On the Self-Purification Cascade during Crystal Growth from Solution

Article pubs.acs.org/crystal

On the Self-Purification Cascade during Crystal Growth from Solution Mike Sleutel*,† and Alexander E. S. Van Driessche*,‡ †

Structural Biology Brussels, Flanders Interuniversity Institute for Biotechnology (VIB), Vrije Universiteit Brussel, Pleinlaan 2, 1050, Elsene, Belgium ‡ Laboratorio de Estudios Cristalograficos, IACT, CSIC - U.Granada, P.T. Ciencias de la Salud, Avenida del conocimiento s/n, 18100 Armilla (Granada), Spain S Supporting Information *

ABSTRACT: We report on the failure of the Cabrera− Vermilyea (CV) step pinning model to reproduce the elementary step kinetics for the case of tetragonal lysozyme crystals growing from contaminated solutions. We measured the supersaturation dependency of the step velocity using confocal microscopy for three different commercially available lysozyme batches with varying levels of impurity content, that is, Seikagaku, Fluka, and Sigma. Strong nonlinear dependencies are obtained in the high to intermediate supersaturation range and near-linear dependencies at lower driving forces. The clear absence of a dead zone for the Fluka and Seikagaku data is in direct contradiction to the CV model. As such, we developed a time-dependent impurity model based on Bliznakov kinetics assuming Langmuir adsorption. Admissible fits are obtained for Fluka and Seikagaku lysozyme corroborating the self-purification interpretation due to the diminishing terrace exposure times at higher supersaturation levels. The steeper recovery toward pure kinetics for Sigma lysozyme than predicted by Langmuir adsorption prompted us to expand the model to allow for impurity− impurity interaction. The resultant kinetic model, which assumes a Kisliuk-like mode of impurity adsorption, did yield acceptable fits with Sigma step kinetics. This Bliznakov−Kisliuk model also predicts clustering of impurity molecules on the surface, which is corroborated by our in situ experimental atomic force microscopy observations.

■

INTRODUCTION Whereas in industrial applications impurities are often tailormade to modulate crystal growth and organisms use them to shape their morphology, protein crystallographers typically regard impurities from a less optimistic perspective. In this work, we focus on crystal growth from such impure solutions, a subject that despite many years of research has numerous unresolved questions and is still subject to a great deal of discussion. Even so, great advances have been made in the past decade by virtue of the application of novel in situ observation techniques, such as atomic force microscopy (AFM)1−3 and laser confocal microscopy combined with differential interference contrast (LCM-DIM),4−6 which allow for high resolution imaging of the step dynamics on crystal surfaces in the presence of impurities. With these techniques, relevant information has been obtained in the fields of biomineralization (e.g., calcite,7−9 calcium oxalate monohydrate (COM)10−12), inorganic (e.g., potassium dihydrogen phosphate (KDP),13,14 paracetamol15), and protein crystallization (e.g., hen egg white lysozyme16,17). The experimental data obtained in these studies were confronted with the classical impurity models to obtain a better understanding of the precise impurity mechanisms. One of the most frequently employed models to predict elementary step kinetics of crystals growing from impure conditions is the Cabrera−Vermilyea (CV) model for step pinning, which was developed more than 50 years ago.18 One of its key © 2013 American Chemical Society

characteristics is the predicted existence of a critical supersaturation, σd, below which virtually no lateral step advancement will occur. This impurity pinning catastrophe is purely of thermodynamic origin; pinned steps become curved when they try to squeeze through the fence of impurities on the surface, and once the critical radius of curvature, rc, becomes larger than the mean impurity spacing, the steps are expected to stop advancing due to the Gibbs−Thomson effect. In initial applications of this theory, it was (implicitly) assumed that the kinetics of impurity adsorption operate on a time scale much smaller than the characteristic exposure time of the impurity binding sites (for an overview, see refs 19 and 20). Later it was realized that for more complex impurities (e.g., biomolecules such as macromolecules) considerable activation barriers may slow the adsorption process creating overlaps between the time scales of adsorption and crystal growth.21,22 A simple steady-state approach using adsorption isotherms becomes invalid in these cases, and one has to develop timedependent modifications of CV in order to accurately capture empiric observations.21 There are however more caveats when using CV to fit experimental kinetic data: (i) For the entire fitting parameter space, there are no scenarios with a nonzero Received: September 18, 2012 Revised: December 27, 2012 Published: January 23, 2013 688

dx.doi.org/10.1021/cg301366q | Cryst. Growth Des. 2013, 13, 688−695

Crystal Growth & Design

Article

Figure 1. Step morphologies of 2D islands on {110} faces of tetragonal lysozyme crystals growing from Fluka (a), Seikagaku (b), and Sigma (c) solutions. The pure step morphology is shown in panel d for comparison.

