Effects of Hydrodynamic Convection and Interionic Electrostatic Forces

Article pubs.acs.org/crystal

Effects of Hydrodynamic Convection and Interionic Electrostatic Forces on Protein Crystallization Published as part of the Crystal Growth & Design virtual special issue on the 14th International Conference on the 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 intimately related to its three-dimensional molecular structure. Although Xray diffraction from single crystals can be employed to solve for the molecular structure, use of this method is often impeded by the slow rate of precipitation of crystals in the pH-buffered, water-based, electrolyte solutions which ordinarily serve as growth media. By taking into account the interionic electrostatic forces that affect protein solubility, nucleation, growth, and Ostwald ripening, we find that the following sequence of growth solution procedures should be effective in producing crystals of any water-soluble protein, which dissolves endothermically. The protein should be dissolved at room temperature in a growth solution, and then the temperature should be lowered to the cold room temperature at 4 °C to establish the supersaturation. To control nucleation, establish a measurable crystallization rate, and limit the number of crystals competing for the dissolved protein, the salt concentration should be minimal, and the pH should be different from the pI. As the rate of decay of the supersaturation approaches zero, Ostwald ripening will commence. If the salt concentration and temperature are maintained as above, and the value of the pH is chosen to be intermediate between the two most widely spaced but numerically adjacent pKa values of any of the ionizable amino acid residues along the protein chain, the number of crystals will decrease and the average crystal size will increase. By taking into account hydrodynamic convection in a growth solution in a gravitational field, we construct a figure of merit, M, that when evaluated using terrestrial measurements, can be used to discriminate between proteins that should benefit from crystallization in microgravity and those that should receive no benefit. The threshold value for the onset of benefits appears to be M ≥ 0.004. Finally, we discriminate between the magnetic field requirements appropriate for the complete levitation of a crystal growth solution in a gravitational field and those appropriate for the suppression of natural convection alone.

1. INTRODUCTION

electrostatic forces and hydrodynamic convection are two phenomena that affect kinetic control. Under conditions where the diffusion of dissolved protein through the growth solution is slow, the growth of a crystal produces a layer of solution depleted in protein next to every growing crystal facet.4 Being depleted in protein, the liquid in the boundary layer is less dense than the surrounding bulk solution. When gravity is acting, the fluid in the boundary layer rises, generating a pattern of flow in the bulk solution which is known as natural or solutal convection.4 This hydrodynamic flow alters the protein concentration gradient within the boundary layer, which affects the rate of protein diffusion to the surface of the crystal and consequently also the rate of growth of the crystal.

The three-dimensional structure of a protein molecule ordinarily serves as the basis for its function in a living organism. If raw protein can be isolated, purified, and precipitated in the form of single crystals of sufficient size and quality, the crystals can be used in X-ray diffraction experiments to determine the molecular structure.1 Recipes for preparing the required crystals are varied but usually involve dissolving the purified protein in a pH-buffered aqueous solution to which a polymer, alcohol, or a salt has been added in order to reduce the protein solubility.2 If the protein concentration in the growth solution is less than the solubility limit, protein crystals will never appear. When such conditions prevail, the crystal growth experiment is said to be under thermodynamic control.3 By contrast, if the concentration of protein in the growth solution exceeds the solubility limit, and the crystals are still slow to appear, the experiment is said to be under kinetic control. Interionic © 2013 American Chemical Society

Received: October 30, 2012 Revised: March 8, 2013 Published: March 18, 2013 1889

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

Crystal Growth & Design

Article

In order to make specif ic our recommendations for the crystallization of proteins, we will limit f urther discussion to those proteins which dissolve endothermically. To begin a crystallization experiment with a newly isolated protein, the protein should be dissolved in the growth solution at room temperature up to its solubility limit. If there is reason to believe, as we have assumed, that the protein dissolves endothermically, a substantial supersaturation can be produced, if the growth solution is then chilled, perhaps to the standard cold room temperature at 4 °C. With growth solution in a supersaturated condition, the experiment will lie outside the region of thermodynamic control and will enter the region of kinetic control.

Protein molecules dissolved in aqueous solution exist as polyelectrolytes. Once a protein polyelectrolyte macro-ion arrives at the surface of a growing crystal, the interactions between its charges and the charges on the protein molecules on the surface of the crystal come into play. Because the sign of the net charge of a protein molecule on the surface is the same as the sign of the net charge on a dissolved protein macro-ion,5 there is a repulsive interionic electrostatic energy barrier which must be surmounted in order to add each macro-ion to the surface. Recently, a bioinformatics approach has been pursued in order to recommend practical methods for controlling the many interactions among the several variables that control the crystallization of proteins.6 By contrast, we describe below some methods based upon a combination of analytic theory and experiment which can be used to surmount thermodynamic control and manage those aspects of kinetic control which depend upon the hydrodynamics and the interionic electrostatic forces.

3. KINETIC CONTROL Crystallization under kinetic control is thought to proceed sequentially through three stages known respectively as nucleation, growth, and Ostwald ripening.18 The nucleation stage begins with an induction period19 during which the supersaturation is essentially constant, while the individual protein macro-ions form dimers, trimers, and higher oligomers up to a critical size beyond which further macro-ion addition leads to crystallization. During the growth stage, which follows upon nucleation, crystals appear, and the supersaturation decays noticeably with time as the linear dimensions of the crystals advance. The Kelvin equation20 predicts that the solubility of a crystal increases with decreasing size.21 As the supersaturation diminishes, the solubility of the smaller crystals will ultimately exceed the ambient protein concentration in the growth solution. This situation marks the onset of the Ostwald ripening phase, where the smaller crystals dissolve, and the material released diffuses through the solution and precipitates onto the larger crystals. As a result, the average crystal size increases. Below, we consider nucleation, growth, and ripening in turn. 3.1. Nucleation. We denote an isolated protein macromolecular 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 ionic valences 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 to depend upon pH. In the example shown in Figure 1, the net charge is zero at pH = pI = 5.2.

