ ≫ 1 condition is better satisfied when the clusters grow as the time increases. A deviation from a linear trend in Figure 5 is more evident at lower q, where a bigger departure from the powerlaw regime condition is expected. The observed increase in the fractal dimension suggests that the shape of the lysozyme oligomers shows a rapid development from chain or rod-like particles toward a fractal and spacefilling structure, which then remains stable at least for the recorded duration of the measurements.43 As indication of a branching of the observed fractal clusters characteristic of the DCLA processes, a modified Kratky plot of the data is shown in Figure 6. As the time increases, the modified Kratky plot shows a small bump, which seems to shift toward the lower momentum transfer q when a plateau is achieved. This behavior is characteristic of nonrandomly branched macromolecules in solutions.44 Long-Term DLS and SLS Results. DLS measurements at later stages of the crystallization process (Figures 8 and 9) reveal a decrease of the amplitude of the autocorrelation function of the lysozyme oligomers despite their increasing hydrodynamic radius. The higher the relative concentration of a particle species, the higher the relative amplitude of the corresponding exponential decay will be in the autocorrelation

Figure 4. Temporal evolution of scattering curves during nucleation measured with static light scattering. The fitting curves show the adaption of the data with the structure factor S(q)red. F



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Figure 8. Long-term time-dependence of the hydrodynamic radius of the observed particles in the solution after mixing the lysozyme and the salt solution. After 4 h, the fixed measuring time of 2 min for recording one DLS curve is too short to make sure that all diffusion constants can be seen in the sample in every curve. When large crystallites swim through the detection volume during the 2 min time window necessary to record the correlation curve, a very large hydrodynamic radius is fitted to the data (see data points with a hydrodynamic radius above 2 μm). If this is not the case, just the normal growth of the fractal oligomers of lysozyme molecules is observed.

Figure 6. Modified Kratky representation of the static light scattering data. A small maximum is observed at a q-value of 0.0075 Å−1.

Figure 7. Temporal evolution of scattering curves during lysozyme crystallization recorded with static light scattering. Determination of the fractal dimension df is realized by fitting the data with the reduced structure factor S(q)red. The time slices at 1 and 5 h indicate no structural changes of the fractals. The time slices at 12 and 48 h demonstrate how the structure factor of these particles exceeds the available momentum transfer range.

Figure 9. Decrease of the relative amplitude of the lysozyme oligomers in the intensity autocorrelation curve as the crystallization proceeds. This indicates a decrease in particle concentration of the lysozyme oligomers since their size still increases.

function. However, this amplitude is also correlated to the hydrodynamic radius. The larger the particle radius, the higher the relative amplitude will be. Figure 9 shows the temporal evolution of the relative amplitude of the lysozyme oligomers during the crystallization of lysozyme. At the beginning, the relative amplitude is rising, which is due to an increase of the respective particle concentration and their hydrodynamic radius. Since we observe a decrease of the relative amplitude despite the particles continuing to grow (see Figure 8), the concentration of these particles is obviously decreasing. The particle number decreases from the initial value of NOligomers = 1 × 106 cm−3 to approximately some hundreds or thousands of particles after 36 h. This also means that fractal intermediates are also present when millimeter-sized crystals have already grown. While it is commonly agreed that for lysozyme the crystallization occurs by consumption of dimers, little is known about the role of the present fractal clusters concerning their participation in crystallization. Previous studies employing AFM imaging of crystal facets suggest that larger clusters and

aggregates are participating as building blocks of crystal growth.5,45 Monitoring the decrease of the concentration of the lysozyme oligomers, which we identified as fractal clusters, gives further evidence to this statement. As the first microcrystals are visible after approximately 24 h we can also prove the coexistence of fractal clusters and microcrystals as illustrated in Figure 10E. SLS measurements at later stages of the crystallization process show that the fractal dimension of the lysozyme oligomers remains constant at the value of approximately 1.73 between 90 min and 5 h after initiation of the crystallization process (data shown in Figure 7). Since the fractal dimension is approximately the same for 1 and 5 h after initiation of crystallization, it can be concluded that there are no significant structural changes in this period of time. Only the upper size limit increases slowly. In this measurement, the fitting with the reduced structure factor S(q)red does not describe the q-values above 1.5 × 10−3 A−1 properly (mostly seen in the curve for t = G



