PHYSICAL REVIEW E MARCH 1997VOLUME 55, NUMBER 3

Crystallization of globular proteins

W. C. K. PoonDepartment of Physics and Astronomy, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom

~Received 29 July 1996!

Globular proteins can be crystallized by the addition of electrolyte or nonadsorbing polymer. Too muchadditive, however, gives rise to amorphous precipitates. By comparing existing data on various proteins withprevious work on a model colloid-polymer mixture, we suggest an explanation for noncrystallization in termsof a hidden gas-liquid binodal inside the equilibrium fluid-crystal region of the phase diagram.@S1063-651X~97!15802-0#

PACS number~s!: 87.15.Da, 82.70.Dd

stondilyo-

telyed

kolsarn

by

om

pto

vet o

otiv

kx-

yl-eut

-is-rs tolye-

ion.l-

I. INTRODUCTION

The crystallization of a protein from solution is the firstep in the determination of its structure by x-ray diffracti@1#. Crystallization is frequently brought about by the adtion of electrolyte or of nonadsorbing polymer such as poethylene glycol~PEG!. In either case, a short-range intermlecular attraction is induced by the added component.

Recent studies of various globular proteins and relamacromolecules@2,3# have shown that crystals are onformed when the second virial coefficient of the inducshort-range intermolecular attraction,B2, becomes suffi-ciently negative. Following Rosenbaum, Zamora, and Zuski @3#, values ofB2 for a variety of short-range potentiacan be reported by mapping onto the Baxter sticky hsphere~SHS! model @4#, in which the interaction betweetwo spherical particles, diameters, is given in terms of astickiness parametert,

USHS

kBT5 lim

d→0H `, r,s

lnF 12td

s1dG , s,r,s1d

0, r.s1d.

~1!

While USHS→2` in the limit d→0, the second virial coef-ficient of the Baxter potential remains finite and is given

B2SHS52pE

0

`

~12e2U/kBT!r 2dr52

3ps3S 12

1

4t D . ~2!

The combined data of George and Wilson@2# and Rosen-baum, Zamora, and Zukoski@3# show that more than 20globular proteins and related macromolecules have a cmon crystallization boundary in thet-f plane, wheref isthe volume fraction of protein molecules, Fig. 1. Both groualso drew attention to the well-known observation thatmuch additive, or, equivalently, too negative aB2, wouldlead to amorphous precipitates. No explanation, howewas given for this rather mysterious but important aspecthe art of protein crystallization.

In this paper we give a possible theoretical explanationthe observed cessation of crystallization at large, nega

551063-651X/97/55~3!/3762~3!/$10.00

--

d

-

d

-

so

r,f

fe

values ofB2 by drawing a parallel with our recent wor@5–7# on a well-characterized model colloid-polymer miture.

II. MODEL COLLOID-POLYMER MIXTURE

Our system consists of sterically stabilized polymethmethacrylate~PMMA! particles and random-coil polystyren~PS! dispersed incis-decalin. Observations were carried oat room temperature, wherecis-decalin is a slightly betterthan theta solvent for PS@8#; i.e., the polymer coils are effectively interpenetrable as far as their mutual interactionconcerned. Previous work@9# suggests that the particles interact as nearly perfect hard spheres; the polymer appeabe nonadsorbing@5# and interacts with the particles mereby excluded volume. The exclusion of polymer from the r

FIG. 1. Phase diagram for particles with short-range attractLine, common crystallization boundary for the four macromoecules in Rosenbaum, Zamora, and Zukoski@3#. The crystallizationboundaries of globular proteins studied by George and Wilson@2#also fall closely around this curve.s, crystallization boundary forthe colloid-polymer mixture in@5#; h, noncrystallization boundaryfor the same system@7#. The vertical bar gives 0.06,t,0.15, theoptimal range for protein crystallization identified in Ref.@3#.

