10358
J. Phys. Chem. 1994, 98, 10358-10367
Colloidal Interactions in Protein Crystal Growth M. L. Grant and D. A. Saville* Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544 Received: May 23, 1994; In Final Form: July 26, 1994@
Large, well-ordered crystals are required for X-ray diffraction studies of proteins, but it is often difficult to obtain suitable crystals. Proteins frequently precipitate as amorphous solids or form crystals that are too small. Larger crystals often lack order, which may reflect a nonspecific character in the protein-crystal interaction. To investigate the assembly process, we studied colloidal interaction forces between a protein crystal and a molecule in solution. Three factors control the interaction potential: the protein charge distribution, the charge on the crystal, and the effective Hamaker constant for the dispersion force. The relative strengths of electrostatic and dispersion forces were determined for a range of surface charges and Hamaker constants for a model protein, lysozyme. We found that the balance between electrostatic repulsion and van der Waals attraction is delicate. The free energy surface of the protein-crystal interaction shows many shallow minima, which is consistent with the protein’s ability to crystallize in different forms. Charge heterogeneity on the crystal surface and the molecule is important in the nucleation of new crystal layers and in the migration of molecules across the surface to growth sites. The Boltzmann-weighted interaction potential of the molecule differs strongly from that of a uniformly charged sphere when the crystal surface is highly charged.
1. Introduction A fundamental axiom of biochemistry declares that the structure of biological macromolecules reflects their function. If the molecule’s function can be determined and the relevant structures identified, a synthetic molecule can be designed to mimic the original molecule. This is the basis of “rational” drug design.’ X-ray diffraction studies of protein crystals (or other crystals of biological macromolecules such as DNA) provide the knowledge of the macromolecular structure required for such endeavors. Crystallographic techniques are sufficiently advanced that the rate-limiting step in protein structure determination is now the growth of suitable crystal^.^-^ A typical X-ray study requires crystals that are large (approximately 1 mm3 in volume) and reasonably well ordered, but few proteins readily form such crystals. Many proteins precipitate as amorphous solids or fine crystals that are too small to use. Some studies suggest that there may be a limiting size beyond which protein crystals cannot g r o ~ . ~Although ,~ this “cessation of growth” phenomenon has become entrenched in the protein crystal growth literature, recent studies under carefully controlled conditions yielded protein crystals much larger than the previously reported terminal size.8-10 Occasionally, large crystals form, but the crystal packing is too disordered to diffract welL4 These observations have combined to give protein crystals a reputation for extreme sensitivity and fragility. Perhaps the crystallographers’difficulties are associated with colloidal interactions between the crystal and molecules in solution and the open nature of the protein crystal. Unlike inorganic or “small molecule” crystals, protein crystals contain many channels of ~ o l v e n t solvent ;~ contents ranging from 27 to 65% by volume have been reported.” Molecules are thought to be held in the crystal lattice by nonspecific electrostatic, dispersive, and hydrogen Two manifestations of the nonspecific nature of the crystal bonds are the existence of several solubility minima and the polymorphism of macromo-
* Corresponding author. @Abstractpublished in Advance ACS Abstracts, September 1, 1994.
