Post-transition state bifurcations gain momentum – current state of the field

Introduction

Once considered a rare phenomenon, reactions involving post-transition state bifurcations (PTSBs) [1], or “ambimodal” [2] transition state structures (TSSs), have been reported more and more frequently in synthetic and even biosynthetic systems (vide infra). A PTSB occurs when a single transition state structure (TSS) leads to multiple products, without intervening minima or secondary barriers to overcome (Fig. 1). Bifurcations can be characterized mathematically [1], [3], [4], [5], [6], [7], by the presence of a valley-ridge inflection point (VRI) [8] on a potential energy surface (PES), or their presence can be inferred when molecular dynamics (MD) simulations produce trajectories leading to more than one product from the region of a single transition state (non-intrinsic reaction coordinate (IRC [4], [9], [10]) dynamic behavior [11]). We favor the former characterization, since multiple products can be formed in MD trajectories when a minimum with multiple low-barrier exit channels is involved or high temperatures are used [12], [13], [14]. On occasion, some have argued for “multifurcations” on the basis of results from dynamics simulations [15], [16], but such results actually appear to be associated with PESs having multiple sequential bifurcations [17] or wide plateaus [18], [19] (in the absence of unusually high TSS symmetry [20]).

Fig. 1:

Schematic representation of a potential energy surface (PES) containing a post-transition state bifurcation (PTSB) using the analytical function z=2x⁵–7x²–5xy+y²+2. Important features of the surface are labeled on the three-dimensional surface, which is projected onto a contour plot in the x-y plane.

The existence of PTSBs has been acknowledged for several decades [21], [22], [23], but an awareness of their prevalence has only recently led to improvements in techniques for analyzing PESs that allow one to make meaningful predictions of experimental outcomes [4], [24]. Traditionally, transition state theory (TST) has been used by organic chemists to relate an activation barrier to a rate constant that can be compared to an experimentally determined rate constant (or a ratio of rate constants, determined by the ratio of products seen experimentally). However, this approach assumes that (1) a single TSS leads to a single product and (2) the shape/curvature of the PES has no bearing on the rate/selectivity of the reaction (although extensions of TST have been developed [25], [26], [27], [28], [29]). In fact, it has been known for several decades that reactions that have a single TSS connected to multiple products without intervening minima cannot be described using statistical factors, and when not taking into account reaction dynamics, the symmetry numbers of the species in a reaction need to be taken into account (though this has rarely been acknowledged; Ref. 37 represents the earliest description of a reaction with a PTSB of which we are aware, describing a generic scenario in which “…both reactions proceed through a common transition state”) [30]. Many of the studies described below highlight the problems associated with these assumptions.

A VRI is defined mathematically as a point on a PES where (1) the eigenvalue of the Hessian matrix is zero and (2) the gradient vector is perpendicular to the corresponding eigenvector [6]. Though there are several methods of locating a VRI on a PES, including variational transition state theory (VTST) [31], reduced gradient following (RGF) [22], [23], Newton trajectories [32], [33], [34], [35] and the artificial force induced reaction (AFIR) method [4], MD simulations are most commonly used for analyzing systems that exhibit unusual post-TS behavior. In particular, MD simulations are frequently used to predict branching ratios for specific reactions for comparison to experimental product distributions. MD simulations for systems with PTSBs are typically initiated from the region of the TSS and propagated forward and backward in time in order to find “productive” trajectories. Notably, doing so assumes a statistical distribution of energy at the TSS from which dynamics trajectories are initiated, making a full analysis of stationary points on a PES necessary to understand what happens prior to the “key” TSS.

