- Email: [email protected]
Effects of thermodynamic nonideality in protein crystal growth
Journal of Crystal Growth 209 (2000) 130}137
E!ects of thermodynamic nonideality in protein crystal growth M.L. Grant* Department of Chemical Engineering, Yale University, P.O. Box 208286, New Haven, CT 06520-8286, USA Received 14 May 1999; accepted 24 August 1999 Communicated by R.W. Rousseau
Abstract The driving force for crystallization is the di!erence in chemical potential between the protein in supersaturated solution and at its solubility. This chemical potential di!erence depends in a complicated manner on protein concentration and solution conditions (temperature, ionic strength and pH). Consequently, driving forces estimated from the nominal ideal solution supersaturation may be in error. In this work, the chemical potential of protein in solution is estimated from a virial expansion in protein concentration. The thermodynamic driving force for the crystallization of hen egg white lysozyme is calculated as a function of temperature, salt concentration, and protein concentration. In all cases examined here, the driving force for crystallization is lower than that for the corresponding ideal solution and the deviation from ideal solution behavior increases with the protein solubility. ( 2000 Elsevier Science B.V. All rights reserved. PACS: 87.15.Nn Keywords: Proteins; Thermodynamics; Activity coe$cients; Supersaturation
1. Introduction Protein crystal growth kinetics are quite complicated, and growth rate experiments are frequently correlated with expressions of the form G&pn,
(1)
where G is the linear growth rate, n is an empirical parameter and p"ln(C/C ) is a measure of the 40supersaturation. Log}log plots of G vs. p are often used to delineate regimes where di!erent growth
* Tel.: #1-203-432-4376; fax: #1-203-432-7232. E-mail address: [email protected] (M.L. Grant)
mechanisms obtain. In some cases, the growth mechanism is inferred from the value of the exponent, n. Crystallization is driven by the di!erence in chemical potential of the solute in solution and in the crystal. Since the solute in the crystal must be in equilibrium with that in solution at the solubility concentration, we have for dilute protein solutions *k"k3#k¹ ln cC!(k3#k¹ ln c C ) 40- 40"k¹ ln(cC/c C ), (2) 40- 40where k3 is the standard-state chemical potential, C is the solute concentration, and c is the thermodynamic activity coe$cient of the solute. In ideal solutions, the activity coe$cients are identically
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equal to unity and the driving force can be expressed simply in terms of the nominal supersaturation, p:
coe$cients. For instance, in an osmotic equilibrium, the osmotic pressure for an ideal solution, P*$, is proportional to the solute concentration:
*k*$"k¹ ln(C/C )"k¹p. (3) 40Thus, crystal growth rates are actually correlated with the thermodynamic driving force estimated from ideal solution theory. Ideal solution theory is not necessarily appropriate for protein crystallization, however. Under typical conditions, proteins are highly charged molecules with long-range interactions that violate the assumptions of ideal solution theory. Moreover, the nonideality of protein solutions is con"rmed by the nonzero values of virial coe$cients obtained from light scattering measurements and osmometry [1}4]. In this paper, thermodynamic activity coe$cients of protein in solution are estimated from published measurements of virial coef"cients and then used to calculate a more accurate chemical potential driving force for crystallization. When the activity coe$cients of Eq. (2) are used to estimate *k, it becomes apparent that the nominal supersaturation, p, does not adequately describe the thermodynamic state of the solution. The presentation is organized as follows: First, we establish the relationship between the thermodynamic activity coe$cient, c, and the virial coe$cients, B . Then the thermodynamic driving force is n calculated as a function of p for three di!erent means of changing the supersaturation: (1) increasing protein concentration, (2) increasing salt concentration, and (3) reducing the temperature. Finally, the implications of thermodynamic nonideality for the interpretation of experimental growth rate data are discussed.
P*$ M "C k¹ N A
2. Virial coe7cients and activity coe7cients The starting point for the analysis of thermodynamic nonideality is the interaction potential between species in solution. In an ideal solution, where each solute molecule is independent of all others, the interaction potential between solutes is identically zero. However, the interactions of real solutions produce deviations from ideal solution behavior which can be quanti"ed in terms of virial
(4)
while for a real solution, we obtain P M "C#B C2#B C3#2, 2 3 k¹ N A
(5)
where M is the molecular weight of the solute, N is A Avogadro's number, and B is the nth virial coe$cn ient. Statistical mechanical expressions for the virial coe$cients are of the form 1N A B "! 2 2 M
P
f
V
12
dr 12
and
A B PP
1 N 2 A B "! 3 3 M
f
f f dr dr , 12 13 23 12 13
(6)
V
where f "[exp(!w /k¹)!1] and w (r) is the ij ij ij potential of mean force between molecules i and j [5}7]; B involves only interactions among n parn ticles. In dilute solution, binary interactions are much more probable than ternary interactions, so it usually su$ces to examine only interactions involving pairs of particles. In this case, we can truncate Eq. (5) after B . 2 The connection between the interaction potential and the chemical potential of the protein is made through a virial expansion in the protein concentration. In a Donnan equilibrium where protein is con"ned on one side of the membrane but the added electrolyte is in equilibrium on both sides of the membrane, the activity coe$cient of the protein can be expressed in terms of the virial coe$cients de"ned in Eq. (5) [8]: ln c"2B C#(3/2)B C2#2+2B C, 2 3 2
(7)
where the standard state for the protein is taken so that cP1 as CP0. Combining Eq. (7) with Eq. (2)
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M.L. Grant / Journal of Crystal Growth 209 (2000) 130}137
gives a simple estimate of the thermodynamic driving force for crystallization: *k"k¹ ln(cC/c C ) 40- 40+k¹[ln (C/C )#2B (C!C )] 402 40"k¹[p#2B (C!C )]. 2 40-
(8)
It is apparent that *k depends not only on the nominal supersaturation, p, but also the magnitudes of C and C . In particular, the solubility is a sensi40tive function of ionic strength and temperature, so *k can vary considerably with solution conditions even at constant p. In this work, we will calculate *k for hen egg white lysozyme based on solubilities reported by Cacioppo and Pusey [9] and virial coe$cients for lysozyme measured by George et al. [4].
