PROTEINS: Structure, Function, and Genetics 31:406–416 (1998) Species Dependence of Enzyme-Substrate Encounter Rates for Triose Phosphate Isomerases Rebecca C. Wade,1* Razif R. Gabdoulline,1 and Brock A. Luty2 Molecular Biology Laboratory, Heidelberg, Germany 2Agouron Pharmaceuticals, San Diego, California 1European ABSTRACT Triose phosphate isomerase (TIM) is a diffusion-controlled enzyme whose rate is limited by the diffusional encounter of the negatively charged substrate glyceraldehyde 3-phosphate (GAP) with the homodimeric enzyme’s active sites. Translational and orientational steering of GAP toward the active sites by the electrostatic field of chicken muscle TIM has been observed in previous Brownian dynamics (BD) simulations. Here we report simulations of the association of GAP with TIMs from four species with net charges at pH 7 varying from -12e to 112e. Computed secondorder rate constants are in good agreement with experimental data. The BD simulations and computation of average Boltzmann factors of substrate–protein interaction energies show that the protein electrostatic potential enhances the rates for all the enzymes. There is much less variation in the computed rates than might be expected on the basis of the net charges. Comparison of the electrostatic potentials by means of similarity indices shows that this is due to conservation of the local electrostatic potentials around the active sites which are the primary determinants of electrostatic steering of the substrate. Proteins 31:406–416, 1998. r 1998 Wiley-Liss, Inc. Key words: electrostatics; Brownian dynamics; triose phosphate isomerase; diffusion-control; similarity index; rate enhancement INTRODUCTION Triose phosphate isomerase (TIM) has been described as a ‘‘perfect enzyme’’1,2 because it is so well optimized that its rate-determining step in the environment in which it functions is not a chemical step, but rather the diffusional association of substrate and enzyme.3 It is a particularly well-characterized example of a diffusion-controlled enzyme. These enzymes have second-order rate constants (kcat/Km) that are viscosity-dependent and are typically very high (,108–109 M-1 s-1) at physiological ionic strength and viscosity. Other enzymes whose rates are dependent, to varying extents, on diffusional encounter with their substrates include superoxide dismutase,4 r 1998 WILEY-LISS, INC. acetylcholinesterase,5 alkaline phosphatase,6 βlactamase I,7,8 glycoxalase II,9 phosphotriesterase,10 and adenosine deaminase.11 A common feature of the rates of diffusioncontrolled enzymes is dependence on ionic strength,4 which indicates the important influence of electrostatic interactions on enzyme-substrate diffusional association. Further evidence of the influence of electrostatic interactions is provided by site-directed mutagenesis. A particularly striking demonstration of this is the mutation of two residues in superoxide dismutase designed with the aid of Brownian dynamics (BD) simulations to enhance electrostatic steering of the substrate to the active sites which resulted in an increased rate constant12 and thus a ‘‘superperfect’’ enzyme.13 BD simulations on several systems show that the electrostatic field of an enzyme generally increases the rate over that for an uncharged model system without electrostatic interactions by 1–2 orders of magnitude at physiological ionic strength14 (for review, see Ref. 15). For TIM, BD simulations of the chicken muscle enzyme show that GAP is translationally and rotationally steered toward the enzyme’s active sites.16,17 The rotational steering is a shorter-range effect and is only detectable when the substrate has diffused within about 5 Å of the position in the active site where it is assumed to react. As this enzyme and the substrate are both negatively charged at pH 7, the electrostatic attraction between them must result from the nonuniformity of the charge distribution over the enzyme and the enzyme’s irregular shape, which can distort its potential according to the curvature of the boundary between low-dielectric protein and high-dielectric solvent. The aim of the present study was to examine which features of the electrostatic potential of TIM give rise to electrostatic steering of the substrate to the active sites and determine the value of the second-order rate con- Abbreviations: BD, Brownian dynamics; GAP, D-Glyceraldehyde 3-phosphate; PGH, phosphoglycolohydroxamate; rmsd, root mean square deviation; TIM, triose phosphate isomerase. *Correspondence to: Rebecca C. Wade, European Molecular Biology Laboratory, Meyerhofstr.1, 69117 Heidelberg, Germany. E-mail: firstname.lastname@example.org Received 10 October 1997; Accepted 9 December 1997 ENZYME-SUBSTRATE ENCOUNTER RATES stant. One approach would be site-directed mutagenesis of TIM. However, for TIM this is complicated by the presence of particularly flexible and highly conserved loops that close over the active sites when substrate binds.18 Although BD simulations suggest that the motions of these loops have little effect on the rate at which substrate reaches the active sites,19,20 loop mutations can alter the enzyme kinetics by destabilizing the enediol phosphate intermediate, thus reducing the efficiency of the chemical steps of the reaction.21 In addition, mutations far from the active site at the dimer interface