CRYSTAL GROWTH & DESIGN 2002 VOL. 2, NO. 6 475-483
Review Delivered at the Crystal Engineering to Crystal Growth: Design and Function Symposium, ACS 223rd National Meeting, Orlando, Florida, April 7-11, 2002
A Model for Tetragonal Lysozyme Crystal Nucleation and Growth Marc L. Pusey*,† and Arunan Nadarajah‡ Biophysics SD46, NASA/Marshall Space Flight Center, Huntsville, Alabama 35812, and Department of Chemical Engineering, 3056 Nitschke Hall, University of Toledo, Toledo, Ohio 43606 Received April 6, 2002
ABSTRACT: Macromolecular crystallization is a complex process, involving a system that typically has five or more components (macromolecule, water, buffer + counterion, and precipitant). Whereas small molecules have only a few contacts in the crystal lattice, macromolecules generally have 10’s or even 100’s of contacts between molecules. Formation of a consistent, ordered, three-dimensional (3D) structure may be difficult or impossible in the absence of any or presence of too many strong interactions. Further complicating the process is the inherent structural asymmetry of monomeric (single chain) macromolecules. The process of crystal nucleation and growth involves the ordered assembly of growth units into a defined 3D lattice. We propose that tetragonal lysozyme crystal nucleation and growth solutions are highly self-associated and that associated species having 43 helix symmetry are the building blocks for the nucleation process. This solution phase self-association carries over into the crystal growth phase, with the aggregated species as the growth units, recapitulating the nucleation process. The symmetry acquired in solution phase self-association facilitates both nucleation and crystal growth. If this model is correct, then fluids and crystal growth models assuming a strictly monodisperse nutrient solution need to be revised. This model has been developed from experimental evidence based upon face growth rate, atomic force microscopy, and fluorescence energy transfer data for the nucleation and growth of tetragonal lysozyme crystals. Introduction The growth of macromolecular crystals in microgravity has been viewed as a means of realizing higher resolution macromolecule structures and as a potentially valuable process for commercial exploitation. The reduction in gravitational acceleration found in an orbiting spacecraft is posited to result in greatly reduced solution flow due to solute density gradients, over what is found on Earth. Model calculations indicate that density gradient driven flows should arise above a crystal size of 10 µm for the model protein lysozyme and that the typical size for onset is approximately an order * To whom correspondence should be addressed. E-mail: Marc.pusey@ msfc.nasa.gov. † NASA/Marshall Space Flight Center. ‡ University of Toledo.
of magnitude larger in microgravity.1,2 While not strong, typically up to 50 µm/s, these flows have been shown to cause growth cessation in crystals previously not exposed to solution flow. Both the magnitude and the duration of the flow are relevant.3,4 Additionally, the crystal morphology is a significant factor. For the case of lysozyme crystals, the growth of the tetragonal form is flow sensitive whereas the higher temperature orthorhombic form is not,4 suggesting that the flow effects are not at the molecular but at a higher structural level. The question of how flows affect macromolecular crystal growth requires some knowledge of how the crystals grow. One common assumption is that poor quality crystals are due to impurities present in the growth medium5-7 and that flows serve to bring more impurities to the growth interface. Experiments with
10.1021/cg0200107 CCC: $22.00 © 2002 American Chemical Society Published on Web 10/04/2002
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lysozyme have shown that many egg white-derived macromolecules that one would commonly expect to be impurities apparently have little or no effect, even at very high concentrations.8 Other species, most notably the naturally occurring dimer, do have an effect, but whereas crystal quality is affected, the incorporation is not affected by growth in microgravity.9 It has been suggested that microgravity may act as a spatial filter for impurities.7,10-13 The lack of density gradient driven solute flow results in a diffusive field about the crystal, limiting growth to that rate, which can be sustained by the diffusive transport of growth units from the bulk solution. Under such conditions, larger impurities would be “filtered” out by virtue of their having lower diffusivities, although potential impurities smaller than the growth unit, such as cystatin (Mr ∼12 700 Da) would have a diffusive advantage. Truly microheterogeneous impurities, such as lysozyme genetic variants or slight conformational changes, would exactly codiffuse with the monomeric protein and not be filtered out. Experimental data give strong evidence that a lysozyme solution supporting crystal growth is not monodisperse but consists of a series of discretely sized self-associated n-mers of the protein.14-24 These n-mers would be a natural consequence of the self-association process, which leads to the nucleation process. We have postulated that the growth process itself occurs by the addition of specific n-mers to the crystal surface. This has been supported by evidence such as the finding that growth step heights are always two molecules high, with only one of two planes being revealed on the (110) face16,18,25,26, the numerous reports of concentration- and time-dependent self-association in oversaturated and undersaturated solution, modeling studies based upon averaged growth rate data, and direct atomic force microscopy (AFM) measurement of the growth step translational movement upon addition of a unit to the crystal face.16-18 This paper presents a self-association-based model for the nucleation and growth of tetragonal lysozyme crystals, reviews the