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PROTEINS Structure, Function, and Genetics 25:300-314 (1996)
Identification and Analysis of Long-Range
Electrostatic Effects in Proteins by Computer
Mode1ing:Aspartate Transcarbamylase
Himanshu Oberoi,*Jaishree Trikha; Xiaoling Yuan,*and Norma M. Allewell
Department of Biochemistry, University of Minnesota, St. Paul, Minnesota 55108
ABSTRACT
While ion pairs are readily
identified in crystal structures, longer range
electrostatic interactions cannot be identified
from the three-dimensional structure alone.
These interactions are likely to be important in
large, multisubunit proteins that are regulated
by allosteric interactions. In this paper, we
show that these interactions are readily detected by electrostatic modeling, using, as an
example, unliganded Escherichia coli aspartate transcarbamylase, a widely studied allosteric enzyme with 12 subunits and a molecular
weight of 310 kD. The Born, dipolar, and sitesite interaction terms of the free energy of protonation of the 810 titratable sites in the holoenzyme were calculated using the multigrid
solution of the nonlinear Poisson-Boltzmann
equation. Calculated titration curves are in
good agreement with experimental titration
curves, and the structural asymmetry observed
in the crystal structure is readily apparent in
the calculated free energies and pKl,z values.
Most of the residues with PK,,~values that differ substantially from those of model compounds are buried in the low dielectric medium
of the protein, particularly at the intersubunit
interfaces. The dependence of the site-siteinteraction free energies on distance is complex,
with a steep dependence at distances less than 5
A and a more shallow dependence at longer distances. Interactions over distances of 6 to 15 A
require a bridging residue and are often not apparent in the structure. The network of interactions between ionizable groups extends across
and between subunits and provides a potential
mechanism for transmitting long-range structural effects and allosteric signals.
lar distances. The transmission of the signal is often
linked to changes in tertiary and quaternary structure; however, the forces that drive these structural
changes and the mechanism of their propagation
through the molecule are poorly understood. A role
for electrostatic forces can readily be envisioned, because they are long range within the low dielectric
environment of the protein and are critical to the
overall structure and solvation of macromolecules.
Moreover, most cellular regulatory mechanisms depend on changes in charge, for example, allosteric
regulation of carbohydrate metabolism by nucleotides and phosphorylated sugars and the regulation
of cell signaling pathways by protein phosphorylation. 1-3
Understanding the role of electrostatic effects in
these processes requires the ability to investigate
the energetics of changes in the protonation state of
individual groups and charge-charge interactions in
detail. Much of the information that one would like
to have is not directly accessible experimentally,
particularly for large macromolecular assemblies.
As theoretical methods have improved and computational power has increased, computer modeling
has become an increasingly attractive alternative.
Approaches based on continuum models have been
used to model enzyme catalysis, ligand binding, protein-protein association, and pK shift^.^-^ However,
methods that are sufficiently rapid to model large
macromolecular assemblies and to make predictions
that can be tested by experiment have not been
available until recently.
We and others have recently developed the multigrid method for solving the nonlinear Poisson-Boltzmann equation (MGPB)and have shown that it produces a substantial increase in the speed and
accuracy with which electrostatic interactions in
0 1996 Wiley-Liss, Inc.
Key words: electrostatic modeling, PoissonBoltzmann equation, finite difference, multigrid, allosteric regulation
INTRODUCTION
Allosteric regulation of protein function requires
that signals be transmitted over long intramolecu0 1996 WILEY-LISS, INC.
Received July 7, 1995; revision accepted January 11, 1996.
Address reprint requests to Norma M. Allewell, Department
of Biochemistry, University of Minnesota, St. Paul, MN 55108.
*Current address: Belmont Research, Inc., 84 Sherman St.,
Cambridge, MA 02140.
'Current address: Department of Biological Chemistry and
Molecular Pharmacology, Harvard Medical School, 240 Longwood Ave., Boston, MA 62115.
*Current address: Incstar, 1990 Industry Blvd., Stillwater,
MN 55082.
LONG-RANGE ELECTROSTATIC EFFECTS
Fig. 1. Alpha carbon trace of the holoenzymeviewed down the
threefold axis. The two catalytic trirners are in the center of the
figure; the three regulatory dimers are on the periphery. The two
catalytic trimers and the r chains in each regulatory dirner are
related by the approximate non-crystallographic twofold axes.
Produced with DAMPS (H. Oberoi and N. Allewell).
proteins can be r n ~ d e l e d .The
~ , ~concurrent development of new approaches for calculating individual
site titration curves1' and treating clusters of
charges (Oberoi and Allewell, in preparation) makes
it possible to calculate pK,,, values and model the
titration behavior of molecules with several hundred ionizable groups in reasonable time on commonly available workstations.
In this paper we describe the application of MGPB
to unliganded aspartate transcarbamylase (ATCase)
from Escherichiu coli. ATCase is a challenging test
case since it is a large molecule with 2,778 residues
and a molecular weight of 310 kD. There are 24
amino and carboxyl terminal groups and 786 titratable side chains, more than have been modeled for
any other system. This widely studied allosteric enzyme is comprised of 12 polypeptide chains: six c
chains, organized as two catalytic trimers, and six r
chains, organized as three regulatory dimers, with
approximate D, symmetry (Fig. 1; reviewed in Refs.
11-14). There are 85 ionizable residues on each c
chain and 46 on each r chain. Both its substrates,
L-aspartate and carbamyl phosphate, and the nucleotides that regulate its affinity for L-aspartate are
polyanions, and a substantial body of experimental
results indicates that electrostatic effects are likely
to be central in catalysis and r e g u l a t i ~ n . l ~ - ~ *
We have previously used the modified TanfordKirkwood model to investigate the effects of assembly, conformation, and ligation on the electrostatic
properties of
These calculations indicated that long-range perturbations of the electro-
301
static properties of the protein result from substrate
and nucleotide binding and assembly of the enzyme
from its subunits and suggested that a more rigorous analysis was warranted.
