transition state theory 
 ( tst ) describes the rates of chemical reactions 
 in terms of the free energy barrier for the process .
although the theory was developed mainly to describe reaction dynamics 
 in the gas phase , it has been applied to activated processes in condensed - phase 
 environments using one of the following forms:1 in eq 1 ,  is 
 the reaction coordinate , and the dynamical factor || corresponds to the equilibrium ensemble 
 average of the absolute crossing velocity ,   
 d/dt , evaluated at the transition state ( ts ) , 
 where  = . the reaction - coordinate 
 velocity is typically described as the velocity of a free particle : 
 || = ( 2/meff ) , where  = ( kbt ) , in which kb is 
 boltzmann s constant and t is the temperature , 
 and meff specifies the effective ( or reduced ) 
 mass for motion along the reaction coordinate .
q is the reduced classical phase - space density at the 
 dividing hypersurface along the reaction coordinate , and qrs is the classical partition function for the reactant 
 state ( rs ) .
the ratio q / qrs is equivalent to the ratio of the equilibrium 
 populations of the system in the ts and the rs , ()/ ( ) 
 d , and represents the probability density that the system will 
 reach . specifically , the integration in 
 the denominator of the last term is over values of the reaction coordinate 
 corresponding to the rs .
it is this probability ratio , or activation 
 factor , that dominates the rate constant and , most importantly , its 
 temperature dependence .
the potential of mean force ( pmf ) , w( ) , is the classical - mechanical free energy profile 
 along the reaction coordinate averaged over all other ( orthogonal ) 
 degrees of freedom .
essential to this theory is the activated 
 complex located at the 
 dividing surface between reactants and products ( i.e. , the ts ensemble ) . 

this activated complex is presumed to be in quasi - equilibrium with 
 the reactant molecules ( i.e. , the complexes concentration 
 is unaffected by the rate of transformation to products or the products 
 concentration ) and is calculated by equilibrium theory . in most flavors 
 of traditional tst , a distinct reaction coordinate is separated out 
 from all other degrees of freedom , and the motion of the system along 
 the reaction coordinate leading to the ts is assumed to be separable 
 from these other degrees of freedom at . if the motion along the reaction coordinate is slow relative to 
 the response of the remaining degrees of freedom , equilibrium solvation 
 is assumed . moreover ,
tst treats barrier crossing at the ts as a free 
 translational motion , and nuclear quantum effects ( nqes ) such as tunneling 
 are ignored .
these inherent limitations of traditional tst are well - known 
 and have been reviewed extensively . the 
 missing ingredients in tst may be added as correction terms 
 in the generalized version of tst , thus in principle providing the 
 true rate constant .
indeed , prefactors accounting for ts recrossing 
 and nqes have been added with great success .
however , the 
 basic assumptions hidden within the tst framework , such as equilibrium 
 solvation and free - particle behavior at the ts , might directly influence 
 the nature of the activated complex .
it is not clear that correction 
 terms relying on the tst activated complex are sufficient to obtain 
 the correct rate constant .
moreover , 
 finding an optimal reaction coordinate is extremely challenging when 
 a large number of degrees of freedom is involved in the reaction , 
 such as reactions in water and in enzymes .
indeed , the choice of reaction coordinate can greatly influence 
 the computed rate constant within tst .
interestingly , a recent work 
 by peters and co - workers suggested that it might be impossible to 
 remove the recrossing phenomenon even in a fairly simple system such 
 as ion dissociation in water .
the 
 above - mentioned question regarding the nature of the activated 
 complex obtained from tst - based approaches becomes particularly acute 
 in complex systems such as enzymes .
although 
 our current understanding of enzyme catalysis relies on decades of 
 groundbreaking work within the tst framework , this understanding assumes in most cases 
 that the environment influences the reaction by means of equilibrium 
 solvation .
moreover , the role of dynamics in condensed - phase reactions 
 is still under intense debate . in this paper
, we scrutinize the nature of the activated complex 
 obtained from standard simulation methods employed in conjunction 
 with tst . in particular , we use umbrella sampling ( us ) and the string method ( sm ) with various reaction coordinates defined on the basis of the reacting 
 species .
the us and sm simulations yield minimum - free - energy paths 
 along a set of reaction coordinates in which the environment provides 
 equilibrium solvation ( i.e. , the effect of the degrees of freedom 
 orthogonal to the reaction coordinate is treated as a boltzmann - averaged 
 influence on the reaction progress ) .
typically , us and sm employ some 
 form of biasing potentials to allow barrier crossings .
these standard 
 methods in computational enzymology are compared with simulations 
 using transition - path sampling ( tps ) , which yields a collection of 
 reactive dynamics trajectories . using tps
, one considers only 
 trajectories connecting the reactant and product basins . as a result 
 of the bias introduced with this requirement , configurations on the 
 transition pathways
therefore , configurations with low weight 
 in the equilibrium ensemble might have a much larger weight in the 
 transition - path ensemble if they belong to regions that must be traversed 
 to cross from reactants to products .
however , the dynamics of the 
 barrier crossings are unperturbed and independent of the definition 
 of the reaction coordinate .
thus , the present work provides an important 
 benchmark comparing condensed - phase reaction simulations using tst - like 
 methods ( i.e. , us and sm ) and dynamics - based methods ( i.e. , tps ) . 

