Kinetics and thermodynamics of enzymatic decarboxylation of α,β-unsaturated acid: a theoretical study

Enzymatic decarboxylation of α,β-unsaturated acid through ferulic acid decarboxylase (FDC1) has been of interest because this reaction has been anticipated to be a promising, environmentally friendly industrial process for producing styrene and its derivatives from natural resources. Because the local dielectric constant at the active site is not exactly known, enzymatic decarboxylation to generate β-methylstyrene (β-MeSt) was studied under two extreme conditions (ε = 1 and 78 in the gas phase and aqueous solution, respectively) using the B3LYP/DZP method and transition state theory (TST). The model molecular clusters consisted of an α-methylcinnamate (Cin) substrate, a prenylated flavin mononucleotide (PrFMN) cofactor and all relevant residues of FDC1. Analysis of the equilibrium structures showed that the FDC1 backbone does not play the most important role in the decarboxylation process. The potential energy profiles confirmed that the increase in the polarity of the solvent could lead to significant changes in the energy barriers, especially for the transition states that involve proton transfer. Analysis of the rate constants confirmed the low/no quantum mechanical tunneling effect in the studied temperature range and that inclusion of the fluctuation of the local dielectric environment in the mechanistic model was essential. Because the computed rate constants are not compatible with the time resolution of the stopped-flow spectrophotometric experiment, the direct route for generating β-MeSt after CO2 elimination (acid catalyst (2)) is unlikely to be utilized, thereby confirming that indirect cycloelimination in a low local dielectric environment is the rate determining step. The thermodynamic results showed that the elementary reactions that involve charge (proton) transfer are affected by solvent polarity, thereby leading to the conclusion that overall, the enzymatic decarboxylation of α,β-unsaturated acid is thermodynamically controlled at high ε. The entropy changes due to the generation of molecules in the active site appeared more pronounced than that due to only covalent bond breaking/formation or structural reorientation. This work examined in detail for the first time the scenarios in each elementary reaction and provided insight into the effect of the fluctuations in the local dielectric environment on the enzymatic decarboxylation of α,β-unsaturated acids. These results could be used as guidelines for further theoretical and experimental studies on the same and similar systems.


Introduction
Decarboxylation has long been known in organic synthesis, in which the formation of a carbanion intermediate and carbon dioxide product controls the reaction rate. 1 Therefore, decarboxylation reactions require organic or metal ion catalysts to stabilize the intermediates. 2 Enzymatic decarboxylation of an a,b-unsaturated acid using ferulic acid decarboxylase (FDC1) has been of interest in recent decades 1,3-5 because the reaction has been anticipated to be a promising, environmentally friendly industrial process for producing styrene and its derivatives from natural resources. Biosynthesis of styrene from this nonoxidative decarboxylation could start from biological sugars (e.g., glucose) to produce L-phenylalanine and trans-cinnamate through the shikimate pathway and coexpression of genes that encode phenylalanine ammonia lyase (PAL), 6 respectively. Enzymatic decarboxylation using FDC1 is accomplished through the 1,3-dipolar cycloaddition reaction between the substrate (an a,b-unsaturated acid, such as a-methylcinnamic acid) and an appropriate enzyme cofactor. 1 Experiments have shown that decarboxylation of aromatic carboxylic acids using FDC1 is reversible. 1 However, in the presence of a hydroxyl (OH) group at the a position of the substrate (e.g., a-hydroxycinnamic acid), the enzyme activity of FDC1 is inhibited, and the reaction becomes irreversible. 5 Although various cofactors have been suggested, e.g., pyridoxal phosphate and pyridine pyrophosphate, 2 biosynthesis of styrene using modied avin cofactors seems to have received special attention, e.g., the prenylated avin mononucleotide (PrFMN). 1,[3][4][5]7,8 Because the mechanisms of the enzymatic decarboxylation of a,b-unsaturated acids have been extensively studied using theoretical and experimental methods, only the results that are relevant to the present study will be discussed in detail. To facilitate discussion, the abbreviations for the molecules that are used in this work are summarized in Table S1. † Several mechanisms for the enzymatic decarboxylation of a,b-unsaturated acids using PrFMN have been reported, among which that proposed by Payne et al. 1 has been widely accepted and further studied in detail. Because the enzyme-catalyzed 1,3dipolar cycloaddition of PrFMN and an a,b-unsaturated acid was unprecedented, the reaction was conrmed using a mechanism-based inhibitor. 4 Based on the theoretical and experimental data in ref. 1, two forms of PrFMN with different ring structures were considered, namely, the iminium and ketimine forms, which are abbreviated PrFMN iminium and PrFMN ketimine , respectively, in Table S1. † It was suggested that through PrFMN iminium , the decarboxylation of cinnamic acid occurs via 1,3-dipolar cycloaddition, whereas the reaction with PrFMN ketimine occurs via Michael addition.
