biomolecular dynamics 
 involves a wide range of time scales ranging 
 from bond fluctuations to slow large - scale motions .
molecular dynamics 
 ( md ) with accurate force fields can in principle be used as a virtual 
 microscope to investigate these motions at atomistic resolution . however , its applicability to problems such as 
 folding or conformational transitions in proteins and rna is limited 
 by the fact that only short time scales ( s ) are directly 
 accessible by straightforward simulation . in spite of the development 
 of ad hoc hardware , many 
 relevant processes are still out of reach for accurate atomistic modeling . 

several different techniques have been developed in the last decades 
 to address this issue .
these techniques can be roughly classified 
 in two groups . in the first group ,
inspired 
 by annealing , ergodicity is achieved by 
 increasing the temperature or by artificially modifying 
 the hamiltonian . in methods of this group
swaps of coordinates between neighboring 
 replicas are periodically attempted and accepted or rejected with 
 a metropolis criterion .
these methods are usually referred to as replica 
 exchange molecular dynamics ( remd ) . whereas temperature is certainly 
 the most adopted control parameter , temperature remd ( t - remd ) is computationally 
 demanding for solvated systems , because replica spacing and interchange 
 probability depends on the system size . moreover , t - remd is ineffective on entropic barriers .
scaling of portions of the hamiltonian ( h - remd ) is a common alternative 
 and could have a better convergence behavior for large systems .
t - remd 
 and h - remd can also be combined , by integrating both schemes on each 
 replica or in a multidimensional framework .
the second group of enhanced sampling techniques 
 includes methods based on importance sampling , where suitable collective 
 variables ( cvs ) are a priori selected and biased . 

this class has its root in umbrella sampling method and includes local elevation , conformational flooding , adaptive biasing 
 force ( abf ) , and metadynamics .
these methods are very efficient but require large a priori information . with the notable exception of bias - exchange metadynamics , approaches based on importance sampling have 
 been traditionally applied to a small number of cvs at a time , due 
 to the difficulties in building a history dependent potential in a 
 high - dimensional cv space . for many systems
it is difficult to find 
 a small number of effective cvs that describe the slow degrees of 
 freedom , and one often has to resort to expensive methods of the first 
 class .
for instance , conformational transitions in unstructured oligonucleotides 
 have been studied with different remd schemes . in these studies ,
the generation of a converged 
 conformational ensemble was proven a cumbersome task , even when a 
 high number of replicas and tens of s of simulated time were 
 employed .
methods bridging between these two classes can be 
 designed by biasing 
 a large number of local cvs ( e.g. , dihedral angles ) , so as to avoid 
 the complication of designing ad hoc cvs .
for example , 
 straatsmaa and mccammon introduced a 
 technique where bias potentials acting on dihedrals was used in a 
 simulated annealing protocol . in that work
a bias was fitted to the 
 potential of mean force of backbone dihedrals and then used to quickly 
 optimize the structure of a polypeptide .
zacharias and collaborators used a similar technique to build bias potentials for dihedrals 
 to be employed in h - remd and successfully applied it to study conformational 
 changes in proteins .
however , these potentials were fitted on a model 
 system and can not account for the specific identity of each residue 
 and for the cross - talk between correlated dihedrals . for nucleic acids , 
 where the complex backbone does not allow a straightforward application 
 of this technique , penalty potentials centered on the stable rotamers 

we propose to use concurrent well - tempered metadynamics ( wt - metad ) to build bias potentials acting on 
 a large number of local cvs .
we then show how to integrate this approach 
 in a h - remd scheme , exploiting the replica ladder to obtain unbiased 
 conformations . in wt - metad the compensation of the underlying free - energy 
 landscape
is modulated by the so - called bias factor . we here 
 change this parameter across the replica ladder , adjusting the ergodicity 
 of each replica .
the final bias could be also used as a static potential 
 so as to completely eliminate any nonequilibrium effect .
since the 
 effect of the bias is that of keeping the chosen cvs at an effectively 
 higher temperature , we refer to the introduced method as replica exchange 
 with collective - variable tempering ( rect ) .
the method is first tested 
 on alanine dipeptide in water and then applied to the conformational 
 sampling of a rna tetranucleotide where it outperforms dihedral - scaling 
 remd and plain md .
the chosen tetranucleotide is a very challenging 
 system that has been recently studied with different variants of remd .
in this 
 section we show how to use wt - metad as an effective method 
 to build bias potentials that allow barriers to be easily crossed . 