The results obtained with the MGPB approach,
which are in good agreement with experimental titration curves and generally yield individual site
pKll2 values in the pH range 2-14, confirm these
conclusions. Groups whose pK,,, values differ most
from those of model compounds are a t the intersubunit interfaces where they are buried in the low dielectric of the protein. The asymmetry observed in
the structure around the approximate non-crystallographic twofold axesz7 is readily apparent in the
calculated electrostatic free energy terms. The sitesite interaction energy has a complex dependence on
the distance between interacting sites, with a number of interactions between groups separated by distances of up to 11A having site-site interaction free
energies greater than 2 kcal . mol-l. Three types of
electrostatic interactions are identified 1)salt linkages; 2) interactions involving a single bridging
group; and 3) networks of interactions extending
over several residues and across interfaces for distances as large as 35 A. The longer range interactions are likely to be important in site-site communication of structural and functional effects.
MATERIALS AND METHODS
The Model
The initial coordinates were those for unliganded
ATCase crystallized in the space group P321 at pH
5.8 (Brookhaven Protein Data Bank entry 6AT1)28
with polar hydrogen atoms added with the HBUILD
option of XPLOR." To idealize the geometry and
minimize the effects of crystal contacts, the 5 ps
structure from a molecular dynamics simulation
was energy minimized with 120 cycles of conjugate
gradient minimization with a 20 kcal . mo1-l constraint on the C" position^.^' The electrostatic terms
on the ionizable residues were switched off to prevent formation of incorrect hydrogen bonds during
minimization. Molecular dynamics calculations
and energy minimization were carried out with
X-PLOR.
The dielectric constant of the interior of the protein was generally set at 4, while the solvent surrounding the protein was treated as a continuum
with a dielectric constant of 80. Other calculations
with a dielectric of 20 yielded interaction energies
and pK,,, values for residues in the interior of the
protein that were on the order of 5 1 0 % smaller.
Partial charges and van der Waals radii were taken
from the CHARMM19 force field.31 Ions in the solvent were assumed to have a Boltzmann distribution and to obey the Debye-Huckel law.,' The ion
exclusion shell was assigned a thickness of 2 A (approximately the radius of the hydrated sodium ion).
302
H. OBEROI ET AL.
All calculations were carried out at an ionic
strength of 0.2 M and a temperature of 298°K.
For the calculations of W,, the entire molecule
was mapped on to a three-dimensional grid, with a
mesh spacing of 0.5-1.4 A, except for residues <20
A from a titrating site where grids with a spacing of
0.25 A centered on each ionizable residue and including all residues enclosed within a box 40 A on
each side were used. AAGB,,, and AAGdipolewere
calculated only for the asymmetric unit, consisting
of one c chain from each catalytic trimer (c, and c,)
and a bridging r subunit dimer (rl and r,); values for
the rest of the molecule were generated by making
use of the threefold molecular symmetry. Calculations carried out on the whole molecule yielded very
similar results.
Poisson-BoltzmannEquation
Electrostatic potentials were calculated with the
MGPB approach' to solving the finite difference
form of the nonlinear Poisson-Boltzmann equation
Vo[(~(r)V+(r)]
- k2(r)sinh[+(r)] + 4~rp(r)= 0
(1)
where r is a vector between a point and the origin,
+(I)is the electrostatic potential, E(r) is the dielectric constant, k is the modified Debye-Huckel parameter, and p(r) is the charge density function.33
The modified Debye-Huckel parameter is given by
(2)
where 1 / ~the
, Debye screening di~tance,~'
is independent of the dielectric constant.
The finite difference form of the nonlinear Poisson-Boltzmann is given by:
k = K &
+ti""
=
Cei+i
+4 ~ d h
C e i + + K~P[I
+ 433 + $251 +....+3(2n + I)!]
each level are carried through only a few iterations.
For intergrid transfer, we have used the method of
local inversion,38 which interpolates from a coarse
to fine grid by locally inverting the discrete operator. The interpolation scheme is that described by
Dougla~.~~.~~
Titrating Charges
Titration curves were calculated by dividing the
charges on the protein into titrating charges, corresponding to ionizable groups on the protein, and
nontitrating or background charges that are taken
from the CHARMM19 parameter set.3' The charge
was assigned to a single atom, based on solvent accessibility41and proximity of other ionizable groups,
i.e., to the most exposed atom, or, when the solvent
accessibilities were equal, to the atom that is closest
t o another titrating atom.' Charges on ionizable residues were adjusted to sum to zero in the uncharged
state.
Free Energies of Protonation and
Titration Curves
The method used to calculate the electrostatic contribution to the titration behavior of proteins has
been described in Bashford and Gerwert4' and is
based on the thermodynamic cycle described in Jorg e n ~ o nThe
. ~ ~finite difference procedure is used to
obtain the three terms that contribute to the free
energy of protonating each ionizable group:
AAG,,,
the interaction term that results from the
polarization of the surrounding medium by the
charge; AAGdipole,the term that results from the interaction of the titrating charge with the nontitrating (fixed)charges on the protein; and W,, the term
that results from the interaction of the titrating
charge with other titrating charges.
1
A A G o r n = - QP[+Pprotein - +Pmodel]
2
(3)
+i
1
- - QU[+%rotein- +Umodel]
where the nonlinear term is represented as a series,
h is the grid spacing in A, +o is the potential a t the
central grid point, q, is the charge at this grid point,
and the summations are over the six neighboring
grid points (i = 1-6).34-36
Multigrid
The MGPB method maps the molecule onto nested
grids of successively smaller spacing and calculates
potentials by iterating through the grids, in both
directions, until convergence is achieved.37 A few
iterations of a smoother (Gauss-Seidel) are applied
a t each level before transferring the correction to a n
adjacent level through a restriction (fine to coarse)
or prolongation (coarse to fine) operator.8 The multigrid approach reduces the time required for a calculation substantially, since most of the iterations are
carried out on coarse grids and the calculations at
2
(4)
A A G =~Cqi[+Pprotein
~ ~ ~ ~ ~- +%rotein]
i
-
xqi[+Pmodel - +Umodel]
(5)
I
W,
=
[QP - Qy](+Pprotein, i
-
+"protein, i) (6)
The model compound is the residue in the solvent in
the conformation in which it exists in the protein. Qp
and Q" are the charges on the titrating group when
the titrating group is protonated and unprotonated,
respectively. Qp has a value of 1for cationic groups
and 0 for anionic groups, while Q" has values of 0
and - 1for cationic and anionic groups, respectively.
qiis the static charge on a nontitrating atom; +p and
are the potentials calculated with the titrating
+"
303
LONG-RANGE ELECTROSTATIC EFFECTS
groups protonated and unprotonated, respectively.
