ARTICLE pubs.acs.org/Langmuir
Effect of PEG End-Group Hydrophobicity on Lysozyme Interactions in Solution Characterized by Light Scattering M. Hamsa Priya,† L. R. Pratt,‡ and M. E. Paulaitis*,† †
William G. Lowrie Department of Chemical and Biomolecular Engineering, Ohio State University, Columbus, Ohio 43210, United States ‡ Department of Chemical and Biomolecular Engineering, Tulane University, New Orleans, Louisiana 70118, United States ABSTRACT: We compare proteinprotein and proteinpolymer osmotic virial coefficients measured by static light scattering for aqueous solutions of lysozyme with low-molecular-weight, hydroxyterminated (hPEG) and methyl-terminated (mPEG) poly(ethylene glycol) at two solution conditions: pH 7.0 and 0.01 M ionic strength, and pH 6.2 and 0.8 M ionic strength. We find that adding PEG to aqueous lysozyme solutions makes a net repulsive contribution to lysozymelysozyme interactions, independent of ionic strength and PEG end-group hydrophobicity. PEG end-group hydrophobicity has a profound effect on the magnitude of this contribution, however, at low ionic strength where mPEG-lysozyme attractive interactions become significant. The enhanced attractions promote mPEG lysozyme preferential interactions at the expense of lysozyme selfinteractions, which leads to lysozymelysozyme interactions that are more repulsive in the presence of mPEG. These preferential interactions also lead to the preferential exclusion of diffusable ions locally around the protein, which results in a pronounced ionic strength dependence of mPEG-mediated lysozymelysozyme interactions.
I. INTRODUCTION Poly(ethylene glycol) (PEG) is a common additive used to manipulate proteinprotein interactions in aqueous solution for the purpose of controlling protein solubility and phase behavior.14 The effect of PEG on proteinprotein interactions is widely accepted to be depletion attraction, an entropically driven process that produces an effective proteinprotein attraction by preferentially excluding the polymer from the interstitial regions between the protein molecules.58 In the primitive model of depletion attraction, the polymerprotein interactions are considered to be purely repulsive, excluded volume interactions.913 However, excluded volumes alone provide only a qualitative description of the effects of PEG.1417 Attractive PEGprotein interactions have also been implicated in analyzing small-angle neutron scattering measurements of PEG-bovine serum albumin osmotic second virial coefficients,18 and static light scattering measurements of PEG-lysozyme osmotic virial coefficients.19,20 Short-range attractions that overcome longerrange steric repulsion have also been reported between streptavidin and grafted, methyl-terminated PEG using the surface force apparatus.21 The attractive interactions, in this case, are attributed to conformational rearrangements of the polymer in the presence of the protein that facilitate attractive contacts between sites on the protein surface and ethylene oxide segments of the polymer. Here, we investigate the effect of PEGprotein attractive interactions on lysozymelysozyme interactions in the r 2011 American Chemical Society
presence of the polymer by comparing osmotic virial coefficients measured for this protein in aqueous solutions with low-molecular-weight, hydroxy-terminated (hPEG) and methyl-terminated (mPEG) poly(ethylene glycol). Previous studies have drawn attention to the hydrophobicity of PEG,22,23 and have shown that end-group hydrophobicity for low-molecular-weight PEG contributes to its aqueous solution behavior and to its partitioning between aqueous and organic phases.2426 Our approach of modifying endgroup hydrophobicity for a given polymer molecular weight allows us to probe the effect of PEGlysozyme attractive interactions on lysozymelysozyme interactions, independent of the polymer molecular weight. An additional contribution of this study is to distinguish the Donnan contribution, which simply accounts for electroneutrality in a multicomponent solution of (poly)electrolytes, from contributions due to proteinprotein and polymerprotein interactions in the osmotic virial coefficients obtained by light scattering from aqueous PEG/protein solutions.
