THE JOURNAL OF CHEMICAL PHYSICS 145, 044903 (2016)

A canonical replica exchange molecular dynamics implementationwith normal pressure in each replica

Emanuel K. Peter,1 Igor V. Pivkin,1 and Joan-Emma Shea2,a)1Institute of Computational Science, Faculty of Informatics, University of Lugano, Switzerland2Department of Chemistry and Biochemistry, Department of Physics, University of California, Santa Barbara,California 93106, USA

(Received 29 April 2016; accepted 28 June 2016; published online 27 July 2016)

In this paper, we present a new canonical replica exchange molecular dynamics (REMD) simulationmethod with normal pressure for all replicas (REMD-NV (p)T). This method is suitable for systemsfor which conventional constant N PT-setups are difficult to implement. In this implementation, eachreplica has an individual volume, with normal pressure maintained for each replica in the simulation.We derive a novel exchange term and validate this method on the structural properties of SPC/E waterand dialanine (Ala2) in the bulk and in the presence of a graphene layer. Compared to conventionalconstant N PT-REMD and NVT-REMD simulations, we find that the structural properties of ournew method are in good agreement with simulations in the N PT-ensemble at all temperatures. Thestructural properties of the systems considered are affected by high pressures at elevated temperaturesin the constant NVT-ensemble, an effect that our method corrects for. Unprojected distributions revealthat essential motions of the peptide are affected by the presence of the barostat in the NPT implemen-tation but that the dynamical eigenmodes of the NV(p)T method are in close quantitative agreementwith the NVT-ensemble. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4958325]

I. INTRODUCTION

Proteins fold into a unique 3-dimensional native structureon time scales ranging from microseconds to even longertime scales.1 The pathways and the kinetics of folding arestrongly influenced by the temperature and the pressure inthe system.2,3 Environmental effects, i.e., solvent effects,the presence of surfaces, and macromolecular crowding canadditionally affect the folding landscape.4–6 At temperaturesabove the characteristic melting temperature, proteins tend tobe destabilized and transition from the folded to the unfoldedensemble.7 Alternatively, high pressure conditions at roomtemperature also lead to a destabilization of proteins, due tochanges in the free energy landscape and smaller possiblepartial volumes of the solvated protein.8–13 Notably, thehydrophobic effect, an important driving force of foldingof proteins, strongly depends on the pressure.14

Computer simulations have become essential for under-standing how protein folding pathways and dynamics inthe native state depend on the pressure and temperature.15

Although the standard molecular dynamics simulationtechnique is widely used to investigate folding pathways,this method suffers from a time scale problem, i.e., relevanttime scales in the microsecond regime can only be reachedat a prohibitive computational cost.16 Enhanced samplingmethods have been developed to address this problem.17–23

Replica exchange molecular dynamics (REMD) simulationshave emerged as one of the most important tools toenhance sampling in protein folding.24,25 A number ofdevelopments have been made to improve the performance

a)Electronic mail: shea@chem.ucsb.edu

of REMD simulations,26–48 and it has also been extendedto the microcanonical ensemble.49 REMD has been widelyused for the simulation of the folding-landscape of smallpeptides,50 protein aggregation,51 and peptide adsorption.11,52

Implementing REMD in simulations of peptide adsorptionto planar surfaces and membranes can be challenging, asthe lateral pressure in the system significantly influences thefolding and aggregation dynamics of the peptide near thesurface.53 Although a REMD method has been developed forthe simulation in the (isothermal-isobaric) N PT-ensemble25

and for REMD at different levels of density,9 the barostat canlead to artifacts if the surface tension of the system is neglectedor sampled incorrectly. Barostats have been implemented toact isotropically, anisotropically, or semi-anisotropically onthe cartesian box-coordinates of the system to maintain theglobal pressure in the system at a reference pressure.54–57 If thedensity within the system is distributed inhomogeneously dueto layering effects, a barostat that correctly accounts for thisinhomogeneity becomes difficult to implement. For example,on surfaces such as graphene, water forms high-density layersnear the surface with a higher local pressure than in the bulk.11

A barostat coupled to this system that neglects the layeringeffect will lead to an increase in the volume in the plane of thewater-layer. This results in an artificial density decrease dueto the inhomogeneity of the density of water perpendicularto the plane of the surface. Additionally, if constraints areapplied on the particle coordinates of the surface, a barostatcan lead to further artifacts, e.g., longitudinal movementsof the surface with respect to the center of mass ofthe system. As a consequence, protein dynamics near thesurface can be affected by the use of a barostat, even inthe case of anisotropic coupling, due to an inhomogeneous

0021-9606/2016/145(4)/044903/11/$30.00 145, 044903-1 Published by AIP Publishing.