Positive values AAG,,,, and AAGdipole correspond to
positive values of the difference in the free energy of
protonation in the protein and the model compound.
The pK of site i is given by
where Q$' is the charge of site j in the unprotonated
state, Oi is the fractional protonation of site i, and
pK,, is the pK, of the residue in the protein with all
other sites unprotonated
-140
5
Two approaches can be used to calculate Oi. Tanford and R o ~ b yuse
~ ~the Henderson-Hasselbach
equation to calculate a set of self-consistent partial
charges, while Bashford and Karplus" take a Boltzmann-weighted sum over all possible protonation
states at each pH. We calculated individual site titration curves using both the Tanford-Roxby
method44 and a hybrid approach45 that uses the
Tanford-Roxby equation for residues that have sitesite interactions lower than 0.9 pH units, and the
Boltzmann-weighted" summation with MonteCarlo sampling for others.46 Although both approaches give similar results when site-site interactions are weak, the Tanford-Roxby equation often
fails to converge when site-site interactions are
strong. The statistical thermodynamic approach,
while slower, converges under these conditions.
Hence the hybrid approach is most efficient.
A calculation for the holoenzyme requires approximately 90 hours on an IBM RISC 6000 workstation.
Calculating pK,,, values with the Tanford-Roxby
method requires about 2 hours; the hybrid scheme
requires about 7 hours. The time required increases
with the number of strongly interacting groups.
Experimental Titration Curves
ATCase was isolated from E . coli strain EK1104
transformed with plasmid pEK2 (a gift from E.
Kantrowitz, Boston College) as previously described.47Purity, assessed by nondenaturing polyacrylamide gel electroph~resis~~,
was greater than
95%)with only c6r4,cg, and aggregates as contaminants. Isoionic, salt-free stock solutions of the protein were prepared by gel filtration and dialyzed
against 0.2 M KC1 overnight at 4°C. Protein concentrations were determined spectrophotometrically a t
280 nm using an extinction coefficient of 0.59
mL . (mg . ~ m 1 - l . ~ ~
Potentiometric titrations were carried out on a
Metrohm E425 pH-stat at 25°C and a pH691 pH
meter calibrated a t pH 4 and pH 10 and 25°C before
each experiment. Solutions were degassed for 20
6
7
8
PH
9
10
Fig. 2. Comparison of calculated and experimental titration
back titration; -, calculated titracurves. 0, forward titration; 0,
tion curves at a grid spacing of 0.6 A; - - - -, titration curves calculated at a grid spacing of 0.3 A. The error bars correspond to the
forward titration, which was the average of four experiments.
minutes before the experiment and the titration vessel was flushed with nitrogen gas (ultrapure carrier
grade) during the experiment. The titrant was standardized at 100 mM HC1 for forward titrations and
100 mM KOH calibrated with potassium acidic
phthalate for back titrations. Titration curves were
obtained by the blank subtraction method of Tanford,5owith the blank solution consisting of 5.0 ml of
the 0.2 M KC1 solution used to dialyze the protein.
Protein concentrations ranged from 5 to 10
mg . m1-l; results were independent of protein concentration, although these experiments were not as
reproducible as those carried out in this laboratory
with monomeric proteins such as ribonuclease A and
lysozyme. Error estimates are shown in Figure 2. In
most experiments, the protein was titrated from pH
6.5-10.5 and back to 6.5. These limits were chosen
because the protein precipitates below 6.5 and loses
its cooperativity (reversibly) above pH 10.X51There
is no dissociation within this pH range, particularly
at the protein concentrations used.
RESULTS
Comparison of X-Ray and Energy
Minimized Structures
Differences in the deposited coordinates, the
structure from the 5 ps molecular dynamics trajectory, and the energy-minimized structure are small,
with root mean square (rms) differences of 0.19
and 0.23 A for backbone and side chain atomic positions, respectively. Thus the results of the electrostatic calculations for all three structures would be
expected to be, and, in fact, are very similar, although the energy minimization did eliminate several unfavorable site-site interactions. All the re-
304
H.OBEROI ET AL.
sults presented below are for the energy-minimized
structure.
Comparison With Experimental Data
Experimental and calculated titration curves are
compared in Figure 2. The enzyme does not dissociate over the pH range examined, although its allosteric properties and hence subunit interactions are a
sensitive function of pH (cf. Refs. 15, 16, and 19).
These curves were obtained with the same sample of
protein and there is a small amount of hysteresis in
the experimental curves so that the forward and
back titrations differ by 4-10 protons. The basis of
this hysteresis warrants further exploration in the
future. The forward titration gives better agreement
with the titration curve calculated with a grid spacing of 0.3 A, while the back titration agrees better
with the curve calculated with a grid spacing of 0.6
b. Both the forward titration and the curve calculated with the finer mesh would be expected to be
more accurate; however, a series of calculations carried out with several grid spacings between 0.3 b
and 0.6 b indicated that there is not a simple correlation between grid spacing and agreement with experiment. This is probably because the calculations
are very sensitive to the relative positions of ionizable atoms and grid points and this is not a simple
function of the grid spacing. Both calculated curves
diverge from the experimental curves a t high pH,
where the average solution structure would not be
expected to be the same as the crystal structure,
since the enzyme is non-cooperative at pH values
above
The agreement of the experimental and
titration curves indicates that the calculations are
able to reproduce the overall titration behavior of
the protein with reasonable accuracy. Exact agreement would not be expected since the sensitivity of
the catalytic and allosteric properties of the enzyme
to pH indicates that the structure is sensitive to pH.
The only other experimental result with which
the calculations can be compared is the pK, of His
134 in the c chain, which has been reported to be
less than 6 in the unliganded catalytic subunit and
ligated with N-phosphonacetyl-L-Asp (PALA),
based on the observation that the proton resonance
to which it was assigned does not titrate over the pH
range 6-8.5.5z The calculated pK,,, values are 5.8
and 6.4for the two c chains in the asymmetric unit.
