Normal mode analysis (NMA) tutorial and lecture notes by K. Hinsen

1. Serkan Apaydın Normal mode analysis (NMA) tutorial and lecture notes by K. Hinsen

2. Protein flexibility Frequency spectrum of a protein Over half of the 3800 known protein movements can be modelled by displacing the studied structure using at most two low-frequency normal modes. Gerstein et al. 2002

3. Outline • NMA • What it is • Vibrational dynamics • Brownian modes • Coarse grained models • Essential dynamics

4. Harmonic approximation Energy (U) 0 Rmin Conformation (r)

5. Harmonic approximation U 0 Rmin r U(r) =0.5(r − Rmin)’ · K(Rmin) · (r − Rmin)

6. NMA U(r) =0.5(r − Rmin)’ · K(Rmin) · (r − Rmin)

7. NMA Normal mode direction 1 U(r) =0.5(r − Rmin)’ · K(Rmin) · (r − Rmin)

8. NMA Normal mode direction 2 -e2 U(r) =0.5(r − Rmin)’ · K(Rmin) · (r − Rmin)

9. NMA (2) U(r) =0.5(r − Rmin)’ · K(Rmin) · (r − Rmin) min min O(n3)

10. Properties of NMA • The eigenvalues describe the energetic cost of displacing the system by one length unit along the eigenvectors. • For a given amount of energy, the molecule can move more along the low frequency normal modes • The first six eigenvalues are 0, corresponding to rigid body movements of the protein

11. 4 ways of doing NMA A. Using minimization to obtain starting conformation, and computing the Hessian K: • Vibrational NMA • Brownian NMA B. Given starting structure: • Coarse grained models C. Given set of conformations corresponding to the motion of the molecule: • Essential Dynamics

12. 1. Vibrational NMA • derived from standard all-atom potentials by energy minimization • time scale: < residence time in a minimum • appropriate for studying fast motions • Useful when comparing to spectroscopic measurements • Requires minimization and Hessian computation

13. 1. Vibrational NMA

14. Vibrational frequency spectrum

15. 2. Brownian NMA • derived from standard all-atom potentials by energy minimization • time scale: > residence time in a minimum • appropriate for studying slow motions • Requires minimization and Hessian computation

16. 2. Brownian NMA

17. The friction coefficients • describe energy barriers between conformational substates • Can be obtained from MD trajectories (<xi2>) • Depend on local atomic density (not a solvent effect) http://dirac.cnrs-orleans.fr/plone/Members/hinsen/

18. 3. Coarse grained models • Around a given structure • time scale: >> residence time in a minimum • appropriate for studying slow, diffusive motions (jump between local minima) • Does not require expensive minimization and Hessian computation

19. 3. Coarse grained models (2) • Capture collective motions • Specific to a protein • Usually related to its function • Largest amplitudes • Atoms are point masses • Springs between nearby points

20. Coarse grained models (3) f can be a step function or may have an exponential dependence. Elastic network model NMA (aka ANM) Find Hessian of V, then eigendecomposition Gaussian network model 

21. Coarse grained models (4) All atom or C-alpha based models… • Or a step function…

22. Equilibrium fluctuations Ribonuclease T1 Disulphide bond facilitator A (DsbA) Gaussian network model: Theory and applications. Rader et al. (2006)

23. Difference between ENM NMA and GNM • GNM more accurate in prediction of mean-square displacements • GNM does not provide the normal mode directions

24. Lower resolution models • Groups of residues clustered into : • unified sites • Rigid blocks (rotation and translation of blocks (RTB) model) • To examine larger biomolecular assemblies G Li, Q Cui - Biophysical Journal, 2002

25. 4. Essential dynamics • Given a set of structures that reflect the flexibility of the molecule • Find the coordinates that contribute significantly to the fluctuations • time scale: >> residence time in a minimum

26. Essential Dynamics(2) <r> = R <(r − R) (r − R)'> =kBT inv(K) Angel E. García, Kevin Y. Sanbonmatsu Proteins. 2001 Feb 15;42(3):345-54.

