1 / 64

Bioinformatics Data Analysis & Tools

Bioinformatics Data Analysis & Tools. Molecular simulations & sampling techniques. Molecular Simulations: Brief History. Protein flexibility. Also a correctly folded protein is dynamic Crystal structure yields average position of the atoms ‘Breathing’ overall motion possible. B-factors.

agnes
Download Presentation

Bioinformatics Data Analysis & Tools

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Bioinformatics Data Analysis & Tools Molecular simulations & sampling techniques Molecular Simulations & Sampling Techniques

  2. Molecular Simulations: Brief History Molecular Simulations & Sampling Techniques

  3. Protein flexibility • Also a correctly folded protein is dynamic • Crystal structure yields average position of the atoms • ‘Breathing’ overall motion possible Molecular Simulations & Sampling Techniques

  4. B-factors • De gemiddelde beweging van atoom rond gemiddelde positie alpha helices beta-sheet Molecular Simulations & Sampling Techniques

  5. Peptide folding from simulation • A small (beta-)peptide forms helical structure according to NMR • Computer simulations of the atomic motions: molecular dynamics Molecular Simulations & Sampling Techniques

  6. unfolded folded Folding and un-folding in 200 ns all different? how different? Unfolded structures 321 1010 possibilities! Folded structures all the same Molecular Simulations & Sampling Techniques

  7. unfolded folded folding equilibrium depends on temperature Temperature dependence 360 K 350 K 340 K 320 K 298 K Molecular Simulations & Sampling Techniques

  8. unfolded folded folding equilibrium depends on pressure Pressure dependence 2000 atm 1000 atm 1 atm Molecular Simulations & Sampling Techniques

  9. Surprising result • Number of relevant non-folded structures is very much smaller than the number of possible non-folded structures • If the number of relevant non-folded structures increases proportionally with the folding time, only 109 protein structures need to be simulated in stead of 1090 structures • Folding-mechanism perhaps simpler after all… Molecular Simulations & Sampling Techniques

  10. Phase Space • Defines state of classical system of N particles: • coordinates q = (x1, y1, z1, x2, … , zN) • momenta p = (px1, py1, pz1, px2, … , pzN) • One conformation (+ momenta) is one point (p,q) in phase space • Motion is a curved line in phase space • trajectory: (p(t),q(t)) Molecular Simulations & Sampling Techniques

  11. Molecular Motions: Time & Length-scales Molecular Simulations & Sampling Techniques

  12. Newton Dynamics Sir Isaac Newton t t + Dt Molecular Simulations & Sampling Techniques

  13. Classical (Newton) Mechanics • A system has coordinates q and momenta p (= mv): p = ( p1, p2, … , pN ) q = ( q1, q2, … , qN ) • This is called the configuration space. • The total energy can be split into two components: • kinetic energy (K): K(p) = ½ mv2 = ½ p2/m • potential energy (V): V(q) depends on interaction(s) • The potential energy is described by • bonded interactions (e.g. bond stretching, angle bending) • non-bonded interactions (e.g. van der Waals, electrostatic) • Non-bonded interactions determine the conformational variation that we observe for example in protein motions. Molecular Simulations & Sampling Techniques

  14. The Hamilton Function • The Hamiltonian function represents the total energy:H(p,q) = K(p) + V(q) • Is the generalised expression of classical mechanics • In two differential expressions: • Newton equations of motion, but in a very elegant way • Use 'generalised coordinates' (p and q): • can use any coordiate system • e.g., Cartesian coordinates or Euler angles dpdHp = ––– = ––– dtdqk dqdHq = ––– = ––– dtdpk . . Molecular Simulations & Sampling Techniques

  15. Hamilton's Principle • "The time derivative of the integral over the energy ofd ( pq - H(p,q) ) dt = 0 • Hamilton's principle is most fundamental • Newton's equation of motion are only one set of equations that can be derived from Hamilton's principle. • The integral is called the 'action‘, meaning: • If we integrate the trajectory of an object in a configuration space given by positions q and momenta p between time points (integration limits) t1 and t2, then the value of the integral (= the 'action') of a 'real‘ trajectory is a minimum (more precisely an extremum) if compared to all other trajectories. • Example: Why does a thrown stone follow a parabolic trajectory? • If you vary the trajectory and calculate the action, the parbolic trajectory will yield the smallest 'action'. . . Molecular Simulations & Sampling Techniques

