simtechprintsissueno2011-35

Postprint Series Issue No. 2011-35 Stuttgart Research Centre for Simulation Technology (SRC SimTech)

SimTech Cluster of Excellence Pfaffenwaldring 7a 70569 Stuttgart [email protected] www.simtech.uni-stuttgart.de

J. Kstner a

Umbrella Sampling Stuttgart, March 2011 a Computational Biochemistry Group, Institute of Theoretical Chemistry, University of Stuttgart, Pfaffenwaldring 55, 70569 Stuttgart, Germany; [email protected] http://www.theochem.uni-stuttgart.de/kaestner Abstract The calculation of free-energy differences is one of the main challenges in computational biology and biochemistry. Umbrella sampling, biased molecular dynamics, is one of the methods that provide the free energy along a reaction coordinate. Here, the method is derived in a historic overview and compared to related methods like thermodynamic integration, slow growth, steered molecular dynamics, or the Jarzynski-based fast growth technique. In umbrella sampling, bias potentials along a (one or more-dimensional) reaction coordinate drive a system from one thermodynamic state to another (e.g. reactant and product). The intermediate steps are covered by a series of windows, at each of which a molecular dynamics simulation is performed. The bias potentials can have any functional form. Often, harmonic potentials are used for their simplicity. From the sampled distribution of the system along the reaction coordinate, the change in free energy in each window can be calculated. The windows are then combined by methods like the weighted histogram analysis method (WHAM) or umbrella integration. If the bias potential is adapted to result in an even distribution between the end states, then this whole range can be spanned by one window (adaptive-bias umbrella sampling). In this case, the free-energy change is directly obtained from the bias. The sampling in each window can be improved by replica exchange methods; either by exchange between successive windows or by running additional simulations at higher temperature.

Advanced Review

Umbrella samplingJohannes Kastner

The calculation of free-energy differences is one of the main challenges in com-putational biology and biochemistry. Umbrella sampling, biased molecular dy-namics (MD), is one of the methods that provide free energy along a reactioncoordinate. Here, the method is derived in a historic overview and is comparedwith related methods like thermodynamic integration, slow growth, steered MD,or the Jarzynski-based fast-growth technique. In umbrella sampling, bias po-tentials along a (one- or more-dimensional) reaction coordinate drive a systemfrom one thermodynamic state to another (e.g., reactant and product). The in-termediate steps are covered by a series of windows, at each of which an MDsimulation is performed. The bias potentials can have any functional form. Often,harmonic potentials are used for their simplicity. From the sampled distributionof the system along the reaction coordinate, the change in free energy in eachwindow can be calculated. The windows are then combined by methods like theweighted histogram analysis method or umbrella integration. If the bias potentialis adapted to result in an even distribution between the end states, then thiswholerange can be spanned by one window (adaptive-bias umbrella sampling). In thiscase, the free-energy change is directly obtained from the bias. The sampling ineach window can be improved by replica exchange methods; either by exchangebetween successive windows or by running additional simulations at highertemperatures. C 2011 John Wiley & Sons, Ltd. WIREs Comput Mol Sci 2011 00 111 DOI:10.1002/wcms.66

INTRODUCTION

T he calculation of free-energy differences is a cen-tral task in computational science. The free-energy difference is the driving force of any pro-cess, such as a chemical reaction. Transition statetheory13 can be used to calculate reaction rates fromenergy barriers, more exactly free-energy barriers.4

The free energy contains the entropy, a measure forthe available space. To map the available space in asystem bigger than a few atoms, extensive sampling isrequired.5,6 Techniques are regularly being reviewedin the literature, with a few recent ones given inJ Comput Chem7 and in a themed issue of J ComputChem in 2009.8,9 Applications range from the solidstate, catalytic reactions, biochemical processes to ra-tional drug design.

The canonical partition function Q of a sys-tem can be calculated via an integral over the wholephase space, i.e., configuration space and momentumspace. If the potential energy E is independent of themomentum, the integral over the latter is a multiplica-

Correspondence to: [email protected]

Computational Biochemistry Group, Institute of TheoreticalChemistry, University of Stuttgart, Stuttgart, Germany

DOI: 10.1002/wcms.66

tive constant to Q, which can be ignored. Then, Q isobtained as

Q=

exp[ E(r )]dNr (1)

with = 1/(kBT), kB being the Boltzmanns constant,T being the absolute temperature, and N being thenumber of degrees of freedom of the system.

