Some Accelerated/Enhanced Sampling Techniques

Transcription

1 Some Accelerated/Enhanced Sampling Techniques The Sampling Problem Accelerated sampling can be achieved by addressing one or more of the MD ingredients Back to free energy differences... Examples of accelerated sampling techniques

2 ( G RT ln e Ek E 0 0 S energy kbt k S Sampling: a Problem?! (Free) energy landscape is complicated and huge ) Obtaining a complete picture of the possible conformations and their probability of occurring is hampered by the sheer number of them and the barriers between local minima in the (free) energy landscape co te na di or 2 e1 t a n i coord Pictures adapted from presentation by Sander Pronk and Van Gunsteren et al. Angew. Chem. Int. Ed. 45, 4064 (2006)

3 Enhanced Sampling Opportunities Multiscaling: Coarse-graining: reducing the number of degrees of freedom, preserving the relevant physics reducing detail in the surroundings leading to effective interactions Biasing: Jumping: adapting interactions to exchanging snapshots reduce phase space and/or between conditions to smoothen the free energy overcome barriers landscape Van Gunsteren et al. Angew. Chem. Int. Ed. 45, 4064 (2006)

4 Enhanced Sampling Opportunities Multiscaling: Coarse-graining: THERE IS NO SUCH THING AS A FREE LUNCH! reducing the number of degrees of freedom, preserving the relevant physics reducing detail in the surroundings leading to effective interactions Biasing: Jumping: adapting interactions to exchanging snapshots reduce phase space and/or between conditions to smoothen the free energy overcome barriers landscape Van Gunsteren et al. Angew. Chem. Int. Ed. 45, 4064 (2006)

5 Q: reaction quotient K: equilibrium constant Free Energy Differences from Simulations Direct by counting 1 2 eq p Q12eq = K12 = 2eq = p1 ( ) kbt ( e ) kbt e Ei E 0 i 2 E j E 0 j 1 ( =e G20 G10 ) RT Boltzmann weights! a2 c2 N 2V1 p2 Reliable value for free energy Q12 = =! is obtained only if the a1 c1 N1 V2 p1 difference statistics are good enough: we A thermodynamic state is in fact a collection of configurations! need many transitions between the states and full sampling of each state to capture the entropy

6 ΔG120 = G20 G10 = RT ln K12 Free Energy from Simulations: ONE of the TRICKS Find extra (also called biasing) potential here ΔU to make probabilities equal better statistics 1 p2eq K12 = eq = p1 2 β Ei e i 2 e βej =e 0 βδg12 j 1 p2 =1= p1 e i 2 e j 1 ( β Ei ΔU βej ) = eβδu e β Ei i 2 βej e j 1 A thermodynamic state is in fact a collection of configurations! =e 0 βδu βδg12 e ΔG120 = ΔU 1 E = 0; β = k BT 0

7 ΔG120 = G20 G10 = RT ln K12 Free Energy from Simulations: ONE of the TRICKS More general: make correction for biasing potential ( e β Ei ΔU 1 2 p2 = p1 i 2 e βej ) =e 0 βδu βδg12 e j 1 eq 2 eq 1 p K12 = = p e β Ei e βej i 2 =e 0 βδg12 j 1 p2 ΔG = ΔU RT ln p A thermodynamic state is in fact a collection of configurations! Input Measure with good statistics

8 ΔG 0 12 = RT ln p eq 2 eq p 1 Free energy differences from Simulations Weighted Histogram Analysis Method Apply a restraining potential at different points PMF ( ξ ) = RT ln p ( ξ ) d=1.0 nm d=1.2 nm gmx wham ( ) p ξ 0 Methanol dimer PMF Potential of Mean Force Add harmonic restraining potential to the distance between centers-of-mass ΔU R ( ξ,d) = K ( 2 ξ d ) 2 The PMF is obtained by reweighting the conformations

9 Biasing Metadynamics Adds penalizing potential for conformations already visited Potential energy landscape is changed on the fly Many kinds of variations are possible depending on how the penalizing potentials are implemented or to which degrees of freedom they are applied True Boltzmann probabilities can be determined afterwards by reweighting You pay by losing the true dynamics of the system and possibly sampling utterly uninteresting and/or irrelevant parts of phase space (you need to have an idea of the relevant degree(s) of freedom) From Review by Bernardi et al. BBA 1850, 872 (2015)

10 Local Elevation Example Biasing Gaussian-shaped penalty function as function of torsion angle Glucose ring puckering MD LE 300 K 2 kj mol -1 penalty increments Glucose ring puckering states characterized by dihedral angles. A normal MD simulation samples only one state; local elevation drives the molecule to the other state and later on leads to multiple transitions Christen et al. J. Comput. Chem. 26, 1719 (2005)

11 The Idea of Replica Exchange Run a number of simulations at the same time (replica s of the system) under different conditions Different temperatures is the most popular Different Hamiltonians is a viable option Every so often, attempts are made to swap conformations between (neighboring) conditions The criterion is based on relative Boltzmann weights, this ensures correct overall Boltzmann statistics at one particular condition Jumping p exchange = min 1, e e E 1 k B T 2 + E 2 k B T 1 E 1 k B T 1 + E 2 k B T 2 From Review by Bernardi et al. BBA 1850, 872 (2015)

