Growth Mechanisms and Kinetics of Trypsin Crystallization - The

Marco Plomp, Alexander McPherson, Steven B. Larson, and Alexander J. Malkin ... Shelley J. Wilkins, Barry A. Coles, and Richard G. Compton , Andrew Co...
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J. Phys. Chem. B 2001, 105, 542-551

Growth Mechanisms and Kinetics of Trypsin Crystallization Marco Plomp, Alexander McPherson, Steven B. Larson, and Alexander J. Malkin* Department of Molecular Biology and Biochemistry, UniVersity of California, IrVine, California 92697-3900 ReceiVed: August 22, 2000

The surface morphologies of orthorhombic and trigonal trypsin crystals grown from solution were investigated by in situ atomic force microscopy. {001} and {100} faces of trigonal and {110} faces of orthorhombic crystals grew strictly by two-dimensional (2D) nucleation. For {101} faces, no 2D nucleation was observed even at relatively high supersaturations. For the {101} face, from the supersaturation dependency of the slope of dislocation hillocks and tangential step rates the surface free energy of step edges R and the tangential kinetic coefficient β were calculated to be 5 ( 1 erg/cm2 and (1.1 ( 0.2) 10-4 cm/s, respectively. The high value of R is thought to be the cause for the absence of 2D nucleation on this face. It was also found that growth of the trigonal {001} face proceeds by alternating deposition of three symmetry-related growth layers with a height equal to one-third of the unit cell parameter c. Using periodic bond chain (PBC) analysis it was demonstrated that the observed morphology as well as the anisotropic growth of the 2D nuclei are controlled by the underlying structure. We believe that this is the first report of growth by symmetry-related layers for a system with a 3-fold screw axis.

1. Introduction Crystallization of macromolecules and their analysis by X-ray diffraction plays an essential role in important areas of modern molecular biology and biotechnology, particularly in the genetic engineering of proteins and rational drug design. In recent years, macromolecular crystallization has also become increasingly important to the pharmaceutical formulation of protein and antibody drugs. Over the past several years theoretical and experimental studies of nucleation phenomena, growth dynamics, mechanisms of defect formation, and impurities effects were performed using light scattering techniques,1-4 interferometry,5,6 X-ray topography,7-9 and atomic force microscopy (AFM).10-20 Despite the progress in understanding the phenomena provided by these efforts, further work is required to achieve a comprehensive appreciation and to be able to rigorously control macromolecular crystallization for practical applications. In this work, crystallization of two polymorphs of bovine pancreas trypsin were investigated. Variation of conditions resulted in growth of different trypsin crystal polymorphs. The principles underlying polymorphism in crystal growth are poorly understood in general, and in macromolecular crystallization in particular. Because polymorphism is responsible for a number of formulation problems in the area of pharmaceutical production, including tabulating failures, variable dissolution rates, and instability of a drugs in their final dosage forms, it has practical consequences as well. In this article, we focus on the differences in growth mechanisms and kinetics of both observed polymorphs, relating these to their growth rate. To understand and predict the three-dimensional morphology of crystals from their structure, Periodic bond chain (PBC) analysis has been increasingly utilized. By this method, which was developed by Hartman and Perdok in 1955,21-23 the crystal structure is reduced to a three-dimensional network of bonds * Author to whom correspondence should be addressed. Fax: +1 949 824 1954. E-mail [email protected].