The free (Helmholtz) energy A is related to Qvia A = 1/ ln Q. The canonical partition func-tion involves a constant number of particles, constantvolume, and a constant temperature. If the pressure,rather than the volume, is kept constant, the Gibbsfree energy (usually denoted as G) is obtained. Apartfrom the change in the ensemble, the following for-malisms and derivations are equivalent for A and G.In the condensed phase, which is relevant for mostapplications, the systems are hardly compressible; soA and G are numerically very similar.

In chemical reactions, one is generally interestedin free-energy differences between two states. If thetwo states differ by geometry (like a reactant andproduct of a reaction) then the integration in Eq. (1)is done over a part of the coordinate space for eachstate.

In many cases, a reaction coordinate ( ), acontinuous parameter which provides a distinction

Volume 00, January /February 2011 1c 2011 John Wi ley & Sons , L td .

Advanced Review wires.wiley.com/wcms

between two thermodynamic states, can be defined.Any order-parameter is possible, even a change in theenergy expression (the Hamiltonian). The reactioncoordinate can be one or more dimensional. Often, is defined on geometric grounds, such as distance,torsion, or the difference between root mean squaredeviations from two reference states.

With defined, the probability distribution ofthe system along can be calculated by integratingout all degrees of freedom but :

Q( ) =

[ (r ) ] exp[(E)dNr ]exp[(E)dNr ] . (2)

Q( ) d can be interpreted as the probability of find-ing the system in a small interval d around . Conse-quently, this allows the calculation of the free energyalong the reaction coordinate; A( ) = 1/ ln Q( ).A( ) is also called potential of mean force (PMF). If is a general, i.e., non-Cartesian, coordinate, or aset of those, a Jacobian term enters Eq. (2). As longas the integration is performed in Cartesian coordi-nates, Q( ) = [ (r) ] exp(E) dNr/Q. If theintegration is done in a different set of coordinates,q, being one of those, an explicit Jacobian determi-nant |J(q)|, with Jij = dqi/drj, has to be taken intoaccount: Q( ) = exp(E)|J(q)|dN1q/Q, wherethe integration is performed over all coordinatesexcept .

In computer simulations, the direct phase-spaceintegrals used in Eqs. (1) and (2) are impossible to cal-culate. However, if the system is ergodic, i.e., if everypoint in phase space is visited during the simulation,Q( ) is equal to

P( ) = limt

1t

t0

[ (t)]dt (3)

that is, the ensemble average Q( ) becomes equal tothe time average P( ) for infinite sampling in an er-godic system. In Eq. (3), t denotes the time and simply counts the occurrence of in a given inter-val (of infinitesimal width in the exact equation andof finite width when calculating a histogram). So, inprinciple, A( ) can be directly obtained from molec-ular dynamics (MD) simulations by monitoring P( ),the distribution of the system along the reaction co-ordinate.

Note that the terms distribution, distributionfunction, frequency, probability density, and possi-bly a few more are sometimes used in the literature ofchemistry and physics in different contexts. Through-

FIGURE 1 | Separation of the reaction coordinate (dashed line)between two states (here represented by two minima on the potentialenergy surface) into distinct windows. The system is mainly sampledperpendicular to the reaction coordinate in each window.

out this article, the term distribution P( ) refers tothe normalized frequency of finding the system in thevicinity of a given value of . If P( ) was obtainedfrom an exact ensemble average rather than a sampledquantity, P( ) would refer to a probability density.

However, simulations are only run for finitetime. Regions in configuration space around a min-imum in E(r) are typically sampled well, whereasregions of higher energy are sampled rarely. Forrare events, those with an energy barrier significantlylarger than kBT, direct sampling is infeasible. To ob-tain a profile A( ), however, also those high-energyregions, those rare events, are required.

Different techniques have been developed tosample such rare events. One can broadly distinguishthree different families of methods: (1) methods thatsample the system in equilibrium, (2) nonequilibriumsampling techniques, and (3) methods that introduceadditional degrees of freedom, along which the freeenergy is calculated. The third family includes -dynamics912 and metadynamics.13 The latter is cov-ered in a different contribution in this series and willnot be discussed here.

