THE JOURNAL OF BIOLOGICAL CHEMISTRY 0 1986 by The American Society of Biological Chemists, Inc.

Vol. 261, No. 25, Issue of September 5, pp. 11544-11549,1986 Printed in U. S. A.

The Effect of Polyethylene Glycol on the Growth and Dissolution Rates of a Crystalline Protein at High Salt Concentration PHOSPHOGLUCOMUTASE*

(Received for publication, December 3, 1985)

William J. Ray, Jr.S and Joseph M. Puvathingal From the Department of Biological Sciences, Purdue University, West Lajayette, Indiana 47907

Although low concentrations of polyethylene glycol (1-5%, w/v) are essential for growing crystals of phos- phoglucomutase from ammonium sulfate solutions (at close to 50% of saturation), the observed rate constant for short-term crystal growth on a defined, microcrys- talline surface is essentially independent of polyethyl- ene glycol concentration under these conditions. But this cosolute produces a substantial increase in the observed rate constant for the dissolution process and thus a corresponding increase in the solubility of the crystalline phase. These obvervations can be rational- ized in terms of a decrease in the thermodynamic ac- tivity of the soluble form of phosphoglucomutase at high salt due to favorable interactions with polyethyl- ene glycol (PEG) at the protein surface, coupled with a difference in accessibility of protein surfaces in the crystalline and solution states. Surfaces with a differ- ential exposure in these two phases likely include both groups that interact favorably with polyethylene gly- col relative to water (nonpolar groups) as well as those that interact unfavorably (ionic groups), but favorable PEG-protein interactions produced on dissolution must outweigh unfavorable ones. A PEG-induced increase in protein solubility at high salt concentration is likely to be general; PEG also may affect the growth of other protein crystals at high salt concentrations as it affects phosphoglucomutase.

The development of crystal growth procedures for proteins has long been considered as more of an art than an exact science. But recent work (McPherson, 1982; Wycoff et al., 1985) suggests that at least the outlines of order are beginning to emerge. Our current studies attempt to further this trend by at least partially defining the role of a cosolute, polyeth- ylene glycol 400 (PEG’), whose presence affects the growth of crystalline phosphoglucomutase at high salt concentration in ways that range from critical to indispensable. Thus, al- though this protein occasionally nucleates and forms tiny crystals at high concentrations of salt and protein in the absence of PEG, the cosolute plays a catalytic role in nuclea- tion that enormously increases its success (Ray and Bracker, 1986). In addition, PEG (or some such cosolute) is indispen-

* The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked ‘‘aduertkement” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

$ Recipient of United States Health Service Grant GM08963. To whom reprint requests should be addressed.

PEG is used for polyethylene glycol 400 unless otherwise specified; per cent ammonium sulfate refers to percentage of a saturated solu- tion.

sable to long-term growth of mm-size crystals.* The present paper describes a study of how PEG affects crystals of phos- phoglucomutase in short-term growth/dissolution experi- ments designed to probe the mechanistic basis of its effects on the overall crystallization process. Since the results show that PEG produces relatively minor effects, if any, on crystal growth rates during time intervals of up to 3 days, the striking effects of PEG on nucleation and long-term growth almost certainly are thermodynamic in origin and likely arise from PEG-protein interactions in solution that may be general at high salt concentration.

EXPERIMENTAL PROCEDURES

Materials and Methods-The major isozyme of rabbit muscle phos- phoglucomutase was obtained according to the procedure described by Ray et al. (1983); two successive chromatographic separations were used as the final steps, and the enzyme was precipitated by dialysis against 60% (NH4),S04 and “crystallization buffer” (Ray, 1986). A microcrystalline suspension of the enzyme was prepared in 53% (NHASO, according to procedure a of Ray and Bracker (1986). The longest dimension of most crystals in such suspensions was <lo pm, and a well-mixed suspension of such crystals could be sampled re- producibly. Just before use, the microcrystals were washed twice by suspension in a fresh buffer/salt solution followed by centrifugation. Protein concentrations were measured either by optical density (Ray et al., 1983) or by a modification (cj. Read and Northcote, 1981) of Bradford‘s procedure (1976). In the modified procedure, the color yield was assessed at 25 ‘C after a 2.0-min incubation, and the sample size used in the assay was varied somewhat so that most optical density changes were between 0.15 and 0.25. For good reproducibility, the protein reagent was stored in the dark and discarded after 2 days, the volume of added sample was minimized (10-25 p1/1.4 ml of reagent), and the assay was conducted in a cuvette equipped with a Teflon stopper. Cuvettes were filled almost to capacity, mixed simi- larly by inversion after addition of the sample, and cleaned with methanol between each measurement. The procedure was calibrated before each run with samples of a stock protein solution of phos- phoglucomutase. Other materials have been described (Ray, 1986).