3762 © 1997 The American Physical Society

as

igeraal

u

a

nfiste

urle

-

edd

e

rdrkde

ionerig

-

obon

de-

th

--n

s-en-gle,oidob-

iduc-

uldtan-ne.ats.d

re

-n

oveow-on-ergypo-

ental-

tion

55 3763BRIEF REPORTS

gion between two nearby particles gives rise to an unbanced osmotic pressure pushing the particles together, reing in an attractive depletion potentialUdep(r ), which canlead to phase separation. The range and strength ofUdep canbe separately tuned by altering the polymer molecular we~and therefore coil size! and concentration, respectively. Thphase behavior depends on the polymer to particle sizej5r g /R. Small j gives an expanded region of colloidfluid-crystal coexistence@5#. For larger j, colloidal gas-liquid coexistence becomes possible@6#.

The experimental boundary between single-phase fland crystal-fluid coexistence regions forj50.08, previouslygiven @5# in terms of the colloid volume fraction (f) and thepolymer weight concentration, is mapped onto thef-t planein Fig. 1 @with B2 calculated fromUdep(r ) using equations in@6##. The deviation of our data from the average crystalliztion boundary taken from Ref.@3#, t;0.15 for 0&f&0.4, iswithin the spread of the two data sets. This gives us codence that the same fundamental physics underlies ourtem and the diverse globular protein solutions investigapreviously in@2# and @3#.

III. CESSATION OF CRYSTALLIZATION

The position of the crystallization boundary in ocolloid-polymer mixture is well accounted for by a simpstatistical mechanical theory@10,5#. In the theory, polymermolecules are treated as an ideal gas~i.e., points, a reasonable approximation for the near-u conditions used in ourexperiments!, and the colloids are hard spheres of radiusR~to model the interaction between sterically stabilizPMMA spheres!. However, a polymer molecule is excludefrom coming closer than a distanced to the surface of acolloidal particle~a simple model of the excluded voluminteraction between polymer and particle!. In practice weexpectd;r g . The phase behavior of this nonadditive hasphere model is then solved within a mean-field framewowhich predicts that the topology of the phase diagrampends on the ratiod/R.

For d/R50.08, this theory predicts crystallization forallpolymer concentrations across a crystal-fluid boundaryt&0.15, which matches well the observed crystallizatboundary in our experimental system, Fig. 1. In the expments, however, crystallization was suppressed at henough polymer concentrations~or, equivalently, att valuessufficiently lower than;0.15) @5,7#. Beyond an experimentally reproduciblenoncrystallization boundary, Fig. 1, vari-ous types of nonequilibrium aggregation behavior wereserved, culminating, at the highest polymer concentrati~or smallestt), in the formation of transient gels@7#. Exten-sive small-angle light scattering investigations of our mocolloid-polymer mixture @7# suggest that the noncrystallization boundary should be identified with ameta-stable gas-liquid binodalburied in the equilibrium fluid-crystal coexistence region, Fig. 2. The presence ofboundary can be understood as follows.

Following our earlier work@10# we write the free energyF of a colloid-polymer mixture as a function off, the col-loid volume fraction, andmp , the polymer chemical potential @11#, F5F(f,mp). F can be calculated within a meanfield framework for a disordered arrangement of colloids a

l-ult-

ht

tio

id

-

-ys-d

,-

at

i-h

-s

l

is

d

polymers, the fluid branch, and an ordered~face-centered-cubic! arrangement of colloids with polymers randomly dipersed, the crystal branch. At low polymer chemical pottials, the fluid and crystal branches each show a sinminimum, Fig. 3~a!. This gives rise to single-phase fluidfluid-crystal coexistence, or single-phase crystal. The collconcentrations in coexisting fluid and crystal phases aretained by the common tangent construction~see, e.g.,@12#!.

