Methods for searching and sampling configurational space

International Spring School Statistical Thermodynamics, Santiago de Chile Tuesday, ovember 28, 2017 Lecture 18 Methods for searching and sampling configurational space Prof. Dr. Wilfred F. van Gunsteren ETH Zürich, Switzerland W.F.van Gunsteren/Santiago de Chile 281117/1

Searching and sampling configuration space A. Types of methods for searching configuration space A. Systematic or exhaustive search B. Heuristic search 1. on-step methods (e.g. Distance Geometry) 2. Step methods: change of a complete configuration (e.g. MC, MD, SD) 3. Step methods: build-up of a configuration (e.g. CBMC) B. Types of search enhancement techniques 1. Deformation or smoothening of the potential energy surface Soft-core non-bonded interaction Local-elevation search Coarse graining of the molecular model 2. Scaling of system parameters Temperature annealing Tight coupling to a heat bath Mass scaling Mean-field approaches 3. Multi-copy searching and sampling Replica-exchange and multi-canonical algorithms Cooperative search: SWARM MD W.F.van Gunsteren/Santiago de Chile 281117/2

Methods for searching configuration space for configurations r with low V(r )=energy I. Molecular coordinates as variables: A. Systematic or exhaustive search - Scan complete space (SS) r cartesian angles W.F.van Gunsteren/Santiago de Chile 281117/3 small molecules B. Heuristic search - Generate tiny set of representative conformers: 1. on-step methods Distance geometry algorithms (DG) Distribution? Solvent? 2. Step methods: change of a complete configuration a. Energy: V(x) EM V b. Gradient: MC x 2 V c. 2 nd Derivative: MD xx d. Random: SD e. Memory: PECT, PEACS 3. Step methods: build-up of a configuration configurational bias MC combinatorial chain growing

Techniques to enhance the searching and sampling power of simulation methods 1. Deformation or smoothening of the potential energy surface a. omission of high-resolution structure factor data in structure refinement based on X-ray diffraction data b. gradual introduction of longer-range distance restraints in variable target structure refinement based on MR OE data c. softening of the hard core of atoms in the non-bonded interaction (soft-core atoms) d. reduction of the ruggedness of the energy surface through a diffusion-equation type of scaling e. avoiding the repeated sampling of an energy well through local potential energy elevation or conformational flooding f. softening of geometric restraints derived from experimental (MR, X-ray) data through time-averaging of these g. circumvention of energy barriers through an extension of the dimensionality of the Cartesian space (4D-MD) h. freezing of high-frequency degrees of freedom through the use of constraints i. coarse-graining the model by reduction of the number of interaction sites 2. Scaling of system parameters a. temperature annealing b. tight coupling to heat bath c. mass scaling d. mean-field approaches 3. Multi-copy searching and sampling a. genetic algorithms b. replica-exchange and multi-canonical algorithms c. cooperative search: SWARM W.F.van Gunsteren/Santiago de Chile 281117/4 Review: J. Comput. Chem. 29 (2007) 157-166

Methods for searching configuration space for configurations r with low V(r )=energy I. Molecular coordinates as variables: A. Systematic or exhaustive search - Scan complete space (SS) r cartesian angles W.F.van Gunsteren/Santiago de Chile 281117/5 small molecules B. Heuristic search - Generate tiny set of representative conformers: 1. on-step methods Distance geometry algorithms (DG) Distribution? Solvent? 2. Step methods: change of a complete configuration a. Energy: V(x) EM V b. Gradient: MC x 2 V c. 2 nd Derivative: MD xx d. Random: SD e. Memory: PECT, PEACS 3. Step methods: build-up of a configuration configurational bias MC combinatorial chain growing

Systematic search of conformational space 1. Scan complete or significant part of space 2. Exclude subspaces based on: - physical/chemical knowledge - solutions obtained so far Exponential growth of computing effort as function of number of degrees of freedom only for small molecules W.F.van Gunsteren/Santiago de Chile 281117/6