lecular systems. Not only can proteins crystallize in many different forms but the differences in energy between some forms are so small that two different forms can grow simultaneously from the same s~lution.‘~ The character of the colloidal interaction potential between the crystal and a nonuniformly charged molecule may explain some of the complexity of protein crystal growth. Systems of biological macromolecules often exhibit complex behavior as a result of nonuniform charge distributions. The enzymatic activity of superoxide dismutase (SOD) is a case in point. Although SOD has a net valence of -414 and its substrate, the superoxide radical, has a valence of -1, electrostatic repulsion between like charges does not greatly limit the enzyme’s activity. Furthermore, enzyme activity decreases with added salt.14 experiment^,'^ calculations of the electrostatic potential near SOD,15316and Brownian dynamics simulation^'^-^^ all indicate that the enzyme activity is controlled by the attraction between the negatively charged superoxide radical and the positive charges that are concentrated near the active site of SOD. Interactions between proteins and charged surfaces also demonstrate the effects of a nonuniform charge distribution. Hodder et a1.21 studied the elution of sperm whale myoglobin in high-performance ion exchange chromatography at pH 9.6. Under these conditions, all the acidic side-chain groups should be ionized, but the results indicated only a portion of the glutamic and aspartic acid residues interacted with the ion exchange resin. A study of the molecule’s structure yielded four sites with interactions that were consistent with experimental data. Theoretical studies also illustrate the effects of anisotropic charge. Yoon and LenhofP2 developed a boundary element method for solving the linearized Poisson-Boltzmann equation. Interactions between a uniformly charged surface and the molecule were incorporated into their formulation, and detailed calculations of protein-surface interaction energies embodied both the complex charge distribution and the highly irregular geometry of ribonuclease A.23 Their results indicate a strong
0022-365419412098-10358$04.50/0 0 1994 American Chemical Society
Colloidal Interactions in Protein Crystal Growth
J. Phys. Chem., Vol. 98, No. 40, 1994 10359
CHART 1: Assumptions Employed in Studying orientation dependence: the molecule is attracted in most Molecule-Surface Interactions orientations but repulsed in a few. This suggests ribonuclease A tends to present its active site to negatively charged surfaces. The electrostatic and dispersion (van der Waals) potentials are additive. Similar computations for lysozyme at pH 7, an ionic strength of 0.1 M, and a surface charged to -10 pC/cm2 indicate the The dispersion potential is independent of molecular interaction is attractive in all orientations. orientation. Roth and Lenhoff used the same boundary element technique The effective Hamaker constant, Aeff, is independent of separation. to compute equilibrium constants for adsorption of lysozyme on surfaces.24 In these calculations, the shape of the molecule The linearized Poisson-Boltzmann equation govems the electrostatics. was represented in two ways: (i) a highly irregular boundary obtained from crystallographic data and (ii) a sphere. The The planar surface can be treated as an insulated, semi-infinite charge distribution in (i) was taken from crystallographic data, region with a uniform surface charge density, a,. while in (ii), the sphere’s charge distribution was either a The protein molecule can be treated as a sphere. monopole (net charge placed in center) or a monopole and a The charge of the protein molecule can be represented by a dipole. Roth and Lenhoff demonstrated that the electrostatic surface charge distribution. interaction potential for the simplified model of lysozyme falls The interior of the protein molecule has a dielectric constant of 2. within the envelope of the more detailed calculations and used the simpler model to examine the effects of ionic strength, orientation of the two bodies. This energy is the reversible work surface charge, and dispersive interactions on the adsorption done against both electrostatic and dispersion or van der Waals equilibrium constant. forces to produce a particular configuration. Nonuniform charge Although some of the physical phenomena of protein crystal effects will be determined from comparison with an otherwise growth are similar to those found in adsorption, there are key identical, uniformly charged particle having the same net charge differences. First, electrostatic interactions in protein crystal as the protein. The charge distribution on the protein is based growth are generally repulsive because the sign of the crystal’s on the globular protein lysozyme. We aim to examine the surface potential is related to the charge on the molecule. At charge effects apart from any geometrical features that give the small separations, however, the configuration of charged groups actual protein its function. Toward that end, we use simplified on the surfaces of the molecule and crystal may produce either geometries for both the molecule and crystal surface. Nevertheattraction or repulsion. The influence of short-scale charge less, the essential features of the molecule-crystal interactions heterogeneity on the crystal surface decays quickly with are preserved. separation, so any attraction of the molecule to the crystal is Our presentation begins with the approximations employed primarily due to van der Waals forces. Second, the nature of to study the molecule-surface interaction potential. Then the the protein crystal differs radically from the packings used for model of the protein is described in detail. Next, the mathHPLC in that the molecules in the crystal are arranged in a ematical statements of the electrostatic subproblems are set out, porous lattice containing roughly 50% solvent by v o l ~ m e . 