one of the input parameters of well - tempered metadynamics is a boosting 
 temperature t = ( 1)t , where  is the bias factor and t is the temperature of the system . in the rest of the paper we will 
 equivalently
this parameter can be used 
 to smoothly interpolate between unbiased sampling (  = 1 , t = 0 ) and flat histogram (    , t   ) .
one can thus 
 introduce a set of replicas using different values of t , ranging from 0 to a value large enough to allow 
 all the relevant barriers to be crossed .
metadynamics relies on the 
 accumulation of a history dependent potential and can not be applied 
 straightforwardly to a large number of cvs . in the next subsection 

we show that this issue can be circumvented by performing many , low - dimensional , 
 concurrent metadynamics simulations .
we then show how to combine many 
 simulations of this kind in a multiple - replica scheme . in well - tempered 
 metadynamics
a history dependent potential v(s , t ) acting on the collective variable s is introduced and evolved according to the following equation 
 of motionhere kb is the 
 boltzmann constant , t is the temperature , b is the characteristic time for the bias 
 evolution , t is a boosting 
 temperature , and k is a kernel function which is 
 usually defined as a gaussian . for simplicity
the variance of the gaussian provides the binning 
 in cv space and is usually chosen based on cv fluctuations or adjusted 
 on the fly . by assuming that the bias 
 is growing uniformly with time one can show rigorously that in the long time limit the bias potential tends to1so 
 that the following probability distribution 
 is sampledthe role of t is that of setting the effective 
 temperature for the cv . 

the explored conformations are thus taken from an ensemble where that 
 cv only is kept at an artificially high temperature , similar to other 
 methods , but has the nice feature that it is obtained with 
 a bias that is quasi - static in the long lime limit .
the bias is usually 
 grown by adding a gaussian every ng steps . 
 as a consequence , to obtain an initial growing rate equal to ( kbt)/(b ) , the initial gaussian height should be chosen equal to ( ( kbt)/(b))ngt where t is the md time step .
we here propose to introduce 
 a separate history - dependent potential on each cvwhere  = 1, ... ,ncv is the index of the cv , and ncv is the number of cvs .
the growth of each of these bias 
 potentials 
 will depend only on the marginal probability for each cvin the long time limit , 
 this potential will 
 tend to flatten the marginal probabilities for every single cv .
since 
 cvs are in general nonorthogonal between each other , one should consider 
 the fact that whenever a bias is added on a cv also the distribution 
 of the other cvs is affected . in the following
we will discuss this 
 issue considering two cvs only , but the argument is straightforwardly 
 generalized to a larger number of cvs .
if two cvs are independent , 
 the joint probability is just the product of the two marginal probabilities , 
 i.e. p(s,s ) = p(s)p(s ) . adding a bias potential on 
 a cv will not affect the distribution of the other . as a consequence , 
 in the long time limit the two bias potentials will converge independently 
 to the predicted fraction of the free energy as in eq 1
. the final bias potential will be completely equivalent to 
 that obtained from a two - dimensional well - tempered metadynamics but 
 will only need the accumulation of two one - dimensional histograms , 
 thus requiring a fraction of the time to converge .
a simple example 
 on a model potential is shown in figure s1 . two identical variables .
we also consider 
 the 
 case of two identical cvs , s = s. this can be obtained for instance by 
 biasing twice the same torsional angle . here
the potentials v and v will grow identically , and the total bias potential acting on s will be vtot = 2v. the total potential will 
 grow asthus , the net effect will be exactly 
 equivalent 
 to that of choosing a doubled t parameter .
in other words , the t parameter acts in an additive way on the selected cvs . 

