Comparison of the Crystallization Kinetics of Canavalin and Lysozyme

Comparison of the Crystallization Kinetics of Canavalin and Lysozyme. Katiuska G. ... Publication Date (Web): March 11, 2006. Copyright ... Using dila...
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CRYSTAL GROWTH & DESIGN

Comparison of the Crystallization Kinetics of Canavalin and Lysozyme Katiuska G.

Caraballo,#

James K.

Baird,*,#

and Joseph D.

2006 VOL. 6, NO. 4 874-880

Ng‡

Departments of Chemistry and Biological Sciences, UniVersity of Alabama in HuntsVille, HuntsVille, Alabama 35899 ReceiVed June 17, 2005

ABSTRACT: Using dilatometry, we examined the crystallization kinetics of the jack bean storage protein, canavalin. We found that the kinetic rate law is first order in the supersaturation. At pH values above the pI ) 5.2, where the protein is negatively charged, the apparent rate constant, k, increases as the pH approaches the pI, decreases with increasing temperature, and increases with increasing salt concentration. The rate constant is essentially independent of the initial protein concentration. Precisely the same patterns of behavior were observed by Kim et al. [Mol. Phys. 2003, 101, 2677-2686] in dilatometer experiments involving the crystallization of lysozyme even though lysozyme molecules are positively charged under crystal growth conditions. By combining dimensional analysis with a simple theory of protein crystal nucleation and growth, we show that k ≈ ω/cs, where ω is the rate coefficient that controls the rate of advance of the crystal facets, and cs is the solubility of the protein in the growth solution. We determined the dependence of ω upon temperature and salt concentration by analyzing the observations of Forsythe and Pusey [J. Cryst. Growth 1994, 139, 89-94] of the rate of advance of the facets of growing lysozyme crystals. To the extent that our data overlap that of Forsythe and Pusey, we are able to show for both canavalin and lysozyme that the dependence of k, ω, and cs upon these variables is consistent with k ≈ ω/cs. 1. Introduction 1.1. Protein Crystals and X-ray Diffraction. Knowledge of the three-dimensional arrangement of the atoms in a protein molecule often serves as the key to understanding the biological function of the protein. Although X-ray diffraction from single crystals can be employed to solve for the molecular structure of proteins, use of this method is sometimes restricted by the limited availability of the protein crystals themselves. Indeed, nature has conspired to make the crystallization of proteins difficult due to the deleterious effects that this event would otherwise cause in a living organism. Nevertheless, in the laboratory it is still possible to crystallize many water-soluble proteins under sufficiently nonphysiological conditions from pH-buffered aqueous solutions of strong electrolytes.1 1.2. Thermodynamics vs Kinetics. From a theoretical point of view, the appearance of protein crystals in a crystal growth solution can be regarded as being controlled either by the thermodynamics or by the kinetics of the crystallization process.2 A protein crystallization experiment is said to be under thermodynamic control when the protein concentration is less than the solubility of the crystals, or when, as commonly occurs, the protein concentration exceeds the solubility of an amorphous solid phase, which precipitates but otherwise lacks the periodic structure that is a prerequisite for X-ray crystallography. The crystallization is said to be under kinetic control when the crystals are slow to appear even though the solution has been made thermodynamically unstable by dissolving enough protein to be in excess of the solubility. Kinetic control involves the subtle interplay between the mechanisms of crystal nucleation and crystal growth. Despite * To whom correspondence should be addressed: Phone (256) 824-6441. Fax: (256) 824-6349. E-mail: [email protected]. # Department of Chemistry. ‡ Department of Biological Sciences.

this complexity, a crystallization trial will be under kinetic control when the relative supersaturation, σ(t), defined by

σ(t) )

( )

c(t) -1 cs

(1)

is greater than zero. In eq 1, c(t) is the protein concentration in the crystal growth solution at time, t, and cs is the protein solubility. Kinetic control is the subject of this paper. Using dilatometry to follow the time rate of decay of the supersaturation, we have studied the kinetics of the crystallization of canavalin. To set the stage for a discussion of our canavalin results, we first provide in the next three sections a theoretical interpretation of some experimental observations obtained by ourselves and others on the rate of nucleation and growth of crystals of lysozyme. A comparison of canavlin with lysozyme then follows. This comparison is apt, because in the case of both proteins, the effects of pH, salt concentration, and temperature have been carefully examined. Moreover, the comparison has general significance because the two proteins are sufficiently different with respect to origin, molecular weight, and macromolecular charge that any shared crystallization features may be representative of the crystallization of water-soluble proteins as a whole. 1.3. Lysozyme Crystal Growth Kinetics Determined by Optical Microscopy. Optical microscopy has been used to observe the nucleation3,4 of crystals of lysozyme and also to measure the linear rate of growth5 of their facets. Applying optical microscopy, Galkin and Vekilov3,4 reported that homogeneous nucleation and heterogeneous nucleation occur simultaneously in crystallizing solutions of lysozyme. On the other hand, Dixit et al.6 have argued that the method of supersaturation control used by Galkin and Vekilov3,4 has the effect of drastically underestimating the rate of nucleation. If so, an underestimation could obscure the distinction between homogeneous and heterogeneous nucleation. In agreement with