3. E4ect of solution conditions on *k The three most common means to set the supersaturation are to (1) control protein concentration in solution, (2) alter the salt concentration in solution, and (3) adjust the temperature in solution. The most straightforward of the three approaches is to increase the protein concentration, which does not a!ect the solubility of the protein or the interactions between pairs of molecules in dilute protein solution. In contrast, increasing the salt concentration alters *k by simultaneously reducing the solubility of the protein and reducing the repulsive contributions to the protein}protein interactions in solution. Finally, reducing the temperature decreases protein solubility and alters the protein}protein interactions. As shown below, each procedure for adjusting supersaturation has a rather di!erent e!ect on the chemical potential driving crystallization. 3.1. Protein concentration The driving force for crystallization is expected to increase with protein concentration. For a dilute ideal solution, the chemical potential of the protein increases logarithmically with protein concentra-
Fig. 1. Thermodynamic driving force for crystallization as a function of nominal supersaturation for lysozyme at pH 4.2 and 2.5% (w/v) NaCl. Each curve shows how *k varies with p as protein is added isothermally to a solution that is initially saturated.
tion. The e!ect of protein concentration on *k at pH 4.2 and 2.5% (w/v) NaCl is shown in Fig. 1, where *k calculated from Eq. (8) is compared with the ideal solution driving force calculated from Eq. (2); the curves begin at the solubility concentration for each temperature. For 2.5% (w/v) NaCl, the solubility is interpolated from the results of Cacioppo and Pusey [9]. Molecular interactions are responsible for deviations from the 453 line expected for ideal solutions. Under these conditions, the deviations are always negative because there is a net intermolecular attraction. Consistent with Eq. (8), the deviation is larger at higher protein concentrations. The most important feature of the calculations is that *k calculated from Eq. (8) increases much more slowly with concentration than p and that the incremental change in *k with ln(C/C ) decreases 40as the nominal supersaturation increases. This re#ects the balance between the logarithmic increase in *k due to the ideal solution contribution and the linear decrease in *k due to the attraction between molecules in solution. The behavior at such relatively high concentrations is of interest because protein crystals are routinely grown from solutions where the concentration is about 10 times the solubility [10]. In such systems, log}log plots of G vs. p show a characteristic curvature with a smaller
M.L. Grant / Journal of Crystal Growth 209 (2000) 130}137
slope at high nominal supersaturation. This behavior is consistent with the shape of the *k curves at low to moderate p. The maxima shown in Fig. 1 indicate an increased stability for protein molecules in solution beyond a critical concentration. This result has serious implications for the analysis of crystal growth rates. However, the existence and location of such maxima are rather tentative because they result from truncating the expression for c after the B term. Additional measurements of virial coe$2 cients at higher protein concentration are required to extend the calculations. Vilker et al. [2] have examined the osmotic pressure of solutions of bovine serum albumin (BSA, molecular weight 69,000) over a concentration range large enough to estimate the next virial coe$cient, B . Under noncrys3 tallizing conditions, they found that B could be 3 estimated by the sum of the three-body excluded volume term and a three-body electrostatic contribution [2,7]; in 0.15 M NaCl solution with 4.5)pH)7.4, they found 7.3 ml/g)B )9.2 ml/g 2 and B +30 (ml/g)2. Because many of the attract3 ive interactions are not well understood in protein systems, similar theoretical estimates are not available for proteins under crystallizing conditions. However, suitable osmotic pressure measurements should yield experimental estimates of B . 3 3.2. Added salt The supersaturation may also be increased by adding salt to the system. There are two mechanisms by which salt alters the chemical potential of the protein. First, the presence of counterions tends to shield the electrostatic repulsion between similar ions. The electrostatic repulsion decays approximately exponentially with distance, with a decay length (Debye length) given by i~1"(e" ek¹/ 2000 N e2)1@2/JI, where e is the elementary A charge, e" is the permittivity of free space, e is the dielectric constant of the solvent, and I" (1/2)+c z2 is the ionic strength; c is the concentrai i i tion (mol/l) of species i and z is its valence. As the i ionic strength increases, the number of counterions near the protein increases, reducing the electrostatic potential at the protein and reducing the
133
strength and range of the repulsion. The diminution of the electrostatic component is re#ected in increasingly negative values of B at higher salt 2 concentrations. Since the electrostatic interaction decays on a length scale proportional to I~1@2, the e!ect of salt is rather weak at the high salt concentrations typical in protein crystal growth studies. The second mechanism by which salt a!ects the *l driving force is through the solubility of the protein: solubility decreases with increasing salt concentration. In the limit that the added salt drives the solubility to near zero and reduces the electrostatic repulsion, *k/k¹ asymptotes to *k lim "ln(C/C )#2B C, (9) 402,.*/ k¹ C40- ?0 where B represents the value of B when elec2,.*/ 2 trostatic repulsion is negligible. Since the protein concentration is assumed constant here, *k is asymptotically parallel to r but with an o!set related to conditions in solution. In Fig. 2, we show the e!ect of added salt on the di!erence in chemical potential. Again, *k/k¹(p because the attractions between protein molecules lower the protein's chemical potential. With an initial salt concentration of 2% (w/v) (protein concentration of 56 mg/ml), *k/k¹ shows a complicated salt dependence re#ecting the competition between the reduction in solubility and the increased attraction between protein molecules. The derivative of the curve is
A
B
L(*k/k¹) 1 LC 40"! #2B 2 LC LC C 4!-5 4!-5 40LB 2 . #2(C!C ) (10) 40- LC 4!-5 Since LC /LC and LB /LC are both negative, 404!-5 2 4!-5 the two terms in Eq. (10) have opposite sign. At the start of the curve, C"C and the slope is positive. 40As the salt concentration increases, the second term dominates due to the (C!C ) term and the deriv40ative changes sign. At still higher salt concentration, 1/C dominates the "rst term and the slope is 40again positive. The general features of the behavior described above apply to proteins that can be salted out of solution. In cases where the e!ect of salt and salt mixtures is qualitatively di!erent from lysozyme,
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M.L. Grant / Journal of Crystal Growth 209 (2000) 130}137
Fig. 2. Thermodynamic driving force as a function of salt concentration for lysozyme at pH 4.2 and 253C. (a) The curves indicate how *k changes as salt is added to a solution that is initially saturated. Each curve is identi"ed by the protein concentration corresponding to solubility at the initial conditions. (b) Same curves as in (a) replotted as functions of p.
