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Theory of protein crystal nucleation and growth controlled by solvent evaporation
Journal of Crystal Growth 204 (1999) 553}562
Theory of protein crystal nucleation and growth controlled by solvent evaporation James K. Baird* Department of Chemistry, University of Alabama in Huntsville, Huntsville, AL 35899, USA Received 16 March 1998; accepted 15 December 1998 Communicated by A. McPherson
Abstract The driving force for protein crystallization is the supersaturation. In the case of crystal growth in a hanging drop, the supersaturation at early times is controlled by the dynamics of solvent evaporation and is largely independent of the rate of appearance of the crystals. This permits the equations of Johnson, Mehl, Avrami, and Kolomogrov to be integrated using the classic model for crystal nucleation and the spiral dislocation model for crystal growth. As results one obtains a formula for the number of crystals in the drop and another formula for their average size. The parameters in these formulae include either explicitly or implicitly the protein mass, temperature, pH, and ionic strength, which are the independent variables known experimentally to in#uence the overall rate of protein crystallization. 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 61.50.Cj; 64.60.Qb; 81.10.Dn; 87.15.Da Keywords: Protein; Nucleation; Spiral dislocation; Hanging drop
1. Introduction The three-dimensional structure of a protein molecule determines its biological function. If the protein can be precipitated in the form of single crystals at least 0.5 mm on a side, the molecular structure can usually be determined by the method of X-ray di!raction. Experimental methods exploiting rotating anode X-ray generators, area de-
* Tel.: #1-256-890-6441; fax: #1-256-890-6349. E-mail address: [email protected] (J.K. Baird)
tectors, and structure solving software have made X-ray crystallography routine. With the advent of these techniques, however, the rate-limiting step in a typical macromolecular structure determination has become the production of the crystals, themselves. Except in the case of hen egg-white lysozyme and a few other proteins, which are commercially available, most proteins are individually isolated and puri"ed in quantities that seldom exceed a few grams. Moreover, the typical crystal growth recipe calls for a pH-bu!ered aqueous solution of strong electrolyte [1], in which the solubility of the protein
0022-0248/99/$ - see front matter 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 9 9 ) 0 0 1 3 4 - 7
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J.K. Baird / Journal of Crystal Growth 204 (1999) 553}562
can be as high as 30 mg/ml [2,3]. Under these conditions, less than 100 ll of solution can be allotted to each of the many experiments which need to be performed in order to "nd conditions under which the protein will crystallize. To cope with this problem, one uses the hanging drop technique. In this method, a droplet containing up to 50 ll of growth solution saturated in protein is deposited onto a glass cover slip [4]. If the slip is carefully inverted, the drop becomes suspended by the action of interfacial tension. The slip is then used as cover over the mouth of a receptacle containing a solution called the reservoir. The concentration of electrolyte in the reservoir solution is ordinarily 2}4 times that in the drop. Once a seal has been made, the two solutions approach vapor pressure equilibrium by isopiestic distillation of water from the drop onto the reservoir. This process, can take from a few hours to a few days, depending upon concentrations, temperature, and the geometry [4}7]. As the isopiestic distillation proceeds, the protein in the drop is concentrated. This stimulates the formation of microscopic nuclei of the crystalline phase. Under favorable conditions, the nuclei grow into crystals large enough to become visible. Although there is a satisfactory theory of droplet evaporation [7], there is yet no theory which combines evaporation, nucleation, and growth in a single formalism. For combining nucleation and growth at constant volume, the accepted procedure is the JMAK theory introduced by Johnson and Mehl [8], Avrami [9}11], and Kolmogorov [12]. The key to extending the JMAK theory to the hanging drop is to observe that at early times, the protein supersaturation driving both nucleation and growth is controlled by the dynamics of solvent evaporation and is largely independent of the rate of appearance of the crystals. This permits the JMAK integrals to be computed to obtain a formula for the number of crystals in the drop and another formula for their average size. We begin in Section 2 by introducing a new simpli"ed theory of the rate of droplet evaporation incorporating only those physicochemical and geometric details which the latest experiments [7] have shown to be most important. In Section 3, we introduce the JMAK formalism. In supersaturation
independent form, we use it to show that at early times in the vapor equilibration process, the rates of nucleation and growth are su$ciently slow as to have no e!ect on the protein supersaturation, which increases monotonically as the volume of the drop decreases. To apply JMAK theory under realistic protein crystal growth conditions, we next select speci"c supersaturation-dependent functions to represent, respectively, the rate of nucleation and the rate of growth. In Section 4, we adopt standard nucleation theory [13] to represent the nucleation step in protein crystallization in a hanging drop. Since there is some evidence that protein crystals grow by spiral dislocation when the relative supersaturation ranges between about 0.5 and 4 [14], we adopt in Section 5 the Burton}Cabrera}Frank (BCF) spiral dislocation model [15] to represent the growth of a protein nucleus into a crystal. In Section 6, we use the results of Sections 2}5 to reduce the JMAK equations to form which can be integrated approximately. These approximate integrals should be applicable to the "rst 10 000 s or so of a hanging drop experiment where the relative supersaturation lies within the range of values of one to four. In Section 7, we show how these integrals can be tested experimentally.
2. Kinetics of isopiestic distillation Experiments have shown that, when the typical hanging drop with radius of about 2 mm is located more than 5 mm from the surface of the reservoir solution, the rate of isopiestic distillation is largely independent of the exact shape of the drop [7]. The relevant geometric parameters are then con"ned to ¸, the length separating the drop from the surface of the solution in the reservoir, and S, the e!ective cross-sectional area of the reservoir. The value of S can be adjusted to "t the theory to the observed rate of evaporation. Although a free parameter, S is known to approach the geometric cross-sectional area of the reservoir as ¸ increases [7]. In discussing the theory of evaporation below, the subscript `1a refers to water, while the subscript `2a refers to the involatile electrolyte (ordinarily a salt) which determines the solution vapor
J.K. Baird / Journal of Crystal Growth 204 (1999) 553}562
pressure. The superscript `Ra refers to the reservoir. The #ux, J , of water vapor from the drop to the reservoir is given by D J "! (P0!P ), ¸R¹
(2.1)
where P0 and P are the reservoir and drop water vapor pressures, respectively, D is the di!usion coe$cient of water vapor through air, R in the denominator is the universal gas law constant, and ¹ is the absolute temperature [4]. Eq. (2.1) assumes that the water vapor behaves as an ideal gas. The vapor pressure of a droplet containing a su$ciently dilute salt solution can be "tted to the equation
n P "P 1!w , (2.2) n where P is the vapor pressure of pure water, n and n are the number of moles of water and salt, respectively, in the drop, and w is an empirically adjusted vapor pressure lowering coe$cient [4}7]. By comparison, in the reservoir, the volume of solution is su$ciently large that its vapor pressure remains sensibly constant no matter how much water distills into it from the drop. Treating P0 as a constant, then, and substituting Eq. (2.2) into Eq. (2.1), we obtain
D(P !P0) P n 1! J " w . (2.3) ¸R¹ P !P0 n The quantity in brackets approaches its equilibrium value of zero asymptotically as tPR [4]. As the salt is involatile, n is a constant, and the num ber of moles of water in the drop at equilibrium is P wn n (R)" . (2.4) (P !P0) In terms of the partial molar volumes < and < of water and salt, respectively, the volume of the drop is <"n < #n < . (2.5) At equilibrium, the volume of water in the drop is < (R)"n (R)< .