10.1021/cg058016u CCC: $33.50 © 2006 American Chemical Society Published on Web 03/11/2006

Crystallization Kinetics of Canavalin and Lysozyme

Crystal Growth & Design, Vol. 6, No. 4, 2006 875

Kuznetsov et al.,7 who assert that heterogeneous nucleation is the rule, rather than the exception, Forsythe and Pusey,5 and also Kim et al.,2 reported that crystals of lysozyme nucleate readily on glass surfaces. Using an automated version of optical microscopy, Forsythe and Pusey5 made extensive observations of the rates of growth of the (110) facets of individual lysozyme crystals nucleated on the glass walls of a cell containing a pH buffered aqueous solution of NaCl. For solution conditions, 22 °C, pH ) 4, and a NaCl concentration of 5% w/v, they plotted in their Figure 2, log L(c) vs log(c/cs), where L(c) is the linear growth rate of the (110) facet of a lysozyme crystal, and c > cs is the protein concentration. For (c/cs) > 7.9, their data lie on a straight line. Where that straight line prevails, we found by curve fitting that

L(σ) ) ω(σ + 1)n

(2)

where n ) 2.2, and ω ) 2 × 10-5 µm/s. If eq 1 is substituted into eq 2, we obtain

L(c) ) (ω/cs)ncn

(3)

Equations 2 and 3 can be used to understand the temperature dependence and the salt concentration dependence of the growth rate data reported by Forsythe and Pusey.5 Because log L is a monotonic function of L, we can draw our conclusions concerning functional dependence of log L upon temperature and salt concentration by examining the functional dependence of L on these variables. In their Figure 3a, Forsythe and Pusey5 plot log L(c) vs c for pH ) 4, 5% w/v NaCl, and various values of the temperature within the range of 4-22 °C. In their Figure 3b, they show the same data plotted in the form log L(σ) vs (σ + 1). On the basis of the substantial separations that are visible between the individual curves in Figure 3a,b, respectively, it is clear that for fixed c, the values of L(c) decrease with increasing temperature, while for fixed σ, the values of L(σ) increase with increasing temperature. The reversal of the temperature dependence of these data when plotted as a function of c, as opposed to (σ + 1), is not coincidental. According to eq 2, if the index n can be assumed to be fixed, as is the case in most kinetic rate laws, the temperature dependence of the curves of log L(σ) vs (σ + 1) should be determined by the temperature dependence of the rate coefficient, ω . In Figure 3b of Forsythe and Pusey, the increase in log L(σ) with increasing temperature at fixed σ can be understood, if ω is an increasing function of temperature, as is to be expected if the growth of lysozyme crystals is a thermally activated process. By contrast, according to eq 3, the temperature dependence of L(c) at a fixed value of c is determined by the temperature behavior of both ω and cs. Lysozyme dissolves endothermically, so cs increases with increasing temperature.8 If cs increases with temperature more strongly than ω, then by virtue of eq 3, we can expect L(c), and also log L(c), at a fixed value of c, to decrease with increasing temperature. This is precisely the behavior of the data plotted in Figure 3a of Forsythe and Pusey. 5 For advancing (110) facets of lysozyme, Forsythe and Pusey have also reported the variation of L with salt concentration at 22 °C and pH ) 4. For growth solutions containing 3, 5, and 7% w/v salt, they show their data plotted in the form, log L(c) vs c, in Figure 4a, and plotted again in the form, log L(σ) vs (σ + 1), in Figure 4b. According to eq 2, the dependence of L(σ) on salt concentration at a fixed value of σ is determined by dependence of ω on