044903-2 Peter, Pivkin, and Shea J. Chem. Phys. 145, 044903 (2016)

distribution of the water density, and the fact that conventionalbarostats assume a homogeneous distribution.58 A barostatthat identifies correct local density fluctuations in the caseof static inhomogeneities in density has, to the best ofour knowledge, yet to be implemented. As a result, REMDsampling of protein adsorption has primarily been performedin the canonical (NVT) ensemble.11,59,60 A limitation of usingthe NVT ensemble in this case is that only the replica forwhich the pressure is normal (1 atm) can be used in theanalysis of the data. Thus, a full description of the foldinglandscape in the full temperature range is not possible asit is affected by higher pressures at elevated temperaturewithin the canonical isochoric-isothermal (constant-NVT)description. This compels the development of a REMDexchange scheme in the NVT ensemble, in which all replicashave a constant normal pressure, which we call NV (p)Ttechnique. In this paper, we present this new methodologywhich extends the NVT-REMD method towards sampling atnormal pressure. In this method, each replica has an individualvolume dependent on the pressure which is sampled in theconstant NVT-ensemble (NV (p)T). Each individual replicais equilibrated and adjusted in its volume to propagate at agiven temperature and normal pressure. We develop a newexchange scheme which considers the volume perturbationaccording to the exchange between two neighboring replicasand derive it from the exchange term in the NPT ensemble.25

We validate the method on SPC/E water and compare thestructural properties of this system with simulations in theconstant N PT and NVT-ensembles. Subsequently, we applyour methodology on the folding landscape of dialanine (Ala2),a non-methylated small model peptide, in the bulk and neara graphene surface. We find that our new methodologyis in good quantitative agreement with simulations in theN PT-ensemble, overcomes the limitations of REMD-NVTsimulations near surfaces, and provides a useful alternativein instances where conventional constant pressure REMDsimulations are difficult to implement.

II. METHODS

A. REMD NV(p)T

In conventional REMD simulations in the NVT ensemble,the pressure increases at elevated temperatures which cansignificantly affect the behavior of proteins and its resultingfolding landscape. Thus, usually only the lowest replica is atnormal pressure and normal density, while the higher replicasare affected by large pressures. In our work, we aim at adescription where each subsystem at elevated temperatureshas normal pressure. In our algorithm, each replica has anindividual and constant box dimension, depending on itsaverage temperature. Thus, the probability of finding a stateis influenced by the volume change at each exchange. Thatmeans, that the N-particle system is perturbed by a smallvolume change at the time of the exchange. Therefore,the detailed balance condition in NVT combined withfinite volume changes has to be reformulated due to thisperturbation. We note that the volume perturbation introducedto the system has to be small enough that there will be

no abrupt changes in the velocities of the system due to theexchange itself. In the following, we derive the novel exchangeterm for this methodology.

Our system consists of M non-interacting replicas.The state of this extended ensemble is described by {X}= {X1,X2, . . . ,XM} and the partition function is written as

Z = Tr{X} exp(−M

m=1

βmH(Xm)) (1)

=

Mm=1

Z(βm,Pm,Vm), (2)

where βm stands for the inverse temperature, Z(βm) is thepartition function of each individual replica, and H is theHamiltonian of each individual replica. Here we formulatedthe state integral of the canonical ensemble as the state integralof the N PT ensemble, following the equivalence of statisticalensemble averages61 (if the state-variables describing thesystem have the same average value (P, V)). Additionally,we mention that we can use the N PT exchange term forsimulations in NVT since the pressure dependent terms wouldbe zero for ∆V = 0.25 We note that we introduce a certainperturbation into the dynamics due to the difference in thevolume for each replica which affects the state function duringeach exchange. In the following we evaluate this perturbationeffect.

The partition function of one single replica m in theNPT-ensemble is specified by

Z(βm,Pm,Vm)NPT =

exp[−βm(U(rm) + PmVm)

+ N ln Vm]dr1mdr2

m . . . drNmdVm, (3)

where P stands for the pressure, V for the volume, β for theinverse temperature, U is the potential energy, and r standsfor the spatial coordinate.25

In our approach, the probability at the time of an exchangebetween m and n to describe the state in replica n is, due to thechange in the volume of ∆Vmn = Vn − Vm (∆Umn = Un −Um),given by

Z(βn,Pn,Vn) −→ Z(βn,Pn,Vm + ∆Vmn) (4)

and

Z(βm,Pm,Vm) −→ Z(βm,Pm,Vn + ∆Vnm). (5)

In the following derivation, we use the exchange betweenreplicas m and n.

