Rationalizing Protein Crystallization Screenings through Water

CRYSTAL GROWTH & DESIGN

Rationalizing Protein Crystallization Screenings through Water Equilibration Theory and Protein Solubility Data† P. M. Martins,*,#,§ J. Pessoa,# Z. Sa`rka`ny,# F. Rocha,§ and A. M. Damas#,⊥ Grupo de Estrutura Molecular, IBMC - Instituto de Biologia Molecular e Celular, Rua do Campo Alegre, 823, 4150-180 Porto, Portugal, Departamento de Engenharia Quı´mica, Faculdade de Engenharia, UniVersidade do Porto, Porto, Portugal, and ICBAS - Instituto de Cieˆncias Biome´dicas Abel Salazar, UniVersidade do Porto, Porto, Portugal

2008 VOL. 8, NO. 12 4233–4243

ReceiVed June 30, 2008; ReVised Manuscript ReceiVed October 13, 2008

ABSTRACT: This work combined water equilibration fundamentals of vapor diffusion crystallization techniques with protein solubility data in order to obtain the variation of protein supersaturation throughout the protein crystallization assays. Once the supersaturation build up profiles (SBUPs) are known, the wide screening space of crystallization conditions is reduced to the key variable for crystal formation and growth, which is supersaturation and its variation with time. Our previous water equilibration model was expanded to include the case of drop evaporation at constant contact area during the hanging drop method. Crystallization experiments of lysozyme were performed under different experimental conditions and the results were interpreted according to the respective SBPUs. In particular, the number and size of the crystals were evaluated at the moment of the SBUP that corresponded to their formation. Following this methodology, two nucleation behaviors were identified depending on the supersaturation levels at which crystal formation occurs. These behaviors, which are believed to be closely linked with the diffracting properties of the crystals, are dictated not only by classic thermodynamic and kinetic factors affecting crystallization and water equilibration, but also by phenomena related to the drop preparation procedures. Introduction The wide screening space of crystallization conditions is recurrently expanded by new methodologies and techniques of growing crystals for the determination of the three-dimensional structure of proteins.1-3 Because of the countless possibilities involved, the selection of the experimental conditions providing good diffracting crystals is frequently a trial and error process.4 An alternative approach to this problem involves the use of phase diagrams and protein solubility curves to optimize the crystallization parameters. The so-called rational approach has been successfully applied by diluting the crystallization drops toward the lower regions of the metastable zone.5 In another example, extensive determinations of protein phase diagrams using microfluidics significantly improved the crystallization results relatively to conventional automation strategies.6 Phase diagrams can be very useful to place the crystallization experiment relatively to the solubility curve,7 but they lack information about how fast the system moves across the diagram. It is known that the number, size, and diffracting properties of the protein crystals are strongly affected by the supersaturation at which nucleation and crystal growth occur. This stands for the absolute value of supersaturation, defined as the quotient of the protein concentration and its solubility (S ) c/c*), and for the rate at which supersaturation evolves (dS/ dt).8 In the case of the most commonly employed techniques of biological macromolecules crystallization, this rate is strongly influenced by water evaporation and by the subsequent step of vapor diffusion across the gas phase. A good understanding of the solvent evaporation kinetics is therefore essential for a † Part of the special issue (Vol 8, issue 12) on the 12th International Conference on the Crystallization of Biological Macromolecules, Cancun, Mexico, May 6-9, 2008. * To whom correspondence should be addressed. E-mail: pmartins@ ibmc.up.pt. # Grupo de Estrutura Molecular, IBMC. § Departamento de Engenharia Quı´mica, Faculdade de Engenharia, Universidade do Porto. ⊥ ICBAS, Universidade do Porto.

rational approach to vapor diffusion techniques. As a result, a comprehensive mathematical treatment describing the hanging drop method was proposed for the first time in 1988,9 and was followed by two other models during the 1990s.10,11 Recently, aspects responsible for discrepancies between the existing theories and the measured evaporation kinetics were successfully addressed in a new model,12 which is briefly described below. In the present work, the applicability of this model is expanded in order to include the case of drop evaporation at constant contact area. Then, the practical application of the water equilibration model is demonstrated by estimating the theoretical supersaturation profiles throughout crystallization assays. Such procedure is believed to be a valuable tool to handle the combined influence of many of the variables constituting the crystallization screening space. A first attempt is made to establish the relationship between the outcome of crystallization experiments and the corresponding supersaturation build up profiles. Finally, the major guidelines of a possible rational approach to crystallization are given with a basis on the presented theory. Water Equilibration Model.12 Figure 1 shows the main geometrical parameters of the water equilibration model previously described by our group. The drop is considered to be a spherical cap centered on the point O, with a radius of curvature R and contact angle RR. The inner diameter of the reservoir and the vertical distance from the coverslip to the solution in the reservoir are represented in Figure 1 by 2a and b, respectively, so that from trigonometric rules one obtains:

Ra ) √a2 + (R sin RR)2

(1)

Rb ) b + R sin RR

(2)

The molar rate of water vapor leaving the droplet (I1) is evaluated at r ) R according to Fick’s first law:

10.1021/cg8006958 CCC: $40.75  2008 American Chemical Society Published on Web 11/07/2008

4234 Crystal Growth & Design, Vol. 8, No. 12, 2008

ΩAR2D dp I1 ) RT dr

Martins et al.