a similar effect can be expected if two cvs are linearly correlated . in realistic applications one can expect the behavior to be somewhere 
 in the middle between these two limiting cases .
the most important 
 consideration here is that the bias potentials will tend to flatten 
 all the marginal probabilities , but there will be no guarantee that 
 the joint probability is flattened .
results for a simple functional 
 form can be seen in figure s2 . in the same 
 figure it is possible to appreciate the importance of using a self - consistent 
 procedure when
( 39 ) two metadynamics were applied on top of each 
 other , namely on the potential energy and on selected cvs , in a non - self - consistent 
 way .
this was possible because the correlation between the potential 
 energy and the selected cvs is small .
the need for a self - consistent 
 solution was also pointed out in a recent paper where a generalization of the abf method was introduced . in that work independent
one - dimensional 
 adaptive forces were applied at the same time to different cvs so 
 as to enhance the sampling of a high multidimensional space . in short
, the novelty of the introduced procedure is that many 
 low - dimensional metadynamics potentials are grown instead of a single 
 multidimensional one .
this allows the bias to converge very quickly 
 to a flattening potential , with the degree of flatness controlled 
 by the parameter t .
the flattening 
 is expected to enhance conformational transitions which are otherwise 
 hindered by free - energy barriers on the biased cvs . when variables 
 are correlated the exact relationship between bias and free energy 
 (
the procedure introduced 
 above produces conformations in an ensemble which is in general difficult 
 to predict .
however , since the bias potential is known , one can in 
 principle reweight results so as to extract conformations in the canonical 
 ensemble . in the case of static bias potentials acting on the cvs , 

this can be done by weighting each frame as e. this can provide in principle correct results 
 even if the joint probability is not flattened . it must be noticed 
 that such a reweighting can provide statistically meaningful results 
 only for small fluctuations of the total biasing potentials , on the 
 order of kbt . however , in a typical setup one would be interested 
 in biasing all the torsional angles of a molecule .
even if each of 
 them contributes with a few kbt , the total fluctuation of the bias would grow with the 
 system size . for similar reasons , also the abf - based scheme introduced 
 in ref ( 40 ) is limited 
 to a relatively low number of cvs .
a more robust and scalable 
 procedure can be designed by introducing a ladder of replicas with 
 increasing values of t , ranging 
 from 0 to a value large enough to enhance the relevant conformational 
 transitions . the first replica (  = 1 , t = 0 ) can be used to accumulate unbiased statistics . 

replicas other than the first one feel multiple biasing potentials 
 on all the cvs . from time to time
an exchange of coordinates between 
 neighboring replicas is proposed and accepted with probability chosen 
 so as to enforce detailed balance with respect to the current biasing 
 potential : here the suffix i = 1, ...
the exchanges allow 
 the bias potential of every single replica to grow as close as possible 
 to equilibrium taking advantage of the enhanced ergodicity of the 
 more biased replicas .
we notice that to reach a quasi - static distribution 
 it is necessary that all the bias potentials converge for all the 
 replicas . since the time scale for convergence is related to the parameter 
 b , it is convenient 
 to use the same b for all the replicas or , equivalently , 
 to choose the initial deposition rate as proportional to t .
the number of replicas required to span a given 
 range in the t parameter 
 is proportional to ( ncv ) , thus allowing for a very large number of cvs .
we underline that , 
 if a very large number of cvs has to be calculated at every step on 
 every replica , the overall performances could be degraded .
this issue 
 could be tackled using e.g. multiple time - step schemes . in the examples presented here this performance issue was not observed .
we notice that in principle one could use the bias potentials built 
 with this protocol to perform a replica - exchange umbrella sampling 
 simulation . in this manner
the final production run would be performed 
 with an equilibrium replica - exchange simulation .
however , we observe 
 that well - tempered metadynamics is designed so that the speed at which 
 the bias grows decreases with time and the potential becomes quasi - static . 
 in the practical cases we investigated ,
alanine dipeptide ( dala ) was modeled 
 with the amber99sb - ildn force field and solvated 
 in a truncated octahedron box containing 599 tip3p water molecules .
the lincs algorithm 
 was used to constrain all bonds , and equations of motion were integrated 
 with a time step of 2 fs .
for each replica the system temperature 
 was kept at 300 k by the stochastic velocity rescaling thermostat . for all nonbonded interactions the direct space 
 cutoff
was set to 0.8 nm , and the electrostatic long - range interactions 
 were treated using the default particle - mesh ewald settings .
we underline that the possibility of running 
 concurrent metadynamics within the same replica is a novelty introduced 
 in plumed 2.0 .
the backbones dihedral angles (  and  ) and the gyration 
 radius ( rg ) were selected 
 as cvs .
we recall that a geometric replica distribution is optimal 
 for constant specific - heat systems . in rect
, this would be true if 
 the exploration of each of the biased cvs were limited to a quasi - parabolic 
 minimum in the free - energy landscape . whereas this is clearly not 
 true in real cases ( e.g. , double - well landscapes ) we found that a 
 geometric schedule was leading to a reasonable acceptance in the cases 
 investigated here .
the possibility of optimizing the replica ladder 
 is left as a subject for further investigation . for the dihedral angles 
 the gaussian width
the initial gaussian height was adjusted to the t of each replica , according to 
 the relation h = ( kbt)/(b)ngt , in order to maintain 
 the same b = 12 ps across the entire replica ladder . 

