metadynamics, · 2014-05-30 · metadynamics, • the,bare,bones,of,metadynamics, –...

Metadynamics

Day 2, Lecture 3 James Dama

Metadynamics

•  The bare bones of metadynamics – Bias away from previously visited configuraAons –  In a reduced space of collecAve variables – At a sequenAally decreasing rate of bias

•  Examples from the literature •  Key thought experiments to build intuiAon – Bias size, shape, and rate – Good and bad collecAve variables

The Essence of Metadynamics Bias adapAvely to escape metastable states Huber, Torda, and Van Gunsteren. Local elevaAon : A method for improving

the searching properAes of molecular dynamics simulaAon. 1994

•  Metastable states are a pervasive feature of real free energy surfaces

•  TransiAons between these states are rare and difficult to observe in simulaAon

The Essence of Metadynamics Bias adapAvely to escape metastable states Huber, Torda, and Van Gunsteren. Local elevaAon : A method for improving

the searching properAes of molecular dynamics simulaAon. 1994

•  Basins are unknown a priori –  Avoid previously sampled conformaAons

–  Add energy ‘hills’ centered on each sample so far

•  Focus on model-‐independent exploraAon

Adding Hills

•  A real equaAon is

•  The idealized equaAon is

•  Makes escaping square wells linear in depth rather than exponenAal in depth

BRIEF ARTICLE

THE AUTHOR

V̇ (s, t) ⇠ !e�V (s,t)/�T e�(F (s)+V (s,t))/T(1)

�V (s, t) = he�(s��(t))2/2�s2(2)

�V (s, t) = he�V (�(t),t)/�T e�(s��(t))2/2�s2(3)

V̇ (s, t) =

Zds0K(s, s0, V (t))�(s0, t)(4)

K(s, s0, V (t)) = he�(s�s0)2/2�s2(5)

K(s, s0, V (t)) = he�V (s0,t)/�T e�(s�s0)2/2�s2(6)

K(s, s0, V (t)) = he�(s�s0)2/2(�s(s0))2(7)

h/�s(8)

�s(9)

�T ⇠ �G‡(10)

V̇ (s, t) = !(t)�(s� �(t))(11)

V̇ (s, t) = !e�V ‡(t)/�T �(s� �(t))(12)

V̇ (s, t) ⇠ !e�V ‡(t)/�T e�(F (s)+V (s,t))/T(13)

e�(F (s)+V (s,1))/T ⇠ C(14)

F (s) = �V (s,1) + C(15)

V̇ (s, t) = !e�V (s,t)/�T (s)�(s� �(t))(16)

V̇ (s, t) ⇠ !e�V (s,t)/�T (s)e�(F (s)+V (s,t))/T(17)

F (s) = �(T/�T (s) + 1)V (s,1)(18)

V̇ (s, t) = !e�f(s,V (s,t))/T �(s� �(t))(19)

(20)

1

2 JAMES DAMA

bias V (s, t). Given a metadynamics-sampled trajectory in collective variables �(t), thehistory-dependent potential is defined as an approximation of

V̇ (s, t) = !�(s� �(t))(1)

where ! is an adjustable rate parameter and � is a delta function on the collective variabledomain. For well-tempered metadynamics, the idealized rule is

V̇ (s, t) = !e

�V (s,t)/�T�(s� �(t))(2)

where �T is a second adjustable parameter referred to as a tempering parameter. Clearly,metadynamics is the limit of well-tempered metadynamics as �T ! 1. This history-dependent bias serves to flatten the sampled distribution of collective variables, pushingfuture sampling away from each point the more often it has already been visited. Inpractice, these rules are approximated by discretization in time and mollification in space,so that the bias is updated at only a discrete set of times and is updated using Gaussianbumps rather than delta functions. Additionally, the trajectory �(t) may be a multi-trajectory composed of the histories of multiple walkers, and in this case � should beunderstood as a sum of delta functions, one per individual walker. Later work (Branduardi,Bussi, and Parrinello, 2012) has introduced functional complexity in !, ! ! !(s), and theidea of selectively tempered metadynamics was to introduce functional complexity in �T ,�T ! �T (s). For the sake of generality, and because it will not increase the complexityof the analysis that follows, I will also consider general bias update rules of the form

V̇ (s, t) = !e

�f(s,V (s,t))/T�(s� �(t))(3)

where f(s, V (s, t)) is a function dependent on both the s-point and the value of the potentialat that point and ! is a constant with units of energy per unit time just as in metadynamicswith non-adaptive Gaussians. Note that this form describes geometry-adaptive Gaussiansonly; when the Gaussians are adapted on-the-fly this form does not hold precisely.

Finally, in addition to local tempering rules, it is possible to imagine tempering rulesin which the entire bias or regions of the bias are used to calculate updates at each singlepoint. For these, one considers update rules of the form

V̇ (s, t) = !e

�f(s,V (t))/T�(s� �(t))(4)

where f(s, V (t)) is a functional of the bias function V (t) at time t instead of a functionof the bias at a point V (s, t) at time t. For this, the same sorts of error bounds canbe formulated, but the evolution of the relative biasing rates towards uniformity changes.These nonlocal rules will be examined in these notes separately from the local temperingrules in a section following the investigation of the local rules.

2.2. Mollified metadynamics bias equations. However, the idealized metadynamicsbiasing rate equation above is not realizable in simulation, so I will also consider metady-namics updates that can be written in the general form

V̇ (s, t) =

Zds

0K(s, s0, V (t))�(s0, t)(5)

The Essence of Metadynamics

•  Local elevaAon was wasteful –  Sampling every state uniformly is expensive

•  Self-‐avoidance changes random walks less and less with more and more variables

•  Focusing on interesAng features is more important with more variables

Bias only specific collecAve variables Laio and Parrinello. Escaping free-‐energy minima. 2002

The Essence of Metadynamics

•  ReacAons are mulAscale – Monitor only state-‐determining variables

–  Leave fast variables alone •  Focus on accomplishing specific exploratory goals

Bias only specific collecAve variables Laio and Parrinello. Escaping free-‐energy minima. 2002

CollecAve Variables

•  Any funcAon of any number of fine-‐grained variables – PosiAon, distance, angle, dihedral – CoordinaAon number, density, crystalline order – Helicity, contact map, NMR spectrum – Strings of configuraAons in another CV space

•  Whatever you would like to explore

The Essence of Metadynamics AdapAvely tune the biasing rule

Barducci, Bussi, and Parrinello. Well-‐tempered metadynamics: A smoothly converging and tunable free-‐energy method. 2008

•  Original metadynamics had limited accuracy –  Errors saturated and never fully disappeared

•  Residual inaccuracy was proporAonal to the rate of hill addiAon

all the underlying profile is filled. This required!500 Gaussiansfor all the profiles reported in Figure 1.We verified that these properties also hold in higher dimen-

sionalities, for virtually any value of w and !G, and for any valueof "s significantly smaller than the system size, as we willspecify in more detail in the following.B. Dependence of the Error on the Metadynamics Pa-

rameters. Since the value of !j does not depend on F(s), weconsider in more detail the flat profile (Figure 1A). In ddimension, we use eq 11 with F(s) ) 0 and reflecting boundaryconditions for |s| ) S/2, and we compute the dependence ofthe error on the metadynamics parameters w, !G and "s and on#, D, and S, which characterize the physical system. We repeatedseveral metadynamics for different values of the parameters and

for d ) 1, d ) 2, and d ) 3. If only one parameter is varied ata time, the dependence of !j on that parameter can beinvestigated. In this manner, we found empirically that !j isapproximately proportional (independently of the dimensional-ity) to the square root of the system size S, of the Gaussianwidth "s and the Gaussian height w, while it is approximatelyproportional to the inverse square root of #, D, and !G. Theseobservations are summarized in Figure 3, in which we plot thelogarithm of !j vs the logarithm of (S"s/D!G)(w/#) when d ) 1and d) 2. Different color codes correspond to different physicalconditions (i.e., different D, #, or S), while dots of the samecolor correspond to different metadynamics parameters. w isvaried in the range between 0.04 and 4, "s between 0.05 and0.4 and !G between 5 and 5000. The continuous line correspondsto the equation

where C(d ) 1) ! 0.5 and C(d ) 2) ! 0.3 in the samelogarithmic scale, and represents a lower bound to the error.For a given value of (S"s/D!G)(w/#), the error obtained withany parameter in the range considered leads to errors at most50% higher than the value given by eq 12.The dependence of the error on the simulation parameters

becomes more transparent if !j is expressed as an explicitfunction of the total simulation time. Consider in fact a freeenergy profile F(s) that has to be filled with Gaussians up to agiven level Fmax (e.g., the free energy of the highest saddle pointin F(s)). The total computational time needed to fill this profilecan be estimated as the ratio between the volume that has to be

Figure 1. Metadynamics results for four different free energyprofiles: (A) F(s) ) - 4; (B) F(s) ) -5 exp(-(s/1.75)2); (C) F(s) )-5 exp(-(s - 2/0.75)2) - 10 exp(-(s + 2/0.75)2); (D) F(s) ) -5exp(-(s - 2/0.75)2) - 4 exp(-(s/0.75)2) - 7 exp(-(s + 2/0.75)2).The average 〈F(s) - FG(s, t)〉 computed over 1000 independenttrajectories is represented as a dashed line, with the error bar given byeq 7.