evidence in support of that model, and suggests an alternative explanation for how microgravity affects macromolecule crystallization in cases where this model holds. Unless specifically mentioned otherwise, the term aggregate refers to a structured assembly (as opposed to an unstructured or amorphous aggregate), which is formed by reversible self-association processes. Starting Assumptions The crystal growth process is one of self-association in an ordered and defined manner. It is logical to assume that the ordered self-association process extends to the nucleation stage or, to put it in proper perspective, that the crystal growth process may be an extension or mechanistic continuation of the nucleation process. Mechanisms that require two distinct processes for nucleation and crystal growth must also incorporate a switching step, with the definition of a critical nucleus then being the point about which this switch is invoked. Unlike “small” molecules, most biological macromolecules have interactions with one or more other molecules as a major aspect of their primary function. In
Review
many cases, these include interactions with other macromolecules, including self-associations. The association state is often a key feature in the expression or regulation of activity or function. Thus, it is important to realize at the outset that participation in associationdissociation interactions is a key feature of macromolecules and that when crystallizing them we are in effect manipulating these interactions. At the simplest level, for a monomeric protein undergoing a defined self-association leading to a dimer
M+MhD
(1)
we define a forward (association) rate constant, k1, for the formation of the dimer (D) by the two monomers (M); a reverse (dissociation) rate constant, k2, for the back reaction where the dimer decomposes; and the net equilibrium for the process, Keq, which is defined as
Keq ) [D]/[M][M]
(2)
Keq ) k1/k2
(3)
Further associations to higher order structures build upon this, giving a pathway for disappearance of dimers to, e.g., trimers
D+MhT
(4)
Keq(trimer) ) [T]/[D][M]
(5)
and so on, with the overall form depending upon the particular association mechanisms, pathways, etc. For the purposes of this discussion, it is sufficient to limit our consideration to the dimerization process only. If crystal nucleation proceeds by the sequential selfassociation of monomeric species, then the first step can be assumed to be a dimerization reaction, and all subsequent steps are assumed to be two body processes, such as dimer + dimer T tetramer, tetramer + monomer T pentamer, monomer + n-mer T n + 1-mer, etc. Processes requiring the simultaneous association of three or more bodies are not likely, and increasingly so for n g 3. In the case of monomeric proteins, this would require that all three molecules come together at the same time, each being correctly oriented about all degrees of rotational and translational freedom. Given the high intermolecular interaction specificity required for protein crystal growth, this is highly unlikely. Keeping the concentration constant and using temperature to control the supersaturation, we find for tetragonal lysozyme that the growth rates progressively increase with decreasing temperature. As lysozyme solubility increases with temperature and decreases with increasing salt concentration, the supersaturation is constantly increasing. However, for a temperaturedependent self-associating system, the actual monomer concentration decreases with decreasing temperature at constant solute concentration. Efforts to derive a linear relationship between the lysozyme monomer concentration and the measured growth rate have not been successful.16,17 The equilibrium distribution between states is concentration-dependent. This results in a progressive shift in the solute n-mer distribution away from the monomer, toward associated species with increasing concen-
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Crystal Growth & Design, Vol. 2, No. 6, 2002 477 Table 1. Crystal Forms of Lysozyme and the Methods by Which They Are Obtained
% of monomeric proteins38
symmetry group
method for lysozyme
refs
36.1 11.1 6.1 5.7 4.9 2.9
P212121, orthorhombic P21, monoclinic C2, monoclinic P43212, tetragonal P3121, trigonal P1, triclinic
higher temps NO3-, I-, SCN-, more chaotropic anions (NH4)2SO4 Cl-, Br-, (NO3-), less chaotropic anions MgSO4, pH ∼8 NO3-
31-34 35-37 39 32, 36, 37 39 36
tration. Note that any subsequent association processes, as would be found in a nucleating solution, would build upon this, with material “passing through” the dimer state and further depleting the monomer concentration. Protein crystallizations are typically carried out using concentrations from ∼10-5 to 5 × 10-3 M solute. A perhaps distinguishing characteristic of macromolecular crystallizations is the large supersaturations often employed. While lysozyme may not be absolutely typical, we find that minimum supersaturations, which are practical, are C/S ∼ 3-4 (C the bulk solution concentration, S the equilibrium solubility concentration), with values as high as C/S ) 70 being employed.27-29 Interestingly, this range narrows and becomes more typical of small molecule crystallizations as one goes to higher (over 100 mg/mL, ∼0.007 M) lysozyme solubilities.30 The crystalline concentration of lysozyme is ∼0.055 M. Protein nucleation is a concentration-dependent selfassociation process, and operations over a large concentration (supersaturation) range mean that there is a correspondingly large range in the association state of the solute. Furthermore, one must bear in mind that the critical transition of a nucleus to an insoluble crystal is only relevant to that particle and does not affect the state of the solution; that is, the event of crystal nucleation does not shut down the self-association process that led to it. Correspondingly, the solubility represents the equilibrium between the solution