the model system for the current study is the enzyme dihydrofolate 
 reductase from escherichia coli ( ecdhfr ) , which serves as an important test system in enzymology . 

the hydride transfer in ecdhfr ( scheme 1 ) has been studied by numerous researchers , both experimentally and computationally .
the current results show that 
 ts ensembles obtained from classical 
 simulations for ecdhfr using standard us or sm techniques 
 with various reaction coordinates are substantially different from 
 those obtained using tps . in particular , the observed donor 
acceptor 
 distances ( dads ) at the ts are considerably longer for tps than us / sm . 
 the present discrepancy between the two sets of approaches warrants 
 further studies to understand the extent to which reaction coordinate 
 dynamics , nonequilibrium solvation , and biasing potentials affect 
 the computed rate constants and kinetic isotope effects .
the initial 
 coordinates used to build the model for the present 
 study were based on the crystal structure of a complex of ecdhfr with folate , nadp , and water molecules 
 ( pdb i d code 1rx2(54 ) ) , where the met20 loop is in the closed 
 conformation .
the original ligands in this structure were exchanged 
 with n5-protonated 7,8-dihydrofolate ( henceforth h3folate ) and nadph to form a model of the reactive michaelis complex . 
 of the 159 amino acid residues , all of the ionizable residues were 
 treated as bearing protonation states corresponding to neutral ph . in particular , asp27 was modeled as deprotonated , and the specific 
 protonation states of the histidine residues were determined on the 
 basis of hydrogen - bonding interactions .
this model was soaked in a 
 pre - equilibrated 65   65   65  cubic 
 water box and thereafter neutralized by adding 14 sodium ions to allow 
 evaluation of electrostatic interactions using the ewald summation 
 scheme .
the potential energy surface 
 ( pes ) in the current study was described by a hybrid quantum mechanics / molecular 
 mechanics ( qm / mm ) hamiltonian .
the qm region consisted of 69 
 atoms , including portions of the substrate and coenzyme in proximity 
 to the reaction center along with two link atoms .
this part was described by a modified am1 semiempirical 
 hamiltonian in which the specific reaction 
 parameters ( srps ) were optimized to treat model reactions involving 
 various derivatives of nicotinamide and pterin compounds ( denoted 
 am1-srp ) .
the mm part contained the protein , 
 the remaining substrate and coenzyme atoms not described by qm , waters , 
 and ions .
the mm atoms were treated using the charmm36 force field with grid - based energy correction maps ( cmap ) for peptide dihedral angles , and water molecules were represented 
 by the three - point - charge tip3p model .
qm / mm interactions were treated by electrostatic embedding , wherein 
 the mm partial atomic charges were included in the one - electron hamiltonian . 

