Control of Nucleation Rate for Tetragonal Hen-Egg White Lysozyme Crystals by Application of an Electric Field with Variable Frequencies H. Koizumi,* K. Fujiwara, and S. Uda
CRYSTAL GROWTH & DESIGN 2009 VOL. 9, NO. 5 2420–2424
Institute for Materials Research, Tohoku UniVersity, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan ReceiVed December 3, 2008; ReVised Manuscript ReceiVed January 24, 2009
ABSTRACT: Control of the nucleation rate is important for the growth of high quality protein crystals; therefore, the development of improved growth methods is required. Crystal growth with an external electric field is an attractive method for control of the nucleation rate. However, mainly a decrease in nucleation rate was observed when an external electric field was applied using DC voltage. By focusing on the chemical potential of the liquid and solid, control of both the increase and the decrease in nucleation rate was successfully achieved by applying an AC electric field. Control of nucleation rate in terms of enhancement and retardation is reported for hen-egg white lysozyme crystals by imposition of an electric field with varying frequencies, which was determined by the electrical permittivity of both the hen-egg white lysozyme solution and crystals. Introduction The most important goal of studies on the crystallogenesis of biological macromolecules is to obtain high quality crystals in order to perform X-ray structure determination of protein molecules. However, it is very difficult to obtain high quality crystals of protein molecules. Therefore, much research has concentrated on obtaining high quality protein crystals by applying external fields, such as electric,1-10 magnetic,11,12 electromagnetic,13-20 and hydrodynamic21,22 fields. Under each of these applied external fields, the electrical permittivity, magnetic permeability, and stiffness are key factors. Effect of external fields is determined by the difference in these parameters between the liquid and solid. Generally, the difference in the electrical permittivity between the liquid and solid is large enough for an electric field to be effective, while the magnetic permeability of the liquid is almost the same as that of the solid. On the other hand, it was reported that a strain field of several GPa is needed to modify the phase transformation.23 Therefore, it is expected that employing an electric field as an external field is the most effective for the control of nucleation as well as growth dynamics. It has also been experimentally confirmed that the presence of an electric field modifies the processes of phase transformation, nucleation, and growth dynamic in some materials,24-28 by controlling the chemical potential of the liquid and solid. Protein crystallization has also been performed by applying an electrostatic1-4,7,8,10 or current-injection5,6,9 field. Almost all studies have used hen-egg white (HEW) lysozyme as a protein material, and it has been reported that the nucleation rate decreases when either an electrostatic or a current-injection field is applied using DC voltage. However, it is important to enhance nucleation for almost all proteins, so it is desirable to increase the nucleation rate by application of an electrostatic or currentinjection field. To date, we have studied the nucleation and growth dynamics of YBa2Cu3O7-x (YBCO) superconductive oxide26,28 and langasite (La3Ga5SiO14),25,27 with focus on the change in chemical potential attributed to the difference in the electrical permittivity * To whom correspondence should be addressed. E-mail: h_koizumi@ imr.tohoku.ac.jp.
Figure 1. Schematic diagram of the predicted dependence of the dielectric permittivity for protein solution and protein crystals on the imposed frequency.
between the liquid and solid. Theoretical analysis suggested that the difference in the electrical permittivity between the liquid and solid determines whether the nucleation rate increases or decreases.25-31 There is a general dependence of the electrical permittivity of both liquid and solid on imposed frequency.32 Much research on the dielectric properties of protein solutions has been carried out, and it has been reported that the electrical permittivity of protein solution is almost constant in the frequency range below 1 MHz.33-35 On the other hand, research concerning the dielectric properties of protein crystals has not often been performed,36 because it is very difficult to handle protein crystals. The dielectric properties of protein powder have been investigated by Takashima,37 and a large electrical dispersion of the electrical permittivity was observed for protein powder in the relatively low frequency range below 1 MHz. The protein powder consists of crystallites, so it is considered that the dielectric property of protein powder reflects that of protein crystals. Furthermore, it is also suggested that the electrical permittivity of protein powder is larger than that of protein solution in the low frequency range (close to zero).37 This means that there is the possibility of a reverse in the electrical permittivity between the liquid and solid. That is, as shown in Figure 1, it is expected that the electrical permittivity of protein crystals is larger than that of protein solution in the low frequency region, and vice versa for the high frequency
10.1021/cg801315p CCC: $40.75 2009 American Chemical Society Published on Web 03/02/2009