the cvs were monitored every 100 steps , and exchanges were attempted 
 with the same frequency .
the simulation was run for 20 ns per replica . a h - remd simulation where the force - field dihedral terms were scaled 
 ( hdih - remd )
was also performed , as implemented 
 in an in - house version of the gromacs code .
the same initial structures , number of replicas , and simulated time 
 as in rect were used .
the scaling factor  for each replica 
 was selected using the relation  = 1/ to allow for a 
 fair comparison of rect and h - remd .
finally a conventional md simulation 
 in the nvt ensemble was run for 120 ns using the same settings .
the initial coordinates were taken 
 from a ribosome crystal structure ( pdb : 3g6e ) , residue 2623 to 2626 .
the system was solvated in a box containing 2502 tip3p water molecules , and the system charge was neutralized 
 by adding 3 na counterions , consistently with previous 
 simulations .
the  ladder was 
 chosen in the range from 1 to 4 following again a geometric distribution . 

the initial structures for the h - remd were taken from a 500 ps md 
 at 600 k , to avoid correlations of the bias during the initial deposition 
 stage of the wt - metad .
as depicted in figure 1 , for each residue the dihedrals of the nucleic 
 acid backbone (  ,  ,  ,  ,  ) , together 
 with the pseudodihedrals angles of the ribose ring ( 1 and 2 ) and the glycosidic torsion angle (  ) 
 were chosen as cvs . to help the free rotation of the nucleotide heterocyclic 
 base around the glycosidic bond , the minimum distance between the 
 center of mass of each base with the other three bases was also biased . 

for the wt - metad we used the same parameters as in the previous system . 

gaussian width for the minimum distance between bases was chosen equal 
 to 0.05 nm .
for this system a hdih - remd , a t - remd , and a plain md simulation were performed in addition 
 to the rect . in the case of hdih - remd 

we used 24 replicas with scaling factors  ranging from 1 to 
 0.25 , so as to cover the same range of the  values chosen for 
 the rect . in the t - remd 24 replicas
were used to cover a temperature 
 range between 300 and 400 k with a geometric distribution . for both 
 methods ,
t - remd and hdih - remd , the simulation 
 length was 200 ns per replica .
all 
 the simulations ( rect , hdih - remd , t - remd , 
 and conventional md ) correspond to the same total simulated time .
schematic 
 representation of the collective variables used for the 
 tetranucleotide simulation . for each nucleotide
, the labeled dihedral 
 angles and the minimum distance between each nucleobase center of 
 mass and the other three nucleobases were biased . as the bias compensates 
 the underlying 
 free energy
the main cvs used in our method are dihedrals angles . to 
 quantify the effect of the hamiltonian modifications on the angle 
 distributions one - dimensional entropies ( s1d )
the calculation procedure was equivalent 
 to the one used in ref ( 56 ) to evaluate the configurational entropy associated with soft degrees 
 of freedom in proteins .
for all the distributions the bandwidth for the kernel 
 density estimation was set to 0.017 rad .
we underline that using this 
 definition we only evaluate the flatness of the individual one - dimensional 
 distributions , and cross - correlation between cvs is ignored .
rna conformations were classified 
 according to the combination of the  angle rotameric state 
 of each nucleotide .
torsion orientations in the range of 0.26 
 to 2.01 rad were considered as syn , while the remaining 
 ones were classified as anti .
the limiting values 
 were chosen according to the position of the barriers in the  
 free - energy profiles of all the residues .
the result of this clustering 
 procedure gave 2= 16 different states that are kinetically 
 well separated by the high torsional barriers .
we observe that the 
 population of these states does not depend only on the torsional potential 
 associated with the  dihedrals but include contributions from 
 base 
in this section we first show the validation 
 of our methodology 
 on a standard model system , dala in water . then we present results 
 for the more challenging case of the conformational sampling of a 
 tetranucleotide .
for all the applications we benchmark against plain 
 md and a h - remd where the dihedral potentials are scaled .
the goal of the introduced method 
 is to enhance conformational sampling in the unbiased replica .
the 
 possibility to explore different metastable conformations in this 
 replica relies on the fact that probability distributions in the biased 
 replicas are flattened and that conformations can travel across the 
 replica ladder .
these conditions can be verified by monitoring the 
 exchange rate and the flatness of the distributions .
the acceptance 
 rate is in the range 6572% for rect and in the range 4353% 
 for hdih - remd , indicating that the former 
 method requires less replicas .
this is likely due to the fact that 
 the total number of scaled dihedrals in hdih - remd is larger than the number of biased cvs in rect . for 
 both remd methods we also verified that all the trajectories in the 
 generalized ensemble sampled the same conformational ensemble ( see figure s3 ) .
a quantitative measure of the 
 flatness of the distribution in the 
 biased replicas can be obtained from the dihedral entropy , shown in 
 figure 2 as a function of scaling factors (  
 and  for rect and hdih - remd , respectively ) . 