Conventions-The concentration of polyethylene glycol is ex- pressed in terms of weight/volume and the concentration of (NH4),S04 is expressed as percentage of the concentration in a saturated aqueous solution.

RESULTS

Effect of Polyethylene Glycol 400 on the Solubility of Phos- phoglucomutase Crystals in Ammonium Sulfate Solutions-A stock suspension of microcrystals of the protein was used for all solubility studies. But even microcrystals dissolve exceed- ingly slowly in solutions of (NH,),SO, sufficiently concen- trated to produce an equilibrium solubility that is convenient for such studies (-1 mg/ml) even when the suspension of crystals is slowly rotated to minimize settling. Hence, suspen- sions of microcrystals containing added soluble protein at

* W. J. Ray, Jr. and J. M. Puvathingal, to be published elsewhere.

11544

GrowthlDissolution of Protein Crystals: Effect of PEG 11545

concentrations both above and below the suspected equilib- rium solubility were utilized. Changes in soluble protein were assessed after slowly rotating the samples for 3 days in small containers in equilibrium with a large, closed reservoir of buffer at the same concentration. Plots of the change in soluble protein uersw the initial concentration of soluble protein were continuous over a range of protein concentra- tions that produced both positive and negative changes; they also were linear (cf. Fig. la ) . Such plots allow estimation of the equilibrium solubility of the protein (uiz. the concentra- tion of soluble protein that would remain unchanged in the presence of added microcrystals) by interpolation, as is shown in Fig la. Concomitant studies were conducted at 0.5, 2, 3, and 4% PEG. Protein solubilities as a function of PEG concentration under otherwise identical conditions are shown in Fig. Ib. Thus, in 50% (NH4)*S04, PEG increases the solubility of the protein; a 4% concentration produces a sol- ubility increase of about %fold.

Effect of Polyethylene Glycol 400 on the Solubility of Small Model Compounds-In some cases, PEG (at a concentration of 20%) produces a modest increase in the water solubility of nonpolar compounds, e.g. toluene, as well as polar compounds with nonpolar side chains, e.g. N-acetyltryptophanamide, i.e. PEG interacts with them more favorably than does water. But the (fractional) increase in solubility produced by PEG at a concentration of 3% is substantially greater in 50% (NH&S04 than in water (data available on request). (Al- though both of the above compounds are substantially less soluble at high salt, the standard Gibbs energy change that accompanies the transfer of a solute from a solution lacking PEG to one that contains PEG is related only to solubility ratios.) In contrast with the above, the partitioning of glucose and glucose 6-phosphate between the salt-rich and PEG-rich phases of the biphasic mixture shown below shows that some polar groups as well as polar-ionic groups interact unfavorably

-0.3 -0.4 ~ 0.6 0.8 1.0 1.2 0 1 2 3 4 0

Ci , mq/ml '10 PEG ( W/V 1 FIG. 1. Effect of polyethylene glycol 400 on the solubility

of phosphoglucomutase crystals. a, change in the concentration of soluble protein, AC, in the presence of a fixed quantity of micro- crystals at 3% PEG. Microcrystals of phosphoglucomutase at 1.6 mg/ ml were suspended in 50% (NH4),S04 in the presence of crystalliza- tion buffer, variable concentrations of soluble protein, Ci, and 3% PEG (w/v). The mixtures were placed in small containers in the presence of a large reservoir (100 ml) of the same solution (without protein) in a larger closed container and rotated slowly (about 1 revolution per 10 min) a t room temperature to minimize the effect of settling of the microcrystals. After 3 days, samples were removed from each container and centrifuged, and the supernatant was assayed for soluble protein. The arrow shows the equilibrium solubility, C,. b, solubility of phosphoglucomutase crystals as a function of PEG concentration. The interpolated solubility was obtained from studies conducted as in a with the same soluble and microcrystalline protein, but at different PEG concentrations.

