490
J . Phys. Chem. 1989, 93, 490-494
is also orthorhombic, and, presumably, the chains are also perpendicular to the lamellar surfaces. The measurements are summarized in Table 111. Relative to the neat components, the n-alkane binary mixture Cs0/C6 at 300 K is conformationally disordered. The disorder is manifest in the form of gauche bonds which occur primarily in the region of the chain ends. The concentration of disorder varies with the composition of the mixture and is at a maximum near the 1:l mixture. In addition to the gauche bonds near the ends of the chains, there is evidence of a low concentration of conformational disorder in the chain interior. This is indicated by the 1300-cm-I infrared band, which is present in the spectra of neat Cso and c 4 6 and in the spectrum of their mixture. For neat Cs0 and C46,this band appears to represent a small fraction of chains that are more or less highly disordered, since the assignment of this band to isolated gtg' kinks seems unlikely on energetic grounds. However, by assuming that the 1300-cm-I band represents isolated kinks, an upper limit to the kink concentration can be established. This limit is found to be 0.30 f 0.05 kinks/molecule for the 1:l mixture. However, only this value can be attributed to the effects of mixing since the melt crystallized neat components have kink concentrations of about 0.15 f 0.02 kinks/molecule. We emphasize that these concentrations are only upper limits and it is likely that the actual isolated kink concentration is much lower. The results for the cso/c46 system indicate that the nature of the conformational disorder induced by mixing is very similar to that found previously for the C21/C19 system.6 In both cases, this
disorder is concentrated in the interlamellar regions and to a good approximation the lamellar interior is not affected. However, the concentration of end-gauche bonds in the C50/C46system is significantly larger for Czl/C19. For the 1:l mixtures in which case conformational disorder is at a maximum, the sum of the gauche concentrations at bond positions 2 and 3 is about 2l/* times greater for csO/c46. Another difference between these systems is that the concentration of gauche bonds at the 3-position is greater for c50/c46. These differences appear to result from the fact that there is a greater chain length mismatch in the c5O/c46 system than in CZ1/Cl9.The fact that the chains in the cso/c46 mixture are longer than those in C21/C19 does not appear to enter as an important factor per se. Our results thus favor a model in which surface irregularities and voids resulting from the chain length mismatch are minimized by a combination of the longitudinal chain translation and the formation of gauche bonds at the chain ends. Our results do not support a model in which the chain-length mismatch is reduced by kinks along the length of the longer chain as has been frequently proposed. The concentration of kinks required for this appears to be much larger than the concentration of kinks estimated for the csO/c46 binary system. Acknowledgment. We gratefully acknowledge support through grants from the National Science Foundation (DMR 87-01586) and the National Institutes of Health (GM 27690). Registry No. n-C5,Hlo2,6596-40-3; n-C46H94, 7098-24-0.
Evaluation of Simple Model Descriptions of the Diffusional Association Rate for Enzyme-Ligand Systems Teresa Head-Gordon and Charles L. Brooks III* Department of Chemistry, Carnegie- Mellon University, Pittsburgh, Pennsylvania 1521 3 (Received: May 31, 1988)
In this report, we assess the ability of simple representations of dielectric properties, enzymatic shape, and charge distribution to determine the qualitative and quantitative features of diffusional association in enzyme-ligand systems. It is shown that these simple model descriptions are reliable for predicting qualitative features of the mechanism of diffusional ligand association for enzymes without deep-cleft geometries. Using such simple models, we present evidence that the reduction of the three-dimensional diffusion space is not a mechanism for the catalysis kinetics but that the linear extent of the attractive potential is largely responsible for the rate enhancement observed for the system SOD/superoxide.
Introduction The enzyme superoxide dismutase (SOD)is a dimer of overall charge 4e- with catalytic sites containing two copper atoms near the protein surface; these atoms participate directly in the reaction in which the superoxide radical 02-is oxidized to hydrogen peroxide and oxygen. The interest surrounding SOD/02- stems from the experimental observation' that the rate of ligand association is only an order of magnitude less than diffusion controlled. This is unusual since the surface area of the free metal is only 1/150thof the total enzyme surface. Furthermore, the binding rate decreases with increasing ionic strength;' consideration of the enzyme and substrate monopole charges alone would predict an increasing rate with increasing ionic strength. Koppeno12 has proposed that positively charged amino acids near the active site, coupled with the overall negative charge of the protein, may facilitate electrostatic steering of 02-into the reactive channel, thus maximizing encounters at the catalytic sites. This scenario (1) Cudd, A.; Fridovich, I. J . Eiol. Chem. 1982, 257, 11443. (2) Koppenol, W. Oxygen and Oxy-radicals in Chemistry and Biology; Rcdgers, M., Powers, E., Eds.; Academic: New York, 1981; p 671.