  16. Harmonic oscillator: • 1-dimensional motion • 2 dimensions in phase-space: • position (1-dimensional) • momentum (1-dimensional) • analytical solution for integration: • q(t) = b · cos (√k/m · t ) • p(t) = -b·√mk· sin ( √k/m·t ) q(t) p(t) Molecular Simulations & Sampling Techniques

  17. Calculating Averages • Integration of phase space: • 1 particle, 2 values per coordinate (e.g. up, down): • 1*6 degrees of freedom (dof); 26 = 64 points • 2 particles: 2*6 dof; 212 = 4.096 points • 3 particles: 3*6 dof; 218 = 262.144 points • 4 particles: 4*6 dof; 224 = 16.777.216 points • Need whole of phase space ? • only low energy states are relevant Molecular Simulations & Sampling Techniques

  18. Solving Complex systems • No analytical solutions • Numerical integration: • by time (Molecular Dynamics) • by ensemble (Monte-Carlo) • Molecular Dynamics:Numerical integration in time • Euler’s approximation: • q(t + Δt) = q(t) + p(t)/m·Δt • p(t + Δt) = p(t) + m·a(t) ·Δt • Verlet / Leap-frog Molecular Simulations & Sampling Techniques

  19. Features of Newton Dynamics • Newton’s equations: • Energy conservative • Time reversible • Deterministic • Numeric integration by Verlet algrorithm: ‘Simulation’r(t + Dt) ~ 2 r(t) - r(t - Dt) + F(t)/mDt2 [ + 2 O(Dt4) ] • In ‘real’ simulation: Rounding errors (cumulative):  not fully reversible  no full energy conservation • Coupling to thermal bath  re-scaling  not fully deterministic • ‘Lyapunov’ instability  trajectories diverge Molecular Simulations & Sampling Techniques

  20. Derivation: Verlet • Taylor expansion: • q(t+Δt) = q(t) + q’(t)Δt + 1/2! q’’(t)Δt2 + 1/3! q’’’(t)Δt3 + … • where: q’(t) = v(t) (1st derivative, velocity) • and: q’’(t) = a(t) (2nd derivative, acceleration) q(t+Δt) = q(t) + q’(t)Δt + 1/2! q’’(t)Δt2 + 1/3! q’’’(t)Δt3 q(t−Δt) = q(t) − q’(t)Δt + 1/2! q’’(t)Δt2 − 1/3! q’’’(t)Δt3+ q(t+Δt) + q(t−Δt) = 2q(t) + 2·1/2! q’’(t)Δt2 • Rearrange: q(t+Δt) = 2q(t) − q(t−Δt) + a(t)Δt2 • 2nd order; but 3rd order accuracy Molecular Simulations & Sampling Techniques

  21. What do we obtain? • Trajectory:q(t) and p(t) • Probability of occurence:P(p,q) = 1/Z e-H(p,q)/kT • Averages along trajectory: <A(p,q)T> = 1/T A(q(t),p(t)) dt (where T denotes total time, and not! temperature) Molecular Simulations & Sampling Techniques

  22. Convergence • Amount of phase-space covered • “Sampling” • Impossible to prove:You cannot know what you don’t know • Energy “landscape” in phase-space • there might be a “next valley” Molecular Simulations & Sampling Techniques

  23. Example: Convergence (1) Molecular Simulations & Sampling Techniques

  24. Example: Convergence (2) Molecular Simulations & Sampling Techniques

  25. Example: Convergence (3) • Apparent Convergenceon all timescales100 ps – 10 ns ! Molecular Simulations & Sampling Techniques

  26. Efficiency • Time step limited by vibrational frequencies • heavy-atom–hydrogen bond vibration 10-14s (10fs) • 10-20 integration steps per vibrational period: • 0.5 fs time step; 2.000.000 steps for 1 ns • Removal of fast vibrations (constraining): • hydrogen atom bond and angle motion • heavy-atom bond motion • out-of-plane motions (e.g. aromatic groups) • In practice: 1-2 fs time step • 5-7 fs maximum Molecular Simulations & Sampling Techniques

  27. Constraining • to remove degrees of freedom, e.g.: • bond i-j vibrations  keep distance i-j constant • angle i-j-k vibrations  keep distance i-k constant • Constraint Algorithms • SHAKE • iterative adjustment of lagrange multipliers • LINCS • Taylor expansion of matrix inversion • non-iterative (more stable) • no highly connected constraints • SETTLE • Analytical Solution • for symmetric 3-atom molecules (like water) Molecular Simulations & Sampling Techniques