Fig. 3. Thermodynamic driving force as a function of temperature for lysozyme at pH 4.2 and 2.5% (w/v) NaCl. (a) The curves indicate how *k changes as the temperature of an initially saturated solution is reduced. Each curve is identi"ed by the protein concentration corresponding to solubility at the initial conditions. (b) Same curves as in (a) replotted as functions of p.
*k will also have a di!erent behavior. While such ion-speci"c interactions are not included in general theories such as the Derjaguin}Landau}Verwey} Overbeek (DLVO) theory of colloidal interactions, they are included in the experimentally determined virial coe$cients. Osmotic pressure measurements, then, provide a means to quantify not only general salt e!ects but also speci"c ion}protein interactions that a!ect protein crystallization.
Fig. 3, *k is calculated for the case of a 2.5% (w/v) NaCl solution that is saturated at its initial temperature. As the temperature decreases, *k/k¹ increases more slowly than p, with the largest deviations from ideal solution behavior at high protein concentration. As in the case of added salt, the deviation from ideal solution behavior is bounded because *k/k¹ asymptotes to
3.3. Temperature
lim *k/k¹"ln(C/C )#2B C, 402 C40- ?0
Supersaturation in crystal growth experiments is often controlled by adjusting the temperature. In
which has the same form as Eq. (10). If B ap2 proaches a constant value as temperature
(11)
M.L. Grant / Journal of Crystal Growth 209 (2000) 130}137
135
decreases, the curve predicted from Eq. (11) will parallel the ideal solution curve with an o!set determined by the protein concentration in solution.
4. Discussion This analysis indicates that the chemical potential di!erences produced by controlling protein concentration, adjusting ionic strength, and reducing the temperature are di!erent, even when the nominal supersaturation, p, is the same. By altering the chemical potential from the ideal solution value, thermodynamic nonideality has some implications for protein crystal growth that are discussed below. 4.1. Heats of crystallization Thermodynamic nonideality must be included in any estimate of thermodynamic properties. For instance, the determination of *H of crystallization from a van't Ho! analysis makes use of a free energy di!erence of the form *G "k !k3" 95!95!k¹ ln(c C ). The temperature dependence gives 40- 40*H according to 95!L k !k3 *G L *H 95!95!- " 95!" L(1/¹) k¹ k¹ L(1/¹) k
A
B
A
B
L " [ln(c C )], 40- 40L(1/¹)
(12)
from which it follows that L *H "k [ln C #2B C ]. 95!402 40L(1/¹)
(13)
Since B (0, the assumption of ideal solution be2 havior leads to an overestimate of *H , as in95!dicated in Fig. 4. Over the temperature range 5}253C, a van't Ho! analysis based on solubilities at pH 4.2 and 2.5% (w/v) NaCl overestimates *H by about 20% on average compared with 95!*H estimated from Eq. (13): !87.3 kJ/mol vs. 95!!71.4 kJ/mol. Note that the primary e!ect of the activity coe$cient is to reduce the slope of the curve rather than to change its shape. Any residual curvature re#ects contributions from higher-order virial coe$cients and the inherent temperature de-
Fig. 4. A van't Ho! analysis of *H of crystallization. Open symbols represent the ideal solution approximation to *G /k¹+ln(C ) while the "lled symbols represent a "rst95!40order approximation to *G /k¹+ln(c C ). The slope of 95!40- 40the line is proportional to *H of crystallization. The ideal solution approximation gives *H "!87 kJ/mol compared 95!with !71 kJ/mol when nonidealities are included.
