the calculation of 
 free energies is critical in understanding many 
 biological phenomena , for example in the molecular recognition between 
 a small molecule and its receptor . computational approaches for obtaining 
 such free energies are of particular interest in drug design and in 
 understanding fundamental enzymatic mechanisms . absolute binding 
 free energies can be computed at low computational 
 cost , for example by docking scoring functions , but generally lack 
 accuracy and therefore can not be directly compared to experimental 
 values . more accurate and computationally 
 expensive approaches redefine the hamiltonian of a molecular dynamics 
 ( md ) simulation as a function of both the atomic coordinates and a 
 ligand coupling parameter  , and , in this way , compute binding 
 free energies using estimators like free energy perturbation ( fep ) , 
 thermodynamic integration ( ti ) , or the multistate bennett acceptance 
 ratio ( mbar ) . even these computationally costly efforts , however , 
 can yield grossly inaccurate free energies due to inadequate sampling 
 of the phase space of the system . a clever 
 way to avoid inadequate sampling is to introduce external 
 forces that guide sampling during the free energy calculation . this 
 has been done , for example , by coupling ti with umbrella sampling , 
 or in the confine - and - release method , in which binding free energy 
 calculations are performed independently for discrete conformational 
 minima known to interchange slowly . one way 
 to facilitate conformational transitions without knowledge 
 of the relevant conformational minima a priori is to simply add a 
 bias that effectively raises energy minima of the hamiltonian . with the added bias known , the equilibrium ensemble average can 
 be recovered by reweighting . in our group , we use a variation of this 
 approach called accelerated md ( amd ) , which has been used as a means 
 to more rapidly explore conformational space of biomolecules and improve 
 free energy calculations . modifying the underlying hamiltonian in complex systems , 
 however , usually results in a great deal of statistical noise upon 
 reweighting , making it impractical to recover equilibrium ensemble 
 averages . numerous other free energy methods 
 utilize a replica - exchange framework 
 in which multiple replicas , each with different values of  
 or otherwise modified hamiltonians , are simulated in parallel and 
 exchange their configurations periodically , to improve convergence 
 of free energy calculations . in such methods , the number of replicas scales as the square root 
 of the total number of degrees of freedom , again hindering the applicability 
 of such approaches to complex biomolecular systems , although recent 
 efforts have shown that specifically tempering the solute degrees 
 of freedom can ameliorate this problem . similar 
 to a replica - exchange framework , the adaptive integration 
 method ( aim ) enhances mixing in  space by adaptively changing 
  throughout a single simulation . by introducing an additional biasing factor that improves the likelihood 
 of transitions between  values , aim promotes coverage of  
 space and allows conformations explored at one value of  to 
 inform the conformational sampling at other values of . the 
 aim approach , however , does not prevent the system from being trapped 
 by high barriers that may persist regardless of the value of . here we show that by introducing a modified hamiltonian only at 
  values between the  = 0 and  = 1 end points , 
 we are able to accelerate the exploration of phase space as well as 
 of  space , without the need for additional cpu resources or 
 any statistical reweighting . we evaluate this approach , which we call 
 accelerated aim ( acclaim ) , using three alchemical transformations : 
 ( i ) a symmetric self - transformation of bromocyclohexane ( brc ) , ( ii ) 
 an asymmetric transformation of charge removal from brc , and ( iii ) 
 a calculation of the relative binding free energy of two ligands that 
 bind to thrombin . the adaptive integration 
 method ( aim ) can help overcome some of the convergence problems encountered 
 with traditional sampling protocols in free energy calculations . the 