II. LIGHT SCATTERING THEORY The turbidity of a solution that arises solely due to composition fluctuations, expressed in concentration units of component Received: August 11, 2011 Revised: September 27, 2011 Published: September 29, 2011 13713
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molar densities Fi, is given by τ¼
Aij 32π3 n2 kB T ψi ψj 4 jaij j 3λ NA i g 1 j g 1
∑ ∑
ð1Þ
where n is the refractive index of the solution, λ is the wavelength of incident light, kBT is the thermal energy, NA is Avogadro’s number, ψi = ∂n/∂Fi is the derivative of the refractive index with respect to molar density of component i at constant temperature and pressure, |aij| is the determinant of the coefficients ! ! ∂μ~j ∂μ~i aij ¼ ¼ ð2Þ ∂Fj ∂Fi T, P, Fk6¼j
T, P, Fk6¼i
with μ~i the dimensionless chemical potential of component i (μ~i t μi/kBT), and Aij is the cofactor of the element aij.27 The summation in eq 1 excludes the pure solvent (component 0). Consider a four-component solution consisting of water (component 0), NaCl (component 1), lysozyme (component 2), and PEG (component 3). Lysozyme, designated here simply as P, carries a positive charge, z. Therefore, we define the protein component as PClz. All components are electrically neutral. The concentrations of free HO and H+ ions in solution are assumed to be negligible. The chemical potentials of the salt, protein, and polymer, respectively, take the form μ~1 ¼ μ~°1 þ ln F1 þ lnðF1 þ zF2 Þ þ μ~ex 1
ð3Þ
μ~2 ¼ μ~°2 þ ln F2 þ z lnðF1 þ zF2 Þ þ μ~ex 2
ð4Þ
μ~3 ¼ μ~°3 þ ln F3 þ μ~ex 3
ð5Þ
~ex where μ ~°i is a function only of temperature, and μ i is the dimensionless excess chemical potential of component i. The ideal contribution to the chemical potential of the salt and the protein—the second and third terms on the right side of eqs 3 and 4, respectively—is the logarithm of the mean ionic molar concentration of the two components.28 For the protein and polymer components μ~ex i ¼ 2
3
3
∑ Bij Fj þ 3 j, k∑¼ 1 Cijk FjFk j¼1
ð6Þ
The osmotic second and third virial coefficients—Bij and Cijk, respectively—are functions of the potentials of mean force between/among the components designated by the subscripts. The cross coefficients, B23 and C223, are related to the polymer protein and polymer-mediated proteinprotein interactions of interest here. We write the following expression for light scattering from a PEG/lysozyme solution relative to that from the PEG solution without the protein HF2 =Δτ ¼ α þ βF2
ð7Þ
where Δτ is the excess Rayleigh ratio—i.e., difference in Rayleigh ratios for the aqueous polymer solution with and without the protein—and H t (2πn(∂n/∂F2))2/λ4NA is an optical constant. Defining ξi as the ideal contribution to the chemical potential for ~i μ ~ex component i (ξi t μ i ), we make the following reasonable simplifying assumptions for the partial derivatives of the dimensionless chemical potentials with respect to component
concentrations ! ∂μ~i ∂Fj
ξij þ μ~ex ij
ð8Þ
T, P, Fk6¼j
which we represent in matrix form as 2
ξ11 6 6ξ 4 12 0
ξ12 ξ22 þ μ~ex 22 μ~ex 23
3 0 7 7 μ~ex 5 23 ξ33 þ μ~ex 33
Retaining only the linear terms in polymer concentration, these assumptions lead to the following expressions for α and β (eq 7) α ¼ 1 þ 4mB23 F3 β¼
2B22 "
z2 þ 2F1
ð9Þ
!
þ 6ð1 þ 2mÞC223 þ 4mB23 2B22
z2 þ 2F1
!