would also explain the observed ionic strength dependence. In an attempt to verify this prediction, a number of simulation studies3-' have evaluated the diffusional rate of association for SOD/superoxide by modeling the long-range electrostatic forces as an interplay between complex charge distributions, enzyme shape, and dielectric properties of the enzyme and solvent. One such model, employed by Allison et aL3 in their initial studies of the SOD system, used a spherical geometry with two 10' polar angle surface active sites and five judiciously placed point charges. This reduced charge distribution3 reproduced the repulsive monopole, (vanishing) dipole, and attractive quadrupole of the electrostatic potential due to the 76 charged groups de(3) Allison, s.;Ganti, G.; McCamrnon, J. J. Chem. Phys. 1986,89, 3899. (4) Head-Gordon, T.; Brooks, C. L., 111 J. Phys. Chem. 1987, 91, 3342. There was a typographical error in eq A9. The corrected equation appears in eq 7 of this paper. (5) Klapper, I.; Hagstrorn, R.; Fine, R.; Sharp, K.; Honig, B. Proteins 1986, I , 47. (6) Sharp, K.; Fine, R.; Schulten, K.; Honig, B. J . Phys. Chem. 1987, 91, 3624. (7) Sharp, K.; Fine, R.; Honig, B. Science (Washington,D.C.)1987, 236(4807), 1460.
0022-3654/89/2093-0490$01.50/0 0 1989 American Chemical Society
Diffusional Rates for Enzyme-Ligand Systems
The Journal of Physical Chemistry, Vol. 93, No. 1, 1989 491 diffusional ligand association for SOD. In the present report, we extend the simple model to address the question of dimensional reduction of the diffusion space and steering of the ligand into the active site. Combining these new results with those from earlier ~tudies,"~we provide insight into the interplay of enzyme shape and electrostatic fields on ligand association.
Methods and Models In this section we review the methsddogygJOused to determine the rate constant for the diffusional association reaction and describe the model of SOD3v4used in this work. All entries in parentheses define the value of a quantity used in this study. For more explicit detail concerning the methods and models, see ref 4, 9, and 10. Brownian Dynamics. The Ermak and McCammong algorithm for the relative motion of two Brownian particles is R,(t + At) = Rmo(t) + j3D,,lAtCFn" + Qmo(f) (1) n
W Figure 1. Superoxide dismutase model. The enzyme is modeled as a
sphere of radius 28.5 A with two surface active sites whose centers are defined by the i z-axis and a polar angle, 6 , of loo. The fixed, enzymatic point charges are denoted by x. R is the vector from the origin to the superoxide ligand, r is the vector from the origin to a fixed enzymatic charge, and 0 is the angle between R and c. termined from the 2-A resolution crystallographic structure.* The forces were evaluated assuming a continuous dielectric (CD), where the macroscopic dielectric constant of the protein interior is the same as that of the solvent. It was found that the rate of association of the monopole/quadrupole system was larger than the monopole alone, thus verifying the importance of positive charges near the reactive portion of the surface. However, the CD model was unable to reproduce the correct ionic strength dependence. One reason is that the calculated rate at zero salt concentration was less than the diffusion-controlled rate. A correct model should converge to a diffusion-controlled rate in the limit of infinite salt concentration. Furthermore, the model predicted an increasing rate with increasing ionic strength for salt concentrations up to 0.1 M, in direct disagreement with what was observed experimentally.' Subsequent simulation studies have included a dielectric discontinuity (DD) at the protein-solvent interface, both with and without the presence of salt exterior to the p r ~ t e i n . ~We - ~ have recently completed a study4 of the effects of four different analytic dielectric models, including the DD models, on the rate of ligand association using the simple spherical SOD system described in ref 3 (also see Figure 1). Other studiess-' have determined the electrostatic potential by solving the linearized Poisson-Boltzmann equation for the full, discrete representation of the protein, in order to analyze the effect of a more realistic enzyme shape and charge distribution, in addition to the dielectric discontinuity. The inclusion of the DD was found to dramatically increase the rate compared to the C D model, due to a focusing of field lines at the