Thermodynamic Properties of Supersaturated Protein Solutions

Thermodynamic Properties of Supersaturated Protein Solutions Dragutin

Knezic,†

Julien

Zaccaro,‡

and Allan S.

Myerson*,†

Illinois Institute of Technology, Department of Chemical Engineering, Chicago, Illinois, USA, and Laboratoire de Cristallographie CNRS, Grenoble, France Received May 5, 2003;

CRYSTAL GROWTH & DESIGN 2004 VOL. 4, NO. 1 199-208

Revised Manuscript Received July 21, 2003

ABSTRACT: The purpose of this work is to investigate water and protein (lysozyme) activities in supersaturated aqueous protein solutions. An electrodynamic levitation trap (ELT) is used to suspend a single charged droplet of aqueous protein solution without a container. This technique allows investigation of homogeneous nucleation and measurements of water activity deep into the metastable zone of a supersaturated droplet. The system was treated as a ternary one (protein + solvent + “ideal” salt), and the protein activity is calculated employing the GibbsDuhem equation. The trend of logarithm (aprotein/a/protein) with protein concentration shows initially a sharp increase, but then at higher protein concentrations the trend starts to level off. The same trend is seen in salt-water systems and organic-water systems and follows the expected behavior of supersaturated solutions. As the system is brought closer and closer to the spinodal curve, the second derivative of the Gibbs free energy of the solution with respect to protein concentration moves toward zero. Due to the small values of mole fractions of protein used for the integration of the Gibbs-Duhem equation as well as the unknown dependence of salt activity on protein concentration, the results show exceptionally high values for aprotein/a/protein. Introduction Crystallization of macromolecules and their analysis by X-ray diffraction play an essential role in important areas of modern molecular biology and biotechnology, particularly in the genetic engineering of proteins and rational drug design. Protein crystals are very difficult to make without knowing the right crystallizing conditions. The difficulty arises from the high sensitivity of the proteins to the environment in which they are crystallized. Ionic strength, temperature, pH, type of salt, and concentration of organic solvent are just few of the factors that might influence the outcome of the crystallization experiment.1 One of the major problems facing scientists and engineers is the lack of thermodynamic, kinetic, and material property data required for the growth of large, high-quality crystals of biological macromolecules. Protein molecules attach themselves to one another via specific binding sites to form aggregates that serve as nucleating sites for the crystal growth. The probability for solute collision increases with concentration, so supersaturated conditions enhance the chance of creating a critical-size nucleus. Higher supersaturation levels require a smaller minimal thermodynamically stable aggregate (critical nucleus) for the onset of nucleation. Increase in supersaturation level increases the probability of nucleation.2 In this nonequilibrium state of high supersaturation, molecules continually combine to form clusters and dissociate back into solution. When the critical size is reached and the energy barrier for forming a nucleus is broken, the aggregate starts to grow spontaneously.3 * To whom correspondence should be addressed: Allan S. Myerson, 3301 S. Dearborn, Seigel Hall 103, Chicago, IL 60616. Tel: (312) 567 3163. Fax: (312) 567 7018. E-mail: [email protected]. † Illinois Institute of Technology. ‡ Laboratoire de Cristallographie CNRS.

Since protein crystals are grown for structure determination mainly by trial and error, the investigation of the thermodynamic properties of the supersaturated protein solutions is indispensable in improving our understanding of nucleation and crystal growth. Crystallization from solution involves a phase change from the solution to the solid state where the crystallizing solute goes through the metastable region. The governing force for crystallization in terms of supersaturation is the difference between the chemical potential of the supersaturated solution and the chemical potential of the saturated solution at the same experimental conditions. The difference in chemical potentials can be expressed as

(a*a )

∆G ) µ - µ* ) RT ln

(1)

where “*” denotes saturated condition, and a represents activity of the species. In real solutions, activity is a function of temperature, pressure, and composition. It is used to characterize the deviation of the solution’s behavior from the ideal model. For a complete understanding and description of the nucleation processes in the solution, the driving forces for nucleation should be known. Knowledge of the thermodynamic activity of a protein in a solution offers a way of calculating the ratio of activity coefficients that can be used to describe the behavior of supersaturated solutions prior to the onset of crystallization. Experiments performed with the spherical void electrodynamic levitation trap (SVELT)4-6 apparatus provide a direct measurement of water activity as a function of protein concentration. From this information, the activity of the protein in solution can be determined from the Gibbs-Duhem equation. Extraction of activity coefficients from protein activity data

10.1021/cg034072o CCC: $27.50 © 2004 American Chemical Society Published on Web 08/27/2003

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Figure 1. Schematic representation of SVELT apparatus.

can be used to analyze the effect of salt concentrations, salt types, pH, and temperature as well as protein solubility. Experimental Section Experimental Apparatus - SVELT. Figure 1 shows the schematic representation of the SVELT apparatus. The levitation trap provides a clean and simple environment for crystal formation. SVELT configuration allows suspension of a single charged microdroplet inside an electrical field. Prior to injection into the trap, the solution is filtered through a 0.8 µm Millipore filter unit (Millex-PF) to remove any contamination. A piezoelectric droplet generator filled with the prefiltered solution forms the droplets of the solution to be analyzed. The diameter of the orifice in the glass tip of the droplet generator is approximately 120 microns, which allows the creation of the droplets 40-50 microns in diameter. Prior to the entrance of the trap, the drop passes through a charging ring (VDC applied to the charging ring ) 86-92), and as a result the surface of droplet gets charged. The spherical internal configuration provides symmetrical distribution of electrical potential inside the trap with the center of the trap experiencing zero or a very small AC field. The trap consists of three electrodes with isolation placed between each electrode. The middle electrode provides a moderate AC voltage (350-400 V) that oscillates the particle vertically along the axis of the trap, which is at the geometric center of the device. In addition to trapping the droplet in the electrodynamic levitation system, its weight must also be balanced out against the gravitational pull. The top and bottom electrodes supply a DC voltage to counteract the pull of gravity and to stabilize and prevent the droplet from drifting from the trap. Both trapping and levitation are independently achieved in the ELT. The droplets can be suspended in this system for long periods of time and can be localized to a fraction of their diameter. The trap sits inside a stainless steel chamber and is isolated from its surroundings. Relative humidity inside the chamber is measured and used in calculating water activity data. Water activity data are necessary for calculation of solute (protein) activity data from the Gibbs-Duhem equation. This activity data are used to characterize the protein’s behavior in the metastable region. This levitation technique allows examination of homogeneous nucleation since there is no contact with the container walls and the chance of a solution sample to contain impurities is considerably reduced due to the small size of the experimental droplet. While the droplet is suspended in a mid-air,