(2.6)
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After introducing Eqs. (2.4)}(2.6) into Eq. (2.3) with the partial molar volumes taken as constants, we obtain
D(P !P0) < (R) 1! J " . (2.7) ¸R¹
< (R) d< "!< SJ "!U 1! , (2.8) dt
(2.9)
Letting < be the initial value of <, Eq. (2.8) has the solution
<(t) <(R) <(t) Ut"< 1! !< (R) ln ! < < < <(R) #< (R) ln 1! , (2.10) < where
<(R)"< (R)#n < . (2.11) Eq. (2.10) predicts the time required for the drop to reach any volume within the range, < ) <(t)(<(R). Ordinarily the salt concentrations in the drop and the reservoir are such that <(R)K< /2. This permits us to use the expansion, ln xK(x!1) to convert Eq. (2.10) into a polynomial form, <(t)"< (1!it) (2.12) which is appropriate at early times. In Eq. (2.12), the time coe$cient, U , (2.13) i" < #< (R) Eq. (2.12) will "nd application in subsequent sections. Eq. (2.5) permits us to convert Eq. (2.8) to the reciprocal
dt 1 < (R) "! 1# U
which will prove useful in Section 6.
(2.14)
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J.K. Baird / Journal of Crystal Growth 204 (1999) 553}562
As an example of the application Eqs. (2.10) and (2.12), consider the case of a < "30 ll drop con taining aqueous 1 M NaCl over a reservoir having the geometry ¸"1 cm and S"200 mm containing aqueous 4 M NaCl at 295 K. Under these conditions, the following parameter values also apply: D "0.26 cm/s, P "0.0277 atm, w"1.875, P !P0"4.07;10\ atm, < "18.0 cm/mol, and < "19.9 cm/mol [7]. On the basis of Eqs. (2.4) and (2.6), we obtain < (R)"6.90 ll, and us ing n "(1 M);(30;10\ l) and Eq. (2.11), we ob tain <(R)"9.29 ll. Using Eq. (2.9), we compute U"1.57;10\ cm/s, and using Eq. (2.13), we compute i"4.26;10\ s\. This permits us to evaluate Eqs. (2.10) and (2.12) which are shown plotted in Fig. 1. For times less than 10 s, the dashed line in the "gure represents the full curve to an accuracy of better than 10%.
3. Time dependence of protein supersaturation Nuclei which achieve a certain critical size go on to become crystals. The rate of nucleation depends upon the relative supersaturation p(q)"(c(q)/c )!1, (3.1) where c is the protein solubility and c(q) is the protein concentration in the drop at time, q. Let I(p(q)) be the rate of formation of critical nuclei per
unit volume of the drop at time, q. During a time interval, dq, the total number of critical nuclei which are generated in the drop is <(q)I(p(q)) dq where <(q) is the drop volume. The most common crystalline form of the easily crystallized protein, hen-egg-white lysozyme, for example, is tetragonal, so we shall for convenience take the critical nucleus to have the shape of a rectangular solid. We let the average linear growth rate of one of the faces of the tetragonal solid be R(p). Since opposite faces of a rectangular nucleus must be parallel to the same crystallographic plane, a linear dimension of the nucleus grows at the rate, 2R(p). If o is the density of the crystal, then the total mass, k(t), of protein in crystalline form at time, t, is
m "c (p(t)#1)<(t)#k(t). (3.3) For realistic forms for I(p) and R(p), there appears no way to solve Eqs. (3.2) and (3.3) simultaneously for p(t) [16]. To go beyond this point analytically, an approximate form for p(t) is required. In terms of c(t), Eq. (3.3) can be rewritten k(t) m . c(t)" ! <(t) <(t)
Fig. 1. The curve represents Eq. (2.10) evaluated with U"1.57;10\ cm/s, < "30 ll, <(R)"9.29 ll, and < (R)" 6.90 ll, while the dashed line represents Eq. (2.12) evaluated with < "30 ll and i"4.26;10\ s\.
R R k(t)"o dq<(q)I(p(q)) 2 dh R(p(h)) . (3.2) O Eq. (3.2) is a JMAK integral. The time dependence of p, required to evaluate Eq. (3.2), is constrained by the principle of conservation of mass. If m is the mass of protein dis solved in the drop at t"0, this principle implies
(3.4)
According to Eq. (3.4), the time dependence of c(t) is determined by that of both <(t) and k(t). If the volume of the drop, <(t), decreases faster than the mass, k(t), of crystals increases, then c(t) and p(t) are controlled principally by the rate of evaporation of the drop. According to Fig. 1, the most rapid decrease in drop volume occurs at early times when Eq. (2.12) applies. Given the limited range of values of p expected in an experiment of short duration, it is possible to "nd constant values which serve as upper bounds on the functions, I(p) and R(p). Substituting constants for I and R into Eq. (3.2), one "nds that
J.K. Baird / Journal of Crystal Growth 204 (1999) 553}562
k(t)"O(t) [17]. For it(1, 1/<(t)K (1/< ) (1#it); hence, it is always possible to "nd su$ciently early times when the "rst term on the right of Eq. (3.4) increases linearly, while the second, which is O(t), is much smaller. Since during such period, the mass of protein going into crystals has little e!ect on c(t), Eq. (3.4) can be rewritten c (p(t)#1)"c(t)Km /<(t) (3.5) approximately. Eq. (3.5) is an approximation to p(t) which permits the evaluation of JMAK integrals.
4. Nucleation theory In the classical theory of nucleation [13], molecules accumulate one by one until a nucleus of critical size forms. If we let P be a molecule of the protein, the agglomeration mechanism is P # P P # P ) )
P P P P )
)
)
)
)
)
)
(4.1)
P # PH P PH L \ L where nH is the critical size. P H not only has the L maximum free energy of formation, and hence minimum population, but it also has the slowest rate of formation. This forces all the reactions preceding the last of Eq. (4.1) to be at steady state. The rate of formation of critical nuclei per unit volume of the drop is [13,18] I(nH)"k(nH)c(g(nH)/2pk ¹) (N /M) ;exp(!*G(nH)/k ¹).
(4.2)
In Eq. (4.2), c is the protein concentration in the drop in g/ml, M is the protein molecular weight, N is Avagadro's number, k is Boltzmann's con stant, and ¹ is the absolute temperature. The free energy of formation of the critical nucleus from molecules is [18] 32c c c v ? @ A *G(nH)" , (k ¹)(ln(1#p))
(4.3)
557
where c , c , and c are the interfacial tensions in the ? @ A three mutually perpendicular directions of a rectangular solid, and v is the molecular volume of a protein molecule in the nucleus. The quantity,
R*G(n) Rn
(k ¹ ln(1#p)) " (4.4) 64c c c v LH ? @ A is the negative curvature *G(n) evaluated at n"nH. The average speci"c rate of formation of a critical nucleus is k(nH). This quantity depends upon the areas of the faces in contact with the solution and the rates of attachment of protein molecules to those faces. Assuming a rectangular shape, the distance from the center of a critical nucleus to the center of its face having a normal pointing in the j-direction is g(nH)"!