can alter the dimer stability, which in turn affects the stability of the active site and thus enzymatic catalysis.22 Thus, we have taken the alternative approach of analyzing different species of TIMs by choosing four enzymes which, although about 50% identical in sequence to each other,23,24 have net charges at pH 7 ranging from -12e to 112e. The crystal structures of each of these enzymes are available in an unliganded form with the active site loop open and their rate constants have been measured. Despite the wide variation in net charge, the rates of these enzymes show only modest differences. We show that the rates can be reproduced by BD simulation and, by analyzing electrostatic potentials with similarity indices, that electrostatic steering is due to conservation of the electrostatic potential in the vicinity of the active sites. MATERIALS AND METHODS Protein Structures The crystal structures of four TIMs were used in which at least one subunit was in the unliganded form with the flexible loop that closes over the active site on substrate binding in an open conformation. With the exception of chicken muscle TIM, all the structures are available in the Brookhaven Protein Data Bank (PDB).25 Their identifier codes are: 1tre for the Escherichia coli TIM,24 5tim for the Trypanosoma brucei brucei TIM,26 and 1ypi for the Saccharomyces cerevisiae TIM.27 A 2.4 Å resolution coordinate set for chicken muscle TIM refined to an R-factor of 0.17 was kindly provided by Dr. P. Artymiuk. Note that this differs slightly in amino acid sequence from the older coordinate set in the PDB (1tim) used for our previous studies,20 which has a net charge of zero. The residue numbering scheme for chicken muscle TIM is used throughout this article. Missing nonhydrogen atoms and polar hydrogen atoms were added to each of the four crystal structures using version 6.03 of the SYBYL molecular modeling software package (Tripos Assn., St. Louis, MO). Their positions were optimized using 3,000 steps of the Powell minimization with the AMBER united atom force field28 in the presence of the crystallographically observed water molecules represented by the TIP3P model.29 The pKas of the ionizable sites were calculated using a continuum dielec- 407 tric approach30 and the same procedure was used to assign the tautomeric states of histidine residues. At pH 7, all titratable residues were found to be in their usual protonation state. Assignments of histidine tautomers were consistent with that expected on the basis of geometric analysis of potential hydrogen bonds and with measurements of histidine pKas.31–33 TIM is a dimer composed of two identical monomers. In the crystal structures, the conformations of the monomers in each dimer differ due to factors such as crystal contacts. Therefore, to model completely symmetric TIM dimers for the BD simulations, each TIM was built from one subunit which was duplicated and the copy superimposed on the position of the other subunit in the crystal structure. The superposition was performed using the INSIGHT software package (Molecular Simulations, San Diego, CA) for the Ca atoms of residues 45–55, 80–85, 107–118, and 241–248 (as an example, the rmsd of the superimposed atoms is 0.16 Å for T. brucei TIM). The subunits used for building the symmetric TIMs were: chicken muscle, B; yeast, B; E. coli, A; T. brucei, A. They were chosen as being least affected by crystal contacts and having the flexible loop in an open conformation. The interface contacts were checked in the constructed dimers and no unfavorable van der Waals contacts were found. In order to define reaction criteria to determine when the substrate reaches the active site during the BD simulations, the PGH inhibitor was docked in the active sites of each protein. Distances from the positions of its phosphate atom (P0) and its C1 carbon (which are 3.93 Å apart) were used to define the criteria for docking a two-subunit model of the substrate in the simulations. The A subunit of the crystal structure (7tim) of yeast TIM with PGH bound34 was used as a template for docking PGH into the active sites of the unliganded TIMs. This was done by superimposing Ca, Cβ, and Cg of His95, Ca of Glu165, C and N of Ser211, and all nonhydrogen atoms of Gly232 and Gly233 (14 atoms per monomer) of each of the four unliganded TIMs onto the corresponding atoms of the yeast TIM-PGH complex, and then adding the PGH coordinates to the unliganded TIM. The superimposed atoms make close contacts with the bound inhibitor and adopt similar positions in the different structures. The resultant symmetric dimers with docked inhibitor coordinates were overlaid on the chicken muscle symmetric dimer by superimposing the active site atoms used for docking the inhibitor (28 atoms for the two active sites) on the corresponding atoms in the chicken muscle enzyme. The rmsd of these atoms from their positions in the chicken muscle TIM is 0.4 Å for yeast TIM, 0.8 Å for E. coli TIM, and 0.4 Å for T. brucei TIM. Although no energy minimization of the atoms observed crystallographically in the four structures was performed, the ‘‘open’’ positions of the active site loops in the four 408 R.C. WADE ET AL. TIMs are similar. The maximum deviation of a loop Ca atom from its position in the chicken muscle structure is less than 2 Å for yeast and T. brucei TIMs. For