pendence of *H , which is not a consequence of 95!nonideality. An analysis of the van't Ho! plot accounting for the temperature dependence of *H 95!yields essentially the same result. 4.2. Membrane yux of proteins Protein}protein interactions also a!ect mass transport rates and the state of aggregation in solution. Pusey and coworkers estimated the dimerization equilibrium constant from classical light scattering and membrane #ux experiments [11}13]. In the analyses, ideal solutions were assumed and deviations from dilute monomer were interpreted in terms of aggregation. The concentration dependence of light scattering intensity, however, can also be interpreted in terms of solute}solute interactions that are quanti"ed by virial coe$cients [14]. As shown below, thermodynamic nonideality a!ects mass transport in a similar manner. Di!usion is driven by gradients in the chemical potential as [15] D D j"! C+k"! C+ [k3#k¹ ln(Cc)], (14) k¹ k¹
136
M.L. Grant / Journal of Crystal Growth 209 (2000) 130}137
where D is the di!usion coe$cient. For one-dimensional di!usion through a membrane of thickness d, we have dC j"!D[1#2B C] 2 dx
(15)
which is readily integrated from (x"0, C"C ), to 0 (x"d, C"0) to give D D j" (C #B C2 )" C (1#B C ). 2 0 2 0 d 0 d 0
(16)
Thus, the #ux is linearly proportional to C at low 0 concentrations but varies quadratically at higher concentrations. The term D/d is an empirical constant for each protein}membrane system, but is independent of concentration. The data of Wilson and Pusey [12] were re-analyzed to account for thermodynamic nonideality. Wilson and Pusey actually measured the protein concentration in solution outside the membrane as a function of time and then di!erentiated the data to obtain the instantaneous rate of change of the concentration, dC/dt, at the start of the experiment. In terms of the membrane #ux calculated in Eq. (16), dC A AD "j " C (1#B C )"KC (1#B C ), 2 0 0 2 0 dt < < d 0 (17) where A is the area of the membrane through which protein can di!use, < the volume of dialysis solution outside the membrane, K"AD/
Fig. 5. Analysis of membrane #ux experiments accounting for thermodynamic nonideality. The plot symbols are data from Wilson and Pusey [12] at pH 4 while the curves are "ts that account for nonideal solution behavior. Squares correspond to 1% (w/v) NaCl and circles to 3% (w/v) NaCl. Solid curves: calculated from Eq. (17) with B estimated from George et al. 2 [4] and K estimated from linear regression; dashed curves: B and K estimated simultaneously from linear regression of 2 #ux data.
results at 3% (w/v) NaCl (circles). As a sensitivity check, K and B were estimated simultaneously 2 from linear regression of Wilson's results for 1% (w/v) NaCl; the values are K"6.8]10~7 s~1 and B "2.8 ml/g (dashed curve through the squares in 2 Fig. 5). The two values of K agree within 20%, although the discrepancy is larger for B , probably 2 because of the large uncertainty associated with extrapolating B outside the range of measure2 ment. This higher value of K was then used in a linear regression to estimate B "!14.4 ml/g at 2 3% NaCl (dashed curve through the circles), where aggregation is reported. The di!erence in the two values of B is less than 25% and the two curves are 2 not that di!erent, so it seems that binary interactions are able to account for lysozyme behavior well above its solubility. In general, the predicted #uxes compare favorably with the experimental data, suggesting that thermodynamic interactions may account for behavior that is often attributed to aggregated protein solutions. In large part, this is because the interaction potential governs the behavior of the system. As a measure of the average interaction between molecules in solution, virial coe$cients
M.L. Grant / Journal of Crystal Growth 209 (2000) 130}137
provide the means to characterize macroscopic properties of the system, such as osmotic pressure and activity coe$cients. However, it is impossible to infer the shape of the interaction potential from these measurements; a net attraction (B (0) may 2 be due to either a shallow long-range interaction or a strong short-range interaction that leads to physical aggregation. The latter is presumably the case since proteins form relatively high-density crystals. Nevertheless, it is apparent that protein}protein interactions are responsible for signi"cant deviations from thermodynamically ideal behavior, even if it is not possible to determine whether the deviations are due to aggregation of protein or interactions between monomers. 5. Summary The calculations above demonstrate that the nominal supersaturation, p"ln(C/C ), is inad40equate to describe the di!erence in chemical potential that drives protein crystallization. A more detailed thermodynamic description of the system that includes e!ects of temperature and added salt on the protein shows that the attractive interactions between molecules reduce the thermodynamic driving force *k. The growth rate and *k are not linearly proportional, however, so the growth mechanism cannot be inferred from a plot of growth rate vs. *k. Nevertheless, thermodynamic nonideality accounts for the characteristic decrease in the growth exponent as protein supersaturation
137
increases at constant temperature and ionic strength. The extension of the work to higher supersaturations requires additional measurements of virial coe$cients covering a wider range of conditions. Finally, this work reemphasizes the central role of thermodynamics in protein crystal growth.
References [1] C.A. Haynes, K. Tamura, H.R. KoK rfer, H.W. Blanch, J.M. Prausnitz, J. Phys. Chem. 96 (1992) 905. [2] V.L. Vilker, C.K. Colton, K.A. Smith, J. Colloid Interface Sci. 79 (1981) 548. [3] A. George, W.W. Wilson, Acta Crystallogr. D 50 (1994) 3651. [4] A. George, Y. Chiang, B. Guo, A. Arabshahi, Z. Cai, W.W. Wilson, Methods Enzymology 276 (1997) 100. [5] T.L. Hill, An Introduction to Statistical Thermodynamics, Dover, Mineola, NY, 1986. [6] D.A. McQuarrie, Statistical Mechanics, Harper and Row, New York, 1976. [7] J.W. Tester, M. Modell, Thermodynamics and its Applications, 3rd edition, Prentice-Hall, Upper Saddle River, NJ, 1997. [8] D. Stigter, T.L. Hill, J. Phys. Chem. 63 (1959) 551. [9] E. Cacioppo, M.L. Pusey, J. Cryst. Growth 114 (1991) 286. [10] E. Forsythe, M.L. Pusey, J. Cryst. Growth 139 (1994) 89. [11] M.L. Pusey, J. Cryst. Growth 110 (1991) 60. [12] L.J. Wilson, M.L. Pusey, J. Cryst. Growth 122 (1992) 8. [13] L.J. Wilson, L. Adcock-Downey, M.L. Pusey, Biophys. J. 71 (1996) 2123. [14] P.C. Hiemenz, R. Rajagopalan, Principles of Colloid and Surface Chemistry, 3rd edition, Marcel Dekker, New York, 1996. [15] E.L. Cussler, Di!usion, Cambridge University Press, London, 1984.