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region. Thus, we report the control of nucleation rate of henegg white lysozyme crystals with an external electric field using AC voltage. Thermodynamic Effect of an External Electric Field. The effect of an externally applied electric field on nucleation rate was theoretically analyzed by Kashchiev29,30 and Isard,31 and the critical Gibbs free energy for nucleation with an external electric field is obtained by the following equation:26,28
∆GE∗ )
[
(
1 - (εS/εL) 4 3 (aσ∗)3 ∆µ VcE2εL 27 8π 2 + (εS/εL)
)]
-2
(1)
where εδ is the electrical permittivity of the δ-phase (δ ) L or S), E is the electric field, a is a geometric constant given by a )(4π)1/3(3Vc)2/3, σ* is the specific surface energy of the cluster for nucleation, ∆µ is the free energy difference between the liquid and solid, and Vc is the mean atomic volume of a cluster for nucleation. Thus, the increase or decrease in the nucleation rate for a single component system due to an electric field depends on the difference between the electrical permittivity of the liquid and the solid. However, for a binary system, the derivative of the electrical permittivity with respect to the concentration of the jth species must also be considered, because the composition between solid and liquid in equilibrium is different and chemical potential should be used instead of free energy. The chemical potential of the jth species in a liquid, µLj , and that of a solid, µSj , modified by an external electric field, can be expressed as25 j µLj ) µ0L + RT ln(γLj XLj ) + NAzLj eφLj +
∂ 1 j j2 ΩL(εLEL) ∂XLj 2 (2)
[
]
and j + RT ln(γSj XSj ) + NAzSj eφSj + µSj ) µ0S
∂ 1 j j2 ΩS(εSES) ∂XSj 2 (3)
[
]
where for the jth species, µj0δ (δ ) L or S) is the standard state chemical potential, R is the gas constant, γδj is the activity coefficient, Xδj is the mole fraction of the concentration, NA is Avogadro’s number, zδj is the valence of the ion, φδ is the local j δ is the strength of the external electric field on the potential, E δ-phase and Ωδj is the molar volume. The second term on the right-hand side is a mixed energy term, the third is the charge potential, and the fourth is the derivative of the electrostatic energy induced by the electric field with respect to the concentration of the jth species. In the derivative of the electrostatic energy, the electric field is assumed to be constant, while the electrical permittivity varies with composition. Thus, eqs 2 and 3 can be rewritten as j µLj ) µ0L
Figure 2. Molar free energy diagram of the liquid and solid phases. (a) Solid-liquid equilibrium, (b) driving force for nucleation with no electric field, (c) driving force for nucleation with application of an electric field. j j and µS(E) , are Thus, the field-modified chemical potentials, µL(E) obtained by adding the electrostatic term to the zero-field j j and µS(0) : chemical potential, µL(0)
1 j 2 ∂εL j j µL(E) ) µL(0) + ΩLj E L 2 ∂Xj
(6)
1 j 2 ∂εS j j ) µS(0) + ΩSj E µS(E) S 2 ∂Xj
(7)
L
and
S
From eqs 6 and 7, the driving force, ∆µ, for the nucleation can be expressed as j j j j ∆µ(E) ) µL(E) - µS(E) ) ∆µ(0) +
1 j 2 ∂εL + RT ln(γLj XLj ) + NAzLj eφLj + ΩLj E L 2 ∂Xj
L
(4) and
1 j 2 ∂εS j µSj ) µ0S + RT ln(γSj XSj ) + NAzSj eφSj + ΩSj E S 2 ∂Xj
S
(5)
(
∂ε 1 j j 2 ∂εL j S2 S ΩLEL j - ΩSj E 2 ∂X ∂Xj L
S
)
(8)
where ∆µj(E) and ∆µj(0) are the driving forces for nucleation, with and without an external electric field, respectively. Let us consider the molar free energy of the liquid and solid in the binary system, which is illustrated in Figure 2 as a function of the molar ratio of component B, XδB (δ ) L or S), combined with the field-modified molar free energy of the liquid and solid.
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Here, GS(0) and GL(0) are the G curves of the solid and liquid without an external electric field, respectively. As seen in Figure 2a, the components XLB* and XSB* at the common tangent to each G curve without an external electric field represent the equilibrium compositions of the liquid and solid, respectively. Therefore, the driving force for the nucleation is zero, so that the chemical potentials of component A for the liquid and solid B* at XB ) XB* L and XS , respectively, are equal and are represented by the intersection of the common tangent with the vertical axis (XB ) 0). However, if the composition of the liquid deviates slightly from the equilibrium composition, XLB*, to a certain position, XBL(0), which yields some supersaturation, the chemical potential of the liquid changes, resulting in a new common tangent at XBL(0). Accordingly, the chemical potential of the solid is associated with the common tangent at XBS(0), which is parallel to the common tangent of the liquid, so that the maximum j , is available (Figure 2b) for nucleation. driving force, ∆µ(0) Under application of an electric field, the G curves of the liquid and solid change according to the change in the chemical potential of each phase, given by eqs 6 and 7. Under the external B B B B goes to XL(E) , while XS(0) goes to XS(E) , as electric field, XL(0) shown in Figure 2c. There are two key factors to consider in the change of the G curve of the liquid and solid; (i) the difference in the electrical permittivity between the liquid and solid, and (ii) the sign of the electrical permittivity derivative with respect to the composition which will be discussed next. The conservation of electron flux holds at the interface between the liquid and solid:
j S ) εLE jL εSE
Figure 3. Schematic illustration of the “containerless” batch arrangements with electrodes on both sides of a protein droplet.