  28. Improving Performance • Pairwise potential: Fij = − Fji • Potential E(r) ~ 0 at large r : cut-off • Coulomb: ~ 1/r • Lennard-Jones: ~1/r6 • Atoms move little in one step: pair-list • Evaluating r is expensive: r = √|rj−ri| • Large distances change less: twin-range • short-range each step; long range less often • Multiple time-step methods • Many Processor/Compiler/Language specific optimizations: • use of Fortran vs. C • optimize cache performance • arrays of positions, velocities, foces, parameters are very large • compiler optimizations Molecular Simulations & Sampling Techniques

  29. Ignoring Degrees of Freedom • Internal: • bonds, angles → Constraint algorithm • larger time steps • External: • “Solvent” → Langevin dynamics • less (explicit) particles • Inertia & “solvent” → Brownian dynamics • larger time steps Molecular Simulations & Sampling Techniques

  30. Trajectory on Energy Surface Molecular Simulations & Sampling Techniques

  31. Sampling in Conformational Space • Most of the computational time is spent on calculating(local, harmonic) vibrations. DE >> KT Energy vibration Entropy Molecular Simulations & Sampling Techniques

  32. Barriers • Kitao et al. (1998) Proteins 33, 496-517. Molecular Simulations & Sampling Techniques

  33. Psychology of Theorists 100% “In theory, there should be no difference between theory and practice. In practice, however, there is always a difference...“ (Witten and Frank) “For every complex question there is a simple and wrong solution.” (Albert Einstein) “All models are wrong, but some are useful.” (George Box) 0% OPTIMIST SCALE Molecular Simulations & Sampling Techniques

  34. Monte Carlo Sampling • Ergodic hypothesis: • Sampling over time (Molecular Dynamics approach); and • Ensemble averaging (Monte Carlo approach) • Yield the same result: r (r) = < ri(r) >NVE • Detailed Balance condition: p(o) p(on) = p(n) p(no) Molecular Simulations & Sampling Techniques

  35. Metropolis Selection Scheme • Metropolis acceptance rule that satisfies detailed equilibrium:acc(on) = p(n)/p(o) = e-DE/kT if p(n) < (o)acc(on) = 1 if p(n)  (o)  Metropolis Monte Carlo • Ergodic probability density for configurations around rN e-E/kTp(rN) = ––––––S e-E/kT Molecular Simulations & Sampling Techniques

  36. Search Strategies Molecular Simulations & Sampling Techniques

  37. Leaps Molecular Simulations & Sampling Techniques

  38. Computational Scheme • Readuction of the leaps will lead to classical dynamics • Control parameter: • RMSD • Angle deviation Molecular Simulations & Sampling Techniques

  39. Computational Load: Solvation • Most computational time (>95%) spent on calculating (bulk) water-water interactions Molecular Simulations & Sampling Techniques

  40. Implicit Solvation Molecular Simulations & Sampling Techniques

  41. POPS • Solvent accessible area • fast and accurate area calculation • resolution: • POPS-A (per atom) • POPS-R (per residue) • parametrised on 120000 atoms and 12000 residues • derivable -> MD • Free energy of solvationDGsolvi = areai·si • POPS is implemented in GROMOS96 • parameters 'sigma' from simulations in water: • amino acids in helix, sheet and extended conformation • peptides in helix and sheet conformation Molecular Simulations & Sampling Techniques

  42. POPS server Molecular Simulations & Sampling Techniques

  43. Test molecules: alanine dipeptide Molecular Simulations & Sampling Techniques

  44. Test molecules: BPTI / Y35G-BPTI Classical MD Leap-dynamics Essential dynamics Molecular Simulations & Sampling Techniques

  45. Calmodulin domains • Apparent unfolding temperatures (CD) • C-domain : 315 K (42 ° C) • N-domain : 328 K (55 °C) • LD simulations: • 3 ns • 4 trajectories • 290 K • 325 K • 360 K Molecular Simulations & Sampling Techniques

  46. Snapshots Molecular Simulations & Sampling Techniques

  47. Trajectories Molecular Simulations & Sampling Techniques

  48. Example: Protein & Ligand Dynamics Molecular Simulations & Sampling Techniques

  49. Example: Essential Dynamics Analysis Cyt-P450BM37 x 10ns “free” MD simulations Molecular Simulations & Sampling Techniques

  50. CD Molecular Simulations & Sampling Techniques

More Related