( )

(3)

r)R

In this equation, ΩA is the surface area shape factor of the droplet (2π(1 - sin RR)), D is the diffusion coefficient of water vapor in air, R is the gas constant, and T is the absolute temperature. The derivative dp/dr can be calculated from the vapor pressure profile p(r) along the air space of the crystallization chamber (not shown here). The molar rate I1 can also be expressed in terms of the variation of the drop volume with time, dV/dt, normalized by the molar volume of pure water, j 1: V

I1 ) -

1 dV j 1 dt V

(4)

If the evaporation rate assumed to be exclusively determined by the vapor diffusion step across the air space, the evolution of the drop volume with time would result from the equality given by eqs 3 and 4. However, experimental data on water equilibration rates during the hanging drop method indicate that the mass transfer resistance within the drop is also significant for the whole evaporation process.12 Depending on the magnitude of the mass transfer coefficient of water in the solution (kc) a concentration gradient is established in the interior of the drop. The relationship between the precipitant molar fractions at the interface (x2i) and in the bulk of the drop (x2d) can also be expressed as a function of the current I1:

the contact angle RR. The role of temperature is explicitly given in terms of the variable T and, in an implicit way, in the values of the diffusion coefficient D and of the vapor pressure of pure water p*. Equation 7 shows that the evaporation process will be faster for higher molar fractions of precipitant in the reservoir (x2b) and higher water activity coefficients in solution (γ1) since they correspond to lower values of the time constant. Among the constants present in the equilibration curve (eq 6), yc is the one related to the geometry of the crystallization chamber. Different definitions of yc arise depending on the droplet-toreservoir distance and its relation to the reservoir diameter; in the case of droplet-to-reservoir distances small enough to have Ra g Rb, one obtains

yc ) -

Rb ln(1 - sin RR) R0 sin RR

whereas for Ra e Rb, yc ) -

R2a(1 + sin Ra) ln(1 - sin RR) ⁄ (R0 sin RR) 2Ra(1 + sin Ra) - Rb(1 + sin Rb) + Ra(arcsin(a ⁄ Ra) - arcsin(a ⁄ Rb)) (9)

The angles Ra and Rb are represented in Figure 1 and result from the following trigonometric relationship evaluated for r ) Ra and r ) Rb, respectively:

sin R )

2

I1 )

ΩAR kc (x2i - x2d) j1 V

(5)

This relationship is important for the vapor pressure profile in the gas phase, when defining the boundary condition at r ) R. Equations 3, 4, and 5 provide three definitions of I1 that can be combined together to obtain the ordinary differential equation (ODE) of the variation of the droplet size with time. As it happens in previous models, the contact angle RR is assumed to remain constant as the drop evaporates;9 this assumption will be discussed later taking into account evaporation rate measurements. For the present case, the dimensionless solution of the equilibration curve is

t 1 1 ) - (1 - y3) + (1 - y2) + (1 - y)β τ 3yc 2

(

)

(

)

(

(

)

)

2 (y2 + yy∞ + y∞2 )(1 - y∞)2 β y∞ + ln y∞ 6 (1 + y∞ + y∞2 )(y - y∞)2

(

y∞3 (y2 + yy∞ + y∞2 )(y - y∞) ln 3yc (1 + y∞ + y∞2 )(1 - y∞)

)

3ΩVRTR20 (1 - sin RR) ln(1 - sin RR) τ)sin RR j Ω Dγ p∗x V 1

2

(10)

To conclude the list of variables with a role in the water equilibration kinetics, the constant y∞ in eq 6 is related to the dilution factor when preparing the drop:

y∞ )

( )

R∞ x2,0 ) R0 x2b

1⁄3

(11)

where x2,0 is the initial molar fraction of the precipitant in the drop; and finally the constant β measures the relative weight of the mass transfer resistance within the drop:

β)

3ΩV R0 ΩA x2bkcτ

(12)

Experimental Section

(6)

The size of the drop is here expressed in terms of the dimensionless radius y ) R/R0, where R0 is the drop size at the beginning of the experiment (t ) 0). The constants τ, yc, y∞ and β embody the effect of the several operational variables on the variation of y with time. The definition of the time constant τ,

A

√ ( ar ) 1-

When the theoretical equilibration model was presented, it was envisaged that the next step should be the determination of the supersaturation progress during the hanging drop method with the goal of identifying the supersaturation profiles that lead to high quality protein crystals.12 In compliance with this methodology, we report here recent results obtained during crystallization experiments of lysozyme.

√3(y - 1)y∞ √3 β 2 1y arctan 2 + 3 y∞ ∞ 2y∞ + y∞(y + 1) + 2y 1+

(8)

(7)

2b 1

includes the parameters related with the drop shape, such as the volume and surface area shape factors (ΩV and ΩA), and

Materials. Egg-white lysozyme was purchased from Merck (lot K29564981 205) and was used without further purification. In the preparation of the sodium acetate buffer, sodium acetate trihydrate from Fluka (Microselect, >99.5%) and glacial acetic acid from Merck were used. Sodium chloride (NaCl) was obtained from Merck (pro analysi). The three stock solutions were (i) 0.2 M sodium acetate buffer at pH 4.7 filtered through 0.2 µm cellulose acetate filters, (ii) 50 mg/mL lysozyme in the 0.2 M sodium acetate buffer, and (iii) 10% (w/v) sodium chloride in the 0.2 M sodium acetate buffer. Evaporation Rate Measurements. Two methods were developed for the evaporation rate measurement in our vapor diffusion experiments. In the first method, the evaporation of a hanging drop was followed using a Nikon SMZ800 stereomicroscope equipped with a Nikon DS-5M digital camera. Top-view images of the drop-coverslip interface were periodically captured and then analyzed in order to determine the liquid-solid contact area and contact radius. An image

Rationalizing Protein Crystallization Screenings

Crystal Growth & Design, Vol. 8, No. 12, 2008 4235 Table 1. Experimental Details of the Hanging Drop Method Used on the Crystallization of Lysozyme plate #