-1.5 I -3.5 I/

I' I -3.0 -2.0 -1.0 0

In IC+-C,I FIG. 2. Effect of PEG concentration on the rate of crystal

growth close to the equilibrium solubility. axperiments were conducted as for Fig. 1. A concentration of microcrystals substantially larger than the change in concentration of protein during the exper- iment was employed. Logarithmic plots involving the initial (CJ, the final (Ct) , and the equilibrium (C,) concentrations of soluble protein were constructed according to Equation 8. Filled and open symbols show results obtained at initial concentrations that produced an

PEG; triangle, 2% P E G inverted triangle, 3% PEG; circle, 4% PEG. increase or a decrease in soluble protein, respectively: square, 0.5%

The best correlation of the data by means of a straight line with a unit slope is shown.

with polyethylene glycols in aqueous solutions, as is expected PEG-rich phase, 50% (NH4)2S04 and 5% PEG-400 (<0.2% PEG-8000); salt-rich phase, 33% PEG-8000, 17% PEG-400, and 5% (NH4),S04. Since the surface of a protein is a mosaic of nonpolar, polar, and ionic groups (Arakawa and Timasheff, 1984, 1985b, and references therein), the interaction of PEG with proteins, especially at high salt, must be considered in terms of both favorable and unfavorable interactions with surface groups.

Effect of Polyethylene Glycol on the Rate of Growth of Phos- phoglucomutase Crystals from Ammonium Sulfate Solutions- The extent of growth or dissolution of a suspension of micro- crystals of phosphoglucomutase was assessed as in the pre- ceding solubility studies under conditions where the change in the soluble protein, AC, was substantially smaller than the total concentration of microcrystalline protein that was used, so that the surface area of the crystalline phase would remain essentially constant during the e~periment.~ (Because of this requirement, the change in the soluble protein concentration also was kept small relative to the initial concentration, Ci; the scatter in the data obtained reflects the small size of AC/ Ci.) Since the time was kept constant in these studies, the variables were the initial concentration of soluble protein and its value at time, tC,. (Both are expressed in milligrams/ milliliter.) Hence, the results were plotted in Fig. 2 according to Equation 8 (see the Miniprint4), uiz. as InlC, - C,l uersus lnlCi - Gel, whether the observed process involved crystal growth (filled symbols) or dissolution of the crystal phase (open symbols). According to Equation 8, the slope of such a

Growth rates with a given batch of seed crystals are found to be proportional to the concentration of seed crystals employed, but essentially independent of the extent to which growth occurs, in several studies with relatively insoluble salts (c.f Nancollas and Purdie, 1964).

Equations 1-10 are presented in miniprint at the end of this paper. Miniprint is easily read with the aid of a standard magnifying glass. Full size photocopies are available from the Journal of Biolog- ical Chemistry, 9650 Rockville Pike, Bethesda, MD 20814. Request Document No. 85M-3933, cite the authors, and include a check or money order for $1.20 per set of photocopies. Full size photocopies are also included in the microfilm edition of the Journal that is available from Waverly Press.

11546 GrowthlDissolution of Protein Crystals: Effect of PEG

line should be 1, and a line with unit slope was constructed through the center of gravity of the data (see Fig. 2). The observation that the data for both growth and dissolution of microcrystals are correlated reasonably well by such a line supports the assumptions made in obtaining Equation 8, i.e. that rates are proportional to the first power of the supersa- turation or subsaturation (or that m = n = 15; see the Mini- print). The extrapolated y intercept of the line in Fig. 2 a t h l C i - C,l = O is equal to kihst. Since a single line correlates data from studies conducted at PEG concentrations of 0.5- 4%, hi”‘ is relatively independent of PEG concentrating over this range.

Effect of Polyethylene Glycol 400 on Dissolution Rates of Phosphoglucomutase Crystals in Ammonium Sulfate Solu- tions-The rates of dissolution of protein microcrystals in a stirred cell a t a protein concentration much less than its equilibrium solubility were assessed in terms of changes in light scattering (monitored at 340 nm) that accompany the dissolution process. Although this procedure does not produce data that can be readily quantitated in terms of milligrams of protein solubilized per unit time, relative rates with a standard slurry of crystals should be inversely related to the time required to produce a given optical density change (here a decrease to between 0.95 and 0.90 of the initial optical den- sity). The advantage of this procedure is that it can be applied to dilute suspensions of microcrystals (about 0.1 mg/ml) so that relatively little soluble protein is generated during the measurement. In these experiments, it proved critical to min- imize friction between the stirrer and the cell because friction tended to denature the soluble protein and thus interfered with light scattering measurements. But with the stirring assembly shown in Fig. 3, all of the added microcrystals could be dissolved without generating a significant amount of resid- ual light scattering due to denatured protein (see Fig. 4a). In all dissolution studies (except those reported in Fig. 4 4 , the medium in which crystals were dissolved was the same as that in which they were suspended prior to the study so that initiating the dissolution process did not alter the concentra- tion of any nonprotein component.

