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Characterization of Conformational Equilibria Through Hamiltonian and Temperature Replica-Exchange Simulations: Assessing Entropic and Environmental Effects

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Characterization of Conformational Equilibria ThroughHamiltonian and Temperature Replica-ExchangeSimulations: Assessing Entropic and Environmental Effects
JOSÉ D. FARALDO-GÓMEZ,
1
BENOÎT ROUX
1,2
1
Department of Pediatrics, Institute for Molecular Pediatric Sciences, Gordon Center for Integrative Science, University of Chicago, Chicago, Illinois 60637
2
Department of Biochemistry and Molecular Biology, University of Chicago, Chicago, Illinois 60637 Received 14 April 2006; Revised 16 June 2006; Accepted 2 July 2006 DOI 10.1002/jcc.20652Published online 6 March 2007 in Wiley InterScience (www.interscience.wiley.com).
Abstract:
Moleculardynamicssimulationsbasedonthereplica-exchangeframework(REMD)areemergingasausefultool to characterize the conformational variability that is intrinsic to most chemical and biological systems. In this work,it is shown that a simple extension of the replica-exchange method, known as Hamiltonian REMD, greatly facilitatesthe characterization of conformational equilibria across large energetic barriers, or in the presence of substantial entropiceffects,overcomingsomeofthedifﬁcultiesofREMDbasedontemperaturealone.Inparticular,acomparativeassessmentoftheHREMDandTREMDapproacheswasmade,throughcomputationofthegas-phasefree-energydifferencebetweentheso-called
D
2d
and
S
4
statesoftetrabutylammonium(TBA),anioniccompoundoffrequentlyusedinbiophysicalstudiesof ion channels. Taking advantage of the greater efﬁciency of the HREMD scheme, the conformational equilibrium of TBA was characterized in a variety of conditions. Simulation of the gas-phase equilibrium in the 100–300K rangeallowed us to compute the entropy difference between these states as well as to describe its temperature dependence.Through HREMD simulations of TBA in a water droplet, the effect of solvation on the conformational equilibriumwas determined. Finally, the equilibrium of TBA in the context of a simpliﬁed model of the binding cavity of theKcsA potassium channel was simulated, and density maps for
D
2d
and
S
4
states analogous to those derived from X-raycrystallography were constructed. Overall, this work illustrates the potential of the HREMD approach in the context of computational drug design, ligand-receptor structural prediction and more generally, molecular recognition, where one of the most challenging issues remains to account for conformational ﬂexibility as well for the solvation and entropic effectsthereon.© 2007 Wiley Periodicals, Inc. J Comput Chem 28: 1634–1647, 2007
Key words:
replica-exchange; conformational change; entropy; solvation effects; drug design
Introduction
In trying to characterize the microscopic srcins of chemical andbiological processes, one of the greatest challenges is to account forthe conformational variability that in most cases is intrinsic to theunderlying molecular system, and may even be indispensable forthe process to occur. Computer simulation methods such as molec-ular dynamics (MD) have great potential for such investigations,since their framework relies on a physically realistic description of the energetic and dynamic properties of the system under study.Nevertheless, the scope of standard MD simulations is limited inpractice; for example, the simulation of a chemical or biologicalprocess during which the system explores two or more conforma-tionalstatesseparatedbysubstantialfree-energybarriers(
≥
10
k
B
T
)may require an enormous computational cost, simply because thesampling of conﬁgurational space accomplished by this method issuchthatregionsofhighfreeenergyarerarelyvisited.Insuchcases,the conformational transitions that are characteristic of the processconsidered will certainly be poorly described.Toovercomesamplinglimitationsbecauseoflargeenergeticbar-riers, advanced simulation methods, such as umbrella sampling,
1,2
meta-dynamics,
3
or accelerated dynamics,
4
typically rely on theintroduction of biasing potentials that modify the free-energy sur-face along speciﬁc multidimensional coordinates, as well as onsuitable unbiasing schemes to recover the appropriate statistical
Correspondence to:
B. Roux; e-mail: roux@uchicago.edu© 2007 Wiley Periodicals, Inc.
Replica-Exchange Simulations of Conformational Equilibria
1635
weight of the resulting ensemble of conﬁgurations. Similarly, theexploration of complex and rugged free-energy landscapes such asthose underlying protein folding can be enhanced greatly by gener-ating a large number of unbiased dynamical trajectories by meansof multiple standard MD simulations using world-wide computingresources,
5
orthroughalgorithmsbasedongeneralizedMonteCarloschemes, such as transition path sampling.