entropy grows faster as a function of the scaling factor when using 
 rect , indicating that free - energy barriers on the dala isomerization 
 transition are more effectively compensated by the bias potentials . 
 with hdih - remd entropy of  angle 
 saturates and
apparently the distribution can not be further flattened 
 by decreasing . in the case of the  angle , the dihedral 
 entropy does not grow monotonically when  is decreased .
this 
 behavior indicates that the relevant free - energy barriers are not 
 only originating from the dihedral force - field terms .
the conformational 
 transitions involve indeed also changes in water coordination , reorganization 
 of hydrogen bonds , nonbonded interactions , etc . on the contrary
, rect 
 achieves an almost flat distribution for both dihedral angles at the 
 highest value of the  factor .
the conformations sampled on each replica are shown projected on 
 the , free - energy landscape in figure 
 s5 , where it can be appreciated that all the relevant basins 
 (  ,  , and r ) are explored 
 and connected by points close to the minimum - action pathways ( see 
 refs ( 40 and 57 ) ) . to assess the efficiency and the 
 accuracy of the introduced enhanced 
 sampling technique
the free - energy difference f between the states [,0 ] and [0 , 
 ( )/(2 ) ] was calculated from the distribution of the unbiased 
 replica .
results are shown as a function of time in figure 3 , for the two remd schemes and for the reference 
 conventional md .
both h - remd methods converge to the right value with 
 a similar behavior , whereas plain md needs several tens of ns for 
 the first transition to be observed .
the similarity in the convergence 
 of rect and hdih - remd indicates that for 
 this system the moderate flattening of the distribution induced by 
 hdih - remd is sufficient to achieve ergodicity 
 on this time scale . in order to better evaluate differences between 
 the performance of rect and hdih - remd 
 we applied these methodologies to a more complex system .
entropies are shown as a function of 1/ 
 and  for hdih - remd and rect , respectively . 