# 4B223
F3
ð10Þ where m t (∂n/∂F3)/(∂n/∂F2) is the ratio of refractive index increments for PEG and lysozyme. For aqueous protein solutions without the polymer (F3 = 0), the experimentally derived intercept α is unity, and the slope becomes β ¼ 2B22 þ
z2 2F1
ð11Þ
Note that this slope contains the term, z2/2F1, which arises solely due to electroneutrality and conveys no information about intermolecular interactions. The contribution also arises in the osmotic pressure of electrolyte solutions in Donnan equilibrium,29,30 and although no semipermeable membrane is present here, implies that the composition fluctuations considered in eq 1 produce no net charge within a volume defined by the wavelength of light— i.e., electroneutrality holds within this volume, as well as in the electrolyte solution outside this volume.27 Clearly, this Donnan contribution must be taken into account in order to obtain information about the proteinprotein potential of mean force embodied in the second virial coefficient, B22. If polymer is added under the assumption of no net polymer protein interactions; i.e., B23 = 0, then polymer-mediated protein protein interactions arise from the additional polymer concentration-dependent C223 term in eq 10 βðB23 ¼ 0Þ ¼ 2B22 þ
z2 þ 6ð1 þ 2mÞC223 F3 2F1
ð12Þ
The experimental manifestation of these polymer-mediated interactions for a particular polymer concentration is either an increase or decrease in slope, depending on the sign of C223, with the change in slope proportional to the polymer concentration. Moreover, taking the positive Donnan term into account reduces the C223 derived from the experimental value of the slope. When polymerprotein interactions cannot be neglected, additional polymer concentrationdependent contributions to the slope arise " ! # z2 B23 F3 ð13Þ β ¼ βðB23 ¼ 0Þ þ 4B23 m 2B22 þ 2F1 13714
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Figure 1. Zimm plots (eq 14) of light scattering from aqueous solutions containing lysozyme (2) and hPEG (3) as a function of lysozyme concentration at 7.0 pH and 0.01 M ionic strength (including 10 mM Bis-Tris buffer) adjusted using NaCl relative to light scattering from hPEG solutions at the same solution conditions. Symbols correspond to hPEG concentrations (bottom to top) of 0, 30, 50, 70, 90, and 110 mg/mL. Lines are linear fits to the data. The data at each hPEG concentration, starting with 30 mg/mL, are shifted upward by 2 units for visual clarity.
Assuming a moderately negative B22, as one would expect for favorable protein crystallization conditions,31 and net attractive polymerprotein interactions, such that B23 < 0, these additional contributions (excluding the additional Donnan term) will increase or decrease β, depending on the relative magnitude of the protein protein and polymerprotein interactions, as expected. Interestingly, the additional Donnan term is proportional to the polymer concentration, as well as the magnitude of polymerprotein interactions, such that the overall Donnan contribution to the slope β is z2 1 þ 4mB23 F3 2F1 The physical interpretation of this polymer concentration dependence is that adding polymer displaces diffusable ions from some “local” volume around the protein when the polymerprotein interactions are attractive, which affects the ion binding equilibria. It is customary to report light scattering measurements in concentration units of component mass densities, ci = FiMi, with Mi the molecular weight of component i. In such cases, the Zimm plot (eq 7) takes the form Hc2 =Δτ ¼ α0 þ β0 c2
ð14Þ
where α0 ¼ 1=M2 þ 4mB23 c3
ð15Þ
and β0 ¼
2B22 þ "
z2 M1 2c1 M22
!
þ 6ð1 þ 2mÞC223 þ 4mB23 M2 2B22 þ
z2 M1 2c1 M22
!