6
Replica-exchangemoleculardynamics(REMD)isanalternativeapproach that combines biasing methods that alter the free-energylandscape and a Monte Carlo-based, multiple-simulation strategy.
7
In its most common implementation to date, the free-energy hyper-surface of the molecular system is modiﬁed through temperature,which gradually increases across the ensemble of simulations (orreplicas), in the hope that regions of the conﬁgurational space thatare inaccessible at room temperature, for a given computationaltime, will be more frequently visited at high temperatures.
8,9
How-ever, while temperature-REMD may be useful in speciﬁc cases,its general applicability is clearly limited by the possibility thatthe free-energy landscape at high temperature is drastically dif-ferent from that at the temperature of interest, to the point thatone or more of the conformational states involved in the equilib-rium under study is no longer statistically signiﬁcant. Moreover, alarge number of replicas is required to simulate equilibria involv-ing large free-energy barriers, or systems with many degrees of freedom,
10
which makes this method impractical for many sys-tems of interest (see also Zuckerman and Lyman
11
for furtherdiscussion).In this work, we show that a simple extension of the replica-exchangemethod,knownasHamiltonianREMD,
12,13
greatlyfacil-itates the characterization of conformational equilibria across largeenergetic barriers, and in a variety of environments, overcomingsome of the difﬁculties of REMD based on temperature alone. Theability to carry out such analyses is of great interest not only fromthe methodological viewpoint, for example, as a means to improvethe quality of atomistic or coarse-grained forceﬁelds; it is also of considerable importance in a variety of applications, notably com-putational drug design, ligand-receptor structural prediction, andmore generally, molecular recognition, where one of the most chal-lenging issues remain to account for conformational ﬂexibility, aswell for the solvation and entropic effects thereon.To illustrate the performance of the HREMD method, whichmakes use of a gradual perturbation of the potential energy func-tion across the system of replicas, we simulate the conformationalequilibrium of tetrabutylammonium (TBA), an ionic compoundwith particular relevance in the ﬁelds of zeolites and biologicalion channels, and illustrate the inﬂuence of entropic and environ-mental effects on this equilibrium. More speciﬁcally, we computethe room-temperature free-energy difference between the
D
2d
and
S
4
conformers in several environments, namely in the gas phase,in water, and in association with a reduced model of the potas-sium channel KcsA. In addition, we also characterize the gas-phaseequilibrium in the 100–300K temperature range, which allows usto quantify the entropic contribution to the equilibrium energet-ics. On the basis of our observations, we anticipate that this orsimilar approaches will greatly assist future computational analy-ses of chemical and biological processes where the conformationalvariability of the underlying molecular systems is a determinantfactor.
Methods
Hamiltonian and Temperature Replica-ExchangeSimulation Framework
Molecular dynamics (MD) simulations typically aim to quantifysome structural or dynamical property of a molecular system atequilibrium, for a given thermodynamic ensemble. For example,in the NVT or canonical ensemble, the equilibrium average valueof a property
A
that may be dependent on the coordinates
X
andmomenta
P
of the particles in the sytem is
A
=
d
X
d
P
A
(
X
,
P
)ρ(
X
,
P
)
(1)
ρ(
X
,
P
)
∝
exp
−
H
0
(
X
,
P
)
k
B
T
0
(2)where
ρ(
X
,
P
)
denotes the probability density of a given conﬁg-uration of the phase space, and
H
0
and
T
0
are the Hamiltonianand temperature of the system. In practice, however, this average iscomputed simply as the mean value of
