Metadynamics
Day 2, Lecture 3 James Dama Metadynamics
• The bare bones of metadynamics – Bias away from previously visited configura ons – In a reduced space of collec ve variables – At a sequen ally decreasing rate of bias • Examples from the literature • Key thought experiments to build intui on – Bias size, shape, and rate – Good and bad collec ve variables The Essence of Metadynamics Bias adap vely to escape metastable states Huber, Torda, and Van Gunsteren. Local eleva on : A method for improving the searching proper es of molecular dynamics simula on. 1994
• Metastable states are a pervasive feature of real free energy surfaces • Transi ons between these states are rare and difficult to observe in simula on The Essence of Metadynamics Bias adap vely to escape metastable states Huber, Torda, and Van Gunsteren. Local eleva on : A method for improving the searching proper es of molecular dynamics simula on. 1994
• Basins are unknown a priori – Avoid previously sampled conforma ons – Add energy ‘hills’ centered on each sample so far • Focus on model- independent explora on BRIEF ARTICLE
THE AUTHOR
Adding Hills V (s,t)/ T (F (s)+V (s,t))/T (1) 2JAMESDAMAV˙ (s, t) !e e • A real equa on is ⇠ (s (t))2/2 s2 (2) bias V (s, t). Given a metadynamics-sampled V (s, t)=he trajectory in collective variables (t), the history-dependent potential• The idealized equa on is is definedV ( (t) as,t)/ an T approximation(s (t))2/2 s2 of (3) V (s, t)=he e V˙ (s, t)=! (s (t))(1) ˙ where ! is an adjustableV• rate(Makes escaping square wells linear in depth s, t)= parameterds0K and(s, s 0,Vis a(t delta)) (s function0,t)(4) on the collective variable domain. For well-temperedrather than exponen al in depth metadynamics,Z the idealized rule is (s s )2/2 s2 (5) K(s, s0,V(t)) = he 0 ˙ V (s,t)/ T V (s, t)=!e (s 2 (t2))(2) (6) K(s, s ,V(t)) = he V (s0,t)/ T e (s s0) /2 s where T is a second adjustable0 parameter referred to as a tempering parameter. Clearly, (s s )2/2( s(s ))2 (7) metadynamics is the limitK(s, of s0 well-tempered,V(t)) = he metadynamics 0 0 as T . This history- dependent bias serves to flatten the sampled distribution of collective!1 variables, pushing (8) h/ s future sampling away from each point the more often it has already been visited. In (9) practice, these rules are approximated by discretization in times and mollification in space, so that the bias is updated at only a discrete set of times and is updated using Gaussian (10) T G‡ bumps rather than delta functions. Additionally, the⇠ trajectory (t)maybeamulti- trajectory composed of the historiesV˙ (s, of t)= multiple!(t) ( walkers,s (t))(11) and in this case should be understood as a sum of delta functions, oneV ( pert)/ individualT walker. Later work (Branduardi, V˙ (s, t)=!e ‡ (s (t))(12) Bussi, and Parrinello, 2012) has introduced functional complexity in !, ! !(s), and the V (t)/ T (F (s)+V (s,t))/T ! (13) idea of selectively temperedV˙ (s, t) metadynamics!e ‡ e was to introduce functional complexity in T , T T (s). For the sake⇠ of generality, and because it will not increase the complexity ! (F (s)+V (s, ))/T (14) of the analysis that follows, I will alsoe consider general1 biasC update rules of the form ⇠ (15) ˙ F (s)=f(s,V (s,tV))(/Ts, )+C V (s, t)=!e 1(s (t))(3) V (s,t)/ T (s) V˙ (s, t)=!e (s (t))(16) where f(s, V (s, t)) is a function dependent on both the s-point and the value of the potential at that point and ! is a constantV with(s,t)/ units T (s) of energy(F (s)+V per(s,t)) unit/T time just as in metadynamics (17) V˙ (s, t) !e e with non-adaptive Gaussians.⇠ Note that this form describes geometry-adaptive Gaussians F (s)= (T/ T (s) + 1)V (s, )(18) only; when the Gaussians are adapted on-the-fly this form1 does not hold precisely. Finally, in addition to local temperingf(s,V ( rules,s,t))/T it is possible to imagine tempering rules V˙ (s, t)=!e (s (t))(19) in which the entire bias or regions of the bias are used to calculate updates at each single (20) point. For these, one considers update rules of the form 1 f(s,V (t))/T V˙ (s, t)=!e (s (t))(4) where f(s, V (t)) is a functional of the bias function V (t) at time t instead of a function of the bias at a point V (s, t) at time t. For this, the same sorts of error bounds can be 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 tempering rules in a section following the investigation of the local rules. 2.2. Mollified metadynamics bias equations. However, the idealized metadynamics biasing rate equation above is not realizable in simulation, so I will also consider metady- namics updates that can be written in the general form
V˙ (s, t)= ds0K(s, s0,V(t)) (s0,t)(5) Z The Essence of Metadynamics Bias only specific collec ve variables Laio and Parrinello. Escaping free-energy minima. 2002
• Local eleva on was wasteful – Sampling every state uniformly is expensive • Self-avoidance changes random walks less and less with more and more variables • Focusing on interes ng features is more important with more variables The Essence of Metadynamics Bias only specific collec ve variables Laio and Parrinello. Escaping free-energy minima. 2002
• Reac ons are mul scale – Monitor only state- determining variables – Leave fast variables alone • Focus on accomplishing specific exploratory goals Collec ve Variables
• Any func on of any number of fine-grained variables – Posi on, distance, angle, dihedral – Coordina on number, density, crystalline order – Helicity, contact map, NMR spectrum – Strings of configura ons in another CV space • Whatever you would like to explore The Essence of Metadynamics Adap vely tune the biasing rule Barducci, Bussi, and Parrinello. Well-tempered metadynamics: A smoothly converging and tunable free-energy method. 2008 Assessing the Accuracy of Metadynamics J. Phys. Chem. B, Vol. 109, No. 14, 2005 6717
• Original metadynamics had limited accuracy – Errors saturated and never fully disappeared • Residual inaccuracy was propor onal to the rate of hill addi on
Laio, Rodriguez-Fortea, Gervasio, Ceccarelli and ParrinelloFigure. Assessing the accuracy of 1. Metadynamics results formetadynamics four different free. 2005 energy profiles: (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) )-5 exp(-(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 independent trajectories is represented as a dashed line, with the error bar given by eq 7.
Figure 3. Error as a function of the metadynamics parameters w, τG and δs and of #, D and S in d ) 1 (upper panel) and d ) 2 (lower panel). The continuous and the dashed lines correspond to eq 12 for d ) 1 and d ) 2, respectively.