12 Jumping Replica Exchange Attempts to make trajectories jump between different conditions Temperatures: at higher temperatures, minima are relatively less likely and the free energy landscape is smoother, thus barriers are more easily overcome Hamiltonians: by switching off/on interactions, the landscape may be smoothened or biased towards interesting regions of phase space True Boltzmann probabilities determined by collecting the parts of trajectories at one temperature or hamiltonian You pay by losing the dynamical continuity at one temperature/ hamiltonian and by simulating irrelevant (high temperature) or non-existing (non-physical hamiltonian) systems

13 gmx rms RMSD: structural similarity Root-Mean-Square Deviation average over all particles at one point in time Extensively used in Protein Modeling 1 r k t N p k ( ( ) ref r ) 2 k Here, Np is the number of particles (atoms/beads) in the molecule ref r is the position of particle k k in the reference structure r k ( t) is the position of particle k at time t cf Basic Exercise

14 Replica Exchange Example The Trp-cage peptide is a designed fast folder 64 replica s (13 ns) K T-REMD does not require previous knowledge about interesting degrees of freedom; in fact, it can be used to discover them! Clusters found in REMD simulation (left) characterized by overall rmsd and helix rmsd (rmsdhx) compared to native structures (a: PDB, b: OPLS, c: GROMOS) btw: this is an example of projecting the phase space on a few degrees of freedom (cf PCA) Juraszek and Bolhuis P. N. A. S. 103, (2006)

15 Replica Exchange Example The Trp-cage peptide is a designed fast folder 64 replica s (13 ns) K From folded The free energy landscape is NOT CONVERGED because REMD simulations starting from different conformations do not yield the same probabilities Free energy landscapes characterized by radius of gyration (rg), ratio of native contacts (ρ), solvent accessible surface (sas), helix rmsd (rmsd_hx) as contours From unfolded Juraszek and Bolhuis P. N. A. S. 103, (2006)

16 Coarse-graining Coarse-graining Retaining the relevant degrees of freedom Interesting physics often does not need all details: similar phenomena occur on different length and time scales justifying reduced units and reduced models Coarse-grained models come in many more flavors than atomistic models because they can be specific or generic and tailored to a particular field of research Coarse-graining within a model identifies the slow but overall large motions: these take a system between different states You pay by losing detail and chemical specificity: the model represents one or more (classes of) molecules and you may sometimes miss particularities of the intended (real) system

17 Coarse-graining Example: Martini Model Marrink et al. J. Phys. Chem. B 108, 750 (2004); J. Phys. Chem. B 111, 7812 (2007) Check out

18 Computational gain CG models Effective interactions between beads Reduction in no. degrees of freedom Fewer interactions to be calculated Lower density of particles allows longer lengthscales Fastest motions on ~10 fs time-scale iso ~1 fs Allows larger time-step Speed-up of up to a factor ~1,000 for 4-to-1 heavy-atom mapped particle-based models

19 Comparison Martini to GROMOS Lipid bilayer in liquid-crystalline state Electron density profile across the bilayer shows how the components of the system are distributed (dashed is CG) Snapshots, side view and looking on top of the bilayer Marrink et al. J. Phys. Chem. B 108, 750 (2004)

20 gmx anaeig Characterizing a collection of structures Are conformations realistic? Time scale of sampling Schlitter s formula for configurational entropy Upper bound Approximation for harmonic oscillator S true S = k B 2 ln 1 + k BTe 2! 2 D Mass-weighted covariance matrix Procedure Fit (part of) the structure to remove translation (and rotation) J. Schlitter Chem. Phys. Lett. 215, 617 (1993)

21 Configurational Entropy Build-up S true S = k B 2 ln 1+ k B Te2! 2 D For atomistic hexadecane and lipid tail in a bilayer Baron et al J. Phys. Chem. B 110, (2006)

22 Conf. Entropy Shows Enhanced Sampling For phospholipid DPPC in bilayer Average structures S true S = k B 2 ln 1+ k B Te2 D! 2 Build-up of sampling after mapping sn1 sn2 Baron et al. J. Phys. Chem. B 110, (2006)

23 Multiscaling Multiscaling Combining the best of different level models Concurrent multiscaling: combining two or more levels in a single simulation, e.g. QM-MM, AA-CG Sequential multiscaling: generating snapshots at less detailed or cheaper level and using these at more detailed level to obtain proper statistics or distribution Active developments ongoing in combining different level molecular models You pay by doing a lot of work to make sure that the less detailed model is (thermodynamically) compatible with the more detailed model or accept artifacts around the boundary between the different models

24 Antimicrobial Peptides Induce Pores Atomistic simulations show the pore formation Pore seen in simulation looks unexpected barrel stave torroidal Proposed pore models showed regular arrangements with membrane spanning peptides Leontiadou et al. J. Am. Chem. Soc. 128, (2006)