salt concentration. Although the data in Figure 4b of Forsythe and Pusey5 overlap a bit, L(σ) appears to increase with increasing salt concentration. We can thus conclude that ω also increases with increasing salt concentration. At pH ) 4, where Forsythe and Pusey5 collected their growth rate data, the facets of growing lysozyme crystals adsorb H+ from the growth solution.2,9,10 Individual lysozyme macromolecular ions in the growth solution have a charge of about +12 at this pH.11 The presence of dissolved salt in the growth solution serves to provide Debye-Huckel plasma screening of these charges.2,9-14 The screening reduces the mutual electrostatic repulsion opposing the addition of each successive protein macromolecular ion to the surface of the crystal. Since a fully charged macromolecular ion cannot be incorporated into a unit cell without some charge compensation, a charge neutralization mechanism may also intervene, in addition to Debye-Huckel plasma screening, to reduce the effects of electrostatic repulsion.13 For example, using careful X-ray crystallography, Sauter et al.15 located one sodium ion and five chloride ions per lysozyme macromolecular ion in lysozyme crystals. Dauter et al.16 found as many as eight Cl- per macromolecular ion in their lysozyme crystals. Since Dauter et al.16 grew their crystals at pH ) 5, where a lysozyme molecule is a macromolecular cation having a charge of about +8, eight chloride ions would be just sufficient to neutralize all of the H+ that would otherwise serve as a source of electrostatic repulsion between macromolecular ions and crystals. Since the effects of both Debye-Huckel screening and charge neturalization increase with increasing salt concentration, the specific rate of macromolecular ion addition, ω, should also increase with increasing salt concentration. This is the pattern of behavior followed by the data plotted in Figure 4b of Forsythe and Pusey.5 In contrast to the apparent behavior of ω, the solubility of lysozyme, cs, decreases with increasing salt concentration.8,17 According to eq 3, if cs decreases, while ω increases, their combined effect on log L(c) at a fixed value of c should be to cause log L(c) to increase with increasing salt concentration. This is precisely the behavior of the data plotted in Figure 4a by Forsythe and Pusey.5 From the above analysis of the data of Forsythe and Pusey,5 it is clear that when the protein composition variable is chosen to be the supersaturation, σ, the effective rate coefficient is ω, whereas when it is chosen to be the concentration, c, the effective rate coefficient is equal to ω/cns , which is a composite of ω and cs. We argue below that the rate coefficient which determines the time rate of decay of the supersaturation in a dilatometer experiment is also a composite of ω and cs. 1.4. Lysozyme Crystal Growth Kinetics Determined by Dilatometry. In our recently completed set of investigations of lysozyme crystallization kinetics, we used a dilatometer to follow the time decay of the lysozyme supersaturation in crystallizing, pH buffered aqueous solutions of sodium chloride. We found that our measurements of σ(t) could be described by the equation,2

ln

( )

σ(t) ) -kt σ0

(4)

where σ0 ) (c0/cs) - 1 is the initial value of the supersaturation in the growth solution, c0 is the initial value of the protein concentration, and k is the apparent rate coefficient. To construct a model from which eq 4 could be derived, we assumed that the dilatometer contained a fixed number, N, of

876 Crystal Growth & Design, Vol. 6, No. 4, 2006

Caraballo et al.

heterogeneous nucleation sites per unit volume of growth solution. From each of these nucleation sites, a crystal could be expected to grow at a rate that depended upon the supersaturation, σ. Although power laws of the form, σn (n g 4 ) have sometimes been proposed18-20 to represent the supersaturation dependence of the rate of protein crystal growth, we accept here the opinion of the majority of investigators,21-25 who regard that rate to be directly proportional to σ. This permits us to adopt the Ataka and Asai representation,26 V(σ) ) νσ, for the volume rate of growth of a crystal, where ν is a constant. If the mass density of a protein crystal is F, then the rate of increase of the mass of a crystal growing from the solution at time, τ, is equal to FV(σ(τ)) ) Fνσ(τ) . Taking into account the N heterogeneous nucleation sites per unit volume of solution, the rate of increase of crystalline mass per unit volume of growth solution at time, τ, is NFνσ(τ). The total mass of crystals per unit volume of solution at a time, t, is then NFν∫t0 σ(τ)dτ. Since the volume concentration of the dissolved protein is c(t), and since there are no crystals in the solution at time, t ) 0, conservation of mass dictates that

c0 ) c(t) + NFν

∫0t σ(τ)dτ

(5)

where c0 is the initial concentration of dissolved protein. Notice in eq 5 that the concentrations, c0 and c(t), which are measured in the units, mg/mL, appear together with the supersaturation, σ, which is dimensionless. Should we want to cast eq 5 entirely in terms of the concentrations, c0 and c(t), or alternatively, entirely in terms of the supersaturations, σ(t) and σ0, the solubility, cs, must be introduced. Choosing σ(t) as the dependent variable, we divide eq 5 through by cs, and subtract unity from both sides. The result is

σ0 ) σ(t) + (NFν/cs)

∫0t σ(τ)dτ

(6)