for d ) 1, d ) 2, and d ) 3. If only one parameter is varied at atime,thedependenceofj! on that parameter can be investigated. In this manner, we found empirically that j! is approximately proportional (independently of the dimensional- ity) to the square root of the system size S, of the Gaussian width δs and the Gaussian height w, while it is approximately proportional to the inverse square root of #, D, and τG. These observations are summarized in Figure 3, in which we plot the logarithm of j! vs the logarithm of (Sδs/DτG)(w/#) when d ) 1 Figure 2. Average error (eq 8) as a function of the number of and d ) 2. Different color codes correspond to different physical Gaussians for the four F(s)ofFigure1.Thebumpintheerrorobserved conditions (i.e., different D, #, or S), while dots of the same for the functional forms C and D is due to the fact that in a long part color correspond to different metadynamics parameters. w is of the metadynamics one of the free energy wells is already completely varied in the range between 0.04 and 4, δs between 0.05 and filled, while the other is being filled. The error is measured using 0.4 and τG between 5 and 5000. The continuous line corresponds definition 8 on both the wells, but the filling level is not the same until to the equation the full profile is filled. Sδs w j)! all the underlying profile is filled. This required 500 Gaussians C(d ) (12) DτG # for all the profiles reported in Figure 1. ∼ We verified that these properties also hold in higher dimen- where C(d ) 1) 0.5 and C(d ) 2) 0.3 in the same ∼ " ∼ sionalities, for virtually any value of w and τG,andforanyvalue logarithmic scale, and represents a lower bound to the error. of δs significantly smaller than the system size, as we will For a given value of (Sδs/DτG)(w/#), the error obtained with specify in more detail in the following. any parameter in the range considered leads to errors at most B. Dependence of the Error on the Metadynamics Pa- 50% higher than the value given by eq 12. rameters. Since the value of j! does not depend on F(s), we The dependence of the error on the simulation parameters consider in more detail the flat profile (Figure 1A). In d becomes more transparent if j! is expressed as an explicit dimension, we use eq 11 with F(s) ) 0andreflectingboundary function of the total simulation time. Consider in fact a free conditions for s ) S/2, and we compute the dependence of energy profile F(s) that has to be filled with Gaussians up to a | | the error on the metadynamics parameters w, τG and δs and on given level Fmax (e.g., the free energy of the highest saddle point #, D,andS,whichcharacterizethephysicalsystem.Werepeated in F(s)). The total computational time needed to fill this profile several metadynamics for different values of the parameters and can be estimated as the ratio between the volume that has to be The Essence of Metadynamics Adap vely 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 BRIEF ARTICLE
THE AUTHOR
2JAMESDAMA bias V (s, t). Given a metadynamics-sampledV (s,t trajectory)/ T (F (s in)+V collective(s,t))/T variables (t), the (1) V˙ (s, t) !Tempering e e history-dependent potential is defined⇠ as an approximation of (s (t))2/2 s2 (2) • A real equa on is V˙ (s, t)= V! (s,(s t)= he(t))(1) V ( (t),t)/ T (s (t))2/2 s2 (3)where ! is an adjustable rate V parameter(s, t)=he and is a deltae function on the collective variable domain. For well-tempered• The idealized equa on is metadynamics, the idealized rule is ˙ V (s, t)=V (s,tds)/0 KT (s, s0,V(t)) (s0,t)(4) V˙ (s, t)=!e (s (t))(2) Z (s s )2/2 s2 (5)where T is a second• adjustable Error disappears instead of satura ng parameterK(s, s0 referred,V(t)) = tohe as a tempering 0 parameter. Clearly, metadynamics is the• limit The bias increases logarithmically rather than of well-tempered metadynamics as 2 T 2 . This history- V (s0,t)/ T (s s0) /2 s !1 (6)dependent bias serves tolinearly; escape is exponen al in depth again K flatten(s, s0,V the(t)) sampled = he distributione of collective variables, pushing 2 2 future sampling away from each point the more often(s s it0) has/2( s already(s0)) been visited. In (7) – the exponent decreases by a tunable scalar factor K(s, s0,V(t)) = he practice, these rules are approximated by discretization in time and mollification in space, (8)so that the bias is updated at only a discrete set of times and ish/ updated s using Gaussian (9)bumps rather than delta functions. Additionally, the trajectory s(t)maybeamulti- trajectory composed of the histories of multiple walkers, and in this case should be (10) T G‡ understood as a sum of delta functions, one per individual walker.⇠ Later work (Branduardi, Bussi, and Parrinello, 2012) has introducedV˙ functional(s, t)=! complexity(t) (s ( int))(11) !, ! !(s), and the idea of selectively tempered metadynamics was to introduce functional complexity! in T , V (t)/ T T T (s). For the sake of generality,V˙ (s, t)= and!e because ‡ it will(s not ( increaset))(12) the complexity ! of the analysis that follows, I will also considerV (t) general/ T ( biasF (s)+ updateV (s,t))/T rules of the form (13) V˙ (s, t) !e ‡ e ˙ ⇠ f(s,V (s,t))/T V (s, t)=!e (F (s )+(sV (s, (t))))(3) /T (14) e 1 C where f(s, V (s, t)) is a function dependent on both the s-point and⇠ the value of the potential (15) F (s)= V (s, )+C at that point and ! is a constant with units of energy per unit1 time just as in metadynamics V (s,t)/ T (s) with non-adaptive Gaussians. NoteV˙ (s, that t)= this!e form describes (s geometry-adaptive (t))(16) Gaussians only; when the Gaussians are adapted on-the-flyV (s,t)/ T this(s) form(F (s)+ doesV (s,t not))/T hold precisely. (17) V˙ (s, t) !e e Finally, in addition to local tempering⇠ rules, it is possible to imagine tempering rules in which the entire bias or regionsF of(s the)= bias(T/ are usedT (s) to + calculate 1)V (s, updates)(18) at each single 1 point. For these, one considers update rules off( thes,V ( forms,t))/T V˙ (s, t)=!e (s (t))(19) f(s,V (t))/T V˙ (s, t)=!e (s (t))(4) (20) where f(s, V (t)) is a functional of the bias function1 V (t) at time t instead of a function of the bias at a point V (s, t) at time t. For this, the same sorts of error bounds can be 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 tempering rules in a section following the investigation of the local rules. 2.2. Mollified metadynamics bias equations. However, the idealized metadynamics biasing rate equation above is not realizable in simulation, so I will also consider metady- namics updates that can be written in the general form
V˙ (s, t)= ds0K(s, s0,V(t)) (s0,t)(5) Z Examples From the Literature
• Chemical reac on • Phase transi on • Interfacial chemistry • Protein func on • Protein folding View Article Online
Chemical Weapon Remedia on
Gee, Kuo, Chinn, and Raber. First-principles molecular dynamics simula ons of View Article Online View Article Online condensed-phase V-type nerve agent reac on pathways and energy barriers. 2012 we performed FPMD simulationswe performed where FPMD the free-energy simulations surface where the free-energy surface is sampled via metadynamicsis sampled on a gas-phasevia metadynamics cluster containing on a gas-phase cluster containing Figs. 1-3 View Article Online a single water and VXa molecule, single water similar and to VX that molecule, shown similar in to that shown in Fig. 8a, as well as a fullyFig. condensed-phase 8a, as well as a simulation fully condensed-phase using simulation using Fig. 3 Illustration of the transition state structure of ClOÀ reaction the core ensemble described in section III, above. Two CVs Fig. 1 Chemical structure ofFig. VX 1 (left)Chemical and R-VX structureat (right). the of sulfur VX (left) atomic andthe R-VXcenter core (right). for ensemble R-VX. The described transition in section state is mediated III, above. Two CVs were chosen to study the autocatalytic hydrolysis of VX; the by the presence ofwere a single chosen water to molecule study the (the autocatalytic single water molecule hydrolysis is of VX; the (herein referred to as the(herein ‘‘core-ensemble’’). referred to as The the V-type ‘‘core-ensemble’’). nerve The V-type nerve CVs are defined as the distance between the phosphorus (CV ) part of the transitionCVs state are definedstructure as found the distance for the S-centered between the reaction phosphorus (CV1) 1 agents studied included either MeP(O)(OR)(SCH CH NR ), agents studied included either MeP(O)(OR)(SCH2CH2NR02), 2 2 02 or the nitrogen (CV2) atomic centers, and selected oxygen of a with ClOÀ). Theor water the nitrogen molecule (CV as2 shown) atomic facilitates centers, the and eff selectedective oxygen of a VX (R = C H ;R = i-C H ) or its isomeric analog R-VX VX (R = C2H5;R0 = i-C3H7) or its2 isomeric5 0 analoglowering3 7 R-VX of the sulfur/hypochloritewater molecule proximal reactionwater barrier. to the molecule VX molecule proximal (see to theFig. VX 8a). molecule (see Fig. 8a). (‘‘Russian VX,’’ R = i-C H ;R0 =CH ) (see Fig. 1). (‘‘Russian VX,’’ R = i-C4H9;R0 =C2H5) (see4 Fig.9 1). Parameters2 5 for the biasingParameters potential (hill for the height, biasingH;width, potentialo, (hilletc.) height, H;width,o, etc.) Protonated VX and R-VXProtonated counterparts VX and were R-VX also counterparts studied. were also studied. were the same as those defined in section IIIa. + were+ the same as those defined in section IIIa. The protonated species (VXHThe protonated+ or R-VXH species+)werestudied (VXH or R-VXH )werestudied because of the propensitybecause for either of of the these propensity species for to protonate either of these species to protonate (at the nitrogen atom) in neutral or basic solutions (pKa of VX = IV. Results (at the nitrogen atom) in neutral or basic35 solutions (pKa of VX = IV. Results 35 8.6 @ 25 C). Each of the condensed-phase systems studied here 8.6 @ 25 1C). Each of the condensed-phase1 systems studied here were initially equilibrated with classical moleculara. VX and dynamics R-VX hypochloritea. VX decontamination and R-VX hypochlorite reactions decontamination reactions were initially equilibrated with classical molecular dynamics36 utilizing36 the COMPASS force field before any FPMD simulations The main focus of this paper was to demonstrate the feasibility of utilizing the COMPASS force field before any FPMD simulations The main focus of this paper was to demonstrate the feasibility of were performed. Following the ‘‘classical’’ equilibration, each employing FPMD simulation approaches to study condensed- were performed. Following the ‘‘classical’’ equilibration, each employing FPMD simulation approaches to study condensed- system studied was further equilibrated in FPMD simulations phase decontamination reactions of V-type nerve agents via system studied was further equilibrated in FPMD simulations phase decontamination reactions of V-type nerve agents via with CP2K for 5 ps beforeFig. any 4 metadynamicsSolvent-mediated simulations transition were state structure of ClOÀ reaction Fig. 2 Free energy profile as a function of the CV as determinedFig. 3 fromIllustration of the transition state structure of ClOhypochloriteÀ reaction (ClOÀ) oxidation. Specifically, condensed-phase with CP2K for 5 ps before any metadynamics simulations were hypochlorite (ClOÀ) oxidation. Specifically, condensed-phase performed toat the study sulfur the atomic condensed-phaseto the center sulfur for atomic R-VX. chemical center The reactions. transition for R-VX state (right).FPMD is mediated The metadynamics presence of a and IT-NEB simulations of VX metadynamics simulationsperformed of the tocondensed-phase study the condensed-phase reaction of VX chemicaland reactions. FPMD metadynamics and IT-NEB simulations of VX by the presencesterically of a single hindered water molecule transition (the state single structure water(or ismolecule illustrated VXH+ is)andR-VX(orR-VXH for VX (left). +)withthehypochloriteion R-VX with ClOÀ. Black (grey) lines representa. a CV Oxidation defined by ofV-type the nerve agents with(or hypochlorite VXH+)andR-VX(orR-VXH anion +)withthehypochloriteion distance between thea. hypochlorite Oxidation oxygen of V-type and nerve the nitrogen agentspart with (sulfur) of hypochlorite the transition anion state structure found for the S-centered(ClO reactionÀ)wereperformedtoultimatelyidentifydecontamination (ClOÀ)wereperformedtoultimatelyidentifydecontamination atomic center of VX (dotted) and R-VX (solid);All top oxidation panel. Thewith reactions lower ClOÀ). The of V-type water nervemolecule agents as shown studied facilitates here thedegradation effective pathways and oxidation reaction energy barriers.