4 3 ~ ~ 8 ~ ~and 9 ~ ~the implementation of the solution scheme is discussed. This introduces some uncertainty in the value of the effective Results for the electrostatic free energy at contact are used to Hamaker constant used to compute the van der Waals potential map out the free energy surface for the lysozyme-surface because the crystal may not behave as a uniform body. In system. The “average” behavior of lysozyme is compared with addition, protein crystals are usually grown in highly concenan equivalent, uniformly charged sphere to quantify the effects trated salt solutions; ionic strengths of 1-1.5 M are not of charge heterogeneity. Finally, the attractive van der Waals unusua1.6~10~26~27 The electrostatic and dispersion forces are potential is included and the balance between the electrostatic nearly matched under these conditions. In the somewhat lower and dispersive forces is discussed. Our calculations show that ionic strength typical of HF’LC,23the forces are more disparate. while colloidal forces may not alter the diffusive flux of protein molecules toward the growing crystal significantly,the influence Here, we employ a boundary element method to calculate of charge heterogeneities on the crystal surface can be dramatic. the colloidal interactions between a protein molecule (a simplified model of lysozyme) and a protein crystal, accounting for 2. Method the heterogeneous charge distribution on the protein molecule. Our representation of the molecule as a sphere with a nonuni2.1. Assumptions in the Model. The assumptions used to form surface charge distribution lies between the detailed simplify the calculations are listed in Chart 1. First, the two structure of Yoon and LenhofP3 and the much simpler model interactions are presumed additive; this is the cornerstone of used by Roth and L e n h ~ f f .The ~ ~ surface charge on the crystal the Derjaguin-Landau-Verwey -0verbeek (DLV0)-theory and is treated as a parameter since we do not know which values is justified by numerous s t u d i e ~ . ~Dispersion ~-~~ forces are are appropriate for a given crystal. Not only is it likely that modeled using an “effective” Hamaker constant,A e ~ .Frequency different crystal faces hold different surface charges but it is spectra for a material’s dielectric behavior are required for a also unclear what uniform charge density best represents the detailed calculation using the Lifshitz theory2*so, in the absence effects of a surface made of heterogeneously charged particles. of such data for proteins, we take A,H as given. We assess the In our approach, we express the electrostatic potential relative importance of dispersion forces by varying A,a but produced by the protein molecule and crystal in terms of neglect effects due to restructuring of the solvent, e.g., hydration solutions of simpler subproblems. One subproblem accounts forces. for the anisotropic charge distribution on the protein molecule, Linearization of the Poisson-Boltzmann equation simplifies while the others describe the contribution of the charged crystal the electrostatics problem. The linearization is consistent with surface. The effect of the crystal’s charge density can be studied the low net charge on the molecule and, moreover, allows us systematically by recombining the solutions of the different to readily assess the significance of nonuniform charge distribusubproblems in an appropriate manner. tions in protein-surface interactions. The average surface We will calculate the free energy of interaction between potential of lysozyme in 1 M NaCl in water ( E = 80) is about molecule and crystal as a function of the separation and 16 mV based on a net valence of +10.5 at pH 4.731and a
Grant and Saville
10360 J. Phys. Chem., Vol. 98, No. 40, 1994 hydrodynamic radius of 20 Since the linearization is valid for potentials up to approximately 100 mV,28 it is more than adequate except when the particle and surface are almost in contact. Employing uniform dielectric constants and the other approximations inherent in the Debye-Huckel model certainly represents a considerable simplification of matters at small separations. Nevertheless, these computations represent a first step toward understanding the phenomena, and the magnitudes of the electrostatic effects suggest that more refined calculations will be justified. The disparate sizes of the molecule and surface justify treating the crystal as a semi-infinite region with a surface charge. Screening by counterions limits the range of the electrostatic forces to about 5 - l (15 8, based on a 1 M 1:l electrolyte solution and K - ~FZ 3 A), so the surface appears infinite when the molecule first “senses” it. Nevertheless, the use of a uniformly charged flat plate to represent the surface is one of the more severe assumptions because the length scales of surface roughness and charge heterogeneity are the same as those of the molecule. Although we recognize the limitations of this approximation, the intricacies caused by such effects are beyond the scope of these calculations. Since mobile ions in solution screen the influence of