curved (but not necessarily spherical) boundary defining the low dielectric medium of the protein (eP = 2) and the high dielectric solvent (e, = 78-80) at zero ionic strength. These studies have also confirmed that the electrostatic potential determined from Debye-Hiickel theory, which incorporates the effect of salt exterior to the protein in addition to the dielectric discontinuity (DD-DH model), correctly predicts the qualitative features of the ionic strength dependence as well. The complementary nature of the DD-DH studiese7 now permits elucidation of the interplay of dielectrics, protein shape, and full charge distributions in determining the mechanism of
where R specifies the position of the Brownian particle m and the superscript degree denotes evaluation of the function at the previous time step. The quantities At, T, and j3 are the time step, temperature (298 K), and (kT)-', respectively. We have employed the variable time step algorithm previously described by Allison et aL3 Slip boundary conditions were used to define the relative diffusion constant (hydrodynamic interactions are neglected) Drel = (01 + a 2 ) / 4 ~ ~ 2 (2) where al and a2 are the radii of the spherical Brownian particles and 7 is the solvent viscosity (1 cP). The random force, Qmo(t), defines the stochastic nature of the process and physically represents the random displacements of the Brownian particle due to collisions with the surrounding solvent medium. Brownian motion theory is found to be a very good theoretical framework for systems in which the stochastic process can be modeled as a white noise distribution; the mean and variance of the Gaussian distribution used in this study are (Qi) = 0 (QiQj) = 2Dre1Af (3) Finally, the quantity Fno represents the systematic force, which in this work is the analytical derivative of the electrostatic potentials presented previously4 (see eq 7 below). Trajectory We use the trajectory analysis method of Northrup et al.,1° which allows the rate to be determined from simulations carried out in a finite domain of configuration space. In this method, trajectories are initiated on a sphere defined by R = b (200 A), where the flux of particles through that surface is isotropic. The rate, k(b), from infinity to this surface may be evaluated analytically." The full rate is then a product of k(b) times the probability, B,, that the particles that have reached R = b diffusing in an infinite domain will then react (the activation energy upon reaction is assumed to be negligible).
K = k(b)B, (4) To determine B,, one must simulate in an infinite domain. This is not computationally feasible; hence, trajectories are truncated when the ligand either (1) reaches the defined reactive site on the enzyme and reacts or (2) reaches an outer truncation sphere defined by R = q (400 A), where q > b. The probability B, will then depend on 6, the probability that the particles react from an initial separation R = b, and (1 - y), the probability that once the separation distance of the particles reaches R = q, the particles will escape to infinity. With use of probability branching arguments to account for the possibility that the ligand does not escape to infinity, but may recross R = q repeatedly, the final expression for B , islo B, = 6/(1 - (1 - 6 ) y ) (5) where y = k ( b ) / k ( q )and 6 is the ratio of successfully reactive (9) Ermak, D.; McCammon, J. J . Chem. Phys. 1978,69, 1352. (10) Northrup, S.; Allison, S.; McCammon, J. J . Chem. Phys. 1984, 80,
(8) Bernstein, F.; Koctzle, T.; Williams, G.; Meyer, E.; Brice, M.; Rogers, J.; Kennard, 0.;Shimanouchi, T.; Tasumi, M. J . Mol. Biol. 1977,112,535.
1517. (1 1) Debye, P. Trans. Electrochem. Soe. 1942, 82, 265.
Head-Gordon and Brooks 10 50
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Figure 2. Diffusional association rate versus ionic strength dependence on a log-log plot. The solid line denotes the rates calculated in ref 4, and the dashed line, the rates calculated in ref 5 and 6 . The rates are given in units of M-' s-I in this figure.
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(12) Allison, S.; Baquet, R.; McCammon, J. Biopolymers 1988, 27, 251.
0
Figure 3. Potential energy for the SOD dielectric discontinuity model at 0.0 M concentration. Potential energy contours for the 5 pint charge model' are plotted versus y and z, in units of angstroms. Each contour energy is normalized by kT,and each contour represents 0.2 kT.