2c H hH" , H k ¹ ln(1#p)
(4.5)
where c is the surface tension on that face. To H construct k(nH), we suppose that in the ( j"a) direction, which has surface area (2h )(2h ), molecules @ A attach with a speed, m . The rate of increase of ? crystal volume on this face is thus 4h h m . Using @ A ? Eq. (4.5) to evaluate the sum of these rates over all six faces, one obtains
32c c c v m m m ?# @# A , ? @ A k(nH)" (4.6) (k ¹ ln(1#p)) c c c ? @ A Protein crystals ordinarily grow from supersaturated solutions where p<1. Upon substitution of Eqs. (4.3), (4.4) and (4.6) in Eq. (4.2), we obtain
c c @ A m ? c ? c c c c # ? A m# ? @ m @ A c c @ A !32c c c v ? @ A ;exp , (k ¹)(ln(p))
I(nH)"
N 8 cvp pk ¹ M
(4.7)
where we have replaced everywhere p#1 by p.
5. Theory of spiral dislocation growth The use of surface microprobes has revealed that crystals of biological macromolecules can grow by
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J.K. Baird / Journal of Crystal Growth 204 (1999) 553}562
the spiral dislocation mechanism [19]. The mechanism was "rst treated theoretically by Burton et al. [15]. For the linear rate of growth, R , of a crystalH line face in the j-direction, they obtained the expression [20]
Dck ¹ H H f(p)p ln(1#p) cj H H 19 cv H ;tanh , (5.1) 2 j k ¹ ln(1#p) H In Eq. (5.1), c is the equilibrium surface concentraH tion of protein molecules attached to the growing face, D is their coe$cient for surface di!usion, H while j is the di!usive jump distance on the surH face. The function
2 R (p)" H 19
D q f(p)" 1# H H tanh aj H H
19 cv H 2 j k ¹ ln(1#p) H
\ (5.2)
is called the retardation factor. It takes into account the fact that the concentration of protein molecules adsorbed on the growing edge of a step of the spiral is di!erent than on the surface of the tread of the step [20]. In Eq. (5.2), a is the distance H advanced in the direction of the riser per protein molecule incorporated, while q is the lifetime for H desorption of such a molecule onto the tread. In the case of protein crystal growth, we are interested ordinarily in the case p<1. In this limit, f(p) becomes unity and Eq. (5.1) simpli"es to cv R (p)K H p (5.3) H q H where we have used j"D q [20] and q is the H H H H lifetime for desorption of a molecule on the tread back into the solution.
6. Crystal number and average crystal size The number, N(t), of crystals formed after time, t, in a hanging drop is directly proportional to the nucleation rate and is given by the JMAK formula,
R N(t)" dq I(p(q))<(q).
(6.1)
The integral in Eq. (6.1) is most easily evaluated by using p as the variable of integration. If at t"0, the supersaturation is p , and if at t"t, it is p, then Eq. (6.1) can be expressed as
N(p)"
N
dp
d< dq <(p )I(p ). dp d<
(6.2) N We combine the results of Sections 2}4 to evaluate this integral. Using Eq. (3.5), the volume, <, which is required in Eq. (6.2), can be rewritten in the p<1 limit as
(6.3)
d
(6.4)
dq 1 ap "! 1# d< U 1!bp
(6.5)
which can be seen by substituting Eq. (6.3) into the right-hand side of Eq. (2.14) and replacing t by q on the left. The coe$cients of p in Eq. (6.5) are de"ned by a"< (R)/(p #1)< , (6.6) b"<(R)/(p #1)< , (6.7) It is helpful to write Eq. (4.7) in the simpli"ed form I(p)"Ap exp(!B/(ln p)),
(6.8)
where the coe$cients, A and B, are, respectively,
c c @ A m ? c ? c c c c # ? A m# ? @ m @ A c c @ A
A"
N 8 cv pk ¹ M
(6.9)
and B"32c c c v/(k ¹). (6.10) ? @ A After substitution of Eqs. (6.3)}(6.5) and (6.8) into Eq. (6.2), we obtain
m N 1 a dp # c p 1!bp N B ;exp ! . (ln p )
N(p)"
A U
(6.11)
J.K. Baird / Journal of Crystal Growth 204 (1999) 553}562
Eq. (6.11) can be integrated by parts and the remaining integral evaluated to "rst order in p!p by a Taylor series expansion of its integrand. The result is N(p)"(A/U)(m /c )F(p, p )Q(p, p ) ;exp(!B/(ln p )), where
F(p, p )"ln
p p
1!bp ? @ 1!bp
(6.12)
(6.13)
559
If we divide Eq. (6.17) by Eq. (6.12), we obtain 1H (p)2"(m /2c U)(c v/q )F(p, p ) (6.18) H H H which is an expression for the average length of the crystals in the drop. The function, F(p, p ), de"ned by Eq. (6.13) and shared by Eqs. (6.12) and (6.18) has been plotted in the case of typical values of a and b in Fig. 2. In the "gure, the evaporation of the drop limits the accessible values of the supersaturation to p )p((1/b).
and where
7. Discussion and conclusions
2B(p!p ) (6.14) Q(p, p )"1# p (ln p ) is a correction factor resulting from the use of the Taylor series. Since each nucleus of critical size advances to become a crystal, N(p) is the number of crystals appearing in a drop during the time required for the supersaturation to grow from p to p. Another measurable quantity is the average size of the crystals in the drop. In this regard, we consider the length, 2h , of a crystal parallel to the H j-direction. This length extends at a rate 2R (p). The H length at time, t, of a crystal nucleated at time, q, is the integral of 2R (p) from q to t. Since the nucleaH tion rate at time, q, is I(p(q))<(q), the total of the j-lengths of all the crystals in the drop is
The model in this paper is based upon equations which appear logically sound and which have also been con"rmed to various degrees by experiment. These are Eq. (2.10) (droplet volume as a function of time), Eq. (3.2) (crystal mass as a function of time), Eq. (3.3) (conservation of total protein mass), Eq. (4.7) (volume rate of nucleation), Eq. (5.3) (linear rate of growth of a crystal by screw dislocation), Eq. (6.1) (number of crystals as a function of time), and Eq. (6.15) (crystal size as a function of time). Ideally, one would like to "nd the form of the supersaturation function, p(t), which satis"es all of these equations simultaneously for the widest range of values of p and t. A complete knowledge of this form, be it analytical or numerical, would be useless,
R R H (t)"2 dq I(p(q))<(q) dh R (p(h)) (6.15) H H O which is a JMAK integral. Eq. (6.15) can be converted to
N
d< dq dp <(p )I(p ) dp d< N N d< dh ; dp R (p ) (6.16) dp d< H NY which involves integration over the supersaturation. Using Eqs. (5.3), (6.3)}(6.5) and (6.8), Eq. (6.16) can be integrated by parts. The result is H (p)"2 H
m cv H (F(p, p ))Q(p, p ) c q H ;exp(!B/(ln p )). (6.17)
A H (p)" H 2U
Fig. 2. F(p, p ) as de"ned by Eq. (6.13) plotted as a function of the supersaturation, p. The initial value of the supersaturation is p "3, while b"0.0775 and a/b"0.743 have been computed using Eqs. (6.6) and (6.7) with < (R)"6.90 ll and <(R)" 9.29 ll, as in the caption to Fig. 1.