connecting the centers of mass of the growth units. From this crystal graph, periodic bond chains and connected nets with various orientations are derived, and these are then used for morphology prediction (see, e.g, refs 24,25). Here, we applied PBC analysis to explain experimental observations of the surface morphology of trigonal trypsin crystals. 2. Experimental Section Bovine pancreas trypsin was purchased from Sigma Chemical Co (St. Louis, Mo). Trypsin seed crystals of 0.7-1 mm were grown in a week by the vapor diffusion method.26 In these experiments, a protein solution consisting of 40-60 mg/mL trypsin, 10 mg/mL benzamidine, and 3 mM CaCl2 in water was mixed with an equal amount of reservoir solution which consisted of 1.8-2.2 M (NH4)2SO4, 0.1 M Tris, pH ) 8.5. The inhibitor benzamidine was added in order to prevent autocleavage of trypsin, while Ca2+ stabiles the trypsin structure.27 Trypsin (Mr ) 23 kD, molecular diameter approximately 3 nm) has been reported to crystallize in three different polymorphs: low-density orthorhombic,27 high-density orthorhombic,28 and trigonal.29 In our experiments, X-ray diffraction analysis revealed that trypsin crystals grown by the vapor diffusion method were of the low-density orthorhombic space group P212121 with a ) 63.28 Å, b ) 63.80 Å, c ) 69.24 Å, Z ) 4. Crystals were elongated along one axis, which, from X-ray analysis, was found to be the b-axis. The majority of the crystals were bounded by {101} faces parallel to the elongated direction and capped by relatively small {110} faces, as indicated in Figure 1a. In the course of AFM experiments (see section 3.3) the appearance of 25-50 µm crystals of hexagonal shape with {001} and {100} faces roughly equal in size was observed (Figure 1b). From the height of growth steps and the growth morphology of {001} faces (section 3.3) it was determined that these crystals were of the previously reported trigonal space group P3121, with a ) b ) 54.82 Å, c ) 109.8 Å, and Z ) 6.29

10.1021/jp003007b CCC: $20.00 © 2001 American Chemical Society Published on Web 12/13/2000

Trypsin Crystallization

Figure 1. Morphology of orthorhombic (a) and trigonal (b) trypsin crystals.

For in situ AFM experiments, a seed crystal of trypsin in a droplet of mother liquor was transferred and mounted on a Perspex substrate. The substrate with seed crystal was immediately placed in the sealed fluid cell of a Nanoscope AFM (Digital Instruments, Santa Barbara, CA), which was then filled with a 1:1 mixture of 0-60 mg/mL trypsin solution (3 mM CaCl2, 10 mg/mL benzamidine) and 3.6 M (NH4)2SO4, 0.1 M Tris buffer, pH ) 8.5, leading to a final concentration of 1.8 M (NH4)2SO4 and a concentration of one-half the initial trypsin concentration. Images were collected both in contact mode and tapping mode. Cantilevers from both Digital Instruments and Park Scientific were used. In all cases, the set-point voltage was adjusted at the point were the tip was almost released from the surface, to minimize the force applied to the crystal. The diffusion coefficient of trypsin was measured by quasielectric light scattering (QELS). For this, a Malvern 4700c submicron particle analyzer (Malvern Instruments, Inc, Southborough, MA) was used. Prior to QELS measurements, trypsin solutions were centrifuged for 40 min at 12 000 g, and filtered through a 0.2 µm syringe filter (Millipore Co, Bedford, MA). 3. Results and Discussion 3.1. Surface Morphology and Growth Kinetics of Orthorhombic {101} Faces. Atomic force microscopy (AFM) images of trypsin crystallization were obtained from developing {101} faces of low-density orthorhombic crystals. Seed crystals grown by vapor diffusion were always mounted with a {101} face on the support, with the opposite {101} face being horizontal and readily suitable for imaging. Growth of the {101} faces of orthorhombic crystals proceeded strictly by a dislocation mechanism. As seen in Figure 2, growth steps originated from single, double, or complex dislocation sources. The step height is 4.6 ( 0.5 nm, which is consistent with d101 ) 4.67 nm as calculated from the measured unit cell parameters. No two-dimensional (2D) nucleation was observed even at relatively high supersaturation of σ ) 1.1, where σ is defined as σ ) ln(c/ce), c and ce being actual and equilibrium concentration, respectively. At this supersaturation terrace widths of steps on dislocation hillocks are about 1 µm. One may expect that the absence of 2D nucleation on such relatively narrow terraces is strictly due to the diffusion-field overlap.30 However, at this supersaturation 2D nucleation was not even observed on the relatively large 20 µm terraces on top of multilayer stacks

