effects of hydrodynamic convection and interionic electrostatic forces on protein crystallization

Effects of Hydrodynamic Convection and Interionic ElectrostaticForces on Protein CrystallizationPublished as part of the Crystal Growth & Design virtual special issue on the 14th International Conference onthe Crystallization of Biological Macromolecules (ICCBM14).

James K. Baird* and Robert L. McFeeters

Department of Chemistry and Material Science Graduate Program, University of Alabama in Huntsville, Huntsville, Alabama 35899,United States

ABSTRACT: The biological function of a protein is intimatelyrelated to its three-dimensional molecular structure. Although X-ray diffraction from single crystals can be employed to solve forthe molecular structure, use of this method is often impeded bythe slow rate of precipitation of crystals in the pH-buffered,water-based, electrolyte solutions which ordinarily serve asgrowth media. By taking into account the interionic electrostaticforces that affect protein solubility, nucleation, growth, andOstwald ripening, we find that the following sequence of growthsolution procedures should be effective in producing crystals ofany water-soluble protein, which dissolves endothermically. Theprotein should be dissolved at room temperature in a growthsolution, and then the temperature should be lowered to the coldroom temperature at 4 °C to establish the supersaturation. To control nucleation, establish a measurable crystallization rate, andlimit the number of crystals competing for the dissolved protein, the salt concentration should be minimal, and the pH should bedifferent from the pI. As the rate of decay of the supersaturation approaches zero, Ostwald ripening will commence. If the saltconcentration and temperature are maintained as above, and the value of the pH is chosen to be intermediate between the twomost widely spaced but numerically adjacent pKa values of any of the ionizable amino acid residues along the protein chain, thenumber of crystals will decrease and the average crystal size will increase. By taking into account hydrodynamic convection in agrowth solution in a gravitational field, we construct a figure of merit, M, that when evaluated using terrestrial measurements, canbe used to discriminate between proteins that should benefit from crystallization in microgravity and those that should receive nobenefit. The threshold value for the onset of benefits appears to be M ≥ 0.004. Finally, we discriminate between the magneticfield requirements appropriate for the complete levitation of a crystal growth solution in a gravitational field and thoseappropriate for the suppression of natural convection alone.

1. INTRODUCTION

The three-dimensional structure of a protein moleculeordinarily serves as the basis for its function in a livingorganism. If raw protein can be isolated, purified, andprecipitated in the form of single crystals of sufficient sizeand quality, the crystals can be used in X-ray diffractionexperiments to determine the molecular structure.1 Recipes forpreparing the required crystals are varied but usually involvedissolving the purified protein in a pH-buffered aqueoussolution to which a polymer, alcohol, or a salt has been addedin order to reduce the protein solubility.2

If the protein concentration in the growth solution is lessthan the solubility limit, protein crystals will never appear.When such conditions prevail, the crystal growth experiment issaid to be under thermodynamic control.3 By contrast, if theconcentration of protein in the growth solution exceeds thesolubility limit, and the crystals are still slow to appear, theexperiment is said to be under kinetic control. Interionic

electrostatic forces and hydrodynamic convection are twophenomena that affect kinetic control.Under conditions where the diffusion of dissolved protein

through the growth solution is slow, the growth of a crystalproduces a layer of solution depleted in protein next to everygrowing crystal facet.4 Being depleted in protein, the liquid inthe boundary layer is less dense than the surrounding bulksolution. When gravity is acting, the fluid in the boundary layerrises, generating a pattern of flow in the bulk solution which isknown as natural or solutal convection.4 This hydrodynamicflow alters the protein concentration gradient within theboundary layer, which affects the rate of protein diffusion to thesurface of the crystal and consequently also the rate of growthof the crystal.

Received: October 30, 2012Revised: March 8, 2013Published: March 18, 2013

Article

pubs.acs.org/crystal

© 2013 American Chemical Society 1889 dx.doi.org/10.1021/cg3015833 | Cryst. Growth Des. 2013, 13, 1889−1898

James K.BairdRobert L.McFeetersDepartment of Chemistry and MaterialScience GraduateProgram, University of Alabama in Huntsville, Huntsville, Alabama 35899, United States

pubs.acs.org/crystal

Protein molecules dissolved in aqueous solution exist aspolyelectrolytes. Once a protein polyelectrolyte macro-ionarrives at the surface of a growing crystal, the interactionsbetween its charges and the charges on the protein moleculeson the surface of the crystal come into play. Because the sign ofthe net charge of a protein molecule on the surface is the sameas the sign of the net charge on a dissolved protein macro-ion,5

there is a repulsive interionic electrostatic energy barrier whichmust be surmounted in order to add each macro-ion to thesurface.Recently, a bioinformatics approach has been pursued in

order to recommend practical methods for controlling themany interactions among the several variables that control thecrystallization of proteins.6 By contrast, we describe belowsome methods based upon a combination of analytic theoryand experiment which can be used to surmount thermody-namic control and manage those aspects of kinetic controlwhich depend upon the hydrodynamics and the interionicelectrostatic forces.

2. THERMODYNAMIC CONTROLA number of experimental measurements of the temperaturedependence of the solubility of protein crystals in aqueousgrowth media have been reported. The most extensiveinvestigations have involved lysozyme,7−9 canavalin,10 bovinepancreatic trypsin inhibitor,11 glucose/xylose isomerase,9,12

porcine insulin,13 concanavalin A,14 thaumatin, complexedeither with L-tartrate15 or D-tartrate,15 and finally ovalbumin.16

The results of these investigations are summarized in Table 1.

In all cases with the exception of thaumatin D-tartrate andovalbumin, the solubility is reported to be mostly increasing asthe temperature increases. This implies that the dissolution isendothermic (positive heat of solution). In addition to theexperiments listed in the table, Christopher, Phipps, and Cary17

have reported a few limited measurements of the temperaturedependence of the solubility of 19 additional proteins. Amongthese, they found 9 that appeared to exhibit endothermicdissolution and 10 that appeared to exhibit exothermicdissolution (negative heat of solution). Although the evidenceprovided in Table 1 suggests that endothermic dissolution iscommon, the work of Christopher et al.17 cautions that it is notto be expected in every case. Indeed, the addition of particularpolymers, alcohols, and salts as “crystallizing” agents to a water-based crystal growth solution may have the effect of changingendothermic dissolution into exothermic dissolution and viceversa.