The rate at which the crystals dissolved was independent of stirring rate from about 150 to 600 rpm. Indeed, the dissolution rate seemed to be essentially unaffected by discon- necting the stirrer (Fig. 4b) (although settling of the micro- crystals eventually became a problem). Both theory and ob- servation indicate that changing the stirring rate can alter dissolution rates for a variety of processes where diffusion (transport) is rate-limiting (Bircumshaw and Riddiford, 1952; Howard et al., 1960; Jones, 1964; Lam et al., 1978). But the size of the crystals used in these experiments may not be large enough to use this approach in ruling out a diffusion-limited dissolution rate (Purcell, 1978).

Increased dissolution rates with increased PEG concentra- tion (Fig. 4c) are in accord with,the observations above since hihs = kzh”/Ce, hihS is essentially independent of PEG (Fig. 2), and C, increases with PEG concentration (Fig. lb). In fact, the PEG-induced increase in kzbs of about 6-fold (0-4’36 PEG) was twice as large as the corresponding increase in C, (-3- fold) (Fig. lb). This difference probably is caused by the far- from-equilibrium conditions used in Fig. 4c that can alter the surface function, ( s ) ~ , in Equation 4 (i.e. can produce a kinetic surface roughening (cf. Gilmer and Jackson, 1977)).

Dissolution rates also increase markedly with decreasing (NHJ2S04 concentration. A semilog plot of relative dissolu- tion rates uersus ammonium sulfate concentration is only

Possible values of m and n other than 1 were tested, but all failed to provide nearly as good a correlation as did m = n = 1.

~ ”” - “”y

FIG. 3. Photograph of the stirring unit used for assessing relative dissolution rates of microcrystals. The unit is held in a 1-cm inner diameter Pyrex tube by the O-ring in the Teflon base plate. A Teflon-covered magnetic stirrer is held by the vertical Teflon rotor into which are pressed two support rods (No. 6 gauge stainless steel wire). The support arm, No. 18 gauge stainless steel wire appropriately flattened in places, was drilled to provide a housing for one of the support rods; a hole drilled in the base plate provided the other housing. The assembly was fixed above a belt-driven, alnico, magnetic wheel. When the cuvette plus stirrer was in place, the top of the support arm extended to just below the bottom of the slits in the sample compartment of a Perkin-Elmer Model 570 spectropho- tometer. Two ml of solution was required to extend the liquid column above the top of the slits.

slightly curved (Fig. 4d) and resembles the type of plot ex- pected for equilibrium solubility as a function of salt concen- tration (Arawaka and Timasheff, 1985a), uiz. as though the variation in equilibrium solubility were determined solely by variations in k$h* as opposed to variations in kpohfi. But the possibility of kinetic roughening casts doubt on the possibility that the observed dissolution rate at C << C, can be readily related to the observed dissociation rate a t equilibrium, kSb”. Nevertheless, this study suggests that a slow-growing protein crystal will also dissolve slowly, even a t “zero” protein con- centration, as well as at concentrations of precipitant well below those a t which the crystal grew initially.

DISCUSSION

During initial attempts to grow millimeter-size crystals of phosphoglucomutase for diffraction studies (Lin et al., 1986), a number of problems were encountered that were so dramat- ically alleviated by including low concentrations of polyeth- ylene glycol 400 in the crystal growth medium that studies were initiated to determine the mechanism of the PEG- induced effect in this system and whether such effects might be general. The present study, which focused on kinetic and equilibrium effects in 50% ammonium sulfate, provided three somewhat unexpected results: PEG substantially increased the solubility of the crystalline phase, even when it was used at relatively low concentration (1-4%, w/v); PEG did not substantially alter the observed rate constant for incorporat- ing protein into the lattice, although the rate of incorporation under otherwise constant conditions was decreased and PEG substantially increased the rate constant for dissolution of the lattice. Aside from the interesting implications about protein-cosolute interactions, these observations suggest that