Figure 2. Zimm plots (eq 14) of light scattering from aqueous solutions containing lysozyme (2) and mPEG (3) as a function of lysozyme concentration at 7.0 pH and 0.01 M ionic strength (including 10 mM Bis-Tris buffer) adjusted using NaCl relative to light scattering from mPEG solutions at the same solution conditions. Symbols correspond to mPEG concentrations (bottom to top) of 0, 30, 50, 70, 90, and 110 mg/ mL. Lines are linear fits to the data. The data at each mPEG concentration, starting with 30 mg/mL, are shifted upward by 2 units for visual clarity. used without further purification. The solution pH was maintained at 6.2 and 7.0 using sodium phosphate buffer and bis-Tris buffer, respectively. The ionic strength was adjusted to 0.01 and 0.8 M (including buffer ions) using NaCl. All solutions were prepared with filtered deionized water from a Barnstead Nanopure Ultraviolet water filter system. Buffer solutions were filtered using a Millipore vacuum filtration unit with a 0.1 μm PVDF membrane. Stock solutions contained ∼4 mg/mL of lysozyme at different PEG concentrations ranging from 0 to 110 mg/mL. To minimize protein aggregation, the protein solutions were prepared less than two hours before each experiment. Prior to an experiment, the solutions were filtered using a 0.22 μm Millipore syringe top filter, and centrifuged at 3500 g for 15 min to remove any microscopic air bubbles. All experiments were carried out at 25 °C. Scattered light was collected at an angle of 90° using miniDawn Treos (Wyatt Technology, CA) equipped with a GaAs laser, operating at a wavelength of 658 nm. Protein concentrations were measured using Varian Prostar 325 UV spectrophotometer. UV absorption of lysozyme solutions was measured at 280 nm, and the protein concentration was determined using the extinction coefficient of 2.64 L/g cm.32,33 The solution composition during the experiment was controlled by adjusting the relative injection volumes of PEGlysozyme solution and PEG solution in different 1 mL syringe pumps of a Calypso composition gradient pump system. Before each experiment, the system was flushed with buffer solution for two hours, followed by flushing with the PEG solution for an additional half hour until a constant baseline in the lightscattering detector was observed. Refractive index increments (∂n/∂c3) of 0.131 mL/g and 0.126 mL/g were measured for hPEG and mPEG, respectively, using a Rudolph J57 Automatic Refractometer. A refractive index increment (∂n/∂c2) of 0.185 mL/g was used for lysozyme.19
# 4B223 M3 c3
ð16Þ
III. MATERIALS AND METHODS Hen egg white lysozyme (3 crystallized, L6876), hPEG (Mn = 1000), mPEG (Mn = 1000) were purchased from Sigma Aldrich and
IV. RESULTS AND DISCUSSION Zimm plots (eq 14) at different constant concentrations of hPEG and mPEG are shown in Figures 1 and 2, respectively, for aqueous lysozyme solutions at 7.0 pH and 0.01 M ionic strength. Each data point shown in these plots is an average of five independent measurements. All least-squares fitting and analyses of the corresponding statistical uncertainties were carried out using the statistical software JMP 9.0. The lysozyme osmotic second virial coefficient, B22, obtained from the slope of the 13715
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Table 1. Osmotic Virial Coefficient B22, Cross Coefficients B23 and C223, and the Donnan Contribution,a z2M1/2c1M22, for Aqueous Solutions of Lysozyme (2) and Hydroxy-Terminated (hPEG) or Methyl-Terminated (mPEG) Poly(ethylene glycol) (3) at Ionic Strengths of 0.01 and 0.8 M NaCl (1), and T = 25 °C polymer
ionic strength
(z2M1)/(2c1M22) 104 2
B22 104 2
mol mL/g
B23 104 2
terminal group
M NaCl
mol mL/g
mol mL/g
hPEG
0.01
156.5
50.4 ( 0.6
∼0
hPEG
0.01
156.5
19.8 b
—
hPEG hPEG
0.8 0.8
mPEG
0.01
mPEG
0.8
C223 104 mol ml2/g3
ref
hPEG-Lysozyme Solutions
8.8 ( 0.9 4.5 ( 0.3
2.0 2.0
3.3 ( 1.3 12.9 ( 1.2
35.9 ( 22.6 — 18.6 ( 2.7 16.1 ( 1.3
this work 33 this work 19
mPEG-Lysozyme Solutions
a
156.5
50.4 ( 0.6
2.0
8.8 ( 0.9
10.3 ( 2.8 ∼0
180.9 ( 10.4
this work
13.7 ( 10.3
this work
Charge z carried by lysozyme is obtained from titration experiments.41 b Interpolated from data at ionic strengths of 0.007 and 0.05 M.