Many of the systems discussed below exhibit IRCs that involve two (or more) distinct chemical (bond-forming or bond-breaking) events following a TSS. The events can occur in a step-wise fashion, where an intermediate on the PES separates events, or concertedly (i.e. with no PES minimum found between events). When the two events are concerted, they can occur synchronously or asynchronously [36], [37], [38], [39], [40]. In many cases, the scenario in which these events occur asynchronously is referred to as a “two-step, no intermediate” mechanism [41], but we prefer not to say “two-step” to avoid confusion with “step-wise” and favor instead “two-event”. One can distinguish between step-wise and concerted mechanisms on the basis of the existence of a minimum on the relevant PES corresponding to an intermediate structure, but the assumption that product distributions seen experimentally will reflect the existence of said minimum is not always valid. The simple existence of an intermediate does not demand that products be statistically distributed; rather, the timescale of intramolecular vibrational redistribution (IVR, the time it takes for energy to be distributed statistically across a molecule’s vibrational degrees of freedom) must be faster than the time necessary to overcome the barrier following the intermediate. This means an energetically shallow intermediate might not lead to the expected distribution of products predicted by transition state theory [5], [42]. A reaction involving a PTSB can be thought of as one with a species whose geometry and electronic structure resemble that of a putative intermediate connected to two possible products but with no barriers for formation of each. As such, the PES for a reaction involving a PTSB generally contains a so-called “hidden intermediate” [43], [44]. It seems reasonable to suggest that the product formed via the exit channel that involves vibrations most closely related to (coupled to) those activated in the post-TSS region will predominate (an application of the concept known as “dynamic matching”, i.e. the activated vibrational modes in a TSS are more likely to lead to a product that resides closer to its momentum vector) [12], [45], [46], [47], but this hypothesis has yet to be thoroughly tested for various reaction types. In addition, a recent study has shown that dynamic matching is not necessarily limited to vibrational activation, but rotational activation can also influence the outcome of the product distribution of a reaction containing a PTSB [11].

Another important concept that is gaining momentum ^[1] is the idea that a substrate’s environment can have a huge impact on its distribution of internal energy. The lack of a secondary barrier to formation of the two products following a PTSB seems to increase the sensitivity of such systems to local environment. We see this as a particularly important area of research, ^[2] and several examples of recent studies are described below.

Our review builds on the seminal review on PTSBs published by Ess et al. in [1], with the aim of not only summarizing recent examples of reactions purported to involve PTSBs, but to highlight the relevance of PTSBs to reactions of interest to those working in the fields of synthetic organic chemistry and natural products biosynthesis. For each example we discuss below (a reasonably thorough compilation of examples published since the 2008 review, although not an exhaustive one), we highlight in bold the key concept(s) revealed or extended from the results described. In figures, we use structure numbering from the original papers to simplify comparisons to published work. The reader should refer to the original literature for details about the level(s) of theory used to conduct calculations on each system, but by and large, the methods used in the studies below were standard DFT methods for organic and organometallic systems. ^[3] Our hope is that this new compilation of systems found to involve PTSBs, along with the factors identified to influence selectivity associated with passing through regions of PESs with PTSBs, will facilitate the identification of PTSBs and the design of methods for controlling selectivity on such PESs.

Transition metal-free systems

In 2014, Pham and Houk found a case of a bis-pericyclic [46], [49], [50], [51] reaction (i.e. a reaction proceeding via a transition state structure that has the features of two different pericyclic reactions) that can produce (4+2) or (2+2) cycloaddition products (Fig. 2) [2]. Comprehensive investigations using multiconfigurational methods (CASSCF and CASPT2) revealed a PTSB for one of the systems examined – butadiene+allene. The associated ambimodal TSS is connected to (4+2) product 19 and a diradical intermediate that can subsequently recombine to afford formal (2+2) product 20. These results show that an ambimodal TSS can be connected to both closed- and open-shell products. ^[4]

Fig. 2:

The reaction network of the bis-pericyclic reaction of butadiene+allene as detailed by Pham and Houk [2].

Partial bonds and torsions, where relevant, of the ambimodal TSS are colored such that red corresponds to forming/breaking bonds that lead to one product, blue corresponds to the second, and purple corresponds to partial bonds/torsions in the TSS that are common to both possible products. Resultant red and blue bonds are also shown in the products.