That means that at the time of the exchange, a perturbationexists for the system volume at time t ′ of the exchange between2 neighboring replicas. In other words, the probability offinding the new state after the exchange decreases proportionalto the change in the volume. We describe this new state as

Zt′ = Z(βn,Pn,Vm + ∆Vmn) (6)

and

Zt′ =

exp[−βn(Um + ∆Umn + Pm(Vm + ∆Vmn))

+ N ln(Vm + ∆Vmn)]dr1mdr2

m . . . drNmdVm. (7)

We also can write this expression as

044903-3 Peter, Pivkin, and Shea J. Chem. Phys. 145, 044903 (2016)

Zt′ =

exp[−βnUn − βnPmVn + N ln Vn]dr1

mdr2m . . . drNmdVm. (8)

In the NPT-ensemble, the probability Φ of finding a state is described by25

ΦNPTm (Xm, βm,Pm,Vm) = exp[−βm(U(rm) + PmVm) + N ln Vm]/Zm(βm,Pm,Vm). (9)

In our case, we write that the probability of finding this state is reduced by the change in volume and the general shift in thecoordinates, i.e., the probability to find the state is reduced proportional to the volume fluctuation between canonical ensembles,

Φm(Xm, βm,Pm,Vm)′ = ΦNPTm − ∆Φm(∆Vmn), (10)

where ∆Φm = ΦNPTn . Using a Taylor-expansion of lnΦNPT

m

and ln∆Φm, (since there is a linear dependency of thelogarithm of Φ from the energy, pressure, and volume) weobtain the relations61

lnΦm(Xm, βm,Pm,Vm)NPT

= Cα − βm(Um + PmVm) + N ln Vm + Cβ, . . . (11)

and

ln∆Φm(Xm, βn,Pn,∆Vmn)= Cα − βnUn − βnPnVn + N ln Vn + Cβ, . . . . (12)

With the condition ∞

−∞. . .

∞

−∞Φm(Xm, βm,Pm,Vm)dX ′mdβ′mdP′mdV ′m = 1,

(13)

we identify

exp(Cα) = Z−1t′ . (14)

Thus, the probabilityΦ′ of finding this perturbed state at t = t ′

is then given by

Φm(Xm, βm,Pm,Vm)′ = exp[−βm(Um + PmVm)+ N ln Vm]/Zt′(βn,Pn,Vn)− exp[−βnUn − βnPnVn

+ N ln Vn]/Zt′(βn,Pn,Vn), (15)

while

Φ({X, β})NPT =

Mm

ΦNPTm (16)

and

∆Φ({X, β})m =Mn

ΦNPTn . (17)

We note that ∆Umn = Un −Um = −∆Unm, Pm∆Vmn = Pm(Vn

− Vm) = −Pn∆Vnm (if Pm ≈ Pn due to the implementationpresented in this paper), ln(Vn/Vm) ≈ ln(Vm/Vn) ≈ 0, and∆βnm = −∆βmn. In order to maintain the detailed balancefor the transition matrix W , the following equation must befulfilled:

Φ(. . . ; X, βm,Pm,Vm; . . . ; X ′, βn,Pn,Vn; . . .)×W (X, βm,Pm,Vm|X ′, βn,Pn,Vn)

= Φ(. . . ; X ′, βm,Pm,Vm; . . . ; X, βn,Pn,Vn; . . .)×W (X ′, βm,Pm,Vm|X, βn,Pn,Vn). (18)

Using Equation (15), the property of the probabilitydistribution for M replicas

Φ({X, β}) =Mm

ΦNPTm −

Mn

ΦNPTn (19)

and

Mm

exp[−βm(Um + PmVm) + N ln Vm]/Zt′(βn,Pn,Vn) −Mn

exp[−βnUn − βnPnVn + N ln Vn]/Zt′(βn,Pn,Vn)

=

Mm

exp[−βm(Um + PmVm) + N ln Vm]/Zt′(βn,Pn,Vn)(1 − exp[−(βm − βn)(Un −Um) + (βmPm − βnPn)(Vm − Vn)

+ N ln(Vn/Vm)]/Zt′(βn,Pn,Vn)) (20)

we obtain

W (X, βm,Pm,Vm|X ′, βn,Pn,Vn)W (X ′, βm,Pm,Vm|X, βn,Pn,Vn)= exp(−∆1)(1 − exp(−∆1)). (21)

We then can write

∆1 = (βm − βn)(Un −Um) + (βmPm − βnPn)(∆Vmn) (22)

and

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W (X, βm,Pm,Vm|X ′, βn,Pn,Vn)W (X ′, βm,Pm,Vm|X, βn,Pn,Vn) = exp(−(βm − βn)(Un −Um) + (βmPm − βnPn)∆Vnm)

× (1 − exp(−(βm − βn)(Un −Um) + (βmPm − βnPn)∆Vnm)). (23)

Finally, we note that Equation (23) converges approximately to the exchange probability of the canonical ensemblewhether ∆Vmn = 0 with all replicas with the same volume.24 This probability is then written for ∆Vmn = 0 (with(1 − exp(−(βm − βn)(Un −Um)) ≈ 1))

W (X, βm,Pm,Vm|X ′, βn,Pn,Vn)W (X ′, βm,Pm,Vm|X, βn,Pn,Vn) ≈ exp(−(βm − βn)(Un −Um)). (24)

For the NPT-ensemble, the detailed balance condition is fulfilled with25

W (X, βm,Pm,Vm|X ′, βn,Pn,Vn)W (X ′, βm,Pm,Vm|X, βn,Pn,Vn) = exp(−(βm − βn)(Un −Um) + (βmPm − βnPn)∆Vnm). (25)

Expression (23) does not converge to NPT-expression (25),but fulfills the condition that Φm < ΦNPT

m , because volumefluctuations are explicitly considered in the exchange betweencanonical ensembles in our development.