step velocity below σd. This is in direct contradiction with experimental observations.23,24 (ii) The validity of the Gibbs− Thomson effect to calculate growth rates of curved steps is under dispute; there is at least one case where its invalidity is clearly demonstrated,25 and adjustments need to be applied accordingly.26 (iii) CV is based purely on thermodynamic arguments, any kinetic (thermal) fluctuations against free energy gradients are ignored, that is, any step f ragment with subcritical length is disallowed to advance. (iv) CV-type models can only be applied in the high kink density limit (kink formation energy < kT) where steps have sufficient f lexibility to curve around obstructions.27 In the low kink density limit (kink formation energy > kT), creating local step curvature comes at a too high energetic cost (high step edge free energy), and as such, steps will tend to remain elongated. In this case, not CV but rather kink blocking models (e.g., Bliznakov) should be considered to interpret kinetic data. In this work, we present additional experimental data, obtained from in situ observations of step dynamics of the model protein lysozyme in the presence of impurity cocktails, that are inconsistent with the model formulated by Cabrera and Vermilyea. Detailed theoretical analysis of this experimental data lead to the formulation of a more general model taking into account time-dependent adsorption, impurity clustering, and a nonzero desorption rate of impurities from the surface. Although these findings are based on protein crystallization, this general model should also be useful to explain the effects of biomolecules during the crystallization of inorganic crystals (i.e., biomineralization).

■

qualitatively by J.A. Gavira29 and consisted mainly of ovalbumin, and only trace amounts of the dimer and ovotransferrin were found. Importantly, all these commercial lysozyme samples contain large amounts of NaCl (remnants from the production process),29 especially in the case of Fluka where the NaCl concentration can reach up to 9%. Extensive dialysis after solubilization is therefore paramount to obtain equal NaCl concentrations for all three batches. Highly purified lysozyme (99.99%) was obtained from Maruwa Food Industries; see previous works for details.17,30 Step Velocity Measurement and Surface Imaging. Growing {011} faces of tetragonal hen egg white lysozyme (HEWL) were observed in situ by laser confocal interference contrast microscopy (LCM-DIM),4 which provides a significant contrast level for elementary steps of nanometer height but allows one to observe the entire crystal surface. This advanced optical technique is implemented using a confocal system (FV300, Olympus) attached to an inverted optical microscope (IX70, Olympus) with a 20× objective lens (LUCplan FLN 20×, Olympus) and equipped with a Nomarski prism and a partially coherent superluminescent diode (Amonics Ltd., model ASLD68-050-B-FA, 680 nm) to eliminate interference fringes. The observation cell used in this work was made of two sandwiched glass plates of 0.17 mm thickness separated by a Teflon spacer and was carefully washed by ultrasonic cleaning in Milli-Q water. More details about this experimental setup can be found in previous works.17,30 AFM observations were performed with the setup described previously.31 Tetragonal crystals of model protein hen egg-white lysozyme were grown at 20.0 ± 0.1 °C from a solution containing 70 mg/mL Seikagaku lysozyme (98.5% purity, 6× recrystallized, Seikagaku Co.), 25 mg/mL NaCl, and 50 mM sodium acetate (pH 4.5) buffer. After the seed crystals were transferred to the observation cell, the solution inside the cell was replaced with a growth solution of desired concentration. The observation cell was mounted on a temperaturecontrolled stage with Peltier elements, and two-dimensional (2D) islands formed on the {110} faces were observed by LCM-DIM. The temperature of the observation cell was changed at given time intervals to change the supersaturation of the lysozyme solution inside the cell: the accuracy of the temperature control was ±0.1 °C. The solubility was calculated from the data reported by Sazaki et al.32 Numerical Simulations. The recurrent system of eqs 2, 3, or 6 and 4 was solved numerically using the Mathcad 14.0 software package. In brief, for a given temperature, the impurity surface coverage was set to zero, the supersaturation was calculated from an experimentally determined solubility curve, which was subsequently fed into eqs 2 and 3. From the obtained value of the exposure time τ, θ was calculated using either eq 3 or eq 6 from which a new step velocity could be calculated using eq 2, thus completing the cycle. The cycle was repeated until a steady state was reached from which the final values were extracted for the production of the figures.

MATERIALS AND METHODS

Protein Solutions. Three commercially available lysozyme solutions were used for the step velocity measurements by LCMDIM. Step velocity measurements in the presence of protein impurities were carried out in the supersaturation range C−Ce = 0−31 mg·mL −1 (with C and C e the bulk and equilibrium protein concentration). The impurity levels of Seikagaku (98.5%) and Sigma samples (94.5%) were previously identified by Thomas and coworkers.28 They used sodium dodecylsulfate polyacrylamide gel electrophoresis (SDS-PAGE) with enhanced silver staining (detection limit ≤0.01%) and found that the Seikagaku sample mainly contained 0.5% covalently bounded lysozyme dimer (dimer), 1.0% 18 kDa polypeptide (18 kDa), and less than 0.1% of a 39 kDa polypeptide, while the Sigma impurity cocktail was composed of 0.2% of ovotransferrin, 3.8% ovalbumin, 0.7% dimer, and 1.0% 18 kDa. The protein impurities present in Fluka lysozyme were identified 689