E. coli TIM, the maximum deviation is 4 Å, but this is compensated by a corresponding shift of the other side of the active site leaving the active site opening of similar size. (The deviation of the Ca of residue 211 is 2.4 Å, as compared to 0.7 Å for the other two TIMs). Electrostatic Calculations Electrostatic potentials were calculated by numerically solving the finite difference linearized Poisson Boltzmann equation using an incomplete Cholesky preconditioned conjugate gradient method, as implemented in the University of Houston Brownian Dynamics (UHBD) program, version 4.1.35,36 The OPLS nonbonded parameter set37 was used to assign partial atomic charges and radii. The radii of hydrogen atoms were set to zero to give a solute model consistent with the spherical probe model of a water molecule used to define solvent-accessible surfaces.38 Other atomic radii were multiplied by 1.122 (21⁄6) to correspond to the minimum in the Lennard-Jones potential.39,40 The dielectric constant of the solvent was set at 78.5. The dielectric constant of the solute was set to 2.0. The solvent–solute dielectric boundary was determined from the protein 3D structure by the method of Shrake and Rupley41 using a 1.0 Å radius rolling probe. This radius gave the best correspondence between buried volumes of solvent dielectric and the positions of internal ordered water molecules.23,26 Dielectric boundary smoothing42 was used. The ionic strength of the bulk solvent was set to 100 mM and the solute was surrounded by a 2 Å-thick Stern layer. The temperature was set to 300K. The electrostatic potentials were calculated using cubic grids with 150 points along each axis. For each protein, the electrostatic potential was first calculated with a coarse grid centered on the protein with a grid spacing of 2.5 Å. The potential at the boundary of this grid was calculated analytically by treating each charged atom as an independent ion-excluding sphere in a high dielectric medium. A second electrostatic potential grid surrounding the whole protein was then calculated using a grid spacing of 0.8 Å and boundary potential values interpolated from the coarse grid. Finally, two ‘‘high resolution’’ electrostatic potential grids were calculated for the regions surrounding the two active sites using grids with spacings of 0.2 Å centered on the phosphate atoms (P0) of the docked inhibitors (see Fig. 1). The latter three grids were used for the BD simulations. The use of three grids rather than just one grid over the whole protein improves convergence in the BD simulations, not only because the electrostatic description is improved but also because the protein surface is smoothed, thus improving the treatment of exclusion forces. Fig. 1. Diagram showing the electrostatic potential grids computed and used for the BD simulations. A schematic section through a TIM dimer is shown with the dumbbell model of the GAP substrate docked in the active sites with the phosphate moiety represented by the filled circle. Three cubic potential grids were used for the BD simulations: one encompassing and centered on the whole protein with a grid spacing of 0.8 Å and an edge length 120 Å, and two smaller grids over the active sites with grid spacings of 0.2 Å and an edge length of 30 Å centered on the phosphate moieties of the docked substrates. A further electrostatic grid with edge length of 375 Å was used to derive values of the potential at the boundary of the enclosed grid with 0.8 Å grid spacing. BD Simulations Simulations were carried out with a modified version of the University of Houston Brownian Dynamics (UHBD) program, version 435,36 on a Silicon Graphics Power Challenge Computer. As in our previous simulations,17,20 the substrate was modeled as a dumbbell consisting of two subunits of radius 2.0 Å interacting hydrodynamically via the modified Oseen tensor with stick boundary conditions. They are connected by a pseudobond that is constrained using a modified SHAKE algorithm17 to a length of 4.0 Å with a tolerance of 0.01 Å. Charges of -0.3e and -1.7e were assigned to the dumbbell subunits to represent the glyceraldehyde and phosphate moieties, respectively, of GAP. The protein was treated as rigid and its excluded volume was defined using the same atomic radii as were used for the electrostatic calculations except that the radii of the atoms in 11 residues (11, 13, 95, 165, 169, 170, 171, 211, 212, 232, 233) in the active site were reduced to two-thirds of their normal values in order to implicitly mimic the mobility of the active site on ligand binding. The diffusion constant of the protein was calculated with stick boundary conditions assuming that the protein could be considered as a sphere with a 409 ENZYME-SUBSTRATE ENCOUNTER RATES hydrodynamic radius of 40 Å, which corresponds to about half of the largest dimension of the enzyme. The electrostatic forces on the substrate were calculated from the electrostatic potential grids. When a substrate subunit approached within 10 Å of either of the P0 positions in the active sites, the forces on it were computed from the high-resolution grid calculated for the active site region. When the substrate subunit then diffused more than 14 Å from the nearest P0 position, forces were again calculated with the larger focused grid covering the whole of the protein. A variable timestep of 1 ps was used when the center of the substrate was within 60 Å of the center of the protein and 5 ps when it was further away. At the start of each trajectory, the center of geometry of the substrate was positioned 60 Å from the center of the protein on the ‘‘b-surface.’’ At this distance, the average value of the potential in 0.1 M salt for all four TIMs studied is less than or equal to 0.01 kcal/mol/e and its fluctuation (difference between maximum and minimum values) is less than 0.08 kcal/mol/e. The simulation was continued until the substrate reached either of the active sites or diffused away from the protein to the ‘‘m-surface’’ at a distance of 80 Å from the center of the protein. At the m-surface, the probability of escape from the influence of the protein was calculated using the Numerov algorithm43 and the trajectory was accordingly terminated or continued with the substrate repositioned at an appropriate location on the bsurface in a random orientation. The distance between the b-surface and the m-surface was chosen to be greater than the distance the substrate travels in its rotational relaxation time. The occurrence of a reaction was monitored by means of reaction criteria. Loose reaction criteria require only one of the substrate subunits to approach within a specified distance of the P0 positions and distances of 5–9 Å were tested for this criterion. Tight reaction criteria require the phosphate moiety of the substrate to approach within a specified distance of the P0 position and, simultaneously, for the glyceraldehyde moiety of the substrate to approach within the same distance of the C1 position. Distances of 1–3 Å were tested for this criterion. For each set of conditions, 20,000 trajectories were generated for the charged substrate and 100,000 trajectories were generated for the substrate without charges. Each was started with the substrate positioned at a randomly assigned position and orientation on the b-surface. Trajectories were stopped if they exceeded 2,000,000 ps or 200,000 steps; the fraction of trajectories stopped was insignificant. Model System Simulations BD simulations were also performed for GAP diffusing toward a model target consisting of a low dielectric sphere with a point charge at its center and two reaction sites at opposite points on its surface (180 degrees apart). Rates were monitored for two reaction criteria (loose and tight) which required either subunit of GAP to approach within a specified distance of one of the reaction sites. As no orientational requirement was applied, the rates obtained with the tight criterion were halved for comparison with the rates computed for the protein. The sphere was assigned a radius of 25 Å and point charges corresponding to the net charges on the four TIMs. Boltzmann Factor Estimation of Electrostatic Rate Enhancement The electrostatic rate enhancement was computed as proposed by Zhou et al.14,44 as ,exp(-U/kbT). where U is the interaction energy, kb is Boltzmann’s constant, T is temperature, and the average is over the region of the active site defined by the reaction criteria. The interaction energy was computed at positions of the substrate sampled within the active sites as defined by the same reaction criteria as used for the BD simulations. Sampling was performed systematically by placing the center of the substrate dumbbell at each point on the electrostatic potential grid in the active sites and computing the interaction energy at 3,000 randomly chosen orientations. Calculation of Similarity Indices Electrostatic potentials can be compared quantitatively by calculating similarity indices.45 Similarity indices have been developed and are usually applied to the comparison of small molecules.46–48 They are, however, also useful for the analysis of macromolecules, as Demchuk et al.49 showed when they examined the self-similarity of one protein and the similarity and optimal orientation for superposition of two proteins. The Hodgkin index50 is commonly used to measure the similarity of two molecular potentials. It detects differences in sign, magnitude, and spatial behavior in the potentials, and is given by: SIH 5 2(f1, f2)/(00f1002 1 00f2002). (1) The product (f1,f2) may be defined in two different ways to compare molecular potentials on grids of points. To use the index to provide a global measure of the overall similarity of the full potential fields of two molecules, it is defined as the scalar product of the two grids, i.e., the sum of products of values of electrostatic potentials on all grid points (i,j,k): 2· SIH 5 o f (i, j, k)f (i, j, k) 1 2 ijk o (f (i, j, k)f (i, j, k) 1 f (i, j, k)f (i, j, k)) 1 ijk 1 2 2 . (2) 410 R.C. WADE ET AL. To use the index to describe how the similarity of the potentials varies with position, (f1,f2) is defined as the product of two potential values at a given point (i,j,k) in space: SIH 5 SIH(i, j, k) 5 2f1(i, j, k)f2(i, j, k) f1(i, j, k)f1(i, j, k) 1 f2(i, j, k)f2(i, j, k) . (3) In this article, we will adopt this latter definition in which the similarity index is a function of coordinates because this enables the regions where two potentials are most similar or dissimilar to be identified. In this study, we must compare four electrostatic potentials and, thus, we extended the definition of the Hodgkin index, used to compare two potentials, to compare