the qm / mm interaction energies between the reacting fragments ( qm ) 
 and the protein ( mm ) were fine - tuned by modifying the van der waals 
 parameters of the qm hydrogen atoms .
this 
 combined potential energy was shown to yield accuracy comparable to 
 that of density functional theory ( dft ) , giving accurate results for 
 the hydride transfer reaction in ecdhfr .
periodic boundary conditions 
 were applied , and the ewald method was employed for reciprocal - space 
 summations between mm sites as well as for the qm / mm interactions 
 ( 64  64  64 fft grid ,  = 0.340  ) . a 13.0  group - based cutoff 
 was applied for van der waals and electrostatic interactions .
all 
 atoms were gradually relaxed at the mm level of theory to remove close 
 contacts in the initial protein  ligand 
isobaric ( npt ) ensemble was 
 employed at 298 k and 1 atm using the extended pressure / temperature 
 ( cpt ) algorithm with the hoover thermostat .
newton s equations of motion were integrated 
 with a time step of 1 fs , and the shake 
 algorithm was applied to constrain all 
 mm bonds involving hydrogen atoms .
the system was heated in a stepwise 
 fashion from 48 k to 298 k over 25 ps and thereafter equilibrated 
 at the target temperature ( 298 k ) over the course of 1 ns at the mm 
 level of theory , with a further 200 ps of equilibration using the 
 qm(am1-srp)/mm potential .
further details of the molecular dynamics 
 ( md ) simulations are available in ref ( 48 ) .
the classical - mechanical 
 pmf was 
 determined using the us technique in order to sample the high - energy 
 regions of the pes .
the reaction coordinate 
 (  ) was defined geometrically as the difference between the 
 lengths of the breaking ( c4nnadph  h4n ) and forming 
 ( h4n  c6h3folate ) bonds ( henceforth 
 denoted as 1dasym ) .
a total of 17 discrete regions along 
 the reaction coordinate (  windows  ) were defined with 
 uniform spacing of 0.25  .
each simulation was performed with 
 the addition of a biasing potential ( roughly the negative of the computed 
 pmf ) and a harmonic restraint centered at each window .
the sm 
 simulations were carried out using the implementation by ovchinnikov 
 et al . in charmm .
the first one used two distances ( the 
 forming and breaking bond distances ) , and the second one used three 
 distances ( the forming and breaking bond distances and the dad ) . for 
 each set of cvs ,
the entire path ( called the string ) connecting the 
 reactant basin to the transition state and to the product basin was 
 represented by 48 discretized images , with one md replica assigned 
 to each image .
for example , image 0 represented the reactant - state 
 md replica and image 47 the product - state md replica .
first , starting from an initial path 
 generated by the us simulation , an iterative path optimization was 
 carried out for 200 ps .
each iteration was composed of a short ( 0.5 
 ps ) md simulation , during which the force on each cv was evaluated , 
 and an update of the cvs .
the md simulation was carried out with a 
 harmonic restraining potential applied to each cv with a force constant 
 of 250 kcal mol  .
the 
 cv update was carried out by evolving the path in the direction of 
 the negative gradient of the pmf and then reparametrizing the cvs 
 to enforce approximately equal arc lengths between neighboring images . 

then the final pmf was evaluated over a 200 ps md simulation without 
 updating the cvs . during the final pmf simulation ,
the error of 
 the final pmf values was computed by block - averaging ( 50 ps ) of the 
 entire 200 ps results ( figure s1 in the supporting 
 information ) .
the tps 
 simulations followed 
 the implementation by the group of schwartz and co - workers .
the initial step in the tps algorithm is 
 to define the reactant and product basins .
herein we defined the reactant 
 basin as h3folate + nadph and the product basin 
 as h4folate + nadp ( scheme 1 ) .
the reactive bond lengths were employed as order parameters 
 for each basin , and a bond was considered formed if the distance between 
 the donor or acceptor carbon and the transferring hydride was 1.5 
  .
an order parameter of 1.3  was also tested and 
 had no qualitative impact .
initially , a single reactive trajectory 
 was generated using a biasing potential with a gentle harmonic restraint 
 centered at the ts , which was obtained from us simulations .
this resulted in a 500 fs reactive trajectory connecting the reactant 
 and product basins .
subsequently , a random point along this trajectory 
 was chosen as a seed point for a new trajectory .
the momenta of all 
 of the atoms in the system were perturbed by a small amount to generate 
 a new set of velocities .
the perturbations were chosen from a zero - mean 
 gaussian distribution multiplied by a scaling factor in the range 
 0.100.25 .
the perturbation was then rescaled to ensure that 
 the total energy was conserved and that the system did not acquire 
 net linear or angular momenta .
the random seed point with the new 
 momenta was then propagated forward and backward in time for 250 fs , 
 yielding a 500 fs trajectory . if the new trajectory connected the 
 reactant and product basins , it was selected as a new reactive trajectory . 