J. Phys. Chem. B, Vol. 105, No. 2, 2001 543 formed by the lateral expansion of three-dimensional nuclei (Figure 3). At supersaturations > σ ) 1.1 intense amorphous precipitation was observed upon mixing of trypsin and salt solutions. Evident from Figure 2a-c is that growth steps on the {101} face of orthorhombic trypsin are very rough. Step roughness is typically caused by impurities adsorbed on crystalline surfaces.31-33 Growth steps stop at sites of contact with impurity particles that are firmly attached to the surface. At the same time, portions of steps between neighboring stoppers continue to grow. This results in pinning of growth steps as seen in Figure 2. In Figure 4a the adsorption of a relatively small particle of about 10 nm is seen on the crystalline surface. This results in the pinning of a growth step as seen in Figures 4b,c. In the same series of images a number of pinning sites on a step front caused by smaller particles, which themselves were not detected by the AFM, can be seen. The nature of these impurity particles is not known. They may be traces of other proteins in trace amounts, or modified trypsin molecules, or their aggregates. To exclude the presence of trypsin aggregates as an impurity, in a control experiment the protein solution prior to the incubation in the AFM cell was heated to 50 °C for 10 min in order to dissolve potential aggregates and then filtered. This had no effect on the occurrence of undulated steps. The tangential step velocity V depends on concentration according to34

Vstep ) ωβ(c - ce)

(1)

where β is a measure of the kinetics of incorporation of the molecules, ω is the specific molecular volume of 54 nm3, and c and ce are the actual and equilibrium concentrations of trypsin in solution. To determine the kinetic coefficient β, the tangential step velocity was measured as a function of the protein concentration c (Figure 5). In all experiments a final concentration of 1.8 M (NH4)2SO4 and a pH of 8.5 were utilized. The supersaturation was changed by varying the final trypsin concentration in a range from 0 to 22 mg/mL. When the supersaturation was changed, the AFM cell was typically flushed with a 1:1 mixture of protein and precipitant of twice the volume of the cell. This was done in order to make sure that the new solution occupied the total interior of the cell. The tangential velocity of the steps was measured by inspecting sequential images and calculating the advancement of steps during the elapsed time between images. As seen from Figure 5, dissolution takes places for c < 4 mg/mL ) 1 × 1017 cm-3, growth takes place for c >10 mg/ mL ) 2.5 × 1017 cm-3, while for intermediate protein concentrations neither significant growth nor dissolution takes place. This concentration range is known as the “dead zone” (see, e.g., refs 30,35). The occurrence of such a dead zone is typically ascribed to impurity blocking.30,34 In our case, the rather large width of the dead zone supports the view of a relatively large amount of impurities present as derived from the observed pinning of the steps (Figure 4).The equilibrium concentration is taken to be that concentration in the middle of the dead zone, which is ce ) 2 × 1017 cm-3. However, for the determination of β only the (linear) growth portion of the figure is used (similar to the approach used in ref 36, i.e., the concentration of onset of growth, c* ) 2.75 × 1017 cm-3, is used instead of ce). From the linear part of the curve in Figure 5, the kinetic coefficient β is estimated as β ) (1.1 ( 0.2) 10-4 cm/s. Similar values for the kinetic coefficient β were found for other macromolecular systems.15 Lower values of β for macromolecular crystallization compared with those estimated

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Figure 2. Dislocation outcrops on {101} faces of orthorhombic trypsin crystals. (a) Double dislocation, (b) complex dislocation source, (c) single dislocation, and (d) array of dislocations forming a low angle grain boundary of 3.2° as measured along the fault line.

in inorganic crystallization15 are typically attributed to the considerably lower probability of a proper orientation for the incoming molecule potentially incorporating into the crystal.37 Significantly lower diffusivity as well as larger barriers to adsorption can also contribute to relatively low values of β for macromolecular crystallization. The translation diffusion coefficient D of trypsin molecules was measured by QELS to be 1.4 × 10-6 cm2/s. For trypsin crystals used in experiments with typical sizes of L ) 0.3 mm, β L/D ≈ 0.2. This indicates34 that growth of trypsin crystals is exclusively limited by neither bulk diffusion, nor kinetics or surface processes, but rather proceeds in a mixed regime. Indeed, we also observed that the tangential rate of macrosteps is lower than the rate of individual growth steps, which further indicates the overlap of diffusion fields at the steps. To determine the surface free energy of the step edge, R, we measured the slope of dislocation hillocks in the supersaturation range σ ) ln(c/ce) ) 0.45 - 1.1. Because the generation of new steps at the center of a growth spiral is dependent on R, the following relation can be derived for the slope of a complex dislocation source spiral:38

p)

mh 19rc + 2L

(2)

where mh is the dislocation’s Burgers vector, h is the step height, L is the length of the circumference connecting all individual dislocation outcrops of the source, and rc is the critical nucleus defined by34

rc ) Rω/kTσ

(3)