Methods for searching configuration space for configurations r with low V(r )=energy I. Molecular coordinates as variables: A. Systematic or exhaustive search - Scan complete space (SS) r cartesian angles W.F.van Gunsteren/Santiago de Chile 281117/7 small molecules B. Heuristic search - Generate tiny set of representative conformers: 1. on-step methods Distance geometry algorithms (DG) Distribution? Solvent? 2. Step methods: change of a complete configuration a. Energy: V(x) EM V b. Gradient: MC x 2 V c. 2 nd Derivative: MD xx d. Random: SD e. Memory: PECT, PEACS 3. Step methods: build-up of a configuration configurational bias MC (CBMC) combinatorial chain growing

Generating spatial structures or searching conformational space Step methods: change of a complete configuration 1. Energy minimization: V x pot cons tant x 2. Monte Carlo: x random Only local minimum is found, no escape from it Good for liquids of small molecules, not for folded long chain molecules acceptance probability V (x x) V (x) / kt pot pot e 3. Molecular Dynamics (ewton): Surmounts energy barriers ~ k B T 2 1 Vpot dx Vpot vnew vold t m 2 m x dt x xnew xold vnewt W.F.van Gunsteren/Santiago de Chile 281117/8

4. Stochastic dynamics (Langevin): MD + randomisation 2 d x Vpot dx m random force m 2 dt x dt friction 5. Modified molecular dynamics (PEACS): Enhances barrier crossing Potential Energy Annealing Conformational Search dv pot(t) 1 MD plus V reference V pot(t) dt v slowly lowered R.C. van Schaik et al., J. Comp.-Aided Mol. Des. 6 (1992) 97-112 W.F.van Gunsteren/Santiago de Chile 281117/9

Use of soft-core non-bonded interactions Thomas Beutler et al. Chem. Phys. Letters 222 (1994) 529-539 Use of non-physical potential energy terms Physical non-bonded term: van der Waals Coulomb qq i r ij j C 12 12 ij r V(r) r on-physical softer non-bonded term that allows atoms to pass through each other: V(r) V m r 0 r Conditions: V(0) V V '(0) 0 m V(r ) 0 V '(r ) 0 0 0 1. V(r) is a function of r: f(r) a br cr dr r r V(r) V m 1 3 2 r0 r0 f '(r) 6V r m 2 r0 2 3 r 1 r0 2 3 W.F.van Gunsteren/Santiago de Chile 281117/11

2. V(r) is a function of r 2 : g(r) a br cr 2 4 r V(r) V m 1 ro r r g'(r) 4V m 1 2 r0 r0 2 2 2 3. V(r) is general van der Waals plus Coulomb plus reaction-field form: V( r) qq C C 1 12 6 i j 6 2 6 2 1/2 r r 4 ij ij 0 r c ri j 2 0.5Crf r 1 0.5C 2 1/ 2 R c Rrf rf rf lim 0 standard form 0: V (0) finite V '( 0) 0 V(r) r W.F.van Gunsteren/Santiago de Chile 281117/12

Methods to search conformational space Idea: Include information obtained so far during the simulation into the search scheme: memory function A. Characterize molecular conformations using: - cartesian coordinates too many - torsional angles φ, ψ, χ - dihedral angles spanning residues: R i+1 R i+2 R i R i+3 Review searching: M. Christen & W.F. van Gunsteren J. Comput. Chem. 29 (2007) 157-166 W.F.van Gunsteren/Santiago de Chile 281117/14

B. Penalize the visited conformations by changing the energy function V as function of time V r i V - potential energy term that pushes molecule out of the current conformation V emory phys i m i 0 i 2 i i 0 2 0 i V c umber at e memory i i /2 V 0 0 of conformations for which i i i i i V phys V memory Local elevation search (in 2002 called meta-dynamics) Thomas Huber et al. J. Comp. Aided Mol. Design 8 (1994) 695 W.F.van Gunsteren/Santiago de Chile 281117/15