In the remaining two families of methods, globalsampling can be approximated by two techniques,schematically illustrated in Figure 1. On the one hand,the path is split into windows. Each window coversonly a small part of the range of . The windows aresampled individually. In postprocessing, the resultsof the different windows are combined to result in aglobal free-energy profile A( ). On the other hand,

2 Volume 00, January /February 2011c 2011 John Wi ley & Sons , L td .

WIREs Computational Molecular Science Umbrella sampling

one can run multiple simulations. In each of those,the system is driven from one state of interest (A) tothe other state (B), taking a different path each time.The postprocessing in this case includes averagingover the different simulations.

This review is organized as follows. In Accel-erated Sampling Techniques, the context of umbrellasampling is described by briefly introducing differ-ent techniques to sample free-energy profiles in MDsimulations. In Umbrella Sampling: Method, the me-thodical details of umbrella sampling are derived anddescribed. Bias Potentials deals with different bias po-tentials, whereas Sampling Techniquesmentions tech-niques to accelerate the MD sampling itself.Methodsto Analyze Umbrella Sampling Simulations discussesthe most common methods to analyze umbrella sam-pling simulations, i.e., to extract a free-energy profilefrom the sampled data. Conclusion finally summa-rizes the topic.

ACCELERATED SAMPLINGTECHNIQUES

Accelerated Sampling Techniques Basedon Equilibrium PropertiesAll of the acceleration techniques discussed in this re-view have the goal of calculating the free-energy dif-ference, an equilibrium property. However, in eachsimulation, the system can either be in equilibrium,or one can measure the response to some pertur-bation and derive the free-energy change from that.The former case will be discussed here, the latter inFree-Energy Differences from Nonequilibrium Simu-lations.

To drive a system over an energy barrier, onecan either (1) modify the energy expression in orderto reduce the barrier, or (2) restrict the sampling spaceto all degrees of freedom, but the reaction coordinatedescribing the transition over the barrier. The formeris known as biased MD or umbrella sampling.14,15

Because this is the main focus of the present review,it will be discussed in detail in Umbrella Sampling:Method.

In thermodynamic integration,1621 a tech-nique sometimes also referred to as blue moonsampling,18,22 the transition over a barrier is simu-lated by freezing the reaction coordinate at differ-ent values in a number of windows and samplingthe system perpendicular to . The constraint freez-ing the reaction coordinate has to be implemented inan energy-conserving manner. Generally, the methodof Lagrange multipliers (Shake algorithm23) is used.

The force on the frozen reaction coordinates is sam-pled. The resulting mean force is the derivative of thefree energy with respect to the reaction coordinate.Integration of the mean force results in the PMF.

It should be noted that there is some confusionin the older literature over the term PMF.24 Especiallyin the field of thermodynamic integration, one oftenreferred to PMF as a quantity directly obtained byintegrating the mean force, which differs from the freeenergy by neglect of a correction of the metric tensor.If the reaction coordinate is a spatial coordinate or acombination of those, constraining it to a fixed valuealso changes the momentum sampling. There is onecomponent of the momentum canonically conjugatedto each component of the spatial coordinates. If isfrozen, the associated momentum is frozen (zero) aswell. This can lead to a change in the metric tensor ofthe system. For simple reaction coordinates, metric-tensor corrections have been derived.21,2531

Rather than keeping the reaction coordinatefixed in a number of windows, one can also varya constraint slowly from one state to another in anapproach termed slow growth.3234 Average and in-tegration of the force on the constraint results in thefree energy.

In umbrella sampling,14,15 the reaction coor-dinate is not constrained, but only restrained andpulled to a target value by a bias potential. There-fore, the full momentum space is sampled. Usually,umbrella sampling is done in a series of windows,which are finally combined either with the weightedhistogram analysis method (WHAM)35,36 or usingumbrella integration.37

The bias potential can be varied to pull the sys-tem from one state to another rather than keepingit fixed. If that variation is slow as compared withthe relaxation time of the system, the analysis canbe performed by assuming an equilibrium state of thesystem, i.e., the mean force on the reaction coordinatecan be sampled and integrated. This approach gainedpopularity under the name of steeredMD (SMD).3841

SMD directly simulates the influence of an atomic-force microscope cantilever acting, e.g., on a protein.

Free-Energy Differences fromNonequilibrium SimulationsJarzynski42 demonstrated the equivalence of the free-energy change and an exponential average over theworkW along nonreversible paths originating from acanonic ensemble:

exp(A) = exp(W). (4)



This can be exploited in practical simulations bymoving a constraint on the reaction coordinate rel-atively fast from an equilibrated system to the tar-get system.43,44 This method became known asfast growth. It is related to Bennetts acceptanceratio method.45 The changes in the energy alongthese paths are averaged according to Eq. (4). Thecomputational tradeoff is that the faster the constraintis varied, the larger is the statistical spread, and thus,more trajectories have to be calculated.