Villoutreix et al.53reported a value of -3.5 pH units
for this residue in the Bacillus subtilis ATCase
structure obtained by structure-based homology
modeling of a n ATCase catalytic trimer ligated with
PALA.
Potential Surfaces
Figure 3 shows the electrostatic potential surface
at a contour level of 1.5kT/e for a pair of c:r chains
within the molecule at pH 2,6, and 11. Although the
net charge on the molecule is negative, substrate
and nucleoside triphosphate binding sites are positive, as reported previouslyz6 and remain positive
over the entire pH range of the titration curve. Since
both of the substrates (carbamyl phosphate and
L-Asp) are negatively charged, as are the regulatory
nucleoside triphosphates, the regions of positive potential will assist in docking and electrically polarizing these ligands a t their binding sites, while the
negative potential around other regions of the molecule will tend to prevent nonproductive binding a t
other sites.
Contributions of AAG,,,
and AAG&ipole
AAGBorn and AAGdipole for all ionizable residues of
the cl and r l chains are plotted in Figure 4.The
form of Equations 4 and 5 is such that positive
AAGB,,, terms indicate that the solvent accessibility of the titrating group is lower in the protein than
in the model compound while positive AAGdipole
terms indicate an unfavorable interaction between
the protonated species and the static charges of the
protein relative to the model compound in solution,
for both cationic and anionic groups.
Most of the large AAGdipole terms are negative and
are paired with positive AAGBo,, terms. The large
positive AAGB,,, terms indicate that the solvent accessibility of these anionic groups is low, as expected. The large negative AAGdipol, terms indicate
that they are stabilized by the static charges of the
protein. As a result of this compensation, the pKadj
values of 500 of the 810 ionizable groups differ by
less than 3 pH units from the values for the model
compounds. Most of the large negative AAGdipole
terms (and hence large positive AAGB,,, terms) correspond to anionic groups. The most likely reason
for anionic groups having larger AAGdiPle and
AAGBorn terms is that they are selectively positioned
in the protein so that they can interact favorably
with the dipoles of the secondary structural elements. In addition, because the side chains of Glu
and Asp are relatively short, they will tend to be
closer to the surface of the protein and hence more
affected by the dielectric discontinuity.
Long-Range Interactions
While intuitively one might expect the relationship between the magnitudes of the W, and distance
to be hyperbolic, effects such as desolvation and a
non-uniform dielectric may lead to a more complex
dependence in complex systems.54As shown in Figure 5a, the dependence of W, on distance is roughly
biphasic, with a steep dependence over short distances and a shallow dependence over long distances. However, there is a great deal of scatter
across the entire range. The calculated points fall
below the curve calculated using Coulomb’s law and
a dielectric constant of 4, while curves calculated
with dielectric constants 25-40 pass through the
points. The damping of site-site interactions in the
LONG-RANGE ELECTROSTATIC EFFECTS
305
Fig. 3. Potential surfaces around a c:r pair within the molecule
at pH 2 (a), 6 (b),and 11 (c). The catalytic chain is in gold; the
regulatory chain in green. Blue, positive potentials; red, negative
potentials. Potential surfaces were calculated at a contour level of
1.5 kT/e and are shown for only one c and one r chain for clarity.
Produced with DAMPS (H. Oberoi and N. Allewell).
protein is the result of several factors including interactions of titrating charges with static charges on
the protein, screening of interacting charges by the
titrating charges that lie between them, and the irregular dielectric boundary. These results support
the suggestion that use of a dielectric constant
greater than 20 may improve agreement with experiment. 55
All of the interactions with W, values 2 2
kcal . mol-' are listed in Table I, along with distances between the atoms to which charges were assigned and solvent accessibilities. The error in W, is
estimated as being in the range of about 1
kcal . mol-l, based on the error in the calculated
potential^.^^ Most of the residues involved in these
interactions are clustered in four regions of the molecule: in a zone that extends from the active site to
the cl:c2 interface, a t the c1:rl and cl:c4 interfaces, and within the r chain. A number of these
residues are involved in site-site interactions with
several other residues, often over distances greater
than 10 A. These strong, long-range electrostatic interactions are of particular interest because they
provide a potential mechanism for long-range structural interactions and functional communication
within and between subunits.
In each c:r pair there are 31 site-site interactions over distances greater than 5 A that result
in W, values greater than 2 kcal mol-l. Nine of
these strong, long-range interactions (W, 2 2
kcal . mol-l; distance 2 5A) involve ion pairs or
three-center clusters (Fig. 6a, i and ii) that are not
involved in other strong, long-range interactions.
The remainder are part of a network involving 15
residues whose connectivity is mapped in Figure 6a,
iii).
This network includes Arg c54, Arg c105, and His
c134 a t the active site, Glu c109 at the cl-c4 interface, Glu r142 a t the cl-rl interface, and Tyr 98 in
c2. The interactions in this network involve a bridging residue; for example, His c134 acts as a bridge
between Arg c296 and Asp c129; Asp c129 bridges
Glu c109 and Arg c167 in the catalytic chain and
Glu r142 in the regulatory chain. As shown in Figure 6b, many of these interactions are complex; for
example, Asp c129 interacts with Glu r142 in the
regulatory chain and with the active site both directly (with His c134, Arg c105) and indirectly
through bridging interactions involving Glu c109
and Arg c167. In many cases, there are multiple
connectivities between sites; for example, between
His c134 and Asp c129. The extent to which different networks of interactions are used may be influenced by variables such as pH and state of ligation.
The site-site interactions of His c134 with neighboring residues are of particular interest since this
residue may be involved in the catalytic mechanism.13 His c134 is involved in short, medium, and
long interactions with other residues. The shortrange interactions include not only the interaction
3
306
20
H.OBEROI ET AL.
I
c chain
M
I
I
Residue
Fig. 4. AAG,,,,
and r l chains.
(shaded bars) and AAGdipale(hatched bars) for the ionizable residues in the c1
with Arg c105 identified from the crystal structure,
but also strong interactions with Tyr c98 (in an adjacent chain of the same trimer), Asp c129, Asp
c141, Arg c167, Tyr c226, and Arg c296. If all these
residues were fully charged, the net effect of these
interactions would be to decrease the pK,, of His
c134 by 18 pH units. However, the interactions of
these residues with other titrating groups results in
a net shift of +10.6 pH units, bringing the calculated pK,,, to 5.8.