In order to make specif ic our recommendations for thecrystallization of proteins, we will limit further discussion to thoseproteins which dissolve endothermically. To begin a crystallizationexperiment with a newly isolated protein, the protein should bedissolved in the growth solution at room temperature up to itssolubility limit. If there is reason to believe, as we have assumed,that the protein dissolves endothermically, a substantialsupersaturation can be produced, if the growth solution isthen chilled, perhaps to the standard cold room temperature at4 °C. With growth solution in a supersaturated condition, theexperiment will lie outside the region of thermodynamiccontrol and will enter the region of kinetic control.

3. KINETIC CONTROL

Crystallization under kinetic control is thought to proceedsequentially through three stages known respectively asnucleation, growth, and Ostwald ripening.18 The nucleationstage begins with an induction period19 during which thesupersaturation is essentially constant, while the individualprotein macro-ions form dimers, trimers, and higher oligomersup to a critical size beyond which further macro-ion additionleads to crystallization. During the growth stage, which followsupon nucleation, crystals appear, and the supersaturationdecays noticeably with time as the linear dimensions of thecrystals advance. The Kelvin equation20 predicts that thesolubility of a crystal increases with decreasing size.21 As thesupersaturation diminishes, the solubility of the smaller crystalswill ultimately exceed the ambient protein concentration in thegrowth solution. This situation marks the onset of the Ostwaldripening phase, where the smaller crystals dissolve, and thematerial released diffuses through the solution and precipitatesonto the larger crystals. As a result, the average crystal sizeincreases. Below, we consider nucleation, growth, and ripeningin turn.

3.1. Nucleation. We denote an isolated protein macro-molecular ion by (1)Z1, a dimer consisting of two such ions by(2)Z2, and a j-mer containing j such ions by (j)Zj. The ionicvalences of these oligomers are Z1, Z2, and Zj, respectively.These ion valences, which are associated with the weak acid/base functionalities along the protein chain, can be expected todepend upon pH. In the example shown in Figure 1, the netcharge is zero at pH = pI = 5.2.

Table 1. Sign of the Heat of Solution, ΔsolnH, for theDissolution of Various Proteins in Water

protein ΔsolnH ref

hen egg white lysozyme + 7−9canavalin + 10bovine pancreatic trypsin inhibitor + 11glucose/xylose isomerase + 9, 12porcine insulin + 13concanvalin A + 14thaumatin tartrate 15L-tartrate +D-tartrate −ovalbumin − 16

Figure 1. A plot of the net charge, Z1, on a canavalin trimer crystalgrowth unit as a function of pH.

Crystal Growth & Design Article

dx.doi.org/10.1021/cg3015833 | Cryst. Growth Des. 2013, 13, 1889−18981890

The nucleation mechanism can be summarized by the massbalance equations21

+ ⇄ + + −α

β+Z Z Z(1) (1) (2) ( )HZ Z Z

1 1 21 1

2

12

(1a)

+ ⇄ + + −α

β+Z Z Z(1) (2) (3) ( )HZ Z Z

1 2 31 2

3

23

(1b)

+ − + + −α

β

−+−

−X Yoooj j Z Z Z(1) ( 1) ( ) ( )HZ Z Z

j j1 1j

j

jj1 1

1

(1c)

where

βκ

γ=

′*−−

jj

j1

1

(2)

is the rate coefficient for formation of the j-mer from the (j−1)-mer, while

ακ

γ=

″*

+ −+ −a( )j

j

jH

Z Z Zj j1 1

(3)

is the rate coefficient for decomposition of the j-mer back intothe (j−1)-mer, γj* is the thermodynamic activity coefficient ofthe transition state, and aH+ is the activity of H+. In the absenceof added strong electrolyte, the corresponding rate coefficientsfor these processes are κj−1′ and κj″, respectively. This nucleationmechanism takes into account three physical effects. We turn tothese below:First because matter in bulk is electrically neutral, the protein

molecules that wind up in the interior of a nucleus must beuncharged; hence, a process must exist for discharging proteinmacro-ions as they reach the surface of a nucleus. In the typicalnucleation mass balance step in eqs 1, say eq 1c, for example,(1)Z1 and (j − 1)Zj−1 are regarded as colliding to form a

transition state, (j)Zj*, (not shown), which by virtue of chargeconservation, has ionic valence, Zj* = Z1 + Zj−1. After its

formation, the transition state, (j)Zj*, decays into the j-mer, (j)Zj,plus Z1 + Zj−1 − Zj hydrogen ions, as required to the balancethe equation with respect to charge. By including hydrogen ionson the right-hand side of each step on the nucleation reactionmechanism above, we imagine that this discharge processconsists of dumping H+ to the buffer. Free solution capillaryelectrophoresis experiments5 involving lysozyme crystalssuspended in pH-buffered aqueous solutions of strongelectrolytes demonstrate, however, that the surface electrostaticpotential of lysozyme crystals depends not only upon the pHbut also on the nature and the charge of any dissolved anions.This suggests the possibility that anions are involved in thedischarge process. To take this mechanism into account, thestoichiometry of the left-hand sides of eqs 1 would need to bemodified to include the number of anions assumed toparticipate in each elementary discharge reaction.Second, in a solution containing strong electrolyte, the

mutual repulsion of the reacting species, (1)Z1 and (j−1)Zj−1 ineq 1c, taken as an example, is weakened by the Debye−Huckelplasma screening of the supporting electrolyte. The effect is toaccelerate the net rate of reaction. In the case of the transitionstate, the effect of plasma screening is reflected in the transitionstate activity coefficient, γj*, which can be represented in theDebye−Huckel limiting law approximation by

γ* = − *Z A Iln j j2

(4)

In eq 4, I is the ionic strength of the supporting electrolyte, andA is a parameter which depends upon the temperature and thestatic dielectric constant of water.21 In the case of a 1−1electrolyte, such as NaCl, the ionic strength and the molarconcentration of the electrolyte are identical.Third, in the generalized scheme of chemical kinetics

applicable to highly nonideal solutions, such as aqueous strongelectrolytes, the activity replaces concentration in the law ofmass action;22 because of the presence of the buffer in a proteincrystal growth solution, the pH = −log aH+ is constant. Thisfixes the value of aH+ in eq 3, which makes αj a constant in abuffered solution and causes the rate law for reaction in thereverse direction in each of eqs 1 to be pseudo-first-order.Although independent of ionic strength, the rate coefficients, κj′and κj−1″ , depend implicitly upon the {Zj} and are thus alsofunctions of the pH.In analogy with classical nucleation theory, the mechanism

summarized in eqs 1 is assumed to end in a rate determiningstep in which a spherical nucleus of critical size, j = n, is formed.The radius of this critical nucleus is given by21