Figure 2. Average error (eq 8) as a function of the number ofGaussians for the four F(s) of Figure 1. The bump in the error observedfor the functional forms C and D is due to the fact that in a long partof the metadynamics one of the free energy wells is already completelyfilled, while the other is being filled. The error is measured usingdefinition 8 on both the wells, but the filling level is not the same untilthe full profile is filled.

Figure 3. Error as a function of the metadynamics parameters w, !Gand "s and of #, D and S in d ) 1 (upper panel) and d ) 2 (lowerpanel). The continuous and the dashed lines correspond to eq 12 for d) 1 and d ) 2, respectively.

!j ) C(d )"S"sD!G

w# (12)

Assessing the Accuracy of Metadynamics J. Phys. Chem. B, Vol. 109, No. 14, 2005 6717

Laio, Rodriguez-‐Fortea, Gervasio, Ceccarelli and Parrinello. Assessing the accuracy of metadynamics. 2005

The Essence of Metadynamics AdapAvely tune the biasing rule

Barducci, Bussi, and Parrinello. Well-‐tempered metadynamics: A smoothly converging and tunable free-‐energy method. 2008

•  Slowing the bias rate increases final accuracy –  Bias progressively more slowly

–  Tune using only intrinsic state variables

•  Focus on model-‐independent accuracy

Tempering

•  A real equaAon is

•  The idealized equaAon is

•  Error disappears instead of saturaAng •  The bias increases logarithmically rather than linearly; escape is exponenAal in depth again –  the exponent decreases by a tunable scalar factor

BRIEF ARTICLE

THE AUTHOR


�V (s, t) = he�(s��(t))2/2�s2(2)

�V (s, t) = he�V (�(t),t)/�T e�(s��(t))2/2�s2(3)

V̇ (s, t) =

Zds0K(s, s0, V (t))�(s0, t)(4)

K(s, s0, V (t)) = he�(s�s0)2/2�s2(5)


K(s, s0, V (t)) = he�(s�s0)2/2(�s(s0))2(7)

h/�s(8)

�s(9)

�T ⇠ �G‡(10)

V̇ (s, t) = !(t)�(s� �(t))(11)

V̇ (s, t) = !e�V ‡(t)/�T �(s� �(t))(12)

V̇ (s, t) ⇠ !e�V ‡(t)/�T e�(F (s)+V (s,t))/T(13)

e�(F (s)+V (s,1))/T ⇠ C(14)

F (s) = �V (s,1) + C(15)

V̇ (s, t) = !e�V (s,t)/�T (s)�(s� �(t))(16)


F (s) = �(T/�T (s) + 1)V (s,1)(18)

V̇ (s, t) = !e�f(s,V (s,t))/T �(s� �(t))(19)

(20)

1

2 JAMES DAMA

bias V (s, t). Given a metadynamics-sampled trajectory in collective variables �(t), thehistory-dependent potential is defined as an approximation of

V̇ (s, t) = !�(s� �(t))(1)

where ! is an adjustable rate parameter and � is a delta function on the collective variabledomain. For well-tempered metadynamics, the idealized rule is

V̇ (s, t) = !e

�V (s,t)/�T�(s� �(t))(2)

where �T is a second adjustable parameter referred to as a tempering parameter. Clearly,metadynamics is the limit of well-tempered metadynamics as �T ! 1. This history-dependent bias serves to flatten the sampled distribution of collective variables, pushingfuture sampling away from each point the more often it has already been visited. Inpractice, these rules are approximated by discretization in time and mollification in space,so that the bias is updated at only a discrete set of times and is updated using Gaussianbumps rather than delta functions. Additionally, the trajectory �(t) may be a multi-trajectory composed of the histories of multiple walkers, and in this case � should beunderstood as a sum of delta functions, one per individual walker. Later work (Branduardi,Bussi, and Parrinello, 2012) has introduced functional complexity in !, ! ! !(s), and theidea of selectively tempered metadynamics was to introduce functional complexity in �T ,�T ! �T (s). For the sake of generality, and because it will not increase the complexityof the analysis that follows, I will also consider general bias update rules of the form

V̇ (s, t) = !e

�f(s,V (s,t))/T�(s� �(t))(3)

where f(s, V (s, t)) is a function dependent on both the s-point and the value of the potentialat that point and ! is a constant with units of energy per unit time just as in metadynamicswith non-adaptive Gaussians. Note that this form describes geometry-adaptive Gaussiansonly; when the Gaussians are adapted on-the-fly this form does not hold precisely.

Finally, in addition to local tempering rules, it is possible to imagine tempering rulesin which the entire bias or regions of the bias are used to calculate updates at each singlepoint. For these, one considers update rules of the form

V̇ (s, t) = !e

�f(s,V (t))/T�(s� �(t))(4)

where f(s, V (t)) is a functional of the bias function V (t) at time t instead of a functionof the bias at a point V (s, t) at time t. For this, the same sorts of error bounds canbe formulated, but the evolution of the relative biasing rates towards uniformity changes.These nonlocal rules will be examined in these notes separately from the local temperingrules in a section following the investigation of the local rules.

2.2. Mollified metadynamics bias equations. However, the idealized metadynamicsbiasing rate equation above is not realizable in simulation, so I will also consider metady-namics updates that can be written in the general form

V̇ (s, t) =

Zds

0K(s, s0, V (t))�(s0, t)(5)

Examples From the Literature

•  Chemical reacAon •  Phase transiAon •  Interfacial chemistry •  Protein funcAon •  Protein folding

Chemical Weapon RemediaAon

This journal is c the Owner Societies 2012 Phys. Chem. Chem. Phys., 2012, 14, 3316–3322 3319

we employed the IT-NEB approach to determine the S-centeredhypochlorite reaction PEBs for VX and R-VX. Specifically, thereplicas used in the IT-NEB calculations were taken directlyfrom the previously described metadynamics simulations. TheS-centered reaction PEBs obtained from the NEB simulationsfor VX and R-VX are shown in Fig. 5, and are found to beDH E 15 and DH E 11 kcal/mol, respectively. As a final note,reactions involving either of the two carbon atoms along thebackbone of the V-type nerve agent molecules were found to beenergetically prohibitive, where the reaction PEBs were foundto be greater than 40 kcal/mol (see Table 1 for further details).

We next considered nucleophilic substitution reactions involvinghypochlorite at the phosphorus atomic center (P-centered).The PEBs for reaction of hypochlorite with VX and R-VX, aswell as the protonated VXH+ and R-VXH+ counterparts weredetermined. Reactions of protonated VX and R-VXwere deemedto be important for two reasons: (1) experiments have shown thatthe solubility of VX is increased under acidic conditions, whereincreased solubility is hypothesized to be due to protonation ofnitrogen,16,17,19,20 and (2) during equilibration, many of the

simulations showed spontaneous protonation at the nitrogencenter of VX and R-VX under ‘‘neutral’’ pH conditions.Our simulations also show spontaneous protonation of thehypochlorite ion (the pKa of hypochlorite is reported asbeingB7.5,41 consistent with this observation). P-centered nucleo-philic substitution reactions were found to proceed via an SN2type mechanism, where tetra-coordinated phosphorus forms a

Fig. 2 Free energy profile as a function of the CV as determined from

metadynamics simulations of the condensed-phase reaction of VX and

R-VX with ClO!. Black (grey) lines represent a CV defined by the

distance between the hypochlorite oxygen and the nitrogen (sulfur)

atomic center of VX (dotted) and R-VX (solid); top panel. The lower

panel is a snapshot of the simulation near the reaction barrier crossing

for the VX/nitrogen reaction (the point labeled ‘‘1’’ in the upper

panel). As is apparent, the structure at the crossing consists of a single

water molecule reacting with the incoming hypochlorite anion, which

forms an OH! anion before reacting at the nitrogen atomic center.

Fig. 3 Illustration of the transition state structure of ClO! reaction

at the sulfur atomic center for R-VX. The transition state is mediated

by the presence of a single water molecule (the single water molecule is

part of the transition state structure found for the S-centered reaction

with ClO!). The water molecule as shown facilitates the e!ective

lowering of the sulfur/hypochlorite reaction barrier.

Fig. 4 Solvent-mediated transition state structure of ClO! reaction

to the sulfur atomic center for R-VX (right). The presence of a

sterically hindered transition state structure is illustrated for VX (left).

Fig. 5 Total energy along the IT-NEB optimized path for the reaction

of OCl!anion with the sulfur atomic center of VX (solid) and R-VX

(dotted). Converged elastic bands with 32 movable replicas are shown.

Note that the VX peak only appears to be wider than the R-VX peak;

the x-axis (replica number) is not uniformly spaced along the reaction

coordinate.

Dow

nloa

ded

by U

nive

rsity

of C

hica

go o

n 19

/04/

2013

18:

42:0

3.

Publ

ished

on

12 Ja

nuar

y 20

12 o

n ht

tp://

pubs

.rsc.

org

| doi

:10.