T (°C)

dilution factor

coverslip material

drop shape

1 2 3 4 5

20 4 20 4 20

1:1 1:1 1:1 1:3 1:1

siliconized glass siliconized glass plastic siliconized glass siliconized glass

regular regular regular regular elongated

6 µL of the 50 mg/mL lysozyme solution to 2 µL of the reservoir solution. In plate 5, the typically round drops were spread over the coverslip using the pipet tip to induce an ellipsoidal shape. In the modified microbatch method, the drops were prepared with 4 µL of a NaCl solution twice as concentrated as the reservoir solution, plus 4 µL of the 50 mg/mL lysozyme solution. This way, drop and reservoir solutions have the same precipitant concentration and the evaporation of the drop is avoided. Both in the hanging drop method and in the modified microbatch method, triplicate experiments were carried out for each precipitant concentration. All the plates were periodically checked for the presence of crystals, in order to determine their number and when were they formed. A Nikon SMZ800 stereomicroscope equipped with a Nikon DS-5M digital camera was used. The checking frequency was higher at the beginning of the experiments (∼2 in 2 h during the day followed by an 8 h interval overnight) and gradually decreased to twice a day, 17 days after their start. A final verification of the drops was done 3 months after their preparation.

Results and Discussion Figure 1. Representation of the hanging drop method as considered in the mathematical model derivation. analysis package (Visilog 5.4, Noesis, les Ulis, France) was used, following the algorithm previously employed on the determination of mean crystal sizes in the industrial sugar crystallization.13 If the drop shape can be approximated to either a spherical cap or an ellipsoidal cap, then its volume is proportional to the corresponding volume shape factor and to the third power of the contact radius rd.14 Estimations of the variation of V/V0 with time can therefore be calculated from the obtained values of (rd/rd,0)3, provided that the drop contact angle remains constant. Because of its simplicity, this method could be applied in situ to the hanging drop crystallization setup described below. The reservoir solution consisted of 1 mL of 3% (w/v) NaCl in sodium acetate buffer. The effect of the presence of protein on the evaporation rates was investigated by preparing 8 µL drops with 4 µL of the reservoir solution plus (i) 4 µL of the 50 mg/ml lysozyme solution in sodium acetate buffer or (ii) 4 µL of the sodium acetate buffer alone. Experiments were conducted in controlled-temperature rooms at 4 and 20 °C. In all cases the drops showed high circularity, but, as it will be discussed later, the condition of constant contact angle evaporation was not always verified. For that reason, a second method was developed in which the drop evaporation is followed sideways using a videobased optical contact angle meter (OCA 15, Dataphysics Instruments GmbH). The volume and contact angle of the drop are estimated by axisymmetric drop shape analysis using an ellipse fit. Hanging drops were set up on 18 mm siliconized coverslips (Hampton Research, HR3239) over 4.4 mL of the reservoir solution (4.5% (w/v) NaCl in sodium acetate buffer) in a 20 × 20 × 20 mm optical cuvette sealed with clear tape. As in the top-view method, the drops were prepared with 4 µL of the reservoir solution plus (i) 4 µL of the 50 mg/mL lysozyme solution or (ii) 4 µL of the sodium acetate buffer alone. In the sideview method, the experiments were conducted in a temperaturecontrolled room at 20.7 °C. Crystallization Experiments. Lysozyme crystals were produced at 4 and 20 °C by the hanging drop method and a modified microbatch method, using different concentrations of the precipitant (NaCl). In both methods, 24-well VDX plates (Hampton Research) and 22 mm siliconized coverslips (Hampton Research, HR3-231) were used. The reservoir solutions contained different concentrations of NaCl in 0.2 M sodium acetate buffer at pH 4.7. In the hanging drop method, the drops were prepared by adding 4 µL of the 50 mg/mL lysozyme solution to 4 µL of the reservoir solution. Variations to this procedure are summarized in Table 1. For example, in plate 3 the siliconized coverslips were replaced by 22 mm plastic coverslips (Hampton Research, HR8-082), and in plate 4, the drops were prepared by adding

Evaporation Rates. Figure 2 illustrates the different types of results obtained by the top-view method. When the drops evaporated in the absence of protein at 20 °C (Figure 2a), the evolution of the dimensionless volume followed the expected behavior for 1:1 dilution factors, with the inferred volume at the equilibrium being 1/2 of the one at the beginning of the experiment. Moreover, the experimental results show a good reproducibility and confirm the theoretical equilibration curve predicted by eq 6, using the set of parameters listed in Table 2 for 293 K and VDX plates. The measured value of the contact angle RR was close to 0°. The experimental results of Figures 2b,c show a lower reproducibility than in the case of Figure 2a, and the behavior of the measured equilibration curves is not consistent with the water equilibration theory. The explanation for this lies in the limitations of the top-view method in providing reliable volume estimations under the different modes by which drops evaporate. As it was described, the application of this method is limited to the cases where evaporation takes place at a constant contact angle. In the experiment at 4 °C (Figure 2b), as the drops evaporate they become more flattened, which makes the topview method unsuitable to be applied. Likewise, Figure 2c shows that in the presence of lysozyme the drop contact area remained constant after an eventual increase in the first equilibration moments, and so the drop volume decrease can only be explained by the variation of its contact angle. It is well-known from the literature on interface science that drops evaporate according to two pure modes: one at constant contact angle (CCAn), with a decreasing contact area, and the other at constant contact area (CCAr), accompanied by the flattening of the drop.16,17 The CCAn mode is reported to occur predominantly when the drop contact angle θ, defined as θ ) 90° RR, is greater than 90°, whereas the CCAr mode is predominant for θ < 90°.17,18 One can now explain the drastic change of the experimental equilibration curves from Figure 2a to Figure 2c by the surfactant properties of lysozyme and their effect on the evaporation regime. In fact, proteins are known to adsorb in the air-water interface causing the values of surface tension and contact angle θ to decrease.19-21 This effect was visible in