any number, N, electrostatic potentials. Given the electrostatic potentials fl, l 5 1,2,...,N, we constructed a similarity measure, SI, to satisfy the essential properties of similarity indices (either for the SI at a point or for the SI of a grid volume): 1) SI 5 11, when the potentials are all identical: fl 5 fm for all l Þ m, 2) SI 5 0, when the potentials are mutually orthogonal: (fl,fm) 5 0 for all l Þ m. 3) SI 5 -1, when there are two opposite potentials: fl 5 -fm, for l Þ m and N 5 2 only. Thus, SI 5 1 N21 · o (f , f )9 o00f 00 . l m 2 n lÞm (4) n This index measures the sum of the squared differences between the functions compared: o 00f 2 f l m00 2 o00f 00 5 2(N 2 1) · n lÞm 2 · (1 2 SI). (5) n An alternative representation of the SI is SI 5 1 2 N N21 · o 00f n n 9o 2 f002 00fn002 (6) n where f is an average over N potentials. This representation shows that for small deviations of fl from f, the decrease in the SI from its maximum (51) is proportional to the square of the deviations. For example, an SI of 85% for four potentials corresponds approximately to an average deviation of 33.5% from the mean potential f. Similarity indices were computed (using software written for the purpose) for all points on the electrostatic potential grids of the four TIM proteins outside their combined van der Waals volume as defined with atomic radii set at twice their normal values. TABLE I. Electrostatic Properties and Measured Rates for Four Triose Phosphate Isomerases E. coli Yeast Chicken muscle T. brucei pIa n.d. 5.4 n.d. Net charge calculated at neutral pH (e) 212 26 22 kcat/Km (108 M21s21 )b 2.3 1.0–5.4 2.4–3.7 No. of measurements 1 6 3 (kcat/Km) relc 1.0 0.4–2.3 1.0–1.6 9.8 112 6.6–8.4 2 2.9–3.6 aMeasurements given in Reference 56; n.d.: not determined. was determined at 0.1 M ionic strength, 25–30°C, pH 7.4–7.9. The values given are corrected for the hydration equilibrium of the substrate59 and are given per monomeric subunit. Literature sources: E. coli: 60; yeast: 52, 61–66; chicken muscle: 59, 67, 68; T. brucei: 52, 69. cRates relative to that for E. coli TIM. bk /K cat m RESULTS AND DISCUSSION Comparison of BD Simulations to Experimental Data Experimental rates are given in Table I. Although the kcat and kM values for the four TIMs studied here were measured by a number of authors, they were all measured under ‘‘standard’’ (0.1M, pH 7.6, 30°C)51 or similar (0.05–0.1M, pH 7.0-7.9, 25–30°C) conditions. The measured values depend on the source of the enzyme, e.g., yeast TIM can be extracted from Bakers’ yeast or cloned in E. coli. While the different measured rate constants (kcat/kM) are rather similar for each enzyme, there are larger differences (up to approximately a factor of 5) for the individual kcat and kM values.52 These differences tend to compensate each other, resulting in less variation in kcat/kM. The computed rates are shown in Tables II and III. For ease of comparison, rates are also given relative to those computed for E. coli TIM, which has the most negative net charge. After testing several distances for the reaction criteria, the results are presented for one distance for each of the criteria: 9 Å for the loose criterion and 2 Å for the tight criterion. The reaction criteria are equivalent to those used in our previous work20 when correction for the different position of the docked substrate is made. The tight criterion results in rates in reasonable agreement with those observed experimentally. For the four TIMs, they range from 0.98 to 2.3 times the fastest rate measured for each TIM (see Table II). Thus, with this reaction criterion there is a tendency to overestimate the rate. This can be expected for a number of reasons: 1) GAP is assumed to have a charge of -2e and changes in protonation state are neglected. 2) GAP is treated with a simplified rigid model. Any internal energy barriers that may need to be overcome for the substrate to adopt its binding conformation are neglected. 411 ENZYME-SUBSTRATE ENCOUNTER RATES TABLE II. Diffusion-Controlled Second-Order Rate Constants Calculated From Brownian Dynamics Simulations for Four Triose Phosphate Isomerases Substrate Reaction charge(e) criteriona Net charge calculated at neutral pH(e) Exptal kcat/Kmb Computed kcat/Kmc 22 22 0 0 Electrostatic rate enhancementd loose tight loose tight loose tight E. coli Yeast Chicken muscle 212 26 22 2.3 1.0 25.1 1.00 5.3 1.0 16.6 0.66 0.05 0.010 1.5 105 1.025.4 0.422.3 2.423.7 1.021.6 29.4 1.17 44.2 1.76 5.3 1.0 6.9 1.3 16.4 0.65 15.0 0.60 0.07 0.014 0.016 0.003 1.8 2.9 76 431 T. brucei 112 6.628.4 2.923.6 57.0 2.27 10.6 2.0 15.5 0.62 0.01 0.002 3.7 883 aThe reaction criteria are: Loose: either substrate subunit approaches within 9 Å of the phosphate moiety of the docked substrate; Tight: both of the substrate subunits simultaneously approach within 2 Å of their corresponding moieties in the docked substrate. bMeasured rates are given in 108 M21 s21 followed by rates relative to that measured for E. coli TIM in italics; see Table I. cCalculated rates are given in 108 M21 s21 followed, in italics, by rates relative to those computed for E. coli with a charged substrate for the particular reaction criterion. The standard deviations in the relative rate constants are about 0.01 for the loose reaction criterion and 0.1 for the tight reaction criterion. dThe electrostatic rate