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J.K. Baird / Journal of Crystal Growth 204 (1999) 553}562
however, because experiment suggests that the screw dislocation model fails for relative supersaturation values above about 4 (where an island growth mechanism sets in). Such complete information is also unnecessary, because the values of both of the relevant nucleation parameters (A and B), as well as the value of the relevant growth parameter (c v/q ) can all be found by comparing the approxH H imate analytical results presented here with the outcomes of experiments limited to the range, t:10 s and 1(p(4. Once known, the values of the parameters, A, B, and c v/q , can be used to H H evaluate Eqs. (6.8) and (5.3), respectively, for any time t when p assumes a value where these equations are valid. After incorporation of these parameters, Eqs. (6.1) and (6.15) can be integrated numerically to produce an engineering model of crystallization in a hanging drop which is valid for long times. For accuracy in analyzing data with Eq. (6.12), the correction factor, Q(p, p ), de"ned by Eq. (6.14) should be less than 2. This is always possible, since the value of p can be made arbitrarily close to that of p . To determine how this restricts the range of p, we use Eq. (6.10) to evaluate B. Taking lysozyme as an example, the molecular weight is 14 400. The density of its crystals is 1.24 g/cm [21]. From this we compute v"1.9;10\ cm. The surface tension of protein crystals has been estimated as 0.2 erg/cm [22], which we use in place of each of the c's in Eq. (6.10). We thus "nd that at room temperature, B"1.6. Adopting p "3, the coe$c ient of p!p in Eq. (6.14) becomes 0.65. This implies that for Q(p, p ) to be accurate, p!p should be less than 1/0.65"1.5. It is important to note that N(p), given by Eq. (6.12), does not involve R (p) de"ned by Eq. (5.3). H That is, Eq. (6.12) is sensitive only to the rate of nucleation and is completely insensitive to the details of the mechanism by which the nuclei develop into crystals. Eq. (6.12) can be tested experimentally simply by counting the number of crystals that appear in the drop as the supersaturation advances from p to p . According to Eq. (6.12), the values of the nucleation parameters A and B can be determined by noting how the number of crystals formed depends upon p . A plot of ln(N(p)/F(p, p )Q(p, p )) on the
ordinate with (ln p )\ on the abscissa should be a straight line having slope, !B, and intercept equal to ln[(A/U)(m /c )], from which the value of A can be computed. Since Q(p, p ) also depends upon B, the method of successive approximations may have to be exploited to achieve this line. Whereas N(p) given by Eq. (6.12) depends upon Q(p, p ), 1Hj(p)2 given by Eq. (6.18) is completely independent of this correction factor. There is thus no mathematical limitation, other than p ) p((1/b), on the applicability of Eq. (6.18). However, for our hypothesis of spiral dislocation growth to be applicable, experiments on lysozyme [14] suggest that p "3 and p"4 might be appropriate values. By assuming no change in supersaturation due to crystallization, we can set a lower bound on the time required to go from p to p by using the formula
t"
p!p 1 p#1 i
(7.1)
which follows from Eqs. (2.12) and (3.1). An evaluation of 1H (p)2 by summing the lengths of all H crystals in the drop and dividing by their number should be su$cient to determine experimentally the value of the parameter group, c v/q , which H H describes addition of protein molecules to the crystal. Whereas H (p) depends upon both of the nuH cleation parameters, A and B, the average length, 1H (p)2, is insensitive to the values of these paramH eters, since the factors involving nucleation cancel when Eq. (6.17) is divided by Eq. (6.12). To achieve good results in comparing Eqs. (6.12) and (6.18) with experiments, the crystal growth droplets should be observed under a microscope. Even so, the observer may miss some small crystals that have not yet grown to visible dimensions. Although this problem may introduce some error in the determination of N(p), it should not a!ect H (p), since small crystals contribute little to this H quantity. Batch experiments in which p(t) decays freely from an initial value, have shown that the rate of crystal formation in the case of lysozyme depends upon the initial protein concentration [23], the temperature [23}25], the pH [24], and the ionic strength [25]. Eqs. (6.12) and (6.17) depend both
J.K. Baird / Journal of Crystal Growth 204 (1999) 553}562
explicitly and implicitly on these same variables. Indeed both explicitly involve the initial amount of protein, m . Ignoring U, which depends solely on the properties of the hanging drop, the parameters expected to be the strongest implicit functions of ¹, pH, and ionic strength are c , m (j"a, b, c), and q . H H Our analytical model contains some parameter information, which is independent of any given set of experimental results. For example, when one multiplies both numerator and denominator on the right of Eq. (6.7) by c to obtain c <(R)/ c (p #1)< , it is clear that b can be interpreted as the mass of protein still in solution at isopiestic equilibrium divided by the initial mass of protein in the drop. Also, according to Eqs. (6.6) and (6.7), the exponent, a/b in Eq. (6.13), equals the volume fraction of the drop occupied by water at isopiestic equilibrium. Finally when (p!p );1, Eq. (6.13) can be rewritten as
1 a F(p, p )" # (p!p ) p 1!bp
(7.2)
which demonstrates that in this limit the number of crystals in the drop increases linearly with p!p . This is apparent from the plot of F(p, p ) in Fig. 2. Being general JMAK integrals, Eqs. (6.2) and (6.16) can be used with any functional forms for I(p) and R (p). In the case of I(p), this allows the incorH poration of recently discussed revisions of nucleation theory that permit critical nuclei to be as small as dimers [26,27]. Of more importance, perhaps, is the possibility for introducing models [28] for R (p), that take into account the other growth H mechanisms that have been observed experimentally [22,29}32]. In summary, our result, Eq. (6.12), serves as a test of classical nucleation theory, while our other result, Eq. (6.18), serves as a test of the BCF theory of crystal growth by spiral dislocation.
Acknowledgements The author would like to acknowledge helpful conversations concerning the theory with Professors G. Wilemski and D.T. Wu. This research was
561
sponsored by the National Institute of General Medical Sciences of the National Institutes of Health through grant 1 R15 GM51018 and in part by the Naval Research Laboratory in Washington, DC under a grant N00014-94-1-GO16 from the O$ce of Naval Research.
References [1] A. McPherson, Preparation and Analysis of Protein Crystal, Wiley, New York, 1982. [2] S.B. Howard, P.J. Twigg, J.K. Baird, E.J. Meehan, J. Crystal Growth 90 (1988) 117. [3] E.L. Forsythe, M.L. Pusey, J. Crystal Growth 168 (1996) 112. [4] W.W. Fowlis, L.J. DeLucas, P.J. Twigg, S.B. Howard, E.J. Meehan, J.K. Baird, J. Crystal Growth 90 (1988) 94. [5] L. Sibille, J.K. Baird, J. Crystal Growth 110 (1991) 72. [6] L. Sibille, J.C. Clunie, J.K. Baird, J. Crystal Growth 110 (1991) 80. [7] J.R. Luft, D.T. Albright, J.K. Baird, G.T. DeTitta, Acta Crystallogr. D 52 (1996) 1098. [8] W.A. Johnson, R.F. Mehl, Trans. Am. Inst. Min. Metall. Engrs. 135 (1939) 416. [9] M. Avrami, J. Chem. Phys. 7 (1939) 1103. [10] M. Avrami, J. Chem. Phys. 8 (1940) 212. [11] M. Avrami, J. Chem. Phys. 9 (1941) 177. [12] A.N. Kolmogorov, Izv. Akad. Nauk. SSR. Ser. Math. 3 (1937) 355. [13] K.F. Kelton, Solid State Phys. 45 (1991) 75. [14] P.B. Vekilov, Prog. Cryst. Growth and Charact. 26 (1993) 25. [15] W.K. Burton, N. Cabrera, F.C. Frank, Phil. Trans. Roy. Soc. 243 (1951) 299. [16] M.R. Riedel, S. Karato, Geophys. J. Int. 125 (1996) 397. [17] W.A. Tiller, The Science of Crystallization: Microscopic Interfacial Phenomena, Cambridge University Press, Cambridge, 1991, pp. 376}377. [18] A.G. Walton, Nucleation in liquids and solutions, in: A.C. Zettlemoyer (Ed.), Nucleation, Marcel Dekker, New York, 1969, p. 225. [19] S.D. Durbin, G. Feher, Ann. Rev. Phys. Chem. 47 (1996) 171. [20] P. Bennema, G.H. Gilmer, Kinetics of crystal growth, in: P. Hartman (Ed.), Crystal Growth: An Introduction, NorthHolland, Amsterdam, 1973, p. 263. [21] G.M. Barrow, Physical Chemistry for the Life Sciences, McGraw-Hill, New York, 1981, p. 422. [22] A.J. Malkin, Yu G. Kuznetsov, W. Glantz, A. McPherson, J. Phys. Chem. 100 (1996) 11736. [23] M. Ataka, M. Asai, Biophys. J. 58 (1990) 807. [24] Y. Bessho, M. Ataka, M. Asai, T. Katsure, Biophys. J. 66 (1994) 310.