Downloaded by University of Chicago on 19/04/2013 18:42:03. All oxidation reactions of V-type nerve agents studied here degradation pathways and oxidation reaction energy barriers. panel is a snapshot of the simulation near the reactionwere performed barrierlowering crossing by adding of the sulfur/hypochlorite a single hypochlorite reaction anion barrier. (ClOÀ) Furthermore, when possible, the results of our simulations were were performed by adding a single hypochlorite+ anion (ClOÀ) Published on 12 January 2012 http://pubs.rsc.org | doi:10.1039/C2CP23126C Furthermore, when possible, the results of our simulations were for the VX/nitrogen reaction (the point labeled+and ‘‘1’’ sodium in the cation upper (Na ) to the ensemble described in section compared to experiment to validate the computational approach. and sodium cation (Na ) to the ensemble described in section compared to experiment to validate the computational approach. panel). As is apparent, the structure at the crossingIII. consists The CV of for a single the metadynamics simulations used to study We began by looking at hypochlorite reaction with VX or III. The CV for the metadynamicsthe oxidation simulations reactions used of to VX study (or VXHWe+) began and byR-VX looking at hypochlorite reaction with VX or water molecule reacting with the incoming hypochlorite anion, which + R-VX at both the nitrogen (N-centered) or sulfur (S-centered) the oxidation reactions of VX (or+ VXH ) and R-VX (or R-VXH ) with the hypochlorite ionR-VX (ClOÀ) at was both chosen the nitrogen (N-centered) or sulfur (S-centered) forms an OHÀ anion before reacting+ at the nitrogen atomic center. atomic centers. Fig. 2 shows the free energy as a function of (or R-VXH ) with the hypochloriteas the distance ion between (ClOÀ) the was oxygen chosen atomatomic of the hypochlorite centers. Fig. 2 showsdistance the between free energy the as hypochlorite a function of oxygen and nitrogen or as the distance between the oxygen atom of the hypochlorite we employed the IT-NEB approach to determineanion the and S-centered the atom center of interest in thedistance V-type nerve between agent the hypochloritesulfur distance oxygen as determined and nitrogen by metadynamics or simulations anion and the atom center of interest in the V-type nerve agent sulfur distance as determined by metadynamics simulations hypochlorite reaction PEBs for VX and R-VX.molecule Specifically, (e.g., phosphorus, the sulfur, carbon, etc.). In addition, (see section IIIa). The resultant N-centered reaction potential molecule (e.g., phosphorus, sulfur, carbon, etc.). In addition, (see section IIIa). The resultant N-centered reaction potential replicas used in the IT-NEB calculationsseveral were taken different directly metadynamics parameters (hill height and energy barriers (PEBs) for the hypochlorite reaction for both
several differentDownloaded by University of Chicago on 19/04/2013 18:42:03. metadynamicswidth) were parameters tested, where(hill height the final and parametersenergy were barriers chosen (PEBs) forVX the and hypochlorite R-VX were found reaction to be for similar, both 11.5 and 10 kcal/mol, from the previously described metadynamics simulations. The3 Downloaded by University of Chicago on 19/04/2013 18:42:03. width) were tested, where the final parameters were chosen Published on 12 January 2012 http://pubs.rsc.org | doi:10.1039/C2CP23126C as: H = 1.0 10À Hartree, o = 0.1 Bohr,VX and and where R-VX hills, wereH, foundrespectively. to be similar, However, 11.5 and the S-centered 10 kcal/mol, hypochlorite reaction PEBs S-centered reaction PEBs obtained from3 the NEB simulationsFig. 4 Solvent-mediated transition state structure of ClOÀ reaction Fig. 2 Free energy profileasPublished on 12 January 2012 http://pubs.rsc.org | doi:10.1039/C2CP23126C a functionas: H of= the 1.0 CV as10 determinedÀ Hartree,were fromo added= 0.1 everyBohr, 15 and MD where steps hills, (timeH, steprespectively. = 0.5 fs). However, thefor S-centered VX and hypochlorite R-VX were reaction found to PEBs differ by 410–15 kcal/mol, for VX and R-VX are shown in Fig. 5, and are foundto the to be sulfur atomic center for R-VX (right). The presence of a metadynamics simulations of the condensed-phasewere added every reaction 15 MDof VX stepsMany and (time of the step reactions = 0.5 assessed fs). using metadynamicsfor VX and simulations R-VX werefavoring found to the diff R-VX/hypochloriteer by 410–15 kcal/mol,reaction; compare the dashed DH E 15 and DH E 11 kcal/mol, respectively. As a finalsterically note, hindered transition state structure is illustrated for VX (left). R-VX with ClOÀ. Black (grey) lines representMany ofthe a CV reactions defined assessedwere by the also using studied metadynamics using theFig. simulations IT-NEB 5 Total approach energyfavoring along to improve the the IT-NEB R-VX/hypochlorite our optimizedand solid pathreaction; grayfor the curves reaction compare of Fig. the 2.dashed Upon securitization of the reactions involving either of the two carbon atoms along the distance between the hypochlorite oxygenwere also and studied the nitrogen using the (sulfur)confidence IT-NEB in approach obtaining to appropriate improveof OClÀanionreaction our with energyand the solid sulfur barriers gray atomic in curves the center ofS-centered of Fig. VX 2. (solid) ‘‘transition-state Upon and securitization R-VX structure,’’ of the it was found that a single 37 atomic center ofbackbone VX (dotted) of the and V-type R-VXconfidence (solid); nerve in agent top obtaining panel. molecules The appropriatetransition-state lower were foundreaction region. to energy be Replicas barriers(dotted). (atomic in Converged the coorS-centereddinates) elastic bands used ‘‘transition-state in with the 32 movablewater structure,’’ molecule replicas it are was mediates shown. found the that reaction a single of the R-VX molecule Downloaded by University of Chicago on 19/04/2013 18:42:03. 37 panel is a snapshotenergetically of the simulation prohibitive,transition-state near the where reaction the region. barrier reactionReplicas crossingIT-NEB PEBs (atomic weresimulations found coordinates) were takenNote used that di inrectly the the VX fromwater peak the only metadynamics molecule appears mediates to be wider(see the Fig. than reaction 3); the no R-VX suchof the peak; structure R-VX molecule was identified for the S-centered Published on 12 January 2012 http://pubs.rsc.org | doi:10.1039/C2CP23126C for the VX/nitrogento be greater reaction than (the 40IT-NEB point kcal/mol labeled simulations (see ‘‘1’’ Table in were the 1trajectories takenfor upper further directly along details). from the identifiedthe metadynamicsthe rxeaction-axis (replica coordinate(see number) Fig. of the 3); is reaction.not no uniformlysuch structureVX spaced reaction. was along identified Twothe reaction hypotheses for the S-centered are put forward to understand panel). As is apparent,We next the structure considered attrajectories the nucleophilic crossing along consists sub thestitution identified of aSpecifically, single reactions reaction involving coordinate 32 replicas ofcoordinate. thewere reaction. extractedVX from reaction. the resultant Two hypothesesthis finding: are put (1) the forward development to understand of a solvent mediated transition- water moleculehypochlorite reacting with the at incoming theSpecifically, phosphorus hypochlorite 32 atomic replicas anion, centermetadynamics