all but a small portion of the surface, the effect of a nonuniform surface charge distribution on the crystal can be estimated by calculating the electrostatic potential energy for different amounts of (uniform) surface charge, a,. Detailed calculations which take account of the structure of the surface charge will eventually be needed to appreciate the subtleties of the interaction. The main reason for treating the protein molecule as a sphere is simplicity. Crystallographic data indicate that most globular proteins are roughly spherical when folded, and a table of 20 proteins of known structure33shows that the ratio of maximum to minimum dimensions is seldom greater than 2. Hen egg white lysozyme is variously described as an ellipsoid of or dimensions 23 8, x 28 8, x 40 30 8, x 30 8, x 45 30 8, x 30 8, x 45 A “with a wedge-shaped piece All these are consistent with a spherical ap roximation and a hydrodynamic radius of approximately 20 .32 Charged residues are almost invariably found at the surface of the protein molecule, where they can interact with the polar water molecule^.^^^^^ Other researcher^^^-^* model the charge distribution with point charges lying just inside the surface of the protein that is accessible to a “probe sphere”, e.g., a water molecule 1.4 8, in r a d i ~ s . ~We ~ , represented ~~ the protein’s charge using a smoothed surface charge distribution to make the computations with both the uniformly charged particle and protein as similar as possible. The method for translating the charge distribution obtained from X-ray crystallography into a surface charge distribution is described later. Hydrophobic residues and side chains tend to be buried in the interior of the molecule, at least when the molecule is soluble in ~ a t e r . Calculations ~~,~ of packing density give values around 0.75,33.34indicating that the interior is relatively uniform. Thus, the dielectric constant in the interior of the molecule is often taken to be in the range between 2 and 4.22,23,39-42 Dao-pin et al.37calculated the electrostatically induced shift in pK, for two systems as a sensitivity check and found the shift was insensitive to values chosen in the range 2 5 €1 5 8, except at low ionic strength. The purpose of these approximations is to simplify the problem while preserving its essential features. Results for a molecule with an anisotropic charge distribution can then be compared with those for a uniformly charged molecule and differences in behavior ascribed to the anisotropic particle
81
TABLE 1: Coordinates of Charged Groups in Tetragonal Hen Egg White Lysozyme number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
residue 1 1 5
13 14 15 18 21 33 25 45 48 52 61 66 68 73 87 96 97 101 112 114 116 125 128 129
type
charge $1 $1 +1 +1 +1
+OS -1 +1 +1 -1 $1 -1 -1 $1 -1 +1 +l -1 $1 +1 -1 +1 +1 +1 +1 +1 -1
x
Y
Z
3.28 -3.80 -6.31 -17.40 -12.20 -9.67 +14.73 -11.82 3.17 4.38 18.46 14.17 8.98 13.02 11.50 16.16 1.83 -5.45 -11.68
10.16 10.48 24.40 21.25 9.25 11.01 24.29 23.93 23.88 24.84 15.42 22.91 21.08 20.81 12.56 12.96 16.91 7.45 16.57 14.40 24.43 33.64 30.5 1 36.55 31.20 17.79 21.78
10.35 8.18 2.84 11.10 14.63 17.86 14.83 29.27 5.62 18.14 23.99 29.22 22.43 31.91 27.61 24.25 39.96 16.81 22.82 30.09 32.74 23.22 10.82 22.86 -1.00 0.09 6.41
-5.50 -2.00 5.25 6.61 - 1.79 -10.80 -18.86 -17.14
charge. The level of approximation employed here is appropriate for such an investigation. 2.2. “Representative” Protein Molecule. The approximate coordinates of the charged groups on hen egg white lysozyme (taken from crystallographicdata for the tetragonal space group at 1.4 M NaCl in 0.02 M sodium acetate buffer at pH 4.7) are listed in Table l.31 We assume that all the charged groups lie on a spherical surface and seek the location and size of the sheath which comes closest to all the charged groups. The sphere is defined by the location of its origin (xo, yo, z,) and its radius, a. The origin of the best-fit sphere is the point such that the lengths of radii from the origin to the charges have minimum scatter about the mean value R = a. In the coordinate system specified in Table 1, the best-fit sphere is centered at (-1.67,20.91, 17.91) and has aradius of 16.48 8,. These values compare favorably with the center of mass (- 1.08,20.00, 18.35) determined by inspection31 and the hydrodynamic radius of approximately 20 A.32 Next, we set up the sphere’s coordinate system centered at the origin of the best-fit sphere and aligned with the coordinate system described by the coordinates in Table 1. This (2, j , 2) system is shown in Figure 1. Once the sphere’s coordinate system is defined, locations on the surface can be identified by their latitude a (-90” Ia 5 90’) and longitude B (-180” 5 B 5 180”). For the boundary element calculations, the surface of the molecule is divided into spherical triangular elements, as sketched in Figure 1, each with constant values of surface charge density, surface potential, and normal flux. Each edge of an element is the arc of the great circle connecting the vertices. The element’s node lies at the intersection of the sphere’s surface and the ray connecting the sphere’s origin with the centroid of the element. The convergence of the boundary element method was tested by computing the interaction potential for a uniform sphere discretized into N elements; the interaction potential changes only slightly for N 2 100. Our calculations for lysozyme were performed with 240 elements on the sphere’s surface to compensate for the nonuniform surface charge.