Rewriting eq 2 with the help of eq 3 and with p ) W/h with W being the terrace width gives

W)

19ωR 1 2L + mkT σ m

(4)

In Figure 6, the terrace width W of a double dislocation (m ) 2, L ) 0) is depicted as a function of 1/σ. From the slope of W (1/σ) and eq 4, the surface free energy of the step edge was estimated to be Rortho ) (5 ( 1) × 10-3 J/m2 ) 5 ( 1 erg/cm2. This value is lower than those for solution growth of inorganic crystals,15 but considerably higher than measured for

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Figure 3. Series of images showing the outgrowth of a multilayer stack on a {101} face of an orthorhombic crystal. 2D nucleation is absent even on the very flat top of the stack. σ ) 0.75. The step velocity of the stack is 8 nm/s, while single-step velocity is 11 nm/s.

other macromolecular crystals (Table 1). The relatively high value of R for the major face of low-density orthorhombic trypsin crystals makes formation of 2D nuclei significantly less favorable energetically compared with other macromolecular systems. This explains why 2D nucleation was not observed on the {101} face even at relatively high supersaturations. Similarly, for the protein canavalin, with R ) 2 erg/cm2, 2D nuclei were only observed on large terraces at relatively high supersaturations.15 The trend of Table 1 confirms the expected correlation between R and the occurrence of 2D nucleation, which was already found for a smaller number of protein crystals.15 Differences in values for R for different proteins or crystal faces are due to the differences in binding and number of molecular contacts. 3.2. Surface Morphology of Orthorhombic {110}. Given the absence of 2D nucleation on the dominating {101} faces, it was considered useful to examine the growth of the much smaller, and hence much faster growing, {110} top faces. As seen in Figure 1a, dominating faces on the orthorhombic crystals are the {101} faces. With a {101} face on the support, {110} top faces exhibit an angle of 62° with respect to the {101} faces.

AFM scanning of these faces is virtually impossible because of the finite angle of the AFM tip. To image them, small crystals ( 0.7 spiral at low σ; 2D at σ > 1.1-1.851 mostly spiral; 2D at σ > 2.5 only spiral; max. σ ) 1.1

min. and max. σ refer to minimal and maximum applied σ, respectively.

Figure 7. Sequential images showing 2D nucleation on the {110} faces of orthorhombic trypsin crystals. Trypsin concentration is 15 mg/mL. The AFM slow scan direction is indicated with a white arrow, recording time of images is 45 s.

which could be either {100} or {110}, were imaged. Under the same growth conditions as for the orthorhombic crystals, intense 2D nucleation was observed on both faces, as shown in Figure 8. Steps on the prismatic faces were of height 4.6 ( 0.5 nm,