27. Essential dynamics(3) • Cannot capture the fine level intricacies of the motion • Freezing the small dofs make small energy barriers insurmountable • Need to run MD for a long time in order to obtain sufficient samples 38, 150, 199 dofs

28. Applications of normal modes • Use all modes or a large subset • Analytical representation of a potential well • Limitations: • approximate nature of the harmonic approximation • Choice of a subset • Properties of individual modes • Must avoid overinterpretation of the data • E.g., discussing differences of modes equal in energy • No more meaningful than discussing differences between motion in an arbitrarily chosen Cartesian coord. system

29. Applications of normal modes (2) • Explaining which modes/frequencies are involved in a particular domain’s motion • Answered using projection methods: • Normal modes form a basis of the config. space of the protein • Given displacement d, pi= d · ei • Contribution of mode i to the motion under consideration • Cumulative contribution of modes to displacement

30. Cumulative projections of transmembrane helices in Ca-ATPase

31. Amplitude Time scale Starting structure Practical Vibrational Small Short By Minimization N Brownian Large Long By Minimization Y/N Coarse grained Large Long Given Y Essential Large Long Given N Comparison chart

32. Summary NMA: • no sampling problem • computational efficiency, especially for coarse-grained models • simplicity in application • Predicts experimental quantities related to flexibility, such as B-factors, well.

33. http://igs-server.cnrs-mrs.fr/elnemo/ (all atom)

34. WebNM: (C-alpha based) http://www.bioinfo.no/tools/normalmodes

35. http://promode.socs.waseda.ac.jp/pages/jsp/index.jsp (all-atom)

36. Ignm (C-alpha based): http://ignm.ccbb.pitt.edu/

37. http://molmovdb.org/nma/ (C-alpha based)

38. http://lorentz.immstr.pasteur.fr/nomad-ref.php (all atomic or just C-alpha)

39. Serkan Apaydın Protein Flexibility Predictions Using Graph TheoryJacobs, Rader, Kuhn and ThorpeProteins: Structure, function and genetics 44:150-165 (2001)

40. Characterizing intrinsic flexibility and rigidity within a protein • Compares different conformational states Limited by the diversity of the conformational states

41. Characterizing intrinsic flexibility and rigidity within a protein • Simulates molecular motion using MD Limited by the computational time

42. Characterizing intrinsic flexibility and rigidity within a protein • Identifies rigid protein domains or flexible hinge joints based on a single conformation Can provide a starting point for more efficient MD or MCS

43. Outline • The main idea: constraint counting • Brute force algorithm • Rigidity theory • Pebble game analysis • Rigid cluster decomposition • Flexibility Index • Examples

44. Overview of FIRST • Floppy Inclusion and Rigid Substructure Topography • Given constraints: • Covalent bonds • hydrogen bonds • Salt bridges • Evaluate mechanical properties of the protein: Find regions that are: • rigid • move collectively • move independently of other regions Compute a relative degree of flexibility for each region

45. Rigidity in Networks – a history • 1788: Lagrange introduces constraints on the motions of mechanical systems • 1864: Maxwell determined whether structures are stable or deformable applications in engineering, such as the stability of truss configurations in bridges • 1970: Laman’s theorem: determines the degrees of freedom within 2D networks and allow rigid and flexible regions to be found extended to bond-bending networks in 3D http://unabridged.m-w.com

46. Brute force algorithm to test rigidity INDEPENDENT ORACLE REDUNDANT

47. Brute force algorithm to test rigidity INDEPENDENT ORACLE REDUNDANT • Compute normal modes w/ and w/o the constraint • If the number of zero eigenvalues remains constant, then the constraint is redundant. O(n2 .n3) O(n5) Complexity?

48. Laman’s theorem accelerates constraint counting • Constraint counting to all the subgraphs • Applying directly, complexity is O(exp(n)) • Applying recursively, pebble game algorithm. Complexity is O(n2), O(n) in practice.

49. Pebble Game • 3 pebbles per node • Each edge must be covered by a pebble if it is independent • Pebbles remaining with nodes are free and represent DOFs of the system • An edge once covered should stay covered but pebbles can be rearranged.

50. Mykyta Chubynsky and M. F. Thorpe Arizona State University The Pebble Game: A Demonstration