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J. Phys. Chem. B 1999, 103, 8628-8638
Quantum Mechanical Study of β-Lactam Reactivity: The Effect of Solvation on Barriers of
Reaction and Stability of Transition States and Reaction Intermediates
Irina Massova and Peter A. Kollman*
Department of Pharmaceutical Chemistry, UniVersity of California, San Francisco, California 94143-0446
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β-Lactams are widely used in medicine as potent antibacterials and in chemistry as synthetic intermediates.
The investigation of the intrinsic reactivity of β-lactams is important for the understanding of the mechanisms
of action and inactivation of β-lactamases and penicillin-binding proteins. Ab initio quantum mechanical
calculations using a polarizable continuum model to estimate solvation effects have been utilized to analyze
the hydrolysis and methanolysis of selected β-lactams and simple amides. The roles of four-membered ring
strain, reduced amide resonance, substituent and ring fusion effects on hydrolysis, and methanol-mediated
hydrolysis of these compounds have been studied by reconstructing the corresponding reaction pathways in
gas and solution. Strong correlations have been found between the calculated kinetic, structural, and electronic
properties of these compounds and the experimental data. The findings in this paper shed light on the
contributions of the various structural elements to the reactivity of β-lactams. Insights into this reactivity
could prove very useful in the design of novel potent antimicrobials.
Introduction
The antimicrobial function of the β-lactam antibiotics is
determined by their ability to bind specifically and covalently
to the bacteria cell-wall-synthesizing enzymes (penicillin-binding
proteins or PBPs). PBPs catalyze the cross-linking reaction of
the peptidoglycans, a major structural component of the bacteria
cell wall, rendering it into a three-dimensional rigid network
which allows the bacteria to survive in a harsh environment.
Deprived of active PBPs, bacteria cannot sustain the rigidity of
their cell walls and are ruptured by osmotic forces. To defend
against β-lactam antibiotics, such as penicillins and cephalosporins, bacteria have developed β-lactamases. These enzymes
protect the bacteria from β-lactam antibiotics by hydrolyzing
their β-lactam ring and converting them into inactive compounds. The products of β-lactam hydrolysis by β-lactamases
have no noticeable affinity for the PBPs. Thus, β-lactamases
destroy the antimicrobial function of the antibiotic and provide
the bacteria with resistance to it. The widespread use of β-lactam
antibiotics in hospitals, agriculture, and household products has
caused the emergence of new multiple antibiotic resistant strains
of bacteria that carry β-lactamase genes (MAR).1-4 Nevertheless, β-lactam pharmaceuticals still remain extremely useful for
many infections.5 One of the solutions to the MAR problem is
to develop new β-lactamase inhibitors.6,7 These molecules can
then be administered together with older antibiotics to protect
them from hydrolytic inactivation by β-lactamases. Full knowledge about the mechanism of β-lactamase catalysis is important
for understanding the function and interactions of these enzymes
with substrates and inhibitors. This kind of information should
assist the development and evaluation of novel β-lactams to
overcome the resistance problem.
The hydrolytic reactivity of β-lactams is related to their
activity with β-lactamases. Nonenzymatic methanol-mediated
* To whom correspondence should be addressed:, Peter A. Kollman,
Department of Pharmaceutical Chemistry, University of California, San
Francisco, California 94143-0446. Phone: (415) 476-4637. FAX: (415)
476-0688. E-mail: [email protected]
hydrolysis (e.g. methanolysis followed by hydrolysis) of β-lactams mimics the enzymatic pathway and could be studied in
the gas phase and solution as a model reaction for hydrolysis
of β-lactams by β-lactamases. The methanol molecule corresponds to a truncated version of the catalytic serine residue of
β-lactamases. The energy profile for this reaction is expected
to be similar to the enzyme catalyzed reaction with some
differences in relative depths of the wells and heights of the
hills on the potential surface and with the exception that there
is no analogue for the Michaelis complex in β-lactam methanolmediated hydrolysis reactions.
It has been suggested that the strain in the four-membered
β-lactam ring increases the reactivity of β-lactams.8 β-Lactams
of slightly basic amines display 30-500-fold rate increase and
β-lactams of basic amines show ∼10 times rate acceleration
for hydroxide ion catalyzed hydrolysis compared to corresponding acyclic amides.9,10 However, as it has been pointed out in
refs 9 and 10, this rate enhancement cannot be solely attributed
to the ring strain. Another contribution to β-lactams increased
reactivity is the reduced amide resonance as a result of ring
fusion.11 Furthermore, the change in the carbonyl acidity and
ring nitrogen basicity caused by deviation of their geometries
from their equilibrium values in unstrained structures can also
lead to a decrease in β-lactam ring stability.9,10 In addition, an
ambiguity exists about the direction of nucleophilic attack.10
The theory of stereoelectronic control predicts an approach of
the nucleophile from the β-side of β-lactam, requiring the lone
pair of the ring nitrogen and the lone pair of the nucleophile to
be in an anti-periplanar orientation. However, several observations, such as the nonplanar shape of the fused ring system,
steric hindrance of the β-side, and the absence of the product
for intramolecular aminolysis of penicillins containing a primary
amino group at C-6β side chain, suggest that the nucleophilic
approach is from the R-side, at least for penicillins.10 This issue
remains controversial. More planar cephalosporins with a
primary amino group in their C-7β side chain undergo intramolecular aminolysis,12 and a similar reaction has been observed
10.1021/jp9923318 CCC: $18.00 © 1999 American Chemical Society
Published on Web 09/15/1999
Quantum Mechanical Study of β-Lactam Reactivity
J. Phys. Chem. B, Vol. 103, No. 40, 1999 8629
SCHEME 1
in 6-epi-ampicillin, a penicillin.13 In any case, the approach of
the nucleophile occurs from the R-side of β-lactams during the
enzymatic catalysis. We have investigated all these and other
β-lactam-activity related issues in our current work.
β-Lactams, known from the early 1940s, always were a
subject of thorough study not only for their antimicrobial
properties but also for their importance as synthetic intermediates. One of the first semiempirical molecular modeling studies
of β-lactams was conducted by Boyd. His work helped one to
understand the major steps of enzymatic catalysis, transition
state structures, and the effect of the leaving group on the
inhibitory activity of cephalosporins.14,15 Several semiempirical
calculations using a supermolecular approach and polarizable
continuum dielectric models implemented in AMSOL have
recently been done by Frau et al.16 to study the hydroxide-ion
attack on the carbonyl of N-methyl-2-azetidinone. That group
has also evaluated the reaction barriers for the hydroxide-ion
attack on the carbonyl of 2-azetidinone in gas using ab initio
calculations. Frau et al. have observed the preferential pathway
for the proton abstraction rather than an addition of the hydroxyl
to the carbonyl, which has forced Frau et al. to constrain the
N-H distance during the ab initio calculations. This problem
has been later overcome by using the reaction field method
(SCRF).17 Similar work has been conducted by Pitarch et al.18
on neutral and alkaline hydrolyses of N-methyl-2-azetidinone.
Del Rı́o et al.19 have studied the first step of methanolysis of
2-azetidinone as a part of the reverse reaction of enolate-imine
condensation using high level ab initio calculations and the
polarizable continuum dielectric model. All of the above studies
considered only the first stepsan addition of a nucleophilesin
the reaction pathway of methanolysis/hydrolysis of azetidinones
in solution. The common limitation of these previous studies
was that they did not study the subsequent ring-opening step
of the β-lactam by protonation, which is the accepted mechanism
for this step in solution and in β-lactamases and PBPs.10
The work presented in this manuscript is the first comprehensive attempt to study a more complete reaction pathway for
hydrolysis/methanolysis of a variety of β-lactams and variants
and correlate the experimental results with their structural and
physicochemical data. The series of reactions (1-16) have been
designed to evaluate the effect of ring strain, substituents and
loss of amide resonance on the reactivity of selected β-lactams
(Scheme 1). The reaction pathways in gas and in solution for
the reactions in Scheme 1 have been modeled to investigate
these issues. The differences in the barriers of activation for
the reactions of Scheme 1 as well as relative stability of the
reactants, intermediates, and products helped to correlate
reactivity with structures.
Methods
The GAUSSIAN94 and 98 ab initio modeling software have
been used for the geometry optimization and the energy
evaluation for each compound (reactions in Scheme 1). The HF/
8630 J. Phys. Chem. B, Vol. 103, No. 40, 1999
Massova and Kollman
TABLE 1.
reaction
1/2 (H2O + H2O f HO + H3O )
-
+
∆Gsolv, kcal/mol,
used for ∆Gwater
calculations
(calcd)b (exptl)
basis set
∆Ggas,
kcal/mol
gas (calcd)
HF/6-31+G*
HF/6-31+G*
HF/6-31+G*
G1
G2
G2MP2
CBS-Q
CBS-4a
CBS-Q
115.96
115.96
115.96
112.28
112.71
112.95
113.97
113.63
113.97
-100.54
HF/6-31+G*
41.54
-23.61
HF/6-31+G*
rank of
pKa,
∆Greaction,
∆Gwater,
kcal/mol
∆Gwater pKa
kcal/mol pH units,
water (calcd)
water
water
calcd water
∆Ggas + ∆Gsolv (exptl) (calcd) (exptl)
(exptl)
2(3)
11.30
12.65
15.88
8.61
8.92
9.09
9.84
9.59
14.56
19.11
14.01
-94.1d
15.42
17.26
21.66
11.74
12.17
12.41
13.43
13.09
19.87
-23.0
17.93
3(2)
13.14
12.62
9.25
222.05
-187.65 -187.6
34.40
4(4)
25.22
∼21
∼15.5
HF/6-31+G*
203.96
-164.17
39.79
5(5)
29.17
∼34
∼25
HF/6-31+G*
185.03
-170.41 -168
14.62
1(1)
10.72
-98.7c
-94.3d
-100.54
-100.54
-100.54
-100.54
-100.54
NA
6.49
4.75
a Foresman, J. B.; Frisch, Æ. Exploring Chemistry with Electronic Structure Methods; Gaussian, Inc: Pittsburgh, PA 1996; p 159. b Calculated
as described in the Methods section. c The experimental values were taken from Barone, V.; Cossi, M.; Tomasi, J. J. Chem. Phys. 1997, 107,
3210-3221. d ∆Gsolv calculated from the experimental data (Franks, F. Water: A ComprehensiVe Treatise. Aqueous Solutions of Simple Electrolytes;
Plenum Press: New York, 1973; Vol. 3, pp 55, 56. ∆Gsolv(H+) ) -260.5 kcal/mol; ∆Gsolv(HO-) ) -90.6 kcal/mol). The rest was calculated from
the reaction H3O+aq f H2Oaq + H+aq; pKa ) -1.7 which was adjusted for an activity of water ) 1. The free energy for H+ was calculated as
∆G(H+) ) RT - ST, whereas S ) Rln (e5/2kBT)/(p°Λ3); Λ ) (h2/2πmkBT)1/2, h is Planck’s constant and kB is the Boltzmann constant.
6-31+G* basis set has been used. The first step, the formation
of tetrahedral intermediate during the nucleophilic attack on
carbonyl, usually has a late transition state (TS) with the energy
comparable to that of the intermediate (ITS).20 The application
of the Hammond’s rule for the reactions in Scheme 1 leads to
the assumption that any energy change in the barrier heights
for transition species and a change in the free energies of their
corresponding tetrahedral intermediates are approximately linearly related. Therefore, the energies of tetrahedral species
(ITS1) were used to estimate the relative changes in the
activation barriers for the first step along the reaction pathways.
All products and most of the starting materials and intermediates
have several rotatable bonds. Only a few conformations have
been evaluated for each reaction in order to reduce the
computational time. Calculations showed that the species ITS2
represented merely a point on the energy surface rather than a
stationary state. Energy minimization of ITS2 resulted in a
structure which could be characterized as being obtained from
the species ITS1 by following the “reaction coordinate path”
and representing the structural conformer of a product. Thus,
we produced an additional conformation of a product by energyminimized the structure of ITS1 protonated at the ring nitrogen
(ITS2). The TS2 structure is expected to be similar to that of
ITS1. It has not been determined for the following reasons: (i)
The formation of the tetrahedral intermediate is a rate-limiting
step for alkaline hydrolysis of β-lactams.9,10 (ii) On the other
hand, nucleophilic hydrolysis of penicillin G catalyzed by
weakly acidic alcohols showed that a breakdown of the
tetrahedral species is a slower step.21 (iii) Furthermore, Bakowies
and Kollman20 demonstrated that TS1 and TS2 are similar in
structure and energy and both have a barrier height very close
to the energy of ITS1. The products (PR) of the acylation step
of the reactions had their starting structures chosen as close as
possible to the conformation of penicillin G in the acyl-enzyme
complex of Glu166Asn mutant of TEM-1 β-lactamase from
Escherichia coli.22 The polarizable continuum dielectric model
(PCM)23,24 currently implemented in GAUSSIAN98 was used
to evaluate the solvation free energies. The structures 17, 18,
and 19 were energy-minimized in gas and solution to estimate
the effect of the solvation on geometries. The structures
determined in gas and in water were essentially identical with
the standard deviation of 0.07, 0.03, and 0.04 Å, respectively.
These results are in a good agreement with the findings of
Barone et al.23,25 Furthermore, it has been shown in ref 25 for
the ring-opening reaction of cyclobutene radical that the values
for the barriers and the free energies of all the species which
were energy-minimized in solution were never higher than 0.4
kcal/mol from those determined for the structures energyminimized in gas and reevaluated by single-point calculation
in solution. Moreover, all values shifted in the same direction.
To estimate the possible systematic errors due to imperfection
of the PCM model, we evaluated the pKa’s for five small
molecules using different levels of ab initio calculations (Table
1). The PCM/UAHF model with Icomp ) 4 has been employed
to calculate the free energy changes in water. The differences
vary for each case, although overall trends were predicted
correctly (cf. rank column in Table 1). The free energy
difference of 4.4 kcal/mol (per one water molecule) was
observed when the free energy of formation of HO- and H3O+
from 2 H2O was calculated using the experimental values
reported in refs 23 and 26 (the second and the third entries in
Table 1). A considerable discrepancy has been found for the
experimental free energy of hydration of hydroxide ion reported
in ref 26 (-90.6 kcal/mol) and used in ref 23 (-106 kcal/mol)
to derive the new definition cavity parameters for ab initio
calculations of the solvent effect. Even though this potentially
dramatic difference could result in shift of as much as 15.4 kcal/
mol in the free energy values, the relative differences between
energies would still be the same because the same species (e.g.,
HO-) occurs in all reactions.
The effect of the geometry change in solution, electron
correlation, and the basis set were further investigated in
evaluation of the pKa of water using different protocols listed
in Table 1 and two additional ones. The structures of a water
Quantum Mechanical Study of β-Lactam Reactivity
molecule, hydroxyl anion and hydronium ion were first energy
minimized in water using PCM/UAHF parameter set and Icomp
) 2 for two basis sets HF/6-31+G* and B3LYP/6-31+G* then
the single point calculations have been carried out on the
optimized structures with Icomp ) 4, which has been shown
better to account for the solute escaped electron density outside
the cavity.23 None of the methods was able to achieve the
precision of the approach used by Barone et al.23 to develop
UAHF parameters. Similar results have been found by Schüürmann et al.27 who used an additional four basis sets with
polarization and/or diffuse functions (6-31G**, 6-31+G**,
6-311G(2d,2p), 6-311+G(2d,2p)) and with and without inclusion of the correlation effect (HF and MP2 levels). None of the
basis sets was able to predict the pKa’s of 16 carboxylic acids
better than the protocol developed by Barone et al.23 Therefore,
the following strategy has been used in this work: first, the
geometry was optimized in gas with HF/6-31+G* basis set,
and second the zero-point frequencies and the thermal energy
corrections were calculated in gas phase for 298.15 K; all the
structures of SM, ITS1, ITS2, and PR species were found to
have no imaginary frequencies and were true minimum structures; ultimately, a single-point energy calculation in water using
PCM/UAHF was conducted to evaluate the solvation free
energies with HF/6-31+G* basis set for anions and HF/6-31G*
basis set for cations and neutral molecules.
J. Phys. Chem. B, Vol. 103, No. 40, 1999 8631
SCHEME 2
Results and Discussion
In general, amides require harsher conditions for their
hydrolysis than esters, which is due to an amide resonance
stabilizing a ground state of the starting materials. This
stabilization can be lost in β-lactams, since four-membered ring
strain can partially destroy the amide resonance.11 However,
the crystallographic data shows planarity of the ring nitrogen
in monobactams, which is an indication that amide resonance
still exists in monocyclic β-lactams. Both penicillins and
cephalosporins have two fused ring systems. One is a β-lactam
ring and another, a thiazolidine or dihydrothiazine ring, respectively. The rings share a common amide nitrogen and a carbon
atom. Additional strain along the ring fusion causes the amide
torsion angle to be nonplanar, and this can contribute to the
loss of amide resonance. It can result in a lower activation
barriers for methanolysis/hydrolysis reactions of β-lactams than
those for reactions of similar acyclic amides. The four-membered
β-lactam ring could also have an effect on ring nitrogen basicity
and carbonyl acidity because of the deviation of bond angles
from their unstrained geometries, which could, in turn, effect
the stability of β-lactams. The reactions we studied (Scheme
1) have been designed to evaluate all these factors. Reactions
in Scheme 1 shows progressive increase in complexity of the
system, with the penicillin G shown under 14. Reactions 1,
2, 6, and 7 vs 3 and 8 give insight into ring strain issue,
since they compare analogous peptide and β-lactam reactants.
Reactions 8 and 9, 11 and 12 show the effect of nitrogen
basicity. Comparing reactions 9 with 10 and 12 with 13 helps
one estimate the influence of the adjacent five-membered
ring and substituents on this ring on the stability of the reactant and tetrahedral intermediate during hydrolysis or methanolysis.
The methanol-mediated hydrolysis of β-lactams mimics the
enzymatic pathway, with methanol acting as a catalyst and
representing a surrogate side chain of the catalytic serine residue
of β-lactamases. The acylation step has been studied in the
current work. The hypothetical energy profile is shown in
Scheme 2, whereas SM are the starting materials, TS is a
transition state, ITS1 is a tetrahedral intermediate, and PR are
the products. Assuming a linear dependency between the
activation barriers (TS1) and the free energies of ITS1, we have
used the latter as an estimate for the relative heights of the
activation barriers. The calculations showed that ITS2 was not
a stationary point on energy surface but rather was merely a
point on the reaction pathway between the transition state 2
(TS2) and the products. The ITS2 geometry optimization
resulted in one of the rotamers of the product.
It is impractical to do a full search for the transition state
along the reaction coordinate for such complicated models as
4, 5, and 11-16. Thus, we rely on the experimental and
theoretical studies that the structure and the free energy of TS1
are very close to that of ITS1.10,20 Therefore, we compare the
barriers for the wide variety of molecules using the ITS1 free
energies.
Formation of the Tetrahedral Intermediate. The PCM
formalism has been used to evaluate the free energy changes in
solution for the reactions discussed above. The results are
presented in Tables 2-4. Table 2 shows the activation barriers
for the first step of acylation of methanol by β-lactam in gas
and solution and the hydration free energies. For base-catalyzed
hydrolysis of β-lactams, the first step, the formation of
tetrahedral intermediate, is a rate-limiting step. In contrast,
protonation of the nitrogen is a rate-limiting step for hydrolysis
in noncyclic amides.10 Therefore, the activation barrier for that
step (∆G2,water) can be related to the observed kinetic rates of
the reactions. Interestingly, the numbers ∆G2,water are within the
range of the experimental values of 20-30 kcal/mol for the
activation barriers reported for simple amides,28,29 and the
experimental value of 16.7 kcal/mol has been determined for
the hydroxide-ion attack on 2-azetidinone.30 These numbers are
comparable to the barriers given in Table 2. The data in Table
8632 J. Phys. Chem. B, Vol. 103, No. 40, 1999
Massova and Kollman
TABLE 2: Barriers for the First Step of Neutral Hydrolysis/Methanolysis of β-Lactams
∆G1
∆G2
SM + ROH + H2O 98 SM + RO- + H3O+ 98 ITS1+ H3O+
a Two conformations of ITS1 with the different torsion angles around C-N rotatable bond have been considered. b The Boltzmann weighted
thermodynamic average ∆G2,water energy for 3 is ) 24.04 kcal/mol and for 17 is 21.05 kcal/mol. c The energies (∆G2,water) are less than the values
of the previous column (∆G1,water + ∆G2,water) by 30.84 kcal/mol for hydrolysis and 34.40 kcal/mol for methanolysis (Table 1). The barrier for the
hydrolysis of water is underestimated by 3.69 kcal/mol (Table 1; the first line). This could give rise a systematic error of as much as 7.38 kcal/mol
in overestimation of the values of ∆G2,water.
2 show several important trends. First, a 30-500-fold increase10
in rates of hydrolysis of β-lactams compared to the corresponding acyclic amides has been reproduced by our calculations.
That change corresponds to a 2.0-3.7 kcal/mol decrease in the
activation barrier. Our calculations gave a 4.2 kcal/mol decrease
in the barrier for hydrolysis (reactions 2 and 3) and a 3.5 kcal/
mol decrease in barrier for methanolysis (reactions 8 and 7) in
water. This effect was a combination of two changes: a decrease
the gas-phase barrier for β-lactam reaction and in the solvation
free energies, as the starting materials are converted to
tetrahedral intermediates. The result was not due to the relief
of a ring strain, because the ring was still closed in the ITS1
Quantum Mechanical Study of β-Lactam Reactivity
J. Phys. Chem. B, Vol. 103, No. 40, 1999 8633
TABLE 3: Free Energy Changes for the Second Step of the
Reactions
reaction number
ITS1+H3O+ f PR+H2O
∆Ggas
kcal/mol
∆Gsolv
kcal/mol
∆Gwater
kcal/mol
1a
2a
6b
7b,c
7b
8b,d
8b
9b,d
9b
10b,d
10b
10b,e
11b,d
11b
12b,d
12b
13b,d
13b
14b
15b
16b,d
16b
-226.43
-229.18
-229.62
-232.13
-231.61
-232.72
-231.75
-303.01
-301.90
-293.01
-296.45
-298.52
-217.34
-216.79
-283.86
-286.91
-278.45
-283.29
-284.52
-300.21
-281.19
-283.29
171.63
167.58
171.08
163.49
168.85
171.37
171.58
239.22
240.56
238.01
235.07
238.60
157.78
158.11
221.24
223.94
221.56
222.85
229.20
230.7
219.88
224.65
-54.80
-61.59
-58.54
-68.64
-62.76
-61.35
-60.17
-63.79
-61.34
-55.00
-61.38
-59.92
-59.56
-58.68
-62.62
-62.97
-56.89
-60.44
-55.32
-69.51
-61.31
-58.64
TABLE 4: Free Energies of the Reactions for the Selected
β-Lactams
a Hydrolysis reaction. b Methanolysis reaction. c See footnote a of
Table 2. d Several PR conformations have been considered, as discussed
in the Methods section. The top entry corresponds to the structure
resulting after energy minimization of the ITS1 species protonated at
nitrogen and the next entry corresponds to the structure similar to the
conformation of penicillin G in the acyl-enzyme complex.22 e An
additional conformer with the different thiazolidine ring puckering has
been investigated.
species. It was caused by relief of the strain on the carbonyl
carbon as it converts from a three-coordinated sp2-hybridized
carbon to a four-coordinated sp3-center.10 The introduction of
an electron-rich group at the nitrogen side increases the
activation barrier. In contrast, when one esterifies the carboxylate
group, the rate accelerates by 130-fold (2.9 kcal/mol decrease
in the activation barrier).31 The data in Table 2 show the
substantial barrier increase of 69.8-72.3 kcal/mol in the gas
phase when the carboxylic group is introduced at the nitrogen
side, (reactions 8 and 9; 11 and 12). Though this group
contributes favorably to the solvation energy, the activation
barrier in solution is calculated to increase by 1.5-5.5 kcal/
mol. The electron-rich carboxylate group destabilizes the
growing negative charge on the center of the nucleophilic attack,
resulting in an increased activation barrier.
The introduction of the second ring increases the rates of
hydrolysis 280-3900-fold which corresponds to 3.3-4.9 kcal/
mol decrease in the activation barriers.10 That was supported
by our calculations, which showed barriers decreased by 8.713.8 kcal/mol in the gas phase and 1.6-8.8 kcal/mol in solution
(reactions 9 and 10; 12 and 13; 16 and 14). The second ring
introduces an additional strain at the ring juncture resulting in
the elevated energy of the starting materials and therefore in
the smaller barriers to hydrolysis.
An electron-withdrawing group next to the carbonyl carbon
is known to have an inductive effect on reaction center, which
results in rate enhancement.10 Modeling these reactions in the
gas-phase did show a significant rate enhancement (reactions 8
and 11; 9 and 12; 10 and 13; a 14.8-17.7 kcal/mols decrease in the activation barriers). However, the opposite trend
for the solvation free energies of the species canceled this effect, resulting in almost the same or sometimes even higher
barriers for the reactions in water. These discrepancies with the
experimental observations could be due to a number of factors.
a Hydrolysis reaction. b Methanolysis reaction. c See footnote d of
Table 3. d See footnote e of Table 3.
First, it could be caused by inaccuracies in the solvent model.
A second possible source of error was that all possible
conformers were not sampled. We have chosen the conformation
of our compounds to be as close as possible to that of penicillin
G in the crystal structure of the acyl-enzyme intermediate.22 A
third possible source of the discrepancy is that a change in the
reaction mechanism could occur when a substituent was
introduced. The approach of the nucleophile could be from the
less crowded R-side for β-lactams. The approach for the less
substituted compounds may occur from the β-side as suggested
by the theory of stereoelectronic control.
8634 J. Phys. Chem. B, Vol. 103, No. 40, 1999
Massova and Kollman
Figure 1. The reaction profiles for the attack of the hydroxide ion on carbonyl of N-methyl-3β-hydroxy-2-azetidinone (17) for syn-periplanar (gray
lines) and anti-periplanar (black lines) approaches in the gas phase (A, C) and in water (B, D). All energies are shown relative to the sum of the
electronic energy of the starting materials in the gas phase and their solvation free energies in water. The tick mark on the vertical axis at 22.16
kcal/mol corresponds to the total Gibbs free energy for 18.
To investigate this issue we modeled reactions 3 and 17 in
gas and solution accounting for all possible conformers. The
hydroxyl group has been chosen as a model substituent at carbon
side of 17 because of its small size and similarity to an amide
nitrogen in terms of its hydrogen bond donating ability and the
inductive and field effects on the reaction center. The Boltzmann
weighted thermodynamic average energies were calculated based
on all conformers for 3 and 17. The free energy change for the
first step of the reaction was 21.05 kcal/mol for 17 and 24.04
kcal/mol for 3. That corresponds to a 160-fold rate enhancement
for 17, in agreement with the trends seen in the experiments
for similar β-lactams. The analysis of the tetrahedral intermediate revealed that the C-3β-hydroxyl substituent of 17 participated
in hydrogen bonding to the carbonyl oxygen, thus stabilizing
the growing negative charge on it. Interestingly, this hydrogen
bond formed in all tetrahedral species which had a β-acylamino
substituent at the carbon neighboring to the carbonyl of the
β-lactam ring. This hydrogen bond provided by the amide of
C-3(6,7)β-side chain of β-lactams/monobactams is reminiscent
of the oxyanion hole in proteins, with the difference that only
one amide is available to hydrogen bond to the oxygen on the
tetrahedral center. Hence, these results for 3 and 17 suggest
that the most likely explanation for the discrepancies between
the calculated free energies for 8 vs 11, 9 vs 13, 10 vs 13, and
experiment is incomplete conformational sampling.
The Approach of Nucleophile. We have also studied the
direction of nucleophilic approach, exploring the attack for all
possible conformers of 17. We started our study with the
approach from the less hindered R-side of 17. Compound 17
can serve as a minimal model for a β-lactam with a β-side chain.
It has an almost planar nitrogen and the hydroxyl at C-3β creates
the preference for the nucleophile to attack from the opposite
side. The theory of stereoelectronic control suggests that the
incipient C-O bond should be anti-periplanar to the nitrogen
lone pair. The ITS1 product of such approach is 18, though
this conformation is slightly less thermodynamically favorable
than 19 (Table 2, 0.61 kcal/mol difference) which can be formed
in a syn-periplanar approach of the hydroxide ion. The direction
of the nucleophilic approach for 18 and 19 is from the less
hindered R-side. It is analogous to the direction of the attack
on β-lactams with fused ring systems for which the approach
is dictated by the shape of the molecules, with the difference
that structures such as 18 cannot form for such lactams because
of the rigidity of their nitrogen at the ring fusion juncture. The
energy profiles for the reaction of hydroxide ion with 17
converting to 18 and 19 in the gas phase and in water are shown
in Figure 1,A-D. The current PCM solvation model implemented in GAUSSIAN was not ideal for the calculations of
free energies of solvation for the transition states and intermediate structures between the stationary states because of their
transient nature and mixed hybridization on the reaction center
and adjacent atoms. This made it difficult to fully validate such
models. The spike seen in Figure 1B is probably related to a
reassignment of cavity parameters as the assumed hybridization
on atoms changes during transition state formation. Nevertheless,
the plots shown in Figures 1,A-D seemed to be reasonable
and applicable for further analysis. The energy profiles for hydroxide-ion addition to 17 (Figures 1A,B) are very similar to
ones calculated for hydroxide ion addition to formaldehyde in
the gas phase and aqueous solution.32 The calculation showed
Quantum Mechanical Study of β-Lactam Reactivity
J. Phys. Chem. B, Vol. 103, No. 40, 1999 8635
Figure 2. Stereoviews of the energy-minimized structures of the starting material 17, transition states, and tetrahedral intermediates for 18 and 19.
The two right structures could be viewed with stereoglasses and the left combination represents the crossed eye projections.
the preference in the reaction pathway leading to the formation
of structure 18 (Figures 1,A-D). The investigation of both
pathways suggested that the barrier for the anti-periplanar
approach (18) was 2.6 kcal/mol lower in energy than for the
syn-periplanar attack (19). The relatively small difference in
barrier heights for 18 and 19 indicates that structure 19 will
form under thermodynamically controlled conditions and 18 will
be the major product for kinetically controlled reactions. The
transition state corresponded to the structure with a C-O bond
length of 1.85 Å for both pathways (Figure 2). Some interesting
structural changes could be seen. Thus, the C-O bond of the
former CdO bond in TS1 and ITS1 species originated during
syn-periplanar approach of nucleophile has more single bond
character than that of similar structures formed during anti attack
(TS: 1.245 Å (19) vs 1.233 Å (18); Figure 2).
We studied the hysteresis of the nucleophile approach/
departure by reconstructing the direct approach from the starting
materials and gradually decreasing the distance between hydroxide ion and the carbonyl carbon. The modeling in the
direction from the starting materials to the tetrahedral intermediates always resulted in species 18. We also modeled the reaction
pathway in the opposite direction from the tetrahedral intermediates 18 and 19, when the distance between the carbonyl carbon
and the nucleophilic oxygen was gradually increased. As the
distance got larger, both the carbonyl carbon and the nitrogen
became more planar, and at a distance of 2.1 Å the structure
8636 J. Phys. Chem. B, Vol. 103, No. 40, 1999
that originated from 19 collapsed to that one from 18, as a
demonstration for the preference of the anti-periplanar approach.