if the trajectory was not reactive , it was rejected and a new point 
 along the previous reactive trajectory was chosen . in this way ,
we simulated eight separate tps runs until each yielded 400 reactive 
 trajectories , giving a total of 3200 reactive trajectories . following the generation of the reactive trajectories
, we then turned 
 to locating the ts along each trajectory using committor analysis . 

the dividing surface was defined as a point in phase space with equal 
 probabilities of ending in the reactant basin and the product basin . 
 for each point along the reactive trajectory , a set of activated dynamics 
 simulations with random initial velocities chosen from a maxwell  boltzmann 
 distribution were initiated , and these trajectories were followed 
 as they settled into the reactant or product basins .
the numbers of 
 trajectories settling in the reactant basin ( nr ) and the product basin ( np ) were 
 then collected . in practice
, we performed a bracketing search in the 
 vicinity of the minimum dad for each trajectory and initiated 30 activated 
 dynamics runs .
to determine the 
 momentum distribution of the transferring 
 hydride atom at the dividing surface , we employed the quantum  classical 
 open - chain path integral ( qcopi ) method .
the tss were defined on the basis of the pmf in the case of the 
 us simulations and the committor analysis in the case of the tps trajectories . 

the open paths were represented by 33 beads , and the simulations were 
 performed with isotropic sampling .
approximately 10 classical 
 configurations and 100500 monte carlo staging steps were employed 
 in computing the momentum distribution for the ts ensembles .
a representative tps reactive trajectory is presented in figure 1 . the reactive event is followed by tracing the 
 distances between the transferring hydride and the donor and acceptor 
 atoms as well as by monitoring the dad . as the reaction occurs , the 
 c4n 
h4n distance increases while the c6h4n distance 
 is reduced . toward the ts the reactive c
 h vibration develops 
 large amplitudes , similar to those of the product c  h shortly 
 after the reaction .
this is indicative of thermally hot vibrations , 
 which lie at the edge of the maxwell  boltzmann kinetic energy 
 distribution and only dissipate after hundreds of femtoseconds .
concomitant 
 with these changes , a dip in the dad curve is observed , which is typical 
 for h / h / h transfer reactions 
 ( also see the comparison of rs and ts in table 1 ) .
each chemical event is rapid and occurs within the time frame 
 of a single donor 
acceptor symmetric vibration ( i.e. on the 
 order of a few hundred fs ) .
it is therefore 
 unlikely that the current tps simulations are biased toward short 
 transitions .
the very fast hydride transfer is facilitated by favorable 
 reactive 6n - dimensional phase - space states ( 3n for the configuration and 3n for the 
 corresponding momentum , i.e. , dynamics ) . during such fast chemical 
 events
it is unlikely that the enzyme environment has sufficient time 
 to fully relax , and quasi - equilibrium is not obeyed .
illustrative geometric 
 features from a reactive event during a 
 transition - path sampling trajectory . to quantify the convergence of the tps simulations
, we used 
 the 
 minimum dad during the reactive event . employing this metric , we followed 
 eight independent trajectories as they evolved in trajectory space . 