Knezic et al. solvent is slowly evaporated until the solute crystallizes. With the absence of a container, a substrate, or an impurity, creation of higher degrees of supersaturations is possible compared to supersaturations attained in bulk solutions. In bulk solutions, the energy barrier for nucleation is lowered by the presence of foreign substances or container walls, which act as nucleating centers. Fletcher7 and Millikan8 used levitation technique to find the charge of an electron and the value of Avogadro’s number. Since Millikan’s classic oil drop experiment exhibited instability because it was unable to trap the droplet indefinitely, Straubel9 added an AC ring electrode between other electrodes creating a dynamic force that pulled the droplet toward the center of the trap. Gucker and Rowell10 modified Milikan’s trap to scatter light from a levitated droplet and used it to validate Mie theory of light scattering from a sphere of arbitrary size. The electrodynamic levitation trap lends itself to many different applications such as vapor-liquid equilibrium,11 finding optical properties of spherical and nonspherical particles,12,13 determining evaporation rates,14-16 deliquescence crystallization,17 measuring water activities at high supersaturations,18,19 and crystal nucleation from an aqueous solution.20 In addition, many spectroscopic techniques have been used, such as excitation, phosphoretic, infrared and fluorescence spectroscopy, Raman scattering, and a Fourier transform infrared spectroscopy method for determining the molecular composition and phase transition of a suspended particle.21-26 Na27 and Na, Arnold, and Myerson28,29 examined the thermodynamics of aqueous solutions of electrolytes and nonelectrolytes. Water activity data were used to calculate critical cluster size and the interfacial energy of the solute-solution system. Bohenek et al.30 used SVELT to determine the activity of supersaturated solutions of KDP, ADP, and TGS and to calculate the spinodal curve, critical cluster, and degree of association for each system. Some work has been done on investigating the thermodynamic properties of electrolytes with additives31 and organic solutions with additives.32 Mohan and Myerson31 found that additions of impurities of low concentration (0.2-20 ppm) increase the deviation from pure solution behavior in the higher supersaturated regions. A levitation trap has been used to grow protein crystals by controlling the protein saturation level in a programmed way through a computer with the database of balancing voltage data versus vapor content of the chamber.33 Chung and Trinh34 extended the work of Rhim and Chung33 using the hybrid ultrasonic-electrostatic levitator by successfully growing thaumatin and lysozyme crystals. Santesson et al.35 used an acoustic levitator to develop a precipitation-screening method for rapid detection of the regions of phase diagrams where the proteins precipitate. In recent years, a number of investigators have studied the properties of protein solutions, the effects of thermodynamic nonideality on macromolecular interactions, and methods to model solid-liquid equilibrium. Generally, these effects are analyzed in terms of composition-dependent activity coefficients. Knowledge of protein solubility is vital for an understanding of nucleation and the crystal growth process. Protein solutions consist, in simple terms, of protein-saltsolvent components. There are few models describing solidliquid equilibrium. Some investigators used a statistical thermodynamic theory for multicomponent systems to describe the osmotic pressure and the salt distribution in terms of an expansion in powers of the colloid concentration and to calculate second and third virial coefficients.36,37 Melander and Horvath38 described protein solubility as a function of salt concentration and electrostatic and hydrophobic interactions, while Przybycien and Bailey39 showed inconsistency in their results and indicated that an optimum salt (in terms of a favorable salt-protein interactions for crystallization) might exist for a particular protein. Concentrated protein solutions often produce liquid-liquidphase separation, consisting of an aqueous phase dilute in protein and a protein-rich phase. A number of studies were aimed at understanding the driving forces for phase separation. Some used liquid-liquid-phase transitions to obtain