When Bogle and Singleton investigated, computationally and experimentally, the nucleophilic substitution of dichlorovinyl ketone (Fig. 3a), they discovered an extremely simple predictor for the selectivity of the reaction: whether or not a particular vibrational mode was activated [52]. The reaction is a nucleophile substitution at an sp² carbon where the final products are E (2) or Z (3) isomers (either stereoisomer also can react with another equivalent of nucleophile to afford a disubstituted product 4). In this case, the same TSS leads to both stereoisomers of monosubstituted product. The authors describe two main (but not mutually exclusive) factors affecting the selectivity of this reaction: (1) the shape of PES, which would presumably affect the “average” or “total” selectivity of the reaction for a set of trajectories and (2) the momentum of a particular structure coming off of the TSS region, which would affect the product formed in a particular trajectory. This second factor is an example of “dynamic matching” [46], [53], [54]. Here, the direction (positive or negative) of a particular vibrational mode (“mode 8”) appeared to determine the product selectivity: when mode 8 had a “positive” sign (Fig. 3a), 29/93 trajectories led to 3, but when it had a “negative” sign, 0/92 trajectories led to 3. Since there is an equal chance that the velocity of this mode is positive or negative, the overall selectivity was an average of these two scenarios. It was therefore concluded that the possibility of forming 3 was only viable when mode 8 had a “positive” sign, a very unusual selectivity control element for a synthetically relevant organic reaction. Similarly, the heavy-atom motion of a particular vibrational mode was also implicated in the selectivity of a bis-pericyclic hetero-Diels-Alder reaction studied by the same group – the dimerization of methacrolein (Fig. 3b) [55]. Newtonian kinetic isotope effects (KIEs) were found to control the location of isotopic labels in the products. Unlike a typical KIE, which originates from a difference in zero-point energies of an isotopically labeled versus unlabeled molecule, a Newtonian KIE is dynamically controlled, originating from the difference in masses of isotopes leading to different momenta. It was found that the product that resulted from heavier atoms moving the least and lighter atoms moving more predominated. While qualitative in nature, this study points to the idea that the motion of particular atoms in the region of the TSS can influence product selectivity.

Fig. 3:

(a) A PTSB-containing reaction discovered by Bogle and Singleton [52]. It was found that the selectivity of this reaction observed in MD simulations was determined by the phase of vibrational mode 8 (top right). (b) Proposed mechanism for the dimerization of methacrolein investigated by Andujar-de Sanctis and Singleton using kinetic isotope effects [55].

See legend from Fig. 2.

Two studies carried out by Carpenter, Ezra and Wiggins determined the influence of various factors on the behavior of a reaction involving a PTSB: first, for a purely theoretical model potential [6] and second, for the case of the experimentally relevant cyclopropyl radical ring-opening [56]. In the cyclopropyl radical ring-opening, the difference between bifurcation branches is an issue of torquoselectivity [57]. One methylene group of the cyclopropyl ring rotates before the other (called tilt-CH₂, whose angle with respect to the plane of the ring is labeled ϕ_/ in Fig. 4), and the sense of rotation of the second (called perp-CH₂, whose angle with respect to the plane of the ring is labeled ϕ⟂ in Fig. 4) can either rotate toward (conrotatory) or away from (disrotatory) the first methylene. One of the key take-away messages from these studies is that we should be considering molecules in phase space rather than coordinate space, which is essentially saying that momentum needs to be considered in reactivity models. In order to crudely account for the possibility of energy transfer to other degrees of freedom in the system (e.g. to solvent molecules), a “dissipation” term was included in both the model potential and in the dynamics simulations of the cyclopropyl radical system. It was concluded that the dissipation parameter had a strong, but complicated and “highly non-monotonic”, influence on the ratio of products seen in the dynamics trajectories, an issue discussed further below.

Fig. 4:

The cyclopropyl radical ring-opening reaction investigated by Kramer et al. [56] with key angles labeled on the detailed TSS picture (right).

See legend from Fig. 2.

Yamataka and co-workers reported experimental and theoretical studies of substituent effects on the selectivity of Beckmann [58] and Schmidt [59], [60] rearrangements (Fig. 5), describing a substituent-dependence in the IRC behavior of the relevant TSSs, which they attributed to PTSBs on the PESs for such systems. Two routes for generating an imine-derivative bearing a leaving group (Y in Fig. 5) were examined, and in both cases, the reaction path was found to bifurcate after the rearrangement TSS (to afford either a nitrilium cation or benzylic cation+nitrile fragments). For both reactions with R¹ as a substituted phenyl ring, substrates with electron-donating groups (X) had IRCs leading to the fragmentation products, while substrates with electron-withdrawing groups had IRCs leading to the rearrangement product. Additionally, the IRC behavior was found to be dependent on the level of theory in “borderline” cases (i.e. those with substituents that were not strongly electron-donating nor -withdrawing). MD simulations were run on the differently substituted TSSs and it was found that, in cases where the phenyl ring substituent was extremely electron-donating or -withdrawing, the products seen dynamically were almost exclusively the IRC products. Again, in “borderline” cases, both products were formed in significant amounts (implying the presence of a PTSB, although VRIs on these PESs were not specifically located in all cases). Of particular note, the results of this study imply that selectivity for reactions with PTSBs can be modulated by simple electronic substituent effects.