G r o ~ t ~ ~ ~ ~ s o l ~ t i o n of Protein Crystals: Effect of PEG 11547 Tim. hrs Time, min

0 2 0 4 0 Tim, min (NH,),SO.,, X of Saturation

FIG. 4. Results of dissolution studies with microcrystals. Except for the study in d, the stock suspension of microcrystals contained about 20 mgfml protein and the suspending medium was the same as that in which the crystals were to be dissolved; in d, the concentration of (NH4f2S04 in the stock suspension was maintain^ at 50%. a, plot of the change in optical density at 340 nm as a function of time during the complete dissolution of a sample of microcrystals equivalent to approximately 0.2 mg of protein in 2 ml of 50% (NH4)2SOI that contained crystallization buffer and 3% PEG. The stirring device in Fig. 3 was used. Points show optical density values taken from the recorder trace at 5-min intervals. b, results of three successive dissolution studies analogous to those in a except that 48% (NH,),SO, was used and only the initial phase of studies conducted at different stirring speeds was recorded: top, -600 rpm; middle, -150 rpm; bottom, no stirring after the first 5 s. A photograph of the recorder output is shown. c, traces of recorder outputs are reproduced that show the effect of PEG concentration on dissolution rates of microcrystals in 50% ammonium sulfate and crystallization buffer. d, semilog plot of the effect of ammonium sulfate concentration on the initial dissolution rate of microcrystals (linear phase of the process, as in e). Rates are expressed in arbitrary units, and studies were conducted in the presence of crystallization buffer and 2% PEG.

the important catalytic role that PEG plays in the nucleation of phosphoglucomutase crystals (Ray and Bracker, 1986) and the indispensable role it plays in long-term growth2 do not depend on a PEG-protein lattice interaction that is unique to the present system, i.e. that the critical interactions represent solution effects, not lattice effects. The observation that PEG can be removed from millimeter-size crystals of phosphoglu- comutase by a simple washing procedure without significantly altering their x-ray diffraction patterns (Lin et al., 1986) also indicates that PEG is not an essential component of the lattice.

The PEG-induced increase in protein solubility at a high salt concentration observed here contrasts with its effect at low ionic strength (McPherson, 1982; Wycoff et al., 1985). At low ionic strength, the overall effect of the polyether is dom- inated by its exclusion volume (Lee and Lee, 1981; Atha and Ingham, 1981; Knoll and Hermans, 1983; Arakawa and Ti- masheff, 1985b). Thus, a large exclusion volume produces a reduction in protein solubility if the cosolute interacts with the protein either indifferently or less favorably than water

(cf. Arakawa and Timasheff, 1985b). But especially in aqueous solutions that elevate its activity coefficient (e.g. high salt), PEG interacts favorably with at least two models for nonpolar side chains of proteins, relative to other solution components (see “Results”).G On the other hand, in the same solutions, PEG will interact unfavorably with ionic groups and some nonpoiar groups (see “Results”; see also Arakawa and Ti- masheff, 198513). This difference provides a basis for ration- alizing the precipitating effect of PEG on proteins at low salt concentration and the solubilizing effect at high salt since the surface of a protein must be considered as a mosaic of non- polar, polar, and ionic groups in evaluating its interactions with cosolutes (cf. Arakawa and Timasheff, 1984, 198513; Ar- akawa and Goddette, 1985, and references therein). But any rationale that involves cosolute-induced changes in the solu- bility of a protein must treat the effect of the protein: cosolute interactions in the crystal as well as the solution phases (Rupley, 1969), especially in the case of a crystal such as that of phospho- glucomutase where approximately 60% of the volume is “sol- vent” (approximated from parameters from Lin et d., 1986). Thus, added PEG will alter protein solubility only if the net interaction of the cosolute with surface groups that become accessible on dissolution of the crystal (or become inaccessible when crystals form) is favorable or unfavorable, thermody- namically. In the present case, the favorable interactions, presumably with nonpolar groups, produced by dissolution in the presence of PEG must be more important than the unfa- vorable ones simultaneously produced.

