The PLUMED plugin and free energy methods in electronic-structure-based molecular dynamics Davide Branduardi, Theoretical Molecular Biophysics Group Max Planck for Biophysics, Frankfurt am Main, Germany Tuesday, September 4, 2012 TyOutline What is free energy and why is important Traditional QC approach Explicit sampling Enhanced sampling PLUMED plugin for free energy calculations 2 Tuesday, September 4, 2012 TyOutline What is free energy and why is important Traditional QC approach Explicit sampling Enhanced sampling PLUMED plugin for free energy calculations 3 Tuesday, September 4, 2012 TyWhat is free energy and why is important s= state descriptor (bound/unbound, reactant/products, F (s)= kBT ln P (s) phaseA/phaseB) −V (x) k T e− b δ(s(x) s)dx P (s)= − Q conformational transitions and equilibria ligand binding mechanism of transporters or channels chemical reactions and catalysis phase transitions ..... P (out) P (near) P (in) 4 Tuesday, September 4, 2012 TyOutline What is free energy and why is important Traditional QC approach Explicit sampling Enhanced sampling PLUMED plugin for free energy calculations 5 Tuesday, September 4, 2012 TyTraditional QC approach Rigid rotor/ harmonic approximation is well known [1] F (s)=H(s) TS(s) − ∂ ln q (s)q (s)q (s)q (s) S(s)=R + R ln (q (s)q (s)q (s)q (s)) + RT t e r v t e r v ∂T trasl, elect, rot, vib partition functions where the partition functions are obtained from the calculated vibrational spectrum (Hessian), moments of inertia, electronic structure H(s)=Hel(s)+Ht(s)+Hv(s)+Hr(s) It requires electronic structure and Hessian in the local minima and transition states: 1 single conformation per state. [1] McQuarrie, Simon “Physical Chemistry, a molecular approach” 6 Tuesday, September 4, 2012 Ty Finding the RC (I) Static approach: look for saddle points by using chemical intuition and TS optimization. m1 m2 1079 Tuesday, September 4, 2012 Ty Finding the RC Static approach: look for saddle points by using chemical intuition and TS optimization. m1 Verify then by IRC: move forward and backward to verify to end in minima. 11108 Tuesday, September 4, 2012 Ty Finding the RC Static approach: look for saddle points by using chemical intuition and TS optimization. m1 m2 Verify then by IRC: move forward and backward to verify to end in minima. 14109 Tuesday, September 4, 2012 Ty Finding the RC Static approach: look for saddle points by using chemical intuition and TS optimization. m1 m2 Verify then by IRC: move forward and backward to verify to end in minima. ∆‡G = ∆‡H T ∆‡S − kbT ∆‡G/RT [2] k(T )= e− Rates may be derived from Eyring equation : hc0 [2] Eyring, Chem. Rev. 1935 vol. 17 pp 65 1014 Tuesday, September 4, 2012 TyTraditional QC approach PRO: one calculation per point (3 single points calculations per rate and free energy differences) CONS: optimizing TS is not trivial (redundant coordinates), the landscape must be very simple (Free) energy (Free) ortho coor Reaction coor 11 Tuesday, September 4, 2012 TyProblems with traditional QC approach Many problems are not smooth as before: many parallel valleys (conformers), solvent induced roughness 12 Tuesday, September 4, 2012 TyProblems with traditional QC approach Many problems are not smooth as before: many parallel valleys (conformers), solvent induced roughness A TS B Perfectly parallel basins, Not too bad (small correction over static approaches). Typical case of symmetry in a reaction 12 Tuesday, September 4, 2012 TyProblems with traditional QC approach Many problems are not smooth as before: many parallel valleys (conformers), solvent induced roughness 12 Tuesday, September 4, 2012 TyProblems with traditional QC approach Many problems are not smooth as before: many parallel valleys (conformers), solvent induced roughness Perfectly parallel basins but energetically asymmetric, Can be done with static QC but need to sample all the paths 12 Tuesday, September 4, 2012 TyProblems with traditional QC approach Many problems are not smooth as before: many parallel valleys (conformers), solvent induced roughness 12 Tuesday, September 4, 2012 TyProblems with traditional QC approach Many problems are not smooth as before: many parallel valleys (conformers), solvent induced roughness Rough paths: vibrations are meaningless, paths from IRC are not representative of reactions: USE MD BASED METHODS 12 Tuesday, September 4, 2012 TyProblems with traditional QC approach Many problems are not smooth as before: many parallel valleys (conformers), solvent induced roughness 12 Tuesday, September 