562 [25] [26] [27] [28]
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J.K. Baird, J.C. Clunie, Phys. Chem. Liquids 37 (1999) 285. G. Wilemski, J. Chem. Phys. 103 (1995) 1119. D.T. Wu, Solid State Phys. 50 (1997) 38. M. Ohara, R.C. Reid, Modeling Crystal Growth Rates from Solution, Prentice-Hall, Englewood Cli!s, NJ, 1973. [29] S.D. Durbin, G. Feher, J. Mol. Biol. 212 (1990) 763.
[30] S.D. Durbin, W.E. Carlson, J. Crystal Growth 122 (1992) 71. [31] P.G. Vekilov, M. Ataka, T. Katsura, J. Crystal Growth 26 (1993) 25. [32] J.H. Konnert, P. D'Antonio, K.B. Ward, Acta Crystallogr. D 50 (1994) 603.
Theory of protein crystal nucleation and growth controlled by solvent evaporation James K. Baird* Department of Chemistry, University of Alabama in Huntsville, Huntsville, AL 35899, USA Received 16 March 1998; accepted 15 December 1998 Communicated by A. McPherson
Abstract The driving force for protein crystallization is the supersaturation. In the case of crystal growth in a hanging drop, the supersaturation at early times is controlled by the dynamics of solvent evaporation and is largely independent of the rate of appearance of the crystals. This permits the equations of Johnson, Mehl, Avrami, and Kolomogrov to be integrated using the classic model for crystal nucleation and the spiral dislocation model for crystal growth. As results one obtains a formula for the number of crystals in the drop and another formula for their average size. The parameters in these formulae include either explicitly or implicitly the protein mass, temperature, pH, and ionic strength, which are the independent variables known experimentally to in#uence the overall rate of protein crystallization. 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 61.50.Cj; 64.60.Qb; 81.10.Dn; 87.15.Da Keywords: Protein; Nucleation; Spiral dislocation; Hanging drop
1. Introduction The three-dimensional structure of a protein molecule determines its biological function. If the protein can be precipitated in the form of single crystals at least 0.5 mm on a side, the molecular structure can usually be determined by the method of X-ray di!raction. Experimental methods exploiting rotating anode X-ray generators, area de-
* Tel.: #1-256-890-6441; fax: #1-256-890-6349. E-mail address: [email protected] (J.K. Baird)
tectors, and structure solving software have made X-ray crystallography routine. With the advent of these techniques, however, the rate-limiting step in a typical macromolecular structure determination has become the production of the crystals, themselves. Except in the case of hen egg-white lysozyme and a few other proteins, which are commercially available, most proteins are individually isolated and puri"ed in quantities that seldom exceed a few grams. Moreover, the typical crystal growth recipe calls for a pH-bu!ered aqueous solution of strong electrolyte [1], in which the solubility of the protein
0022-0248/99/$ - see front matter 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 9 9 ) 0 0 1 3 4 - 7
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can be as high as 30 mg/ml [2,3]. Under these conditions, less than 100 ll of solution can be allotted to each of the many experiments which need to be performed in order to "nd conditions under which the protein will crystallize. To cope with this problem, one uses the hanging drop technique. In this method, a droplet containing up to 50 ll of growth solution saturated in protein is deposited onto a glass cover slip [4]. If the slip is carefully inverted, the drop becomes suspended by the action of interfacial tension. The slip is then used as cover over the mouth of a receptacle containing a solution called the reservoir. The concentration of electrolyte in the reservoir solution is ordinarily 2}4 times that in the drop. Once a seal has been made, the two solutions approach vapor pressure equilibrium by isopiestic distillation of water from the drop onto the reservoir. This process, can take from a few hours to a few days, depending upon concentrations, temperature, and the geometry [4}7]. As the isopiestic distillation proceeds, the protein in the drop is concentrated. This stimulates the formation of microscopic nuclei of the crystalline phase. Under favorable conditions, the nuclei grow into crystals large enough to become visible. Although there is a satisfactory theory of droplet evaporation [7], there is yet no theory which combines evaporation, nucleation, and growth in a single formalism. For combining nucleation and growth at constant volume, the accepted procedure is the JMAK theory introduced by Johnson and Mehl [8], Avrami [9}11], and Kolmogorov [12]. The key to extending the JMAK theory to the hanging drop is to observe that at early times, the protein supersaturation driving both nucleation and growth is controlled by the dynamics of solvent evaporation and is largely independent of the rate of appearance of the crystals. This permits the JMAK integrals to be computed to obtain a formula for the number of crystals in the drop and another formula for their average size. We begin in Section 2 by introducing a new simpli"ed theory of the rate of droplet evaporation incorporating only those physicochemical and geometric details which the latest experiments [7] have shown to be most important. In Section 3, we introduce the JMAK formalism. In supersaturation
independent form, we use it to show that at early times in the vapor equilibration process, the rates of nucleation and growth are su$ciently slow as to have no e!ect on the protein supersaturation, which increases monotonically as the volume of the drop decreases. To apply JMAK theory under realistic protein crystal growth conditions, we next select speci"c supersaturation-dependent functions to represent, respectively, the rate of nucleation and the rate of growth. In Section 4, we adopt standard nucleation theory [13] to represent the nucleation step in protein crystallization in a hanging drop. Since there is some evidence that protein crystals grow by spiral dislocation when the relative supersaturation ranges between about 0.5 and 4 [14], we adopt in Section 5 the Burton}Cabrera}Frank (BCF) spiral dislocation model [15] to represent the growth of a protein nucleus into a crystal. In Section 6, we use the results of Sections 2}5 to reduce the JMAK equations to form which can be integrated approximately. These approximate integrals should be applicable to the "rst 10 000 s or so of a hanging drop experiment where the relative supersaturation lies within the range of values of one to four. In Section 7, we show how these integrals can be tested experimentally.