Although mass spectroscopic data cannot differentiate these two forms, because the reaction is stereospecic, the enzyme activity was suggested to be higher using PrFMN iminium ; therefore, the reaction using PrFMN iminium has been further studied in detail. 1 Based on the density functional theory (DFT) method with the Becke, 3-parameter, and Lee-Yang-Parr hybrid functionals and 6-311++G(d,p) basis set (abbreviated B3LYP/6-311++G(d,p)), 1 the equilibrium structures of PrFMN ketimine with bent-down and bent-up forms are À41.5 and À36.9 kJ mol À1 more stable, respectively, than those of PrFMN iminium . The mechanism that was proposed by Payne et al. 1 consists of four consecutive elementary steps (Scheme 1): (I) 1,3-dipolar cycloaddition, (II) Grob-type decarboxylation, (III) protonation and (IV) retro 1,3-dipolar cycloaddition. Based on spectroscopic methods and kinetic isotope effects, Ferguson et al. studied the enzyme activity of FDC1 by following the depletion of the substrate. 3 The results showed that cycloelimination (IV) could represent the rate-determining step.
The roles of residues in FDC1 in PrFMN oxidative maturation, cofactor isomerization and enzyme catalysis were studied using Aspergillus niger FDC1 as a model system. 9 Analysis of the highresolution crystal structures, mass spectrometric and kinetic data indicated that the isomerization of PrFMN iminium to PrFMN ketimine is an irreversible light-dependent process and is independent of the Glu277-Arg173-Glu282 residue network. Most importantly, while irreversible isomerization leads to loss of enzyme activity, the efficiency of enzymatic decarboxylation through the PrFMN iminium cofactor is dependent on the conserved Glu277-Arg173-Glu282 residue network; the network was suggested to facilitate the oxidative maturation of the PrFMN iminium cofactor and to act as a key acid-base during catalysis. The need for the Glu277 and Glu282 acid residues in the enzymatic decarboxylation of a,bunsaturated acid was conrmed using NMR spectroscopy.
To study the stability of the transient intermediates, Kaneshiro et al. 10 performed kinetic experiments to investigate the formation of the PrFMN iminium -styrene cycloadduct that accumulated on the FDC1 enzyme in 0.1 M potassium phosphate buffer (pH ¼ 7.0). Analysis of the stopped-ow UV-vis spectrophotometric results at 277 K and the half-of-sites model revealed that in the active site, a PrFMN iminium -cinnamic acid cycloadduct is formed with k ¼ 131 s À1 and is converted to a PrFMN iminiumstyrene cycloadduct with k ¼ 75 s À1 . These results led to the suggestion that both cycloelimination (IV) of the PrFMN iminiumstyrene cycloadduct and diffusion from the active site represent the rate-determining step, with k ¼ 11 s À1 .
The enzymatic decarboxylation of an a,b-unsaturated acid using FDC1 was theoretically studied by the quantum mechanics/molecular mechanics (QM/MM) method, 8 in which the original crystal structure in the protein data bank (PDB), including the FDC1 enzyme, a-methylcinnamic acid, PrFMN iminium and PrFMN ketimine , was used as the model system. In QM/ MM simulations, the substrate, a part of PrFMN and the side chains of Glu282 and Arg173 were included in the QM region, whereas the remaining part of FDC1 and 8103 TIP3 water molecules were included in the MM region. To study the effect of hydrogen bonding (H-bonding), two water molecules were included in the QM region. The results that were obtained by scanning over the potential energy curves (B3LYP/6-311++G(2d,2p)) suggested that the Michael addition through PrFMN ketimine involves a rather high energy barrier (DE ‡ ¼ 166 kJ mol À1 ), whereas PrFMN iminium can easily form the cofactor-substrate adduct (DE ‡ ¼ 65 kJ mol À1 ) in the 1,3-dipolar cycloaddition process and, therefore, is more relevant to the enzymatic reaction. Although decarboxylation of the pyrrolidine adduct intermediate and subsequent protonation involve low energy barriers (58 and 27 kJ mol À1 , respectively), the overall energy barrier is DE ‡ ¼ 98 kJ mol À1 .