(9)
Thus, if εS , εL, then the effect of an external electric field on the G curve of the solid is larger than that on the liquid, because ES . EL. That is, the G curve of the solid has a larger shift than that of the liquid. On the other hand, the sign of the derivative plays an important role in determining which direction the liquid and solid G curves shift. If the sign of the derivative j 2L(∂εL)/(∂XjL) - ΩjLE j 2S(∂εS)/ is negative for both phases, then ΩjLE j (∂XS) is larger than zero. Taking a similar approach to evaluation j , for nucleation in the case of E ) 0, of the driving force, ∆µ(0) j , increases under such a the driving force for nucleation, ∆µ(E) condition (εS , εL, (∂εL)/(∂XLj ) < 0 and (∂εS)/(∂XSj ) < 0) when an external electric field is applied. The mechanism is shown in Figure 2c. It should be noted that the difference in permittivity between the liquid and solid determines the degree of change in the driving force for nucleation due to an external electric field, while the sign of the derivative dominates the direction of shift for the liquid and solid G curves. Experimental Procedures The HEW lysozyme (Seikagaku Kogyo Co.) was purified by sixtimes crystallization and lyophilized. 114 mg/mL HEW lysozyme and 1 M NaCl were prepared, and both solutions of equal quantity were mixed. The mixed solution was passed through a 0.20 µm pore size filter to remove foreign matter and large protein aggregates. Finally, the solution of 57 mg/mL HEW lysozyme and 0.5 M NaCl at pH 4.5 was employed. Moreover, it is expected that the difference in the electrical permittivity between the liquid and solid would change by varying precipitants. Thus, NiCl2 0.8 M was also prepared, and the solution of 57 mg/mL HEW lysozyme and 0.4 M NiCl2 at pH 4.3 was also used. All crystallization experiments were conducted at 17 ( 0.2 °C using the “containerless” batch method developed by Chayen.38 As shown in Figure 3, a crystallization drop is suspended between two oils. The bottom layer contains high-density oil, while the top layer contains
Figure 4. (a-c) Distribution of crystals in a drop, with and without application of an external electric field, using NaCl as a precipitant.
low-density oil, so that the crystallization drop does not touch the container walls. Fluorinert (F ) 1.68 g/m3) and paraffin (F ) 0.86-0.89 g/m3) were used as high- and low-density oils, respectively. The distance between the electrodes was 0.5 cm and the size of the crystallization drop was 10 µL. In the case of the precipitant, NaCl, an external electric field of 860 V/cm was applied to the crystallization drops at frequencies of 1 MHz, 500 kHz, and 10 kHz. In the case of the other precipitant, NiCl2, an external electric field of 660 V/cm was applied to the crystallization drops at frequencies of 1 MHz and 500 kHz. In both
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Figure 5. HEW lysozyme crystals in drops nucleated in the presence of (a) no electric field, (b) an electric field at 500 kHz, and (c) an electric field at 1 MHz using NiCl2 as a precipitant. cases, the crystallization drops with and without application of an external electric field were observed using an optical microscope after 24 h.