We have focused on a subset of site-site interactions that have energies greater'than 2 kcal . mol-'
and distances greater than 5 A and hence are
likely to be involved in long-range structural and
functional efforts. However, many other interactions are likely to have structural andlor functional significance. For example, Asp 236 (c4), Glu 109
(cl),Asp 129 (cl),His 134 (cl),and Tyr 98 (c2) trace
a pathway with interactions greater than 2
kcale mol-' that links all six c chains in the two
catalytic trimers. In addition, an interaction between Glu c109 and Lys r143 with a calculated
W, of 6.8 kcal . mol-' links this pathway with the
regulatory subunit. Hence the complete network of
electrostatic interactions is considerably more com-
plex than the restricted set of interactions diagrammed in Figure 6.
Asymmetry
While the molecular symmetry of ATCase about
the crystallographic threefold axis in the P321 crystal form must be exact, there are small but significant differences between the chains related by the
approximate non-crystallographic twofold axesz7
(reviewed in Refs. 13 and 14). This asymmetry may
be functionally significant since both carbamyl
phosphate and nucleoside triphosphates bind with
negative c o ~ p e r a t i v i t y . ~
The
~ , ~magnitude
~
of the
asymmetry in the electrostatic properties of the two
halves of the molecule induced by the structural
asymmetry can be assessed from the calculated
pK,,, values listed in Table I1 for residues a t the
subunit interfaces and active sites of the c l , c6, and
rl, r6 chains (the asymmetric unit). Lys c31, Lys
c40, Arg c56, Arg c65, Lys c84, Tyr c98, Arg c167,
Lys c178, Lys c244, Asp c278, and Asp r39, Asp r55,
Lys r139 have pK,,, values that differ by at least 2
pH units. Residues a t the cl:c2 interface have the
highest asymmetry with 6 of the 14 ionizable resi-
307
LONG-RANGE ELECTROSTATIC EFFECTS
0
I
I
I
5
10
15
Distance (A)
a.
150
5 100
C
Q)
3
CT
F
LL
50
0
b.
Distance (A)
Fig. 5. a: Dependence of absolute values of W, (kcal . mol-’)
on distance (A). The site-site interactions are shown as open
squares. The ewes were calculated with the equation W, = Ckr,
where C = 331 and r is the distance in A, and dielectric constants
(E) of 4 (-),
25 (- - - -), and 40 (. - . - . -). b: Histograms showing
the distribution of distances between interacting sites for various
ranges of W,.
dues a t this interface having electrostatic terms that
differ between the two halves of the molecule by
more than 2 pK units. On the other hand, many
other residues show very little asymmetry. These
calculations therefore allow the ionizable groups a t
the active site and subunit interfaces t o be divided
into two groups: those that show significant electrostatic asymmetry and are likely to be involved in the
negatively cooperative binding of carbamyl phosphate and nucleoside triphosphates, and those that
TABLE I: Site-Site Interactions (WJ2 2 kcal-mol-'
SA
SA
Residue
(A2)
Arg 17 cl
36.2
Glu 37 c l
Lys 40 cl
Glu 50 c l
Arg 54 c l
43.1
9.5
20.5
7.4
His 64 cl
Arg 65 cl
Glu 86 cl
2.5
8.4
0.3
Asp 90 c l
Tyr 98 cl
3.0
1.0
Asp 100 cl
Arg 105 c l
21.0
18.9
Glu 109 el
3.4
Arg 113 cl
22.3
Glu 117 cl
12.0
Asp 129 cl
0.0
His 134 c l
0.1
Asp 141 c l
0.0
Glu 149 cl
Lys 164 cl
1.8
2.5
Tyr 165 c l
0.0
His 170 c l
Tyr 185 c l
0.4
0.1
Glu 216 cl
10.4
Glu 239 c l
7.8
Tyr 240 cl
Lys 258 c l
His 265 c l
1.5
12.6
0.0
Arg 269 c l
1.4
Glu 272 cl
0.0
Asp 278 cl
7.9
Arg 296 c l
2.9
Arg 41 r l
Lys 56 r l
17.8
2.9
Arg 102 r l
Arg 128 r l
Tyr 140 r l
Lys 143 r l
58.8
41.4
0.8
0.0
Residue
Asp 153 cl
Asp 180 cl
Lys 40 c2
Glu 37 c3
Arg 167 c l
Glu 86 c2
Tyr 98 c2
His 64 c2
Asp 100 c3
Asp 90 c l
Arg 54 c3
Asp 278 c3
Arg 54 c3
Asp 141 c3
His 134 c3
Lys 42 cl
Glu 50 cl
Arg 167 c l
Arg 296 c l
Lys 143 r l
Asp 236 c4
Asp 129 cl
Asp 14 cl
Glu 142 r l
Lys 139 rl
Tyr 140 r l
His 134 cl
Arg 167 cl
Glu 142 r l
Arg 105 c l
Asp 141 c l
Arg 167 cl
Tyr 226 cl
Arg 296 cl
Tyr 98 c2
Tyr 226 cl
Tyr 98 c2
Asp 223 cl
Tyr 165 c l
Tyr 165 c4
Glu 239 c4
Arg 234 el
Tyr 165 c4
Glu 239 c4
Tyr 197 cl
Glu 221 cl
Arg 183 cl
His 156 cl
His 255 c l
Asp 253 cl
Tyr 165 c4
Lys 164 c4
Asp 271 cl
Asp 223 c l
Tyr 226 cl
Glu 272 c l
Asp 278 e l
Tyr 285 cl
Asp 90 c2
Tyr 240 c l
Asp 271 el
Tyr 285 cl
Asp 90 c2
Tyr 98 c2
Tyr 226 cl
Asp 141 cl
Glu 62 r l
Gly 8 r l
Asp 19 rl
Asp 104 r l
Glu 144 r l
Lys 139 r l
Asp 236 cl
Asv 236 c4
(A2)
12.5
46.3
9.5
43.1
11.8
0.3
1.0
2.5
21.0
3.0
7.4
7.9
7.4
0.0
0.1
24.5
20.5
11.8
2.9
0.0
8.0
0.0
23.8
0.4
13.8
0.8
0.1
11.8
0.4
18.9
0.0
11.8
0.0
2.9
1.0
0.0
1.0
5.9
0.0
0.7
8.0
28.1
0.7
8.0
8.8
37.7
55.8
4.0
6.0
15.1
0.7
1.4
3.8
5.9
0.0
0.0
7.9
0.2
3.0
1.5
3.8
0.2
3.0
1.0
0.0
0.0
11.1
0.0
20.0
7.8
12.1
13.8
5.4
8.0
w,
Distance
(kcal . mol-l)
(A)
2.0
2.9
2.6
2.7
2.7
7.1
2.8
2.0
6.1
4.8
7.2
2.1
2.9
2.2
2.3
2.6
3.1
4.4
2.3
6.8
2.3
2.3
4.1
3.2
5.1
8.2
2.7
3.6
5.0
5.0
2.8
3.2
2.4
5.7
2.3
17.5
2.2
3.1
7.2
2.0
5.8
2.4
2.1
5.4
2.8
3.0
3.0
2.8
2.6
2.0
6.8
4.1
4.7
6.4
2.8
8.1
8.8
4.3
5.5
2.0
3.4
11.0
2.1
4.2
2.2
3.1
4.5
6.6
7.6
2.4
3.4
7.6
3.8
4.2
3.3
5.4
3.5
3.5
4.8
3.1
7.7
10.0
3.3
5.1
3.1