Thermodynamics of Supersaturated Protein Solutions

Crystal Growth & Design, Vol. 4, No. 1, 2004 201

protein phase diagrams,40,41 turbidity measurements for studies of nucleation,42 and kinetic model for nucleation of a crystalline phase.43 Others applied statistical-mechanical theory (random-phase approximation)44-46 to depict liquidliquid phase separations in solutions of globular proteins or used atomic force microscopy (AFM) and light scattering techniques for in situ crystallization of protein from solution.47,48 In addition, liquid-liquid-phase separation was used in determining homogeneous nucleation rates and was shown that for lysozyme crystals the rate passes through a maximum in the vicinity of the liquid-liquid phase boundary.49-51 Timasheff and co-workers52-57 developed a preferential interaction theory of protein’s stability in a three-component aqueous solution and calculated chemical potentials. It was found that developed preferential interaction parameter could explain the effect of salt on protein stability when associated with the protein. Wills et al.58,59 discussed the expressions used in osmometry, isopiestic measurements, equilibrium dialysis, gel chromatography, and sedimentation equilibrium studies and pointed out the manner in which the thermodynamic activity of macromolecular solutes should be defined for different circumstances. Grant60 used the same approach and estimated the chemical potential of protein in solution from a virial expansion in protein concentration. In all cases, the driving forces estimated from the ideal solution behavior (supersaturation taken as the ratio of concentration of protein over the concentration of protein solubility) were lower than that for the corresponding ideal solution estimates. In general, since protein solutions deviate greatly from ideal solutions, properties such as protein solubility are expressed through activity coefficients. Many studies included data from sedimentation equilibrium experiments and osmotic pressure measurements and were based on virial expansion. Ross and Minton61 used both techniques to calculate the activity coefficients of hemoglobin by treating the solutions as a twocomponent (solvent and protein) solution. Agena et al.62 used the original UNIQUAC model for the representation of protein activity coefficients in four-component protein solutions. The experimental activity coefficients from osmotic measurements were used to obtain interaction parameters for the UNIQUAC equation. The calculated protein activity data agreed qualitatively with the experimental solubility behavior. Guo et al.63 explored the nonideality of protein solution and derived the relationship between the second osmotic coefficient and a protein’s solubility. Rupert et al.64 used the classical thermodynamic approach of equating the fugacities in equilibrium between solid protein and protein in the solution. Their approach provided theoretical support for the empirical relationship between the second osmotic virial coefficient and the solubility of proteins. Petsev et al.65 by determining the binodal and spinodal lines along with the second virial coefficient of the protein calculated protein chemical potential and osmotic pressure for concentrated protein solutions. It was proposed that a pair of parameters (molecular volume and second virial coefficient) might be enough to predict the phase behavior of solutions protein with simple interaction potentials. Equations Used for Calculation of Protein Activity Data. For a stationary particle suspended in the center (null point) of a trap, the weight of the particle is balanced with the applied DC field. It can be expressed as a function of other properties of the system by the following relationship

Vdc mg ) Cq w mdroplet ) AVDC 2zo

(2)

where m is the mass of the particle, g is the standard gravitational acceleration (9.81 m/s2), q is the surface charge on the particle (∼104 e), zo is the characteristic length of the cell (for SVELT it is the radius of the spherical void ∼ 0.5 cm), Vdc is the DC voltage applied, and C is a geometric constant (C ) 1 for spherical void). For a particle whose charge remains unaltered during an experiment and for a nonvolatile solute, the relative mass changes of the solution droplet can be easily determined by

measuring the DC balancing voltage. From eq 2, the mass of the particle is proportional to the DC voltage measured. The value of a constant A ) (C q/2z0g) is not required for relative mass measurements and is only needed for calculation of the mass of the suspended particle. The solute concentration (w) at any point during an experiment is easily calculated with eq 3.

w)

DRY mDRY VDC ) mWET VWET DC

(3)

To calculate concentrations in mg of protein/mL of solution following equations can be used (assuming density of the solution equals density of the water ) 1 g/cm3, M is the molecular weight of the species):

for mole fractions (y ) mole fraction) c) FsolutionMprotein(1000) Mprotein + Mwater(1/yprotein - 1 - ysalt/yprotein) + Mwaterysalt/yprotein (4)

for weight fractions (x ) weight fraction) c ) Fsolutionxprotein(1000)

(5)

From eq 2 and the reasons stated above, any change in the particle’s mass must be due to a change in the solvent content in the droplet. Evacuating the chamber results in a reduction of the surrounding vapor pressure whereupon the droplet tries to equilibrate by evaporating solvent (water). The water activity of a charged solution is considered to be18

aW )

PW Psat W

exp

(

)

2q2MS( - 1) 4σM FRTd πFRTd4

(6)

The first term of the exponential in eq 6 represents the dependence of activity on vapor pressure elevation due to the droplet’s curvature (Kelvin effect), while the second term represents vapor pressure depression due to the surface charge of a suspended droplet. These effects are known to be insignificant for small droplets and can be treated as negligible.17 So the eq 6 can be reduced to

aw )

PW Psat W

) RH

(7)

where RH is the relative humidity of the vapor phase. Thus, by measuring the saturation ratio in vapor phase, the activity of the solvent can be determined. The thermodynamics of aqueous protein solutions with salt added can be derived from the corresponding Gibbs-Duhem equation:

xwdµw + xpdµp + xsdµs ) 0

(8)

where x is the mole fraction, µ is the chemical potential and the subscripts w, p, and s represent the solvent (water), solute (protein), and salt, respectively. The solution temperature T, and static pressure P, in eq 8 are assumed to be constant. From eqs 1-6, the resulting equation is

xwd ln aw + xpd ln ap + xsd ln as ) 0

(9)

Solving for the activity of protein, the following equation is obtained:

ap ln / ) ap

∫

as

as/

()

xs d ln as + xp

∫

aw

aw/

()

xw d ln aw xp

(10)

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The NaCl salt was assumed to behave as an ideal solute. With this assumption, the activity of salt can be represented by its mole fraction because for ideal solute the activity coefficient equals one.

a ) γx

(11)

Equation 10 is modified to

ap ln / ) ap

∫

xs

x/s

()

xs d ln xs + xp

∫

()

aw

a/w

xw d ln aw xp

(12)