Zimm plot in the absence of PEG is reported in Table 1. Linear fits of the data at each PEG concentration give the cross virial coefficients, B23 and C223, from the polymer concentration dependence of the intercepts (eq 15) and slopes (eq 16), respectively, of the Zimm plots. The uncertainty in C223 is computed using a propagation of uncertainty analysis. These virial coefficients are also reported in Table 1. The observed linear dependence on protein concentration for the lysozyme and PEG concentrations considered justifies retaining only the linear term in eq 14. The greater scatter in the Zimm plot at the highest mPEG concentration (Figure 2) is attributed to the high viscosity of the polymer solution at this mPEG concentration. The solution viscosity affects mixing and equilibration times, as well as the time required to thoroughly purge the sample cell between measurements. Although these times could be increased accordingly, the polymer solution viscosity in essence sets a practical upper limit to the polymer concentrations that can be studied. The PEG concentrations considered here are also below the crossover concentration from dilute to semidilute polymer solution behavior; thus, the nonmonotonic PEG concentration dependence of the slope β0 (eq 16) observed by others20 for aqueous lysozyme solutions at higher PEG concentrations can be ruled out. The molecular weight of lysozyme obtained from the intercept of the Zimm plot in the absence of PEG is 14 434 Da, in agreement with the molecular weight calculated from the amino acid sequence: 14 300 Da. Note that this Zimm plot for c3 = 0 in Figures 1 and 2 has a positive slope, but the osmotic second virial coefficient for lysozyme derived from the slope is negative, indicating net attractive proteinprotein interactions. The difference is attributed to the Donnan term, z2M1/2c1M22, which is large and also positive at the low ionic strength (Table 1). The B22 derived here is also more negative than the lysozyme osmotic second virial coefficient obtained from previous light scattering measurements at the same pH and low ionic strengths, using the same bis-Tris buffer.33 This difference is attributed to differences in the lysozyme supplied by Sigma Aldrich—the specific product line used in the earlier study was discontinued—as well as lot to lot variations within the same product line. The lysozyme used in the earlier study was also supplied as an essentially salt-free lyophilized powder, whereas the lysozyme used in this study was supplied as a dialyzed and lyophilized powder containing small, but unspecified quantities of buffer salts. Larger amounts of dissolved ions in the lysozyme
solution would lead to a B22 that is more negative, especially at low ionic strength, as observed here. The corresponding Zimm plot in the absence of PEG at 6.2 pH and 0.8 M ionic strength (not shown) gives a lysozyme molecular weight of 13 977 Da and an osmotic second virial coefficient for lysozyme (Table 1) that is likewise negative, but smaller in magnitude, in agreement with previously reported measurements of the lysozyme osmotic second virial coefficient at the same pH and ionic strength, using the same sodium phosphate buffer.19 The Donnan term at this high ionic strength (Table 1) is also much smaller in magnitude. Osmotic cross virial coefficients obtained from the polymer concentration dependence of the Zimm plots are also compiled in Table 1. At the high ionic strength, the small positive B23 for hPEG-lysozyme and B23 ≈ 0 for mPEGlysozyme solutions reflect the weak polymer concentration dependence of the corresponding intercepts (eq 15), while the small positive values of C223 for both PEGlysozyme solutions reflect the weak polymer concentration dependence of the corresponding slopes (eq 16). Second and third cross virial coefficients for the hPEGlysozyme solutions at high ionic strength are also compared with virial coefficients reported in an earlier, similar light scattering study.19 The B23 obtained in both studies is positive, indicating net repulsive polymerprotein interactions, although the value derived here is much smaller in magnitude. The small, positive C223 obtained here is within the experimental uncertainty of the previously reported value. At low ionic strength, the strong mPEG concentration dependence stands out in contrast to the weak dependence on hPEG concentration. The difference is reflected in both the negative B23 for mPEG compared to B23 ≈ 0 for hPEG, indicating greater attractive polymerprotein interactions for the methyl-terminated polymer, and the large, positive C223 for mPEGlysozyme solutions compared to the hPEGlysozyme solutions. The positive values for C223 indicate that the addition of either hPEG or mPEG to aqueous lysozyme solutions stabilizes protein dissolution by making a net repulsive contribution to lysozymelysozyme interactions in the presence of the polymer. This stabilizing effect is opposite to that predicted by the primitive model of depletion attraction. The magnitude of this effect, however, is much more pronounced for mPEG at low ionic strength, which is distinguished from hPEG at this ionic strength by greater mPEGlysozyme attractive interactions (B23 < 0). 13716