2. THERMODYNAMIC CONTROL A number of experimental measurements of the temperature dependence of the solubility of protein crystals in aqueous growth media have been reported. The most extensive investigations have involved lysozyme,7−9 canavalin,10 bovine pancreatic trypsin inhibitor,11 glucose/xylose isomerase,9,12 porcine insulin,13 concanavalin A,14 thaumatin, complexed either with L-tartrate15 or D-tartrate,15 and finally ovalbumin.16 The results of these investigations are summarized in Table 1. Table 1. Sign of the Heat of Solution, ΔsolnH, for the Dissolution of Various Proteins in Water protein

ΔsolnH

ref

hen egg white lysozyme canavalin bovine pancreatic trypsin inhibitor glucose/xylose isomerase porcine insulin concanvalin A thaumatin tartrate L-tartrate D-tartrate ovalbumin

+ + + + + +

7−9 10 11 9, 12 13 14 15

+ − −

16

In all cases with the exception of thaumatin D-tartrate and ovalbumin, the solubility is reported to be mostly increasing as the temperature increases. This implies that the dissolution is endothermic (positive heat of solution). In addition to the experiments listed in the table, Christopher, Phipps, and Cary17 have reported a few limited measurements of the temperature dependence of the solubility of 19 additional proteins. Among these, they found 9 that appeared to exhibit endothermic dissolution and 10 that appeared to exhibit exothermic dissolution (negative heat of solution). Although the evidence provided in Table 1 suggests that endothermic dissolution is common, the work of Christopher et al.17 cautions that it is not to be expected in every case. Indeed, the addition of particular polymers, alcohols, and salts as “crystallizing” agents to a waterbased crystal growth solution may have the effect of changing endothermic dissolution into exothermic dissolution and vice versa.

Figure 1. A plot of the net charge, Z1, on a canavalin trimer crystal growth unit as a function of pH. 1890



Article

ln γj* = −Z*j 2A I

The nucleation mechanism can be summarized by the mass balance equations21

In eq 4, I is the ionic strength of the supporting electrolyte, and A is a parameter which depends upon the temperature and the static dielectric constant of water.21 In the case of a 1−1 electrolyte, such as NaCl, the ionic strength and the molar concentration of the electrolyte are identical. Third, in the generalized scheme of chemical kinetics applicable to highly nonideal solutions, such as aqueous strong electrolytes, the activity replaces concentration in the law of mass action;22 because of the presence of the buffer in a protein crystal growth solution, the pH = −log aH+ is constant. This fixes the value of aH+ in eq 3, which makes αj a constant in a buffered solution and causes the rate law for reaction in the reverse direction in each of eqs 1 to be pseudo-first-order. Although independent of ionic strength, the rate coefficients, κ′j and κ″j−1, depend implicitly upon the {Zj} and are thus also functions of the pH. In analogy with classical nucleation theory, the mechanism summarized in eqs 1 is assumed to end in a rate determining step in which a spherical nucleus of critical size, j = n, is formed. The radius of this critical nucleus is given by21

β1

(1)Z1 + (1)Z1 ⇄ (2)Z 2 + (Z1 + Z1 − Z 2)H+ α2

(1a)

β2

(1)Z1 + (2)Z 2 ⇄ (3)Z3 + (Z1 + Z 2 − Z3)H+ α3

(1b)

βj − 1

(1)Z1 + (j − 1)Zj−1 XoooY (j)Zj + (Z1 + Zj − 1 − Zj)H+ αj

(1c)

where κ′ j − 1 γ*

βj − 1 =

j

(2)

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

κ″ j (a H +)Z1+ Zj−1− Zj γ* j

(4)

(3)

rn =

is the rate coefficient for decomposition of the j-mer back into the (j−1)-mer, γj* is the thermodynamic activity coefficient of the transition state, and aH+ is the activity of H+. In the absence of added strong electrolyte, the corresponding rate coefficients for these processes are κ′j−1 and κ″j , respectively. This nucleation mechanism takes into account three physical effects. We turn to these below: First because matter in bulk is electrically neutral, the protein molecules that wind up in the interior of a nucleus must be uncharged; hence, a process must exist for discharging protein macro-ions as they reach the surface of a nucleus. In the typical nucleation 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 charge conservation, has ionic valence, Zj* = Z1 + Zj−1. After its formation, the transition state, (j)Z*j , decays into the j-mer, (j)Zj, plus Z1 + Zj−1 − Zj hydrogen ions, as required to the balance the equation with respect to charge. By including hydrogen ions on the right-hand side of each step on the nucleation reaction mechanism above, we imagine that this discharge process consists of dumping H+ to the buffer. Free solution capillary electrophoresis experiments5 involving lysozyme crystals suspended in pH-buffered aqueous solutions of strong electrolytes demonstrate, however, that the surface electrostatic potential of lysozyme crystals depends not only upon the pH but also on the nature and the charge of any dissolved anions. This suggests the possibility that anions are involved in the discharge process. To take this mechanism into account, the stoichiometry of the left-hand sides of eqs 1 would need to be modified to include the number of anions assumed to participate 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 in eq 1c, taken as an example, is weakened by the Debye−Huckel plasma screening of the supporting electrolyte. The effect is to accelerate the net rate of reaction. In the case of the transition state, the effect of plasma screening is reflected in the transition state activity coefficient, γj*, which can be represented in the Debye−Huckel limiting law approximation by

b2 + 4ac 2a

−b +

(5)

where the parameters, a, b, c are defined by a = 16πz 2vkBTA I

(6)

b = kBT ln S

(7)

c = 2v(γ + 2.303zkBT (pH) − 2zZ1kBTA I )

(8)