In this way, an exchange between neighboring states X ′

and X occurs if the following equation with a generateduniform random number ξ ∈]0; 1] is fulfilled:

ξ ≤ e−(βm−βn)(Un−Um)+(βmPm−βnPn)∆Vnm

× (1 − e−(βm−βn)(Un−Um)+(βmPm−βnPn)∆Vnm). (26)

B. Implementation

We note that the computational cost of our approach isonly negligibly higher than conventional NPT-REMD. Unlikein conventional NPT-REMD, there is no exchange probabilityequal to 1, so that the exchange term has to be calculated perpair of replica for each exchange attempt. We note that theperiodic boundary conditions are updated after the exchange ofcoordinates, so that no particle overlaps occur at any exchangebetween replicas with different volumes. The volumes of eachreplica are kept constant at their initial value. We testedthe exchange efficiency of NPT-REMD, NV(p)T-REMD, andNVT-REMD in simulations of dialanine in the bulk. Wefound that the average number of exchange attempts neededfor one complete cycle in temperature space is approximately14 times larger for NV(p)T-REMD than for NVT-REMDor NPT-REMD (see Figure S162). For the dialanine systemin the bulk, the acceptance ratio for NPT was 60.2%, forNVT was 62.4%, and for the new technique at constantpressure was 8.2%. Thus, the implementation needs a higheramount of replicas than conventional NVT-REMD, while thevolume variance between neighboring replicas has to be inthe range of 1 × 10−9 nm3, corresponding to a variation of onebox diameter of 1 × 10−3 nm in a box with volume 1 nm3.With this variation, the volume perturbation is small andlinear, which means that there are no atomic overlaps at theexchange. We emphasize that variations of the box-diameterlarger than 1 × 10−2 nm would not be possible with thisapproach.

C. Program and system preparation

We used the GROMACS 4.5.5 simulation package forall simulations and the implementation of the exchangeterm.63,64 For the simulations of the water behavior atdifferent temperatures, we centered 6540 SPC/E waters intoa box with dimensions 4.050 22 × 4.050 22 × 4.050 22 nm3

for the simulations of constant NVT and constant N PT . Theconstant NVT setups were equilibrated with suitable box-volumes to obtain normal average pressures at 298 K. For thesimulations of dialanine in the bulk, we centered the peptidein a box with dimensions 2.760 33 × 2.776 33 × 2.769 80 nm3

and filled it with 691 SPC/E waters. For the simulationsof dialanine near graphene, we centered the peptide in abox with dimensions 2.76 × 2.76 × 2.6225 nm3 and filled thesystem with 577 SPC/E waters. For the description of thepeptide interactions, we used the AMBER99-SB forcefield.65

For the interactions of graphene with water and the peptide,we applied parameters from a previous study.11 We usedthe Nosé-Hoover thermostat with tc = 1.0 ps to maintain thetemperature in the individual replicas and used the Parrinello-Rahman barostat with tp = 1.0 ps to maintain a pressureof 1 atm in the N PT ensemble. For the simulation ofdialanine on graphene in the NPT ensemble, we used thestochastic canonical velocity rescaling sampling method incombination with the Berendsen barostat, which we coupledsemi-isotropically in the xy-dimension to the system.66 Weused a compressibility value of 4.5 × 10−5 bar−1. For thecalculation of the non-bonded Coulomb interaction, we usedthe particle mesh Ewald algorithm with a cutoff of 1.0 nm,with a neighborlist cutoff of the same length. The van derWaals interactions were described using a Lennard-Jones 12-6 shift function with a cutoff of 1.0 nm. For the simulationswith the NV (p)T method, we equilibrated each single replicaand varied the individual box dimensions in x, y, and zuntil an average pressure of approximately 1.0 atm wasreached. We simulated all systems using a time step of1.0 fs. The replica exchange MD simulations were performedin a temperature range from 298 to 372 K in 23 replicas. Wesimulated the dialanine system in the bulk and on graphenefor 100 ns, while the water system has been integrated over

044903-5 Peter, Pivkin, and Shea J. Chem. Phys. 145, 044903 (2016)

5 ns. We determined the free energies using the relation∆F = ln P/Pmin, where P stands for the probability and Pmin

for its corresponding minimum value. For the determinationof the unprojected free-energy landscapes of dialanine, weperformed a principal component analysis (PCA) to determinethe first 8 eigenvalues of principal motions of the peptide. Forthe fit and the determination of the average structure, weused all atomic coordinates of dialanine.67 We then projectedthe trajectory on the first and the 8th eigenvector obtainedfrom the PCA. An unprojected FEL has the advantage ofrevealing the hidden complexity of essential motions in proteindynamics.68–70