At higher polymer chemical potentials, however, the flubranch of the free energy shows a double minimum strture, Fig. 3~b!. The points on theequilibriumphase boundaryat this polymer chemical potential,f f andfc ~cf. Fig. 2!, arestill obtained by constructing the~lowest! common tangentbetween the fluid and crystal branches. However, one coeasily discard the crystal branch and perform the doublegent contruction on the fluid branch of the free energy aloIn this way, a gas-liquid binodal is obtained, which liesfg andf l at this polymer chemical potential; compare Fig2 and 3~b!. This metastable gas-liquid binodal is burie

FIG. 2. Schematic phase diagram of a colloid-polymer mixtuin which the ratio of the polymer to colloid size is&0.25. Thevariables are the colloid volume fractionf and the polymer chemi-cal potentialmp . A sample in theF (C) region remains a singlephase colloidal fluid~crystal!. The equilibrium phase boundary is ibold. Above this boundary, in theF1C region, a sample shouldseparate into coexisting colloidal fluid and crystal phases. Abthe dotted curve, which is the metastable gas-liquid binodal, hever, the growth of colloidal crystals is disrupted, and various nequilibrium aggregation behaviors are observed. The free endiagrams associated with the two indicated polymer chemicaltentialsmp

low and mphigh are shown in Figs. 3~a! and 3~b!, respec-

tively.

FIG. 3. Schematic free energy diagramsF(f,mp). ~a! Lowpolymer chemical potentialmp

low ~see the vertical axis in Fig. 2!.Any homogeneous colloidal fluid with a volume fraction betwef f and fc will phase separate into coexisting fluid and crysphases with those volume fractions~the common tangent construction!. ~b! At high polymer chemical potentialmp

high, the fluid branchhas a double minimum structure. The common tangent construccan be used to trace out both the equilibrium phase boundaryf f

andfc , and the metastable gas-liquid binodalfg andf l .

e

ry

hene

e

uxt

d--bleaith

deoaasetbltas

est-chrly

fnd-the

ib-nceofhein-

-ere

utofetedrchn-arch

3764 55BRIEF REPORTS

within the equilibrium fluid-crystal coexistence region in thphase diagram, Fig. 2@13#. The position of the gas-liquidbinodal predicted by our mean-field theory@10# correspondswell with the experimental noncrystallization boundafound in our colloid-polymer mixture@7#.

Our noncrystallization boundary lies very close to tt;0.06 boundary identified by Rosenbaum, Zamora, aZukoski beyond which proteins will not crystallize. Wtherefore suggest that the failure of proteins to crystallizealso due to a hidden gas-liquid binodal. The double-wstructure in the fluid branch of the free energy at lowtfavors local density fluctuations which produce amorphostructures. The metastable liquid-gas spinodal for the BaSHS model was plotted in Fig. 1 of@3#, buried inside thecrystallization region. This spinodal~and thus the associatebinodal! lies at values oft a little higher than the noncrystallization boundary att;0.06. Note, however, that the precise location of the equilibrium crystal-fluid and metastagas-liquid boundaries for the SHS system are still uncertMoreover, Stell warned that the procedure of mapping toSHS model viaB2 is likely to be increasingly invalid forvalues oft less than unity@14#.

IV. CONCLUSION

Based on a comparison with existing work on a mocolloid-polymer mixture, we have associated the failureglobular proteins with large, short-range intermoleculartractions to crystallize with the presence of a hidden gliquid binodal in the phase diagram. Indeed, in complanalogy to the case of colloid-polymer mixtures, metastaamorphous precipitates are formed rather than crys~which is the prediction of equilibrium thermodynamic!

e

D.

s

ss

en

.

rthca

d

isll

ser

n.e

lft--ee,ls

when the noncrystallization boundary,t;0.06, is crossed.The practice of relating nonequilibrium behavior~in this

case the failure to crystallize! to hidden, metastable phasboundaries within equilibrium regions of two-phase coexience is a common one in metallurgy. The possibility of sua boundary was pointed out explicitly by Cahn in an eareview@15# ~see also Sec. 10.1.6 in@12#!, and forms the basisof a metallurgical rule of thumb — the Ostwald rule ostages@16#. The presence of such a metastable phase bouary inside an equilibrium two-phase region is related totriple-minima structure of the free energy, Fig. 3~b!. Thelowest two minima give rise to two coexisting equilibriumphases~via the common tangent construction!. The third,metastable, minimum, though of no significance for equilrium phase behavior, nevertheless can decisively influephase transitionkinetics, as demonstrated by the cessationcrystallization in the systems discussed in this paper. Ttheoretical basis of such effects is the subject of activevestigation@17#.