In these equations, z is the pH-dependent surface charge density on a nucleus, v is the volume occupied by a protein molecule in the bulk nucleus phase, and γ is the interfacial tension acting at the boundary separating the nucleus from the growth solution. The supersaturation ratio is S = a1/as1, where a1 is the activity of the protein monomer, and as1 is its activity at the solubility limit. The absolute temperature is T, while Boltzmann’s constant is kB. On the basis of this model, the steady state nucleation rate, J, can be calculated in the form,21 ⎡ v(b2 + 4ac)1/2 ⎤1/2 ⎛ 1 ⎞3/2 ⎛ c ⎞ ⎥ ⎜ ⎟ κ′n ⎜ 1 ⎟ J=⎢ ⎝ c̃ ⎠ 8π 2kBT ⎦ ⎝ rn ⎠ ⎣ ⎤ ⎡ − πr 3 n exp⎢ {b + 3(b2 + 4ac)1/2 }⎥ ⎦ ⎣ 6vkBT

(9)

where in the exponential, the value shown for the exponent on rn corrects a misprint which appeared in an earlier publication.21 In eq 9, the concentration of protein monomer in the growth solution is c1, and the concentration of protein in the thermodynamic standard state is c̃. In the absence of strong electrolyte, the rate coefficient for forming the nucleus of critical size, j = n, by collision of a monomer with an (n − 1)mer is equal to κ′n. The units of J are the same as κ′n, which can be 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 its definition, κ′n, is the rate coefficient for adding a protein macroion to a charged nucleus. If attachment of a protein macro-ion to a crystal nucleus can be regarded as an “elementary” process, 1891



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κ′n can be represented by the Arrhenius equation,23 κ′n = κ′n̅ exp(−En/RT), where κ′n̅ is a weakly temperature-dependent factor with the same units as κn′ , En > 0 is the activation energy, R is the gas law constant, and T is the Kelvin temperature. As such, the value of κ′n should increase as the temperature increases. Turning next to the pH dependence, we note that since the sign of the net charge on the critical nucleus is the same as that on the protein macro-ion, their mutual electrostatic repulsion should diminish, and the value of κ′n 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 strength has been evaluated at fixed supersaturation ratio, S.21 This evaluation indicates that at f ixed pH and ionic strength, the value of J increases rapidly as the temperature increases from 4 to 25 °C. Evaluation of eq 9 at f ixed pH and temperature shows that J increases with increasing ionic strength. For pH < 6, the ultimate rise in J with increasing ionic strength equals that which can be achieved by increasing the temperature alone. Evaluation of eq 9 as a function of pH at f ixed temperature and ionic strength shows that J has a global maximum at pH = 2 and a smaller local maximum at pH = 11. At values of the pH between these two extremes, J is substantially smaller. Although we have evaluated J only in the case of lysozyme,21 it may still be possible to extract from this one example some trends that may be helpful in understanding the nucleation of crystals of other 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 should be 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 small crystals. In order to reduce the rate of nucleation while maintaining a high supersaturation, the behavior exhibited by lysozyme suggests that the temperature should be below room temperature, the pH should be far from the pI, and the salt concentration 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 competition between the rate of diffusion of protein macro-ions through the solution and their rate of attachment to the surface of a growing facet.24 Because of the interionic electrostatic forces acting between the various ions in the growth solution, the diffusive flux of protein is coupled to the diffusive flux of salt. In a three -component system consisting of water, salt, and protein,25 the diffusion fluxes of protein, Jv1, and salt, Jv2, in the x-direction, for example, are linked to the respective concentration gradients, ∂c1/∂x and ∂c2/∂x by the equations,26 v J1v = −D11

∂c1 v ∂c 2 − D12 ∂x ∂x

(10a)

v J2v = −D21

∂c1 v ∂c 2 − D22 ∂x ∂x

(10b)

Table 2. Elements of the Diffusion Coefficient Matrix in the Volume Fixed Frame of Reference for the Ternary Mixture, H2O + Lysozyme Chloride + NaCla Dv11

Dv12 −9

Dv21 −14

0.1102 × 10

8.6 × 10

Dv22 −9

19.8 × 10

1.461 × 10−9

a

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

In a protein crystal growth experiment, the electrolyte, which is part of the growth solution, ordinarily has little solubility in the solid crystal. Consequently its diffusive flux Jv2 is zero at the surface of the crystal. If we substitute Jv2 = 0 into eqs 10 and solve for Jv1, we find v ⎛ ∂c1 ⎞ ⎜ ⎟ J1v = −Deff ⎝ ∂x ⎠

(11)

where the effective diffusion coefficient, v Deff =

v v D11 D22

− v D22

Dveff,

is given by

v v D12 D21

(12)

In diffusion-controlled growth, (∂c1/∂x) > 0, so the minus sign in eq 11 indicates that the diffusion flux is directed in the negative x-direction, which is toward the surface of the crystal. When the data in Table 2 are substituted into eq 12, we find that v v Deff = 0.99D11

(13)

Dveff

Dv11

The coefficient of proportionality linking and in eq 13 is nearly unity. This observation is of technical importance because it permits the experimentally elaborate optical method25 for measuring diffusion coefficients to be replaced by the much simpler diaphragm cell method,27,28 in which the cell volume above the sintered glass diaphragm is loaded with aqueous salt solution, while the diaphragm and the cell volume below it are loaded with this same salt solution but with protein added. Equations 10−13 apply to diffusion in a crystal growth solution which is at rest with respect to the center of volume of the container. In the presence of gravitational convection, diffusion should be reckoned in a center of mass frame of reference. Transformation equations are available for converting the elements of the diffusion coefficient matrix, {Dvij}, which is appropriate in the center volume frame of reference to the elements of the diffusion coefficient matrix, {Dmij }, which is appropriate in the center of mass frame of reference.29 Even in the absence of external forces which cause convection, the center of mass moves by a process called advection which has its origin in the interdiffusion of components, such as protein and salt, which have substantially different molar masses. Advection alters the fluxes of protein and salt and can affect the rate of crystal growth.30 3.2.2. Role of Attachment. The rate of growth of a macroscopic crystal is determined by the competition between the rate of transport of protein marco-ions through the solution and their rate of attachment to the surface of a crystal. Several different models have been proposed to represent the kinetic coefficient for attachment of protein macro-ion to a protein crystal surface.31−37 Little is known for certain,20,31,38,39 about this attachment coefficient, except that like κn′, which governs

where i = 1 refers to the protein and i = 2 refers to salt. In the laboratory, the diffusive fluxes are measured optically25 with respect to the center of volume of the container through which the solutes diffuse. The superscript “v” identifies this as the “center of volume” frame of reference. The elements of the diffusion coefficient matrix are denoted by {Dijv} . The experimentally determined numerical values25 for the {Dvij} for lysozyme chloride in aqueous sodium chloride are listed in Table 2. 1892