1039

/C2C

P231

26C

View Article Online

This journal is c the Owner Societies 2012 Phys. Chem. Chem. Phys., 2012, 14, 3316–3322 3319

we employed the IT-NEB approach to determine the S-centeredhypochlorite reaction PEBs for VX and R-VX. Specifically, thereplicas used in the IT-NEB calculations were taken directlyfrom the previously described metadynamics simulations. TheS-centered reaction PEBs obtained from the NEB simulationsfor VX and R-VX are shown in Fig. 5, and are found to beDH E 15 and DH E 11 kcal/mol, respectively. As a final note,reactions involving either of the two carbon atoms along thebackbone of the V-type nerve agent molecules were found to beenergetically prohibitive, where the reaction PEBs were foundto be greater than 40 kcal/mol (see Table 1 for further details).

We next considered nucleophilic substitution reactions involvinghypochlorite at the phosphorus atomic center (P-centered).The PEBs for reaction of hypochlorite with VX and R-VX, aswell as the protonated VXH+ and R-VXH+ counterparts weredetermined. Reactions of protonated VX and R-VXwere deemedto be important for two reasons: (1) experiments have shown thatthe solubility of VX is increased under acidic conditions, whereincreased solubility is hypothesized to be due to protonation ofnitrogen,16,17,19,20 and (2) during equilibration, many of the

simulations showed spontaneous protonation at the nitrogencenter of VX and R-VX under ‘‘neutral’’ pH conditions.Our simulations also show spontaneous protonation of thehypochlorite ion (the pKa of hypochlorite is reported asbeingB7.5,41 consistent with this observation). P-centered nucleo-philic substitution reactions were found to proceed via an SN2type mechanism, where tetra-coordinated phosphorus forms a

Fig. 2 Free energy profile as a function of the CV as determined from

metadynamics simulations of the condensed-phase reaction of VX and

R-VX with ClO!. Black (grey) lines represent a CV defined by the

distance between the hypochlorite oxygen and the nitrogen (sulfur)

atomic center of VX (dotted) and R-VX (solid); top panel. The lower

panel is a snapshot of the simulation near the reaction barrier crossing

for the VX/nitrogen reaction (the point labeled ‘‘1’’ in the upper

panel). As is apparent, the structure at the crossing consists of a single

water molecule reacting with the incoming hypochlorite anion, which

forms an OH! anion before reacting at the nitrogen atomic center.

Fig. 3 Illustration of the transition state structure of ClO! reaction

at the sulfur atomic center for R-VX. The transition state is mediated

by the presence of a single water molecule (the single water molecule is

part of the transition state structure found for the S-centered reaction

with ClO!). The water molecule as shown facilitates the e!ective

lowering of the sulfur/hypochlorite reaction barrier.

Fig. 4 Solvent-mediated transition state structure of ClO! reaction

to the sulfur atomic center for R-VX (right). The presence of a

sterically hindered transition state structure is illustrated for VX (left).

Fig. 5 Total energy along the IT-NEB optimized path for the reaction

of OCl!anion with the sulfur atomic center of VX (solid) and R-VX

(dotted). Converged elastic bands with 32 movable replicas are shown.

Note that the VX peak only appears to be wider than the R-VX peak;

the x-axis (replica number) is not uniformly spaced along the reaction

coordinate.

Dow

nloa

ded

by U

nive

rsity

of C

hica

go o

n 19

/04/

2013

18:

42:0

3.

Publ

ished

on

12 Ja

nuar

y 20

12 o

n ht

tp://

pubs

.rsc.

org

| doi

:10.

1039

/C2C

P231

26C

View Article Online

Gee, Kuo, Chinn, and Raber. First-‐principles molecular dynamics simulaAons of condensed-‐phase V-‐type nerve agent reacAon pathways and energy barriers. 2012

3318 Phys. Chem. Chem. Phys., 2012, 14, 3316–3322 This journal is c the Owner Societies 2012

(herein referred to as the ‘‘core-ensemble’’). The V-type nerveagents studied included either MeP(O)(OR)(SCH2CH2NR02),VX (R = C2H5; R

0 = i-C3H7) or its isomeric analog R-VX(‘‘Russian VX,’’ R = i-C4H9; R0 = C2H5) (see Fig. 1).Protonated VX and R-VX counterparts were also studied.The protonated species (VXH+ or R-VXH+) were studiedbecause of the propensity for either of these species to protonate(at the nitrogen atom) in neutral or basic solutions (pKa of VX=8.6 @ 25 1C).35 Each of the condensed-phase systems studied herewere initially equilibrated with classical molecular dynamicsutilizing the COMPASS force field36 before any FPMD simulationswere performed. Following the ‘‘classical’’ equilibration, eachsystem studied was further equilibrated in FPMD simulationswith CP2K for 5 ps before any metadynamics simulations wereperformed to study the condensed-phase chemical reactions.

a. Oxidation of V-type nerve agents with hypochlorite anion

All oxidation reactions of V-type nerve agents studied herewere performed by adding a single hypochlorite anion (ClO!)and sodium cation (Na+) to the ensemble described in sectionIII. The CV for the metadynamics simulations used to studythe oxidation reactions of VX (or VXH+) and R-VX(or R-VXH+) with the hypochlorite ion (ClO!) was chosenas the distance between the oxygen atom of the hypochloriteanion and the atom center of interest in the V-type nerve agentmolecule (e.g., phosphorus, sulfur, carbon, etc.). In addition,several di!erent metadynamics parameters (hill height andwidth) were tested, where the final parameters were chosenas:H= 1.0 " 10!3 Hartree, o= 0.1 Bohr, and where hills,H,were added every 15 MD steps (time step = 0.5 fs).

Many of the reactions assessed using metadynamics simulationswere also studied using the IT-NEB approach to improve ourconfidence in obtaining appropriate reaction energy barriers in thetransition-state region.37 Replicas (atomic coordinates) used in theIT-NEB simulations were taken directly from the metadynamicstrajectories along the identified reaction coordinate of the reaction.Specifically, 32 replicas were extracted from the resultantmetadynamics trajectories; 16 replicas each from both sidesof the reaction potential energy barrier.

b. Hydrolysis of VX in basic solutions

FPMD condensed-phase metadynamics simulations of basehydrolysis were performed by adding a single hydroxide anion(OH!) to the core-ensemble of VX described in section III. Herethe CVwas chosen as the bond distance between sulfur and carbon(S–C) atoms in the VX molecule. The metadynamics parameters(hill height, H; width, o, etc.) were chosen as: H = 5.0 " 10!4

Hartree, o = 0.1 Bohr, and where hills, H, were added every30 MD steps (15 fs).38

c. Autocatalytic hydrolysis of VX

It has been proposed that VX or R-VXmay ‘‘autocatalytically’’hydrolyze.16 To study the plausibility of such a mechanism,

we performed FPMD simulations where the free-energy surfaceis sampled via metadynamics on a gas-phase cluster containinga single water and VX molecule, similar to that shown inFig. 8a, as well as a fully condensed-phase simulation usingthe core ensemble described in section III, above. Two CVswere chosen to study the autocatalytic hydrolysis of VX; theCVs are defined as the distance between the phosphorus (CV1)or the nitrogen (CV2) atomic centers, and selected oxygen of awater molecule proximal to the VX molecule (see Fig. 8a).Parameters for the biasing potential (hill height,H; width, o, etc.)were the same as those defined in section IIIa.

IV. Results

a. VX and R-VX hypochlorite decontamination reactions

The main focus of this paper was to demonstrate the feasibility ofemploying FPMD simulation approaches to study condensed-phase decontamination reactions of V-type nerve agents viahypochlorite (ClO!) oxidation. Specifically, condensed-phaseFPMD metadynamics and IT-NEB simulations of VX(or VXH+) and R-VX (or R-VXH+) with the hypochlorite ion(ClO!) were performed to ultimately identify decontaminationdegradation pathways and oxidation reaction energy barriers.Furthermore, when possible, the results of our simulations werecompared to experiment to validate the computational approach.We began by looking at hypochlorite reaction with VX or

R-VX at both the nitrogen (N-centered) or sulfur (S-centered)atomic centers. Fig. 2 shows the free energy as a function ofdistance between the hypochlorite oxygen and nitrogen orsulfur distance as determined by metadynamics simulations(see section IIIa). The resultant N-centered reaction potentialenergy barriers (PEBs) for the hypochlorite reaction for bothVX and R-VX were found to be similar, 11.5 and 10 kcal/mol,respectively. However, the S-centered hypochlorite reaction PEBsfor VX and R-VX were found to di!er by 410–15 kcal/mol,favoring the R-VX/hypochlorite reaction; compare the dashedand solid gray curves of Fig. 2. Upon securitization of theS-centered ‘‘transition-state structure,’’ it was found that a singlewater molecule mediates the reaction of the R-VX molecule(see Fig. 3); no such structure was identified for the S-centeredVX reaction. Two hypotheses are put forward to understandthis finding: (1) the development of a solvent mediated transition-state structure for VX is sterically hindered by the presence of themore bulky isopropyl nitrogen moieties of VX versus the lessbulky ethyl moieties of R-VX, or (2) the simulations are none-rgodic (i.e., simulations were not run long enough to allow fordevelopment of a ‘‘single water molecule mediated transition statestructure’’ in VX (again in part due to the presence of the morebulky isopropyl groups in VX).39 To this end, additional S-centeredVX/hypochlorite simulations were performed, where the additionalsampling eventually resulted in a water mediated transition-statestructure (see right panel of Fig. 4).40 The additional VX samplingrequired to identify the water mediated transition-state structurelends credence to the hypothesis that the more prominent sterice!ect found in VX ultimately impedes the development of solventmediation e!ects important in chemical reactions.Starting from the water mediated transition-state structures

yielded from the above S-centered metadynamic simulations,

Fig. 1 Chemical structure of VX (left) and R-VX (right).