5.7
7.7
10.7
11.4
4.9
4.7
4.5
7.7
3.7
4.7
8.4
3.7
4.7
4.0
3.2
7.3
5.3
6.2
4.6
9.2
5.9
9.3
6.0
11.4
3.5
10.7
7.8
3.2
8.2
3.1
4.5
8.0
4.5
5.0
3.2
4.8
4.2
4.8
4.6
2.9
3.9
3.1
2.9
7.2
7.0
3.0
5.3
4.6
5.5
3.8
4.5
5.7
6.1
9.8
7.8
4.8
6.6
7.6
3.5
3.3
3.2
4.9
3.6
All the interactions that fulfill this criterion for a given residue in column 1 are listed in column 2.
Distances are between ionizable atoms (see methods section). SA, solvent accessibility for residue.
309
LONG-RANGE ELECTROSTATIC EFFECTS
cl Arg 17
cl Glu 149
cl Arg 269
c1 His 64
-
cl Arg 54
cl Asp 180
cl Asp 223
0)
cl Tyr 285
C2 His 64
I
c2 Tyr
Tir 98
cl Arg 296
cl Asp 141
I
Cl Arg 105
-cl
cl Lys 164
cl Tyr 165
r l Gly0
His 134
cl Arg 167
r l Asp 19
(li)
PI
cl A
C l Tyr 226
12-rI
cl G u 109
cl His 265
c1 Glu 272
-
Cl Tyr 240
a.
(iii)
Fig. 6. (Legend on p. 310.)
Glu 142
310
H. OBEROI ET AL.
do not show electrostatic asymmetry and are not
likely to be directly involved.
The possibility that this asymmetry is the result
of grid artifacts is ruled out in three ways. Each
electrostatic term is calculated three times for the
whole molecule, with the titrating group in a different position in the grid each time. Differences in the
free energy terms between symmetry-related chains
(e.g., c l and c2) were -1% in these calculations.
Similarly, calculations in which the position of the
molecule in the grid was varied differed by 2-4%.
Finally, the structural variability generated by the
molecular dynamics run results in differences in the
Born and dipole terms and calculated pK,,, values
that generally do not exceed one pH unit. In contrast, free energy terms and pK,,, values for the
same residue in the two halves of the molecule related by the non-crystallographic twofold axis often
differ by several pH units.
Effects of Mutations
These calculations also make it possible to analyze quantitatively the electrostatic effects in various single site mutations that have been shown to
affect function. Two examples are presented below
for illustrative purposes.
Mutating Asp cl00 to Ala or Asn has been shown
to decrease the stability of the catalytic trimer of the
enzyme and increase the affinity of the holoenzyme
for L - A s ~ . ~This
' was attributed to the elimination
of a hydrogen bond to Arg c65 that was identified in
the crystal structure. We calculate a value of 6
kcal . mol-' for the W, term for this interaction.
However, the calculations also indicate that there is
a second interaction with Lys c42 (W, = 2.5
kcal . mol-l, distance = 4.9 A) and suggest that it
would be of interest to explore the functional consequences of mutating Lys c42.
Arg c54, whose side chain interacts with Glu c86
and whose backbone carbonyl group interacts with
the OH of Tyr c98 a t the cl:c2 interface, has also
been implicated in the catalytic mechanism, since
mutation to Ala results in a significant decrease in
activity." The calculated pK,,, for Arg c54 in cl is
6.4 (5.8) and the magnitude of the Arg c54-Glu c86
electrostatic interaction is 7.1 kcal . mol-l. However, the interaction of its sidechain with Glu c86 is
absent at the comparable c4:c5 interface in the second trimer, suggesting that the structural basis of
its role in catalysis may be more complex. These
Fig. 6. a: Connectivities between sites separated by distances
of greater than 5 A with W,, terms greater than 2.0 kcal . mol-'. i:
Pairwise interactions. ii: Bridging interactions. iii: Long-range interactions. Residues that are involved in long-range interactions
also form paiwise interactions with neighboring residues. b: Mapping of residues in a on the molecule. Pairwise interactions (a, i)
are shown with dashed lines; three-center interactions (a, ii) with
solid lines and circles; and interactions in the network (a, iii) with
solid lines.
differences in site-site interactions between subunits related by the non-crystallographic symmetry
axis may be the cause of the differences in the Born
and dipole terms for Glu c86.
DISCUSSION
ATCase is the largest protein to which the finite
difference method has been applied and one of the
few for which a complete set of pK,,, values have
been calculated. The multigrid method developed in
this laboratory allows calculations t o be carried out
several orders of magnitude more rapidly than is
possible with other methods. Comparison of calculated and experimental titration curves provides a
test of the validity of the calculations, and for larger
molecules is often the only experimentally accessible quantitative comparison available. The results
presented here illustrate the potential of calculations of this type for identifying long-range electrostatic effects that cannot be identified directly from
the crystal structure and for analyzing the effects of
single site mutants. Calculations based on structures in different states of ligation and different
quaternary states are under way, and comparison of
these with the results for the unliganded structure
will allow us to assess the role of these effects in the
catalytic and allosteric mechanisms.
