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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
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
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
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:
Received 10 October 1997; Accepted 9 December 1997
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.
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-
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
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
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
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
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
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).
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):
o f (i, j, k)f (i, j, k)
o (f (i, j, k)f (i, j, k) 1 f (i, j, k)f (i, j, k))
. (2)
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)
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.
SI 5
o (f , f )9 o00f 00 .
This index measures the sum of the squared differences between the functions compared:
o 00f 2 f
o00f 00
5 2(N 2 1) ·
· (1 2 SI).
An alternative representation of the SI is
SI 5 1 2
o 00f
2 f002
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
muscle T. brucei
Net charge calculated
at neutral pH (e)
kcat/Km (108 M21s21 )b
2.3 1.0–5.4 2.4–3.7
No. of measurements
(kcat/Km) relc
1.0 0.4–2.3 1.0–1.6
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
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.
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
Electrostatic rate
E. coli
Chicken muscle
2.3 1.0
25.1 1.00
5.3 1.0
16.6 0.66
0.05 0.010
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
T. brucei
6.628.4 2.923.6
57.0 2.27
10.6 2.0
15.5 0.62
0.01 0.002
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*
24.9 0.99
0.21 0.04
32.6 1.30
0.42 0.08
37.9 1.51
0.66 0.12
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
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
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
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
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-
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
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.
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.
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.
Localized potentials can also be expected to be
important for the bimolecular association rates of
macromolecules, such as two proteins.
We are grateful to Peter Artymiuk for supplying
coordinates for chicken muscle TIM and to Rik
Wierenga for the provision of E. coli TIM coordinates
and helpful discussions about kinetic measurements. We thank Eugene Demchuk for assistance
with initial modeling and electrostatics calculations.
We are grateful to Andy McCammon, Jan Antosiewicz, and Michael Gilson for providing utilities for
calculating pKas with the UHBD program. We thank
Xiang Zhou for helpful discussions regarding Boltzmann factor estimation of the electrostatic enhancement of rates.
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