4, 2012 TyOutline What is free energy and why is important Traditional QM approach Explicit sampling Enhanced sampling PLUMED plugin for free energy calculations 13 Tuesday, September 4, 2012 TyThe ingredients of the probability picture Free energy is connected to the probability of a given state to occur in a single measurement F (s)= k T ln P (s) − B Probability State descriptor V (x) k T (bond length, folded/ e− B δ(s(x) s)dx P (s)= V (x) − unfolded descr., liquid/ e− kB T dx crystal/amorphous) 14 Tuesday, September 4, 2012 TyExplicit sampling via molecular dynamics F (s)= k T ln P (s) − B a real system a set of independent (cuvette) objects (ensemble) a b a a a a b a a a a a a a a a b b MD: evolve in time with Newton’s eq. of motion use ergodic hypothesis a b a a a a b a a F (s)= k T ln P (s) ∆t 2∆t 3∆t ... − B all the available states can be sampled with t->infinite 15 Tuesday, September 4, 2012 a TyLimitations Transition of 8 kcal/mol can take up to milliseconds at 300K ν = ν exp ( β∆V ‡) 0 − 9 1 ν = 5 10 s− ∆V ‡ 1 0 k T 2 B Free Energy Free React Prod RC 16 Tuesday, September 4, 2012 TyLimitations Transition of 8 kcal/mol can take up to milliseconds at 300K ν = ν exp ( β∆V ‡) 0 − 9 1 ν = 5 10 s− ∆V ‡ 1 0 k T 2 B Free Energy Free React Prod RC MD (DFT) can reach hundreds of ps. Lucky if you see one single event! 16 Tuesday, September 4, 2012 TyLimitations Transition of 8 kcal/mol can take up to milliseconds at 300K ν = ν exp ( β∆V ‡) 0 − 9 1 ν = 5 10 s− ∆V ‡ 1 0 k T 2 B Free Energy Free React Prod RC MD (DFT) can reach hundreds of ps. Lucky if you see one single event! property time 16 Tuesday, September 4, 2012 TyLimitations Transition of 8 kcal/mol can take up to milliseconds at 300K ν = ν exp ( β∆V ‡) 0 − 9 1 ν = 5 10 s− ∆V ‡ 1 0 k T 2 B Free Energy Free React Prod RC MD (DFT) can reach hundreds of ps. Lucky if you see one single event! 100 ps property time 16 Tuesday, September 4, 2012 TyLimitations Transition of 8 kcal/mol can take up to milliseconds at 300K ν = ν exp ( β∆V ‡) 0 − 9 1 ν = 5 10 s− ∆V ‡ 1 0 k T 2 B Free Energy Free React Prod RC MD (DFT) can reach hundreds of ps. Lucky if you see one single event! 100 ps property time If you want thermodynamics you must average over events 16 Tuesday, September 4, 2012 TyLimitations Transition of 8 kcal/mol can take up to milliseconds at 300K ν = ν exp ( β∆V ‡) 0 − 9 1 ν = 5 10 s− ∆V ‡ 1 0 k T 2 B Free Energy Free React Prod RC MD (DFT) can reach hundreds of ps. Lucky if you see one single event! 100 ps property time If you want thermodynamics you must average over events property time 16 Tuesday, September 4, 2012 TyLimitations Transition of 8 kcal/mol can take up to milliseconds at 300K ν = ν exp ( β∆V ‡) 0 − 9 1 ν = 5 10 s− ∆V ‡ 1 0 k T 2 B Free Energy Free React Prod This is not RCyour solution MD (DFT) can reach hundreds of ps. Lucky if you see one single event! 100 ps property time If you want thermodynamics you must average over events property time 16 Tuesday, September 4, 2012 TyOutline What is free energy and why is important Traditional QM approach Explicit sampling Enhanced sampling PLUMED plugin for free energy calculations 17 Tuesday, September 4, 2012 TyThe ancestor: Torrie & Valleau[3] Free energy: Free energy for a biased system: use biasing potential to overcome barriers and retrieve free energies [3] Torrie, Valleau J. Comp. Phys, 1977 vol. 23, pp 187 18 Tuesday, September 4, 2012 TyLessons from Torrie & Valleau Biasing potentials: 1 k T 2 B Free Energy Free React Prod RC [3] Torrie, Valleau J. Comp. Phys, 1977 vol. 23, pp 187 19 Tuesday, September 4, 2012 TyLessons from Torrie & Valleau Biasing potentials: biasing pot 1 k T 2 B Free Energy Free React Prod RC [3] Torrie, Valleau J. Comp. Phys, 1977 vol. 23, pp 187 19 Tuesday, September 4, 2012 TyLessons from Torrie & Valleau Biasing potentials: Metastabilities are removed biasing pot 1 k T 2 B resulting potential 1 k T 2 B Free Energy Free React Prod Energy Free React Prod RC RC [3] Torrie, Valleau J. Comp. Phys, 1977 vol. 23, pp 187 19 Tuesday, September 4, 2012 TyLessons from Torrie & Valleau Biasing potentials: Metastabilities are removed biasing pot 1 k T 2 B resulting potential 1 k T 2 B Free Energy Free React Prod Energy Free React Prod RC RC allow to measure free energies: useful allow to overcome barriers: cheap need only an additional term to standard DFT forces: easy [3] Torrie, Valleau J. Comp. Phys, 1977 vol. 23, pp 187 19 Tuesday, September 4, 2012 TyAdaptive sampling methods Adaptive Biasing Force Darve & Pohorille, J.