Fig. 5:

The general form of the Beckmann and Schmidt rearrangements, and the specific PTSB-containing reactions analyzed by Yamataka et al. [58], [59], [60], [61].

See legend from Fig. 2.

In a study of the ene reaction of singlet molecular oxygen (¹O₂) and tetramethylethylene (Fig. 6), Sheppard and Acevedo examined the influence of reaction environment on the nature of the PES [62]. The gas-phase behavior of this system had previously been investigated by Singleton et al. who concluded that this system proceeds via a “two-step, no intermediate mechanism” involving a PTSB [41]. Perhaps the most important finding by Sheppard and Acevedo was that in QM/MM studies using explicit solvent, no VRI was found; that is, the mechanism went from a concerted mechanism with a PTSB in the gas phase to a traditional two-step mechanism with a perepoxide intermediate when the influence of explicit solvent molecules on the reacting system was taken into account.

Fig. 6:

The addition of singlet oxygen to tetramethylethylene investigated by Sheppard and Acevedo [62].

See legend from Fig. 2.

Carpenter et al. examined the effects of solvent – specifically, the ability of a solvent to redistribute energy – on the enantioselectivity of the ring-opening of 2,3-difluoro-2,3-dimethyldiazocyclopropane (Fig. 7) [63], which involves a PTSB leading to two enantiomers. The solvents investigated were CHCl₃, CHFClBr, and H₃C–CH(OH)–CF₃ (TFIPA). Though two of the solvents are chiral, only TFIPA seemed to induce enantioselectivity in the dynamics simulations to a statistically significant extent. The authors predicted that TFIPA could induce about 15% enantiomeric excess, an effect that is approximately an order of magnitude larger than that observed experimentally for reactions where each enantiomer is formed via its own separate TSS. It was hypothesized that hydrogen bonding in the TFIPA system might be the cause of such strong enantioinduction. The results of this study suggest that small changes in solvent composition can have a large effect on product distributions for systems that contain PTSBs.

Fig. 7:

PTSB-containing ring-opening reaction leading to two substituted allenyl enantiomers studied by Carpenter et al [63].

See legend from Fig. 2.

Nieves-Quinones and Singleton investigated the selectivity for the nitration of toluene and found several ambimodal TSSs (Fig. 8; five TSSs were optimized for this system, three examples of which are shown) [15]. MD trajectories led to multiple products in both the gas phase and implicit solvent for each TSS. An analysis of the relative energies of neither the TSSs nor the trajectories led to a satisfactory rationalization of experimental product ratios. MD simulations using TSSs that involved single BF₄⁻ counterions or single molecules of H₂SO₄ with an implicit solvent model also were not useful in predicting product ratios. The system was then examined using a sphere of 101 explicit CH₂Cl₂ solvent molecules. Interestingly, when the free energy surface for NO₂⁺ addition to toluene was determined using potential of mean force calculations, no barrier was found for the addition. In the end, running MD simulations including explicit solvent molecules was crucial for observing reactivity that coincided with experimental results. Importantly, a benefit of including explicit solvent molecules in such a computational model is that, as long as sampling of the configurations of the system has been conducted properly, entropy is directly accounted for; appropriately determining entropy corrections when modeling solution-phase reactions is a long-standing problem. In addition, as was the case in this study, sometimes the timescale of reorganization of solvent molecules is slower than would be captured by MD trajectories in implicit solvent, an issue that is not treated in a traditional stationary point analysis. Particularly in “sensitive” systems, such as those containing PTSBs, this factor could make a large difference in reproducing experimental product ratios.