 sampling protocols we discuss here could in principle be combined 
 with any of a number of robust free energy estimators ; in this work , 
 we specifically use thermodynamic integration ( ti ) to obtain free energies . with ti , the free energy 
 is calculated using1where f is the change 
 in free energy , u is the potential energy , and  
 is a parameter which links the initial state , where  = 0 , to 
 the final state , where  = 1 . the sampling protocol commonly 
 used in free energy calculations 
 consists of running multiple md simulations at fixed  values 
 and then numerical integration is applied to eq 1 . two commonly used methods of numerical integration are trapezoidal 
 and cubic spline interpolation , where the latter has been shown to 
 improve results . running multiple 
 simulations at fixed  values can be inefficient 
 because even though some simulations may cover extensive regions of 
 phase space at particular values of  , other simulations may 
 remain trapped in their local minima . aim improves sampling by changing  throughout a single 
 simulation , 
 which allows conformations sampled at one  to seed sampling 
 at other values of . this is similar 
 to -dynamics , where  is 
 propagated as the coordinate of a fictitious particle , changing its 
 value during a single simulation . however , instead of propagating 
  as the coordinate of this particle , aim uses a periodic monte 
 carlo ( mc ) step to change in  during the simulation . for a 
 trial move from i to i+1 , the acceptance probability is given by2where u = u(i+1 ) u(i ) is the difference 
 in the potential energy , and f = fii+1 is a biasing factor which can help the simulation sample  
 values more efficiently by overcoming barriers in u( ) . the efficiency of moving 
 between  values thus depends on the difference e between the potential energy and the biasing factor given 
 by3 we explore two biasing factors , denoted as mc1 and mc2 . mc1 
 is 
 the biasing factor originally used with aim , which uses the free energy as calculated with ti4the brackets indicate averages over samples 
 at a given  value . in practice , discrete values are used for  and u/ values are stored for each  
 value so that numerical integration can be used to evaluate eq 4 . mc2 is a modified biasing factor which approximates 
 the free energy 
 between  and  + 1 using5where n is the number of 
 attempted moves between the adjacent  values . this approximation 
 assumes that the system essentially stays in equilibrium at adjacent 
  values . if this assumption does 
 not hold for a particular system , then at long sampling times the 
 mc2 biasing factor will not sample from a flat distribution , which 
 may reduce the efficiency of moving between  values . the motivation for defining an alternative biasing factor was to 
 assess whether considering the difference in the potential energy 
 could improve the rate at which the system accepts transitions between 
 different  values . importantly , both eqs 4 and 5 follow the relation6which is necessary for the mc step 
 to satisfy 
 balance . note that while the biasing factors given by eqs 4 and 5 change at each step , 
 they do converge given enough sampling time . although the addition 
 of this biasing term does not satisfy detailed balance , sampling will 
 converge to the boltzmann distribution asymptotically . dynamically 
 changing the value of  during the simulation augments the overlap 
 of phase space among all values of . such an approach effectively 
 resolves any inadequacy in sampling , provided that all of the relevant 
 conformational minima are accessible in at least some of the values 
 of  visited . as illustrated in figure 1a , however , the aim method can not resolve issues in sampling in 
 which high barriers separate conformational minima regardless of the 
 value of . to facilitate sampling when such large barriers 
 are present at all  values , the potential energy can be modified 
 according to7where 
  is an adjustable parameter that 
 controls the maximum level of scaling and  is the scaling factor . then u/ is 
 modified accordingly8 importantly , at 
  equal to zero or one , the scaling factor 
  is equal to one , resulting in the original unbiased potential . 