In this work, because the information on the kinetic and thermodynamic aspects was limited, the proposed elementary reactions of the enzymatic decarboxylation of a,b-unsaturated acid were further studied using the DFT method with B3LYP functionals and transition state theory (TST). While previous theoretical studies focused only on potential energy proles in low local dielectric environments, 5,7 this theoretical study focused on the scenarios (progress) in the elementary reactions and on the kinetic and thermodynamic properties in two extreme local dielectric environments, namely, the gas phase and aqueous solution with 3 ¼ 1 and 78, respectively. This theoretical study began with geometry optimizations of the proposed model molecular clusters, 5 which consisted of the amethylcinnamate (Cin) substrate, PrFMN iminium cofactor (abbreviated PrFMN hereaer) and all relevant residues in the active site of FDC1. The kinetic and thermodynamic aspects of the elementary reactions at 3 ¼ 1 and 78 were analyzed in detail based on the TST results and were included in the proposed mechanisms. The results were discussed in comparison with the reported theoretical and experimental data.

Quantum chemical methods
Because the FDC1 enzyme is exceedingly large for high-level ab initio methods and because our previous studies showed that the mechanisms for proton transfer in heterocyclic aromatic systems can be studied reasonably well using the B3LYP method with the DZP basis set, 11-13 the B3LYP/DZP method was used in this study; our benchmark calculations on bifunctional proton transfers in poly(benzimidazole) (PBI) H-bond systems 14 conrmed that the B3LYP/DZP method yields approximately the same equilibrium and transition structures and relative interaction energies as the B3LYP/TZP method with reasonable computational resources. In this study, the model molecular clusters that were hypothesized in ref. 5 were chosen as model systems, which consist of all important active site residues, the substrate and the cofactor. The model molecular clusters were constructed by substituting the carbon atoms of FDC1 Backbone that connect the residues with methyl (CH 3 ) groups (Table S2 †); for example, C R Glu277 is the carbon atom of the CH 3 group that substitutes the carbon atom of the FDC1 Backbone that connects the Glu277 residue (Scheme 1). Because our previous studies showed that the local dielectric environment (microenvironment) can affect the structures and energetics of elementary processes 11,13,14 and because enzymatic decarboxylation occurs in aqueous solution, the conductor-like screening model (COSMO) was used to simulate the effect of the aqueous environment. COSMO was used successfully in our previous studies on proton transfer processes in H-bond systems. 11,13,14 Previous theoretical studies used 3 ¼ 4 (ref. 5) and 5.7 (ref. 7) to model the local dielectric environment at the active site of FDC1. In this work, because the local dielectric constant was not exactly known and we wanted to study the elementary reactions in extreme local dielectric conditions, the lowest and highest possible values (and uctuation) were used, namely, 3 ¼ 1 and 78, in the gas phase and bulk water, respectively. All B3LYP/DZP calculations were performed using the TURBOMOLE 7.50 soware package. 15 Equilibrium structures and potential energy curves The six model molecular clusters that were hypothesized in the previous study 5 were considered in this work (Table S2 †), in which a-methylcinnamate (Cin) was chosen as the substrate for generating b-methylstyrene (b-MeSt). The active site of FDC1 consists of Glu277, Arg173 and Gln190, in which the salt bridge between protonated Arg173 (Arg173H + ) and the carboxylate group (COO À ) of Cin is responsible for the enzyme-substrate interaction (the docking site); Glu277 is the proton source of Arg173H + . The interaction between PrFMN and the Gln190 residue is an N-H/O À H-bond, e.g., in the precursor React; Cin only weakly interacts with PrFMN through a p-p interaction. The ve hypothesized elementary reactions, which are presented in Scheme 1, are (I) 1,3-dipolar cycloaddition, React / TS1 / Int1; (II) decarboxylation, Int1 / TS2 / Int2; (III) acid catalyst (1), Int2b / TS3 / Int3; (IV) cycloelimination, Int3 / TS4 / Prod; and (V) acid catalyst (2), Int2b / TS3b / Prod.
The symbols that are used in this manuscript (e.g., React, TS1 and Int1) correspond to ref. 5.
The structures of the molecules in the model molecular clusters were fully optimized without any geometrical constraint using the B3LYP/DZP method before they were included in the model molecular clusters (Table S1 †). The Newton-Raphson method was used with the converge criterion for the total energies and energy gradients, 1.0 Â 10 À6 and 1.0 Â 10 À4 au, respectively. Then, the structures of the six model molecular clusters (including the residues, cofactor and substrate) were reoptimized using the same method (Table S2 †). The equilibrium structures of the model molecular clusters were employed in the elementary reaction path optimizations using the nudged elastic band (NEB) method with the limitedmemory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimizer in the ChemShell soware package. 16 In the elementary reaction path optimizations, fourteen structures that connected the precursor, transition structure and product were optimized. The relative energies with respect to the precursor (DE Rel ) along the optimized reaction path were plotted. The effect of the aqueous environment was studied using the solvation energy (DE Solv ), which was dened as the difference between the total energies of the model molecular clusters at 3 ¼ 78 and 3 ¼ 1, namely, E Total,3 and E Total .

