Metadynamics Metainference with PLUMED Max Bonomi University of Cambridge [email protected] Outline • Molecular Dynamics as a computational microscope: - sampling problems - accuracy of force fields • Enhanced sampling with biased MD: - metadynamics - recent developments • Combining simulations with experiments: Metainference • Addressing sampling and accuracy issues: M&M • The open source library PLUMED A computational microscope Molecular Dynamics (MD) evolves a system in time under the effect of a potential energy function How? By integrating Newton’s equations of motion m R¨ = V i i rRi The potential (or force field) is derived from - Higher accuracy calculations - Fitting experimental observables Limitations: • time scale accessible in standard MD • accuracy of classical force fields The time scale problem In MD, sampling efficiency is limited by the time scale accessible in typical simulations: ★ Activated events B ★ Slow diffusion A Dimensional reduction It is often possible to describe a physical/chemical process in terms of a small number of coarse descriptors of the system: S = S(R)=(S1(R),...,Sd(R)) Key quantity of thermodynamics is the free energy as a function of these variables: 1 1 F (S)= ln P (S) where β = −β kBT βU(R) canonical dβRF (Sδ()S S(R)) e− PF((SS))=e− − ensemble dR e βU(R) / R − R Examples Isomerization: dihedral angle Protein folding: gyration radius, number of contacts, ... Phase transitions: lattice vectors, bond order parameters, ... Rare events simplified likely unlikely likely How can we estimate a free energy difference if we never see a transition? Biased sampling The idea is to add a bias potential that acts on the collective variables: U(R) U(R)+V (S(R)) ! What is a good choice of bias potential? Biased sampling The idea is to add a bias potential that acts on the collective variables: U(R) U(R)+V (S(R)) ! What is a good choice of bias potential? Metadynamics History-dependent bias potential acting on selected degrees of freedom or Collective Variables (CVs) S =(S1(R),...,Sd(R)) t0<t d 2 (S S (R(t0))) V (S,t)=W exp i − i G − 2σ2 i=1 i ! t0=⌧GX,2⌧G,. X V (S,t )= F (S)+C G !1 − Laio & Parrinello PNAS 2002 REVIEW: Barducci, Bonomi, Parrinello WIREs Comput Mol Sci 2011 Pros and Cons Advantages • Enhanced sampling along the CVs • Reconstruction of the FES: V (S,t )= F (S)+C Bussi, Laio, Parrinello PRL 2006 G !1 − • A priori knowledge of the landscape not required Disadvantages • Lack of convergence in a single run • Overfilling • The choice of the CVs is not trivial Well-Tempered Metadynamics The initial Gaussian height w 0 is rescaled during the simulation: V (s,t) k ∆T w = w0 e− B where T + ∆ T is a fictitious CV temperature. • Convergence and overfilling issues solved: ∆T V (s,t) F (s) !T + ∆T • ∆ T used to tune the extent of exploration Barducci, Bussi, Parrinello PRL 2008 Choosing the right CVs A good set of CVs for metadynamics (and other biasing techniques) should: • Discriminate between initial and final states • Be as small as possible • Include all the slow modes of a process Metadynamics is inefficient with a large number of CVs. Possible strategies: • devise automatic protocols to find good CVs • improve metadynamics to deal with a large number of CVs • couple metadynamics with other methods, such as REM Parallel Bias Metadynamics Biasing a large number of CVs with WTMetaD is inefficient In PBMetaD we apply multiple low-dimensional bias potentials: V (S1,t),...,V(SN ,t) one at a time: P (R, ⌘) exp β U(R)+ ⌘ V (S ,t) t / − i i " i !# X where ⌘ =( ⌘ 1 ,..., ⌘ N ) switches on and off (and allows updating) one bias potential at a time Each bias potential converges to ∆T V (S ,t) F (S ) the corresponding free energy: i !T + ∆T i Pfaendtner & Bonomi JCTC 2015 Parallel Bias Metadynamics Since we are not interested in the ⌘ -distribution, we can marginalize this variable: P (R)= d⌘P (R, ⌘) exp [ β (U(R)+V (S,t))] t t / − PB where: Z 1 N V (S,t)= log exp [βV (S ,t)] PB −β − i i=1 X In order for each bias potential to converge to the corresponding free energy, we need a new rescaling rule: V (Si,t) ! = ! e− kB ∆Ti P (⌘ =1R) i 0,i i | where: exp [ βV (Si,t)] P (⌘i =1R)= − | N exp [ βV (S ,t)] j=1 − j Pfaendtner & Bonomi JCTC 2015 P Benchmark on a model system A B F (kBT) 2 s s1 s1 C D s1 2 T) ε B t * F (k s1 s2 2 T) ε B t * F (k s2 t (103 MC steps) Pfaendtner & Bonomi JCTC 2015 Improving the accuracy of force fields A more accurate description of a system can be achieved if we combine all the sources of information available