 at intermediate values of  , however , the scaling factor is 
 between  and 1 , decreasing the potential energy and the size 
 of barriers , as illustrated in figure 1b . the idea of modifying the potential energy to improve conformational 
 sampling has been explored previously with hamiltonian replica exchange 
 ( hrex ) and the method of hidden restraints . however , this is the first time that this idea 
 has been combined with aim , alleviating the requirements for multiple 
 replicas ( needed with hrex ) and predefined target states ( needed with 
 the hidden restraints method ) . this specific construction of a -dependent 
 biasing approach results in no bias applied at the end points ,  
 = 0 and  = 1 , such that statistical reweighting is avoided 
 entirely , in contrast to methods like accelerated md . example potential energy 
 surface showing the difference between 
 ( a ) the adaptive integration method ( aim ) and ( b ) the accelerated 
 adaptive integration method ( acclaim ) . when using aim , barriers in 
 the potential energy surface that are present at all  values 
 can hinder conformational sampling . when using acclaim , barriers in 
 the potential energy are lowered at intermediate  values by 
 the scaling factor  , which allows the system to explore more 
 conformations . the pmemd module 
 in amber 12 has been modified to include alchemical transformation 
 calculations . similar 
 parameters were used for the model systems and any model - specific 
 parameters are detailed below . small molecules 
 were parametrized 
 using the generalized amber force field ( gaff ) , and the protein was modeled using the amber 99sb - ildn force 
 field . atomic 
 charges for the small molecules were calculated using resp to fit the electrostatic potentials determined 
 using gaussian03 at the hartree  fock/6 - 31 g * 
 level of theory . a truncated octahedral box was used with a minimum 
 distance of 10  to the solute . particle mesh ewald ( pme ) with a 1  grid was used to calculate long - range 
 electrostatics . an 8  cutoff for short - range nonbonded interactions a 2 fs time step was used , with the shake and settle algorithms used 
 to constrain bonds to hydrogen atoms . equilibrated systems for all 
 the simulations were prepared using a similar protocol . then the systems were heated 
 to 300 k over 500 ps using a langevin thermostat with a 2 ps collision frequency . the density 
 was adjusted using a berendsen barostat , with the pressure set to 1 bar and a 2 ps coupling constant . finally , 
 the system was equilibrated for 500 ps at constant temperature and 
 pressure . the 
 following protocol 
 was used to determine the spacing of  values for the aim and 
 acclaim simulations . first ,  was set to 0 and the equilibrated 
 structures were used as a starting point for both aim and acclaim 
 simulations . a short 110 ps simulation was run starting with 11 equally 
 spaced  values . every 50 steps , the biasing factor f ( either mc1 or mc2 , where relevant ) was recorded , along 
 with the difference e ( eq 3 ) between the potential energy and biasing factor , which is 
 related to the probability of accepting an attempted mc move by9every 5 ps ,  was 
 changed to its adjacent 
 value . this continued until  = 1 at which point  was 
 decreased until  = 0 . the average e value describes the difficulty in moving from one  to the 
 next . if e for any pair of adjacent  
 windows exceeded the cutoff of 2 ( kcal / mol ) , n additional 
 values were added linearly , where n is given by10the number of  values resulting from 
 this method are summarized in tables s4 and s5 in the supporting information . long 55 ns simulations were 
 run with the new set of  windows , starting from the equilibrated 
 structure . for the brc transformations , changes in were attempted 
 every 50 steps , while for the thrombin ligand transformation changes 
 in  were attempted every 500 steps . the use of scaling in acclaim 
 is generalizable to any combination of the components of the potential 
 energy and can be limited to designated regions of the system . some 
 components of the potential energy that are of particular relevance 
 in binding free energy calculations include the dihedrals ( d ) , the 
 14 van der waals and 14 electrostatic interactions 
 ( 4 ) , which are the interactions between atoms connected by three bonds , 
 as well as the short - range nonbonded van der waals and electrostatic 
 interactions ( n ) , which are between atoms not bonded to each other 
 but within the short - range cutoff . scaling the bond or angle terms , 
 for instance , would not lead to improved conformational sampling , 
 but instead would distort the geometry of the molecules . in 
 both cases of brc and thrombin examined here , the d , 4 , and n components 
 of the potential energy were scaled , as these terms are relevant in 
 the problematic chair - flipping and ring - flipping conformational changes 
 in these systems . we did , however , verify that various other combinations 
 of scaled d , 4 , or n terms would also be adequate for a simple model 