Dow

nloa

ded

by U

nive

rsity

of C

hica

go o

n 19

/04/

2013

18:

42:0

3.

Publ

ished

on

12 Ja

nuar

y 20

12 o

n ht

tp://

pubs

.rsc.

org

| doi

:10.

1039

/C2C

P231

26C

View Article Online

3318 Phys. Chem. Chem. Phys., 2012, 14, 3316–3322 This journal is c the Owner Societies 2012

(herein referred to as the ‘‘core-ensemble’’). The V-type nerveagents studied included either MeP(O)(OR)(SCH2CH2NR02),VX (R = C2H5; R

0 = i-C3H7) or its isomeric analog R-VX(‘‘Russian VX,’’ R = i-C4H9; R0 = C2H5) (see Fig. 1).Protonated VX and R-VX counterparts were also studied.The protonated species (VXH+ or R-VXH+) were studiedbecause of the propensity for either of these species to protonate(at the nitrogen atom) in neutral or basic solutions (pKa of VX=8.6 @ 25 1C).35 Each of the condensed-phase systems studied herewere initially equilibrated with classical molecular dynamicsutilizing the COMPASS force field36 before any FPMD simulationswere performed. Following the ‘‘classical’’ equilibration, eachsystem studied was further equilibrated in FPMD simulationswith CP2K for 5 ps before any metadynamics simulations wereperformed to study the condensed-phase chemical reactions.

a. Oxidation of V-type nerve agents with hypochlorite anion

All oxidation reactions of V-type nerve agents studied herewere performed by adding a single hypochlorite anion (ClO!)and sodium cation (Na+) to the ensemble described in sectionIII. The CV for the metadynamics simulations used to studythe oxidation reactions of VX (or VXH+) and R-VX(or R-VXH+) with the hypochlorite ion (ClO!) was chosenas the distance between the oxygen atom of the hypochloriteanion and the atom center of interest in the V-type nerve agentmolecule (e.g., phosphorus, sulfur, carbon, etc.). In addition,several di!erent metadynamics parameters (hill height andwidth) were tested, where the final parameters were chosenas:H= 1.0 " 10!3 Hartree, o= 0.1 Bohr, and where hills,H,were added every 15 MD steps (time step = 0.5 fs).

Many of the reactions assessed using metadynamics simulationswere also studied using the IT-NEB approach to improve ourconfidence in obtaining appropriate reaction energy barriers in thetransition-state region.37 Replicas (atomic coordinates) used in theIT-NEB simulations were taken directly from the metadynamicstrajectories along the identified reaction coordinate of the reaction.Specifically, 32 replicas were extracted from the resultantmetadynamics trajectories; 16 replicas each from both sidesof the reaction potential energy barrier.

b. Hydrolysis of VX in basic solutions

FPMD condensed-phase metadynamics simulations of basehydrolysis were performed by adding a single hydroxide anion(OH!) to the core-ensemble of VX described in section III. Herethe CVwas chosen as the bond distance between sulfur and carbon(S–C) atoms in the VX molecule. The metadynamics parameters(hill height, H; width, o, etc.) were chosen as: H = 5.0 " 10!4

Hartree, o = 0.1 Bohr, and where hills, H, were added every30 MD steps (15 fs).38

c. Autocatalytic hydrolysis of VX

It has been proposed that VX or R-VXmay ‘‘autocatalytically’’hydrolyze.16 To study the plausibility of such a mechanism,

we performed FPMD simulations where the free-energy surfaceis sampled via metadynamics on a gas-phase cluster containinga single water and VX molecule, similar to that shown inFig. 8a, as well as a fully condensed-phase simulation usingthe core ensemble described in section III, above. Two CVswere chosen to study the autocatalytic hydrolysis of VX; theCVs are defined as the distance between the phosphorus (CV1)or the nitrogen (CV2) atomic centers, and selected oxygen of awater molecule proximal to the VX molecule (see Fig. 8a).Parameters for the biasing potential (hill height,H; width, o, etc.)were the same as those defined in section IIIa.

IV. Results

a. VX and R-VX hypochlorite decontamination reactions

The main focus of this paper was to demonstrate the feasibility ofemploying FPMD simulation approaches to study condensed-phase decontamination reactions of V-type nerve agents viahypochlorite (ClO!) oxidation. Specifically, condensed-phaseFPMD metadynamics and IT-NEB simulations of VX(or VXH+) and R-VX (or R-VXH+) with the hypochlorite ion(ClO!) were performed to ultimately identify decontaminationdegradation pathways and oxidation reaction energy barriers.Furthermore, when possible, the results of our simulations werecompared to experiment to validate the computational approach.We began by looking at hypochlorite reaction with VX or

R-VX at both the nitrogen (N-centered) or sulfur (S-centered)atomic centers. Fig. 2 shows the free energy as a function ofdistance between the hypochlorite oxygen and nitrogen orsulfur distance as determined by metadynamics simulations(see section IIIa). The resultant N-centered reaction potentialenergy barriers (PEBs) for the hypochlorite reaction for bothVX and R-VX were found to be similar, 11.5 and 10 kcal/mol,respectively. However, the S-centered hypochlorite reaction PEBsfor VX and R-VX were found to di!er by 410–15 kcal/mol,favoring the R-VX/hypochlorite reaction; compare the dashedand solid gray curves of Fig. 2. Upon securitization of theS-centered ‘‘transition-state structure,’’ it was found that a singlewater molecule mediates the reaction of the R-VX molecule(see Fig. 3); no such structure was identified for the S-centeredVX reaction. Two hypotheses are put forward to understandthis finding: (1) the development of a solvent mediated transition-state structure for VX is sterically hindered by the presence of themore bulky isopropyl nitrogen moieties of VX versus the lessbulky ethyl moieties of R-VX, or (2) the simulations are none-rgodic (i.e., simulations were not run long enough to allow fordevelopment of a ‘‘single water molecule mediated transition statestructure’’ in VX (again in part due to the presence of the morebulky isopropyl groups in VX).39 To this end, additional S-centeredVX/hypochlorite simulations were performed, where the additionalsampling eventually resulted in a water mediated transition-statestructure (see right panel of Fig. 4).40 The additional VX samplingrequired to identify the water mediated transition-state structurelends credence to the hypothesis that the more prominent sterice!ect found in VX ultimately impedes the development of solventmediation e!ects important in chemical reactions.Starting from the water mediated transition-state structures

yielded from the above S-centered metadynamic simulations,

Fig. 1 Chemical structure of VX (left) and R-VX (right).

Dow

nloa

ded

by U

nive

rsity

of C

hica

go o

n 19

/04/

2013

18:

42:0

3.

Publ

ished

on

12 Ja

nuar

y 20

12 o

n ht

tp://

pubs

.rsc.

org

| doi

:10.

1039

/C2C

P231

26C

View Article Online

Figs. 1-‐3

Water to Ice Phase TransiAon

leading to structures of varying stability, in agreement with theanalysis of ref. 8. The free energy of the most stable defectstructure (B) is 6.9 kcal/mol higher than that of the ideal crystal,which we assume as the reference zero value. The free energybarriers for the defect formation and recombination are estimatedto be 8.9 and 2.0 kcal/mol, respectively, which, assuming acharacteristic frequency for the reaction coordinates of !5 THzand using the classical transition state theory, gives an estimatedlifetime of the defect of !0.5 ns at 120 K. The second defectstructure (C) explored during the metadynamics run is less stable(8.4 kcal/mol) and can either recombine or transform intostructure B through a barrier of 0.9 kcal/mol. When thetemperature is raised to 270 K, the height of the free energyminimum corresponding to structure B and its formation andrecombination barriers remain unchanged, whereas the recom-bination barrier for structure C goes to zero. These free energycalculations extend the results of ref 8 to finite temperature andconfirm the relevance of these topological defects also at themelting temperature.We found that the “5+7” defects play an even greater role,

since they are responsible for the shallow minimum in the FESindicated as premelt in Figure 3. From metadynamics, thetransition barrier to this local minimum can be estimated atroughly 12 kcal/mol. We analyzed the nature of this localminimum by performing an inherent structure analysis24 of themetadynamics trajectory, quenching to zero K in 50 ps framesof the MD trajectory taken every 2 ps. As shown in Figure 4,the inherent structures display a relevant quantity of five-membered rings and a smaller number of four-membered ringsand coordination defects. In fact these structures correspond toa condensation of topological defects involving about 50 H2Omolecules in an otherwise perfect ice Ih lattice. As the tetrahedralcoordination is preserved, five-membered rings are accompaniedby an equal number of seven-membered ones. One typical defectcluster thus obtained is shown in Figure 5. The energy of theparticles, either belonging to smaller rings or under-coordinated,ranges from 0.60 to 0.95 kcal/mol relative to the energy of iceIh.These defective structures were then embedded in a crystalline