Free-energy perturbation46 (FEP) can be re-garded as a limiting case of methods based onJarzynskis equation:

exp(A) = exp(E)a (5)with E being the difference of the initial state andthe final state, and the ensemble averaged over theinitial state a. It should be noted, of course, that FEPwas proposed and usedmany decades before the moregeneral Eq. (4). In FEP, the instantaneous change fromone state to another is sampled over a canonical en-semble. Thus, it corresponds to fast growth with theconstraint immediately moved to the target value. Theexponential average of the change results in the free-energy difference. The term perturbation is mislead-ing because the method is exact and does not corre-spond to a perturbation theory in the usual sense.

A special challenge for free-energy simula-tions are quantum mechanics/molecular mechanics(QM/MM) setups, in which a small part of the systemis described by comparatively expensive QM calcula-tions, whereas most of the system is handled by classi-cal force fields (MM).47 A variant of FEP4850 can beused to restrict the sampling to the computationallycheaper force field part.

UMBRELLA SAMPLING: METHOD

Umbrella sampling was developed by Torrie andValleau14,15 based on related previous work.51,52 Abias, an additional energy term, is applied to the sys-tem to ensure efficient sampling along the whole re-action coordinate. This can either be aimed at in onesimulation or in different simulations (windows), thedistributions of which overlap. The effect of the biaspotential to connect energetically separated regions inphase space gave rise to the name umbrella sampling.

In this section, the formalism of recovering unbi-ased free-energy differences from biased simulationswill be discussed. The next section describes differentforms of bias potentials used in the literature.

The bias potential wi of window i is an addi-tional energy term, which depends only on the reac-

tion coordinate:

Eb(r ) = Eu(r ) + wi ( ). (6)

The superscript b denotes biased quantities, whereasthe superscript u denotes unbiased quantities. Quan-tities without superscripts are always unbiased.

In order to obtain the unbiased free energyAi( ),we need the unbiased distribution, which is, accordingto Eq. (2):

Pui ( ) =exp[E(r )] [ (r ) ] dNr

exp[E(r )] dNr . (7)

MD simulation of the biased system provides the bi-ased distribution along the reaction coordinate Pbi .Assuming an ergodic system,

Pbi ( )=exp{[E(r )+i ( (r ))]}[ (r ) ]dNr

exp{[E(r ) + i ( (r ))]}dNr .

(8)

Because the bias depends only on and the in-tegration in the enumerator is performed over all de-grees of freedom but ,

Pbi ( ) = exp[i ( )]

exp[E(r )][ (r ) ]dNrexp{[E(r ) + i ( (r ))]}dNr . (9)

Using Eq. (7) results in

Pui ( ) = Pbi ( ) exp[i ( )]

exp { [E(r ) + i ( (r ))]}dNr

exp [E(r )]dNr= Pbi ( ) exp[i ( )]

exp[E(r )] exp{ i [ (r)]}dNr

exp[E(r )]dNr= Pbi ( ) exp[i ( )] exp[i ( )]. (10)

From Eq. (10), Ai( ) can be readily evaluated. Pbi ( )is obtained from an MD simulation of the bi-ased system, wi( ) is given analytically, and Fi =(1/) lnexp[ i ( )] is independent of :

Ai ( ) = (1/) ln Pbi ( ) wi ( ) + Fi . (11)

This derivation is exact. No approximation entersapart from the assumption that the sampling in eachwindow is sufficient. This is facilitated by an appro-priate choice of umbrella potentials wi( ).



FIGURE 2 | Global free energy (thick solid curve) and thecontributions Ai of some of the windows (thin dashed curves). Forclarity, only every third window is shown. At the bottom, the biaseddistributions Pbi as obtained from the simulation are shown (thin solidcurves). Relatively few bins (100) have been used to generate thisfigure.

As long as one window spans the whole rangeof to be studied, Eq. (11) is sufficient to unbias thesimulation. A( ) is in any case only defined up to anadditive constant; so in this case, Fi can be chosenarbitrarily.