Kinetics of elementary reactions
Characteristic structures of the model molecular clusters on the potential energy curves were used in the calculations of the rate constants based on the TST method. [17][18][19] To study the effect of quantum mechanical tunneling, the classical (k Class ) and quantized-vibrational (k Q-vib ) rate constants were initially computed over the temperature range of 200-371 K. k Class was calculated using eqn (1): 20 Q R and Q ‡ are the partition functions of the precursor and transition structures, respectively. DE ‡ is the energy barrier obtained from the NEB method. k B and ħ are the Boltzmann and Planck constants (ħ ¼ h/2p), respectively. k Q-vib was obtained with the zero point energy-corrected energy barrier (DE ‡,ZPC ): Q R,ZPC and Q ‡,ZPC in eqn (2) are the partition functions of the precursor and transition structures, respectively, that were obtained with ZPC. DE ‡,ZPC was obtained by including the zeropoint correction energy (DE ZPE ) to DE ‡ . The denitions and methods to calculate the energy barriers in eqn (1) and (2) are illustrated as an example in ESI. † The temperatures below which quantum mechanical tunneling dominates were approximated using the crossover temperature (T c ): 21,22 U ‡ in eqn (3) is the imaginary frequency of the transition structure. To approximate the effect of quantum mechanical tunneling, the Wigner corrections were made by multiplying k Qvib (T) by the Wigner transmission coefficient (k F-Wig ) in eqn (4): 21,22 In this work, k F-Wig is regarded as the full Wigner transmission coefficient. Because k F-Wig diverges near T c , without a theoretical foundation, the simple Wigner transmission coefficient (k S-Wig ) in eqn (5) is recommended to avoid the divergence: 17 k S-Wig is a Taylor series expansion of k F-Wig around 1/k B T ¼ 0, maintaining only the rst two terms. The Wigner corrected rate constants (k F-Wig and k S-Wig ) were computed using eqn (6): k F-Wig and k S-Wig equal to 1 at the classical limit (ħ ¼ 0). The activation free energies (DG ‡ ) were computed from the rate constant using k(T) ¼ (k B T/ħ)e ÀDG ‡ /RT . To correlate k S-Wig with the experimental data, 10 the Eyring equation (eqn (7)) was primarily used to calculate the activation enthalpy (DH ‡ ): 20 DS ‡ is the activation entropy and R is the gas constant. DH ‡ in eqn (7) was obtained from the linear relationship between ln k S-Wig (T) and 1000/T. DG ‡ obtained from the TST method were used to determine DS ‡ using DG ‡ ¼ DH ‡ À TDS ‡ . All the kinetic and thermodynamic calculations were performed using the DL-FIND program 23 included in the ChemShell package. 16

Results and discussion
To facilitate discussion, additional character codes are used. To characterize the scenarios (progress) in the elementary reactions, lowercase letters in parentheses are used. For example, for 1,3-dipolar cycloaddition (React / TS1 / Int1) in Fig. 1, the three consecutive steps, namely, p-p stacking, dipolarophile iminium pair and pyrrolidine cycloadduct formations, are labeled (a), (b) and (c), respectively. The properties/processes with superscript "3" correspond to a high local dielectric environment. For example, TS1 3 in and (a) 3 in Fig. 1 and S1b † are the transition structure and p-p stacking, respectively, that were observed on the potential energy curve at 3 ¼ 78.

Equilibrium structures of the model molecular clusters
The equilibrium structures and total energies of the model molecular clusters that are involved in the elementary reactions are presented in Table S2. † The B3LYP/DZP results show that the equilibrium structures of the model molecular clusters and the shapes of the active sites therein are not signicantly different at 3 ¼ 1 versus 78. The average residue-to-residue distances (Table S3 †) reveal small standard deviations (SD) for all elementary reactions; the average residue-to-residue distances were approximated using the distances between the carbon atoms of the CH 3 groups that substituted the carbon atom of FDC1 Backbone (Scheme 1). For example, for React / TS1 / Int1 A. It appears that the highest SD are for elementary reactions that involve proton transfer, in which formation of p-p stacking leads to an increase in R C Glu277 .46 AE 1.94 and 13.43 AE 1.86Å are for acid catalyst (1) (III) and acid catalyst (2) (V), respectively.