ice Ih supercell containing 4608 molecules and brought to 270K, to observe their stability. Several MD simulations wereperformed in the NPT ensemble with different random initialvelocities. Remarkably, the average lifetime of the cluster oftopological defects is 0.4 ( 0.1 ns. This relatively stable

accumulation of defects in a restricted region of the crystal isa nucleus for further disorder and melting as the number ofcoordination defects and smaller rings grows. In Figure 4, twodistinct regimes in the premelting inherent structures can beobserved. When the system is dragged out of the free-energybasin corresponding to the cluster of topological defects, arelevant number of small rings and coordination defects appearsand the energy suddenly increases. Roughly speaking, thissignals the watershed between configurations belonging to thebasin of attraction of ice Ih and to that of the liquid.Since the topological defects cannot migrate, the mobility of

the defective droplet is related only to defect formation andrecombination at the interface with the crystal, and no relevantmotion of its center of mass was observed. We computed themomentum of inertia of the defective region and found that theinertia tensor has two eigenvalues I1 and I2 of similar size anda smaller one I3, with an asphericity ratio (I1 + I2)/2 - I3/(I1 +I2)/2 + I3 = 0.4. This indicates that the cluster of defects has asomewhat elongated shape. The analysis of the eigenvectorsshows that the defective region is roughly aligned along the(33h1) crystallographic direction.Another feature of the FES that is worth commenting upon

is the shallow basin that appears just before the larger liquidbasin and corresponds to a liquid-solid interface.In summary, our simulations reveal that topological defects

where five- and seven-membered rings are formed play a crucial

Figure 3. Two-dimensional projection of the FES into the space ofthe two collective variables energy and five-membered rings. Themetadynamics run has been interrupted before the basin correspondingto the liquid state was explored.

Figure 4. Defect statistics and energy of the inherent structures visitedby the system in the premelt region. The zero of this graph correspondsto the beginning of the melting transition in a metadynamics run of amodel with 576 water molecules.

Figure 5. Cluster of topological defects obtained during the inherentstructure analysis of the premelting phase. H2O molecules forming eitherfour- or five-membered rings are represented by colored sticks. Greylines represent the ice Ih structure embedding the cluster of defects.

Letters J. Phys. Chem. B, Vol. 109, No. 12, 2005 5423Donadio, Raiteri, and Parrinello. Topological defects and melAng of hexagonal ice. 2005

Fig. 3

Membrane PermeaAon

Exploring the Free Energy Landscape of Solutes Embedded in LipidBilayersJoakim P. M. Jam̈beck* and Alexander P. Lyubartsev*

Division of Physical Chemistry, Arrhenius Laboratory, Stockholm University, Stockholm, SE-10691, Sweden

*S Supporting Information

ABSTRACT: Free energy calculations are vital for our understanding of biologicalprocesses on an atomistic scale and can offer insight to various mechanisms. However, insome cases, degrees of freedom (DOFs) orthogonal to the reaction coordinate have highenergy barriers and/or long equilibration times, which prohibit proper sampling. Here weidentify these orthogonal DOFs when studying the transfer of a solute from water to amodel membrane. Important DOFs are identified in bulk liquids of different dielectricnature with metadynamics simulations and are used as reaction coordinates for thetranslocation process, resulting in two- and three-dimensional space of reactioncoordinates. The results are in good agreement with experiments and elucidate thepitfalls of using one-dimensional reaction coordinates. The calculations performed hereoffer the most detailed free energy landscape of solutes embedded in lipid bilayers to dateand show that free energy calculations can be used to study complex membranetranslocation phenomena.SECTION: Biophysical Chemistry and Biomolecules

Thermodynamics is one of the cornerstones in physicalchemistry, and the concept of free energy is arguably the

most important aspect of thermodynamics.1 Many chemicaland biological processes are governed by the change in freeenergy such as solvation phenomena,2 protein!ligand associ-ation,3,4 enzymatic reactions5 and membrane-water partition-ing.6 The ability to predict free energies has been a long-termgoal in several areas (pharmaceutical research to mention one),and endeavors in the molecular modeling field during three lastdecades1,7!11 have made it possible to perform thesecalculations relatively routinely. Currently, due to these efforts,we can discriminate between different methods based on theproblem we have at hand, and this can bring us closer to a morecomplete understanding of biological processes.12

Major efforts have been put into understanding the effect ofdrugs, toxins, anesthetics, and other solutes on cells from amolecular perspective, especially with the focus on freeenergies.13!17 Despite this, the mechanisms for some of thesewidely used compounds are yet not fully understood.18 Still freeenergy calculations offer a great deal of insight as a fullunderstanding of the passive diffusion phenomena of thesecompounds over the membrane requires a detailed view of theunderlying free energy surface (FES). As already mentioned,free energy calculations are used in computer-aided drugdesign.19 Issues that often render drug candidates unfit forclinical testing are related to slow translocation of thecompounds across the membranes, resulting in poorpharmacokinetic properties and poor bioavailability. Despitethe fact that major efforts have been made to decrease thefailure attribute of these two properties, the progress of manycandidate compounds is still tampered by these factors. To

measure the permeability and partitioning of compounds isdifficult from a high-throughput perspective, and thereforecomputer simulations can be used as an alternative.6 Inparticular, molecular dynamics (MD) simulations have beensuccessful when the interactions between small solutes andmodel membranes have been studied.14,16,17,20!23 These studiesoffer detailed free energy profiles of transferring the solute fromthe surrounding water phase to the center of themembrane.24!26

As MD simulations sample parts of phase space according toa Boltzmann distribution, important regions separated by larger(free) energy barriers may therefore not be properly sampledduring the simulations. Several methods have been developedwith the aim to allow proper sampling of less probable parts ofphase space such as umbrella sampling (US),7 the Wang!Landau algorithm,8 the adaptive biasing force method9,10 andmetadynamics.11 As membrane partitioning studies goes, theUS method is the most employed technique, and the reactioncoordinate/collective variable (CV) is (often) trivial to define:the distance between the membrane midplane and solute alongthe membrane normal for which the probability distribution isbiased. From these biased probability distributions, a freeenergy profile or potential of mean force (PMF) of thetranslocation can be obtained with, e.g., the weighted histogramanalysis method.27 However, other degrees of freedom (DOFs)that are, per definition, orthogonal to the biased DOF(s), areoften of importance, and if these DOFs are not properly

Received: April 15, 2013Accepted: May 10, 2013

Letter

pubs.acs.org/JPCL

© XXXX American Chemical Society 1781 dx.doi.org/10.1021/jz4007993 | J. Phys. Chem. Lett. 2013, 4, 1781!1787

Jämbeck and Lyubartsev. Exploring the Free Energy Landscape of Solutes Embedded in Lipid Bilayers. 2013

sampled, the obtained FES will contain errors that in the worsecase could lead to wrong conclusions. Often one can mistakenlyassume convergence of the PMF along the reaction coordinatebased on long correlation and relaxation times of other DOFs.While transferring a molecule from water to the membranecenter, one important DOF orthogonal to the reactioncoordinate is the membrane disruptions as the solute istransferred toward the hydrophobic core of the membrane.24 Ifone considers the conformational freedom of the solute, thenumber of important DOFs that have to be properly sampledgrows fast, and these DOFs can be difficult to define a priori.Yet they remain vital in certain cases in order for simulations tobe able to reproduce and ultimately predict experiments. Theability of a solute to form intramolecular hydrogen bonds is oneexample of these DOFs.28!31 Procedures such as Hamiltonianreplica exchange US (HREX-US) can decrease the correlationbetween sampled conformations and speed up the convergencein some cases; however, when probable states are separated bylarge energetic barriers, 1D sampling often has to be abandonedfor biased sampling in several dimensions. 2D HREX-UScalculations are robust and can be performed;32!34 however,they are extremely demanding from a computational point ofview and require a lot of efforts before the simulations are eveninitiated. These calculations are further complicated when morethan two DOFs are needed to be biased. An alternative methodto this is metadynamics,11 which has been demonstrated to bean efficient and potent tool in the exploration of complex FESsin many studies,3,35!38 especially in the case where more thantwo DOFs are of interest.39!41

In the present Letter we show that the more traditional one-dimensional (1D) US approach is not able to properly describepartitioning between water and a model membrane of threetypical drug compounds (aspirin, diclofenac, and ibuprofen)and that more extensive sampling is required. First, importantDOFs of the solutes are identified in bulk liquids of differentdielectric nature. This allows for a more complete picture of thesolute’s conformational space before proceeding with largescale membrane simulations were the reaction coordinates aretwo- (2D) or three-dimensional (3D) instead of a 1D reactioncoordinate. With this scheme we are able to sample relevantregions of phase space, and the resulting PMFs and standardbinding free energies are in good agreement with experimentscompared to the more naive approach of running 1D USsimulations without any consideration of the conformationalspace of the solutes.Aspirin, diclofenac, and ibuprofen were chosen, as the two

former have the possibility to form an intramolecular hydrogenbond, and they all have carboxyl groups that have a history ofbeing difficult to sample properly.29,30 In Figure 1, theintramolecular CVs are shown, and for all simulations withthe membrane, an additional CV was added: the center of massdistance between the solute and lipid bilayer. The torsion angle! is not needed as a CV for ibuprofen, as there are nopossibilities for this solute to form intramolecular hydrogenbonds. Before the membrane simulations the underlying FES ofthe torsion angles " and ! for aspirin and diclofenac (Figure 1)were explored in gas phase, water, and n-hexane. As thehydrophobic core of a lipid bilayer behaves similarly to a bulkalkane, the latter solvent was used to mimic this region. Thesesimulations were performed using the well-tempered (WT)metadynamics procedure described by Barducci et al.42 Foribuprofen, the same simulations were performed but using USsimulations to sample along the torsion angle ".