enhancement is given as the ratio of the rates computed with charged and uncharged substrate. Standard deviations are on the order of 1022 for the loose reaction criterion and 102 for the tight reaction criterion. TABLE III. Diffusion-Controlled Rate Constants Calculated From Brownian Dynamics Simulations for a Model System Representing Some of the Features of the TIM Dimers* Reaction criterion 212 Loose Tight 24.9 0.99 0.21 0.04 Charge(e): 26 22 32.6 1.30 0.42 0.08 37.9 1.51 0.66 0.12 112 67.8 2.70 3.15 0.59 *The model system consists of a 25 Å radius low-dielectric sphere with a point charge at the center and two reaction sites on the surface 180° apart. The same model of the substrate was used as for the all-atom protein simulations. Rates are given in 108 M21 s21 followed, in italics, by rates relative to those computed for E. coli. TIM with the real all-atom model and a charged substrate with the same reaction criterion. The standard deviations in the relative rate constants are about 0.01 for the loose reaction criterion and 0.07 for the tight reaction criterion. 3) Protein motions are not accounted for explicitly. In particular, the flexible loop that closes over the active site is kept in an open position. However, previous theoretical19 and experimental53 analyses have shown that any gating of the active site by the flexible loop would have little effect on the rate of substrate binding in the active site. 4) Hydrodynamic interactions between GAP and TIM are neglected but can be expected to have a small effect on the computed rates. While hydrodynamic interactions will tend to lower the rate of translational diffusion of the substrate toward its binding site, hydrodynamic torques will favor binding of the substrate in a binding site of complementary shape. Studies with model systems indicate that for enzyme–substrate interactions at physiological ionic strength, the effects of these two hydrodynamic interactions largely can- cel out, each individually causing an approximately 15% difference in rate.54,55 5) The linearized Poisson-Boltzmann equation is used. 6) The choice of atomic radii also has an effect on the computed rates, and use of radii that are unscaled except in the active site for the BD simulations resulted in rates closer to the experimental values (ranging from 0.5–1.3 times the fastest rate measured for each TIM). Electrostatic Steering BD simulations Rates were computed with both a charged and an uncharged substrate. These calculations show that for all the TIMs the enzyme’s electrostatic field steers the substrate toward the active sites and increases the rate constants computed. This is the case for both loose and tight reaction criteria, although the effect of electrostatic steering is much greater for the tight reaction criterion. Note that electrostatic steering is present even for the E. coli enzyme, which is the enzyme with the largest net charge of the same sign as the charge on the substrate. In the absence of electrostatic interactions, the association rates are determined by the shapes and sizes of the diffusing molecules. The probability of reaction with tight criteria during the trajectories generated without charges on the substrate is very low and, consequently, even though five times as many trajectories were generated with uncharged as with charged substrate, rates are determined less accurately than for trajectories with the charged substrate. Differences in the rates with uncharged substrate for the four enzymes are due to small 412 R.C. WADE ET AL. Fig. 2. Electrostatic potential contours superimposed on ribbon representations (yellow) of triose phosphate isomerases from four species: (A) E. coli, (B) yeast, (C) chicken muscle, and (D) T. brucei. Potentials are contoured at -0.4 (red) and 10.4 (blue) kcal/mol/e. The pink spheres in the two active sites show the docked positions of the dumbbell model of the substrate used in the BD simulations. differences in the shape of the active site due to small backbone displacements and sidechain rotations. If flexibility were to be fully accounted for, it is quite possible that these differences would diminish or even disappear. These uncertainties in the calculation of the rates for uncharged substrate mean that electrostatic rate enhancements for tight reaction criteria can only be estimated roughly. They are computed as ,100–900-fold, but if the rates with uncharged substrate were to be the same for all enzymes, they would be ,150–300-fold. In both cases, the rate for T. brucei TIM shows the greatest rate enhancement, indicating some influence of net charge on the rate. The presence of electrostatic steering regardless of the sign of the net charge on the enzyme shows, however, that while the magnitude of the enzyme’s net charge may influence the association rate, perhaps more important for substrate steering is how the charge is distributed over the enzyme. To distinguish between the effects on association rates of a monopole point charge and a dispersed charge distri- bution, simulations were carried out for a model system, as described in Table III. The association between the GAP dumbbell and a target spherical molecule with a point charge at the center and two reactive patches on the surface on opposite sides of the sphere was simulated. The radius of the model target was assigned after