which were extracted(P-centered). trajectories; from the 16 resultant replicas eachthis from finding: both (1) sides the developmentstate structure of a solvent for VX mediated is sterically transition- hindered by the presence of the forms an OHÀ Theanion PEBs before for reacting reactionmetadynamics at the of hypochloritenitrogen trajectories; atomic with center.of VX the 16 and replicas reaction R-VX, each potential as from energysimulations both barrier. sides showedstate structure spontaneous for VX protonation is stericallymore bulky hindered at theisopropyl nitrogen by thenitrogen presence moieties of the of VX versus the less of the reaction+ potential+ energy barrier. bulky ethyl moieties of R-VX, or (2) the simulations are none- well as the protonated VXH and R-VXH b.counterparts Hydrolysis of were VX in basiccenter solutions of VXmore and R-VX bulky isopropyl under ‘‘neutral’’nitrogen moieties pH conditions. of VX versus the less we employed thedetermined. IT-NEB Reactionsapproach ofto protonateddetermine the VX S-centered and R-VX were deemed Our simulationsbulky alsoethyl showmoieties spontaneous of R-VX,rgodic protonation or(i.e. (2),simulationswerenotrunlongenoughtoallowfor the simulations of the are none- hypochlorite reaction PEBs for VXb. and Hydrolysis R-VX. Specifically, of VX in basicFPMD the solutions condensed-phase metadynamics simulations of base development of a ‘‘single water molecule mediated transition state to be important for two reasons: (1) experiments have shown that hypochlorite ionrgodic (the (i.e. pK,simulationswerenotrunlongenoughtoallowfora of hypochlorite is reported as replicas used in the IT-NEB calculationsFPMD condensed-phase were taken directlyhydrolysis metadynamics were performed simulations by of adding base41 a singledevelopment hydroxide of anion a ‘‘singlestructure’’ water molecule in VX mediated (again in transition part due state to the presence of the more the solubility of VX is increased under acidic conditions, where being B7.5, consistent with this observation). P-centered nucleo- 39 from the previously described metadynamicshydrolysis were simulations. performed(OH by TheÀ adding)tothecore-ensembleofVXdescribedinsectionIII.Here a single hydroxide anion bulky isopropyl groups in VX). To this end, additional S-centered increased solubility is hypothesized to be due to protonation of philic substitutionstructure’’ reactions in VX were (again found in to part proceed due tovia thean presence SN2 of the more the CV was chosen as the bond distance between sulfur and carbon VX/hypochlorite39 simulations were performed, where the additional S-centered reactionnitrogen, PEBs16,17,19,20 obtainedand(OH fromÀ (2))tothecore-ensembleofVXdescribedinsectionIII.Here during the NEB equilibration, simulations many of the type mechanism,bulky whereisopropyl tetra-coordinatedgroups in VX). phosphorusTo this end, forms additional a S-centered for VX and R-VX are shown inthe Fig. CV 5, was and chosen are as found the bond(S–C) to be distance atoms betweenin the VX sulfur molecule. and carbon The metadynamicsVX/hypochlorite parameters simulationssampling were performed, eventually where resulted the in additional a water mediated transition-state 4 40 (S–C) atoms in the VX molecule.(hill height, TheH metadynamics;width,o, etc. parameters)werechosenas:samplingH =5.0 eventually10À resultedstructure in a water(see right mediated panel of transition-state Fig. 4). The additional VX sampling DH E 15 and DH E 11 kcal/mol,c respectively. As a final note,  This journal is the Owner Societies 2012 Hartree, Fig.= 0.1 5 Bohr,Total andenergy where along hills,4 thePhys. IT-NEBH, Chem. were optimized added Chem. every Phys., path2012, for the14 reaction,3316–332240 3319 o À required to identify the water mediated transition-state structure reactions involving either of the(hill two height, carbonH;width, atoms alongo, etc.)werechosenas: the H38 =5.0 10 structure (see right panel of Fig. 4). The additional VX sampling 30 MD stepsof (15 OCl fs).Àanion with the sulfur atomic center of VX (solid)lends and R-VX credence to the hypothesis that the more prominent steric backbone of the V-type nerve agentHartree, moleculeso = were 0.1 Bohr, found and to be where hills, H, were added every required to identify the water mediated transition-state structure 38 (dotted). Converged elastic bands with 32 movable replicase areffect shown. found in VX ultimately impedes the development of solvent energetically prohibitive, where the30 MD reaction steps PEBs (15 fs). were foundc. Autocatalytic hydrolysis of VX lends credence to the hypothesis that the more prominent steric Note that the VX peak only appears to be wider than the R-VXmediation peak; effects important in chemical reactions. effect found in VX ultimately impedes the development of solvent to be greater than 40 kcal/mol (seec. Table Autocatalytic 1 for further hydrolysis details).It has of been VX proposedthe x-axis that (replica VX number) or R-VX is may not ‘‘autocatalytically’’ uniformly spaced along theStarting reaction from the water mediated transition-state structures mediation effects important in chemical reactions. We next considered nucleophilic substitution reactions involvinghydrolyze.16coordinate.To study the plausibility of such a mechanism, yielded from the above S-centered metadynamic simulations, hypochlorite at the phosphorusIt atomic has been center proposed (P-centered). that VX or R-VX may ‘‘autocatalytically’’ Starting from the water mediated transition-state structures hydrolyze.16 To study the plausibility of such a mechanism, yielded from the above S-centered metadynamic simulations, The PEBs for reaction of hypochlorite with VX and R-VX,3318 as Phys.simulations Chem. Chem. showed Phys., 2012, spontaneous14,3316–3322 protonation at the nitrogen This journal is c the Owner Societies 2012 well as the protonated VXH+ and R-VXH+ counterparts were center of VX and R-VX under ‘‘neutral’’ pH conditions. determined. Reactions of protonated3318 VXPhys. and R-VX Chem. wereChem. deemed Phys., 2012, Our14,3316–3322 simulations also show spontaneous protonationThis journal of the is c the Owner Societies 2012 to be important for two reasons: (1) experiments have shown that hypochlorite ion (the pKa of hypochlorite is reported as the solubility of VX is increased under acidic conditions, where being B7.5,41 consistent with this observation). P-centered nucleo- increased solubility is hypothesized to be due to protonation of philic substitution reactions were found to proceed via an SN2 nitrogen,16,17,19,20 and (2) during equilibration, many of the type mechanism, where tetra-coordinated phosphorus forms a
This journal is c the Owner Societies 2012 Phys. Chem. Chem. Phys., 2012, 14,3316–3322 3319 Water to Ice Phase Transi on
DonadioLetters, Raiteri, and Parrinello. Topological defects and mel ng of J. Phys. Chem. B, Vol. 109, No. 12, 2005 5423 hexagonal ice. 2005
Fig. 3
Figure 4. Defect statistics and energy of the inherent structures visited by the system in the premelt region. The zero of this graph corresponds Figure 3. Two-dimensional projection of the FES into the space of to the beginning of the melting transition in a metadynamics run of a the two collective variables energy and five-membered rings. The model with 576 water molecules. metadynamics run has been interrupted before the basin corresponding to the liquid state was explored.