the results are displayed in figure 2 . as is 
 readily clear from the figure , the trajectory search converged after 
 ca .
this suggests that considerable trajectory searches are required until 
 enzyme states that are optimal for hydride transfer are found .
the 
 final 50 trajectories of each of the eight independent tps runs were 
 employed in the ts analysis .
minimum donor  acceptor distances ( dads ) 
 from eight separate 
 tps reactive trajectory series .
the block - averaged ( ba ) minimum dads 
 are shown in black . in table 1
we 
 have collected the average values of key geometric parameters for 
 the rs and ts obtained from the tps simulations .
these are compared 
 with values obtained from us and sm simulations following previously 
 described approaches .
further details 
 of the sm results are available in figure s1 in the supporting information .
analysis of the geometrical parameters 
 of the ts ensembles revealed significant differences between the tps 
 simulations on the one hand and the us and sm simulations on the other 
 hand . for simplicity , in the following
we compare only the tps and 
 us results , as the us and sm results are similar . at the ts , the average 
 c4n 
h4n distance from tps is 1.37  0.05  , while 
 the average c6h4n distance is 1.41  0.04  . the 
 tps average value for the antisymmetric stretch coordinate ( i.e. , rc4n  h4n 
rc6h4n ) , which is often employed as a reaction coordinate 
 in standard us simulations , is 0.04  0.07  .
the 
 corresponding values obtained from us ( 1dasym ) are 1.30 
  0.03 , 1.40  0.03 , and 0.10  for rc4n  h4n , rc6h4n , and the antisymmetric stretch coordinate . in our previous work
, 
 we defined a rehybridization coordinate and applied it in multidimensional free energy simulations of several 
 enzyme - catalyzed reactions involving a hydrogen transfer step .
this coordinate quantifies the 
 difference between the orbital hybridization states of the acceptor 
 and donor carbons in geometric means and 
 is independent of the position of the transferring hydrogen .
the average 
 ts value of the rehybridization coordinate obtained from tps is 0.09 
  0.12  , compared to 0.10  0.20  from 
 us .
these values suggest that the ts obtained using tps is similar 
 to that obtained using us .
however , the average tps dad is 2.75  
 0.06  , which is considerably longer than that obtained with 
 us ( 2.64  0.06  ) .
the dad distributions from us and tps , 
 which are presented in figure s2 in the supporting 
 information , show a wider distribution in the tps simulations 
 than in the us simulations .
in addition , the tps ts is more linear , 
 with an average c4n  h4n 
c6 angle of 164.2  6.4 , 
 while an angle of 159.8  7.8 was obtained with us .
we 
 note that an in vacuo model bimolecular ts complex calculated with 
 the same am1-srp hamiltonian has a saddle - point angle of 162.4. the van der waals contact / proximity between 
 the sulfur atom of the met20 side chain and the n5 and n7n positions 
 of the reactive complex allows the lone pair of the sulfur to maintain 
 interactions with the bound ligands via either hydrogen bonding or 
 dative bonding .
together with the hydrogen bonding between the n7n 
 atom and the carbonyl oxygen of ile14 , these support the nicotinamide 
 ring in its binding site toward the catalytic event . according to 
 the present tps simulations , the interaction between the pterin n5 
 nitrogen and the met20 sulfur atom is slightly weakened during the 
 course of the reaction because the positive charge on the nitrogen 
 is neutralized as the hydride transfer occurs .
this latter finding 
 is in contrast to that of luk and co - workers , who suggested a stronger 
 interaction with met20 in the ts on the basis of us simulations .
indeed , this subtle difference was also not observed in the current 
 us simulations .
the hydrogen bonds of the nadph / nadp amide 
 group with the backbones of ala7 and ile14 are shortened upon reaching 
 the ts , suggesting tighter binding of the ts according to the tps 
 simulations . again , this trend was not fully observed in our us simulations . 