 system like brc ( supporting information figures s3s6 ) . in the brc transformations , all solute 
 atoms were scaled , whereas 
 only a select group of relevant atoms in the thrombin ligand transformation 
 were scaled . scaling the entire potential energy surface of a biomolecule 
 like thrombin would reduce the overlap in the potential energy between 
 adjacent  values , decreasing the exchange probability and thus 
 hindering convergence of these simulations . for the brc transformations , 
 the scaling term was set to 
 zero for the d and 4 terms and was set to 0.1 for the n terms . for 
 the thrombin ligand transformation ,  was set to 0.168  16 
 for all terms , which was used in previous work that examined this 
 transformation using a hrex method . although 
 we did test both more and less aggressive acceleration parameters 
 (  = 0.05 or 0.25 , respectively ) for the more complicated case 
 of thrombin ligand binding ( supporting information figures s7 and s8 ) , we found no dependence of the free energies 
 on the choice of scaling parameter in this range . we note that in 
 all cases we avoided scaling the nonbonded terms to zero , to prevent 
 issues that occur when linearly scaling the potential for nonbonded 
 terms . these issues are similar to those that led to the development 
 of softcore potentials . in principle , softcore scaling 
 could be used instead of the linear scaling that we used , but this 
 scheme was simpler for the initial implementation . a suggested protocol 
 for choosing an appropriate value for  is included in the results and discussion section . to specifically 
 address the effect of introducing the scaling function , 
 we compared results from acclaim simulations to those obtained from 
 aim simulations , in which no component of the potential energy is 
 scaled . in all of the production simulations , 
 we used the alternative biasing factor , mc2 , given by eq 5 at the mc steps . we separately evaluated the impact of the 
 biasing factor by running aim simulations of the brc transformations 
 with mc1 and mc2 ( supporting information figures s3s6 ) . we also examined how the biasing factor affects 
 the number of full round - trips ( i.e. ,  = 0   
 = 1   = 0 ) over time in aim and acclaim solvation free 
 energy simulations . the first alchemical 
 transformation we considered was the self - transformation of bromocyclohexane 
 ( brc ) . in this transformation , both  = 0 and = 1 describe 
 brc , but they differ in the position of the br substituent , such that 
 the equatorial or axial conformations , separated by a high barrier , 
 were coupled at  = 0 and  = 1 , respectively ( see figure 2a ) . the free energy of the transformation is 0 ( kcal / mol ) , 
 since this is a transformation to the same molecule , but this result 
 depends on the accurate sampling of both conformations at both end 
 points . ( a ) bromocyclohexane 
 self - transformation , where the carbon atoms are offset in the  
 = 0 and  = 1 states . the transparent hydrogen and bromine atoms 
 interact fully at  = 0 and the solid hydrogen and bromine atoms 
 interact fully at  = 1 such that  = 0 corresponds to 
 the equatorial chair conformation and  = 1 corresponds to the 
 axial chair conformation . two sets 
 of simulations were run , starting from either the equatorial ( top ) 
 or axial ( bottom ) conformations . the 
 p1 pyridine ring , which is highlighted in yellow , can favor either 
 the in conformation ( shown ) or the out conformation depending on its 
 substitution . for this model , the relative binding free energy between 
 cda and cdb was calculated . two sets of simulations were run , starting 
 from either the in or out conformations . a modified softcore potential was used for the perturbed 
 h and 
 br atoms . as implemented in amber , softcore transformations decouple 
 the energy of the perturbed part of the molecule . however , the goal of this calculation is to determine 
 how the free energy relates to conformational sampling of brc . in 
 order to approximate the change in the free energy due to the conformational 
 change in brc from  = 0 to  = 1 , the dihedral and 14 
 terms were scaled linearly and were not decoupled from the rest of 
 the system . this modification was tested by comparing the resulting 
 free energy to the result from an umbrella sampling ( us ) calculation 
 as described below . preparation of the equilibrated structures 
 for us was similar to 
 that described above , except a restraining potential was enabled . the 
 reaction coordinate  for this system is given by11where d1 is the dihedral between c1 , c2 , c3 , 
 and c4 ; d2 is the dihedral between c2 , c3 , c4 , and c5 ; and so on . this type of restraining potential was added to pmemd so that us calculations could be carried out . thirty - three windows 
 were used , equally spaced between 80 and 80. the 
 force constant for this calculation was 131 ( kcal / molrad ) . to assess acclaim in 
 an asymmetric transformation that results in a nonzero change in free 
 energy , we considered the free energy associated with the removal 
 of charges from brc . to probe convergence , we compared the results 