preliminary calculations with radii from 20 to 40 Å showed that 25 Å produced the best agreement between the computed rates for the sphere with a -12e charge with the loose reaction criterion and for E. coli TIM with the same reaction criterion. For the loose reaction criterion, the relative rates for the four model systems with the same net charges as the four TIMs studied are remarkably similar to those computed for the all-atom, all-charge TIM models. These relative rates are also in good agreement with the relative rates observed experimentally, although they are about an order of magnitude greater than the experimental rates. Thus the relative rate of diffusion up to a distance of 9 Å without any restrictions on substrate orientation can be mimicked by rela- ENZYME-SUBSTRATE ENCOUNTER RATES 413 Fig. 3. (A) The average electrostatic potential of four triose phosphate isomerases from different species (chicken muscle, E. coli, T. brucei, yeast). Contours are at -0.4 (solid) and 10.4 (line) kcal/mol/e. (B) The similarity index multiplied by the sign of the potential (contoured at 0.85, - solid, 1 line) for the four proteins. Regions of the potentials enclosed in the contours are the most similar. The twofold symmetry of the dimeric proteins is readily apparent. A large region of similarity over each substrate binding site is clearly discernible, within which the potential is positive for all four proteins and can attract the negatively charged substrate. The pincer-like regions correspond to small negative values of the potentials and the other smaller regions correlate with the locations of conserved charged residues. (C) The regions with conserved sign for the electrostatic potentials. Blue regions enclose the points where electrostatic potentials from all four TIMs are positive. In red regions, the electrostatic potentials of all four TIMs are negative. tively nonspecific interactions with the net charges of the molecules. It is important to note though that the rates for the negatively charged model spheres are slower than that for an uncharged sphere (,40 3 108 M-1s-1) due to electrostatic repulsion. The opposite is the case for the all-atom TIMs with the loose reaction criterion, due to electrostatic attraction. For the tight reaction criteria, the rates computed with the model system are much lower than those computed for the real system and measured experimentally. The relative rates for the four model systems also differ from those computed for the all-atom, all-charge TIM models and from those observed experimentally. The ratio of the rates of the E. coli and T. brucei enzymes is computed to be 2.0 and experimentally observed to be 2.9–3.6. For the model systems, the ratio of the rates for the target spheres with -12e and 112e charges is 15. These differences show that the enzyme’s charge distribution is such that the four TIMs appear much more similar to each other at short substrate-active site distances than their net charges would suggest. The electrostatic potential must thus be more conserved in the region around the active site than elsewhere. This can be seen from a quantitative comparison of the electrostatic potentials of the four TIM enzymes by means of similarity indices (see below). Boltzmann factor analysis Zhou44 has proposed that the rate enhancement due to interaction potentials in a diffusion-controlled bimolecular reaction is given by the average Boltzmann factor of the interaction energy of the substrate in the reactive region when the latter is small. Zhou et al.14 found this relation to hold true for acetylcholinesterase, for which a 240-fold increase in substrate binding rate due to electrostatic steering was found in BD simulations and a 1,000-fold enhancement was computed from the average Boltzmann factor. When applied to TIM, Boltzmann factor analysis also shows electrostatic rate enhancement. The trends in the extent of rate enhancement due to electrostatic steering among the four TIMs are similar to those in the BD simulations. The average Boltzmann factor, however, provides an overestimate of the extent of rate enhancement by a factor ranging from 3 to 18 for the different species and criteria. The maximum enhancements are computed 414 R.C. WADE ET AL. for T. brucei TIM and are 68-fold for loose reaction criteria and 2,600-fold for tight reaction criteria. The accuracy of the estimate of the rate enhancement by the average Boltzmann factor is expected to decrease the greater the magnitude of the interaction energy.44 Thus, a reason for the larger overestimate for TIM than for acetylcholinesterase is likely to be the difference in the magnitude of the substrate charge, which is twice as large for TIM as for acetylcholinesterase. Electrostatic potential similarity analysis The electrostatic potentials of the four TIMs studied are shown in Figure 2. The wide variation in charge distribution between these four proteins is clearly apparent. The average potential is shown in Figure 3A. On average, these potentials confine the negatively charged substrate to a ring of positive potential around the proteins, which includes the two active sites. Figure 3B shows contours of the similarity index for the four potentials, colored according to the sign of the potential. The most conserved regions of the electrostatic potentials are enclosed within