Note added in proof:Recently, two publications have appeared that throw further light on the issues discussed h@18,19#.

ACKNOWLEDGMENTS

The author thanks Henk Lekkerkerker for pointing oRefs.@15# and@16#, and Peter Pusey for a critical readingthe manuscript. The colloid-polymer work, which forms thbackground of this paper, was funded partly by the UniKingdom Biotechnology and Biological Sciences ReseaCouncil ~formerly the Agriculture and Food Research Coucil! and the Engineering and Physical Sciences ReseCouncil.

min

nchse,em.

r, H.

ns,

v.

ett.

@1# P. C. Weber, Adv. Protein Chem.41, 1 ~1991!.@2# A. George and W. Wilson, Acta. Crystallogr. D50, 361

~1994!.@3# D. Rosenbaum, P. C. Zamora, and C. F. Zukoski, Phys. R

Lett. 76, 150 ~1996!.@4# R. J. Baxter, J. Chem. Phys.49, 2770~1968!.@5# W. C. K. Poon, J. S. Selfe, M. B. Robertson, S. M. Ilett, A.

Pirie, and P. N. Pusey, J. Phys.~France! II 3, 1075~1993!.@6# S. M. Ilett, A. Orrock, W. C. K. Poon, and P. N. Pusey, Phy

Rev. E51, 1344~1995!.@7# W. C. K. Poon, A. D. Pirie, and P. N. Pusey, Faraday Discu

101, 65 ~1995!.@8# G. C. Berry, J. Chem. Phys.44, 4550~1966!.@9# See, e.g., S. M. Underwood, J. R. Taylor, and W. van Meg

Langmuir10, 355 ~1994!.@10# H. N. W. Lekkerkerker, W. C. K. Poon, P. N. Pusey, A

Stroobants, and P. B. Warren, Europhys. Lett.20, 559 ~1992!.@11# The chemical potentialmp , which determines the polyme

fugacity, must be equal in any coexisting phases. Usingvariable reduces the number of equations to be solved inculating the phase diagram. See@10# for the relation betweenmp and the polymer concentration.

v.

.

.

,

isl-

@12# R. T. DeHoff,Thermodynamics in Material Science~McGraw-Hill, New York, 1992!.

@13# At larger polymer to colloid size ratios, this double minimucan give rise to a region of colloidal gas-liquid coexistencethe equilibrium phase diagram@10,6#.

@14# G. Stell, J. Stat. Phys.63, 1203~1991!.@15# J. W. Cahn, Transc. Metall. Soc. AIME242, 166 ~1968!, Fig.

6. In particular, compare Figs. 6~c! and 6~b! with Figs. 2~b!

and 2~c! of this paper.@16# This rule states that the first phase to emerge in a que

experiment is often not the thermodynamically stable phabut rather a metastable one. See W. Ostwald, Z. Phys. Ch22, 289 ~1897!.

@17# For a nonconserved order parameter, see J. BechhoefeLowen, and L. S. Tuckerman, Phys. Rev. Lett.67, 1266~1991!; for a conserved order parameter, see R. M. L. EvaW. C. K. Poon, and M. E. Cates~unpublished!.

@18# M. L. Broide, T. M. Tominic, and M. D. Saxowsky, Phys. ReE 53, 6325~1996!.

@19# N. Ashene, A. Lomakin, and G. B. Benedek, Phys. Rev. L77, 4832~1996!.