and the crystalline phases and again has no bearing on events in the solution except as they affect that equilibrium through removal or release of solute. Thus, the occurrence of self-association in an oversaturated solution means that it is also present in an undersaturated solution. Whether the nucleation and growth mechanisms proposed below for tetragonal lysozyme are valid for other crystal forms of the protein, or other proteins, must be determined on a case by case basis. The lysozyme crystal form obtained is dependent upon the precipitating salt species and the temperature (Table 1). Monomeric proteins typically crystallize in a limited number of space groups,38 and lysozyme crystallizes in the six most common forms. Five of these space groups are dependent upon the precipitating anion. For all cases studied so far, simple warming of the protein solution, either in the presence or in the absence of the precipitant, results in the appearance of the orthorhombic form. For five of the crystal forms, then,we look to anion effects for determination of the specific initial intermolecular interactions giving the final crystallographic symmetry. Weak interactions, such as hydrogen-bonding and van der Waals interactions, are too numerous and promiscuous to define crystallographic packing. It is the strong interactions that define the crystallographic packing, and these must be carefully controlled.
Review of the Data A number of methods have been used to study the self-association process in crystallizing protein solutions, including light, neutron, and X-ray scattering, ultracentrifugation, and diffusion measurements. The simplest and perhaps most intuitively easy to understand is the method of dialysis kinetics, where the rate of passage of a macromolecule through a membrane having defined pore sizes is followed to determine the monomeric solute concentration.14,40 The flux of the protein through the membrane was followed by measuring the increase in absorbance, in this case at 280 nm, in the dialysis solution as a function of time. Protein under noncrystallization conditions had a flux rate through the dialysis tubing that was linear with concentration. When crystallization conditions were employed, the flux rate was no longer linear but curved; the rate fell off with increasing concentration. The differences could be clearly observed even in undersaturated solutions, as expected for a concentration-dependent self-association process. Several membrane pore sizes were employed, up to 100 K molecular weight cut off (lysozyme MW ∼14 400), where n-mer sizes up to a hexamer would be expected to pass through freely. In all cases, the use of crystallization conditions showed that there was a progressive self-association occurring with a resultant nonlinear flux rate through the dialysis membrane, indicating that species of MW g 100 000 were present in the solution and that their relative population increased with increasing lysozyme concentration. Wilson et al.41 have recently used the slope of the equilibrium constants derived from dialysis kinetics plotted vs the square root of the ionic concentration, to show that dimer formation would be favored by increasing salt concentration and by pH values above 4.5. They also point out that the latter finding is well-supported by experimental evidence. If supersaturated lysozyme solutions are highly associated, then what is the growth unit concentration? More importantly, what is the growth unit? On the basis of the high supersaturations required for crystal nucleation and growth, electron microscopy, and then AFM measurements, have shown the growth step of the (110) faces to be two molecules high.18,25,42 On the basis of this, we had postulated that the actual growth unit was an octamer.16,17 Recently, the presence of small monomer high growth steps has been shown, attributed to aggregate incorporation from the bulk solvent.43 Interestingly, this report also states that the (101) faces always grew by aggregate addition, the minimum size of which would have to be a tetramer to give a single molecule step height.19 However, regardless of the nature of the growth unit, all solubility data are made with the measured concentrations being of the total soluble protein concentration as monomer, which it
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demonstrably is not. Thus all supersaturations, and indeed all growth unit concentrations based upon measured protein concentrations, are apparent only and not true concentrations. Modeling of the measured lysozyme growth rate data was carried out assuming a simple monomer T dimer T tetramer T octamer T 16-mer distribution, with the growth units being one of these species.16 The growth then occurs by the addition of this growth unit to the crystal face by the screw dislocation growth or the two-dimensional nucleation growth process. The association model and the presumed equilibrium constants were fit to the growth rate data and the enthalpies as derived from the solubility data. The best fits were obtained when octamers were the presumed growth units, while monomers gave the worst fits. Furthermore, when the data were used to predict dialysis kinetics results, a remarkably close fit to the actual data at the same conditions was obtained.40 It was subsequently shown, through AFM measurement of growth unit attachment and detachment from the crystal face, that in fact no single-sized species is the growth unit. Rather, a range of growth units is found, with the apparent sizes being multiples of a tetramer, comprising a single turn about the 43 helix, up to a ∼24-mer. No evidence of single molecule attachment was found.18 However, in light of the recent report describing single molecule regions, we cannot discount them as growth units as well.43 Instead, we suggest that the preferred growth unit size is weighted toward integer multiples of 43 helix-based structures and not structures having n e 3 