trajectories to the total number of trajectories simulated. The quantity 6 is determined from the Brownian dynamics simulation. Superoxide DismutaselSuperoxide Model System. SOD is modeled as a sphere of radius 28.5 %I, with two surface active sites defined by a 10' polar angle centered about the z-axis (Figure 1) and using the reduced charge distribution3 described above. The superoxide radical was represented by a uniformly reactive sphere of radius 1.5 %I and charge 1e-. Simple Models: Influence of Enzyme Topology on Ligand Association The complementary ~tudies,"~which now exist for the SOD/superoxide system, provide a means of evaluating the usefulness of simple model representations of the components determining the electrostatic field: dielectric properties of the enzyme and solvent, enzyme shape (which formally defines the dielectric boundary), and the enzyme charge distribution. The assumption of a spherical enzyme shape allows analytical evaluation of the DD-DH electrostatic forces4 This idealization could be suspect because a change in the shape of the dielectric boundary would change the electrostatic field lines to some degree; in actuality, SOD has a nonsmooth surface and is roughly ellipsoidal in shape, with the active-site coppers buried in shallow clefts. Realistic representations of this shape were incorporated in the numerical studies by Sharp et aL5-' and more recently by Allison et a1.I2 and yielded results in qualitative agreement with those found using the idealized spherical shape! This is illustrated in Figure 2 where the calculated rates are plotted versus ionic strength on a log-log plot for the studies by Head-Gordon and Brooks4 and Honig and co-worker~.~-~ In this figure it is evident that similar qualitative trends are predicted by both models. In the following discussion, we analyze these similarities, and differences, in terms of the three components that each study has used to define the DD-DH electrostatic surface: charge distribution, dielectric properties, and enzyme topology. A reduced charge distribution that adequately reproduces the lowest moments of the multipole expansion is consistent with the coarse-grained, phenomenological description (Brownian motion) of the diffusional process; Le., at large, relative separations these long-range components of the electrostatic force dominate the diffusional association. The higher moments describe the electrostatic field close to the protein surface, the region where the Brownian description begins to break down and incorporation of atomic detail becomes necessary. The differences between the multipole expansions of the analytical4 and numerical studies5-'
-50
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Y Figure 4. Potential energy for the SOD continuous dielectric model at 0.0 M concentration. All details are given in the caption of Figure 3.
begin at fourth order; thus, these differences should be largely unimportant. A further explanation for the similarities between the DD-DH electrostatic surfaces of ref 4-7 is that the interplay between the charge distribution and the topology of the dielectric boundary is dominated by the former. In other words, the reduced charge distribution3used in the analytical DD-DH study4 has incorporated some gross aspects of "shape", and enclosing that distribution of charges within a low dielectric sphere, as opposed to a more complicated topology, does not qualitatively change the features of the electrostatic potential. This point is exemplified when the continuous dielectric model is compared, where a discussion of shape with respect to dielectric effects is meaningless, to the dielectric discontinuity case (Figures 3 and 4). The electrostatic potential generated by other relatively simple, low-dielectric geometries, such as the ellipsoidal topology of the SOD system, should be dominated largely by the charge distribution and, therefore, differ very little, qualitatively, from that of the sphere. For the case of more extreme geometries, such as clefts, further consideration should be given to the role of shape. The importance of cleft geometries for diffusion-controlled reactions has been d i s c ~ s s e d . ' ~ - ' The ~ work by Zauhar and (13) Samson, R.; Deutch, J. J . Chem. Phys. 1978, 68, 285.
Diffusional Rates for Enzyme-Ligand Systems Morgan14 is most relevant for the problem considered here. In this study,14 the three-dimensional polarization effect due to the juxtaposition of linear dielectric media is mapped onto a twodimensional representation, namely an appropriate distribution of induced polarization charge at the dielectric interface. The numerical problem of finding the electrostatic potential for an arbitrary geometry was then solved by finite elements for twodimensional (circular) model proteins with a cleft and a threedimensional study of the cleft-enzyme, lysozyme. The presence of the cleft geometry was found to increase the linear extent of the potential around the buried active site. When intermolecular forces are considered, the linear extent is a measure of the effective target radius R2-7v17
The Journal of Physical Chemistry, Vol. 93, No. 1, 1989 193 I