Colloidal Interactions in Protein Crystal Growth
J. Phys. Chem., Vol. 98, No. 40, 1994 10361
(a) 90
J d
0. The perpendicular line f “ the crystal surface through the center of the sphere is an axis of rotational
symmetry. Since the crystal face is uniformly charged, a rotation of the molecule about the perpendicular line from the surface through the center of the sphere (Figure 2) does not change the electrostatic interactions. Thus, the free energy of the system depends only on the molecule-crystal separation and the latitude and longitude (a, B) of the point closest to the surface. The first step in constructing the surface charge distribution on the molecule is to project each charge onto the surface of the sphere along its radius vector; the surface charge distribution is diagrammed in Figure 3a. Rather than represent the charged groups on the surface as point charges with infinite charge densities, we employ the method of “local averaging” to create a patchwork of charge smooth enough to ensure that the governing equations remain valid hut “lumpy” enough to exhibit behavior unique to anisotropically charged proteins. Here the averaging is done by centering a spherical cap at the point of interest and summing the point charges which lie within the cap. The local charge density is the net charge within the cap divided by the area of the cap. We used a local averaging area of 100 A2, which is about 3% of the surface area of the sphere (3400 AZ);a spherical cap this size would extend to a latitude of a = +70.3’ if placed at the north pole. For our conditions, an averaging area of 100 Azensures the validity of the linearized Poisson-Boltzmann equation as the molecule moves away from the plate.z6 The smoothed charge distribution, scaled by a, = d T / e a % 1.4 x pUcmZ, is shown in Figure 3b. Once a smooth charge distribution is available, it must he translated into a form consistent with our “constant element”
d : c . z m -2110(1 -i(xxI (1 iixxi ?IHXI .25m Figure 3. Map of the surface charge distribution for an idealized ( ~ ) + l / 2 All . charges given in lysozyme sphere: (a) (.)+I; (.)-I; units of the proton charge (1.6 x L O W C); (b) local (smoothed) surface charge distribution when the charged groups are averaged over I00 A2; charge densities are scaled by u, zz 1.4 x IO-* @/an2; (c) appmximation for the surface charge density in (b) with 240 constant
elements. The legend at the bottom of the figure shows the charge level. formulation wherein each element has uniform potential and charge. The (constant) charge density, q,assigned to an element is calculated by averaging the smooth charge density over the area of the element, AI: uj = (I/Aj)JudA
(1)
Ai
The actual surface charge distribution used in our computations is shown in Figure 3c. The discretized charge distribution retains much of the general character exhibited by the smoothed charge distribution, although the range of surface charge densities is slightly smaller because of the additional averaging described by eq 1. The uniformly charged sphere has a scaled charge density, a*G uede&T, of 358 based on a net valence of +10.531 and a surface area of 3400 A2. Several layers of approximation have been employed to create a realistic model of a protein molecule. The choice of sampling area is dictated by the linearization of the goveming equation, while the idealization of the molecule as a sphere and the corresponding mapping of the charged groups reflect a desire to keep the problem as simple as possible. Despite these simplifications the resulting surface charge distribution appears able to capture the essential features of the electrostatic phenomena. 2.3. Solution for the Electrostatic Potential. Figure 4 depicts the particleesurface system. The scaled electrostatic
10362 J. Phys. Chem.,Vol. 98, No. 40, 1994
Grant and Saville us Le., sohere-sphere).