AAG,,,
and hAGdipolemay be considered to be
the static components of electrostatic effects because
they depend largely on the local environment of individual ionizable residues rather than their dynamic interactions with other ionizable groups.
Hence AAGBornand AAGdipolecan be used to examine the electrostatic consequences of specific structural features. AAG,,,
is a measure of the change
in the solvation of the charge of the ionizable group
that results from incorporating it into the folded protein while AAGdipoleis a measure of the strength of
the interaction of the ionizable group with the static
(nontitrating) charges in the protein. The relative
magnitudes of these terms and their signs yield information about both stability and function. For example, a buried charge with an energetically unfavorable AAG,,,
that would normally destabilize
the protein and have a highly perturbed pK value
may have a AAGdipolethat compensates for the
burial so that the group neither destabilizes the protein nor has an altered pK,,,.
While the AAGB,
and AAGdipoleterms are primarily a measure of the local environment, the sitesite (W,) interaction terms often depend on longrange interactions that are linked to changes in
tertiary and quaternary structure. Compensatory
changes in the apparent pK,,, of residues such as
those illustrated by His c134 and Asp el29 require
cooperative changes in the titration behavior of distant residues, which in turn may induce structural
changes over several residues. Such cooperative interactions are likely to be important in the trans-
311
LONG-RANGE ELECTROSTATIC EFFECTS
TABLE 11. (Continued)
TABLE 11. AAG,,
and AAGaipOlein Units of pH
and Calculated P K ~Values
,~
for Residues at the
Active Site and Subunit Interfaces*
AAGBorn
Interdomain interface, c chain
Arg c 17
1.3
Lys c 31
-6.4
-3.0
Arg c 54
-4.3
Arg c 105
-4.1
Asp c 129
10.1
Asp c 141
11.3
9.8
Asp c 153
4.1
Arg c 167
-4.7
-4.1
Lys c 178
-3.2
-2.5
Asp c 180
1.3
Asp c 223
5.6
Tyr c 226
14.1
13.1
Lys c 258
-3.0
Lys c 262
-4.2
His c 265
-8.0
Arg c 269
-4.8
Asp c 278
6.2
5.7
His c 282
-4.6
Tyr c 285
12.2
Interdomain interface, r chain
5.7
Tyr r 77
Arg r 102
-0.8
4.2
Asp r 104
cl:r4 interface
-2.4
Lys c 244
-0.1
-7.7
Lys r 143
Glu r 144
3.4
rl:r6 interface
0.6
Glu r 10
0.3
Asp r 39
4.7
-2.4
Arg r 41
-1.0
Arg r 55
-1.4
cl:c4 interface
-4.2
Lys c 164
7.5
Tyr c 165
3.0
Asp c 236
4.4
Glu c 239
cl:c2 interface
1.6
Glu c 37
0.8
Lys c 40
-0.8
-1.3
His c 41
-4.3
Arg c 54
Arg c 56
-3.0
-4.0
His c 64
-5.7
-3.8
Arg c 65
-2.9
AAGdipole
PK1/2
1.3
0.8
1.2
-2.2
1.0
-20.0
-18.8
-7.9
-1.0
-0.3
1.1
1.1
2.3
0.7
-2.7
-24.8
-18.7
0.5
2.4
-10.8
0.8
-1.8
-3.7
1.0
-14.9
12.7
4.5
8.5
6.4
10.3
7.2
6.4
6.7
7.1
8.1
10.3
8.2
10.3
5.6
6.6
-8.2
-0.4
-0.5
7.7
11.5
7.7
4.6
8.1
8.5
10.2
9.2
5.8
2.3
7.8
-1.2
-1.1
-12.0
-2.6
7.0
9.4
-0.8
0.4
0.3
0.4
0.8
2.5
3.8
4.8
8.9
10.7
12.6
-6.1
-10.3
-6.6
-8.6
0.7
-0.2
0.0
0.4
-2.2
3.9
-3.0
1.6
4.1
4.1
5.4
8.1
0.6
4.9
4.6
9.4
5.2
6.4
11.8
1.4
13.3
(Continued)
Lys c 84
Glu c 86
Asp c 90
Tyr c 98
Asp c 100
Lys c 232
Arg c 269
c1:rl interface
Glu c 109
Arg c 113
Glu c 117
Glu c 204
Glu r 119
Arg r 130
Lys r 139
Tyr r 140
Glu r 142
Lys r 143
Active site
Arg c 54
Lys c 84
Arg c 105
His c 134
Arg c 167
Arg c 229
Glu c 233
-1.9
-0.1
7.9
5.2
4.7
7.5
4.0
-1.0
-2.3
-4.8
-2.2
-1.6
-4.4
-2.5
-1.6
-9.0
0.3
-2.2
-1.6
0.8
6.3
-3.0
4.3
0.7
6.3
-0.6
-3.0
-1.9
8.8
6.3
-7.7
-16.9
-1.5
-9.6
-0.6
-10.8
0.4
-5.5
-3.4
-16.0
-11.1
-12.0
-4.3
-1.9
0.1
-4.1
-7.3
-4.7
-4.1
-2.3
2
-2.2
-2.2
-1.6
1.0
-3.8
-0.3
1.1
-1.4
-3.6
6.3
8.9
8.6
6.9
7.0
9.1
8.0
7.3
9.3
10.2
8.7
4.5
12.0
2.2
5.2
3.8
6.4
6.3
8.9
10.3
6.4
8.1
10.3
9.3
2.8
*1 kcal = 1.4pH units. Errors in free energies are on the order
of 1 kcal . m-'. Values for the two halves of the molecule related by the approximate non-crystallographic twofold axes
are shown when values of AAG,,,,, AAGdipoler
and pK,,, differ
by more than 2 pH units. In these cases, the upper value corresponds to cl or rl and the lower value to c4 or r6. Calculated
pK,,, values outside the range 2-14 are not listed.
mission of signals over long distances in ATCase
and other multisubunit proteins. While pK,,, values
are very sensitive to how charges are assigned and
small changes in side chain positions, W, values are
less sensitive t o charge assignment and hence are a
more reliable parameter for analysis.