Article

region. Rayleigh-Bernard cells can be prevented from forming if the fluid container is narrower than the sum of the widths of the rising and falling regions required to maintain a cell. This observation 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 the facet pushes the edge of the depletion zone into the bulk solution, which has the effect of changing the protein concentration gradient which drives the rate of diffusive transport to the crystal surface.35 This moving boundary effect is negligible, however, when the volume occupied by the solid crystals is small compared with the volume of the fluid in the container.32 An analytical treatment of solutal convection in a protein crystal growth solution shows that dimensionless groups can be used to relate the facet growth rate to the thermophysical properties of the growth solution.33 Assuming that the growth solution is sufficiently dilute in protein as to make diffusion independent of the frame of reference, these dimensionless groups, 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 its mass density, the Schmidt number is defined by33

the addition of a protein macro-ion to the surface of a crystal nucleus, the attachment coefficient refers to an elementary process. As such, the attachment coefficient should be thermally activated23 and should increase with any change in buffer that carries 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 to generate significant solutal convection.34 At a level of gravity equal to 10−6 go, which is available on the International Space Station, for example, a crystal must be at least 1000 μm in size to generate convection.34 Solutal convection alters the value of the protein concentration gradient, ∂c1/∂x, at the surface of the crystal. Along with the attachment coefficient, this gradient determines the rate of crystal growth.33 The effects of solutal convection on crystal growth have been analyzed in several different hydrodynamic models, and the results have been applied to a variety of proteins.4,31−37 Table 3 summarizes these theories, their mathematical methods, and the proteins to which they have been applied. Table 3. Summary of Theories of Solutal Convection in Protein Crystal Growth protein

crystal shape

method

ref

generic lysozyme lysozyme ferritin lysozyme generic generic ferritin lysozyme

vertical flat plate orthorhombic solid cylindrical solid sphere horizontal flat plate inclined plate sphere tetragonal solid tetragonal solid

analytical numerical numerical analytical numerical numerical dimensional analysis numerical/dimensional numerical

4, 33 32 31 31 35 36 37 34 34

v Sc = (μ/ρDeff )

(14)

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

Gr = g0αh3ρ2 /4μ2

(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 growth solution with respect to the protein concentration, c1. The Grashoff number expresses the ratio of the buoyancy to the viscous flow. If kG is the linear rate of growth of the facet, the Sherwood number is defined by

Under all conditions of solutal convection analyzed so far, the rate of transport of protein to the crystal and its attachment to a point on a crystal facet depend upon the position of the point in the gravitational field.4,31−37 This observation, we believe, explains the positional dispersion in measured growth rates which has been detected experimentally by the continuous microscopic optical observation of the linear growth rates of various spots on the surface of lysozyme crystals.40 The spatial dispersion in growth rates caused by convection can lead to crystals assuming shapes which they would not otherwise assume if they were in thermodynamic equilibrium with the quiescent growth solution.4,36 These equilibrium shapes are governed 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 line distance from the facet to the center of the crystal, must be the same for all facets.41 The pattern of solutal convection depends upon the angle between the gravity acceleration vector, go⃗ , and the outward normal, n̂, to the crystal facet. In the extreme case where n̂ and go⃗ are antiparallel, the depletion zone is then everywhere below a more dense growth solution which is above it. This gravitationally unstable arrangement resolves itself by generating a pattern of flow known as Rayleigh-Benard convection,35 in which the fluid over an area of the growing facet rises, while the fluid over the adjacent areas falls. A rising region when coupled with a falling region, both of which are needed to maintain the continuity of flow, is termed a Rayleigh-Bernard cell. The linear rate of advance of a crystal facet due to growth should be fast over the falling region and slow over the rising

v Sh = k Gh/Deff

(16)

The Sherwood number expresses the ratio of the linear growth rate to the effective diffusion velocity, Dveff/h. Finally, if cs1 is the protein solubility limit, and c01 is the protein concentration in the bulk of the solution, the kinematic supersaturation is defined by ϕs =

c10 − c1s c10

(17)

An analytical solution to the equations of convective diffusion is possible when the growing crystal facet is a semi-infinite flat plate with normal, n̂, perpendicular to g0⃗ .33 If the rate of attachment of protein molecules to the surface of the facet is much greater than the rate of diffusion, the Sherwood number satisfies the equation, (Sh) = 0.9((Sc)(Gr )ϕs)1/4

(18)

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

Sh (ScGrϕs)1/4

(19)

When M is of the order of unity, the rate of attachment is much greater than the rate of molecular transport through the solution. The depletion layer has its maximum extent, and 1893



Article

convection is well developed. By contrast, when M ≪ 1, transport of protein to the facet is fast compared with the rate of attachment, the depletion layer is limited, and convection is minimal. We suggest that somewhere between these two extremes lies a threshold value of M, above which convection is just sufficient to produce the growth rate anomalies42 that degrade crystal quality. For growth solution conditions where the value of M substantially exceeds this threshold value, exposure of a growing crystal and its growth solution to microgravity 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 the thermophysical properties of the growth solution and its crystals. For example, the kinematic viscosity, ν = μ/ρ, of the solution can be determined by Ostwald viscometry,43 the density, ρ, by pyncnometry,44 and the effective diffusion coefficient, Dveff, can be determined by use of the diaphragm cell technique.27,28 The height, h, and linear growth rate, kG, of a crystal facet can be determined by optical microscopy.40 Should experiments confirm the hypothesis that certain crystal defects have their origin in convective diffusion, then it follows from hydrodynamic theory33 that crystals of different proteins growing from different solutions should exhibit similar defects when the growth rates and the values of the thermophysical properties of the solutions combine to produce identical values for the figure of merit. This argument permits us to use the observation of growth plume convection and the known thermophysical properties in the case of lysozyme to estimate the threshold value of M for the onset of crystal defects. Those thermophysical properties are summarized in Table 4. We proceed as follows: We combine the value of Dv11