Fig. 8:

The nitration of toluene reaction investigated by Nieves-Quinones and Singleton using explicit solvent MD simulations [15].

See legend from Fig. 2.

Transition metal-catalyzed reactions

Gold (I)

Of particular interest to several groups in recent years are Au-catalyzed reactions involving PTSBs. Rather than describing these systems as involving bifurcations, most authors describe these Au(I)-catalyzed transformations as proceeding via “two-step, no intermediate” reaction mechanisms. For example, in 2010 Garayalde et al. disclosed the results of density functional theory (DFT) calculations on the cyclopropyl ring-opening step of the transformation of XIVa (Fig. 9) [64]. The authors concluded that TS_XIVa(1) could lead directly to either the six-membered ring product (XVIa) or the five-membered ring product (XVa). This makes sense in that this TSS resembles a complex of a carbocation with a π-bond and shifting of these two fragments relative to one another should lead to the two products. A second TSS connecting these two products to each other was located (TS_XIVa(2)) and, like other TSSs for 1,2-alkyl shifts of carbocations, even more closely resembles a complex of a carbocation with a π-bond, consistent with the fact that PTSBs are associated with a high energy TSS whose exit channel leads to the region of a second TSS. In that products derived from XVIa were not detected experimentally, it was concluded that there must be an inherent preference for formation XVa. This product was also calculated to be thermodynamically favored over XVIa by about 20 kcal/mol. This study set the stage for several subsequent studies on PTSBs in Au-promoted reactions.

Fig. 9:

Au(I)-catalyzed transformation containing a PTSB analyzed by Garayalde et al. [64].

See legend from Fig. 2.

Later the same year, Wang et al. [65] described a similar PES for the Au(I)-catalyzed intermolecular hydroamination of allenes (Fig. 10). Again, two TSSs were found along the reaction coordinate connecting the gold-allene complex A with hydroamination product B without intervening minima. Additionally, experimental mechanistic studies led the authors to the conclusion that higher nucleophile concentrations lead to a preference for the “two-step no intermediate” pathway to B, while lower concentrations of nucleophile favor a pathway that goes through planar intermediate 6, which is associated with stereochemical scrambling. These results suggest that the relative concentration of species in a bimolecular reaction can influence which branch of a bifurcating pathway is followed.

Fig. 10:

Au(I)-catalyzed transformation containing a PTSB analyzed by Wang et al. [65].

The Au(I)-catalyzed formation of tetracyclic indolines was also found to potentially involve PTSBs (Fig. 11) [66]. Two ambimodal TSSs were located: one connected to products of exo addition (14) and another connected to products of endo addition (15). In both cases, these TSSs again resemble complexes of carbocations with π-bonds. Interestingly, it appeared as though the endo TSS led to three possible products with no additional intermediates, an apparent “trifurcation”. As described above, however, this outcome can be rationalized by the presence of sequential ambimodal TSSs, although TSSs for the interconversion of products 20–22 were not located. Exo TSS 14 was found to be lower in energy than endo TSS 15, and it was concluded that, since TSS 14 leads to product 17, an analog of the precursor to the experimental product, these results were consistent with experiment.

Fig. 11:

Au(I)-catalyzed transformation containing a PTSB analyzed by Noey et al. [66].

See legend from Fig. 2.

Several additional examples of Au(I)-catalyzed cyclization reactions involving ambimodal TSSs resembling complexes of carbocations with π-bonds have been described (Fig. 12) and studied by various theoretical methods. These include: (a) The “dual activation” reaction mechanism for the Au(I)-catalyzed diyne cyclizations discovered by the Zhang and Hashmi groups [67], [68], [69], [70], [71]. Atom-centered density matrix propagation (ADMP) MD calculations predicted an inherent dynamical preference for the experimentally observed products in some of Zhang’s reactions, thermodynamic control was suggested to rationalize the selectivity in some of Hashmi’s reactions, and gain of resonance stabilization that skews the PES in favor of one type of product was suggested to be a general factor in controlling selectivity. (b) The Au-catalyzed reactions of 2-propargyl-β-tetrahydrocarbolines studied by Yu and co-workers, where chemoselectivity was proposed to be controlled either kinetically or dynamically depending on the substituents on the substrate [72].