For ibuprofen, the resulting FESs are shown in Figure S1(Supporting Information, SI), and for aspirin and diclofenac,they are shown in Figure S2. When the dielectric constant ofthe solvent is decreased by going from water to n-hexane, thehydrogen of the carboxyl group prefers to be in transconformation (" " 0) with respect to the carbonyl oxygenfor diclofenac and ibuprofen. In aqueous solution, the freeenergy minima is found when the hydroxyl hydrogen is in thetrans conformation (|"| " 180) meaning that the hydrogen isoriented away from the carbonyl in order to participate in ahydrogen bond network with water instead of the 1,4intramolecular interaction. The mentioned features are similarto the findings presented by Paluch et al.30 For both solutes, theFESs in n-hexane and gas phase are similar, with more of ashallow minimum for the former when compared to the latter.The free energy barriers between these two states are high("14kBT), resulting in extremely long simulations beingneeded in order to sample these transitions. As the conforma-tional preference for these two molecules differ significantlybetween a solvent with # " 78 (water) and # " 2 (n-hexane), atransition is to be expected when the solute is transferred fromwater to the hydrophobic core of the membrane. By merelybiasing the sampling along the distance between the solute andmembrane, this transition is likely to never occur. Even if thesimulations reach a microsecond time scale, it is plausible thatthe resulting PMF will be incorrect due to the lack of sampling,which can lead to the wrong conclusions. This is shown later onfor ibuprofen. For aspirin, the FESs are more complicated and

Figure 1. Intramolecular CVs in the present study.

The Journal of Physical Chemistry Letters Letter

dx.doi.org/10.1021/jz4007993 | J. Phys. Chem. Lett. 2013, 4, 1781!17871782

Aspirin

Diclofenac

Figs. 0 & 1

Cytoskeleton Protein FuncAon

efficient route to exploring those states in a single simulation.Moreover, a defining feature of metadynamics lies in the factthat under certain conditions the history-dependent potential ofGaussian functions provides a good estimate of the free-energyof the system projected into the CVs (28, 29). Metadynamics hasbeen successfully used so far to study several aspects of proteinfolding (30–32).

Herein an investigation of nucleotide state-dependent effectson the conformational states of the actin monomer and trimeris presented. Application of metadynamics to MD simulations ofactin shows that the state of the nucleotide bound to the cleftinfluences both the width of the cleft and the conformation ofthe DB loop. The effect of the bound nucleotide on the relativefree-energy differences between the stable conformationalstates has also been estimated. Metadynamics simulations havebeen performed on the ATP, ADP-Pi, and ADP states of actinin both the monomer and trimer. The metadynamics simulationsalso reveal a distinct allosteric relationship between the width ofthe bound nucleotide and the conformation of the DB loop.

ResultsOpening and Closing of the Actin Nucleotide Binding Cleft. Metady-namics simulations were first performed to investigate therelationship between the width of the nucleotide binding cleftand the state of the bound nucleotide as depicted schematicallyin Fig. 1. The results for monomeric actin are shown in Fig. 2 asa function of the bound nucleotide. The most stable state, i.e.,the region with the lowest free-energy, is the darkest color. Theplots show a clear trend in the cleft width as a function of thebound nucleotide. In the ATP-bound state, the most stable stateis a closed conformation, with strong contact between thenucleotide’s P! atom and the protein backbone. After ATPhydrolysis (ADP-Pi phase), the phosphate portion of the nucle-otide loses its contact with the protein, and the most stable statebecomes slightly more closed with respect to the distance definedin the methods section (i.e., the COM distance between residuenos. 14–16 and 156–158). The binding cleft of the ADP-Pi formbecomes more closed than the ATP form because of the loss ofcontacts between phosphate groups and the protein backbone,as seen by the shift in the value of the contact number. Incontrast, in the ATP form, the !- and "-phosphates are posi-tioned between each side of the binding cleft. Finally, after the

release of the Pi group, the cleft is more stable in an openconformation. It can be seen that bound Pi group is responsiblefor the stabilization of the closed state, given that both theADP-Pi and ADP phases are most stable when there is nocontact between the !-phosphate group of the nucleotide andthe binding cleft. The metadynamics simulations completely andreversibly explore the phase space defined by the full-range ofthe opening and closing CVs and, in the case of ATP and ADPmonomeric actin, a few other metastable minima are located.Although the other minima found in these simulations are higherin free-energy by only a few kcal/mol, these regions are separatedfrom any other metastable minima by relatively large barriers. Itis particularly important that the ADP-Pi state is found in aclosed conformation as was proposed in ref. 33, although not yetsupported by high-resolution experimental structures or com-puter modeling.

It is also interesting to compare the most stable states foundby metadynamics simulation with the experimental values avail-

Fig. 1. Systems and biological events investigated using MD and metady-namics. (A) The actin trimer with the binding pocket and DB loop labeled as# and !, respectively. The monomer shown in blue is the monomer to whichmetadynamics was applied. (B) Shown is the proposed conformational changeof the DB loop. (C) Shown is the nucleotide binding pocket. The arrows in Cillustrate the collective variables used to study the binding pocket.

Fig. 2. Free-energy surfaces for the opening and closing of the nucleotidebinding cleft in monomeric actin. The isolines are drawn using a 1 kcal/molspacing, and the energy scale is in kcal/mol. Based on block-averaging analysis,the uncertainty is !1 kcal/mol. The collective variables used in the opening andclosing simulations are described in Results.

12724 ! www.pnas.org"cgi"doi"10.1073"pnas.0902092106 Pfaendtner et al.

Pfaendtner, Barducci, Parrinello, Pollard, and Voth. NucleoAde-‐dependent conformaAonal states of acAn. 2009

Fig. 1

Protein Folding

a different metadynamics (18) history-dependent potential actingon a different CV (Methods and SI Text). Three CVs act at thesecondary structure level by quantifying, respectively, the frac-tion of !-helical, antiparallel, and parallel "-sheet content of theprotein. Three other CVs act at the tertiary structure level bybiasing the number of hydrophobic contacts and the orientationof the side-chain dihedral angles #1 and #2 for hydrophobic andpolar side chains. The seventh CV, called “CamShift,” measuresthe difference between the experimental and calculated chemicalshifts, which were obtained using the CamShift method (37)(Methods and SI Text). Our results indicate that in the approachwe present here, this last variable is essential to fold GB3 andreach convergence readily in the free-energy calculations.

Folding of GB3 Using Chemical Shifts as CVs. The method that weintroduce in this work makes it possible to visit efficiently a widerange of structures, ranging from extended to compact. Repre-sentative examples are shown in Fig. 1A. Native-like conformationsare visited multiple times, reaching a backbone rmsd of 0.5 Å fromthe reference structure (PDB ID code 2OED). In these native-likestructures, the internal packing of hydrophobic side chains ispractically identical to that observed in the reference structure (Fig.1C). In the calculations that we performed, this level of accuracycould be reached only by using a bias-exchange scheme inwhich theCamShift CV is included in the CV set (Methods and SI Text). Todemonstrate this point, we performed another simulation withthe same setup, using the six CVs discussed above that describethe secondary and tertiary structures, but not the CamShift CV.The difference between the two simulations is substantial. In thesimulation without the CamShift CV, the closest configuration to

the reference structure has an rmsd of 2.7 Å (Fig. 2, Inset B). After50 ns, the rmsd starts increasing progressively (red line) and thefolded state is not explored at all. By contrast, the simulation withthe CamShift CV visits the folded state several times, with severalunfolding–refolding events. During the first 50 ns, the latter sim-ulation not only performed better, reaching an rmsd of 2.5 Å, but italso formed the correct secondary and tertiary contacts, particu-larly the ones involved in forming the first "-hairpin (Fig. 2, InsetA), which is critical for the folding of this protein (38, 39). Thefraction of native contacts also was systematically higher in thesimulation using the CamShift CV (Fig. 2, Inset C). These resultsindicate that the folding events observed later in the simulation area result of the systematic bias induced by the CamShift CV towardthe correct local topology in the native state.

Thermodynamics of GB3 Folding. The molecular dynamics simu-lations that we performed using the approach presented in thiswork reached convergence after !240 ns, because at this pointthe bias potentials acting on all the replicas started to becomestationary (40). We then continued the simulations for another140 ns to reconstruct the free-energy landscape of the protein(Methods). In Fig. 3A, the free-energy landscape is representedas a function of three CVs: the fraction of antiparallel "-sheet,the fraction of parallel "-sheet, and the coordination numberbetween the hydrophobic side chains (Fig. 3A). This represen-tation reveals the organization of the free-energy landscape, witha deep minimum corresponding to native-like structures, sepa-rated by a relatively high barrier from other minima. The lowestfree-energy minimum (Methods) includes configurations very sim-ilar to those of the reference structure (on average, at 1.3 Å rmsd).