If we then differentiate both sides of eq 6 with respect to the time, t, we obtain the differential equation,

dσ(t) ) -kσ(t) dt

(8)

(9)

for the protein concentration, and

m(t) ) (c0 - cs)(1 - e-kt)

(11)

If we then combine eqs 8 and 11, we find

k ≈ NFl2ω/cs

(12)

Equation 12 suggests that the apparent rate constant for decay of the supersaturation, k, is a composite of ω and cs. 2. Experimental Section

Representing the initial value of σ(t) as σ0, we can integrate eq 7 to obtain eq 4. All of the properties of the crystal growth solution that depend on σ(t) can be derived from eq 4. For example, by combining eqs 1 and 4, we obtain

c(t) ) cs + (c0 - cs)e-kt

ν ≈ l 2ω

(7)

where the effective rate coefficient, k, is given by

k ) NFν/cs

microscopy growth rate data of Forsythe and Pusey,5 are by comparison rather much larger than the values of the supersaturation explored by Kim et al.2 in their dilatometer experiments. Nevertheless, the Forsythe and Pusey5 linear growth rate function, L(σ) ) ω(σ + 1)n, can be made consistent with the Ataka and Asai26 volume growth rate function, V(σ) ) νσ, if we expand L(σ) in a Taylor series about zero supersaturation. After discarding an unphysical constant term, which has its origin in the large values of σ covered by eq 2, and which interferes with the approach of L(σ) to zero for small values of σ, we obtain the first power equation, L(σ) ≈ nωσ, which can be compared with the first power equation, V(σ) ) νσ, of Ataka and Asai.26 By finding a connection between ω and ν, we can make L(σ) and V(σ) consistent. Such a connection must necessarily exist, because in the late stages of the growth of a crystal, when the numbers of molecules forming the corners and edges of the facets are small compared with the number of molecules in the crystal as a whole, the rate of growth of the volume of the crystal must necessarily be proportional to the average rate of growth of its facets. In the absence of a completely satisfactory theory of protein crystal nucleation and growth,27 we turn to dimensional analysis12 to develop a plausible link between ω and ν. We begin by noting that ω has the dimensions, length/s, while ν has the dimensions, (length)3/s. Since Forsythe and Pusey5 and Kim et al.2 agree that lysozyme crystals nucleate heterogeneously on glass, the missing dimensioned quantity linking ω and ν may very well be the linear size, l, of the typical nucleation site. If so, then we propose on dimensional grounds that

(10)

for the mass, m(t) ) c0 - c(t), of protein crystals per unit volume of growth solution. 1.5. Relation Between Rate Coefficients Observed in Microscopy and Dilatometry. We have seen that optical microscopy can be used to determine the rate of growth of crystal facets, while dilatometry can be used to determine the rate of growth of crystal volume. The values of the supersaturation, σ, covered by eq 2, which describes the optical

2.1. Choice of Canavlin. The experiments of Forsythe and Pusey5 and Kim et al.2 on lysozyme revealed the effects of a positive macromolecular charge on the rate of protein crystallization. For comparison with these effects, we sought to crystallize a protein, which when in the form of a macromolecular ion carried a negative charge under crystallization conditions. The rate of decay of the supersaturation during the crystallization of this protein should also be easy to follow in a dilatometer as a function of temperature, pH, and salt concentration. The jack bean (CanaValia ensiformis) storage protein, canavalin, satisfactorily met these requirements. Before canavalin can be crystallized, however, the protease, trypsin, must be used to clip a peptide bond in each of two loops in the primary structure. The result is a canavalin macromolecule having three terminal carboxyl groups and three terminal amino groups,28 a molecular weight of 47 kDa, and an isoelectric point (pI) at pH ) 5.2.29 The value of the pI is within the range of many common buffers. Below neutral pH, three canavalin molecules self-assemble into a trimer of molecular weight about 142 kDa.29 On the basis of the primary structure of canavalin, we calculated the charge, Z1, on a canavalin monomer macromolecular ion as a function of pH. A description of our method can be found in the appendix. Using for our calculations the Scatchard equation, which assumes no interaction between ionizable groups, we found the pI to be located at pH ) 4.7, which is about 0.5 pH units below the measured value at pH ) 5.2. A plot of our calculated values of Z1 as a function of pH can be found in Figure 1. At a given value of pH < 7, the charge on the trimer should be about three times that on the monomer.

