Department of Mechanical Engineering Bioinformatics: Practical Application of Simulation and Data Mining Protein Folding II Prof. Corey O’Hern Department of Mechanical Engineering Department of Physics Yale University

What did we learn about proteins? Many degrees of freedom; exponentially growing # of energy minima/structures Folding is process of exploring energy landscape to find global energy minimum Need to identify pathways in energy landscape; # of pathways grows exponentially with # of structures Coarse-graining/clumping required energy minimum transition Transitions are temperature dependent

Coarse-grained (continuum, implicit solvent, C) models for proteins J. D. Honeycutt and D. Thirumalai, “The nature of folded states of globular proteins,” Biopolymers 32 (1992) 695. T. Veitshans, D. Klimov, and D. Thirumalai, “Protein folding kinetics: timescales, pathways and energy landscapes in terms of sequence-dependent properties,” Folding & Design 2 (1996)1.

3-letter C model: B9N3(LB)4N3B9N3(LB)5L B=hydrophobic N=neutral L=hydrophilic Number of sequences for Nm=46 Nsequences= 3 ~ 1022 Number of structures per sequence Np ~ exp(aNm)~1019

and dynamics different mapping?

Molecular Dynamics: Equations of Motion Coupled 2nd order Diff. Eq. How are they coupled? for i=1,…Natoms

(iv) Bond length potential

Pair Forces: Lennard-Jones Interactions Parallelogram rule force on i due to j -dV/drij > 0; repulsive -dV/drij < 0; attractive

‘Long-range interactions’ BB LL, LB NB, NL, NN V(r) hard-core attractions -dV/dr < 0 r*=21/6 r/

Bond Angle Potential 0=105 ijk k i j ijk=[0,]

Dihedral Angle Potential Vd(ijkl) Successive N’s Vd(ijkl) ijkl

Bond Stretch Potential for i, j=i+1, i-1 i j

Equations of Motion Constant Energy vs. Constant Temperature velocity verlet algorithm Constant Energy vs. Constant Temperature (velocity rescaling, Langevin/Nosé-Hoover thermostats)

T0=5h; fast quench; (Rg/)2= 5.48 Collapsed Structure T0=5h; fast quench; (Rg/)2= 5.48

T0=h; slow quench; (Rg/)2= 7.78 Native State T0=h; slow quench; (Rg/)2= 7.78

start end

Total Potential Energy native states

Radius of Gyration unfolded Tf native state slow quench

2-letter C model: (BN3)3B (1) Construct the backbone in 2D N B (2) Assign sequence of hydrophobic (B) and neutral (N) residues, B residues experience an effective attraction. No bond bending potential. (3) Evolve system under Langevin dynamics at temperature T. (4) Collapse/folding induced by decreasing temperature at rate r.

Energy Landscape E/C E/C end-to-end distance end-to-end distance 5 contacts 4 contacts 3 contacts

Rate Dependence 2 contacts 3 contacts 4 contacts 5 contacts

Misfolding

Reliable Folding at Low Rate

Slow rate

Fast rate

So far… Uh-oh, proteins do not fold reliably… Quench rates and potentials Next… Thermostats…Yuck! More results on coarse-grained models Results for atomistic models Homework Next Lecture: Protein Folding III (2/15/10)