Although the inherent asymmetry of the structure would suggest that its electrostatic properties
would be asymmetric, the magnitude of the asymmetry in the calculated electrostatic free energy
terms is striking. These differences persist and are
even more striking in the PALA-liganded structure
(Oberoi et al., in preparation). Differences in the calculated pK,,, values of individual residues in cl-c6
or rl-r6 related by the approximate non-crystallographic twofold axis may result from differences in
AAGBorn,AAGdipole(and therefore pK,,) or the sitesite interaction (W,) term. The fact that this electrostatic asymmetry is demonstrable in two struc-
312
H. OBEROI ET AL.
tures with very different quaternary states and
charge configurations attests to the robustness and
reliability of the calculations. It also suggests that
comparisons of the electrostatic properties of the
liganded and unliganded structure may provide additional insights into the functional significance of
the asymmetry.
Although subject to the limitations of continuum
models,4.6,60grid methods allow considerable refinement of the physical model over previous approaches. Several additional steps can be taken to
reduce the error further.
The effect of varying the dielectric constant either
throughout the grid or within regions of it remains
to be investigated. While results obtained with the
modified Tanford-Kirkwood model suggested that
the exact value of the dielectric constant within the
protein did not have much effect:’
evidence that a
value of 20, rather than the generally accepted
value of 4, yields better agreement with experiment
has recently been presented.55 Since direct experimental determinations and simulations yield values
near 4,60 the higher value may compensate for neglecting structural features such as bound solvent,
protein motion, and ion binding. It is also likely that
the increased screening due to the higher dielectric
constant may compensate for side chain mobility
and resonance, since neglecting these effects would
effectively exaggerate the magnitude of the interaction terms and the calculated pK,,, values. In other
studies the effect of changing the protein dielectric
constant to 10 was shown to have a modest effect on
calculated pK shifts while a dielectric constant of 78
produced a reduction of nearly 50%in the calculated
pK shifts.62Such effects may in part be responsible
for the discrepancy between the calculated and experimental titration curves.63
Standard continuum models generally assign a
high dielectric constant to all regions of the molecule accessible to a spherical probe. Since this includes regions internal to the molecule that may
contain bound water, the dielectric constant in these
regions may be lower. Developing methods to include water in the calculations, as was done in Yang
et al.45will improve the accuracy of the calculations.
We find that including bound solvent in the internal
cavity of adipocyte lipid binding protein produces
significant changes in calculated pKlI2 values
(Oberoi et al., in preparation).
A second significant source of error is the neglect
of side chain motion. The first systematic attempt to
examine the consequence of side chain motion using
a continuum model was by Wendoloski and Matthew,64 who used the modified Tanford-Kirkwood
model and the Tanford-Roxby equation to model pH
dependence. Differences in calculated pK,,, values
among several tuna cytochrome c structures from a
molecular dynamics run were of the same order of
magnitude as those between finite-difference and
Tanford-Kirkwood calculations. You and B a ~ h f o r d ~ ~
have recently used a finite difference solution of the
Poisson-Boltzmann solution to study the effects of
conformational flexibility. We have shown for hen
egg white lysozyme that crystal structures derived
by different refinement protocols also have calculated pK,,, values that differ by > 10 pH units.’ The
substantial differences in the results reported here
for the two halves of ATCase are a third indication
of the sensitivity of these calculations to side chain
positions. Although computationally intensive, the
incorporation of side chain motion is in principle
straightforward.
A third issue is the method used to assign charges
to individual atoms. We obtained good agreement
with experiment when charges were assigned to a
single atom on the basis of solvent accessibility and
proximity of neighboring charges, Although this approach has been reasonably successful with the molecules to which we have applied it to date, methods
with stronger theoretical justification that yield
equally good agreement with experiment are clearly
desirable.
The results presented here suggest a role for three
classes of electrostatic interactions in intramolecular communication. One class of interactions, over
distances of 3-5 A, involves pairs of residues that
interact directly and is generally discernible from
direct observation of the crystal structure. A second
class of interactions requires a “bridging” residue
and therefore involves three residues and extends
over distances of 6-15 A. These interactions are not
readily apparent from the structure since at least
one pair of residues may be 6-8
apart. The complete set of ion pair and bridging interactions forms
a third type of pathway, a network that extends
across and between subunits and provides a potential mechanism for generating long-range structural
changes and transmitting allosteric effects. When
calculations for various liganded structures become
available for comparison with these calculations, it
should be possible to assess the role of these longrange interactions in the catalytic and allosteric
mechanisms of the enzyme in greater detail.
The actual network of long- and short-range interactions within the molecule is complex, with multiple pathways between many ionizable groups. The
extent to which any given pathway is used will depend on pH, since the magnitudes of site-site interactions vary with pH, and also with the state of ligation of the protein. In addition, the complexity of
the network suggests that communication between
different parts of the molecule may be plastic, so
that a mutation in one pathway may be compensated for by communication through a different
pathway. This would in turn mean that failure of a
mutation to alter function cannot be interpreted to
mean that that residue does not have a functional
role.
LONG-RANGE ELECTROSTATIC EFFECTS
Networks of interactions over long intramolecular
distances such as we have described here may serve
the dual purpose of maintaining subunit interactions and channeling allosteric and cooperative signals. To validate and understand the role of the
types of interactions illustrated in this paper will
require the design of new sets of single site mutants.
Most single site mutations to date have been designed to investigate the roles of residues that are
either at the active or effector site or are involved in
subunit-subunit interactions. Electrostatic modeling has the potential to identify residues that may
be equally important from a structural and/or functional point of view, but whose role is less obvious.
ACKNOWLEDGMENTS
This work was supported by NIH grant DK-17335
and the University of Minnesota. The facilities of
the Minnesota Supercomputer Institute were used
for some calculations. We thank Drs. Vince J. LiCata and Victor Bloomfield for critical discussion,
and Hiroki Morizono and Dr. Meena Hariharan for
help with the figures.
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