Physics Statistics ? Experimental data How can we properly combine them? The challenges of data modeling a) Input b) Errors + Measurement+Prior Mixing Replica-Averaged Modelling +Camilloni et al. JACS 2012 Random Bayesian Modelling* Systematic *Rieping et al. Science 2005 Forward model c) Output Metainference: Ensemble of models Addressing these challenges a) Input b) Errors Measurement+Prior Mixing Random Metainference* *Bonomi et al. Science Advances 2016 Systematic Forward model c) Output Metainference: Ensemble of models To produce ensemble of models and determine their populations a) Input b) Errors Measurement+Prior Mixing Random Systematic Forward model c) Output Metainference: Ensemble of models 30% 20% 45% 5% Metainference Inspired by replica-averaged modelling, we consider a finite sample of the distribution of models (N replicas): 1 N f(X)= f(X ) N r r=1 X The Metainference energy function (or score) is: N N d 1 E (X, σ)=k T log p(X )+ (d f (X))2 + log σ log p(σ ) MI B · − r i − i 2σ2 r,i − r,i r=1 ( i=1 r,i ) X X where σ r,i includes all sources of errors: errors are negligible Replica-Averaged Modelling SEM 2 B 2 σr,i = (σr,i ) +(σr,i) data is not generated q by an ensemble SEM p and σr,i 1/ N Bayesian Modelling / Integrative Dynamical Biology We compare Metainference and replica- averaged modeling with real experimental data collected on ubiquitin: • Chemical Shifts + RDCs We also compare the Metainference ensemble with single structures: • X-ray (1UBQ) • NMR (1D3Z) Cα-RMSD = 0.52 Å and with the ensemble generated by standard MD Models are evaluated by fit with other exp data (RDCs, J3) Bonomi et al. Science Advances 2016 Ubiquitin ensembles a) Input b) Ensemble α βHB+ D52 D52 E24 G53 G53 E24 β ~35% βHB- βHB+ G53 φ α ~65% α ψ D52 d (E24 NH - D52 CO) (nm) d (E24 sc - G53 NH) (nm) c) Validation 3 3 JHNC JHNHA RDCs set 2 RDCs set 3 RMSD (Hz) Bonomi et al. Science Advances 2016 Chemical Shifts weights prior error = 0.08 PDF k [kJ/mol] Bonomi et al. Science Advances 2016 Metadynamic Metainference & Bonomi et al. Scientific Reports. 2016 Metadynamics Metainference Ensemble Metainference PBMetaD of replicas energy function bias (X1, σ1) VPB(S(X1),t) (X2, σ2) VPB(S(X2),t) }EMI(X, σ)+{ (XN , σN ) VPB(S(XN ),t) with these additional tricks: • replicas share the bias, as in multiple-walkers MetaD* • need to reweigh to calculate averages in the unbiased ensemble *Raiteri et al. JPCB 2006 Benchmark Our favorite test case: alanine dipeptide in vacuum The prior information is the AMBER99SB-ILDN force field* A Prior B Exact C Gaussian noise F (kJ/mol) rad) ( ψ Φ (rad) Φ (rad) Φ (rad) *Lindorff-Larsen et al. Proteins 2010 Benchmark We assume that the prior is inaccurate and that in the real distribution the relative weight of the two minima is different: A Prior B Exact C Gaussian noise F (kJ/mol) rad) ( ψ Φ (rad) Φ (rad) Φ (rad) We introduce synthetic experimental data as average distances between heavy atoms, calculated in the exact ensemble, + noise Results Gaussian noise Outliers noise A B C T) B F (kJ/mol) F RMSD (k Φ (rad) # data # data CD DE F T) B F (kJ/mol) F RMSD (k ψ (rad) # data # data Bonomi et al. Scientific Reports. 2016 Gaussian noise Outliers noise B σ1 σB σB 0 2 B σ3 B noerrors σ4 pdf Noise inference A B Gaussianabsence of errorsnoise Outlierssystematic noise errors B systematicerrors σ B B B 1 σ σ1 B σ B 0 σ0 2 B σ2 σ3 σB 3 noerrors σB B 4 σ4 pdf pdf pdf A B C D B B noise level systematicerrors B noiseσ level σ σ1 σB σB 0 2 B σ3 B σ4 Bonomi et al. Scientific Reports. 2016 pdf C D σB σB A Prior B Exact C Gaussian noise F Metainference alone (kJ/mol) rad) ( with PBMetaD ψ A Φ (rad) Φ (rad) Φ (rad) rad) ( Φ without PBMetaD 1 2 3 4 rad) ( Φ B 5 6 7 8 rad) ( ψ rad) ( Φ time (ns) time (ns) time (ns) time (ns) time (ns) The implementation depending on the physical problem: depending on physical distances, angles, ... problem/type of machine/... evaluate collective variables compute forces correct forces move atoms several possible algorithms e.g. umbrella sampling, metadynamics, ... PLUMED PLUGIN MD code evaluate collective variables compute forces correct forces move atoms Bonomi et al. CPC 2008 Tribello et al. CPC 2014 PLUMED PLUGIN MD codes gromacs compute forces lammps move atoms evaluate collective variables compute forces move atoms correct forces compute forces move atoms compute forces move atoms namd One open source pluging q-espresso for several MD codes! PLUgin for MEtaDynamics Why PLUMED? PLUgin for free-energy MEthoDs Bonomi et al.