Results and Discussion Figure 4 shows the crystal distributions in drops with and without an external electric field applied, and with NaCl as the precipitant. Comparison of the results from different frequencies indicates the nucleation rate increased when the external electric field was applied at 1 MHz, as shown in Figure 4a. On the other hand, the nucleation rate with the external electric field at 500 kHz was almost the same as that without the external electric field, as shown in Figure 4b. Furthermore, the nucleation rate decreased when the external electric field was applied at 10 kHz, as shown in Figure 4c. This indicates that the frequency of the external electric field varied the chemical potentials of the liquid and solid by changing the electrical permittivity of each phase. When the precipitant was changed to NiCl2, the nucleation rate changed significantly for applied external electric fields at 1 MHz and 500 kHz which has been verified by carrying out 10 time measurements. Figure 5 shows the optical micrographs of HEW lysozyme crystals in drops nucleated in the presence of (a) no electric field, (b) an electric field at 500 kHz, and (c) an electric field at 1 MHz, using NiCl2 as a precipitant. The nucleation rate increased significantly when applying an external electric field at 1 MHz, as seen in Figure 5c, but decreased when applying an external electric field at 500 kHz, as seen in Figure 5b. Let us consider how the nucleation rate increases or decreases when applying an external electric field. The difference in the electrical permittivity between the liquid and solid determines which of the liquid or solid G curve changes due to an external electric field is larger, while the sign of the derivative dominates the direction of the G curve shift for the liquid and solid. Therefore, the sign of the derivative of the liquid and solid electrical permittivity must be known to understand whether the nucleation rate increases or decreases when applying an external electric field. The dependence of the electrical permittivity on the concentration of protein molecules in the solution was determined by Pennock et al. using horse hemoglobin,39 and it was reported that the sign of the liquid derivative is negative. On the other hand, it was reported that the electrical permittivity of protein powder increases with increasing water content in the protein powder.37 Thus, it is predicted that the sign of the derivative for the solid is also negative. In the high frequency region (1 MHz), the electrical permittivity of the liquid may be larger than that of the solid (εL . εS); therefore, the change in the G curve of the solid due to the external electric field is larger than that of the liquid (ES . EL), according to eq 9. According to eq 8, the driving force for
j nucleation, ∆µ(E) , with an external electric field is larger than that without, as shown in Figure 2c. In the case of the NaCl and NiCl2 precipitants, therefore, the nucleation rate was observed to increase when applying an external electric field at 1 MHz. On the other hand, in the low frequency region (10 j L2(∂εL)/(∂XLj ) kHz: NaCl case and 500 kHz: NiCl2 case), ΩLj E j S2(∂εS)/(∂XSj ) is negative, so that the driving force for ΩLj E j , with an external electric field is smaller than nucleation, ∆µ(E) that without. Accordingly, the nucleation rate decreased when applying an external electric field at 10 kHz and 500 kHz in the case of the NaCl and NiCl2 precipitants, respectively. Finally, in the case of the NaCl precipitant, the nucleation rate with an external electric field at 500 kHz was almost the same as that without an external electric field (Figure 4b). This suggests that the electrical permittivity of the liquid and solid are almost the same under this condition; therefore, the modified chemical potential may also be almost the same between the liquid and solid. That is, control of the increase or decrease in nucleation rate can be performed by imposing an external electric field with an appropriate frequency. Finally, it should be noted that the change in nucleation rate in the case of the NiCl2 is significantly larger when compared to that in the case of the NaCl. The electrical permittivity of liquid and solid would vary with the precipitant. Thus, the difference in the electrical permittivity between the liquid and solid may be larger when employing NiCl2 as a precipitant. For the same reason, moreover, the frequency region for which the nucleation rate changes from decrease to increase would shift to higher frequency.
Conclusion The nucleation rate of HEW lysozyme was successfully controlled in terms of both increase and decrease in nucleation, by application of an AC electric field. There are two key factors for control of the nucleation rate: (i) the difference in the electrical permittivity between the liquid and solid, and (ii) the sign of electrical permittivity derivative with respect to the composition. The first factor determines which of the liquid or solid G curves exhibit a larger change due to application of the external electric field, while the second factor dominates the direction of shift of the liquid and solid G curves. Therefore, control of the electric permittivity of the liquid and solid by changing the frequency of the external electric field is a key to regulate the nucleation rate in terms of enhancement as well as retardation. Acknowledgment. This work was partly supported by the Iketani Science and Technology Foundation (0201050-A).
References (1) Taleb, M.; Didierjean, C.; Jelsch, C.; Mangeot, J. P.; Capelle, B.; Aubry, A. J. Cryst. Growth 1999, 200, 575–582.