lysozyme crystals,46 we suggest that these improvements may have 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 1/4 power, the value of M is not likely to vary strongly with the nature and concentrations of the components used to prepare the growth solution. Exceptions may occur in special cases where thickeners, such as polyhydric alcohols or polyethylene glycol, have been added to increase the shear viscosity. Given this weak dependence of M on the growth solution conditions, plus the fact that lysozyme crystals seem to benefit from the exposure to microgravity,46 the value of M in the vicinity of 0.004 may be a universal boundary. Experimentation with a variety of proteins will be required to confirm this hypothesis. If confirmed, the figure of merit concept can be used to discriminate between proteins that are likely to benefit from exposure to microgravity and those which are not. 3.2.4. Time Dependence of the Supersaturation During Growth. In the absence of convection, the time dependence of the protein concentration in the depletion zone around a growing protein crystal can be obtained by solving Fick’s laws of diffusion analytically.26 In the presence of convection, the time dependence must be calculated numerically.31,32,34−36,38,39 These numerical methods are capable of providing the shapes of the protein concentration and fluid velocity profiles over only a limited time span. In the absence of theoretical methods applicable over the entire time span of growth, one can substitute bulk solution experimental methods, which although they respond only to the spatial average properties of the solution, are nonetheless, sensitive to the entire time course of the crystallization. Dilatometry is one such experimental method.3,47 A dilatometer consists of an enclosed volume connected to the atmosphere through a capillary side arm. To start an experiment, the enclosed volume plus the side arm are filled with the supersaturated crystal growth solution. Since the capillary has a uniform inside diameter, any change in the volume of the contents of the dilatometer is reflected in a proportional 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 decreases with time as the crystals appear. If Δh(t) is the change in height of the fluid in the side arm at time t, and Δh(∞) is the change in height at equilibrium (which corresponds to t → ∞), theory3 indicates that

Table 4. Thermophysical Properties of an Aqueous Solution of Lysozyme Containing a Lysozyme Crystal of Height h = 0.5 mm Growing at a Linear Rate of 8.75 × 10−9 m/sa c01 (mg/mL) Ref 45

cs1 (mg/mL) Ref 45

μ (Ns/m2) Ref 43

(dρ/dc1)

11.7

1.2

0.094

a

Ref 43

h (mm) Ref 45

kG (m/s) Ref 40

0.3032

0.5

8.75 × 10−9

c01,

The bulk lysozyme concentration is the solubility of lysozyme is cs1, the shear viscosity is μ, the mass density is ρ, and the height of the crystal is h.

listed in Table 2 with eq 13 and compute Dveff = 0.11 × 10−9 m2/s. By combining the value of ρ found in Table 2 with the value of μ in Table 4, we use eq 14 to compute Sc = 8230. Next using 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 growth plume above a growing lysozyme crystal,45 the height of the crystal 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 the slope of the typical distance vs time plot shown in Figure 1 of ref 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 16 then gives Sh = 0.040. Finally, by combining the values of c01 and cs1 listed in Table 4, we can use eq 17 to compute ϕs = 0.9 According to eq 19, under these conditions, M = 0.004. Schlieren photography45 of the growth plume has confirmed the existence of well-developed solutal convection around a lysozyme crystal growing under conditions similar to those summarized in Table 4. Since exposure to microgravity has been reported to lead to improvements in the X-ray quality of

⎛ Δh(t ) ⎞ σ(t ) = σ(0)⎜1 − ⎟ Δh(∞) ⎠ ⎝

(20)

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

c1(t ) −1 c1s

(21)

where c1(t) is the instantaneous value of the concentration of dissolved protein, and cs1 is the thermodynamic solubility of the protein. Experiments carried out on crystallizing lysozyme3 and canavalin47 solutions show that Δh(t ) = Δh(∞)(1 − exp(−kt ))

(22)

where k is the specific crystallization rate. When eq 22 is substituted into eq 20, we find σ(t ) = σ(0) exp( −kt ) 1894

(23)



Article

shells are diamagnetic.49−52 In a diamagnetic material, application of an external magnetic field, H, induces a magnetic dipole moment, M, which is opposite in direction to H. As a consequence, the diamagnetic susceptibility, χ = M/H, is negative.53 When the magnet is designed so as to produce a constant magnetic field gradient, (∂H/∂y), in the y-direction, there will be a volume force μ0χH(∂H/∂y)acting on the center of 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 also experience a volume force, ρg0. Any diamagnetic material in the form of a rigid solid or a homogeneous liquid will be completely levitated when these forces balance. Levitation is the analog of microgravity. The condition for magnetic levitation is49

Although empirical, eq 23 is consistent with an analytic theory of growth of crystals that have nucleated all at the same time.48 The mass, m(t), of the crystals per unit volume of the dilatometer is then given by47 m(t ) = (c1(0) − c1s)(1−e−kt )