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where R, is proportional to the association rate. For attractive forces, R, is greater than the radius of the enzyme and the observed binding rate is found to be greater than the diffusion-controlled limit. The presence of the cleft was thought to enhance the linear extent of the target potential around the active site14even further than the attractive forces alone. The electrostatic DD-DH potential maps for SOD, calculated analytically4 with no cleft, are similar to those from numerical studies using the explicit protein shapes7*'* that includes a shallow cleft. The results found for the rate of association in ref 4-6 (Figure 2) are also indicative of the similarities between the surfaces, especially at high ionic strengths. However, at low ionic strengths, the rate of association found in the numerical study is larger than that found in the analytical case. There is room for possibility that the differences in association rate at low ionic strengths are due to the cleft geometry present in the numerical study. But, as was pointed out in our previous the truncation scheme used in the numerical study (a boundary condition that the potential go to zero at 95 A) tended to distort the potential so that the linear extent of the attractive potential around the active site was overemphasized, resulting in a larger binding rate. The artifact present in the numerical calculation probably explains the difference in the magnitudes of the association rate found in the two and, instead, we will focus on the similar shape of the two curves. To explain such a similarity, it seems likely that the active-site depression in SOD is too shallow for the predicted cleft enhan~ement.'~ It may also be. possible that the cleft feature is indirectly incorporated into the reduced charge d i s t r i b u t i ~ n ;a~ similar .~ effect was not ruled out in the lysozyme study.14 A further consideration is whether a cleft enhancement was indirectly incorporated into the analytical study4 by a fortuitous choice of the polar angle, 0 (see Figure I), defining the surface area of the active site; generally, the larger the surface area of the active site, the greater should be the association rate. In the case of SOD, the linear extent of the attractive forces (Figure 3) is significant for all polar angles less than 20°; choosing the value of the polar angle to be 5 O instead of IOo should result in little quantitative difference in the rates of association between the two cases. Electrostatic Field and Salt Effects on Ligand Association for
SOD The comparison of the DD-DH ~ t u d i e s ~on- ~SOD in the previous section have shown that the simple electrostatic models are adequate in qualitatively describing the ligand binding process of this system. In an effort to understand the role of the electrostatic potential in more detail, we modified the analytic DD-DH field4 of the native model3 of SOD. Recalling that the form of Zauhar, R.; Morgan, R. J . Mol. Eiol. 1985, 186, 815. (15) Warwicker, J.; Watson, H. J. Mol. Biol. 1982, 157, 671. (16) Chou, K.; Li, T.; Forsen, S.Eiophys. Chem. 1980, 12, 265. (17) Berg, 0.; yon Hippel, P. Annu. Rev.Eiophys. Eiophys. Chem. 1985, 14, 131. (14)
-100
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1 -100
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Y Figure 5. (a) Elimination of the repulsive potential for the DD model at 0.0 M salt concentration. All details are given in the caption of Figure 3. (b) Elimination of the attractive potential for the DD model at 0.0 M salt concentration. All details are given in the caption of Figure 3. TABLE I: Rates of Association for the Unmodified and Modified DD-DH Potentials for Three Different Salt Concentrations (All Rates Normalized by ka) rate of association (k/kd) ionic
strength, M
native
no repulsion
no attraction
0.00 0.10 0.30
1.34 f 0.03 1.43 f 0.08 1.25 f 0.05
1.30 f 0.01 1.37 f 0.05 1.28 f 0.01
0.99 f 0.05 0.94 f 0.02 0.93 i 0.02
the electrostatic potential for a single charge along the z-axis in the DD-DH model is4 qr"(2n
+ I)K,(KR) Pn(cos O)e-K(R-o)
\k=cR"+'[q(n + I)K,+,(Ka) + nK,(Ka)(t, n
- e,)]
(7)
where a is the radius of the protein, R is the distance between the origin and test particle, r is the distance between the origin and the single charge embedded in the sphere, and 0 is the angle made by the vectors R and r. The functions P,,(cos 0) and K,(KR) are the Legendre polynomials and modified Bessel functions, respectively. The full electrostatic potential for the anisotropic reduced charge distribution3 is given by a superposition of the potential due to each charged site, following the appropriate rotation of axes. We have modified the electrostatic potential by (i) eliminating the repulsive potential outside the protein, Le., setting all values
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The Journal of Physical Chemistry, Vol. 93, No. 1, 1989
Head-Gordon and Brooks
TABLE 11: Calculated Surface Area and Linear Extent for Several Energy Levels at Three Different Salt Concentrations linear extent, A surface area, A2 energy, kT 0.0 M 0.1 M 0.3 M 0.0 M 0.1 M 1960.2 12.3 14.1 -0.3 1352.7 3236.6 1073.6 11.6 13.4 -0.4 1090.3 2278.5 703.3 11.1 12.3 -0.5 846.6 1264.2 536.6 10.6 12.0 -0.6 604.1 947.2 250.8 9.9 11.3 -0.7 462.7 656.3 0.0 9.2 10.4 -0.8 285.5 429.4