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Langmuir The enhanced mPEGlysozyme attractions are attributed to hydrophobic interactions between the mPEG end-groups and hydrophobic regions on the solvent accessible surface of the protein.34 The hydrophobic attractions, in turn, promote mPEGlysozyme preferential interactions. The preferential exclusion of lysozyme, and consequently, lysozymelysozyme interactions that are more repulsive, follows directly from the enhancement of mPEGlysozyme preferential interactions. The effect of polymer end-group hydrophobicity on PEGlysozyme interactions is not surprising in light of experimental evidence showing that polymer partitioning between water and organic phases is sensitive to end-group hydrophobicity for PEG molecular weights less than 2000 Da.2426 The pronounced effect of polymer end-group hydrophobicity on PEG-mediated lysozyme lysozyme interactions is attributed to the spatial heterogeneity of PEGlysozyme preferential (hydrophobic) interactions across the surface of the protein. We speculate that mPEGlysozyme preferential interactions involve local interactions with specific, solvent-accessible hydrophobic regions on the protein surface that substantially alters the configurational complementarity of important, short-range proteinprotein interactions.35,36 Since these short-range, preferential hydrophobic interactions are favorable, the loss of configurational complementarity, which will be greater in the presence of the methyl-terminated polymer, effectively reduces lysozymelysozyme attractive interactions. Preferential interactions between mPEG and lysozyme also leads to the preferential exclusion of diffusable ions from some “local” volume around the protein, as suggested by a Donnan contribution z 2 M1 ½1 þ 4mB23 M2 c3 2c1 M22 that depends on the polymer concentration and polymer protein interactions through B23. This contribution depends on ionic strength directly through the salt concentration, c1, and indirectly through the effect of ionic strength on B23. Thus, increasing mPEGlysozyme attractive interactions for a given ionic strength reduces the Donnan contribution, which in effect is equivalent to reducing the charge carried by the protein. The ionic strength dependence of the Donnan contribution is manifested in the effect of ionic strength on C223 for mPEG compared to hPEG (Table 1). Increasing ionic strength reduces the impact of adding either polymer to the aqueous protein solution, as expected,17 but the effect is much more pronounced for mPEG. Specifically, C223 decreases by more than an order of magnitude with increasing ionic strength for the mPEG lysozyme aqueous solutions, compared to a smaller decrease in C223 of only a factor of 2 with increasing ionic strength for the hPEGlysozyme aqueous solutions. The significantly greater ionic strength dependence obtained for mPEG is related directly to the enhanced mPEGlysozyme attractive interactions. Other potential contributing factors, such as the formation of mPEG clusters produced by interchain hydrophobic interactions, can be ruled out, since our measurements of light scattering from the aqueous polymer solutions detected no clustering for either hPEG or mPEG, as has been reported for higher-molecular-weight PEG.37 Loop closure due to intrachain hydrophobic interactions between the mPEG end-groups is also expected to be negligible for these low-molecular-weight polymers.38
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As a practical matter, other PEG end-group functionalities with different, weak (i.e., noncovalent) affinities for specific sites on protein surfaces, including mixtures of low-molecular-weight PEGs with multiple end-group functionalities, may enable “finetuning” of proteinprotein interactions as a means to control phase behavior. Manipulating hydrophobic interactions alone— for example, by using bulky fluorinated end-groups39,40—is one interesting possibility for directing site-specific PEGprotein hydrophobic interactions.
V. CONCLUSIONS The effect of PEG end-group hydrophobicity on lysozyme interactions in aqueous solution is to induce attractive intermolecular interactions between the protein and the methylterminated polymer that are not observed for the hydroxyterminated polymer. The stronger mPEGlysozyme attractions are attributed to hydrophobic interactions between the polymer end-groups and hydrophobic patches on the surface of the protein. These attractions promote preferential interactions between lysozyme and mPEG at the expense of lysozyme selfinteractions, which leads to lysozymelysozyme interactions that are more repulsive in the presence of the methyl-terminated polymer. Preferential interactions between mPEG and lysozyme also lead to the preferential exclusion of diffusable ions locally around the protein. Thus, an additional effect of PEG end-group hydrophobicity is to enhance the ionic strength dependence of PEGmediated lysozymelysozyme interactions. Specifically, reducing the ionic strength has a much greater impact on the solution thermodynamics for aqueous lysozyme solutions containing mPEG compared to hPEG. Finally, our analysis of polymer-mediated proteinprotein interactions emphasizes the importance of accounting for the Donnan contribution to proteinprotein and proteinpolymer osmotic virial coefficients derived from light scattering measurements. Distinguishing this contribution from contributions related to intermolecular interactions was recognized in our earlier light scattering studies of proteinprotein interactions33,36 as essential to modeling the interaction part of the osmotic second virial coefficient for proteins in aqueous solution. We extended that analysis here to interpret the osmotic cross virial coefficients in terms of proteinpolymer interactions and the effect of those interactions on proteinprotein interactions in the presence of the polymer. ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT Financial support from the National Science Foundation (BES-0555281) and the Department of Energy (DE-FG0204ER25626) is gratefully acknowledged. We also thank Dilip Asthagiri (Johns Hopkins) and David Vanderah (NIST) for their helpful comments on the work. ’ REFERENCES (1) McPherson, A. Introduction to protein crystaliization. Methods 2004, 34, 254–265. 13717
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