molecules. Also, high-resolution surface analysis showed that the 43 helices are apparently packed more tightly than expected on the (110) face, based upon the bulk crystallographic structure, and that adjacent surface helices are likely not in contact.44 Current work is now focused on fluorescence resonance energy transfer measurements to obtain the structure of the associated species in solution. The data obtained for the initial dimerization step are consistent with the results expected based upon the corresponding crystallographic structure for a two molecule dimer of the 43 helix, the postulated basis of the growth unit.45 As fluorescence data are typically collected at low fluorescence probe concentrations, e10-5 M, then the average distances between molecules in the solution is 50 nm or more. Resonance energy transfer measurements in dilute solution gave an N-terminal amine to N-terminal amine distance of 4.57 nm, as compared to an expected distance of 4.29 nm based upon the crystallographic structure for two adjacent molecules of the 43 helix. Measurements using fluorescence anisotropy decay to determine the rotational diffusivity of the species in solution also clearly indicate a progressive shift to larger species in the solution with increasing concentration (work in progress). Thus, we find that there is structured solute self-association even in dilute solution, exactly as would be predicted by a progressive self-association process. This also follows the basis for the second virial coefficient measurements of Wilson et al.46,47 Other laboratories have also shown the presence of associated species in solution.20-24,48 Most notably, the
Review
Figure 1. Salt driven hydrophobic interactions, demonstrated by hydrophobic interaction chromatography. A column (1 × 10 cm) is packed with BioGel Methyl HIC resin and equilibrated with the indicated salt solution in 0.1 M sodium acetate buffer, pH 4.6. At t ) 0, a 25 µL aliquot of protein solution is injected onto the column and eluted at a constant flow rate of 1 mL/min. The calculated volume (time) to emergence of the peak is plotted against the salt concentration. Legend: +, NaCl; ), NaBr; 4, (NH4)2SO4.
work of Ataka and co-workers20-24 has shown that species of at least a tetramer in size are formed and that the driving forces are predominately hydrophobic interactions. The role of hydrophobic interactions was not clear at first, particularly since lysozyme is a highly charged protein. However, salt driven hydrophobic interactions can be demonstrated, as shown in Figure 1. Here, a short column of methyl-substituted chromatography media (BioGel Methyl HIC, BioRad, CA) was equilibrated at a given salt concentration. At t ) 0 s, a 25 µL aliquot of protein solution was injected onto the column under constant flow, and the time to peak elution was determined and converted to elution volume. The data were reproducible to (0.02 mL. Progressively higher concentrations of sodium chloride resulted in greater retardation of the lysozyme on the column, i.e., stronger hydrophobic interactions. Sodium bromide has the same effect but at lower concentrations, whereupon a maximum retention value was reached and then plateaued. Ammonium sulfate initially disrupted the slight, presumably buffer driven interactions, reached a minimum, and then caused a progressive increase in retention. Interestingly, lysozyme crystallization from ammonium sulfate was only achieved at salt concentrations below ∼0.6 M, with the best results at ∼0.3 M.39 When the crystallization solution is viewed in light of the above, we can surmise that the role of the salt is to suppress the proteins charge-charge interactions and to promote the weaker hydrophobic interactions. In keeping with the second virial coefficient theory of Wilson et al.,46,47 the net effects are to weaken the strongest and strengthen the weaker interactions to arrive at a state where the protein-solvent interactions are mildly repulsive and the protein-protein interactions are mildly attractive. Overly strong proteinprotein interactions are postulated to lead to rapid desolubilization and formation of amorphous precipitate. Several earlier studies following the enzymatic activities and other properties of lysozyme in solution at neutral pH are also worth mentioning here. The pres-
Review
ence of aggregates in solution is followed through the changes in lysozyme activity in solution, which also allows the structure of the aggregates to be inferred by the progressive blocking of the active site cleft. These studies indicate that the aggregates form “head-to-side” interactions, which correspond to the 43 helices.17,49-54 A number of publications have indicated that observed scattering data are due to nonideal solute behavior and not self-association. In light of the mechanism proposed below, and the experimental evidence that it is based upon, it is important to briefly examine this topic and try to resolve the differences. The study of initial macromolecule interactions with respect to the crystal nucleation and growth process was initiated by Feher.55-57 This work made use of Gibbs58 concepts to form a mechanistic basis for understanding light scattering experiments on nucleating lysozyme solutions. The crystallization process was characterized through the ratio of K∞/K1, where K1 is the dimerization equilibrium constant and K∞ is that for addition of a monomer to the crystal. This ratio was deemed to be useful for distinguishing a structured (crystalline) from a nonstructured (amorphous) aggregation process. In the latter case, subsequent equilibrium constants past the dimer stage would be approximately equivalent to the dimerization equilibrium constant, while for crystallization the free energy barriers of dimerization and crystal growth would be markedly different. While Feher et al. conducted their analysis within the framework of equilibrium constants, Wilson invoked the concept of the importance of