■

Article

RESULTS AND DISCUSSION

Impurity Effect on Step Kinetics, Assuming Nonequilibrium Langmuir Adsorption. Step morphology and kinetics were measured as a function of supersaturation in the ⟨110⟩ direction of {011} faces of tetragonal lysozyme crystals growing from Fluka and Sigma (94.5%) solutions (Figures 1 and 2a). Strong nonlinear dependencies of the lateral step velocity, vstep, on C−Ce were observed. This is in line with previous observations for another commercial lysozyme batch24 (Seikagaku) that has an impurity content of ±2.5%. Kinetic data for Seikagaku, Sigma, and Fluka are summarized in Figure 2a. Step morphologies shown in Figure 1 demonstrate the progressive rounding (Fluka and Seikagaku) and pinning (Sigma) of the steps for these three batches, with the pure morphology shown as a reference (Figure 1a−c,d). Confronting the experimental data with time-dependent adaptations of the CV model did not yield any satisfactory fits (see ref 24 for the Seikagaku example). We attribute this failure to the following fact: CV step dependencies (and nonequilibrium modifications thereof) are characterized by a clear dead-zone followed by a steep increase to pure step velocities at higher supersaturations (dashed line in Figure 2a). From this, we conclude that the impurity species that dominate the step retardation are not immobile step pinners that maintain an equilibrium surface concentration throughout the entire supersaturation range (the limit initially considered by CV). This still leaves open a plethora of other possibilities. Let us consider mobile impurities that diffuse two-dimensionally across the surface as considered by Voronkov and Rashkovich.33 In the low supersaturation range (low step velocities), the adsorbed impurities will be averaged over all possible adsorption sites, and steps will not have sufficient time to curl around the impurities. In this scenario, simply no step pinning will occur. At higher supersaturation values, the mean step velocity will outpace the characteristic diffusive velocity of the adsorbents (i.e., the relative time scales will have switched), and progressive step segmentation will set in. Hence, for mobile surface bound impurities that exhibit overlapping time scales of diffusion and step advancement, there is a progressive increase in the pinning of the steps toward higher driving forces. Again, this is not in line with our experimental observations. We observe a clear recovery at higher supersaturations: at low C− Ce values, there is a linear dependency with a lower step kinetic coefficient β compared with βpure, followed by a transition zone toward unhindered step advancement at high C−Ce values. This leaves open two possibilities: impurities enter into (and desorb from) kinks directly from solution or from the surface on a time scale short compared with the time required for the steps to advance multiple lattice dimensions, typically referred to as kink blocking. The Bliznakov model proposed in the 1950s is a general kink blocking model developed for either terrace, step, or kink impurities with short residence times that lower the effective kink density, with or without subsequent incorporation.34,35 The Bliznakov growth rate expression for impurity hindered growth has nonzero growth rates even at low supersaturations, that is, no clear dead zone is present.36 The general form is as follows (see also eq 69 in ref 36): vi = vpure(1 − θ ) + vimax θ ≈ vpure(1 − θ )

Figure 2. (a) Fitting of the time-dependent Bliznakov model with Kisliuk-type impurity adsorption to experimentally obtained step velocities in the ⟨110⟩ direction on {110} faces of tetragonal lysozyme crystals growing from three different commercially available batches: Fluka (■), Seikagaku (▲), and Sigma (●). Fluka and Seikagaku data were fitted using the Langmuir−Bliznakov model (black and gray lines), and Sigma data with the Kisliuk−Bliznakov model (light gray line). Quasi-pure kinetics (99.99% purified Seikagaku; □ and dotted line) and theoretical CV kinetics with nonequilibrium adsorption (dashed line) are shown for comparison. (b) Comparison between the time-dependent Bliznakov (full lines) and Cabrera−Vermilyea (dotted line) model for elementary step velocities, assuming Langmuir adsorption of impurities onto the crystal surface at a fixed τi value of 800 s. For complete saturation of all impurity binding sites (θmax = 1), the modified Bliznakov model also predicts a dead-zone supersaturation, σd, below which no step advancement occurs. For θmax < 1 (i.e., non-negligible rates of impurity desorption), the Bliznakov-type model predicts a reduced step kinetic coefficient for lower C−Ce values (long exposure times), followed by a gradual transition toward near-pure kinetics at higher driving forces (short exposure times). (c) Comparison between the Langmuir (α = 0) and Kisliuk (α = 5) modes of adsorption, assuming time-dependent Bliznakov kinetics to account for the impurity effect: for τ ≫ τi, both models are identical, however, for τ ≤ τi, Kisliuk adsorption predicts an earlier and steeper recovery toward the pure mode of growth (gray and black lines, respectively).