Implementation 1. Use torsion angles, f i 2. Each conformation f 1, f 2, f 3,,f n =f n 3. Discretise to M parts M n grid points f n 0 4. Gaussian function at grid points: n f -f - n 2 V f = k e 2w mem mem f n 0 n 2 0 5. V = V + V total phys mem A toy application Pentane (two torsional angles) Complete space can be mapped out W.F.van Gunsteren/Santiago de Chile 281117/16

potential energy Test case: pentane Thomas Huber et al. J. Comp. Aided Mol. Design 8 (1994) 695 2 dihedral angles (3 minima each) 9 low V phys conformers free SD-simulation (united atoms) simulation time 100 ps, T=300 K, GROMOS force field trans - trans lowest V phys trans gauche+ gauche- trans -180-120 -60 0 60 120 180 dihedral angle [ ] cis - cis gauche+ gauchegauche+ gauche+ Higher-energy conformers are not (yet) sampled in 100 ps normal MD(SD) simulation highest low higher W.F.van Gunsteren/Santiago de Chile 281117/17

Local elevation search: pentane Local-elevation simulation of pentane (united atoms) T=300 K, Gaussian local-elevation function with k=5kj/mol per MD step simulation time 20ps simulation time 100ps Higher-energy conformations are sampled in 20 ps local-elevation MD simulation Almost all conformations are sampled in 100 ps LE-MD simulation W.F.van Gunsteren/Santiago de Chile 281117/18

The local elevation simulation method ormal simulation: relevant properties - Many conformers few visited - Compact representation should be possible Local-elevation simulation: - run simulation - store visited conformations (using compact representation) - push system away when old conformation is seen V(r) visited V(r) memory normal (MD) r Local elevation (MD) r W.F.van Gunsteren/Santiago de Chile 281117/19

Cyclosporin A Sar 3 O Melen 4 O O H Abu 2 Val 5 H MeBmt 1 Val 11 Melen 10 HO O O O H O H O Ala 7 D-Ala 8 Melen 6 - Amid bond (fixed to trans) ω-dihedral - central bond of φ-dihedral - central bond of ψ-dihedral O O cis! Melen 9 W.F.van Gunsteren/Santiago de Chile 281117/20

Cyclosporin A: potential energy 300 K MD 600 K MD local elevation 300K LE-MD W.F.van Gunsteren/Santiago de Chile 281117/21

Cyclosporin A: similarity of conformations Criterion: Δφ i 30 (upper) 45 (lower) SD simulation at 300 K for each of the 11 φ-angles 164 = average number of visits of same conformer W.F.van Gunsteren/Santiago de Chile 281117/22

Cyclosporin A: similarity of conformations Criterion: Δφ i 30 (upper) 45 (lower) for each of the 11 φ-angles SD simulation at 600 K 26 visits on average W.F.van Gunsteren/Santiago de Chile 281117/23

Cyclosporin A: similarity of conformations Criterion: Δφ i 45 (upper) 60 (lower) for each of the 11 φ-angles 1.6 visit on average Local elevation simulation at 300 K W.F.van Gunsteren/Santiago de Chile 281117/24

Ribonuclease A: RMS fluctuations in the loop region Loop search in vacuo dashed: MD solid: LE-MD in solution solid: LE-MD Local-elevation MD searches a much larger conformational space W.F.van Gunsteren/Santiago de Chile 281117/25

Ribonuclease A: loop conformations Standard MD Local-elevation MD Local-elevation MD searches a much larger conformational space W.F.van Gunsteren/Santiago de Chile 281117/26 Scott et al., J. Phys. Chem. A103 (1999) 3596-3607

Coarse-grained versus fine-grained models AL (l=0) All-atom model (non-hydrogen) 16 (CH 2 or CH 3 ) atoms liquid alkanes: hexadecane MAP mapped all-atom configurations CG (l=1) Coarse-grained model 4 atoms Centre of mass A 1 A 4 A Centre of mass B 1 B 4 B Centre of mass C 1 C 4 C Centre of mass D D 1 D 4 Compare: - structural characteristics - energetic / entropic characteristics W W.F.van Gunsteren/Santiago de Chile 281117/28 M. Christen & WFvG, J. Chem. Phys. 124 (2006) 154106