the value for the osmotic second virial coefficient, or B22, as a dilute solution parameter for characterizing crystallization conditions.46,47 It was experimentally shown that there is a so-called crystallization slot for the value of B22 and that this corresponded to moderately poor solvation conditions for the macromolecule. The interactions of the macromolecule with the solvent directly affect the equilibria and kinetics of the interactions between macromolecules, and the results showed that B22 values below the crystallization slot led overly strong interactions and amorphous precipitation. Thus, the strength of the macromolecule interactions, as measured in a dilute solution and reflected by a measure of its nonideal behavior, could be related to the outcome in a crystallization experiment. Ducruix et al.59 presented an alternative approach to the analysis of scattering data. They rejected interpretion of their X-ray scattering data in terms of progressive oligomer formation as this would lead to erroneous values for the normalized scattering at high angles. The normalization was based upon the solute concentration, presumed to be all monomeric. This approach was maintained in subsequent work, where the data were used to determine interaction potentials, leading to the conclusion that crystallization is under the control of strong and short range (a few Ångstroms) attractions.60 Thus, while it was recognized that regular contact formation was critical to making crystalline macromolecule assemblies, the role of concentration-dependent self-association as a means of nucleating those assemblies was rejected and the problem was recast solely in terms of solution nonideality and short-range interactions. Short-range, ∼3 Å, attractive potentials are
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invoked to account for salt effects. Note that this distance is approximately that for hydrogen bonds and may reflect the nearer portion of the average separation distance between molecules within a macromolecular crystal. Resolution between the two approaches to explaining solute behavior comes when we realize that the notion that nonideal solute behavior is totally separate from association is not correct and that indeed the relationship between the two concepts has been well-studied. The coefficients for the osmotic pressure virial expansion are parameters definable as physical interactions between molecules.61,62 Wills et al.63 point out that longrange forces are nonassociative, while short-range attractive forces (see above) are due to specific group interactions, thus giving us a basis for the origin of structure in macromolecule crystals (see below). They suggest that the stability of the aggregates formed determines whether they should be treated as separately identifiable species or as a special manifestation of thermodynamic nonideality.63-65 Finally, we note that the protein lysozyme itself has often been a model material for the study of protein self-association.53,66-70 Much of the theory resolving nonideal behavior and solute self-association has been developed using lysozyme data showing a pH dependence of association; see for example, refs 65, 68, and 71. The association of lysozyme in relatively low ionic strength solutions with increasing pH has been well-documented, and it is perhaps reasonable to expect that similar behavior would be part of the crystal nucleation and growth process. Developed Model Most monomeric proteins crystallize in the same few space groups, the most common being, in descending order, P212121, P21, C2, P43212, and P3121, which together account for over 60% of the monomeric protein structures.38 Wukovitz and Yeates38 suggest that the uniqueness of the order of preference is their allowing more rigid body degrees of freedom. They point out that only a few such degrees of freedom are available when first assembling a nucleus, and the internal crystal structure is quickly realized after just a few assembly steps. In the case of the P43212 space group, only two sets of unique interactions are required to form the crystal: one to form the 43 helix and a second to form the helix-helix interactions. Detailed analysis of these interactions has shown that the interaction set responsible for the formation of the 43 helix is by far the stronger of the two.17 On the basis of the above, we can postulate a model for the nucleation and growth process of tetragonal lysozyme. The first step is the protein’s interaction with the added precipitant anion, which binds to both basic side chains and other sites on the protein molecule.72-74 These interactions reduce protein-solvent interactions, shield charged residues, and promote protein-protein interactions, which are in turn accompanied by the subsequent loss of many of the bound anions. Only one anion remains in an intermolecular contact region,72 which is the strong bond required for guiding formation of the 43 helix. Additional anions remain bound at noncrystallographic contact regions. The first step in the assembly process is the association of two monomers to form a dimer, with the nascent
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Figure 2. Proposed helix-helix assembly process. The helices grow along the c-axis by addition to the ends, giving the structure(s) shown in panel a. The arrows show the helix orientation. In panel b, one helix rotates, to give a unit that is “upside down”, as shown in panel c. This enables a linear array of - (leu129 C terminus) and + (lys13 side chain) charges to line up, as shown in panel c, and join together (panel d). At this point, all eight positions of the P43212 unit cell are present. Note that either octamer in panel c can be rotated 90° about the c-axis and, after translation along c, again correctly mate to the other octamer. Note that the (110) faces are perpendicular to the c-axis, while the (101) faces are composed of the exposed ends of the 43 helices.
