Dynamics EventTime Bond stretch ~1 to 20 fs Elastic domain modes 100 fs to several ps Water reorientation 4 ps Inter-domain bending 10 ps to 100 ns Globular protein tumbling 1 to 10 ns Aromatic ring flipping 100 µs to several sec Allosteric shifts 2 µs to several sec Local denaturation 1 ms to several sec Time-scales comparison of 2 static structures clustering of rotation vectors extraction of hinges, axes

Elastic Network Models couple Ca’s together with springs (coarse-graining) –connectivity based on side-chain contacts (<7A), not backbone simulate dynamics, extract modes from B-factors to local unfolding to domain rotations Atilgan,A.R., Durell,S.R., Jernigan,R.L., Demirel,M.C., Keskin,O., and Bahar,I. (2001). Anisotropy of fluctuation dynamics of proteins with an elastic network model. anisotropic motions spatial clustering lysozyme k and u k : eigenvalues and eigenvectors of   k  k ) 1/2

a) first mode of vibration of RBP – note motion in loops b) loops are where transthyretin binds

The conformational plasticity of protein kinases (Huse and Kruiyan, Cell, 2002)

Normal modes analysis Brooks and Karplus (1983) – BPTI Bruccoleri, Karplus, McCammon (1986) - lysozyme Dobbins Lesk, Sterberg (2008) – NMA analysis of induced-fit changes at protein-protein interfaces approximate potential around minimum with harmonic simulations: all-atom MD vs. elastic network among contacts diagonalization rotates vibrations into orthogonal linear-combination dimensions (decoupled)

Levitt Sander Stern (1985) –methodology (normalization condition) (solve equations of motion) (from Lagrange’s equation) ribonuclease

Jianpeng Ma: F1-ATPase motions

Relationship of Normal Modes Vibrations to Allostery D. Ming and M. Wall (2005). Quantifying Allosteric Effects in Proteins. ligand-binding often causes subtle changes that transmit to remote sites hypothesis: dynamic effects, rather than conformational use NMA to compare vibrations with and without ligand bound can identify functional interaction sites; try different surface positions CHARMM modules: VIBRAN, DIAG define allosteric potential Dx –Kullback-Leibler divergence of conformational PDFs –must compute marginal distr of bound confs to unbound –simulate with harmonic vibrations around joint states equilibrium z0=(x0,y0) via MD –approximate potential with Hessian; diagonalize

distribution of D x (m) for 4859 atoms on surface of lysozyme, complex with NAG fits extreme-value distribution