Algorithm for mixed FG/CG simulation Multigraining MD fine-grained topology, configuration: r fg coarse-grained topology, configuration: r cg 2a. calculate l-dependent energies and forces (according to coupling parameter l) 4a. propagate velocities and positions using the leap-frog scheme 1. update coarse-grained configuration (using virtualgrain definition) 3. distribute forces from coarsegrained on fine-grained particles using virtual-grain definition g(r fg ) and add potential energy terms 2b. calculate l -dependent energies and forces (according to coupling parameter l) 4b. propagate velocities and positions of nonmapped solvent particles using the leap-frog scheme W.F.van Gunsteren/Santiago de Chile 281117/29 M. Christen & WFvG, J. Chem. Phys. 124 (2006) 154106

Multi-grained simulation of 25 hexadecanes in water M. Christen & W.F. van Gunsteren, J. Chem. Phys, 124 (2006) 154106 CG + FG CG CG CG + FG Time: 0 ps 8ps 25ps 100ps FG 8.5ps FG 25.5ps CG level simulation with occasional switching to FG level enhances exploration of FG conformational space W.F.van Gunsteren/Santiago de Chile 281117/30

Search enhancement techniques Scale system parameters 1. Search at high temperature, T annealing V kt 2.5 kj/mol T 10 higher 2 v 3 hi k T=mv B V /k T higher T B e lower V in principle: equivalent in practice: T global V local x t 3 smaller gher CPU 3 more More barrier crossings: 1. Wanted O.K. 2. Unwanted e.g. chirality V chir larger more rigidity less barrier crossing annealing: lower T infinitely slow W.F.van Gunsteren/Santiago de Chile 281117/32

Scaling of system parameters 2. Tight coupling to a temperature bath: velocities are maintained going up hill more barrier crossings? 3. Scaling of atomic masses: Equilibrium properties are independent of masses increase some masses more inertia more barrier crossings? 4. Enhanced sampling via a mean-field approach: Example: Huber et al., Biopolymers 39 (1996) 103-114 Optimization methods for conformational sampling using a Boltzmannweighted mean-field approach W.F.van Gunsteren/Santiago de Chile 281117/33

Metropolis Monte Carlo method Algorithm: DO large number of steps n=1,2,. 1.Make a step r in configuration space: 2.Calculate the change in potential energy ΔE: r n r r 1 E V r n n n1 n E E E n ED DO 3. IF (ΔE 0) THE Accept new configuration ELSE IF ( Accept ELSE Take ED IF r n B e E k T r n 1 again. r n 1 > random number ε(0,1)) THE 1 always accept 1 reject accept e E k B T 0 0 E W.F.van Gunsteren/Santiago de Chile 281117/35

Rationale: consider the probability of transiton Equilibrium: Metropolis Monte Carlo method P t P t 2 21 1 12 P P e ( E2E1) kbt 2 1 1 or P P 2 1 e e E 2 E 1 k k B B T T Each configuration occurs with Boltzmann probability. t2 1 1 ote: - Assumption: detailed balance between states 1 tx y 2 E from state x to state y E 2 1 E2 E1 / kbt 1 1 1 t e t e 12 E E k T 2 1 / B - MC steps must cover Cartesian space uniformly: dxdydz - Using polar coordinates, sample r 2 sindrd d : non-uniform sampling - Unphysical steps are possible: exchange of particles W.F.van Gunsteren/Santiago de Chile 281117/36

Replica-exchange simulation Idea: - A number of replicas of a system that do not interact with each other, is simulated simultaneously. - The replicas are a) at different thermodynamic state points e.g. different temperatures (T s ), or b) characterised by different Hamiltonians, e.g. interatomic interactions, H(p,r ;λ s ) - From time to time, after time period τ ex, a Monte Carlo exchange of configurations r m and r n between two replicas m and n is attempted with exchange probability (assumption: detailed balance) p( m n) = W.F.van Gunsteren/Santiago de Chile 281117/37 with (T-REMD) = with (H-REMD) = 1 for mn 0 min(1,exp( - mn )) exp( mn) for mn 0 mn mn T T H H m m m m ( p, r ) ( p, r ) ( p, r ) ( p, r ) ( p, r ) ( p, r ) ( p n n T m m m m T n n n n H m m m, r m n n n ) ( p, r ) and probability density ( p, r ) exp(-h ( p, r ) / k T ) H n n n B