These average residue-to-residue distances (Table S3a and b †) are in good agreement with the PDB crystallographic data (code Similar results were obtained from the analysis of the average residue-toresidue distances per each model molecular structure on the optimized reaction paths. They are also not signicantly different; for example, for structure 1 of elementary reactions (I) and (II) ( Table S3c † The above results suggest that in the enzymatic decarboxylation reaction, the active site structure and volume do not signicantly change. These results also imply that the motion of FDC1 Backbone can be neglected in the model systems. These ndings are in accordance with the results in ref. 9, in which the Glu277-Arg173-Glu282 residue network was suggested to be conserved in the enzymatic decarboxylation reaction; the residues help immobilize the substrate and cofactor in the active site. In addition, because the computed residue-to-residue distances are in good agreement with the PDB crystallographic data (code 4ZA7), one can conclude that the model molecular clusters are appropriate for representing the active site of FDC1.

Elementary reactions
The structures and energetics of the model molecular clusters on the potential energy curves of elementary reactions (I)-(V) that were obtained via the NEB method at 3 ¼ 1 and 3 ¼ 78 are included in Fig. S1-S5, † together with the relative total energies (DE Rel and DE Rel,3 ). The transition structures on the potential energy curves are summarized in Table S4. † To keep the manuscript concise, the scenarios, interactions among molecules, and energetic effect of the local dielectric environment on the elementary reactions are explained in detail in ESI. † Only the simplied potential energy proles are included in the manuscript ( Fig. 1-5).
1,3-Dipolar cycloaddition (I). For 1,3-dipolar cycloaddition (I), the potential energy prole in Fig. 1 and potential energy curve Fig. S1c † reveal that at 3 ¼ 1, React / TS1 is a two-step process, in which the formation of p-p stacking (a) occurs rst (DE ‡ ¼ 24 kJ mol À1 ), followed by dipolarophile-iminum pair formation (b) in the transition structure TS1 (DE ‡ ¼ 50 kJ mol À1 ). TS1 is characterized by the a,b-double bond of Cin staying exactly above the iminium ion (C 29 PrFMN -N 35 (Fig. S1a †). It appears that pyrrolidine cycloadduct formation (c) and relaxation of p-p stacking (d) occur instantly in TS1 / Int1, thereby leading to the transformation of the enolate anion to a C]O group at the O 30 PrFMN atom.
At 3 ¼ 78, the potential energy prole in Fig. 1 and potential energy curve in Fig. S1c † are almost the same as those at 3 ¼ 1.
The energy barriers for p-p stacking (a) 3 and TS1 3 formation (b) 3 are slightly different, namely, DE ‡ ¼ 26 and 48 kJ mol À1 , respectively. This could be because cycloadduct formation (React 3 / TS1 3 / Int1 3 ) does not involve direct charge (proton) transfer (Fig. S1b †). Therefore, the electric eld that is induced by the aqueous solvent (3 ¼ 78) does not have a strong inuence on the energy barriers.
Decarboxylation (II). At 3 ¼ 1, the potential energy prole in Fig. 2  The scenario is slightly different at 3 ¼ 78 (Fig. 3, S3b and c †), in which proton transfer from the COOH group of Glu282 to C a Cin (a) 3 instantly produces the transition state (TS3 3 ); for (1)   whereas b-MeSt leaves the iminium ion (TS4 / Prod) on a barrierless potential curve; the model molecular cluster Prod consists of free b-MeSt and the regenerated PrFMN, Glu277, Arg173H + and Gln190 (Fig. S4a †), as in React.