Crystallization Kinetics of Canavalin and Lysozyme

Crystal Growth & Design, Vol. 6, No. 4, 2006 877

Figure 1. A plot of the net charge, Z1, on a canavalin monomer as a function of pH. According to Figure 1, the net negative charge on the canavalin monomer decreases monotonically from -14 to -16 over the pH range 6.6 to 7.2, which was sampled by our dilatometer experiments. This pH range lies on the basic side but still relatively close to the pI at pH ) 5.2. By contrast, our dilatometer experiments employing lysozyme covered the pH range from 4.2 to 5.1 lying rather far to the acid side of the pI that occurs at pH ) 11.1.11 Over this pH range, a lysozyme macromolecular ion carries a net positive charge of about +10. If for pH values between 6.6 and 7.2, canavalin crystals, like canavalin macromolecular ions, are negatively charged, then we can expect, as in the case of lysozyme,2 that electrostatic repulsion between the macromolecular ions and the crystals will play a prominent role in the crystallization kinetics. To make the time scales for the crystallization of the two proteins roughly comparable, the NaCl concentration in our canavalin experiments was restricted to the range of 0.4% w/v to 1% w/v. 2.2. Protein Preparation. Canavalin was isolated from whole jack beans purchased from Sigma-Aldrich Corporation (St. Louis, MO) following the procedure of Sumner and Howell30 that involves bean grinding, defatting, and protein extraction.31 After dissolution with distilled water containing trace amounts of ammonium hydroxide, the extracted protein was recrystallized at least three times by dialysis against 2% w/v sodium chloride in 0.5 M sodium phosphate buffer at pH ) 6.8. The final purified protein sample, which appeared as a mass of rhombohedral crystals, was redissolved to a concentration of 30 mg/ mL at 20 °C in an aqueous solution of sodium chloride containing a mono/dibasic sodium phosphate buffer. The sum of the concentrations of the two buffer salts was in all cases equal to 0.05 M. We found that rhombohedral single crystals of canavalin appeared in this solution when the pH was in the range 6.6 to 7.2 and the temperature was 15 °C and below. 2.3. Dilatometry. To start a kinetics experiment, the canavalin crystal growth solution was used to fill a 12 mL dilatometer having a capillary sidearm with an inside diameter 0.3 mm. The dilatometer was hand blown from Pyrex glass.2 The filled dilatometer was maintained isothermal by immersing it in a 14 L water bath, the temperature of which could be controlled to about (0.01 °C over a two-week period. An operating temperature below room temperature was achieved by cooling the water bath with a coil of copper tubing carrying chilled water. As the crystals appeared, the level of the fluid in the dilatometer sidearm fell. This column of liquid was under continuous surveillance by a video camera connected to a videocassette recorder.31 The height, h(0), at time t ) 0, and the height, h(t), at time t, were combined to compute the change in height ∆h(t) ) h(0) - h(t). From the long-time asymptote of the height, h(∞), the value ∆h(∞) ) h(0) - h(∞) was computed. The values of ∆h(t) and ∆h(∞) were substituted into the dilatometer formula2

∆h(t) σ(t) )1σ0 ∆h(∞)

(13)

to compute the ratio σ(t)/σ0.

3. Results and Discussion 3.1. Experimental Results. Theory suggests that the independent variables that may affect the rate of decay of the

Figure 2. First-order decay of the supersaturation ratio, σ/σ0 as a function of kt for various values of the pH. The rate constant is k, and the time is t. The initial protein concentration was c0 ) 20 mg/mL, temperature ) 7.9 °C, and NaCl concentration ) 1.0% w/v.

Figure 3. First-order decay of supersaturation ratio, σ/σ0, as a function of kt for various NaCl concentrations. The initial protein concentration was c0 ) 20 mg/mL, pH ) 6.6, and temperature ) 7.9 °C. Table 1. Rate Constant, k, for Canavalin Supersaturation Decay as a Function of pH for the Crystallization Conditions Initial Protein Concentration c0 ) 20 mg/mL, Temperature ) 7.9 °C, and NaCl Concentration ) 1.0 % w/v pH

k (day-1)

6.60 6.80 7.00 7.20

0.91 0.86 0.82 0.80

canavalin supersaturation are likely to be pH, salt concentration, temperature, and initial protein concentration.2,11,14,32 With three of these four variables fixed, k was measured as a function of the fourth. The applicability of eq 4 to the data could best be demonstrated by plotting ln(σ(t)/σ0) vs kt. When data plotted in this dimensionless form closely follow a straight line of slope, -1, it is an indication that σ(t) decays exponentially with t regardless of the conditions; any systematic deviations from this line illustrates the failure of the exponential law, while random deviations reflect the experimental error. In the case of canavalin, the line could be clearly traced up to a time equal to about four half-lives, after which further changes in the height of the column of fluid in the sidearm became difficult to discern. Figure 2 shows a plot of this type for various values of the pH. The dependence of the rate constant, k, upon pH under these conditions is summarized in Table 1. The error associated with each value of k in this and subsequent tables was estimated to be about (0.006 day-1. Figure 3 shows a plot of ln(σ(t)/σ0) vs kt for crystallizing canavalin solutions for various concentrations of NaCl. The dependence of the rate constant upon NaCl concentration is shown in Table 2. Figure 4 shows a plot of ln(σ(t)/σ0) vs kt for various values of the temperature. The