43 helix dimer being the preferred species. Other species may be formed but may not be competent to continue to an ordered macroscopic structure, i.e., species other than the crystallographic dimer may be formed, and the deciding factor is which, if any, can continue the selfassociation process to the formation of a crystal. Kinetics are important, with overly fast desolubilization potentially leading to less than ideal structured solids, i.e., amorphous precipitates. In some cases, metastable structures are possible. For example, one can grow tetragonal lysozyme crystals from high protein and low nitrate salt concentrations, which within a few days to weeks will redissolve as monoclinic or triclinic crystals appear (work in progress, this lab). Helix growth can be through the addition of monomers, dimerization, or a mixture of methods. Helix elongation is a simple linear growth process, and such isodesmic self-associations have been shown for lysozyme.71 Figure 2 shows a block representation of the 43 helix, which runs in the direction of the c-axis. The side chain amine of Lys13 and the C-terminal carboxyl (Leu129) form a linear alternating array of positive and negative charges. Rotation of a helix by 180° about the c-axis results in a helix running the opposite direction, and this enables the Lys13 amine and Leu129 carboxyl to form symmetry-related double salt bonds that join the two helices together. At this point, all eight molecular positions of the unit cell are present. What is also clear is that the symmetry of the 43 helices assists in this process, as one can rotate any correctly oriented helix by 90° about the c-axis and have an equivalent set of bonds.
Review
To continue the nucleation process, the side-by-side addition of subsequent helices can occur at one of six locations, two of which elongate the planar structure and four of which result in a corner being formed (Figure 3). Note that in this figure each box represents an endon view of a 43 helix, with the colors indicating whether they are oriented in an “up” or “down” direction. Filling of the corner results in a four helix box, with each being bound to two others, and thus, a structure is formed that is likely to be more stable than those preceding it. At this point, any helix addition once again results in formation of a corner, a higher stability binding site. Thus, we find a seamless transition between the nucleation and the crystal growth process, where the primary determinant of whether a given assembly continues to grow or not is the competition between the forward association rate kinetics, the backward dissociation processes, and the overall stability of the particle to that point. The more stable the particle, the more likely it will be to proceed to the next stage of the growth process, and in line with Gibbs theory, there will come a point where the stability is equal to or greater than the time between dissolution events and the particle preferentially grows. It is clear that there are many rate processes involved in protein crystal growth. However, this does not mean that they all occur at the same rate. As mentioned above, the interaction set responsible for the formation of the 43 helix is much stronger than that responsible for the attachment of the aggregate growth unit to the crystal face. This leads to the rapid formation and accumulation in solution of the aggregates corresponding to the 43 helices, followed by slow attachment of the growth units to the crystal face. This can be expressed by the following equation: strong
weak
monomer 9 8 growth units 9 8 crystal interaction interaction Moreover, this means that the attachment of the growth unit to the crystal, that is the crystal growth rate, becomes the rate-determining step. This also means that the relatively rapid and reversible self-association reactions in solution will be at equilibrium during crystal growth. This is the basis for the equilibrium reactions discussed in the previous section during the crystallization process. High-resolution AFM measurements showed that the (110) surface layer was formed by the molecular arrangement corresponding to complete 43 helices only, in agreement with the above mechanism.44 This study also indicated that the surface molecules were more tightly packed about the 43 helix axes than in the bulk crystal. The latter observation is a significant one as it suggests that the lysozyme aggregates in solution may also resemble, but do not exactly correspond to, the structure of 43 helices in the bulk crystal. If these structures were identical, it would suggest that there is no free energy barrier for proteins to crystallize (i.e., they do not require critical nuclei) and that it may not be a first-order phase transition. However, all observations made to date on this phase transition suggest that it is in fact a first-order one, including the existence of free energy barriers for crystallization and critical nuclei. The observation that there is a structural change
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Figure 3. Proposed assembly pathway for lysozyme 43 helices leading to crystal nucleation and subsequent growth. Two different colored squares are used, representing “up”- and “down”-oriented helices. The positions of the symmetry-related salt bridges between the helices are shown as heavy lines. Helix dimerization (A) is shown in greater detail in Figure 2. Addition of a third helix (B) can be at any one of six locations. The four helix structure (C) is more stabilized as each helix has bonds to two neighbors. Helix addition past this point now repeats one of two processes, initiation of a new layer or the filling in of that layer, the crystal growth process (D). At some point (E), interior helices are sufficiently shielded from the bulk solution that they can relax to the final crystallographic coordinates.