E!ects of thermodynamic nonideality in protein crystal growth M.L. Grant* Department of Chemical Engineering, Yale University, P.O. Box 208286, New Haven, CT 06520-8286, USA Received 14 May 1999; accepted 24 August 1999 Communicated by R.W. Rousseau
Abstract The driving force for crystallization is the di!erence in chemical potential between the protein in supersaturated solution and at its solubility. This chemical potential di!erence depends in a complicated manner on protein concentration and solution conditions (temperature, ionic strength and pH). Consequently, driving forces estimated from the nominal ideal solution supersaturation may be in error. In this work, the chemical potential of protein in solution is estimated from a virial expansion in protein concentration. The thermodynamic driving force for the crystallization of hen egg white lysozyme is calculated as a function of temperature, salt concentration, and protein concentration. In all cases examined here, the driving force for crystallization is lower than that for the corresponding ideal solution and the deviation from ideal solution behavior increases with the protein solubility. ( 2000 Elsevier Science B.V. All rights reserved. PACS: 87.15.Nn Keywords: Proteins; Thermodynamics; Activity coe$cients; Supersaturation
1. Introduction Protein crystal growth kinetics are quite complicated, and growth rate experiments are frequently correlated with expressions of the form G&pn,
(1)
where G is the linear growth rate, n is an empirical parameter and p"ln(C/C ) is a measure of the 40supersaturation. Log}log plots of G vs. p are often used to delineate regimes where di!erent growth
* Tel.: #1-203-432-4376; fax: #1-203-432-7232. E-mail address: [email protected] (M.L. Grant)
mechanisms obtain. In some cases, the growth mechanism is inferred from the value of the exponent, n. Crystallization is driven by the di!erence in chemical potential of the solute in solution and in the crystal. Since the solute in the crystal must be in equilibrium with that in solution at the solubility concentration, we have for dilute protein solutions *k"k3#k¹ ln cC!(k3#k¹ ln c C ) 40- 40"k¹ ln(cC/c C ), (2) 40- 40where k3 is the standard-state chemical potential, C is the solute concentration, and c is the thermodynamic activity coe$cient of the solute. In ideal solutions, the activity coe$cients are identically
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equal to unity and the driving force can be expressed simply in terms of the nominal supersaturation, p:
coe$cients. For instance, in an osmotic equilibrium, the osmotic pressure for an ideal solution, P*$, is proportional to the solute concentration:
*k*$"k¹ ln(C/C )"k¹p. (3) 40Thus, crystal growth rates are actually correlated with the thermodynamic driving force estimated from ideal solution theory. Ideal solution theory is not necessarily appropriate for protein crystallization, however. Under typical conditions, proteins are highly charged molecules with long-range interactions that violate the assumptions of ideal solution theory. Moreover, the nonideality of protein solutions is con"rmed by the nonzero values of virial coe$cients obtained from light scattering measurements and osmometry [1}4]. In this paper, thermodynamic activity coe$cients of protein in solution are estimated from published measurements of virial coef"cients and then used to calculate a more accurate chemical potential driving force for crystallization. When the activity coe$cients of Eq. (2) are used to estimate *k, it becomes apparent that the nominal supersaturation, p, does not adequately describe the thermodynamic state of the solution. The presentation is organized as follows: First, we establish the relationship between the thermodynamic activity coe$cient, c, and the virial coe$cients, B . Then the thermodynamic driving force is n calculated as a function of p for three di!erent means of changing the supersaturation: (1) increasing protein concentration, (2) increasing salt concentration, and (3) reducing the temperature. Finally, the implications of thermodynamic nonideality for the interpretation of experimental growth rate data are discussed.
P*$ M "C k¹ N A
2. Virial coe7cients and activity coe7cients The starting point for the analysis of thermodynamic nonideality is the interaction potential between species in solution. In an ideal solution, where each solute molecule is independent of all others, the interaction potential between solutes is identically zero. However, the interactions of real solutions produce deviations from ideal solution behavior which can be quanti"ed in terms of virial
(4)
while for a real solution, we obtain P M "C#B C2#B C3#2, 2 3 k¹ N A
(5)
where M is the molecular weight of the solute, N is A Avogadro's number, and B is the nth virial coe$cn ient. Statistical mechanical expressions for the virial coe$cients are of the form 1N A B "! 2 2 M
P
f
V
12
dr 12
and
A B PP
1 N 2 A B "! 3 3 M
f
f f dr dr , 12 13 23 12 13
(6)
V
where f "[exp(!w /k¹)!1] and w (r) is the ij ij ij potential of mean force between molecules i and j [5}7]; B involves only interactions among n parn ticles. In dilute solution, binary interactions are much more probable than ternary interactions, so it usually su$ces to examine only interactions involving pairs of particles. In this case, we can truncate Eq. (5) after B . 2 The connection between the interaction potential and the chemical potential of the protein is made through a virial expansion in the protein concentration. In a Donnan equilibrium where protein is con"ned on one side of the membrane but the added electrolyte is in equilibrium on both sides of the membrane, the activity coe$cient of the protein can be expressed in terms of the virial coe$cients de"ned in Eq. (5) [8]: ln c"2B C#(3/2)B C2#2+2B C, 2 3 2
(7)
where the standard state for the protein is taken so that cP1 as CP0. Combining Eq. (7) with Eq. (2)
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M.L. Grant / Journal of Crystal Growth 209 (2000) 130}137
gives a simple estimate of the thermodynamic driving force for crystallization: *k"k¹ ln(cC/c C ) 40- 40+k¹[ln (C/C )#2B (C!C )] 402 40"k¹[p#2B (C!C )]. 2 40-
(8)
It is apparent that *k depends not only on the nominal supersaturation, p, but also the magnitudes of C and C . In particular, the solubility is a sensi40tive function of ionic strength and temperature, so *k can vary considerably with solution conditions even at constant p. In this work, we will calculate *k for hen egg white lysozyme based on solubilities reported by Cacioppo and Pusey [9] and virial coe$cients for lysozyme measured by George et al. [4].