(1)

where vpure, vi, and vmax are lateral step velocities under pure, i impure, and maximum impurity coverage conditions, respectively, and θ is the fractional surface coverage of impurities. The 690



Article

approximation on the right-hand side entails discounting displacement of adsorbed impurity particles by incoming growth units. A modification of the Bliznakov model was presented by Kubota and Mullin who introduced an effectiveness parameter to allow for nonzero step velocities even at full surface coverage.37 Equation 1 combined with Chernov’s38 expression for elementary step velocity leads to the interpretation that the apparent step kinetic coefficient βpure is modulated by the presence of impurities adsorbing into the kinks:

τ=

(4)

with J being the rate of 2D nucleation, ω* the frequency of attachment of molecules to the critical 2D nucleus, Γ the Zeldovich factor, Z the steady-state admolecule concentration, κ the specific edge free energy of the step, and s the surface area of a single molecule in the critical 2D cluster. A similar expression for time-dependent impurity adsorption was derived by Weaver and co-workers who considered the case of dislocation-driven growth.21 Both vi and θ(τ) and τ are interdependent. This can be understood as follows: the supersaturation determines the lateral step velocity (eq 2) and 2D nucleation rate, which in turn define the average time a freshly created terrace is exposed to the bulk (eq 4). If this exposure time is equal to or larger than the average adsorption time of an impurity molecule, a nonzero surface concentration of impurity molecules will set in (eq 3). This value for θ feeds back into eq 2 resulting in a deceleration of the steps and concurrent increase of the terrace exposure time (eq 4). This cycle will be repeated until a steady state is achieved for a given driving force. We therefore solve these recurrent equations numerically and plot the step velocity as a function of supersaturation for varying impurity concentrations (Figure 2b). For θmax = 0 or τ ≪ τi, the pure growth rate is obtained, whereas for θmax = 1 or τ ≫ τi, we observe a dead-zone supersaturation, σd, below which no step advancement occurs with a steep increase toward the pure curve above σd. This is in line with the Cabrera−Vermilyea model18 (dashed line), which defines Θ as (1−2rc(nmaxθ)1/2)1/2. Note that for the entire parameter space {θmax, τi, Θ > 0}, there are no scenarios for the CV model with a nonzero step velocity below σd, which is in clear contradiction with our experimental observations (Figure 2a). In Figure 2b, it can be seen that the model presented in eqs 2 and 3 displays a wide range of possible scenarios of partial impediment of advancing steps, characterized by a rapid return to near-pure kinetics for overlapping time scales of τ and τi. This recovery is the result of a self-purifying cascade: the reduction in impurity surface concentration accelerates the steps leading to a shorter terrace exposure time. The 2D nucleation rate and by extension τ are strong functions of T and Ce (two parameters typically used to modify the supersaturation). Hence, minor variations in C−Ce can lead to drastic changes in θ and vi. This effect will be maximal for specific driving forces where the time scales of impurity binding and exposure of the binding site are comparable. Consequently, at high supersaturations, even if bulk liquid concentrations of the impurity remain constant, the relatively slow process of impurity incorporation becomes incommensurable with the crystal growth mechanism, and the crystal effectively ignores the presence of impurities, a process termed self-purification. However, the recovery of impurity poisoning predicted by this model occurs slower than what we witness for the least pure lysozyme batch, Sigma, that is, no combination of the fitting parameters {θmax, τi} can satisfactorily reproduce the experimentally obtained dependence of the growth rate. Impurity−Impurity Clustering, Assuming Nonequilibrium Kisliuk Adsorption. We put forward two possible

(2)

where Ω is the volume of the growth species. The factor Θ = (1 − θ) is a simple measure of the impurity effect and regulates the step velocity from zero (maximum coverage θ = 1) to the value reached for pure solutions (no coverage θ = 0). If impurities first adsorb onto the surface and diffuse twodimensionally across the terraces to finally enter into the kinks, the Bliznakov expression needs to be adjusted accordingly.26 The modulator in this case becomes (1 − a(nθmax)1/2n−1 k ) with a being the lattice constant, nmax the impurity surface number density at full saturation and nk the kink density for the pure case. Typically, it is assumed that the adsorption of (impurity) molecules onto the crystal surface follows a Langmuir-type binding.34 If the kinetics of impurity adsorption are comparable to the rate of growth of the crystal surface, a steady-state concentration of adsorbed impurity molecules will not be reached. As such, a nonequilibrium expression must be employed for the impurity surface coverage:24

τi = (C ikA + kD)−1

with

θmax =

C ikA C ikA + kD

[πvstep2J ]1/3

−1/3 ⎡ ⎞⎤ ⎛ πκ 2s 2 ⎟⎥ = ⎢π (Ωβ(C − Ce)) ω*ΓZ exp⎜ − 2 2 ⎢⎣ ⎝ k T ln(C /Ce) ⎠⎥⎦

vi = Ω(C − Ce)βi = Ω(C − Ce)βpureΘ = vpure(1 − θ )

θ(τ ) = θmax(1 − e−τ / τi)