Acid catalyst (2) (V). Based on the potential energy proles and potential energy curves that have been discussed up to this point, the highest energy barrier at 3 ¼ 1 is for cycloelimination (IV) (DE ‡ ¼ 81 kJ mol À1 ), whereas that at 3 ¼ 78 is for acid catalyst (1) (III) (DE ‡ ¼ 137 kJ mol À1 ). To complete the discussion on the potential energy proles and potential energy curves of the elementary reactions, the route for generating Prod directly from Int2b without the formation of pyrrolidine cycloadduct (Fig. 5 and S5 †) is discussed. At 3 ¼ 1, the proton transfer from the COOH group of Glu282 to C a Cin (a) instantly leads to C a Cin -

The effect of a high local dielectric environment
The potential energy proles for the enzymatic decarboxylation of a,b-unsaturated acid that were obtained in this and previous studies are presented in Fig. 6. To verify the theoretical results, our potential energy proles at 3 ¼ 1 are compared with proles at 3 ¼ 4 (ref. 5) (Fig. 6a) that were obtained from B3LYP/6-311+G(2d,2p)//6-31G(d,p) calculations with the intrinsic reaction coordinate (IRC) and conductor-like polarizable continuum model (CPCM) methods. Because React and Int2b possess different number of atoms (115 and 126 atoms, respectively), the elementary reactions are categorized into two groups, namely, the decarboxylation/CO 2 elimination (React / TS1 / Int1 / TS2 / Int2) and b-MeSt formation/cofactor regeneration on the indirect (Int2b / TS3 / Int3 / TS4 / Prod) and direct routes (Int2b / TS3b / Prod). Comparison of the potential energy proles in Fig. 6a reveals similar energy barriers at 3 ¼ 1 and 4, 5 except for acid catalyst (1) (III), for which DE ‡ at 3 ¼ 4 is $17 kJ mol À1 higher than that at 3 ¼ 1, thereby implying that a slight increase in 3 could result in a signicant change in the energy barrier for the elementary reaction involving proton transfer. The potential energy proles in Fig. 6b conrm the above observation by showing that the increase in the polarity of the solvent from 3 ¼ 1 to 78 leads to signicant changes in DE ‡ , especially for the transition states that involve proton transfer; DE ‡ for acid catalyst (1) (III) increases from 42 to 137 kJ mol À1 , whereas that of acid catalyst (2) (V) decreases from 73 to 47 kJ mol À1 . It appears that due to the regeneration of the positive and negative charges at PrFMN, the Glu277, Arg173H + and Gln190 residues, the end-product cluster (Prod) is more stable at 3 ¼ 78 than at 3 ¼ 1.

Kinetics and thermodynamics of the elementary reactions
All the kinetic and thermodynamic results at 3 ¼ 1 and 78 that were obtained based on the TST method are presented in Tables S5 and S6, † respectively. The emphasis will be on the results at 277 K in Tables 1 and 2, which is the temperature at which the stopped-ow spectrophotometric experiment 10 was performed. Comparison of the rate constants that were obtained using different methods reveals considerable differences only for k Class f/r . This conrms that for large biological molecules, at least the zero-point vibrational energies must be included in TST calculations. The values for T c ¼ 3-123 K suggest a low/no quantum mechanical tunneling effect in the studied temperature range.   .60 Â 10 À14 s À1 , whereas for the same reaction at 3 ¼ 1 (Int2b / TS3 / Int3), k S-Wig f ¼ 1.71 Â 10 8 s À1 , which indicates that acid catalyst (1) (III) is kinetically favorable in a low local dielectric environment. This is in accordance with our previous work, 11,14,24 in which the uctuation of the local dielectric environment was conrmed to govern the kinetics of proton transfer processes; based on this analysis, React / TS1 / Int1 (1,3-dipolar cycloaddition (I)) is kinetically more favorable than React 3 / TS1 3 / Int1 3 (k S-Wig f ¼ 3.44 Â 10 3 and k S-Wig,3 f ¼ 2.44 Â 10 3 s À1 , respectively). Attempt was made to correlate the rate constants obtained from the TST method with the experimental data. 10 Because the experiments on enzyme kinetics are complex due to several factors, such as experimental conditions (e.g., temperature, pH and ionic strength), sensitivity of the spectroscopic equipment and measurement timescale (time resolution), it is not straightforward to compare our theoretical results with the experimental data. In this work, the Arrhenius rate constants (k Arr ) were calculated in terms of DG ‡ , which were obtained from the TST method (Tables S5 and S6 †), using k Arr ¼ Ae ÀDG ‡ /k B T .
Because the pre-exponential constant (A) in the Arrhenius equation is not known for this enzyme system, the value was tentatively approximated using the highest rate constants   Tables 1 and 2 will be used in further discussion.
To correlate k Arr f and k Arr,3 f with the experimental rate constants, 10 the elementary reactions that occur within the time resolution of stopped-ow spectrophotometry ($10 À3 s) are considered. 10 Based on the assumption that the two active sites on FDC1 react with different rates (denoted (a) and (b) for the fast and slow sites, respectively), 10 the stopped-ow spectrophotometric results at 277 K and the half-of-sites model suggested that for the fast site (a), the PrFMN iminium -cinnamic acid cycloadduct is formed with k 1(a) ¼ 131 s À1 and is converted to the PrFMN iminium -styrene cycloadduct with k 2(a) ¼ 75 s À1 . However, cycloelimination to generate the styrene product and free FDC1 appeared to be the slowest process, with k cat ¼ 11 s À1 . Because the observed rete constants were reported to be in the range of k obs ¼ 0.75-2.0 Â 10 2 s À1 , only the elementary reactions with k Arr f larger than k obs are included in the proposed mechanism. Based on the analysis of all the rate constants (k Arr f and k Arr,3 f ) and activation free energies (DG ‡ ) in Tables 1 and 2, the kinetically controlled paths for the enzymatic decarboxylation of a,b-unsaturated acid (long rightwards blue arrows) are proposed in Fig. 7a.