3. E4ect of solution conditions on *k The three most common means to set the supersaturation are to (1) control protein concentration in solution, (2) alter the salt concentration in solution, and (3) adjust the temperature in solution. The most straightforward of the three approaches is to increase the protein concentration, which does not a!ect the solubility of the protein or the interactions between pairs of molecules in dilute protein solution. In contrast, increasing the salt concentration alters *k by simultaneously reducing the solubility of the protein and reducing the repulsive contributions to the protein}protein interactions in solution. Finally, reducing the temperature decreases protein solubility and alters the protein}protein interactions. As shown below, each procedure for adjusting supersaturation has a rather di!erent e!ect on the chemical potential driving crystallization. 3.1. Protein concentration The driving force for crystallization is expected to increase with protein concentration. For a dilute ideal solution, the chemical potential of the protein increases logarithmically with protein concentra-
Fig. 1. Thermodynamic driving force for crystallization as a function of nominal supersaturation for lysozyme at pH 4.2 and 2.5% (w/v) NaCl. Each curve shows how *k varies with p as protein is added isothermally to a solution that is initially saturated.
tion. The e!ect of protein concentration on *k at pH 4.2 and 2.5% (w/v) NaCl is shown in Fig. 1, where *k calculated from Eq. (8) is compared with the ideal solution driving force calculated from Eq. (2); the curves begin at the solubility concentration for each temperature. For 2.5% (w/v) NaCl, the solubility is interpolated from the results of Cacioppo and Pusey [9]. Molecular interactions are responsible for deviations from the 453 line expected for ideal solutions. Under these conditions, the deviations are always negative because there is a net intermolecular attraction. Consistent with Eq. (8), the deviation is larger at higher protein concentrations. The most important feature of the calculations is that *k calculated from Eq. (8) increases much more slowly with concentration than p and that the incremental change in *k with ln(C/C ) decreases 40as the nominal supersaturation increases. This re#ects the balance between the logarithmic increase in *k due to the ideal solution contribution and the linear decrease in *k due to the attraction between molecules in solution. The behavior at such relatively high concentrations is of interest because protein crystals are routinely grown from solutions where the concentration is about 10 times the solubility [10]. In such systems, log}log plots of G vs. p show a characteristic curvature with a smaller
M.L. Grant / Journal of Crystal Growth 209 (2000) 130}137
slope at high nominal supersaturation. This behavior is consistent with the shape of the *k curves at low to moderate p. The maxima shown in Fig. 1 indicate an increased stability for protein molecules in solution beyond a critical concentration. This result has serious implications for the analysis of crystal growth rates. However, the existence and location of such maxima are rather tentative because they result from truncating the expression for c after the B term. Additional measurements of virial coe$2 cients at higher protein concentration are required to extend the calculations. Vilker et al. [2] have examined the osmotic pressure of solutions of bovine serum albumin (BSA, molecular weight 69,000) over a concentration range large enough to estimate the next virial coe$cient, B . Under noncrys3 tallizing conditions, they found that B could be 3 estimated by the sum of the three-body excluded volume term and a three-body electrostatic contribution [2,7]; in 0.15 M NaCl solution with 4.5)pH)7.4, they found 7.3 ml/g)B )9.2 ml/g 2 and B +30 (ml/g)2. Because many of the attract3 ive interactions are not well understood in protein systems, similar theoretical estimates are not available for proteins under crystallizing conditions. However, suitable osmotic pressure measurements should yield experimental estimates of B . 3 3.2. Added salt The supersaturation may also be increased by adding salt to the system. There are two mechanisms by which salt alters the chemical potential of the protein. First, the presence of counterions tends to shield the electrostatic repulsion between similar ions. The electrostatic repulsion decays approximately exponentially with distance, with a decay length (Debye length) given by i~1"(e" ek¹/ 2000 N e2)1@2/JI, where e is the elementary A charge, e" is the permittivity of free space, e is the dielectric constant of the solvent, and I" (1/2)+c z2 is the ionic strength; c is the concentrai i i tion (mol/l) of species i and z is its valence. As the i ionic strength increases, the number of counterions near the protein increases, reducing the electrostatic potential at the protein and reducing the
133
strength and range of the repulsion. The diminution of the electrostatic component is re#ected in increasingly negative values of B at higher salt 2 concentrations. Since the electrostatic interaction decays on a length scale proportional to I~1@2, the e!ect of salt is rather weak at the high salt concentrations typical in protein crystal growth studies. The second mechanism by which salt a!ects the *l driving force is through the solubility of the protein: solubility decreases with increasing salt concentration. In the limit that the added salt drives the solubility to near zero and reduces the electrostatic repulsion, *k/k¹ asymptotes to *k lim "ln(C/C )#2B C, (9) 402,.*/ k¹ C40- ?0 where B represents the value of B when elec2,.*/ 2 trostatic repulsion is negligible. Since the protein concentration is assumed constant here, *k is asymptotically parallel to r but with an o!set related to conditions in solution. In Fig. 2, we show the e!ect of added salt on the di!erence in chemical potential. Again, *k/k¹(p because the attractions between protein molecules lower the protein's chemical potential. With an initial salt concentration of 2% (w/v) (protein concentration of 56 mg/ml), *k/k¹ shows a complicated salt dependence re#ecting the competition between the reduction in solubility and the increased attraction between protein molecules. The derivative of the curve is
A
B
L(*k/k¹) 1 LC 40"! #2B 2 LC LC C 4!-5 4!-5 40LB 2 . #2(C!C ) (10) 40- LC 4!-5 Since LC /LC and LB /LC are both negative, 404!-5 2 4!-5 the two terms in Eq. (10) have opposite sign. At the start of the curve, C"C and the slope is positive. 40As the salt concentration increases, the second term dominates due to the (C!C ) term and the deriv40ative changes sign. At still higher salt concentration, 1/C dominates the "rst term and the slope is 40again positive. The general features of the behavior described above apply to proteins that can be salted out of solution. In cases where the e!ect of salt and salt mixtures is qualitatively di!erent from lysozyme,
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M.L. Grant / Journal of Crystal Growth 209 (2000) 130}137
Fig. 2. Thermodynamic driving force as a function of salt concentration for lysozyme at pH 4.2 and 253C. (a) The curves indicate how *k changes as salt is added to a solution that is initially saturated. Each curve is identi"ed by the protein concentration corresponding to solubility at the initial conditions. (b) Same curves as in (a) replotted as functions of p.