= − + +r

b b aca

42n

2

(5)

where the parameters, a, b, c are defined by

π=a z vk TA I16 2B (6)

=b k T SlnB (7)

γ= + −c v zk T zZ k TA I2 ( 2.303 (pH) 2 )B 1 B (8)

In these equations, z is the pH-dependent surface chargedensity on a nucleus, v is the volume occupied by a proteinmolecule in the bulk nucleus phase, and γ is the interfacialtension acting at the boundary separating the nucleus from thegrowth solution. The supersaturation ratio is S = a1/a1

s , wherea1 is the activity of the protein monomer, and a1

s is its activity atthe solubility limit. The absolute temperature is T, whileBoltzmann’s constant is kB.On the basis of this model, the steady state nucleation rate, J,

can be calculated in the form,21

πκ

π

= + ′

−+ +

⎜ ⎟⎡⎣⎢

⎤⎦⎥

⎛⎝⎜

⎞⎠⎟

⎛⎝

⎞⎠

⎡⎣⎢

⎤⎦⎥

Jv b ac

k T rcc

rvk T

b b ac

( 4 )8

1

exp6

{ 3( 4 ) }

nn

n

2 1/2

2B

1/2 3/21

3

B

2 1/2

(9)

where in the exponential, the value shown for the exponent onrn corrects a misprint which appeared in an earlier publication.

21

In eq 9, the concentration of protein monomer in the growthsolution is c1, and the concentration of protein in thethermodynamic standard state is c. In the absence of strongelectrolyte, the rate coefficient for forming the nucleus ofcritical size, j = n, by collision of a monomer with an (n − 1)-mer is equal to κn′. The units of J are the same as κn′, which canbe mol/L s or molecules/cm−3 s−1 as required.In evaluating eq 9, an implicit dependence of κn′ on

temperature and pH must be assumed. By virtue of itsdefinition, κn′, is the rate coefficient for adding a protein macro-ion to a charged nucleus. If attachment of a protein macro-ionto a crystal nucleus can be regarded as an “elementary” process,



κn′ can be represented by the Arrhenius equation,23 κn′ = κn′exp(−En/RT), where κn′ is a weakly temperature-dependentfactor with the same units as κn′, En > 0 is the activation energy,R is the gas law constant, and T is the Kelvin temperature. Assuch, the value of κn′ should increase as the temperatureincreases. Turning next to the pH dependence, we note thatsince the sign of the net charge on the critical nucleus is thesame as that on the protein macro-ion, their mutualelectrostatic repulsion should diminish, and the value of κn′should increase as the pH approaches the pI.In the case of lysozyme, which has pI = 11, the explicit

dependence of eq 9 upon temperature, pH, and ionic strengthhas been evaluated at fixed supersaturation ratio, S.21 Thisevaluation indicates that at f ixed pH and ionic strength, the valueof J increases rapidly as the temperature increases from 4 to 25°C. Evaluation of eq 9 at f ixed pH and temperature shows that Jincreases with increasing ionic strength. For pH < 6, theultimate rise in J with increasing ionic strength equals thatwhich can be achieved by increasing the temperature alone.Evaluation of eq 9 as a function of pH at f ixed temperature andionic strength shows that J has a global maximum at pH = 2 anda smaller local maximum at pH = 11. At values of the pHbetween these two extremes, J is substantially smaller. Althoughwe have evaluated J only in the case of lysozyme,21 it may stillbe possible to extract from this one example some trends thatmay be helpful in understanding the nucleation of crystals ofother proteins. We now turn to a discussion of those trends.If adequate mass is to be made available for the formation of

the crystalline phase, a protein crystal growth solution shouldbe substantially supersaturated in protein. According to eq 9,however, a high value of S will produce a high nucleation rate,which can lead to the formation of a large number of smallcrystals. In order to reduce the rate of nucleation whilemaintaining a high supersaturation, the behavior exhibited bylysozyme suggests that the temperature should be below roomtemperature, the pH should be far from the pI, and the saltconcentration should be limited.3.2. Crystal Growth. 3.2.1. Role of Diffusion. The rate of

growth of a protein crystal is determined by the competitionbetween the rate of diffusion of protein macro-ions through thesolution and their rate of attachment to the surface of a growingfacet.24 Because of the interionic electrostatic forces actingbetween the various ions in the growth solution, the diffusiveflux of protein is coupled to the diffusive flux of salt. In a three-component system consisting of water, salt, and protein,25 thediffusion fluxes of protein, J1

v, and salt, J2v, in the x-direction, for

example, are linked to the respective concentration gradients,∂c1/∂x and ∂c2/∂x by the equations,26

= −∂∂

−∂∂

J Dcx

Dcx

v v v1 11

112

2(10a)

= −∂∂

−∂∂

J Dcx

Dcx

v v v2 21

122

2(10b)

where i = 1 refers to the protein and i = 2 refers to salt. In thelaboratory, the diffusive fluxes are measured optically25 withrespect to the center of volume of the container through whichthe solutes diffuse. The superscript “v” identifies this as the“center of volume” frame of reference. The elements of thediffusion coefficient matrix are denoted by {Dij

v} . Theexperimentally determined numerical values25 for the {Dij

v}for lysozyme chloride in aqueous sodium chloride are listed inTable 2.

In a protein crystal growth experiment, the electrolyte, whichis part of the growth solution, ordinarily has little solubility inthe solid crystal. Consequently its diffusive flux J2

v is zero at thesurface of the crystal. If we substitute J2

v = 0 into eqs 10 andsolve for J1

v, we find

= −∂∂

⎛⎝⎜

⎞⎠⎟J D

cx

v v1 eff

1

(11)

where the effective diffusion coefficient, Deffv , is given by

=−

DD D D D

Dv

v v v v

veff11 22 12 21

22 (12)

In diffusion-controlled growth, (∂c1/∂x) > 0, so the minus signin eq 11 indicates that the diffusion flux is directed in thenegative x-direction, which is toward the surface of the crystal.When the data in Table 2 are substituted into eq 12, we findthat

=D D0.99v veff 11 (13)