Martins et al. Table 2. Geometrical and Physicochemical Parameters of the Hanging Drop Experiments at 277 and 293 K T(K) VDX plates optical cuvette

a b a b

277 8.5 12 N/A N/A 0.239 6.101 1.752 1 × 10-5

(mm) (mm) (mm) (mm)

D (cm2/s)9 p* (Torr) γ1 (-) kc (cm/s)b

293 8.5 12 10 9.0 0.26a 17.5 1.752 2 × 10-5

a The vapor diffusion coefficient reported for 298 K was not considered to change significantly at 293 K. b The mass transfer coefficients were estimated by using tabled values of the liquid water diffusion coefficients and the approximate diffusion path within the drop.12,15

contact angle meter, a decrease in the contact angle was measured at 20 °C from θ ∼ 90° to θ ∼ 80° due to the presence of the protein. The observed variation of the interfacial properties is high enough to cause the change of the evaporation mode from CCAn in Figure 2a to about CCAr in Figure 2c. The results obtained at 4 °C (Figure 2b) are likely to correspond to a mixed regime of the two modes. It was also observed that drops containing lysozyme were much more susceptible to spread out due to mechanical movement and plate handling. This partially explains the observed increase of the contact area that led to the results of Figure 2c. In addition, the fact that drops undergo different spreading kinetics depending on their interfacial properties may have affected differentially the results of Figure 2.22,23 The top-view method demonstrated to be suitable for in situ evaluation of the evaporation modes and for the equilibration curves measurement when the CCAn mode is observed. The evaporation of drops by other regimes was followed by the sideview method. In this case, the variation of the drop contact angle is also taken into account on the computation of the drop volume. Figure 3 reports the obtained V/V0(t) profiles and the corresponding variations of the dimensionless contact radius, rd/rd,0, and contact angle, θ. Figure 3a illustrates the behavior for the CCAr mode since the variation of the drop volume occurs at constant contact radius and decreasing θ values. This represents a change relatively to the result obtained by the top-view method in the absence of protein (Figure 2a), meaning that different evaporation regimes can result from small variations in the experimental conditions. Concerning the drop evaporation in the presence of lysozyme, the CCAr mode anticipated from Figure 2c is in general confirmed by the results shown in Figure 3b. After an equilibration period of about 20 h, a reduction of the solid-liquid interfacial area can be noticed to occur simultaneously to the contact angle decrease. The results obtained by the top-view and side-view methods show that the premise of fixed contact angle assumed on the derivation of water equilibration kinetics is not always verified. To evaluate the implications arising from this fact, eqs 3 and 4 were rewritten as a function of the drop contact angle θ and contact radius rd:24 Figure 2. Variation with time of the dimensionless volume of the drop inferred using the top-view method. Evaporation of drops containing 1.5% (w/v) of NaCl at (a) 20 °C and (b) 4 °C, in absence of protein, and at (c) 20 °C containing 25 mg/mL of lysozyme. Symbols correspond to different experiments and lines correspond to equilibration curves expected by eq 6.

the shape of the drops containing 25 mg/mL of lysozyme, since they were more spread out over the coverslips than the same drops in the absence of lysozyme. By using a video-based optical

I1 ) -

( )( )

ΩAD rd RT sin θ

I1 ) -

πr3d

2

dp dr

r)R

dθ 2 dt j V1(1 + cos θ)

(13) (14)

The derivative dp/dr in eq 13 is evaluated at the air-liquid interface using the vapor pressure profile along the air space (see ref 12 for details). In this work, only the cases of small


Crystal Growth & Design, Vol. 8, No. 12, 2008 4237

Figure 4. Theoretical influence of the initial contact angle on the water equilibration curves according to the CCAn (dashed lines) and CCAr (solid lines) evaporation modes at different initial concentrations of NaCl in the drop.

Figure 3. Variation with time of the dimensionless volume, dimensionless contact radius and contact angle of the drop measured by the sideview method. Evaporation of drops at 20.7 °C containing 2.25% (w/v) of NaCl (a) in absence of protein and (b) containing 25 mg/mL of lysozyme.

droplet-to-reservoir distance were considered. The variation of θ with time was determined by numerically solving the ODE that results from eqs 13 and 14. The corresponding drop volume variation was computed from the geometrical definition:

V)

πr3d(1 - cos θ)2(2 + cos θ) 3 sin3 θ

(15)

The parameters used on the simulations of the CCAr evaporation mode are given in Table 2 for the experiments at 20 °C using VDX crystallization plates; the obtained curves are represented in Figure 4 for different initial contact angles and precipitant concentrations. In the same figure, the CCAn evaporation profiles are plotted for the same initial conditions using eq 6 and subsequent definitions. For the sake of simplicity, no concentration gradients were considered within the drop by admitting the mass transfer resistance in the liquid phase to be negligible (β ) 0). For values of θ greater or equal to ∼90°, there are no significant differences between the expected evaporation rates under the two regimes. As illustrated in Figure 4 for the case of θ ) 80°, these differences start to be significant for more hydrophilic surfaces. For a given contact angle, the differences between the equilibration curves were not found to vary significantly with the precipitant concentration. In Figure 5 the equilibration curves measured by the sideview method at 20.7 °C are compared with the corresponding numerical solutions of the CCAr equilibration curve. With the

Figure 5. Comparison between the equilibration curves measured by the side-view method (solid lines) and the theoretical curves assuming the CCAr evaporation mode (dashed lines). Evaporation of drops containing 2.25% (w/v) of NaCl and 25 mg/mL of lysozyme (θi ) 77.9°) and 2.25% (w/v) of NaCl in absence of lysozyme (θi ) 87.4°).