2. Kinetics of isopiestic distillation Experiments have shown that, when the typical hanging drop with radius of about 2 mm is located more than 5 mm from the surface of the reservoir solution, the rate of isopiestic distillation is largely independent of the exact shape of the drop [7]. The relevant geometric parameters are then con"ned to ¸, the length separating the drop from the surface of the solution in the reservoir, and S, the e!ective cross-sectional area of the reservoir. The value of S can be adjusted to "t the theory to the observed rate of evaporation. Although a free parameter, S is known to approach the geometric cross-sectional area of the reservoir as ¸ increases [7]. In discussing the theory of evaporation below, the subscript `1a refers to water, while the subscript `2a refers to the involatile electrolyte (ordinarily a salt) which determines the solution vapor
J.K. Baird / Journal of Crystal Growth 204 (1999) 553}562
pressure. The superscript `Ra refers to the reservoir. The #ux, J , of water vapor from the drop to the reservoir is given by D J "! (P0!P ), ¸R¹
(2.1)
where P0 and P are the reservoir and drop water vapor pressures, respectively, D is the di!usion coe$cient of water vapor through air, R in the denominator is the universal gas law constant, and ¹ is the absolute temperature [4]. Eq. (2.1) assumes that the water vapor behaves as an ideal gas. The vapor pressure of a droplet containing a su$ciently dilute salt solution can be "tted to the equation
n P "P 1!w , (2.2) n where P is the vapor pressure of pure water, n and n are the number of moles of water and salt, respectively, in the drop, and w is an empirically adjusted vapor pressure lowering coe$cient [4}7]. By comparison, in the reservoir, the volume of solution is su$ciently large that its vapor pressure remains sensibly constant no matter how much water distills into it from the drop. Treating P0 as a constant, then, and substituting Eq. (2.2) into Eq. (2.1), we obtain
D(P !P0) P n 1! J " w . (2.3) ¸R¹ P !P0 n The quantity in brackets approaches its equilibrium value of zero asymptotically as tPR [4]. As the salt is involatile, n is a constant, and the num ber of moles of water in the drop at equilibrium is P wn n (R)" . (2.4) (P !P0) In terms of the partial molar volumes < and < of water and salt, respectively, the volume of the drop is <"n < #n < . (2.5) At equilibrium, the volume of water in the drop is < (R)"n (R)< .
(2.6)
555
After introducing Eqs. (2.4)}(2.6) into Eq. (2.3) with the partial molar volumes taken as constants, we obtain
D(P !P0) < (R) 1! J " . (2.7) ¸R¹
< (R) d< "!< SJ "!U 1! , (2.8) dt
(2.9)
Letting < be the initial value of <, Eq. (2.8) has the solution
<(t) <(R) <(t) Ut"< 1! !< (R) ln ! < < < <(R) #< (R) ln 1! , (2.10) < where
<(R)"< (R)#n < . (2.11) Eq. (2.10) predicts the time required for the drop to reach any volume within the range, < ) <(t)(<(R). Ordinarily the salt concentrations in the drop and the reservoir are such that <(R)K< /2. This permits us to use the expansion, ln xK(x!1) to convert Eq. (2.10) into a polynomial form, <(t)"< (1!it) (2.12) which is appropriate at early times. In Eq. (2.12), the time coe$cient, U , (2.13) i" < #< (R) Eq. (2.12) will "nd application in subsequent sections. Eq. (2.5) permits us to convert Eq. (2.8) to the reciprocal
dt 1 < (R) "! 1# U
which will prove useful in Section 6.
(2.14)
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J.K. Baird / Journal of Crystal Growth 204 (1999) 553}562
As an example of the application Eqs. (2.10) and (2.12), consider the case of a < "30 ll drop con taining aqueous 1 M NaCl over a reservoir having the geometry ¸"1 cm and S"200 mm containing aqueous 4 M NaCl at 295 K. Under these conditions, the following parameter values also apply: D "0.26 cm/s, P "0.0277 atm, w"1.875, P !P0"4.07;10\ atm, < "18.0 cm/mol, and < "19.9 cm/mol [7]. On the basis of Eqs. (2.4) and (2.6), we obtain < (R)"6.90 ll, and us ing n "(1 M);(30;10\ l) and Eq. (2.11), we ob tain <(R)"9.29 ll. Using Eq. (2.9), we compute U"1.57;10\ cm/s, and using Eq. (2.13), we compute i"4.26;10\ s\. This permits us to evaluate Eqs. (2.10) and (2.12) which are shown plotted in Fig. 1. For times less than 10 s, the dashed line in the "gure represents the full curve to an accuracy of better than 10%.
3. Time dependence of protein supersaturation Nuclei which achieve a certain critical size go on to become crystals. The rate of nucleation depends upon the relative supersaturation p(q)"(c(q)/c )!1, (3.1) where c is the protein solubility and c(q) is the protein concentration in the drop at time, q. Let I(p(q)) be the rate of formation of critical nuclei per
unit volume of the drop at time, q. During a time interval, dq, the total number of critical nuclei which are generated in the drop is <(q)I(p(q)) dq where <(q) is the drop volume. The most common crystalline form of the easily crystallized protein, hen-egg-white lysozyme, for example, is tetragonal, so we shall for convenience take the critical nucleus to have the shape of a rectangular solid. We let the average linear growth rate of one of the faces of the tetragonal solid be R(p). Since opposite faces of a rectangular nucleus must be parallel to the same crystallographic plane, a linear dimension of the nucleus grows at the rate, 2R(p). If o is the density of the crystal, then the total mass, k(t), of protein in crystalline form at time, t, is
m "c (p(t)#1)<(t)#k(t). (3.3) For realistic forms for I(p) and R(p), there appears no way to solve Eqs. (3.2) and (3.3) simultaneously for p(t) [16]. To go beyond this point analytically, an approximate form for p(t) is required. In terms of c(t), Eq. (3.3) can be rewritten k(t) m . c(t)" ! <(t) <(t)
Fig. 1. The curve represents Eq. (2.10) evaluated with U"1.57;10\ cm/s, < "30 ll, <(R)"9.29 ll, and < (R)" 6.90 ll, while the dashed line represents Eq. (2.12) evaluated with < "30 ll and i"4.26;10\ s\.
R R k(t)"o dq<(q)I(p(q)) 2 dh R(p(h)) . (3.2) O Eq. (3.2) is a JMAK integral. The time dependence of p, required to evaluate Eq. (3.2), is constrained by the principle of conservation of mass. If m is the mass of protein dis solved in the drop at t"0, this principle implies
(3.4)
According to Eq. (3.4), the time dependence of c(t) is determined by that of both <(t) and k(t). If the volume of the drop, <(t), decreases faster than the mass, k(t), of crystals increases, then c(t) and p(t) are controlled principally by the rate of evaporation of the drop. According to Fig. 1, the most rapid decrease in drop volume occurs at early times when Eq. (2.12) applies. Given the limited range of values of p expected in an experiment of short duration, it is possible to "nd constant values which serve as upper bounds on the functions, I(p) and R(p). Substituting constants for I and R into Eq. (3.2), one "nds that
J.K. Baird / Journal of Crystal Growth 204 (1999) 553}562
k(t)"O(t) [17]. For it(1, 1/<(t)K (1/< ) (1#it); hence, it is always possible to "nd su$ciently early times when the "rst term on the right of Eq. (3.4) increases linearly, while the second, which is O(t), is much smaller. Since during such period, the mass of protein going into crystals has little e!ect on c(t), Eq. (3.4) can be rewritten c (p(t)#1)"c(t)Km /<(t) (3.5) approximately. Eq. (3.5) is an approximation to p(t) which permits the evaluation of JMAK integrals.