III. RESULTS AND DISCUSSION

A. Simulations of water

We begin by comparing simulations performed on SPC/Ewater with 3 different REMD methods: N PT , NV (p)T , andNVT (see Figure 1). We measured the average pressures andfind that the pressure in the NVT ensemble increases from40 atm at 298 K to 1106 atm at constant volume at 372 K.For NV (p)T , we observe that the average pressure remainsin the range from −2 to 45 atm, while the pressure in theN PT ensemble remains constant at 1 atm (see Figure 1(e)).In a comparison of the radial distribution functions (RDFs)between the water oxygens (OW-OW), we observe absolutequantitative agreement between the RDFs from the simulationin the N PT ensemble and the NV (p)T method at 3 differenttemperatures (see Figures 1(a)–1(c)). In contrast, the NV (p)T

and NPT results differ from the NVT for the replicas wherethe pressure departs significantly from normal, and agreementis only observed at 298 K. We also see that the quantitativedecrease in the first maximum and second maximum of theRDF in NV (p)T is in agreement with the N PT simulation,while the probabilities in the coordination-minimum betweenthe first, second, and third coordination shells are increasing atelevated temperatures. In a comparison of the first maximumprobability density as function of temperature, we notice thatthere is a very good agreement between N PT and NV (p)T ,while the probability density g(r) in NVT decreases moredrastically with increasing temperature (see Figure 1(d)).Expressed in free energies, all three simulation methods resultin a interaction energy of −1.14 kT at 298 K. At 372 K,the potential of mean force for N PT is 0.966 82 kT andfor NV (p)T is equal to 0.9656 kT, while we obtain a valueof 0.946 53 kT for NVT . We do note that this results in adifference of 0.02 kT at this elevated temperature of 372 K.

Our simulations of water in N PT , NV (p)T , and NVTare in agreement with prior simulations on SPC/E water.71

The temperature dependence of pressure and the hydrationmaximum observed in our simulations is in agreement withprior simulations as well.72 Our results also agree with theobservation of a stronger decrease in the probability maximumin the RDF in NVT than in N PT with increasing temperature,which depends on both pressure and temperature.73 We notethat the behavior of SPC water at high pressures in NVT leadsto a stronger desolvation in the first hydration shell, whichcould affect the hydrophobic effect and pressure induceddenaturation of proteins.14 Importantly, we note that the

FIG. 1. Results from simulations on properties of SPC/E water using the NV (p)T method, constant NPT , and constant NVT . (a) Radial distributionfunctions (RDFs) between water-oxygen, water-oxygen (OW-OW) from constant NPT , and NV (p)T simulations at 298 (a), 346 (b), and 372 K (c). (d) Thefirst maximum of the RDF plotted as function of replica temperature from 3 different REMD simulations in constant NPT , NV (p)T , and NVT . (e) Pressuresfrom the same simulations of SPC/E water as function of temperature. We observe that the maximum value in the RDF behaves equally in NPT and NV (p)T ,while the maximum from NVT -REMD deviates from the constant pressure methods. This is a result from the increasingly high pressure in NVT .

044903-6 Peter, Pivkin, and Shea J. Chem. Phys. 145, 044903 (2016)

solvation thermodynamics of water are influenced by thechosen ensemble. In NVT , the high pressure lowers theoverall interaction energy between water-oxygens. This canaffect the resulting free energy landscape of proteins andtRNA simulated in this ensemble.74,75 We conclude that forsimulations of water, our new method NV (p)T is in verygood agreement with results from simulations in the N PTensemble.

B. Simulations of dialanine peptide

We continue with a case study of dialanine (Ala2), wherewe compare the 3 REMD methods: N PT , NV (p)T , and

NVT . In Figure 2, we show the free energy landscapes (FEL)as function of the backbone dihedral angles Ψ and Φ at372 K (upper panels) and the pressure distributions at 298 K,348 K and 372 K (lower panels). We note that the FELs,projected onto the Φ and Ψ coordinates (see also Figures 3(d)and S2(a)–S2(c)62), are nearly identical at room temperature,where the pressure is on average the same in all methods.The pressure distribution is equal for all 3 methods at room-temperature, where the average pressure is at 1 atm in NVT,NPT, and NV(p)T (see Figure 2(g)). The pressure in the NVTsimulation rises to 712 atm at 348 K and reaches a value of1143 atm at 372 K (see Figures 2(g)–2(i)), while the averagepressure remains approximately 1 atm for NV (p)T and N PT

FIG. 2. Free energy landscapes of dialanine as function of backbone dihedral angles Φ and Ψ at 372 K sampled in (a) NPT -REMD, (b) NV (p)T -REMD, and(c) NVT -REMD. While we observe an approximately identical distribution for Ala2 in NPT and NV (p)T , we notice a deviation for NVT in the regions nearΦ=−80◦ - Ψ= 150◦ and Φ=−140◦ - Ψ= 150◦. ((d)-(f)) Free energy landscapes obtained from a projection of the trajectory on the first and the 8th eigenvectorobtained from PCA. (d) NPT-REMD, (e) NV(p)T-REMD, and (f) NVT-REMD. ((g)-(i)) Pressure distributions for all 3 different methods at 298 K, 348 K, and372 K. The regions in the free energy landscapes are labeled with small letters used in the main text.