(24)

where c1(0) is the initial value of c1(t). For crystals of lysozyme3 and canavalin,47 growing in pH -buffered aqueous electrolyte growth solutions, k has been determined as a function of pH, temperature, and sodium chloride concentration. The trends in k, which are the same in the case of both proteins, can be summarized as follows: At constant temperature and pH, k increases with increasing salt concentration. At constant temperature and salt concentration, k increases with any change in the buffer which carries the pH closer to the pI. At constant pH and salt concentration, k decreases with increasing temperature. Theory47 suggests that k = NρωS2 /cs, where N is the number of nucleation sites per unit volume, S2 is the cross sectional area of a nucleation site, cs is the solubility of the protein, and ω is the rate coefficient for the elementary process of attachment of a protein macro-ion to a crystal facet (analogous to κn′ in nucleation theory). Like κn′, we can represent the temperature dependence of ω using the Arrhenius equation,23 ω = ωL exp(−EL/RT), where ωL is a temperature independent constant with dimensions, cm/s, and EL > 0 is the activation energy. We can represent cs by the van’t Hoff equation,47 cs = b exp(−ΔHs/RT), where b is a constant with the same dimensions as cs, and ΔHs = ΔsolnH is the heat of solution. Assuming that the other parameters determining k do not depend exponentially on the temperature, we find that the apparent activation energy of k is the composite, E = EL − ΔHs. If protein dissolution is endothermic, and if ΔHs > EL, then the apparent activation energy, E, is negative and k decreases with increasing temperature as observed. As proteins, lysozyme and canvalin are quite different. Lysozyme3 is an animal protein consisting of 129 amino acid residues, molecular weight of 12 kDa, and pI = 11. By contrast, canavalin47 is a plant protein, which crystallizes as a trimer consisting of 1095 amino acids, molecular weight 125.8 kDa, and pI = 5.2. The common dependence of the value of k upon pH and salt concentration for these proteins finds its basis in the fact that both go into aqueous solutions as polyelectrolytes. Any change in buffer that carries the pH closer to the pI will decrease the macro-ion charge, while an increase in electrolyte concentration will enhance the Debye−Huckel plasma screening of these charges. Either change will weaken the repulsive interionic electrostatic force acting between the macro-ions which will in turn accelerate the rate of formation of a crystal. The temperature dependence of k for the two proteins is the same presumably because both dissolve endothermically.47 If for a given protein, the pH, salt concentration, and temperature dependence of k are the same as that observed in the case of lysozyme3 and canvalin,47 the following rules of thumb may prove useful: Because the value of k increases with decreasing temperature, the temperature should be low. The value of k can be increased by increasing the salt concentration and by changing the acidity to bring the pH as close as possible to the pI. The advantage gained should be readily noticeable.3,47 3.2.5. Quenching of Gravitational Affects in an Applied Magnetic Field. Protein crystals and protein crystal growth solutions containing ions and atoms with closed electronic

⎛ ∂H ⎞ ρg μ0 μ0 2 H ⎜ ⎟= 0 χ ⎝ ∂y ⎠

(25)

In the case of water, for example, which has magnetic susceptibility,53 χ = −9 × 10−9 and density ρ = 1000 kg/m3, levitation requires μ02H(∂H/∂y) = −1370 T/m2.49,52 This condition specifies a magnetic field intensity that lies near the upper limit of performance of the best available superconducting magnets.50 Should the material be a dispersion of a solid in a liquid, for example, protein crystals suspended in growth solution, the acceleration due to gravity will be canceled simultaneously for the crystals and the growth solution, if both have 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 concentration, density, and magnetic susceptibility, the condition for suppression of convection requires that in eq 25, ∂ρ/∂y be substituted for ρ, and ∂χ/∂y be substituted for χ. After introducing the mass fraction, φ, and use of the chain rule, the result can be written in the form51 ⎛ ∂H ⎞ (∂ρ /∂φ)g0μ0 μ0 2 H ⎜ ⎟= (∂χ /∂φ) ⎝ ∂y ⎠

(26)

In the case of lysozyme, experiments51 show that (∂ρ/∂φ) = 5.65 kg/m3, while (∂χ/∂φ) = 9.97 × 10−9. Upon substitution of these results into eq 26, we find that the condition for suppression of convection is μ02H(∂H/∂y) = 6.98 T/m2. This condition is substantially less demanding than the condition required to achieve levitation; moreover, it requires a magnetic field gradient with a positive, rather than a negative sign. 3.2.6. Hall Effect. By virtue of the presence of the protein macromolecular ions and inert electrolyte dissolved in a protein crystal growth solution, a protein crystal growth solution is an ionic conductor. If a volume element of the solution has charge density q, and convection velocity, v,⃗ it will experience a volume Lorentz force,

FL⃗ = q(vxB ⃗ ⃗)

(27)

when subjected to an applied magnetic flux density, B⃗ . The Lorentz force will give rise to a Hall effect drift current in a direction parallel to F⃗L.54 The Hall effect in electrolytes has been investigated within the context of the theory of nonequilibrium thermodynamics.54 This theory suggests that the Hall current should be orders of magnitude smaller than the diffusion current and consequently should have a minimal effect on the rate of growth of a crystal. 1895



Article

3.3. Ostwald Ripening. A successful protein crystallization experiment can produce a large number of crystals of varying sizes.55 This nonequilibrium situation can be improved by waiting for the crystal size distribution to advance toward larger sizes by virtue of the surface tension (surface free energy density) driven process called Ostwald ripening.20 The crystal surface tension is dominated by the acid/base sites which occur whenever the combination of the protein molecular structure and the crystal structure permits ionizable groups to occupy positions on the surface of the crystal. The ionizable groups include the molecular C-terminus, the N-terminus, and also any amino 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. The ionizable groups make two contributions to the surface free energy. These are (1) the entropy associated with the mixing of protonated sites and empty sites on the surface of the crystal and (2) the entropy and energy stored in the Debye−Huckel ion plasma which surrounds the crystal and which depends upon the nonzero surface charge density on the crystal.20 Although both of these contributions serve to reduce the surface tension and reduce the average crystal size in Ostwald ripening, the first dominates. In the case of the first effect, when the pH is far to the acid side of the pKa of a particular type of surface site, all of the sites of this type will be protonated. There will be no entropy associated with the exchange of H+ between empty and occupied sites. In this situation, the particular type of site in question will contribute only to the “background”, pH independent part of the surface tension. As the pH becomes more basic, however, the fraction of the total sites of this type which ionize (i.e., donate H+ to the solvent) will grow, empty sites will be created, and the surface entropy of mixing will develop. The effect of this entropy of mixing is to reduce the surface tension. As the pH becomes still more basic, the fraction of ionized sites of this particular type will approach unity. When the fraction of ionized sites reaches unity, there is no longer an entropy of mixing associated with this type of site; the site again merges with the “background” and no longer contributes to the pH dependence of the surface tension. The most favorable condition for the surface tension of the protein crystal in Ostwald ripening occurs when the ionizable sites of all possible types are forced into the “background”. In principle, this can be achieved for the crystal as whole by adjusting the pH to the acid side of the pKa of the most acidic ionizable group or alternatively by adjusting the pH to the basic side of the pKa of the least acidic ionizable group. Depending upon the protein, these extremes may be as acidic as pH = 2 or as basic as pH = 13. As either of these extremes is likely to denature the protein molecule, neither would seem to be practical. As a compromise, we can rank order the pKa values of the ionizable groups in the order, pKa1 < pKa2 < pKa3, etc., and then choose from among adjacent pairs, a pair for which the difference ΔpKa = |pKa1 − pKa2| is largest. If the pH is set equal to pH = (1/2)(pKa1 + pKa2), then both pKa1 and pKa2, as well as the pKa values of other ionizable groups more distant from this mean, should lie in one of the ranges, pKa ≪ pH or pKa ≫ pH. The contribution made to the surface tension by each type of ionizable group lying in either of these ranges should be restricted to the background. In addition to the surface energy, the Ostwald ripening process involves the diffusion of protein macromolecular ions through the growth solution and their attachment to the surface of a crystal facet.20,56 As we have noted above, the dependence