A
throughout the simulation
A
≈
A
sim
=
1
N
s N
s
s
=
1
A
sim
(
s
)
(3)where
N
s
is the number of conﬁgurations generated, and
A
(
s
)
is thevalue of
A
for each of them. The extent to which the approximationin eq. (3) is reasonable depends on two factors: ﬁrst, the simulationalgorithm must be such that the ensemble of conﬁgurations gener-atedisstatisticallyconsistentwiththeprobabilitydensity
ρ(
X
,
P
)
ineq. (2); second, the ensemble of conﬁgurations must include thoseregions of the phase space
(
X
,
P
)
that contribute the most to theintegral in eq. (1).Modern MD algorithms are designed to satisfy the ﬁrst of theseconditions;however,insufﬁcientcomputationalpowerseverelylim-its the extent of the conﬁgurational sampling that MD simulationscan accomplish for many systems of interest. Typically, regions of the conﬁgurational space that may contribute signiﬁcantly to theintegralineq.(1)areseparatedfromthestartingconﬁgurationofthesimulation by substantial free-energy barriers, which by construc-tion the MD algorithm rarely overcomes; thus, the approximationin eq. (3) is compromised.To tackle this problem, methodologies such as umbrella sam-pling implement biasing potentials that force the system to explorerelevantregionsoftheconﬁgurationalspace,andsubsequentlymakeuseofunbiasingschemestorecovertheconsistencyofthesamplingwith eq. (2). In replica-exchange molecular dynamics (REMD), bycontrast, the simulation is coupled to an ensemble of simulationsof the same molecular system, which are designed to more easilyaccess relevant regions of the conﬁgurational space. In particular,theseadditionalsimulations,orreplicas,mayhaveadifferentHamil-tonian
H
, a different temperature
T
, or both, which are chosen soas to reduce the free-energy barriers characteristic of the systemwith the Hamiltonian and temperature of interest,
H
0
and
T
0
. Byallowing replicas to exchange their conﬁgurations every so often,the latter will have a much better chance of exploring all relevantJournal of Computational Chemistry DOI 10.1002/jcc
1636
Faraldo-Gómez and Roux
•
Vol. 28, No. 10
•
Journal of Computational Chemistry
regions of the conﬁgurational space contributing to the integral ineq. (1), even though in actuality it has not climbed the free-energybarriers separating those regions.Asinumbrellasampling,however,itiscrucialinREMDtomain-tainconsistencyoftheconﬁgurationalsamplingwiththeprobabilitydensity characteristic of the system under study. For a system of
N
replicas,andassumingtheHamiltoniancanbeseparatedintopoten-tial and kinetic energy functions, i.e.,
H
(
X
,
P
)
=
U
(
X
)
+
K
(
P
)
,this overall probability density function can be written as
ρ(
X
1
,
X
2
,
...
,
X
N
,
P
1
,
P
2
,
...
,
P
N
)
=
χ(
X
1
,
X
2
,
...
,
X
N
)ξ(
P
1
,
P
2
,
...
,
P
N
)
(4)
χ(
X
1
,
X
2
,
...
,
X
N
)
∝
N
i
=
1
exp
{−
β
i
U
i
(
X
i
)
}
(5)
ξ(
P
1
,
P
2
,
...
,
P
N
)
∝
N
i
=
1
exp
{−
β
i
K
i
(
P
i
)
}
(6)where
β
i
is 1
/
k
B
T
i
. Although the MD algorithm is designed to sam-plethephasespaceaccordingtothe
χ
and
ξ
distributionsinthetimeperiodsbetweenexchanges,oneneedstoensurethattheexchangeof conﬁgurationsbetweenreplicasisconsistentwiththesedistributionsas well. With respect to the spatial part
χ
, this can be achieved byallowingexchangesbetweenreplicasonlywithacertainprobabilty,
p
ex
,whichisconﬁguration-dependent.Toderivethisexchangeprob-ability, we can think of the replica-exchange process as a Markovchain of states of the replica ensemble. That is, two states
m
and
n
can be understood as the state of the ensemble before and after apairofreplicas
(
i
,
j
)
haveexchangedtheirrespectiveconﬁgurations.Thus, the transition probability between them,
π
mn
, will be relatedto the exchange probability between replicas by the expressions
π
mn
=
p
ex
(
i
,
j
)π
nm
=
1
−
p
ex
(
i
,
j
)
(7)For this Markov process to have a limiting distribution equalto