The “remainder” wtential is u‘. The total potekal is their sum u = u‘p
+ us=+ ur
The equations for the sphere-sphere problem are
solvcnt
region 1:
v2u: = 0
region 2:
vzuy= ( a K ) 2 U y
K-’
potential (u = VI&; q0= kT/e) is a solution of the linearized Debye-Hiickel equation:28
V2Ut = 0
region 2 (electrolyte solution):
(2)
v2u2 = ( a K ) 2 U z
(3)
where a = particle radius, K~ = 2n~,z~e~le,e2kT for a z-z electrolyte; nb = bulk ion number density, e = elementary charge, eo = permittivity of free space, k = Boltzmann’s constant, and T = absolute temperature. All lengths in the problem have been scaled by the particle radius, a. The boundary conditions at the surface of the panicle are the continuity of potential, U I= u2
(4)
and the jump in electric displacement,
e,Vu,.n,
+ E2Vu2n2= u*
(5)
where the surface charge density has been scaled on uo = eo&/ a. Note that nl and n2 point out of their respective regions. The boundary conditions in the electrolyte are
Z-m:
u,-0
e,Vu;Sn,
+ e,Vu;tn,
= a*
-
z--: Figure 4. Definition sketch. Lengths are scaled by the particle radius, a, and surface charge densities by a,,= c,$,,/a.
.; =“
auyiaz = o uy 0
z = 0:
region 1 (molecule):
I
sphere surface:
while u‘ satisfies region 1:
02.;
= -(aK)2ufP
region 2
v2.;
= (aK)’u;
= -@*
sphere surface: u; = u;
{e,Vu;n,
z = 0: Z-m:
+ e2vu>n2= -(e,
- e2)VuQ*n,
au;iaz = o u;-0
Since both us$and ur must satisfy a no-flux condition at z = 0, boundary element solutions for each can be constructed using the method of images. The starting point for the boundary element formulation is a more general form of the DebyeHuckel equation:
v2u - (aK)ZU = -e*
(22)
where e’ is afixed charge density. Equation 22 governs the potential in the electrolyte (region 2) when e‘ = 0, in region 1 of the “remainder” problem when a K = 0, and in region 1 of the sphere-sphere problem when both a K and e’ = 0. The fundamental solution (Green’s function) of the homogeneous form of eq 22 is
(7)
The solution of the problem specified by eqs 2-7 is complicated, but its linearity allows it to be split into readily solved subproblems. We write the electrostatic potential as the sum of three potentials describing the following subproblems: (i) a uniformly charged plate in an electrnlyte solution, (ii) a sphere with fixed surface charge near an insulated flat plate, and (iii) the “remainder” when the solutions to the first two problems are subtracted from u. The potential in problem (i) is ufp, the flat plate potential that satisfies eqs 3, 6, and 7:28-30
Since the potential of problem (ii) is identical to that produced by the charged sphere and its mirror image,”-45 we denote it
where r is measured from the “source” point x, (Le., r = Ix %I). The boundary element method involves integrating the fundamental solution and its normal derivative over the bounding surfaces in the system to obtain a system of equations describing the distribution of sources. Techniques for obtaining boundary element solutions of eq 22 can be found elsewhere,22.23.26.45 2.4. Interaction Potential. We are interested in the electrostatic potential energy and its dependence on particlesurface separation. This energy, or interaction potential, is defined in terms of the work performed to bring two bodies to a particular configuration from some reference state.46 When performed isothermally and reversibly, the free energy change for the system can be expressed in terms of the work required to assemble all the charges from i ~ ~ f i n i t y : ~ ~ “ ~ , ~ ~
Colloidal Interactions in Protein Crystal Growth
Here R denotes the volume of the system and the f superscript indicates a fixed charge density (not subject to thermal randomization). The expression on the far right is obtained when the charge density is proportional to the potential and all the fixed charges reside on the surfaces of the system. r represents all the surfaces of the system, and u is the fixed surface charge. The change in free energy can he divided into two parts:
AGelc, = AGec+ Qelec(d.a.B)
(25)
where AGecis the change in free energy required to "charge up" the molecule and plate at infinite separation, and Qelz(d,a& is the change in free energy as the molecule and plate are brought together in a given orientation (a,b). In terms of scaled electrostatic variables,
Here, d = gap (in particle radii) between the plate and the surface of the sphere (see Figure 4). u(d,a,p) = potential when the particle and plate are separated by d, and u(m) = potential when particle is infinitely far from plate; dA is dimensionless. Since rotary Brownian motion enables a molecule to sample many orientations, we also evaluate the "average" interaction potential between the molecule and the plate. The proper orientation average is46"8.49
.Ihl,
lii
,,t,
~
,
?
0,
.I%
,#I,
11%
180
/J i d c j r c c i )
u*:
