J. Phys. Chem. B 2007, 111, 6507-6519
6507
The Hydrolysis of Amides and the Proficiency of Amidohydrolases. The Burden Borne by kw Guillermina Estiu and Kenneth M. Merz, Jr.* Department of Chemistry and the Quantum Theory Project, 2328 New Physics Building, P.O. Box 118435, UniVersity of Florida, GainesVille, Florida 32611-8435 ReceiVed: NoVember 20, 2006; In Final Form: March 2, 2007
The hydrolysis of small amides has garnered major attention due to its relevance to peptide hydrolysis, one of the most fundamental reactions of biology. Both experimental and theoretical research efforts have studied the reaction in different media, and a consensus has been reached regarding the specific acid- and basecatalyzed reaction pathways. Nevertheless, for the water reaction, large discrepancies between theoretical and experimental results are found in the literature. Herein, we report the results of theoretical calculations of formamide and urea hydrolysis at different pH values. Model systems have been built clustering one and two water molecules with the reactive amide. A careful analysis of the reaction pathways at different temperatures has allowed us to accurately reproduce available experimental data and to separate the water reactions from their acid and base counterparts. The relevance of the results in providing an accurate definition of the proficiency of amidohydrolases is discussed.
Introduction The reaction mechanisms for acid- and base-catalyzed hydrolysis of simple amides have received much attention due to their relevance to various biological processes.1-17 After intensive experimental and theoretical research, these reactions are among the best understood of any chemical process. At high pH, the system follows the “base-catalyzed” route, in which a hydroxide ion performs a nucleophilic attack on the carbonyl carbon atom next to an amide bond, forming a tetrahedral intermediate.4,5,10,15-23 At low pH, protonation of the carbonyl oxygen atom followed by nucleophilic attack carried out by a watermoleculeisknownasthe“acid-catalyzed”pathway.5,10,16,18-20,24-29 However, the noncatalyzed “water reaction”, which is more relevant for physiological pH, remains somewhat controversial.1,3-5,7,8,10-12,17-20,30-32 The reaction is known to be a very slow process at room temperature, and in spite of limited experimental evidence of its occurrence,10,18 a base-catalyzed mechanism seems to be favored, even in neutral aqueous solution. The base-catalyzed hydrolysis of formamide is by far the most extensively studied reaction of this type.4,5,9,10,17-20,22,26,33,34 The activation energy has been accurately determined, and theoretical calculations have reproduced the experimental values within acceptable errors.4,10,17,18,22,26,33,34 From these studies, the need to properly model the aqueous environment has become evident, because the formation of the tetrahedral intermediate, that follows hydroxide attack at the carbonyl carbon, proceeds with no apparent barrier in the gas phase.4,12 Moreover, in spite of its associated exothermicity, this gas-phase reaction is unlikely to occur due to the preference for hydrogen abstraction from an amine group by the reactant hydroxide. Monte Carlo simulations that account for the influence of solvent correct this trend and lead to a calculated free energy barrier of 27 kcal/ mol.4 This number is in close agreement with the one derived by Warshel et al.33 A better agreement with experiment has been attained by Klein and co-workers35,36 (deviation of only 1 kcal/ mol from the experimental free energies) from the computation
of 100 ps biased Car-Parrinello MD trajectories. The same authors evaluated the role of water participating actively in the reaction mechanism, which can act as the nucleophile after transferring a proton to the reacting OH. These two theoretical models attain the best agreement with the available experimental observations.5,10,18 The very recent work of Xiong and Zhan34 also attained near quantitative agreement with experiment. The mechanism of the acid hydrolysis of amides has been extensively studied as well,5,10,12,18-20,24,25 but theoretical approaches including the influence of the aqueous environment are recent, and most of them use N-methylacetamide (NMA) as the model system.24,25 For this substrate, Car-Parrinello molecular dynamics (CPMD) simulations support a mechanism in which a proton is detached from the incoming nucleophile (water) while it approaches the protonated carbonyl group. The released proton is solvated by an additional water molecule and protonates the amine group of NMA.24,25 This mechanism has been previously proposed for formamide on the basis of experimental studies,5 and the CPMD calculated activation free energy for NMA is in good agreement with these experiments.24 A similar mechanism has been analyzed as a possible reaction pathway for the uncatalyzed hydrolysis of NMA but was found to be unlikely using CPMD.30 Instead, a concerted mechanism was proposed, in which a hydroxide of a two-water ensemble attacks the carbonyl carbon while a proton of the second water is donated to the amine end (free energy ∼35), yielding a water assisted addition across the C-N bond.30 The same intermediate has been previously proposed by Antonczak et al. after quantum mechanical (QM) calculations that only incorporated two water molecules3 (compared with the CPMD approach, which included 56 water molecules). The analysis of these results demonstrates that the inclusion of more than two water molecules in the theoretical model does not open new mechanistic routes. Several water molecules have also been considered in recent QM calculations of the uncatalyzed hydrolysis of formamide.8 The QM-based mechanism relies on the assumption that formamide is stabilized in water as a solvated structure built from the
10.1021/jp0677086 CCC: $37.00 © 2007 American Chemical Society Published on Web 05/18/2007
6508 J. Phys. Chem. B, Vol. 111, No. 23, 2007 simultaneous hydrogen bond coordination of five water molecules. The hydrogen bond interactions are rearranged along the reaction pathway, but the number of molecules in the first coordination sphere is retained throughout. Nonetheless, only two water molecules are mechanistically relevant, leading to CO addition.8 The concerted mechanism is at odds with the results of steered CPMD simulations, which concluded that there is no need to disrupt the water coordination sphere of formamide, nor that of the incoming water molecule during reaction.17 From the comparison of the different theoretical approaches employed to date, it becomes evident that modeling solvent is crucial. Through the years, the application of different levels of theory and different solvent approaches to the study of the uncatalyzed hydrolysis of amides has resulted in reported free energy values ranging from 57 to 35 kcal/mol.3,4,7-9,11,12,15,17,30 Nevertheless, even the smaller calculated value does not lead to the experimental rate constant reported for the water reaction, which was estimated to be 10-10 s-1 from a two-point Arrhenius plot that uses data determined at high temperature (329 and 393 K).18 A value closer to the experimental value has been derived through the application of QM methodologies in a recently published article, but the calculated results were, we believe (Vide infra), obtained using an inappropriate kinetic analysis.8 A rate constant of the same order of magnitude has been estimated for the uncatalyzed hydrolysis of urea, by analogy to the experimentally determined value for tetramethylurea (10-11 s-1). The latter has been extrapolated from measurements at temperatures higher than 400 K, in solutions buffered at pH 6.8.31 Theoretical calculations have not yet addressed this system to our knowledge. In spite of the overall lack of success with quantum mechanical methods to date in explaining the uncatalyzed hydrolysis reaction, there is still a significant experimental and theoretical interest in explaining the mechanism of amide water hydrolysis, because the uncatalyzed reaction defines the reference state used to quantify the proficiency of hydrolytic enzymes.37-46 Moreover, the neutral hydrolysis of amide bonds is complicated by the potential availability of the acid and/or base reaction channels. This added mechanistic complication makes theoretical study even more difficult as the observed rate constant can be a superposition of several reaction channels. A rigorous experimental study of formamide hydrolysis has still left this issue as an open question.18 After a thorough analysis of the reaction at different pHs and temperatures (56 °C (329 K) and 120 °C (393 K)), Slebocka-Tilk et al. concluded that the water reaction can never contribute more than 40% to the observed rate at the respective pH minima.18 In principle, theoretical calculations, at an appropriate level of theory, should be capable of reproducing the experimental data and from there predict the kinetic data that cannot be determined experimentally. This kind of analysis would be certainly helpful for a full understanding of this deceptively simple reaction. This article describes a detailed analysis of the acid-catalyzed, base-catalyzed, and uncatalyzed hydrolysis of formamide and urea using theoretical tools. Our aim is to compare our results with the available experimental kinetic data. To this end, we have analyzed the energy associated with the first stage of nucleophilic attack, which defines the rate-determining step. Calculations have been done as a function of the temperature, in order to allow a rigorous comparison with the available experimental data. The hydrolysis of tetramethylurea is also analyzed, using kinetic descriptors derived using the inverse of the analogy applied by Callahan, Yuan, and Wolfenden.31
Estiu and Merz Methodology All of the calculations have been performed at the MP2/6311++G** level of theory, using the Gaussian 03 suite of programs.47 Structural parameters and energies for reactants, products, stable intermediates, and transition states were obtained using full geometry optimizations, with no imposed constraints. The search for stationary points on the potential energy surface followed gradient-based algorithms and quadratic synchronous transit (QST2) approaches. Critical points were further characterized by determination of the vibrational frequencies at the same level of theory. Bulk solvent effects (aqueous solution) were modeled via a polarizable continuum model.48,49 This approach does not allow us to explicitly model the desolvation or shedding of explicit water molecules as the reaction proceeds (as has been noted by other authors),17 but through the explicit incorporation of solvent molecules, we have captured some of these effects. For example, the influence of a second water molecule was analyzed by modeling it in a discrete manner and then comparing the results with those previously reported.3,11 Moreover, we find that the rate-determining step in all cases is the formation of the tetrahedral intermediate, so at least in our hands, desolvation effects appear to be less important that bond formation. The thermodynamic functions (free energy (G), entropy (S), and enthalpy (H)) were obtained from the appropriate partition functions, calculated at the temperature of interest using MP2/6-311++G** frequencies. The vibrational frequencies were obtained using the harmonic approximation, and at some of the higher temperatures examined herein, this approximation is less satisfactory. However, no indications that this approximation suffered catastrophic failures were observed in the results reported herein. Results and Discussion The Hydrolysis of Formamide. The Base-Catalyzed Reaction. The mechanism of the base-catalyzed reaction has been extensively studied both experimentally and theoretically.4,5,10,17-20,22,26,33,34 The reaction is first order in hydroxide concentration, with a free energy of activation of 21.5 kcal/ mol.18 Recent experimental studies have analyzed the reaction at different temperatures using kinetic isotope effects and supported a mechanism involving the direct nucleophilic attack of hydroxide on the carbonyl carbon atom to form a tetrahedral intermediate.18 As previously discussed, inclusion of the aqueous environment in the calculations prevents proton abstraction from formamide by the hydroxide ion.4 The latter is favored by 26 kcal/mol in the gas phase relative to nucleophilic attack at the carbonyl carbon. In order to account for this, different solvent approaches have been used to model the aqueous environment, leading to calculated activation free energies that bracket those derived from experiment.4,17 For example, steered CPMD simulations gave a value of 15 kcal/mol,17 while Monte Carlo simulations predicted a value of 27 kcal/mol.4 We have chosen to use a continuum approach, and to analyze in detail the different steps of the experimental study of the base-catalyzed hydrolysis.18 A similar study and approach has been reported recently by Xiong and Zhan, and they were able to obtain excellent agreement with experiment.34 With this goal in mind, we have evaluated the thermodynamic quantities associated with this reaction at 25, 56, and 120 °C. The solvent corrected optimized geometry of the transition state has a C-Ohyd distance of 2.04 Å. At this internuclear distance, the C-Ohyd-Hhyd angle is 112° (see TSFORN; see Figure 1). These structural characteristics closely resemble those obtained using steered CPMD simulations, which gave reported
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Figure 1. Computed geometry for the formamide anion/water hydrogenbonded complex (FORNWAT) and the TS for the addition of hydroxide to formamide (TSFORN). Color code: red, oxygen; blue, nitrogen; cyan, carbon; white, hydrogen.
TABLE 1: Free Energies for the Stable Species Associated with Base-Catalyzed Formamide Hydrolysisa 298 K 329 K 398 K
FOR + OH
FORN + WAT
FORNWAT
TSFORN
3.57 4.51 7.28
-6.46 -4.20 -0.50
0.0 0.0 0.0
21.39 21.46 22.40
a Energies (kcal/mol) are relative to FORNWAT (see more details in the text). Positive values indicate less stable species. Formamide (FOR), formamide anion (FORN), hydroxide ion (OH), and water (WAT). For FORNWAT and TSFORN, see Figure 1.
values of 2.2 Å and 110°, respectively.17 In order to properly determine the energy of the reaction, we have analyzed in detail the relative stability of the different species that define the initial state. Formamide and hydroxide are known to coexist in equilibrium with the formamide anion and water. We have found a third minimum on the potential energy surface, which can be described as a water molecule hydrogen bonded to the formamide anion (water/formamide anion adduct, FORNWAT, see Figure 1). Both the formamide anion and the FORNWAT complex are more stable than formamide and hydroxide (see Table 1). Nevertheless, complex formation after the approach of the nucleophile positions the reactive species in an optimum configuration to proceed to the next stage of the reaction, with no need to separate water from the formamide anion. In other words, because the equilibrium for proton transfer is fast and reversible, the most favored reaction channel follows FORNWAT formation, and the reaction barrier will be determined by the difference between its energy and the transition state (TS) energy (TSFORN). For this reaction pathway, the calculated activation free energy (21.4 kcal/mol (calcd), see Table 1) is in excellent agreement with experiment (21.5 kcal/ mol (expt) at 298 K).18 To better highlight the agreement between our data and those reported in ref 18, we used the Eyring equation to calculate the rate constants at the same temperatures used in the experiments. We obtain rate constants of 4 s-1 (398 K), 0.032 s-1 (329 K), and 0.0011 s-1 (298 K), versus the experimentally derived values of 3.2 s-1 (398 K) and 0.032 s-1 (329 K).18 The comparison considers pseudo-first-order kinetics, and therefore, the free energies are computed under standard conditions ([OH-] ) 1). The Acid-Catalyzed Reaction. The experimental study of the acid-catalyzed hydrolysis of formamide has led to a mechanism that is initiated by protonation of the carbonyl oxygen of the amide, which activates it toward nucleophilic attack by water.5 Nucleophilic attack is assisted by a second water molecule, that captures a proton which yields the neutral tetrahedral intermediate and H3O+, thereby avoiding formation of the unstable O-protonated tetrahedral intermediate. H3O+ also plays a role in the second step of the reaction, assisting in the release of
Figure 2. Computed geometry for protonated formamide hydrogen bonded with one and two water molecules (FORPW, FORPW2). Color code as in Figure 1.
TABLE 2: Free Energies for the Species Associated with Acid-Catalyzed Formamide Hydrolysisa 298 K 329 K 398 K
FOR + H3O
FORP + WAT
FORPW
TSFP3
TSFP4
1.51 3.64 8.09
-14.49 -13.51 -9.16
0.0 0.0 0.0
44.99 44.80 44.11
51.64 51.20 49.00
a Energies (kcal/mol) are relative to FORPW (see more details in the text). Positive values indicate less stable species. Formamide (FOR), protonated formamide (FORP), hydronium ion (H3O), and water (WAT). For FORPW, see Figure 2, and for TSFP, see Figure 3.
NH3 through protonation of the nitrogen atom. The reaction has been studied by solvent kinetic isotope effects, which show that the oxygen atoms in the tetrahedral intermediate are in equilibrium between the protonated and deprotonated states.5 The analysis of the kinetics as a function of temperature resulted in an activation free energy of 22.8 kcal/mol.18 The proposed mechanism has been studied theoretically for NMA using CPMD.25 The simulations used the path sampling algorithm of Chandler and co-workers,50 for the reactants defined as the O-protonated amide and water, and supported the transfer of a proton of the reactive water to an adjacent one. No reaction barriers were computed due to the computational expense of the calculations, but previous constrained CPMD simulations gave a free energy of activation of 19 kcal/mol,24 in close agreement with the experimental value 22.8 kcal/mol.13,14 We have analyzed different possible reaction pathways, searching for transition states and comparing the stability of the species that define the initial state of the reaction. Proton transfer from H3O+ to formamide is computed to be favored by 16 kcal/mol at room temperature. Analogous to the previous study in basic media, an additional minimum was found in which a water molecule is hydrogen bonded to protonated formamide through both the oxygen and the nitrogen atoms of the molecule (FORPW, see Figure 2). This adduct is nearly isoenergetic with the system consisting of formamide and H3O+ at room temperature (FOR + H3O, see Table 2) but is stabilized as the temperature increases, becoming 8 kcal/mol more stable at 398 K (see Table 2). However, it remains, at this temperature, 9 kcal/mol less stable than protonated formamide plus a water molecule (FORP + WAT, see Table 2). Moreover, in order to account for the experimentally proposed mechanism, a second water molecule has to be modeled in the reaction pathway, which will assist the nucleophilic attack through the release of H3O+. This model uses a H2O/H3O+ cluster as the reactive species, responsible for protonating the carbonyl oxygen and forming the gem-diol intermediate after H3O+ release. The optimized structure of the two-water adduct is also shown in Figure 2 (FORPW2). The relative energies are listed in Table 3. Following the same reasoning as that for the basecatalyzed reaction, we have chosen the adducts as the states that define the “reactants”, since once they are formed as part
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TABLE 3: Energies for the Species Associated with Acid-Catalyzed Formamide Hydrolysis, Following the Two-Water Mechanisma FOR + H3O + WAT FORPW + WAT FORPW2 TSFP2 298 K 329 K 398 K
-4.89 -2.76 -0.88
-7.20 -6.40 -7.19
0.0 0.0 0.0
23.84 22.84 23.72
a Energies (kcal/mol) are relative to FORPW2 (see more details in the text). Positive values indicate less stable species.
Figure 3. Computed geometries for several transition states identified in this work. Color code as in Figure 1.
of the proton transfer reaction, the reaction channels of lowest energy are directly available. In the search for the TS, we first modeled the experimentally proposed mechanism: concerted addition of two water molecules to protonated formamide with the second water molecule capturing a proton to stabilize the gem-diol intermediate. We identified two almost isoenergetic TS structures (TSFP1 and TSFP2, see Figure 3) in which the proton of the reactive water has been transferred to the second one, and can be formally described as a H3O+ species hydrogen bonded to an incoming OH-. The latter is forming an elongated bond with the carbonyl carbon. TSFP2 is 2 kcal/mol lower in energy than TSFP1 and 24 kcal/mol higher in free energy than the reactants, defined as FORPW2. TSFP2 allows the releasing H3O+ to assist the second step of the reaction, transferring a proton to the amide end. From the analysis of the structure of TSFP2, it can be determined that the same H3O+ released after proton subtraction can be involved in the second step of the reaction. The calculated free energy barrier (23.8 kcal/mol) is in excellent agreement with the experimental value (22.8 kcal),18 and, hence, supports the previously proposed mechanism. The previous mechanism has been suggested as a way to overcome the fact that H3O+ is not a strong nucleophile per se. As a consequence, the initial protonation of formamide becomes necessary to assist the reaction to proceed through a nucleophilic pathway. Nevertheless, water is still a poor nucleophile, and even in the case of protonated formamide, where the electrophilicity of the carbonyl carbon atom has been enhanced by protonation, water addition to either the CN (TSF3, see Figure 3) or the CO bond (TSF4, see Figure 3) leads to activation free energies of 45 and 51 kcal/mol, respectively.
As for the base-catalyzed reaction, we calculated the kinetic rate constants using the Eyring equation and the free energy values for the most favored pathway at different temperatures. The calculated rate constants can again be compared with the experimentally reported values. The calculated values of 0.77 s-1 (398 K), 3.8 × 10-3 s-1 (329 K), and 0.2 × 10-4 s-1 (298 K) are in good accord with the experimental values of 0.15 s-1 (398 K) and 3 × 10-3 s-1 (329 K).18 The Water Reaction. The interest in the uncatalyzed hydrolysis of amides lies in the quantitative determination of the proficiency of hydrolytic enzymes.7,18,44,51 Most of the work in this area has been carried out by Wolfenden and co-workers, who studied the hydrolysis of small dipeptides, reporting rate constants close to 10-11 s-1.43 The simplest amide, formamide, has been frequently used as the de facto model of the peptide bond. However, its neutral hydrolysis had not been considered in detail until 1981, when Hine estimated the rate constant for this reaction from kinetic measurements at high temperature (80 °C).10 Hine’s research has been followed by Slebocka-Tilk et al., who measured the kinetics of the hydrolysis of formamide as a function of pH at temperatures of 56 and 120 °C.18 In both cases, the observed kinetics has been fitted to a general expression, kobsd ) kH+[H3O+] + kOH-[OH-] + kw, from which kw was derived. In this way, rate constants for kw values of 3.6 × 10-9 s-1 (56 °C) and 1.1 × 10-6 s-1 (120 °C) have been estimated. In addition, a value of kw ) 1.1 × 10-10 s-1 has been estimated at 25 °C from a two-point fitting to an Arrhenius plot. These studies taken together have led to the conclusion that it is unlikely that conditions will be found that allow for the isolation of the water term from the corresponding acid and base reactions.18 In spite of the advances on the experimental front, the mechanism of the water reaction has not been uniquely determined. Theoretical studies have been numerous, and have certainly contributed since 1990, following the development of improved solvent treatments. Nevertheless, neither the modeling of the solvent using a continuum approach11 nor the definition of discrete water molecules that participate in the reaction,3,11 or even the treatment of 27 solvent molecules at the QM level using Car-Parinello MD simulations,17 has succeeded in reproducing the experimentally determined rate constant but resulted in calculated free energies close to 50 kcal/mol. The results of quantum chemical calculations have been recently thoroughly reviewed and compiled by Gorb et al.,8 and will not be re-examined in detail here. An inspection of the data published to date shows that the primary origin of the difference between the values reported by different groups relies on the definition of the initial state.8,11,17 This choice is complicated by the fact that the formamide molecule in solution is solvated, and the reactive water can either belong to the solvation shell or to bulk solvent. In the case of the base- and acid-catalyzed reactions, a water molecule properly positioned for nucleophilic attack can be viewed as helping the hydrolytic pathway. However, the socalled “adducts” are lower in energy than the separated species in the catalyzed reactions (making it clear that they need to be included in the analysis) but higher in energy in the uncatalyzed one.11 At the present level of theory, when a polarizable continuum model (PCM) approach is used to model the solvent, water and formamide are stabilized as separated species by solvation, mainly due to the contribution of the entropic term to the free energy. In the base- and acid-catalyzed reactions, the interactions between the nucleophile and electrophile overcome this effect. In this way, in a neutral aqueous environment
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TABLE 4: Free Energies for the Stable Species Associated with Uncatalyzed Formamide Hydrolysisa 298 K 329 K 398 K
FOR + H2O
FORWAT
TSFOR
0.0 0.0 0.0
9.76 8.72 7.09
60.42 59.24 57.22
a Formamide (FOR), formamide/water complex (FORWAT), and water (WAT). For TSFOR, see Figure 5. Energies given in kcal/mol, relative to the separated species. See more details in the text.
TABLE 5: Free Energies for the Species Associated with Uncatalyzed Formamide Hydrolysisa FOR + 2H2O FORWAT2N FORWAT2O TSFOR2CN TSFOR2CO 298 K 329 K 398 K
0.0 0.0 0.0
23.65 21.96 18.03
20.27 19.13 15.25
60.11 58.35 53.78
64.07 62.25 57.48
a Formamide (FOR), formamide/water complex (FORWAT), and water (WAT). For FORWAT2N and FORWAT2O, see Figure 4. For TSFOR2CN and TSFOR2CO, see Figure 5. Energies in kcal/mol are relative to the separated species.
and at the present level of theory, separated formamide and water are 9.7 kcal/mol lower in energy than a formamide/water adduct, and 20 kcal/mol lower than a formamide/two-water complex (Tables 4 and 5). Nevertheless, it has been recently proposed, on the basis of the results of MD simulations with QM/molecular mechanical (MM) methods, that formamide is stabilized in water through hydrogen bond interactions with five water molecules,6 and the entire system yields a TS 35 kcal/ mol higher in energy than the original cluster. In this mechanism, only two water molecules out of the five change their coordination on the way to the TS, defining a reaction pathway that can be described as a two-water CO addition.8 This description is somewhat arbitrary, because it does not consider how the hydrogen bonds of the nonreacting water molecules change along the ascent to the transition state. The TS associated with a two-water CO addition, as well as a similar one involving a two-water assisted CN addition, have been previously described by Antonczek et al.3 and by Kallies and Mitzer.11 A free energy of 38 kcal/mol was calculated in the first case, with no model for the solvent, which led to the overstabilization of the formamide/two-water adduct.3 Solvent corrections included in the work of Kallies et al. increased this value to 48 kcal/mol.11 A different description of the neutral hydrolysis has been derived from steered Car-Parrinello MD simulations, which favored a reaction pathway in which the reacting water does not belong to the formamide solvation shell but to the bulk water.17 Within this framework, the water/formamide approach does not induce relevant changes in either of the reactant’s solvation shells. These results gave a free energy of 40 kcal/mol, and have been criticized for being based on the definition of a constrained reaction coordinate that does not allow one to capture concerted reactions.30 A similar mechanism was supported by Monte Carlo simulations, leading to a calculated free energy of 27.3 kcal/ mol.4 In spite of the proximity of this value to the experimental one, it has to be noted that the associated TS was obtained using a CNO angle constrained to the value optimized for the tetrahedral intermediate, and may correspond to a late TS.4 Our results are summarized in Tables 4 and 5 and the structures shown in Figures 4 and 5 (adducts and TS’s, respectively). The results in Table 4 show that one-water CO addition leads to a calculated free energy of 50.7 kcal/mol relative to FORWAT and 60.4 kcal/mol relative to the separated species.
Figure 4. Computed geometries for several hydrogen-bonded complexes identified in this work. Color code as in Figure 1.
When a second water is considered to assist the reaction, inspection of the data in Table 5 shows a preference for CN addition over CO addition. The free energies are 36.5 and 43.8 kcal/mol relative to the two-water/formamide adducts but 60 and 64 kcal/mol relative to the separated species. When measured relative to the separated species, the activation free energy decreases at 398 K to 54 and 57 kcal/mol, respectively. Our results can be compared with those of Gorb et al.,8 who obtained a similar activation free energy for the CO addition (35 kcal/mol) at the B3LYP/6-31(G) level of theory, for the energies corrected for solvent effects (PCM) after gas-phase optimization. This level of theory was unable to identify a TS for CN addition. Moreover, in the TS for CO addition reported in ref 8, proton transfer from a water molecule to the carbonyl oxygen is more advanced than in ours, a fact that we associate with the lack of solvent modeling during the optimization. The kinetic rate constants derived from all of the calculated free energies are slower than the experimental values regardless of the definition of the reactants.18 Nevertheless, at this point, the comparison between experimental and theoretically derived kinetic parameters deserves further discussion. In any comparison between experimental and calculated rate constants, one should note that the experimental values correspond to pseudo-first-order constants (k′); that is, they include the concentration of water, according to
rate ) k[form][wat] ) k′[form]
(1)
Hence, if we compare the rate constant derived from the free energy difference between the separated species and the TS, we have to multiply the resulting k value by 55 M. However, if we consider the reaction starting from the adduct (FORWAT or FORWAT2), the concentration of water term no longer has to be included in the equation. For the reaction involving two water molecules, the kinetic expression to be used is52,53
rate ) ka[FORWAT2] ) kaKeq1[FORWAT][wat] ) kaKeq1Keq2[form][wat]2 (2) where Keq1 is the equilibrium constant for the reaction
FORMWAT + WAT w FORWAT2
(3)
and Keq2 corresponds to
FORM + WAT w FORWAT
(4)
The rate constant calculated for the reaction starting from the adduct (ka) has to be corrected by the equilibrium constant if the water concentration is included in the kinetic expression.
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Figure 5. Computed geometries for several transition states identified in this work. TSFOR represents a one-water transition state, while TSFOR2CN and TSFOR2CO represent two-water transition states where water is added across the CN and CO bonds, respectively. Color code as in Figure 1.
These constants are associated with free energy values according to
ka ) A exp(-∆Ga†/RT)
(5)
Keq ) exp(-∆Geq)/RT
(6)
The combination of eqs 5 and 6 leads to
ln ka + ln K ) ln A - (∆Ga† + ∆Geq)/RT ) ln k
(7)
where, according to eq 2, k is the rate constant associated with the reaction starting from the separated species. Using an appropriate treatment of the kinetic equations implies the use of k rather than ka when the concentration of water is included in the equation. This procedure was not followed by Gorb et al., who multiplied their calculated ka value by the concentration of water to the second power.8 Even after correction by the water concentration, we obtain rate constants of the order of 10-30 or 10-28 s-1 for the reactions involving one or two water molecules, respectively, relative to the separated species. A value closer to experiment (6.6 × 10-14 Versus 10-10 s-1) is obtained when the reaction is considered to start from the two-water/formamide adduct, assuming that the species have reached equilibrium when the kinetic experiment starts. However, our ability to match experiment quite closely for the acid- and base-catalyzed reactions strongly suggests that we still do not have the correct model for this reaction mechanism. In the case of the acid- and base-catalyzed reactions, we worked from the adducts (or the initially formed complexes after protonation or deprotonation) rather than the separated species. This was because the adducts were more stable. Moreover, it was essential to work from these adducts rather than the separated species in order to obtain near quantitative agreement with experiment. In the case of the uncatalyzed reaction, the complexes are significantly less stable than the first formed adducts, suggesting that these reactions should have their thermodynamic quantities evaluated from the separated species. When this is done, very large free energies of activations for the uncatalyzed reaction are obtained, and even if one starts the thermodynamic analysis from the two-water/ formamide complexes, only qualitatively correct agreement with experiment is obtained. As a physical chemist, one is trained to start from the lowest energy reactants, which in this case are the separated species; however, the model we are working from involves a continuum solvation shell that has one or two additional explicit water molecules brought into the proximity of formamide. Potentially the appropriate starting state is really the one- or two-water adduct, since upon the dissolution of formamide into water an entropic penalty is paid to bring water molecules into the solvation shell of formamide. Hence, by including the largely entropic penalty in our calculations, we may be double-counting this when we compare our results to
experiment. Furthermore, if we assume that the water molecules are in equilibrium with the adducts, it can be shown using standard kinetics analysis that the barrier is defined from the adduct to the transition state and not from the separated species (see the Supporting information). Regardless, further analysis of this reaction is necessary to reconcile the differences between theory and experiment. We now have all of the necessary information to address this in detail below. The ObserVed Water Rate Constant. In order to compare our results with those determined experimentally,18 we need to realize that the experimentally observed value is built from three contributions:
kobsd ) kH+[H3O+] + kOH-[OH-] + kw
(8)
kw is determined as the observed rate constant for the pH at which the other contributions are minima.18 It can be easily demonstrated (zeroing the derivative of kobsd as a function of pH) that this minimum will occur when
[H+] ) (kOH-Kw/kH+)1/2
(9)
kobsd ) 2(kH+kOH-Kw)1/2 + kw
(10)
For this pH,
where Kw is the water autoprotolysis constant. If we insert our calculated values into eqs 9 and 10 [Kw ) 10-11.9 at 393 K, Kw ) 10-13.1 at 293 K; see Slebocka-Tilk et al. (2002)], we obtain a minimum pH of 5.64 and an observed rate constant of 3.5 × 10-6 s-1 for the reaction at 393 K, in excellent agreement with experiment (pH 5.4 and k ) 3 × 10-6 s-1, respectively).18 The resulting value is not dependent on whether we use the kw value calculated from the separated species (1.2 × 10-17 s-1, one-water reaction; 6.6 × 10-14 s-1, two-water reaction) or from the adduct (2.2 × 10-15 s-1, one water molecule; 1.7 × 10-7 s-1, two water molecules), because the first term in eq 10 is at least 1 order of magnitude larger. For the reaction at 329 K, the calculated values (pH 6.04, pkobsd ) 8.16) are also in excellent agreement with experiment (pH 6.1, pkobsd ) 8.1).18 Nor in this case does kw affect the value of kobsd, as the calculated values (1.0 × 10-25 s-1, onewater reaction; 2.1 × 10-23 s-1, two-water reaction; 1.2 × 10-21 s-1, one-water adduct; 3.2 × 10-12 s-1, two-water adduct) are almost 3 orders of magnitude smaller than the first term of eq 10 (7 × 10-9 s-1). This comparison indicates that the value of kobsd that matches the experimental data is determined by the rate constants for the acid- and base-catalyzed hydrolysis, and does not depend on the value of kw. Moreover, the agreement between theory and experiment is excellent. For the reaction at room temperature, the minimum pH would be 6.13, and the observed rate constant, kobsd ) 1.4 × 10-11 s-1. The value of kw again does not affect kobsd either, because it is computed to
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Figure 6. Red line: NLLSQ fit of the data reported in ref 18 for 120 °C (black dots), using the equation kobsd ) kH+[H3O+] + kOH-KW/[H3O+] + kw(Chi2 ) 1.6217 × 10-12, R2 ) 0.99012).
be 1 × 10-14 s-1 (two-water adduct), 3.2 × 10-25 s-1 (onewater adduct), 1.7 × 10-30 s-1 (one-water reaction), and 1.3 × 10-28 s-1 (two-water reaction). Going one step further, we have analyzed the fitting procedure used by Slebocka-Tilk et al. to derive the rate constant for the water reaction. The authors derived kw by fitting of the experimental values to eq 8, with and without the kw term. We have reproduced their analysis for the data reported at 120 °C, obtaining the results shown in Figure 6. From this fitting (red line, Figure 6, all rate constants are s-1), the values kH+ ) 0.1730 ( 0.0276, kOH- ) 3.9466 ( 0.1983, and kw ) (1.2411 ( 0.7606) × 10-6 have to be compared with kH+ ) 0.1500 ( 0.0200, kOH) 3.2000 ( 0.2400, kw ) (1.0900 ( 0.2900) × 10-6, reported in ref 18. The green lines represent the values corresponding to the fitting without the last term of the equation. The difference between the two lines at minimum pH has been associated with the contribution of the water reaction. However, if the errors associated with the fit are examined in more detail, the blue lines in Figure 6 are obtained after addition and subtraction of the associated errors. It is easily seen that the green curve lies within the estimated error. Because of this, we do not consider that this mathematical treatment of the experimental data allows for the direct determination of kw. The error associated with the numerical fitting of the experimental data was also noted by Slebocka-Tilk et al., where they noted that “it is unfortunate that our best-fit water constants must have a 10-30% error which limits our ability to make conclusions and predictions”. However, our fitting of the same data suggests that the errors are large enough to obscure the true value for kw. In light of this, we can make computational predictions regarding the value of kw, and these are, at 298 K, 1 × 10-14 s-1 (two-water adduct), 3.2 × 10-25 s-1 (one-water adduct), 1.7 × 10-30 s-1 (one-water reaction), and 1.2 × 10-28 s-1 (two-water reaction). These previous analyses lend further support to the accuracy of the computational results presented in this article. From this analysis, we believe that the experimental value for kw has not been realized for formamide. Indeed, in order to
reconcile theory and experiment, the unambiguous determination of kw is critical and this is the challenge faced by experiment. The Hydrolysis of Urea. Urea is unusually stable due to its resonance stabilization (estimated to be 30-40 kcal/mol),54 making the hydrolytic decomposition a very slow reaction. Its experimental determination is further complicated by the existence of a competitive reaction, in which urea decomposes unimolecularly, to give isocyanic acid and the ammonium ion.1,2,16,55-63 Because the latter is favored at any pH,58-63 the hydrolytic decomposition of urea has never been observed. In spite of this, research has been carried out,7,11,31,55-57,64,65 in part to determine the proficiency of urease and related amidohydrolases. As part of this effort, the kinetic parameters of the water reaction have been recently extrapolated for urea from the values determined for tetramethylurea, using the methylation of the amino groups to block the elimination pathway.31 The experimental results were not in good agreement with those obtained theoretically for the uncatalyzed hydrolysis of urea.7,64 Recently, Alexandrova and Jorgensen66 have reported a combined QM/MM simulation of the elimination reaction and neutral hydrolysis of urea in water using the PM3/PDDG Hamiltonian. While their computed activation energy for elimination (∼37 kcal/mol) was higher than experiment (4.59 kcal/mol depending on the experimental value used), neutral hydrolysis was only 3 kcal/mol higher (∼40 kcal/mol), giving a rate ratio of the hydrolysis Versus elimination pathway that is in good agreement with experiment on TMU.31 In the following sections, we discuss the kinetics of the base-, acid-, and water-catalyzed hydrolysis of urea, following the same approach as that used for formamide, whose accuracy relative to experiment has been demonstrated already. We show the results of calculations for temperatures between 298 and 498 K, aimed at comparing the kinetics with the experimental values reported for tetramethylurea. In doing so, we are aware of the fact that N-substitution not only prevents the elimination mechanism but also changes the electrophilicity of the carbonyl carbon atom. However, as noted by Wolfenden and co-workers,
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Figure 7. Computed geometry for the urea/hydroxide hydrogen-bonded complex (URNWAT) and the transition state for the addition of hydroxide to urea (TSURN).
Figure 8. Computed geometries for several hydrogen-bonded complexes of urea identified in this work.
TABLE 6: Free Energies for the Stable Species Associated with Base-Catalyzed Urea Hydrolysisa
TABLE 7: Free Energies for the Species Associated with Acid-Catalyzed Urea Hydrolysisa
298 K 398 K 498 K
UR + OH
URN + WAT
URNWAT
TSURN
-2.01 2.57 4.83
-9.66 -2.82 0.72
0.00 0.00 0.00
24.85 26.54 25.85
a Energies (kcal/mol) are relative to URNWAT (see more details in the text). Positive values indicate less stable species. Urea (UR), urea/ hydroxide complex (URNWAT), hydroxide + urea addition TS (TSURN), and hydroxide ion (HO). See Figure 7 for URNWAT and TSURN.
tetramethylurea should be a good model for urea hydrolysis and Vice Versa. In our efforts, we take advantage of the same principle espoused by Wolfenden and co-workers, but in our case, it reduces the computational cost of doing detailed calculations on tetramethylurea. Base-Catalyzed Hydrolysis. The experimental studies of the decomposition of urea in basic media indicated that the reaction follows an elimination mechanism.58,62,63 This reaction is first order in urea, with no nucleophilic attack involved. The hydrolytic decomposition has never been captured, nor has it been theoretically studied to our knowledge. We found the reaction to follow a similar pathway as the one described for formamide. In this case, the transition state shows a C-Ohyd distance of 2.05 Å and a C-Ohyd-Hhyd angle of 98° (see TSURN, Figure 7). The bending of the OH allows a weak hydrogen bond between the hydroxyl hydrogen atom and the carbonyl oxygen atom. As in the case of formamide, we found that proton abstraction by hydroxide is favored at temperatures between 298 and 498 K (see Table 5). A third stable species was also identified on the potential surface, in which hydroxide is stabilized by hydrogen bond coordination with both amino ends of urea (see Figure 7). Proton transfer to the hydroxide ion is not completed in this adduct (URNWAT, see Figure 7), as was found in the case of formamide (see FORNWAT, Figure 1), in agreement with the lower acidity of urea.67 This OH/UREA adduct is 9.6 kcal/mol higher in energy than the anion of urea plus water at 298 K but nearly isoenergetic with the same species at 498 K. In this way, at high temperature, it offers a more favorable reaction pathway for the reaction to proceed to the TS (see Table 6). The experimental study of the hydrolysis of tetramethylurea, from which the kinetic parameters for neutral hydrolysis of urea have been estimated, was performed between 400 and 525 K.31 At these temperatures, the kinetic parameters are determined by the free energy difference between the TS and the adduct (URNWAT). For these conditions, we calculate a free energy of activation of 26 kcal/mol (498 K), which leads to kinetic rate constants of 3.2 × 10-6 s-1 (298 K), 2.1 × 10-2 s-1 (398K), and 48 s-1 (498 K). At room temperature, the separated species are more stable than the adduct. However, as previously
UR + URP + H3O+ WAT URPWN URPWO TSURPWN TSURPWO 298 K 18.19 398 K 21.21 498 K 24.15
-4.95 -1.57 1.13
0.00 0.00 0.00
12.24 12.15 11.42
71.53 70.28 68.96
71.90 70.72 69.59
a Energies (kcal/mol) are relative to URPWN (see more details in the text). Positive values indicate less stable species. Urea (UR), protonated urea (URP), water hydrogen-bonded complex to amino hydrogens (URPWN), water hydrogen-bonded complex to the protonated urea carbonyl (URPWO), hydronium ion (H3O+), and water (WAT). For URPWN and URPWO, see Figure 8.
discussed for the case of formamide, we pointed out that the adduct can be formed during the (fast) proton transfer step, and the mechanism can then proceed from this adduct before water release. Under this assumption, the reactant is defined by the adduct (URNWAT), even at lower temperatures. Regardless of the temperature, the rate constants are smaller than the corresponding ones for formamide. This observation is in agreement with the lower resonance stabilization of the latter, which results in a smaller energy penalty for pyramidalization of the carbonyl carbon. Acid-Catalyzed Hydrolysis. Similar to the base-catalyzed reaction, the acid-catalyzed hydrolysis of urea follows an elimination reaction to give isocyanic acid and NH4+.58,59,61-63,68 No experimental or theoretical studies have been reported for the hydrolytic decomposition pathway to our knowledge. In our theoretical analysis, we found that protonation of the carbonyl forming URP + WAT stabilizes the system by ∼23 kcal/mol relative to urea and H3O+ at any of the temperatures considered. Nevertheless, the water molecule can be stabilized by hydrogen bond coordination to protonated urea (see URPWN and URPWO, Figure 8). Bifurcated coordination with both amino ends gave a structure that is only 5 kcal/mol less stable than protonated urea and water at 298 K and is more stable by ∼1 kcal/mol at 498 K (see Table 7). Another coordination mode is possible, which includes the protonated carbonyl oxygen and one of the amino ends (URPWO). The data in Table 7 show that the resulting adduct is 12.2 kcal/mol higher in energy than the first one at room temperature and 11.4 kcal/mol higher at 498 K. We have considered the possibility of a second water assisting the attack on protonated urea by capturing a proton of the reactive water in a manner similar to what was described for formamide. For a consistent model, the adduct involved in proton transfer, from which the reaction proceeds, has to include two water molecules coordinated to protonated urea. Two possible coordination modes were considered, which are shown in Figure 9. Their relative energies are reported in Table 8. In spite of the lack of experimental information on the possible mechanism of the acid-catalyzed hydrolysis of urea,
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Figure 9. Computed geometries for hydrogen-bonded complexes of protonated urea and two water molecules identified in this work.
TABLE 8: Free Energies for the Stable Species Associated with Acid-Catalyzed Urea Hydrolysisa UR + H3O + URPWN + WAT WAT URPW2N URPW2O TSURP1 298 K 329 K 398 K
-4.55 1.69 8.53
-22.90 -21.08 -15.62
0.00 0.00 0.00
-0.56 -0.69 -0.81
36.02 35.01 34.39
a
Urea (UR), two-water hydrogen-bonded complex to the amino hydrogens (URPW2N), two-water hydrogen-bonded complex to the protonated urea carbonyl (URPW2O), hydronium ion (H3O+), and water (WAT). For URPW2N and URPW2O, see Figure 9. For TSURP1, see Figure 10. Energies (kcal/mol) are relative to URPWN (see more details in the text). Positive values indicate less stable species.
we have searched for transition state structures similar to those found for formamide. However, we cannot support unambiguously one mechanism over another. For the case of formamide, there were experimental data to which our theoretical model could be compared, while in the case of urea (or tetramethylurea) this is lacking.5,10,18,26 For the case of urea, one can assume that a similar mechanism would be followed, but there is no experimental evidence that may help to further discern the most favored reaction pathway. Moreover, even assuming a similar reaction pathway, there is an additional uncertainty for the case of urea, which is associated to whether URPW2N or URPW2O would define the first step of the reaction. Nevertheless, unlike the one-water adducts, the two-water adducts are nearly isoenergetic. On the basis of our experience with formamide, we were able to identify a transition state similar to TSFORP1 (see TSURP1, Figure 10). However, in the case of urea, the H-bond interactions do not clearly describe a proton abstraction from the reactive water. The H-O distance is only elongated to 1.07 Å when the O(water)-C distance is 1.54 Å (see Figure 10). This TS is 36.6 kcal/mol higher in energy than the most stable two-water adduct (URPW2O) at 298 K. The different characteristics of the TS structures for urea and formamide hydrolysis can be understood on the basis of the higher charge (or enhanced electrophilic character) on the carbonyl carbon in the former relative to the latter (calculated Mulliken populations on the carbon atom: 0.46 for urea, 0.24 for formamide). This higher charge appears to allow water attack without further assistance of a second water molecule. Thus, we have also analyzed the reaction pathways for one-water addition in detail for urea. Two TS’s were identified, associated with water addition across the C-N bond of urea protonated on the carbonyl oxygen atom, and with water addition across the CO bond of urea protonated on an amino nitrogen atom. The TS structures are shown in Figure 10, and the calculated free energies are reported in Table 8. The activation free energy is again dependent on the state that we associate with reactants.
Figure 10. Computed TS geometries for the acid-catalyzed hydrolysis of urea. Color code as in Figure 1.
From our computed results, the comparison of the total energies of TSURP1 (-377.644 hartrees) and TSURPWO + WAT (-377.6233 hartrees) shows that the former is more stable by 13 kcal/mol. This characteristic is independent of the reactant reference state. The lower energy of the two-water TS relative to the one-water TS is certainly indicative of a preference for a two-water reaction pathway in the acid catalysis of urea. The present calculations demonstrate that, in the case of acid catalysis, two water molecules participate actively as reactants, and have to be included explicitly in the model (regardless of the electrophilic nature of the carbonyl carbon). In this way, assuming that the free activation energy is determined by the two-water assisted reaction starting with URPW2N (Table 8, 36.6 kcal/mol), we can calculate the rate constants for acid catalysis as a function of the temperature. We obtain values of 2 × 10-14 s-1 at 298 K, 4.7 × 10-7 s-1 at 398 K, and 8.8 × 10-3 s-1 at 498 K. We predict that the acid-catalyzed hydrolysis of urea is much slower than the corresponding one for formamide. We associate the different reactivity with the larger resonance stabilization of urea relative to formamide. Water Reaction. The decomposition of urea has been extensively studied since the early 1950s,1,2,59,62,68,69 and the kinetic parameters for the elimination reaction have been determined.1,2,68 For this decomposition mechanism, experimental and theoretical data agree on a rate constant of 10-7 s-1.1,2,7,68 The low barrier for the elimination reaction has prevented the experimental detection of the hydrolytic pathway. However, the rate constant for the water reaction has been estimated from the one determined for tetramethylurea (Me4Ur) at neutral pH.31 For Me4Ur, the rate constant at room temperature has been extrapolated from experimental determinations between 400 and 520 K. As a further approximation, a correction factor of 2.8 has been applied assuming that N-dimethylation will have the same effect for urea as for acetamide.70 On this basis, a rate constant of 1.17 × 10-11 s-1 at 298 K has been estimated.31 This reaction has also been studied in great detail using modern theoretical tools, resulting in slower predicted kinetics and activation free energies close to 50 kcal/mol (rate constants on the order of 10-22).7,11 The water environment was simulated using continuum models, and one7 to three11 water molecules
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TABLE 9: Free Energies for the Stable Species Associated with Water-Catalyzed Urea Hydrolysisa 298 K 398 K 498 K
UR + H2O
URWATO
TSUR
0.00 0.00 0.00
8.15 5.46 1.25
56.09 52.71 48.75
a For URWATO, see Figure 11. For TSUR, see Figure 12. Energies (kcal/mol) are relative to the separated species (see more details in the text). Positive values indicate less stable species.
TABLE 10: Free Energies for the Stable Species Associated with Two-Water-Catalyzed Urea Hydrolysisa UR + 2H2O URWAT2N URWAT2O TSUR2CN TSUR2CO 298 K 398 K 498 K
0.0 0.0 0.0
22.46 15.37 5.40
21.88 14.87 7.22
75.5 68.14 59.92
65.05 59.11 51.64
a For URWAT2N and URWAT2O, see Figure 11. For TSUR2CN and TSUR2CO, see Figure 12. Energies (kcal/mol) are relative to the separated species (see more details in the text). Positive values indicate less stable species.
Figure 12. Computed TS geometries for the water-catalyzed hydrolysis of urea. Color code as in Figure 1.
Figure 11. Computed geometries for hydrogen-bonded complexes of urea and two water molecules identified in this work. Color code as in Figure 1.
have been added as explicit reactants. As discussed above for the case of formamide, the theoretical calculations always indicate that separated species are more stable than the water/ urea ensembles when solvent is accurately modeled with a continuum approach. We have already discussed, in the case of formamide, the uncertainties in the selection of the initial state. The same considerations apply for the case of urea and lead us to leave this issue open. Following this thought process, we report the results obtained when different initial states are considered, modeling one and two water molecules in the reaction with urea. This reaction has been thoroughly analyzed by Estiu and Merz.7 The results are reviewed here for the geometries fully optimized in a continuum solvent model at the same level as that used by Estiu and Merz. The energies are reported in Tables 9 and 10, and the associated structures are shown in Figures 11 and 12. As for the case of protonated urea, there are two possible coordination modes of water (see URWATN and URWATO in Figure 11). URWATN is 2 kcal/mol lower in energy than URWATO. However, as was discussed in detail by Estiu and Merz,7 urea protonation on the amine nitrogen atom (URWATN) favors an elimination mechanism. Hence, URWATO was used
to define the water/adduct reactant for the hydrolytic mechanism. The results in Table 8 show that the stability of the water/urea adduct relative to the separated species increases with the temperature, becoming isoenergetic at 498 K. The hydrolytic reaction proceeds through CO addition, with a calculated free energy of 47 kcal/mol relative to URWATO. If the separated species are used as reactants, the activation free energy increases to 56 kcal/mol at 298 K. Nevertheless, it decreases with temperature, becoming 52.7 kcal/mol at 398 K and 48.5 kcal/ mol at 498 K. For the reaction proceeding from URWATO, these free energies lead to rate constants of 3.2 × 10-23 s-1 (298 K), 0.9 × 10-13 s-1 (398 K), and 1.5 × 10-8 s-1 (498 K). The reaction starting from the separated species results in rate constants of 2 × 10-27 s-1 (298 K), 4.9 × 10-15 s-1 (398 K), and 2.4 × 10-7 s-1 (498 K). The values for the separated species are already corrected by the concentration of water. For the case of the water assisted reaction, inspection of the data in Table 10 shows a preference for CO addition over CN addition. The two-water adducts URWAT2N and URWAT2O (Figure 11) are almost isoenergetic, but the addition to the CO bond is favored by 9.5 kcal/mol at room temperature. The free energies of activation are 43.8 and 53 kcal/mol for CO and CN addition, relative to the two-water/urea adducts. If the separated species are taken as the reactants, the associated free energies of activation are 65 and 75.5 kcal/mol, at 298 K, for the CO and CN addition, respectively. These values decrease to 59 and 68 kcal/mol at 398 K and to 51.6 and 60 kcal/mol at 498 K. Thus, if the reaction is considered to start from the water adduct, the calculated rate constants for CO addition have values of 1 × 10-19 s-1 at 298 K, 3.9 × 10-12 s-1 at 398 K, and 3.4 × 10-7 s-1 at 498 K. If the free energy is measured relative to the separated species, which we feel is incorrect from the perspective of both kinetics and our continuum-based computational approach (see formamide discussion and the Supporting Information), rate constants of 2.5 × 10-32 s-1 (298 K), 8 × 10-17 s-1 (398 K), and 7 × 10-7 s-1 (498 K) are obtained. Nonetheless, it is interesting to note that both reaction pathways are almost equally favored at 498 K, the temperature range at which the reaction kinetics of tetramethylurea has been experimentally analyzed.31 A TS similar to TSUR2CN has been previously proposed by Kallies and Mitzner as the one of lowest energy, using a B3LYP
Hydrolysis of Amides functional and the 6-31G* basis set, with solvent corrections at the gas-phase geometries.11 At this level of theory, a structure similar to TSUR2CO has been found to be 4 kcal/mol higher in energy. The authors considered three-water/urea adducts as the species involved in hydrolysis, but only the coordinates of two waters were found to have negative eigenvalues in the TS Hessian matrix.11 This finding confirms the fact, previously discussed for formamide, that the inclusion of more than two waters in the model does not change the reaction mechanism. A reaction pathway following CN addition has also been found using CPMD to study the neutral hydrolysis of NMA.30 The associated TS features a water molecule attacking the amide carbonyl carbon while at the same time donating a hydrogen atom to a second one that plays a donor/acceptor role, helping to protonate the terminal amine. The calculated C-O distance of 1.65 Å matches that of TSUR2CN (see Figure 12). No other TS was identified in the Car-Parrinello MD simulation,30 which may be a limitation associated with the definition of the reaction coordinate. The ObserVed Water Rate Constant. We can insert the calculated values of the rate constants in eq 10 and predict the kobsd value that would be measured if the elimination pathway was suppressed. At 298 K, we obtain a value of kobsd ) 5 × 10-17 + kw s-1. There are several possible values of kw, depending on the initial state that is involved in the kinetic analysis, but our best estimate is 10-19, and does not influence the kobsd value. At 398 K, the equation has the form kobsd ) 6.3 × 10-11 + kw s-1, with kw no larger than 10-12, leading to a similar situation. At temperatures as high as 498 K, we obtain the following expression: kobsd ) 1 × 10-6 + kw s-1. In this case, the calculated values of kw lie between 0.7 × 10-6 and 1.5 × 10-8 s-1. Hence, it is possible that an experiment can be designed that allows one to uniquely identify the nature of the reactants and to determine the value of kw provided the reaction starts from the two-water adduct. Moreover, the experimental determination of kobsd, together with the theoretical values, affords a way to discern the nature of the reactants. However, the experimental procedure has to be able to discriminate between values of the order of 10-6. This is not an easy task, as the kinetics of a hydrolytic reaction at 500 K is not easily determined with the accuracy required to differentiate between values of the same order of magnitude, which may lie within the experimental error. Leaving this discussion to the section entitled “Comparision with Available Data”, we can go back to the case of formamide and examine whether a temperature increase to 498 K would allow the separation of the water reaction from the acid and base contributions. Using the calculated free energies, we derive rate constants of kH+ ) 5 × 102 s-1, kOH- ) 5 × 103 s-1, and kw ) 3 × 10-3 s-1. Using these values in eq 10 yields kobsd ) 10-2 + 3 × 10-3. Therefore, we conclude that, even for this temperature, the water reaction cannot be isolated from the acid and base contributions. Comparison with AVailable Data. The hydrolytic decomposition of urea has never been detected experimentally, because the elimination pathway is strongly favored at all temperatures. The hydrolytic mechanism has been examined for 1,1,3,3tetramethylurea and the results reported for temperatures between 400 and 525 K.31 Furthermore, the rate constant at 25 °C has been extrapolated from the plot of the logarithm of the rate of Me4U as a function of the reciprocal of the absolute temperature.31 From the resulting value (4.2 × 10-12 s-1), the rate constant for urea hydrolysis has been estimated to be 1.17 × 10-11 s-1. In order to do this, a correction factor of 2.8 has been applied, which was derived from the effect of
J. Phys. Chem. B, Vol. 111, No. 23, 2007 6517 N-methylation on the hydrolysis rate constant of acetamide.70 Although the rate of hydrolysis is said to be invariant between pH 4 and 10, only the data for pH 6.8 (phosphate buffer) were reported. Under these conditions, a rate constant of k ) 10-4.5 s-1 has been measured at 498 K,31 from which a value of kobsd ) 8.86 × 10-5 s-1 can be derived for urea. This value is not close to any value calculated by us for water hydrolysis at 498 K. The closest one, derived from the assumption that the kinetics is determined by a two-water reaction differs by 2 orders of magnitude (7 × 10-7 s-1). The experimental data, reported only as a brief communication,31 do not allow a thorough understanding of the origin of the lack of agreement. Nevertheless, several points are worth consideration: In the treatment of the kinetic data, the dependence of Kw with the temperature is not discussed. For the temperatures used in the experimental determinations (between 400 and 500 K), Kw ) 10-11.2.71 In this way, buffering the system at pH 6.8 implies [OH-] ) 10-4.4, that is, basic media. According to eq 9, the pH at which the contributions of acid and base terms are minima, and the detection of the water reaction becomes feasible, is close to 4, that is, acidic conditions. The pH of the buffer is assumed to be stable at temperatures as high as 525 K. It is not clear, from the reading of the communication, if the pH of the Me4U solution has been measured at 298 K. According to the Henderson-Hasselbach equation, the pH of the buffer depends linearly on the pKa of the conjugate acid. Although we are not aware of any determination of the phosphate pKa at 500 K, it is known that the pKa for acetic acid increases from 4.75 at room temperature to 6.0 at 500 K.71 The possibility of general acid and base catalysis was not eliminated at higher temperatures by carrying out the reaction at differing buffer concentrations. On the basis of these considerations, we believe that the values reported by Callahan, Yuan, and Wolfenden at 498 K correspond to kobsd at this temperature and at a pH not fully characterized but that we can certainly estimate. Using the experimental k value determined at 498 K, corrected by the factor 2.8, and our values of kH+, kOH-, and kw, calculated at the same temperature, we can solve a second-order equation in [H+], derived from eq 8, from which the pH value can be obtained.
8.86 × 10-5 ) 1.5 × 10-3[H3O+] + 41Kw/[H3O+] + 3 × 10-6 (11) This equation leads to two possible values: pH 5.5 or pH 1.9. This analysis, based on theoretical calculations, predicts a shift in the pH of phosphate buffer of one unit when the temperature increases to 500 K. If this correction is applied to the experimental values, we are able to reproduce the experimentally determined values. In doing this, it has to be kept in mind that the value reported by Callahan, Yuan, and Wolfenden corresponds to kobsd, which we have differentiated from kw. Equation 11 does not allow us to explain the stability of the rate constant between pH 4 and 10, reported by these authors. This observation needs a thorough analysis of the behavior of the different buffers as a function of temperature and an examination of the rate on the concentration of buffer itself to eliminate the possibility of general acid/base catalysis, which requires a detailed experimental characterization of the system. The previous discussion supports the experimental values reported at high temperature. However, a similar analysis cannot
6518 J. Phys. Chem. B, Vol. 111, No. 23, 2007 be applied to the treatment of the data at room temperature. The disagreement between experimental and theoretical values does not originate exclusively from a shift of the pH value. We predict for urea a kobsd value of 5 × 10-17 s-1 (3.4 × 10-17 s-1 for Me4U) and a value of kw no larger than 10-19 s-1, more than 7 orders of magnitude slower than the experimentally reported value of 10-11 s-1, derived by extrapolation from high temperatures. Our ability to reproduce the values at high temperature (experimentally determined) but not the values at room temperature (extrapolated) suggests that a reanalysis of the kinetic data is necessary in order to test the validity of what has been assumed in the derivation of the latter. This revision has to account for possible changes in the reaction mechanism, an appropriate treatment of the Eyring plots, where the change of the pre-exponential factor with the temperature is accounted for as well as the temperature dependence of the pH with the temperature for the different buffers. Conclusions The hydrolysis of simple amides, like formamide and urea, has been extensively studied from both theoretical and experimental viewpoints.1-11,13,14,16,17,19,20,24-26,30 Significant effort in this area is justified by the need to better understand peptide hydrolysis, for which these simple amides offer good models.17,18,43-46 Going a step further, the solvent-phase reaction has been used to estimate the proficiency of amidohydrolases.43 In this regard, the kinetic rate constant of the hydrolytic reaction has been associated with the term “knon” in the definition of the proficiency of an enzyme. The term knon refers to the rate constant of the reaction proceeding spontaneously in dilute aqueous solutions in the absence of a catalyst.44,45 It is wellknown that a reference state involving catalytic groups, acids, bases, and metal ions in water will give a lower apparent proficiency.44 The proficiency of a enzyme has been defined as the ratio of the second-order rate constant of the enzyme-catalyzed process (kcat/Km) and the rate constant of the uncatalyzed reaction (knon).41,44,45,51,72 From a detailed analysis of the two terms that contribute to this definition, it has been found that the reaction rates of uncatalyzed reactions span 19 orders of magnitude, in contrast to the relatively narrow range of values of kcat/Km observed for enzyme-catalyzed reaction rates.73 Thus, differences in enzyme proficiency tend to reflect differences in the rate of the uncatalyzed reactions, and the determination of the reference state becomes relevant. The definition of knon, as the reaction occurring in the absence of any catalyst, associates this term with kw in the case of hydrolytic reactions. Within this framework, the uncertainty in the definition of the proficiency of amidohydrolases relies on the, yet unsuccessful (in our opinion), isolation of the water reaction from the acid and base contributions. For the enzymecatalyzed hydrolysis of peptide bonds, proficiency values of 1014-1017 have been obtained on the basis of knon values determined for the hydrolysis of representative dipeptides.43 Using the same approach, the knon value for the hydrolysis of tetramethylurea has been used to estimate the proficiency of the enzyme urease. We have shown in the present work that the rate constants that have been used as knon to estimate the proficiency of amidohydrolases are complicated with both uncatalyzed and catalyzed reaction channels. In general, we can reproduce all of the experimental characteristics of the hydrolysis of formamide, while in the case of urea due to the lack of detailed data sets and an examination of the acid- and base-catalyzed reactions
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