χ
, the transition probability
π
mn
must satisfy the condition of microscopicreversibility
χ
m
π
mn
=
χ
n
π
nm
.FollowingMetropolis,
14
a computationally efﬁcient choice of
π
mn
would be
π
mn
=
1
χ
n
≥
χ
m
π
mn
=
χ
n
/χ
m
χ
n
< χ
m
(8)Combining eq. (5) and eqs. (7 and 8) lead to a suitable deﬁnition of the exchange probability
p
ex
(
i
,
j
)
p
ex
(
i
,
j
)
=
1
(
i
,
j
)
≤
0
p
ex
(
i
,
j
)
=
exp
(
−
(
i
,
j
)) (
i
,
j
) >
0 (9)
(
i
,
j
)
=
β
i
[
U
i
(
X
j
)
−
U
i
(
X
i
)
]+
β
j
[
U
j
(
X
i
)
−
U
j
(
X
j
)
]
(10)which will guarantee that the sampling of the coordinate space inevery replica will be consistent with
χ
, as if no exchanges weretaking place.Concerning the kinetic part of the canonical distribution,
ξ
, anefﬁcient and convenient strategy is to reinitialize the velocities of all particles in the system upon every exchange of conﬁgurations,extracting their new values from a Maxwell-Boltzmann distributionat temperature
T
i
, which is by deﬁnition consistent with eq. (6). Itcanbeshownthatthisapproachisasaccurateormoreinreproducingthe canonical ensemble (e.g., for a system of harmonic oscillators)as other schemes where velocity scaling factors of the form
β
i
/β
j
are introduced after each exchange, or where the full Hamiltonianfunction is included in the Metropolis test in eqs. (9 and 10).All simulations in this report implement the scheme out-lined above or variations thereof. For example, in the constant-temperature Hamiltonian REMD simulations (HREMD) presentedhereafter, only some of the terms of the potential energy func-tion
U
(
X
)
are modiﬁed across replicas (see the following sectionfor further details), speciﬁcally through a scaling parameter, i.e.,
U
i
=
U
A
+
λ
i
U
B
(a similar scheme has been recently reported byAffentranger et al.
15
); thus eq. (10) reduces to
(
i
,
j
)
=
β (λ
i
−
λ
j
)
[
U
B
(
X
j
)
−
U
B
(
X
i
)
]
(11)In temperature REMD (TREMD), by contrast, the Hamiltonian isthe same for all replicas and therefore
(
i
,
j
)
=
(β
i
−
β
j
)
[
U
(
X
j
)
−
U
(
X
i
)
]
(12)Fromeqs.(10–12),itisalsoapparentthatthereplica-exchangealgo-rithm may in practice involve either the exchange of conﬁgurationsbetween replicas, i.e., the exchange of atomic coordinates for allparticles in the system, while each replica retains its temperatureand/or Hamiltonian, or the exchange of temperatures and/or Hamil-tonian functions, while each replica preserves the conﬁguration of thesimulationsystem.Inthiswork,thelatterapproachwasadopted,since it was found to be more convenient computationally.In all cases, it is important to realize that the efﬁciency of theREMD scheme relies on
(
i
,
j
)
being a sufﬁciently small num-ber, on average, to yield a large enough acceptance probability of exchange,andthusensurethatthereplicasareactuallycoupled.Forthisreason,exchangesaretypicallyattemptedonlybetweenreplicaswhoseHamiltonianortemperatureareadjacentintheensemble,i.e.,
H
j
=
H
i
+
1
and
T
j
=
T
i
+
1
, and the temperatures
T
i
and couplingparameters
λ
i
are generally exponentially distributed. Similarly, thefrequency at which exchanges are attempted should be as large aspossible,butnotsolargethatitprecludesthereplicasfromsamplingsigniﬁcantly diverse regions of the conﬁgurational space betweenexchanges.
Description of the Simulation System
We have analyzed the conformational equilibrium of tetrabutylam-monium (TBA) in a variety of conditions, including the interior of a simpliﬁed model of the KcsA K
+
channel protein. TBA is aninteresting example because it primarily exists in either of two con-formations, known as
D
2d
and
S
4
, which are separated by a largeenergy barrier and have different degeneracy (six and three, respec-tively);
16
additionally, it is expected that entropic and solvationJournal of Computational Chemistry DOI 10.1002/jcc
Replica-Exchange Simulations of Conformational Equilibria
1637
Figure 1.