878 Crystal Growth & Design, Vol. 6, No. 4, 2006

Caraballo et al. Table 4. Rate Constant, k, for Canavalin Supersaturation Decay as a Function of Initial Protein Concentration, c0, for the Crystallization Conditions pH ) 6.6, Temperature ) 7.9 °C, and NaCl Concentration ) 1.0 % w/v

Figure 4. First-order decay of supersaturation ratio, σ/σ0, as a function of kt for various values of the temperature. The initial protein concentration was c0 ) 20 mg/mL, pH ) 6.6, and NaCl concentration ) 1.0% w/v. Table 2. Rate Constant, k, for Canavalin Supersaturation Decay as a Function of the NaCl Concentration for the Crystallization Conditions Initial Protein Concentration c0 ) 20 mg/mL, pH ) 6.6, and the Temperature ) 7.9 °C % NaCl

k (day-1)

0.4 0.6 0.8 1

0.67 0.77 0.81 0.91

Table 3. Rate Constant, k, for Canavalin Supersaturation Decay as a Function of Temperature for the Crystallization Conditions Initial Protein Concentration c0 ) 20 mg/mL, pH ) 6.6, and NaCl Concentration ) 1.0 % w/va T (°C)

1/T (10-3 K-1)

k (day-1)

15.0 10.3 7.9 5.0

3.47 3.52 3.55 3.59

0.67 0.75 0.91 1.16

a An Arrhenius plot of ln k vs 1/T (10-3 K-1) makes a straight line of positive slope indicating that the apparent activation energy is negative.

Figure 5. First-order decay of the supersaturation ratio, σ/σ0, as a function of, t, for various values of the initial protein concentration. The pH ) 6.6, temperature ) 7.9 °C, and NaCl concentration ) 1.0% w/v.

dependence of the rate constant on temperature is summarized in Table 3. The decrease in k with increasing temperature that is apparent in Table 3 implies that the effective activation energy is negative. In Figure 5, we have plotted ln σ(t)/σ0 vs t for various values of c0 to illustrate the weak dependence of k upon initial protein concentration. The corresponding values of k are summarized in Table 4. Although these values of k differ by more than the estimated error of (0.006 day-1, the dependence of k upon c0

c0 (mg/mL)

k (day-1)

10 15 20 25

0.88 0.94 0.91 0.96

is neither monotonic nor is it quite as large as the dependence of k upon pH, NaCl concentration, and temperature as is evidenced in Tables 1-3. We suggest that k is at most a weak function of c0 and may indeed be independent of this variable. In our canavalin experiments, we observed micron-sized crystals of the protein to form not only on the walls of the dilatometer but also to collect in a pile at the bottom. The material on the bottom likely arrived there by gravitational sedimentation of crystals produced by heterogeneous nucleation on foreign particles suspended in the bulk solution. Heterogeneous nucleation in the bulk solution cannot be discounted, because although we employed distilled water, and the dilatometer was thoroughly washed between runs, we exerted no special effort to screen out microscopic foreign particles from our crystal growth solutions. Since we used the same dilatometer for each run and the water came from the same source, we can assume, as in section 3, that N is a constant, independent of solution conditions. 3.2. Effect of the Solubility on the Apparent Rate Constant. De Mattei and Feigelson33 measured the solubility of canavalin in 1% w/v aqueous NaCl solutions as a function of pH. They found that at 9 °C the solubility started from less than 1 mg/mL at pH ) 5 and increased monotonically, approaching 30 mg/mL at pH ) 7.5. By interpolation of the data in Figure 1 of De Mattei and Feigelson,33 we concluded that the solubility increases by about 10% over the pH range 6.6 to 7.2 investigated by us. As in the case of most proteins, the solubility of canavlin increases as the pH departs from the isoelectric point. Although the temperature was 9 °C in the solubility experiments of De Mattei and Feigelson,33 we can probably assume at the temperature of 7.9 °C, where our kinetic measurements were made, that cs depends more strongly upon pH than does ω; then, as is the case with lysozyme, the decrease in k recorded in Table 1 over this pH range is consistent with eq 12 and the increase in cs that occurs as the pH gets further from the isoelectric point. There appear to be no measurements of the solubility of canavalin as a function of salt concentration. Nevertheless, if the solubility, cs, of canavalin decreases with increasing salt concentration, as it does in the case of most proteins, and if ω increases with increasing salt concentration, as in the case of lysozyme, then we can conclude on the basis of eq 12 that k should increase with increasing salt concentration. This is precisely the behavior exhibited by the values of k summarized in Table 2. De Mattei and Feigelson33 reported that the heat of crystallization of canavalin averaged about -20 kJ/mol over the pH range 6.6 to 7.2. Since dissolution is the reverse of crystallization, this implies a positive heat of solution. We can represent ω in eq 12 by the Arrhenius equation, ω ) ωL exp(-EL/RT), and cs by the van’t Hoff equation cs ) b exp(-∆Hs/RT) . Here, R is the universal gas law constant, and T is the absolute temperature. In the Arrhenius equation, ωL is the preexponential factor, and EL is the activation energy for adding a protein molecule to the surface of the crystal, while in the van’t Hoff