1

and (3)

with Ci being the bulk impurity concentration, kA and kD the rate constants for adsorption and desorption to and from the surface, θmax the steady-state surface impurity coverage, and τi the characteristic time for impurity adsorption (Figure 2b). Note that by introducing kA and kD, the model is not restricted to any single type of impurity (in terms of mean residence time on the surface): strongly bound immobile impurities can be characterized by a large kA/kD ratio, whereas weakly bound impurities that bind only briefly will have a small kA/kD ratio. The surface coverage is determined by the ratio of τi and the time that the impurity binding site is exposed to the bulk liquid. If we assume predominant binding to the terraces (see refs 17, 24, and 30 and Supporting Information), the latter is effectively the average delay between the creation of a fresh surface and the burying of the existing surface by the formation of a new layer, that is, the mean terrace exposure time τ. Different growth mechanisms have different characteristic times, which depend on extrinsic factors such as supersaturation, temperature, type and concentration of precipitant, etc. If layer generation is limited to homogeneous and heterogeneous 2D nucleation and assuming that steps do not interact (no diffusion field overlap, no entropic repulsion, etc), the characteristic exposure time τ of the terraces is given by24 691



Article

factors that may contribute to the enhancement of the impurity effect and increased sensitivity of the crystal growth process to impurities (as observed for Sigma lysozyme). As suggested in the original paper of Bliznakov, the line tension κ could be an increasing function of the impurity concentration. Impurities that reside in the step will expose different surfaces to the solution and most likely have a different number of dangling bonds potentially leading to a higher or lower line tension. In the former case, this would result in a suppression of the 2D nucleation rate at lower C−Ce thus accelerating the negative feedback loop of impurity poisoning. This assumption can be verified experimentally by measuring the 2D nucleation rate with and without impurities in the growth solution. We obtain near-identical dependencies of the nucleation rate on [T2 ln(C/ Ce)]−1 demonstrating that κ is constant (within experimental error) and therefore unaltered by the presence of impurities. Secondly, eq 3 assumes that the adsorbed molecules do not interact, that is, all adsorption sites on the surface are considered equal. If admolecules do interact favorably, then molecules that are already adsorbed onto the surface could function as local hotspots of adsorption for future impurity molecules. To account for this interaction, a sticking coefficient θkA′, which is admolecule concentration dependent, is introduced into the mass balance. A modified Kisliuk adsorption model is obtained:39 dθ = C ikA(1 − θ ) + C ikA(1 − θ )kA ′θ − kDθ dt = C ikA(1 − θ )(1 + kA ′θ ) − kDθ

(5)

with kA′ = 0 being in agreement with the Langmuir model and kA′ > 0 and kA′ < 0 corresponding to favorable and unfavorable interimpurity interactions. If kA > 0, impurity molecules will tend to group in stabilized clusters at intermediate coverage levels. Integration of eq 5 gives ⎛ 1 − e−τ / τi ⎞ ⎟ θ(τ ) = θmax ⎜ ⎝ 1 + αe−τ / τi ⎠

Figure 3. (a, b) Dependence of the fractional surface coverage θ assuming Kisliuk adsorption, as a function of driving force C−Ce (a) and terrace exposure time τ (b) for increasing values of the clustering factor α (note that α = 0 corresponds to Langmuir adsorption). Increasing the strength of impurity−impurity interaction leads to a delayed (lower C−Ce) but considerably steeper response in θ.

(6)

with α being a fitting parameter that is a nontrivial function of kA, kA′, kD, and Ci (see Supporting Information). As a consistency check, we evaluate eq 6 at infinite exposure times, which indeed returns limτ→∞ θ(τ) = θmax, that is, the coverage at steady state (analogous to the Langmuir solution). If we consider desorption to be negligible (kD ≪ kA), the general solution in eq 6 simplifies to 40

θ (t ) =

1 − e−τ / τi 1 + kA ′e−τ / τi

values of τi and α. For any given value of θmax, both models predict identical step velocities for low values of C−Ce. At intermediate C−Ce values, the two models diverge; that is, the Kisliuk model predicts a more rapid recovery from surface poisoning, which is in line with the initial assumptions. The onset of self-purification coincides with the overlapping of the time scale of impurity adsorption and layer burying. At higher driving forces, that is, very short exposure times, quasi-pure growth rates are obtained. Finally, we confront the experimental growth rates with the time-dependent Kisliuk−Bliznakov model presented in this work taking into account that there are four fitting parameters: the surface coverage at infinite exposure time θmax, the parameter α, the characteristic time for impurity adsorption τi, and σd, the dead zone supersaturation. All other parameters in the model are intrinsic system constants and have been determined experimentally (see Table 1 in ref 24). The overall shape of the supersaturation dependency of the step velocity for Fluka, Seikagaku, and Sigma batches are predicted by the Bliznakov model. The Fluka and Seikagaku fits were obtained using α = 0, that is, no impurity clustering had to be invoked to capture the experimental trends. To fit the sharp switch

(7) −1

with θmax = 1 and τi = [(1 + kA′)CikA] . For θmax = 1, a clear dead-zone is predicted (Figure 2c), which is not observed in the experimental data. From this, we conclude that desorption of impurity molecules from the surface cannot be neglected, and we will therefore use the more general eq 6 to calculate θ(t). Additionally, the impact of impurity clustering on the supersaturation and time-dependence of the fractional surface coverage is shown in Figure 3a,b. For increasing values of α, θ(t) will approach θmax more steeply, but at lower C−Ce values (larger exposure times). The expression for time-dependent Kisliuk adsorption (eq 6) is subsequently inserted into the Bliznakov growth rate expression. For comparison, in Figure 2c, both the time-dependent Langmuir model (eq 3) and the modified Kisliuk model (eq 6) are shown for identical and fixed 692