The coefficient of proportionality linking Deffv and D11

v in eq 13is nearly unity. This observation is of technical importancebecause it permits the experimentally elaborate opticalmethod25 for measuring diffusion coefficients to be replacedby the much simpler diaphragm cell method,27,28 in which thecell volume above the sintered glass diaphragm is loaded withaqueous salt solution, while the diaphragm and the cell volumebelow it are loaded with this same salt solution but with proteinadded.Equations 10−13 apply to diffusion in a crystal growth

solution which is at rest with respect to the center of volume ofthe container. In the presence of gravitational convection,diffusion should be reckoned in a center of mass frame ofreference. Transformation equations are available for convert-ing the elements of the diffusion coefficient matrix, {Dij

v}, whichis appropriate in the center volume frame of reference to theelements of the diffusion coefficient matrix, {Dij

m}, which isappropriate in the center of mass frame of reference.29 Even inthe absence of external forces which cause convection, thecenter of mass moves by a process called advection which hasits origin in the interdiffusion of components, such as proteinand salt, which have substantially different molar masses.Advection alters the fluxes of protein and salt and can affect therate of crystal growth.30

3.2.2. Role of Attachment. The rate of growth of amacroscopic crystal is determined by the competition betweenthe rate of transport of protein marco-ions through the solutionand their rate of attachment to the surface of a crystal. Severaldifferent models have been proposed to represent the kineticcoefficient for attachment of protein macro-ion to a proteincrystal surface.31−37 Little is known for certain,20,31,38,39 aboutthis attachment coefficient, except that like κn′, which governs

Table 2. Elements of the Diffusion Coefficient Matrix in theVolume Fixed Frame of Reference for the Ternary Mixture,H2O + Lysozyme Chloride + NaCla

D11v D12

v D21v D22

v

0.1102 × 10−9 8.6 × 10−14 19.8 × 10−9 1.461 × 10−9

aThe data are taken from ref 25. The units are m2/s. Theconcentration properties of the solution are c1 = 8.58 mg/mL (0.6mM), c2 = 0.90 M, and ρ = 1.03558 g/cm3. The subscripts, 1, and 2,stand for lysozyme chloride and salt, respectively.



the addition of a protein macro-ion to the surface of a crystalnucleus, the attachment coefficient refers to an elementaryprocess. As such, the attachment coefficient should be thermallyactivated23 and should increase with any change in buffer thatcarries the pH closer to the pI.3.2.3. Hydrodynamics. In earth’s gravity, go, crystals larger

than about 50 μm in linear dimension can be expected togenerate significant solutal convection.34 At a level of gravityequal to 10−6 go, which is available on the International SpaceStation, for example, a crystal must be at least 1000 μm in sizeto generate convection.34 Solutal convection alters the value ofthe protein concentration gradient, ∂c1/∂x, at the surface of thecrystal. Along with the attachment coefficient, this gradientdetermines the rate of crystal growth.33 The effects of solutalconvection on crystal growth have been analyzed in severaldifferent hydrodynamic models, and the results have beenapplied to a variety of proteins.4,31−37 Table 3 summarizes thesetheories, their mathematical methods, and the proteins towhich they have been applied.

Under all conditions of solutal convection analyzed so far,the rate of transport of protein to the crystal and its attachmentto a point on a crystal facet depend upon the position of thepoint in the gravitational field.4,31−37 This observation, webelieve, explains the positional dispersion in measured growthrates which has been detected experimentally by the continuousmicroscopic optical observation of the linear growth rates ofvarious spots on the surface of lysozyme crystals.40 The spatialdispersion in growth rates caused by convection can lead tocrystals assuming shapes which they would not otherwiseassume if they were in thermodynamic equilibrium with thequiescent growth solution.4,36 These equilibrium shapes aregoverned by Wulff’s theorem, which states that the ratio, γ/s,where γ is the surface tension of a facet, and s is the straight linedistance from the facet to the center of the crystal, must be thesame for all facets.41

The pattern of solutal convection depends upon the anglebetween the gravity acceleration vector, go, and the outwardnormal, n , to the crystal facet. In the extreme case where n andgo are antiparallel, the depletion zone is then everywhere belowa more dense growth solution which is above it. Thisgravitationally unstable arrangement resolves itself by generat-ing a pattern of flow known as Rayleigh-Benard convection,35

in which the fluid over an area of the growing facet rises, whilethe fluid over the adjacent areas falls. A rising region whencoupled with a falling region, both of which are needed tomaintain the continuity of flow, is termed a Rayleigh-Bernardcell. The linear rate of advance of a crystal facet due to growthshould be fast over the falling region and slow over the rising

region. Rayleigh-Bernard cells can be prevented from forming ifthe fluid container is narrower than the sum of the widths ofthe rising and falling regions required to maintain a cell. Thisobservation recommends crystal growth in narrow capillaries.A growing facet separating the crystal from the growth

solution serves as a moving boundary.35 The advance of thefacet pushes the edge of the depletion zone into the bulksolution, which has the effect of changing the proteinconcentration gradient which drives the rate of diffusivetransport to the crystal surface.35 This moving boundary effectis negligible, however, when the volume occupied by the solidcrystals is small compared with the volume of the fluid in thecontainer.32

An analytical treatment of solutal convection in a proteincrystal growth solution shows that dimensionless groups can beused to relate the facet growth rate to the thermophysicalproperties of the growth solution.33 Assuming that the growthsolution is sufficiently dilute in protein as to make diffusionindependent of the frame of reference, these dimensionlessgroups, Schmidt number, Grashof number, Sherwood number,and the kinematic supersaturation, can be described as follows:If μ is the shear viscosity of the growth solution, and ρ is itsmass density, the Schmidt number is defined by33

μ ρ=Sc D( / )veff (14)

The Schmidt number represents the ratio of viscous drag rateto the rate of diffusion. If g0 is the magnitude of accelerationdue to gravity, the Grashoff number is defined by

α ρ μ=Gr g h /403 2 2

(15)

where h is the height of the crystal facet and α = (c1/ρ)(∂ρ/∂c1)is the logarithmic increment of the density of the growthsolution with respect to the protein concentration, c1. TheGrashoff number expresses the ratio of the buoyancy to theviscous flow. If kG is the linear rate of growth of the facet, theSherwood number is defined by

=Sh k h D/ vG eff (16)

The Sherwood number expresses the ratio of the linear growthrate to the effective diffusion velocity, Deff

v /h. Finally, if c1s is the

protein solubility limit, and c10 is the protein concentration in

the bulk of the solution, the kinematic supersaturation isdefined by

ϕ =−c ccs

10

1s

10

(17)

An analytical solution to the equations of convective diffusion ispossible when the growing crystal facet is a semi-infinite flatplate with normal, n , perpendicular to g0.