the maximum entropy value corresponding to a flat distribution is 
 represented with a straight line .
also in this case we monitor the average 
 exchange ratio ( 7683% for hdih - remd , 2532% for t - remd , and 6080% for rect ) . in figure
s7 the variation of the exchange ratios 
 in time is shown for the exchanges between the first 2 and the last 
 2 replicas of each method .
as it can be appreciated in figure s6 , for hdih - remd and rect the empirical distribution of rmsd is very similar 
 for all the trajectories in the generalized ensemble , indicating that , 
 for each method , all of them sampled the same conformational space . 
 on the contrary , in the case of t - remd , agreement among the distributions 
 of rmsd
the same behavior was obtained in ref ( 26 ) where several t - remd simulations were performed 
 on the same system , with the same number of replicas and a similar 
 temperature range . in that work divergence among the obtained generalized 
 ensembles was observed even for a simulated time as long as 2 s 
 per replica . for hdih - remd and rect round - trip 
 times are shown on figure s8 .
the average 
 round - trip time is 0.5 ns for hdih - remd , 1.8 ns for t - remd , and 1.2 ns for rect .
estimate of 
 the free - energy difference between the two metastable 
 minima in alanine dipeptide .
data are shown for both replica - exchange 
 methods ( hdih - remd and rect ) and for conventional 
 md as a function of the total simulated time . in figure 4
we show the sum of the 
 entropies 
 for the 32 dihedrals used as cvs . in this respect , rect is clearly 
 more effective than hdih - remd in flattening 
 the dihedral distributions , consistently with what was observed for 
 dala .
notably , the entropic increment observed in rect is close to 
 the one observed in t - remd when using an equivalent temperature .
this 
 confirms that rect has an effect comparable to that of raising the 
 temperature of the biased cvs by a factor . the significance 
 of this entropic values could be appreciated on 
 the time series and related histograms for all the dihedral angles 
 shown in figures s9s12 for the 
 most and least ergodic replica of hdih - remd and rect .
it is clear that for rect , at the most ergodic replica , 
 all the accessible torsional range is sampled . on the contrary , in 
 the highest replica of hdih - remd
total entropy of backbone , puckering , 
 and glycosidic dihedral angles 
 in the tetranucleotide for both replica - exchange methods .
entropies 
 are shown as a function of 1/ and  , for hdih - remd and rect , respectively . for t - remd
, temperature 
 is chosen as t =  300 k. as the entropy increases 
 the dihedral distributions become more flat .
the maximum entropy value 
 corresponding to a flat distribution is represented with a straight 
 line .
entropies obtained for the unbiased replicas in the three methods 
 are consistent within their error bars ( error not shown ) .
the transition around the glycosidic bond , from anti to syn , is among the slowest relaxation 
 times in 
 rna dynamics . to evaluate the convergence 
 of the unbiased replica we analyzed the population of the anti rotamer for each nucleotide  angle .
populations 
 are shown in figure 5 as a function of the 
 total simulated time .
the guanosine is the nucleotide with 
 the highest syn proportion , and the cytidines are 
 the ones with the smallest ( < 2% ) , as correspond to their rotameric 
 preferences .
values from both h - remd 
 approaches seem well consistent , except for the population of the 
 first nucleotide . from the time behavior of these populations ,
it 
 is clear that for all the remd approaches the guanosine proportion 
 of anti is the most difficult to converge . here
rect 
 can reach values close to a longer reservoir - remd simulation , while both hdih - remd 
 and t - remd show results closer to those obtained from conventional 
 md , with a higher occupation of the anti conformer .
estimated 
 glycosidic angle anti population for 
 each nucleotide as a function of the total simulated time .
data are 
 shown for hdih - remd , t - remd , and rect 
 unbiased replicas and for conventional md .
reference values taken 
 from ref ( 26 ) are shown 
 as dashed lines .
we observe that our method 
 is enforcing the exploration of both anti and syn conformations in the biased 
 replicas for each nucleotide independently .
this however does not 
 guarantee that all 16 combinations of anti and syn conformations are explored .
figure 6 shows the free energy of the rna structures grouped by the 
 combination of the  angle anti(a)/syn(s ) rotamers . 

all 16 combinations , except for ssss and asss , are sampled in the unbiased replica from rect .
on 
 the contrary , the unbiased replica from t - remd and hdih - remd explores respectively 13 and 8 of the states and plain 
 md only 5 of them .
the most populated cluster corresponds to an all - anti conformation , followed by the saaa .
then , the three clusters asaa , ssaa , and sasa appear with similar population . in the same figure
the free - energy values for the ergodic replica 
 show that all the 16 combinations are populated in rect within a range 
 of 6 kbt . in the case 
 of hdih - remd
the most ergodic replica 
 visits only 9 combinations with a population that is very close to 
 that of the unbiased replica .
the most ergodic replica in t - remd explores 
 14 clusters , but their populations have large statistical errors . 