If the free-energy curves Ai( ) of more windowsare to be combined to one global A( ), see Figure 2,the Fi have to be calculated. They are associatedwith introducing the bias potential and connect thefree-energy curves Ai( ) obtained in the differentwindows:

exp(Fi ) = exp[ i ( )]

=

Pu( ) exp[ i ( )]d

=

exp{[A( ) + i ( )]}d (12)

with Pu( ) being the global unbiased distribution. TheFi cannot directly be obtained from sampling. Meth-ods to Analyze Umbrella Sampling Simulations willdeal with methods to calculate them, i.e., to com-bine the results of different windows in umbrellasampling.

BIAS POTENTIALS

Ideally, the bias potential is chosen such that samplingalong the whole range of the reaction coordinate is uniform. Therefore, the optimal bias potential iswopt = A( ). This would lead to a truly uniformdistribution Pbi ( ). However, A( ) is obviously not

known; it is what we aim to calculate with umbrellasampling. Therefore, two main families of bias po-tentials have emerged: harmonic biases in a series ofwindows along , and an adaptive bias, which is ad-justed to match A( ) in only one window spanningthe whole range of .

Harmonic Bias PotentialsTo ensure sampling in all regions of , the range ofinterest of is split into a number of windows. In eachwindow, a bias function is applied to keep the systemclose to the reference point refi of the respective win-dow i. Often, a simple harmonic bias of strength K isused:

i ( ) = K/2( refi

)2. (13)

After the simulations, the free-energy curves arecombined with techniques discussed in Methods toAnalyze Umbrella Sampling Simulations (typicallyWHAM or umbrella integration). The form of thebias given in Eq. (13) is appealing because it containsonly few parameters: K (which in principle can bewindow dependent), the number of images, and refi .The latter are usually chosen uniformly distributedalong . The higher the number of images, the smalleris generally the statistical error relative to CPU time.53

However, the CPU time needed for equilibration, onthe contrary, increases with the number of images.The MD simulations of the images are completely in-dependent and thus, can run in parallel.

The choice of K, the strength of the bias, is theonly critical decision. It has to be made before simula-tions are run. By contrast, additional windows couldalways be inserted if the first series of windows resultsin too large gaps between the distributions. Overall,K has to be large enough to drive the system over thebarrier. Too large K, however, will cause very nar-row distributions Pbi ( ). Sufficient overlap betweenthe distributions is required for WHAM, whereas itis not required, but still advantageous in umbrellaintegration.54 Increasing K at constant time step alsoleads to increasing errors in the numerical integra-tion of the equations of motions. If the time step istoo large (or K is too large), configurations with highenergies will be overrepresented.20

For umbrella integration analysis, analytic ex-pressions for the statistical error can be derived, whichallow an estimate of an ideal K based on quantities,which can often be estimated prior to sampling.54

It has also been suggested that the location of thenext window to be sampled ( i+1ref) can be chosenfrom the location and the widths of the previous



window to match their estimated half maxima.55 Analternative is to use data from the experiment to definethe most promising bias parameters.56

Adaptive Bias Umbrella SamplingThe aim of adaptive bias umbrella sampling13,5760 isto cover the whole range of interest of the reactioncoordinate in one simulation. In principle, this canbe achieved by choosing a bias w( ) = A( ). Thisexactly flattens the energy surface and leads to a uni-form sampling along . Because A( ) is, of course, notknown a priori, one typically starts out with an initialguess of w( ) and iteratively improves it to achieve auniform distribution.

Specialized Umbrella PotentialsThe local elevation method61 adds, similar to meta-dynamics, a history-dependent (and thus, time-dependent) bias to the potential energy. This hasrecently62,63 been combined with umbrella samplingby building up a local elevation bias in a compara-tively short simulation and then sampling the distribu-tion in that bias to reconstruct the free energy. Otherspecial forms of umbrella potentials were used.64

SAMPLING TECHNIQUES

In each window of an umbrella sampling run, thephase space has to be sampled as good as possible.Overlap between windows is required for WHAManalysis (see below) and is desirable for umbrella inte-gration. The quality of the sampling can be enhancedby Hamiltonian replica exchange.6567 In specified in-tervals, the geometry of window i is used to calculatethe total biased energy of a neighboring window j (i.e.,the bias wj of window j is used), and additionally, theenergy of geometry j with the bias wi is calculated.If the sum of these energies is smaller than the sumof the original energies, the two sets of coordinatesare exchanged. If the sum is larger, exchange is stillpossible based on the Metropolis criterion. Then thesimulations continue. This is done at regular intervalswith all pairs of images. Replica exchange betweenumbrella sampling windows enhances the quality ofthe sampling without additional computational cost.