Fig. 12:

(a) The “dual activation” reaction mechanism for Au(I)-catalyzed diyne cyclization discovered by the Zhang and Hashmi groups [67], [68], [69], [70], [71]. (b) Au-catalyzed reactions of 2-propargyl-β-tetrahydrocarbolines studied by Yu and co-workers [72].

See legend from Fig. 2.

Rhodium(II)

PTSBs have also been described for reactions involving rhodium catalysts. The first PTSB involving a rhodium catalyst was disclosed by Hansen et al. in 2011 for a “combined C–H activation/Cope rearrangement” (CHCR, Fig. 13) [73]. In this case, the CHCR competes with a direct C–H insertion reaction, where both possible products arise from the same TSS and are interconverted by a [3,3]-sigmatropic shift TSS. The pathway to the CHCR product via the ambimodal TSS is characterized by two events that occur in a concerted, but very asynchronous, fashion. First, a hydride shift to the carbenoid center occurs, followed by C–C bond formation and C–C bond breakage to afford the final CHCR product.

Fig. 13:

The Rh(II)-catalyzed combined C–H insertion/Cope rearrangement investigated experimentally and computationally by Hansen et al. [73].

See legend from Fig. 2.

Very recently, Hare and Tantillo described PTSBs for various Rh(II)-catalyzed intramolecular C–H insertion reactions that afford β-lactone products (Fig. 14) [24]. Similar to the CHCR reaction above, the β-lactone formation pathway involved two concerted but highly asynchronous events: a hydride shift to the carbenoid center followed by C–C bond formation. The ambimodal TSS also led to products of fragmentation with an associated pathway in which the hydride shift was followed by C–O bond cleavage. Several factors were found to influence the minimum energy path (MEP) from the ambimodal TSS to products, including the relative stereochemistry and the conformation of the TSS. As a result, it was suggested that these factors may be markers of the presence of PTSBs. In addition, the presence of a PTSB provided a rationalization for formation of unexpected byproducts, a scenario that may be relevant to other systems.

Fig. 14:

Rh(II)-catalyzed C–H insertion reaction found to involve a PTSB by Hare and Tantillo [24].

See legend from Fig. 2.

Biosynthetically relevant systems

PTSBs have also been found in biosynthetically relevant transformations, leading to questions as to how nature is able to control reactivity for reactions with complex PESs. Reports of PTSBs have been particularly prevalent for carbocation rearrangements leading to terpene natural products, likely a result of the tendency of carbocation rearrangements to involve PESs that are extremely flat (note also the prevalence of PTSBs for Au-triggered carbocation reactions described above) [74], [75], [76], [77], [78], and several of these will be described first (those for which PTSBs were proposed but not examined in detail can be found in Refs. [74], [79], [80], [81]). Because these reactions often involve the conversion of acyclic, achiral starting materials into complex (poly)cyclic structures with numerous stereogenic centers, a deep mechanistic understanding of the factors influencing selectivity could be useful for the design of new stereoselective reactions.

The first biosynthetically relevant system containing a PTSB was described in 2009 by Hong and Tantillo [82]. The conversion of a pimarenyl cation to abietadienyl cation (Fig. 15) in the absence of enzyme was found to involve a PTSB following a TSS appearing to correspond to an intramolecular proton transfer [83]. Following proton transfer, there are two possibilities for the next event, neither of which has a barrier on the PES: (1) methyl migration to afford product 4 (the abietadienyl cation), and (2) ring expansion to afford product 5, a structure with a terpene skeleton not yet reported in products of abietadiene synthase or found in any reported natural product. After this initial PES analysis of the abietadienyl cation system, it was concluded that either inherent dynamic effects precluded formation of “unnatural” product 5 in favor of the abietadienyl cation 4, or abietadiene synthase was specifically responsible for avoiding formation of this side product. Two follow-up studies by Siebert et al. [84], [85] tested the inherent dynamical tendencies of both a truncated version of the reacting substrate and the full substrate by running MD simulations initiated from the region of the ambimodal TSS. Though formation of product 4 was preferred slightly over formation of product 5 in these simulations, suggesting that nature may take advantage of dynamical tendencies inherent in enzyme substrates, product 5 was still formed in large enough quantities to conclude that inherent dynamical tendencies were not solely to blame for exclusive formation of product 4.