We have investigated the possibility of β-side attack of HOon 17 and our calculations demonstrated that the nature of this
attack for the simple compound was dominated by steric
hindrance. The β-side approach resulted in proton abstraction
rather than the addition of the hydroxyl to the carbonyl in the
gas phase. It is likely that β-side approach in water would not
abstract the 3β-OH proton, but still would be involved hydrogen
bond formation between the proton on the hydroxyl and the
nucleophile. This process would lower the basicity of the
nucleophile, which should result in a higher barrier for β-side
attack. This would also occur in monobactams with the C-3β
side chain and penicillins and cephalosporins with the C-6(7)β
side chain where the amide proton on the side chain can act as
a proton donor, stabilize the nucleophile, and inhibit it from
attacking the carbonyl bond.
The addition of the hydroxyl to the carbonyl bond of 17
occurred without barrier in the gas-phase resulting in an ITS1
species with the energy of 34.0 kcal/mol (syn-periplanar
approach) and of 37.6 kcal/mol (anti-periplanar approach) lower
than that of the SM. A small hill was observed at a 2.1 Å C-O
distance due to the HO- interaction with the proton at C-4. The
desolvation of the hydroxide ion in water resulted in a barrier
of 27.3 kcal/mol for the reaction in solution. The energies plotted
at vertical axes of Figures 1B and 1D are the sum of the
electronic energies and the solvation free energies (∆Ee +
∆Gsolv, whereas Ee is an electronic energy, and Gsolv is a
solvation free energy). The total Gibbs free energy differs from
this term by the sum of zero point energies and thermal
corrections: Gtotal ) Ee + Gsolv + E1; E1 ) ZPE + Ee298 +
Ev298 + Er298 + RT - ST, whereas ZPE is a zero point energy,
Ee298, Ev298, and Er298 are the thermal corrections to electronic,
vibrational and rotational energies and S is an entropy. The
inclusion of the ∆E1 will result in an increase for the free energy
of the tetrahedral intermediate by 14.25 kcal/mol for the synperiplanar approach and by 13.67 kcal/mol for the anti-attack.
We illustrated this in Figure 1B by a the mark at the vertical
axis at 22.16 kcal/mol which corresponds to the total free energy
of 18. Most of this change, ∼12.5 kcal/mol, is contributed by
the entropy loss that occurs at the latest stages of the addition
of the hydroxide making the value of the free energy of the
intermediate even more comparable to that of the transition state.
This quantitative evaluation supports the validity of the application of Hammond’s rule to the reactions considered here,
and, as a consequence, the use of the free energies of the
tetrahedral intermediates for the analysis of the activation
barriers and reaction rates discussed in this work.
The preference for the R-side approach of the hydroxide ion
for 17 leads to the more general support for R-side approach of
the nucleophile in reactions with penicillins with C-6β-side
chains and monobactams with C-3β-side chains, in which the
side-chain amide can participate in a hydrogen bond with the
negatively charged oxygen. This process will stabilize the
tetrahedral species. Another aspect favoring R-side approach is
the repulsion between the negatively charged oxygen on the
tetrahedral center and the carboxylate at C-3 of the thiazolidine
ring of penicillins (or C-4 of the dihydrothiazine ring of
cephalosporins) and the sulfonate group on nitrogen of some
monobactams. All these factors support the R-side approach of
the nucleophile for the basic and neutral methanolysis and
hydrolysis of β-lactams.
The Breakdown of the Tetrahedral Species. The energies
for the second step of the reaction are given in Table 3. The
Massova and Kollman
cleavage of the β-lactam ring of the cyclic compounds in the
gas phase is accompanied by the release of slightly more energy
than C-N bond cleavage in amides because of the relief of
ring strain. But the ring strain adds only fractions of a kcal/mol
to the energy released in the gas phase (reactions 7 and 8) and
this effect is masked in solution because of the opposite trend
in solvation free energies (Table 3). These numbers are quite
different from the estimate of 26-29 kcal/mol of ring strain,33
in agreement with findings of ref 10. Many β-lactams actually
release even less energy than N,N-dimethylacetamide upon C-N
bond cleavage. The cleavage of the monobactams is more
favorable than the ring-fused systems, despite one’s expectation
that the strain energy is larger in the latter. The second step is
not a rate-limiting step for the reactions of β-lactams; therefore,
it does not influence the overall rate of the reaction.
The Thermodynamics of the Overall Reaction. The total
free energy changes for reactions 1-16 are given in Table 4.
Calculations showed that most of the analyzed reactions of
β-lactams are exothermic, in contrast to what is observed for
the acetamide reaction, which is endothermic. The β-lactams
are less stable than peptides. The free energy of the reaction is
correlated with the barrier height for the first step. Monobactam
8 has a 3.5-kcal/mol lower barrier than 7 (Table 2). The total
difference in solubilities of the products and starting materials
for reactions 7 and 8 is about 1 kcal/mol (3.89 vs 4.77/4.98
kcal/mol, respectively, Table 4). As a result the methanolysis
of monobactam 8 releases 2 kcal/mol more free energy than
that of acetamide 7. This is the direct consequence of the
destabilization of the starting materials because of strain energy
in 8.
Another effect on reaction free energies is ring strain because
of ring fusion. The addition of the ring(s) lowers the free energy
of reaction (reactions 10, 13, and 15). The fusion of the second
ring increases the strain along the ring juncture which is relieved
when the β-lactam ring opens, and the reaction of monobactams
on average has a less negative/more positive ∆G(reaction) than
that of bicyclic penicillins (reactions 9 and 10; 12 and 13). One
can thus relate the release of ring strain to the calculated free
energy for the reaction. In addition, the methanolysis reactions
involve a less negative/more positive ∆G(reaction) than hydrolysis (compare reactions 6 and 1; 7 and 2).