Fig. 3. Thermodynamic driving force as a function of temperature for lysozyme at pH 4.2 and 2.5% (w/v) NaCl. (a) The curves indicate how *k changes as the temperature of an initially saturated solution is reduced. Each curve is identi"ed by the protein concentration corresponding to solubility at the initial conditions. (b) Same curves as in (a) replotted as functions of p.
*k will also have a di!erent behavior. While such ion-speci"c interactions are not included in general theories such as the Derjaguin}Landau}Verwey} Overbeek (DLVO) theory of colloidal interactions, they are included in the experimentally determined virial coe$cients. Osmotic pressure measurements, then, provide a means to quantify not only general salt e!ects but also speci"c ion}protein interactions that a!ect protein crystallization.
Fig. 3, *k is calculated for the case of a 2.5% (w/v) NaCl solution that is saturated at its initial temperature. As the temperature decreases, *k/k¹ increases more slowly than p, with the largest deviations from ideal solution behavior at high protein concentration. As in the case of added salt, the deviation from ideal solution behavior is bounded because *k/k¹ asymptotes to
3.3. Temperature
lim *k/k¹"ln(C/C )#2B C, 402 C40- ?0
Supersaturation in crystal growth experiments is often controlled by adjusting the temperature. In
which has the same form as Eq. (10). If B ap2 proaches a constant value as temperature
(11)
M.L. Grant / Journal of Crystal Growth 209 (2000) 130}137
135
decreases, the curve predicted from Eq. (11) will parallel the ideal solution curve with an o!set determined by the protein concentration in solution.
4. Discussion This analysis indicates that the chemical potential di!erences produced by controlling protein concentration, adjusting ionic strength, and reducing the temperature are di!erent, even when the nominal supersaturation, p, is the same. By altering the chemical potential from the ideal solution value, thermodynamic nonideality has some implications for protein crystal growth that are discussed below. 4.1. Heats of crystallization Thermodynamic nonideality must be included in any estimate of thermodynamic properties. For instance, the determination of *H of crystallization from a van't Ho! analysis makes use of a free energy di!erence of the form *G "k !k3" 95!95!k¹ ln(c C ). The temperature dependence gives 40- 40*H according to 95!L k !k3 *G L *H 95!95!- " 95!" L(1/¹) k¹ k¹ L(1/¹) k
A
B
A
B
L " [ln(c C )], 40- 40L(1/¹)
(12)
from which it follows that L *H "k [ln C #2B C ]. 95!402 40L(1/¹)
(13)
Since B (0, the assumption of ideal solution be2 havior leads to an overestimate of *H , as in95!dicated in Fig. 4. Over the temperature range 5}253C, a van't Ho! analysis based on solubilities at pH 4.2 and 2.5% (w/v) NaCl overestimates *H by about 20% on average compared with 95!*H estimated from Eq. (13): !87.3 kJ/mol vs. 95!!71.4 kJ/mol. Note that the primary e!ect of the activity coe$cient is to reduce the slope of the curve rather than to change its shape. Any residual curvature re#ects contributions from higher-order virial coe$cients and the inherent temperature de-
Fig. 4. A van't Ho! analysis of *H of crystallization. Open symbols represent the ideal solution approximation to *G /k¹+ln(C ) while the "lled symbols represent a "rst95!40order approximation to *G /k¹+ln(c C ). The slope of 95!40- 40the line is proportional to *H of crystallization. The ideal solution approximation gives *H "!87 kJ/mol compared 95!with !71 kJ/mol when nonidealities are included.