we highlight the fact that results from t - remd could be affected by 
 the lack of convergence of trajectories across the temperature space 
 ( see figure s6 ) .
this could lead to an 
 underestimation of the errors as evaluated from block analysis .
the introduced method allows for building bias 
 potentials for a 
 hamiltonian replica - exchange scheme using concurrent well - tempered 
 metadynamics on several cvs .
replicas are simulated using a ladder 
 of well - tempered bias factors . when cvs are correlated , the 
 self - consistency among the bias potentials is crucial to achieve flat 
 sampling in each individual cv , as illustrated in figure s2 . in this case
we also remark that here flattening is not 
 complete but modulated by the value of . this is useful since 
 it avoids sampling very high energy states ( e.g. , with steric clashes ) 
 that would have a very low chance of being accepted in the unbiased 
 replica .
the method slightly outperforms 
 t - remd , where the entire system is heated , indicating that for these 
 small systems there is not a substantial advantage in schemes where 
 part of the system is biased . however , rect can be straightforwardly 
 generalized to large systems since the acceptance only depends on 
 the size of the biased portion .
results from both dala and tetranucleotide 
 simulations show that 
 the bias potentials constructed with concurrent wt - metad are able 
 to gradually scale the free - energy barriers .
we notice that only barriers 
 in the one - dimensional free - energy profiles are compensated , which 
 means that some regions in the multidimensional space of all the cvs 
 might not be explored . in principle
estimated free energies for the tetranucleotide 
 conformations clustered 
 according to the  angle anti / syn rotameric 
 combinations ( circles ) .
free energies are computed as kbtlogpi , where pi is the normalized population of each cluster on the indicated 
 replica .
confidence intervals are shown as bars and span the range [ kbtlog(pi+pi),kbtlog(pipi ) ] , where pi is the standard deviation of the average pi as obtained from four blocks . 

clusters which are observed in only one of the four blocks have an 
 infinite upper bound .
the second application on which we tested rect , namely conformational 
 sampling of a tetranucleotide , is particularly challenging .
crick 
 pairings , and ergodic sampling is out of reach of conventional md 
 simulations .
so far , converged ensembles have 
 been obtained only through highly expensive multidimensional remd 
 simulations , corresponding to a total simulated time of several tens 
 of s .
one of the reasons for this difficult 
 convergence is the long relaxation time for the anti to syn transitions , which could be additionally 
 hindered by an incorrect force - field description of base 
figure 6 illustrates the ability 
 of rect 
 to accelerate conformational transitions among the  angle anti / syn rotamers .
although the conformational space of 
 the more biased replicas is highly expanded , the convergence in the 
 unbiased replica is not affected . on the contrary
the method facilitates 
 the sampling of glycosidic rotamer conformations that otherwise would 
 not be explored by md simulations of the same overall length .
we finally 
 remark that our procedure can be combined with weighted histogram so as to include the statistics of the biased 
 replicas .
rect 
 is based on the idea of building a replica ladder where a large set 
 of selected cvs is progressively heated .
we 
 first discuss the possibility of using methods other than well - tempered 
 metadynamics to build the replica ladder .
these methods could be used 
 in a rect scheme provided they are suitably extended so as to sample 
 a partially flattened distribution .
we also observe that other methods 
 aimed at keeping selected cvs at a given temperature have been proposed 
 based on coupling thermostats to cvs directly .
these techniques 
 have been mostly used in the past with an exploration purpose relying 
 on additional calculations so as to provide free energies ( see , e.g. , 
 ref ( 62 ) ) , but it is 
 not clear if they can be integrated in a rect scheme . in the 
 following
we discuss the comparison of rect with related methods that 
 are not directly based on cv tempering .
our method is closely related 
 to the one introduced in ref ( 31 ) .
there , a bias potential 
 aimed at disfavoring the most probable rotamers is manually constructed 
 and applied on several replicas using a scaling factor .
this bias 
 disfavors the major minima but does not ensure a proper compensation 
 of the free - energy barriers , as their positions and magnitudes are 
 not a priori known . the main advantage of rect is 
 that several low - dimensional bias potentials are built with a self - consistent 
 procedure so that the technique can be straightforwardly applied to 
 a large number of degrees of freedom .
every replica performs an independent metadynamics 
 simulation so that one cv at a time is feeling the flattening potential . 