in the lysozyme aggregates upon entering the crystalline phase resolves this contradiction, suggesting that despite the aggregation process lysozyme crystallization remains a first-order phase transition. The above result means that the surface layer of molecules is more closely packed along the 43 axes, with subsequent layers gradually relaxing to the bulk crystallographic arrangement. At some point, the interior of the crystal must be buried sufficiently for the required relaxation to proceed. The completeness of the required coverage is not known but may be as few as five helices or as many as nine or more. Ataka et al.75 calculated a critical nuclei size from three to seven. Although this was assuming monomers, the numbers are reasonable if we assume them to be 43 helices instead. In this, we view critical nucleus formation in terms of the number of helices present and not as an absolute number of molecules. The role of salt concentration in helix relaxation is supported by recent osmotic shock studies.76 Soaking of a crystal in a hypertonic salt solution resulted in a linear salt concentration difference-dependent decrease in the a- and b-axes, which are perpendicular to the 43 helices, and had little effect on the c-axis, parallel to the helices. Hypotonic salt solutions had little effect on the crystallographic unit cell dimensions.
Consequences of This Model This model for the nucleation and growth of tetragonal lysozyme crystals involves a significant level of solution phase assembly to occur prior to solute incorporation into a larger structure. This self-association is concentration-dependent, occurs on either side of the solubility concentration, and does not end with the appearance of nuclei. However, the measured soluble concentrations make the assumption that all protein is monomeric and does not reflect the solution distribution of associated species. Thus, the first major consequence of this model is that all supersaturation ratios are apparent and, at best, only approximately reflect the actual supersaturation. The second, and perhaps more interesting consequence, is the effect of solution phase assembly of growth units on the crystal growth process. Assembly by monomers requires that each be correctly positioned about all degrees of rotational and translational freedom as it comes into the attachment site. Molecules that are misoriented when they attach cannot easily adjust in place. The energy barriers between conformations of associated species mean that they cannot rearrange in situ but instead must dissociate and then reassociate.77 All bonds must first be broken to enable an incorrectly positioned molecule to correctly realign. This argument
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also holds for why protein crystals probably do not nucleate by first forming randomly associated species that subsequently rearrange while still in the solid state. By acquiring structure in solution, in this case one or more turns about a 43 helix, one increases the numbers of symmetry-related steering interactions; that is, each will assist in directing the growth unit to a correct attachment orientation. Rotation of the unit 90° about the helix axis results in the same sets of interactions, and thus, symmetry facilitates the growth process. The presence of structured solute in solution may also affect its motion about the growth interface. Lysozymes’ small size and shape preclude significant effects on its orientation when in a flowing solution. However, preassembly of growth units in a crystallizing solution progressively changes these conditions. The overall particle sizes go up, and the axial ratios may become elongated (i.e., an extended 43 helix) or planar (i.e., two or more helices side by side), with a corresponding increase in particle mass. Flow, even the typically slow flow velocities measured about growing lysozyme crystals, may now begin to exert some preferential orientation effects on these structures. In this, depending upon the relative orientations, they may exert a slightly increased impedance or assistance to the growth units orientating to the crystal lattice. The overall effects may not be strong, but only a small percentage of units need to attach misaligned to cause structure-degrading defects. The presence and effects of flow on aggregate orientation are currently under active investigation. Another consequence may be a revision in current thinking about how microgravity affects the macromolecular crystallization process. One current theory is that by the absence of convective flows, the resulting diffusion-controlled transport gives a “filtering” effect, whereby larger impurities diffuse more slowly than monomeric