Fig. 1. (A) Representation of the conformationalsampling achieved by the approach introduced inthis work. The conformations visited are shown asa function of the CamShift collective variable (CV)and of the backbone rmsd from the referencestructure (PDB ID code 2OED). (B) Structure with thelowest rmsd (0.5 Å) from the reference structure. (C)Detail of the side chain packing of the structure in B.

2 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1218350110 Granata et al.

Granata, Camilloni, Vendruscolo, and Laio. CharacterizaAon of the free-‐energy landscapes of proteins by NMR-‐guided metadynamics. 2013

Fig. 1

IntuiAon-‐Building Thought Experiments

•  Divided into three parts: – Adding individual hills – Adding ensembles of hills – Adding series of hills

Adding One Hill •  How does a new hill push a simulaAon? – What determines how hard the hill pushes? – How does this impact the design of hills?

•  How does each hill flafen the surface? – What does it mean to ‘fill a well’? – How does this impact the design of hills?

•  What is the best hill to add? – What is the worst?

Adding One Hill

•  The untempered equaAon is

•  The tempered equaAon is

BRIEF ARTICLE

THE AUTHOR


�V (s, t) = he�(s��(t))2/2�s2(2)

�V (s, t) = he�V (�(t),t)/�T e�(s��(t))2/2�s2(3)

V̇ (s, t) =

Zds0K(s, s0, V (t))�(s0, t)(4)

K(s, s0, V (t)) = he�(s�s0)2/2�s2(5)


K(s, s0, V (t)) = he�(s�s0)2/2(�s(s0))2(7)

h/�s(8)

�s(9)

�T ⇠ �G‡(10)

V̇ (s, t) = !(t)�(s� �(t))(11)

V̇ (s, t) = !e�V ‡(t)/�T �(s� �(t))(12)

V̇ (s, t) ⇠ !e�V ‡(t)/�T e�(F (s)+V (s,t))/T(13)

e�(F (s)+V (s,1))/T ⇠ C(14)

F (s) = �V (s,1) + C(15)

V̇ (s, t) = !e�V (s,t)/�T (s)�(s� �(t))(16)


F (s) = �(T/�T (s) + 1)V (s,1)(18)

V̇ (s, t) = !e�f(s,V (s,t))/T �(s� �(t))(19)

(20)

1

BRIEF ARTICLE

THE AUTHOR


�V (s, t) = he�(s��(t))2/2�s2(2)

�V (s, t) = he�V (�(t),t)/�T e�(s��(t))2/2�s2(3)

V̇ (s, t) =

Zds0K(s, s0, V (t))�(s0, t)(4)

K(s, s0, V (t)) = he�(s�s0)2/2�s2(5)


K(s, s0, V (t)) = he�(s�s0)2/2(�s(s0))2(7)

h/�s(8)

�s(9)

�T ⇠ �G‡(10)

V̇ (s, t) = !(t)�(s� �(t))(11)

V̇ (s, t) = !e�V ‡(t)/�T �(s� �(t))(12)

V̇ (s, t) ⇠ !e�V ‡(t)/�T e�(F (s)+V (s,t))/T(13)

e�(F (s)+V (s,1))/T ⇠ C(14)

F (s) = �V (s,1) + C(15)

V̇ (s, t) = !e�V (s,t)/�T (s)�(s� �(t))(16)


F (s) = �(T/�T (s) + 1)V (s,1)(18)

V̇ (s, t) = !e�f(s,V (s,t))/T �(s� �(t))(19)

(20)

1

Adding One Hill

•  How does a new hill push a simulaAon? – What determines how hard the hill pushes? – How does this impact the design of hills?

¥¥¥¥

Adding One Hill

•  How does the hill push a simulaAon? –  No push at first –  No push at the peak – Max push ager half down

•  Maximum force occurs with a delay

•  No preferred direcAon •  Hill size limited by slope, not height

¥¥¥¥

¥¥

¥¥

Adding One Hill

•  How does each hill flafen the surface? – What does it mean to ‘fill a well’? – How does this impact the design of hills?

•  What is the best hill to add in metadynamics? – What is the worst?

¥¥¥¥

¥¥¥¥

Adding One Hill

•  How does each hill flafen the surface? –  Smooth + smooth = smooth –  Smooth + rough = rough –  Rough + rough = ?

•  Hills should match the roughness of a surface –  Sets an opAmal lengthscale

•  As a surface flafens, each hill seems rougher –  The lengthscale changes

¥¥

¥¥

¥¥

Adding One Hill

•  What is the best hill? –  the exact opposite of the free energy surface

•  What is the worst? –  anything else that doesn’t depend on the sample

•  Robustness is essenAal •  It’s too easy to come up with brifle tricks

¥¥

¥¥

¥¥

Adding Ensembles of Hills

•  Is my final desired sampling state stable? –  If I have flat sampling, will I add a flat bias? –  If there is noise, will I eventually average it out?

•  What happens at the edges of a simulaAon? –  If I have flat sampling up to an edge, will I add a flat bias?

•  Can I flafen a feature thinner than my hills? – What sampling pafern leads to that?


•  Is the final sampling state stable? –  If I have flat sampling, will I add a flat bias? –  If there is noise, does it eventually average out?

¥ ¥ ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥ ¥ ¥ ?


•  Is the final sampling state stable? –  Even sampling leads to an even bias, but..

–  StaAsAcal fluctuaAons are not fully damped

•  With infinite walkers, any metadynamics is exact

•  With finite walkers, metadynamics requires tempering

¥ ¥ ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥ ¥ ¥


•  What happens at the edges of a simulaAon? –  If I have flat sampling up to an edge, will I add a flat bias?

¥¥ ¥ ¥ ¥?


•  What happens at the edges of a simulaAon? –  Even sampling leads to an uneven bias

–  The added bias is 50% too small near each wall

•  Wells appear near the edges of a simulaAon

•  To avoid trapping requires correcAons

¥¥ ¥ ¥ ¥

¥¥ ¥ ¥ ¥


•  Can I flafen a feature thinner than my hills? – What sampling pafern leads to that?

from ?


•  Can I flafen a feature thinner than my hills? –  Gaussians are a complete set of basis funcAons

–  However, hills can only be added, never subtracted

•  Filling thin wells works, takes a long Ame

•  A rough guess for hill size can be good enough

Adding Series of Hills

•  What can happen if I add hills too quickly? – What does “too quickly” mean?

•  What happens if my collecAve variables are not the slowest in a system?


•  What if I add hills too quickly? –  SimulaAon is pushed by a growing wave of hills

–  “Hill surfing”

•  Nonequilibrium arAfacts can be very dangerous

•  Power input should be lower than a system’s natural fluctuaAons

¥¥¥¥ ¥¥

¥¥¥¥ ¥¥

Adding Series of Hills •  What if the collecAve variables are not slow? –  If there are rare events orthogonal to them?

•  What if I chose x instead of y?


•  What if the collecAve variable is not slow? –  SimulaAon equilibrates in each metastable state

–  Bias oscillates between various constrained PMFs

•  Orthogonal variables can sink any method

•  Metadynamics fails instrucAvely

¥¥¥¥

¥¥

¥¥

Adding Series of Hills •  The concept of equilibrium is subtle •  There are always orthogonal variables •  Great example: water liquid-‐liquid coexistence

214504-3 D. T. Limmer and D. Chandler J. Chem. Phys. 138, 214504 (2013)

flipping” during coarsening, giving the transient impressionof a non-equilibrium barrier between low-density and high-density states that changes with N.

These finite-size effects are fundamental to the nature ofphase transitions. Establishing the existence of a phase transi-tion requires studying system-size dependence, for example,by computing changes in free energy barriers with respect tochanging N. No such computations have yet been performedfor putative liquid-liquid transitions in models of water thatexhibit water-like structure of the liquid and crystal phases.To do so requires algorithms that can attend to the collectivenature of systems undergoing phase transitions. Free energymethods are among the tools that are suitable for the task,provided they are combined with trajectory algorithms thatare appropriately efficient and reversible.29

In Sec. III, we detail how pertinent free energies canbe computed for supercooled water, and we consider differ-ent variants of the ST2 model as applications. Juxtapositionof free energy surfaces for three different variants indicatesthat reasonable changes in electrostatic boundary conditionsdo not change general phase behaviors. Section IV presentsfree energy surfaces obtained for other systems: the mW ofwater, the TIP4P/2005 model of water,30 and the Stillinger-Weber (SW) model of Si. In every case, the models are foundto exhibit one stable or metastable liquid phase plus ice-like crystal phases. Coexistence between two distinct liquidphases does not occur. A summary of our findings is given inSec. IV, and Appendices A, B, and C present further detailsand results.