which corresponds to the height of a complete unit cell for the {100} faces, which is 1/2x3 a ) 4.75 nm. (The step height for {110} faces would be equal to a ) 5.5 nm). 2D nucleation on the (001) face occurs by formation of elongated nuclei parallel to the crystal edges, i.e., along 〈100〉 directions, with an average width/length ratio of 1:4 (Figure 8bcd), which could indicate anisotropy of R (see section 3.4). The shape of the nuclei forming each successive growth layer of thickness 3.2 ( 0.5 nm is related by 3-fold symmetry to the shape of the initiating nuclei of the preceding layer. Although no complete σ-dependent step velocity curve was systematically obtained for the trigonal crystal faces, several measurements of step velocities indicated that the tangential growth velocity Vstep for trigonal crystals was about an order of magnitude less than those for orthorhombic crystals. Thus, for example, at c ) 4 ×1017 cm-3, Vstep,ortho{101} ) 10 nm/s, while Vstep,trig{001} varies from 1.5 nm/s for the fast direction to only 0.4 nm/s for the slow direction of the 2D nuclei. The isotropic step velocity of the prismatic {100} face is equally low: 1 nm/s at the same growth conditions. However, this lower Vstep is compensated by high rates of 2D nucleation, resulting in a normal growth rate Rtrig ) 0.01d001,trig/s ) 0.1 nm/s, compared to Rortho ) 0.01d101,ortho/s ) 0.05 nm/s as measured at growth spirals on orthorhombic {101} faces at the same concentration. The rate of 2D nucleation J for the trigonal {001} face at c ) 4 × 1017 cm-3 was estimated accordingly using eq 5 to be 2 × 1010 cm-2 s-1. This is considerably higher than rates of 2D nucleation measured for thaumatin (J ) 5 × 103 - 2 × 106 cm-2 s-1 for σ ) 0.7-1.6),39 catalase (J ) 25-750 cm-2 s-1 for σ ) 0.9-2.3),37 and the {110} face of orthorhombic trypsin (J ) 1 × 108 cm-2 s-1 at c ) 6 × 1017 cm-3). This indicates that the surface free energy of the step edge R for this trigonal crystal is lower than for the above-mentioned protein crystals, as well as for the orthorhombic {110} and especially {101} face of orthorhombic trypsin crystals This could account for their observed higher growth rate as compared to the orthorhombic crystals. 3.4. d003 Growth Layers of Trigonal {001} Faces. The observed 3-fold symmetric growth behavior of the trigonal {001} face can be explained by the presence of the 3-fold screw axes parallel to c in the P3121 crystal structure. The six molecules in the unit cell with a height of c ) 10.9 nm can, in accordance with the space group symmetry, be sorted into three groups consisting of two molecules each, that are mutually related by a 3-fold screw axis. This results in three triad-related growth layers within a unit cell, each with a height of d003 ) |1/3 c| ) 3.6 nm, which corresponds well to the experimentally observed growth steps of height 3.2 ( 0.5 nm. Symmetry-related growth layers have been observed in AFM studies of the crystallization of beef liver catalase40 and barite;41 however, to our knowledge this is the first report for a system with a 3-fold screw axis. The observation that the trigonal crystals grow by related layers of thickness d003 can be explained

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Figure 8. Surface morphology on trigonal crystals. (a) High-density, isotropic 2D nuclei on prismatic {100} faces. (b, c) High-density, anisotropic 2D nuclei on top {001} faces. The 120° rotation of the orientation of subsequent layers is emphasized in (d), where layers with the same orientation in (c) have been shaded identically as those in the inset. This inset shows the shape of 2D nuclei on successive layers as predicted by PBC analysis (see text).

by symmetry considerations. There is a direct relation between the X-ray diffraction-based reflection conditions,42 which cause systematic extinctions for centered cells, glide planes and screw axes, and the thickness dhkl of the growth layers in associated crystallographic orientations. This relation was developed by Bravais, Friedel, Donnay, and Harker,43 and embedded in PBC theory (e.g., ref 44). Trigonal trypsin follows the rule formulating that the existence of an n-fold screw axis in a crystal structure leads to the formation of growth layers of thickness dhkl/n ) dnhnknl on the face (hkl) perpendicular to that axis. Because the structure of these symmetry-related layers is the same, formation of these three growth layers is energetically equally favorable. As demonstrated below, structure considerations permit two different divisions of the six molecules into three layers. To delineate the division of molecules into the three layers that occurs in the course of crystallization, a basic periodic bond chain (PBC) analysis was performed with the help of atomic coordinate data 29 from the Protein Database.45 From these data, all crystal contacts between neighboring molecules in the unit

cell within a range less than 4D were tabulated. As a rough estimate, their number was taken as a measure for the strength of the interaction between the molecules. According to the X-ray diffraction data, the molecules, numbered 1..6 according to their general Wyckoff positions,42 are stacked in the c-direction in the order ..3..5..1..4..2..6..3..5..(Figure 9). Molecules in the positions 1, 2, and 3 are directly symmetry-related by the 3-fold screw axis as are those in positions 4, 5, and 6. Every molecule has crystal contacts with seven other molecules, which can be either molecules within the same unit cell or within neighboring cells. For sake of simplicity, all crystal contacts between two molecules are taken together and are designated a bond. The crystal contacts are arranged in four different types of bonds, denoted A..D, ranging from 75 to 16 crystal contacts (see Table 2). Thus, for example, a molecule in position 1 has three of its seven bonds with molecules in the neighboring positions 5 (two bonds A and one bond D), while only two of them (of type C) are with neighboring molecules in positions 4 (see Table 2, Figure 10ac). Furthermore, a molecule in position 1 has a bond B with molecules in positions 2 and 3.