of the protein diffusion coefficient and attachment coefficient upon salt concentration and temperature is not entirely clear. In contrast to the growth phase, where our lack of knowledge does not prevent us from speculation concerning the effects of diffusion and attachment on the rate of growth, these two parameters enter the theory of the Ostwald ripening phase in a fashion which is sufficiently intricate56 as to preclude predictions of the effect of salt and temperature on the rate of coarsening.

4. TIME DEPENDENCE OF THE SUPERSATURATION INCLUDING NUCLEATION, GROWTH, AND RIPENING The nucleation stage begins with an induction period19 during which the relative supersaturation, σ(t), is essentially constant, and the individual protein molecules agglomerate to form nuclei of critical size. The induction time required to establish this steady state depends upon the temperature, the pH and the salt concentration.57 As mentioned above, the results of dilatometer experiments show that during the growth period, which follows the induction period, the relative supersaturation diminishes exponentially with time according to eq 23. Finally, in the Ostwald ripening stage during which the smaller crystals dissolve while the larger crystals grow, the time dependence of the relative supersaturation converts from an exponential to an inverse time power law, where σ(t) is proportional to t−δ and δ is a positive rational number less than unity.20 Although we have 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, which attempts to cover the entire time scale, been able to plot only the qualitative features of the time decay of the relative supersaturation.

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

5. SUMMARY In Table 5, we have collected together our predictions of the effects of supersaturation ratio, pH, temperature, and salt concentration on the rate of nucleation, J; the specific rate of crystal growth, k; and average crystal size, ⟨a⟩, the number of crystals, per unit volume, N, and the absolute value of the rate of decay of the relative supersaturation, |dσ(t)/dt|, during the Ostwald ripening phase. 6. CONCLUSIONS By distinguishing between thermodynamic control and kinetic control and by combining the theory and the experimental observations discussed above, we can make the following recommendations for coping with the effects of hydrodynamics 1896



Article

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

increasing S increasing T increasing I pH → pI pH = (1/2)(pKa1 + pKa2)

nucleation

growth

J

k

Ostwald Ripening N

|dσ/dt|

increases increases increases increases N/A

N/A decreases increases increases N/A

N/A unknown unknown unknown increases

N/A unknown unknown unknown decreases

increases unknown unknown unknown decreases

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

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

and electrostatics in the crystallization of proteins from pH buffered aqueous solutions of strong electrolytes. Our conclusions with respect to hydrodynamics are summarized by the definition of M in eq 19 and the discussion in Section 3.2.3. Our recommendations with respect to electrostatics, which follow, are restricted to proteins that dissolve endothermically. The comments referring to thermodynamic control are based upon Table 1, while the comments referring to kinetic control 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 the crystal growth medium at room temperature up to its solubility limit. The growth solution should then be chilled below room temperature in order to increase the supersaturation. This will make the largest possible excess of protein available for crystallization. 2. Kinetic Control (a) Nucleation: To avoid the creation of a large number of competing nuclei, the nucleation rate should be limited by maintaining the temperature below room temperature, the salt concentration should be minimal, and the pH should not be equal to the pI. (b) Growth: To encourage rapid growth of existing crystals, the temperature should be as low as feasible, the salt concentration should be large, and the pH should be equal to the pI. Note that these conditions with respect to salt and pH contrast 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 during the ripening phase, if the pH = (1/2)(pKa1 + pKa2), where pKa1 and pKa2 are numerically adjacent pKa values for which ΔpKa = |pKa1 − pKa2| is large. Because the effects of temperature and salt concentration upon the protein macro-ion diffusion coefficient and attachment coefficient are not well-known, we cannot make a recommendation concerning the appropriate temperature and salt concentration for Ostwald ripening. Suffice it to say, however, the slower the rate of decay of the supersaturation, the larger will be the crystals. (d) Compromise Conditions: Because increasing the salt concentration and adjusting the pH to be close to the pI serves to increase both the rate of nucleation and the rate of growth, optimal conditions of salt and pH are not to be found if the goal of a crystallization trial is to suppress the rate of nucleation while increasing the rate of growth. Some compromise is necessary. Since the rate of nucleation is more sensitive to the salt concentration than is the rate of growth, and the effect of salt on ripening is unknown, we recommend the minimum salt concentration necessary to produce an adequate supersaturation. Likewise since the rate of nucleation is more sensitive to pH than are the rates of either growth or ripening, the pH value

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors thank Dr. Hana McFeeters and Mary Hames for useful discussion and manuscript editing.