Reversibility is particularly important to the issues ad-dressed here and in Paper I. Distinct reversible phases canbe interconverted, with properties that are independent of thepaths by which they are prepared. Reversible liquid phasesare thus not the same as amorphous solids or glasses. Theformer are reversible and ergodic, so their measured station-ary behaviors are independent of history. The latter, like high-density or low-density amorphous ices (HDA and LDA), arenot ergodic, so their behaviors depend much on history (i.e.,preparation protocols). Observed transitions between HDAand LDA phases,31 therefore, are necessarily different than

reversible liquid-liquid transitions. Melting amorphous ice toproduce a non-equilibrium liquid that then crystalizes is dif-ferent too.32

Crystallization following the melting of glass33 and crys-tallization following the rapid quench of water into the liq-uid’s “no-man’s land”34 are much like non-equilibrium dy-namics evolving from low to high Q6 on the middle and leftfree energy surfaces pictured in Fig. 1, an observation wor-thy of future study. But these interesting non-equilibrium pro-cesses and the transitions between different amorphous solidsof water are not our focus in this work. Rather, we are con-cerned with whether water-like systems when constrained tonot freeze can exhibit two distinct liquid phases. If such re-versible polymorphism were possible, these systems couldalso exhibit a second critical point as Stanley and many of hisco-workers have proposed.1, 8, 10, 11, 15, 16, 19, 35–38 If, instead, re-versible molecular simulation models exhibit only ice and oneliquid, then the symmetry differences between ice and liquidexclude the possibility of an associated critical point. We be-lieve the systematic evidence provided herein and in Paper Iindicates that there is only one liquid and no low-temperaturecritical point.

II. THEORY OF COARSENING AND ARTIFICIALPOLYAMORPHISM IN COMPUTER SIMULATIONOF WATER

This section provides a quantitative theoretical analysisshowing the difficulty in obtaining correct reversible free en-ergy surfaces of supercooled water. We do so by examiningthe effects of time-scale separation for dynamics on a re-versible free energy surface. The particular surface we em-ploy is the free energy F(!, Q6) derived in Paper I for a vari-ant of the ST2 model. This free energy is shown in Fig. 2(a).The methods used to obtain that surface are the subject ofSec. III, but here we only need to assume that there is such asurface, and that it is qualitatively like the surface shown inFig. 2(a).

Free energy surfaces for several models are derived inSecs. III and IV. The generic features of the free energy

FIG. 2. Slow relaxation behavior and its consequences for free energy calculations. (a) The reversible free energy surface for 216 molecules with the ST2apotential energy function at temperature T = 235 K and pressure p = 2.2 kbar. (See text for definition of the ST2a potential.) Contour lines are separated by1.5 kBT, and statistical uncertainties are about 1 kBT. (b) Negative logarithm of the non-equilibrium distribution for crystal order, Q6, as it relaxes from theliquid state. It is computed from the Fokker-Planck equation with the free energy surface given in Panel (a) under the assumption that the density, !, remains atequilibrium with the instantaneous value of Q6. (c) and (d) Non-equilibrium pseudo free energy surfaces computed from Eq. (1) at two intermediate stages ofrelaxation, t = 10 "Q6 and t = 1000 "Q6 . The unit of time, "Q6 , is the autocorrelation time for Q6 fluctuations in the liquid basin (i.e., at small Q6). The reduceddensity is !̃ = (! ! !xtl)/#!, where !xtl is the mean density of the crystal basin (i.e., at large Q6), and #! is the difference between the mean densities of theliquid and crystal basins. Contour lines are separated by 1 kBT and statistical uncertainties are about 1 kBT.

Downloaded 06 Jun 2013 to 128.135.100.114. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

214504-3 D. T. Limmer and D. Chandler J. Chem. Phys. 138, 214504 (2013)

flipping” during coarsening, giving the transient impressionof a non-equilibrium barrier between low-density and high-density states that changes with N.

These finite-size effects are fundamental to the nature ofphase transitions. Establishing the existence of a phase transi-tion requires studying system-size dependence, for example,by computing changes in free energy barriers with respect tochanging N. No such computations have yet been performedfor putative liquid-liquid transitions in models of water thatexhibit water-like structure of the liquid and crystal phases.To do so requires algorithms that can attend to the collectivenature of systems undergoing phase transitions. Free energymethods are among the tools that are suitable for the task,provided they are combined with trajectory algorithms thatare appropriately efficient and reversible.29

In Sec. III, we detail how pertinent free energies canbe computed for supercooled water, and we consider differ-ent variants of the ST2 model as applications. Juxtapositionof free energy surfaces for three different variants indicatesthat reasonable changes in electrostatic boundary conditionsdo not change general phase behaviors. Section IV presentsfree energy surfaces obtained for other systems: the mW ofwater, the TIP4P/2005 model of water,30 and the Stillinger-Weber (SW) model of Si. In every case, the models are foundto exhibit one stable or metastable liquid phase plus ice-like crystal phases. Coexistence between two distinct liquidphases does not occur. A summary of our findings is given inSec. IV, and Appendices A, B, and C present further detailsand results.

Reversibility is particularly important to the issues ad-dressed here and in Paper I. Distinct reversible phases canbe interconverted, with properties that are independent of thepaths by which they are prepared. Reversible liquid phasesare thus not the same as amorphous solids or glasses. Theformer are reversible and ergodic, so their measured station-ary behaviors are independent of history. The latter, like high-density or low-density amorphous ices (HDA and LDA), arenot ergodic, so their behaviors depend much on history (i.e.,preparation protocols). Observed transitions between HDAand LDA phases,31 therefore, are necessarily different than

reversible liquid-liquid transitions. Melting amorphous ice toproduce a non-equilibrium liquid that then crystalizes is dif-ferent too.32

Crystallization following the melting of glass33 and crys-tallization following the rapid quench of water into the liq-uid’s “no-man’s land”34 are much like non-equilibrium dy-namics evolving from low to high Q6 on the middle and leftfree energy surfaces pictured in Fig. 1, an observation wor-thy of future study. But these interesting non-equilibrium pro-cesses and the transitions between different amorphous solidsof water are not our focus in this work. Rather, we are con-cerned with whether water-like systems when constrained tonot freeze can exhibit two distinct liquid phases. If such re-versible polymorphism were possible, these systems couldalso exhibit a second critical point as Stanley and many of hisco-workers have proposed.1, 8, 10, 11, 15, 16, 19, 35–38 If, instead, re-versible molecular simulation models exhibit only ice and oneliquid, then the symmetry differences between ice and liquidexclude the possibility of an associated critical point. We be-lieve the systematic evidence provided herein and in Paper Iindicates that there is only one liquid and no low-temperaturecritical point.

II. THEORY OF COARSENING AND ARTIFICIALPOLYAMORPHISM IN COMPUTER SIMULATIONOF WATER

This section provides a quantitative theoretical analysisshowing the difficulty in obtaining correct reversible free en-ergy surfaces of supercooled water. We do so by examiningthe effects of time-scale separation for dynamics on a re-versible free energy surface. The particular surface we em-ploy is the free energy F(!, Q6) derived in Paper I for a vari-ant of the ST2 model. This free energy is shown in Fig. 2(a).The methods used to obtain that surface are the subject ofSec. III, but here we only need to assume that there is such asurface, and that it is qualitatively like the surface shown inFig. 2(a).

Free energy surfaces for several models are derived inSecs. III and IV. The generic features of the free energy

FIG. 2. Slow relaxation behavior and its consequences for free energy calculations. (a) The reversible free energy surface for 216 molecules with the ST2apotential energy function at temperature T = 235 K and pressure p = 2.2 kbar. (See text for definition of the ST2a potential.) Contour lines are separated by1.5 kBT, and statistical uncertainties are about 1 kBT. (b) Negative logarithm of the non-equilibrium distribution for crystal order, Q6, as it relaxes from theliquid state. It is computed from the Fokker-Planck equation with the free energy surface given in Panel (a) under the assumption that the density, !, remains atequilibrium with the instantaneous value of Q6. (c) and (d) Non-equilibrium pseudo free energy surfaces computed from Eq. (1) at two intermediate stages ofrelaxation, t = 10 "Q6 and t = 1000 "Q6 . The unit of time, "Q6 , is the autocorrelation time for Q6 fluctuations in the liquid basin (i.e., at small Q6). The reduceddensity is !̃ = (! ! !xtl)/#!, where !xtl is the mean density of the crystal basin (i.e., at large Q6), and #! is the difference between the mean densities of theliquid and crystal basins. Contour lines are separated by 1 kBT and statistical uncertainties are about 1 kBT.

Downloaded 06 Jun 2013 to 128.135.100.114. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Limmer and Chandler. The putaAve liquid-‐liquid transiAon is a liquid-‐solid transiAon in atomisAc models of water. II. 2013

Fig. 2

IntuiAon-‐Building: Conclusion

•  Carefully limit the power input due to the bias – Control the slope of the hills and rate of addiAon – Look out for heterogeneous fluctuaAon

•  Hills must adapt as the simulaAon progresses – Offset staAsAcal fluctuaAon and avoid overwork

•  Always look out for orthogonal variables – Look for pulsing around convergence – Define early what is fast and what is forbidden

Things leg out for Ame

•  Concrete implementaAons – We discuss one code, PLUMED, this agernoon

•  Current fronAers – Lots of work on how to use more CVs at once – Lots of work on how to choose and design CVs – ConAnuing work on efficient convergence

•  Placement among other methods – Unique efficiency tradeoffs – ComplementariAes to others

Summary •  Metadynamics is theoreAcally simple: – Bias away from previously visited states –  In a reduced space of collecAve variables – At a sequenAally decreasing rate of bias

•  Metadynamics is pracAcally convenient: – Few parameters to choose –  ImplementaAon is comparaAvely simple – Fails early and fails instrucAvely

metadynamics, · 2014-05-30 · metadynamics, • the,bare,bones,of,metadynamics, –...

Documents