044903-7 Peter, Pivkin, and Shea J. Chem. Phys. 145, 044903 (2016)

FIG. 3. Backbone dihedral angle Φ at 298 K ((a) and (d)), 348 K ((b) and (e)), and 372 K ((c) and (f)). In plots a-c the whole Φ-dependency is shown, while inplots ((d)-(f)) we show the same function in a smaller range. While there is an agreement between 3 methods (NPT , NV (p)T , NVT ) at 298 K, we observethat NVT deviates from the other 2 methods at elevated temperatures.

at elevated temperatures above room temperature T > 298 K.Due to an increase in pressure (and a decrease in the specificmolar volumes), the probabilities, i.e., free energies, getslightly shifted from the trans- to the cis-configuration inthe NVT ensemble for T > 298 K in comparison to the 2other cases, NV(p)T and NPT. For the sake of clarity, weonly show the distributions at 372 K, where the pressure nowdiffers between the NVT case and the NV(p)T and NPT cases,and the FEL differences are hence more pronounced. Here,we observe 5 different minima with a primary maximumat Φ = −140◦ - Ψ = 150◦ (Region (a)). We see a secondminimum in the probability distribution at Φ = −80◦ -Ψ = 150◦ (Region (b)), while we measure minor minimaat Φ = −140◦ - Ψ = −75◦ (Region (c)), Φ = −80◦ - Ψ = −75◦

(Region (e)), and Φ = 50◦ - Ψ = 150◦ (Region (d)) (seeFigures 2(a)–2(c)). We see that the minimum at Φ = −140◦

- Ψ = 150◦ (Region (a)) represents the trans-configuration,while the maximum at Φ = −80◦ - Ψ = 150◦ (Region (b))represents a tilted trans-configuration which is less stablethan the trans-configuration. The configuration at Φ = −140◦

- Ψ = −75◦ (Region (c)), Φ = −80◦ - Ψ = −75◦ (Region (e)),and Φ = 50◦ - Ψ = 150◦ (Region (d)) represents the cis-configuration which is less stable than the trans-configurationof Ala2. We observe that there is a good quantitative agreementin the probability distribution in the simulations at 372 K forN PT and NV (p)T (see Figures 2(a) and 2(b)). In contrastto the results from N PT and NV (p)T , we observe that theminima at Φ = −140◦ - Ψ = 150◦ (Region (a)) and Φ = −80◦

- Ψ = 150◦ (Region (b)) are shifted in their magnitude inNVT simulations. In addition the probability is slightlyincreased at Ψ = 100◦ and Ψ = −175◦ (Region (a)). In otherwords, the tilted trans-configurations of Ala2 have a higher

probability in NVT than in N PT and NV (p)T , where thetrans-configuration is more favorable. In terms of the finiteenergy difference, the trans-configuration of Ala2 is 0.1-0.5 kThigher for NVT than the same configuration in N PT andNV (p)T .

These differences are better visualized in Figure 3,where we show the probability as function of Φ at 3different temperatures. At 298 K, we observe 3 maximain the distribution at Φ = −140◦ - Φ = −80◦ and Φ = 50◦

(Regions (a), (b), and (d)). We see that the main maximumcan be found for the trans-configuration at Φ = −140◦,and we find that all 3 methods agree quantitatively in themagnitude of the probability and the shape of the function(see Figures 3(a) and 3(d)). At elevated temperatures of 348 Kand 372 K, we observe that the probability distribution ofΦ inNV (p)T agrees well with the N PT result. In agreement withour result shown in Figure 2, we find that the population of thetrans-configuration at Ψ = −140◦ decreases for NVT , whilethe tilted trans-configurations and the cis-configuration aremore populated than in N PT and NV (p)T (see Figures 3(b),3(c), 3(e), and 3(f)). As in the case of the simulations on water,we observe that the pressure increases in the NVT ensembleat elevated temperatures, which contribute to shifting of thepopulation of Ala2 towards tilted and cis-configurations.

In addition to the projected free energy landscapes, weanalyzed the unprojected distributions as function of thefirst and the 8th eigenvector obtained from PCA taken at372 K (see Figure 2(d)-2(f)). The first eigenvector correspondsto the slowest mode, which is equally distributed in all3 cases, while the fastest of 8 modes (corresponding tothe 8th eigenvector) is strongly influenced by the presenceof the barostat in NPT. Clearly, we observe that the

044903-8 Peter, Pivkin, and Shea J. Chem. Phys. 145, 044903 (2016)

unprojected landscapes are identical in both NVT-cases,with some different behaviors of the minima at eigenvector1at 0.6 nm and the intermediate region of eigenvector8 inthe range between 0.1 and 0.1 nm. This indicates thatthe NV(p)T-simulation is in good agreement with NPTin the static structural properties (Φ and Ψ-distribution),while NV(p)T has the dynamic characteristics of the NVTsimulation.