Crystallization Kinetics of Canavalin and Lysozyme

equation, b is related to the entropy of dissolution, and ∆Hs is the heat of solution of the protein. When these representations for ω and cs are substituted into eq 12, the apparent activation energy, E, that governs k is found to be equal to EL - ∆Hs. Should the canavlin crystal, like the canavalin macromolecular ion, carry a net negative charge, the electrostatic repulsion between the ion and the crystal should make EL positive. Because canavalin dissolves endothermically (∆HS > 0), the decrease in k with increasing temperature, which is apparent in Table 3, can be explained if ∆Hs > EL. 4. Conclusions When we compare the kinetics of lysozyme crystallization2 with the kinetics of canavalin crystallization,31 we find in both cases firm evidence that the time decay of the supersaturation, σ, is governed by first-order kinetics and eq 4. That is to say, plots such as Figures 2-5, were found in the case of both proteins. In the case of both proteins, the rate coefficient, k, increased with increasing salt concentration, decreased with increasing temperature, and was independent of the initial protein concentration, c0. In our experiments with lysozyme,2 the range of pH investigated was below the value of pI, where the protein was positively charged, while in the case of canavalin,31 the range of pH was above the pI, where the protein was negatively charged. Nevertheless, in both cases, the value of k increased as the pH moved closer to the pI. Since for both proteins, the value of k increased with increasing salt concentration, it appears that increasing salt concentration serves to reduce cs and to increase ω. We can summarize the effects of salt and pH upon k by stating that k increases with any change in solution conditions that serves to neutralize the electrostatic effects of the macromolecular charge. In the case of both lysozyme and canvalin, k decreased with increasing temperature, from which we can conclude that the effective activation energy that governs k is negative. Since both proteins dissolve endothermically (∆Hs > 0), however, it is possible to explain the apparent negative activation energy in terms of eq 12, if ∆Hs > EL. The general agreement between our dilatometer data sets for lysozyme and canvalin persisted even though in the case of lysozyme2 the value of k ranged between 0.34 and 2.88 day-1 under the conditions tried, while in the case of canavalin,31 it was restricted to the more limited range between 0.67 and 1.16 day-1. This agreement is even more significant in the face of the fact that lysozyme is an animal protein with a molecular weight of 14 kDa and a positive charge under crystallization conditions, whereas canavalin is a plant protein with a molecular weight of 142 kDa and a negative charge under crystallization conditions. Our dilatometer experiments can be used to predict the time scale governing protein crystal growth experiments under batch conditions. Equations 9 and 10 demonstrate that the disappearance of the dissolved protein and the build-up of crystalline protein mass are both determined by the same rate constant, k, that governs the time decay of the supersaturation. Since the kinetics are first order, the half-life for the disappearance of dissolved protein and for the appearance of the crystals is given by t1/2 ) 0.693/k. Assuming that cs is more strongly dependent upon solution conditions than is ω, then according to eq 12, the half-life should be shortest where the solubility is a minimum. We have previously shown that it is possible to follow the time decay of the protein concentration in a crystallizing

Crystal Growth & Design, Vol. 6, No. 4, 2006 879 Table 5. Ionizable Amino Acid Residues and Their pKa Values for the Canavalin Monomera

aThe

amino acid

pKa

n

aspartic acid glutamic acid cysteine tyrosine carboxyl group lysine arginine histidine amino group

3.86 4.25 8.33 10.07 2.34 10.53 12.48 6.0 9.69

20 29 2 14 3 16 20 5 3

number of times a residue occurs in the monomer is n.