Fig. 15:

A biosynthetic carbocation isomerization that involves a PTSB, discovered by Hong and Tantillo [82].

See legend from Fig. 2.

Subsequently, Hong and Tantillo discovered that PTSBs likely play important roles in a related terpene synthase promoted reaction, the formation of miltiradiene (Fig. 16) [17]. In this case, an isomeric pimarenyl cation (A2) can react by an ambimodal TSS (TS10_re), but here, multiple sequential ambimodal TSSs are found to be connected to each other. When MD simulations were initiated from the regions of different ambimodal TSSs, different distributions of products were predicted, with the predicted major products differing in some cases. This discovery can be seen as a caution to computational/theoretical chemists to consider all chemical events in a reaction mechanism, to ensure that the “history” of the reacting substrate is not forgotten. When MD simulations were initiated from the region of the initial ambimodal TS10_re, the carbocation precursor to miltiradiene (D2/D3) was predicted to be the major product, but again not with high enough selectivity to explain the absence of other products. It seems that nature does take advantage of inherent dynamical tendencies, but enhances them. For this reaction, it was proposed that a particular bond torsion is connected directly to product outcome, similar to the scenario found for the substitution reaction shown in Fig. 3a. Additionally, the sensitivity of the dynamics results to the location from which particular MD trajectories are initiated may hint at an enzyme’s means of steering a reaction toward particular products. That is, the shape of the active site could influence not just the initial conformation of the substrate (which also was shown in this report to play a key role in dynamic control of product distribution, i.e. results were very different when starting from conformer A1/TS2_si), but also the vibrations accessible to the substrate throughout the reaction, preventing access to some reaction pathways that are barrierless in the gas phase.

Fig. 16:

Reaction network for the formation of miltiradiene (deprotonation product of D2/D3) and related carbocations, mapped out by Hong and Tantillo [17].

See legend from Fig. 2.

On the basis of gas phase calculations, Major and Weitman and Hong and Tantillo suggested that a PTSB also may be involved in the proposed biosynthetic mechanism for formation of bornyl diphosphate by bornyl diphosphate synthase (BPPS), with bornyl cation, which is a TSS and not an intermediate, serving as branching point between BPP and camphyl cation (Fig. 17) [86], [87], [88]. Closely related carbocations have also been found to rearrange via PTSBs [74], [89]. Major and Weitman’s combined quantum mechanics/molecular mechanics (QM/MM) MD studies on the entire enzyme-substrate system predicted the formation of multiple products, with a preference for formation of BPP. However, the predicted preference was not large enough to preclude formation of various side products, consistent with the experimentally observed promiscuity of BPPS. The preference for formation of BPP as a major product was explained as arising from a combination of dynamic effects and electrostatic guidance by pyrophosphate (OPP) in the active site. The conclusion that the pyrophosphate ion was directing the reaction towards formation of BPP is consistent with the studies discussed above; simple, non-covalent, electrostatic interactions can have a large influence on product ratios in systems with PTSBs. This finding could have significant implications for the role of terpene synthases (and other catalysts) in modulating carbocation reactions in general [3].

Fig. 17:

A portion of the carbocation rearrangement reaction investigated by Major and Weitman and Hong and Tantillo, which contains a PTSB in the gas phase and in the enzyme (investigated using QM/MM calculations) [86], [87]. The ambimodal TSS in the gas phase was found to be the bornyl cation, but in the enzyme was a TSS between the bornyl cation and the pinyl cation.

See legend from Fig. 2.