The better the sampling, the more important isa proper choice of the reaction coordinate. If the reac-tion coordinate misses important structural changes,it can lead to artificial lowering or raising of the resultobtained by umbrella sampling.8 A too high barriermay be the result of an unfavorable path being taken.

A too low barrier may be the result of discontinuitiesin the path: the change from one window i to the nextwindow i+ 1 may be reflected by only a small changein , but a larger change in other degrees of freedomwhich are not included in . Such artificial behaviorresults in jumps in the root mean square difference be-tween the average structures of subsequent umbrellasampling windows.8

METHODS TO ANALYZE UMBRELLASAMPLING SIMULATIONS

Weighted Histogram Analysis Method(WHAM)Numerous methods have been proposed for an es-timation of Fi,68,69 a promising one being70 theWHAM.35,36 It aims to minimize the statistical er-ror of Pu( ). The global distribution is calculated by aweighted average of the distributions of the individualwindows:

Pu( ) =windows

i

pi ( )Pui ( ). (14)

The weights pi are chosen in order to minimize thestatistical error of Pu:

2(Pu)pi

= 0 (15)

under the conditionpi = 1. This leads to35,36:

pi = aij a j

, ai ( ) = Ni exp[ i ( ) + Fi ] (16)

with Ni being the total number of steps sampled forwindow i. The Fi are calculated by Eq. (12):

exp(Fi ) =

Pu( ) exp[wi ( )] d. (17)

Because Pu enters Eq. (17) and Fi enters Eq. (14) viaEq. (16), these have to be iterated until convergence.For many bins, this convergence can be slow.

Umbrella IntegrationAn alternative toWHAM for combining the windowsin umbrella sampling simulations with harmonic bi-ases is umbrella integration.37 The problem of calcu-lating Fi is avoided by averaging the mean force ratherthan the distribution P. The unbiased mean force isindependent of the Fi:

Aui

= 1

ln Pbi ( )

dwid

. (18)



The distribution Pbi is expanded into a cumulant ex-pansion, which is truncated after the second term (i.e.,approximating Pbi by a normal distribution). This isequivalent to truncating a power series of Aui ( ) afterthe quadratic term. Because A( ) can be assumed tobe smooth and because each window is only supposedto cover a small part of , truncating such a powerexpansion is well justified:

Pbi ( ) =1

bi

2

exp

1

2

( bi

bi

)2 . (19)Thus, and with a bias in the form of Eq. (13), Eq. (18)now reads

Aui

= 1

bi( bi)2 K( refi ) (20)

which only depends on the mean value bi and the vari-ance ( bi )

2 of in each window. These two quantitiescan easily be sampled. For one window, before com-bining the different windows, its integration yields:

Aui ( ) =( bi

)22

(1

( bi)2 K

)

+( bi

)K( refi bi

)+ Ci . (21)

bi shifts Aui ( ) along the axis and determines its

slope, whereas ( bi )2 determines the curvature of Aui ,

and Ci is just the integration constant.The curves of the mean forces of the different

windows can directly be averaged to result in a globalmean force:

A

=windows

i

pi ( )Aui

. (22)

This is conveniently done with (normalized) weightsproportional to Pbi :

pi ( ) = aij a j

, ai ( ) = Ni Pbi ( ). (23)

The resulting global mean force can be numericallyintegrated.

The difference between WHAM and umbrellaintegration is threefold: (1) The unbiased distribu-tions of the images are averaged in WHAM, whereasthe mean force is averaged in umbrella integration.(2) The biased distributions are approximated by nor-mal distributions in umbrella integration, but not in

FIGURE 3 | Weights of weighted histogram analysis method(WHAM) and umbrella integration of three windows in a realsimulation of the enzyme para-hydroxybenzoate hydroxylase (PHBH).49

A maximum in the free energy is found between the second and thirdwindows.

WHAM. (3) The (non-normalized) weights for com-bining the windows are different: ai( ) = Ni exp(wi( ) + Fi) = NiPbi /Pui in WHAM and ai( ) = NiPbiin umbrella integration.