 from independent simulations that were started from either axial or 
 equatorial conformations ( see figure 2b ) , as 
 simulations starting from different initial states should converge 
 to the same free energy value in the limit of infinite sampling . as 
 in the previous transformation , the acclaim simulations only scaled 
 the parts of the potential involving brc . additional us calculations 
 were run using the same protocol as described above , except that all 
 of the charges on brc were removed , to obtain appropriate reference 
 values . the final test transformation 
 considered was the relative binding free energy to thrombin for the 
 ligands 2-(6-chloro-3-[2,2-difluoro-2(2-pyridinyl)ethyl]amino-2-oxo-1(2h)-pyrazinyl)-n-[(2-fluoro-6-pyridinyl)methyl]acetamide 
 and 2-(6-chloro-3-[2,2-difluoro-2(2-pyridinyl)ethyl]amino-2-oxo-1(2h)-pyrazinyl)-n-[(2-fluoro-3-methyl-6-pyridinyl)methyl]acetamide , 
 abbreviated as cda or cdb , respectively . this transformation has been 
 studied previously using replica exchange with solute tempering ( rest ) 
 an hrex method . the only difference between 
 cda and cdb is the addition of a methyl group on the p1 pyridine ring 
 ( see figure 2c ) . this causes cdb to favor the 
 conformation where the p1 pyridine ring is pointing in that is , where 
 the methyl group is pointing in toward the protein . as with the previous 
 study , acclaim was applied to the p1 pyridine ring and the conformational 
 sampling of this ring was examined . as with the brc charge transformation , separate simulations were started with the p1 pyridine ring in the 
 in and out states to assess convergence . also , as with the previous 
 study , the f atom in the p1 pyridine ring was set to be softcore to 
 reduce the effective size of the ring and enhance sampling . initial coordinates came from 
 the crystal structure for thrombin bound to cda ( pdb 1mu6 ) and the structure 
 for cdb came from pdb 1mu8 . the loop comprising 
 of residues 146 to 150 near the ligand binding site was modeled from 
 a more complete structure ( pdb 4mlf ) . the solvation 
 free energy ( sfe ) was calculated by running aim and acclaim simulations 
 for the ligand transformation in solution . the relative binding free 
 energy was calculated using12 we performed production 
 simulations using the us , aim and acclaim simulations four times each . the potential of mean force ( pmf ) was calculated from the us simulations 
 using the weighted histogram analysis method ( wham ) as implemented in wham version 2.0.7 by alan grossfield 
 ( http://membrane.urmc.rochester.edu/content/wham/ ) . the error for each trial 
 was determined using a mc bootstrap analysis , and the error was propagated according to13 free energies for aim and acclaim were 
 calculated using cubic spline interpolation with subsampling as implemented 
 in pymbar ( https://simtk.org/home/pymbar ) . for the calculation of free energy as a function of simulation time , 
 the free energy was calculated over all u/ values collected up to that point 
 using pymbar . the free energy was averaged across the four simulations 
 and the error was propagated according to eq 13 . the probability of finding the model system in each of its 
 major 
 conformations at  = 0 or 1 was also calculated as a function 
 of simulation time . for brc , the cutoff values 
 for each conformation were determined from the location of the barriers 
 in the pmf ( see supporting information figure 
 s2 ) . for thrombin , the dihedral angle of the ligand atoms n6 , c15 , 
 c16 , and c17 was used to classify the in and out states , where states 
 above 0 were classified as in and states below 0 were 
 classified as out . the conformational probabilities were averaged 
 over the four trials , and the standard deviation was used as an estimate 
 of the error . the brc self - transformation 
 is simple , as the conformational landscape of the system is easily 
 characterizable in one dimension by the  coordinate , yet it 
 is an illustrative example of the problems encountered in free energy 
 calculations when the conformational space contains kinetically trapped 
 states . the reference probabilities for the equatorial and axial conformations 
 were obtained from the potential of mean force calculations , using 
 the  coordinate , as shown in supporting 
 information figure s2a . the population of the equatorial conformation 
 was calculated as a function of simulation time at  = 0 and 
  = 1 . the underlying potential is symmetric with respect to 
  , so the populations at  = 0 and  = 1 should 
 converge to the same value . the population probabilities are 
 summarized in figure 3a . for the acclaim simulations , 
 the populations at both end points converged , within error , to the 
 reference population . the populations from the aim simulations , on 
 the other hand , in which no scaling was applied , did not overlap between 
 the two end points nor with the reference value . because a high 10 
 ( kcal / mol ) barrier is associated with chair flipping , the molecule 
 was essentially trapped in the same chair conformation , regardless 
 of the value of . since the br was fully coupled in an axial 