4. Nucleation theory In the classical theory of nucleation [13], molecules accumulate one by one until a nucleus of critical size forms. If we let P be a molecule of the protein, the agglomeration mechanism is P # P P # P ) )
P P P P )
)
)
)
)
)
)
(4.1)
P # PH P PH L \ L where nH is the critical size. P H not only has the L maximum free energy of formation, and hence minimum population, but it also has the slowest rate of formation. This forces all the reactions preceding the last of Eq. (4.1) to be at steady state. The rate of formation of critical nuclei per unit volume of the drop is [13,18] I(nH)"k(nH)c(g(nH)/2pk ¹) (N /M) ;exp(!*G(nH)/k ¹).
(4.2)
In Eq. (4.2), c is the protein concentration in the drop in g/ml, M is the protein molecular weight, N is Avagadro's number, k is Boltzmann's con stant, and ¹ is the absolute temperature. The free energy of formation of the critical nucleus from molecules is [18] 32c c c v ? @ A *G(nH)" , (k ¹)(ln(1#p))
(4.3)
557
where c , c , and c are the interfacial tensions in the ? @ A three mutually perpendicular directions of a rectangular solid, and v is the molecular volume of a protein molecule in the nucleus. The quantity,
R*G(n) Rn
(k ¹ ln(1#p)) " (4.4) 64c c c v LH ? @ A is the negative curvature *G(n) evaluated at n"nH. The average speci"c rate of formation of a critical nucleus is k(nH). This quantity depends upon the areas of the faces in contact with the solution and the rates of attachment of protein molecules to those faces. Assuming a rectangular shape, the distance from the center of a critical nucleus to the center of its face having a normal pointing in the j-direction is g(nH)"!
2c H hH" , H k ¹ ln(1#p)
(4.5)
where c is the surface tension on that face. To H construct k(nH), we suppose that in the ( j"a) direction, which has surface area (2h )(2h ), molecules @ A attach with a speed, m . The rate of increase of ? crystal volume on this face is thus 4h h m . Using @ A ? Eq. (4.5) to evaluate the sum of these rates over all six faces, one obtains
32c c c v m m m ?# @# A , ? @ A k(nH)" (4.6) (k ¹ ln(1#p)) c c c ? @ A Protein crystals ordinarily grow from supersaturated solutions where p<1. Upon substitution of Eqs. (4.3), (4.4) and (4.6) in Eq. (4.2), we obtain
c c @ A m ? c ? c c c c # ? A m# ? @ m @ A c c @ A !32c c c v ? @ A ;exp , (k ¹)(ln(p))
I(nH)"
N 8 cvp pk ¹ M
(4.7)
where we have replaced everywhere p#1 by p.
5. Theory of spiral dislocation growth The use of surface microprobes has revealed that crystals of biological macromolecules can grow by
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J.K. Baird / Journal of Crystal Growth 204 (1999) 553}562
the spiral dislocation mechanism [19]. The mechanism was "rst treated theoretically by Burton et al. [15]. For the linear rate of growth, R , of a crystalH line face in the j-direction, they obtained the expression [20]
Dck ¹ H H f(p)p ln(1#p) cj H H 19 cv H ;tanh , (5.1) 2 j k ¹ ln(1#p) H In Eq. (5.1), c is the equilibrium surface concentraH tion of protein molecules attached to the growing face, D is their coe$cient for surface di!usion, H while j is the di!usive jump distance on the surH face. The function
2 R (p)" H 19
D q f(p)" 1# H H tanh aj H H
19 cv H 2 j k ¹ ln(1#p) H
\ (5.2)
is called the retardation factor. It takes into account the fact that the concentration of protein molecules adsorbed on the growing edge of a step of the spiral is di!erent than on the surface of the tread of the step [20]. In Eq. (5.2), a is the distance H advanced in the direction of the riser per protein molecule incorporated, while q is the lifetime for H desorption of such a molecule onto the tread. In the case of protein crystal growth, we are interested ordinarily in the case p<1. In this limit, f(p) becomes unity and Eq. (5.1) simpli"es to cv R (p)K H p (5.3) H q H where we have used j"D q [20] and q is the H H H H lifetime for desorption of a molecule on the tread back into the solution.
6. Crystal number and average crystal size The number, N(t), of crystals formed after time, t, in a hanging drop is directly proportional to the nucleation rate and is given by the JMAK formula,
R N(t)" dq I(p(q))<(q).
(6.1)
The integral in Eq. (6.1) is most easily evaluated by using p as the variable of integration. If at t"0, the supersaturation is p , and if at t"t, it is p, then Eq. (6.1) can be expressed as
N(p)"
N
dp
d< dq <(p )I(p ). dp d<
(6.2) N We combine the results of Sections 2}4 to evaluate this integral. Using Eq. (3.5), the volume, <, which is required in Eq. (6.2), can be rewritten in the p<1 limit as
(6.3)
d
(6.4)
dq 1 ap "! 1# d< U 1!bp
(6.5)
which can be seen by substituting Eq. (6.3) into the right-hand side of Eq. (2.14) and replacing t by q on the left. The coe$cients of p in Eq. (6.5) are de"ned by a"< (R)/(p #1)< , (6.6) b"<(R)/(p #1)< , (6.7) It is helpful to write Eq. (4.7) in the simpli"ed form I(p)"Ap exp(!B/(ln p)),
(6.8)
where the coe$cients, A and B, are, respectively,
c c @ A m ? c ? c c c c # ? A m# ? @ m @ A c c @ A
A"
N 8 cv pk ¹ M
(6.9)
and B"32c c c v/(k ¹). (6.10) ? @ A After substitution of Eqs. (6.3)}(6.5) and (6.8) into Eq. (6.2), we obtain
m N 1 a dp # c p 1!bp N B ;exp ! . (ln p )
N(p)"
A U
(6.11)
J.K. Baird / Journal of Crystal Growth 204 (1999) 553}562
Eq. (6.11) can be integrated by parts and the remaining integral evaluated to "rst order in p!p by a Taylor series expansion of its integrand. The result is N(p)"(A/U)(m /c )F(p, p )Q(p, p ) ;exp(!B/(ln p )), where
F(p, p )"ln
p p
1!bp ? @ 1!bp
(6.12)
(6.13)
559
If we divide Eq. (6.17) by Eq. (6.12), we obtain 1H (p)2"(m /2c U)(c v/q )F(p, p ) (6.18) H H H which is an expression for the average length of the crystals in the drop. The function, F(p, p ), de"ned by Eq. (6.13) and shared by Eqs. (6.12) and (6.18) has been plotted in the case of typical values of a and b in Fig. 2. In the "gure, the evaporation of the drop limits the accessible values of the supersaturation to p )p((1/b).