finally , the tps simulations predict the angle between the amide moiety 
 and the pyridine ring of the nicotinamide to be 1520 
 in the rs , while it is ca .
6 at the ts ( similar to that observed 
 in the crystal structure of the dhfr : folate : nadp ternary 
 complex ) , significantly increasing the 
 overlap between the  systems within the nicotinamide .
thus , 
 tps seems to predict that dhfr is an enzyme designed to bind the ts 
 more tightly while enhancing orbital conjugation , although differences 
 between many of the structural parameters are within the standard 
 deviations .
we further note that the classical recrossing transmission 
 coefficients 
 (  ) computed using us and tps are considerably different ( figure 3 ) . using us with an antisymmetric stretch coordinate , 
 we obtained a recrossing factor of 0.63 , corresponding to a free energy 
 error of 0.3 kcal / mol in the barrier height .
in contrast , using tps we obtained a transmission coefficient of 
 0.82 , as there is much less recrossing .
similarly , using a 3d coordinate , 
 we obtain transmission coefficients of 0.76 and 0.87 with us and tps , 
 respectively .
time - dependent transmission coefficients , (t ) , for the hydride transfer reaction in ecdhfr computed 
 on the basis of trajectories produced with one- and three - dimensional 
 umbrella sampling simulations with an antisymmetric stretch and collective 
 reaction coordinates ( us 1d and 3d , gray ) and from transition - path 
 sampling simulations ( tps 1d and 3d , black ) . in the tps simulations
, 
 the transition states were determined using a committor analysis and 
 (t ) was computed using the 1d and 3d coordinates 
 as a metric .
we note that using a 
 three - dimensional reaction coordinate composed 
 of an antisymmetric stretch , the dad , and a rehybridization coordinate 
 gave nearly identical ts geometries as the simple one - dimensional 
 antisymmetric stretch coordinate .
additionally , 
 sm with an additional collective reaction coordinate gave similar 
 results as us , albeit with a slightly longer dad .
we also note that 
 employing an environmental reaction coordinate based on potential 
 energy in conjunction with a standard antisymmetric stretch coordinate 
 yielded results similar to the current us results .
taken together , the results of the present work suggest 
 that simulations incorporating tst - like assumptions are unable to 
 properly sample long dads , whereas the dynamics - based tps method accesses 
 long dads during barrier crossing . finally , we compared the 
 quantum momentum distributions ( obtained 
 using the recently developed qcopi method ) for the transferring hydride at the us and tps dividing surfaces . 

the momentum distribution is a highly sensitive reporter of the potential 
 experienced by a particle . in figure 4
we compare 
 the momentum distributions obtained for dividing - surface configurations 
 from us simulations restrained to different dads with the ts obtained 
 from tps simulations .
it is clear from inspection of the us curves 
 in figure 4 that as the dad increases at the 
 ts , the momentum distribution becomes increasingly narrow , reflecting 
 a wider position distribution .
interestingly , the momentum distribution 
 obtained for the tps dividing surface is significantly different from 
 all of those attained with us .
careful inspection of the tails of 
 the momentum distributions suggests that the potential experienced 
 by the hydride using us resembles a single - well potential , whereas 
 the potential using tps corresponds to a small double - well potential . 

this suggests that the potentials experienced by the transferring 
 hydride at the ts obtained using different techniques are rather different .
momentum 
 distributions for the transferring hydride at transition 
 states obtained using umbrella sampling at different donor 
tps simulations have been applied to various 
 enzyme systems , although presently 
 it is not a household method in computational enzymology . in the present 
 work
, we performed a comparison between the activated complexes obtained 
 for the enzyme dhfr using the tps method and two other widely used 
 approaches , us and sm .
we found that the tss obtained using the two 
 families of methods are significantly different .
in particular , the 
 tss obtained using tps have longer dads and are more linear than those 
 observed using us and sm .
additionally , the classical recrossing transmission 
 coefficients using us and tps were found to be significantly different . 
 with us
there is considerable recrossing of the ts , whereas with tps 
 there is much less recrossing . in the current study
, we did not compare 
 the free energy profiles associated with the transition - path ensemble . 