33 If the rate ofattachment of protein molecules to the surface of the facet ismuch greater than the rate of diffusion, the Sherwood numbersatisfies the equation,

ϕ=Sh Sc Gr( ) 0.9(( )( ) )s1/4

(18)

On the basis of this relation, we can define a Figure of Merit

ϕ=M

ShScGr( )s

1/4(19)

When M is of the order of unity, the rate of attachment is muchgreater than the rate of molecular transport through thesolution. The depletion layer has its maximum extent, and

Table 3. Summary of Theories of Solutal Convection inProtein Crystal Growth

protein crystal shape method ref

generic vertical flat plate analytical 4, 33lysozyme orthorhombic solid numerical 32lysozyme cylindrical solid numerical 31ferritin sphere analytical 31lysozyme horizontal flat plate numerical 35generic inclined plate numerical 36generic sphere dimensional analysis 37ferritin tetragonal solid numerical/dimensional 34lysozyme tetragonal solid numerical 34



convection is well developed. By contrast, when M ≪ 1,transport of protein to the facet is fast compared with the rateof attachment, the depletion layer is limited, and convection isminimal. We suggest that somewhere between these twoextremes lies a threshold value of M, above which convection isjust sufficient to produce the growth rate anomalies42 thatdegrade crystal quality. For growth solution conditions wherethe value of M substantially exceeds this threshold value,exposure of a growing crystal and its growth solution tomicrogravity should have beneficial effects.The dimensionless groups (Sc), (Gr), and (Sh) in eq 19 can

all be evaluated on the basis of terrestrial measurements of thethermophysical properties of the growth solution and itscrystals. For example, the kinematic viscosity, ν = μ/ρ, of thesolution can be determined by Ostwald viscometry,43 thedensity, ρ, by pyncnometry,44 and the effective diffusioncoefficient, Deff

v , can be determined by use of the diaphragm celltechnique.27,28 The height, h, and linear growth rate, kG, of acrystal facet can be determined by optical microscopy.40

Should experiments confirm the hypothesis that certaincrystal defects have their origin in convective diffusion, then itfollows from hydrodynamic theory33 that crystals of differentproteins growing from different solutions should exhibit similardefects when the growth rates and the values of thethermophysical properties of the solutions combine to produceidentical values for the figure of merit. This argument permitsus to use the observation of growth plume convection and theknown thermophysical properties in the case of lysozyme toestimate the threshold value of M for the onset of crystaldefects. Those thermophysical properties are summarized inTable 4. We proceed as follows: We combine the value of D11

v

listed in Table 2 with eq 13 and compute Deffv = 0.11 × 10−9

m2/s. By combining the value of ρ found in Table 2 with thevalue of μ in Table 4, we use eq 14 to compute Sc = 8230. Nextusing the values of c1 and ρ found in Table 2 and the value of(∂ρ/∂c1) listed in Table 4, we compute α = (c1/ρ)(∂ρ/∂c1) =2.51 × 10−3. In experiments designed to observe the growthplume above a growing lysozyme crystal,45 the height of thecrystal was approximately h = 0.5 mm, and since g0 = 9.8 m/s2,we can use eq 15 to compute Gr = 0.933. By calculating theslope of the typical distance vs time plot shown in Figure 1 ofref 40, we obtain the value, kG = ((16 − 9) × 10−6/800)m/s =8.75 × 10−9 m/s, which is listed in Table 4. Evaluation of eq 16then gives Sh = 0.040. Finally, by combining the values of c1

0 andc1s listed in Table 4, we can use eq 17 to compute ϕs = 0.9According to eq 19, under these conditions, M = 0.004.

Schlieren photography45 of the growth plume has confirmedthe existence of well-developed solutal convection around alysozyme crystal growing under conditions similar to thosesummarized in Table 4. Since exposure to microgravity hasbeen reported to lead to improvements in the X-ray quality of

lysozyme crystals,46 we suggest that these improvements mayhave their onset when the value of M ≥ 0.004.How reliable is this estimate? Because the dimensionless

groups in the denominator of eq 19 are raised to the 1/4power, the value of M is not likely to vary strongly with thenature and concentrations of the components used to preparethe growth solution. Exceptions may occur in special caseswhere thickeners, such as polyhydric alcohols or polyethyleneglycol, have been added to increase the shear viscosity. Giventhis weak dependence of M on the growth solution conditions,plus the fact that lysozyme crystals seem to benefit from theexposure to microgravity,46 the value of M in the vicinity of0.004 may be a universal boundary. Experimentation with avariety of proteins will be required to confirm this hypothesis. Ifconfirmed, the figure of merit concept can be used todiscriminate between proteins that are likely to benefit fromexposure to microgravity and those which are not.

3.2.4. Time Dependence of the Supersaturation DuringGrowth. In the absence of convection, the time dependence ofthe protein concentration in the depletion zone around agrowing protein crystal can be obtained by solving Fick’s lawsof diffusion analytically.26 In the presence of convection, thetime dependence must be calculated numerically.31,32,34−36,38,39

These numerical methods are capable of providing the shapesof the protein concentration and fluid velocity profiles overonly a limited time span. In the absence of theoretical methodsapplicable over the entire time span of growth, one cansubstitute bulk solution experimental methods, which althoughthey respond only to the spatial average properties of thesolution, are nonetheless, sensitive to the entire time course ofthe crystallization.Dilatometry is one such experimental method.3,47 A

dilatometer consists of an enclosed volume connected to theatmosphere through a capillary side arm. To start anexperiment, the enclosed volume plus the side arm are filledwith the supersaturated crystal growth solution. Since thecapillary has a uniform inside diameter, any change in thevolume of the contents of the dilatometer is reflected in aproportional change in height of the fluid in the side arm.Because protein crystals are denser than their growth solution,the total volume of the contents of the dilatometer decreaseswith time as the crystals appear. If Δh(t) is the change in heightof the fluid in the side arm at time t, and Δh(∞) is the changein height at equilibrium (which corresponds to t→∞), theory3

indicates that

σ σ= − ΔΔ ∞

⎛⎝⎜

⎞⎠⎟t

h th

( ) (0) 1( )

( ) (20)

In eq 20, σ(t) is the relative supersaturation defined by

σ = −tc t

c( )

( )11

1s

(21)

where c1(t) is the instantaneous value of the concentration ofdissolved protein, and c1

s is the thermodynamic solubility of theprotein. Experiments carried out on crystallizing lysozyme3 andcanavalin47 solutions show that