exception of the value of vapor pressure, the parameters of Table 2 listed for 293 K were considered not to change significantly * at 293.7 K and were used in the simulations (p293.7K ) 18.8 Torr). A good agreement is observed between the experimental and theoretical curves represented in Figure 5. As the dashed lines in the figure show, drops containing lysozyme are supposed to evaporate slightly faster than in the absence of protein. This is due to the different initial contact angles in both cases. The expected differences in the equilibration curves are within the experimental error and could not be confirmed by the side-view measurements. The closeness of evaporation rates confirms a previous result obtained with drops containing ammonium sulfate in the absence and presence of 5 mg/mL lysozyme.25 As a practical conclusion to be drawn, the analytical equilibration curve provided by eq 6 is suitable to characterize the hanging drop experiments when hydrophobic coverslips are used and the protein surfactant properties are moderate (θ > 90°). In all other cases, the numerical solution of the CCAr equilibration curve should be preferably adopted. Estimation of Supersaturation Build up Profiles (SBUPs). One of the applications envisaged for the water equilibration theory is the estimation of the supersaturation


Martins et al.

( )

∗ c20◦C ) 0.354 exp

8.45 cNaCl

(16)

which was obtained by curve fitting to the published experimental data for this system.26 In a similar way, SBUPs at 4 °C were computed using the following lysozyme solubility curve:

( )

∗ c4◦C ) 0.130 exp

Figure 6. Steps involved on the construction of a SBUP. (a) Theoretical equilibration curve and variation of the lysozyme solubility with time. (b) Theoretical variation of supersaturation with time.

evolution during vapor diffusion crystallization techniques. The SBUPs result from the combined influence of water equilibration variables and thermodynamic variables, and are believed to determine decisively the kinetics of crystal nucleation and growth. In this section, the computation of a SBUP will be exemplified and then the interpretation of crystallization results in the light of the respective SBUPs will be given. The first step of the computation of SBUPs consists of the determination of the theoretical equilibration curve for the experimental conditions of the crystallization assay. As concluded from the preceding discussion, drops evaporate with fixed contact area during the crystallization of lysozyme and therefore the numerical solution of the CCAr equilibration curve will be used. Figure 6a shows the simulation of a crystallization experiment for the conditions of plate 1 in Table 1 using a concentration of NaCl in the reservoir of 2%. As the evaporation occurs, the concentrations of protein and precipitant increase in the same proportion to the drop volume decrease (not shown in Figure 6). By using the information of the instantaneous precipitant concentration, the value of the protein solubility can be inferred from the respective solubility curve. In the present case, the influence of the NaCl concentration on the lysozyme solubility at 20 °C and pH 4.7 is given by the following equation:

6.73 cNaCl

(17)

In the previous equations, cNaCl is expressed in percentage units (w/v). Because of the exponential dependence of c* on -1 cNaCl , the increase of the precipitant concentration in the drop causes a strong decrease in the protein solubility (Figure 6a). Recalling the definition of supersaturation as S ) c/c*, one can now compute the variation of S with time from the respective profiles of lysozyme concentration and lysozyme solubility. The obtained SBUP is represented in Figure 6b. As the drop volume decreases to one-half of the initial volume, the lysozyme supersaturation increases by several orders compared to its initial value. This shows that small variations on the water equilibration kinetics can have marked implications on the conditions for crystal formation and growth. The SBUPs thus calculated do not take into account possible variations on the drop pH that may occur when volatile crystallization agents are used. The SBUPs and Crystal Formation. Supersaturation profiles such as the one represented in Figure 6b are intrinsically theoretical since they would be altered by the occurrence of crystal nucleation and growth. In fact, supersaturation has been observed to increase due to water evaporation until the first crystals are formed (primary nucleation); from that point onward, the protein concentration in solution decreases due to crystal growth, and supersaturation gradually evolves until saturation is attained (S ) 1).27,28 Despite that, it is known that the nucleation moment and the supersaturation trajectory preceding it, determine decisively the final number and size of the protein crystals.8,27,29,30 In principle, high supersaturation levels lead to extensive nucleation and numerous small-sized crystals are obtained; at low supersaturation fewer crystals are expected to be formed and they can grow more until saturation is attained. In each case, the crystals are produced at different growth rates and having a different crystalline order.5 In what concerns the particular vapor diffusion experiment reproduced in Figure 6, no crystals were obtained after a period of 3 months. Accordingly, the SBUP calculated and represented in Figure 6b is consistent all through the equilibration period. Because of the importance of the supersaturation history on the final quality of crystals, microbatch and vapor-diffusion crystallization experiments were designed to approach the nucleation phase by different pathways. The modified microbatch technique allowed the determination of the minimum levels at which crystals form under stationary supersaturation. Early studies on the nucleation of lysozyme do not fit this purpose since they were carried out either at too high supersaturation levels or under changing supersaturation.31-34 Moreover, for the conclusions drawn under stationary conditions to be applied on the interpretation of vapor diffusion results, both techniques should be carried out under close conditions, avoiding for example the employment of oils in the microbatch technique. The results represented in Figure 7 were obtained by a modified microbatch technique by keeping the same precipitant concentration in the reservoir and in the drop in a typical vapor diffusion apparatus. The measured values of apparent induction time (AIT) correspond to the time elapsed since the preparation of the drop until the first crystal to form is detected by optical microscopy.