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Figure 10. Crystallization model based on the experimental observations: (A) aggregate free lysozyme solution after filtration (0.02 μm) prior to mixture with the electrolyte solution. Lysozyme monomers are the only particle fraction present at this point. (B) After mixing with the electrolyte solution, dimers form instantly: this process occurs far too rapidly to be observed by the applied scattering techniques. These stable dimers maintain their size and shape during the whole experiment, whereas their concentration decreases until the mother liquor reaches equilibrium with the crystals having been formed. (C) Under the chosen conditions, oligomers are detectable after approximately 2 min. Their initial shape is chain or rod-like. (D) As time elapses, the oligomers continue growing. They exhibit a self-similar structure leading to fractals. Their concentration is at its maximum after about 20 min. (E) First microcrystals are visible to the eye after more than 24 h. At this time there are still some lysozyme fractals present together with the stable dimers. However, both the dimer and the fractal concentration have by this stage decreased significantly.

should contribute more at high q, resulting in a poor overlap between the SLS and SANS experimental data, which is not observed.

5 h), which appears to be due to a smooth constant background present in the data. On closer inspection of the time slice at 48 h, this background can be identified by a small increase in the Rayleigh ratio for larger q-values. As the fractals continue growing, the structure factor shifts to lower q-values. The time slices for 12 and 48 h demonstrate how the structure factor of these particles runs out of the available momentum transfer range of the employed static light scattering instrument and the fractal dimension of later times could not be determined. Umbach et al.46 also investigated the fractal dimension of lysozyme solution at similar protein and salt concentrations by means of small angle SLS showing a peak for df after 2 h. The change in behavior as opposed to our results could be ascribed to a different temperature in their experiment, which strongly affects the kinetics of the nucleation process. An important issue related to high protein concentration solutions, which are essential for the nucleation process, is the potential multiple scattering contribution. Using a mean field approximation, Berry and Percival47 showed that the multiple scattering for N small refracting and absorbing spherules, each of radius ro and coagulated into random clusters with fractal dimension df and an exponential cutoff, is negligible if 2π /λr0 ≪ 1 and df < 2, with λ = 664 nm being the wavelength of the incoming laser light of the Wyatt static light scattering device. If df > 2 the multiple scattering can become significant and grows as N increases. The critical value of N for df > 2 was found at N≈

IV. DISCUSSION One of the key questions concerning protein nucleation is whether the observed nucleation is homogeneous or heterogeneous. Even for the most studied protein lysozyme, there is an ongoing debate about this issue.49 In the case of homogeneous nucleation the probability of a given fluctuation occurring is identical over the whole volume of the system. This condition assumes an extremely pure solution and the fact that the supersaturation is sufficiently high to overcome the activation barrier. In order to reproduce such a situation, we filtered both the lysozyme and the salt solution before mixing. In a real experiment external perturbation of the homogeneous nucleation condition, such as the existence of foreign surfaces or dust particles cannot be avoided. This could lead to heterogeneous nucleation, where the probability of nucleation is locally higher in some positions with respect to other ones. Since we observe the formation of crystals on the surface of the sample container it would stand to reason that the elapsing nucleation is heterogeneous. However, if the nucleation is heterogeneous, it occurs on impurities or surfaces, and there is no reason to expect the surfaces of the container walls and the surfaces of the impurities to be completely uniform. Following classical nucleation theory (CNT), in the case of heterogeneous nucleation, a range of barriers would be expected, instead of a single nucleation barrier. If this was the case, no monodisperse growth behavior would be observed, since different nucleation barriers would imply a variety of nucleation rates with distinct sizes of the critical nuclei. The DLS data shows very little deviation from exponential decays, indicating very little size distribution present in the observed oligomers. This means that our experiments show that the temporal formation of the nuclei is uniform, which contradicts heterogeneous nucleation. So we

⎛ 2π ⎞(−d f )/(d f − 2) ⎜ r⎟ ⎝ λ 0⎠

Nelson et al.48 also found that in case df < 2 the multiple scattering contribution tends to a constant. As in this study we observe DLCA fractals with typical fractal dimensions of less than 2, we assume that the multiple scattering does not contribute significantly. Moreover, if this was not true, this H