leading to structures of varying stability, in agreement with the analysis of ref. 8. The free energy of the most stable defect structure (B) is 6.9 kcal/mol higher than that of the ideal crystal, which we assume as the reference zero value. The free energy barriers for the defect formation and recombination are estimated to be 8.9 and 2.0 kcal/mol, respectively, which, assuming a characteristic frequency for the reaction coordinates of 5THz and using the classical transition state theory, gives an estimated∼ lifetime of the defect of 0.5 ns at 120 K. The second defect structure (C) explored during∼ the metadynamics run is less stable (8.4 kcal/mol) and can either recombine or transform into structure B through a barrier of 0.9 kcal/mol. When the temperature is raised to 270 K, the height of the free energy minimum corresponding to structure B and its formation and Figure 5. Cluster of topological defects obtained during the inherent recombination barriers remain unchanged, whereas the recom- structure analysis of the premelting phase. H2Omoleculesformingeither bination barrier for structure C goes to zero. These free energy four- or five-membered rings are represented by colored sticks. Grey calculations extend the results of ref 8 to finite temperature and lines represent the ice Ih structure embedding the cluster of defects. confirm the relevance of these topological defects also at the melting temperature. accumulation of defects in a restricted region of the crystal is We found that the “5+7” defects play an even greater role, a nucleus for further disorder and melting as the number of since they are responsible for the shallow minimum in the FES coordination defects and smaller rings grows. In Figure 4, two indicated as premelt in Figure 3. From metadynamics, the distinct regimes in the premelting inherent structures can be transition barrier to this local minimum can be estimated at observed. When the system is dragged out of the free-energy roughly 12 kcal/mol. We analyzed the nature of this local basin corresponding to the cluster of topological defects, a minimum by performing an inherent structure analysis24 of the relevant number of small rings and coordination defects appears metadynamics trajectory, quenching to zero K in 50 ps frames and the energy suddenly increases. Roughly speaking, this of the MD trajectory taken every 2 ps. As shown in Figure 4, signals the watershed between configurations belonging to the the inherent structures display a relevant quantity of five- basin of attraction of ice Ih and to that of the liquid. membered rings and a smaller number of four-membered rings Since the topological defects cannot migrate, the mobility of and coordination defects. In fact these structures correspond to the defective droplet is related only to defect formation and a condensation of topological defects involving about 50 H2O recombination at the interface with the crystal, and no relevant molecules in an otherwise perfect ice Ih lattice. As the tetrahedral motion of its center of mass was observed. We computed the coordination is preserved, five-membered rings are accompanied momentum of inertia of the defective region and found that the by an equal number of seven-membered ones. One typical defect inertia tensor has two eigenvalues I1 and I2 of similar size and cluster thus obtained is shown in Figure 5. The energy of the a smaller one I3, with an asphericity ratio (I1 + I2)/2 - I3/(I1 + particles, either belonging to smaller rings or under-coordinated, I2)/2 + I3 = 0.4. This indicates that the cluster of defects has a ranges from 0.60 to 0.95 kcal/mol relative to the energy of ice somewhat elongated shape. The analysis of the eigenvectors Ih. shows that the defective region is roughly aligned along the These defective structures were then embedded in a crystalline (33h1) crystallographic direction. ice Ih supercell containing 4608 molecules and brought to 270 Another feature of the FES that is worth commenting upon K, to observe their stability. Several MD simulations were is the shallow basin that appears just before the larger liquid performed in the NPT ensemble with different random initial basin and corresponds to a liquid-solid interface. velocities. Remarkably, the average lifetime of the cluster of In summary, our simulations reveal that topological defects topological defects is 0.4 ( 0.1 ns. This relatively stable where five- and seven-membered rings are formed play a crucial Letter
pubs.acs.org/JPCL
Exploring the Free Energy Landscape of Solutes Embedded in Lipid Bilayers Joakim P. M. Jambeck̈ * and Alexander P. Lyubartsev* Division of Physical Chemistry, Arrhenius Laboratory, Stockholm University, Stockholm, SE-10691, SwedenMembrane Permea on
S Jämbeck and Lyubartsev. Exploring the Free Energy Landscape of Solutes * Supporting Information The Journal of Physical Chemistry Letters Letter Embedded in Lipid Bilayers. 2013 sampled, the obtained FES will contain errors that in the worse case could lead to wrong conclusions. Often one can mistakenly Aspirin ABSTRACT: Free energy calculations are vital for our understanding ofassume biological convergence of the PMF along the reaction coordinate processes on an atomistic scale and can offer insight to various mechanisms. However,based on long in correlation and relaxation times of other DOFs. While transferring a molecule from water to the membrane some cases, degrees of freedom (DOFs) orthogonal to the reaction coordinatecenter, have one high important DOF orthogonal to the reaction energy barriers and/or long equilibration times, which prohibit proper sampling.coordinate Here we is the membrane disruptions as the solute is transferred toward the hydrophobic core of the membrane.24 If identify these orthogonal DOFs when studying the transfer of a solute fromone water considers to a the conformational freedom of the solute, the model membrane. Important DOFs are identified in bulk liquids of differentnumber dielectric of important DOFs that have to be properly sampled nature with metadynamics simulations and are used as reaction coordinatesgrows for fast, the and these DOFs can be difficult to define a priori. Yet they remain vital in certain cases in order for simulations to translocation process, resulting in two- and three-dimensional space ofbe able reaction to reproduce and ultimately predict experiments. The Diclofenac coordinates. The results are in good agreement with experiments and elucidateability of a solute the to form intramolecular hydrogen bonds is one example of these DOFs.28−31 Procedures such as Hamiltonian pitfalls of using one-dimensional reaction coordinates. The calculations performedreplica exchange here US (HREX-US) can decrease the correlation offer the most detailed free energy landscape of solutes embedded in lipid bilayersbetween to sampled date conformations and speed up the convergence in some cases; however, when probable states are separated by and show that free energy calculations can be used to study complexlarge membrane energetic barriers, 1D sampling often has to be abandoned translocation phenomena. for biased sampling in several dimensions. 2D HREX-US Figs. 0 & 1 calculations are robust and can be performed;32−34 however, SECTION: Biophysical Chemistry and Biomolecules they are extremely demanding from a computational point of view and require a lot of efforts before the simulations are even initiated. These calculations are further complicated when more than two DOFs are needed to be biased. An alternative method to this is metadynamics,11 which has been demonstrated to be hermodynamics is one of the cornerstones in physical measure thean permeability efficient and potent and tool in partitioning the exploration of complexof compounds FESs is ffi in many studies,3,35−38 especially in the case where more than chemistry, and the concept of free energy is arguably the di cult from a high-throughput39−41 perspective, and therefore T two DOFs are of interest. 6 1 most important aspect of thermodynamics. Many chemical computer simulationsIn the present Letter can we be show used that the as more an traditional alternative. one- In and biological processes are governed by the change in free particular, moleculardimensional (1D) dynamics US approach (MD) is not able simulations to properly describe have been 2 partitioning between water and a model membrane of three energy such as solvation phenomena, protein−ligand associ- successful whentypical drug the compounds interactions (aspirin, between diclofenac, and small ibuprofen) solutesFigure and 1. Intramolecular CVs in the present study. 3,4 5 and that more extensive sampling is14,16,17,20 required. First,−23 important ation, enzymatic reactions and membrane-water partition- model membranes have been studied.fi Theseff studies 6 ff DOFs of the solutes arefi identi ed in bulk liquids of di erent ing. The ability to predict free energies has been a long-term o er detaileddielectric free energy nature. This pro allowsles for of a more transferring complete picture the of solute the fromFor ibuprofen, the resulting FESs are shown in Figure S1 goal in several areas (pharmaceutical research to mention one), the surroundingsolute’s conformational water phase space before to proceeding the center with large of(Supporting the Information, SI), and for aspirin and diclofenac, 24scale−26 membrane simulations were the reaction coordinates are they are shown in Figure S2. When the dielectric constant of and endeavors in the molecular modeling field during three last membrane. two- (2D) or three-dimensional (3D) instead of a 1D reaction the solvent is decreased by going from water to n-hexane, the 1,7−11 As MD simulationscoordinate. With sample this scheme parts we of are phase able to sample space relevant accordinghydrogen to of the carboxyl group prefers to be in trans decades have made it possible to perform these regions of phase space, and the resulting PMFs and standard conformation (θ ∼ 0) with respect to the carbonyl oxygen calculations relatively routinely. Currently, due to these efforts, a Boltzmannbinding distribution, free energies important are in good agreement regions with separated experiments by largerfor diclofenac and ibuprofen. In aqueous solution, the free ff (free) energycompared barriers to the may more therefore naive approach not of be running properly 1D US sampledenergy minima is found when the hydroxyl hydrogen is in the we can discriminate between di erent methods based on the simulations without any consideration of the conformational trans conformation (|θ| ∼ 180) meaning that the hydrogen is problem we have at hand, and this can bring us closer to a more during the simulations.space of the solutes. Several methods have been developedoriented away from the carbonyl in order to participate in a complete understanding of biological processes.12 with the aim toAspirin, allow diclofenac, proper and sampling ibuprofen were of chosen, less probable as the two partshydrogen of bond network with water instead of the 1,4 former have the possibility to form an intramolecular hydrogen7 intramolecular interaction. The mentioned features are similar Major efforts have been put into understanding the effect of phase spacebond, such and as they umbrella all have carboxyl sampling groups that (US), have a historythe of Wangto the− findings presented by Paluch et al.30 For both solutes, the ffi8 29,30 9,10 drugs, toxins, anesthetics, and other solutes on cells from a Landau algorithm,being di cultthe to adaptive sample properly. biasing forceIn Figure method 1, the FESsand in n-hexane and gas phase are similar, with more of a intramolecular11 CVs are shown, and for all simulations with shallow minimum for the former when compared to the latter. molecular perspective, especially with the focus on free metadynamics.the membrane,As membrane an additional CV partitioning was added: the center studies of mass goes,The the free energy barriers between these two states are high 13−17 distance between the solute and lipid bilayer. The torsion angle (∼14kBT), resulting in extremely long simulations being energies. Despite this, the mechanisms for some of these US methodψ isis the not most needed employed as a CV for technique,ibuprofen, as there and are the no reactionneeded in order to sample these transitions. As the conforma- 18 fi widely used compounds are yet not fully understood. Still free coordinate/collectivepossibilities for variable this solute (CV) to form is intramolecular (often) trivial hydrogen to de tionalne: preference for these two molecules differ significantly ff bonds. Before the membrane simulations the underlying FES of between a solvent with ε ∼ 78 (water) and ε ∼ 2(n-hexane), a energy calculations o er a great deal of insight as a full the distancethe between torsion angles theθ membraneand ψ for aspirin midplane and diclofenac and (Figure solute 1) alongtransition is to be expected when the solute is transferred from understanding of the passive diffusion phenomena of these the membranewere normal explored infor gas which phase, the water, probability and n-hexane. distribution As the water is to the hydrophobic core of the membrane. By merely hydrophobic core of a lipid bilayer behaves similarly to a bulk biasing the sampling along the distance between the solute and compounds over the membrane requires a detailed view of the biased. From these biased probability distributions, a free fialkane, the latter solvent was used to mimic this region. These membrane, this transition is likely to never occur. Even if the underlying free energy surface (FES). As already mentioned, energy pro simulationsle or potential were performed of using mean the well-tempered force (PMF) (WT) ofsimulations the reach a microsecond time scale, it is plausible that metadynamics procedure described by Barducci et al.42 For the resulting PMF will be incorrect due to the lack of sampling, free energy calculations are used in computer-aided drug translocation can be obtained with, e.g., the weighted histogram ibuprofen,27 the same simulations were performed but using US which can lead to the wrong conclusions. This is shown later on design.19 Issues that often render drug candidates unfit for analysis method.simulationsHowever, to sample along other the torsion degrees angle ofθ. freedom (DOFs)for ibuprofen. For aspirin, the FESs are more complicated and fi clinical testing are related to slow translocation of the that are, per de nition, orthogonal to the biased DOF(s),1782 are dx.doi.org/10.1021/jz4007993 | J. Phys. Chem. Lett. 2013, 4, 1781−1787 compounds across the membranes, resulting in poor often of importance, and if these DOFs are not properly pharmacokinetic properties and poor bioavailability. Despite the fact that major efforts have been made to decrease the Received: April 15, 2013 failure attribute of these two properties, the progress of many Accepted: May 10, 2013 candidate compounds is still tampered by these factors. To
© XXXX American Chemical Society 1781 dx.doi.org/10.1021/jz4007993 | J. Phys. Chem. Lett. 2013, 4, 1781−1787 Cytoskeleton Protein Func on Pfaendtner, Barducci, Parrinello, Pollard, and Voth. Nucleo de-dependent conforma onal states of ac n. 2009
Fig. 1
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 which metadynamics was applied. (B) Shown is the proposed conformational change of the DB loop. (C) Shown is the nucleotide binding pocket. The arrows in C illustrate the collective variables used to study the binding pocket.