Figure 7. Apparent induction times measured by the modified microbatch technique at 4 °C (2) and 20 °C (0) represented as a function of (a) the precipitant concentration and (b) lysozyme supersaturation. The symbols represent (a) individual measurements of the AIT in each triplicate and (b) the corresponding average values for each precipitant concentration. In the first case, the number of crystals obtained in the drops is given next to each symbol, whenever this number is below 100. Results marked with an asterisk (*) were obtained more than 400 h after the drop preparation and were not used to compute the average AIT at the corresponding precipitant concentration (9).

Because of the experimental limitations, this is a longer period than the fundamental induction time for nucleation defined as the time needed for the appearance of the first stable nuclei.8 Figure 7a shows that the AIT strongly decreases as the precipitant concentration increases. At 4 °C no crystals were obtained for NaCl concentrations in the reservoir below 1.66%, while at 20 °C this concentration limit was of 2.8%. The results of Figure 7a indicate that the nucleation step occurred in a great extent in all conditions of the 4 °C experiments, with the number of obtained crystals being more than 100 in all but one of the drops analyzed. Larger and fewer crystals per drop were obtained at 20 °C for NaCl concentrations below 4%. In general, the results obtained for each precipitant concentration were shown to be reproducible both in terms of the time required for crystal formation and the nucleation extent. The average AIT for each triplicate is expressed as a function of the lysozyme supersaturation in Figure 7b. The lysozyme solubility was determined for each precipitant concentration using the solubility curves at 4 °C (eq 17) and 20 °C (eq 16). According to Figure 7b, the influence of the supersaturation on the AIT follows a common trend at 4 °C and at 20 °C. For both temperatures, a critical supersaturation range for crystal formation is possible


to be identified between ∼3.5 and ∼7, in which the AIT decreases from above 100 h to around 10 h. Supersaturation values below 3.5 are not high enough to induce crystal formation, whereas for S > 7 the crystals are expected to be formed in less than 10 h and in great number. The critical nucleation range found by the modified microbatch method is an important reference to interpret vapor diffusion experiments and the respective theoretical supersaturation profiles. Considering the example previously given in Figure 6b, the supersaturation value at the end of the equilibration period is still below the identified limit for crystal formation. The clear drop result obtained at end of the experiment could in this way be anticipated from the theoretical supersaturation profile. In further crystallization assays, higher supersaturation levels were covered as a consequence of different evaporation kinetics. The hanging drop method was employed at 4 and 20 °C by changing the precipitant concentration and the technique parameters summarized in Table 1. At 20 °C, the changes in the technique were at the level of the drop shape so as to have small variations on the water equilibration rates. Compared to plate 1, more hydrophilic coverslips were used in plate 3, which led to drops with a lower contact angle θ. In plate 5, the drops were prepared with an elongated shape with the help of the pipet tip. At 4 °C, different dilution factors were used in the preparation of plates 2 and 4. In all cases, the drops were closely followed by optical microscopy until the first crystal to form was detected. The corresponding AIT was registered based on the time elapsed since the drop preparation. In Figure 8, the symbols place the moment of the crystals appearance along the respective SBUPs. The supersaturation variation prior to the primary nucleation events corresponds to the section of the theoretical curve that precedes each symbol. Next to the symbols, the nucleation extent is given in terms of the number of well developed crystals at the end of the crystallization experiment. All the SBUPs represented in Figure 8 were calculated assuming the drop shape to be of a spherical cap with initial contact angle θi ) 77.9°. As previously referenced, this does not apply to the plates with plastic coverslips (plate 3) and elongated drops (plate 5), meaning that the supersaturation curves plotted in Figure 8a,b for plate 1 is a rough estimation of the profiles of plates 3 and 5. While in Figure 8a,b the plates 1, 3 and 5 are represented by the same theoretical profile, in Figure 8c,d each plate is represented by a specific SBUP. Triplicate experiments were carried out for each precipitant concentration so that three experimental results per plate are associated with every SBUP. An immediate conclusion to be drawn from Figure 8 is that the majority of crystals were formed at supersaturation levels above the critical range for crystal formation indicated by the grid box. The exceptions to this trend were observed in plate 3 at the highest precipitant concentration (Figure 8b), and in two pair of drops in plate 4 for 3% and 4% of NaCl in the reservoir (Figures 8c,d, respectively). In the other cases, primary nucleation occurred at supersaturation values above those expected by the modified microbatch method, with the nucleation events being frequently observed at S > 15! The different behaviors involved are well illustrated by the experiments carried out at 20 °C for a precipitant concentration in the reservoir of 3% (Figure 8a). In this case, the vapor-liquid equilibrium is attained after a ∼80 h period at a supersaturation level of ∼9.4. At this supersaturation, it would be expected from the microbatch results shown in Figure 7b that the crystals would be formed within a few hours period. Nevertheless, only one of the drops produced crystals (after an


Martins et al.

Figure 8. Lysozyme crystallization experiments following the hanging drop method at (a, b) 20 °C and (c, d) 4 °C. The lines represent the theoretical SBUPs and the symbols indicate time of the first crystals appearance obtained in triplicate measurements. The number of crystals obtained in each drop is given next to each symbol. The precipitant concentration in the reservoir and the crystallization plate identification are indicated in the legend to the figures. The grid box represents the critical supersaturation range for crystal formation measured by the modified microbatch method.

additional ∼60 h period that followed the evaporation equilibrium) and the other 8 drops remained clear until the last plate checking, 3 months after their preparation. An analogous result was obtained with plate 2 for a concentration of NaCl in the reservoir of 2% (Figure 8c): after an equilibration period of ∼250 h, the supersaturation was considerably high (∼13.6), yet no crystals were observed in the drops even after a period of 400 h. The results of Figures 7 and 8 diverge in two additional aspects that are concerned with the nucleation extent and the AIT reproducibility. While in the modified microbatch method the number of well developed crystals is more than 100 for supersaturation levels above ∼5 (4 °C) and ∼9 (20 °C), in the majority of the vapor diffusion results the number of crystals is below 10 and they are obtained at much higher supersaturation levels. On the other hand, the high reproducibility of the AIT results in Figure 7a was not observed in Figure 8, where the symbols indicating the nucleation events are generally scattered over the SBUPs. It was not possible as well to establish a clear relationship between the number of obtained crystals and the maximum supersaturation achieved during the drop evaporation. At this point, two nucleation behaviors are possible to be identified according to the crystallization technique employed: by the modified microbatch method, crystals are formed in a great number, even at moderate supersaturation levels, and in a reproducible way. By the vapor diffusion method, the

theoretical SBUPs indicate that nucleation only occurs at high supersaturations, the number of crystals produced per drop is small and the time required for crystal formation is not easily predictable. The explanation for such different scenarios is not immediate and is certainly related to the mechanisms by which nucleation occurs. Other molecular interactions that take place in solution during the often long periods preceding the formation of the nuclei also deserve close attention. It could be argued that different lysozyme concentrations used in both methods could lead to different crystallization results. In fact, although the initial lysozyme concentration was in most of the cases 25 mg/mL, during the vapor diffusion method this value increases, while in the microbatch method the concentration remains constant until the crystals are formed. Accordingly, different regions of the phase diagram would be covered by each technique and that would lead to different nucleation mechanisms.27,35,36 We believe that this justification is not sufficient to explain our crystallization results in the experiments starting from the grid box or above, as in the cases represented in Figure 8b for all the conditions except plate 3 at 6% NaCl, in Figure 8c for plate 2 at 3% NaCl and in Figure 8d for plate 2. Although these experiments also depart from the critical region for nucleation, the observed nucleation behaviors remain clearly distinct from those observed in the microbatch


method. The case of plate 4 cannot be used as an example because the initial lysozyme concentration was above 25 mg/ mL due to the dilution factor used (1:3). In all other appointed conditions, the initial concentrations of protein and precipitant are close to the ones used in the modified microbatch method. Consequently, the different nucleation mechanisms are not likely to be explained by different regions of the phase diagram covered by each method. Another possible explanation could be related with the concentration gradients formed within the drop while the evaporation occurs. These gradients are supposed to be absent during the microbatch method, but could be responsible for important supersaturation variations inside the drop during the vapor diffusion techniques. Once more, the results of Figure 8 and the particular cases of Figure 8a for both precipitant concentrations, Figure 8b for plate 1 at 5% NaCl and Figure 8c for plate 2 at 2% NaCl do not confirm this hypothesis. In these cases, the crystals were observed to form many hours after the vapor-liquid equilibrium was achieved, when the internal gradient concentrations due to evaporation were vanished. Since this did not bring any visible change on the nucleation mechanism that could resemble the modified microbatch method, we conclude that the concentration gradients do not explain satisfactorily our results. We believe that the explanation for the identified differences could be related with the adsorption equilibrium at the air-liquid and solid-liquid interfaces. Similar to what happens with metastable phases such as dense fluids and gels,31,35,36 the adsorbed state is believed to be a precursor to the formation of stable nuclei due to its considerable degree of molecular organization. Also, because of the high surface area to volume ratio of the drops, a great amount of protein is expected to be in the adsorbed state at the drop interfaces.37 Moreover, protein adsorption is expected to be strongly affected by small variations in the drop preparation procedures and by the sample conditions. Illustrating this, Langmuir and Schaefer, in their classical paper in 1938,38 reported large differences in the amount of insulin adsorbed onto a plate depending on the followed sequence of events: if the plate was dipped in a 1% insulin solution and then washed with water, and dried, the thickness of the adsorbed layer was about the same as if this procedure was followed with 0.8% sodium chloride added to the insulin solution. However, if the plate was dipped in a 1% insulin solution and then washed with a 0.8% sodium chloride solution instead of water, and dried, the film thickness was many times greater than the one obtained in the first cases. In a different example, the time required for the lysozyme adsorption equilibrium in the air/water interface was found to be affected in many hours by the age of the lysozyme solutions.39 Our experiments should not be affected differently by aging factors since the crystallization plates were all prepared at the same day, using the same freshly prepared solutions. Nevertheless, the questions related with the drop preparation may partially explain the different nucleation behaviors documented in Figures 7 and 8, since the microbatch and vapor diffusion crystallization plates were set up by different operators. Although the drops were always prepared by adding the lysozyme solution to the salt solution, the degree of mixture within the drop could have differed in both techniques. As a possible consequence, very distinct adsorption dynamics may result, which would affect differently the time, extent and mechanism of crystal formation. A considerable amount of work has been done on the mechanisms of globular protein adsorption, but little is known about the relationship between protein adsorption