The Ground-State Effect. The geometrical parameters for
compounds 1-16 are shown in Table 5. The calculations with
HF/6-31+G* basis set result in overestimated vibrational
frequencies.34 All the frequencies in Table 5 have been scaled
to 87% to represent the experimental value for acetamide (1).
The frequencies were then in reasonable agreement with
experiment (Table 6). To evaluate the ground state and kinetic
effects on hydrolysis of β-lactams, we determined the correlation
of their geometrical parameters and energies with the experimental kinetic data10 available only for compounds 3, 4, 5, and
14 of those molecules studied by us. An excellent correlation
was found (99%, r2 ) 0.978; Figure 3) between the β-lactam
CdO bond length and the observed second-order rate constants
for hydroxide ion catalyzed hydrolysis of β-lactams, thus relating
the stability of the starting materials to the activation barrier
height for the first step of the reaction. The CdO bond length
represents the cumulative effect of the various factors such as
ring strain, ring fusion, substituent on nitrogen and carbon sides
on the stability of SM. Interestingly, the monobactams have
the CdO bond length comparable to that of noncyclic amides,
in contrast to the bicyclic β-lactams, which is clear indication
that amide resonance still exists in the monocyclic system. The
mechanisms for the hydroxide ion and the methoxide-ion
Quantum Mechanical Study of β-Lactam Reactivity
J. Phys. Chem. B, Vol. 103, No. 40, 1999 8637
TABLE 5: Some Geometrical Parameters and Vibrational Frequencies of the Selected β-Lactams
TABLE 6: Experimental and Calculated Amide Vibrational
Frequencies
compound
acetamide
primary amides
secondary amides
tertiary amides
β-lactam (monocyclic)
β-lactam (multiple-ring system)
amide I band CdO amide I band CdO
stretching frequency stretching frequency
(experiment, cm-1) (calculated, cm-1)
1694a,b
1650a,b
1640a,b,c
1680-1630a,b
1760-1730d
1810-1750d
1695
na
1680-1662
1658
1739-1699
1742-1727
a
Solid. b Silverstein, R. M.; Bassler, C. G.; Morrill, T. C.; Robert
M. Spectrometric Identification of Organic Compounds; Wiley: New
York, 1991; p 419. c 1700-1680 in solution. d Various conditions of
measurements. Page, M. I. The Chemistry of β-Lactams; Blackie
Academic & Professional: New York, 1992; p 351.
catalyzed hydrolysis reactions should be similar. Therefore, we
expect to see a linear relationship between the activation barrier
heights and the β-lactam CdO bond length for methoxide-ion
Figure 3. The dependence of the second-order rate constants for
hydroxide-ion catalyzed hydrolysis of β-lactams on the ground-state
effect. Numbers near diamonds correspond to the compounds. kobs is
the observed second-order rate constant (M-1 s-1, experiment).10
catalyzed reactions just like we observed for hydroxide-ion
catalyzed hydrolysis. Probably the largest source of error in our
models is the imperfections of the solvation model used. The
deviation from the linearity between the calculated activation
barriers and the calculated β-lactam CdO bond lengths could
8638 J. Phys. Chem. B, Vol. 103, No. 40, 1999
Massova and Kollman
References and Notes
Figure 4. The correlation of the barriers for the methoxide-ion
catalyzed hydrolysis of β-lactams with the ground-state effect. ∆Gcalc,
is the calculated activation barrier for the first step of the reaction in
water (∆G2,water from Table 2).
serve as a measure for how well the PCM/UAHF method
reproduces the real solvation energies. The correlation of 77%
(r2 ) 0.598, Figure 4) has been found, which is indicative of a
reasonably good solvent model. As has been pointed out in ref
10, the pyramidal shape of amide nitrogen does not necessary
mean the loss of an amide resonance in the case when the
overlap of the lone pair orbital on the nitrogen still can be
achieved with the π orbital on the carbonyl. The length of the
β-lactam CdO bond is a good estimate of the degree of such
loss of amide resonance, which may be caused by ring strain,
ring fusion, substituent effects and other factors. Figure 3 shows
the strong correlation between the reactivity of β-lactams and
the loss of amide resonance (ground-state effect). The PCM
model has provided reasonable estimates for the solvation
energies in qualitative agreement with the experimental data
(Figure 4).
Conclusions
This work represents the first attempt to correlate the
experimental data to the geometrical and electronic structures
of a broad set of the important β-lactams as well as related
amides. The study has been done with the use of ab initio
methods and the PCM/UAHF solvent model. Generally good
agreement has been found between calculated barriers of
reactions and the kinetic parameters for the selected compounds.
The validity of the application of the PCM/UAHF model for
the estimate of the solvent contribution has been demonstrated.
The CdO bond length has been shown to correlate to the
β-lactam hydrolytic stability. It can be used as a criteria for
evaluation of the β-lactam susceptibility toward hydrolysis/
methanolysis and, as a consequence, toward hydrolysis by
β-lactamases and may relate to the β-lactam inhibitory activity
with PBPs. The findings in this paper shed light on the
contributions of the various structural elements to the reactivity
of β-lactams. That information can prove very useful in the
design of novel potent antimicrobials.
Acknowledgment. The authors thank the National Institutes
of Health for research support (Grant GM-29072 to P.A.K. and
Grant AI-09998 to I.M.). We thank Jed Pitera and Jim Caldwell
for their helpful comments on this manuscript.
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