25 Sequential Multiscaling & Backmapping Use CG model to explore phase space Put in atomistic details in interesting states A membrane is attacked by peptides (UA ~100 ns) and forms a toroidal pore The peptides become membrane-spanning in a 24 µs CG simulation After back-mapping, this pore is seen to be stable also in the atomistic force field 24 µs 50 ns Rzepiela et al. Faraday Discussions 144, 431 (2010)

26 backward.py User Friendly Backmapping Routine Define rules for initial placing of finer detail based on positions of coarser model Cook 3-bead lipids Martini 12-bead lipids GROMOS 50 UAs Being able to easily connect three quite different resolution lipid models, structures that take very long to build or self-assemble at atomistic level can be generated in no time at all... Wassenaar et al. J. Chem. Theory Comput. 10, 676 (2014)

27 Concurrent Multiscaling Example OPLS-AA/L Valine tripeptide in CG polarizable Martini water (Advanced Tutorial) Interaction of peptide with water through virtual sites at center of mass of groups of atoms according to Martini model Forces on virtual sites are redistributed to the atoms Polarizable Martini water (blue 3-particle model) OPLS-AA peptide: licorice Virtual sites: green spheres

28 Adaptive Resolution Simulation (AdResS) Resolution changes depending on the position Simulation done with Espresso++ Available in GROMACS; recent improvements not yet in standard release Protein in water surrounded by Martini water - molecules diffusing into or out of a pre-defined sphere around the protein change between atomistic and Martini models Zavadlav et al. J. Chem. Phys. 140, (2014)

29 Multiscaling is an Exciting Field Cool technology, but many issues still to be resolved or investigated to fully deliver the promise Artifacts around transition regions because of different chemical potentials of the models Putting in atomistic details smoothly and efficiently Combine with multiple time-stepping Combine with QM region

30 Summary Coarse-graining: The Best Way to Enhance Sampling Depends on Your System and Research Question Multiscaling: reducing the number of degrees of freedom, preserving the relevant physics reducing detail in the surroundings leading to effective interactions Biasing: Jumping: adapting interactions to exchanging snapshots reduce phase space and/or between conditions to smoothen the free energy overcome barriers landscape Van Gunsteren et al. Angew. Chem. Int. Ed. 45, 4064 (2006)

31 Thank you for your attention Recent Reviews: Bernardi et al., Enhanced sampling techniques in molecular dynamics simulations of biological systems, BBA 1850, 872 (2015) Cavalli et al., Investigating Drug Target Association and Dissociation Mechanisms Using Metadynamics-Based Algorithms, Acc. Chem. Res. 48, 277 (2015) Ingolfsson et al., The power of coarse graining in biomolecular simulations, WIRE Comput. Mol. Sci., 4, 225 (2014)

32 Hamiltonian Exchange Example A torsional barrier for rotation was varied between 40% (λ=0.6) and 100% (λ=0) of the nominal value 11 replica s 500 molecules all trans start 273 K At 273 K trans-gauche isomerization across the nominal barrier does not occur (the barrier is ~18 kj mol -1 ) Paths of the 11 replica simulations in λ-space are indicated by different colors Christen et al. J. Comput. Chem. 26, 1719 (2005)

33 Hamiltonian Exchange Example A torsional barrier for rotation was varied between 40% (λ=0.6) and 100% (λ=0) of the nominal value 11 replica s 500 molecules all trans start 273 K λ=0 replica RMSD of torsional angle as a function of time shows faster sampling for the λ=0 replica than for reference simulation normal MD Christen et al. J. Comput. Chem. 26, 1719 (2005)

34 D = Mass-weighted covariance matrix 1 K m 1 K ( r k 1 r ) K k k! "! ( r k N r )( N r k 1 r ) 1 # k gmx covar S true S = k B 2 ln 1 + k BTe 2! 2 D ( r k 1 r )( 1 r k N r ) N m N K k ( r k N r ) 2 N Here, K is the number of conformations k r 1 is the position of atom/bead number 1 in frame k m 1 is the mass of atom/bead 1 Note similarity to variance! Baron et al J. Phys. Chem. B 110, (2006)

35 gmx rmsf RMSF: structural mobility/flexibility Root-Mean-Square Fluctuation average over time for each atom (or residue) 1 N f k ( r i r ref ) 2 k i Here, Nf is the number of frames in the trajectory ref r i is the position of particle i in the reference structure i r k is the position of particle i in frame k

36 D = S true S = k B 2 ln 1 + k BTe 2 D! 2 Mass-weighted covariance matrix 1 K m 1 K ( r k 1 r ) K k k! "! ( r k N r )( N r k 1 r ) 1 # k ( r k 1 r )( 1 r k N r ) N m N K k ( r k N r ) 2 N 23 Here, K is the number of conformations k r 1 is the position of atom/bead number 1 in frame k m 1 is the mass of atom/bead 1 Note similarity to variance! Baron et al J. Phys. Chem. B 110, (2006)