growth units, and are thus partitioned by the diffusive depletion zone.7,10,78 The diffusivity of an impurity dimer is ∼70% that of a monomer, and how this works for impurities smaller than the growth unit is unclear. Whether this difference is sufficient to significantly affect lysozyme dimer incorporation remains to be shown, with evidence showing that it both does not8,9 and does7 in the literature. Regardless, the above theory would only work for a monodisperse (monomeric growth units only) growth solution. Reversible self-association to higher order species, resulting in a polydisperse mixture with one or more of the higher order species being the growth unit, obviates any diffusive impurity partitioning process. However, a derivative of the filtering argument may still be valid. On Earth, solute density gradient driven flows maintain an interfacial concentration approximately equal to that of the bulk solution,1,2 with concomitantly high growth rates. Solute flow may lead to attachment of misoriented growth units, which become trapped by the high growth rate, leading to more mosaic crystals. Growth in microgravity results in a diffusive boundary layer and lower interfacial concentrations and growth rates. Lower concentrations, relative to the bulk, mean that associated species formed there will tend to dissociate as they approach the crystal surface, with the net result being smaller growth units.
Review
This would be particularly true for larger units such as helix dimers and trimers as shown in Figure 3A,B. Lower concentrations will also mean lower growth rates, reducing the possibility that any attached misoriented units become trapped before they can dissociate. Support for this comes from Yoshizaki et al.,79 who presented data indicating that lysozyme crystal quality is affected by increased molecular misorientation at higher supersaturations. Growth by units formed by self-association in the bulk solution has a second effect, which may be particularly relevant to growth in microgravity. Assuming spherical particles, diffusivity is approximately proportional to mass-1/2. Thus, an octamer has eight times the mass but ∼35% the diffusivity of a monomer. When growth is by preformed associated species, on a per particle basis, octameric growth units bring mass to the crystal surface faster than monomers. This makes no assumptions about the relative distribution of octamers vs monomers or the lifetimes of the octamers relative to the diffusive process. Depletion of the boundary layer would be for associated species, in this case octamers, leaving behind the monomers. The interfacial solution would then need to replenish the supply of growth units by association as well as by diffusive flux of material from the bulk solution. Regardless, the shape of the solute concentration gradient about the growing crystal will not be the same as for simple monomer diffusion. Whether tetragonal lysozyme is typical of monomeric proteins or an isolated case still remains to be shown. However, it does have the salient characteristics of monomeric proteins, such as asymmetry in shape and charge distribution. Also, all other monomeric proteins must be induced to overcome the same barriers to crystallization: the balancing of strong and weak attractive and repulsive forces such that a consistent pattern of interactions is obtained, resulting in a crystal. In this regard, it is unlikely that lysozyme is atypical. In many cases, there is evidence that the crystallographic structure contains postulated associated structures of the solution phase. Formation of dimers, trimers, ..., n-mers, etc., has one other effect; once formed, the contact regions are now removed from any subsequent interactions. Acknowledgment. We acknowledge the skillful assistance of Ms. Adrianne Pannell in figure preparation. Support for this work has come from a series of NASA grants over the past decade. The able technical assistance of Ms. Elizabeth Forsythe is also gratefully acknowledged. References (1) Pusey, M. L.; Snyder, R. S.; Naumann, R. J. Biol. Chem. 1986, 261, 6524-6529. (2) Pusey, M.; Naumann, R. J. Cryst. Growth 1986, 76, 593599. (3) Pusey, M. L.; Witherow, W.; Naumann, R. J. Cryst. Growth 1988, 90, 105-111. (4) Nyce, T. A.; Rosenberger, F. J. Cryst. Growth 1991, 110, 52-59. (5) Chernov, A. A.; Komatsu, H. In Science and Technology of Crystal Growth; van der Eerden, J. P., Bruinsma, O. S. L., Eds.; Kluwer Academic Publishers: Dordrecht, Boston, London, 1995; pp 327-353. (6) Caylor, C.; Dobrianov, I.; Lemay, S.; Kimmer, C.; Kriminski, S.; Finkelstein, F.; Zipfel, W.; Webb, W.; Thomas, B. R.; Chernov, A.; Thorne, R. Proteins 1999, 36, 270-281.
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