Article

Prompted by these results, we performed atomic force microscopy on lysozyme seed crystals growing from a Sigmalysozyme growth solution. The tested absolute supersaturation was C−Ce = 13 mg·mL−1, which is in the regime where the surface should be saturated with impurities (see Figure 3c). A deflection image of the {110} face clearly reveals the presence of surface adsorbed clusters on the terraces between the advancing elementary steps (Figure 4a). Interestingly, two types of clusters can be discerned: circular patches of maximum 0.5 μm in diameter and extended linear aggregates measuring 200 to 2000 nm in length (Figure 4b,c). The circular patches are 7−50 nm in height, which is considerably higher than an elementary step (∼5.7 nm), and the fiber-like structures measure 1.2−2.7 nm in height, which is substantially smaller. This suggests that these clusters are not composed of lysozyme molecules, at least not in a native folded state. As such, their constituents can be regarded as impurities. To what extent they have an effect on the step kinetics remains unclear and will be the subject of further investigation. Additionally, given that we used a zero-dimensional model in terms of spatial coordinates, we cannot differentiate between isotropic and anisotropic clustering of the impurities. A more specific model would have to be developed to predict/analyze the kinetic effects of the different modes of aggregation. In any case, these images show that foreign surface features can arise on protein crystals within the typical time frame of layer generation and terrace burying. Similar observations were made for COM, where peptide clusters were observed that impede step advancement.41 Final Impurity Model: Immobile Step Pinners Combined with Rapidly Exchanging Kink Blockers. As illustrated by the previous two sections, the effects of impurity cocktails on crystal growth kinetics can be far from trivial. The combinatorial (and potential synergetic) effects of different

between pure and impure modes of growth observed for Sigma, Langmuir adsorption no longer suffices and Kisliuk adsorption needs to be incorporated into the model (αSigma ≈ 5 > 0). Additionally, we observe a small dead zone in the Sigma data, which suggests that for this particular batch, immobile impurities create an impenetrable impurity fence in the low supersaturation regime (C−Ce < 9 mg·mL−1). To account for the quasi-zero growth rates below a critical supersaturation, we also incorporate into eq 2 the modified CV model put forward by Weaver and co-workers.26 This is achieved by multiplying the right-hand side by [1 − (eσd − 1)/(eσ − 1)] where σd and σ are the dead zone and effective supersaturation. The fitting parameters are summarized in Table 1. From the θmax values, Table 1. Fitting Parameters Used for Reproducing the Experimentally Obtained Curves in Figure 2a σd (−) α (−) θmax (−) τi (s)

Fluka

Seikagaku

Sigma

0 0 0.7 103

0 0 0.8 2 × 102

1.3 5 0.85 1.5 × 102

we conclude that saturation of all impurity binding sites will not be achieved for these systems even at infinite exposure times. Interpreting this using Langmuir binding (for Fluka and Seikagaku), this simply means that kD is not negligible compared with CikA. By increasing the bulk impurity content, one could potentially reach a state of full coverage at larger exposure times. Note also that the measure of step inhibition (i.e., strength of the impurity effect) is inversely proportional to τi: we observe the strongest retardation for Sigma, which has the fastest impurity binding to the surface (smallest τi).

Figure 4. (a) Deflection image of a {110} face of tetragonal lysozyme crystal exposed to a Sigma-lysozyme growth solution. On the terraces between elementary steps, two types of impurity clusters are observed, circular patches (white arrows) and linear aggregates (black arrows). (b,c) Enlargements of the surface adsorbed clusters. (d) The 99.99% purified Seikagaku as a reference. 693



■

Article

CONCLUDING REMARKS AND FUTURE PERSPECTIVES Now what can we extract from these experimental observations and fitting exercises? Our results indicate that impurity interaction could be an important factor in modulating the action of impurities (additives) on crystal growth. This is very much a realistic scenario for protein crystallization of nonmodel systems, where a mother liquor that contains a significant amount of an impurity mixture is a rule rather than an exception. But this process is likely to be of relevance in the field of biomineralization as well, where crystallization usually takes place in the presence of impurity (or additive) cocktails. Hints in the direction of interimpurity effects have already been provided in the case of calcite were polypeptides enhance the incorporation rate of Mg.42 In the pharmaceutical industry, similar processes occur as well, when several additives are used to control the morphology of crystals. Hence, a logical next step for future efforts is to reproduce such complex instances of crystallization in the laboratory to try to dissect the individual impurity effects and look for possible synergetic consequences. We conclude by pointing out that from a “fundamental crystal growth” point of view one could indeed interpret the diminished impurity effects at higher supersaturation as a mechanism of self-purification that is intrinsic to the system. Structural biologists will typically lean toward a less optimistic interpretation: from their experience, crystals nucleate at high supersaturations and then automatically seek up a state of equilibrium. The crystal will therefore inescapably attain a regime of slow kinetics rendering itself more susceptible to impurities present in solution. As such, the self-induced poisoning cascade might be a more suitable term to describe this process.