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(2) Taleb, M.; Didierjean, C.; Jelsch, C.; Mangeot, J. P.; Aubry, A. J. Cryst. Growth 2001, 232, 250–255. (3) Nanev, C. N.; Penkova, A. J. Cryst. Growth 2001, 232, 285–293. (4) Charron, C.; Didierjean, C.; Mangeot, J. P.; Aubry, A. J. Appl. Crystallogr. 2003, 36, 1482–1483. (5) Mirkin, N.; Frontana-Uribe, B. A.; Rodrı´guez-Romero, A.; Herna´ndezSantoyo, A.; Moreno, A. Acta Crystallogr. 2003, D59, 1533–1538. (6) Moreno, A.; Sazaki, G. J. Cryst. Growth 2004, 264, 438–444. (7) Penkova, A.; Gliko, O.; Dimitrov, I. L.; Hodjaoglu, F. V.; Nanev, C.; Vekilov, P. G. J. Cryst. Growth 2005, 275, e1527-e1532. (8) Penkova, A.; Pan, W.; Hodjaoglu, F.; Vekilov, P. G. Ann. N. Y. Acad. Sci. 2006, 1077, 214–231. (9) Hammadi, Z.; Astier, J.-P.; Morin, R.; Veesler, S. Cryst. Growth Des. 2007, 7, 1472–1475. (10) Al-Haq, M. I.; Lebrasseur, E.; Choi, W.-K.; Tsuchiya, H.; Torii, T.; Yamazaki, H.; Shinohara, E. J. Appl. Crystallogr. 2007, 40, 199–201. (11) Sazaki, G.; Yoshida, E.; Komatsu, H.; Nakada, T.; Miyashita, S.; Watanabe, K. J. Cryst. Growth 1997, 173, 231–234. (12) Astier, J.; Veesler, S.; Boistelle, R. Acta Crystallogr. 1998, D54, 703– 706. (13) Tyndall, J. Philos. Mag. 1896, 37, 384–394. (14) Tam, A.; Moe, G.; Happer, W. Phys. ReV. Lett. 1975, 35, 1630–1633. (15) Garetz, B.; Aber, J.; Goddard, N.; Young, R.; Myerson, A. Phys. ReV. Lett. 1996, 77, 3475–3476. (16) Zaccaro, J.; Matic, J.; Myerson, A.; Garetz, B. Cryst. Growth Des. 2001, 1, 5–8. (17) Garetz, B.; Matic, J.; Myerson, A. Phys. ReV. Lett. 2002, 89, 175501. (18) Okutsu, T.; Nakamura, K.; Haneda, H.; Hiratsuka, H. Cryst. Growth Des. 2004, 4, 113–115. (19) Okutsu, T.; Furuta, K.; Terao, T.; Hiratsuka, H.; Yamano, A.; Ferte, N.; Veesler, S. Cryst. Growth Des. 2005, 5, 1393–1398.
Koizumi et al. (20) Veesler, S.; Furuta, K.; Horiuchi, K.; Hiratsuka, H.; Ferte´, N.; Okutsu, T. Cryst. Growth Des. 2006, 6, 1631–1635. (21) Nagatoshi, Y.; Sazaki, G.; Suzuki, Y.; Miyashita, S.; Ujihara, T.; K.; Fujiwara, K. N., N.; Usami, J. Cryst. Growth 2003, 254, 188–195. (22) Asai, T.; Suzuki, Y.; Sazaki, G.; Tamura, K.; Sawada, T.; Nakajima, K. Cell. Mol. Biol. 2004, 50, 329–334. (23) Chen, C.; Presnall, D. Am. Mineral. 1975, 60, 398–406. (24) Saban, K. V.; Thomos, J.; Varughese, P. A.; Varghese, G. Cryst. Res. Technol. 2002, 37, 1188–1199. (25) Uda, S.; Huang, X.; Koh, S. J. Cryst. Growth 2005, 281, 481–491. (26) Huang, X.; Uda, S.; Yao, X.; Koh, S. J. Cryst. Growth 2006, 294, 420–426. (27) Uda, S.; Koh, S.; Huang, X. J. Cryst. Growth 2006, 292, 1–4. (28) Huang, X.; Uda, S.; Koh, S. J. Cryst. Growth 2007, 307, 432–439. (29) Kashchiev, D. Philos. Mag. 1972, 25, 459–470. (30) Kashchiev, D. J. Cryst. Growth 1972, 13/14, 128–130. (31) Isard, J. O. Philos. Mag. 1977, 35, 817–819. (32) Fro¨ hlich, H. Theory of Dielectrics: Dielectric Constant and Dielectric Loss; Clarendon Press: Gloucestershire, U. K., 1949. (33) Grant, E.; Keefe, S.; Takashima, S. J. Phys. Chem. 1968, 72, 4373– 4380. (34) Laogun, A.; Sheppard, R.; Grant, E. Phys. Med. Biol. 1984, 29, 519– 524. (35) Pething, R. Annu. ReV. Phys. Chem. 1992, 43, 177–205. (36) Ataka, M.; Tanaka, S. Biopolymers 1980, 19, 669–679. (37) Takashima, S. J. Polym. Sci. 1962, 62, 233–240. (38) Chayen, N. E. Protein Eng. 1996, 9, 927–929. (39) Pennock, B.; Schwan, H. J. Phys. Chem. 1969, 73, 2600–2610.
CG801315P