lysozyme solution by sampling the solution periodically and determining the optical density of the aliquots.12,13 Although at the time we were performing these experiments, we did not yet understand that the kinetics of decay of the concentration were first order, we did notice, nevertheless, that the half-life increased with increasing temperature and decreased with increasing salt concentration. This behavior is just the inverse of what we observed for k in the case of our dilatometer experiments, but since t1/2 and k are inversely proportional, the results of the aliquot and dilatometer experiments are entirely consistent. When comparing the aliquot and dilatometer methods as to accuracy and ease of operation in the laboratory, one should note that the observation of the height of the fluid in a dilatometer sidearm is a non-invasive measurement, whereas the harvest of an aliquot for optical density measurement necessarily disturbs the growth solution. So far as comparison of experimental results with theoretical growth laws is concerned, the dilatometer method gives σ(t)/σ0 directly through eq 13, whereas in the aliquot method, this quantity must be calculated from separate measurements of c(t) and cs. Acknowledgment. This research was sponsored by the National Institute of General Medical Sciences of the National Institutes of Health under Grant No. 1 R15 ΒΜ 510018 and is offered by K.G.C. in partial satisfaction of the requirements for the M.S. in Materials Science from the University of Alabama in Huntsville. K.G.C. would like to thank the Southeastern Universities Research Association for summer stipends during 2000 and 2001. The authors also extend their thanks to Dr. Javier Fuentes and Dr. Yeong Woo Kim for help with the experimental apparatus. Appendix For a canavalin monomer, we list in Table 5, the pKa value for each ionizable side chain as well as the number of times, n, that the associated amino acid residue occurs in the canavalin monomer. Shown also are the pKa values for the acid forms of the terminal carboxyl and amino groups. To calculate the net charge, Z1, on the canavalin monomer as a function of pH, we substituted the pKa values and the amino acid residue populations in Table 5 into the Scatchard equation11

Z1 )

∑j nj

10-pH

Kj + 10-pH

-

∑i ni

Ki

Ki + 10-pH

(A.1)

In eq A.1, Kj is the ionization constant, and nj is the number of positively charged groups of the type, j, while Ki is the ionization constant and ni is the number of negatively charged groups of type, i. The results have been plotted in Figure 1. References (1) McPherson, A. F. Crystallization of Biological Macromolecules; Cold Spring Harbor Laboratory Press: New York, 1999.

880 Crystal Growth & Design, Vol. 6, No. 4, 2006 (2) Kim, Y. W.; Barlow, D. A.; Caraballo, K. G.; Baird, J. K. Mol. Phys. 2003, 101, 2677-2686. (3) Galkin, O.; Vekilov, P. G. J. Phys. Chem. B 1999, 103, 1096510971. (4) Galkin, O.; Vekilov, P. G. J. Am. Chem. Soc. 2000, 122, 156-163. (5) Forsythe, E.; Pusey, M. L. J. Cryst. Growth 1994, 139, 89-94. (6) Dixit, N. M.; Kulkarni, A. M.; Zukoski, C. F. Colloids Surf. A 2001, 190, 47-60. (7) Kuznetsov, Yu. G.; Malkin, A. J.; Lucas, R. W.; McPherson, A. Colloids Surf. B 2000, 19, 333-346. (8) Forsythe, E. L.; Judge, R. A.; Pusey, M. L. J. Chem. Eng. Data 1999, 44, 637-640. (9) Holmes, A., Holliday, S. G.; Clunie, J. C.; Baird. J. K. Acta Crystallogr. 1997, D53, 456-457. (10) Lee, H.-M.; Kim, Y. W.; Baird, J. K. J. Cryst. Growth 2001, 232, 294-300. (11) Baird, J. K.; Kim, Y. W. Mol. Phys. 2002, 100, 1855-1866. (12) Baird, J. K.; Hill, S. C.; Clunie, J. C. J. Cryst. Growth 1999, 196, 220-225. (13) Baird, J. K.; Clunie, J. C. Phys. Chem. Liq. 1999, 37, 285-295. (14) Baird, J. K.; Scott, S. C.; Kim, Y. W. J. Cryst. Growth 2001, 232, 50-62. (15) Sauter, C.; Otalora, F.; Gavira, J.-A.; Vidal, O.; Giege, R.; GarciaRuiz, J. M. Acta Crystallogr. 2001, D57, 1119-1126. (16) Dauter, Z.; Dauter, M.; de La Fortelle, E.; Bricogne, G.; Sheldrick, G. M. J. Mol. Biol. 1999, 289, 83-92. (17) Howard, S. B.; Twigg, P. J.; Baird, J. K.; Meehan, E. J. J. Cryst. Growth 1988, 90, 94. (18) Durbin, S. D.; Feher, G. J. Cryst. Growth 1986, 76, 583-592.

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