efficient route to exploring those states in a single simulation. Moreover, a defining feature of metadynamics lies in the fact that under certain conditions the history-dependent potential of Gaussian functions provides a good estimate of the free-energy of the system projected into the CVs (28, 29). Metadynamics has been successfully used so far to study several aspects of protein folding (30–32). Herein an investigation of nucleotide state-dependent effects on the conformational states of the actin monomer and trimer is presented. Application of metadynamics to MD simulations of actin shows that the state of the nucleotide bound to the cleft influences both the width of the cleft and the conformation of the DB loop. The effect of the bound nucleotide on the relative free-energy differences between the stable conformational states has also been estimated. Metadynamics simulations have been performed on the ATP, ADP-Pi, and ADP states of actin in both the monomer and trimer. The metadynamics simulations also reveal a distinct allosteric relationship between the width of the bound nucleotide and the conformation of the DB loop. Fig. 2. Free-energy surfaces for the opening and closing of the nucleotide binding cleft in monomeric actin. The isolines are drawn using a 1 kcal/mol Results spacing, and the energy scale is in kcal/mol. Based on block-averaging analysis, the uncertainty is Ϸ1 kcal/mol. The collective variables used in the opening and Opening and Closing of the Actin Nucleotide Binding Cleft. Metady- closing simulations are described in Results. namics simulations were first performed to investigate the relationship between the width of the nucleotide binding cleft and the state of the bound nucleotide as depicted schematically release of the Pi group, the cleft is more stable in an open in Fig. 1. The results for monomeric actin are shown in Fig. 2 as conformation. It can be seen that bound Pi group is responsible afunctionoftheboundnucleotide.Themoststablestate,i.e., for the stabilization of the closed state, given that both the the region with the lowest free-energy, is the darkest color. The ADP-Pi and ADP phases are most stable when there is no plots show a clear trend in the cleft width as a function of the contact between the -phosphate group of the nucleotide and bound nucleotide. In the ATP-bound state, the most stable state the binding cleft. The metadynamics simulations completely and is a closed conformation, with strong contact between the reversibly explore the phase space defined by the full-range of nucleotide’s P atom and the protein backbone. After ATP the opening and closing CVs and, in the case of ATP and ADP hydrolysis (ADP-Pi phase), the phosphate portion of the nucle- monomeric actin, a few other metastable minima are located. otide loses its contact with the protein, and the most stable state Although the other minima found in these simulations are higher becomes slightly more closed with respect to the distance defined in free-energy by only a few kcal/mol, these regions are separated in the methods section (i.e., the COM distance between residue from any other metastable minima by relatively large barriers. It nos. 14–16 and 156–158). The binding cleft of the ADP-Pi form is particularly important that the ADP-Pi state is found in a becomes more closed than the ATP form because of the loss of closed conformation as was proposed in ref. 33, although not yet contacts between phosphate groups and the protein backbone, supported by high-resolution experimental structures or com- as seen by the shift in the value of the contact number. In puter modeling. contrast, in the ATP form, the -and␥-phosphates are posi- It is also interesting to compare the most stable states found tioned between each side of the binding cleft. Finally, after the by metadynamics simulation with the experimental values avail-
12724 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0902092106 Pfaendtner et al. a different metadynamics (18) history-dependent potential acting the reference structure has an rmsd of 2.7 Å (Fig. 2, Inset B). After on a different CV (Methods and SI Text). Three CVs act at the 50 ns, the rmsd starts increasing progressively (red line) and the secondary structure level by quantifying, respectively, the frac- folded state is not explored at all. By contrast, the simulation with tion of α-helical, antiparallel, and parallel β-sheet content of the the CamShift CV visits the folded state several times, with several protein. Three other CVs act at the tertiary structure level by unfolding–refolding events. During the first 50 ns, the latter sim- biasing the number of hydrophobic contacts and the orientation ulation not only performed better, reaching an rmsd of 2.5 Å, but it of the side-chain dihedral angles χ1 and χ2 for hydrophobic and also formed the correct secondary and tertiary contacts, particu- polar side chains. The seventh CV, called “CamShift,” measures larly the ones involved in forming the first β-hairpin (Fig. 2, Inset the difference between the experimental and calculated chemical A), which is critical for the folding of this protein (38, 39). The shifts, which were obtained using the CamShift method (37) fraction of native contacts also was systematically higher in the (Methods and SI Text). Our results indicate that in the approach simulation using the CamShift CV (Fig. 2, Inset C). These results we present here, this last variable is essential to fold GB3 and indicate that the folding events observed later in the simulation are reach convergence readily in the free-energy calculations. a result of the systematic bias induced by the CamShift CV toward the correct local topology in the native state. Folding of GB3 Using Chemical Shifts as CVs. The method that we introduce in this work makes it possible to visit efficiently a wide Thermodynamics of GB3 Folding. The molecular dynamics simu- range of structures, ranging from extended to compact. Repre- lations that we performed using the approach presented in this sentative examples are shown in Fig. 1A. Native-like conformations work reached convergence after ∼240 ns, because at this point are visited multiple times, reaching a backbone rmsd of 0.5 Å from the bias potentials acting on all the replicas started to become the reference structure (PDB ID code 2OED). In these native-like stationary (40). We then continued the simulations for another structures, the internal packing of hydrophobic side chains is 140 ns to reconstruct the free-energy landscape of the protein practically identical to that observed in the reference structure (Fig. (Methods). In Fig. 3A, the free-energy landscape is represented 1C). In the calculations that we performed, this level of accuracy as a function of three CVs: the fraction of antiparallel β-sheet, could be reached only by using a bias-exchange scheme in which the the fraction of parallel β-sheet, and the coordination number CamShift CV is included in the CV set (Methods and SI Text). To between the hydrophobic side chains (Fig. 3A). This represen- demonstrate this point, we performed another simulation with tation reveals the organization of the free-energy landscape, with the same setup, using the six CVs discussed above that describe a deep minimum corresponding to native-like structures, sepa- the secondary and tertiary structures, but not the CamShift CV. rated by a relatively high barrier from other minima. The lowest The difference betweenProtein Folding the two simulations is substantial. In the free-energy minimum (Methods)includesconfigurations very sim- fi Granatasimulation, Camilloni without the, Vendruscolo CamShift CV, the, and closestLaio con. Characteriza on of the free-guration to ilar to those of the reference structure (on average, at 1.3 Å rmsd). energy landscapes of proteins by NMR-guided metadynamics. 2013
Fig. 1
Fig. 1. (A) Representation of the conformational sampling achieved by the approach introduced in this work. The conformations visited are shown as a function of the CamShift collective variable (CV) and of the backbone rmsd from the reference structure (PDB ID code 2OED). (B) Structure with the lowest rmsd (0.5 Å) from the reference structure. (C) Detail of the side chain packing of the structure in B.