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Figure 9. P3121 trigonal trypsin crystal structure as viewed along [120]. The molecules are numbered according to their general Wyckoff positions, the unit cell is indicated in white. Molecules in positions 1, 2, and 3 are symmetry-related by the 3-fold screw-axis, as those in positions 4, 5, and 6. There are two ways to distribute the molecules in three layers of d003, which are layers formed by molecules in positions ..3-5..1-4..2-6..3-5.. or 6-3..5-1..4-2..6-3...

TABLE 2: Bonds in Trigonal P3121 Trypsin Crystals bonds involving molecule 1[000] a

bond

no. of crystal contacts

1[000] - 5[000] 1[000] - 5[100] 1[000] - 2[000] 1[000] - 3[000] 1[000] - 4[01h0] 1[000] - 4[100] 1[000] - 5[01h0]

A A B B C C D

75 75 36 36 16 16 16

a Superscripts indicate cell; e.g., molecule 5[000] is in the same cell as 1[000], while 5[100] is in the neighboring cell in the +x direction.

From the molecular stacking ..3..5..1..4..2..6..3..5.., two different divisions into d003 growth layers are possible. These are layers formed by molecules in positions ..3-5..1-4..26..3-5..or..6-3..5-1..4-2..6-3.., respectively. In this notation, molecules joined by a dash are in the same d003 layer. From a PBC point of view, these two different divisions are competing c-oriented connected nets, built up out of different combinations of PBCs in a slice d003.24 These PBCs are defined as uninterrupted chains of bonds between growth units with a crystal lattice periodicity.24 For example, the bonds of a molecule in position 1 of one unit cell to the molecule 1 of an [100]translated cell (via molecule in position 5 of the [100]-translated cell, see Figure 10d) form an [100]-directed PBC. From two or more (nonparallel) PBCs a two-dimensional connected net with an orientation (hkl) can be constructed. According to PBC theory, for a given slice dhkl, all crystal interactions forming the crystal energy Ecryst can be divided into bonds inside the slice adding up to the slice energy Eslice, and bonds interconnecting these slices, forming the attachment energy Eattach. Based on the basic connected net criterion, the net with the highest Eslice, i.e. with the strongest bonds inside the layer, will be the one forming growth layers during the crystallization process.24 In Table 3 the bonds involved in the two possible connected nets, together with their number of crystal contacts inside and outside the slice d003, representing Eslice and Eattach, are given. From these data, which are represented graphically in Figure 10bd, it can be seen that the 1-5 type growth layers have a