The first point is the main difference betweenthe methods. The second difference can be changedin either of the methods. If umbrella integration isapplied on the whole distribution, its noise level in-creases generally above the one obtained by WHAM.The additional differentiation adds to the noise. Also,its convergence properties with the bin width (num-ber of bins) are lost. On the contrary, WHAM wasmeanwhile used with Pbi approximated by normaldistributions.37,71 This leads to free-energy profiles assmooth as those obtained from umbrella integration.

The weights used by the different methods aresomewhat difficult to transform between the methodsbecause they weight different quantities. However,weights used in real simulations can be compared asdepicted in Figure 3. In both cases, analytic quanti-ties, not directly dependent on histograms, are usedin ai. Thus, the weights are smooth curves even ifthe distributions are noisy. It is clear from Figure 3that the weights used in WHAM are broader than theones used in umbrella integration. Using the weightsof umbrella integration ai( ) = NiPbi in WHAM ispossible, but it results in noisier curves because Pbi isdirectly obtained from histograms in WHAM. It alsoleads to slightly deteriorated free-energy profiles asthe windows effectively overlap less. Strong overlap



between the windows is more important in WHAMthan in umbrella integration.

As a special case of umbrella integration, onecan truncate the power series of A( ) already afterthe linear term19,37,7275:

Aui

= K(bi refi

). (24)

Comparison with Eq. (20) shows that this is accurate

for = bi . However, it has also been used for = refi .

The expressions of umbrella integration allowfor an estimate of the statistical error in A fromMD simulation data.54 This, in turn, can be used tochose the parameters of the simulation, such as thestrength of the bias K and the number of windows, inorder to minimize the statistical error while keepingthe requirement for CPU time at bay.

Umbrella integration can also be performedin multidimensional reaction coordinates.76,77 How-ever, the necessary integration step becomes more dif-ficult (and prone to statistical error) in higher dimen-sions, whereas the alternative WHAM analysis canmore straight forwardly be extended to more dimen-sions.

The main advantages of umbrella integrationover WHAM are the independence of the number ofgrid points (bins) and the availability of an error es-timate. The fact that only and 2 enter the analysisof umbrella integration can be used to test the MDruns for equilibration78 of these two quantities. Thiscannot directly be done for WHAM, where the wholedistribution enters the analysis. However, in princi-ple, one could test for equilibration of and 2, andwhen these are equilibrated, assume that the wholedistribution is equilibrated. Additionally, umbrella in-tegration is noniterative, which speeds up the analysis.However, the CPU time required for the analysis is, ingeneral, negligible as compared with the time neededto acquire the MD sampling data. The reduction ofAi( ) to second order in reduces noise significantly.For cases with very few windows, however, this canbecome a source of inaccuracies.

Estimation of the Sampling Error BarThe error bar from finite sampling can be estimatedusing umbrella integration analysis.54 This, in turn,allows to set up guidelines as to how the necessarysimulation parameters should be set. The most im-portant parameter is K, the strength of the bias. Ingeneral, K should be chosen as small as possible toallow for much overlap between the images. Let usintroduce as the negative second derivative of thefree energy with respect to at the main barrier. Then,K > is necessary to ensure a unimodular distribu-tion in all images. This is necessary in umbrella inte-gration because these distributions are approximatedby normal distributions. In WHAM, sampling overthe barrier is necessary, resulting in K > kBT. Ofcourse, is not known a priori, but sometimes it canbe estimated.54

In general, it is preferable to sample manywindows for shorter times than fewer windows forlonger.79 This leads to a smaller statistical error be-cause of the better overlap between the windows andis better parallelizable.

CONCLUSION

The question whether umbrella sampling or one of itsrelated methods discussed in Accelerated SamplingTechniques is to be used cannot be answered in gen-eral. It may depend on the particular system. Someauthors have compared the applicability of some ofthese methods.44,8082 Although umbrella samplingmight be preferred over thermodynamic integrationbecause of errors in the integration80 (which reducewith more windows), the additional free parameterK, which has to be chosen in umbrella sampling, wasused as an argument against the latter.82 Additionally,the availability of a correction of the metric tensor forthe particular choice of the reaction coordinate mightbe an argument in favor of umbrella sampling, wheresuch a correction is unnecessary.

Overall, umbrella sampling is meanwhile a ma-ture and broadly accepted method for calculatingfree-energy differences.

ACKNOWLEDGMENTS

The author thanks the German Research Foundation (DFG) for financial support within theCluster of Excellence in Simulation Technology (EXC 310/1) and grant SFB716/C.6, both atthe University of Stuttgart.



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