2of6 | www.pnas.org/cgi/doi/10.1073/pnas.1218350110 Granata et al. Intui on-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 simula on? – What determines how hard the hill pushes? – How does this impact the design of hills?
• How does each hill fla en 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? BRIEF ARTICLE
THE AUTHORBRIEF ARTICLE
THE AUTHOR
Adding One Hill V (s,t)/ T (F (s)+V (s,t))/T (1) V˙ (s, t) !e e • The ⇠untempered equa on is V (s,t)/ T (F2(s)+V2(s,t))/T (1) V˙ (s, t) !e (s e( t)) /2 s (2) V (s,⇠ t)=he (s (t))2/2 s2 (2) • The tempered equa on is V ( (t),t) / VT(s, t)=(s he(t)) 2/ 2 s2 (3) V (s, t)=he e V ( (t),t)/ T (s (t))2/2 s2 (3) V (s, t)=he e V˙ (s, t)= ds0K(s, s0,V(t)) (s0,t)(4) V˙ (s, t)= ds0K(s, s0,V(t)) (s0,t)(4) Z 2 2 Z (s s0) /2 s (5) K(s, s0,V(t)) = he (s s )2/2 s2 (5) K(s, s0,V(t)) = he 0 V (s ,t)/ T (s s )2/2 s2 (6) K(s, s0,V(t)) = he 0 V (es ,t)/ T0 (s s )2/2 s2 (6) K(s, s0,V(t)) = he 0 e 0 2 2 (s s0) /2((s ss(s)20))/2( s(s ))2 (7) (7) K(s, s0,VK(t())s, s=0,Vhe(t )) = he 0 0 (8) (8) h/ s h/ s (9) (9) s s
(10) (10) T TG‡ G‡ ⇠ ⇠ V˙ (s, t)=!(t) (s (t))(11) V˙ (s, t)=!(t) (s (t))(11) ˙ V ‡(t)/ T V (s, tV)=(t)!/ e T (s (t))(12) V˙ (s, t)=!e ‡ (s (t))(12) V (t)/ T (F (s)+V (s,t))/T (13) V˙ (s, t) !e ‡ e ˙ V ‡(⇠t)/ T (F (s)+V (s,t))/T (13) V (s, t) !e e (F (s)+V (s, ))/T (14) ⇠ e 1 C (F (s)+V (s, ))/T ⇠ (14) (15) e F (s)=1 V (s,C )+C ⇠ 1 (15) F (s)= VV(s,t()s,/ T (s)+) C V˙ (s, t)=!e (s (t))(16) 1 ˙ V (s,tV)(/s,t )T/ (sT)(s) (F (s)+V (s,t))/T (17) V (s,V t˙)=(s, t!) e !e (se (t))(16) ⇠ (17) V˙ (s, t) !e V (s,tF)(/s )=T (s)e(T/(F (sT)+(sV)(s,t +)) 1)/TV (s, )(18) 1 ⇠ f(s,V (s,t))/T F (s)=V˙ (s,(T/ t)= T!e( s) + 1)V (s, (s )(18) (t))(19) 1 (20) f(s,V (s,t))/T V˙ (s, t)=!e (s (t))(19) 1 (20) 1 Adding One Hill
¥ ¥
• How does a new hill push a simula on? – What determines how hard the hill pushes? – How does this impact the design of hills? Adding One Hill
¥ ¥ ¥ ¥
• How does the hill push a • Maximum force occurs simula on? with a delay – No push at first • No preferred direc on – No push at the peak • Hill size limited by slope, – Max push a er half down not height Adding One Hill
¥ ¥
¥ ¥
• How does each hill fla en 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 fla en • Hills should match the the surface? roughness of a surface – Sets an op mal lengthscale – Smooth + smooth = smooth • As a surface fla ens, each – Smooth + rough = rough hill seems rougher – Rough + rough = ? – The lengthscale changes Adding One Hill ¥
¥ ¥
• What is the best hill? • Robustness is essen al – the exact opposite of the free energy surface • It’s too easy to come • What is the worst? up with bri le tricks – anything else that doesn’t depend on the sample 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 simula on? – If I have flat sampling up to an edge, will I add a flat bias?
• Can I fla en a feature thinner than my hills? – What sampling pa ern leads to that? Adding Ensembles of Hills
¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ?
• 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? Adding Ensembles of Hills
¥ ¥ ¥ ¥ ¥ ¥
¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥
• Is the final sampling • With infinite walkers, any state stable? metadynamics is exact – Even sampling leads to an even bias, but.. • With finite walkers, – Sta s cal fluctua ons metadynamics requires are not fully damped tempering Adding Ensembles of Hills
? ¥ ¥ ¥ ¥ ¥
• What happens at the edges of a simula on? – If I have flat sampling up to an edge, will I add a flat bias? Adding Ensembles of Hills
¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥
• What happens at the • Wells appear near the edges of a simula on? edges of a simula on – Even sampling leads to an uneven bias • To avoid trapping – The added bias is 50% requires correc ons too small near each wall Adding Ensembles of Hills
from ?
• Can I fla en a feature thinner than my hills? – What sampling pa ern leads to that? Adding Ensembles of Hills
• Can I fla en a feature • Filling thin wells works, thinner than my hills? takes a long me – Gaussians are a complete set of basis func ons • A rough guess for hill – However, hills can only be size can be good enough added, never subtracted Adding Series of Hills
• What can happen if I add hills too quickly? – What does “too quickly” mean?
• What happens if my collec ve variables are not the slowest in a system? Adding Series of Hills ¥ ¥ ¥
¥ ¥ ¥
• What if I add hills too • Nonequilibrium ar facts quickly? can be very dangerous – Simula on is pushed by • Power input should be a growing wave of hills lower than a system’s – “Hill surfing” natural fluctua ons Adding Series of Hills • What if the collec ve variables are not slow? – If there are rare events orthogonal to them?
