Metadynamics Day 2, Lecture 3 James Dama Metadynamics • The bare bones of metadynamics – Bias away from previously visited configuraons – 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 elevaon : A method for improving the searching properAes of molecular dynamics simulaon. 1994 • Metastable states are a pervasive feature of real free energy surfaces • TransiAons between these states are rare and difficult to observe in simulaon The Essence of Metadynamics Bias adapAvely to escape metastable states Huber, Torda, and Van Gunsteren. Local elevaon : A method for improving the searching properAes of molecular dynamics simulaon. 1994 • Basins are unknown a priori – Avoid previously sampled conformaons – Add energy ‘hills’ centered on each sample so far • Focus on model- independent exploraon BRIEF ARTICLE THE AUTHOR Adding Hills V (s,t)/∆T (F (s)+V (s,t))/T (1) 2JAMESDAMAV˙ (s, t) !e− e− • A real equaon is ⇠ (s σ(t))2/2δs2 (2) bias V (s, t). Given a metadynamics-sampled∆V (s, t)=he− trajectory− in collective variables σ(t), the history-dependent potential• The idealized equaon is is definedV (σ(t) as,t)/ an∆T approximation(s σ(t))2/2δs2 of (3) ∆V (s, t)=he− e− − V˙ (s, t)=!(s σ(t))(1) ˙ − where ! is an adjustableV• rate(Makes escaping square wells linear in depth s, t)= parameterds0K and(s, sδ0,Vis a(t delta))δ(s function0,t)(4) on the collective variable domain. For well-temperedrather than exponenAal in depth metadynamics,Z the idealized rule is (s s )2/2δs2 (5) K(s, s0,V(t)) = he− − 0 ˙ V (s,t)/∆T V (s, t)=!e− δ(s 2 σ(t2))(2) (6) K(s, s ,V(t)) = he V (s0,t)/∆T e (s s0)−/2δs where ∆T is a second adjustable0 parameter− referred− − to as a tempering parameter. Clearly, (s s )2/2(δs(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/δs future sampling away from each point the more often it has already been visited. In (9) practice, these rules are approximated by discretization inδ 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 σ(t)maybeamulti- trajectory composed of the historiesV˙ (s, of t)= multiple!(t)δ( walkers,s σ(t))(11) and in this case δ should be − understood as a sum of delta functions, oneV ( pert)/∆ individualT walker. Later work (Branduardi, V˙ (s, t)=!e− ‡ δ(s σ(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− − δ1(s σ(t))(3) V (s,t)/∆T (s) − V˙ (s, t)=!e− δ(s σ(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− δ(s σ(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− δ(s σ(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 collecAve variables Laio and Parrinello. Escaping free-energy minima. 2002 • Local elevaon 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 The Essence of Metadynamics Bias only specific collecAve variables Laio and Parrinello. Escaping free-energy minima. 2002 • ReacAons are mulAscale – Monitor only state- determining variables – Leave fast variables alone • Focus on accomplishing specific exploratory goals CollecAve Variables • Any funcAon of any number of fine-grained variables – PosiAon, distance, angle, dihedral – Coordinaon number, density, crystalline order – Helicity, contact map, NMR spectrum – Strings of configuraons 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 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 proporAonal to the rate of hill addiAon 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).