Δ = Δ ∞ − −h t h kt( ) ( )(1 exp( )) (22)

where k is the specific crystallization rate. When eq 22 issubstituted into eq 20, we find

σ σ= −t kt( ) (0) exp( ) (23)

Table 4. Thermophysical Properties of an Aqueous Solutionof Lysozyme Containing a Lysozyme Crystal of Height h =0.5 mm Growing at a Linear Rate of 8.75 × 10−9 m/sa

c10

(mg/mL)Ref 45

c1s

(mg/mL)Ref 45

μ(Ns/m2)Ref 43

(dρ/dc1)

Ref 43

h(mm)Ref 45

kG(m/s)Ref 40

11.7 1.2 0.094 0.3032 0.5 8.75 × 10−9

aThe bulk lysozyme concentration is c10, the solubility of lysozyme is c1

s ,the shear viscosity is μ, the mass density is ρ, and the height of thecrystal is h.



Although empirical, eq 23 is consistent with an analytic theoryof growth of crystals that have nucleated all at the same time.48

The mass, m(t), of the crystals per unit volume of thedilatometer is then given by47

= − − −m t c c( ) ( (0) )(1 e )kt1 1

s(24)

where c1(0) is the initial value of c1(t). For crystals of lysozyme3

and canavalin,47 growing in pH -buffered aqueous electrolytegrowth solutions, k has been determined as a function of pH,temperature, and sodium chloride concentration. The trends ink, which are the same in the case of both proteins, can besummarized as follows: At constant temperature and pH, kincreases with increasing salt concentration. At constanttemperature and salt concentration, k increases with anychange in the buffer which carries the pH closer to the pI. Atconstant pH and salt concentration, k decreases with increasingtemperature. Theory47 suggests that k = Nρω 2/cs, where N isthe number of nucleation sites per unit volume, 2 is the crosssectional area of a nucleation site, cs is the solubility of theprotein, and ω is the rate coefficient for the elementary processof attachment of a protein macro-ion to a crystal facet(analogous to κn′ in nucleation theory). Like κn′, we canrepresent the temperature dependence of ω using theArrhenius equation,23 ω = ωL exp(−EL/RT), where ωL is atemperature independent constant with dimensions, cm/s, andEL > 0 is the activation energy. We can represent cs by the van’tHoff equation,47 cs = b exp(−ΔHs/RT), where b is a constantwith the same dimensions as cs, and ΔHs = ΔsolnH is the heat ofsolution. Assuming that the other parameters determining k donot depend exponentially on the temperature, we find that theapparent activation energy of k is the composite, E = EL − ΔHs.If protein dissolution is endothermic, and if ΔHs > EL, then theapparent activation energy, E, is negative and k decreases withincreasing temperature as observed.As proteins, lysozyme and canvalin are quite different.

Lysozyme3 is an animal protein consisting of 129 amino acidresidues, molecular weight of 12 kDa, and pI = 11. By contrast,canavalin47 is a plant protein, which crystallizes as a trimerconsisting of 1095 amino acids, molecular weight 125.8 kDa,and pI = 5.2. The common dependence of the value of k uponpH and salt concentration for these proteins finds its basis inthe fact that both go into aqueous solutions as polyelectrolytes.Any change in buffer that carries the pH closer to the pI willdecrease the macro-ion charge, while an increase in electrolyteconcentration will enhance the Debye−Huckel plasma screen-ing of these charges. Either change will weaken the repulsiveinterionic electrostatic force acting between the macro-ionswhich will in turn accelerate the rate of formation of a crystal.The temperature dependence of k for the two proteins is thesame presumably because both dissolve endothermically.47

If for a given protein, the pH, salt concentration, andtemperature dependence of k are the same as that observed inthe case of lysozyme3 and canvalin,47 the following rules ofthumb may prove useful: Because the value of k increases withdecreasing temperature, the temperature should be low. Thevalue of k can be increased by increasing the salt concentrationand by changing the acidity to bring the pH as close as possibleto the pI. The advantage gained should be readilynoticeable.3,47

3.2.5. Quenching of Gravitational Affects in an AppliedMagnetic Field. Protein crystals and protein crystal growthsolutions containing ions and atoms with closed electronic

shells are diamagnetic.49−52 In a diamagnetic material,application of an external magnetic field, H, induces a magneticdipole moment, M, which is opposite in direction to H. As aconsequence, the diamagnetic susceptibility, χ = M/H, isnegative.53 When the magnet is designed so as to produce aconstant magnetic field gradient, (∂H/∂y), in the y-direction,there will be a volume force μ0χH(∂H/∂y)acting on the centerof mass of any material located in the region of the gradient,where μ0 is the permeability of free space. In terrestrial gravity,the center of mass of a material with mass density, ρ, will alsoexperience a volume force, ρg0. Any diamagnetic material in theform of a rigid solid or a homogeneous liquid will becompletely levitated when these forces balance. Levitation isthe analog of microgravity. The condition for magneticlevitation is49

μρ μ

χ∂∂

=⎛⎝⎜

⎞⎠⎟H

Hy

g0

2 0 0

(25)

In the case of water, for example, which has magneticsusceptibility,53 χ = −9 × 10−9 and density ρ = 1000 kg/m3,levitation requires μ0

2H(∂H/∂y) = −1370 T/m2.49,52 Thiscondition specifies a magnetic field intensity that lies near theupper limit of performance of the best available super-conducting magnets.50 Should the material be a dispersion ofa solid in a liquid, for example, protein crystals suspended ingrowth solution, the acceleration due to gravity will be canceledsimultaneously for the crystals and the growth solution, if bothhave the same value of the ratio, ρ/χ.If the material in the magnetic field is a liquid solution in

which convection is producing spatial gradients of concen-tration, density, and magnetic susceptibility, the condition forsuppression of convection requires that in eq 25, ∂ρ/∂y besubstituted for ρ, and ∂χ/∂y be substituted for χ. Afterintroducing the mass fraction, φ, and use of the chain rule, theresult can be written in the form51

μρ φ μ

χ φ∂∂

=∂ ∂

∂ ∂

⎛⎝⎜

⎞⎠⎟H

Hy

g( / )

( / )02 0 0

(26)

In the case of lysozyme, experiments51 show that (∂ρ/∂φ) =5.65 kg/m3, while (∂χ/∂φ) = 9.97 × 10−9. Upon substitution ofthese results into eq 26, we find that the condition forsuppression of convection is μ0

2H(∂H/∂y) = 6.98 T/m2. Thiscondition is substantially less demanding than the conditionrequired to achieve levitation; moreover, it requires a magneticfield gradient with a positive, rather than a negative sign.