Comparison of the rate constants of the proposed elementary reactions (long rightwards blue arrows in Fig. 7a) with those that were obtained in the experiment suggests that within the time resolution of stopped-ow spectrophotometry, k Arr f of 1,3dipolar cycloaddition (I) is compatible (associated) with k 1(a) ; for React 3 / TS1 3 / Int1 3 and React / TS1 / Int1, k Arr,3 f ¼ 1.50 Â 10 2 and k Arr f ¼ 2.75 Â 10 2 s À1 at 3 ¼ 78 and 1, respectively. However, because decarboxylation (II) at 3 ¼ 1 is slower than the time resolution of stopped-ow spectrophotometry (k Arr f ¼ 1.02 Â 10 À3 s À1 ), decarboxylation (II) is likely to occur in a high local dielectric environment. Likewise, although the direct route for To examine whether the proposed kinetically controlled (favorable) mechanisms in Fig. 7a (long rightward blue arrows) are also thermodynamically controlled, the standard free energy changes (DG 0 and DG 0,3 ) of each elementary reaction were calculated from the difference between the activation free energies (DG ‡ ) in the forward and reverse directions. In addition, because the entropic effect has been suggested to play an important role in enzymatic reactions, 25 an attempt was made to study the entropy changes of the system (the model molecular clusters); although several known and unknown factors contribute to the entropy change, e.g., the entropy change of the surrounding, we tentatively consider only the entropy change in the system. The standard entropy changes of each elementary reaction (DS 0 and DS 0,3 ) were computed in the studied temperature range (200-371 K). These thermodynamic data are listed in Table 3, and the values at 277 K are presented in Fig. 7b.
The results reveal similar trends for DG 0 and DG 0,3 (Table 3), except for acid catalyst (1) (III), in which DG 0,3 is negative, whereas DG 0 is positive; at 277 K, DG 0 and DG 0,3 for 1,3-dipolar cycloaddition (I) are both positive, whereas those for decarboxylation (II), cycloelimination (IV) and acid catalyst (2) (V) are all negative. Analysis of the scenarios in the elementary reactions in Fig. S1-S5 † suggests that at least three factors affect the standard free energy and entropy changes of the systems, namely, the disorder/order due to breaking/formation of covalent bonds, Table 2 Thermodynamics and kinetics of the elementary reactions of the enzymatic decarboxylation of a,b-unsaturated acid in 3 ¼ 78 at 277 K. Rate constants, temperatures and energies are in s À1 , K and kJ mol À1 , respectively; DE ‡,3 ¼ energy barrier on the optimized reaction path; DE ‡,ZPE,3 ¼ difference between E ZPE,3 of the transition structure and precursor; DE ‡,ZPC,3 ¼ zero point energy-corrected energy barrier; DH ‡,3 ¼ activation enthalpy; T c ¼ crossover temperature; k S-Wig,3  Fig. 7 (a) The kinetically controlled paths (long rightwards blue arrows) for the enzymatic decarboxylation of a,b-unsaturated acid at 277 K, proposed based on the potential energy profiles (Fig. 6), Arrhenius rate constants (k Arr f and k Arr,3 f ) and activation free energies (DG ‡ and DG ‡,3 ) obtained from the TST method. Energies and rate constants are in kJ mol À1 and s À1 , respectively. Long rightwards blue dashed line arrow is an alternative kinetically controlled path, which is too fast to be monitored using the stopped-flow spectroscopic method. It appears that the entropy changes for the elementary reactions that generate molecules, e.g., decarboxylation (II) and cycloelimination (IV), are more pronounced than those for the reactions that involve only charge (proton) transfer, covalent bond breaking/formation and structural reorientation, e.g., 1,3dipolar cycloaddition (I) and catalyst (1) (III) at 3 ¼ 78. Based on the total free energy changes (DG 0,Total and DG 0,Total,3 in Fig. 7b), the decarboxylation/CO 2 elimination reaction ((I) and (II)) at 3 ¼ 78 is slightly more favorable than at 3 ¼ 1 (DG 0,Total,3 ¼ À73 and DG 0,Total ¼ À66 kJ mol À1 ). Likewise, the b-MeSt formation/ cofactor regenerations in the indirect route ((III) and (IV)) at 3 ¼ 78 are signicantly more favorable than at 3 ¼ 1, (DG 0,Total,3 ¼ À11 and DG 0,Total ¼ 10 kJ mol À1 ). These results lead to the conclusion that elementary reactions that involve charge (proton) transfer favor a high local dielectric environment. The proposed thermodynamically favorable paths are illustrated in Fig. 7b (long rightwards red arrows).