Temperature replica-exchange SD-simulation of 11 replicas (ΔT=10K) of a box with 512 n-butanes exchange frequency: τ ex = 1ps -1 ot all replicas are exchanging ΔT too large W.F.van Gunsteren/Santiago de Chile 281117/38

Temperature replica-exchange MD simulation of 11 replicas (ΔT=5K) of a box with Ca 2+ and SO 4 2- ions in water (2002 molecules) exchange frequency: τ ex = 2ps -1 1. Are the potential energy distributions overlapping? YES T D (Ca 2+ ) D (SO 4 2- ) K nm/ns nm/ns 300 1.06 1.23 325 1.17 1.31 350 1.63 1.84 2. Are the replicas exchaging sufficiently often? YES 3. Is the ion diffusion enhanced by more than a factor replicas = 11? O: T-replica-exchange MD is not more efficient than standard MD in this case W.F.van Gunsteren/Santiago de Chile 281117/39 A.P.E. Kunz & WFvG, J. Phys. Chem B 115 (2011) 2931-2936

Coarse-grained versus fine-grained models AL (l=0) All-atom model (non-hydrogen) 16 (CH 2 or CH 3 ) atoms liquid alkanes: hexadecane MAP mapped all-atom configurations CG (l=1) Coarse-grained model 4 atoms Centre of mass A 1 A 4 A Centre of mass B 1 B 4 B Centre of mass C 1 C 4 C Centre of mass D D 1 D 4 Compare: - structural characteristics - energetic / entropic characteristics W M. Christen & WFvG, J. Chem. Phys. 124 (2006) 154106 W.F.van Gunsteren/Santiago de Chile 281117/40

grain level (l) Multi-grained simulation of liquid octane grain level of the 24 replicas during 300 replica exchange steps CG FG W.F.van Gunsteren/Santiago de Chile 281117/41 replica exchange trials FG/CG replica-exchange simulation enhances sampling M. Christen & WFvG, J. Chem. Phys. 124 (2006) 154106

Multi-copy search techniques: the SWARM method Idea: combine a swarm of molecules with molecular trajectories into a cooperative system that searches conformational space (like a swarm of insects) Implementation: each molecule is, in addition to the physical forces, subject to (artificial) forces that drive the trajectory of each molecule toward an average of the trajectories of the swarm of molecules Huber and van Gunsteren: J.Phys.Chem. A102 (1998) 5937-5943 SWARM-MD: Searching configurational space by cooperative molecular dynamics W.F.van Gunsteren/Santiago de Chile 281117/44

Methods for searching configuration space for configurations r with low V(r )=energy I. Molecular coordinates as variables: A. Systematic or exhaustive search - Scan complete space (SS) r cartesian angles W.F.van Gunsteren/Santiago de Chile 281117/45 small molecules B. Heuristic search - Generate tiny set of representative conformers: 1. on-step methods Distance geometry algorithms (DG) Distribution? Solvent? 2. Step methods: change of a complete configuration a. Energy: V(x) EM V b. Gradient: MC x 2 V c. 2 nd Derivative: MD xx d. Random: SD e. Memory: PECT, PEACS 3. Step methods: build-up of a configuration configurational bias MC combinatorial chain growing

Problem: If W.F.van Gunsteren/Santiago de Chile 281117/47 Biased sampling V( r ) is large for given regions in configurational space, the sampling in MC, MD, SD simulations will be poor. BIAS Remedy: Add a biasing potential energy term V r to the (physical) Hamiltonian H r, p, possibly based on experimental data, that focuses the sampling on a given region of configurational space. Qr ( ) Ensemble average of a quantity : H ( r, p ) kt B Q( r ) e dp dr e H ( r, p ) BIAS BIAS BIAS V kt H V kbt B H V kbt Qe e dr dp e dr dp BIAS BIAS H V k BIAS BT V kt H V k B BT e dr dp e e dr dp Q V BIAS BIAS Unbiased ensemble average of Q can be obtained from two biased ensemble averages. B B kt B Qe e V k kt T dp dr BIAS BIAS

Spatial distribution of licences GROMOS biomolecular simulation software GROMOS = Groningen Molecular Simulation + GROMOS Force Field Generally available: http://www.gromos.net W.F.van Gunsteren/Santiago de Chile 281117/49