impurities with varying modes of action and even diverging characteristic time scales of adsorption lead to complex supersaturation dependencies of the elementary step kinetics. To clarify matters, we recapitulate the final model and summarize the different kinetic contributions in Figure 5. In

Figure 5. Final kinetic impurity model (red curve), which combines (i) a step pinning induced dead-zone by surface bound impurities in the low supersaturation range (full gray line), (ii) blocking of kinks by mobile impurities that exhibit nonequilibrium adsorption in the medium to high supersaturation regime (full light gray line), and (iii) clustering of the impurities on the surface. The classical limits (i.e., steady-state impurity concentration due to fast impurity adsorption) of both the CV (dashed line) and kink blocking (dotted line) models are shown for comparison. By setting σd = 0, τi = ∞, and α = 0, step pinning, kink blocking, and clustering, respectively, can be easily removed from the model.

■

ASSOCIATED CONTENT

S Supporting Information *

Derivation of eqs 6 and 7 and two-dimensional nucleation rates. This material is available free of charge via the Internet at http://pubs.acs.org.

■

the low supersaturation range, the reduction of the step velocity is dominated by thermodynamics, that is, step bending due to tightly bound obstacles on the surface. Increasing the supersaturation leads to a rapid increase of the step velocity. This partial recovery is due to breaking through the impurity fence and a reduction of the number density of bound step pinners. At higher supersaturation values, the impurity effect becomes kinetically dominated. Here, step advancement is limited mainly by the blocking of kinks through impurities that rapidly exchange with either the terraces or bulk liquid. The final recovery to the intrinsic kinetics of the system occurs asymptotical due to the increasing inavailability of the impurity binding sites, that is, exposure times of both the terraces and the kinks decrease as a function of supersaturation. By setting σd = 0, τi = ∞, and α = 0, step pinning, kink blocking, and clustering, respectively, can be easily removed from the model. Note that this model builds upon the work done by Weaver et al.26 Most notable changes are (i) allowing for impurities to desorb from the surface and (ii) clustering of impurities into aggregates on the surface. Of course, this model is still quite restrictive: it does not allow for step pinners to have a diffusive velocity comparable to the advancement rate of the step. Ideally one should be able to tune the mobility of the impurity species by adjusting a single fitting parameter.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; [email protected]. Phone: +32 2 629 1932. Fax: +32 2 629 1963. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS M.S. is grateful for the support by the Belgian PRODEX Programme under contract number ESA AO-2004-070. A.E.S.V.D. acknowledges the support by Grant No. AYA2009-10655 from the Ministry of Science and Innovation, Spain and the Consolider-Ingenio 2010 project “Factoriá Española”.

■

REFERENCES

(1) Malkin, A. J.; Kuznetsov, Y. G.; McPherson, A. J. Struct. Biol. 1996, 117, 124−137. (2) Yau, S. T.; Thomas, B. R.; Vekilov, P. G. Phys. Rev. Lett. 2000, 85, 353−356. (3) McPherson, A.; Kuznetsov, Y. G.; Malkin, A.; Plomp, M. J. Struct. Biol. 2003, 142, 32−46. (4) Sazaki, G.; Matsui, T.; Tsukamoto, K.; Usami, N.; Ujihara, T.; Fujiwara, K.; Nakajima, K. J. Cryst. Growth 2004, 262, 536−542.

694



Article

(39) Kisliuk, P. J. Phys. Chem. Solids 1957, 3, 95−101. (40) Henderson, A. P.; Seetohul, L. N.; Dean, A. K.; Russell, P.; Pruneanu, S.; Ali, Z. Langmuir 2009, 25, 931−938. (41) Friddle, R. W.; Weaver, M. L.; Qiu, S. R.; Wierzbicki, A.; Casey, W. H.; De Yoreo, J. J. Proc. Natl. Acad. Sci. U.S.A. 2010, 107, 11−15. (42) Stephenson, A. E.; De Yoreo, J. J.; Wu, L.; Wu, K. J.; Hoyer, J.; Dove, P. M. Science 2008, 322, 724−727.