Of course, the groups that become accessible during the dissolution of a crystal may or may not be representative of the entire protein surface. But since the transfer of phos- phoglucomutase from salt-rich to PEG-rich phases is ther- modynamically favorable (Ke > 20; Ray and Bracker, 1986), it is likely that favorable PEG-protein interactions exceed unfavorable ones at high salt concentrations over the entire surface of the protein, not just in the area that is exposed during dissolution of the crystalline phase. Moreover, a net favorable PEG-protein interaction at high salt may be a general phenomenon since a number of common proteins partition between PEG-rich and protein-rich phases in a similar manner (see Ray and Bracker, 1986). In addition, polyethylene glycols, even at low salt concentrations, exhibit net favorable interactions with ~-lactoglobulin (at high PEG concentration; Arakawa and Timasheff, 198513) and lower the thermal denaturation curves of a number of protein^.^ The lowering is greatest for those that might be classified as hydrophobic proteins and thus is in accord with the observa- tions reported here since the work of Arakawa and Timasheff (1984 and references therein) has verified the close relation- ship between denaturants that lower thermal melting curves and “solubilizers” that exhibit preferential interaction with proteins. On the other hand, phosphoglucomutase is much too soluble to be classified as hydrophobic, at least with regard to its surface residues.’

Kinetic studies show that growth and dissolution rates for crystals of phosphoglucomutase depend on the difference between the concentration of protein in solution and the concentration at the equilibrium solubility, i.e. on the extent of supersaturation or subsaturation, respectively (see the Min- iprint; see also Feher and Kam, 1985). When treated in this

Polyethers such as polypropylene glycol also interact favorably (relative to water) with the nonpolar side chains of amino acids, even at low ionic strength (Gekko and Koga, 1984).

I

’ J. C. Lee, personal communication. W. J. Ray, unpublished results.

11548 GrowthlDissolution of Protein Crystals: Effect of PEG

way, two unexpected features of the growth/dissolution proc- ess were a first power dependence of growth rate on the supersaturation and a dissolution rate that increased with increasing PEG concentration.

The lack of a substantial PEG-induced effect (Fig. 2) on the rate constant for lattice growthg can be rationalized in terms of a rate equation which includes the following PEG- dependent activity coefficients for the protein: ys (in solution), Y$, (in the “transition state” leading to incorporation into the lattice), and y1 (in the lattice itself), as in Equations 9 and 10. The observation that the decrease in crystal growth rate produced by increasing PEG concentration under otherwise constant conditions arises primarily from the increase it pro- duces in crystal solubility, [C,], means that the ratio ys/yg is effectively unaltered by PEG. This requires that the transition state for the incorporation step be “solution-like” in terms of its net solvation by PEG. Hence, it is the subsequent forma- tion of specific lattice contacts that produces the net decrease in favorable PEG-protein interactions characteristic of the lattice (see above). Most of these late-developing specific contacts may well be nonpolar, but the only thermodynamic requirement is that their formation produce a net unfavorable contribution to PEG-protein interactions (see above).

Since added PEG does not substantially alter the rate constant for crystal growth, it must increase crystal solubility by increasing the rate constant for the dissolution process. This conclusion was verified directly (Fig. 4c). An induced increase in the dissociation rate constant can be rationalized in terms of the mechanism for ligand-induced dissociation of a metal ion from a multidentate complex. Such dissociations, like the dissociation of a protein molecule from the surface of a crystal, are multistep processes, and interactions with the acceptor ligand can stabilize early intermediates and thus lower the energy of the transition state for the rate-limiting bond-breaking step (cf. Kula and Rabenstein, 1967; Janes and Margerum, 1966; Ray, 1967). Thus, the increased value of yl/ ys (increased solubility) produced by added PEG is reflected in (increase in &&) since the transition state is solution- like in terms of favorable PEG-protein interactions (see above). This means that a cosolute, PEG, which is indispen- sable for long-term crystal growth in the present system, acts primarily to increase the rate constant for dissolution, not for growth. A rationale for this observation will be described elsewhere?