A. Molecular representation of tetrabutylammonium (TBA) in its most probable conformations,known as
D
2d
and
S
4
. The transition between these two states, which are three- and six-fold degenerate,respectively,involvesaconcerted120
◦
rotationofthecentraltorsions
1
(
C
α
,N,C
α
,C
β
)
.BandC.Potential-energy surface along two of the torsions
1
(
C
α
,N,C
α
,C
β
)
, for the energy function given in eq. (14) with
λ
=
1.0 and
λ
=
0.10, respectively. D. Top panel: one-dimensional potential-energy proﬁle connecting the
D
2d
and
S
4
states, ﬁrst moving along the
x
axis in (B), and then along the
y
axis. Bottom panel, for the samepathway, evolution in the root-mean-squared deviation,
η
i
, with respect to each of the three degenerate
D
2d
ideal conformations; as can be seen, the minimum of the three curves (in this example,
η
1
) discriminatesunambiguously the
D
2d
and
S
4
states, as well as their intermediate state.
effects will also inﬂuence its conformational equilibrium, reﬂect-ing the alternate packing of the hydrocarbon chains and differentexposure of the core of the molecule (Fig. 1A).Structurally, the ideal geometries of the
D
2d
and
S
4
con-formers differ in the arrangement of the 12 central torsions
1
(
C
α
,N,C
α
,C
β
)
, of which only four are independent, and thusthese two conformations are also characteristic of TBA analoguessuch as tetraethylammonium (TEA) and tetrapentylammonium(TPA). To model the conformational equilibrium of this molecule,we employed the CHARMM27 forceﬁeld,
17
corrected so as toexactly reproduce the quantum-mechanical potential-energy dif-ference between the
D
2d
and
S
4
conformations of TEA, namely0.9kcal/mol in favor of
D
2d
.
18
Because all bonded terms of theclassical potential-energy function are identical in the ideal
D
2d
and
S
4
geometries, this correction is applied to the four pseudoangles
θ (
C
β
,N,C
β
)
, via the Saxon-Wood ﬂat-bottom potential:
U
corr
=
h
1
+
exp
p
2
−
θ
−
θ
ref
p
1
−
1
(13)where
h
=
1.4kcal/mol,
θ
ref
=
90.0,
p
1
=
0.03, and
p
2
=
0.3. Theresulting potential-energy surface connecting the
D
2d
and
S
4
con-formationsofTBA,showninFigure1B,yieldsanenergydifferenceJournal of Computational Chemistry DOI 10.1002/jcc
1638
Faraldo-Gómez and Roux
•
Vol. 28, No. 10
•
Journal of Computational Chemistry
of 1.1kcal/mol in favor of
D
2d
, with energy barriers on the order of 10kcal/mol separating these and their intermediate states.
Simulation Details and Statistical Analysis
All simulations were carried out with a modiﬁed version of the CHARMM c32a2 molecular simulation software,
19
using itsLangevin-equationintegratorwithacollisionfrequency
β
=
5ps
−
1
,and full electrostatics and van der Waals interactions. In the Hamil-tonianREMDsimulations,theCHARMMpotentialenergyfunctionwas modiﬁed across replicas through a single scaling factor
λ
i
ranging from 0.10 to 1.00, that is,
U
i
=
U
bonds
+
U
(
core
)
angles
+
λ
i
U
elec
+
U
vdw
+
U
torsions
+
U
(
other
)
angles
+
(
1
−
0.75
(
1
−
λ
i
))
U
corr
≡
U
A
+
λ
i
U
B
(14)The modiﬁed potential energy surface of TBA, according toeq. (14) and
λ
i
=
0.10, is shown in Figure 1C; while the relativeenergy of the
D
2d
and
S
4
conformations is changed only slightlywith respect to that with
λ
i
=
1.00 (0.8kcal/mol in favor of
D
2d
),the energy barriers separating these and the intermediate state aredramatically reduced (from more that 10 to less than 2kcal/mol). Inthe replica simulation of TBA in water, the additional contributionsto the potential energy, namely water–water bonded and nonbondedterms as well as TBA-water interactions, were not scaled and thuswere identical across replicas; by contrast the TBA-protein interac-tions in the simulation of the simpliﬁed model of KcsA were scaleddown to facilitate the conformational search within the cavity.To assess the convergence of the REMD method as a functionof simulation time, our strategy has been to set up all replicas ineach simulation in either the