(5) Van Driessche, A. E. S.; Otálora, F.; Sazaki, G.; Sleutel, M.; Tsukamoto, K.; Gavira, J. A. Cryst. Growth Des. 2008, 8, 4316−4323. (6) Sleutel, M.; Maes, D.; Van Driessche, A. E. S. Adv. Chem. Phys. 2012, 151, 223−276. (7) Davis, K. J.; Dove, P. M.; De Yoreo, J. J. Science 2000, 290, 1134− 1137. (8) Elhadj, S.; De Yoreo, J. J.; Hoyer, J. R.; Dove, P. M. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 19237−19242. (9) Pina, C. M.; Merkel, C.; Jordan, G. Cryst. Growth Des. 2009, 9, 4084−4090. (10) Qiu, S. R.; Wierzbicki, A.; Salter, E. A.; Zepeda, S.; Orme, C. A.; Hoyer, J. R.; Nancollas, G. H.; Cody, A. M.; De Yoreo, J. J. J. Am. Chem. Soc. 2005, 127, 9036−9044. (11) Wang, L.; Qiu, S. R.; Zachowicz, W.; Guan, X.; De Yoreo, J. J.; Nancollas, G. H.; Hoyer, J. R. Langmuir 2006, 22, 7279−7285. (12) Qiu, S. R.; Orme, C. A. Chem. Rev. 2008, 108, 4784−4822. (13) Land, T. A.; Martin, T. L.; Potapenko, S.; Palmore, G. T.; De Yoreo, J. J. Nature 1999, 399, 442−445. (14) Thomas, T. N.; Land, T. A.; Johnson, M.; Casey, W. H. J. Colloid Interface Sci. 2004, 280, 18−26. (15) Thompson, C.; Davies, M. C.; Roberts, C. J.; Tendler, S. J.; Wilkinson, M. J. Int. J. Pharm. 2004, 280, 137−150. (16) Nakada, T.; Sazaki, G.; Miyashita, S.; Durbin, S. D.; Komatsu, H. J. Cryst. Growth 1999, 196, 503−510. (17) Van Driessche, A. E. S.; Sazaki, G.; Dai, G.; Otalora, F.; Gavira, J. A.; Matsui, T.; Yoshizaki, I.; Tsukamoto, K.; Nakajima, K. Cryst. Growth Des. 2009, 9, 3062−3071. (18) Cabrera, N.; Vermileya, D. A. Growth and Perfection of Crystals; Wiley: New York, 1958. (19) Sangwal, K. Prog. Cryst. Growth Charact. Mater. 1996, 32, 3−43. (20) Sangwal, K. Additives and Crystallization Processes: From Fundamentals to Applications; John Wiley & Sons: Chichester, England, 2007. (21) Weaver, M. L.; Qiu, S. R.; Friddle, R. W.; Casey, W. H.; De Yoreo, J. J. Cryst. Growth Des. 2010, 10, 2954−2959. (22) Dai, G.; Sazaki, G.; Matsui, T.; Tsukamoto, K.; Nakajima, K.; Kang, Q.; Hu, W. Cryst. Growth Des. 2011, 11, 88−92. (23) Ristic, R. I.; De Yoreo, J. J.; Chew, C. M. Cryst. Growth Des. 2008, 8, 1119−1122. (24) Sleutel, M.; Sazaki, G.; Van Driessche, A. E. S. Cryst. Growth Des. 2012, 12, 2367−2374. (25) Teng, H. H.; Dove, P. M.; Orme, C. A.; De Yoreo, J. J. Science 1998, 282, 724−727. (26) Weaver, M. L.; Qiu, S. R.; Hoyer, J. R.; Casey, W. H.; Nancollas, G. H.; De Yoreo, J. J. J. Cryst. Growth 2007, 306, 135−145. (27) De Yoreo, J. J.; Zepeda-Ruiz, L. A.; Friddle, R. W.; Qiu, S. R.; Wasylenki, L. E.; Chernov, A. A.; Gilmer, G. H.; Dove, P. M. Cryst. Growth Des. 2009, 9, 5135−5144. (28) Thomas, B. R.; Vekilov, P. G.; Rosenberger, F. Acta Crystallogr., Sect D: Biol. Crystallogr. 1996, D52, 776−784. (29) Gavira, J. A. Protein crystal growth in gel by counterdiffusion methods. Ph.D. thesis, University of Granada, Spain, 2000. (30) Van Driessche, A. E. S.; Sazaki, G.; Otalora, F.; Gonzalez-Rico, F. M.; Dold, P.; Tsukamoto, K.; Nakajima, K. Cryst.Growth Des. 2007, 7, 1980−1987. (31) Sleutel, M.; Vanhee, C.; Van de Weerdt, C.; Decanniere, K.; Maes, D.; Wyns, L.; Willaert, R. Cryst. Growth Des. 2008, 8, 1173− 1180. (32) Sazaki, G.; Kurihara, K.; Nakada, T.; Miyashita, S.; Komatsu, H. J. Cryst. Growth 1996, 169, 355−360. (33) Voronkov, V.; Rashkovich, L. J. Cryst. Growth 1994, 144, 107− 115. (34) Davey, R. J. Cryst. Growth 1976, 34, 109−119. (35) Myerson, A. S. Handbook of Industrial Crystallization; Butterworth-Heinemann: Boston, MA, 2002. (36) Bliznakov, B. Fortschr. Min. 1958, 149−191. (37) Kubota, N.; Mullin, J. J. Cryst. Growth 1995, 152, 203−208. (38) Chernov, A. A. Modern Crystallography III. Crystal Growth; Springer-Verlag: Berlin, 1984. 695


On the Self-Purification Cascade during Crystal Growth from Solution

Recommend Documents