 orientation at  = 1 and fully coupled in an equatorial orientation 
 at  = 0 , the resulting populations were inaccurate when scaling 
 was not applied ( aim ) . as expected , the disparities in the configurational 
 sampling at 
 the end points observed with conventional aim translated into inaccurate , 
 i.e. nonzero , free energies for the brc self - transformation , whereas 
 the free energies obtained with acclaim converged close to zero ( figure 3b ) . for the aim simulations , the resulting free 
 energy was 0.8 to 0.9 ( kcal / mol ) , which is consistent 
 with the difference in the free energy between the equatorial and 
 axial conformations , according to the calculated pmf ( supporting information figure s2a ) . this agreement 
 between us and aim calculations not only highlights the inadequate 
 sampling but also shows that the alternative softcore potential implemented 
 for these simulations is valid for the determination of the free energy 
 difference between different conformations . the brc self - transformation 
 demonstrates clearly how the inclusion of the scaled potential at 
 intermediate  values in the aim framework can improve sampling 
 of kinetically trapped conformations and correspondingly result in 
 accurate free energies in cases where the aim method alone is insufficient . 
 a nearly identical brc self - transformation , though without any partial 
 atomic charges , has been used in previous work , in which hamiltonian replica exchange was used to enhance 
 sampling . the charges included in the present work both improve the 
 representation of brc in water and are expected to make the test transformation 
 more challenging . nevertheless , we found that the acclaim method achieved 
 comparable accuracy in the calculated free energy as the replica exchange 
 approach . the latter approach requires multiple replicas at each  
 value whereas the aim framework permits the calculation of free energy 
 from a single simulation that adaptively samples different hamiltonians 
 and  values . ( a ) population of equatorial conformation 
 sampled as a function of simulation time . populations 
 are shown at both  = 0 ( purple series ) and  = 1 ( red 
 series ) . due to the symmetry in this self - transformation ( b ) free energy 
 as a function of simulation time . the free energy for a self - transformation 
 should converge to 0 ( kcal / mol ) , as indicated by the horizontal green 
 line . error bars represent the uncertainty , which was propagated from 
 each trial using eq 13 . in addition to the self - transformation 
 of brc , we considered the asymmetric transformation , with an associated 
 nonzero change in free energy , of removing the charges from brc . because 
 equilibrium properties are independent of the initial state of the 
 system , the conformational populations and the final free energy calculated 
 from simulations starting from different initial conformations should 
 converge to the same value , and we consider this overlap as a metric 
 for convergence . the population of the equatorial conformation , calculated 
 at  = 0 , is shown in figure 4a ( corresponding 
 data from  = 1 is provided in supporting 
 information figure s5 ) . at both end points , as observed in 
 the symmetric self - transformation , the equilibrium population was 
 correctly sampled with acclaim , but the brc molecule was confined 
 to its initial conformation with the conventional aim approach . for the acclaim simulations , the free energy 
 values calculated using simulations starting from the two different 
 conformations converged , within the estimated error , whereas the results 
 from the aim simulations depended on their initial conformations , 
 demonstrating the impact of the kinetically trapped states . these 
 results show that acclaim improves conformational sampling , which , 
 in turn , can improve convergence of the free energy for alchemical 
 transformations . values 
 are shown 
 for simulations started from the equatorial ( purple series ) or axial 
 ( red series ) conformations , which should converge to the same value . ( a ) population of equatorial conformation at  = 0 as a function 
 of time , from simulations with either initial conformation . ( b ) free 
 energy as a function of time , from simulations with either initial 
 conformation . the final alchemical 
 transformation , determining the relative binding free energy of the 
 ligands cda and cdb bound to thrombin , involves a more complex system 
 with substantially more degrees of freedom than brc . on the basis 
 of previous work , it was assumed that 
 sampling the two conformations of the p1 pyridine ring was critical 
 in obtaining an accurate relative binding free energy . the population 
 of the p1 pyridine ring in the in conformation was used to determine 
 convergence of conformational sampling . the results for  = 
 0 are shown in figure 5a and for  = 
 1 in supporting information figure s7 . these results show that the aim simulations do not sample both conformations 
 of the p1 pyridine ring , because they stay in their initial conformations . 