Conclusions
Enzymatic decarboxylation of a,b-unsaturated acid through ferulic acid decarboxylase (FDC1) has been of interest because the reaction is anticipated to be a promising, environmentally friendly industrial process for producing styrene and its derivatives from natural resources. In this study, the proposed mechanisms for the enzymatic decarboxylation of a,b-unsaturated acid were theoretically studied using the B3LYP/DZP method and TST. The present study began with geometry optimizations of the proposed model molecular clusters in extreme local dielectric environments (3 ¼ 1 and 78). The model molecular clusters consisted of the Cin substrate, PrFMN cofactor and all relevant residues of FDC1 at the active site. These moderate model molecular clusters made it possible to calculate kinetic and thermodynamic properties with reasonable computational resources.
Analysis of the B3LYP/DZP results showed that the active site structure and volume are not signicantly changed in the enzymatic decarboxylation reaction, which suggested that the FDC1 backbone does not play the most important role in enzymatic decarboxylation processes. These ndings are in accordance with the experimental result that the Glu277-Arg173-Glu282 residue network was conserved in the enzymatic decarboxylation reaction. These ndings conrmed that the selected model molecular clusters (including the active site) are reasonable. Comparison of the potential energy proles that were obtained via the NEB method revealed similar energy barriers at 3 ¼ 1 and 4, 5 except for acid catalyst (1), for which DE ‡ at 3 ¼ 4 is higher than that at 3 ¼ 1, thereby implying that an increase in the local dielectric environment could result in a signicant change in the energy barrier for the elementary reaction that involves proton transfer. The potential energy proles at 3 ¼ 78 conrmed that the increase in the polarity of the solvent could lead to signicant changes in DE ‡ , especially for the transition states that involve charge (proton) transfer.
Comparison of the rate constants that were obtained based on various methods revealed that the zero-point vibrational energies are important and cannot be neglected in TST calculations.
Although the values of the crossover temperatures suggested a low or no quantum mechanical tunneling effect on the Table 3 Standard free energies and entropies of the elementary reactions in 3 ¼ 1 and 78, obtained from TST calculations. Energies and temperatures are in kJ mol À1 and K, respectively. DG 0 and DG 0,3 ¼ standard free energies; DS 0 and DS 0,3 ¼ standard entropies enzymatic decarboxylation of a,b-unsaturated acid, it is advisable to include this effect in the theoretical study on every enzymatic reaction to assure that the effect can be neglected at least in the studied temperature range. Analysis of the rate constants at 3 ¼ 1 and 78 conrmed that the inclusion of the uctuation of the local dielectric environment in the mechanistic model is essential; otherwise, some of the hypothesized elementary reactions are too slow to be monitored using the stopped-ow spectroscopic method. Because the rate constants at 3 ¼ 1 and 78 are not compatible with the time resolution of stopped-ow spectrophotometry, the direct route for generating Prod through acid catalyst (2) is unlikely to be utilized, whereas the cycloelimination that occurs in the indirect route in a low local dielectric environment is the rate determining step.
To examine the entropic effect and determine whether the proposed kinetically controlled (favorable) mechanisms are also thermodynamically controlled, the standard free energy and entropy changes of the elementary reactions were calculated. The results showed that at 277 K, the thermodynamic properties of the elementary reactions that involve charge (proton) transfer ((III) and (IV)) are strongly affected by a high local dielectric environment, which led to the conclusion that overall, the enzymatic decarboxylation of a,b-unsaturated acid is thermodynamically controlled in a high local dielectric environment. It appeared that the factors that affect the standard entropy changes are the disorder/order due to breaking/formation of covalent bonds and charge (proton) transfer in the active site; the standard entropy changes due to generation of molecules are the most signicant (pronounced). The results that are reported in this work illustrate for the rst time scenarios in each elementary reaction and provide insight into the effect of the local dielectric environment on the kinetics and thermodynamics of the enzymatic decarboxylation process of a,b-unsaturated acid, which could be used as guidelines for further theoretical and experimental studies on the same and similar systems.

Conflicts of interest
There are no conicts to declare.