Although the effect of the ammonium sulfate precipitant on the rate constant for lattice growth was not assessed, the rate constant for dissolution exhibits an inverse relationship to salt concentration (Fig. 4d), i.e. exhibits a relationship that likely is proportional to some function of water activity. An increase in dissolution rate at lower salt concentration prob- ably is produced by a more effective solvation of the transition state for dissolution (as well as prior intermediates) that exceeds the more effective solvation of the surface of the protein that is accessible to solvent in the crystalline lattice. In fact, the increase in equilibrium solubility with decreasing ammonium sulfate concentration could be caused by this

@ The observed effect of PEG on the morphology of phosphoglu- comutase crystals shows that protein molecules can be added to the growing crystal in a t least three different ways. Each mode of addition involves different types of protein-lattice interactions (see Footnote 2). Hence, any attempt to describe overall growth/dissolution rates

that likely will depend somewhat on protein concentration and the in quantitative terms will provide only a composite rate “constant”

extent of growth (cf. Gilmer, 1980). But in practice, an effective

could be obtained during a limited growth period. (average) rate constant for incorporating protein into lattice, kBObs,

effect alone. If so, ammonium sulfate, like PEG, would affect growth rates solely by its effect on the equilibrium solubility of the protein, i.e. solely by altering the extent of supersatu- ration (as expressed by the factor (1 - CJC), Equation 3), as does the cosolute, PEG, and not by altering the rate constant for the growth process. But additional studies will be required to verify this possibility.

Since the growth rates of very few proteins have been measured as a function of conditions (however, see Feher and Kam, 1985), it is not possible to draw broad generalizations by comparing the present results with those from other stud- ies. However, the present studies have identified features of the growth process that will be considered in a comparative manner in other studies.

A c k n o ~ ~ d g ~ n ~ - W e are indebted to G. Feher and 2. Kam for a copy of their manuscript prior to publication and to G. Feher and S. Durbin for helpful discussions.

REFERENCES Awakawa, T., and Goddette, D. (1985) Arch. Biochem. Biophys. 240,

Arakawa, T., and Timasheff, S. N. (1984) Biochemistry 23, 5924-

Arakawa, T., and Timasheff, S. N. (1985a) Methods Enzymol. 114 ,

Arakawa, T., and Timasheff, S. N. (1985b) Biochemistry 24, 6756-

Atha, D. J., and Ingham, K. C. (1981) J. Biol. Chem. 2 5 6 , 12108-

Bircumshaw, L. L., and Riddiford, A. C. (1952) Q. Reu. (London) 6 ,

Bradford, M. M. (1976) Anal. Biochem. 72,248-254 Davies, C. W., and Jones, A. L. (1985) Trans. Faraday SOC. 51,812-

Feher, G., and Kam, Z. (1985) Methods EnzymoL 141, 77-131 Gekko, K., and Koga, S. (1984) Biochim. Biophys. Acta 786,151-160 Gilmer, G. H. (1980) Science 208,355-363 Gilmer, G. H., and Jackson, K. A. (1977) in Crystal Growth and

Materials (Kaldis, E., and Schuel, H. J., eds) p. 80, North-Holland Biomedical Press, Amsterdam

Howard, J. R., Nancollas, G. H., and Purdie, N. (1960) Trans. Faraday

Janes, D. L., and Margerum, D. W. (1966) Znorg. Chem. 6,1135-1140 Jones, A. L. (1964) Trans. Famday Sac. 59,2355-2361 Knoll, D., and Hermans, J. (1983) J. Biol. Chem. 258,5710-5715 Kula, R. J., and Rabenstein, D. L. (1967) J. Am. Chem. Soc. 89,552-

Lam, C.-Y., Nancollas, G. H., and KO, S.-J. (1978) Znuest. Urol. 15,

Lee, J. C., and Lee, L. L. Y. (1981) J. Biol. Chem. 256,625-631 Lin, 2.-J., Konno, M., Abad-Zapatero, C., Wierenga, R., Murthy, M.

R. N., Ray, W. J., Jr., and Rossmann, M. G. (1986) J. Biol. Chem.

McPherson, A., Jr. (1982) P~paratioR and Analysis of Protein Crys-

Nancollas, G. H., and Purdie, N. (1961) Trans. Faraday SOC. 57,

Nancollas, G. H., and Purdie, N. (1964) Q. Reu. Chem. Sac. Lo&. 18 ,

Purcell, E. M. (1978) J , Fluid Mech. 84,551-559 Ray, W. J., Jr. (1967) J. Biol. Chem. 2 4 2 , 3737-3744 Ray, W. J., Jr. (1986) J. BioL Chem. 261,275-278 Ray, W. J., Jr., and Bracker, C. E. (1986) J. Cryst. Growth, in press Ray, W. J., Jr., Hermodson, M., Puvathingal, J. M., and Mahoney,