 with acclaim , we were able to obtain convergence in the sampled populations 
 between simulations , regardless of their different initial conformations . the corresponding free energy as a function of time is shown in 
 figure 5b . with aim , there is a difference 
 of 2 ( kcal / mol ) in the relative binding free energy , depending on 
 the initial conformation . with acclaim , interestingly , the free energies obtained 
 here are in agreement with the experimental value reported for this 
 transformation and are quite close to the free energy obtained using 
 replica exchange with solute tempering ( rest ) , despite the difference 
 in force fields used . if the relevant 
 conformations are unknown a priori , we find that the computational 
 efficiency of this type of free energy calculation is greater with 
 acclaim than with rest ( supporting information figure s7 ,  = 0.168  16 , compared to figure 4b of ref ( 24 ) ) ; however , the use of multiple initial configurations in 
 a rest calculation would diminish such a difference . we found 
 that the convergence of the free energy was independent 
 of the parameter  , within a tested range of 0.050.25 
 ( supporting information figure s8 ) . interestingly , 
 we observed this marked improvement in the relative binding free energy 
 in spite of a fairly large variance in the fraction of in population 
 sampled . these results show that for the relative binding transformation 
 of cda and cdb to thrombin , even a somewhat modest enhancement in 
 the exploration of configurational space can translate to dramatic 
 improvements in the accuracy of the free energy , and we anticipate 
 this to be true for other biomolecular systems as well . values are shown for simulations started with the ligand p1 pyridine 
 ring starting in the in conformation ( purple series ) or in the out 
 conformation ( red series ) . ( a ) population of the p1 pyridine ring 
 in the in conformation at  = 0 as a function of time . ( b ) relative 
 binding free energy as a function of simulation time , calculated using 
 eq 12 . when applying acclaim to other systems , we would recommend 
 starting 
 with a lower scaling factor such as  = 0.05 and scaling the 
 d , 4 , and n terms . then , the protocol described in distribution of  values can be used to determine the 
 number of  values . if the number of windows is too large then 
 the protocol should be repeated after increasing the scaling factor 
 by 0.05 . if additional information about the barriers in the system 
 is known , selective scaling of the d , 4 , or n terms could also be 
 used to reduce the number of windows , as was done with the brc transformations 
 ( supporting information figures s3s6 ) . based on these results , we would recommend a maximum number of  
 values to be between 13 and 15 . a larger number of  values 
 would indicate that the perturbation is too large , which will hinder 
 convergence of the free energy . we introduced an alternative 
 biasing factor mc2 ( eq 5 ) for the mc step in 
 acclaim . the motivation for modifying the biasing factor was that 
 the difference in free energy between adjacent values of  is 
 expected to be more pronounced when the hamiltonians are differentially 
 scaled according to . initial estimates of the free energy 
 difference using ti , as with mc1 ( eq 4 ) , can 
 be inaccurate , which causes a large difference between the potential 
 energy and the biasing factor u  f . this makes moves between adjacent values of  more 
 difficult , decreasing the total number of  round - trips . by 
 estimating the free energy difference directly from the potential 
 energy , as with mc2 , the discrepancy between the potential energy 
 and the biasing factor is reduced , which improves the probability 
 of accepting mc moves for short sampling times . at longer sampling 
 times , the efficiency of switching between  values will depend 
 on how well the system stays in equilibrium , following from the approximation 
 inherent to the alternative biasing factor . to evaluate the 
 impact of the biasing factor on the number of round - trips in  
 space , we performed aim and acclaim simulations of the relative solvation 
 free energy of cda to cdb with the original and alternative biasing 
 factors . as is shown in figure 6a , the number 
 of round - trips in  space was not affected by the choice of 
 biasing factor in conventional aim simulations . however , as shown 
 in figure 6b , with the acclaim simulations 
 we found a significant increase in the number of round - trips in  
 space due as a result of the alternative biasing factor . for these 
 simulations , the  values were determined for mc2 , and the same 