PTSBs have been proposed to be involved in several other classes of enzymatic reactions. For example, nearly concurrently with the report of a PTSB in the abietadiene-forming reaction described above, Tao, Gatti and Schlegel described a PTSB involved in the formation of 3-deoxy-D-manno-octulosonate 8-phosphate [90]. QM/MM calculations were conducted to map out the PES of the reaction of arabinose-5-phosphate and phosphoenolpyruvate (A5P and PEP, respectively; Fig. 18) plus a water molecule within the enzyme where this reaction occurs, KDO8PS (PDB ID: 2NWS). Several possible mechanisms were investigated and the lowest energy pathway was found to correspond to syn addition of water to the si side of PEP and PEP to the re side of A5P. This path exhibited a single TSS (TS1, Fig. 18) that could lead to two possible intermediates: Z_INT, a zwitterionic intermediate along a stepwise mechanism, or T_INT, a tetrahedral intermediate along a concerted mechanism (with respect to nucleophilic addition of PEP and water to A5P). This discovery led the authors to conclude that this mechanism has both stepwise and concerted characteristics, and since there is a low barrier to convert Z_INT to the lower in energy intermediate T_INT, the lifetime of Z_INT would be short. This study highlights the need to compute as many possible reasonable reaction paths that one can imagine for a biological reaction and incorporation of the enzyme in calculations is often necessary to produce results consistent with experiment (although much of the inherent carbocation reactivity described above is consistent with results from experiments with enzymes).

Fig. 18:

A portion of the reaction investigated by Tao et al. [90]. A single TSS leads to both tetrahedral and zwitterionic intermediates, which are along pathways that are either concerted or stepwise with respect to nucleophilic addition of PEP and water to A5P, respectively.

See legend from Fig. 2.

PTSBs have also been implicated in several biosynthetic cycloaddition reactions. Patel et al. proposed a PTSB for an enzymatic transannular Diels-Alder reaction involved in the biosynthesis of spinosyn A (Fig. 19) [91]. Here, a bis-pericyclic [46], [49], [50], [51]. TSS (TS7) that could lead to either a [4+2]-cycloaddition product or a previously unreported [6+4]-cycloaddition product was found; this ambimodal TSS was followed by a TSS by which the [6+4] product interconverts with the [4+2] product via a [3,3]-sigmatropic shift (TS8). The IRC for the ambimodal TSS led to the [6+4]-cycloadduct, and 353 quasiclassical direct dynamics trajectories initiated from the region of this TSS led to a predicted product ratio of 2.5:1 in favor of the [6+4] product. The “time gap” [92], [93] between bond-forming events in the dynamics simulations was unusually long, on average 173 and 139 fs for 5 and 6, respectively (other unsymmetrical Diels-Alder reactions have been shown to have a median time gap on the order of 25 fs or less). This high asynchronicity was hinted at in an earlier study by Smentek and Hess on the PES [94]. Thus, it was concluded that, though this is an “energetically concerted” reaction, the large time gap between bond-forming events marks this reaction as “dynamically stepwise” and involving what is sometimes referred to as an “entropic intermediate” [53]. To verify this, canonical variational transition state theory (VTST) was used to determine the location of the transition state structure on the free energy surface leading to the “dynamical bottleneck”. A closely related bispericyclic [6+4]/[4+2] reaction has been described for formation of heronamide A [95], and PTSBs also have been proposed for other biosynthetic Diels-Alder reactions thought to occur in the absence of enzymes [96], [97].

Fig. 19:

The bis-pericyclic reaction that may occur within the active site of SpnF investigated by Patel et al. [91].

See legend from Fig. 2.

Conclusions and outlook

PTSBs often occur in systems with a TSS that involves two distinct chemical “events” that, if occurring in two separate chemical steps, would be separated by a “reasonable” intermediate – an intermediate that would also be expected to have a low barrier for a different reaction. Though PTSBs have been proposed to be involved in various reactions for many years, useful guidelines for (1) definitively detecting PTSBs, (2) accurately predicting product ratios, and (3) manipulating selectivity are finally emerging. Though an exhaustive list of factors that control product distributions in systems containing PTSBs cannot yet be formulated, we have found the following points exemplified by the studies above to be the most pertinent: (a) the energy distribution of particular vibrational modes (atom motion/momentum), (b) the effects of non-covalent interactions between the reacting molecule(s) and explicit solvent molecules, enzymes, or other molecules present in the system, and (c) the structure (e.g. substituents, relative stereochemistry, and conformation) of the ambimodal TSS. These factors are by no means mutually exclusive, and likely work in tandem to achieve a specific distribution of products. Our hope is that the collection of results described above will convince readers that PTSBs are common enough to be considered when analyzing unexpected selectivity for organic, organometallic and bioorganic reactions and will inspire future studies aimed at controlling selectivity for reactions with PTSBs.