• What if I chose x instead of y? Adding Series of Hills ¥ ¥
¥ ¥
• What if the collec ve • Orthogonal variables variable is not slow? can sink any method – Simula on equilibrates in each metastable state • Metadynamics fails – Bias oscillates between instruc vely various constrained PMFs 214504-3 D. T. Limmer and D. Chandler J. Chem. Phys. 138,214504(2013)
flipping” during coarsening,214504-3 giving the D. T. transient Limmer and impression D. Chandlerreversible liquid-liquid transitions. Melting amorphous ice J. Chem. to Phys. 138,214504(2013) of a non-equilibrium barrier between low-density and high- produce a non-equilibrium liquid that then crystalizes is dif- density states that changesflipping” with N. during coarsening, giving theferent transient too.32 impression reversible liquid-liquid transitions. Melting amorphous ice to These finite-size effectsof a are non-equilibrium fundamental to barrier the nature between of low-densityCrystallization and high- followingproduce the melting a non-equilibrium of glass33 and liquid crys- that then crystalizes is dif- phase transitions. Establishingdensity the states existence that changes of a phase with transi-N. tallization following the rapidferent quench too.32 of water into the liq- 33 tion requires studying system-sizeThese dependence, finite-size effects for example, are fundamentaluid’s “no-man’s to the nature land” of 34 are muchCrystallization like non-equilibrium following the dy- melting of glass and crys- by computing changes inphase free energy transitions. barriers Establishing with respect the to existencenamics of a evolving phase transi- from lowtallization to high Q followingon the middle the rapid and left quench of water into the liq- 6 34 changing N.Nosuchcomputationshaveyetbeenperformedtion requires studying system-size dependence,free energy for surfacesexample, pictureduid’s in “no-man’s Fig. 1,anobservationwor- land” are much like non-equilibrium dy- for putative liquid-liquidby transitions computing in changes models of in water free energy that barriersthy of future with respect study. But to thesenamics interesting evolving non-equilibrium from low to high pro- Q6 on the middle and left exhibit water-like structurechanging of theN liquid.Nosuchcomputationshaveyetbeenperformed and crystal phases. cesses and the transitions betweenfree energy different surfaces amorphous pictured solids in Fig. 1,anobservationwor- for putative liquid-liquid transitions in models of water that thy of future study. But these interesting non-equilibrium pro- To do so requires algorithms that can attend to the collective of water are not our focus in this work. Rather, we are con- exhibit water-like structure of the liquid and crystal phases. cesses and the transitions between different amorphous solids nature of systems undergoing phase transitions. Free energy cerned with whether water-like systems when constrained to To do so requires algorithms that can attend to the collective of water are not our focus in this work. Rather, we are con- methods are among the tools that are suitable for the task, not freeze can exhibit two distinct liquid phases. If such re- nature of systems undergoing phase transitions. Free energy cerned with whether water-like systems when constrained to provided they are combined with trajectory algorithms that versible polymorphism were possible, these systems could methods are among the tools that are suitable for the task, not freeze can exhibit two distinct liquid phases. If such re- are appropriately efficient and reversible.29 also exhibit a second critical point as Stanley and many of his provided they are combined with trajectory algorithms that versible1, 8, 10, 11, 15 polymorphism, 16, 19, 35–38 were possible, these systems could In Sec. III,wedetailhowpertinentfreeenergiescanare appropriately efficient and reversible.co-workers29 have proposed.also exhibit a secondIf, critical instead, point re- as Stanley and many of his be computed for supercooledIn water, Sec. andIII,wedetailhowpertinentfreeenergiescan we consider differ- versible molecular simulationco-workers models exhibit have proposed. only ice and1, 8, 10 one, 11, 15, 16, 19, 35–38 If, instead, re- ent variants of the ST2 modelbe computed as applications. for supercooled Juxtaposition water, andliquid, we consider then the differ- symmetryversible differences molecular between simulation ice and liquid models exhibit only ice and one of free energy surfaces forent three variants different of the variants ST2 model indicates as applications.exclude Juxtaposition the possibility of anliquid, associated then the critical symmetry point. differences We be- between ice and liquid that reasonable changes inof electrostatic free energy boundarysurfaces for conditions three differentlieve variants the systematic indicates evidenceexclude provided the possibility herein and of in an Paper associated I critical point. We be- do not change general phasethat reasonable behaviors. SectionchangesIV inpresents electrostaticindicates boundary that conditions there is onlylieve one liquid the systematic and no low-temperature evidence provided herein and in Paper I free energy surfaces obtaineddo not for change other generalsystems: phase the mW behaviors. of critical Section point.IV presents indicates that there is only one liquid and no low-temperature water, the TIP4P/2005 modelfree energy of water, surfaces30 and obtained the Stillinger- for other systems: the mW of critical point. Weber (SW) model of Si.water, In every the case, TIP4P/2005 the models model are found of water,30 and the Stillinger- to exhibit one stable orWeber metastable (SW) liquidmodel phaseof Si. In plus every ice- case,II. the THEORY models are OF found COARSENING AND ARTIFICIAL POLYAMORPHISM IN COMPUTER SIMULATION like crystal phases. Coexistenceto exhibit between one stable two distinct or metastable liquid liquid phase plus ice- II. THEORY OF COARSENING AND ARTIFICIAL OF WATER POLYAMORPHISM IN COMPUTER SIMULATION like crystal phases. Coexistence between two distinct liquid phases does not occur. A summary of our findings is given in OF WATER Sec. IV, and AppendicesphasesA, B,and doesC notpresent occur. further A summary details of our findingsThis section is given provides in a quantitative theoretical analysis and results. Sec. IV, and Appendices A, B,andC presentshowing further the difficulty details in obtainingThis correct section reversible provides free a quantitative en- theoretical analysis Reversibility is particularlyand results. important to the issues ad- ergy surfaces of supercooledshowing water. the We difficulty do so by in examining obtaining correct reversible free en- dressed here and in Paper I.Reversibility Distinct reversible is particularly phases can importantthe effectsto the issues of time-scale ad- ergy separation surfaces for of dynamics supercooled on a water. re- We do so by examining be interconverted, with propertiesdressed here that are and independent in Paper I. of Distinct the reversibleversible free phases energy can surface.the The effects particular of time-scale surface separation we em- for dynamics on a re- be interconverted, with properties that are independent of the versible free energy surface. The particular surface we em- paths by which they are prepared. Reversible liquid phases ploy is the free energy F(ρ, Q6)derivedinPaperIforavari- are thus not the same aspaths amorphous by which solids they or are glasses. prepared. The Reversibleant of the liquid ST2 phases model. Thisploy free is energy the free is shown energy inF( Fig.ρ, Q2(a)6)derivedinPaperIforavari-. ant of the ST2 model. This free energy is shown in Fig. 2(a). former are reversible andare ergodic, thus not so their the same measured as amorphous station- solidsThe methods or glasses. used The to obtain that surface are the subject of former are reversible and ergodic, so their measured station- The methods used to obtain that surface are the subject of ary behaviors are independent of history. The latter, like high- Sec. III, butAdding Series of Hills here we only need to assume that there is such a ary behaviors are independent of history. The latter, like high- Sec. III, but here we only need to assume that there is such a density or low-density amorphous ices (HDA and LDA), are surface, and that it is qualitatively like the surface shown in density or low-density amorphous ices (HDA and LDA), are surface, and that it is qualitatively like the surface shown in not ergodic, so their behaviors depend much on history (i.e., Fig. 2(a). not ergodic, so their behaviors depend• much on history (i.e., Fig. 2(a). The concept of equilibrium is subtle Free energy surfaces for several models are derived in preparation protocols). Observedpreparation transitions protocols). between Observed HDA transitions between HDA Free energy surfaces for several models are derived in 31 Secs. III and IV. The generic features of the free energy and LDA phases, therefore,and LDA are necessarily phases,31 therefore, different arethan• necessarilyThere are always orthogonal variables different than Secs. III and IV. The generic features of the free energy • Great example: water liquid-liquid coexistence
Fig. 2 Limmer and Chandler. The puta ve liquid-liquid transi on is a liquid-solid FIG. 2. Slow relaxation behaviorFIG. and 2. its Slow consequences relaxation forbehavior free energy and its calculations. consequences (a) for The free reversible energy free calculations. energy surface (a) The for reversible 216 molecules free energy with the surfaceST2a for 216 molecules with the ST2a potential energy function at temperaturepotential energyT 235 function K and pressure at temperaturep 2.2T kbar.235 (See K and text pressure for definitiontransi on in atomis c models of water. II. 2013 p 2.2 of the kbar. ST2a (See potential.) text for definition Contour lines of the are ST2a separated potential.) by Contour lines are separated by 1.5 k T,andstatisticaluncertaintiesareabout1= = = k T.(b)Negativelogarithmofthenon-equilibriumdistributionforcrystalorder,= Q , as it relaxes from the 1.5 kBT,andstatisticaluncertaintiesareabout1B kBT.(b)Negativelogarithmofthenon-equilibriumdistributionforcrystalorder,B Q6, as it relaxes from the 6 liquid state. It is computed fromliquid the Fokker-Planck state. It is computed equation from with the the Fokker-Planck free energy surface equation given with in Panel the free (a) energy under the surface assumption given in that Panel the (a) density, underρ the, remains assumption at that the density, ρ, remains at equilibrium with the instantaneous value of Q6.(c)and(d)Non-equilibriumpseudofreeenergysurfacescomputedfromEq.(1) at two intermediate stages of equilibrium with the instantaneous value of Q6.(c)and(d)Non-equilibriumpseudofreeenergysurfacescomputedfromEq.(1) at two intermediate stages of relaxation, t 10 τQ6 and t 1000 τQ6 . The unit of time, τQ6 , is the autocorrelation time for Q6 fluctuations in the liquid basin (i.e., at small Q6). The reduced relaxation, t 10 τQ6 and t 1000 τQ6 . The= unit of time, τQ=6 , is the autocorrelation time for Q6 fluctuations in the liquid basin (i.e., at small Q6). The reduced = = density isρ ˜ (ρ ρxtl)/#ρ,whereρxtl is the mean density of the crystal basin (i.e., at large Q6), and #ρ is the difference between the mean densities of the density isρ ˜ (ρ ρxtl)/#ρ,whereρxtl is= the mean− density of the crystal basin (i.e., at large Q6), and #ρ is the difference between the mean densities of the = − liquid and crystal basins. Contour lines are separated by 1 kBT and statistical uncertainties are about 1 kBT. liquid 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 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 Intui on-Building: Conclusion
• Carefully limit the power input due to the bias – Control the slope of the hills and rate of addi on – Look out for heterogeneous fluctua on • Hills must adapt as the simula on progresses – Offset sta s cal fluctua on and avoid overwork • Always look out for orthogonal variables – Look for pulsing around convergence – Define early what is fast and what is forbidden Things le out for me
• Concrete implementa ons – We discuss one code, PLUMED, this a ernoon • Current fron ers – Lots of work on how to use more CVs at once – Lots of work on how to choose and design CVs – Con nuing work on efficient convergence • Placement among other methods – Unique efficiency tradeoffs – Complementari es to others Summary
• Metadynamics is theore cally simple: – Bias away from previously visited states – In a reduced space of collec ve variables – At a sequen ally decreasing rate of bias
• Metadynamics is prac cally convenient: – Few parameters to choose – Implementa on is compara vely simple – Fails early and fails instruc vely