much higher Eslice, which makes them, and not the 1-4 type layers, the obvious choice for the actual growth steps. Another fundamental feature of PBC theory is the classification of faces in K (kinked), S (stepped) and F (flat) faces, for corresponding connected nets containing zero, one, and at least two intersecting, nonparallel PBCs, respectively.24 Because S faces contain only one PBC, they are stabilized in only one direction (along the PBC) and will be roughened in the other directions, which will lead to a stepped surface. In contrast, an F face contains PBCs in more directions, leading to a stabilization of the whole face and subsequent layer-by-layer growth. In the present case the 1-4 layers include only one PBC in the [110] direction (see Figure 10b), making it an S (stepped)-type net, while the 1-5 net layers include at least three easily recognizable PBCs (see Figure 10d: along [100] with two A bonds, along [010] with an A and D bond, and along [110] with an A and D bond), making this net an F (flat)-type net. The latter corresponds to the experimentally observed smooth {001} face. This also confirms that the 1-5 type layers are the observed growth layers. From the strength of the PBCs in the 1-5 connected net the two-dimensional island shape occurring in these layers can be predicted by applying two-dimensional PBC analysis.46 In threedimensional PBC analysis, Eattach(hkl) is used as an estimate for the growth rate R of the corresponding face (hkl). By plotting vectors Rhkl of a crystal’s major faces and erecting faces (hkl) perpendicular to them (a so-called Wulff construction 47), an estimate of the 3D crystal form can be made.24 For twodimensional PBC analysis, the 2D equivalent of Eattach(hkl) can be used to obtain Rhk, and from that a 2D Wulff plot can be made by erecting (hk) steps. In the trigonal trypsin {001} 1-5 net, the above-mentioned [100] directed PBC consisting of only strong A bonds (see Figure 10d) is by far the strongest and will dominate the 2D growth form. From the analysis, it shows that this growth form is bounded by [100], [120], [210], and [1h10] directed steps/PBCs. This predicted growth form, which should be reflected by the shape of 2D nuclei, is depicted in the inset of Figure 8d. Indeed, the predicted 1:4.7 width/length ratio closely matches the observed 1:4 ratio of the 2D nuclei. This indicates that the assumption made, namely that the number of crystal contacts between two molecules is a rough measure for the strength of that interaction, is valid for this case. Dislocation growth spirals were not observed on the trigonal {001} face, probably because of the intense 2D nucleation at relative high supersaturations utilized in our experiments. It might be interesting to create lower supersaturation conditions for the trigonal form. At these conditions dislocation growth is expected to compete with 2D nucleation. In this case, a step produced by a screw dislocation would be composed of three (or a multitude of three) d003 layers showing 120° rotated anisotropy. The fast and the slow directions of each d003 layer differ, which would ultimately lead to hindering of the fast layer by the two other, slower layers in any given direction. These would result in an overall slowing reduction of dislocation growth, as is observed for inorganic crystals with faces perpendicular to a 21 screw axis,41 and a transition to growth by 2D nucleation at relatively low supersaturation. 4. Conclusions {101} faces of orthorhombic trypsin crystals grew strictly by a dislocation spiral mechanism. Even at relatively high supersaturations no 2D nucleation was observed. Growth of the much smaller {110} faces proceeded by 2D nucleation in a wide

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Figure 10. (a, c) View along the [001] direction of the crystal contacts between the molecule in position 1 and its nearest neighboring molecules in positions 4 (a) and 5 (b). Bonds A, C, and D are indicated in green. (b, d) Periodic bond chains and connected nets resulting from the crystal contacts in (a, c). Here, the molecules are reduced to centers of mass, bonds are drawn in green. Thicker lines indicate stronger bonds. Both the number and the strength of bonds in the 1-5 connected net containing molecules in positions 1 and 5 are larger than those for the net with molecules in positions 1-4.

TABLE 3: Connected Nets in c Direction

molecules in layer

bonds in net per d003

bonds between nets per d003

no. of contacts in layer (∝ Eslice)

no. of contacts between layers (∝ Eattach)

..1-4..2-6..3-5.. ..5-1..4-2..6-3..

2C 2A + D

2B + 2A + D 2B + 2C

32 166

238 104

supersaturation range. For the {101} face of orthorhombic trypsin, from supersaturation dependencies of tangential step rates and the slope of dislocation spirals, the kinetic coefficient of steps β and the surface free energy R were determined to be respectively β ) (1.1 ( 0.2) 10-4 cm/s, and R ) 5 ( 1 erg/ cm2. This value of R is exceptionally high if compared to other macromolecular crystals. Trigonal trypsin crystals grew strictly by 2D nucleation. The difference in growth mechanism between orthorhombic {101} faces and the faces of the trigonal crystals (i.e., dislocation growth versus 2D nucleation), which is likely caused by a large difference in R and differences in supersaturation for orthorhombic and trigonal crystal forms, is the main cause for the different growth rate of both polymorphs. It was found that growth of the {001} face of trigonal trypsin crystals proceeds by sequential deposition of three growth layers with a height equal to one-third of the unit cell parameter c. These growth layers were related by 3-fold symmetry along the [001] direction, which is caused by the 3-fold screw axis present in the crystal structure. Applying periodic bond chain (PBC) analysis it was shown how the underlying crystal

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