of the potential calculated to be greater than 0 kT to zero, and (ii) eliminating the attractive region. The resulting potentials are shown in Figure 5a,b. Table I presents the normalized association rate calculated from the unmodified and modified potentials at salt concentrations of 0.0, 0.1, and 0.3 M. It is evident for all salt concentrations that the elimination of the repulsive potential has no effect on the rate. For the repulsive lobes to be important in reducing the dimensional diffusion space, the ligand must encounter a very large barrier (>2 kT) at very large distances from the protein surface. If such a scenario were to hold in the case of SOD, one would expect the repulsive lobes to reduce the dimension of diffusion space to something significantly less than three, thereby “steering” the ligand into the reaction sink to increase the rate of association. In order for the repulsive potential to play an important role in the dimensional reduction of diffuson space, this barrier would have to spatially extend much farther than most realistic potentials would permit. For the case of SOD, the repulsive potential extends only 5-10 A (Figures 3 and 5b) from the protein surface, an extent that is too small to be effectual in biasing the ligand‘s trajectory. Thus, we observe only a small effect on removal of these regions of the potential. Elimination of the attractive channel into the reactive site, on the other hand, reduces the rate to essentially that of diffusion controlled. The attractive potential found for realistic potentials is capable of reducing the dimensional diffusion space by enhancing the linear extent of the target. An analysis similar to that used in the repulsive case above would establish a critical energetic depth and spatial extent for this attractive region beyond which the ligand cannot escape rea~tion.~’In an effort to quantify the effective target site for the analytic DD-DH model, we have numerically integrated the surface area and evaluated, approximately, the linear extent at several levels of attractive energy contour for salt concentrations of 0.0, 0.1, and 0.3 M (Table 11) and compared these with the trend of increasing association rate 0.3 M < 0.0 M < 0.1 M. We have chosen to use both the surface area and linear extent to measure the correlation of attractive energy with rate. The former is the measure appropriate for diffusion to a plane surface, while the latter is appropriate for diffusion to a spherical surface. The critical depth is found to lie somewhere between -0.4 and -1.0 kT. This makes physical sense because the probability of escape should be roughly proportional to the Boltzmann factor, which becomes increasingly small for depths of order kT. Thus, the distinguishing electrostatic feature between the different dielectric models (and between salt concentrations) developed for SOD is the ability to further extend
0.3 M 13.4 12.3 11.3 10.2 8.5 0.0
the attractive target sink that traps the ligand energetically. This point has been suggested in earlier work by Sharp et al.>’ Again, the repulsive potential typically found for protein systems is incapable of reducing the dimensional diffusion space and appears not to be the mechanism of diffusional association of SOD/ superoxide in free s~lution.~-~J’
Conclusion The complementary studies” have provided a means of assessing the reliability of models simply representing dielectric properties, protein shape, and charge distributions for determining the rate of enzyme-ligand binding. The simplified geometries and charge distributions appear to be adequate in determining the qualitative character of the diffusional encounter rate relative to the dielectric model used. However, a more quantitative description of the electrostatic field and diffusional association will most certainly require a greater level of detail in describing the enzyme topology. Simple representations, which permit analytic electrostatic potential solutions, have proven to be adequate in the qualitative understanding of the mechanism of diffusional association for those enzymes whose active sites are not deeply buried. For enzyme systems with deeply recessed active sites, these simple models will most likely fail to qualitatively describe the binding mechanism. Thus, it appears that the use of numerical solutions should be important for the qualitative description of enzyme-ligand association for deep-cleft geometries and for quantitative descriptions, in general, where inclusion of a complex geometry is required. Simple models have proven to be sufficient for elucidating the mechanism of diffusional association in the system SOD/superoxide. Using these simple models, we have presented new evidence that the repulsive electrostatic features generated by the analytical DD-DH model of SOD4 are not sufficiently long ranged to “steer” the superoxide ligand into the active site and, thereby, reduce its three-dimensional diffusion space. We anticipate that most enzyme systems will not have sufficiently long-range repulsion for “steering” to be important. Instead, the sole electrostatic feature responsible for the observed rate dependence on salt is the linear extent of the attractive target potential around the active site, where the ligand is trapped energetic all^.^-^ Acknowledgment. We thank the N I H for support through Grant No. GM37554-02 and the Pittsburgh Supercomputing Center for use of computational resources. Registry No. SOD,9054-89-1; 02-, 11062-77-4.