Experimental Procedure. The model protein chosen for this work is lysozyme (MW ∼ 14300 g/mol). Lysozyme is relatively easy to crystallize from aqueous solutions, and experimental data (solubility, density, viscosity) are readily available in the literature.66-70 To enhance phase separation and to help crystallization, an ionic salt (NaCl) is dissolved together with lysozyme. Sodium chloride was used as a precipitating agent because the NaCl does not denature the protein in solution and has little effect on the conformational change of lysozyme with its concentration.66 Rosenberger et al.71 state and support a hypothesis through observation of crystallizing conditions of 20 proteins that monodispersity is a prerequisite for crystallization. To facilitate crystallization from solution, the working pH was chosen to be 4.0 based on a study by Bruzessi et al.72 that reports that protein monomer goes through reversible association into dimers at pH > 4.5. In addition, Kim66 reports that pH values of the lysozyme solutions in a 0.1 M sodium acetate buffer solution (pH ) 4.0) were independent of the lysozyme concentration. All of the experimental solutions were made in aqueous 0.1 M sodium acetate buffer and pH ) 4.0. Chicken egg white lysozyme (L-6876, LOT 65H7025, 3× crystallized, dialyzed, and lyophilized, approximately 95% protein) was used to create solutions for experiments. All other nonprotein materials used were reagent grade. A Fisher Scientific (Accumet Portable AP63) pH meter was used to adjust the solution’s pH by the addition of HCl solution (50% V/V). Salt (NaCl) was added to the buffer solution (100 g) to achieve a concentration of salt slightly above desired concentration. Lysozyme was added to 60-80 mL of the buffered salt solution in an appropriate quantity to achieve the final protein-to-salt ratio. The solutions prepared varied from 1 to 6 to 6-1 mass ratio of protein to salt. At the end, a final pH check was performed and the pH was adjusted to 4.00 ( 0.01 by addition of small quantities of HCl or NaOH solution. Prior to the injection of the drop, the chamber was humidified to a high level of relative humidity. This was done to ensure minimal loss of water upon the injection of the drop. The droplet was then injected inside the trap. Upon catching a droplet inside the AC field, a DC field created by a superimposing DC voltage was applied, allowing the particle to be pulled toward the center of the spherical void where the AC field is at its null point. The DC voltage was noted because it is proportional to the particle’s mass (eq 2). Laser light scattered 90° to an incident beam allowed the trapped particle in the center to be observed. Experiments performed with this SVELT configuration did not have any temperature control. The ambient temperature in the lab was the determining factor in the degree of control of the system. As a result, the temperature of the system has not remained constant during a particular experiment. The temperature of the chamber varied from 24-29 °C during consecutive experiments in a day. The individual runs varied 0.6-1.6 °C from the starting temperature. A vacuum pump was connected to the system to evacuate air from the chamber and to ensure evaporation of any unwanted residues before the introduction of water vapor from the water reservoir. Once the chamber-containing trap was sealed tight, the pump was switched on allowing slow evacuation. During evacuation, water vapor was removed from the chamber. The droplet equilibrated with its surrounding at-

mosphere by evaporating water. By slowly removing the solvent, the droplet became more and more supersaturated. By varying the relative humidity of the chamber, the relative weight of a suspended particle changed. The droplet’s mass loss was compensated for by manually adjusting the DC voltage to maintain the drop in the center of the trap. Visual inspection of the droplet’s position was made through a microscope from the side window of the trap for the adjustment of the DC voltage. The trapping AC voltage was reduced along with the pressure to retain the stability of the droplet inside an electric field and to prevent the droplet from being knocked out of the null point. Evacuating continued until 48-54% relative humidity where an abrupt change in the droplet’s mass occurred, as witnessed by a large voltage jump, which was attributed to the formation of a protein crystal. The same behavior of sudden voltage drop (loss of mass) is seen in simple salt-water systems in the crystallization step. For some droplets in testing runs, the evacuation of the chamber continued to very low levels of relative humidity, and the voltage remained the same as the one after crystallization. If there were a L-L separation, the mass change would probably be due to evaporation of the liquid from a less dense phase. Then the evaporation (voltage reduction) from the droplet would continue until crystallization occurred which would cause a second voltage drop. The second voltage drop was never observed. It might be possible that the L-L separation occurs just prior to the crystallization seen as a voltage drop, and the two cannot be distinguished. The data taken for further analysis is the experimental data prior to the onset of crystallization. The crystallization process is exothermic. Once the solute crystallizes, the heat of crystallization heats the solvent causing an increase in the rate of evaporation. Consequently, the droplet loses mass very rapidly after the point of crystallization. After the droplet crystallized, the DC voltage was noted and taken to be proportional to the mass of the protein’s crystal. The relative humidity was lowered a few percent from a crystallization relative humidity, and the evacuation was stopped as not to disturb the crystal lattice by evaporating solvent from the protein crystal. A temperature controlled water reservoir served as a source for reintroducing the water vapor into the chamber to dissolve the crystal. Slightly above the deliquescence humidity, at which the crystalline phase and a liquid solution have the same thermodynamic stability, the droplet starts to take up water from the vapor phase and quickly forms a solution. Water vapor was released beyond the point of deliquescence to gather relative humidity data for a larger set of different protein and salt concentrations. Once the high level of humidity (93% and above) was reached the evacuating of the chamber began. On one suspended drop, it was possible to carry out multiple runs of crystallization and dissolving steps. An example of the experimental data for a few runs of the 1-6 Ly-NaCl condition (relative humidity vs time and voltage vs time) can be seen in Figure 2.

Results and Discussion Nine experimental conditions were chosen to work with, mass ratios of protein to salt ranging from 1:6 to 6:1 Ly-NaCl. Table 1 shows the concentrations of each solution. Solutions with these concentrations were used to fill the injector reservoir and were used to create droplets. The bulk of the data recorded fell into three temperature regimes (25, 26, and 27 °C). To extract data at constant temperature, raw data for further analysis were divided into those three temperature regions. Table 1 shows the number of runs and temperature ranges of the runs chosen for further analysis at 25 °C. The humidity of the chamber was recorded simultaneously with the DC voltage throughout the experiment. The evacuation rate of the chamber was very slow to ensure that the system was in the state of equilibrium and the activity of the water vapor phase equaled the


Figure 2. Experimental data for a 1-6 Ly-NaCl condition: relative humidity vs time (upper plot) and voltage vs time (lower plot). As the water vapor is released into the system, the relative humidity (RH) increases until the point of deliquescence. At that point, the water vapor reservoir is closed. Evacuation of the chamber via vacuum pump results in reduction in the RH, decreasing the mass (VDC) of the droplet until the solute crystallizes.


Figure 3. Relative humidity (%) vs voltage (volts): example run for 1-4 Ly-NaCl mass ratio at 25 °C. Decrease in relative humidity leads to a decrease in the droplet’s mass (VDC). Voltage jump indicates the point of solute crystallization.

Table 1. Concentrations for Each Experimental Solution,a Number of Runs, and Temperature Ranges of the Runs Taken for Analysis at 25 °C mass ratio of protein to salt 1-6 1-4 1-1 1.5-1 2-1 3-1 4-1 5-1 6-1 a

conc of solutions Ly [%] NaCl [%] 0.5 0.5 0.5 1.5 3 3 4 2.5 3

3 2 0.5 1 1.5 1 1 0.5 0.5

no. of runs

temp range

2 7 8 3 5 3 3 2 4

25.5-26.7 24.5-26.1 24.6-26.2 25.1-26.3 25.0-26.4 25.2-26.0 24.6-26.1 24.8-26.1 25.0-26.0

% is given in g of solute/g of solution.

activity of water in the droplet. Because the rate was slow and the readings were taken every second, a large number of data points were obtained. The data (humidity vs DC voltage) were reduced from thousands of points to 50-90 points per run to help ease the calculation procedure. Figure 3 is an example plot of the raw data (humidity vs DC voltage) recorded at 25 °C for a run of 1:4 Ly-NaCl mass ratio. Reduction in relative humidity causes the decrease in droplet’s mass (VDC) until the solute crystallizes (voltage jump). Included in the Figure 3 are the estimates of the errors associated with the automatic acquisition of the experimental data. Some causes of the errors could be the noise in the signal transmission or temperature fluctuation. Error percentages ranged from 0.2 to 2.0%, with most of the larger errors occurring at lower ranges of relative humidity. Water Activities in Protein Solutions. During the evacuation of the chamber, the only volatile component of the droplet is water. The mass of the protein and salt remain unchanged during the experiment. Consequently, the ratio of the mass of protein to salt is fixed during the run. Unlike salt crystals, which upon crys-

Figure 4. Water activity vs mass fraction of protein for different mass ratios of protein to salt at 25 °C.

tallization contains trace amounts of solvent inside a crystal lattice, protein crystals contain a large amount of solvent (bound and bulk water). To get a starting point for the calculation of concentrations, a crude assumption was made that the last reading of the voltage represented the dry mass of the insolubles. This was used to calculate the relative weight fractions from eq 3. Since the “dry” particle contains solvent, the relative weight fractions have to be adjusted. Steinrauf73 reports that the water content of the tetragonal form of lysozyme crystals is 33.5 wt %. An assumption was made that the crystals formed are of tetragonal form and contain 33.5% solvent. With this correction, the mass fraction of protein and salt in the drops was adjusted. Figure 4 is the plot of the water activity vs mass fraction of

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Knezic et al.

Table 2. Data Plotted in Figure 4 with Percent Errors at Each Point of the Grapha ratio: 1 to 6

1 to 1

2 to 1

wprotein

aw

std

% error

wprotein

aw

std

% error

wprotein

aw

std

% error

0.01 0.02 0.02 0.03 0.03 0.04 0.04 0.05 0.05

0.9376 0.9280 0.8798 0.8296 0.7682 0.7119 0.6307 0.5739 0.5319

0.0000 0.0000 0.0000 0.0234 0.0260 0.0214 0.0325 0.0326 0.0000

0.00 0.00 0.00 2.82 3.38 3.01 5.15 5.68 0.00

0.06 0.08 0.10 0.13 0.15 0.17 0.18 0.19 0.20

0.9311 0.9005 0.8326 0.7465 0.6591 0.6155 0.5919 0.5328 0.5228

0.0166 0.0374 0.0509 0.0397 0.0397 0.0269 0.0129 0.0152 0.0000

1.78 4.15 6.11 5.31 6.02 4.37 2.18 2.86 0.00

0.06 0.08 0.10 0.12 0.14 0.19 0.22 0.24 0.28 0.30

0.9731 0.9729 0.9554 0.9283 0.8927 0.8000 0.7366 0.6719 0.5682 0.5226

0.0000 0.0077 0.0173 0.0118 0.0161 0.0132 0.0152 0.0075 0.0164 0.0000

0.00 0.80 1.81 1.27 1.81 1.65 2.06 1.12 2.89 0.00

4 to 1

a

6 to 1

wprotein

aw

std

% error

wprotein

aw

std

error

0.20 0.22 0.25 0.27 0.28 0.30 0.32 0.35 0.38 0.40 0.42

0.9503 0.9146 0.8499 0.8326 0.8151 0.7912 0.7519 0.7084 0.6445 0.5792 0.5579

0.0000 0.0000 0.0224 0.0229 0.0188 0.0292 0.0054 0.0249 0.0333 0.0451 0.0282

0.00 0.00 2.63 2.75 2.31 3.69 0.72 3.52 5.17 7.79 5.06

0.20 0.24 0.28 0.30 0.36 0.38 0.40 0.42 0.45 0.46

0.9451 0.9177 0.8690 0.8364 0.7588 0.7218 0.6897 0.6411 0.5561 0.5436

0.0078 0.0021 0.0118 0.0109 0.0099 0.0179 0.0220 0.0135 0.0309 0.0000

0.82 0.22 1.36 1.31 1.31 2.49 3.18 2.11 5.55 0.00

std represents the standard deviation. Table 3. Coefficients of the Fit for 3-D Plot of Mass Fraction of Protein (x) vs Mass Fraction of Salt (y) vs Relative Humidity (z) at 25 °C equation: z ) y0 + ax + by + cx2 + dy2