D
2d
or the
S
4
conformation, and sim-ulate the replica system until approximately the same populationsof each state,
P
(
D
2d
)
and
P
(
S
4
)
, were achieved in the ensemblecorresponding to
T
=
300K and
λ
=
1.00. To compute these pop-ulations,werecordedthevalueofaconformationalcoordinate
η
thatunambiguouslydistinguishesthesetwoconformationsaswellasthetransitionstatebetweenthem(Fig.1D).Speciﬁcally,thiscoordinateis deﬁned as
η
=
min
{
η
1
,
η
2
,
η
3
}
(15)where
η
i
denotes the root-mean-square deviation of a given con-formation of TBA during the simulation with respect to the idealgeometries of each of the degenerate
D
2d
conformations. From theprobabilitydensity
ρ(η)
,wecanderivethepopulationsofeachstate,and thus their relative free energy, via the expression
F
=−
k
B
T
ln
P
(
D
2d
)
P
(
S
4
)
=−
k
B
T
ln
η
∈
D
2d
d
ηρ(η)
η
∈
S
4
d
ηρ(η)
(16)wheretheintegrationrangeforthe
D
2d
and
S
4
statesare
η
∈[
0,0.3
]
and
η
∈[
0.85,1.20
]
,respectively.Inpractice,thereportedvaluesof the free-energy difference derive from a block average comprisingthe latest 80% of the sampling obtained at
T
=
300K and with
λ
=
1.00 from each simulation, that is
F
=
1
m
m
i
=
1
F
i
(17)with
m
=
4blocksforalltime-scales.Theerrorofthisblockaverageis deﬁned as the standard error of the mean
e
=
σ(
F
)
√
m
=
mi
=
1
(
F
i
−
F
)
2
m
2
(18)Lastly, the vibrational analysis used for the calculation of con-formational entropies of TBA in the
D
2d
and
S
4
conformations wascarried out with the VIBRAN module of CHARMM.
20
Results and Discussion
Hamiltonian Vs. Temperature REMD
We ﬁrst compare the efﬁciency of the temperature and HamiltonianREMD(TREMDandHREMD)approachesinreproducingthecon-formational equilibrium of TBA, for an equivalent computationalcost. To this end, we carried out two independent 13-replica simu-lations for each method, where the starting conformation of TBA inall replicas,
X
init
, was set to be either
D
2d
or
S
4
; the sampling timeperreplicainallfoursimulationswas100ns.ThetemperaturerangeusedintheTREMDsimulationswas300–600K,whilethepotential-energy scaling factor
λ
in the HREMD simulations ranged between1.00and0.10,andthetemperaturewas300Kforallreplicas;inbothcases,thevaluesoftemperatureand/or
λ
assignedtoeachreplicafol-lowed an exponential distribution. Exchanges were attempted onlybetweenrandomlychosenreplicasthatwereadjacentintemperatureor scaling parameter
λ
, every 1ps of simulation.The proportion of replica exchanges that were accepted duringthese simulations are shown in Table 1, for those pairs within whichexchange attempts were made. By virtue of the exponential dis-tribution of the values of temperature and
λ
, the acceptance ratioswere found to be uniform across replica pairs in the TREMD sim-ulations, and slightly less so for the HREMD, because of the factthat the scaling parameter is applied to only a subset of the potentialenergy function. At any rate, these acceptance ratios, around 74%in TREMD and in the 50–65% range for HREMD, are expected tobe more than acceptable. To validate this expectation, we analyzedthe fraction of the simulation time during which each replica had agiven temperature, or a given scaling parameter. The range of theseresidence times across all replicas are shown in Table 2. As can beseen, in all four simulations, all replicas were found to run at alltemperatures or with all scaling parameters, for a fraction of thesimulation time that is close to the ideal value, i.e., 7.7% or 1/13 of the simulation time.Although the analysis above demonstrates that the REMDscheme is coupling all replicas efﬁciently and uniformly, it remainsto be determined whether the ensemble of conﬁgurations obtainedat each temperature, or with a given Hamiltonian, are correctfrom the statistical viewpoint, i.e., whether they remain within theJournal of Computational Chemistry DOI 10.1002/jcc

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