3.2.6. Hall Effect. By virtue of the presence of the proteinmacromolecular ions and inert electrolyte dissolved in a proteincrystal growth solution, a protein crystal growth solution is anionic conductor. If a volume element of the solution has chargedensity q, and convection velocity, v, it will experience a volumeLorentz force,

= F q vxB( )L (27)

when subjected to an applied magnetic flux density, B. TheLorentz force will give rise to a Hall effect drift current in adirection parallel to FL.

54 The Hall effect in electrolytes hasbeen investigated within the context of the theory ofnonequilibrium thermodynamics.54 This theory suggests thatthe Hall current should be orders of magnitude smaller than thediffusion current and consequently should have a minimal effecton the rate of growth of a crystal.



3.3. Ostwald Ripening. A successful protein crystallizationexperiment can produce a large number of crystals of varyingsizes.55 This nonequilibrium situation can be improved bywaiting for the crystal size distribution to advance toward largersizes by virtue of the surface tension (surface free energydensity) driven process called Ostwald ripening.20 The crystalsurface tension is dominated by the acid/base sites which occurwhenever the combination of the protein molecular structureand the crystal structure permits ionizable groups to occupypositions on the surface of the crystal. The ionizable groupsinclude the molecular C-terminus, the N-terminus, and also anyamino acid residue having a side chain capable of ionization.The catalog of the latter includes arginine, aspartic acid,glutamic acid, histidine, lysine, tyrosine, and cysteine. Theionizable groups make two contributions to the surface freeenergy. These are (1) the entropy associated with the mixing ofprotonated sites and empty sites on the surface of the crystaland (2) the entropy and energy stored in the Debye−Huckelion plasma which surrounds the crystal and which dependsupon the nonzero surface charge density on the crystal.20

Although both of these contributions serve to reduce thesurface tension and reduce the average crystal size in Ostwaldripening, the first dominates. In the case of the first effect, whenthe pH is far to the acid side of the pKa of a particular type ofsurface site, all of the sites of this type will be protonated. Therewill be no entropy associated with the exchange of H+ betweenempty and occupied sites. In this situation, the particular typeof site in question will contribute only to the “background”, pHindependent part of the surface tension. As the pH becomesmore basic, however, the fraction of the total sites of this typewhich ionize (i.e., donate H+ to the solvent) will grow, emptysites will be created, and the surface entropy of mixing willdevelop. The effect of this entropy of mixing is to reduce thesurface tension. As the pH becomes still more basic, the fractionof ionized sites of this particular type will approach unity. Whenthe fraction of ionized sites reaches unity, there is no longer anentropy of mixing associated with this type of site; the site againmerges with the “background” and no longer contributes to thepH dependence of the surface tension.The most favorable condition for the surface tension of the

protein crystal in Ostwald ripening occurs when the ionizablesites of all possible types are forced into the “background”. Inprinciple, this can be achieved for the crystal as whole byadjusting the pH to the acid side of the pKa of the most acidicionizable group or alternatively by adjusting the pH to the basicside of the pKa of the least acidic ionizable group. Dependingupon the protein, these extremes may be as acidic as pH = 2 oras basic as pH = 13. As either of these extremes is likely todenature the protein molecule, neither would seem to bepractical. As a compromise, we can rank order the pKa values ofthe ionizable groups in the order, pKa1 < pKa2 < pKa3, etc., andthen choose from among adjacent pairs, a pair for which thedifference ΔpKa = |pKa1 − pKa2| is largest. If the pH is set equalto pH = (1/2)(pKa1 + pKa2), then both pKa1 and pKa2, as wellas the pKa values of other ionizable groups more distant fromthis mean, should lie in one of the ranges, pKa ≪ pH or pKa ≫pH. The contribution made to the surface tension by each typeof ionizable group lying in either of these ranges should berestricted to the background.In addition to the surface energy, the Ostwald ripening

process involves the diffusion of protein macromolecular ionsthrough the growth solution and their attachment to the surfaceof a crystal facet.20,56 As we have noted above, the dependence

of the protein diffusion coefficient and attachment coefficientupon salt concentration and temperature is not entirely clear. Incontrast to the growth phase, where our lack of knowledge doesnot prevent us from speculation concerning the effects ofdiffusion and attachment on the rate of growth, these twoparameters enter the theory of the Ostwald ripening phase in afashion which is sufficiently intricate56 as to precludepredictions of the effect of salt and temperature on the rateof coarsening.

4. TIME DEPENDENCE OF THE SUPERSATURATIONINCLUDING NUCLEATION, GROWTH, ANDRIPENING

The nucleation stage begins with an induction period19 duringwhich the relative supersaturation, σ(t), is essentially constant,and the individual protein molecules agglomerate to formnuclei of critical size. The induction time required to establishthis steady state depends upon the temperature, the pH and thesalt concentration.57 As mentioned above, the results ofdilatometer experiments show that during the growth period,which follows the induction period, the relative supersaturationdiminishes exponentially with time according to eq 23. Finally,in the Ostwald ripening stage during which the smaller crystalsdissolve while the larger crystals grow, the time dependence ofthe relative supersaturation converts from an exponential to aninverse time power law, where σ(t) is proportional to t−δ and δis a positive rational number less than unity.20 Although wehave some idea of the time scales for nucleation induction,19,57

and for growth3,47 the time scale, for ripening is unknown.Because of this uncertainty, we have in Figure 2, whichattempts to cover the entire time scale, been able to plot onlythe qualitative features of the time decay of the relativesupersaturation.

5. SUMMARYIn Table 5, we have collected together our predictions of theeffects of supersaturation ratio, pH, temperature, and saltconcentration on the rate of nucleation, J; the specific rate ofcrystal growth, k; and average crystal size, ⟨a⟩, the number ofcrystals, per unit volume, N, and the absolute value of the rateof decay of the relative supersaturation, |dσ(t)/dt|, during theOstwald ripening phase.