pendence of *H , which is not a consequence of 95!nonideality. An analysis of the van't Ho! plot accounting for the temperature dependence of *H 95!yields essentially the same result. 4.2. Membrane yux of proteins Protein}protein interactions also a!ect mass transport rates and the state of aggregation in solution. Pusey and coworkers estimated the dimerization equilibrium constant from classical light scattering and membrane #ux experiments [11}13]. In the analyses, ideal solutions were assumed and deviations from dilute monomer were interpreted in terms of aggregation. The concentration dependence of light scattering intensity, however, can also be interpreted in terms of solute}solute interactions that are quanti"ed by virial coe$cients [14]. As shown below, thermodynamic nonideality a!ects mass transport in a similar manner. Di!usion is driven by gradients in the chemical potential as [15] D D j"! C+k"! C+ [k3#k¹ ln(Cc)], (14) k¹ k¹
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M.L. Grant / Journal of Crystal Growth 209 (2000) 130}137
where D is the di!usion coe$cient. For one-dimensional di!usion through a membrane of thickness d, we have dC j"!D[1#2B C] 2 dx
(15)
which is readily integrated from (x"0, C"C ), to 0 (x"d, C"0) to give D D j" (C #B C2 )" C (1#B C ). 2 0 2 0 d 0 d 0
(16)
Thus, the #ux is linearly proportional to C at low 0 concentrations but varies quadratically at higher concentrations. The term D/d is an empirical constant for each protein}membrane system, but is independent of concentration. The data of Wilson and Pusey [12] were re-analyzed to account for thermodynamic nonideality. Wilson and Pusey actually measured the protein concentration in solution outside the membrane as a function of time and then di!erentiated the data to obtain the instantaneous rate of change of the concentration, dC/dt, at the start of the experiment. In terms of the membrane #ux calculated in Eq. (16), dC A AD "j " C (1#B C )"KC (1#B C ), 2 0 0 2 0 dt < < d 0 (17) where A is the area of the membrane through which protein can di!use, < the volume of dialysis solution outside the membrane, K"AD/
Fig. 5. Analysis of membrane #ux experiments accounting for thermodynamic nonideality. The plot symbols are data from Wilson and Pusey [12] at pH 4 while the curves are "ts that account for nonideal solution behavior. Squares correspond to 1% (w/v) NaCl and circles to 3% (w/v) NaCl. Solid curves: calculated from Eq. (17) with B estimated from George et al. 2 [4] and K estimated from linear regression; dashed curves: B and K estimated simultaneously from linear regression of 2 #ux data.
results at 3% (w/v) NaCl (circles). As a sensitivity check, K and B were estimated simultaneously 2 from linear regression of Wilson's results for 1% (w/v) NaCl; the values are K"6.8]10~7 s~1 and B "2.8 ml/g (dashed curve through the squares in 2 Fig. 5). The two values of K agree within 20%, although the discrepancy is larger for B , probably 2 because of the large uncertainty associated with extrapolating B outside the range of measure2 ment. This higher value of K was then used in a linear regression to estimate B "!14.4 ml/g at 2 3% NaCl (dashed curve through the circles), where aggregation is reported. The di!erence in the two values of B is less than 25% and the two curves are 2 not that di!erent, so it seems that binary interactions are able to account for lysozyme behavior well above its solubility. In general, the predicted #uxes compare favorably with the experimental data, suggesting that thermodynamic interactions may account for behavior that is often attributed to aggregated protein solutions. In large part, this is because the interaction potential governs the behavior of the system. As a measure of the average interaction between molecules in solution, virial coe$cients
M.L. Grant / Journal of Crystal Growth 209 (2000) 130}137
provide the means to characterize macroscopic properties of the system, such as osmotic pressure and activity coe$cients. However, it is impossible to infer the shape of the interaction potential from these measurements; a net attraction (B (0) may 2 be due to either a shallow long-range interaction or a strong short-range interaction that leads to physical aggregation. The latter is presumably the case since proteins form relatively high-density crystals. Nevertheless, it is apparent that protein}protein interactions are responsible for signi"cant deviations from thermodynamically ideal behavior, even if it is not possible to determine whether the deviations are due to aggregation of protein or interactions between monomers. 5. Summary The calculations above demonstrate that the nominal supersaturation, p"ln(C/C ), is inad40equate to describe the di!erence in chemical potential that drives protein crystallization. A more detailed thermodynamic description of the system that includes e!ects of temperature and added salt on the protein shows that the attractive interactions between molecules reduce the thermodynamic driving force *k. The growth rate and *k are not linearly proportional, however, so the growth mechanism cannot be inferred from a plot of growth rate vs. *k. Nevertheless, thermodynamic nonideality accounts for the characteristic decrease in the growth exponent as protein supersaturation
137
increases at constant temperature and ionic strength. The extension of the work to higher supersaturations requires additional measurements of virial coe$cients covering a wider range of conditions. Finally, this work reemphasizes the central role of thermodynamics in protein crystal growth.
References [1] C.A. Haynes, K. Tamura, H.R. KoK rfer, H.W. Blanch, J.M. Prausnitz, J. Phys. Chem. 96 (1992) 905. [2] V.L. Vilker, C.K. Colton, K.A. Smith, J. Colloid Interface Sci. 79 (1981) 548. [3] A. George, W.W. Wilson, Acta Crystallogr. D 50 (1994) 3651. [4] A. George, Y. Chiang, B. Guo, A. Arabshahi, Z. Cai, W.W. Wilson, Methods Enzymology 276 (1997) 100. [5] T.L. Hill, An Introduction to Statistical Thermodynamics, Dover, Mineola, NY, 1986. [6] D.A. McQuarrie, Statistical Mechanics, Harper and Row, New York, 1976. [7] J.W. Tester, M. Modell, Thermodynamics and its Applications, 3rd edition, Prentice-Hall, Upper Saddle River, NJ, 1997. [8] D. Stigter, T.L. Hill, J. Phys. Chem. 63 (1959) 551. [9] E. Cacioppo, M.L. Pusey, J. Cryst. Growth 114 (1991) 286. [10] E. Forsythe, M.L. Pusey, J. Cryst. Growth 139 (1994) 89. [11] M.L. Pusey, J. Cryst. Growth 110 (1991) 60. [12] L.J. Wilson, M.L. Pusey, J. Cryst. Growth 122 (1992) 8. [13] L.J. Wilson, L. Adcock-Downey, M.L. Pusey, Biophys. J. 71 (1996) 2123. [14] P.C. Hiemenz, R. Rajagopalan, Principles of Colloid and Surface Chemistry, 3rd edition, Marcel Dekker, New York, 1996. [15] E.L. Cussler, Di!usion, Cambridge University Press, London, 1984.