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lysozyme. The experiments reflect not only a decelerating growth, but also a change in the particle shape, as the lysozyme oligomers are filling the space leading to a more densely packed structure. The following relation between the mass M of a fractal object and its radius r can serve as a general interpretation of the fractal dimension:42 M(r ) ∝ r d f , with df < d. The fractal dimension is always smaller than the embedding dimension d, which is three in our case. The fractal dimension of the oligomers seems to achieve a maximum of df = 1.73, which then stays more or less constant with time. This value lies between the fractal dimension of df = 1.72−1.81 reported for aggregates following the diffusion limited cluster aggregation (DLCA).19,20,60 Long time measurements, carried out under the same conditions, confirmed these observations, but failed to give additional information on the evolution of the particle shape. This is because the clusters are still growing, and thus their scattering footprint exceeds the accessible momentum transfer covered by SLS. The measured fractal dimension of 1.73 is close to the one found in selfavoiding polymers if one thinks of a protein molecule as the monomer. For these polymers one finds a fractal dimension of 1.67.61 Another explanation of the measured fractal dimension would be a branching with random distances between branch points.44 In principle the two situations cannot be distinguished based on scattering techniques alone. In the latter case a modified Kratky plot of the data should show a peak at the low q-region, which is present when the branching is not purely random.62 In Figure 6 one can make out a small peak in the Kratky representation of the data, which is indicative of the branching case. Figure 10 attempts to summarize our finding and describes a preliminary model of the crystallization process based on the experimental results. Prior to its mixing with electrolyte solution, the lysozyme solution remains monodisperse containing only lysozyme monomers (see Figure 10A). After establishing the mother liquor through the addition of the electrolyte, dimerization takes place almost immediately. This step appears to be far too rapid to be resolved by the utilized scattering methods (see Figure 10B). The dimers are arranged in an elongated structure after a few minutes (see Figure 10C), followed by the creation of a fractal structure. Further growth of these fractals could be attested by means of DLS measurement until 36 h, but with no change in the fractal dimension. Lysozyme dimers as well as fractals are present right up to the point at which small crystals are visible to the eye. The step from C to D depicted in Figure 10 describes the evolution of the fractal aggregates when the fractal dimension goes from 1 to 1.73. The branching of the fractal cluster described in step D resembles the typical transmission electron micrographs measured for colloid aggregates in the DLCA regime.37 The DLCA fractal clusters are rather less compact than in the case of RCLA, as can be inferred from their df, which is lower than 2. As the sticking probability is almost equal to unity, one cluster does not have a high probability to deeply penetrate another cluster, but is just added to the first cluster after a single collision controlled by diffusion. However, restructuring is difficult to prove experimentally.16 Further studies concerning the fractal dimension of subnuclear clusters may be feasible utilizing small angle light scattering or light microscopy. As a consequence of crystal growth both particle concentrations dropped markedly, indicating that the crystal

assume predominantly homogeneous nucleation in our experiment. As shown previously, stable lysozyme dimers form immediately after mixing the two solution parts. This goes along with the structure factor peak disappearing, which is found in the lysozyme stock solution. 38 Adding salt immediately screens the charges on the lysozyme monomers leading to the formation of dimers with a hydrodynamic radius of 2.7 nm. These dimers exhibit a stable size and shape coexisting with larger lysozyme fractal clusters during the crystallization process. This constant coexistence between these two species can be inferred only from the SANS measurement during the nucleation process. Long time measurements up to 72 h indicate that the concentration of these stable dimers decreases with time, suggesting that they may be consumed during the formation of clusters and crystals. The presence of intermediate particles following a growth pattern described by diffusion limited cluster aggregation has also been observed by means of the DLS measurements. Indeed, as the screening of the lysozyme surface charges due to the NaCl is complete at the selected salt concentration, it is reasonable that the aggregation kinetics is just limited by diffusion. When the aggregation process is described by a diffusion limited process, the sticking probability α between two objects is ideally equal to unity (instead of a value close to zero in the case of the RLCA regime), and the hydrodynamic radius follows a power-law behavior (instead of an exponential trend). The fractal dimension df obtained by fitting the kinetic data with the power-law dependence described by eq 3 is equal to 1.85 ± 0.07 (df = 2.1 in the case of RLCA). In particular, this value suggests that the kinetics is ruled by a cluster−cluster diffusion aggregation process and not by the particle−cluster diffusion aggregation process described by Witten et al.,50 which would result in a higher value of the fractal dimension (df = 2.5 instead of df = 1.78).51 In the former regime, monomeric units are able to penetrate more deeply into the cluster bringing up more compact fractal structures. The fractal dimensionality alone can just give limited information about the kinetic properties of the aggregation process.52 Indeed, the kinetics strongly affects the cluster-size distribution ns(t) as a function of the size s and the time t. The cluster size distribution depends on several parameters such as the degree of irreversibility in fractal constituent bonds,53 the diffusion coefficient behavior as a function of the cluster size52 and the size-dependent sticking probability.54 The reaction probability between two fractal constituents is described by the Kernel Kjk of the Smoluchowski equation.55 A Kernel defining the aggregation between a big cluster and a smaller one seems to be a good description of the diffusionlimited regime.56 In such a case, a monodisperse fractal cluster distribution is expected.56−58As the autocorrelation function obtained by our DLS measurements are well fitted by single exponential functions, showing the presence of a monodisperse distribution, kinetics controlled by a DLCA regime is corroborated. By means of the SLS experiments on the same sample, we also observed that not only the size of the clusters is changing with time but also their shape evolves as displayed by the evolution of the fractal dimension of these particles shown in Figure 5. At the beginning their fractal dimension equals one, which indicates a rod-like shape of the forming aggregates.43 This shape could also be observed by electron microscopy59 during the early stages of the crystallization of hen-egg white I