thus , it is typically used with a relatively small number of ad hoc designed cvs capable of describing the relevant conformational 
 transitions . on the other hand ,
rect is designed to be used with a 
 very large number of dummy cvs with little a priori information and to bias them concurrently to exploit their cooperation 
 in enhancing conformational sampling .
for this reason , the two approaches 
 are complementary and could even be combined in a multidimensional 
 replica exchange suitable for a massively parallel environment .
comparison with solute tempering and related methods . in replica exchange with solute tempering the solute hamiltonian 

is scaled so as to obtain an effect equivalent to a rise in the simulation 
 temperature .
any set of atoms can be identified 
 as solute , giving the opportunity to enhance sampling in a region 
 localized in space .
this requires modifying charges 
 of the enhanced region , with long - range effects and sometimes affecting 
 fundamental properties such as hydrophobicity . in our method , the 
 bias potentials act on precisely selected degrees of freedom minimally 
 perturbing their coupling with the rest of the system .
moreover , the 
 bias is adaptively built so as to compensate the free energy and not 
 the potential energy , so that with properly chosen cvs it could be 
 used to compensate entropic barriers .
the potential energy of the system is modified so 
 as to decrease the probability to sample minima on the potential energy . on the contrary
, rect employs 
 a bias which is related with the free energy so as to achieve a flatter 
 histogram on the selected cvs .
we finally remark that rect , 
 although formally based on the a priori choice of 
 a set of cvs , typically requires the 
 same amount of information as methods not based on cvs . indeed ,
as 
 we have shown , the method can be easily applied to a very large number 
 of cvs , virtually including by construction all the slow degrees of 
 freedom of the system .
additionally , when a few relevant cvs can be 
 identified based on chemical intuition , rect can be straightforwardly 
 combined with standard metadynamics similarly to parallel tempering or solute tempering .
replica exchange with collective - variable 
 tempering ( rect ) has 
 been here proposed as a novel and flexible enhanced - sampling method . 

rect takes advantage of the adaptive nature of well - tempered metadynamics 
 to build bias potentials that compensate free - energy barriers .
the 
 flattening of the barriers is modulated by the well - tempered factor 
  , and the chosen collective variables ( cvs ) are effectively 
 kept at a higher temperature .
the biasing potentials are built combining 
 concurrent low - dimensional metadynamics protocols so as to be usable 
 on a very large number of cvs .
multiple replicas are then used so 
 as to smoothly interpolate between a highly biased , ergodic simulation 
 and an unbiased one (  = 1 ) . the number of required replicas 
 scales with the square root of the number of chosen cvs for a fixed 
 range of  factors .
this allows a very large number of cvs to 
 be biased , so that virtually all the relevant transitions can be accelerated . 

the cvs used here were mostly dihedral angles , which exhibit relevant 
 barriers in many biomolecular conformational transitions , but the 
 method can be used with any cv .
the application of this technique 
 to the dala in water shows that the cv probability distributions are 
 effectively flattened by the action of the bias potentials and unbiased 
 statistics is correctly recovered . in the case of the tetranucleotide 
 conformational sampling
is greatly enhanced since rect effectively 
 overcomes the high free - energy barriers of the  angle transitions 
 that hindered the conformational sampling at room temperature .
rect 
 is available in plumed , and a sample input file is provided in figure s13 .
rect is a promising tool to enhance 
 the exploration of the conformational space in highly flexible biomolecular 
 systems such as rna , proteins , or rna / protein complexes .
 the computational study of conformational 
 transitions in rna and 
 proteins with atomistic molecular dynamics often requires suitable 
 enhanced sampling techniques . we here introduce a novel method where 
 concurrent metadynamics are integrated in a hamiltonian replica - exchange 
 scheme . 
 the ladder of replicas is built with different strengths of 
 the bias potential exploiting the tunability of well - tempered metadynamics . 
 using this method , 
 free - energy barriers of individual collective variables 
 are significantly reduced compared with simple force - field scaling . 
 
 the introduced methodology is flexible and allows adaptive bias potentials 
 to be self - consistently constructed for a large number of simple collective 
 variables , such as distances and dihedral angles . 
 the method is tested 
 on alanine dipeptide and applied to the difficult problem of conformational 
 sampling in a tetranucleotide . 
Introduction
Methods
Results
Discussion
Conclusion