Figure 5. Water activity vs mass fraction of protein at 25 °C. Five different protein to salt ratios of the runs presented in Figure 3. Standard deviations at each point are plotted for water activity.

protein for all of the experimental conditions at 25 °C. The end points for the mass fraction of protein were calculated from the VDC just prior to crystallization. A decrease in relative humidity results in droplet mass loss and an increase in the protein concentration, as seen in the Figure 4. Figure 5 is the plot of five different protein-to-salt ratios of the runs presented in Figure 4 for 25 °C. Included on the graph are experimental errors based on different values of water activity at particular mass fraction of protein. Table 2 presents percent errors at each point of the graph. These variations are most probably due to the fluctuation of temperature during a particular run. Calculating solute activity from the Gibbs-Duhem equation involves an integral with the lower limit being

temp [°C]

coefficient

value

std error

R2

25

y0 a b c d

100 4.1037 -97.7526 -179.1527 -258.8023

0.47 3.23 6.24 7.39 21.37

0.866

the saturated condition (eq 12). The standard state is chosen to be saturated protein solution at constant temperature of the system. Droplets in the SVELT apparatus constantly lose water until they crystallize. As a result of water evaporation, the concentrations of protein and salt continually increase. Lysozyme solubility decreases with increasing salt concentration. The data obtained for a particular mass ratio of protein to salt cannot be applied directly in the Gibbs-Duhem equation because the solubility point is constantly changing. All of the experimental data (nine different mass ratios of protein to salt) for 25 °C were plotted on a 3-D graph (relative humidity vs mass fraction of protein vs mass fraction of salt) and fitted to a paraboloidal fit. The fit was forced to go through the RH ) 100% point at zero concentration of salt and protein, which is the RH of pure water. Figure 6 shows the fit and Table 3 lists the parameters and equation of the fit. In this way, by keeping the salt concentration constant, the solubility point is fixed. Water activity vs protein concentration can be obtained from the fit at constant salt levels and used in the Gibbs-Duhem equation. Results for Protein Activity Data. Since the data for salt activity, as a function of protein concentration, are not available, the system was treated as a ternary system with salt taken as an ideal solute. Seven different mass fractions of salt (2, 3, 4, 5, 7, 8, and 10%) were held constant. Water activity vs protein concentra-



Figure 6. 3-D fit of the experimental data (mass fraction of protein vs mass fraction of salt vs relative humidity) at 25 °C. / Figure 7. Salt as an ideal solute: logarithm (aprotein/aprotein ) vs mole fraction of protein, 25 °C for different levels of salt.

Table 4. Experimental and Calculated Data for Lysozyme Solubility at 25 °Ca % NaCl

% Ly

fit

R2

2 3 4 5 7 8 10

4.790 1.409 0.445 0.304 0.135 0.073 0.039

27.21*(% salt)-2.847 27.21*(% salt)-2.847

0.97 0.97

experimental data extracted data experimental data extracted data extracted data data from power fit data from power fit a

Taken from ref 74.

tion was extracted from the 3-D fit at constant salt concentration. The saturated concentration for each experimental solution was calculated from the solubility curve obtained from Forsythe et al.74 Forsythe et al.74 determined the solubility of chicken egg white lysozyme in sodium chloride solutions (1-7% w/v) buffered with 0.1 M sodium acetate in the pH range 4.0-5.4 from 1.630.7 °C. Table 4 shows the fits for the experimental data at 25 °C and the solubilities calculated from those fits. The solubility data at 8 and 10% salt were calculated from the fitted power equation to the solubility data vs % salt. Figure 7 shows the logarithm of (aprotein/a/protein) vs mole fraction of protein at different levels of salt at 25 °C. Figure 8. shows the plot from Figure 7 for four different salt levels along with the estimated errors. Table 5 lists the percent error for each point plotted in the Figure 7. Data in Figure 7 show a sharp increase of logarithm (aprotein/a/protein) initially with protein concentration but then at higher protein concentrations the trend starts to level off. Supersaturation increases with the increase in the protein concentration driving the system more and more into the metastable region. The system becomes unstable when the spinodal curve is reached. At the spinodal, the second derivative of Gibbs free energy of solution with respect to concentration equals zero.

|

∂2∆G ∂xi2

T,P,[xj]

)0

(13)

Figure 8. Salt as an ideal solute with error bars: logarithm / (aprotein/aprotein ) vs mole fraction of protein, 25 °C at four different constant salt levels.

where ∆G is the free energy of the solution and x is the mole fraction. As expected, the trend on the plot follows the stability criterion from eq 13, since the slope of the curve decreases and approaches zero (spinodal). The same type of trend for logarithm (asolute/a/solute) is observed in simple salt-water systems and glycine-water systems investigated with the SVELT apparatus as seen in the Figure 9. The salt-water and glycine-water data were taken from Na.27 Applying the ternary system approach that treats salt as an ideal solute (eq 12) gives values for logarithm (aprotein/a/protein) 2 orders of magnitude larger than in small molecule systems previously studied, as seen in Figures 7 and 9. The high values of the logarithm of

206


Knezic et al.