W. C. (1983) J. Biol. Chem. 258,9166-9174 Read, S. M., and Northcote, D. H. (1981) Anal. Biochem. 116, 53-

64 Rupley, J. A. (1969) in Structure and Stability of Biological Macro-

molecules (Timasheff, s. N., and Fasman, G. D., eds) p. 291, Marcel Dekker, Inc., New York

Wycoff, H. W., Hirs, C. H. W., and Timasheff, S. N. (eds) (1985) Methods in Enzymology, vol. 114, part a, Academic Press, New York

21-32

5929

49-77

6762

12117

157-185

817

SOC. 56,278-283

556

473-477

261,264-274

tals, Wiley, New York

2272-2279

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Growth/Dissolution of Protein Crystah: Effect of PEG 11549

Tb. I&?f.ct of Wlyothylone Dlysol on tho crovth and Supploeentary Itataria1 to

st sigh salt concontration! PhOsphoqlUCDeutafe Oissolution Patea of a Ctptallino Protmin

William J . Ray, Jr. and Joseph M. Puvathingail

of Protein Plus W e to the -. The rate of growth of rslatively in8oluble ionic crystals from aqueous eolution, e.-,., silver chloride, barium sulfate, or eaqnesiue oxalate, varies with sou0 power, m, of the extent of =rsaturation, i..., 111th (c-c~J', whora ce is the aquilibriw eolubility of

Correspondingly, diesolution rates vary with s01e power, n, 02 the extent or tho malt, c is its conc~ntration at a qiven time, and c > ce.

- aubeaturation, i..., vith (c,-cI~ ( D ~ V ~ O S and Jones, 1 9 5 8 ) J Nancollas and Purdie, 1961 and i964i LaD et al.. 1978) where n frequently is not equal to a:

-dC/dt - k;(s)g(C-C,,)m (11

dC/dt - k;t(s)d(Ce-c)n (21

mra, the subscripts. q and d, refw to growth and dissolution proces8es.i the rate constants, k; and k&& arm p.t rate Constant= while (e)-, and are surface functions. Undar fixed oonditions. (a ) in proportional to the amount of added e m d crystab; it need not be the same in growth and dissolution .xperiments conductad undsr substantially different conditions, but nust be emsontially the sa- close to equilihriusi (if the Deed crystals are the same).

can w m i t t a n in t e r u of rate constants:' Whbn a - n - 1, em in the praaent 8yst.m (erne Results), aquati0ns I end 2

-dC/dt - kpfm)-,(C) (I-CJC) (3)

dC/dt - kd(s)d(l-C/CO) ( 0

On integrating equation. 3 and 4 , one obtains

1n(ct-ce)/(ci-c0) - -tg(o)-,t - -robat (5)

ln(ce-ct)/(ca-ci) - -td(s)dt/ce - -k:bst/ce (6 )

Harm. subscripts i and t specify the initial concentration of solublm C and th. concentration at a given ti=. c1o.s to apuilibrlul,

k p - Lobe 4 /ce (71

%&&&&a mhL?U%x is aonitorsd during either p r o m stud1.s (c > eo) or H*nc?., W a t i o n s 5 and 6 are the earno, Lib' is Hanutad vhsn +ppuunp &

diSsO1utiOn studiaa (C < CO) (sme meults), and both equation. can be combined to give

lnice-ctl - -ktb% + 1nlc,,-ciI 68)

protoin in the cryatallin. lattico (a1 -Ti), and the "transition stat." for Thm thaz8odynauic activities of th0 woluble protein (ae - Ceye) , the

incorporating protein molecules into the lattice (a provlda a 80- rigorous uprresion analogous to equdionCfj?This exprsseion

can be u m d to

allow ana to lomiat0 a rational for tha effect Of added cosolutse on the lattia growth procus:

kzba - kq181,(yJ&) ( 9 )

kqYs - k4WC. (10)

If .I;"' is indapudont of add& oasolutm (e. obsarzed: Oee Rssults). the cosoluta eithar does not altar the activity CMffiCimts of the solution and transition stabs ( ys and y+) or it altors both to eB#Ontially the extent (equation 9 ) . If Co i a eltorod by the cosolute (a. observed: .*e R.su1c.s). ys and yl muat b. effmotad to diffarent sxtant.,