and where
7. Discussion and conclusions
2B(p!p ) (6.14) Q(p, p )"1# p (ln p ) is a correction factor resulting from the use of the Taylor series. Since each nucleus of critical size advances to become a crystal, N(p) is the number of crystals appearing in a drop during the time required for the supersaturation to grow from p to p. Another measurable quantity is the average size of the crystals in the drop. In this regard, we consider the length, 2h , of a crystal parallel to the H j-direction. This length extends at a rate 2R (p). The H length at time, t, of a crystal nucleated at time, q, is the integral of 2R (p) from q to t. Since the nucleaH tion rate at time, q, is I(p(q))<(q), the total of the j-lengths of all the crystals in the drop is
The model in this paper is based upon equations which appear logically sound and which have also been con"rmed to various degrees by experiment. These are Eq. (2.10) (droplet volume as a function of time), Eq. (3.2) (crystal mass as a function of time), Eq. (3.3) (conservation of total protein mass), Eq. (4.7) (volume rate of nucleation), Eq. (5.3) (linear rate of growth of a crystal by screw dislocation), Eq. (6.1) (number of crystals as a function of time), and Eq. (6.15) (crystal size as a function of time). Ideally, one would like to "nd the form of the supersaturation function, p(t), which satis"es all of these equations simultaneously for the widest range of values of p and t. A complete knowledge of this form, be it analytical or numerical, would be useless,
R R H (t)"2 dq I(p(q))<(q) dh R (p(h)) (6.15) H H O which is a JMAK integral. Eq. (6.15) can be converted to
N
d< dq dp <(p )I(p ) dp d< N N d< dh ; dp R (p ) (6.16) dp d< H NY which involves integration over the supersaturation. Using Eqs. (5.3), (6.3)}(6.5) and (6.8), Eq. (6.16) can be integrated by parts. The result is H (p)"2 H
m cv H (F(p, p ))Q(p, p ) c q H ;exp(!B/(ln p )). (6.17)
A H (p)" H 2U
Fig. 2. F(p, p ) as de"ned by Eq. (6.13) plotted as a function of the supersaturation, p. The initial value of the supersaturation is p "3, while b"0.0775 and a/b"0.743 have been computed using Eqs. (6.6) and (6.7) with < (R)"6.90 ll and <(R)" 9.29 ll, as in the caption to Fig. 1.
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however, because experiment suggests that the screw dislocation model fails for relative supersaturation values above about 4 (where an island growth mechanism sets in). Such complete information is also unnecessary, because the values of both of the relevant nucleation parameters (A and B), as well as the value of the relevant growth parameter (c v/q ) can all be found by comparing the approxH H imate analytical results presented here with the outcomes of experiments limited to the range, t:10 s and 1(p(4. Once known, the values of the parameters, A, B, and c v/q , can be used to H H evaluate Eqs. (6.8) and (5.3), respectively, for any time t when p assumes a value where these equations are valid. After incorporation of these parameters, Eqs. (6.1) and (6.15) can be integrated numerically to produce an engineering model of crystallization in a hanging drop which is valid for long times. For accuracy in analyzing data with Eq. (6.12), the correction factor, Q(p, p ), de"ned by Eq. (6.14) should be less than 2. This is always possible, since the value of p can be made arbitrarily close to that of p . To determine how this restricts the range of p, we use Eq. (6.10) to evaluate B. Taking lysozyme as an example, the molecular weight is 14 400. The density of its crystals is 1.24 g/cm [21]. From this we compute v"1.9;10\ cm. The surface tension of protein crystals has been estimated as 0.2 erg/cm [22], which we use in place of each of the c's in Eq. (6.10). We thus "nd that at room temperature, B"1.6. Adopting p "3, the coe$c ient of p!p in Eq. (6.14) becomes 0.65. This implies that for Q(p, p ) to be accurate, p!p should be less than 1/0.65"1.5. It is important to note that N(p), given by Eq. (6.12), does not involve R (p) de"ned by Eq. (5.3). H That is, Eq. (6.12) is sensitive only to the rate of nucleation and is completely insensitive to the details of the mechanism by which the nuclei develop into crystals. Eq. (6.12) can be tested experimentally simply by counting the number of crystals that appear in the drop as the supersaturation advances from p to p . According to Eq. (6.12), the values of the nucleation parameters A and B can be determined by noting how the number of crystals formed depends upon p . A plot of ln(N(p)/F(p, p )Q(p, p )) on the
ordinate with (ln p )\ on the abscissa should be a straight line having slope, !B, and intercept equal to ln[(A/U)(m /c )], from which the value of A can be computed. Since Q(p, p ) also depends upon B, the method of successive approximations may have to be exploited to achieve this line. Whereas N(p) given by Eq. (6.12) depends upon Q(p, p ), 1Hj(p)2 given by Eq. (6.18) is completely independent of this correction factor. There is thus no mathematical limitation, other than p ) p((1/b), on the applicability of Eq. (6.18). However, for our hypothesis of spiral dislocation growth to be applicable, experiments on lysozyme [14] suggest that p "3 and p"4 might be appropriate values. By assuming no change in supersaturation due to crystallization, we can set a lower bound on the time required to go from p to p by using the formula
t"
p!p 1 p#1 i
(7.1)
which follows from Eqs. (2.12) and (3.1). An evaluation of 1H (p)2 by summing the lengths of all H crystals in the drop and dividing by their number should be su$cient to determine experimentally the value of the parameter group, c v/q , which H H describes addition of protein molecules to the crystal. Whereas H (p) depends upon both of the nuH cleation parameters, A and B, the average length, 1H (p)2, is insensitive to the values of these paramH eters, since the factors involving nucleation cancel when Eq. (6.17) is divided by Eq. (6.12). To achieve good results in comparing Eqs. (6.12) and (6.18) with experiments, the crystal growth droplets should be observed under a microscope. Even so, the observer may miss some small crystals that have not yet grown to visible dimensions. Although this problem may introduce some error in the determination of N(p), it should not a!ect H (p), since small crystals contribute little to this H quantity. Batch experiments in which p(t) decays freely from an initial value, have shown that the rate of crystal formation in the case of lysozyme depends upon the initial protein concentration [23], the temperature [23}25], the pH [24], and the ionic strength [25]. Eqs. (6.12) and (6.17) depend both
J.K. Baird / Journal of Crystal Growth 204 (1999) 553}562
explicitly and implicitly on these same variables. Indeed both explicitly involve the initial amount of protein, m . Ignoring U, which depends solely on the properties of the hanging drop, the parameters expected to be the strongest implicit functions of ¹, pH, and ionic strength are c , m (j"a, b, c), and q . H H Our analytical model contains some parameter information, which is independent of any given set of experimental results. For example, when one multiplies both numerator and denominator on the right of Eq. (6.7) by c to obtain c <(R)/ c (p #1)< , it is clear that b can be interpreted as the mass of protein still in solution at isopiestic equilibrium divided by the initial mass of protein in the drop. Also, according to Eqs. (6.6) and (6.7), the exponent, a/b in Eq. (6.13), equals the volume fraction of the drop occupied by water at isopiestic equilibrium. Finally when (p!p );1, Eq. (6.13) can be rewritten as
1 a F(p, p )" # (p!p ) p 1!bp
(7.2)
which demonstrates that in this limit the number of crystals in the drop increases linearly with p!p . This is apparent from the plot of F(p, p ) in Fig. 2. Being general JMAK integrals, Eqs. (6.2) and (6.16) can be used with any functional forms for I(p) and R (p). In the case of I(p), this allows the incorH poration of recently discussed revisions of nucleation theory that permit critical nuclei to be as small as dimers [26,27]. Of more importance, perhaps, is the possibility for introducing models [28] for R (p), that take into account the other growth H mechanisms that have been observed experimentally [22,29}32]. In summary, our result, Eq. (6.12), serves as a test of classical nucleation theory, while our other result, Eq. (6.18), serves as a test of the BCF theory of crystal growth by spiral dislocation.
Acknowledgements The author would like to acknowledge helpful conversations concerning the theory with Professors G. Wilemski and D.T. Wu. This research was
561
sponsored by the National Institute of General Medical Sciences of the National Institutes of Health through grant 1 R15 GM51018 and in part by the Naval Research Laboratory in Washington, DC under a grant N00014-94-1-GO16 from the O$ce of Naval Research.
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