Ala2 has been investigated in a number of computationalstudies.17,76,77 In these studies, the methylated form Ace-Ala-2-NHMe was simulated. In the case of this methylated form,the cis-configuration at Φ = 50◦ - Ψ = 150◦ is energeticallyless favorable, and another FEL minimum appears for themethylated form at Φ = 50◦ - Ψ = −50◦, which appears tobe lowly populated in our simulations of pure dialanine.In agreement with simulations on the methylated form, weobserve that the trans- and the tilted trans-configurationsare the highly populated configurations, while the cis-configuration is less favorable. In our non-methylated formthe cis configuration occurs, which is not observed for themethylated form due to steric repulsion. In an experimentalstudy, it has been found that partial molar volume effectscontribute to the conformational changes of dialanine at higherpressures.8 This is in agreement with our NVT simulations,where a partial increase of the probability of the tilted trans-

and the cis-conformers is observed for replicas correspondingto high pressures. These configurations are not observed inN PT and NV (p)T simulations at normal pressure. Theseconfigurations have a slightly lower partial molar volume andare favored at elevated pressures, which is in agreement withthe experiment and simulations.8,9

C. Simulations of dialanine peptide on graphene

We now turn to a study of dialanine on a graphene surface.As mentioned in the Introduction, the presence of interfacescan be problematic in NVT and NPT implementations ofREMD. At room temperature, the pressures are equal ineach simulation in the NVT ensemble and NV(p)T atapproximately 1 atm, while the pressure in the NPT ensembleis approximately at 250 atm (see Figure S4(i)62). At thistemperature, the free energy landscapes of dialanine asfunction of Φ and Ψ are identical (see Figures S2(d)–S2(f)62).In Figure 4, we show the conformational distributions at372 K, where the pressure in NVT reaches a value of1264 atm, and the differences in the probability distributionsbecome the most significant. In the free energy landscapes asfunction of Φ and Ψ, we observe that dialanine populates 3minima at Φ − 60◦ - Ψ − 110◦, Ψ − 70◦ (Regions (b) and(d)) and Φ − 120◦ - Ψ − 70◦ (Regions (b), (d), and (e)),

FIG. 4. ((a)-(c)) Free energy landscapes of dialanine on graphene as a function of backbone dihedral angles Φ and Ψ at 372 K sampled in (a) NPT -REMD,(b) NV (p)T -REMD, and (c) NVT -REMD. The regions in the free energy landscapes are labeled with small letters used in the main text. ((d)-(f)) Free energylandscapes obtained from a projection of the trajectory on the first and the 8th eigenvector obtained from PCA. (d) NPT-REMD, (e) NV(p)T-REMD, and (f)NVT-REMD.

044903-9 Peter, Pivkin, and Shea J. Chem. Phys. 145, 044903 (2016)

in contrast to dialanine in the bulk, where we find oneminimum at Φ = −140◦ - Ψ = −75◦ (see Figure 4(a)-4(c),Region (c)). The minimum at Φ − 120◦ - Ψ − 70◦ (Region(b)) is slightly higher populated in NV(p)T and NVT with acorresponding free energy value of −4 kBT , while dialanineaccesses this minimum with approximately −3.5 kBT in theNPT simulation. We observe a slight difference between NPTand the other 2 simulations in the population of the minimumcis-configuration at Φ = 50◦ - Ψ = 150◦ (Region (c)), wherethe distribution ranges fromΦ = 0 toΦ = 60◦, which is not thecase for NVT and NV(p)T, where the distributions are definedwithin Φ = 20◦ - Φ = 55◦ (Region (d)). The main minimum atΦ = −150◦ - Ψ = 150◦ (Regions (a) and (a′)) representing thetrans-configuration is populated equally in all 3 simulations.In the analysis of the free energy landscapes as function ofΨ and the distance between graphene and dialanine, we seethat dialanine populates the regions between Ψ = −135◦ -Ψ = −45◦ (Region (b)) and Ψ = 80◦ - Ψ = 180◦ (Region (a))with a free energy ranging from −4.5 to −6 kBT adsorbedon graphene in all 3 simulations (see Figures S3(a)–S3(c)62).However, configurations of dialanine away from the surfaceare populated differently at d ≥ 0.4 nm (Regions (e) and(f)). For NV(p)T, we find that the distribution of Ψ with afree energy value lower than 3.5 kBT ranges from 110◦ to155◦ (Region (e)) for NV(p)T at d ≥ 0.4 nm, which is alsothe case at Ψ = −80◦ with d ≥ 0.4 nm (Region (f)), wherethe distribution has a width of approximately 30◦. This isin contrast to NVT and NPT, where the distribution of Ψ iswider by approximately 10◦ to 15◦. We find the same behaviorfor Φ as function of the distance, where the distribution ofis broader in the case of NVT and NPT (Region (e)) (seeFigures S3(d)–S3(f)62).