the contours. Two highly conserved regions of positive potential are clearly discernable over the active sites. The potential is positive in this region for all four proteins and attracts the negatively charged substrate into the active sites. The conserved positive potential around the active site is in part due to the conservation of two lysine residues in all four TIMs: the catalytic Lys13 in the active site (conserved in all known TIM sequences) and Lys237, whose sidechain nitrogen atom is 13 Å from that of Lys13. This conserved region of positive potential is a major determinant of the computed rates. To define the dimensions of the substrate steering potential at the binding sites, we computed a more direct measure of potential conservation. Namely, we assigned a value 1 to a grid point if the four potentials have a positive sign at this point, a value of -1 if all four potentials were negative, and zero elsewhere, i.e., when there was no correlation in the sign of the four potentials. The resulting map is shown in Figure 3C. The blue region correlates highly with the same color region in the similarity index map and corresponds to the regions where the potentials of all four enzymes are positive and can facilitate substrate steering to the active site. Each region has maximum dimensions of 20 3 30 3 25 Å3 and a volume of about 5,600 Å3. The extension of the region from the position of the P0 atom of the bound substrate is less than 25 Å. The substrate is steered by the electrostatic force, i.e., the gradient of the potential. Similarity indices for the force vectors are highly correlated with those of the potentials (data not shown). The electrostatic potential up to a distance of about 25 Å from the active site would thus appear to be the cause of electrostatic steering in TIMs as it is conserved among the TIMs studied, which have association rates within a factor of 4 of each other. There are other highly conserved regions of the electrostatic potentials. The two large pincer-like regions correspond to negative values of the potentials and while they might have a role as deflectors, which could be valuable for allowing the product to leave the active site, the magnitude of the potentials in these regions is small (see Fig. 3A). These conserved regions of negative potential could also indicate other conserved functionality or arise as a consequence of conserved interactions that maintain the common fold. The pincer-like regions of negative potential may be partially attributed to residue 23, which is a glutamic acid in all four TIMs studied here and a number of others, but is not conserved in all known TIM sequences. Smaller regions of the conserved negative potential on the upper part of Figure 3B correspond to larger magnitudes of the potential and may be partially explained by the conservation of three clusters of charged residues nearby. The two negative regions closest to the active site are contributed to by the catalytic Glu165, Glu97, which is hydrogen-bonded to the catalytic Lys13, and a bit further from the active site, residues 129 and 133. The first three of these residues are conserved in all known TIM sequences. Residues 104, 106, and 107 contribute to the conserved negative potential further from the active site, with 104 being conserved in all known TIM sequences. In between are residues 98 and 99, two arginines conserved in all known TIM sequences. Their sidechains do not point to the protein surface and this probably explains the small size of the positive conserved region at this location in Figure 3B. CONCLUSIONS Fast diffusional association of substrate and enzyme is an important factor in TIM’s ability to function ‘‘perfectly.’’ Here, we find that this can be achieved by electrostatic steering of the substrate to the enzyme that is primarily dependent on the potential in the close vicinity of the active sites. This potential will be dependent on the nature of the residues at considerable distance (at least 10 Å) from the active site and also the three-dimensional structure of the protein, e.g., a helix dipole points to the active site. Nevertheless, the relative unimportance of the potential remote from the active site permits TIM to adopt different charge distributions elsewhere in the enzyme according to its needs in the given organism. For example, many trypanosomal glycosomal glycolytic enzymes have been observed56 to be highly basic, with higher pI values than their mammalian cytosolic counterparts, and the T. brucei TIM is no exception. The observation that association rates are primarily determined by electrostatic steering from the potential close to the active site can be expected to be a general feature of diffusion-controlled enzymes. ENZYME-SUBSTRATE ENCOUNTER RATES Sergi et al.57 used Brownian dynamics simulations to calculate rate constants for six different superoxide dismutases. These enzymes have similar catalytic rate constants, apparently because they have very similar electrostatic potential distributions in their active sites.58 A similarity index analysis of the electrostatic potentials of superoxide dimutases from five different species70, corroborates this observation, showing that a confined region of positive electrostatic potential is conserved around each active site. 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