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REFERENCES

(1) Stout, G. H.; Jensen, L. H. X-ray Structure Determination: A Practical 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. Data 1999, 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. Growth 1997, 173, 132−140. (12) Vuolanto, A.; Uotila, S.; Leisola, M.; Visuri, K. J. Cryst. Growth 2003, 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. 1897



Article

(17) Christopher, G. K.; Phipps, A. G.; Gray, R. J. J. Cryst. Growth 1998, 191, 820−826. (18) Penrose, O. In Sotchastic Process in Nonequilibrium Systems; Garrido, L.; Seglar, P.; Shepherd, P. S., Eds.; Springer: New York, 1978. (19) Wu, D. T. J. Chem. Phys. 1992, 97, 2644−2650. (20) Rowe, J. D.; Baird, J. K. Int. J. Thermophys. 2007, 28, 855−864. (21) Baird, J. K.; Kim, Y. W. Mol. Phys. 2002, 100, 1855−1866. (22) Baird, J. K. J. Chem. Educ. 1999, 76 (8), 1146. (23) Moore, J. W.; Pearson, R. G., Kinetics and Mechanism, 3rd ed.; John Wiley and Sons: New York, 1981; pp 177−181. (24) Otalora, F.; Novella, M. L.; Gavira, J. A.; Thomas, B. R.; Garcia Ruiz, J. M. Acta Crystallogr. D 2001, 57, 412−417. (25) Albright, J. G.; Annunziata, O.; Miller, D. G.; Paduano, L.; Pearlstein, A. J. J. Am. Chem. Soc. 1999, 121 (14), 3256−3266. (26) Annunziata, O.; Albright, J. G. Ann. N.Y. Acad. Sci. 2002, 974, 610−623. (27) Leaist, D. G. J. Phys. Chem. 1986, 90, 6600−6602. (28) Burns, N. L.; Clunie, J. C.; Baird, J. K. J. Phys. Chem. 1991, 95, 3801−3804. (29) Tyrrell, H. J. V.; Harris, K. R. Diffusion in Liquids: A Theoretical and Experimental Study; Butterworths: London, 1984. (30) Westphal, G. H.; Rosenberger, F. J. Crystal Growth 1978, 43, 687. (31) Lee, C. P.; Chernov, A. A. J. Cryst. Growth 2002, 240, 531−544. (32) Lin, H.; Rosenberger, F.; Alexander, J. I. D.; Nadarajah, A. J. Cryst. Growth 1995, 151, 153−162. (33) Baird, J. K.; Guo, L. J. Chem. Phys. 1998, 109, 2503−2508. (34) Carotenuto, L.; Cartwright, J. H. E.; Castagnolo, D.; García Ruiz, J. M.; Otálora, F. Microgravity Sci. Technol. 2002, 13, 14−21. (35) Lima, D.; De Wit, A. Phys. Rev. E 2004, 70, 021603−021603−7. (36) Zhu, Z.-H.; Hong, Y.; Ge, P.-W.; Yu, Y.-D. Chin. Phys. 2004, 13, 1982−1991. (37) Riahi, D. N.; Obare, C. W. Appl. Appl. Math. 2009, 4, 13−25. (38) Poodt, P. W. G.; Christiansen, P. C. M.; Van Enckevort, W. J. P.; Maan, J. C.; Vlieg, E. Cryst. Growth Des. 2008, 8, 2194−2199. (39) Lappa, M. Phys. Rev. E. 2005, 71, 031904. (40) Judge, R. A.; Forsythe, E. L.; Pusey, M. L. Cryst. Growth Des. 2010, 10, 3164−3168. (41) DeFay, R.; Prigogine, I.; Bellemans, A. Surface Tension and Adsorption; John Wiley: New York, 1966; p 296. (42) Poodt, P. W. G.; Heijna, M. C. R.; Schouten, A.; Gros, P.; van Enckevort, W. J. P.; Vlieg, E. Cryst. Growth Des. 2009, 9, 885−888. (43) Fredericks, W. J.; Hammonds, M. C.; Howard, S. B.; Rosenberger, F. J. Cryst. Growth 1994, 141, 183−192. (44) Clunie, J. C.; Baird, J. K. Phys. Chem. Liq. 1999, 37, 357−371. (45) Pusey, M. L.; Witherow, W.; Naumann, R. J. Cryst. Growth 1988, 90, 105−111. (46) Judge, R. A.; Snell, E. H.; van der Woerd, M. J. Acta Crystallogr. D 2005, 61, 763−771. (47) Caraballo, K. G.; Baird, J. K.; Ng, J. D. Cryst. Growth Des. 2006, 6, 874−880. (48) Barlow, D. A. J. Cryst. Growth 2009, 311, 2480−2483. (49) Qi, J.; Wakayama, N. I.; Ataka, M. J. Cryst. Growth 2001, 232, 132−137. (50) Yin, D. C.; Wakayama, N. I.; Harata, K.; Fujiwara, M.; Kiyoshi, T.; Wada, H.; Niimura, N.; Arai, S.; Huang, W. D.; Tanimoto, Y. J. Cryst. Growth 2004, 270, 184−191. (51) Ramachandran, N.; Leslie, F. W. J. Cryst. Growth 2005, 274, 297−306. (52) Moreno, A.; Yokaichiya, F.; Dimasi, E.; Stojanoff, V. Ann. N. Y. Acad. Sci. 2009, 1161, 429−436. (53) Hinchcliffe, A.; Munn, R. Molecular Electromagnetism; John Wiley and Sons: New York, 1985. (54) Lielmezs, J.; Musbally, G. M. Electrochim. Acta 1972, 17, 1609− 1613. (55) Ng, J. D.; Lorber, B.; Witz, J.; Theobald-Dietrich, A.; Kern, D.; Giege, R. J. Cryst. Growth 1996, 168, 50−62. (56) Marquesee, J. A.; Ross, J. J. Chem. Phys. 1983, 79, 373−378.

(57) Biscans, B.; Laguerie, C. J. Phys. D: Appl. Phys. 1993, 26, B118− B112.

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Effects of Hydrodynamic Convection and Interionic Electrostatic Forces

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