In addition to the projected static Φ-Ψ distributions, weanalyzed the unprojected dynamic distributions as functionof the first and the 8th eigenvector obtained from PCA,taken from the same trajectories (see Figures 4(d)-4(f)). Asin the bulk dialanine case, we observe a strong influenceof the barostat in the NPT simulation. The unprojectedlandscapes are nearly identical in both NVT and NV(p)Timplementations, with some difference observed for theminima at eigenvector1 at 0.6 nm and the intermediate regionof eigenvector8 in the range between 0.1 and 0.1 nm. Hencewhile the NV(p)T-simulation agrees well with NPT in thestatic structural properties (Φ and Ψ-distribution), NV(p)Thas approximately similar dynamic characteristics than theNVT simulation.

In the analysis of the water density as function ofthe box diameter, we observe water layering on grapheneat a distance of 0.2 away from the graphene plane (seeFigures S4(a)–S4(c)62). We find that the density maximumof this water layer decreases with temperature in all 3simulations, while the variability of the density maximumwith temperature is the lowest for the NVT simulation,where the density value varies from 3243 kg/m3 at 298 K to2974 kg/m3 at 372 K. The decrease in density in NPT andNV(p)T has approximately the same slope, with a decreasein density from 3365 kg/m3 to 2572 kg/m3 for NPT and3279 kg/m3 to 2406 kg/m3 for NV(p)T (see Figure S4(d)62).The peptide is mainly adsorbed on the graphene surface, as

we see in the density profiles of dialanine averaged over thetrajectory (see Figures S4(f)–S4(h)62). The location of thedensity maximum of the peptide in z is at the position of thedensity maximum of water. Finally, we compared the surfacetension as a function of temperature and find that NPT andNV(p)T show a similar decay of this quantity with increasingtemperature. In contrast, the surface tension increases in NVTat elevated temperature (see Figure S4(e)62). We measuredthe average pressures for dialanine on graphene as a functionof temperature and found an increase in the pressure fromapproximately 1 atm at 298 K to 1264 atm in the NVT-ensemble at 372 K. In the NV(p)T simulation, the pressureranges between −50 and 50 atm in the whole temperaturespace (see Figure S4(i)62). In the NPT-ensemble, a pressureof 1 atm can only be achieved using a high compressibilityvalue in the xy-direction, which can lead to artifacts dueto a high semi-isotropic volume fluctuation. We note thatpressure coupling in simulations on surfaces is problematic,if constraints are applied on the positions of the surface, andthe NV(p)T technique has clear advantages in this case. TheNV(p)T technique reaches pressures close to 1 atm and avoidsartifacts caused by constraints on the surface or semi-isotropicvolume fluctuations.

The effect of water layering and surface hydrophobicityon folding of TrpZip2, GB1, and TrpCage near grapheneand graphite has been investigated recently.11,59 The effectof water layering and adsorption of the peptide in the highdensity layer is in agreement with these simulations. In atheoretical work, Yancey et al. have shown that the freeenergy of adsorption for a peptide to a hydrophobic surfaceis dramatically influenced by the pressure.58 In addition, wefound that the correct density fluctuations are important tocorrectly sample the configuration space of peptide systemsadsorbed on surfaces. This pressure dependence of peptideadsorption has been also demonstrated in chromatographyexperiments by Szabelski et al.78

IV. CONCLUSION

In this paper, we presented a new method for the REMDsimulation in the NVT ensemble at normal pressure forall replicas. We developed an expression which considersthe volume perturbation for each exchange and maintainsdetailed balance. Using this new exchange criterion, wevalidated our new methodology on SPC/E water and ondialanine. We compared our new method with N PT-REMDcalculations and found it to be in good agreement withrespect to structural properties over all temperatures. Inconventional NVT-REMD, the structural properties of waterand the peptide system are affected by high pressures,and the population densities are shifted. In the unprojectedfree energy analysis, we find that the dynamical behaviorof the NV(p)T simulation is nearly identical to the NVTensemble, while dynamical modes of dialanine are affectedby the barostat in NPT. Considering a system of a peptidein the presence of a graphene surface, we see that ourmethod avoids artifacts caused by barostats and constraintsapplied on the surface atoms that can arise in a conventional

044903-10 Peter, Pivkin, and Shea J. Chem. Phys. 145, 044903 (2016)

NPT implementation. We emphasize that the importance ofpressure-effects increases with protein size. Although the rateof convergence in REMD NV (p)T is slightly slower than inREMD NVT and NPT, the REMD NV (p)T method has thebenefit of being able to be applied in situations where normalpressure is required at all temperatures, and barostats difficultto implement.

ACKNOWLEDGMENTS

We acknowledge Yuji Sugita for fruitful discussionsand critical reading of the manuscript. E.K.P. and I.V.P.acknowledge support by a grant from the Swiss Platformfor Advanced Scientific Computing. Simulations were carriedout at Swiss National Supercomputer Center (CSCS) underthe Project Nos. u4 and s583. J.E.S. acknowledges support ofthe NSF Grant No. MCB-1158577.

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