 values were used for mc1 so that the only difference was the biasing 
 factor used in the mc step . to verify that the gain in number of round - trips 
 was not simply due to the distribution of  values , additional 
 simulations were run , in which mc1 was used in the protocol for determining 
 the number of  values . with these simulations , the total number 
 of  values increased , and the total number of  round - trips 
 decreased , suggesting that using the  values determined with 
 mc2 was more appropriate . it appears that for this transformation , 
 the approximation underlying the alternative biasing factor was reasonable , 
 because we did not see a decrease in the number of  round - trips 
 at longer times . thus , for cases in which this approximation holds , 
 the alternative biasing factor introduced here improves the effectiveness 
 of scaling the potential at intermediate values of  by improving 
 the resulting number of  round - trips . for cases in which this 
 approximation does not hold , moves between adjacent values of  
 could become more difficult , and thus for these systems we would recommend 
 using the original biasing factor . using 
 both a simple model system , bromocyclohexane , as well as 
 more complex biomolecule , thrombin , we have demonstrated that the 
 introduction of a scaled potential at intermediate  values 
 to the aim method results in improved sampling of relevant conformational 
 states at the end points of an alchemical transformation . importantly , 
 we have shown that for the simple case of brc , the acclaim method 
 is effective for both symmetric and asymmetric alchemical transformations . the challenge in using acclaim to study binding to complex biomolecules , 
 however , remains being able to identify the slow ( and most relevant ) 
 degrees of freedom that should be accelerated in the system . number of  
 round - trips as a function of time for the solvation 
 free energy of the ligands that bind to thrombin . shown here are the 
 results for ( a ) aim and ( b ) acclaim simulations run using the original 
 ( mc1 ) or new biasing factors ( mc2 ) . we have also considered an alternative biasing method in 
 aim and 
 acclaim that improves the likelihood of transitions in  space 
 when intermediate  values are being scaled ( e.g. , in acclaim ) , 
 and is still valid in the absence of scaling ( e.g. , in aim ) . in practice , the mixing in  space , e.g. the frequency of round - trips made 
 in an acclaim simulation , can be conveniently used to gauge the level 
 of acceleration applied to the chosen atoms in the system . furthermore , 
 the ability to specifically accelerate a ligand or residue with slowly 
 exchanging conformational states , without having to identify an optimal 
 physical reaction coordinate , is a clear strength of the acclaim approach . 
 because the potential energy at the end points is unbiased , the resulting 
 free energies require no statistical reweighting , although the issue 
 of poor exchanges between  values still makes overacceleration 
 a relevant issue . future work may consider alternative methods 
 of determining  
 values as well as the degrees of freedom that would benefit from enhanced 
 sampling . future consideration of alternative forms for the scaling 
 factor may also optimize transitions in the presence of scaled potentials .
conformational changes that occur 
 upon ligand binding may be too 
 slow to observe on the time scales routinely accessible using molecular 
 dynamics simulations . 
 the adaptive integration method ( aim ) leverages 
 the notion that when a ligand is either fully coupled or decoupled , 
 according to  , barrier heights may change , making some conformational 
 transitions more accessible at certain  values . 
 aim adaptively 
 changes the value of  in a single simulation so that conformations 
 sampled at one value of  seed the conformational space sampled 
 at another  value . 
 adapting the value of  throughout 
 a simulation , however , does not resolve issues in sampling when barriers 
 remain high regardless of the  value . in this work , 
 we introduce 
 a new method , called accelerated aim ( acclaim ) , in which the potential 
 energy function is flattened at intermediate values of  , promoting 
 the exploration of conformational space as the ligand is decoupled 
 from its receptor . 
 we show , with both a simple model system ( bromocyclohexane ) 
 and the more complex biomolecule thrombin , that acclaim is a promising 
 approach to overcome high barriers in the calculation of free energies , 
 without the need for any statistical reweighting or additional processors .