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growth proceeds not only by addition of small dimers, but also via large fractal aggregates.63 Moreover, the simultaneous decrease of both dimers and fractal cluster may support an irreversible kinetic aggregation process in the bulk solution, where the fractal clusters do not break up again into dimers. In principle these fractal aggregates can be dissolved to dimers in the proximity of lysozyme crystals and thereby consumed, as observed by Georgalis et al. with high resolution light microscopy under different crystallization conditions after a temperature quench.64 This would be an alternative mechanism which cannot be distinguished from a direct addition of the fractals to the existing crystals. However, the latter case is more consistent with our data since the DLS data shows hardly any polydispersity of the fractals at later times. These results confirm the recent suggestion about the cluster contribution to crystal growth in the context of nonclassical mechanism of layer generation during crystal growth. Sleutel and Van Driessche showed that the cluster aggregation to the crystal can even lead to a cleansing of an impurity-poisoned crystal surface.65 In light of a recent publication from the group of Dominique Maes, the fractal objects found in this study can be identified as the “dense liquid clusters” from which the crystal seeds arise.66 This is supported by the long time DLS measurements shown in Figure 8 where one observes the growth of the fractal clusters first. Only after some time large objects are seen in some measurements, which can be interpreted as the formation of the first small crystals. This points to a two-step process of crystallization:67 First the precrystalline dense liquid clusters form and grow in density as their fractal dimension increases. Second, from some of those dense liquid clusters first crystal nuclei arise which grow then to the crystals seen by the naked eye. In order to deepen the knowledge about the model of the two-step nucleation mechanism, similar attempts have been performed by Schubert et al. using dynamic light scattering and visualization methods, which allowed them to be the first to observe the nuclei of protein clusters by means of transmission electron microscopy with a size between 200 nm and around 1 μm.68 In summary, with the wealth of our data we can draw a very distinct picture of the crystallization process of lysozyme. Our findings will therefore help to improve more advanced models of crystal growth69 and serve as a basis for simulations.70 On the basis of these promising results, it could be of value to use the same experimental setup to study the nucleation process of a different protein in order to compare our results with the ones stemming from a potentially different nucleation mechanism. Moreover, we suggest that advanced nucleation theories should comprise the structure of the nuclei.24

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

T. E. Schrader: 0000-0001-5159-0846 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This project received funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. 654000. In the early phase, this project received funding from the European Union’s 7th Framework Programme for research, technological development and demonstration under the NMI3-II Grant Number 283883. The authors thank Ralf Biehl and Emmanuel Kentzinger for their support with the small-angle X-ray characterization of the sample. Additionally, the authors are thankful for the possibility to test their setup at the instrument KWS-2 at MLZ in Garching. The work of Stefan Rustler and Simon Lechelmayr at a very early stage of the project is also gratefully acknowledged. The authors are grateful to Prof. Dr. Dieter Richter for his continuous support and for helpful discussions. The authors would like to thank Clair Ryalls for carefully proof-reading the manuscript.

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S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.cgd.7b01433. Sample preparation; Figure S1. UV-absorption signal recorded during size exclusion chromatography. Figure S2. Temporal evolution of the hydrodynamic radius of the lysozyme oligomers recorded by several successive insitu light scattering measurements. (PDF) J



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