Predictions of protein flexibility: first-order measures

1. Predictions of protein flexibility: first-order measures Julio A. Kovacs, Pablo Chacon, and Ruben Abagyan Proteins: Structure, Function, and Bioinformatics56:661-668 (2004)

2. the study of molecular flexibility connection between structure and function. Two different approaches: • 1- methods based on comparative analysis of two or more conformational states. i.e. with NMR, crystallography • 2- methods which attempt to forecast flexibility from a single starting conformation i.e. MD, MC, coarse-grain methods (more interesting biologically due to predictive power)

3. recently an elegant coarse-grain approach was developed for estimating protein flexibility using graph theory (FIRST) • but this method only makes a distinction between rigid and non-rigid residues, fast but needs careful validation

4. computationally intensive molecular dynamics and Monte Carlo vs. much faster coarse-grain methods • midway between simulations and modelling approaches is NMA which allows the study of the dynamics of proteins • using NMA in conjunction with conformal vector field theory, Kovacs et al. have defined a function which measures the capability of a molecule to deform at each residue • correlation of observed protein motions with motions analyzed by NMA can be found in the molecular movements database (MolMovDB, Echols et al., Nucleic Acids Res., 2003) http://www.molmovdb.org/molmovdb/

5. normal mode analysis (NMA): • separation of frequencies • decomposition of different vibrational modes of a given structure • any displacement can be described by a linear combination of the modes • considering the modes as vector fields, we can define a “deformability function”

6. method • the structure is represented as a network of point masses (Cα atoms), interconnected with springs • the strength of each spring is dependent on the distance between the masses and on the residue contact areas • the spring strengths are set by: rij = the distance between the α carbons of residues i and j sij = normalized residue contact areas computed by ICM (Shrake and Ruptey, J Mol Biol, 1973) ro = the mean distance between consecutive α carbons a = determined to optimize the correlation of the prediction with the experimental data

7. this model is then subjected to NMA calculations (LAPACK linear algebra library) yielding eigenvalues (λn) and eigenvectors (un) • each eigenvector determined by NMA can be viewed as a vector field over the structure • vector fields associate a vector with every point in space i.e. each Cα

8. given a vector field u: a tensor function S is defined which represents rigid motions of the vector field. (Weber and Goldberg, Queen’s papers in pure and applied mathematics, 1969) • S detects changes in shape produced when following the flow of u (each mode of NMA)

9. deformability: du = || S || denotes the norm of a tensor function which quantifies the deformation of the vector field on the molecule • frequencies of vibrations are ωn= sqrt(λn) • thus by combining the deformation measures du with frequencies of vibrations of each mode, we can get dM, which describes how much, in average, the molecule can deform at each residue

10. the resulting modes and frequencies are merged to give a measure of the deformation of each residue in the molecule • by means of normal modes, all possible ways in which the molecule can deform are measured

11. deformability gives a measure of the flexibility of the protein, not mobility (i.e. GNM) i.e. hinge points can be detected

12. results • 10 kinases of various shapes and sizes were used, each having two conformations (A and B) • deformation functions dM were calculated for each A • deformability predictions were compared with the dihedral angle difference (DAD) between both conformations of kinases deformability (dM) DAD

13. terminal regions, hinge regions, and external loops have high deformability values • well correlated with experimental data • comparison with all-atom models show accurate reproduction of large-scale motions

14. deformability values and B-factors were compared and all are different measures • generally there is low correlation between them but in some structures there are similarities

15. conclusions • introduction of the concept of deformability – capability of a molecule to deform at each of its residues • deformability predictions were compared with experimental data by measuring the DAD of two atomic conformations and good agreement found • a method for estimating flexibility with low computational cost and wide applicability

16. conclusionscontinued … • using a simplified harmonic interaction potential and by reducing spatial detail, large systems can be studied • without atomic resolution, there is good correlation between experimentally observed functional motions and the modes observed in this method • still working to improve the algorithm

17. Cij = (ro/rij)6 + a sij