/ Figure 9. Simple binary system: logarithm (asolute/asolute ) vs molality of solute. / Figure 11. Salt as an ideal solute: logarithm (γprotein/γprotein ) vs mole fraction of protein, 25 °C for different levels of salt.

/ Figure 10. Salt as an ideal solute: logarithm (aprotein/aprotein ) vs ratio of mole fraction of protein to mole fraction of protein at saturation, 25 °C for different levels of salt.

protein activity are predictable, since the mole fractions of protein are very small. Figure 10 is the plot of the logarithm (asolid/a/solid) vs the ratio of mole fraction of protein to mole fraction of protein at saturation. As expected, an increase in the supersaturation ratio results in higher values of activity ratios. The increase is sharp initially but at higher levels of supersaturation starts to level off. The effect of salt concentration can be clearly seen. Higher levels of supersaturation are achieved at higher levels of salt, which causes the system to approach the spinodal sooner. At constant activity ratio, the supersaturation increases with the salt level. Activity measures the difference in chemical potentials. For lower constant levels of salt achieving a particular difference in chemical potentials requires less supersaturation than for a higher level of salt since the solubility of lysozyme decreases with an increase in salt

concentration. Due to the polynomial 3-D fit of the experimental data and the uncertainty in the solubility point at higher salt concentrations, the logarithm of activity showed negative values at low protein concentrations, especially at higher constant salt levels. Those points were omitted from the graph. The ending points (crystallization points) of integration are not known because the data are extracted from 3-D plot. The final condition for all salt levels was taken as the highest mass fraction of protein concentration prior to crystallization achieved from the experiments for the condition 6:1 Ly-NaCl (wprotein ) 0.486). The logarithm of protein activity coefficients was calculated from activity using eq 11. Figure 11 shows the logarithm of (γprotein/γ/protein) vs mole fraction of protein at different levels of salt at 25 °C. The values of the logarithm are very high and follow the trend of the activity data. Since the values of the activity coefficients are extremely high, the values of ln(γprotein/γ/protein) are calculated up to the calculating limit of the software used. Table 6 shows coefficients and ranges for the third degree polynomial fits of the data plotted in Figures 7 and 11. Conclusion Electrodynamic levitation of single solution droplets allows investigation of homogeneous nucleation. ELT provided a practical and effective way of measuring the water activities deep into the metastable zone. High levels of supersaturations are achieved in the droplets since each droplet is very small and is not in the contact with a container wall. The metastable zone was observed to be very wide for lysozyme solutions. Even though the solubility point is constantly changing with the decrease in relative humidity, droplets continue to be supersaturated until 48-54% relative humidity. For the calculation of lysozyme activity, the Gibbs-Duhem equation was employed. For a ternary treatment (pro-



Table 5. The Percent Error for Each Point Plotted in the Figure 7 2%

4%

(ap/a/p)

std dev

% error

0.000067 0.000142 0.000226 0.000320 0.000427 0.000550 0.000692 0.000858 0.001055 0.001222

4.146 112.615 226.094 339.512 451.897 563.353 674.462 786.169 899.858 984.061

1.32 24.49 40.43 53.62 65.55 76.96 88.39 100.32 113.34 123.84

31.78 21.75 17.88 15.79 14.51 13.66 13.11 12.76 12.60 12.58

xprotein

ln(ap/a/p)

std dev

0.000070 0.000148 0.000235 0.000335 0.000448 0.000579 0.000732 0.000912 0.001128 0.001312

13.046 54.789 162.837 272.079 380.853 489.008 597.020 705.831 816.863 899.461

115.47 123.65 140.30 154.20 166.85 179.06 191.38 204.37 218.73 230.51

xprotein

ln

xprotein

ln(ap/a/p)

std dev

% error

0.000068 0.000144 0.000230 0.000326 0.000435 0.000561 0.000707 0.000902 0.001083 0.001256

26.693 131.492 242.652 354.305 465.080 575.041 684.732 808.427 907.463 990.820

67.82 91.23 107.40 120.82 132.99 144.66 156.37 170.19 182.10 193.02

254.09 69.38 44.26 34.10 28.59 25.16 22.84 21.05 20.07 19.48

% error

xprotein

ln(ap/a/p)

std dev

% error

885.07 225.68 86.16 56.67 43.81 36.62 32.06 28.95 26.78 25.63

0.000183 0.000238 0.000338 0.000453 0.000588 0.000740 0.000924 0.001168 0.001332

20.016 84.707 193.268 301.498 411.332 516.768 625.198 747.267 818.475

164.40 174.02 188.12 200.98 213.65 225.98 239.26 255.61 266.16

821.34 205.43 97.33 66.66 51.94 43.73 38.27 34.21 32.52

7%

10%

Table 6. Fits of the Third Degree Polynomial Expression for the Data Presented in Figures 6 and 10a equation: y ) a + bx + cx2 + dx3 [T ) 25 °C]

a

% salt

coefficient

value for a

range (×104)