this is not a trivial task , and additional information is required 
 to determine reaction probabilities .
however , 
 the free energy profiles from tps are not expected to be significantly 
 different from the us and sm profiles because the reaction barrier 
 is less sensitive to the exact location of the ts .
indeed , the free 
 energy surface for dhfr is rather flat in the ts region with respect 
 to the dad .
a related careful comparison 
 between us and tps on the rotational barrier in a disaccharide in 
 vacuo showed similar results with the two methods , although additional 
 pathways were found using tps . however , 
 certain kinetic parameters such as the kinetic isotope effect and 
 dynamic recrossing at the ts are sensitive to the geometry of the 
 ts .
in particular , 
 the nqes responsible for the kinetic isotope effect are highly sensitive 
 to the fine details of the ts .
it is not clear that simple corrections 
 to the rate constant via various prefactors can rectify the errors 
 introduced into the computation of kinetic isotope effects using an 
 incorrect ts ensemble .
we stress that the difference between 
 the us / sm and tps approaches 
 is not merely one of computational strategy .
instead , it is likely 
 that underlying physical assumptions inherent to the methods , such 
 as equilibrium solvation and the nature of the reaction coordinate 
 and its motion , give rise to the different results .
the traditional 
 tst - based us and sm techniques assume that the barrier - climbing process 
 is slow relative to the relaxation of the environment , and therefore , 
 at every value of the reaction coordinate the environment is assumed 
 to be fully relaxed to achieve thermodynamic equilibrium throughout 
 the entire system for all degrees of freedom .
in contrast , the actual 
 barrier crossing is very rapid in tps , and the environment is largely 
 unchanged during the reaction .
this is consistent with a reaction 
 dynamics model where the system spends most of its time searching 
 for a reactive configuration and set of momenta .
once the proper 6n - dimensional state ( 3n for the configuration 
 and 3n for the momentum , where n is the number of atoms in the system ) suitable for reaction is attained , 
 the reaction occurs rapidly . in this case , if certain protein dynamics 
 are coupled with the chemical step , the effects of such dynamics are 
 naturally taken into account in tps , whereas they are ignored in the 
 tst - based methods .
an additional fundamental difference between 
 the two families of 
 methods concerns the motion along the reaction coordinate . in us and 
 sm , the motion along the reaction coordinate is unphysical because 
 of the applied bias .
the reaction coordinate is assumed to be separable 
 from the other degrees of freedom , and the velocity of the reactive 
 mode is assumed to approach that of a free particle . on the other 
 hand , the reactive trajectories produced from tps correspond in principle 
 to a true barrier - climbing process . indeed , in tps simulations there 
 is a kinetic energy associated with the barrier crossing , similar 
 to what one would expect in the true chemical step .
this motion along 
 the reaction coordinate , combined with the lack of time for environmental 
 reorganization during hydride transfer , generates a friction force 
 that influences the nature of the activated complex in multidimensional 
 systems .
we note that additional differences in the details 
 of the current 
 simulations exist .
the us and sm simulations were performed in the 
 isothermal  isobaric ( npt ) ensemble using a 
 thermostat , while the tps simulations employed a microcanonical ( nve ) ensemble ( albeit conforming with nvt conditions ) .
however , us in the nve ensemble yielded 
 dads identical to those in the npt simulations , so 
 this is unlikely to be of significance here .
we believe that 
 it could be of interest to compare biased and unbiased 
 simulations for other activated processes and to explore the validity 
 of the equilibrium solvation assumption inherent to traditional tst 
 as well as the effects of nonequilibrium dynamics on the reaction . 

additionally , more advanced tools for analysis of the dividing surface 
 could be employed .
we further note that 
 a quantum - dynamical analogue of classical tps simulations should be 
 used to identify the ts more precisely .
finally , as noted above , direct rates or kinetic isotope effects 
 were not computed with tps in the current work . a direct comparison 
 of computed and experimental absolute rates and kinetic isotope effects 
 and their temperature dependence will be necessary to further scrutinize 
 the current tps and us results .
 the rate expression of traditional 
 transition state theory ( tst ) 
 assumes no recrossing of the transition state ( ts ) and thermal quasi - equilibrium 
 between the ground state and the ts . 
 currently , it is not well understood 
 to what extent these assumptions influence the nature of the activated 
 complex obtained in traditional tst - based simulations of processes 
 in the condensed phase in general and in enzymes in particular . here 
 
 we scrutinize these assumptions by characterizing the tss for hydride 
 transfer catalyzed by the enzyme escherichia coli dihydrofolate reductase obtained using various simulation approaches . 
 
 specifically , we compare the tss obtained with common tst - based methods 
 and a dynamics - based method . using a recently developed accurate hybrid 
 quantum mechanics / molecular mechanics potential , we find that the 
 tst - based and dynamics - based methods give considerably different ts 
 ensembles . this discrepancy , which could be due equilibrium solvation 
 effects and the nature of the reaction coordinate employed and its 
 motion , raises major questions about how to interpret the tss determined 
 by common simulation methods . 
 we conclude that further investigation 
 is needed to characterize the impact of various tst assumptions on 
 the ts phase - space ensemble and on the reaction kinetics . 
Introduction
Methods
Results
Discussion