and protein crystallization. The analysis of Figures 2-5 already demonstrated that the adsorption of lysozyme molecules at the air-water interface could induce distinct evaporation regimes and affect the results of vapor diffusion techniques. From the results of Figures 7 and 8, it is also likely that distinct nucleation behaviors can result from the procedure adopted during the preparation of the drops and from its effect on the kinetics of protein adsorption. These issues will continue to be investigated in our future work. A Possible Approach to Rational Crystallization of Proteins. Although there is much to be learned about protein crystallization science, at this point we venture to sketch the possible guidelines of a practical approach based on the theory presented thus far. Let us consider the case of a protein with unknown structure, for which traditional methods of blind screening did not provide the good diffracting crystals for X-ray crystallography studies. Initially, a selection of a limited number of precipitants should be made among the chemicals most commonly used for this purpose and, whenever possible, taking into account specific interactions with the protein that are known to be promising. The protein solubility should then be measured in a range of concentrations of the chosen crystallizing agents and, at least, at two different temperatures. New methods of protein solubility estimation have been proposed showing important advantages in terms of the time required for the measurements,40 and in some cases obviating the use of protein crystals or precipitates.41 Analysis of the solubility curves will indicate whether vapor diffusion techniques are suitable for a given proteinprecipitant system. Situations may occur where the protein solubility is little changed or even increased by the presence of the precipitant, for which alternative techniques such as dialysis or microbatch by the cooling method would be more appropriate. On the contrary, if the protein solubility is highly decreased by the precipitant, crystallization experiments by vapor diffusion should be used and carefully designed prior to their execution in the laboratory. In this case, all the steps described in this manuscript for the construction of theoretical SBUPs should be followed, simulating the most diverse experimental setups and the influence of the main variables involved in the hanging drop method. Variables such as temperature, precipitant concentration and dilution factor during drop preparation are expected to have the stronger influence on the supersaturation profiles. A first selection of the conditions to be experimentally tested should include those providing smoother approaches to the supersaturation range for nucleation, without entering in regions of too high supersaturation associated to extensive crystal formation and/ or fast growth. The huge investment in the indiscriminate crystallization trials is expected to be drastically decreased by working within the set of conditions known to be the most promising. Should some conditions deserve finer tuning, a priori studies can also be performed using SBUPs in order to address the effect of other parameters such as the crystallization chamber geometry, the total volume of the drop, the shape of the drop as different coverslip materials are used, the volume of solution in the reservoir (and the droplet-to-reservoir distance), the use of oils to slow down evaporation, etc. Seeding is certainly a practice to be considered to promote crystal growth from a very early stage and at low supersaturations. After a new set of laboratory tests, the worst scenario would be to still not be able to get well-diffracting crystals and there different crystallizing agents have to be investigated. Recurring failures may suggest


that other alternatives are needed, such as the use of detergents, additives or even proteolysis and genetic protein engineering. In every case in which the rational approach is restarted and new solubility curves are measured, it is assured that the capacity to produce crystals in the previous protein-precipitant systems was thoroughly explored using reduced experimental means. On the contrary, the blind crystallization screenings can be a very demanding process with great possibilities of overlooking successful conditions for crystal formation. Conclusions We put forward the concept of supersaturation build up profiles as a valuable tool to organize the screening space of crystallization conditions and design vapor diffusion experiments. In the hanging drop method, the drops can evaporate at fixed contact angle or fixed contact area depending on their initial contact angle, and different theoretical SBUPs result from each case. The construction of SBUPs emphasized the importance of the protein solubility curves in rational crystallogenesis. Our experimental work shows that the nucleation behavior of lysozyme significantly varies with the supersaturation history. Different induction times, nucleation extents and nucleation mechanisms were observed in the microbatch method (stationary supersaturation) and in the vapor diffusion method (dynamic conditions). To understand these differences, further work is required on the phenomena of interface adsorption of proteins, which are believed to link the drop preparation procedures and the protein nucleation behavior.

LIST OF SYMBOLS a, b, c, c*, cNaCl, D, I1, k c, p, p*, R, R0, R ∞, R a, Rb, r, rd, rd,0, S, T, t, V, V0, j 1, V x2,0, x2b, x2d, x2i, y, y∞, yc, R, Ra,

inner radius of the reservoir (mm) vertical distance from the coverslip to the surface of the solution in the reservoir (mm) protein concentration (mg/mL) protein solubility (mg/mL) concentration of NaCl (%, w/v) vapor diffusion coefficient in air (cm2/s) molar rate of water leaving the droplet (mol/s) mass transfer coefficient in the drop (cm/s) vapor pressure (Torr) vapor pressure of pure water (Torr) radius of the drop (mm) initial radius of the drop (mm) radius of the drop after equilibration (mm) radius of the sphere intercepting the reservoir walls at the level of the coverslip (mm) - Figure 1 radius of the sphere tangent to the surface of the solution in the reservoir (mm) - Figure 1 spherical radial coordinate (mm) contact radius of the drop (mm) initial contact radius of the drop (mm) supersaturation (-) temperature (K) time (s) volume of the drop (mm3) initial volume of the drop (mm3) molar volume of pure water (mm3/mol) initial molar fraction of the precipitant in the drop (-) precipitant molar fraction in the reservoir solution (-) precipitant molar fraction in the drop (-) precipitant molar fraction at the drop interface (-) dimensionless radius of the drop (-) dimensionless radius of the drop after equilibration (-) dimensionless radius defined by eqs 8 and 9 or (-) angle defined in Figure 1 for a given radius r (°) angle R for a radius r ) Ra (°)

Martins et al. Rb, RR, β, γ1, τ, θ, ΩA, ΩV, R,

angle R for a radius r ) Rb (°) angle R for a radius r ) R (°) dimensionless parameter defined by eq 12 (-) water activity coefficient in solution (-) time constant defined by eq 7 (s) drop contact angle (°) surface area shape factor of the drop (-) volume shape factor of the drop (-) gas constant (62.364 dm3 Torr K-1 mol-1)

Acknowledgment. The authors thank Instituto de Engenharia Biome´dica (INEB) for allowing the use of the video-based optical contact meter. P.M.M. gratefully acknowledges Grant SFRH/BPD/34556/2007 from Fundaçaõ para a Cieˆncia e a Tecnologia (FCT), Portugal. This work was supported by Projects FEDER&FCT POCI/SAU-NEU/69123/2006, POCI/ V.5/A0117/2005, and PTDC/EQU-FTT/81496/2006 from FCT, Portugal, and Project EURAMY(FP6-LIFESCIHEALTH-6) from EU.

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Rationalizing Protein Crystallization Screenings through Water

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