-103.012384 1653747.438 -948682794.7 264396670920.84 0.99999

0.644-12.22

-94.128719 1657490.248 -965181063 270973343615.50 0.99999

0.644-7.232

2

a b c d R2

-45.499682 1562593.465 -824684127.3 208483797209.66 0.999911

0.236-12.39

-29.982863 1447466.608 -407444820.2 -200395519609.88 0.999797

0.236-6.690

3

a b c d R2

-78.641202 1584816.045 -883842329.4 238014904917.47 0.999989

0.652-12.56

-68.831587 1573924.958 -850142552 198833950377.63 0.999982

0.652-7.074

4

a b c d R2

-97.225612 1550930.396 -851883740.6 225340345293.06 0.999988

0.657-12.74

-87.647985 1544429.759 -837959540.4 209209835671.63 0.99998

0.657-7.816

5

a b c d R2

-150.233214 1502858.135 -815188148.5 211992044175.66 0.999989

1.156-13.12

-146.019106 1541804.805 -910293509 273607456313.69 0.999999

1.156-9.003

7

a b c d R2

-190.435508 1471007.387 -784649305 200209244866.50 0.999988

1.490-13.32

-187.520498 1513857.89 -878652390.8 256593484326.31 0.999999

1.490-10.07

8

a b c d R2

-222.868223 1461938.358 -772832045 195546521351.59 0.999988

1.831-13.32

-220.935702 1506597.28 -863655471.9 247296114120.06 0.999998

1.831-10.51

10

a b c d R2

value for γ

range (×104)

/ / x is the mole fraction of protein and y is the ln(aprotein/aprotein ) and ln(γprotein/γprotein ) for Figures 6 and 10, respectively.

tein + solvent + “ideal” salt), the trend shows initially a sharp increase of logarithm (aprotein/a/protein) with protein concentration, but then at higher protein concentrations the trend starts to level off. The same behavior

is observed in simple salt-water systems and organicwater systems investigated with SVELT apparatus. The trend seen with this approach follows the expected behavior of supersaturated solutions. As the

208


system is brought closer and closer to the spinodal point, the second derivative of the Gibbs free energy of the solution with respect to protein concentration is moving toward zero. Integration using the Gibbs-Duhem equation involves a ratio of mole fraction of solvent (number close to unity) to mole fraction of protein (very small) due to protein’s high molecular weight, which in this case results in large values for protein activity. Treating the salt component as an ideal solute is an assumption that also affects high values of the protein activity. Assumptions used coupled with the unknown function of salt activity vs protein concentration resulted in the very high and unreasonable values for aprotein/a/protein and γprotein/γ/protein. Acknowledgment. We express gratitude to Dr. Stephen Arnold and Neil Wotherspoon for many helpful discussions regarding SVELT operation. This work was supported by NASA Grant NAG8-1370. References (1) Blundell, T. L.; Johnson, L. N. Protein Crystallography; New York: Academic Press: 1976. (2) Rosenberger, F.; Muschol, M.; Thomas, B. R.; Vekilov, P. G. J. Cryst. Growth 1996, 168, 1. (3) Kam, Z.; Shore, H. B.; Feher, G. J. Mol. Biol. 1978, 123, 539. (4) Arnold, S.; Hessel, N. B. Rev. Sci. Instrum. 1985, 56, 2066. (5) Arnold, S.; Folan, L. M. Rev. Sci. Instrum. 1987, 58, 1732. (6) Arnold, S. Rev. Sci. Instrum. 1991, 62, 3025. (7) Fletcher, H. A. Phys. Rev. 1914, 4, 440. (8) Millikan, R. A. The Electron; The University of Chicago Press: Chicago, US, 1917. (9) Straubel, H. Z. Z. Electrochem. 1956, 60, 1033. (10) Gucker, F. T.; Rowell, R. L. Discuss. Faraday Soc. 1960, 30, 185. (11) Davis, E. J.; Ray, A. K.; Chang, R. AIChE Symp. 1978, 74 (175), 190. (12) Massoli, P.; Beretta, F.; D’Alessio, A. Appl. Opt. 1989, 28(6), 1200. (13) Tang, I. N.; Munkelwitz, H. R.; Wang, N. J. Colloid. Interface Sci. 1978, 63, 297. (14) Davis, E. J.; Ravindran, P. Aerosol. Sci. Technol. 1982, 1, 337. (15) Richardson, C. B.; Hightower, R. L.; Pigg, A. L. Appl. Opt. 1986, 25, 7, 1226. (16) Chang, R.; Davis, E. J. J. Colloid Interface Sci. 1976, 54, 352. (17) Cohen, M. D.; Flagan, R. C.; Seinfeld, J. H. J. Phys. Chem. 1987, 91, 4563. (18) Rubel, G. O. Aerosol Sci. 1981, 12, 551. (19) Tang, I. N.; Munkelwitz, H. R.; Wang, N. J. Colloid Interface Sci. 1986, 114 (2), 409. (20) Tang, I. N.; Munkelwitz, H. R.; Wang, N. J. Colloid Interface Sci. 1984, 98(2), 430. (21) Arnold, S. J. Aerosol Sci. 1978, 10, 49. (22) Arnold, S.; Philip, M. A.; Gelbrad, F. J. Colloid Interface Sci. 1983, 91(2), 507. (23) Seinfeld, J. H.; Grader, G. S.; Arnold, S.; Flagan, R. L. J. Chem. Phys. 1987, 86, 11, 5897. (24) Cohen, M. D.; Flagan, R. C.; Seinfeld, J. H. J. Phys. Chem. 1987, 91, 4583. (25) Grader, G. S.; Flagen, R. C.; Seinfeld, J. H.; Arnold, S. Rev. Sci. Inst. 1987, 58, 584. (26) Widmann, J. F.; Aardahl, C. L.; Davis, E. J. Trends Anal. Chem. 1998, 17(6), 339. (27) Na, H. S., Ph.D. Dissertation, Polytechnic University, New York, 1993. (28) Na, H. S.; Arnold, S.; Myerson, A. S. J. Cryst. Growth 1994, 139, 104. (29) Na, H. S.; Arnold, S.; Myerson, A. S. J. Cryst. Growth 1995, 149, 229. (30) Bohenek, M.; Myerson, A. S.; Sun, W. M. J. Cryst. Growth 1997, 179, 213.

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CG034072O

Thermodynamic Properties of Supersaturated Protein Solutions

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