6. CONCLUSIONSBy distinguishing between thermodynamic control and kineticcontrol and by combining the theory and the experimentalobservations discussed above, we can make the followingrecommendations for coping with the effects of hydrodynamics

Figure 2. Qualitative graphical representation of the decay of theprotein relative supersaturation, σ(t), in a crystallizing protein solutionas a function of time, t, where δ is a positive number less than unity.



and electrostatics in the crystallization of proteins from pHbuffered aqueous solutions of strong electrolytes. Ourconclusions with respect to hydrodynamics are summarizedby the definition of M in eq 19 and the discussion in Section3.2.3. Our recommendations with respect to electrostatics,which follow, are restricted to proteins that dissolve endothermi-cally. The comments referring to thermodynamic control arebased upon Table 1, while the comments referring to kineticcontrol are based upon Section 3 and Table 5.1. Thermodynamic Control. When attempting to crystallize a

newly isolated protein, the protein should be dissolved in thecrystal growth medium at room temperature up to its solubilitylimit. The growth solution should then be chilled below roomtemperature in order to increase the supersaturation. This willmake the largest possible excess of protein available forcrystallization.2. Kinetic Control(a) Nucleation: To avoid the creation of a large number of

competing nuclei, the nucleation rate should be limited bymaintaining the temperature below room temperature, the saltconcentration should be minimal, and the pH should not beequal to the pI.(b) Growth: To encourage rapid growth of existing crystals,

the temperature should be as low as feasible, the saltconcentration should be large, and the pH should be equal tothe pI. Note that these conditions with respect to salt and pHcontrast with the conditions required to limit nucleation.(c) Ostwald Ripening: The crystal size distribution prevailing

at the end of the growth stage will coarsen significantly duringthe ripening phase, if the pH = (1/2)(pKa1 + pKa2), where pKa1and pKa2 are numerically adjacent pKa values for which ΔpKa =|pKa1 − pKa2| is large. Because the effects of temperature andsalt concentration upon the protein macro-ion diffusioncoefficient and attachment coefficient are not well-known, wecannot make a recommendation concerning the appropriatetemperature and salt concentration for Ostwald ripening.Suffice it to say, however, the slower the rate of decay of thesupersaturation, the larger will be the crystals.(d) Compromise Conditions: Because increasing the salt

concentration and adjusting the pH to be close to the pI servesto increase both the rate of nucleation and the rate of growth,optimal conditions of salt and pH are not to be found if thegoal of a crystallization trial is to suppress the rate of nucleationwhile increasing the rate of growth. Some compromise isnecessary. Since the rate of nucleation is more sensitive to thesalt concentration than is the rate of growth, and the effect ofsalt on ripening is unknown, we recommend the minimum saltconcentration necessary to produce an adequate supersatura-tion. Likewise since the rate of nucleation is more sensitive topH than are the rates of either growth or ripening, the pH value

should be chosen so as to minimize the nucleation rate. That isto say, it should not be equal to the pI. The choice pH = (1/2)(pKa1 + pKa2), where pKa1 and pKa2 are widely spaced butnumerically adjacent values of the amino acid residue ionizationconstants, can serve as a compromise. In the case oftemperature control, there are no inconsistencies. If thetemperature is kept low, the rate of nucleation will besuppressed, while the rate of growth will be enhanced.

■ AUTHOR INFORMATION

Corresponding Author*E-mail: [email protected].

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

The authors thank Dr. Hana McFeeters and Mary Hames foruseful discussion and manuscript editing.

■ REFERENCES(1) Stout, G. H.; Jensen, L. H. X-ray Structure Determination: APractical Guide, 2nd ed.; John Wiley: New York, 1989.(2) McPherson, A. F. Crystallization of Biological Macromolecules;Cold Spring Harbor Laboratory Press: New York, 1999.(3) Kim, Y. W.; Barlow, D. A.; Caraballo, K. G.; Baird, J. K. Mol. Phys.2003, 101, 2677−2686.(4) Baird, J. K.; Meehan, E. J.; Xidis, A. L.; Howard, S. B. J. Cryst.Growth 1986, 76, 694−700.(5) Lee, M.-H.; Kim, Y. W.; Baird, J. K. J. Cryst. Growth 2001, 232,294−300.(6) Sanchez-Puig, N.; Sauter, C.; Lorber, B.; Giege, R.; Moreno, A.Protein Peptide Lett. 2012, 19, 725−731.(7) Howard, S. B.; Twigg, P. J.; Baird, J. K.; Meehan, E. J. J. Cryst.Growth 1988, 90, 94−104.(8) Forsythe, E. L.; Judge, R. A.; Pusey, M. L. J. Chem. Eng. Data1999, 44, 637−640.(9) Van Driessche, A. E. S.; Gavira, J. A.; Lopez, L. D. P.; Otalora, F.J. Cryst. Growth 2009, 311, 3479−3484.(10) DeMattei, R. C.; Feigelson, R. S. J. Cryst. Growth 1991, 110, 34−40.(11) Lafont, S.; Veesler, S.; Astier, J. P.; Boistelle, R. J. Cryst. Growth1997, 173, 132−140.(12) Vuolanto, A.; Uotila, S.; Leisola, M.; Visuri, K. J. Cryst. Growth2003, 257, 403−411.(13) Bergeron, L.; Filobelo, L. F.; Galkin, O.; Vekilov, P. G. Biophys.J. 2003, 85, 3935−3942.(14) Mikol, V.; Giege, R. J. Cryst. Growth 1989, 97, 324−332.(15) Asherie, N.; Ginsberg, C.; Blass, S.; Greenbaum, A.; Knafo, S.Cryst. Growth Des. 2008, 8, 1815−1817.(16) Judge, R. A.; Johns, M. R.; White, E. T. J. Chem. Eng. Data 1996,41, 422−424.

Table 5. Summary of Effects of Supersaturation Ratio, S; Temperature, T; Ionic Strength, I; and pH on Nucleation, Growth andOstwald Ripening of a Protein Crystala

nucleation growth Ostwald Ripening

J k <a> N |dσ/dt|

increasing S increases N/A N/A N/A increasesincreasing T increases decreases unknown unknown unknownincreasing I increases increases unknown unknown unknownpH → pI increases increases unknown unknown unknownpH = (1/2)(pKa1 + pKa2) N/A N/A increases decreases decreases

aThe dependent quantities are the nucleation rate, J (eq 9); crystallization rate, k (eq 23); the average crystal size, ⟨a⟩; the number of crystals perunit volume, N; and the absolute value of the time rate of change of the relative supersaturation |dσ/dt|. N/A means “not applicable.”.



mailto:[email protected]

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effects of hydrodynamic convection and interionic electrostatic forces on protein crystallization

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